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+{"text":"\\section{Introduction}\\label{sec:introduction}\nLet $q\\geq 2$ be a positive integer. Then every real $\\theta\\in[0,1)$\nadmits a unique expansion of the form\n\\[\\theta=\\sum_{k\\geq1}a_kq^k\\quad(a_k\\in\\{0,\\ldots,q-1\\})\\]\ncalled the $q$-ary expansion. We denote by\n$\\mathcal{N}(\\theta,d_1\\cdots d_\\ell,N)$ the number of occurrences of the block\n$d_1\\cdots d_\\ell$ amongst the first $N$ digits, \\textit{i.e.}\n\\[\\mathcal{N}(\\theta,d_1\\cdots d_\\ell,N):=\\#\\{0\\leq i< n\\colon\na_{i+1}=d_1,\\ldots,a_{i+\\ell}=d_\\ell\\}.\\] Then we call a number normal\nof order $\\ell$ in base $q$ if for each block of length $\\ell$ the\nfrequency of occurrences tends to $q^{-\\ell}$. As a qualitative\nmeasure of the distance of a number from being normal we introduce for\nintegers $N$ and $\\ell$ the discrepancy of $\\theta$\nby \\[\\mathcal{R}_{N,\\ell}(\\theta)=\\sup_{d_1\\ldots\n  d_\\ell}\\lvert\\frac{\\mathcal{N}(\\theta,d_1\\cdots\n  d_\\ell,N)}{N}-q^{-k}\\rvert,\\] where the supremum is over all blocks\nof length $\\ell$. Then a number $\\theta$ is normal to base $q$ if for\neach $\\ell\\geq1$ we have that $\\mathcal{R}_{N,\\ell}(\\theta)=o(1)$ for\n$N\\to\\infty$. Furthermore we call a number absolutely normal if it is\nnormal in all bases $q\\geq2$.\n\nBorel \\cite{borel1909:les_probabilites_denombrables} used a slightly\ndifferent, but equivalent (\\textit{cf.} Chapter 4 of \\cite{bugeaud2012:distribution_modulo_one}), definition of normality to show that almost\nall real numbers are normal with respect to the Lebesgue\nmeasure. Despite their omnipresence it is not known whether numbers\nsuch as $\\log2$, $\\pi$, $e$ or $\\sqrt2$ are normal to any base. The\nfirst construction of a normal number is due to Champernowne\n\\cite{champernowne1933:construction_decimals_normal} who showed that\nthe number\n\\begin{align*}\n0.1\\,2\\,3\\,4\\,5\\,6\\,7\\,8\\,9\\,10\\,11\\,12\\,13\\,14\\,15\\,16\\,17\\,18\\,19\\,20\\dots\n\\end{align*}\nis normal in base $10$.\n\nThe construction of Champernowne laid the base for a class of\nnormal numbers which are of the form\n\\begin{gather*}\n\\sigma_q=\\sigma_q(f)=\n  0.\\left\\lfloor f(1)\\right\\rfloor_q\\left\\lfloor f(2)\\right\\rfloor_q\\left\\lfloor f(3)\\right\\rfloor_q \\left\\lfloor f(4)\\right\\rfloor_q \\left\\lfloor f(5)\\right\\rfloor_q \\left\\lfloor f(6)\\right\\rfloor_q \\dots,\n\\end{gather*}\nwhere $\\left\\lfloor\\cdot\\right\\rfloor_q$ denotes the expansion in base $q$ of the integer\npart. Davenport and Erd{\\H o}s\n\\cite{davenport_erdoes1952:note_on_normal} showed that $\\sigma(f)$ is\nnormal for $f$ being a polynomial such that $f(\\mathbb{N})\\subset\\mathbb{N}$. This\nconstruction was extended by Schiffer\n\\cite{schiffer1986:discrepancy_normal_numbers} to polynomials with\nrational coefficients. Furthermore he showed that for these\npolynomials the discrepancy $\\mathcal{R}_{N,\\ell}(\\sigma(f))\\ll (\\log\nN)^{-1}$ and that this is best possible. These results where extended\nby Nakai and Shiokawa\n\\cite{nakai_shiokawa1992:discrepancy_estimates_class} to polynomials\nhaving real coefficients. Madritsch, Thuswaldner and Tichy\n\\cite{madritsch_thuswaldner_tichy2008:normality_numbers_generated}\nconsidered transcendental entire functions of bounded logarithmic\norder. Nakai and Shiokawa\n\\cite{nakai_shiokawa1990:class_normal_numbers} used\npseudo-polynomial functions, \\textit{i.e.} these are function of the\nform\n\\begin{gather}\\label{mani:pseudopoly}\n  f(x)=\\alpha_0 x^{\\beta_0}+\\alpha_1x^{\\beta_1}+\\cdots+\\alpha_dx^{\\beta_d}\n\\end{gather}\nwith $\\alpha_0,\\beta_0,\\alpha_1,\\beta_1,\\ldots,\\alpha_d,\\beta_d\\in\\mathbb{R}$,\n$\\alpha_0>0$, $\\beta_0>\\beta_1>\\cdots>\\beta_d>0$ and at least one\n$\\beta_i\\not\\in\\mathbb{Z}$. Since we often only need the leading term we write\n$\\alpha=\\alpha_0$ and $\\beta=\\beta_0$ for short. They were also able\nto show that the discrepancy is $\\mathcal{O}((\\log N)^{-1})$. We refer\nthe interested reader to the books of Kuipers and Niederreiter\n\\cite{kuipers_niederreiter1974:uniform_distribution_sequences}, Drmota\nand Tichy \\cite{drmota_tichy1997:sequences_discrepancies_and} or\nBugeaud \\cite{bugeaud2012:distribution_modulo_one} for a more complete\naccount on the construction of normal numbers.\n\nThe present method of construction by concatenating function values is in\nstrong connection with properties of $q$-additive functions. We call a\nfunction $f$ strictly $q$-additive, if $f(0)=0$ and the function\noperates only on the digits of the $q$-ary representation, i.e.,\n\\[\n  f(n)=\\sum_{h=0}^\\ell f(d_h)\\quad\\text{ for }\\quad n=\\sum_{h=0}^\\ell d_hq^h.\n\\]\nA very simple example of a strictly $q$-additive function is the sum of digits\nfunction $s_q$, defined by\n\\[\n  s_q(n)=\\sum_{h=0}^\\ell d_h\\quad\\text{ for }\\quad n=\\sum_{h=0}^\\ell d_hq^h.\n\\]\n\nRefining the methods of Nakai and Shiokawa\n\\cite{nakai_shiokawa1990:class_normal_numbers} the author obtained the\nfollowing result.\n\\begin{thm}[{\\cite[Theorem 1.1]{madritsch2012:summatory_function_q}}]\n  Let $q\\geq2$ be an integer and $f$ be a strictly $q$-additive\n  function. If $p$ is a pseudo-polynomial as defined in\n  (\\ref{mani:pseudopoly}), then there exists $\\eta>0$ such that\n\\begin{gather*}\n  \\sum_{n\\leq N}f\\left(\\left\\lfloor p(n)\\right\\rfloor\\right)\n  =\\mu_fN\\log_q(p(N))\n  +NF\\left(\\log_q(p(N))\\right)\n  +\\mathcal{O}\\left(N^{1-\\eta}\\right),\n\\end{gather*}\nwhere\n\\[\n\\mu_f=\\frac1q\\sum_{d=0}^{q-1}f(d)\n\\]\nand $F$ is a $1$-periodic function depending only on $f$ and $p$.\n\\end{thm}\n\nIn the present paper, however, we are interested in a variant of\n$\\sigma_q(f)$ involving primes. As a first example, Champernowne\n\\cite{champernowne1933:construction_decimals_normal} conjectured and\nlater Copeland and Erd{\\H o}s\n\\cite{copeland_erdoes1946:note_on_normal} proved that the number\n\\begin{align*}\n0.2\\,3\\,5\\,7\\,11\\,13\\,17\\,19\\,23\\,29\\,31\\,37\\,41\\,43\\,47\\,53\\,59\\,61\\,67\\dots\n\\end{align*}\nis normal in base $10$. Similar to the construction above we want to\nconsider the number\n\\begin{gather*}\n\\tau_q=\\tau_q(f)=0.\\left\\lfloor f(2)\\right\\rfloor_q \\left\\lfloor f(3)\\right\\rfloor_q \\left\\lfloor f(5)\\right\\rfloor_q \\left\\lfloor f(7)\\right\\rfloor_q \\left\\lfloor f(11)\\right\\rfloor_q \\left\\lfloor f(13)\\right\\rfloor_q \\dots,\n\\end{gather*}\nwhere the arguments of $f$ run through the sequence of primes.\n\nThen the paper of Copeland and Erd{\\H o}s corresponds to the function\n$f(x)=x$. Nakai and Shiokawa\n\\cite{nakai_shiokawa1997:normality_numbers_generated} showed that the\ndiscrepancy for polynomials having rational coefficients is\n$\\mathcal{O}((\\log N)^{-1})$. Furthermore Madritsch, Thuswaldner and\nTichy\n\\cite{madritsch_thuswaldner_tichy2008:normality_numbers_generated}\nshowed, that transcendental entire functions of bounded logarithmic\norder yield normal numbers. Finally in a recent paper Madritsch and\nTichy \\cite{madritsch_tichy2013:construction_normal_numbers}\nconsidered pseudo-polynomials of the special form $\\alpha x^\\beta$\nwith $\\alpha>0$, $\\beta>1$ and $\\beta\\not\\in\\mathbb{Z}$.\n\nThe aim of the present paper is to extend this last construction to\narbitrary pseudo-polynomials. Our first main result is the following\n\\begin{thm}\\label{thm:normal}\nLet $f$ be a pseudo-polynomial as in (\\ref{mani:pseudopoly}). Then\n\\[\n\\mathcal{R}_N(\\tau_q(f))\\ll(\\log N)^{-1}.\n\\]\n\\end{thm}\n\n\nIn our second main result we use the connection of this construction\nof normal numbers with the arithmetic mean of $q$-additive functions\nas described above. Known results are due to Shiokawa\n\\cite{shiokawa1974:sum_digits_prime} and Madritsch and Tichy\n\\cite{madritsch_tichy2013:construction_normal_numbers}. Similar\nresults concerning the moments of the sum of digits function over\nprimes have been established by K\\'atai\n\\cite{katai1977:sum_digits_primes}.\n\nLet $\\pi(x)$ stand for the number of primes less than or equal to\n$x$. Then adapting these ideas to our method we obtain the following\n\\begin{thm}\\label{thm:summatoryfun}\nLet $f$ be a pseudo-polynomial as in (\\ref{mani:pseudopoly}). Then \n\\[\n\\sum_{p\\leq P}s_q(\\left\\lfloor f(p)\\right\\rfloor)=\\frac{q-1}2\\pi(P)\\log_qP^\\beta+\\mathcal{O}(\\pi(P)),\n\\]\nwhere the sum runs over the primes and the implicit $\\mathcal{O}$-constant may\ndepend on $q$ and $\\beta$.\n\\end{thm}\n\n\\begin{rem}\nWith simple modifications Theorem \\ref{thm:summatoryfun} can be extended to\ncompletely $q$-additive functions replacing $s_q$.\n\\end{rem}\n\n\nThe proof of the two theorems is divided into four parts. In the\nfollowing section we rewrite both statements in order to obtain as a\ncommon base the central theorem -- Theorem \\ref{mani:centralthm}. In\nSection~\\ref{sec:proof-prop-refm1} we start with the proof of this\ncentral theorem by using an indicator function and its Fourier\nseries. These series contain exponential sums which we treat by\ndifferent methods (with respect to the position in the expansion) in\nSection \\ref{sec:expon-sum-estim}. Finally, in\nSection~\\ref{sec:proof-prop-refm2} we put the estimates together in\norder to proof the central theorem and therefore our two statements.\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n\nThroughout the rest $p$ will always denote a prime. The implicit\nconstant of $\\ll$ and $\\mathcal{O}$ may depend on the\npseudo-polynomial $f$ and on the parameter\n$\\varepsilon>0$. Furthermore we fix a block $d_1\\cdots d_\\ell$ of\nlength $\\ell$ and $N$, the number of digits we consider.\n\nIn the first step we want to know in the\nexpansion of which prime the $N$-th digit occurs. This can be seen as\nthe translation from the digital world to the world of blocks. To this\nend let $\\ell(m)$ denote the length of the $q$-ary\nexpansion of an integer $m$. Then we define an integer $P$ by\n\\begin{gather*}\n\\sum_{p\\leq P-1}\\ell\\left(\\lfloor f(p)\\rfloor\\right) <N\\leq\n\\sum_{p\\leq P}\\ell\\left(\\lfloor f(p)\\rfloor\\right),\n\\end{gather*}\nwhere the sum runs over all primes. Thus we get the following relation\nbetween $N$ and $P$\n\\begin{equation}\\label{mani:NtoP}\n\\begin{split}\nN&=\\sum_{p\\leq P}\\ell(\\left\\lfloor f(p)\\right\\rfloor)+\\mathcal{O}(\\pi(P))+\\mathcal{O}(\\beta \\log_q(P))\\\\\n&=\\frac{\\beta}{\\log q}P+\\mathcal{O}\\left(\\frac{P}{\\log P}\\right).\n\\end{split}\\end{equation}\nHere we have used the prime number theorem in the form (\\textit{cf.}\n\\cite[Th\\'eor\\`eme 4.1]{tenenbaum1995:introduction_la_theorie}) \n\\begin{gather}\\label{pnt}\n  \\pi(x)=\\mathrm{Li}\\, x+\\mathcal{O}\\left(\\frac x{(\\log x)^G}\\right),\n\\end{gather}\nwhere $G$ is an arbitrary positive constant and\n\\[\n  \\mathrm{Li}\\,x=\\int_2^x\\frac{\\mathrm{d}t}{\\log t}.\n\\]\n\nNow we show that we may neglect the occurrences of the block\n$d_1\\cdots d_\\ell$ between two expansions. We write\n$\\mathcal{N}(f(p))$ for the number of occurrences of this block in the\n$q$-ary expansion of $\\lfloor f(p)\\rfloor$. Then \\eqref{mani:NtoP}\nimplies that\n\\begin{gather}\\label{mani:Ntrunc}\n  \\lvert\\mathcal{N}(\\tau_q(f);d_1\\cdots d_\\ell;N)-\\sum_{p\\leq\n    P}\\mathcal{N}(f(p))\\rvert\\ll\\frac N{\\log N}.\n\\end{gather}\n\nIn the next step we use the polynomial-like behavior of $f$. In\nparticular, we collect all the values having the same length of\nexpansion. Let $j_0$ be a sufficiently large integer. Then for\neach integer $j\\geq j_0$ there exists a $P_j$ such that\n\\[\n  q^{j-2}\\leq f(P_j)<q^{j-1}\\leq f(P_j+1)<q^j\n\\]\nwith\n\\[\n  P_j\\asymp q^{\\frac j\\beta}.\n\\]\nFurthermore we set $J$ to be the greatest length of the $q$-ary\nexpansions of $f(p)$ over the primes $p\\leq P$, i.e.,\n\\begin{gather*}\nJ:=\\max_{p\\leq P}\\ell(\\lfloor f(p)\\rfloor)=\\log_q(f(P))+\\mathcal{O}(1)\\asymp\\log\nP.\n\\end{gather*}\n\nNow we show that we may suppose that each expansion has the same\nlength (which we reach by adding leading zeroes). For $P_{j-1}<p\\leq\nP_j$ we may write $f(p)$ in $q$-ary expansion, i.e.,\n\\begin{gather}\\label{mani:expansionoffp}\nf(p)=b_{j-1}q^{j-1}+b_{j-2}q^{j-2}+\\dots+b_{1}q+b_{0}+b_{-1}q^{-1}+\\dots.\n\\end{gather}\nThen we denote by $\\mathcal{N}^*(f(p))$ the number of occurrences of the block\n$d_1\\cdots d_\\ell$ in the string $0\\cdots0b_{j-1}b_{j-2}\\cdots b_1b_0$, where we\nfilled up the expansion with leading zeroes such that it has length $J$. The error of\ndoing so can be estimated by \n\\begin{align*}\n0&\\leq\\sum_{p\\leq P}\\mathcal{N}^*(f(p))-\\sum_{p\\leq P}\\mathcal{N}(f(p))\\\\\n&\\leq\\sum_{j=j_0+1}^{J-1}(J-j)\\left(\\pi(P_{j+1})-\\pi(P_{j})\\right)+\\mathcal{O}(1)\\\\\n&\\leq\\sum_{j=j_0+2}^{J}\\pi(P_{j})+\\mathcal{O}(1)\\ll\\sum_{j=j_0+2}^{J}\\frac{q^{j\/\\beta}}j\n\\ll\\frac P{\\log P}\\ll\\frac N{\\log N}.\n\\end{align*}\n\nIn the following three sections we will estimate this sum of indicator\nfunctions $\\mathcal{N}^*$ in order to prove the following theorem.\n\\begin{thm}\\label{mani:centralthm}\nLet $f$ be a pseudo polynomial as in \\eqref{mani:pseudopoly}. Then\n\\begin{gather}\\label{mani:centralthm:statement}\n\\sum_{p\\leq\n  P}\\mathcal{N}^*\\left(\\left\\lfloor f(p)\\right\\rfloor\\right)=q^{-\\ell}\\pi(P)\\log_qP^\\beta+\\mathcal{O}\\left(\\frac{P}{\\log\n  P}\\right)\n\\end{gather}\n\\end{thm}\n\nUsing this theorem we can simply deduce our two main results.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:normal}]\nWe insert \\eqref{mani:centralthm:statement} into \\eqref{mani:Ntrunc}\nand get the desired result.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:summatoryfun}]\nFor this proof we have to rewrite the statement. In particular, we use that the\nsum of digits function counts the number of $1$s, $2$s, etc. and\nassigns weights to them, i.e.,\n\\[\ns_q(n)=\\sum_{d=0}^{q-1}d\\cdot\\mathcal{N}(n;d).\n\\]\nThus\n\\begin{align*}\n\\sum_{p\\leq P}s_q(\\left\\lfloor p^\\beta\\right\\rfloor)\n&=\\sum_{p\\leq P}\\sum_{d=0}^{q-1}d\\cdot\\mathcal{N}(p^\\beta)\n=\\sum_{p\\leq\n  P}\\sum_{d=0}^{q-1}d\\cdot\\mathcal{N}^*(p^\\beta)+\\mathcal{O}\\left(\\frac{P}{\\log\n    P}\\right)\\\\\n&=\\frac{q-1}2\\pi(P)\\log_q(P^\\beta)+\\mathcal{O}\\left(\\frac{P}{\\log\n    P}\\right)\n\\end{align*}\nand the theorem follows.\n\\end{proof}\n\n\nIn the following sections we will prove Theorem \\ref{mani:centralthm}\nin several steps. First we use the ``method of little glasses'' in\norder to approximate the indicator function by a Fourier series having\nsmooth coefficients. Then we will apply different methods (depending\non the position in the expansion) for the estimation of the\nexponential sums that appear in the Fourier series. Finally we put\neverything together and get the desired estimate.\n\n\\section{Proof of Theorem \\ref{mani:centralthm}, Part I}\\label{sec:proof-prop-refm1}\nWe want to ease notation by splitting the pseudo-polynomial $f$ into a polynomial and the rest. Then\nthere exists a unique decomposition of the following form:\n\\begin{gather}\\label{pseudo:poly:split}\nf(x)=g(x)+h(x)\n\\end{gather}\nwhere $h\\in\\mathbb{R}[X]$ is a polynomial of degree $k$ (where we set\n$k=0$ if $h$ is the zero polynomial) and\n$$g(x)=\\sum_{j=1}^r\\alpha_jx^{\\theta_j}$$\nwith $r\\geq1$, $\\alpha_r\\neq0$, $\\alpha_j$ real, $0<\\theta_1<\\cdots<\\theta_r$\nand $\\theta_j\\not\\in\\mathbb{Z}$ for $1\\leq j\\leq r$.\n\nLet $\\gamma$ and $\\rho$ be two parameter which we will frequently\nuse in the sequel. We suppose that\n\\begin{gather*}\n0<\\gamma<\\rho<\\min\\left(\\frac1{4(k+1)},\\frac{\\theta_r}{2}\\right).\n\\end{gather*}\n\nThe aim of this section is to calculate the Fourier transform of\n$\\mathcal{N}^*$. In order to count the occurrences of the block\n$d_1\\cdots d_\\ell$ in the $q$-ary expansion of $\\lfloor f(p) \\rfloor$\n($2\\le p \\le P$) we define the indicator function\n\\begin{align*}\n\\mathcal{I}(t)=\\begin{cases}\n  1, &\\text{if }\\sum_{i=1}^\\ell d_iq^{-i}\\leq t-\\lfloor t\\rfloor\n     <\\sum_{i=1}^\\ell d_iq^{-i}+q^{-\\ell};\\\\\n  0, &\\text{otherwise;}\n     \\end{cases}\n\\end{align*}\nwhich is a $1$-periodic function. Indeed, we have\n\\begin{gather}\\label{position}\n\\mathcal{I}(q^{-j}f(p)) = 1 \\Longleftrightarrow d_1\\cdots d_\\ell =\nb_{j-1}\\cdots b_{j-\\ell},\n\\end{gather}\nwhere $f(p)$ has an expansion as in (\\ref{mani:expansionoffp}). Thus\nwe may write our block counting function as follows\n\\begin{gather}\\label{mani:NthetatoNstar}\n\\mathcal{N}^*(f(p))=\\sum_{j=\\ell}^J\\mathcal{I}\\left(q^{-j}f(p)\\right).\n\\end{gather}\n\n\nIn the following we will use Vinogradov's ``method of little glasses''\n(\\textit{cf.} \\cite{vinogradov2004:method_trigonometrical_sums}). We want to approximate\n$\\mathcal{I}$ from above and from below by two $1$-periodic functions\nhaving small Fourier coefficients. To this end we will use the\nfollowing\n\\begin{lem}[{\\cite[Lemma\n    12]{vinogradov2004:method_trigonometrical_sums}}]\\label{vin:lem12}\nLet $\\alpha$, $\\beta$, $\\Delta$ be real numbers satisfying\n\\begin{gather*}\n0<\\Delta<\\frac12,\\quad\\Delta\\leq\\beta-\\alpha\\leq1-\\Delta.\n\\end{gather*}\nThen there exists a periodic function $\\psi(x)$ with period $1$,\nsatisfying\n\\begin{enumerate}\n\\item $\\psi(x)=1$ in the interval $\\alpha+\\frac12\\Delta\\leq x\n  \\leq\\beta-\\frac12\\Delta$,\n\\item $\\psi(x)=0$ in the interval $\\beta+\\frac12\\Delta\\leq x\n  \\leq1+\\alpha-\\frac12\\Delta$,\n\\item $0\\leq\\psi(x)\\leq1$ in the remainder of the interval\n  $\\alpha-\\frac12\\Delta\\leq x\\leq1+\\alpha-\\frac12\\Delta$,\n\\item $\\psi(x)$ has a Fourier series expansion of the form\n  $$\n  \\psi(x)=\\beta-\\alpha+\\sum_{\\substack{\\nu=-\\infty\\\\\\nu\\neq0}}^\\infty\n   A(\\nu) e(\\nu x),\n  $$\n  where\n  \\begin{gather}\\label{mani:A}\n  \\lvert A(\\nu)\\rvert \\ll \\min \\left( \\frac 1\\nu,\n  \\beta-\\alpha,\\frac{1}{\\nu^2\\Delta} \\right).\n  \\end{gather}\n\\end{enumerate}\n\\end{lem}\n\nWe note that we could have used Vaaler polynomials\n\\cite{vaaler1985:some_extremal_functions}, however, we do not gain\nanything by doing so as the estimates we get are already best\npossible. Setting\n\\begin{equation}\\label{mani:abd}\n\\begin{split}\n\\delta=P^{-\\gamma},\\quad\n\\begin{aligned}\n\\alpha_-&=\\sum_{\\lambda=1}^\\ell d_\\lambda q^{-\\lambda}+(2\\delta)^{-1},&\n\\beta_-&=\\sum_{\\lambda=1}^\\ell d_\\lambda q^{-\\lambda}+q^{-\\ell}-(2\\delta)^{-1},\\\\\n\\alpha_+&=\\sum_{\\lambda=1}^\\ell d_\\lambda q^{-\\lambda}-(2\\delta)^{-1},&\n\\beta_+&=\\sum_{\\lambda=1}^\\ell d_\\lambda q^{-\\lambda}+q^{-\\ell}+(2\\delta)^{-1}.\n\\end{aligned}\n\\end{split}\n\\end{equation}\nand an application of Lemma \\ref{vin:lem12} with\n$(\\alpha,\\beta,\\delta)=(\\alpha_-,\\beta_-,\\delta)$ and\n$(\\alpha,\\beta,\\delta)=(\\alpha_+,\\beta_+, \\delta)$,\nrespectively, provides us with two functions $\\mathcal{I}_-$ and\n$\\mathcal{I}_+$. By our choice of\n$(\\alpha_\\pm,\\beta_\\pm,\\delta)$ it is immediate that\n\\begin{equation}\\label{uglI}\n\\mathcal{I}_-(t)\\leq\\mathcal{I}(t)\\leq\\mathcal{I}_+(t) \\qquad\n(t\\in\\mathbb{R}).\n\\end{equation}\nLemma \\ref{vin:lem12} also implies that these two functions have\nFourier expansions\n\\begin{align}\\label{mani:Ifourier}\n\\mathcal{I}_\\pm(t)=q^{-\\ell}\\pm P^{-\\gamma}+\n  \\sum_{\\substack{\\nu=-\\infty\\\\\\nu\\neq0}}^\\infty A_\\pm(\\nu)e(\\nu t)\n\\end{align}\nsatisfying\n\\begin{gather*\n\\lvert A_\\pm(\\nu)\\rvert\n\\ll\\min(\\lvert\\nu\\rvert^{-1},P^{\\gamma}\\lvert\\nu\\rvert^{-2}).\n\\end{gather*}\nIn a next step we want to replace $\\mathcal{I}$ by $\\mathcal{I}_+$\nin (\\ref{mani:NthetatoNstar}). For this purpose we observe, using \\eqref{uglI},\nand \\eqref{mani:Ifourier} that\n\\begin{gather*}\n\\lvert\\mathcal{I}(t)-q^{-\\ell}\\rvert\n  \\ll P^{-\\gamma} + \\sum_{\\substack{\\nu=-\\infty\\\\\\nu\\neq0}}^\\infty\n  A_\\pm(\\nu)e(\\nu t).\n\\end{gather*}\nThus setting $t=q^{-j}f(p)$ and summing over $p\\leq P$ yields\n\\begin{gather}\\label{mani:0.5}\n\\lvert\\sum_{p\\leq P}\\mathcal{I}(q^{-j}f(p))-\\frac{\\pi(P)}{q^{\\ell}}\\rvert\n\\ll\\pi(P)P^{-\\gamma}+\\sum_{\\substack{\\nu=-\\infty\\\\\\nu\\neq0}}^\\infty\nA_{\\pm}(\\nu)\\sum_{p\\leq P}e\\left(\\frac{\\nu}{q^j}f(p)\\right).\n\\end{gather}\n\nNow we consider the coefficients $A_\\pm(\\nu)$. Noting\n\\eqref{mani:A} one observes that\n\\begin{gather*}\nA_\\pm(\\nu)\\ll\\begin{cases}\n  \\nu^{-1},       &\\text{for }\\lvert\\nu\\rvert\\leq P^{\\gamma};\\\\\n  P^{\\gamma}\\nu^{-2}, &\\text{for }\\lvert\\nu\\rvert>P^{\\gamma}.\n          \\end{cases}\n\\end{gather*}\nEstimating all summands with $\\lvert\\nu\\rvert>P^{\\gamma}$ trivially we get\n\\begin{gather*\n\\sum_{\\substack{\\nu=-\\infty\\\\\\nu\\neq0}}^\\infty\n  A_\\pm(\\nu)e\\left(\\frac{\\nu}{q^j}f(p)\\right)\n\\ll\\sum_{\\nu=1}^{P^{\\gamma}}\\nu^{-1}e\\left(\\frac{\\nu}{q^j}f(p)\\right)+P^{-\\gamma}.\n\\end{gather*}\nUsing this in \\eqref{mani:0.5} yields\n\\begin{gather*}\n\\lvert\\sum_{p\\leq P}\\mathcal{I}(q^{-j}f(p))-\\frac{\\pi(P)}{q^{\\ell}}\\rvert\n\\ll\\pi(P)P^{-\\gamma}+\\sum_{\\nu=1}^{P^{\\gamma}}\n\\nu^{-1}S(P,j,\\nu),\n\\end{gather*}\nwhere we have set \n\\begin{gather}\\label{S_Pjnu}\nS(P,j,\\nu):=\\sum_{p\\leq P}e\\left(\\frac{\\nu}{q^j}f(p)\\right).\n\\end{gather}\n\n\\section{Exponential sum estimates}\\label{sec:expon-sum-estim}\nIn the present section we will focus on the estimation of the sum\n$S(P,j,\\nu)$ for different ranges of $j$. Since $j$ describes the\nposition within the $q$-ary expansion of $f(p)$ we will call these\nranges the ``most significant digits'', the ``least significant\ndigits'' and the ``digits in the middle'', respectively.\n\nNow, if $\\theta_r>k\\geq0$, \\textit{i.e} the leading coefficient of $f$ origins\nfrom the pseudo polynomial part $g$, then we consider the two ranges\n$$1\\leq q^j\\leq P^{\\theta_r-\\rho}\\quad\\text{and}\\quad\nP^{\\theta_r-\\rho}<q^j\\leq P^{\\theta_r}.$$\nFor the first one we will apply Proposition \\ref{prop:least_significant} and\nfor the second one Proposition \\ref{prop:most_significant}.\n\nOn the other hand, if $k>\\theta_r>0$, meaning that the leading coefficient of\n$f$ origins from the polynomial part $h$, then we have an additional\npart. In particular, in this case we will consider the three ranges\n$$1\\leq q^j\\leq P^{\\theta_r-\\rho},\\quad\nP^{\\theta_r-\\rho}<q^j\\leq P^{k-1+\\rho},\\quad\\text{and}\\quad\nP^{k-1+\\rho}<q^j\\leq P^{k}.$$ We will, similar to above, treat the\nfirst and last range by Proposition~\\ref{prop:least_significant} and\nProposition \\ref{prop:most_significant}, respectively. For the middle\nrange we will apply Proposition \\ref{prop:middle_digits}. Since\n$2\\rho<\\theta_r$, we note that the middle range is empty if $k=1$.\n\nSince the size of $j$ represents the position of the digit in the\nexpansion (\\textit{cf.} \\eqref{position}), we will deal in the\nfollowing subsection with the ``most significant digits'', the ``least\nsignificant digits'' and the ``digits in the middle'', respectively.\n\n\\subsection{Most significant digits}\nWe start our series of estimates for the exponential sum $S(P,j,\\nu)$\nfor $j$ being in the highest range. In particular, we want to show the\nfollowing\n\\begin{prop}\\label{prop:most_significant}\nSuppose that for some $k\\geq1$ we have $\\lvert\nf^{(k)}(x)\\rvert\\geq\\Lambda$ for any $x$ on $[a,b]$ with\n$\\Lambda>0$. Then\n$$S(P,j,\\nu)\\ll\\frac1{\\log P}\\Lambda^{-\\frac1k}+\\frac{P}{(\\log P)^G}.$$\n\\end{prop}\n\nThe main idea of the proof is to use Riemann-Stieltjes integration\ntogether with \n\\begin{lem}[{\\cite[Lemma 8.10]{iwaniec_kowalski2004:analytic_number_theory}}]\n\\label{ik:lem8.10}\nLet $F\\colon[a,b]\\to\\mathbb{R}$ and suppose that for some $k\\geq1$\nwe have $\\lvert F^{(k)}(x)\\rvert\\geq\\Lambda$ for any $x$ on $[a,b]$\nwith $\\Lambda>0$. Then\n\\[\n\\lvert\\int_a^be(F(x))\\mathrm{d}x\\rvert\n\\leq k2^k\\Lambda^{-1\/k}.\n\\]\n\\end{lem}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:most_significant}]\nWe rewrite the sum into a Riemann-Stieltjes integral:\n\\begin{align*}\nS(P,j,\\nu)=\\sum_{p\\leq P}e\\left(\\frac{\\nu}{q^j}f(p)\\right)\n=\\int_{2}^{P}e\\left(\\frac{\\nu}{q^j}f(t)\\right)\\mathrm{d}\\pi(t)+\\mathcal{O}(1).\n\\end{align*}\nThen we apply the prime number theorem in the form\n\\eqref{pnt} to gain the usual integral back. Thus\n\\begin{align*}\nS(P,j,\\nu)\n=\\int_{P(\\log P)^{-G}}^{P}\ne\\left(\\frac{\\nu}{q^j}f(t)\\right)\n\\frac{\\mathrm{d}t}{\\log t}\n+\\mathcal{O}\\left(\\frac{P}{(\\log P)^G}\\right).\n\\end{align*}\nNow we use the second mean-value theorem to get\n\\begin{equation}\\label{mani:res:most}\n\\begin{split}\nS(P,j,\\nu)\\ll\\frac1{\\log P}\\sup_{\\xi}\n  \\lvert\\int_{P(\\log P)^{-G}}^{\\xi}e\\left(\\frac{\\nu}{q^j}f(t)\\right)\\mathrm{d}t\\rvert\n  +\\frac{P}{(\\log P)^G}.\n\\end{split}\n\\end{equation}\nFinally an application of Lemma \\ref{ik:lem8.10} proves the lemma.\n\\end{proof}\n\n\\subsection{Least significant digits} Now we turn our attention to the\nlowest range of $j$. In particular, the goal is the proof of the\nfollowing\n\\begin{prop}\\label{prop:least_significant}\nLet $P$ and $\\rho$ be positive reals and $f$ be a pseudo-polynomial as in\n\\eqref{pseudo:poly:split}. If $j$ is such that\n\\begin{gather}\\label{mani:gammarange}\n  1\\leq q^j\\leq P^{\\theta_r-\\rho}\n\\end{gather}\nholds, then for $1\\leq\\nu\\leq P^\\gamma$ there exists $\\eta>0$\n(depending only on $f$ and $\\rho$) such that\n\\begin{gather*}\n  S(P,j,\\nu)=(\\log P)^8P^{1-\\eta}.\n\\end{gather*}\n\\end{prop}\n\n\nBefore we launch into the proof we collect some tools that will be\nnecessary in the sequel. A standard idea for estimating exponential\nsums over the primes is to rewrite them into ordinary exponential sums\nover the integers having von Mangoldt's function as weights and then\nto apply Vaughan's identity. We denote by\n\\[\n\\Lambda(n)=\\begin{cases}\n\\log p,&\\text{if $n=p^k$ for some prime $p$ and an integer $k\\geq1$;}\\\\\n0,&\\text{otherwise}.\n\\end{cases}\n\\]\nvon Mangoldt's function. For the rewriting process we use the following\n\\begin{lem}\n\\label{mr:lem11}\nLet $g$ be a function such that $\\lvert g(n)\\rvert\\leq 1$ for all\nintegers $n$. Then\n\\[\n\\lvert\\sum_{p\\leq P}g(p)\\rvert\\ll\\frac1{\\log P}\\max_{t\\leq P}\n\\lvert\\sum_{n\\leq t}\\Lambda(n)g(n)\\rvert+\\mathcal{O}(\\sqrt{P}).\n\\]\n\\end{lem}\n\n\\begin{proof}\nThis is Lemma 11 of \\cite{mauduit_rivat2010:sur_un_probleme}. However,\nthe proof is short and we need some piece later.\n\nWe start with a summation by parts yielding\n\\[\\sum_{p\\leq P}g(p)=\\frac1{\\log P}\\sum_{p\\leq\n  x}\\log(p)g(p)+\\int_2^P\\left(\\sum_{p\\leq\n    t}\\log(p)g(p)\\right)\\frac{\\mathrm{d}t}{t(\\log t)^2}.\\]\n\nNow we cut the integral at $\\sqrt{P}$ and use Chebyshev's inequality\n(\\textit{cf.} \\cite[Th\\'eor\\`eme 1.3]{tenenbaum1995:introduction_la_theorie})\nin the form $\\sum_{p\\leq\n  t}\\log(p)\\leq\\log(t)\\pi(t)\\ll t$ for the lower part. Thus\n\\begin{align*}\n\\lvert\\sum_{p\\leq P}g(p)\\rvert\n&\\leq\\left(\\frac1{\\log\n    P}+\\int_{\\sqrt{P}}^P\\frac{\\mathrm{d}t}{t(\\log t)^2}\\right)\n  \\max_{\\sqrt{P}<t\\leq P}\\lvert\\sum_{p\\leq\n    P}\\log(p)g(p)\\rvert+\\mathcal{O}(\\sqrt{P})\\\\\n&=\\frac2{\\log P}\\max_{\\sqrt{P}<t\\leq P}\\lvert\\sum_{p\\leq t}\\log(p)g(p)\\rvert+\\mathcal{O}(\\sqrt{P}).\n\\end{align*}\n\nFinally we again use Chebyshev's inequality $\\pi(t)\\ll t\/\\log(t)$ to obtain\n\\begin{gather}\\label{mani:log_Mangoldt_equivalence}\n\\lvert\\sum_{n\\leq t}\\Lambda(n)g(n)-\\sum_{p\\leq t}\\log(p)g(p)\\rvert\n\\leq\\sum_{p\\leq\\sqrt{t}}\\log(p)\\sum_{a=2}^{\\left\\lfloor\\frac{\\log(t)}{\\log(p)}\\right\\rfloor}1\n\\leq\\pi(\\sqrt{t})\\log(t)\\ll\\sqrt{t}.\n\\end{gather}\n\\end{proof}\n\nIn the next step we use Vaughan's identity to subdivide this weighted\nexponential sum into several sums of Type I and II.\n\\begin{lem}[{\\cite[Lemma\n    2.3]{bergelson_kolesnik_madritsch+2014:uniform_distribution_prime}}]\n\\label{bkmst:lem23}\nAssume $F(x)$ to be any function defined on the real line, supported on $[P\/2, P]$\nand bounded by $F_0$. Let further $U,V,Z$ be any parameters satisfying $3\n\\leq U < V < Z < P$, $Z \\geq 4U^2$, $P \\geq 64 Z^2 U$, $V^3 \\geq 32 P$ and\n$Z-\\frac12\\in\\mathbb{N}$. Then\n$$\\left| \\sum_{P\/2< n\\leq P} \\Lambda(n) F(n)  \\right| \\ll K\n\\log P + F_0 +  L (\\log P)^8 ,$$\nwhere $K$ and $L$ are defined by\n\\begin{align*}\nK&=\\max_M\\sum_{m=1}^\\infty d_3(m)\\left\\vert\\sum\\limits_{Z<n\\leq M} F(mn)\\right\\vert,\\\\\nL&=\\sup\\sum_{m=1}^\\infty d_4(m)\\left\\vert\\sum\\limits_{U < n < V} b(n) F(mn)\\right\\vert,\n\\end{align*}\nwhere the supremum is taken over all arithmetic functions $b(n)$ satisfying $|b(n)| \\leq d_3(n).$\n\\end{lem}\n\nAfter subdividing the weighted exponential sum with Vaughan's identity\nwe will use the following lemma in order to estimate the occurring\nexponential sums.\n\\begin{lem}[{\\cite[Lemma\n     2.5]{bergelson_kolesnik_madritsch+2014:uniform_distribution_prime}}]\n\\label{bkmst:lem25}\n   Let $X,k,q\\in \\mathbb{N}$ with $k,q\\geq 0$ and set $K=2^k$ and $Q=\n   2^q$. Let $h(x)$ be a polynomial of degree $k$ with real\n   coefficients. Let $g(x)$ be a real $(q+k+2)$ times continuously\n   differentiable function on $[X\/2 , X]$ such that $\\left| f^{(r)}(x)\n   \\right| \\asymp F X^{-r}$ $( r = 1, \\dots, q+k+2) $.  Then, if $F =\n   o (X^{q+2})$ for $F$ and $X$ large enough, we have\n $$\\left| \\sum_{X\/2 < x \\leq X} e(g(x) + h(x)) \\right| \\ll X^{1 - \\frac{1}{K}} + X \\left( \\frac{\\log^k X}{F} \\right)^{\\frac{1}{K}} + X \\left( \\frac{F}{X^{q+2}} \\right)^{\\frac{1}{(4KQ-2K)}}.$$\n\\end{lem}\n\nNow we have the necessary tools to state the\n\\begin{proof}[Proof of Proposition \\ref{prop:least_significant}]\nAn application of Lemma \\ref{mr:lem11} yields\n\\[S(P,j,\\nu)\\ll\\frac1{\\log P}\\max\\lvert\\sum_{n\\leq\n  P}\\Lambda(n)e\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\\rvert+P^{\\frac12}.\\]\nWe split the inner sum into $\\leq \\log\nP$ sub sums of the form $$\\lvert \\sum\\limits_{X< n \\leq\n  2X}\\Lambda(n)e\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\\rvert$$ with\n$2X \\leq P$ and let $S$ be a typical one of them. We may\nassume that $X \\geq P^{1-\\rho}$.\n\nUsing Vaughan's identity (Lemma~\\ref{bkmst:lem23}) with $U = \\frac{1}{4} X^{1\/5}$, $V= 4 X^{1\/3}$ and $Z$ the\nunique number in $\\frac12+\\mathbb{N}$, which is closest to $\\frac{1}{4}\nX^{2\/5}$, we obtain\n\\begin{gather}\\label{mani:S}\nS \\ll 1+(\\log X)S_1+(\\log X)^8S_2,\n\\end{gather}\nwhere\n\\begin{align*}\nS_1&=\\sum_{x < \\frac{2X}{Z}} d_3(x) \\sum_{y > Z, \\frac{X}{x} < y < \\frac{2X}{x}} e\\left(\\frac{\\nu}{q^j}(g(xy)+h(xy))\\right)\\\\\nS_2&=\\sum_{\\frac XV<x\\leq\\frac{2X}U} d_4(x) \\sum_{U < y < V, \\frac{X}{x} < y \\leq \\frac{2X}{x}} b(y) e\\left(\\frac{\\nu}{q^j}(g(xy)+h(xy))\\right)\\notag\n\\end{align*}\n\nWe start with the estimation of $S_1$. Since $d_3(x)\\ll\nx^{\\varepsilon}$ we have for\n\\begin{align*}\n\\lvert S_1\\rvert\\ll X^\\varepsilon\\sum_{x\\leq\\frac{2X}Z}\n  \\lvert\\sum_{\\substack{\\frac Xx<y\\frac{2X}x\\\\y>Z}}e\\left(\\frac{\\nu}{q^j}(g(xy)+h(xy))\\right)\\rvert.\n\\end{align*}\nFor estimating the inner sum we fix $x$ and denote $Y=\\frac Xx$. Since\n$\\theta_r\\not\\in\\mathbb{Z}$ and $\\theta_r>k\\geq0$, we have that\n\\[\\lvert\\frac{\\partial^\\ell g(xy)}{\\partial y^\\ell}\\rvert\n\\asymp X^{\\theta_r}Y^{-\\ell}.\\]\n\nNow on the one hand, since $q^j\\leq P^{\\theta_r-\\rho}$, we have $\\nu\nq^{-j}X^{\\theta_r}\\gg X^{\\rho}$. On the other hand for\n$\\ell\\geq5(\\lfloor\\theta_r\\rfloor+1)$ we get\n\\[\\frac{\\nu}{q^j}X^{\\theta_r}Y^{-\\ell}\\leq P^\\gamma X^{\\theta_r-\\frac25\\ell}\\ll X^{-\\frac12}.\\] \n\nThus an application of Lemma \\ref{bkmst:lem25} yields the following\nestimate:\n\\begin{equation}\\label{mani:estim:S1}\n\\begin{split}\n\\lvert S_1\\rvert &\\ll X^{\\varepsilon}\\sum_{x \\leq 2X\/Z} Y \\left[\n  Y^{-\\frac{1}{K}} + (\\log Y)^kX^{-\\frac{\\rho}{K}} + X^{-\\frac{1}{2} \\frac{1}{4K \\cdot 8L^5 - 2K}} \\right] \\\\\n&\\ll X^{1+\\varepsilon}(\\log X)\\left(X^{-\\rho} + X^{-\\frac{1}{64L^5-4} } \\right)^{\\frac1K}, \n\\end{split}\\end{equation}\nwhere we have used that $\\frac kK<1$ and $\\rho<\\frac13$.\n\nFor the second sum $S_2$ we start by splitting the interval $(\n\\frac{X}{V} , \\frac{2X}{U} ]$ into $\\leq \\log X$ subintervals of the\nform $(X_1, 2X_1]$. Thus\n\\begin{align*}\n\\lvert S_2\\rvert\n&\\leq (\\log X)X^{\\varepsilon}\\sum_{X_1<x\\leq\n  2X_1}\\lvert\\sum_{\\substack{U<y<V\\\\\\frac\n    Xx<y\\leq\\frac{2X}x}}b(y)e\\left(\\frac{\\nu}{q^j}(g(xy)+h(xy))\\right)\\rvert\n\\end{align*}\n\nNow an application of Cauchy's inequality together with $\\lvert\nb(y)\\rvert\\ll X^\\varepsilon$ yields\n\\begin{align*}\n\\lvert S_2\\rvert^2\n&\\leq (\\log X)^2X^{2\\varepsilon}X_1\\sum_{X_1<x\\leq\n  2X_1}\\lvert\\sum_{\\substack{U<y<V\\\\\\frac\n    Xx<y\\leq\\frac{2X}x}}b(y)e\\left(\\frac{\\nu}{q^j}(g(xy)+h(xy))\\right)\\rvert^2\\\\\n&\\ll (\\log X)^2X^{4\\varepsilon}X_1\\\\\n&\\quad\\times\\left(X_1\\frac{X}{X_1}+\\lvert \\sum_{X_1<x\\leq2X_1} \\sum_{A < y_1 < y_2 \\leq B}e\\left(\\frac{\\nu}{q^j} (g(xy_1)-g(xy_2) + h(xy_1)-h(xy_2))\\right)  \\rvert\\right)\n\\end{align*}\nwhere $A = \\max \\{U, \\frac{X}{x} \\} $ and\n$B = \\min \\{U, \\frac{2X}{x} \\}$. Changing the order of summation, we\nget\n\\begin{multline*}\n|S_2|^2 \\ll (\\log X)^2X^{4\\varepsilon}X_1\\\\\n\\times\\left(X+\n  \\sum_{A < y_1 < y_2 \\leq B}\\lvert \\sum_{X_1<x\\leq2X_1} e\\left(\\frac{\\nu}{q^j} (g(xy_1)-g(xy_2) + h(xy_1)-h(xy_2))\\right)  \\rvert\\right)\n\\end{multline*}\n\nAs above we want to apply Lemma \\ref{bkmst:lem25}. To this end we fix\n$y_1$ and $y_2 \\ne y_1$. Similarly to above we get that\n$$\\lvert\\frac{\\partial^\\ell\\left(g(xy_1)-g(xy_2)+h(xy_1)-h(xy_2)\\right)}{\\partial x^\\ell}\\rvert\\asymp\\frac{\\lvert\n  y_1-y_2\\rvert}{y_1}X^{\\theta_r}X_1^{-\\ell}.$$\nNow, on the one hand we have $\\frac{\\nu}{q^j}\\frac{\\lvert\n  y_1-y_2\\rvert}{y_1}X^{\\theta_r}\\gg X^{\\rho}$ and on the other hand\n\\[\\frac{\\nu}{q^j}\\frac{\\lvert\n  y_1-y_2\\rvert}{y_1}X^{\\theta_r}X_1^{-\\ell}\n\\ll X^{\\gamma+\\theta_r}\\left(\\frac{X}{V}\\right)^{-\\ell}\n\\ll X^{\\gamma+\\theta_r-\\frac23\\ell}\n\\ll X^{-\\frac12}\\]\nif $\\ell\\geq2\\lfloor\\theta_r\\rfloor+3$. Thus again an application of Lemma \\ref{bkmst:lem25} yields\n\\begin{equation}\\label{mani:estim:S2}\n\\begin{split}\n\\lvert S_2\\rvert^2 &\\ll (\\log X)^2X^{4\\varepsilon}X_1\\left(X +\n  \\sum_{A<y_1<y_2\\leq B} X_1\\left( X_1^{-\\frac{1}{K}} + X^{-\\frac{\\rho}{K}} + X^{-\\frac{1}{2} \\frac{1}{4K \\cdot 2L^2 - 2K}}\\right)\\right) \\\\\n&\\ll (\\log X)^2X^{4\\varepsilon}\\left(X^{\\frac53} + X^{2-\\frac{\\rho}{K}} + X^{2- \\frac{1}{16KL^2 - 4K}}\\right).\n\\end{split}\n\\end{equation}\n\nPlugging the two estimates \\eqref{mani:estim:S1} and\n\\eqref{mani:estim:S2} into \\eqref{mani:S} proves the proposition.\n\\end{proof}\n\n\\subsection{The digits in the middle}\n\nNow we are getting more involved in order to consider those $j$\nleading to a position between $\\theta_r$ and $k$. These sums\ncorrespond to the ``digits in the middle'' in the proof of Theorem\n\\ref{mani:centralthm}. We want to prove the following\n\\begin{prop}\\label{prop:middle_digits}\n  Let $P$ and $\\rho$ be positive reals and $f$ be a pseudo-polynomial as in\n  \\eqref{pseudo:poly:split}. If $2\\rho<\\theta_r<k$ and $j$ is such that\n\\begin{gather}\\label{mani:middle_range}\n  P^{\\theta_r-\\rho}< q^j\\leq P^{k-1+\\rho}\n\\end{gather}\nholds, then for $1\\leq\\nu\\leq P^\\gamma$ we have\n\\begin{gather*}\n  S(P,j,\\nu)=\\sum_{p\\leq P}e\\left(\\frac{\\nu f(p)}{q^j}\\right)\\ll P^{1-\\frac{\\rho}{4^k}}.\n\\end{gather*}\n\\end{prop}\n\nThe main idea in this range is to use that the dominant part of $f$\ncomes from the polynomial $h$. Therefore after getting rid of the\nfunction $g$ we will estimate the sum over the polynomial by the\nfollowing\n\\begin{lem}\\label{lem:exponential_sum_primes_poly}\nLet $h\\in\\mathbb{R}[X]$ be a\npolynomial of degree $k\\geq2$. Suppose $\\alpha$ is the leading\ncoefficient of $h$ and that there are integers $a$, $q$ such that\n$$\\lvert q\\alpha-a\\rvert<\\frac1q\\quad\\text{with}\\quad\n(a,q)=1.$$\nThen we have for any $\\varepsilon>0$ and $H\\leq X$\n$$\\sum_{X<p\\leq X+H}\\log(p)e(h(p))\\ll\nH^{1+\\varepsilon}\\left(\\frac1q+\\frac1{H^{\\frac12}}+\\frac{q}{H^k}\\right)^{4^{1-k}}.$$\n\\end{lem}\n\n\\begin{proof}\n  This is a slight variant of \\cite[Theorem\n  1]{harman1981:trigonometric_sums_over}, where we sum over an\n  interval of the form $]X,X+H]$ instead of one of the form\n  $]0,X]$.\n\\end{proof}\n\nNow we have enough tools to state the \n\\begin{proof}[Proof of Proposition \\ref{prop:middle_digits}]\n  As in the Proof of Proposition \\ref{prop:least_significant} we start\n  by an application of Lemma~\\ref{mr:lem11} yielding\n\\[S(P,j,\\nu)\\ll\\frac1{\\log P}\\max\\lvert\\sum_{n\\leq\n  P}\\Lambda(n)e\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\\rvert+P^{\\frac12}.\\]\n\nWe split the inner sum into $\\leq \\log P$ sub sums of the form\n\\[S:=\\sum_{X<n\\leq X+H}\\Lambda(n)e\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\\]\nwith $P^{1-2\\rho}\\leq X\\leq P$ and \\[H=\\min\\left(P^{1-\\theta_r}\\lvert\\nu\\rvert^{-1}q^j,X\\right).\\]\nNow we want to separate the function parts $g$ and $h$. Therefore we define two\nfunctions $T$ and $\\varphi$ by\n$$T(x)=\\sum_{X< n\\leq\n  X+x}\\Lambda(n)e\\left(\\frac{\\nu}{q^j}h(n)\\right)\n\\quad\\text{and}\\quad\n\\varphi(x):=e\\left(\\frac{\\nu}{q^j}g\\left(X+x\\right)\\right)$$\nThen an application of summation by parts yields\n\\begin{equation}\\label{mani:eq_1}\n\\begin{split}\n\\sum_{X< n\\leq X+H}\\Lambda(n)\ne\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\n&=\\sum_{n=1}^{H} \\varphi(n)(T(n)-T(n-1))\\\\\n&=\\sum_{n=1}^{H}T(n)\\left(\\varphi(n)-\\varphi(n+1)\\right)+\\varphi(H-1)T(H)\\\\\n&\\ll\\lvert T(H)\\rvert+\\sum_{n=1}^{H-1}\\left|\\varphi(n)-\\varphi(n+1)\\right|\\lvert T(n)\\rvert\n\\end{split}\n\\end{equation}\n\nLet $\\alpha_k$ be the leading coefficient of $P$. Then by Diophantine\napproximation there always exists a rational $a\/b$ with $b>0$,\n$(a,b)=1$,\n\\[1\\leq b\\leq H^{k-\\rho}\n\\quad\\text{and}\\quad\n\\lvert\\frac{\\nu\\alpha_k}{q^j}-\\frac ab\\rvert\\leq\n\\frac{H^{\\rho-k}}{b}.\\]\nWe distinguish three cases according to the size of $b$.\n\\begin{itemize}\n\\item[] \\textbf{Case 1.} $H^\\rho<b$. In this case we may apply Lemma\n  \\ref{lem:exponential_sum_primes_poly} together with\n  \\eqref{mani:log_Mangoldt_equivalence} to get $$T(h)\\ll\n  H^{1-\\frac{\\rho}{4^{k-1}}+\\varepsilon}.$$\n\\item[] \\textbf{Case 2.} $2\\leq b<H^\\rho$. In this case we get\n  that $$\\lvert\\frac{\\nu\\alpha_k}{q^j}\\rvert\\geq\\lvert\\frac\n  ab\\rvert-\\frac1{b^2}\\geq\\frac1{2b}\\geq\\frac12H^{-\\rho}\\geq\\frac12P^{-\\rho}.$$\n  Since $2\\rho<\\theta_r$, this contradicts our lower bound $q^j\\geq P^{\\theta_r-\\rho}$.\n\\item[] \\textbf{Case 3.} $b=1$. This case requires a further\n  distinction according to whether $a=0$ or not.\n  \\begin{itemize}\n  \\item[] \\textbf{Case 3.1.}\n    $\\lvert\\frac{\\nu\\alpha_k}{q^j}\\rvert\\geq\\frac12$. It follows\n    that $$q^j\\leq2\\lvert\\nu\\alpha_k\\rvert$$ again contradicting our lower\n    bound $q^j\\geq P^{\\theta_r-\\rho}$.\n  \\item[] \\textbf{Case 3.2.}\n    $\\lvert\\frac{\\nu\\alpha_k}{q^j}\\rvert<\\frac12$. This implies that\n    $a=0$ which yields \n    \\begin{gather}\\label{case3.2}\n    q^j\\geq\\lvert\\nu\\alpha_k\\rvert H^{k-\\rho}.\n    \\end{gather}\n    We distinguish two further cases according to\n    whether $P^{1-\\theta_r}\\lvert\\nu\\rvert^{-1}q^j\\leq X$ or not.\n    \\begin{itemize}\n    \\item[] \\textbf{Case 3.2.1}\n      $P^{1-\\theta_r}\\lvert\\nu\\rvert^{-1}q^j\\leq X$. This implies that\n      $q^j\\leq P^{\\theta_r}\\lvert\\nu\\rvert$\n      and \\[H=P^{1-\\theta_r}\\lvert\\nu\\rvert^{-1}q^j\\geq\n      P^{1-\\rho}\\lvert\\nu\\rvert^{-1}\\geq P^{1-2\\rho}.\\] Plugging these\n      estimates into \\eqref{case3.2} gives \\[P^{\\theta_r}\\geq\n      \\lvert\\alpha_k\\rvert P^{(1-2\\rho)(k-\\rho)}.\\] However, since\n      $4(k+1)\\rho<1$, we\n      have \\[(1-2\\rho)(k-\\rho)>k-1+2\\rho\\geq\\theta_r\\] yielding a\n      contradiction.\n    \\item[] \\textbf{Case 3.2.2}\n      $P^{1-\\theta_r}\\lvert\\nu\\rvert^{-1}q^j>X$. Then $H=X\\geq\n      P^{1-2\\rho}$ and \\eqref{case3.2}\n      becomes \\[P^{k-1+\\rho}\\geq\\lvert\\nu\\alpha_k\\rvert\n      P^{(1-2\\rho)(k-\\rho)}\\] yielding a similar contradiction as in\n      \\textbf{Case 3.2.1}.\n    \\end{itemize}\n  \\end{itemize}\n\\end{itemize}\nTherefore \\textbf{Case 1} is the only possible and we may always apply\nLemma \\ref{lem:exponential_sum_primes_poly} together with\n\\eqref{mani:log_Mangoldt_equivalence}. Plugging this\ninto~\\eqref{mani:eq_1} yields\n\\begin{align*}\n\\sum_{X< n\\leq X+H}\\Lambda(n)\ne\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\n&\\ll H^{1-\\frac{\\rho}{4^{k-1}}+\\varepsilon}\\left(1+\\sum_{X< n\\leq X+H}\\left|\\varphi(n)-\\varphi(n+1)\\right|\\right)\n\\end{align*}\n\nNow by our choice of $H$ together with an application of the mean\nvalue theorem we have that\n$$\\sum_{X\\leq n\\leq X+H}\\lvert \\varphi(n)-\\varphi(n+1)\\rvert\n\\ll H\\frac{\\nu}{q^j}P^{\\theta-1}\\ll 1.$$\nThus \n\\begin{align*}\n\\sum_{X\\leq n\\leq X+H} \\Lambda(n) \ne\\left(\\frac{\\nu}{q^j}(g(n)+h(n))\\right)\n\\ll H^{1-\\frac{\\rho}{4^{k-1}}+\\varepsilon}.\n\\end{align*}\n\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{mani:centralthm}, Part II}\\label{sec:proof-prop-refm2}\nNow we use all the tools from the section above in order to estimate\n\n\\begin{gather}\\label{distance_from_mean}\n\\sum_{j=\\ell}^J\\lvert\\sum_{p\\leq P}\\mathcal{I}(q^{-j}f(p))-\\frac{\\pi(P)}{q^{\\ell}}\\rvert\n\\ll\\pi(P)H^{-1}J+\\sum_{\\nu=1}^{H}\n\\nu^{-1}\\sum_{j=\\ell}^JS(P,j,\\nu).\n\\end{gather}\n\nAs indicated in the section above, we split the sum over $j$ into two\nor three parts according to whether $\\theta_r>k$ or not. In any case\nan application of Proposition \\ref{prop:least_significant} yields for the\nleast significant digits that\n\\begin{gather}\\label{estimate:least}\n\\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}\\sum_{1\\leq q^{j}\\leq\n  P^{\\theta_r-\\rho}} S(P,j,\\nu)\n\\ll (\\log P)^9JP^{1-\\eta}.\n\\end{gather}\n\nNow let us suppose that $\\theta_r>k$. Then an application of Proposition\n\\ref{prop:most_significant} yields\n\\begin{equation}\\label{estimate:most_non_integer}\n\\begin{split}\n\\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}&\\sum_{P^{\\theta_r-\\rho}< q^{j}\\leq\n  P^{\\theta_r}}S(P,j,\\nu)\\\\\n&\\ll \\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}\\sum_{P^{\\theta_r-\\rho}< q^{j}\\leq\n  P^{\\theta_r}}\\frac1{\\log\n  P}\\left(\\frac{\\nu}{q^j}\\right)^{-\\frac1{\\left\\lfloor\\theta_r\\right\\rfloor}}+\\frac{P}{(\\log P)^{G-2}}\\\\\n&\\ll \\frac{P}{\\log P}.\n\\end{split}\n\\end{equation}\n\nPlugging the estimates \\eqref{estimate:least} and\n\\eqref{estimate:most_non_integer} into~\\eqref{distance_from_mean} we\nget that\n$$\\sum_{j=\\ell}^J\\lvert\\sum_{p\\leq P}\\mathcal{I}(q^{-j}f(p))-\\frac{\\pi(P)}{q^{\\ell}}\\rvert\n\\ll\\frac{P}{\\log P},$$\nwhich together with \\eqref{mani:NthetatoNstar} proves Theorem\n\\ref{mani:centralthm} in the case that $\\theta_r>k$.\n\nOn the other side if $\\theta_r<k$, then we consider the two ranges\n$$P^{\\theta_r-\\rho}<q^j\\leq P^{k-1+\\rho}\n\\quad\\text{and}\\quad\nP^{k-1+\\rho}<q^j\\leq P^k.$$\nFor the ``digits in the middle'' an application of Proposition\n\\ref{prop:middle_digits} yields\n\\begin{equation}\\label{estimate:middle}\n\\begin{split}\n\\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}\\sum_{P^{\\theta_r-\\rho}< q^{j}\\leq\n  P^{k-1+\\rho}}S(P,j,\\nu)\n&\\ll \\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}\\sum_{P^{\\theta_r-\\rho}< q^{j}\\leq\n  P^{k-1+\\rho}}P^{1-\\frac{\\rho}{4^k}}\\\\\n&\\ll(\\log P)JP^{1-\\frac{\\rho}{4^k}}.\n\\end{split}\n\\end{equation}\n\nFinally we consider the most significant digits. By an application of\nProposition~\\ref{prop:most_significant} we have\n\\begin{equation}\\label{estimate:most_integer}\n\\begin{split}\n\\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}&\\sum_{P^{k-1+\\rho}< q^{j}\\leq\n  P^{k}}S(P,j,\\nu)\\\\\n&\\ll \\sum_{1\\leq \\nu\\leq P^\\gamma}\\nu^{-1}\\sum_{P^{k-1+\\rho}< q^{j}\\leq\n  P^{k}}\\frac1{\\log\n  P}\\left(\\frac{\\nu}{q^j}\\right)^{-\\frac1k}+\\frac{P}{(\\log P)^{G-2}}\\\\\n&\\ll \\frac{P}{\\log P}.\n\\end{split}\n\\end{equation}\n\nPlugging the estimates \\eqref{estimate:least}, \\eqref{estimate:middle} and\n\\eqref{estimate:most_integer} into~\\eqref{distance_from_mean} we\nget that\n$$\\sum_{j=\\ell}^J\\lvert\\sum_{p\\leq P}\\mathcal{I}(q^{-j}f(p))-\\frac{\\pi(P)}{q^{\\ell}}\\rvert\n\\ll\\frac{P}{\\log P},$$\nwhich together with \\eqref{mani:NthetatoNstar} proves Theorem\n\\ref{mani:centralthm} in the case that $\\theta_r<k$.\n\n\\section*{Acknowledgment}\nThe author wants to thank G{\\'e}rald Tenenbaum for many fruitful\ndiscussions and suggestions in connection with the proof of\nProposition \\ref{prop:middle_digits}.\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}\\setcounter{equation}{0}}\n\n\n\\newcommand{h_{\\rm gpi}}{h_{\\rm gpi}}\n\n\\newcommand{{\\rm diag}\\,}{{\\rm diag}\\,}\n\\newcommand{{\\rm sgn}\\,}{{\\rm sgn}\\,}\n\\newcommand{{\\rm sig}\\,}{{\\rm sig}\\,}\n\\newcommand{{\\rm Ran}\\,}{{\\rm Ran}\\,}\n\\newcommand{{\\rm rank}\\,}{{\\rm rank}\\,}\n\\newcommand{{\\rm span}\\,}{{\\rm span}\\,}\n\n\n\n\\begin{titlepage}\n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\n\\rightline{DAMTP-R94\/11}\n\\vspace{0.1in}\n\\LARGE\n\\center{Generalised Point Interactions for the \\\\ Radial\nSchr\\\"{o}dinger Equation\\\\ via Unitary Dilations}\n\\Large\n\\vspace{0.2in}\n\\center{C.J. Fewster\\footnote{E-mail address:\nC.J.Fewster@amtp.cam.ac.uk}}\n\\vspace{0.2in}\n\\large\n\\center{\\em Department of Applied Mathematics and Theoretical Physics,\n\\\\ University of Cambridge,\n\\\\  Silver Street, Cambridge CB3 9EW, U.K.}\n\\vspace{0.2in}\n\\center{July 8, 1994}\n\\small\\center{Revised November 28, 1994}\n\\vspace{0.2in}\n\n\\begin{abstract}\nWe present an inverse scattering construction of generalised point\ninteractions (GPI) -- point-like objects with non-trivial scattering\nbehaviour. The construction is developed for single centre $S$-wave\nGPI models with rational $S$-matrices, and starts from an integral\ntransform suggested by the scattering data. The theory of unitary\ndilations is then applied to construct a unitary mapping between\nPontryagin spaces which extend the usual position and momentum\nHilbert spaces. The GPI Hamiltonian is defined as a multiplication\noperator on the momentum Pontryagin space and its free parameters are\nfixed by a physical locality requirement. We determine the spectral\nproperties and domain of the Hamiltonian in general, and construct\nthe resolvent and M{\\o}ller wave operators thus verifying that the\nHamiltonian exhibits the required scattering behaviour. The physical\nHilbert space is identified. The construction is illustrated by GPI\nmodels representing the effective range approximation. For negative\neffective range we recover a known class of GPI models, whilst the\npositive effective range models appear to be new. We discuss the\ninterpretation of these models, along with possible extensions to our\nconstruction.\n\n\\vspace{0.3truecm}\n{\\noindent {\\bf PACS Numbers} 03.65.Nk, 03.65.Db}\n\\end{abstract}\n\n\\setcounter{footnote}{0}\n\\renewcommand{\\thefootnote}{\\arabic{footnote}}\n\\end{titlepage}\n\n\n\\sect{Introduction and Main Ideas}\n\nGeneralised point interactions (GPI) are solvable models in quantum\nmechanics representing point objects with non-trivial scattering\nbehaviour. The prototype for such models is the class of point\ninteractions (PI), corresponding to Hamiltonians with\n$\\delta$-function potentials essentially introduced by Fermi\n\\cite{Fermi} and rigorously defined as self-adjoint extensions of\n$-\\triangle$ on the domain of smooth functions compactly supported\naway from the interaction centre \\cite{BF}. This construction leads\nto 1-parameter families of PI Hamiltonians in dimensions 2 and 3\nwhich provide the leading order (scattering length) approximation to\nthe scattering behaviour of Schr\\\"{o}dinger operators with short\nrange potentials in the sector of zero angular momentum (see, for\nexample \\cite{KF}). We refer to \\cite{Alb} for an extensive\nbibliography on PI models.\n\nGPI models are employed to treat more general scattering behaviour,\nsuch as higher order corrections to $S$-wave scattering, non-trivial\nscattering in non-zero angular momentum sectors or point objects in\ndimensions $d\\ge 4$. Such models have  been studied for a long time\nfrom the pseudo-potential viewpoint in many body physics -- see\n\\cite{Huang,GWu}. The mathematical study of such generalised point\ninteraction (GPI) models began in the Russian literature with the\nwork of Shirokov \\cite{Shiro}, and more rigorous formulations were\nlater developed by Pavlov \\cite{Pav1,Pav2,Pav3} and Shondin\n\\cite{Shond1,Shond2} (see also \\cite{Diejen}). In contrast to PI\nmodels, GPI Hamiltonians are not defined on the usual Hilbert space\n$L^2({\\rm I\\! R}^d)$, but on an extended space whose inner product may be\nindefinite, in which case one must identify a physical Hilbert space\nof states in order to recover the probability interpretation of\nquantum mechanics.\n\nIn this paper, we introduce and develop a new inverse scattering\nconstruction for single centre GPI models with non-trivial $S$-wave\nscattering. It is currently an open problem to extend this\nconstruction to higher angular momenta and dimensions $d\\ge 4$; we\ndiscuss this further in the Conclusion. Our method is based on the\ntechnique of {\\em unitary dilations} due in origin to Sz.-Nagy\n\\cite{Nagy} and extended by Davis \\cite{Davis}.\n\nThe $S$-wave inverse scattering problem has been partially studied by\nShondin \\cite{Shond1}, as we describe below. First, let us briefly\nmention the two principal non-inverse GPI constructions, which we\ncall the {\\em auxiliary space} and {\\em distributional} methods. In\nthe auxiliary space method, developed by Pavlov and co-workers\n\\cite{Pav2,Pav3}, one starts with a given extended Hilbert space and\nthen seeks the class of GPI Hamiltonians which `live' on this space.\nOn the other hand, the distributional method of Shondin \\cite{Shond2}\nis a direct attempt to define $H=-\\triangle+\\ket{\\omega}\\bra{\\omega}$,\nwhere $\\omega$ is somewhere in the $j<0$ portion of the scale of\nSobolev spaces ${\\cal H}_j=(-\\triangle+1)^{-j\/2}L^2({\\rm I\\! R}^d)$. The\nconstruction leads to a Pontryagin space\\footnote{A Pontryagin space\nis an indefinite (Krein) inner product space with a finite rank of\nindefiniteness -- see Section~2.1.} $\\Pi_m={\\cal H}_0\\oplus\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}^{2m}$,\nwhere $m$ is the unique integer such that\n$\\omega\\in{\\cal H}_{-m-2}\\backslash{\\cal H}_{-m-1}$ and the inner product has\nsignature $(m,m)$ on the finite dimensional part. The GPI Hamiltonian\nis then defined on $\\Pi_m$ using Krein's formula \\cite{Akh}.\n\nReturning to the inverse problem, Shondin \\cite{Shond1} considered\nthe inverse scattering problem in $d=3$ for $S$-wave scattering data\nof form\n\\begin{equation}\n\\cot\\delta_0(k) = k^{-1}r(k^2), \\label{eq:GPIlo}\n\\end{equation}\nwhere $r(z)$ is a rational function with real coefficients. Shondin\nrefers to this class of data as the `$R$ class': it corresponds\nexactly with the class of rational $S$-wave $S$-matrices\nsatisfying the usual analytic continuation property $S(k)=S^*(-k)$\n\\cite{Newt}. In particular, the $R$ class contains truncated low\nenergy expansions of form\n\\begin{equation}\n\\cot\\delta_0(k) = -\\frac{1}{kL} + \\frac{1}{2}kr_0 + k^3r_1+\\cdots\n+k^{2n+1}r_n, \\label{eq:low}\n\\end{equation}\nfor low energy parameters $L,r_0,r_1,\\ldots,r_n$ and any $n\\ge 0$,\nand thus furnishes approximations to the low energy $S$-wave\nbehaviour of Schr\\\"{o}dinger operators with short range potentials to\narbitrary order. Shondin's method starts by writing down a candidate\nresolvent on a (positive definite) extension of the usual Hilbert\nspace. (In this respect it resembles the auxiliary space method).\nVarious free functions in this resolvent are then fixed by requiring\nthat the candidate be the resolvent of a self-adjoint operator.\nHowever, this method is limited to those $r(z)$ with negative\nimaginary part in the upper half plane, which is a somewhat\nrestrictive sub-class: for example, scattering data of\nform~(\\ref{eq:low}) is possible only with $r_1,\\ldots ,r_n=0$ and\n$r_0<0$. As we will see later, in the context of our method, more\ngeneral scattering data corresponds to GPI models defined on\nPontryagin spaces. Thus, in order to apply Shondin's method to such\ndata, one would have to guess not only the appropriate extension to\nthe Hilbert space, but also its inner product, thereby rendering it\nmuch less practical as a construction.\n\nIn contrast, the method proposed here allows one to treat the full\n$R$ class. It proceeds from the simple observation that, for\npoint-like interactions, the scattering data $\\delta_0(k)$\ncompletely specify the $S$-wave continuum eigenfunctions as $u_k(r)\n= (2\/\\pi)^{1\/2}\\sin (kr+\\delta_0(k))$, when scattering normalisation\nis imposed. This is quite different from the usual situation in\ninverse scattering theory, where the scattering data specifies (in\nthe first instance) only the asymptotic form of the $u_k(r)$. As a\nresult, the inverse scattering problem for GPI models can be solved\nwithout recourse to the usual Gel'fand-Levitan machinery.\n\nThe $u_k(r)$ define an integral transform ${\\cal T}$ between\n${\\cal H}_r=L^2((0,\\infty),dr)$ and ${\\cal H}_k=L^2((0,\\infty),dk)$, (i.e., the\nradial position and momentum Hilbert spaces) by\n\\begin{equation}\n({\\cal T} \\psi)(k) =\n\\left(\\frac{2}{\\pi}\\right)^{1\/2}\\int_0^\\infty \\sin\n(kr+\\delta_0(k))\\psi(r) dr,\n\\end{equation}\nWe adopt ${\\cal T}$ as a candidate for an eigenfunction transform\nassociated with our desired GPI Hamiltonian $h_{\\rm gpi}$. Of course, ${\\cal T}$\ncannot be the full eigenfunction transform unless it is unitary, in\nwhich case we can define $h_{\\rm gpi}={\\cal T}^* k^2{\\cal T}$. However, when ${\\cal T}$ is\nnon-unitary, we may `dilate' it to a unitary operator between\nenlarged inner product spaces, using the theory of unitary dilations\nas follows. Firstly, we quantify the departure of ${\\cal T}$ from\nunitarity by the operators  $M_1=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}-{\\cal T}\\Tt^*$ and\n$M_2=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}-{\\cal T}^*{\\cal T}$, and the closures ${\\cal M}_1$ and ${\\cal M}_2$ of their\nrespective ranges. For scattering data in the $R$ class, we will show\nthat the rank and signature of $M_1$ and $M_2$ are finite and given\nin terms of indices reminiscent of Levinson's theorem \\cite{Newt}.\n\nNext, we define indefinite inner products on the ${\\cal M}_i$, together\nwith an operator\n$\\hat{{\\cal T}}:{\\cal H}_r\\oplus{\\cal M}_1\\rightarrow{\\cal H}_k\\oplus{\\cal M}_2$ which is\nunitary with respect to the relevant inner products and satisfies\n$P_{{\\cal H}_k}\\hat{{\\cal T}}|_{{\\cal H}_r}={\\cal T}$, where $P_{{\\cal H}_k}$ is the\northoprojector onto ${\\cal H}_k$. $\\hat{{\\cal T}}$ is said to be a unitary\ndilation\\footnote{See Section~2 for a note on the nomenclature.}  of\n${\\cal T}$. We emphasise that the construction of $\\hat{{\\cal T}}$ and the\nenlarged inner product spaces requires no information beyond that\nencoded in ${\\cal T}$, and is unique up to a unitary equivalence and\nfurther dilation. To complete the construction, we define the GPI\nHamiltonian $h_{\\rm gpi}$ by\n\\begin{equation}\nh_{\\rm gpi} = \\hat{{\\cal T}}^\\dagger\n\\left(\\begin{array}{cc} k^2 & 0 \\\\ 0 & \\Lambda \\end{array}\n\\right)\n\\hat{{\\cal T}},\n\\end{equation}\nwhere the dagger denotes the Pontryagin space adjoint, and we have\nused an obvious block matrix notation.  We will show that $\\Lambda$\nis completely determined by imposing a physical locality requirement:\nthat the `interaction' be localised at the origin. Mathematically,\nthis is expressed by requiring the Hamiltonian agrees with the free\nHamiltonian away from the interaction centre, i.e., $h_{\\rm gpi}(\\psi,0)^T=\n(-\\psi^{\\prime\\prime},0)^T$ if $\\psi\\in C_0^\\infty(0,\\infty)$.\n{}Subject to this locality condition, we have thus constructed both\n$h_{\\rm gpi}$ and its spectral representation. The non-uniqueness in our\nconstruction leads to a family of unitarily equivalent GPI\nHamiltonians with the same spectral and scattering properties.\n\nOur plan is as follows. In Section~2, we briefly describe some\nfeatures of analysis in indefinite inner product spaces, and\nalso describe the construction of unitary dilations, essentially\nfollowing Davis \\cite{Davis}. In addition, we sketch our construction\nin a more abstract setting. Next, in Section~3, we explicitly\nconstruct the operators $M_1$ and $M_2$ (which are of finite rank)\nfor a generic subclass of the $R$ class -- those whose scattering\namplitudes exhibit only simple poles on the physical sheet -- and\ncompute their rank and signature. In Section~4, we construct $h_{\\rm gpi}$ as\ndescribed above. Subject to the locality condition, we show that the\neigenvalues of $h_{\\rm gpi}$ occur at precisely those energies for which the\nscattering amplitude derived from~(\\ref{eq:GPIlo}) exhibits poles on\nthe physical sheet, as is the case for ordinary scattering from `nice'\npotentials. We construct the corresponding eigenfunctions of $h_{\\rm gpi}$,\nand isolate the physical Hilbert space. We also determine the domain\nand resolvent of $h_{\\rm gpi}$, and explicitly construct its M{\\o}ller wave\noperators in a two-space setting, verifying that they exhibit the\nrequired scattering theory.\n\nIn Section 5, we illustrate our procedure by constructing GPI models\nwith scattering behaviour $\\cot\\delta_0(k)=-1\/(kL)+kM$, representing\nthe effective range approximation of low energy scattering theory\n\\cite{Newt}. In the case $M<0$, ${\\cal H}_r$ and ${\\cal H}_k$ are extended to\nlarger Hilbert spaces, and we recover the models of `type $B_2$'\npreviously constructed by Shondin \\cite{Shond1}. These models also\narise as a special case of the auxiliary space construction in\n\\cite{Pav1}. The case $M>0$, for which Pontryagin spaces are\nrequired, appears to be new. Our methods allow the entire  class of\nGPI Hamiltonians to be constructed, along with their spectral\nrepresentations. A particularly interesting subclass of the models\nconstructed corresponds to the case $L=\\infty$, with scattering theory\n$\\cot\\delta_0(k)=kM$. Such models reproduce the leading order\nbehaviour of non-point interactions exhibiting a zero energy\nresonance. We refer to these models as {\\em resonance point\ninteractions} (RPI).\n\nWe also discuss how these GPI models may be used as models for\nSchr\\\"{o}dinger operators with spherically symmetric potentials of\ncompact support. To do this, we employ a general methodology for\ndiscussing the `large scale effects of small objects' developed by\nKay and the author \\cite{KF}. In particular, we develop {\\em fitting\nformulae} (analogous to those given in \\cite{KF}) for matching a\ngiven potential $V(r)$ to the `best fit' GPI model. Finally, in\nSection 6, we conclude by discussing various extensions to our\nmethod.\n\nThe motivation for the present work arose in a consideration of the\nscattering of charged particles off magnetic flux tubes of small\nradius \\cite{FK}, in which it was found that the scattering lengths\nfor spin-$\\frac{1}{2}$ particles generically take the values $0$ or\n$\\infty$ in certain angular momentum sectors. In consequence, the\nanalogue of PI models representing dynamics in\nthe background of an infinitesimally thin wire of flux fails to\ndescribe the leading order scattering theory in these sectors, and\nshould be replaced by models analogous to the RPI models mentioned\nabove. The special nature of this system can be attributed to the\nfact that it is an example of supersymmetric quantum mechanics.\nElsewhere \\cite{F}, we will construct the appropriate class of RPI\nfor this system.\n\n\n\\sect{Preliminaries}\n\\subsection{Unitary Dilations}\n\nWe begin by describing the unitary dilation theory required in the\nsequel. Let ${\\cal H}_1,\\ldots,{\\cal H}_4$ be Hilbert spaces and\n$T\\in{\\cal L}({\\cal H}_1,{\\cal H}_2)$. Then\n$\\hat{T}\\in{\\cal L}({\\cal H}_1\\oplus{\\cal H}_3,{\\cal H}_2\\oplus{\\cal H}_4)$ is called a\n{\\em dilation} of $T$ if\n$T= P_{{\\cal H}_2}\\hat{T}|_{{\\cal H}_1}$\nwhere $P_{{\\cal H}_2}$ is the orthogonal projector onto ${\\cal H}_2$. In\nblock matrix form, $\\hat{T}$ takes form\n\\begin{equation}\n\\hat{T} = \\left(\\begin{array}{cc} T & P \\\\ Q & R \\end{array}\\right).\n\\label{eq:dilfm}\n\\end{equation}\nOur nomenclature follows that of Halmos \\cite{Halmos}. Elsewhere\n(e.g., in the work of Davis \\cite{Davis}), the term `dilation' (or\n`dilatation') often means that $\\hat{T}^n$ is a dilation of $T^n$\nand $(\\hat{T}^*)^n$ is a dilation of $(T^*)^n$ for each $n=1,2,\\ldots$\n(in addition, ${\\cal H}_1={\\cal H}_2$, and ${\\cal H}_3={\\cal H}_4$). We refer to such\noperators as {\\em power dilations}: in the block\nform~(\\ref{eq:dilfm}), this requires $PR^nQ=0$ for each\n$n=0,1,2,\\ldots$.\n\nAccording to a result of Sz.-Nagy \\cite{Nagy}, any contraction $T$\nfrom one Hilbert space to another (i.e., a bounded operator\nsatisfying $\\|T\\|\\le 1$) has a unitary dilation between larger\nHilbert spaces. Subsequently, Davis \\cite{Davis} extended this\nresult to arbitrary closed densely defined operators at the\ncost of introducing indefinite inner product spaces. (It is clear\nthat if $\\|T\\|>1$, no Hilbert space unitary dilation is possible.)\nIn fact, Davis' construction yields a unitary {\\em power} dilation\nof the original operator.  This has no physical relevance in our\nconstruction, and so we use a more economical `cut-down' version of\nDavis' result, described below. First, we briefly review the salient\nfeatures of analysis in indefinite inner product spaces. Full\ntreatments can be found in the monographs of Bogn\\'ar \\cite{Bognar}\nand Azizov and Iokhvidov~\\cite{Azizov}.\n\nWe employ a particular class of indefinite inner product\nspaces known as {\\em $J$-spaces}. Let ${\\cal H}$ be a Hilbert space with\n(positive definite) inner product $\\inner{\\cdot}{\\cdot}$,\nequipped with a unitary involution, $J$. We define a non-degenerate\nindefinite inner product $[\\cdot,\\cdot]$ on ${\\cal H}$ by\n\\begin{equation}\n[x,y]=\\inner{x}{Jy},\n\\end{equation}\nwhich we call the {\\em $J$-inner product}. ${\\cal H}$ equipped with the\n$J$-inner product is called a $J$-space. ${\\cal H}$ admits decomposition\n${\\cal H}={\\cal H}_+\\oplus{\\cal H}_-={\\cal H}_+[+]{\\cal H}_-$ into the eigenspaces ${\\cal H}_\\pm$\nof $J$ with eigenvalue $\\pm 1$, where $[+]$ denotes the orthogonal\ndirect sum in the $J$-inner product. If at least one of the\n${\\cal H}_\\pm$ is finite dimensional, then ${\\cal H}$ is a {\\em Pontryagin\nspace} with respect to $[\\cdot,\\cdot]$ .\n\nThe topology of a $J$-space is determined by the Hilbert space norm;\nhowever, operator adjoints and the notion of unitarity are defined\nrelative to the $J$-inner product. Thus if ${\\cal H}_i$ ($i=1,2$) are\n$J_i$-spaces, and $T\\in{\\cal L}({\\cal H}_1,{\\cal H}_2)$, the {\\em\n$(J_1,J_2)$-adjoint} $T^\\dagger$ of $T$ is defined in terms of the\nHilbert space adjoint $T^*$ by\n\\begin{equation}\nT^\\dagger = J_1T^*J_2.\n\\end{equation}\nEquivalently, $[T^\\dagger x,y]_{{\\cal H}_1}=[x,Ty]_{{\\cal H}_2}$ for all\n$x\\in{\\cal H}_2$, $y\\in{\\cal H}_1$. If $[Ux,Uy]_{{\\cal H}_2}=[x,y]_{{\\cal H}_1}$ for all\n$x,y\\in {\\cal D}\\subset{\\cal H}_1$, $U$ is said to be {\\em\n$(J_1,J_2)$-isometric}; if in addition $U$ is a linear isomorphism of\n${\\cal H}_1$ and ${\\cal H}_2$, and ${\\cal D}={\\cal H}_1$, $U$ is said to be {\\em\n$(J_1,J_2)$-unitary}. Equivalently, $UU^\\dagger = \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal H}_1}$ and\n$U^\\dagger U = \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal H}_2}$. If ${\\cal H}_1={\\cal H}_2$ with $J_1=J_2=J$, terms\nsuch as $(J_1,J_2)$-isometric are abbreviated to $J$-isometric etc.\n\nReturning to the construction of unitary dilations, let $T$ be any\nbounded operator $T\\in{\\cal L}({\\cal H}_1,{\\cal H}_2)$, and define operators\n$M_1 = \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}-TT^*$ and $M_2=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}-T^*T$. It is trivial to show that the\nrespective closures ${\\cal M}_i=\\overline{{\\rm Ran}\\, M_i}$ of their ranges are\n${\\rm sgn}\\, (M_i)$-spaces, and hence that ${\\cal K}_i={\\cal H}_i\\oplus{\\cal M}_i$ are\n$J_i$-spaces, where $J_i=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal H}_i}\\oplus{\\rm sgn}\\,(M_i)$. We now define\na dilation $\\hat{T}$ of $T$ by\n\\begin{equation}\n\\hat{T} = \\left(\\begin{array}{cc} T & -{\\rm sgn}\\,(M_1)|M_1|^{1\/2} \\\\\n                                  |M_2|^{1\/2} & T^*|_{{\\cal M}_1}\n                \\end{array} \\right),\n\\end{equation}\nwhich has $(J_1,J_2)$-adjoint $\\hat{T}^\\dagger$ equal to\n\\begin{equation}\n\\hat{T}^\\dagger = J_1\\hat{T}^*J_2=\n \\left(\\begin{array}{cc}\n                            T^* & {\\rm sgn}\\, (M_2) |M_2|^{1\/2} \\\\\n                            - |M_1|^{1\/2} & T|_{{\\cal M}_2}\n                        \\end{array} \\right).\n\\end{equation}\nHere, we have used the intertwining relations $Tf(T^*T)=f(TT^*)T$ and\n$T^*f(TT^*)=f(T^*T)T^*$, which hold for any continuous Borel function\n$f$. It is now easy to show that $\\hat{T}^\\dagger\\hat{T}=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal K}_1}$\nand $\\hat{T}\\hat{T}^\\dagger=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal K}_2}$, thus verifying that\n$\\hat{T}$ is a $(J_1,J_2)$-unitary dilation of $T$. In our\napplication, $M_1$ and $M_2$ are finite rank, and so the $J$-spaces\nconstructed above are Pontryagin spaces.\n\nWe briefly consider the uniqueness of the unitary dilations\nconstructed above. Suppose ${\\cal N}_i$ are $J_i$\nspaces ($i=1,2$) and that $\\tilde{T}:{\\cal H}_1\\oplus{\\cal N}_1 \\rightarrow\n{\\cal H}_2\\oplus{\\cal N}_2$ is a unitary dilation of $T$ with matrix\nform~(\\ref{eq:dilfm}). Then, provided that the $M_i$ are finite\nrank, one may show that\n\\begin{equation}\nP_{{\\cal H}_2\\oplus {\\cal Q}}\n\\tilde{T}|_{{\\cal H}_1\\oplus {\\cal P}}=\n\\left(\\begin{array}{cc} \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1} & 0 \\\\ 0 & U_2 \\end{array}\\right)\n\\hat{T}\n\\left(\\begin{array}{cc} \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1} & 0 \\\\ 0 & U_1^\\dagger\n\\end{array}\\right),\n\\end{equation}\nwhere ${\\cal P}=P^\\dagger\\overline{{\\rm Ran}\\, M_1}$, ${\\cal Q}=Q\\overline{{\\rm Ran}\\,\nM_1}$, and $U_1$ and $U_2$ are unitaries (with respect to the\n$J$-inner products) from ${\\cal M}_1$ and ${\\cal M}_2$ to ${\\cal P}$ and ${\\cal Q}$\nrespectively. In addition, $P_{{\\cal H}_2\\oplus {\\cal Q}}$ is an orthogonal\nprojection onto ${\\cal H}_2\\oplus{\\cal Q}$ in ${\\cal H}_2\\oplus{\\cal N}_2$.\n\nThus $\\hat{T}$ is unique up to further dilation and unitary\nequivalence of the above form. If the $M_i$ are not of finite rank,\nthis statement also holds if the $M_i$ are strictly positive. More\ngenerally, it is not clear whether ${\\cal Q}$ is necessarily\northocomplemented, and therefore whether $P_{{\\cal H}_2\\oplus {\\cal Q}}$ exists.\n\n\\subsection{Abstract Setting}\n\\label{sect:abs}\n\nIn this section, we sketch our construction in a general setting,\nwhich makes clear how it may be extended. In particular, we show how\nthe domain and action of the Hamiltonian is determined.\n\nLet ${\\cal H}_i$ ($i=1,2$) be Hilbert spaces and let $A$ be a densely\ndefined symmetric operator with domain ${\\cal D}\\subset {\\cal H}_1$. Suppose\nthat $A$ possesses two self-adjoint extensions $A_\\pm$ such that\n\\begin{equation}\nA_\\pm = {\\cal T}_\\pm^* \\tilde{A}{\\cal T}_\\pm\n\\end{equation}\nwhere $\\tilde{A}$ is a self-adjoint operator on ${\\cal H}_2$ with\n$(\\tilde{A}+\\omega)^{-1}$ bounded for some $\\omega\\in{\\rm I\\! R}$, and\n${\\cal T}_\\pm$ are unitary operators ${\\cal T}_\\pm :{\\cal H}_1\\rightarrow{\\cal H}_2$. Let\n$a_+$ and $a_-$ be bounded operators on ${\\cal H}_2$ which commute with\n$\\tilde{A}$ and define\n\\begin{equation}\n{\\cal T} = a_+{\\cal T}_+ + a_-{\\cal T}_-.\n\\end{equation}\nIn our application, $a_\\pm$ are determined by the scattering\ndata. We define $M_1$ and $M_2$ as above, for simplicity\nassuming that they are finite rank (as they are in our application).\nThe unitary dilation $\\hat{{\\cal T}}$ derived above is then used to\ndefine a self-adjoint operator $B$ on the Pontryagin space $\\Pi_1\n={\\cal H}_1\\oplus{\\cal M}_1$ by\n\\begin{equation}\nB = \\hat{{\\cal T}}^\\dagger\n\\left(\\begin{array}{cc} \\tilde{A} & 0 \\\\ 0 & \\Lambda \\end{array}\n\\right)\n\\hat{{\\cal T}},\n\\end{equation}\nwhere $\\Lambda$ is a self-adjoint operator on ${\\cal M}_2$ (with respect\nto its inner product). Thus\n\\begin{equation}\nB\\left(\\begin{array}{c} \\varphi\\\\ \\Phi\\end{array}\\right)=\n\\left(\\begin{array}{c}\n{\\cal T}^*\\tilde{A}({\\cal T}\\varphi-\\Theta)\n+{\\rm sgn}\\, M_2|M_2|^{1\/2}\\Lambda(|M_2|^{1\/2}\\varphi+{\\cal T}^*|_{{\\cal M}_1}\\Phi) \\\\\n-|M_1|^{1\/2}\\tilde{A}({\\cal T}\\varphi-\\Theta)\n+{\\cal T}|_{{\\cal M}_2}\\Lambda (|M_2|^{1\/2}\\varphi+{\\cal T}^*|_{{\\cal M}_1}\\Phi)\n\\end{array}\\right),\n\\label{eq:actB}\n\\end{equation}\nwhere $\\Theta={\\rm sgn}\\, M_1 |M_1|^{1\/2}\\Phi$ (considered as an element of\n${\\cal H}_2$), and $B$ has domain\n\\begin{equation}\nD(B) = \\{ (\\varphi,\\Phi)^T\\mid {\\cal T}\\varphi-\\Theta\n\\in D(\\tilde{A})\\}. \\label{eq:domB}\n\\end{equation}\n{}To gain a more explicit description of $D(B)$, we impose the\nrequirement that $B$ be a self-adjoint extension of the\n{\\em non-densely defined} operator $A\\oplus 0$ on ${\\cal D}\\oplus\n0\\subset\\Pi_1$, i.e., $B(\\varphi,0)^T=(A\\varphi,0)^T$ for all\n$\\varphi\\in{\\cal D}$. Later this will carry the physical interpretation of\na locality condition. It is easy to show that this requirement is\nsatisfied if and only if ${\\cal M}_2$ is invariant under $A^*$ and\n\\begin{equation}\n\\Lambda=(|M_2|^{-1\/2}A^*|_{{\\cal M}_2}|M_2|^{1\/2})^*.\n\\end{equation}\n\nAs a consequence of locality, we note that if  $(\\varphi,\\Phi)^T\\in\nD(B)$ with $B (\\varphi,\\Phi)^T=(\\tilde{\\varphi},\\tilde{\\Phi})^T$,\nthen  $\\varphi\\in D(A^*)$, and $\\tilde{\\varphi}=A^*\\varphi$. For take\nany $\\psi\\in {\\cal D}$. Then\n\\begin{equation}\n\\inner{\\tilde{\\varphi}}{\\psi}_{{\\cal H}_1} =\n\\left[\n\\left(\\begin{array}{c}\\tilde{\\varphi}\\\\\n\\tilde{\\Phi}\\end{array}\\right), \\left(\\begin{array}{c}\\psi\\\\\n0\\end{array}\\right)\\right]_{\\Pi_1} =\n\\left[\n\\left(\\begin{array}{c}\\varphi\\\\ \\Phi\\end{array}\\right),\nB\\left(\\begin{array}{c}\\psi\\\\ 0\\end{array}\\right)\n\\right]_{\\Pi_1}= \\inner{\\varphi}{A\\psi}_{{\\cal H}_1}.\n\\end{equation}\nWe may therefore re-write~(\\ref{eq:domB}) as\n\\begin{equation}\nD(B)=\\left\\{ \\left(\\begin{array}{c} \\varphi \\\\ \\Phi \\end{array}\n\\right) \\mid \\varphi\\in D(A^*), \\quad \\Theta_1 \\in D(\\tilde{A})\n\\right\\},\n\\end{equation}\nwhere $\\Theta_1 =a_+{\\cal T}_+\\chi_+ + a_-{\\cal T}_-\\chi_-+\\Theta$ and\n$\\chi_\\pm=(A_\\pm+\\omega)^{-1}(A^*+\\omega)\\varphi-\\varphi$. The\nadvantage of this expression is that $\\chi_\\pm$ can be shown to be the\nunique element of $\\ker (A^*+\\omega)$ such that $\\varphi+\\chi_\\pm\\in\nD(A_\\pm)$. In our application, $\\chi_\\pm$ may be expressed in terms\nof the value of $\\varphi$ and its first derivative at the origin.\n\n{}To determine the action of $B$ more explicitly, we use the fact that\nthe upper component of the right-hand side of~(\\ref{eq:actB}) is\nequal to $A^*\\varphi$ in order to compute $\\tilde{\\Theta}={\\rm sgn}\\,\nM_1|M_1|^{1\/2} \\tilde{\\Phi}$. We obtain\n\\begin{equation}\n\\tilde{\\Theta}=-M_1\\tilde{A}({\\cal T}\\varphi-\\Theta)+\n{\\cal T}(A^*\\varphi-{\\cal T}^*\\tilde{A}({\\cal T}\\varphi-\\Theta))=\n{\\cal T} A^*\\varphi-\\tilde{A}({\\cal T}\\varphi-\\Theta)\n\\end{equation}\nUsing the fact that $\\Theta_1\\in D(\\tilde{A})$, this becomes\n\\begin{equation}\n\\tilde{\\Theta}=\\tilde{A}\\Theta_1 +\\omega(\\Theta_1-\\Theta) +\n{\\cal T}(A^*+\\omega)\\varphi-(\\tilde{A}+\\omega)\n({\\cal T}\\varphi+\\Theta_1-\\Theta).\n\\end{equation}\nThe last two terms cancel by definition of $\\chi_\\pm$ and we\nconclude that\n\\begin{equation}\nB\\left(\\begin{array}{c}\\varphi\\\\ \\Phi\\end{array}\\right)=\n\\left(\\begin{array}{c} A^*\\varphi \\\\\n({\\rm sgn}\\, M_1 |M_1|^{1\/2})^{-1}\\tilde{\\Theta}\n\\end{array}\\right)\n\\end{equation}\nwhere $\\tilde{\\Theta}=\\tilde{A}\\Theta_1 + \\omega(a_+{\\cal T}_+\\chi_+ +\na_-{\\cal T}_-\\chi_-)$.\n\n\\sect{Determination of $M_1$ and $M_2$}\n\nIn this section, we determine the operators $M_1=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}-{\\cal T}\\Tt^*$ and\n$M_2=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}-{\\cal T}^*{\\cal T}$, where ${\\cal T}$ is an integral transformation\narising from the scattering data in the Shondin $R$ class\n\\cite{Shond1} given by\n\\begin{equation}\n\\cot\\delta_0(k) = k^{-1}\\frac{p(k^2)}{q(k^2)},\\qquad\n\\delta_\\ell(k)\\equiv 0\\quad {\\rm for}~\\ell\\ge 1,\n\\label{eq:GPIlow}\n\\end{equation}\nwhere $p(z)$ and $q(z)$ are coprime polynomials in ${\\rm I\\! R}[z]$, the\nring of polynomials with real coefficients. In particular, we will\nshow how the rank and signature of the $M_i$ are determined by two\n`Levinson indices' defined below. We emphasise that our methods are\nvery different to those of Shondin.\n\nThe scattering amplitude corresponding to $\\delta_0(k)$ is\n\\begin{equation}\nf_0(k) = \\frac{1}{k}e^{i\\delta_0(k)}\\sin\\delta_0(k)=\n\\frac{q(k^2)}{p(k^2) - ikq(k^2)}.\n\\end{equation}\nDefining the polynomial $W(z)$ by\n\\begin{equation}\nW(z) =\\left\\{\\begin{array}{cl} p(-z^2)-zq(-z^2) & p(0)\\not=0 \\\\\n                               p(-z^2)\/z-q(-z^2) & p(0)=0,\n\\end{array}\\right. \\label{eq:W}\n\\end{equation}\nwe note that $f_0(k)$ exhibits poles where $W(ik)=0$. The set\n$\\Omega$ of zeros of $W(z)$ in the left-hand half-plane ${\\rm Re}\\,\nz<0$ corresponds to poles of $f_0(k)$ such that $k^2$ lies on the\nphysical sheet. We refer to the situation where these poles\n(and hence the corresponding zeros of $W(z)$) are simple as the\n{\\em generic case}. In Theorem~\\ref{Thm:local}, we will show\nthat the discrete spectrum of the GPI Hamiltonian is precisely\n$\\{E=-\\omega^2\\mid\\omega\\in\\Omega\\}$ under the requirement of\nlocality.\\footnote{These eigenvalues can be complex: we will return\nto this point in section~5.3.}\n\nThe qualitative features of the scattering data~(\\ref{eq:GPIlow})\nare described by the degrees of $p$ and $q$, two indices $I_L^\\pm$\ndefined below, and the asymptotic behaviour of $\\cot\\delta_0(k)$\ngiven by\n\\begin{equation}\n\\sigma_0 = {\\rm sgn}\\,\\lim_{k\\rightarrow 0^+}\n\\cot\\delta_0(k)\\qquad {\\rm and}\\qquad \\sigma_\\infty\n={\\rm sgn}\\,\\lim_{k\\rightarrow\\infty}\\cot\\delta_0(k),\n\\end{equation}\nwhere the limits are allowed to be $\\pm\\infty$. The indices\n$I_L^\\pm$ are defined by\n\\begin{equation}\nI_L^+= \\frac{\\delta_0(0)-\\delta_0(\\infty)}{\\pi}\\qquad{\\rm and}\n\\qquad\nI_L^-=\\frac{\\zeta(0)-\\zeta(\\infty)}{\\pi},\n\\end{equation}\nwhere the auxiliary scattering data $\\zeta(k)$ is defined\nas a continuous function on ${\\rm I\\! R}^+$ by\n\\begin{equation}\n\\cot\\zeta(k) = -k^{-1}\\frac{p(-k^2)}{q(-k^2)}. \\label{eq:zeta}\n\\end{equation}\nWe refer to $I_L^\\pm$ as the Levinson indices (although Levinson's\ntheorem \\cite{Newt} will not hold in its usual form).\n\nWe now define the integral transform\n${\\cal T}=\\cos\\delta_0(k){\\cal S}+\\sin\\delta_0(k){\\cal C}$, which is suggested by\nthe\nna\\\"{\\i}ve generalised eigenfunctions $u_k(r) =\n(2\/\\pi)^{1\/2}\\sin (kr+\\delta_0(k))$.\nHere, ${\\cal S}$ and ${\\cal C}$ are the sine and cosine transforms,\ndefined by\n\\begin{eqnarray}\n({\\cal S} \\psi)(k) =\n\\sqrt{\\frac{2}{\\pi}}\\int_0^\\infty dr\\, \\psi(r)\\sin kr  & {\\rm and}\n& ({\\cal C} \\psi)(k) =\n\\sqrt{\\frac{2}{\\pi}}\\int_0^\\infty dr\\, \\psi(r)\\cos kr\n\\end{eqnarray}\n(the integrals are intended as limits in $L^2$-norm).\nBoth are unitary maps from ${\\cal H}_r$ to ${\\cal H}_k$;\ntheir inverses have the same form, with $r$ and $k$ exchanged.\nThus ${\\cal T}$ is given explicitly by\n\\begin{equation}\n{\\cal T} = \\frac{p(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}{\\cal S} +\n\\frac{kq(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}} {\\cal C}.\n\\end{equation}\nBecause ${\\cal S}$ and ${\\cal C}$ furnish the spectral representations of\n$-d^2\/dr^2$ on $L^2({\\rm I\\! R}^+)$ with Dirichlet and Neumann boundary\nconditions respectively at the origin, we are in the general situation\nof Section~2.2.\n\nWe now restrict to the generic case and explicitly construct the\n$M_i$ and compute their rank and signature. $M_2$ is given by the\nfollowing proposition, whose proof is given later in this section.\n\\begin{Prop}\n\\label{Prop:M2} In the generic case,\n\\begin{equation}\nM_2 = \\sum_{\\omega\\in\\Omega} \\alpha_\\omega\n\\ket{\\xi_\\omega}\\bra{\\xi_{\\overline{\\omega}}}, \\label{eq:M2}\n\\end{equation}\nwhere $\\xi_\\omega(r) =e^{\\omega r}$, and $\\alpha_\\omega$ is the\nresidue\n\\begin{equation}\n\\alpha_\\omega= {\\rm Res}_\\omega\n2zf_0(-iz). \\label{eq:aw}\n\\end{equation}\nIn addition, ${\\rm Ran}\\,\nM_2={\\rm span}\\,\\{\\xi_\\omega\\mid\\omega\\in\\Omega\\}$,\nand\n\\begin{eqnarray}\n{\\rm rank}\\, M_2 &=& \\frac{1}{2}\\deg W + I_L^+  \\label{eq:M2rnk} \\\\\n{\\rm sig}\\, M_2 &=& \\frac{1}{2}\\left(\\sigma_0^2-\\sigma_\\infty^2\\right)\n-I_L^- . \\label{eq:M2sig}\n\\end{eqnarray}\n\\end{Prop}\n\nNext, define ${\\cal M}_1$ to be the space of all\n$L^2$-vectors of form $Q(k^2)k(p(k^2)^2+k^2q(k^2)^2)^{-1\/2}$,\nsuch that $Q(z)\\in \\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}[z]$ is a polynomial with complex\ncoefficients. Thus\n\\begin{equation}\n{\\cal M}_1 =\n(p(k^2)^2+k^2q(k^2)^2)^{-1\/2}k\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}_{\\mho-1}[k^2]\n\\end{equation}\nwhere $\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}_r[z]$ is the $r+1$-dimensional complex vector space of\npolynomials with complex coefficients and degree at most $r$,\nand $\\mho=\\dim {\\cal M}_1$ is given by\n\\begin{equation}\n\\mho=\\frac{1}{2}\\deg W+\\frac{1}{2}(\\sigma_\\infty^2-\\sigma_0^2) =\n\\max\\{\\deg p ,\\deg q\\}.\n\\end{equation}\n$M_1$ is described by\n\\begin{Prop} \\label{Prop:M1}\nIn the generic case, $M_1$ vanishes on\n${{\\cal M}_1}^\\perp$, and its action on ${\\cal M}_1$ is given by\n$M_1 Q(k^2)k(p(k^2)^2+k^2q(k^2)^2)^{-1\/2}=\\tilde{Q}(k^2)k\n(p(k^2)^2+k^2q(k^2)^2)^{-1\/2}$, where\n\\begin{equation}\n\\tilde{Q}(k^2)=\nQ(k^2)+\\sum_{\\omega\\in\\Omega}\n\\frac{Q(-\\omega^2)\\alpha_\\omega}{q(-\\omega^2)}\n\\frac{p(k^2)-\\omega q(k^2)}{\\omega^2+k^2}.\n\\end{equation}\nMoreover, ${\\rm Ran}\\, M_1={\\cal M}_1$ and\n\\begin{eqnarray}\n{\\rm rank}\\, M_1 & = &\\frac{1}{2}\\deg W +\n\\frac{1}{2}(\\sigma_\\infty^2-\\sigma_0^2) \\label{eq:M1rnk} \\\\\n{\\rm sig}\\, M_1 &=& -(I_L^+ +I_L^-). \\label{eq:M1sig}\n\\end{eqnarray}\n\\end{Prop}\n\nAs an example, let us consider the sub-class of the $R$ class\nconsidered by Shondin \\cite{Shond1}; namely, the case where\n$r(z)=p(z)\/q(z)$ has negative imaginary part in the upper half-plane.\nIn this case, it is easy to show that there can be no solutions to\n$r(-z^2)=z$ and hence to $W(z)=0$ in the left-hand half-plane,\nexcept on the real axis. Moreover, one can show that the residues\n$\\alpha_\\omega$ at these zeros are necessarily positive, so $M_2$ is\na positive operator as a result of~(\\ref{eq:M2}). Accordingly, ${\\cal T}$\nis contractive, and our method yields a unitary dilation defined on\nHilbert spaces. This explains why Shondin was able to construct\nthese GPI models on enlarged {\\em Hilbert} spaces.\n\nWe now prove the above propositions.\n\n{\\noindent\\em Proof of Proposition~\\ref{Prop:M2}:}\n$M_2$ may be written in two equivalent forms:\n\\begin{eqnarray}\nM_2 &=& {\\cal S}^{-1}\\sin^2\\delta_0(k){\\cal S}\n -{\\cal C}^{-1}\\sin^2\\delta_0(k){\\cal C}   \\nonumber \\\\\n& &-{\\cal C}^{-1}\\sin\\delta_0(k)\\cos\\delta_0(k){\\cal S} -\n{\\cal S}^{-1}\\sin\\delta_0(k)\\cos\\delta_0(k){\\cal C} \\label{eq:ker1}\\\\\n&=& {\\cal C}^{-1}\\cos^2\\delta_0(k){\\cal C}  -{\\cal S}^{-1}\\cos^2\\delta_0(k){\\cal S}\n\\nonumber \\\\\n& &-{\\cal C}^{-1}\\sin\\delta_0(k)\\cos\\delta_0(k){\\cal S} -\n{\\cal S}^{-1}\\sin\\delta_0(k)\\cos\\delta_0(k){\\cal C} . \\label{eq:ker2}\n\\end{eqnarray}\n{}To convert this into an integral kernel we use the following Lemma,\nwhich may be proved by standard means (cf. Theorem IX.29 in\n\\cite{RSii}). Here, $v(x)$ and $w(x)$ stand for either $\\sin x$ or\n$\\cos x$, and ${\\cal V}$ and ${\\cal W}$ are the corresponding integral\ntransforms from ${\\cal H}_r$ to ${\\cal H}_k$.\n\\begin{Lem} \\label{Lem:ker}\nLet $g(k)\\in L^2({\\rm I\\! R}^+)\\cap L^\\infty({\\rm I\\! R}^+)$ and define\n$G={\\cal V}^{-1}g(k){\\cal W}$. Then $G$ has integral kernel\n\\begin{equation}\nG(r,r^\\prime) = \\frac{2}{\\pi} \\int_0^\\infty v(kr)w(kr^\\prime)g(k)dk,\n\\end{equation}\n(where the integral is a limit in $L^2$-norm).\n\\end{Lem}\n\nIn the case $\\deg p>\\deg q$, $\\sin^2\\delta_0(k)$ and\n$\\sin\\delta_0(k)\\cos\\delta_0(k)$ are $L^2\\cap L^\\infty$ and so,\napplying Lemma~\\ref{Lem:ker} to~(\\ref{eq:ker1}) and combining\nterms, $M_2$ has integral kernel\n\\begin{equation}\nM_2(r,r^\\prime) =\n\\frac{i}{\\pi}\\int_{-\\infty}^\\infty e^{i\\delta_0(k)}\n\\sin\\delta_0(k) e^{ik(r+r^\\prime)}dk\n=\\frac{1}{\\pi}\\int_{-\\infty}^\\infty\n\\frac{ikq(k^2)e^{ik(r+r^\\prime)}}{p(k^2)-ikq(k^2)} dk.\n\\end{equation}\nMaking the substitution $z=ik$ and closing the contour in the\nleft-hand half-plane, the integrand has a simple pole at each\n$\\omega\\in\\Omega$ and~(\\ref{eq:M2}) follows. If $\\deg q\\ge\\deg p$, we\nargue similarly using~(\\ref{eq:ker2}) to obtain the same result as\nbefore.\n\nBy linear independence of the $\\xi_\\omega$ and non-vanishing of the\n$\\alpha_\\omega$, it follows that ${\\rm Ran}\\,\nM_2={\\cal M}_2={\\rm span}\\,\\{\\xi_\\omega\\mid\\omega\\in\\Omega\\}$, so ${\\rm rank}\\,\nM_2=|\\Omega|$, the cardinality of $\\Omega$.\nUsing residue calculus, one may show that\n\\begin{equation}\n|\\Omega| = \\frac{1}{2}\\deg W +\n\\frac{1}{2\\pi}\\int_{-\\infty}^\\infty \\frac{W^\\prime(ik)}{W(ik)} dk.\n\\end{equation}\nBy rewriting the second term as an integral over $(0,\\infty)$, a\nsmall amount of algebra shows that the integrand is\n$-\\pi^{-1}\\delta^\\prime_0(k)$. Thus~(\\ref{eq:M2rnk}) is established.\n\n{}To compute ${\\rm sig}\\, M_2$, we define the hermitian\nform $m_2(\\varphi,\\psi): {\\cal M}_2\\times{\\cal M}_2\\rightarrow \\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$ by\n$m_2(\\varphi,\\psi)=\\inner{\\varphi}{M_2\\psi}$. Labelling the\nelements of $\\Omega$ as $\\omega_1,\\ldots,\\omega_{|\\Omega|}$, and\nwriting $\\psi=\\sum_i c_i\\xi_{\\omega_i}$, we have\n\\begin{equation}\nm_2(\\psi,\\psi) = \\sum_{i,j,k} \\overline{c_i}\n\\inner{\\xi_{\\omega_i}}{\\xi_{\\omega_j}}\\alpha_j\n\\inner{\\xi_{\\overline{\\omega_j}}}{\\xi_{\\omega_k}}c_k\n= c^\\dagger\\Xi^\\dagger A\\Xi c,\n\\end{equation}\nwhere $A$ and $\\Xi$ are hermitian. $\\Xi$ has components\n$\\Xi_{ij}=\\inner{\\xi_{\\omega_i}}{\\xi_{\\omega_j}}$,\nand is non-singular by\nlinear independence of the $\\xi_\\omega$. By Sylvester's Law of\nInertia \\cite{Cohn}, the signature of $M_2$ equals that of $A$,\nwhich has components\n\\begin{equation}\nA_{ij} = \\left\\{ \\begin{array}{cl}\n\\alpha_{\\omega_i} & \\omega_i=\\overline{\\omega_j} \\\\\n0 & {\\rm otherwise}. \\end{array}\\right.\n\\end{equation}\n$A$ has eigenvalues $\\{\\alpha_\\omega\\mid \\omega\\in{\\rm I\\! R}\\}\\cup \\{\\pm\n|\\alpha_\\omega|\\mid\\omega\\not\\in{\\rm I\\! R}\\}$.\nLabelling the $\\omega_i$ so that $\\omega_1,\\ldots,\\omega_r$ are\nthe real elements of $\\Omega$, we therefore have\n${\\rm sig}\\, M_2={\\rm sig}\\,{\\rm diag}\\, (\\alpha_{\\omega_1}\\ldots,\\alpha_{\\omega_r})$.\n(We have used the fact that\n$\\alpha_{\\overline{\\omega}}=\\overline{\\alpha_\\omega}$, and\nin particular that $\\omega_r\\in{\\rm I\\! R}$ implies\n$\\alpha_r\\in{\\rm I\\! R}$.) Defining $\\zeta(k)$ by~(\\ref{eq:zeta}), it is easy\nto show that $\\cot\\zeta(-\\omega)=1$ for $\\omega\\in\\Omega$, and that\n\\begin{equation}\n\\alpha_\\omega =2\\lim_{z\\rightarrow -\\omega}\n\\frac{z+\\omega}{1-\\cot\\zeta(z)} = \\frac{1}{\\zeta^\\prime(-\\omega)}.\n\\end{equation}\nThus ${\\rm sig}\\,{\\rm diag}\\,(\\alpha_1,\\ldots,\\alpha_r)$ is equal to the number of\ntimes that $\\zeta(k)\\equiv\\pi\/4 \\pmod\\pi$ as $k$\ntraverses ${\\rm I\\! R}^+$, counted according to the sign of\n$\\zeta^\\prime(k)$ at such points. This is related to the Levinson\nindex $I_L^-$ by~(\\ref{eq:M2sig}). $\\vrule height 1.5ex width 1.2ex depth -.1ex $\n\n{\\noindent\\em Proof of Proposition~\\ref{Prop:M1}:} We compute\n\\begin{eqnarray}\nM_1 &=& -\\frac{p(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}{\\cal S}{\\cal C}^{-1}\n\\frac{kq(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}  \\nonumber \\\\\n&& -\\frac{kq(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}{\\cal C}{\\cal S}^{-1}\n\\frac{p(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}},\n\\end{eqnarray}\nwhich vanishes identically on the closure of\n${\\cal D}=(p(k^2)^2+k^2q(k^2))^{1\/2}{\\cal S} C_0^\\infty(0,\\infty)$ as a result\nof elementary properties of the sine and cosine transforms.\nFurthermore, $\\overline{{\\cal D}}^\\perp$ is precisely the space ${\\cal M}_1$\ndefined above, because $\\psi\\perp{\\cal D}$ if and only if\n$(p(k^2)^2+k^2q(k^2))^{1\/2}\\psi$ is the sine transform of a\ndistribution supported at the origin and therefore an odd polynomial\n(cf. Theorem~V.11 in \\cite{RSi}). Hence $M_1$ vanishes on\n${\\cal M}_1^\\perp$ and ${\\rm Ran}\\, M_1\\subset {\\cal M}_1$.\n\nNext, we compute the action of $M_1$ on ${\\cal M}_1$. By contour\nintegration,\n\\begin{equation}\n{\\cal T}^*\\frac{kQ(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}\n=-\\left(\\frac{\\pi}{2}\\right)^{1\/2}\\sum_{\\omega\\in\\Omega}\n\\frac{Q(-\\omega^2)\\alpha_\\omega}{q(-\\omega^2)}\\xi_\\omega(r),\n\\label{eq:Tstr}\n\\end{equation}\nfor polynomials $Q(z)$ such that the operand is in $L^2$. Moreover,\nit is easy to show that\n\\begin{equation}\nT\\xi_{\\omega} = \\left(\\frac{2}{\\pi}\\right)^{1\/2}\n\\frac{k}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}\n\\frac{p(k^2)-\\omega q(k^2)}{\\omega^2+k^2},\n\\label{eq:Txi}\n\\end{equation}\nfrom which the action of $M_1$ can be read off as required.\n\n{}To compute the rank and signature of $M_1$, we use the fact that\n\\begin{equation}\n{\\rm rank}\\, M_1 - {\\rm rank}\\, M_2 = {\\rm sig}\\, M_1 -{\\rm sig}\\, M_2 =\n\\dim\\ker {\\cal T}^*-\\dim\\ker{\\cal T} ,\n\\end{equation}\nwhich follows from the intertwining relations $M_1{\\cal T}={\\cal T} M_2$ and\n$M_2{\\cal T}^*={\\cal T}^*M_1$. It therefore remains to determine the dimensions\nof the relevant kernels. Firstly, note that $\\ker {\\cal T}^*\\subset{\\cal M}_1$\nand that (from~(\\ref{eq:Tstr}))\n$\\psi=Q(k^2)k(p(k^2)^2+k^2q(k^2)^2)^{-1\/2}\\in\\ker {\\cal T}^*$ if and only\nif $\\psi\\in{\\cal M}_1$ and $Q(-\\omega^2)=0$ for each $\\omega\\in\\Omega$.\nThus $\\prod_{\\omega\\in\\Omega} (z+\\omega^2)$ divides $Q(z)$ and so\n\\begin{equation}\n\\dim\\ker{\\cal T}^* = \\min\\{\\mho-|\\Omega|,0\\}.\n\\end{equation}\n\nNow consider $\\ker{\\cal T}$. We note that~(\\ref{eq:Txi}) may be rewritten\n\\begin{equation}\nq(-\\omega_i^2){\\cal T}\\xi_{\\omega_i}=\\left(\\frac{2}{\\pi}\\right)^{1\/2}\n\\frac{k}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}\n\\frac{p(k^2)q(-\\omega_i^2)-p(-\\omega_i^2)q(k^2)}{k^2+\\omega_i^2},\n\\end{equation}\nand apply the following abstract algebraic result:\n\\begin{Lem}\n\\label{Lem:FW}\nLet $Q,R\\in \\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}[z]$ be coprime with $\\max\\{\\deg Q,\\deg R\\}=k\\ge 0$,\nand let $\\lambda_1,\\ldots,\\lambda_m$ be distinct elements of $\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$.\nThen the polynomials $P_1(z),\\ldots P_m(z)$, defined by\n\\begin{equation}\n(z-\\lambda_i) P_i(z) = R(\\lambda_i)Q(z)-Q(\\lambda_i)R(z)\n\\end{equation}\nspan a $\\min\\{k,m\\}$-dimensional subspace of $\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}_{k-1}[z]$.\n\\end{Lem}\n{\\em Proof:} Let  $n=\\min\\{k,m\\}$. Then it is enough to show that\n$P_1,\\ldots,P_n$ are linearly independent. Assuming that $\\deg Q=k$,\nwe note that  $P_i(z)=R(z) \\tilde{Q}_i(z)-Q(z)\\tilde{R}_i(z)$, where\n$\\tilde{Q}_i(z)=(Q(z)-Q(\\lambda_i))\/(z-\\lambda_i)$ and\n$\\tilde{R}_i(z)=(R(z)-R(\\lambda_i))\/(z-\\lambda_i)$. Suppose the $P_i$\nare linearly dependent. Then $R(z) S(z) = Q(z) T(z)$ where $S(z)\n=\\sum_i \\alpha_i \\tilde{Q}_i(z)$ and $T(z) =\\sum_i\\alpha_i\n\\tilde{R}_i(z)$, for some $0\\not= (\\alpha_1,\\ldots,\\alpha_n)^T\\in\n\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}^n$. Because $Q$ and $R$ are coprime, this implies that $S$ and\n$T$ vanish identically. But one may easily show that the\n$\\tilde{Q}_i$ are linearly independent, by explicitly considering\ntheir coefficients. We therefore obtain a contradiction. $\\vrule height 1.5ex width 1.2ex depth -.1ex $\n\nIn our application, $m=|\\Omega|$ with $\\lambda_i=-\\omega_i^2$ for\neach $i=1,\\ldots,m$ and $k=\\max\\{\\deg p,\\deg q\\}=\\mho$. Thus\n$\\dim{\\cal T}{\\rm Ran}\\, M_2=\\min\\{|\\Omega|,\\mho\\}$ and so\n\\begin{equation}\n\\dim\\ker{\\cal T}= \\min\\{|\\Omega|-\\mho,0\\}.\n\\end{equation}\nIt follows that ${\\rm rank}\\, M_1-{\\rm rank}\\, M_2={\\rm sig}\\, M_1-{\\rm sig}\\,\nM_2=\\mho-|\\Omega|$, from which~(\\ref{eq:M1rnk}) and~(\\ref{eq:M1sig})\nfollow. $\\vrule height 1.5ex width 1.2ex depth -.1ex $\n\n\\sect{The GPI Hamiltonian}\n\\label{sect:GHm}\n\\subsection{Locality and Spectral Properties}\n\nThe results of the previous two sections allow the construction of a\nunitary dilation $\\hat{{\\cal T}}$ of the integral transform ${\\cal T}$. Here,\nwe employ $\\hat{{\\cal T}}$ to define a GPI Hamiltonian consistent with\nscattering theory~(\\ref{eq:GPIlow}). We denote\n$\\Pi_r={\\cal H}_r\\oplus{\\cal M}_1$ and $\\Pi_k={\\cal H}_k\\oplus{\\cal M}_2$ with $J$-inner\nproducts specified by $J_r=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal H}_r}\\oplus{\\rm sgn}\\, (M_1)$, and\n$J_k=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{{\\cal H}_k}\\oplus{\\rm sgn}\\, (M_2)$. In terms of our general\ndiscussion in Section~2.2, we set $A=-d^2\/dr^2$ on domain\n$C_0^\\infty(0,\\infty)$, and define ${\\cal T}_+={\\cal S}$, ${\\cal T}_-={\\cal C}$, setting\n$a_+$ and $a_-$ to be multiplication by $\\cos\\delta_0(k)$ and\n$\\sin\\delta_0(k)$ respectively. Thus\n$A_+={\\cal S}^*k^2{\\cal S}$, the self-adjoint extension of $A$ with Dirichlet\nboundary conditions at the origin, whilst $A_-={\\cal C}^*k^2{\\cal C}$ is the\nextension with Neumann boundary conditions at the origin. The\noperators $A_\\pm+1$ both have bounded inverse.\n\nThe $S$-wave GPI Hamiltonian is defined by\n\\begin{equation}\nh_{\\rm gpi} = \\hat{{\\cal T}}^\\dagger\\left(\n\\begin{array}{cc} k^2 & 0 \\\\ 0 & \\Lambda \\end{array}\\right)\n\\hat{{\\cal T}}, \\label{eq:GPIham}\n\\end{equation}\nwhere $\\Lambda$ is a ${\\rm sgn}\\, (M_2)$-self-adjoint operator\n$\\Lambda^\\dagger=\\Lambda$ on ${\\cal M}_2$. To fix $\\Lambda$, we require\nthat $h_{\\rm gpi}(\\psi,0)^T=(-\\psi^{\\prime\\prime},0)^T$ for\nall $\\psi\\in C_0^\\infty(0,\\infty)$ as a locality requirement.\nFor general $\\psi\\in{\\cal M}_2$, we have\n\\begin{equation}\nA^*\\psi = -\\sum_{\\omega\\in\\Omega}\\alpha_\\omega \\omega^2\n\\ket{\\xi_\\omega}\\inner{\\xi_{\\overline{\\omega}}}{M_2^{-1}\\psi},\n\\end{equation}\nso ${\\cal M}_2$ is invariant under $A^*$ and it follows immediately from\nSection~2.2 that\n\\begin{Thm} \\label{Thm:local}\nIn the generic case, the unique choice of $\\Lambda$ consistent with\nlocality is\n\\begin{equation}\n\\Lambda= -\\left({\\rm sgn}\\,(M_2) |M_2|^{1\/2}\\right)^{-1}\n\\sum_{\\omega\\in\\Omega} \\alpha_\\omega \\omega^2\\ket{\\xi_\\omega}\n\\bra{\\xi_{\\overline{\\omega}}} |M_2|^{-1\/2}.\n\\end{equation}\n\\end{Thm}\n\nWe proceed to determine the eigenvectors and eigenvalues of\n$\\Lambda$. First note that\n$\\inner{\\xi_{\\overline{\\omega_j}}}{M_2^{-1}\\xi_{\\omega_i}}=\n\\alpha_{\\omega_{i}}^{-1}\\delta_{ij}$, which follows from the identity\n$\\xi_{\\omega_i}=\\sum\\alpha_\\omega\\ket{\\xi_\\omega}\n\\inner{\\xi_{\\overline{\\omega}}}{M_2^{-1}\\xi_{\\omega_i}}$. It is then\na matter of computation to see that $\\varphi_i=\\left({\\rm sgn}\\,(M_2)\n|M_2|^{1\/2}\\right)^{-1}\\xi_{\\omega_i}$ is an eigenvector of $\\Lambda$\nwith eigenvalue $-\\omega_i^2$ for each $i=1,\\ldots,|\\Omega|$. Because\n$\\Lambda$ has rank $|\\Omega|$, this exhausts the discrete spectrum of\n$h_{\\rm gpi}$. The following is then immediate.\n\\begin{Thm}\nIn the generic case, and  with\n$\\Lambda$ is defined as above, $h_{\\rm gpi}$ has the following spectral\nproperties: $\\sigma(h_{\\rm gpi})=\\sigma_{\\rm ac}(h_{\\rm gpi}) \\cup\\sigma_{\\rm\npp}(h_{\\rm gpi})$ where $\\sigma_{\\rm ac}(h_{\\rm gpi})={\\rm I\\! R}^+$ and $\\sigma_{\\rm\npp}(h_{\\rm gpi})$ consists of the $|\\Omega|$ eigenvalues $-\\omega_i^2$, whose\ncorresponding eigenvectors are\n\\begin{equation}\n\\psi_i = \\hat{{\\cal T}}^\\dagger\\varphi_i=\n\\left(\\begin{array}{c} \\xi_{\\omega_i} \\\\\n{\\cal T} \\left({\\rm sgn}\\,(M_2) |M_2|^{1\/2}\\right)^{-1}\\xi_{\\omega_i}\n\\end{array}\\right).\n\\end{equation}\nThe absolutely continuous subspace is the Hilbert space\n$\\hat{{\\cal T}}^\\dagger{\\cal H}_k$.\n\\end{Thm}\n\nThis bears out our earlier statement that the poles of the\nscattering amplitude on the physical sheet correspond to the discrete\nenergy spectrum, if locality is imposed.\n\nThe physical Hilbert space is required to be a positive definite\ninvariant subspace of $\\Pi_r$ relative to $h_{\\rm gpi}$.\\footnote{An\ninvariant subspace ${\\cal L}$ of a $J$-space ${\\cal K}$ relative to a linear\noperator $A$ on ${\\cal K}$ is a subspace of ${\\cal K}$ such that\n$\\overline{D(A)\\cap{\\cal L}}={\\cal L}$ and ${\\rm Ran}\\, A|_{{\\cal L}}\\subset {\\cal L}$, where\nthe closure is taken in the norm topology of ${\\cal K}$.} In $\\Pi_k$, we\nhave the $[\\cdot,\\cdot]_{\\Pi_k}$-orthogonal decomposition\n$\\Pi_k={\\cal H}_k [+] {\\cal M}_2$, where ${\\cal M}_2$ is spanned by the eigenvectors\n$\\varphi_i$ of $\\Lambda$. We compute\n\\begin{equation}\n[\\varphi_i,\\varphi_j]_{{\\cal M}_2}\n=\\inner{\\xi_{\\omega_i}}{M_2^{-1}\\xi_{\\omega_j}} \\nonumber \\\\\n= \\left\\{\\begin{array}{cl} 0 & \\omega_i\\not=\\overline{\\omega_j} \\\\\n\\alpha_{\\omega_j}^{-1} & \\omega_i=\\overline{\\omega_j}.\n\\end{array}\\right.\n\\end{equation}\nHence $\\Pi_k$ is decomposable as\n$\\Pi_k = {\\cal H}_k [+] E_+ [+] E_- [+] H$\nwhere $E_+$ is spanned by the $\\varphi_i$ with\n$[\\varphi_i,\\varphi_i]_{{\\cal M}_2}>0$ ($\\alpha_{\\omega_i}>0$),\n$E_-$ is spanned\nby those with $[\\varphi_i,\\varphi_i]_{{\\cal M}_2}<0$\n($\\alpha_{\\omega_i}<0$), and\n$H$ is the {\\em hyperbolic invariant subspace} spanned by those\n$\\varphi_i$ with $\\omega_i\\not\\in{\\rm I\\! R}$. Moreover, this is a\ndecomposition into invariant subspaces, because $D(k^2)$ is dense in\n${\\cal H}_k$. The physical Hilbert space ${\\cal H}_{\\rm phys}$ is therefore\ndefined by\n\\begin{equation}\n{\\cal H}_{\\rm phys} = \\hat{{\\cal T}}^\\dagger ({\\cal H}_k[+]E_+).\n\\end{equation}\n\nWe briefly discuss the uniqueness of the GPI Hamiltonian constructed\nin this way. As noted in Section~2.1, $\\hat{{\\cal T}}$ is unique up to\nfurther unitary dilation and unitary equivalence because the $M_i$\nare of finite rank. Further dilation merely corresponds to the\n(trivial) freedom to form the direct sum of $h_{\\rm gpi}$ with the\nHamiltonian of an arbitrary independent system. On the other hand,\nreplacing $\\hat{{\\cal T}}$ by $(\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}\\oplus U_2)\\hat{{\\cal T}}(\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}\\oplus U_1)$\nwhere $U_i$ is a ${\\rm sgn}\\, M_i$-unitary operator on ${\\cal M}_i$ for $i=1,2$,\nit is easy to show that the local GPI Hamiltonian $h_{\\rm gpi}^\\prime$\nobtained is given by\n\\begin{equation}\nh_{\\rm gpi}^\\prime=\n\\left(\\begin{array}{cc} \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1} & 0\\\\0 & U_1\\end{array}\\right)^\\dagger\nh_{\\rm gpi}\\left(\\begin{array}{cc} \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1} & 0\\\\0 & U_1\\end{array}\\right).\n\\end{equation}\nWe have therefore constructed a family of unitarily equivalent GPI\nHamiltonians on $\\Pi_r$ corresponding to the same scattering data. It\nis clearly sufficient to study $h_{\\rm gpi}$ alone in order to determine the\ndomain and scattering properties of $h_{\\rm gpi}^\\prime$.\n\n\\subsection{Domain and Resolvent}\n\nWe now determine the domain and explicit action of the operator\n$h_{\\rm gpi}$ under the locality assumption. Our result is the following:\n\\begin{Thm} \\label{Thm:dom}\nLet $\\Theta_0\n=(2\/\\pi)^{1\/2}k^{2\\mho-1}(p(k^2)^2+k^2q(k^2)^2)^{-1\/2}$. Then in\nthe generic case,\n\\begin{eqnarray}\nD(h_{\\rm gpi})&=&\\left\\{\n\\left( \\begin{array}{c} \\varphi \\\\ \\Phi\\end{array}\\right)\n\\mid \\varphi,\\varphi^\\prime\\in\nAC_{\\rm loc}(0,\\infty),~\\varphi,\\varphi^{\\prime\\prime}\\in L^2;\\quad\n\\Phi\\in{\\cal M}_1, \\right.\\nonumber \\\\\n&&\\qquad\\qquad\n\\left.\\begin{array}{c} \\ \\\\ \\ \\end{array}\n\\Theta -\\lambda[\\varphi]\\Theta_0 \\in\nD(k^2)\\cap {\\cal M}_1\\right\\} ,\n\\end{eqnarray}\nwhere $\\Theta={\\rm sgn}\\, M_1|M_1|^{1\/2}\\Phi$ and\n\\begin{equation}\n\\lambda[\\varphi] = \\left\\{\\begin{array}{cl}\nP\\varphi(0) & \\deg p>\\deg q \\\\\nP\\varphi(0)-Q\\varphi^\\prime(0) & \\deg p=\\deg q \\\\\n-Q\\varphi^\\prime(0) & \\deg p< \\deg q, \\end{array}\\right.\n\\end{equation}\nand $P$ and $Q$ are the leading coefficients of $p(z)$ and $q(z)$\nrespectively. (In the case $M_1=0$,\n$D(h_{\\rm gpi})=\\{\\varphi\\mid\\varphi,\\varphi^\\prime\\in\nAC_{\\rm loc}(0,\\infty),~\\varphi,\\varphi^{\\prime\\prime}\\in\nL^2;~\\lambda[\\varphi]=0\\}$.) Moreover,\n\\begin{equation}\nh_{\\rm gpi}\\left(\\begin{array}{c} \\varphi \\\\ \\Phi\\end{array}\\right)=\n\\left(\\begin{array}{c} -\\varphi^{\\prime\\prime}\\\\ \\tilde{\\Phi}\n\\end{array}\\right) ,\n\\end{equation}\nwhere $\\tilde{\\Phi}$ is given in terms of $\\tilde{\\Theta}={\\rm sgn}\\, M_1\n|M_1|^{1\/2}\\tilde{\\Phi}$ by\n\\begin{equation}\n\\tilde{\\Theta}= k^2(\\Theta-\\lambda[\\varphi]\\Theta_0) +\n\\left(\\frac{2}{\\pi}\\right)^{1\/2}\n\\frac{k(\\lambda[\\varphi] k^{2\\mho} -\\varphi(0)p(k^2)\n+\\varphi^\\prime(0)q(k^2))}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}} .\n\\end{equation}\n\\end{Thm}\n{\\em Proof:} The result is a direct application of the discussion in\nSection~2.2. The key point is that, for each $\\varphi\\in\nD(-d^2\/dr^2|_{C_0^\\infty(0,\\infty)}^*)$, the vectors $\\chi_+$ and\n$\\chi_-$ are given by\n\\begin{equation}\n\\chi_+ = -\\varphi(0)e^{-r}\\qquad {\\rm and} \\qquad\n\\chi_-=\\varphi^\\prime(0)e^{-r},\n\\end{equation}\nwhich follows because $\\chi_+$ ($\\chi_-$) is the unique element of\n$\\ker (-d^2\/dr^2|_{C_0^\\infty(0,\\infty)}^*+1)$ such that\n$\\varphi+\\chi_+$ ($\\varphi+\\chi_-$) is in the domain of the Laplacian\nwith Dirichlet (Neumann) boundary conditions at the origin. $\\vrule height 1.5ex width 1.2ex depth -.1ex $\n\nThe resolvent of $h_{\\rm gpi}$ may be written in the form of Krein's formula\nas\n\\begin{equation}\n(h_{\\rm gpi}-z)^{-1} =\n\\left(\\begin{array}{cc} R_0(z) & 0 \\\\ 0 & R_1(z)\\end{array}\\right)\n+\\frac{q(z)}{p(z)+(-z)^{1\/2}q(z)}F(z)F(\\overline{z})^\\dagger.\n\\label{eq:reso}\n\\end{equation}\nHere, $R_0(z)={\\cal S}^{-1}(k^2-z)^{-1}{\\cal S}$ is the free resolvent\nand the defect element $F(z)\\in\\Pi_r$ is given by\n\\begin{equation}\nF(z)=\\left( \\begin{array}{c} e^{-(-z)^{1\/2}r} \\\\\n({\\rm sgn}\\, M_1|M_1|^{1\/2})^{-1}\\Psi(z) \\end{array}\\right),\n\\end{equation}\nwhere $\\Psi(z)\\in{\\cal M}_1$ is\n\\begin{equation}\n\\Psi(z) = \\left(\\frac{2}{\\pi}\\right)^{1\/2}\n\\frac{k(p(k^2)q(z)-p(z)q(k^2))}{(k^2-z)(p(k^2)^2+k^2q(k^2)^2)^{1\/2}},\n\\end{equation}\nand the operator $R_1(z)$ is defined on ${\\cal M}_1$ by\n\\begin{equation}\nR_1(z)\\Phi=({\\rm sgn}\\, M_1|M_1|^{1\/2})^{-1}\\left(\\frac{2}{\\pi}\\right)^{1\/2}\n\\frac{k(Q(k^2)-Q(z)q(k^2)\/q(z))}{(k^2-z)(p(k^2)^2+k^2q(k^2)^2)^{1\/2}},\n\\end{equation}\nwhere $Q(z)$ is defined in terms of $\\Phi$ by\n\\begin{equation}\n\\Theta={\\rm sgn}\\, M_1|M_1|^{1\/2}\\Phi=\\left(\\frac{2}{\\pi}\\right)^{1\/2}\n\\frac{kQ(k^2)}{(p(k^2)^2+k^2q(k^2)^2)^{1\/2}}.\n\\end{equation}\nThe above expression for $R(z)$ may be verified directly using\nTheorem~\\ref{Thm:dom}, and the fact that\n\\begin{equation}\n[({\\rm sgn}\\, M_1|M_1|^{1\/2})^{-1}\\Psi(\\overline{z}),\\Phi]_{{\\cal M}_1}=\n-\\frac{Q(z)}{q(z)},\n\\label{eq:clm}\n\\end{equation}\nwhich is required when one takes inner products with\n$F(\\overline{z})$. Using this result, it follows that~(\\ref{eq:reso})\nholds for elements of form $(0,\\Phi)^T$ with $Q(z)=0$; direct\ncomputation establishes it for $Q(z)\\equiv 1$ and also for  vectors\nof form $(\\varphi,0)^T$ with $\\varphi\\in{\\cal H}_r$.\\footnote{Here, it is\nuseful to employ the decomposition ${\\cal H}_r= \\overline{{\\rm Ran}\\,\n(-d^2\/dr^2-z)|_{C_0^\\infty(0,\\infty)}}\\oplus \\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}\ne^{-(-\\overline{z})^{1\/2}r}$.} Thus~(\\ref{eq:reso}) holds on the whole\nof $\\Pi_r$. It remains to establish equation~(\\ref{eq:clm}).\nMultiplying through by $q(z)$, the LHS of~(\\ref{eq:clm}) is equal to\n\\begin{equation}\n\\inner{q(\\overline{z})\\Psi(\\overline{z})}{M_1^{-1}\\Theta}=\n\\inner{{\\cal T}^*q(\\overline{z})\n\\Psi(\\overline{z})}{M_2^{-1}{\\cal T}^*\\Theta}+\\inner{q(\\overline{z})\n\\Psi(\\overline{z})}{\\Theta}. \\end{equation}\nUsing the identity\n$\\inner{\\xi_{\\overline{\\omega_j}}}{M_2^{-1}\\xi_{\\omega_i}}=\n\\alpha_{\\omega_{i}}^{-1}\\delta_{ij}$ and the results of Section~3,\nthe first term is\n\\begin{equation}\n\\inner{{\\cal T}^*q(\\overline{z})\n\\Psi(\\overline{z})}{M_2^{-1}{\\cal T}^*\\Theta}=\n\\sum_{\\omega\\in\\Omega}\\frac{p(z)-\\omega\nq(z)}{q(-\\omega^2)(\\omega^2+z)}Q(-\\omega^2)\\alpha_\\omega.\n\\end{equation}\nThe required result then follows from the calculation\n\\begin{eqnarray}\n\\inner{q(\\overline{z})\n\\Psi(\\overline{z})}{\\Theta} &=&\n\\frac{1}{\\pi}\\int_{-\\infty}^\\infty dk\\frac{ikQ(k^2)\n(p(z)-ikq(z))}{(k^2-z)(p(k^2)-ikq(k^2))} \\nonumber \\\\\n&=& -Q(z)-\\sum_{\\omega\\in\\Omega}\\frac{p(z)-\\omega\nq(z)}{q(-\\omega^2)(\\omega^2+z)}Q(-\\omega^2)\\alpha_\\omega.\n\\end{eqnarray}\n\n\n\\subsection{Scattering Theory}\n\\label{sect:GPIsc}\n\nIn this section, we construct M{\\o}ller wave operators for $h_{\\rm gpi}$\nrelative to the free Hamiltonian $h_0={\\cal S}^{-1} k^2{\\cal S}$ on ${\\cal H}_r$ in\norder to check that $h_{\\rm gpi}$ actually exhibits the required scattering\nbehaviour. Because scattering is a function of the continuous\nspectrum only, our results in this section are actually independent\nof the precise form of $\\Lambda$, and therefore of the locality\nrequirement.\n\nWe work in the $S$-wave, and employ a two space setting: let $B$\nbe self-adjoint on ${\\cal H}_1$, $A$ be self-adjoint on ${\\cal H}_2$ and ${\\cal J}$\nbe a bounded operator from ${\\cal H}_1$ to ${\\cal H}_2$. Then the M{\\o}ller\noperators $\\Omega^\\pm(A,B;{\\cal J})$ are defined by\n\\begin{equation}\n\\Omega^\\pm(A,B;{\\cal J}) = \\lim_{t\\rightarrow\\mp\\infty}\ne^{iAt}{\\cal J} e^{-iBt}P_{\\rm ac}(B),\n\\end{equation}\nand are said to be complete if the closure of ${\\rm\nRan}\\Omega^\\pm(A,B;{\\cal J})$ is equal to ${\\rm Ran} P_{\\rm ac}(A)$.\n\nIn the following,\n${\\cal J}_r$ and ${\\cal J}_k$ are the natural embeddings of ${\\cal H}_r$ and\n${\\cal H}_k$ into $\\Pi_r$ and $\\Pi_k$ respectively.\n\n\\begin{Thm} Let ${\\cal J}:{\\cal H}_r\\rightarrow\\Pi_r$ be given by\n${\\cal J}=\\hat{{\\cal T}}^\\dagger {\\cal J}_k{\\cal T}$. Then\n$\\Omega^\\pm(h_{\\rm gpi},h_0;{\\cal J})$ exist, are complete, and given by\n\\begin{equation}\n\\Omega^\\pm(h_{\\rm gpi},h_0;{\\cal J}) = \\hat{{\\cal T}}^\\dagger {\\cal J}_k\ne^{\\pm i\\delta_0(k)}{\\cal S},\n\\label{eq:Mops}\n\\end{equation}\nwhere $\\delta_0(k)$ is given by~(\\ref{eq:GPIlow}).\n \\end{Thm}\n{\\em Proof:} Writing $U_t$ for multiplication by $e^{-ik^2t}$ on\n${\\cal H}_k$, we have\n\\begin{eqnarray}\ne^{ih_{\\rm gpi} t} {\\cal J} e^{-ih_0t}P_{\\rm ac}(h_0) &=& \\hat{{\\cal T}}^\\dagger\n\\left(\\begin{array}{cc} U_{-t} & 0 \\\\ 0 & \\exp i\\Lambda t\n\\end{array}\\right) \\hat{{\\cal T}} {\\cal J}  {\\cal S}^{-1} U_t {\\cal S}\n\\nonumber \\\\\n&=& \\hat{{\\cal T}}^\\dagger\n{\\cal J}_k U_{-t} {\\cal T}  {\\cal S}^{-1} U_t {\\cal S}.\n\\end{eqnarray}\nNow, for any $u(k)\\in C_0^\\infty(0,\\infty)$,\n\\begin{eqnarray}\n\\| U_{-t}{\\cal T}{\\cal S}^{-1}U_{t}u(k)-e^{\\pm i\\delta_0(k)} u(k)\\|^2 & = &\n\\|\\sin\\delta_0(k){\\cal C} ({\\cal C}^{-1}\\pm i{\\cal S}^{-1})U_t u(k)\\|^2\n\\nonumber\\\\ & \\le & \\frac{2}{\\pi}\\int_0^\\infty dr\n\\left|\\int_0^\\infty dk e^{i(\\pm kr-k^2t)} u(k)\\right|^2,\n\\end{eqnarray}\nwhich vanishes as $t\\rightarrow\\mp\\infty$ by (non)-stationary phase\narguments (see the Corollary to Theorem XI.14 in \\cite{RSiii}). Thus\n$U_{-t}{\\cal T}{\\cal S}^{-1}U_t \\rightarrow e^{\\pm i\\delta_0(k)}$ strongly as\n$t\\rightarrow\\mp\\infty$. The existence and form of the M{\\o}ller\noperators are then immediate. One easily checks that they are unitary\nmaps from ${\\cal H}_r$ to $P_{\\rm ac}(h_{\\rm gpi})=\\hat{{\\cal T}}^\\dagger{\\cal J}_k{\\cal H}_k$,\nto establish completeness. $\\vrule height 1.5ex width 1.2ex depth -.1ex $\n\nWe conclude that our construction does indeed yield the required\nscattering theory, and also that -- as a by-product of the\nconstruction -- complete M{\\o}ller operators may easily and\nexplicitly be determined.\n\n\\sect{Examples}\n\nAs an application, we construct the class of GPI models with\nscattering data\n\\begin{equation}\n\\cot\\delta_0(k) = -\\frac{1}{kL}+kM, \\label{eq:ERlow}\n\\end{equation}\nwhere $L$ is the scattering length, and $M$ is twice the effective\nrange. These models therefore represent the effective range\napproximation to the behaviour of a non-point interaction in the\n$S$-wave. This class of models has been partially studied by\nShondin~\\cite{Shond1}, who considered the case $M<0$ (`models of type\n$B_2$') and also appears as a special case of the models considered\nby Pavlov in \\cite{Pav1}.  (We also note that van Diejen and Tip\n\\cite{Diejen} have constructed models of type $\\cot\\delta_0(k) = (ak\n+ bk^3+ck^5)^{-1}$ using the distributional method.) The case $M>0$\ndoes not appear to have been treated before. Our construction\nprovides a unified construction for all models in the above class,\nand also provides the spectral representation such models as a\nby-product of the construction (although we will not state this\nexplicitly).\n\nThe above class of GPI models contains two interesting\nsub-families: the ordinary point interactions ($M=0$) and also the\nresonance point interactions arising formally by setting $L=\\infty$,\ni.e., $\\cot\\delta_0(k)=kM$ with $M\\in{\\rm I\\! R}\\cup\\{\\infty\\}$. Such models\nare required in situations where the scattering length is\ngenerically forced to be infinite, for example in certain systems of\nsupersymmetric quantum mechanics.\n\nWe begin by briefly treating the point interactions, both for\ncompleteness and also to demonstrate how this class arises in our\nformalism. We then turn to the general case, obtaining RPI models\nin the limit $L\\rightarrow -\\infty$.\n\n\\subsection{Point Interactions}\n\nThe required integral transform is\n\\begin{equation}\n{\\cal T} = (1+(kL)^2)^{-1\/2}{\\cal S} - kL(1+(kL)^2)^{-1\/2}{\\cal C}.\n\\end{equation}\nIn the cases $L=0,\\infty$, ${\\cal T}$ reduces to ${\\cal S}$ and ${\\cal C}$\nrespectively, and the Hamiltonian is given immediately by\n${\\cal T}^*k^2{\\cal T}$. We exclude these cases from the rest of our discussion.\n\nWe therefore apply the construction of Section 3, with $p(z)\\equiv\n-L^{-1}$ and $q(z)\\equiv 1$. We find that $\\mho=0$, so $M_1=0$ (i.e.,\n${\\cal T}\\Tt^*=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}$). Straightforward application of\nProposition~\\ref{Prop:M2} yields\n\\begin{equation}\nM_2=\\left\\{\\begin{array}{cl}\n\\ket{\\chi_L}\\bra{\\chi_L} & L>0 \\\\ 0 & L<0, \\end{array}\\right.\n\\end{equation}\nwhere $\\chi_L(r)=(2\/L)^{1\/2}e^{-r\/L}$ is normalised to unity.\nHence if $L<0$, ${\\cal T}$ is unitary and the Hamiltonian is\n$h_L = {\\cal T}^* k^2 {\\cal T}$,\nwith purely absolutely continuous spectrum ${\\rm I\\! R}^+$. In the case\n$L>0$, the momentum Hilbert space is extended to ${\\cal H}_k\\oplus\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$,\nrepresenting a single bound state, and the unitary dilation\n$\\hat{{\\cal T}}:{\\cal H}_r\\rightarrow{\\cal H}_k\\oplus\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$ takes form\n\\begin{equation}\n\\hat{{\\cal T}} =\n\\left(\\begin{array}{c} {\\cal T} \\\\ \\bra{\\chi_L} \\end{array}\\right);\n\\qquad \\hat{{\\cal T}}^*=\n\\left(\\begin{array}{cc} T^* & \\ket{\\chi_L}\\end{array}\\right).\n\\end{equation}\n(${\\cal H}_k\\oplus\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$ has the obvious inner product.)\nThe Hamiltonian is\n\\begin{equation}\nh_L = \\hat{{\\cal T}}^{-1}\\left(\\begin{array}{cc} k^2 & 0 \\\\ 0 & \\lambda\n\\end{array}\\right) \\hat{{\\cal T}} = {\\cal T}^* k^2{\\cal T} +\n\\lambda\\ket{\\chi_L}\\bra{\\chi_L},\n\\end{equation}\nand the locality requirement fixes $\\lambda= -L^{-2}$, which is, of\ncourse, the usual value. Finally, the domain of $h_L$ is given by\nTheorem~\\ref{Thm:dom} as the space of $\\varphi$ with\n$\\varphi,\\varphi^\\prime\\in AC_{\\rm loc}(0,\\infty)$,\n$\\varphi^{\\prime\\prime}\\in L^2$ and satisfying\nthe well known boundary condition\n\\begin{equation}\n\\varphi(0)+L\\varphi^\\prime(0) = 0.\n\\end{equation}\n{}To summarise, all the well known properties of point interactions\nmay be derived within our formalism.\n\n\\subsection{Effective Range Approximation}\n\nIn this section, we maintain $M\\not=0$, $L\\not=0$, setting\n$p(z)=-L^{-1}+zM$ and $q(z)\\equiv 1$. We will not explicitly\nconstruct the dilation (although this follows immediately from our\ndiscussion), but will use the results of Section~4 to read off the\ndomain and action of the GPI Hamiltonian $h_{L,M}$.\n\nUsing the results of Section~3, we find\n\\begin{equation}\n\\mho=1;\\qquad |\\Omega|=\n\\left\\{\n\\begin{array}{cl} 1+\\frac{1}{2}({\\rm sgn}\\, M+{\\rm sgn}\\, L) & L\\not=\\infty \\\\\n\\frac{1}{2}(1+{\\rm sgn}\\, M) & L=\\infty. \\end{array}\n\\right.\n\\end{equation}\nWriting $W(z)=-M(z-\\omega_1)(z-\\omega_2)$, $\\Omega$ is\nthe subset of $\\{\\omega_1,\\omega_2\\}$ lying in the left-hand\nhalf-plane, and we have $\\omega_1+\\omega_2=-M^{-1}$,\n$\\omega_1\\omega_2=(ML)^{-1}$. The residues $\\alpha_\\omega$ are\n\\begin{equation}\n\\alpha_{\\omega_1}=-\\frac{2\\omega_1}{M(\\omega_1-\\omega_2)},\\qquad\n\\alpha_{\\omega_2}=\\frac{2\\omega_2}{M(\\omega_1-\\omega_2)}.\n\\end{equation}\nIn addition, the space ${\\cal M}_1={\\rm Ran}\\, M_1$ is equal to $\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}\\ket{\\eta}$,\nwhere\n\\begin{equation}\n\\eta(k) = {\\cal N} \\frac{k}{(k^2+(k^2M-L^{-1})^2)^{1\/2}},\n\\end{equation}\nand the normalisation constant is\n\\begin{equation}\n{\\cal N}=\\left\\{\\begin{array}{cl}\n(2|M|\/\\pi)^{1\/2} & ML>0 \\\\\n(2|M|\/\\pi)^{1\/2}(1-4ML^{-1})^{1\/4} & ML<0. \\end{array}\n\\right.\n\\end{equation}\nUsing Proposition~\\ref{Prop:M1}, we obtain\n\\begin{equation}\nM_1 = \\lambda\\ket{\\eta}\\bra{\\eta} ;\n\\qquad \\lambda = \\left\\{\\begin{array}{cl}\n+1 & M<0, L<0 \\\\\n-{\\rm sgn}\\, M (1-4ML^{-1})^{-1\/2} & ML<0 \\\\\n-1 & M>0, L>0.\n\\end{array}\\right.\n\\end{equation}\n\nAccordingly, the extended position inner product space is\n$\\Pi_r={\\cal H}_r\\oplus\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$ with $J$-inner product specified by\n$J=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}\\oplus (-{\\rm sgn}\\, M)$. The scalar component is the coefficient of\n$\\ket{\\eta}$ in ${\\cal M}_1$. For all generic cases (i.e., all cases\nother than $L=4M>0$) Theorem~\\ref{Thm:dom} entails that the domain\nof $h_{L,M}$ is\n\\begin{equation}\nD(h_{L,M}) = \\left\\{\\left(\\begin{array}{c} \\varphi \\\\ \\Phi\n\\end{array}\\right) \\mid \\varphi,\\varphi^\\prime\\in AC_{\\rm\nloc}(0,\\infty),~\\varphi,\\varphi^{\\prime\\prime}\\in L^2;\\quad \\Phi =\n-|M|^{1\/2}\\varphi(0)  \\right\\},\n\\label{eq:DhLM}\n\\end{equation}\nand that the action is\n\\begin{equation}\nh_{L,M} \\left(\\begin{array}{c} \\varphi \\\\ -|M|^{1\/2}\\varphi(0)\n\\end{array}\\right) =\n\\left(\\begin{array}{c}\n-\\varphi^{\\prime\\prime}\n\\\\ -{\\rm sgn}\\, M |M|^{-1\/2}(\\varphi^\\prime(0)+L^{-1}\\varphi(0))\n\\end{array}\\right).\n\\end{equation}\nMoreover, one may show that these equations also hold in the\nnon-generic case $L=4M>0$.\n\nIt is worth noting how this domain and action correspond to the\nscattering data~(\\ref{eq:ERlow}). Solving the equation\n$h_{L,M}(\\varphi,\\Phi)^T=k^2(\\varphi,\\Phi)^T$ for the generalised\neigenfunctions of $h_{L,M}$, we find\n$\\varphi(r)\\propto\\sin(kr+d(k))$ for some $d(k)$, and also obtain the\nrelation\n\\begin{equation}\n-{\\rm sgn}\\, M |M|^{-1\/2}(\\varphi^\\prime(0)+L^{-1}\\varphi(0))=\n-k^2|M|^{1\/2}\\varphi(0),\n\\end{equation}\nwhich entails that $k\\cot d(k) = \\varphi^\\prime(0)\/\\varphi(0) =\n-L^{-1}+k^2M$. Thus $d(k)$ is precisely the scattering data\n$\\delta_0(k)$.\n\nThe RPI models, which have scattering data $\\cot\\delta_0(k) =kM$ are\nobtained in the same way. The space ${\\cal M}_1$ is spanned by\n$\\psi_M(k) =(2|M|\/\\pi)^{1\/2}(1+(kM)^2)^{-1\/2}$, and the operator\n$M_1$ is found to be $M_1=-({\\rm sgn}\\, M)\\ket{\\psi_M}\\bra{\\psi_M}$. Thus\nthe inner product space is $\\Pi_r={\\cal H}_r\\oplus\\relax{\\hbox{$\\inbar\\kern-.3em{\\rm C}$}}$ with\n$J=\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}\\oplus(-{\\rm sgn}\\, M)$. They have the domain~(\\ref{eq:DhLM}) and\naction\n\\begin{equation}\nh_{{\\rm rpi},M} \\left(\\begin{array}{c} \\varphi \\\\ -|M|^{1\/2}\\varphi(0)\n\\end{array}\\right) =\n\\left(\\begin{array}{c}\n-\\varphi^{\\prime\\prime}\n\\\\ -{\\rm sgn}\\, M |M|^{-1\/2}\\varphi^\\prime(0)\n\\end{array}\\right).\n\\end{equation}\n\nLet us consider the physical Hilbert space for these models. From\nSection~4, this is constructed by projecting out the hyperbolic\ninvariant subspace, and also those eigenfunctions with negative norm\nsquared (if present). The bound states of $h_{L,M}$ are clearly\nvectors of form $(\\xi_\\omega,|M|^{1\/2})^T$ with norm squared equal to\n$-(2{\\rm Re}\\,\\omega)^{-1}-M$, where $\\omega$ is a root of\n$\\omega^2+M^{-1}\\omega+(ML)^{-1}=0$. There are four cases to consider:\n\n{\\noindent \\em Case (i): $M<0$}. $\\Pi_r$ is positive definite so no\nprojection is required.\n\n{\\noindent \\em Case (ii): $M>0$, $L<0$}. There is a unique bound\nstate with\n\\begin{equation}\n\\omega=\\frac{1+(1-4M\/L)^{1\/2}}{-2M}\n\\label{eq:ome}\n\\end{equation}\nand negative norm squared. Projecting this state out, we obtain\n\\begin{equation}\n{\\cal H}_{\\rm phys} =\n\\left\\{\\left(\\begin{array}{c} \\varphi \\\\\nM^{-1\/2}\\inner{\\xi_\\omega}{\\varphi}\\end{array}\\right)\\mid\n\\varphi\\in{\\cal H}_r \\right\\}.\n\\label{eq:HP}\n\\end{equation}\n\n{\\noindent \\em Case (iii): $M>0$, $0<L<4M$}. There are two bound\nstates with complex conjugate eigenvalues. Accordingly, their\neigenfunctions span a hyperbolic invariant subspace. Projecting\nthis subspace out, we find\n\\begin{equation}\n{\\cal H}_{\\rm phys} =\n\\left\\{\\left(\\begin{array}{c} \\varphi \\\\\nM^{-1\/2}\\inner{\n\\frac{1}{2}(\\xi_\\omega+\\xi_{\\overline{\\omega}})}{\\varphi}\n\\end{array}\\right)\\mid\\varphi\\in{\\cal H}_r~{\\rm s.t.}~\n\\inner{(\\xi_\\omega-\\xi_{\\overline{\\omega}})}{\\varphi}=0 \\right\\},\n\\end{equation}\nwhere $\\omega$ is given by equation~(\\ref{eq:ome}).\n\n{\\noindent \\em Case (iv): $M>0$, $L>4M$}. There are two bound\nstates with real eigenvalues. However, only the state specified\nby~(\\ref{eq:ome}) has negative norm. Projecting this out, we arrive\nat the same expression for ${\\cal H}_{\\rm phys}$ as in case (ii).\n\nRPI models are covered by Case (i) for $M<0$, and have ${\\cal H}_{\\rm\nphys}$ given by~(\\ref{eq:HP}) for $M>0$, with $\\omega=-1\/M$.\n\nThe GPI Hamiltonian acts on ${\\cal H}_{\\rm phys}$ by restriction. For\nexample, in case (ii) above, we have\n\\begin{eqnarray}\nD(h_{L,M}|_{{\\cal H}_{\\rm phys}}) &=& \\left\\{\n\\left(\\begin{array}{c} \\varphi \\\\\nM^{-1\/2}\\inner{\\xi_\\omega}{\\varphi}\\end{array}\\right)\\mid\n\\varphi,\\varphi^\\prime\\in AC_{\\rm loc}(0,\\infty),~\n\\varphi,\\varphi^{\\prime\\prime}\\in L^2;\\right. \\nonumber \\\\\n&&\\qquad\\qquad\\qquad\\qquad\\quad\n\\left.\\begin{array}{c} \\ \\\\ \\ \\end{array}\nM\\varphi(0)=-\\inner{\\xi_\\omega}{\\varphi} \\right\\}\n\\end{eqnarray}\non which $h_{L,M}|_{{\\cal H}_{\\rm phys}}$ acts as before. The restricted\noperator has the same continuum spectrum as $h_{L,M}$, but has no\nbound states in this case. Moreover, the property of locality is\npartially lost: it is clear that vectors of form $(\\varphi,0)^T$\nwith $\\varphi\\in C_0^\\infty(0,\\infty)$ are in ${\\cal H}_{\\rm phys}$ only if\n$\\varphi\\perp\\xi_\\omega$. However, for elements of this form in\n${\\cal H}_{\\rm phys}$, it remains the case that $h_{L,M}|_{{\\cal H}_{\\rm\nphys}}(\\varphi,0)^T= (-\\varphi^{\\prime\\prime},0)^T$. Thus the\nproperties of locality and `positivity' are not entirely compatible.\n\n\n\\subsection{Physical Interpretation}\n\nIn this section, we discuss how the effective range models\nconstructed above may be used to model Schr\\\"{o}dinger operators\n$H=-\\triangle+V$, where $V$ is smooth, spherically symmetric and\ncompactly supported within radius $a$ of the origin. Our methodology\nextends that described in~\\cite{KF}, in which the scattering length\napproximation is discussed.\n\nGiven a smooth spherically symmetric potential $V(r)$ supported\nwithin radius $a$ of the origin, we may find the `best fit' GPI model\n$h_{L,M}$ as follows. Let $u_0$ be the $S$-wave zero energy\neigenfunction, i.e., the solution to $-u_0^{\\prime\\prime}+Vu_0=0$\nwith regular boundary conditions at the origin.  Then the arguments\nof Section 11.2 of \\cite{Newt} give the low energy parameters $L$ and\n$M$ as\n\\begin{equation}\nL= a - \\left.\\frac{u_0}{u_0^\\prime}\\right|_{r=a};\n\\label{eq:L}\n\\end{equation}\nand\n\\begin{equation}\nM = a\\left\\{ 1-\\frac{a}{L}\n+\\frac{1}{3}\\left(\\frac{a}{L}\\right)^{2}-\n\\left( 1-\\frac{a}{L}\\right)^2\n\\frac{\\int_0^a |u_0(r)|^2 dr}{a|u_0(a)|^2} \\right\\}.\n\\label{eq:M}\n\\end{equation}\nThus the scattering behaviour is $\\cot\\delta_0(k)=\n-(kL)^{-1}+kM+O(k^3)$ and the best fit GPI model in our class is\n$h_{L,M}$. We refer to equations~(\\ref{eq:L}) and~(\\ref{eq:M}) as\n{\\em fitting formulae}; equation~(\\ref{eq:L}) is the fitting formula\nemployed in \\cite{KF}. The range of energies for which the\napproximation is valid can be determined by a `believability'\nanalysis analogous to that described in \\cite{KF}. We will not do\nthis here.\n\nNote that $M$ obeys the bound\n\\begin{equation}\n-\\infty \\le M < a\\left\\{ 1-\\frac{a}{L}\n+\\frac{1}{3}\\left(\\frac{a}{L}\\right)^{2}\\right\\}.\n\\end{equation}\nMoreover, this bound is best possible: for any\n$L\\in{\\rm I\\! R}\\cup\\{\\infty\\}$ and any $M$ in the above range, one can\nclearly find a smooth function $u_0(r)$ satisfying regular boundary\nconditions at the origin, $u_0\\propto(1-r\/L)$ for $r>a$ and such\nthat~(\\ref{eq:M}) holds. Then the potential defined by\n$V(r)=u_0^{\\prime\\prime}(r)\/u_0(r)$ has $S$-wave scattering behaviour\napproximated to second order by $h_{L,M}$. The contribution to the\ntotal scattering cross section from the effective range term\ngenerally outweighs that from higher angular momenta, so the $S$-wave\nGPI model provides a second order approximation to the full\nscattering behaviour.\n\nFinally, we discuss the interpretation of the discrete spectrum of\n$h_{L,M}$. We have constructed $h_{L,M}$ so that its\nscattering behaviour matches that of a given Schr\\\"{o}dinger\noperator at low energies, $E$. For larger $|E|$, the approximation\nbreaks down -- in the language of \\cite{KF} we say that it is no\nlonger `believable'. Thus, deeply bound states are unlikely to be\nbelievable. In particular, for $0<L<4M$, $h_{L,M}$ exhibits a complex\nconjugate pair of eigenvalues, which can never be\nbelievable.\\footnote{These are not eigenvalues of $h_{L,M}$\nrestricted to the physical Hilbert space. However, they persist as\npoles in the scattering amplitude and our remarks still apply:\n$h_{L,M}$ does not give a reliable approximation to the scattering\ntheory at those scales.} Such phenomena are artifacts of the\nidealisation process, due to the truncation of the low energy\nexpansion. The issue of believability is discussed in \\cite{KF};\nsimilar comments are made in \\cite{Diejen}.\n\n\\sect{Conclusion}\n\nWe begin by discussing various generalisations of our method. There\nare many situations in which the analysis of Section~2.2 may be\napplied. In two dimensions, for example, one can consider radial GPI\nHamiltonians which agree with\n\\begin{equation}\nh = -\\frac{1}{r}\\frac{d}{dr}r\\frac{d}{dr} + \\frac{\\nu^2}{r^2}\n\\end{equation}\naway from the origin, as models for an infinitesimal `dot' of\nmagnetic flux $\\nu$, with $|\\nu|<1$. In this case, one must employ\nHankel transforms rather than sine and cosine transforms. In \\cite{F}\nwe will implement this programme to construct a models of RPI type\nfor the Dirac equation in the presence of an infinitesimal tube of\nflux. These models provide the leading order approximation to the\nscattering data.\n\nOur method could also be applied to $S$-wave GPI models with a\nCoulombic tail. In this case, the appropriate integral transforms\nwould be based upon Whittaker functions and the scattering data\nwould be specified in terms of Coulomb-modified partial wave shifts.\nIn this case, the dimension of ${\\cal M}_2$ would be countably\ninfinite, due to the countable discrete spectrum of\nsuch models. However, one would expect ${\\cal M}_1$ to remain finite\ndimensional for simple models.\n\nSecondly, it is of interest to generalise the unitary dilation method\nto sectors of higher angular momentum with $\\ell\\ge 1$ (and the\ncorresponding analogues for magnetic flux dots -- i.e., $|\\nu|\\ge 1$\n-- and Coulombic GPI for $\\ell\\ge 1$). This is more problematic,\nbecause the radial Hamiltonian $-d^2\/dr^2+\\ell(\\ell+1)\/r^2$ is\nessentially self-adjoint on $C_0^\\infty(0,\\infty)$ and so the method\nof Section~2.2 does not apply. Here, it might be possible to  obtain\na suitable integral transform by analysing the distributional\nconstruction. We hope to return to this elsewhere.\n\nFinally, we consider applications to the definition of arrays of\npoint scatterers. Here, the most likely use of our methods is to\ngenerate the `monomer' by inverse scattering. By passing to the\nresolvent written in the form of Krein's formula, one can isolate\nthe appropriate `defect element' and proceed to form the array by\nmethods discussed in \\cite{Diejen}, which generalise the procedure\nfor arrays of PI developed in \\cite{Gmann}.\n\n{}To summarise, we have introduced an inverse scattering construction\nfor GPI models using the theory of unitary dilations, and developed\nthe method in detail for the class of single centre $S$-wave GPI\nmodels with rational $S$-matrices. A physical locality requirement\ncompletes the specification of the Hamiltonian, whose scattering,\nspectral and domain properties are explicitly determined from our\nresults.\n\n{\\em Acknowledgements:} The notions of fitting formulae and the\ngeneral methodology of Section~5.3 are due in origin to Bernard Kay\n\\cite{KF}; I thank him and also Graham Allan and Clive Wells for\nuseful conversations. In addition, I thank Churchill College,\nCambridge, for financial support under a Gateway Studentship.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSingle-particle ultrashort pulse X-ray scattering experiments are a promising new approach to the structure determination of biomolecules~\\cite{hajdu_single-molecule_2000, huldt_diffraction_2003, gaffney_imaging_2007, miao_beyond_2015}. \nWhereas most alternative approaches focus on nano-crystals~\\cite{chapman_femtosecond_2006, chapman_femtosecond_2011, boutet_high-resolution_2012, fromme_femtosecond_2011, kirian_femtosecond_2010, schlichting_serial_2015, roedig_high-speed_2017, barends_serial_2022}, X-ray scattering experiments on single molecules can in principle also provide structural information on molecular ensembles and, hence, biomolecular conformational and functional motions. \nIn such 'hit and destroy' experiments, a stream of single molecules is exposed to a beam of high intensity femtosecond X-ray free electron laser (XFEL) pulses (Figure~\\ref{fig: experiment}a), and for each hit the positions of the scattered photons (red dots) on the detector are recorded as a scattering image~\\cite{neutze_potential_2000}. \nImportantly, the ultra-short pulses serve to outrun the subsequent destruction of the particles due to radiation damage, but also\nimply that only very few photons are being recorded for each molecule~\\cite{gaffney_imaging_2007}.\n\nThe feasibility of the method has already been demonstrated by a number of experiments~\\cite{seibert_single_2011, ekeberg_three-dimensional_2015, hosseinizadeh_high-resolution_2014}, but so far only structures of relatively large specimen have been successfully determined, for instance of entire mimivirus particles~\\cite{seibert_single_2011, ekeberg_three-dimensional_2015}  ($450\\,\\text{nm}$ in diameter) and coliphage viruses~\\cite{hosseinizadeh_high-resolution_2014} ($20\\,\\text{nm}$ in diameter). \nWhereas for large specimens many photons are scattered per image, for example $10^7$ for the mimivirus~\\cite{seibert_single_2011, ekeberg_three-dimensional_2015}, for typical proteins only 10-50 coherently scattered photons per image are expected~\\cite{yoon_comprehensive_2016, hantke_condor_2016}, which further complicates structure determination particularly for small molecules. \nSuch images can be obtained at with an intensity of $10^{12}$ photons per pulse at $5\\,\\mathrm{keV}$ and a $1\\,\\mathrm{\\text{\\textmu} m}$ beam diameter~\\cite{von_ardenne_structure_2018}, for example from the XFELs at DESY or SLAC.\n\nMost importantly, the orientation of the molecules at the time of scattering is typically unknown, which poses an additional and substantial refinement challenge.\nAlso, so far it is impossible to systematically include shot noise, incoherently scattered photons, background scattering, or detector noise in the structure refinement.\nThese issues are particularly challenging for the structure refinement of small specimen at high resolution, for example proteins or protein complexes at near atomic resolution. \nA number of methods have been proposed to address these issues, such as orientation determination methods~\\cite{shneerson_crystallography_2008,loh_reconstruction_2009, walczak_bayesian_2014,kassemeyer_optimal_2013,elser_three-dimensional_2011, tegze_atomic_2012, donatelli_reconstruction_2017, flamant_expansion-maximization-compression_2016, ayyer_dragonfly_2016} and manifold embedding algorithms~\\cite{fung_structure_2009,moths_bayesian_2011,schwander_symmetries_2012,giannakis_symmetries_2012}, which, however, typically require 100 to 1000 photons per scattering image.\nAs an alternative, correlation based approaches~\\cite{saldin_structure_2009, saldin_beyond_2010, saldin_new_2011, saldin_structure_2010,kurta_correlations_2017} require, in principle, as little as three photons per image~\\cite{von_ardenne_structure_2018}. \n\nFurther, many biomolecules show structural heterogeneity and conformational dynamics between different distinct structures, which, when resolved, would provide a direct view on biomolecular function. Hence, and similar to the current main challenge in cryogenic electron microscopy~\\cite{guaita_recent_2022}, not one but many structures need to be extracted from the scattering images. Accordingly, in addition to the orientation, also the current conformer for each scattering image is unknown. \nWhereas both orientation determination methods and manifold-based methods have been applied to determine multiple conformational structures~\\cite{zhuang_unsupervised_2022,hosseinizadeh_conformational_2017-1}, the required large number of photons per image precludes their application to single biomolecules.\n\nTo overcome the above issues, we developed and assessed a rigorous Bayesian method for structure determination from single-molecule scattering images. \nWe will demonstrate that this method can not only determine a single structure at high resolution but also determine structural ensembles.\nUnexpectedly, much fewer images are required for refining an ensemble of $n$ conformers consisting of $m$ atoms each than for refining a single structure consisting of $m\\times n$ atoms, which may bring applications to non-synthetic experimental data into reach.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[height = 0.3\\textwidth]{setup.pdf}\n    \\caption{Single molecule scattering experiment. \\textbf{a} A stream of single molecules is hit by femtosecond X-ray pulses, and the scattered photons are recorded as images (image reproduced from von Ardenne et al.~\\cite{von_ardenne_structure_2018}). \\textbf{b} The scattered photons (red dots) are distributed on the Ewald sphere according to the 3D-intensity function $I$ (blue).}\n    \\label{fig: experiment}\n\\end{figure}\n\n\n\\newpage\n\n\\section{Results}\n\n\\paragraph{Summary of the approach.}\n\nFor each scattering image, the positions of the recorded photons specify vectors $\\vec k_1, \\dots, \\vec k_l$ on the Ewald sphere in Fourier space (Figure~\\ref{fig: experiment}b). The probability of observing a photon at a particular position on the detector is proportional to the 3D intensity function $I(\\vec k) \\propto |\\mathcal F\\{\\vec R\\rho_i\\}(\\vec k)|^2$ at the corresponding position $\\vec k$ on the Ewald sphere, which in turn is given by the Fourier transform of the electron density $\\rho_i$ of conformer $i$. Here, $\\vec R$ is the unknown orientation of the molecule for this particular image.\n\nIt follows that the probability of observing an image with photon positions $\\vec k_1, \\dots, \\vec k_l$ is obtained by averaging over both the conformational ensemble $\\boldsymbol\\uprho = \\{\\rho_1, \\dots, \\rho_n\\}$ with weights $\\vec w = \\{w_1, \\dots, w_n\\}$ and all orientations $\\vec R$. Because each scattering image is an independent event, the total probability of the set of all images $\\mathcal I$ reads\n\\begin{equation}\n    P(\\mathcal I \\,|\\, \\boldsymbol\\uprho, \\vec w) \\propto \\prod_{(\\vec k_1, \\dots, \\vec k_l) \\in \\mathcal I} \\sum_{i=1}^n w_i\\int_{\\mathrm{SO}(3)}P(\\vec k_1, \\dots, \\vec k_l \\,|\\, \\vec R\\rho_i)\\, \\mathrm d \\vec R\\,.\n    \\label{eq: likelihood}\n\\end{equation}\nThis probability serves to determine either a single structure or a structural ensemble by sampling from the Bayesian posterior probability $P(\\boldsymbol\\uprho, \\vec w \\,|\\, \\mathcal I) \\propto P(\\mathcal I \\,|\\, \\boldsymbol\\uprho, \\vec w)P(\\boldsymbol\\uprho, \\vec w)$ using a Markov chain Monte Carlo approach. For the prior $P(\\boldsymbol\\uprho, \\vec w)$ the orientations are assumed to be uniformly distributed. To minimize the number of required degrees of freedom, and as a means of regularization, we chose a physically motivated representation of each $\\rho_i$ in terms of a sum of Gaussian functions, which also completes the definition of the prior.\n\n\nFor typical cases, the number of required degrees of freedom remains large and poses a formidable sampling challenge. \nTo address this issue, we adopted a hierarchical simulated annealing approach. \nStarting at very low resolution, the macromolecular structures were sampled in multiple hierarchical stages of increasing resolution. To increase the sampling efficiency, in each of these stages, for each Markov step the previous ensemble of structures of maximal posterior probability was used as a proposal density.\nTo this end, the scattering images that would have been observed for a smoothed low resolution copy of the original molecule were obtained from the original images by rejection sampling using the convolution theorem (see the Methods section). \n\nFurther, we adapted the Bayesian formalism such that it only uses those images which contain new information, that is, that contain photons for which the magnitude of $\\vec k$ is larger than the threshold of the resolution from the previous stage.\nWith increasing resolution, the fraction of such useful images becomes very small, thus markedly enhancing computational efficiency up to two orders of magnitude. The approach is described in detail in the Supplementary Information.\n\n\\paragraph{Sample test refinements.}\nBecause our Bayesian approach uses all available information, it we expect it to require fewer scattering images to achieve a certain resolution than, for example, correlation based methods.\nTo assess this aspect, we first tested our method on the single structure level, using the same 46-residue protein crambin~\\cite{jelsch_accurate_2000} as in the study of von Ardenne et al.~\\cite{von_ardenne_structure_2018}.\nA total of $10^8$ noise-free synthetic images were generated, containing a realistic average of $15$ photons each. \nFrom these, the structure was solved in five hierarchical stages (Fig.~\\ref{fig: crambin}a), increasing the number of degrees of freedom by a factor of two in each stage. \nFor the final stage, a representation consisting of $184$ Gaussian functions was used, which is four times the number of residues. For more details see Supplementary Note~\\ref{sup: parameters}. \nIndeed, a similar Fourier shell correlation resolution~\\cite{van_heel_fourier_2005} of $4.2\\,${\\AA} (Fig.~\\ref{fig: crambin}b) was obtained as for the previous correlation based method, using only half of the total number of scattered photons.\n\nNext, to demonstrate that our method can resolve ensembles of multiple conformers, we used three molecular dynamics trajectories of alanine dipeptide~\\cite{scherer_pyemma_2015} of length $250\\,\\text{ns}$ each to generate $10^6$ scattering images, using a randomly chosen snapshot for each image. \nAs before, an average of $15$ photons per image were generated. \nThen, a weighted ensemble of eight conformers was determined from these images (Fig.~\\ref{fig: dipeptide}), each described by a sum of $10$ Gaussian functions. \nTo obtain sufficient statistics, a total of $10$ independent simulated annealing runs were carried out, using the same images.\n\nTo assess the quality of the obtained ensemble, for each of the eight structures the resolution with respect to its nearest neighbor in the input trajectories was calculated using Fourier shell correlations~\\cite{van_heel_fourier_2005} (Fig.~\\ref{fig: dipeptide}b), resulting in a weighted average resolution of $1.6\\,\\text{{\\AA}}$. \nThis result shows that the obtained eight structures are indeed close to the reference ensemble. \nTo also assess the accuracy of the entire ensemble, for each time step in the input trajectories, the resolution with respect to its nearest neighbor among all the determined structures was calculated (Fig.~\\ref{fig: dipeptide}d). \nAs a main result we found that $90\\%$ of the input trajectories are within $2.1\\,${\\AA} Fourier shell correlation resolution of the determined structures, and that all of the trajectory frames are within $2.5\\,${\\AA} resolution of the determined structures. \nFigure~\\ref{fig: dipeptide}e compares the $10$ obtained ensembles with the reference ensemble using a Ramachandran plot~\\cite{ramachandran_stereochemistry_1963} showing the distribution of the torsion angles $\\phi$ and $\\psi$. \nFor each of the determined structures its nearest neighbor in the input trajectories was used to compute these angles. \nAs can be seen, the reference density is well represented by the determined structures. \n\nNext, we asked if our method is also capable of extracting structural ensembles for the larger mini-protein chignolin~\\cite{honda_crystal_2008}, comprising $10$ residues. \nTo that end, $50$ molecular dynamics trajectories of length $10\\,\\text{\\textmu s}$ were used to generate $1.2\\cdot10^7$ images with, on average, $15$ photons each.\nAs a further challenge, this ensemble also contained unfolded structures. \nFrom the obtained images, we determined multiple stages of weighted structural ensembles of increasing resolution and increasing number of conformers (Fig.~\\ref{fig: 5awl}a). \nAs above, resolutions were computed using Fourier shell correlations (Fig.~\\ref{fig: 5awl}b,c), finding a weighted average resolution of $5\\,${\\AA} for the folded conformers, and $9-10\\,${\\AA} for the unfolded conformers.\nInterestingly, in the final stage one of the six determined weights is nearly zero, suggesting that five conformers suffice for the used number of images at this resolution level. \nIt is also remarkable that the $9\\%$ fraction of unfolded states in the reference structure ensemble was correctly identified.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{crambin.pdf}\n    \\caption{Structure determination of Crambin. \\textbf{a} Hierarchical stages of retrieved electron densities. \\textbf{b} Fourier shell correlation between the retrieved densities and the reference density. \\textbf{c} Retrieved electron density. \\textbf{d} Reference electron density.}\n    \\label{fig: crambin}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{dipeptide_weighted_v2.pdf}\n    \\caption{Structural ensemble determination of the alanine dipeptide. \\textbf{a} Reconstructed conformers (green), the corresponding weights, and the nearest neighbors in the input trajectories (blue) with the corresponding resolutions. \\textbf{b} Fourier shell correlations used to compute these resolutions. \\textbf{c} Weighted resolution distribution for $10$ independent runs from the the same data. \\textbf{d} Resolution distribution over the time steps of the input trajectories relative to their nearest neighbors among the determined structures from all $10$ runs. \\textbf{e} Ramachandran plot for the input trajectories (shown as a density) and the determined structures from all $10$ runs (points, the color indicates the different runs). }\n    \\label{fig: dipeptide}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{5awl_tree.pdf}\n    \\caption{Structural ensemble determination for chignolin. \\textbf{a} Hierarchical stages of retrieved structures (green) and their nearest neighbors (blue) in the input trajectories with the corresponding resolutions. \\textbf{b} Fourier shell correlations of the reconstructed structures relative to their nearest neighbors. \\textbf{c} Resolution distribution over the time steps of the input trajectories relative to their nearest neighbors among the determined structures. }\n    \\label{fig: 5awl}\n\\end{figure}\n\n\n\n\n\\paragraph{Scaling.}\nIn the sample applications described above we observed that resolving $n$ conformational structures consisting of $m$ residues each required much fewer scattering images and photons than resolving a single $n\\times m$ residue structure of the same total size and complexity --- even in cases where the members of the former ensemble are very different from each other.\nTo investigate this counterintuitive observation in more detail, small `structures' consisting of randomly placed Gaussian functions were used.  \nFor each combination of parameters, eight independent structure determination runs were performed, and for each run the achieved resolution was determined. \nThe structure weights $w_i$ where chosen to be uniform and kept fixed during the simulated annealing runs.\n\nFigure~\\ref{fig: scaling}c and~\\ref{fig: scaling}f show for each combination of parameters the smallest number of images for which all of the replicas achieved a resolution better than a given threshold. \nAs can be seen in Fig.~\\ref{fig: scaling}c, for the structure ensemble of $n$ conformations with $m$ residues each, the required number of images is approximately proportional to $n^2$, that is, the square of the number of conformations. \nWhereas somewhat unexpected, this finding is in line with a theoretical argument showing that the information content of a single image is in this case proportional to $1\/n^2$ (Supplementary Note \\ref{sup: information}). \nIn contrast, the number of images required to resolve a single structure of $n\\times m$ residues grows even much faster, resembling a power law $m^c$ with an exponent $c \\approx 5$ (Fig.~\\ref{fig: scaling}f). \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{scaling.pdf}\n    \\caption{Dependence of the resolution on the number of images, the number of conformations, and the size of the structure. \\textbf{a} Resolution as a function of the number of images for various numbers of conformations $n$. \\textbf{b} Resolution as a function of the number of conformations for various numbers of images. \\textbf{c} Required number of images to achieve various resolutions as a function of the number of conformations. For comparison, a quadratic relationship is shown (dashed line). \\textbf{d} Resolution as a function of the number of images for various structure sizes (parameterized by the number of Gaussians $m$). \\textbf{e} Resolution as a function of the number of Gaussians for various numbers of images. \\textbf{f} Required number of images to achieve various resolutions as a function of the number of Gaussians. For comparison, a power law $m^5$ is shown (dashed line). }\n    \\label{fig: scaling}\n   \n\\end{figure}\n\n\n\\clearpage\n\\section{Discussion}\nHere we have developed a rigorous Bayesian method for determining biomolecular structures from single molecule X-ray scattering images in the extreme few photon Poisson regime. \nUsing synthetic scattering images generated from simulated X-ray scattering experiments, we have demonstrated that both single structures as well as structural ensembles of small biomolecules can be resolved to near atomic resolution. \n\nOur results for the globular protein crambin show that a similar resolution of $4.2\\,${\\AA} is obtained compared to previous correlation based methods~\\cite{von_ardenne_structure_2018} that require similarly small numbers of photons per scattering image. Because such correlation based methods disregard higher correlations, whereas the full information content of each image is used in our Bayesian approach, we speculated that the latter should require correspondingly fewer images. \nThis was indeed observed for the above protein, for which the number of images required to obtain a resolution of approximately $4.2\\,${\\AA} was reduced from roughly $2\\cdot 10^8$ to $1\\cdot 10^8$. \n\nOur method should be applicable also to much larger molecules, the main bottleneck currently being computational cost. \nHowever, we expect that the factor by which our method requires fewer photons than correlation based methods will increase with the size of the molecule, due to the larger number of photons expected per image.\nFurther, we think that the computational limitations can be overcome by improved optimization or sampling methods or by utilizing prior structural information, either from structure databases, from AlphaFold~\\cite{jumper_highly_2021}, or guided by molecular dynamics force fields.\n\nFor alanine dipeptide the full conformational ensemble generated by an atomistic simulation was extracted at atomistic resolution of on average $1.8\\,${\\AA} from simulated scattering experiments, in which not only the current orientation of the biomolecule but also its current conformer was unknown.\nFor the 10 amino acid protein chignolin~\\cite{honda_crystal_2008} both the folded and unfolded ensembles were resolved, albeit so far at lower resolution. \nNotably, also the weights corresponding to the folded and unfolded conformers were recovered accurately. \nFurthermore, using weighted ensembles allows the number of conformers to be determined dynamically, as some of the determined weights may be zero.\n\nUnexpectedly few images were required to resolve structural ensembles.\nBecause an ensemble of $n$ conformers consisting of $m$ atoms each has the same number of degrees of freedom as a single structure of $n\\times m$ atoms, a similar number of images should be required. \nHowever, closer analysis suggests that roughly $O(m^5)$ images are required to a resolve a single structure with $m$ atoms. One might therefore expect that $O(n^5m^5)$ images are required for an ensemble of $n$ such structures. \nHowever, our analysis suggests that only $O(n^2m^5)$ images are required for this, which is consistent with a theoretical argument that the expected information content of a single scattering image is in this case proportional to $1\/n^2$.\nThis suggests that determining more complex conformational ensembles may be within reach, and that it may be possible to collect the required amount data from experiments. \nOur Bayesian analysis of structural heterogeneity is similar in spirit to approaches that were successfully applied in cryo-electron microscopy~\\cite{cossio_bayesian_2013,grant_cistem_2018,punjani_cryosparc_2017,lyumkis_likelihood-based_2013,scheres_chapter_2016,tang_eman2_2007,zivanov_new_2018,kinman_uncovering_2022}, which, from a mathematical standpoint, shares some similarities with single molecule X-ray scattering, albeit at a much lower noise level. \n\nFrom a more general perspective, our Bayesian approach represents a systematic and rigorous approach to include shot noise in the extreme Poisson regime characteristic for single molecule X-ray scattering experiments. \nIn contrast to other proposed methods, this Bayesian framework will also allow to include other sources of noise and uncertainty in a conceptually straightforward manner, such as incoherently scattered photons, background scattering, detector noise, or scattering by disordered water at the biomolecular surface. \nPrerequisite is the development and calibration of forward noise models such as those documented in, e.g., Ref.~\\cite{yoon_comprehensive_2016}. \n\n\n\\section{Methods}\n\n\\paragraph{Structure and structure ensemble representation.}\n\\label{sec: representation}\nElectron density functions of the reference structures or conformers were described by a sum of $m$ of Gaussian functions with atomic positions $\\vec y_i$, heights $h_i$ and standard deviations $\\sigma_i$,\n\\begin{equation}\\label{eq: representation}\n    \\rho(\\vec r) = \\sum_{i=1}^m \\frac{h_i}{\\left(\\sigma_i\\sqrt{2\\pi}\\right)^3} \\exp \\left(\\frac1{2\\sigma_i^2}\\lVert \\vec r - \\vec y_i\\rVert^2\\right)\\,.\n\\end{equation}\nElectron density functions of the determined structures were described similarly, with one common height $h = h_i$ and one common standard deviation $\\sigma=\\sigma_i$, which is treated as an unknown and determined together with the positions $\\vec y_i$.\nStructural ensembles were represented by a weighted sum of conformers $\\boldsymbol\\uprho = \\{\\rho_1, \\dots, \\rho_n\\}$ with weights $\\vec w = \\{w_1, \\dots, w_n\\}$.\n\n\n\\paragraph{Synthetic data generation.}\n\\label{sec: datagen}\n\nFor each of the synthetic scattering images, the photon positions on the detector $D$ were drawn from a probability distribution proportional to the intensity function $I(\\vec k) = |F\\{\\rho\\}(\\vec k)|^2$ restricted to the appropriate Ewald sphere.\nSpecifically, generation of each image involved the following steps:\n\\begin{enumerate}\n    \\item A conformation of the molecule is selected randomly from the reference ensemble (for example, consisting of molecular dynamics trajectories),\n    \\item a random orientation $\\vec R$ of the molecule is drawn uniformly from the rotation group $\\mathrm{SO}(3)$,\n    \\item the number of scattered photons is drawn from a Poisson distribution with mean $N\\int_D I(\\vec R\\vec k) \\,\\mathrm d \\vec k$, where $N$ is the incoming beam intensity,\n    \\item the position of each scattered photon is drawn from the probability distribution proportional to $(I\\circ \\vec R)|_D$.\n\\end{enumerate}\nThe last two steps were implemented using rejection sampling. To this end, a von Mises-Fisher distribution $p$ on $D$ was chosen with high enough standard deviation that $I(\\vec k) \\leq p(\\vec k)$ everywhere. Then, for each photon, its position $\\vec k$ was drawn from $p$ and it was accepted with probability $I(R\\vec k) \/ p(R\\vec k)$. The beam intensity $N$ was chosen together with a normalization of $\\rho$ such that this procedure accurately produces a Poisson distribution of the desired expected number of photons per image.\n\n\n\\paragraph{Computation of likelihoods.}\n\\label{sec: distribution}\nThe probability density of observing an image defined by photon positions $\\vec k_1, \\dots, \\vec k_l$ given an electron density function $\\rho$ with is corresponding intensity function $I(\\vec k)$ was computed by averaging over all possible orientations $\\vec R\\in \\mathrm{SO}(3)$ of the molecule, \n\\begin{equation}\n\\label{eq: probability density}\n    \\begin{aligned}\n        P(\\vec k_1, \\dots, \\vec k_l \\,|\\, \\rho\n        &= \\frac{N^l}{l!}\\int_{\\mathrm{SO}(3)} \\exp\\left(-N\\!\\int_D I(\\vec R\\vec k) \\,\\mathrm d \\vec k\\right) \\left(\\prod_{i=1}^l I(\\vec R\\vec k_i)\\right) \\,\\mathrm d \\vec R, \\\\\n       \n    \\end{aligned}\n\\end{equation}\nwhere for each orientation, the probability is a product of the Poisson distribution for the number of photons $l$ in the image and a factor depending on the photon positions. \nThese integrals were approximated by averaging over a discrete set of typically $r\\approx 10^3$ to $r\\approx 10^5$ rotations $\\vec R_i$ with weights $s_i$,\n\\begin{equation}\n    P(\\vec k_1, \\dots, \\vec k_l \\,|\\, \\rho) \\approx \\frac{N^l}{l!}\\sum_{i=1}^r s_i\\exp\\left(-N\\!\\int_D I(\\vec R_i\\vec k) \\,\\mathrm d \\vec k\\right) \\prod_{j=1}^l I(\\vec R_i\\vec k_j).\n\\end{equation}\nThe rotations $\\vec R_i$ and their weights $s_i$ are constructed by combining a Lebedev quadrature rule on $S^2$ with a uniform quadrature rule on $S^1$ via the Hopf map~\\cite{noauthor_sphere_lebedev_rule_nodate,graf_sampling_2009} (Supplementary Note~\\ref{sup: computation}).\n\n\\paragraph{Simulated annealing and hierarchical sampling.}\n\\label{sec: annealing}\nA Monte Carlo simulated annealing approach with the energy function $-\\log P$ was used to sample from or maximize the Bayesian posterior probability, as described in detail in the Supplement.\nTo enhance convergence, Bayesian sampling and maximization were performed in multiple hierarchical resolution stages. \nStarting from a low resolution representation of $\\rho$ with correspondingly few degrees of freedom, the number of Gaussian functions was doubled in each stage and the reduced resolution structure determined by the previous stage was used as a proposal density (see Supplement). \nTo calculate likelihoods for the reduced resolution structures, lower resolution scattering images were generated from the original images by rejection sampling, that is, by removing each photon in the original images with probability $1 - \\exp(-\\sigma^2k^2\/2)$.\nBy construction, this rejection scheme samples from a Fourier transformed density $I_\\rho \\cdot \\exp(-\\sigma^2|\\vec k|^2\/2)$ which, by the convolution theorem, corresponds to a smoothed real space density $\\tilde\\rho = \\rho*\\mathcal N(\\sigma)$ obtained as the convolution of $\\rho$ with a Gaussian kernel with width (resolution) $\\sigma$. \nComputational efficiency was further increased substantially by selecting only those original images for the likelihood computations that actually contain useful information at the respective resolution. As described in the Supplement, the Bayesian formalism allows for removing this selection bias.\n\n\n\\paragraph{Structure alignment and resolution estimate.}\nBecause the orientations of the obtained structures are irrelevant, these were rotationally aligned to each other\nby minimizing the cost function\n\\begin{equation}\n    d(\\vec S) = \\frac1n \\sum_{i=1}^n \\min_{j=1}^m\\, \\lVert\\vec y_i - \\vec S \\vec y_j'\\rVert + \\frac1m \\sum_{j=1}^m \\min_{i=1}^n\\, \\lVert\\vec y_i - \\vec S \\vec y_j'\\rVert.\n\\end{equation}\nHere, the positions $\\vec y_1, \\dots, \\vec y_n$ and $\\vec y_1', \\dots, \\vec y_m'$ define two structures per equation \\eqref{eq: representation} and $\\vec S$ is a rotation matrix $\\vec S \\in \\mathrm{O}(3)$. Both rotations and reflections were included, as X-ray scattering images do not distinguish between mirror images. \n\nThe resolution of the aligned structures was estimated using Fourier shell correlations~\\cite{van_heel_fourier_2005},\n\\begin{equation}\n    \\mathrm{FSC}(k) = \\frac{\\int_{\\lVert\\vec k\\rVert = k} \\hat\\rho_1(\\vec k)^*\\hat\\rho_2(\\vec k)\\,\\mathrm d\\vec k}{\\sqrt{\\int_{\\lVert\\vec k\\rVert = k} |\\hat\\rho_1(\\vec k)|^2\\,\\mathrm d\\vec k}\\sqrt{\\int_{\\lVert\\vec k\\rVert = k} |\\hat\\rho_2(\\vec k)|^2\\,\\mathrm d\\vec k}}\\,,\n\\end{equation}\nwhere $\\rho_1$ are $\\rho_2$ the structures to be compared and $\\hat\\rho$ denotes the Fourier transform of $\\rho$. Accordingly, the achieved resolution was determined as $2\\pi\/k_\\mathrm{fsc}$, where $k_\\mathrm{fsc}$ is the threshold at which the Fourier shell correlation drops below $1\/2$, providing a conservative estimate~\\cite{van_heel_fourier_2005}.\n\n\\paragraph{Molecular dynamics simulations.}\nAll atomistic simulation trajectories were generated using the GROMACS 2018 software package~\\cite{abraham_gromacs_2015} with the Charmm36mm force field~\\cite{huang_charmm36m_2017} and the OPC water model~\\cite{izadi_building_2014}. \nFor chignolin, the starting structure was taken from the Protein Data Bank~\\cite{berman_protein_2000}, entry 5AWL~\\cite{honda_crystal_2008}.\nAll hydrogen atoms were described by virtual sites~\\cite{feenstra_improving_1999}.\nEach protein was placed within a triclinic water box, such that the smallest distance between protein surface and box boundary was larger than $1.5\\,$nm. \nSodium and chloride ions were added to neutralize the system, corresponding to a physiological concentration of $150\\,$mmol\/l. \nEnergy minimization was performed using steepest descent for $5\\cdot 10^4$ steps.\nEach system was subsequently equilibrated for $0.5\\,$ns in the $NVT$ ensemble, and subsequently for $1.0\\,$ns in the $NPT$ ensemble at $1\\,$atm pressure and temperature $300\\,$K using an integration time step of $2\\,$fs. \nThe velocity rescaling thermostat~\\cite{bussi_canonical_2007} and Parrinello-Rahman pressure coupling~\\cite{parrinello_polymorphic_1981} were used with coupling coefficients of $\\tau=0.1\\,$ps and $\\tau=1\\,$ps, respectively.\nAll bond lengths of the solute were constrained using the LINCS algorithm~\\cite{hess_lincs_1997} with an expansion order of 6, and the geometry of the water molecules was constrained using the SETTLE algorithm~\\cite{miyamoto_settle_1992}. \nElectrostatic interactions were calculated using PME~\\cite{darden_particle_1993}, with a real space cutoff of $10\\,${\\AA} and a Fourier spacing of $1.2\\,${\\AA}.\nFor all production runs, a $4\\,$fs integration was used, and the atom coordinates were saved every $100\\,$ps, such that $10^5$ snapshots were available for each trajectory.\nThe trajectories for alanine dipeptide were taken from mdshare \\cite{noauthor_mdshare_nodate-1}. The structure for crambin was taken from PDB entry 1EJG~\\cite{jelsch_accurate_2000}.\n\n\\section{Acknowledgments}\nThis work was financially supported by the Federal Ministry of Education and Research through the joint research project 05K20EGA Fluctuation XFEL, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - CRC 1456\/1 - 432680300. The MD-trajectories were kindly provided by Nicolai Kozlowski. \n\n\n\\printbibliography\n\n\n\n\\clearpage\n\\begin{appendix}\n\n\\section{Introduction}\nSingle-particle ultrashort pulse X-ray scattering experiments are a promising new approach to the structure determination of biomolecules~\\cite{hajdu_single-molecule_2000, huldt_diffraction_2003, gaffney_imaging_2007, miao_beyond_2015}. \nWhereas most alternative approaches focus on nano-crystals~\\cite{chapman_femtosecond_2006, chapman_femtosecond_2011, boutet_high-resolution_2012, fromme_femtosecond_2011, kirian_femtosecond_2010, schlichting_serial_2015, roedig_high-speed_2017, barends_serial_2022}, X-ray scattering experiments on single molecules can in principle also provide structural information on molecular ensembles and, hence, biomolecular conformational and functional motions. \nIn such 'hit and destroy' experiments, a stream of single molecules is exposed to a beam of high intensity femtosecond X-ray free electron laser (XFEL) pulses (Figure~\\ref{fig: experiment}a), and for each hit the positions of the scattered photons (red dots) on the detector are recorded as a scattering image~\\cite{neutze_potential_2000}. \nImportantly, the ultra-short pulses serve to outrun the subsequent destruction of the particles due to radiation damage, but also\nimply that only very few photons are being recorded for each molecule~\\cite{gaffney_imaging_2007}.\n\nThe feasibility of the method has already been demonstrated by a number of experiments~\\cite{seibert_single_2011, ekeberg_three-dimensional_2015, hosseinizadeh_high-resolution_2014}, but so far only structures of relatively large specimen have been successfully determined, for instance of entire mimivirus particles~\\cite{seibert_single_2011, ekeberg_three-dimensional_2015}  ($450\\,\\text{nm}$ in diameter) and coliphage viruses~\\cite{hosseinizadeh_high-resolution_2014} ($20\\,\\text{nm}$ in diameter). \nWhereas for large specimens many photons are scattered per image, for example $10^7$ for the mimivirus~\\cite{seibert_single_2011, ekeberg_three-dimensional_2015}, for typical proteins only 10-50 coherently scattered photons per image are expected~\\cite{yoon_comprehensive_2016, hantke_condor_2016}, which further complicates structure determination particularly for small molecules. \nSuch images can be obtained at with an intensity of $10^{12}$ photons per pulse at $5\\,\\mathrm{keV}$ and a $1\\,\\mathrm{\\text{\\textmu} m}$ beam diameter~\\cite{von_ardenne_structure_2018}, for example from the XFELs at DESY or SLAC.\n\nMost importantly, the orientation of the molecules at the time of scattering is typically unknown, which poses an additional and substantial refinement challenge.\nAlso, so far it is impossible to systematically include shot noise, incoherently scattered photons, background scattering, or detector noise in the structure refinement.\nThese issues are particularly challenging for the structure refinement of small specimen at high resolution, for example proteins or protein complexes at near atomic resolution. \nA number of methods have been proposed to address these issues, such as orientation determination methods~\\cite{shneerson_crystallography_2008,loh_reconstruction_2009, walczak_bayesian_2014,kassemeyer_optimal_2013,elser_three-dimensional_2011, tegze_atomic_2012, donatelli_reconstruction_2017, flamant_expansion-maximization-compression_2016, ayyer_dragonfly_2016} and manifold embedding algorithms~\\cite{fung_structure_2009,moths_bayesian_2011,schwander_symmetries_2012,giannakis_symmetries_2012}, which, however, typically require 100 to 1000 photons per scattering image.\nAs an alternative, correlation based approaches~\\cite{saldin_structure_2009, saldin_beyond_2010, saldin_new_2011, saldin_structure_2010,kurta_correlations_2017} require, in principle, as little as three photons per image~\\cite{von_ardenne_structure_2018}. \n\nFurther, many biomolecules show structural heterogeneity and conformational dynamics between different distinct structures, which, when resolved, would provide a direct view on biomolecular function. Hence, and similar to the current main challenge in cryogenic electron microscopy~\\cite{guaita_recent_2022}, not one but many structures need to be extracted from the scattering images. Accordingly, in addition to the orientation, also the current conformer for each scattering image is unknown. \nWhereas both orientation determination methods and manifold-based methods have been applied to determine multiple conformational structures~\\cite{zhuang_unsupervised_2022,hosseinizadeh_conformational_2017-1}, the required large number of photons per image precludes their application to single biomolecules.\n\nTo overcome the above issues, we developed and assessed a rigorous Bayesian method for structure determination from single-molecule scattering images. \nWe will demonstrate that this method can not only determine a single structure at high resolution but also determine structural ensembles.\nUnexpectedly, much fewer images are required for refining an ensemble of $n$ conformers consisting of $m$ atoms each than for refining a single structure consisting of $m\\times n$ atoms, which may bring applications to non-synthetic experimental data into reach.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[height = 0.3\\textwidth]{setup.pdf}\n    \\caption{Single molecule scattering experiment. \\textbf{a} A stream of single molecules is hit by femtosecond X-ray pulses, and the scattered photons are recorded as images (image reproduced from von Ardenne et al.~\\cite{von_ardenne_structure_2018}). \\textbf{b} The scattered photons (red dots) are distributed on the Ewald sphere according to the 3D-intensity function $I$ (blue).}\n    \\label{fig: experiment}\n\\end{figure}\n\n\n\\newpage\n\n\\section{Results}\n\n\\paragraph{Summary of the approach.}\n\nFor each scattering image, the positions of the recorded photons specify vectors $\\vec k_1, \\dots, \\vec k_l$ on the Ewald sphere in Fourier space (Figure~\\ref{fig: experiment}b). The probability of observing a photon at a particular position on the detector is proportional to the 3D intensity function $I(\\vec k) \\propto |\\mathcal F\\{\\vec R\\rho_i\\}(\\vec k)|^2$ at the corresponding position $\\vec k$ on the Ewald sphere, which in turn is given by the Fourier transform of the electron density $\\rho_i$ of conformer $i$. Here, $\\vec R$ is the unknown orientation of the molecule for this particular image.\n\nIt follows that the probability of observing an image with photon positions $\\vec k_1, \\dots, \\vec k_l$ is obtained by averaging over both the conformational ensemble $\\boldsymbol\\uprho = \\{\\rho_1, \\dots, \\rho_n\\}$ with weights $\\vec w = \\{w_1, \\dots, w_n\\}$ and all orientations $\\vec R$. Because each scattering image is an independent event, the total probability of the set of all images $\\mathcal I$ reads\n\\begin{equation}\n    P(\\mathcal I \\,|\\, \\boldsymbol\\uprho, \\vec w) \\propto \\prod_{(\\vec k_1, \\dots, \\vec k_l) \\in \\mathcal I} \\sum_{i=1}^n w_i\\int_{\\mathrm{SO}(3)}P(\\vec k_1, \\dots, \\vec k_l \\,|\\, \\vec R\\rho_i)\\, \\mathrm d \\vec R\\,.\n    \\label{eq: likelihood}\n\\end{equation}\nThis probability serves to determine either a single structure or a structural ensemble by sampling from the Bayesian posterior probability $P(\\boldsymbol\\uprho, \\vec w \\,|\\, \\mathcal I) \\propto P(\\mathcal I \\,|\\, \\boldsymbol\\uprho, \\vec w)P(\\boldsymbol\\uprho, \\vec w)$ using a Markov chain Monte Carlo approach. For the prior $P(\\boldsymbol\\uprho, \\vec w)$ the orientations are assumed to be uniformly distributed. To minimize the number of required degrees of freedom, and as a means of regularization, we chose a physically motivated representation of each $\\rho_i$ in terms of a sum of Gaussian functions, which also completes the definition of the prior.\n\n\nFor typical cases, the number of required degrees of freedom remains large and poses a formidable sampling challenge. \nTo address this issue, we adopted a hierarchical simulated annealing approach. \nStarting at very low resolution, the macromolecular structures were sampled in multiple hierarchical stages of increasing resolution. To increase the sampling efficiency, in each of these stages, for each Markov step the previous ensemble of structures of maximal posterior probability was used as a proposal density.\nTo this end, the scattering images that would have been observed for a smoothed low resolution copy of the original molecule were obtained from the original images by rejection sampling using the convolution theorem (see the Methods section). \n\nFurther, we adapted the Bayesian formalism such that it only uses those images which contain new information, that is, that contain photons for which the magnitude of $\\vec k$ is larger than the threshold of the resolution from the previous stage.\nWith increasing resolution, the fraction of such useful images becomes very small, thus markedly enhancing computational efficiency up to two orders of magnitude. The approach is described in detail in the Supplementary Information.\n\n\\paragraph{Sample test refinements.}\nBecause our Bayesian approach uses all available information, it we expect it to require fewer scattering images to achieve a certain resolution than, for example, correlation based methods.\nTo assess this aspect, we first tested our method on the single structure level, using the same 46-residue protein crambin~\\cite{jelsch_accurate_2000} as in the study of von Ardenne et al.~\\cite{von_ardenne_structure_2018}.\nA total of $10^8$ noise-free synthetic images were generated, containing a realistic average of $15$ photons each. \nFrom these, the structure was solved in five hierarchical stages (Fig.~\\ref{fig: crambin}a), increasing the number of degrees of freedom by a factor of two in each stage. \nFor the final stage, a representation consisting of $184$ Gaussian functions was used, which is four times the number of residues. For more details see Supplementary Note~\\ref{sup: parameters}. \nIndeed, a similar Fourier shell correlation resolution~\\cite{van_heel_fourier_2005} of $4.2\\,${\\AA} (Fig.~\\ref{fig: crambin}b) was obtained as for the previous correlation based method, using only half of the total number of scattered photons.\n\nNext, to demonstrate that our method can resolve ensembles of multiple conformers, we used three molecular dynamics trajectories of alanine dipeptide~\\cite{scherer_pyemma_2015} of length $250\\,\\text{ns}$ each to generate $10^6$ scattering images, using a randomly chosen snapshot for each image. \nAs before, an average of $15$ photons per image were generated. \nThen, a weighted ensemble of eight conformers was determined from these images (Fig.~\\ref{fig: dipeptide}), each described by a sum of $10$ Gaussian functions. \nTo obtain sufficient statistics, a total of $10$ independent simulated annealing runs were carried out, using the same images.\n\nTo assess the quality of the obtained ensemble, for each of the eight structures the resolution with respect to its nearest neighbor in the input trajectories was calculated using Fourier shell correlations~\\cite{van_heel_fourier_2005} (Fig.~\\ref{fig: dipeptide}b), resulting in a weighted average resolution of $1.6\\,\\text{{\\AA}}$. \nThis result shows that the obtained eight structures are indeed close to the reference ensemble. \nTo also assess the accuracy of the entire ensemble, for each time step in the input trajectories, the resolution with respect to its nearest neighbor among all the determined structures was calculated (Fig.~\\ref{fig: dipeptide}d). \nAs a main result we found that $90\\%$ of the input trajectories are within $2.1\\,${\\AA} Fourier shell correlation resolution of the determined structures, and that all of the trajectory frames are within $2.5\\,${\\AA} resolution of the determined structures. \nFigure~\\ref{fig: dipeptide}e compares the $10$ obtained ensembles with the reference ensemble using a Ramachandran plot~\\cite{ramachandran_stereochemistry_1963} showing the distribution of the torsion angles $\\phi$ and $\\psi$. \nFor each of the determined structures its nearest neighbor in the input trajectories was used to compute these angles. \nAs can be seen, the reference density is well represented by the determined structures. \n\nNext, we asked if our method is also capable of extracting structural ensembles for the larger mini-protein chignolin~\\cite{honda_crystal_2008}, comprising $10$ residues. \nTo that end, $50$ molecular dynamics trajectories of length $10\\,\\text{\\textmu s}$ were used to generate $1.2\\cdot10^7$ images with, on average, $15$ photons each.\nAs a further challenge, this ensemble also contained unfolded structures. \nFrom the obtained images, we determined multiple stages of weighted structural ensembles of increasing resolution and increasing number of conformers (Fig.~\\ref{fig: 5awl}a). \nAs above, resolutions were computed using Fourier shell correlations (Fig.~\\ref{fig: 5awl}b,c), finding a weighted average resolution of $5\\,${\\AA} for the folded conformers, and $9-10\\,${\\AA} for the unfolded conformers.\nInterestingly, in the final stage one of the six determined weights is nearly zero, suggesting that five conformers suffice for the used number of images at this resolution level. \nIt is also remarkable that the $9\\%$ fraction of unfolded states in the reference structure ensemble was correctly identified.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{crambin.pdf}\n    \\caption{Structure determination of Crambin. \\textbf{a} Hierarchical stages of retrieved electron densities. \\textbf{b} Fourier shell correlation between the retrieved densities and the reference density. \\textbf{c} Retrieved electron density. \\textbf{d} Reference electron density.}\n    \\label{fig: crambin}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{dipeptide_weighted_v2.pdf}\n    \\caption{Structural ensemble determination of the alanine dipeptide. \\textbf{a} Reconstructed conformers (green), the corresponding weights, and the nearest neighbors in the input trajectories (blue) with the corresponding resolutions. \\textbf{b} Fourier shell correlations used to compute these resolutions. \\textbf{c} Weighted resolution distribution for $10$ independent runs from the the same data. \\textbf{d} Resolution distribution over the time steps of the input trajectories relative to their nearest neighbors among the determined structures from all $10$ runs. \\textbf{e} Ramachandran plot for the input trajectories (shown as a density) and the determined structures from all $10$ runs (points, the color indicates the different runs). }\n    \\label{fig: dipeptide}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{5awl_tree.pdf}\n    \\caption{Structural ensemble determination for chignolin. \\textbf{a} Hierarchical stages of retrieved structures (green) and their nearest neighbors (blue) in the input trajectories with the corresponding resolutions. \\textbf{b} Fourier shell correlations of the reconstructed structures relative to their nearest neighbors. \\textbf{c} Resolution distribution over the time steps of the input trajectories relative to their nearest neighbors among the determined structures. }\n    \\label{fig: 5awl}\n\\end{figure}\n\n\n\n\n\\paragraph{Scaling.}\nIn the sample applications described above we observed that resolving $n$ conformational structures consisting of $m$ residues each required much fewer scattering images and photons than resolving a single $n\\times m$ residue structure of the same total size and complexity --- even in cases where the members of the former ensemble are very different from each other.\nTo investigate this counterintuitive observation in more detail, small `structures' consisting of randomly placed Gaussian functions were used.  \nFor each combination of parameters, eight independent structure determination runs were performed, and for each run the achieved resolution was determined. \nThe structure weights $w_i$ where chosen to be uniform and kept fixed during the simulated annealing runs.\n\nFigure~\\ref{fig: scaling}c and~\\ref{fig: scaling}f show for each combination of parameters the smallest number of images for which all of the replicas achieved a resolution better than a given threshold. \nAs can be seen in Fig.~\\ref{fig: scaling}c, for the structure ensemble of $n$ conformations with $m$ residues each, the required number of images is approximately proportional to $n^2$, that is, the square of the number of conformations. \nWhereas somewhat unexpected, this finding is in line with a theoretical argument showing that the information content of a single image is in this case proportional to $1\/n^2$ (Supplementary Note \\ref{sup: information}). \nIn contrast, the number of images required to resolve a single structure of $n\\times m$ residues grows even much faster, resembling a power law $m^c$ with an exponent $c \\approx 5$ (Fig.~\\ref{fig: scaling}f). \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = \\textwidth]{scaling.pdf}\n    \\caption{Dependence of the resolution on the number of images, the number of conformations, and the size of the structure. \\textbf{a} Resolution as a function of the number of images for various numbers of conformations $n$. \\textbf{b} Resolution as a function of the number of conformations for various numbers of images. \\textbf{c} Required number of images to achieve various resolutions as a function of the number of conformations. For comparison, a quadratic relationship is shown (dashed line). \\textbf{d} Resolution as a function of the number of images for various structure sizes (parameterized by the number of Gaussians $m$). \\textbf{e} Resolution as a function of the number of Gaussians for various numbers of images. \\textbf{f} Required number of images to achieve various resolutions as a function of the number of Gaussians. For comparison, a power law $m^5$ is shown (dashed line). }\n    \\label{fig: scaling}\n   \n\\end{figure}\n\n\n\\clearpage\n\\section{Discussion}\nHere we have developed a rigorous Bayesian method for determining biomolecular structures from single molecule X-ray scattering images in the extreme few photon Poisson regime. \nUsing synthetic scattering images generated from simulated X-ray scattering experiments, we have demonstrated that both single structures as well as structural ensembles of small biomolecules can be resolved to near atomic resolution. \n\nOur results for the globular protein crambin show that a similar resolution of $4.2\\,${\\AA} is obtained compared to previous correlation based methods~\\cite{von_ardenne_structure_2018} that require similarly small numbers of photons per scattering image. Because such correlation based methods disregard higher correlations, whereas the full information content of each image is used in our Bayesian approach, we speculated that the latter should require correspondingly fewer images. \nThis was indeed observed for the above protein, for which the number of images required to obtain a resolution of approximately $4.2\\,${\\AA} was reduced from roughly $2\\cdot 10^8$ to $1\\cdot 10^8$. \n\nOur method should be applicable also to much larger molecules, the main bottleneck currently being computational cost. \nHowever, we expect that the factor by which our method requires fewer photons than correlation based methods will increase with the size of the molecule, due to the larger number of photons expected per image.\nFurther, we think that the computational limitations can be overcome by improved optimization or sampling methods or by utilizing prior structural information, either from structure databases, from AlphaFold~\\cite{jumper_highly_2021}, or guided by molecular dynamics force fields.\n\nFor alanine dipeptide the full conformational ensemble generated by an atomistic simulation was extracted at atomistic resolution of on average $1.8\\,${\\AA} from simulated scattering experiments, in which not only the current orientation of the biomolecule but also its current conformer was unknown.\nFor the 10 amino acid protein chignolin~\\cite{honda_crystal_2008} both the folded and unfolded ensembles were resolved, albeit so far at lower resolution. \nNotably, also the weights corresponding to the folded and unfolded conformers were recovered accurately. \nFurthermore, using weighted ensembles allows the number of conformers to be determined dynamically, as some of the determined weights may be zero.\n\nUnexpectedly few images were required to resolve structural ensembles.\nBecause an ensemble of $n$ conformers consisting of $m$ atoms each has the same number of degrees of freedom as a single structure of $n\\times m$ atoms, a similar number of images should be required. \nHowever, closer analysis suggests that roughly $O(m^5)$ images are required to a resolve a single structure with $m$ atoms. One might therefore expect that $O(n^5m^5)$ images are required for an ensemble of $n$ such structures. \nHowever, our analysis suggests that only $O(n^2m^5)$ images are required for this, which is consistent with a theoretical argument that the expected information content of a single scattering image is in this case proportional to $1\/n^2$.\nThis suggests that determining more complex conformational ensembles may be within reach, and that it may be possible to collect the required amount data from experiments. \nOur Bayesian analysis of structural heterogeneity is similar in spirit to approaches that were successfully applied in cryo-electron microscopy~\\cite{cossio_bayesian_2013,grant_cistem_2018,punjani_cryosparc_2017,lyumkis_likelihood-based_2013,scheres_chapter_2016,tang_eman2_2007,zivanov_new_2018,kinman_uncovering_2022}, which, from a mathematical standpoint, shares some similarities with single molecule X-ray scattering, albeit at a much lower noise level. \n\nFrom a more general perspective, our Bayesian approach represents a systematic and rigorous approach to include shot noise in the extreme Poisson regime characteristic for single molecule X-ray scattering experiments. \nIn contrast to other proposed methods, this Bayesian framework will also allow to include other sources of noise and uncertainty in a conceptually straightforward manner, such as incoherently scattered photons, background scattering, detector noise, or scattering by disordered water at the biomolecular surface. \nPrerequisite is the development and calibration of forward noise models such as those documented in, e.g., Ref.~\\cite{yoon_comprehensive_2016}. \n\n\n\\section{Methods}\n\n\\paragraph{Structure and structure ensemble representation.}\n\\label{sec: representation}\nElectron density functions of the reference structures or conformers were described by a sum of $m$ of Gaussian functions with atomic positions $\\vec y_i$, heights $h_i$ and standard deviations $\\sigma_i$,\n\\begin{equation}\\label{eq: representation}\n    \\rho(\\vec r) = \\sum_{i=1}^m \\frac{h_i}{\\left(\\sigma_i\\sqrt{2\\pi}\\right)^3} \\exp \\left(\\frac1{2\\sigma_i^2}\\lVert \\vec r - \\vec y_i\\rVert^2\\right)\\,.\n\\end{equation}\nElectron density functions of the determined structures were described similarly, with one common height $h = h_i$ and one common standard deviation $\\sigma=\\sigma_i$, which is treated as an unknown and determined together with the positions $\\vec y_i$.\nStructural ensembles were represented by a weighted sum of conformers $\\boldsymbol\\uprho = \\{\\rho_1, \\dots, \\rho_n\\}$ with weights $\\vec w = \\{w_1, \\dots, w_n\\}$.\n\n\n\\paragraph{Synthetic data generation.}\n\\label{sec: datagen}\n\nFor each of the synthetic scattering images, the photon positions on the detector $D$ were drawn from a probability distribution proportional to the intensity function $I(\\vec k) = |F\\{\\rho\\}(\\vec k)|^2$ restricted to the appropriate Ewald sphere.\nSpecifically, generation of each image involved the following steps:\n\\begin{enumerate}\n    \\item A conformation of the molecule is selected randomly from the reference ensemble (for example, consisting of molecular dynamics trajectories),\n    \\item a random orientation $\\vec R$ of the molecule is drawn uniformly from the rotation group $\\mathrm{SO}(3)$,\n    \\item the number of scattered photons is drawn from a Poisson distribution with mean $N\\int_D I(\\vec R\\vec k) \\,\\mathrm d \\vec k$, where $N$ is the incoming beam intensity,\n    \\item the position of each scattered photon is drawn from the probability distribution proportional to $(I\\circ \\vec R)|_D$.\n\\end{enumerate}\nThe last two steps were implemented using rejection sampling. To this end, a von Mises-Fisher distribution $p$ on $D$ was chosen with high enough standard deviation that $I(\\vec k) \\leq p(\\vec k)$ everywhere. Then, for each photon, its position $\\vec k$ was drawn from $p$ and it was accepted with probability $I(R\\vec k) \/ p(R\\vec k)$. The beam intensity $N$ was chosen together with a normalization of $\\rho$ such that this procedure accurately produces a Poisson distribution of the desired expected number of photons per image.\n\n\n\\paragraph{Computation of likelihoods.}\n\\label{sec: distribution}\nThe probability density of observing an image defined by photon positions $\\vec k_1, \\dots, \\vec k_l$ given an electron density function $\\rho$ with is corresponding intensity function $I(\\vec k)$ was computed by averaging over all possible orientations $\\vec R\\in \\mathrm{SO}(3)$ of the molecule, \n\\begin{equation}\n\\label{eq: probability density}\n    \\begin{aligned}\n        P(\\vec k_1, \\dots, \\vec k_l \\,|\\, \\rho\n        &= \\frac{N^l}{l!}\\int_{\\mathrm{SO}(3)} \\exp\\left(-N\\!\\int_D I(\\vec R\\vec k) \\,\\mathrm d \\vec k\\right) \\left(\\prod_{i=1}^l I(\\vec R\\vec k_i)\\right) \\,\\mathrm d \\vec R, \\\\\n       \n    \\end{aligned}\n\\end{equation}\nwhere for each orientation, the probability is a product of the Poisson distribution for the number of photons $l$ in the image and a factor depending on the photon positions. \nThese integrals were approximated by averaging over a discrete set of typically $r\\approx 10^3$ to $r\\approx 10^5$ rotations $\\vec R_i$ with weights $s_i$,\n\\begin{equation}\n    P(\\vec k_1, \\dots, \\vec k_l \\,|\\, \\rho) \\approx \\frac{N^l}{l!}\\sum_{i=1}^r s_i\\exp\\left(-N\\!\\int_D I(\\vec R_i\\vec k) \\,\\mathrm d \\vec k\\right) \\prod_{j=1}^l I(\\vec R_i\\vec k_j).\n\\end{equation}\nThe rotations $\\vec R_i$ and their weights $s_i$ are constructed by combining a Lebedev quadrature rule on $S^2$ with a uniform quadrature rule on $S^1$ via the Hopf map~\\cite{noauthor_sphere_lebedev_rule_nodate,graf_sampling_2009} (Supplementary Note~\\ref{sup: computation}).\n\n\\paragraph{Simulated annealing and hierarchical sampling.}\n\\label{sec: annealing}\nA Monte Carlo simulated annealing approach with the energy function $-\\log P$ was used to sample from or maximize the Bayesian posterior probability, as described in detail in the Supplement.\nTo enhance convergence, Bayesian sampling and maximization were performed in multiple hierarchical resolution stages. \nStarting from a low resolution representation of $\\rho$ with correspondingly few degrees of freedom, the number of Gaussian functions was doubled in each stage and the reduced resolution structure determined by the previous stage was used as a proposal density (see Supplement). \nTo calculate likelihoods for the reduced resolution structures, lower resolution scattering images were generated from the original images by rejection sampling, that is, by removing each photon in the original images with probability $1 - \\exp(-\\sigma^2k^2\/2)$.\nBy construction, this rejection scheme samples from a Fourier transformed density $I_\\rho \\cdot \\exp(-\\sigma^2|\\vec k|^2\/2)$ which, by the convolution theorem, corresponds to a smoothed real space density $\\tilde\\rho = \\rho*\\mathcal N(\\sigma)$ obtained as the convolution of $\\rho$ with a Gaussian kernel with width (resolution) $\\sigma$. \nComputational efficiency was further increased substantially by selecting only those original images for the likelihood computations that actually contain useful information at the respective resolution. As described in the Supplement, the Bayesian formalism allows for removing this selection bias.\n\n\n\\paragraph{Structure alignment and resolution estimate.}\nBecause the orientations of the obtained structures are irrelevant, these were rotationally aligned to each other\nby minimizing the cost function\n\\begin{equation}\n    d(\\vec S) = \\frac1n \\sum_{i=1}^n \\min_{j=1}^m\\, \\lVert\\vec y_i - \\vec S \\vec y_j'\\rVert + \\frac1m \\sum_{j=1}^m \\min_{i=1}^n\\, \\lVert\\vec y_i - \\vec S \\vec y_j'\\rVert.\n\\end{equation}\nHere, the positions $\\vec y_1, \\dots, \\vec y_n$ and $\\vec y_1', \\dots, \\vec y_m'$ define two structures per equation \\eqref{eq: representation} and $\\vec S$ is a rotation matrix $\\vec S \\in \\mathrm{O}(3)$. Both rotations and reflections were included, as X-ray scattering images do not distinguish between mirror images. \n\nThe resolution of the aligned structures was estimated using Fourier shell correlations~\\cite{van_heel_fourier_2005},\n\\begin{equation}\n    \\mathrm{FSC}(k) = \\frac{\\int_{\\lVert\\vec k\\rVert = k} \\hat\\rho_1(\\vec k)^*\\hat\\rho_2(\\vec k)\\,\\mathrm d\\vec k}{\\sqrt{\\int_{\\lVert\\vec k\\rVert = k} |\\hat\\rho_1(\\vec k)|^2\\,\\mathrm d\\vec k}\\sqrt{\\int_{\\lVert\\vec k\\rVert = k} |\\hat\\rho_2(\\vec k)|^2\\,\\mathrm d\\vec k}}\\,,\n\\end{equation}\nwhere $\\rho_1$ are $\\rho_2$ the structures to be compared and $\\hat\\rho$ denotes the Fourier transform of $\\rho$. Accordingly, the achieved resolution was determined as $2\\pi\/k_\\mathrm{fsc}$, where $k_\\mathrm{fsc}$ is the threshold at which the Fourier shell correlation drops below $1\/2$, providing a conservative estimate~\\cite{van_heel_fourier_2005}.\n\n\\paragraph{Molecular dynamics simulations.}\nAll atomistic simulation trajectories were generated using the GROMACS 2018 software package~\\cite{abraham_gromacs_2015} with the Charmm36mm force field~\\cite{huang_charmm36m_2017} and the OPC water model~\\cite{izadi_building_2014}. \nFor chignolin, the starting structure was taken from the Protein Data Bank~\\cite{berman_protein_2000}, entry 5AWL~\\cite{honda_crystal_2008}.\nAll hydrogen atoms were described by virtual sites~\\cite{feenstra_improving_1999}.\nEach protein was placed within a triclinic water box, such that the smallest distance between protein surface and box boundary was larger than $1.5\\,$nm. \nSodium and chloride ions were added to neutralize the system, corresponding to a physiological concentration of $150\\,$mmol\/l. \nEnergy minimization was performed using steepest descent for $5\\cdot 10^4$ steps.\nEach system was subsequently equilibrated for $0.5\\,$ns in the $NVT$ ensemble, and subsequently for $1.0\\,$ns in the $NPT$ ensemble at $1\\,$atm pressure and temperature $300\\,$K using an integration time step of $2\\,$fs. \nThe velocity rescaling thermostat~\\cite{bussi_canonical_2007} and Parrinello-Rahman pressure coupling~\\cite{parrinello_polymorphic_1981} were used with coupling coefficients of $\\tau=0.1\\,$ps and $\\tau=1\\,$ps, respectively.\nAll bond lengths of the solute were constrained using the LINCS algorithm~\\cite{hess_lincs_1997} with an expansion order of 6, and the geometry of the water molecules was constrained using the SETTLE algorithm~\\cite{miyamoto_settle_1992}. \nElectrostatic interactions were calculated using PME~\\cite{darden_particle_1993}, with a real space cutoff of $10\\,${\\AA} and a Fourier spacing of $1.2\\,${\\AA}.\nFor all production runs, a $4\\,$fs integration was used, and the atom coordinates were saved every $100\\,$ps, such that $10^5$ snapshots were available for each trajectory.\nThe trajectories for alanine dipeptide were taken from mdshare \\cite{noauthor_mdshare_nodate-1}. The structure for crambin was taken from PDB entry 1EJG~\\cite{jelsch_accurate_2000}.\n\n\\section{Acknowledgments}\nThis work was financially supported by the Federal Ministry of Education and Research through the joint research project 05K20EGA Fluctuation XFEL, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - CRC 1456\/1 - 432680300. The MD-trajectories were kindly provided by Nicolai Kozlowski. \n\n\n\\printbibliography\n\n\n\n\\clearpage\n\\begin{appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{section: Introduction}\n\nBy $\\mathcal H$ we will always denote an undirected graph without multiple edges (and by abuse of notation also denote its set of vertices). In this paper we focus on $\\mathcal H$ which is four-cycle free, that is, it is finite, it has no self-loops and the four-cycle, denoted by $C_4$ is not a subgraph of $\\mathcal H$. Fix $d\\geq 2$. The basic object of study is $X_\\mathcal H$, the space of graph homomorphisms from $\\mathbb{Z}^d$ to $\\mathcal H$. Here by $\\mathbb{Z}^d$ we will mean both the group and its standard Cayley graph.\n\nSuch a space of configurations $X_\\mathcal H$, referred to as a hom-shift, can be obtained by forbidding certain patterns on edges of $\\mathcal H^{\\mathbb{Z}^d}$. If $\\mathcal H$ is a finite graph then $X_\\mathcal H$ is a nearest neighbour shift of finite type. In addition it is also `isotropic' and `symmetric', that is, given vertices $a, b \\in \\mathcal H$ if $a$ is not allowed to sit next to $b$ in $X_\\mathcal H$ for some coordinate direction, then $a$ is not allowed to sit next to $b$ in all coordinate directions. Most of the concepts related to shift spaces are introduced in Section \\ref{section: Hom-shifts}.\n\nRelated to a shift space $X$ is its topological entropy denoted by $h_{top}(X)$ which measures the growth rate of the number of patterns allowed in $X$ with the size of the underlying shape (usually rectangular). For a given shift space, its computation is a very difficult task (look for instance in \\cite{pavlovhardsquare2012} and the references within). We will focus on a different aspect: as in \\cite{covensmital}, a shift space $X$ is called entropy minimal if for all shift spaces $Y\\subsetneq X$, $h_{top}(Y)<h_{top}(X)$. Thus if a shift space has zero entropy and is entropy minimal then it is a topologically minimal system.\n\nFor $d=1$, it is well known that all irreducible nearest neighbour shifts of finite type are entropy minimal \\cite{LM}. However not much is known about it in higher dimensions: Shift spaces with a strong mixing property called uniform filling are entropy minimal \\cite{Schraudner2010minimal}, however nearest neighbour shifts of finite type with weaker mixing properties like block-gluing may not be entropy minimal \\cite{boyle2010multidimensional}. If $\\mathcal H$ is a four-cycle free graph then $X_\\mathcal H$ is not even block-gluing (we do not prove this but is implied by our results). There has been some recent work \\cite{lightwoodschraudnerentropy} which describes some conditions which are equivalent to entropy minimality for shifts of finite type. Our first main result is Theorem \\ref{theorem:four cycle free entropy minimal} which states that $X_\\mathcal H$ is entropy minimal for all connected four-cycle free graphs $\\mathcal H$. Our approach to this seemingly combinatorial question will be via thermodynamic formalism, using measures of maximal entropy, more generally `adapted' Markov random fields. They are defined and described in Section \\ref{section:thermodynamic formalism}.\n\nOur second main result is with regard to the pivot property (Section \\ref{section: the pivot property}). A shift space $X$ is said to have the pivot property if for all distinct configurations $x,y\\in X$ which differ at finitely many sites, there exists a sequence $x=x^1, x^2, \\ldots, x^n=y\\in X$ such that successive configurations $x^i, x^{i+1}$ differ exactly at a single site. We will prove that for four-cycle free graphs $\\mathcal H$, $X_{\\mathcal H}$ has the pivot property (Theorem \\ref{theorem: pivot property for four cycle free}). Many properties similar to the pivot property have appeared in the literature (often by the name local-move connectedness) \\cite{brightwell2000gibbs,dominolocal2010,ordentlich2014data,Shiffieldribbon2002}. For instance consider the following problem: Let $G$ be a finite undirected graph without multiple edges and self-loops. Given two graph homomorphisms $x,y$ from $G$ to $H$, can we find graph homomorphisms $x=x^1, x^2, \\ldots, x^n=y$ from $G$ to $H$ such that successive configurations $x^i, x^{i+1}$ differ exactly at a single site? Such a problem is called a homomorphism reconfiguration problem. If $\\mathcal H$ is four-cycle free it was recently shown in \\cite{Marcinfourcyclefree2014} that the homomorphism reconfiguration problem is solvable in time polynomial in the size of the graph $\\mathcal H$. Our interest in such problems comes from the connections between the pivot property and the study of Markov and Gibbs cocycles \\cite{chandgotia2013Markov}.\n\nA critical part of our proofs for both Theorem \\ref{theorem:four cycle free entropy minimal} and \\ref{theorem: pivot property for four cycle free} depends on the identification of the associated height functions (defined in Section \\ref{Section:heights}). Some of these ideas come from \\cite{chandgotia2013Markov}: Let $C_n$ denote the cycle with vertices $0, 1, \\ldots, n-1$. If $n \\neq 1, 2, 4$ then it was proven that $X_{C_n}$ has the pivot property. Further, Lemma 6.7 in \\cite{chandgotia2013Markov} implied that it is entropy minimal as well. We give a brief description of the latter: Given a configuration $x\\in X_{C_n}$ it was proved that there exists a corresponding height function $h_x \\in X_{\\mathbb{Z}}$ such that $h_x\\mod n= x$. (for $n=3$ also look at \\cite{schmidt_cohomology_SFT_1995}) Further given any ergodic measure $\\mu$ (assuming `adaptedness' cf. Section \\ref{section:thermodynamic formalism}) on $X_{C_n}$ it was shown that a well-defined notion of slope (bounded by $-1$ and $1$) exists which measures the average rate of the increase of the height for every direction. If the slope is maximal in any direction it was proven that $\\mu$ is frozen and otherwise it was shown that it is fully supported; frozen meaning that if $x, y\\in supp(\\mu)$ differ only at finitely many sites then $x=y$. From standard results in thermodynamic formalism it follows that $X_{C_n}$ is entropy minimal.\n\nThese ideas do not immediately translate to four-cycle free graphs. For this we will develop a different notion of `heights': Fix a connected four-cycle free graph $\\mathcal H$. For all $u \\in \\mathcal H$ let $D_\\mathcal H(u)$ denote the ball (for the graph distance) of radius $1$ around $u$. As in algebraic topology, $C$ is a covering space of $\\mathcal H$ if there is a graph homomorphism (called the covering map) $f: C\\longrightarrow \\mathcal H$ such that for all $u \\in \\mathcal H$, $f^{-1}(D_{\\mathcal H}(u))$ is a disjoint union of a constant number of subgraphs of $C$ isomorphic to $D_{\\mathcal H}(u)$ via $f$. Given a graph $\\mathcal H$, its universal cover (denoted by $E_\\mathcal H$) is the unique covering space of $\\mathcal H$ which is a tree. (Section \\ref{section:universal covers} and \\cite{Angluin80}) Denote by $\\pi:E_\\mathcal H\\longrightarrow \\mathcal H$ the corresponding covering map. $\\pi$ induces a map from $X_{E_\\mathcal H}$ to $X_\\mathcal H$ (also denoted by $\\pi$). It is not difficult to see that $\\pi(X_{E_\\mathcal H})\\subset X_\\mathcal H$; we will prove that the map is surjective and that the preimage of a configuration in $X_\\mathcal H$ is unique once we fix the `lift' in $X_{E_\\mathcal H}$ at any vertex of $\\mathbb{Z}^d$. Thus for every $x\\in X_\\mathcal H$ we can associate $\\tilde x \\in X_{E_\\mathcal H}$ such that $\\pi(\\tilde x)= x$ and a `height' function $h_x: \\mathbb{Z}^d\\times \\mathbb{Z}^d \\longrightarrow \\mathbb{Z}$ such that $h_x(\\vec i , \\vec j)$ is the graph distance between $\\tilde x_{\\vec{i}}$ and $\\tilde x_{\\vec{j}}$. From here on the steps for the proofs of Theorems \\ref{theorem:four cycle free entropy minimal} and \\ref{theorem: pivot property for four cycle free} are similar to the steps for the corresponding proofs in \\cite{chandgotia2013Markov}, but the proofs for the individual steps are quite different because unlike in \\cite{chandgotia2013Markov} the height functions are not additive but subadditive, meaning\n$$h_x({\\vec{i}},{\\vec{j}})\\leq h_x({\\vec{i}}, \\vec k)+h_x(\\vec k, {\\vec{j}}).$$\n\nTo streamline the proofs, we use graph folding \\cite{nowwinkler}; much of this is discussed in Section \\ref{section:Folding, Entropy Minimality and the Pivot Property}.\n\n\n\n\n\\section{Shifts of Finite Type and Hom-Shifts} \\label{section: Hom-shifts}\nIn this paper $\\mathcal H$ will always denote an undirected graph without multiple edges and single isolated vertices. For such a graph we will denote the adjacency relation by $\\sim_{\\mathcal H}$ and the set of vertices of $\\mathcal H$ by $\\mathcal H$ (abusing notation). We identify $\\mathbb{Z}^d$ with the set of vertices of the Cayley graph with respect to the standard generators $\\vec e_1,\\vec e_2, \\ldots, \\vec e_d$, that is, $\\vec{i}\\sim_{\\mathbb{Z}^d}\\vec{j}$ if and only if $\\|\\vec{i}-\\vec{j}\\|_1=1$ where $\\|\\cdot\\|_1$ is the $l^1$ norm. We drop the subscript in $\\sim_{\\mathcal H}$ when $\\mathcal H=\\mathbb{Z}^d$. Let $D_n$ and $B_n$ denote the $\\mathbb{Z}^d$-balls of radius $n$ around $\\vec{0}$ in the $l^1$ and the $l^\\infty$ norm respectively. The graph $C_n$ will denote the $n$-cycle where the set of vertices is $\\{0, 1, 2, \\ldots, n-1\\}$ and $i\\sim_{C_n} j$ if and only if $i \\equiv j \\pm 1\\!\\!\\mod n$. The graph $K_n$ will denote the complete graph with $n$ vertices where the set of vertices is $\\{1, 2, \\ldots, n\\}$ and $i \\sim_{K_n} j$ if and only if $i \\neq j$.\n\nLet ${\\mathcal A}$ be a finite \\emph{alphabet} (with the discrete topology) and ${\\mathcal A}^{\\mathbb{Z}^d}$ be given the product topology, making it compact. We will refer to elements $x\\in {\\mathcal A}^{\\mathbb{Z}^d}$ as \\emph{configurations}, denoting them by $(x_{\\vec{i}})_{\\vec{i}\\in \\mathbb{Z}^d}$ where $x_{\\vec{i}}$ is the value of $x$ at $\\vec{i}$. For all $\\vec{i}\\in \\mathbb{Z}^d$ the map $\\sigma^{\\vec{i}}:{\\mathcal A}^{\\mathbb{Z}^d}\\longrightarrow {\\mathcal A}^{\\mathbb{Z}^d}$ given by\n$$(\\sigma^{\\vec{i}}(x))_{\\vec{j}}:=x_{\\vec{i}+\\vec{j}}$$\nis called the \\emph{shift map} and defines a $\\mathbb{Z}^d$-action on ${\\mathcal A}^{\\mathbb{Z}^d}$. Closed subsets of ${\\mathcal A}^{\\mathbb{Z}^d}$ which are invariant under the shift maps are called \\emph{shift spaces}. A \\emph{sliding block code} from a shift space $X$ to a shift space $Y$ is a continuous map $f: X\\longrightarrow Y$ which commutes with the shifts, that is, $\\sigma^{{\\vec{i}}}\\circ f= f\\circ \\sigma^{\\vec{i}}$ for all ${\\vec{i}}\\in \\mathbb{Z}^d$. A surjective sliding block code is called a \\emph{factor map} and a bijective sliding block code is called a \\emph{conjugacy}. We note that a conjugacy defines an equivalence relation; in fact, it has a continuous inverse since it is a continuous bijection between compact sets.\n\nThere is an alternate description of shift spaces using forbidden patterns: A \\emph{pattern} is an element of ${\\mathcal A}^F$ for some finite set $F\\subset \\mathbb{Z}^d$. Given a pattern $a\\in {\\mathcal A}^F$, we will often denote both the pattern and its corresponding cylinder set by $[a]_F$ or by $[x]_F$ when $x|_F=a$. For any set $F\\subset \\mathbb{Z}^d$ and $v\\in {\\mathcal A}$, $x|_F=v$ will mean that $x_{\\vec{i}}=v$ for all ${\\vec{i}}\\in F$. For a set of (forbidden) patterns ${\\mathcal F}$ define\n$$X_{\\mathcal F}:=\\{x\\in {\\mathcal A}^{\\mathbb{Z}^d}\\:\\Big{|}\\: \\sigma^{\\vec{i}}(x)|_F\\notin {\\mathcal F} \\text{ for all }F\\subset \\mathbb{Z}^d\\text{ and }{\\vec{i}} \\in \\mathbb{Z}^d\\}.$$\n\nIt can be proved that a subset $X\\subset {\\mathcal A}^{\\mathbb{Z}^d}$ is a shift space if and only if there exists a set of forbidden patterns ${\\mathcal F}$ such that $X_{\\mathcal F}= X$. Note that given two distinct sets of patterns ${\\mathcal F}_1, {\\mathcal F}_2$ it is possible that $X_{{\\mathcal F}_1}=X_{{\\mathcal F}_2}$. A \\emph{shift of finite type} is a shift space $X$ such that $X=X_{{\\mathcal F}}$ for some finite set ${\\mathcal F}$. A \\emph{nearest neighbour shift of finite type} is a shift of finite type $X$ such that $X=X_{{\\mathcal F}}$ for some set ${\\mathcal F}$ consisting of patterns on single edges and vertices of $\\mathbb{Z}^d$. It follows from a simple recoding argument that any shift of finite type is conjugate to a nearest neighbour shift of finite type \\cite{schimdt_fund_cocycle_98}. In this paper, we will focus on a special class of nearest neighbour shifts of finite type where the forbidden patterns are the same in every `direction':\n\n\nGiven a graph $\\mathcal H$ let\n$$X_{\\mathcal H}:=\\{x\\in\\mathcal H^{\\mathbb{Z}^d}\\:|\\: x_{\\vec {i}}\\sim_\\mathcal H x_{\\vec{j}}\\text{ for all }\\vec{i}\\sim \\vec{j}\\}.$$\nSuch spaces will be called \\emph{hom-shifts}. Note that $X_\\mathcal H$ is the space of graph homomorphisms from $\\mathbb{Z}^d$ to $\\mathcal H$. If $\\mathcal H$ is finite and\n$${\\mathcal F}_\\mathcal H:=\\{[v,w]_{\\vec{0},\\vec{e}_j}\\:|\\:v\\nsim_\\mathcal H w, 1\\leq j\\leq d\\}$$\nthen $X_{\\mathcal H}= X_{{\\mathcal F}_{\\mathcal H}}$. These are exactly the nearest neighbour shifts of finite type with symmetric and isotropic constraints. For example if the graph $\\mathcal H$ is given by Figure \\ref{Figure: Hard Square} then $X_\\mathcal H$ is the hard square shift, that is, configurations with alphabet $\\{0,1\\}$ such that adjacent symbols cannot both be $1$. $X_{K_n}$ is the space of $n$-colourings of the graph, that is, configurations with alphabet $\\{1,2, \\ldots, n\\}$ where all adjacent colours are distinct. We note that the properties, symmetry and isotropy, are not invariant under conjugacy.\n\n${\\mathcal F}$ will always denote a set of patterns and $\\mathcal H$ will always denote a graph, there will not be any ambiguity in the notations $X_{\\mathcal F}, X_\\mathcal H$.\n\nA graph $\\mathcal H$ is called \\emph{four-cycle free} if it is finite, it has no self-loops and $C_4$ is not a subgraph of $\\mathcal H$. For instance $K_4$ is not a four-cycle free graph.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[angle=0,\nwidth=.1\\textwidth]{hardsquare.pdf}\\caption{Graph for the Hard Square Shift}\n\\label{Figure: Hard Square}\n\\end{figure}\n\nThe set of \\emph{globally allowed patterns} of a shift space $X$ on a set $A\\subset \\mathbb{Z}^d$ is\n$$\\mathcal L_A(X):=\\{a\\in {\\mathcal A}^{A}\\:\\big{|}\\:\\text{ there exists }x\\in X\\text{ such that }x|_A=a\\}.$$\nIts \\emph{language} is the set of all finite patterns appearing in $X$ that is\n$$\\mathcal L(X):=\\bigcup_{A\\subset \\mathbb{Z}^d \\text{ finite}}\\mathcal L_A(X).$$\nWe comment that this is different from the set of locally allowed patterns: Let $X$ be a shift space with a forbidden list ${\\mathcal F}$. Given a finite set $A$, a pattern $a\\in {\\mathcal A}^A$ is said to be \\emph{locally allowed} if no pattern from ${\\mathcal F}$ appears in $a$. In general it is undecidable for shifts of finite type whether a locally allowed pattern belongs to $\\mathcal L(X)$ \\cite{Robinson1971}; however it is decidable when $X$ is a hom-shift where it is sufficient to check whether the pattern extends to a locally allowed pattern on $B_n$ for some $n$.\n\nThe topological entropy of the shift space $X$ is the log growth rate of the number of allowed patterns in $X$, that is,\n$$h_{top}(X):=\\lim_{n\\longrightarrow \\infty}\\frac{\\log|\\mathcal L_{B_n}(X)|}{|B_n|}.$$\nThe existence of the limit follows from subadditivity arguments via the well-known multivariate version of Fekete's Lemma \\cite{silviofekete}. Moreover the topological entropy is an invariant under conjugacy (for $d=1$ look at Proposition 4.1.9 in \\cite{LM}, the proof extends to higher dimensions). We remark that the computation of this invariant for shifts of finite type in $d>1$ is a hard problem and very little is known \\cite{pavlovhardsquare2012}, however there are algorithms to compute approximating upper and lower bounds of the topological entropy of the hom-shifts \\cite{symmtricfriedlan1997,louidor2010improved}. Further if $\\mathcal H$ is a finite connected graph with at least two edges, then $h_{top}(X_\\mathcal H)>0$:\n\n\\begin{prop}\\label{proposition: hom-space positive entropy}\nLet $\\mathcal H$ be a finite graph with distinct vertices $a, b$ and $c$ such that $a\\sim_\\mathcal H b$ and $b\\sim_\\mathcal H c$. Then $h_{top}(X_{\\mathcal H})\\geq\\frac{\\log{2}}{2}$.\n\\end{prop}\n\\begin{proof}\n\nIt is sufficient to see this for a graph $\\mathcal H$ with exactly three vertices $a$, $b$ and $c$ such that $a\\sim_\\mathcal H b$ and $b\\sim_\\mathcal H c$. For such a graph any configuration in $X_\\mathcal H$ is composed of $b$ on one partite class of $\\mathbb{Z}^d$ and a free choice between $a$ and $c$ for vertices on the other partite class. Then\n$$|\\mathcal L_{B_n}(X_\\mathcal H)|=2^{{\\lfloor\\frac{(2n+1)^d}{2}\\rfloor}}+2^{{\\lceil\\frac{(2n+1)^d}{2}\\rceil}}$$\nproving that $h_{top}(X_{\\mathcal H})=\\frac{\\log{2}}{2}$.\n\\end{proof}\n\nA shift space $X$ is called \\emph{entropy minimal} if for all shift spaces $Y\\subsetneq X$, $h_{top}(X)>h_{top}(Y)$. In other words, a shift space $X$ is entropy minimal if forbidding any word causes a drop in entropy. From \\cite{quastrow2000} we know that every shift space contains an entropy minimal shift space with the same entropy and also a characterisation of same entropy factor maps on entropy minimal shifts of finite type.\n\nOne of the main results of this paper is the following:\n\\begin{thm}\\label{theorem:four cycle free entropy minimal}\nLet $\\mathcal H$ be a connected four-cycle free graph. Then $X_\\mathcal H$ is entropy minimal.\n\\end{thm}\nFor $d=1$ all irreducible shifts of finite type are entropy minimal \\cite{LM}. A necessary condition for the entropy minimality of $X_\\mathcal H$ is that $\\mathcal H$ has to be connected.\n\\begin{prop}\\label{proposition:entropy requires connectivity}\nSuppose $\\mathcal H$ is a finite graph with connected components $\\mathcal H_1, \\mathcal H_2, \\ldots \\mathcal H_r$. Then $h_{top}(X_\\mathcal H)=\\max_{1\\leq i \\leq r}h_{top}(X_{\\mathcal H_i})$.\n\\end{prop}\nThis follows from the observation that\n$$\\max_{1\\leq i \\leq r}|\\mathcal L_{B_n}({X_{\\mathcal H_i}})|\\leq |\\mathcal L_{B_n}({X_\\mathcal H})|= \\sum_{i=1}^r|\\mathcal L_{B_n}({X_{\\mathcal H_i}})|\\leq r\\max_{1\\leq i \\leq r}|\\mathcal L_{B_n}({X_{\\mathcal H_i}})|.$$\n\n\n\n\n\n\\section{Thermodynamic Formalism}\\label{section:thermodynamic formalism}\nHere we give a brief introduction of thermodynamic formalism. For more details one can refer to \\cite{Rue,walters-book}.\n\nBy $\\mu$ we will always mean a shift-invariant Borel probability measure on a shift space $X$. The \\emph{support} of $\\mu$ denoted by $supp(\\mu)$ is the intersection of all closed sets $Y \\subset X$ for which $\\mu(Y)= 1$. Note that $supp(\\mu)$ is a shift space as well.\nThe \\emph{measure theoretic entropy} is\n\\begin{equation*}\nh_\\mu:=\\lim_{i \\rightarrow \\infty}\\frac{1}{|D_i|}H^{D_i}_{\\mu},\n\\end{equation*}\n\\noindent where $H^{D_i}_{\\mu}$ is the Shannon-entropy of $\\mu$ with respect to the partition of $X$ generated by the cylinder sets on $D_i$, the definition of which is given by:\n\\begin{equation*}\nH^{D_i}_{\\mu}:=\\sum_{a\\in \\mathcal L_{D_i}(X)}-\\mu([a]_{D_i})\\log{\\mu([a]_{D_i})},\n\\end{equation*} with the understanding that $0\\log 0=0$.\n\nA shift-invariant probability measure $\\mu$ is a \\emph{measure of maximal entropy} of $X$ if the maximum of $\\nu \\mapsto h_\\nu$ over all shift-invariant probability measures on $X$ is obtained at $\\mu$. The existence of measures of maximal entropy follows from upper-semi-continuity of the function $\\nu \\mapsto h_\\nu$ with respect to the weak-$*$ topology.\n\n\nFurther the well-known \\emph{variational principle} for topological entropy of $\\mathbb{Z}^d$-actions asserts that if $\\mu$ is a measure of maximal entropy $h_{top}(X)=h_\\mu$ whenever $X$ is a $\\mathbb{Z}^d$-shift space.\n\nThe following is a well-known characterisation of entropy minimality (it is used for instance in the proof of Theorem 4.1 in \\cite{meestersteif2001}):\n\\begin{prop}\n\\label{proposition:entropyviamme}\nA shift space $X$ is entropy minimal if and only if every measure of maximal entropy for $X$ is fully supported.\n\\end{prop}\nWe understand this by the following: Suppose $X$ is entropy minimal and $\\mu$ is a measure of maximal entropy for $X$. Then by the variational principle for $X$ and $supp(\\mu)$ we get\n$$h_{top}(X)=h_\\mu\\leq h_{top}(supp(\\mu))\\leq h_{top}(X)$$\nproving that $supp(\\mu)=X$. To prove the converse, suppose for contradiction that $X$ is not entropy minimal and consider $Y\\subsetneq X$ such that $h_{top}(X)= h_{top}(Y)$. Then by the variational principle there exists a measure $\\mu$ on $Y$ such that $h_\\mu= h_{top}(X)$. Thus $\\mu$ is a measure of maximal entropy for $X$ which is not fully supported.\n\nFurther is known if $X$ is a nearest neighbour shift of finite type; this brings us to Markov random fields which we introduce next.\n\nGiven a set $A\\subset \\mathbb{Z}^d$ we denote the \\emph{$r$-boundary} of $A$ by $\\partial_r A$, that is,\n$$\\partial_r A=\\{w\\in \\mathbb{Z}^d\\setminus A\\:\\Big \\vert\\: \\|w-v\\|_1\\leq r \\text{ for some }v\\in A\\}.$$\nThe \\emph{1-boundary} will be referred to as the \\emph{boundary} and denoted by $\\partial A$.\nA \\emph{Markov random field} on $\\mathcal{A}^{\\mathbb{Z}^d}$ is a Borel probability measure $\\mu$ with the property that\nfor all finite $A, B \\subset \\mathbb{Z}^d$ such that $\\partial A \\subset B \\subset A^{c}$ and $a \\in {\\mathcal A}^A, b \\in {\\mathcal A}^B$ satisfying $\\mu([b]_B)>0$\n\\begin{equation*}\n\\mu([a]_A\\;\\Big\\vert\\;[b]_B)= \\mu([a]_A\\;\\Big\\vert\\;[b]_{ \\partial A}).\n\\end{equation*}\n\nIn general Markov random fields are defined over graphs much more general than $\\mathbb{Z}^d$, however we restrict to the $\\mathbb{Z}^d$ setting in this paper.\n\nA \\emph{uniform Markov random field} is a Markov random field $\\mu$ such that further\n\n\\begin{equation*}\n\\mu([a]_A\\;\\Big\\vert\\;[b]_{ \\partial A})=\\frac{1}{n_{A,b|_{\\partial A}}}\n\\end{equation*}\nwhere $n_{A,b|_{\\partial A}}=|\\{a\\in {\\mathcal A}^A\\:|\\: \\mu([a]_A\\cap [b]_{\\partial A})>0\\}|$.\n\nFollowing \\cite{petersen_schmidt1997, schmidt_invaraint_cocycles_1997}, we denote by $\\Delta_X$ the \\emph{homoclinic equivalence relation} of a shift space $X$, which is given by\n\\begin{equation*}\n\\Delta_X := \\{(x,y)\\in X\\times X\\;|\\; x_{\\vec i}=y_{\\vec i} \\text{ for all but finitely many } \\vec i\\in \\mathbb{Z}^d\\}.\n\\end{equation*}\n\nWe say that a measure $\\mu$ is \\emph{adapted} with respect to a shift space $X$ if\n$supp(\\mu)\\subset X$ and\n\\begin{equation*}\nx\\in supp(\\mu) \\Longrightarrow \\{y\\in X\\:|\\: (x,y)\\in \\Delta_X\\}\\subset supp(\\mu).\n\\end{equation*}\n\nTo illustrate this definition, let $X\\subset \\{0,1\\}^{\\mathbb{Z}}$ consist of configurations in $X$ in which at most a single $1$ appears. $X$ is uniquely ergodic; the delta-measure $\\delta_{0^{\\infty}}$ is the only shift-invariant measure on $X$. But\n$$supp(\\delta_{0^\\infty})= \\{0^\\infty\\}\\subsetneq \\{y\\in X\\;|\\; 0^\\infty_i= y_i \\text{ for all but finitely many } i\\in \\mathbb{Z}\\}=X,$$\nproving that it is not adapted. On the other hand, since the homoclinic relation of ${\\mathcal A}^{\\mathbb{Z}^d}$ is minimal, meaning that for all $x\\in {\\mathcal A}^{\\mathbb{Z}^d}$\n$$\\overline{\\{y\\in {\\mathcal A}^{\\mathbb{Z}^d}\\;|\\; y_{\\vec i}=x_{\\vec i} \\text{ for all but finitely many } {\\vec{i}}\\in \\mathbb{Z}^d\\}}= {\\mathcal A}^{\\mathbb{Z}^d},$$\nit follows that a probability measure on ${\\mathcal A}^{\\mathbb{Z}^d}$ is adapted if and only if it is fully supported.\n\n\n\nThe relationship between measures of maximal entropy and Markov random fields is established by the following theorem. This is a special case of the Lanford-Ruelle theorem \\cite{lanfruell,Rue}.\n\n\\begin{thm} All measures of maximal entropy on a nearest neighbour shift of finite type $X$ are shift-invariant uniform Markov random fields $\\mu$ adapted to $X$.\\label{thm:equiGibbs}\n\\end{thm}\n\nThe converse is also true under further mixing assumptions on the shift space $X$ (called the D-condition). The full strength of these statements is obtained by looking at \\emph{equilibrium states} instead of measures of maximal entropy. The measures obtained there are not uniform Markov random fields, rather Markov random fields where the conditional probabilities are weighted via an \\emph{interaction} giving rise to \\emph{Gibbs states}. Uniform Markov random fields are Gibbs states with interaction zero.\n\nWe will often restrict our proofs to the ergodic case. We can do so via the following standard facts implied by Theorem $14.15$ in \\cite{Georgii} and Theorem 4.3.7 in \\cite{kellerequ1998}:\n\n\\begin{thm}\\label{theorem: ergodic decomposition of markov random fields}\nLet $\\mu$ be a shift-invariant uniform Markov random field adapted to a shift space $X$. Let its ergodic decomposition be given by a measurable map $x\\longrightarrow \\mu_x$ on $X$, that is, $\\mu= \\int_X \\mu_x d\\mu$. Then $\\mu$-almost everywhere the measures $\\mu_x$ are shift-invariant uniform Markov random fields adapted to $X$ such that $supp(\\mu_x)\\subset supp(\\mu)$. Moreover $\\int h_{\\mu_x} d\\mu(x)= h_\\mu$.\n\\end{thm}\n\n\nWe will prove the following:\n\\begin{thm}\\label{theorem: MRF fully supported }\nLet $\\mathcal H$ be a connected four-cycle free graph. Then every ergodic probability measure adapted to $X_\\mathcal H$ with positive entropy is fully supported.\n\\end{thm}\n\nThis implies Theorem \\ref{theorem:four cycle free entropy minimal} by the following: The Lanford-Ruelle theorem implies that every measure of maximal entropy on $X_\\mathcal H$ is a uniform shift-invariant Markov random field adapted to $X_\\mathcal H$. By Proposition \\ref{proposition: hom-space positive entropy} and the variational principle we know that these measures have positive entropy. By Theorems \\ref{theorem: ergodic decomposition of markov random fields} and \\ref{theorem: MRF fully supported } they are fully supported. Finally by Proposition \\ref{proposition:entropyviamme}, $X_\\mathcal H$ is entropy minimal.\n\nAlternatively, the conclusion of Theorem \\ref{theorem: MRF fully supported } can be obtained via some strong mixing conditions on the shift space; we will describe one such assumption. A shift space $X$ is called \\emph{strongly irreducible} if there exists $g>0$ such that for all $x, y \\in X$ and $A, B\\subset \\mathbb{Z}^d$ satisfying $\\min_{\\vec i \\in A, \\vec j \\in B}\\|\\vec i - \\vec j\\|_1\\geq g$, there exists $z\\in X$ such that $z|_{A}= x|_A$ and $z|_B= y|_B$. For such a space, the homoclinic relation is minimal implying the conclusion of Theorem \\ref{theorem: MRF fully supported } and further, that every probability measure adapted to $X$ is fully supported. Note that this does not prove that $X$ is entropy minimal unless we assume that $X$ is a nearest neighbour shift of finite type. Such an argument is used in the proof of Lemma 4.1 in \\cite{meestersteif2001} which implies that every strongly irreducible shift of finite type is entropy minimal. A more combinatorial approach was used in \\cite{Schraudner2010minimal} to show that general shift spaces with a weaker mixing property called uniform filling are entropy minimal.\n\n\n\\section{The Pivot Property}\\label{section: the pivot property}\nA \\emph{pivot} in a shift space $X$ is a pair of configurations $(x,y)\\in X$ such that $x$ and $y$ differ exactly at a single site. A subshift $X$ is said to have \\emph{the pivot property} if for all distinct $(x,y)\\in \\Delta_X$ there exists a finite sequence of configurations $x^{(1)}=x, x^{(2)},\\ldots, x^{(k)}=y \\in X$ such that each $(x^{(i)}, x^{(i+1)})$ is a pivot. In this case we say $x^{(1)}=x, x^{(2)},\\ldots, x^{(k)}=y$ is a \\emph{chain of pivots} from $x$ to $y$.\nHere are some examples of subshifts which have the pivot property:\n\n\\begin{enumerate}\n\\item Any subshift with a trivial homoclinic relation, that is, the homoclinic classes are singletons.\n\\item Any subshift with a safe symbol\\footnote{A shift space $X\\subset {\\mathcal A}^{\\mathbb{Z}^d}$ has a \\emph{safe symbol} $\\star$ if for all $x\\in X$ and $A\\subset \\mathbb{Z}^d$ the configuration $z\\in {\\mathcal A}^{\\mathbb{Z}^d}$ given by\n\\begin{equation*}\nz_{\\vec i}:=\\begin{cases}\nx_{\\vec i} &\\text{ if } \\vec i \\in A\\\\\n\\star &\\text{ if } \\vec i \\in A^c\n\\end{cases}\u2022\n\\end{equation*}\nis also an element of $X$.}.\n\\item The hom-shifts $X_{C_r}$. This was proved for $r\\neq 4$ in \\cite{chandgotia2013Markov}, the result for $r=4$ is a special case of Proposition \\ref{proposition: frozenfoldpivot}.\n\\item $r$-colorings of $\\mathbb{Z}^d$ with $r\\geq 2d+2$. (It is well-known, look for instance in Subsection 3.2 of \\cite{chandgotia2013Markov})\n\\item\\label{item: pivot property list number 5} $X_\\mathcal H$ when $\\mathcal H$ is dismantlable. \\cite{brightwell2000gibbs}\n\\end{enumerate}\nWe generalise the class of examples given by (\\ref{item: pivot property list number 5}) in Proposition \\ref{proposition: frozenfoldpivot}. It is not true that all hom-shifts have the pivot property.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[angle=0,\nwidth=.1\\textwidth]{fivecolouring.pdf}\\caption{Frozen Pattern}\n\\label{Figure: Five colour}\n\\end{figure}\nThe following was observed by Brian Marcus: Recall that $K_n$ denotes the complete graph with $n$ vertices. $X_{K_4}, X_{K_5}$ do not possess the pivot property if the dimension is two. For instance consider a configuration in $X_{K_5}$ which is obtained by tiling the plane with the pattern given in Figure \\ref{Figure: Five colour}. It is clear that the symbols in the box can be interchanged but no individual symbol can be changed. Therefore $X_{K_5}$ does not have the pivot property. However both $X_{K_4}$ and $X_{K_5}$ satisfy a more general property as discussed in Subsection \\ref{subsection: Hom-shifts and the pivot property}.\n\nThe following theorem is another main result in this paper.\n\\begin{thm}\\label{theorem: pivot property for four cycle free}\nFor all four-cycle free graphs $\\mathcal H$, $X_\\mathcal H$ has the pivot property.\n\\end{thm}\nIt is sufficient to prove this theorem for four-cycle free graphs $\\mathcal H$ which are connected because of the following proposition:\n\\begin{prop}\\label{proposition: pivot for disconnected}\nLet $X_1, X_2, \\ldots, X_n$ be shift spaces on disjoint alphabets such that each of them has the pivot property. Then $\\cup_{i=1}^n X_i$ also has the pivot property.\n\\end{prop}\nThis is true since $(x, y)\\in \\Delta_{\\cup_{i=1}^n X_i}$ implies $(x, y)\\in \\Delta_{X_i}$ for some $1\\leq i \\leq n$.\n\n\\section{Folding, Entropy Minimality and the Pivot Property}\\label{section:Folding, Entropy Minimality and the Pivot Property} Given a graph $\\mathcal H$ we say that a vertex $v$ \\emph{folds} into a vertex $w$ if and only if $u \\sim_\\mathcal H v$ implies $u \\sim_\\mathcal H w$. In this case the graph $\\mathcal H\\setminus \\{v\\}$ is called a \\emph{fold} of $\\mathcal H$. The folding gives rise to a `retract' from $\\mathcal H$ to $\\mathcal H\\setminus\\{v\\}$, namely the graph homomorphism from $\\mathcal H$ to $\\mathcal H\\setminus \\{v\\}$ which is the identity on $\\mathcal H\\setminus \\{v\\}$ and sends $v$ to $w$. This was introduced in \\cite{nowwinkler} to help characterise cop-win graphs and used in \\cite{brightwell2000gibbs} to establish many properties which are preserved under `folding' and `unfolding'. Given a finite tree $\\mathcal H$ with more than two vertices note that a leaf vertex (vertex of degree $1$) can always be folded to some other vertex of the tree. Thus starting with $\\mathcal H$, there exists a sequence of folds resulting in a single edge. In fact using a similar argument we can prove the following proposition.\n\n\\begin{prop}\\label{proposition:folding trees into other trees}\nLet $\\mathcal H\\subset \\mathcal H^\\prime$ be trees. Then there is a graph homomorphism $f: \\mathcal H^\\prime \\longrightarrow \\mathcal H$ such that $f|_{\\mathcal H}$ is the identity map.\n\\end{prop}\n\nTo show this, first note that if $\\mathcal H\\subsetneq\\mathcal H^\\prime$ then there is a leaf vertex in $\\mathcal H^\\prime$ which is not in $\\mathcal H$. This leaf vertex can be folded into some other vertex in $\\mathcal H^\\prime$. Thus by induction on $|\\mathcal H^\\prime \\setminus \\mathcal H|$ we can prove that there is a sequence of folds from $\\mathcal H^\\prime$ to $\\mathcal H$. Corresponding to this sequence of folds we obtain a graph homomorphism from $\\mathcal H^\\prime$ to $\\mathcal H$ which is the identity on $\\mathcal H$.\n\nHere we consider a related notion for shift spaces. Given a nearest neighbour shift of finite type $X\\subset {\\mathcal A}^{\\mathbb{Z}^d}$, \\emph{the neighbourhood} of a symbol $v\\in {\\mathcal A}$ is given by\n$$N_X(v):=\\{a \\in {\\mathcal A}^{\\partial \\vec 0}\\:|\\: [v]_{\\vec 0}\\cap [a]_{\\partial \\vec 0}\\in \\mathcal L_{D_1}(X)\\},$$\nthat is the collection of all patterns which can `surround' $v$ in $X$. We will say that $v$ \\emph {config-folds} into $w$ in $X$ if $N_X(v)\\subset N_X(w)$. In such a case we say that $X$ \\emph{config-folds} to $X\\cap({\\mathcal A}\\setminus \\{v\\})^{\\mathbb{Z}^d}$. Note that $X\\cap({\\mathcal A}\\setminus \\{v\\})^{\\mathbb{Z}^d}$ is obtained by forbidding $v$ from $X$ and hence it is also a nearest neighbour shift of finite type. Also if $X=X_\\mathcal H$ for some graph $\\mathcal H$ then $v$ config-folds into $w$ in $X_\\mathcal H$ if and only if $v$ folds into $w$ in $\\mathcal H$. Thus if $\\mathcal H$ is a tree then there is a sequence of folds starting at $X_\\mathcal H$ resulting in the two checkerboard configurations with two symbols (the vertices of the edge which $\\mathcal H$ folds into). This property is weaker than the notion of folding introduced in \\cite{chandgotiahammcliff2014}.\n\nThe main thrust of this property in our context is: if $v$ config-folds into $w$ in $X$ then given any $x\\in X$, every appearance of $v$ in $x$ can be replaced by $w$ to obtain another configuration in $X$. This replacement defines a factor (surjective, continuous and shift-invariant) map $f: X\\longrightarrow X\\cap({\\mathcal A}\\setminus \\{v\\})^{\\mathbb{Z}^d}$ given by\n\\begin{equation*}\n(f(x))_{\\vec i}:=\\begin{cases}\nx_{\\vec i}&\\text{ if } x_{\\vec i}\\neq v\\\\\nw&\\text{ if } x_{\\vec i}= v.\n\\end{cases}\u2022\n\\end{equation*}\nNote that the map $f$ defines a `retract' from $X$ to $X\\cap({\\mathcal A}\\setminus \\{v\\})^{\\mathbb{Z}^d}$. Frequently we will config-fold more than one symbol at once (especially in Section \\ref{section: Proof of the main theorems}):\n\nDistinct symbols $v_1, v_2, \\ldots, v_n$ \\emph{config-fold disjointly} into $w_1, w_2, \\ldots, w_n$ in $X$ if $v_i$ config-folds into $w_i$ and $v_i\\neq w_j$ for all $1\\leq i, j \\leq n$. In this case the symbols $v_1, v_2, \\ldots, v_n$ can be replaced by $w_1, w_2, \\ldots, w_n$ simultaneously for all $x \\in X$. Suppose $v_1, v_2,\\ldots v_n$ is a maximal set of symbols which can be config-folded disjointly in $X$. Then $X\\cap({\\mathcal A}\\setminus \\{v_1, v_2, \\ldots, v_n\\})^{\\mathbb{Z}^d}$ is called a \\emph{full config-fold} of $X$.\n\nFor example consider a tree $\\mathcal H:=(\\mathcal V,\\mathcal E)$ where $\\mathcal V:=\\{v_1, v_2, v_3, \\ldots, v_{n+1}\\}$ and $\\mathcal E:=\\{(v_i, v_{n+1})\\:|\\: 1\\leq i \\leq n\\}$. For all $1\\leq i \\leq n$, $\\mathcal V\\setminus \\{v_i, v_{n+1}\\}$ is a maximal set of symbols which config-folds disjointly in $X_\\mathcal H$ resulting in the checkerboard patterns with the symbols $v_i$ and $v_{n+1}$ for all $1\\leq i \\leq n$. Thus the full config-fold of a shift space is not necessarily unique. However it is unique up to conjugacy:\n\n\\begin{prop}\\label{Proposition: Uniqueness of full config-fold}\nThe full config-fold of a nearest neighbour shift of finite type is unique up to conjugacy via a change of the alphabet.\n\\end{prop}\nThe ideas for the following proof come essentially from the proof of Theorem 4.4 in \\cite{brightwell2000gibbs} and discussions with Prof. Brian Marcus.\n\\begin{proof}\nLet $X\\subset {\\mathcal A}^{\\mathbb{Z}^d}$ be a nearest neighbour shift of finite type and\n$$M:=\\{v\\in {\\mathcal A} \\:|\\: \\text{ for all }w\\in {\\mathcal A},\\  v \\text{ config-folds into }w \\Longrightarrow w \\text{ config-folds into }v\\}.$$\nThere is a natural equivalence relation $\\equiv$ on $M$ given by $v\\equiv w$ if $v$ and $w$ config-fold into each other. Let $A_1, A_2, A_3, \\ldots, A_r\\subset M$ be the corresponding partition. Clearly for all distinct $v, w\\in M$, $v$ can be config-folded into $w$ if and only if $v, w\\in A_i$ for some $i$. It follows that $A\\subset A_i$ can be config-folded disjointly if and only if $\\emptyset\\neq A\\neq A_i$.\n\nLet $v\\in {\\mathcal A}\\setminus M$. We will prove that $v$ config-folds to a symbol in $M$. By the definition of $M$ there exists $v_1\\in {\\mathcal A}$ such that $N_X(v)\\subsetneq N_X(v_1)$. If $v_1\\in M$ then we are done, otherwise choose $v_2\\in {\\mathcal A}$ such that $N_X(v_1)\\subsetneq N_X(v_2)$. Continuing this process recursively we can find a sequence $v= v_0, v_1, v_2, \\ldots, v_n$ such that $N_X(v_{i-1})\\subsetneq N_X(v_i)$ for all $1\\leq i \\leq n$ and $v_n\\in M$. Thus $v$ config-folds into $v_n$, a symbol in $M$. Further if $v$ config-folds to a symbol in $A_i$ it can config-fold to all the symbols in $A_i$. Therefore $B$ is a maximal subset of symbols in ${\\mathcal A}$ which can be config-folded disjointly if and only if $B=\\cup_{i=1}^rB_i\\cup ({\\mathcal A}\\setminus M)$ where $B_i\\subset A_i$ and $|A_i\\setminus B_i|=1$. Let $B'\\subset {\\mathcal A}$ be another such maximal subset, ${\\mathcal A}\\setminus B:=\\{b_1, b_2, \\ldots, b_r\\}$ and ${\\mathcal A}\\setminus B':=\\{b'_1, b'_2, \\ldots, b'_r\\}$ where $b_i, b'_i\\in A_i$. Then the map\n$$f: X\\cap ({\\mathcal A}\\setminus B)^{\\mathbb{Z}^d} \\longrightarrow X\\cap ({\\mathcal A}\\setminus B')^{\\mathbb{Z}^d} \\text{ given by } f(x) := y\\text{ where } y_{\\vec i}=b'_j \\text{ whenever }x_{\\vec i}=b_j$$\nis the required change of alphabet between the two full config-folds of $X$. \\end{proof}\n\nLet $X\\cap({\\mathcal A}\\setminus \\{v_1, v_2, \\ldots, v_n\\})^{\\mathbb{Z}^d}$ be a \\emph{full config-fold} of $X$ where $v_i$ config-folds into $w_i$ for all $1\\leq i\\leq n$. Consider $f_X: {\\mathcal A} \\longrightarrow{\\mathcal A}\\setminus \\{v_1, v_2, \\ldots, v_n\\}$ given by\n\\begin{equation*}\nf_X(v):=\\begin{cases}\nv&\\text{ if } v\\neq v_j \\text{ for all }1\\leq j\\leq n\\\\\nw_j&\\text{ if } v= v_j\\text{ for some }1\\leq j \\leq n.\n\\end{cases}\u2022\n\\end{equation*}\n\\noindent This defines a factor map $f_X: X\\longrightarrow X\\cap({\\mathcal A}\\setminus \\{v_1, v_2, \\ldots, v_n\\})^{\\mathbb{Z}^d}$ given by $(f_X(x))_{{\\vec{i}}}:= f_X(x_{\\vec{i}})$ for all ${\\vec{i}} \\in \\mathbb{Z}^d$. $f_X$ denotes both the factor map and the map on the alphabet; it should be clear from the context which function is being used.\n\n\n\nIn many cases we will fix a configuration on a set $A\\subset \\mathbb{Z}^d$ and apply a config-fold on the rest. Hence we define the map $f_{X,A}: X\\longrightarrow X$ given by\n\\begin{equation*}\n(f_{X,A}(x))_{\\vec i}:=\\begin{cases}\nx_{\\vec i}&\\text{ if } \\vec i \\in A\\\\\nf_X(x_{\\vec i})&\\text{ otherwise.}\n\\end{cases}\u2022\n\\end{equation*}\u2022\n\nThe map $f_{X,A}$ can be extended beyond $X$:\n\n\\begin{prop}\\label{prop: folding_fixing_a_set}\nLet $X\\subset Y$ be nearest neighbour shifts of finite type, $Z$ be a full config-fold of $X$ and $y\\in Y$ such that for some $A\\subset \\mathbb{Z}^d$, $y|_{A^c\\cup\\partial (A^c)}\\in \\mathcal L_{A^c\\cup\\partial (A^c)}(X)$. Then the configuration $z$ given by\n\\begin{equation*}\nz_{\\vec i}:=\\begin{cases}\ny_{\\vec i}&\\text{ if } \\vec i \\in A\\\\\nf_X(y_{\\vec i})&\\text{ otherwise}\n\\end{cases}\u2022\n\\end{equation*}\nis an element of $Y$. Moreover $z|_{A^c}\\in \\mathcal L_{A^c}(Z)$.\n\\end{prop}\nAbusing the notation, in such cases we shall denote the configuration $z$ by $f_{X, A}(y)$.\n\nIf $A^c$ is finite, then $f_{X,A}$ changes only finitely many coordinates. These changes can be applied one by one, that is, there is a chain of pivots in $Y$ from $y$ to $f_{X,A}(y)$.\n\nA nearest neighbour shift of finite type which cannot be config-folded is called a \\emph{stiff shift}. We know from Theorem 4.4 in \\cite{brightwell2000gibbs} that all the stiff graphs obtained by a sequence of folds of a given graph are isomorphic. By Proposition \\ref{Proposition: Uniqueness of full config-fold} the corresponding result for nearest neighbour shifts of finite type immediately follows:\n\\begin{prop}\\label{proposition:uniqueness of stiff shifts}\nThe stiff shift obtained by a sequence of config-folds starting with a nearest neighbour shift of finite type is unique up to conjugacy via a change of the alphabet.\n\\end{prop}\n\nStarting with a nearest neighbour shift of finite type $X$ the \\emph{fold-radius} of $X$ is the smallest number of full config-folds required to obtain a stiff shift. If $\\mathcal H$ is a tree then the fold-radius of $X_\\mathcal H$ is equal to\n$$\\left\\lfloor\\frac{diameter(\\mathcal H)}{2}\\right\\rfloor.$$\nThus for every nearest neighbour shift of finite type $X$ there is a sequence of full config-folds (not necessarily unique) which starts at $X$ and ends at a stiff shift of finite type. Let the fold-radius of $X$ be $r$ and $X= X_0, X_1, X_2, \\ldots, X_r$ be a sequence of full config-folds where $X_r$ is stiff. This generates a sequence of maps $f_{X_i}:X_{i}\\longrightarrow X_{i+1}$ for all $0\\leq i \\leq r-1$. In many cases we will fix a pattern on $D_n$ or $D_n^c$ and apply these maps on the rest of the configuration. Consider the maps $I_{X,n}:X\\longrightarrow X$ and $O_{X,n}:X\\longrightarrow X$ (for $n>r$) given by\n\\begin{equation*}\nI_{X,n}(x):=f_{X_{r-1},D_{n+r-1} }\\left(f_{X_{r-2}, D_{n+r-2}}\\left(\\ldots\\left(f_{X_{0}, D_n}(x)\\right)\\ldots\\right)\\right)\\text{(Inward Fixing Map)}\n\\end{equation*}\u2022\nand\n\\begin{eqnarray*}\nO_{X,n}(x):=f_{X_{r-1}, D_{n-r+1}^c }\\left(f_{X_{r-2}, D_{n-r+2}^c}\\left(\\ldots\\left(f_{X_{0}, D_n^c}(x)\\right)\\ldots\\right)\\right)\\text{(Outward Fixing Map)}.\n\\end{eqnarray*}\u2022\nSimilarly we consider maps which do not fix anything, $F_X: X\\longrightarrow X_r$ given by\n\\begin{eqnarray*}\nF_X(x):= f_{X_{r-1}}\\left(f_{X_{r-2}}\\left(\\ldots\\left(f_{X_{0}}(x)\\right)\\ldots\\right)\\right).\n\\end{eqnarray*}\nNote that $D_k\\cup \\partial D_k= D_{k+1}$ and $D_k^c\\cup \\partial (D_k^c)=D_{k-1}^c$. This along with repeated application of Proposition \\ref{prop: folding_fixing_a_set} implies that the image of $I_{X,n}$ and $O_{X,n}$ lie in $X$. This also implies the following proposition:\n\n\\begin{prop}[The Onion Peeling Proposition]\\label{prop: folding_ to _ stiffness_fixing_a_set}\nLet $X\\subset Y$ be nearest neighbour shifts of finite type, $r$ be the fold-radius of $X$, $Z$ be a stiff shift obtained by a sequence of config-folds starting with $X$ and $y^1, y^2\\in Y$ such that $y^1|_{D_{n-1}^c}\\in \\mathcal L_{D_{n-1}^c}(X)$ and $y^2|_{D_{n+1}}\\in \\mathcal L_{D_{n+1}}(X)$. Let $z^1, z^2\\in Y$ be given by\n\\begin{eqnarray*}\nz^1&:=&f_{X_{r-1},D_{n+r-1} }\\left(f_{X_{r-2}, D_{n+r-2}}\\left(\\ldots\\left(f_{X_{0}, D_n}(y^1)\\right)\\ldots\\right)\\right)\\\\\nz^2&:=&f_{X_{r-1}, D_{n-r+1}^c }\\left(f_{X_{r-2},D_{n-r+2}^c}\\left(\\ldots\\left(f_{X_{0}, D_n^c}(y^2)\\right)\\ldots\\right)\\right)\\text{ for }n>r.\n\\end{eqnarray*}\u2022\nThe patterns $z^1|_{D_{n+r-1}^c}\\in \\mathcal L_{D_{n+r-1}^c}(Z)$ and $z^2|_{D_{n-r+1}}\\in \\mathcal L_{D_{n-r+1}}(Z)$. If $y^1, y^2\\in X$ then in addition\n\\begin{eqnarray*}\nz^1|_{D_{n+r-1}^c}&=&F_X(y^1)|_{D_{n+r-1}^c}\\text{ and}\\\\\nz^2|_{D_{n-r+1}}&=&F_X(y^2)|_{D_{n-r+1}}.\n\\end{eqnarray*}\u2022\n\\end{prop}\n\nAbusing the notation, in such cases we shall denote the configurations $z^1$ and $z^2$ by $I_{X, n}(y^1)$ and $O_{X,n}(y^2)$ respectively. Note that $I_{X, n}(y^1)|_{D_n}= y^1|_{D_n}$ and $O_{X,n}(y^2)|_{D_n^c}= y^2|_{D_n^c}$. Also, $O_{X,n}$ is a composition of maps of the form $f_{X,A}$ where $A^c$ is finite; there is a chain of pivots in $Y$ from $y$ to $O_{X,n}(y)$.\nThere are two kinds of stiff shifts which will be of interest to us: A configuration $x\\in {\\mathcal A}^{\\mathbb{Z}^d}$ is called \\emph{periodic} if there exists $n \\in \\mathbb N$ such that $\\sigma^{n \\vec e_{i}}(x)=x$ for all $1\\leq i \\leq d$. A configuration $x\\in X$ is called \\emph{frozen} if its homoclinic class is a singleton. This notion coincides with the notion of frozen coloring in \\cite{brightwell2000gibbs}. A subshift $X$ will be called \\emph{frozen} if it consists of frozen configurations, equivalently $\\Delta_X$ is the diagonal. A measure on $X$ will be called \\emph{frozen} if its support is frozen. Note that any shift space consisting just of periodic configurations is frozen. All frozen nearest neighbour shifts of finite type are stiff: Suppose $X$ is a nearest neighbour shift of finite type which is not stiff. Then there is a symbol $v$ which can be config-folded to a symbol $w$. This means that any appearance of $v$ in a configuration $x\\in X$ can be replaced by $w$. Hence the homoclinic class of $x$ is not a singleton. Therefore $X$ is not frozen.\n\n\n\\begin{prop}\\label{proposition: periodicfoldentropy} Let $X$ be a nearest neighbour shift of finite type such that a sequence of config-folds starting from $X$ results in the orbit of a periodic configuration. Then every shift-invariant probability measure adapted to $X$ is fully supported.\n\\end{prop}\n\\begin{prop}\\label{proposition: frozenfoldpivot}\nLet $X$ be a nearest neighbour shift of finite type such that a sequence of config-folds starting from $X$ results in a frozen shift. Then $X$ has the pivot property.\n\\end{prop}\n\n\\noindent\\textbf{Examples:}\n\\begin{enumerate}\n\\item\n$X:=\\{0\\}^{\\mathbb{Z}^d}\\cup \\{1\\}^{\\mathbb{Z}^d}$ is a frozen shift space but not the orbit of a periodic configuration. Clearly the delta measure $\\delta_{\\{0\\}^{\\mathbb{Z}^d}}$ is a shift-invariant probability measure adapted to $X$ but not fully supported. A more non-trivial example of a nearest neighbour shift of finite type which is frozen but not the orbit of a periodic configuration is the set of the Robinson tilings $Y$ \\cite{Robinson1971}. There are configurations in $Y$ which have the so-called ``fault lines''; they can occur at most once in a given configuration. Consequently for all shift-invariant probability measures on $Y$, the probability of seeing a fault line is zero. Thus no shift-invariant probability measure (and hence no adapted shift-invariant probability measure) on $Y$ is fully supported.\n\n\\item\\label{Example: Safe Symbol}\nLet $X$ be a shift space with a safe symbol $\\star$. Then any symbol in $X$ can be config-folded into the safe symbol. By config-folding the symbols one by one, we obtain a fixed point $\\{\\star\\}^{\\mathbb{Z}^d}$. Thus any nearest neighbour shift of finite type with a safe symbol satisfies the hypothesis of both the propositions.\n\n\\item \\label{Example: Folds to an edge}Suppose $\\mathcal H$ is a graph which folds into a single edge (denoted by $Edge$) or a single vertex $v$ with a loop. Then the shift space $X_\\mathcal H$ can be\nconfig-folded to $X_{Edge}$ (which consists of two periodic configurations) or the fixed point $\\{v\\}^{\\mathbb{Z}^d}$ respectively. In the latter case, the graph $\\mathcal H$ is called \\emph{dismantlable} \\cite{nowwinkler}. Note that finite non-trivial trees and the graph $C_4$ fold into an edge. For dismantlable graphs $\\mathcal H$, Theorem 4.1 in \\cite{brightwell2000gibbs} implies the conclusions of Propositions \\ref{proposition: periodicfoldentropy} and \\ref{proposition: frozenfoldpivot} for $X_\\mathcal H$ as well.\n\\end{enumerate}\u2022\n\n\\begin{proof}[Proof of Proposition \\ref{proposition: periodicfoldentropy}] Let $\\mu$ be a shift-invariant probability measure adapted to $X$. To prove that $supp(\\mu)= X$ it is sufficient to prove that for all $n\\in \\mathbb N$ and $x\\in X$ that $\\mu([x]_{D_n})>0$. Let $X_0=X$, $X_1$, $X_2$$,\\ldots,$ $X_r$ be a sequence of full config-folds where $X_r:=\\{ \\sigma^{\\vec i_1}(p), \\sigma^{\\vec i_2}(p),\\ldots, \\sigma^{\\vec i_{k-1}}(p) \\}$ is the orbit of a periodic point. For any two configurations $z,w\\in X$ there exists $\\vec i\\in \\mathbb{Z}^d$ such that $F_X(z)= F_X(\\sigma^{\\vec i}(w)).$\nSince $\\mu$ is shift-invariant we can choose $y \\in supp (\\mu)$ such that $F_X(x)= F_X(y).$\nConsider the configurations $I_{X,n}(x)$ and $O_{X,n+2r-1}(y)$. By Proposition \\ref{prop: folding_ to _ stiffness_fixing_a_set} they satisfy the equations\n\\begin{eqnarray*}\nI_{X,n}(x)|_{D_{n+r-1}^c}&=&F_X(x)|_{D_{n+r-1}^c}\\text{ and }\\\\\nO_{X,n+2r-1}(y)|_{D_{n+r}}&=&F_X(y)|_{D_{n+r}}.\n\\end{eqnarray*}\n\\noindent Then $I_{X,n}(x)|_{\\partial D_{n+r-1}}= O_{X,n+2r-1}(y)|_{\\partial D_{n+r-1}}$. Since $X$ is a nearest neighbour shift of finite type, the configuration $z$ given by\n\\begin{eqnarray*}\nz|_{D_{n+r}}&:=&I_{X,n}(x)|_{D_{n+r}}\\\\\nz|_{D_{n+r-1}^c}&:=&O_{X,n+2r-1}(y)|_{D_{n+r-1}^c}\n\\end{eqnarray*}\n\\noindent is an element of $X$. Moreover\n\\begin{eqnarray*}\nz|_{D_{n}}&=&I_{X,n}(x)|_{D_{n}}=x|_{D_n}\\\\\nz|_{D_{n+2r-1}^c}&=&O_{X,n+2r-1}(y)|_{D_{n+2r-1}^c}=y|_{D^c_{n+2r-1}}.\n\\end{eqnarray*}\u2022\nThus $(y, z)\\in \\Delta_X$. Since $\\mu$ is adapted we get that $z\\in supp(\\mu)$. Finally\n$$\\mu([x]_{D_n})=\\mu([z]_{D_n})>0.$$\n\\end{proof}\n\nNote that all the maps being discussed here, $f_X$, $f_{X,A}$, $F_X$, $I_{X,n}$ and $O_{X,n}$ are (not necessarily shift-invariant) single block maps, that is, maps $f$ where $\\left(f(x)\\right)_{\\vec i}$ depends only on $x_{\\vec i}$. Thus if $f$ is one such map and $x|_A= y|_A$ for some set $A\\subset \\mathbb{Z}^d$ then $f(x)|_A=f(y)|_A$; they map homoclinic pairs to homoclinic pairs.\n\n\\begin{proof}[Proof of Proposition \\ref{proposition: frozenfoldpivot}] Let $X_0=X$, $X_1$, $X_2$$,\\ldots,$ $X_r$ be a sequence of full config-folds where $X_r$ is frozen. Let $(x, y) \\in \\Delta_X$. Since $X_r$ is frozen, $F_{X}(x)= F_X(y)$. Suppose $x|_{D_n^c}= y|_{D_n^c}$ for some $n\\in \\mathbb N$. Then $O_{X,n+r-1}(x)|_{D_n^c}=O_{X,n+r-1}(y)|_{D_n^c}$. Also by Proposition \\ref{prop: folding_ to _ stiffness_fixing_a_set},\n$$O_{X, n+r-1}(x)|_{D_n}=F_X(x)|_{D_n}=F_X(y)|_{D_n}= O_{X, n+r-1}(y)|_{D_n}.$$\nThis proves that $O_{X,n+r-1}(x)=O_{X,n+r-1}(y)$. In fact it completes the proof since for all $z\\in X$ there exists a chain of pivots in $X$ from $z$ to $O_{X,n+r-1}(z)$.\n\\end{proof}\n\n\n\n\\section{Universal Covers}\\label{section:universal covers}\nMost cases will not be as simple as in the proof of Propositions \\ref{proposition: periodicfoldentropy} and \\ref{proposition: frozenfoldpivot}. We wish to prove the conclusions of these propositions for hom-shifts $X_\\mathcal H$ when $\\mathcal H$ is a connected four-cycle free graph. Many ideas carry over from the proofs of these results because of the relationship of such graphs with their universal covers; we describe this relationship next. The results in this section are not original; look for instance in \\cite{Stallingsgraph1983}. We mention them for completeness.\n\nLet $\\mathcal H$ be a finite connected graph with no self-loops. We denote by $d_\\mathcal H$ the ordinary graph distance on $\\mathcal H$ and by $D_\\mathcal H(u)$, the \\emph{ball of radius 1} around $u$. A graph homomorphism $\\pi:\\mathcal C\\longrightarrow \\mathcal H$ is called a \\emph{covering map} if for some $n \\in \\mathbb N \\cup \\{\\infty\\}$ and all $u \\in \\mathcal H$, there exist disjoint sets $\\{C_i\\}_{i=1}^n\\subset \\mathcal C$ such that $\\pi^{-1}\\left(D_\\mathcal H(u)\\right)= \\cup_{i=1}^n C_i $ and $\\pi|_{C_i}: C_i\\longrightarrow D_\\mathcal H(u)$ is an isomorphism of the induced subgraphs for $1\\leq i\\leq n$. A \\emph{covering space} of a graph $\\mathcal H$ is a graph $\\mathcal C$ such that there exists a covering map $\\pi: \\mathcal C\\longrightarrow \\mathcal H$.\n\nA \\emph{universal covering space} of $\\mathcal H$ is a covering space of $\\mathcal H$ which is a tree. Unique up to graph isomorphism \\cite{Stallingsgraph1983}, these covers can be described in multiple ways. Their standard construction uses non-backtracking walks \\cite{Angluin80}: A \\emph{walk} on $\\mathcal H$ is a sequence of vertices $(v_1, v_2, \\ldots, v_n)$ such that $v_i\\sim_\\mathcal H v_{i+1}$ for all $1\\leq i \\leq n-1$. The \\emph{length} of a walk $p=(v_1, v_2, \\ldots, v_n)$ is $|p|=n-1$, the number of edges traversed on that walk. It is called \\emph{non-backtracking} if $v_{i-1}\\neq v_{i+1}$ for all $2\\leq i \\leq n-1$, that is, successive steps do not traverse the same edge. Choose a vertex $u \\in \\mathcal H$. The vertex set of the universal cover is the set of all non-backtracking walks on $\\mathcal H$ starting from $u$; there is an edge between two such walks if one extends the other by a single step. The choice of the starting vertex $u$ is arbitrary; choosing a different vertex gives rise to an isomorphic graph. We denote the universal cover by $E_\\mathcal H$. The covering map $\\pi: E_\\mathcal H\\longrightarrow \\mathcal H$ maps a walk to its terminal vertex. Usually, we will denote by $\\tilde u, \\tilde v$ and $\\tilde w$ vertices of $E_\\mathcal H$ such that $\\pi(\\tilde u)= u$, $\\pi(\\tilde v)= v$ and $\\pi(\\tilde w)= w$.\n\nThis construction shows that the universal cover of a graph is finite if and only if it is a finite tree. To see this if the graph has a cycle then the finite segments of the walk looping around the cycle give us infinitely many vertices for the universal cover. If the graph is a finite tree, then all walks must terminate at the leaves and their length is bounded by the diameter of the tree. In fact, the universal cover of a tree is itself while the universal cover of a cycle (for instance $C_4$) is $\\mathbb{Z}$ obtained by finite segments of the walks $(1, 2, 3, 0, 1, 2, 3, 0, \\ldots )$ and $(1, 0, 3, 2, 1, 0, 3, 2, \\ldots )$.\n\nFollowing the ideas of homotopies in algebraic topology, there is a natural operation on the set of walks: two walks can be joined together if one begins where the other one ends. More formally, given two walks $p=(v_1, v_2, \\ldots, v_n)$ and $q=(w_1, w_2, \\ldots, w_m)$ where $v_n=w_1$, consider $p\\star q=(v_1, v_2, \\ldots, v_n, w_2, w_3, \\ldots, w_m)$. However even when $p$ and $q$ are non-backtracking $p\\star q$ need not be non-backtracking. So we consider the walk $[p\\star q]$ instead which erases the backtracking segments of $p \\star q$, that is, if for some $i>1$, $v_{n-i+1}\\neq w_{i}$ and $v_{n-j+1}=w_j$ for all $1\\leq j \\leq i-1$ then\n$$[p\\star q]:=(v_1, v_2, \\ldots, v_{n-i+1}, w_{i-1}, w_{i}, \\ldots, w_m).$$\n\nThis operation of erasing the backtracking segments is called \\emph{reduction}, look for instance in \\cite{Stallingsgraph1983}.\nThe following proposition is well-known (Section 4 of \\cite{Stallingsgraph1983}) and shall be useful in our context as well:\n\\begin{prop}\\label{proposition:isomorphism_of_universal_covering_space}\nLet $\\mathcal H$ be a finite connected graph without any self-loops. Then for all $\\tilde{v}, \\tilde w \\in E_\\mathcal H$ satisfying $\\pi(\\tilde v)= \\pi(\\tilde w)$ there exists a graph isomorphism $\\phi: E_\\mathcal H\\longrightarrow E_\\mathcal H$ such that $\\phi(\\tilde v)= \\tilde w$ and $\\pi \\circ \\phi = \\pi$.\n\\end{prop}\n\nTo see how to construct this isomorphism, consider as an example $ (u)$, the empty walk on $\\mathcal H$ and $(v_1, v_2, \\ldots, v_n)$, some non-backtracking walk such that $v_1=v_n=u$. Then the map $\\phi: E_\\mathcal H\\longrightarrow E_\\mathcal H$ given by\n$$\\phi(\\tilde w):= [(v_1, v_2, \\ldots, v_n) \\star \\tilde w].$$\nis a graph isomorphism which maps $(u)$ to $(v_1, v_2, \\ldots, v_n)$; its inverse is $\\psi: E_\\mathcal H\\longrightarrow E_\\mathcal H$ given by\n$$\\psi(\\tilde w):= [(v_n, v_{n-1}, \\ldots, v_1)\\star \\tilde w].$$\n\nThe maps $\\phi, \\pi$ described above give rise to natural maps, also denoted by $\\phi$ and $\\pi$ where $$\\phi:X_{E_\\mathcal H}\\longrightarrow X_{E_\\mathcal H}$$\nis given by $\\phi(\\tilde x)_{\\vec{i}} := \\phi(\\tilde x_{\\vec{i}})$ and\n$$\\pi: X_{E_\\mathcal H} \\longrightarrow X_{\\mathcal H}$$\nis given by $\\pi(\\tilde x)_{\\vec{i}}:=\\pi(\\tilde x_{\\vec{i}})$ for all ${\\vec{i}} \\in \\mathbb{Z}^d$ respectively. A \\emph{lift} of a configuration $x\\in X_\\mathcal H$ is a configuration $\\tilde{x}\\in X_{E_\\mathcal H}$ such that $\\pi \\circ \\tilde{x}= x$.\n\nNow we shall analyse some consequences of this formalism in our context. More general statements (where $\\mathbb{Z}^d$ is replaced by a different graph) are true (under a different hypothesis on $\\mathcal H$), but we restrict to the four-cycle free condition. We noticed in Section \\ref{section:Folding, Entropy Minimality and the Pivot Property} that if $\\mathcal H$ is a tree then $X_{\\mathcal H}$ satisfies the conclusions of Theorems \\ref{theorem: MRF fully supported } and \\ref{theorem: pivot property for four cycle free}. Now we will draw a connection between the four-cycle free condition on $\\mathcal H$ and the formalism in Section \\ref{section:Folding, Entropy Minimality and the Pivot Property}.\n\n\\begin{prop}[Existence of Lifts]\\label{proposition:covering_space_lifting}\nLet $\\mathcal H$ be a connected four-cycle free graph. For all $x\\in X_\\mathcal H$ there exists $\\tilde{x}\\in X_{E_\\mathcal H}$ such that $\\pi(\\tilde{x})=x$. Moreover the lift $\\tilde{x}$ is unique up to a choice of $\\tilde{x}_{\\vec 0}$.\n\\end{prop}\n\n\\begin{proof}\nWe will begin by constructing a sequence of graph homomorphisms $\\tilde{x}^n:D_n \\longrightarrow E_\\mathcal H$ such that $\\pi \\circ\\tilde{x}^n =x|_{D_n}$ and $\\tilde{x}^m|_{D_n}= \\tilde{x}^n$ for all $m>n$. Then by taking the limit of these graph homomorphisms we obtain a graph homomorphism $\\tilde{x}\\in X_{E_\\mathcal H}$ such that $\\pi \\circ\\tilde{x}=x$. It will follow that given $\\tilde{x}^0$ the sequence $\\tilde{x}^n$ is completely determined proving that the lifting is unique up to a choice of $\\tilde{x}_{\\vec {0}}$.\n\nThe recursion is the following: Let $\\tilde{x}^n: D_n \\longrightarrow E_\\mathcal H$ be a given graph homomorphism for some $n\\in \\mathbb N\\cup \\{0\\}$ such that $\\pi \\circ\\tilde{x}^n=x|_{D_n}$. For any ${\\vec{ i}}\\in D_{n+1}\\setminus D_n$, choose a vertex ${\\vec{j}}\\in D_n$ such that $\\vec{j}\\sim \\vec{i}$. Then $\\pi(\\tilde{x}^n_{\\vec{j}})=x_{\\vec{j}}\\sim x_{\\vec{i}}$. Since $\\pi$ defines a local isomorphism between $E_\\mathcal H$ and $\\mathcal H$, there exists a unique vertex $\\tilde v_{\\vec{i}}\\sim \\tilde{x}^n_{\\vec{j}} \\in E_\\mathcal H$ such that $\\pi(\\tilde v_{\\vec{i}})= x_{\\vec{i}}$. Define $\\tilde{x}^{n+1}: D_{n+1}\\longrightarrow E_\\mathcal H$ by\n\\begin{equation*}\n\\tilde{x}^{n+1}_{\\vec{i}}:=\\begin{cases}\\tilde{x}^{n}_{\\vec{i}} &\\text{if } \\vec{i}\\in D_n\\\\\\tilde v_{\\vec{i}} & \\text{if } \\vec{i}\\in D_{n+1}\\setminus D_n.\\end{cases}\n\\end{equation*}\u2022\nThen clearly $\\pi \\circ \\tilde{x}^{n+1}= x|_{D_{n+1}}$ and $\\tilde{x}^{n+1}|_{D_n}=\\tilde{x}^n$. Note that the extension $\\tilde{x}^{n+1}$ is uniquely defined given $\\tilde{x}^n$.\nWe need to prove that this defines a valid graph homomorphism from $D_{n+1}$ to $E_\\mathcal H$. Let $\\vec{i}\\in D_{n+1}\\setminus D_n$ and $\\vec{j}\\in D_n$ be chosen as described above. Consider if possible any $\\vec{j}^\\prime\\neq \\vec{j} \\in D_n$ such that $\\vec{j}^\\prime \\sim \\vec{i}$. To prove that $\\tilde{x}^{n+1}$ is a graph homomorphism we need to verify that $\\tilde{x}^{n+1}_{\\vec{j}^\\prime}\\sim \\tilde{x}^{n+1}_{\\vec{i}}$.\n\nConsider $\\vec{i}^\\prime\\in D_n$ such that $\\vec{i}^\\prime\\sim \\vec{j}$ and $\\vec{j}^\\prime$. Then $\\vec{i}^\\prime, \\vec{j}, \\vec{i}$ and $\\vec{j}^\\prime$ form a four-cycle. Since $\\mathcal H$ is four-cycle free either $x_{\\vec{i}^\\prime}=x_{\\vec{i}}$ or $x_{\\vec{j}^\\prime}= x_{\\vec{j}}$.\n\nSuppose $x_{\\vec{i}^\\prime}=x_{\\vec{i}}$; the other case is similar. Since $\\pi$ is a local isomorphism and $\\tilde{x}^{n+1}_{\\vec{i}},\\tilde{x}^{n+1}_{\\vec{i^\\prime}} \\sim \\tilde{x}^{n+1}_{\\vec{j}}$, we get that $\\tilde{x}^{n+1}_{\\vec{i}}=\\tilde{x}^{n+1}_{\\vec{i}^\\prime}$. But ${\\vec{i}}', {\\vec{j}}' \\in D_n$ and $\\tilde x^{n+1}|_{D_n}= \\tilde x^{n}$ is a graph homomorphism; therefore $\\tilde{x}^{n+1}_{\\vec{i}}=\\tilde{x}^{n+1}_{\\vec{i}^\\prime}\\sim \\tilde{x}^{n+1}_{\\vec{j}^\\prime}$.\n\\end{proof}\n\n\\begin{corollary}\\label{corollary:covering_space_lifting_homoclinic}\nLet $\\mathcal H$ be a connected four-cycle free graph and $x, y\\in X_\\mathcal H$. Consider some lifts $\\tilde{x}, \\tilde{y} \\in X_{E_\\mathcal H}$ such that $\\pi(\\tilde{x})= x $ and $\\pi(\\tilde{y})=y$. If for some $\\vec{i}_0 \\in \\mathbb{Z}^d$, $\\tilde{x}_{\\vec{i}_0}= \\tilde{y}_{\\vec{i}_0}$ then $\\tilde{x}= \\tilde{y}$ on the connected subset of\n$$\\{\\vec{j} \\in \\mathbb{Z}^d\\:|\\: x_{\\vec{j}}= y_{\\vec{j}}\\}$$ which contains $\\vec{i}_0$.\n\\end{corollary}\n\n\\begin{proof}\nLet $D$ be the connected component of $\\{\\vec{i} \\in \\mathbb{Z}^d \\:|\\: x_{\\vec{i}} = y_{\\vec{i}}\\}$ and $\\tilde D$ be the connected component of $\\{\\vec{i} \\in \\mathbb{Z}^d \\:|\\: \\tilde x_{\\vec{i}} = \\tilde y_{\\vec{i}}\\}$ which contain $\\vec{i}_0$.\n\nClearly $\\tilde D \\subset D$. Suppose $\\tilde D\\neq D$. Since both $D$ and $\\tilde D$ are non-empty, connected sets there exist $\\vec{i} \\in D \\setminus \\tilde D$ and $\\vec{j} \\in \\tilde D$ such that $\\vec{i} \\sim \\vec{j}$. Then $x_{\\vec{i}}= y_{\\vec{i}}$, $x_{\\vec{j}}= y_{\\vec{j}}$ and $\\tilde x_{\\vec{j}}= \\tilde y_{\\vec{j}}$. Since $\\pi $ is a local isomorphism, the lift must satisfy $\\tilde x_{\\vec{i}} = \\tilde y_{\\vec{i}}$ implying $\\vec{i} \\in \\tilde D$. This proves that $D= \\tilde D$.\n\\end{proof}\nThe following corollary says that any two lifts of the same graph homomorphism are `identical'.\n\\begin{corollary}\\label{corollary:lift_are_isomorphic}\nLet $\\mathcal H$ be a connected four-cycle free graph. Then for all $\\tilde x^1,\\tilde x^2 \\in X_{E_\\mathcal H}$ satisfying $\\pi(\\tilde x^1) = \\pi(\\tilde x^2)= x$ there exists an isomorphism $\\phi: E_\\mathcal H \\longrightarrow E_\\mathcal H$ such that $\\phi\\circ \\tilde x^1= \\tilde x^2$.\n\\end{corollary}\n\\begin{proof} By Proposition \\ref{proposition:isomorphism_of_universal_covering_space} there exists an isomorphism\n$\\phi: E_\\mathcal H\\longrightarrow E_\\mathcal H$ such that $\\phi(\\tilde x^1_{\\vec{0}})= \\tilde x^2_{\\vec{0}}$ and $\\pi \\circ \\phi = \\pi$. Then $(\\phi\\circ\\tilde x^1)_{\\vec{0}} = \\tilde x^2_{\\vec{0}}$ and $\\pi (\\phi\\circ\\tilde x^1)= (\\pi \\circ \\phi)(\\tilde x^1)= \\pi (\\tilde x^1)= x$. By Proposition \\ref{proposition:covering_space_lifting} $\\phi\\circ\\tilde x^1= \\tilde x^2$.\n\\end{proof}\n\nIt is worth noting at this point the relationship of the universal cover described here with the universal cover in algebraic topology. Undirected graphs can be identified with $1$ dimensional CW-complexes where the set of vertices correspond to the $0$-cells, the edges to the $1$-cells of the complex and the attaching map sends the end-points of the edges to their respective vertices. With this correspondence in mind the (topological) universal covering space coincides with the (combinatorial) universal covering space described above; indeed a $1$ dimensional CW-complex is simply connected if and only if it does not have any loops, that is, the corresponding graph does not have any cycles; it is a tree. The uniqueness, existence and many such facts about the universal covering space follow from purely topological arguments; for instance look in Chapter $13$ in \\cite{MunkresTopology75} or Chapters $5$ and $6$ in \\cite{Masseyanintroduction1977}.\n\n\n\n\\section{Height Functions and Sub-Cocycles}\\label{Section:heights}\nExistence of lifts as described in the previous section enables us to measure the `rigidity' of configurations. In this section we define height functions and subsequently the slope of configurations, where steepness corresponds to this `rigidity'. The general method of using height\nfunctions is usually attributed to J.H.Conway \\cite{ThurstontilinggroupAMM}.\n\nFix a connected four-cycle free graph $\\mathcal H$. Given $x\\in X_\\mathcal H$ we can define the corresponding \\emph{height function} $h_x:\\mathbb{Z}^d\\times \\mathbb{Z}^d\\longrightarrow \\mathbb{Z}$ given by $h_x({\\vec{i}},{\\vec{j}}):=d_{E_\\mathcal H}(\\tilde{x}_{\\vec{i}},\\tilde{x}_{\\vec{j}} )$ where $\\tilde{x}$ is a lift of $x$. It follows from Corollary \\ref{corollary:lift_are_isomorphic} that $h_x$ is independent of the lift $\\tilde{x}$.\n\n\nGiven a finite subset $A\\subset \\mathbb{Z}^d$ and $x\\in X_\\mathcal H$ we define the \\emph{range of $x$ on A} as\n\\begin{equation*}\nRange_A(x):=\\max_{{\\vec{j}}_1, {\\vec{j}}_2\\in A} h_x({\\vec{j}}_1, {\\vec{j}}_2).\n\\end{equation*}\nFor all $x\\in X_\\mathcal H$\n\\begin{equation*}\nRange_A(x)\\leq Diameter(A)\n\\end{equation*}\nand more specifically\n\\begin{equation} \\label{equation:diameter_bounds_height}\nRange_{D_n}(x)\\leq 2n\n\\end{equation}\nfor all $n \\in \\mathbb N$. Since $\\tilde x\\in X_{E_\\mathcal H}$ is a map between bipartite graphs it preserves the parity of the distance function, that is, if $\\vec i, \\vec j \\in \\mathbb{Z}^d$ and $x\\in X_\\mathcal H$ then the parity of $\\|\\vec i - \\vec j\\|_1$ is the same as that of $h_x(\\vec i, \\vec j)$. As a consequence it follows that $Range_{\\partial D_n}(x)$ is even for all $x\\in X_{\\mathcal H}$ and $n \\in \\mathbb N$. We note that\n$$Range_{A}(x)= Diameter(Image(\\tilde x|_{A})).$$\n\nThe height function $h_x$ is subadditive, that is,\n$$h_x(\\vec i,\\vec j)\\leq h_x(\\vec i, \\vec k)+ h_x(\\vec k, \\vec j)$$\nfor all $x\\in X_\\mathcal H$ and $\\vec i ,\\vec j$ and $\\vec k \\in \\mathbb{Z}^d$. This is in contrast with the usual height function (as in \\cite{chandgotia2013Markov} and \\cite{peled2010high}) where there is an equality instead of the inequality. This raises some technical difficulties which are partly handled by the subadditive ergodic theorem.\n\nThe following terminology is not completely standard: Given a shift space $X$ a \\emph{sub-cocycle} is a measurable map $c: X\\times \\mathbb{Z}^d \\longrightarrow \\mathbb N\\cup \\{0\\}$ such that for all $\\vec i, \\vec j \\in \\mathbb{Z}^d$\n$$c(x, \\vec i +\\vec j)\\leq c(x, \\vec i)+ c(\\sigma^{\\vec i}(x), \\vec j).$$\nSub-cocycles arise in a variety of situations; look for instance in \\cite{Hammersleyfirst1965}. We are interested in the case $c(x, \\vec i)= h_x(\\vec 0, \\vec i)$ for all $x\\in X_\\mathcal H$ and $\\vec i \\in \\mathbb{Z}^d$. The measure of `rigidity' lies in the asymptotics of this sub-cocycle, the existence of which is provided by the subadditive ergodic theorem. Given a set $X$ if $f: X\\longrightarrow \\mathbb R$ is a function then let $f^+:=\\max(0,f)$.\n\n\\begin{thm}[Subadditive Ergodic Theorem]\\label{theorem:Subadditive_ergodic_theorem}\\cite{walters-book}\nLet $(X, \\mathcal B, \\mu)$ be a probability space and let $T: X\\longrightarrow X$ be measure preserving. Let $\\{f_n\\}_{n=1}^\\infty$ be a sequence of measurable functions $f_n: X\\longrightarrow \\mathbb R\\cup \\{-\\infty\\}$ satisfying the conditions:\n\\begin{enumerate}[(a)]\n\\item\n$f_1^+ \\in L^1(\\mu)$\n\\item\nfor each $m$, $n \\geq 1$, $f_{n+m }\\leq f_n + f_m \\circ T^n$ $\\mu$-almost everywhere.\n\\end{enumerate}\u2022\nThen there exists a measurable function $f: X\\longrightarrow \\mathbb R\\cup \\{-\\infty \\}$ such that $f^+\\in L^1(\\mu)$, $f\\circ T=f$, $\\lim_{n\\rightarrow \\infty} \\frac{1}{n}f_n =f$, $\\mu$-almost everywhere\nand\n$$\\lim_{n \\longrightarrow \\infty}\\frac{1}{n}\\int f_n d\\mu = \\inf_{n}\\frac{1}{n}\\int f_n d \\mu= \\int f d\\mu.$$\n\\end{thm}\n\nGiven a direction $\\vec{ i} =(i_1, i_2, \\ldots, i_d)\\in \\mathbb R^d$ let $\\lfloor\\vec{ i}\\rfloor=(\\lfloor i_1\\rfloor, \\lfloor i_2\\rfloor, \\ldots, \\lfloor i_d\\rfloor)$. We define for all $x \\in X_\\mathcal H$ the \\emph{slope of $x$ in the direction $\\vec{ i}$} as\n$$sl_{\\vec {i}}(x):= \\lim_{n \\longrightarrow \\infty}\\frac{1}{n} h_x(\\vec 0, \\lfloor n \\vec{ i}\\rfloor)$$\nwhenever it exists.\n\nIf $\\vec i\\in \\mathbb{Z}^d$ we note that the sequence of functions $f_n: X_\\mathcal H\\longrightarrow \\mathbb N\\cup \\{\\vec 0\\}$ given by\n$$f_n(x)=h_x(\\vec 0, n\\vec i)$$\nsatisfies the hypothesis of this theorem for any shift-invariant probability measure on $X_\\mathcal H$: $|f_1|\\leq \\|\\vec i\\|_1$ and the subadditivity condition in the theorem is just a restatement of the sub-cocycle condition described above, that is, if $T= \\sigma^{\\vec i}$ then\n$$f_{n+m }(x)= h_x(\\vec 0, (n+m)\\vec i)\\leq h_x(\\vec 0, n \\vec i)+ h_{\\sigma^{n \\vec i}x}(\\vec 0,m \\vec i ) =f_n(x) + f_m(T^n(x)).$$\nThe asymptotics of the height functions (or more generally the sub-cocycles) are a consequence of the subadditive ergodic theorem as we will describe next. In the following by an ergodic measure on $X_\\mathcal H$, we mean a probability measure on $X_\\mathcal H$ which is ergodic with respect to the $\\mathbb{Z}^d$-shift action on $X_\\mathcal H$.\n\n\n\n\\begin{prop}[Existence of Slopes]\\label{prop:existence_of_slopes}\nLet $\\mathcal H$ be a connected four-cycle free graph and $\\mu$ be an ergodic measure on $X_\\mathcal H$. Then for all $\\vec{ i}\\in \\mathbb{Z}^d$\n$$sl_{\\vec {i}}(x)=\\lim_{n \\longrightarrow \\infty}\\frac{1}{n} h_x({\\vec{0}}, n \\vec{ i})$$\nexists almost everywhere and is independent of $x$. Moreover if $\\vec {i}= (i_1, i_2\\ldots, i_d)$ then\n$$sl_{\\vec {i}}(x)\\leq \\sum_{k=1}^d |i_k| sl_{\\vec {e}_k}(x).$$\n\\end{prop}\n\\begin{proof}\nFix a direction $\\vec{ i}\\in \\mathbb{Z}^d$. Consider the sequence of functions $\\{f_n\\}_{n=1}^\\infty$ and the map $T: X_\\mathcal H\\longrightarrow X_\\mathcal H$ as described above. By the subadditive ergodic theorem there exists a function $f: X_\\mathcal H\\longrightarrow \\mathbb R\\cup \\{-\\infty\\}$ such that\n$$\\lim_{n \\rightarrow \\infty}\\frac{1}{n}f_n=f\\ almost\\ everywhere.$$\nNote that $f= sl_{\\vec{i}}$. Since for all $x\\in X_\\mathcal H$ and $n \\in\\mathbb N$, $0\\leq f_n\\leq n\\|{\\vec{i}}\\|_1$, $0\\leq f(x)\\leq \\|\\vec i\\|_1$ whenever it exists. Fix any $\\vec{j}\\in \\mathbb{Z}^d$. Then\n\\begin{eqnarray*}\nf_n(\\sigma^{\\vec{j}}(x))&=& h_{\\sigma^{\\vec{j}}(x)}({\\vec{0}}, n \\vec{i})= h_x(\\vec{j}, n \\vec{ i}+\\vec{ j})\n\\end{eqnarray*}\nand hence\n\\begin{eqnarray*}\n-h_x(\\vec{j}, {\\vec{0}}) + h_x({\\vec{0}}, n \\vec{i})- h_x(n \\vec{i}, n \\vec{i}+\\vec{j})&\\leq& f_n(\\sigma^{\\vec{j}}(x))\\\\\n&\\leq& h_x(\\vec{j},{\\vec{0}}) + h_x({\\vec{0}}, n \\vec{i})+ h_x(n \\vec{i}, n \\vec{i}+\\vec{j})\n\\end{eqnarray*}\nimplying\n\\begin{eqnarray*}\n-2\\|\\vec{j}\\|_1+ f_n(x)\\leq & f_n(\\sigma^{\\vec{j}}(x))& \\leq 2\\|\\vec{j}\\|_1+ f_n(x)\n\\end{eqnarray*}\u2022\nimplying\n$$f(x)=\\lim_{n \\longrightarrow \\infty} \\frac{1}{n} f_n(x)= \\lim_{n \\longrightarrow \\infty}\\frac{1}{n} f_n(\\sigma^{\\vec{j}} x)= f(\\sigma^{\\vec{j}}(x))$$\nalmost everywhere. Since $\\mu$ is ergodic $sl_{{\\vec{i}}}= f$ is constant almost everywhere. Let $\\vec{i}^{(k)} = (i_1, i_2, \\ldots, i_k, 0, \\ldots, 0)\\in \\mathbb{Z}^d$. By the subadditive ergodic theorem\n\\begin{eqnarray*}\nsl_{\\vec{i}}(x)= \\int sl_{\\vec{i}}(x) d\\mu&=& \\lim_{n \\longrightarrow \\infty}\\frac{1}{n}\\int h_x({\\vec{0}}, n \\vec{i}) d\\mu\\\\\n&\\leq&\\sum_{k=1}^d \\lim_{n \\longrightarrow\\infty}\\frac{1}{n} \\int h_{\\sigma^{n \\vec{i}^{(k-1)}}(x)}({\\vec{0}}, ni_{k}\\vec{e}_k ) d \\mu\\\\\n&=&\\sum_{k=1}^d \\lim_{n \\longrightarrow\\infty}\\frac{1}{n} \\int h_x({\\vec{0}}, ni_{k}\\vec{e}_k ) d \\mu\\\\\n&\\leq&\\sum_{k=1}^d|i_k|\\lim_{n \\longrightarrow\\infty}\\frac{1}{n} \\int h_x({\\vec{0}}, n\\vec{e}_k ) d \\mu\\\\\n&=&\\sum_{k=1}^d |i_k| sl_{\\vec {e}_k}(x).\n\\end{eqnarray*}\u2022\t\nalmost everywhere.\n\\end{proof}\n\\begin{corollary}\\label{corollary: existence_of _slopes_in_reality}\nLet $\\mathcal H$ be a connected four-cycle free graph. Suppose $\\mu$ is an ergodic measure on $X_\\mathcal H$. Then for all $\\vec{i}\\in \\mathbb R^d$\n$$sl_{\\vec{i}}(x)=\\lim_{n \\longrightarrow \\infty}\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{i}\\rfloor)$$\nexists almost everywhere and is independent of $x$. Moreover if $\\vec{i}= (i_1, i_2,\\ldots, i_d)$ then\n$$sl_{\\vec{i}}(x)\\leq \\sum_{k=1}^d |i_k| sl_{\\vec{e}_k}(x).$$\n\\end{corollary}\n\\begin{proof}\nLet $\\vec{i}\\in \\mathbb Q^d$ and $N\\in \\mathbb N$ such that $N \\vec{i} \\in \\mathbb{Z}^d$. For all $n \\in \\mathbb N$ there exists $k \\in \\mathbb N\\cup\\{0\\}$ and $0\\leq m\\leq N-1$ such that $n = kN+m$. Then\nfor all $x\\in X_\\mathcal H$\n$$h_x({\\vec{0}}, k N\\vec{i})- N\\|\\vec{i}\\|_1\\leq h_x({\\vec{0}}, \\lfloor n\\vec{i} \\rfloor)\\leq h_x({\\vec{0}}, k N\\vec{i})+ N\\|\\vec{i}\\|_1$$\nproving\n$$sl_{\\vec{i}}(x)=\\lim_{n\\longrightarrow\\infty}\\frac{1}{n}h_x({\\vec{0}}, \\lfloor n\\vec{ i} \\rfloor) = \\frac{1}{N}\\lim_{k \\longrightarrow \\infty} \\frac{1}{k}h_x({\\vec{0}}, k N\\vec{i}) = \\frac{1}{N}sl_{N\\vec{i}}(x)$$\nalmost everywhere. Since $sl_{N\\vec{i}}$ is constant almost everywhere, we have that $sl_{\\vec{i}}$ is constant almost everywhere as well; denote the constant by $c_{\\vec{i}}$ . Also\n$$sl_{\\vec{i}}(x)\\leq \\frac{1}{N}\\sum _{l=1}^d|N i_l|sl_{\\vec{e}_l}(x)=\\sum _{l=1}^d|i_l|sl_{\\vec{e}_l}(x).$$\n\nLet $X\\subset X_\\mathcal H$ be the set of configurations $x$ such that\n$$\\lim_{n \\longrightarrow \\infty}\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{i}\\rfloor)=c_{\\vec{i}}$$\nfor all ${\\vec{i}} \\in \\mathbb Q^d$. We have proved that $\\mu(X)=1$.\n\nFix $x\\in X$.\nLet $\\vec i, \\vec{j}\\in \\mathbb R^d$ such that $\\|\\vec{i}- \\vec{j}\\|_1<\\epsilon$. Then\n$$\\left|\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{i}\\rfloor)-\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{j}\\rfloor)\\right|\\leq\\frac{1}{n}\\|\\lfloor n \\vec{i}\\rfloor-\\lfloor n \\vec{j}\\rfloor\\|_1\\leq\\epsilon+\\frac{2d}{n}.$$\nThus we can approximate $\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{i}\\rfloor)$ for ${\\vec{i}} \\in \\mathbb R^d$ by $\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{j}\\rfloor)$ for ${\\vec{j}} \\in \\mathbb Q^d$ to prove that $\\lim_{n \\longrightarrow \\infty}\\frac{1}{n} h_x({\\vec{0}}, \\lfloor n \\vec{i}\\rfloor)$ exists for all $\\vec i \\in \\mathbb R^d$, is independent of $x\\in X$ and satisfies\n$$sl_{\\vec{i}} (x)\\leq \\sum _{k=1}^d|i_k|sl_{\\vec{e}_k}(x).$$\n\\end{proof}\n\nThe existence of slopes can be generalised from height functions to continuous sub-cocycles; the same proofs work:\n\\begin{prop}Let $c:X\\times \\mathbb{Z}^d \\longrightarrow \\mathbb R$ be a continuous sub-cocycle and $\\mu$ be an ergodic measure on $X$. Then for all $\\vec{i}\\in \\mathbb R^d$\n$$sl^c_{\\vec{i}}(x):=\\lim_{n \\longrightarrow \\infty}\\frac{1}{n} c(x, \\lfloor n \\vec{i}\\rfloor)$$\nexists almost everywhere and is independent of $x$. Moreover if $\\vec{i}= (i_1, i_2\\ldots, i_d)$ then\n$$sl^c_{\\vec{i}}(x)\\leq \\sum_{k=1}^d |i_k| sl^c_{\\vec{e}_k}(x).$$\n\\end{prop}\n\nLet $C_X$ be the space of continuous sub-cocycles on a shift space $X$. $C_X$ has a natural vector space structure: given $c_1, c_2\\in C_X$, $(c_1 +\\alpha c_2)$ is also a continuous sub-cocycle on $X$ for all $\\alpha\\in \\mathbb R$ where addition and scalar multiplication is point-wise. The following is not hard to prove and follows directly from definition.\n\\begin{prop}\\label{proposition: sub-cocycles under conjugacy}\nLet $X, Y$ be conjugate shift spaces. Then every conjugacy $f: X \\longrightarrow Y$ induces a vector-space isomorphism $ f^\\star: C_Y\\longrightarrow C_X$ given by\n$$f^\\star(c)(x, \\vec {i}):= c(f(x), \\vec{i})$$\nfor all $c\\in C_Y$, $x\\in X$ and $\\vec i \\in \\mathbb{Z}^d$. Moreover $sl^c_{\\vec i}(y)=sl^{f^\\star(c)}_{\\vec i}(f^{-1}(y))$ for all $y\\in Y$ and $\\vec i \\in \\mathbb R^d$ for which the slope $sl^c_{\\vec i}(y)$ exists.\n\\end{prop}\n\\section{Proofs of the Main Theorems} \\label{section: Proof of the main theorems}\n\\begin{proof}[Proof of Theorem \\ref{theorem: MRF fully supported }] If $\\mathcal H$ is a single edge, then $X_\\mathcal H$ is the orbit of a periodic configuration; the result follows immediately. Suppose this is not the case. The proof follows loosely the proof of Proposition \\ref{proposition: periodicfoldentropy} and morally the ideas from \\cite{lightwoodschraudnerentropy}: We prove existence of two kind of configurations in $X_\\mathcal H$, ones which are `poor' (Lemma \\ref{lemma:slope 1 is frozen}), in the sense that they are frozen and others which are `universal' (Lemma \\ref{lemma:patching_various_parts}), for which the homoclinic class is dense.\n\nIdeas for the following proof were inspired by discussions with Anthony Quas. A similar result in a special case is contained in Lemma 6.7 of \\cite{chandgotia2013Markov}.\n\n\\begin{lemma}\\label{lemma:slope 1 is frozen} Let $\\mathcal H$ be a connected four-cycle free graph and $\\mu$ be an ergodic probability measure on $X_\\mathcal H$ such that $sl_{\\vec e_k}(x)=1$ almost everywhere for some $1\\leq k \\leq d$. Then $\\mu$ is frozen and $h_\\mu=0$.\n\\end{lemma}\n\\begin{proof}\nWithout loss of generality assume that $sl_{\\vec e_1}(x)=1$ almost everywhere. By the subadditivity of the height function for all $k, n \\in \\mathbb N$ and $x\\in X_\\mathcal H$ we know that\n$$\\frac{1}{kn}h_x(\\vec 0, kn\\vec{e}_1) \\leq \\frac{1}{kn}\\sum_{m=0}^{n-1}h_x(km\\vec{e}_1, k(m+1)\\vec{e}_1)=\\frac{1}{n}\\sum_{m=0}^{n-1}\\frac{1}{k}h_{\\sigma^{km \\vec e_1} (x)}(\\vec 0, k\\vec{e}_1) \\leq 1.$$\nSince $sl_{\\vec{e}_1}(x)= 1$ almost everywhere, we get that\n$$\\lim_{n\\longrightarrow \\infty} \\frac{1}{n}\\sum_{m=0}^{n-1}\\frac{1}{k}h_{\\sigma^{km \\vec e_1} (x)}(\\vec 0, k\\vec{e}_1)=1$$\nalmost everywhere. By the ergodic theorem\n$$\\int \\frac{1}{k}h_{x}(\\vec 0, k\\vec{e}_1) d \\mu= 1.$$\nTherefore $h_{x}(\\vec 0, k\\vec{e}_1)=k$ almost everywhere which implies that\n\\begin{equation}\nh_x(\\vec i, \\vec i+k\\vec{e}_1)=k \\label{eq:slopeoneheightconstantrise}\n\\end{equation}\u2022\nfor all $\\vec i \\in \\mathbb{Z}^d$ and $k \\in \\mathbb N$ almost everywhere. Let $X\\subset supp(\\mu)$ denote the set of such configurations.\n\nFor some $n \\in \\mathbb N$ consider two patterns $a,b \\in \\mathcal L_{B_n\\cup \\partial_2 B_n}(supp(\\mu))$ such that $a|_{\\partial_2 B_n}= b|_{\\partial_2 B_n}$. We will prove that then $a|_{B_n}= b|_{B_n}$. This will prove that $\\mu$ is frozen, and $|\\mathcal L_{B_n}(supp(\\mu))|\\leq|\\mathcal L_{\\partial_2 B_n}(supp(\\mu))|\\leq |{\\mathcal A}|^{|\\partial_2 B_n|}$ implying that $h_{top}(supp(\\mu))=0$. By the variational principle this implies that $h_\\mu=0$.\n\nConsider $x, y \\in X$ such that $x|_{B_n\\cup \\partial_2 B_n}= a$ and $y|_{B_n\\cup \\partial_2 B_n}= b$. Noting that $\\partial_2 B_n$ is connected, by Corollary \\ref{corollary:covering_space_lifting_homoclinic} we can choose lifts $\\tilde x, \\tilde y\\in X_{E_\\mathcal H}$ such that $\\tilde x|_{\\partial_2 B_n}= \\tilde y|_{\\partial_2 B_n}$. Consider any $\\vec i \\in B_n$ and choose $k\\in - \\mathbb N$ such that $\\vec i + k \\vec e_1, \\vec i + (2n+2+k)\\vec e_1 \\in \\partial B_n$. Then by Equation \\ref{eq:slopeoneheightconstantrise} $d_{E_\\mathcal H}(\\tilde x_{\\vec i + k \\vec e_1}, \\tilde x_{\\vec i + (2n+2+k) \\vec e_1})= 2n+2$. But\n$$(\\tilde x_{\\vec i + k \\vec e_1},\\tilde x_{\\vec i + (k+1)\\vec e_1}, \\ldots,\\tilde x_{\\vec i + (2n+2+k) \\vec e_1} )\\text{ and }$$\n$$(\\tilde y_{\\vec i + k \\vec e_1},\\tilde y_{\\vec i + (k+1)\\vec e_1}, \\ldots,\\tilde y_{\\vec i + (2n+2+k) \\vec e_1} )$$\nare walks of length $2n+2$ from $\\tilde x_{\\vec i + k \\vec e_1}$ to $\\tilde x_{\\vec i + (2n+2+k) \\vec e_1}$. Since $E_\\mathcal H$ is a tree and the walks are of minimal length, they must be the same. Thus $\\tilde x|_{B_n}=\\tilde y|_{B_n}$. Taking the image under the map $\\pi$ we derive that\n$$a|_{B_n}=x|_{B_n}=y|_{B_n}= b|_{B_n}.$$\n\\end{proof}\n\nThis partially justifies the claim that steep slopes lead to greater `rigidity'. We are left to analyse the case where the slope is submaximal in every direction. As in the proof of Proposition 7.1 in \\cite{chandgotia2013Markov} we will now prove a certain mixing result for the shift space $X_\\mathcal H$.\n\n\\begin{lemma}\\label{lemma:patching_various_parts} Let $\\mathcal H$ be a connected four-cycle free graph and $|\\mathcal H|= r$. Consider any $x\\in X_\\mathcal H$ and some $y \\in X_\\mathcal H$ satisfying $Range_{\\partial D_{(d+1)n+3r+k}}(y)\\leq 2k$ for some $n \\in \\mathbb N$. Then\n\\begin{enumerate}\n\\item\\label{case:not_bipartite}\nIf either $\\mathcal H$ is not bipartite or $x_{\\vec 0}, y_{\\vec 0}$ are in the same partite class of $\\mathcal H$ then there exists $z\\in X_\\mathcal H$ such that\n$$z_{\\vec i}=\n\\begin{cases}\nx_{\\vec i}& \\ if \\ \\vec i \\in D_n\\\\\ny_{\\vec i} &\\ if \\ \\vec i \\in D_{(d+1)n+3r+k}^c.\n\\end{cases}$$\n\\item \\label{case:bipartite}\nIf $\\mathcal H$ is bipartite and $x_{\\vec 0}, y_{\\vec 0}$ are in different partite classes of $\\mathcal H$ then there exists $z\\in X_\\mathcal H$ such that\n$$z_{\\vec i}=\n\\begin{cases}\nx_{\\vec i+\\vec e_1} &if \\ \\vec i \\in D_n\\\\\ny_{\\vec i} & if \\ \\vec i \\in D_{(d+1)n+3r+k}^c.\n\\end{cases}$$\n\\end{enumerate}\u2022\n\\end{lemma}\nThe separation $dn+3r+k$ between the induced patterns of $x$ and $y$ is not optimal, but sufficient for our purposes.\n\\begin{proof} We will construct the configuration $z$ only in the case when $\\mathcal H$ is not bipartite. The construction in the other cases is similar; the differences will be pointed out in the course of the proof.\n\\begin{enumerate}\n\\item\n\\textbf{Boundary patterns with non-maximal range to monochromatic patterns inside.}\nLet $\\tilde y$ be a lift of $y$ and $\\mathcal T^\\prime$ be the image of $\\tilde y|_{ D_{(d+1)n+3r+k+1}}$. Let $\\mathcal T$ be a minimal subtree of $E_\\mathcal H$ such that\n$$Image(\\tilde y|_{\\partial D_{(d+1)n+3r+k}})\\subset \\mathcal T\\subset \\mathcal T^\\prime.$$\nSince $Range_{\\partial D_{(d+1)n+3r+k}}(y)\\leq 2k$, $diameter(\\mathcal T)\\leq 2k$. By Proposition \\ref{proposition:folding trees into other trees} there exists a graph homomorphism $f:\\mathcal T^\\prime \\longrightarrow \\mathcal T$ such that $f|_\\mathcal T$ is the identity. Consider the configuration $\\tilde y^1$ given by\n$$\\tilde y^1_{\\vec i}= \\begin{cases}\nf(\\tilde y_{\\vec i}) &\\text{ if }\\vec i \\in D_{(d+1)n+3r+k+1}\\\\\n\\tilde y_{\\vec i} &\\text{ otherwise.}\n\\end{cases}\u2022$$\nThe pattern\n$$\\tilde y^1|_{D_{(d+1)n+3r+k+1}}\\in \\mathcal L_{D_{(d+1)n+3r+k+1}}(X_{\\mathcal T})\\subset \\mathcal L_{D_{(d+1)n+3r+k+1}}(X_{E_\\mathcal H}).$$\nMoreover since $f|_\\mathcal T$ is the identity map,\n$$\\tilde y^1|_{D_{(d+1)n+3r+k}^c}=\\tilde y|_{D_{(d+1)n+3r+k}^c}\\in \\mathcal L_{D_{(d+1)n+3r+k}^c}(X_{E_\\mathcal H}).$$\nSince $X_{E_\\mathcal H}$ is given by nearest neighbour constraints $\\tilde y^1\\in X_{E_\\mathcal H}$.\n\nRecall that the fold-radius of a nearest neighbour shift of finite type (in our case $X_\\mathcal T$) is the total number of full config-folds required to obtain a stiff shift. Since $diameter(\\mathcal T)\\leq 2k$ the fold-radius of $X_{\\mathcal T}\\leq k$. Let a stiff shift obtained by a sequence of config-folds starting at $X_{\\mathcal T}$ be denoted by $Z$. Since $\\mathcal T$ folds into a graph consisting of a single edge, $Z$ consists of two checkerboard patterns in the vertices of an edge in $\\mathcal T$, say $\\tilde v_1$ and $\\tilde v_2$. Corresponding to such a sequence of full config-folds, we had defined in Section \\ref{section:Folding, Entropy Minimality and the Pivot Property} the outward fixing map $O_{X_\\mathcal T, (d+1)n+3r+k}$. By Proposition \\ref{prop: folding_ to _ stiffness_fixing_a_set} the configuration $O_{X_\\mathcal T,(d+1)n+3r+k}(\\tilde y^1)\\in X_{E_{\\mathcal H}}$ satisfies\n\\begin{eqnarray*}\nO_{X_\\mathcal T,(d+1)n+3r+k}(\\tilde y^1)|_{D_{(d+1)n+3r+1}}\\in \\mathcal L_{D_{(d+1)n+3r+1}}(Z) \\\\\nO_{X_\\mathcal T,(d+1)n+3r+k}(\\tilde y^1)|_{D_{(d+1)n+3r+k}^c}=\\tilde y^1|_{D_{(d+1)n+3r+k}^c}=\\tilde y|_{D_{(d+1)n+3r+k}^c}.\n\\end{eqnarray*}\n\\noindent Note that the pattern $O_{X_\\mathcal T,(d+1)n+3r+k}(\\tilde y^1)|_{\\partial D_{(d+1)n+3r}}$ uses a single symbol, say $\\tilde v_1$. Let $\\pi (\\tilde v_1)= v_1$. Then the configuration $y^\\prime= \\pi(O_{X_\\mathcal T,(d+1)n+3r+k}(\\tilde y^1))\\in X_\\mathcal H$ satisfies\n\\begin{eqnarray*}\ny^\\prime|_{\\partial D_{(d+1)n+3r}} &=& v_1\\\\\ny^\\prime|_{D_{(d+1)n+3r+k}^c}&=&y|_{D_{(d+1)n+3r+k}^c}.\n\\end{eqnarray*}\u2022\n\\item\n\\textbf{Constant extension of an admissible pattern.}\nConsider some lift $\\tilde x$ of $x$. We begin by extending $\\tilde x|_{B_n}$ to a periodic configuration $\\tilde x^1\\in X_{E_\\mathcal H}$. Consider the map $f: [-n, 3n]\\longrightarrow [-n, n]$ given by\n\\begin{equation*}\nf(k)=\\begin{cases}\nk &\\text{ if } k \\in [-n,n]\\\\\n2n-k &\\text{ if }k \\in [n,3n].\n\\end{cases}\u2022\n\\end{equation*}\n\\noindent Then we can construct the pattern $\\tilde a\\in \\mathcal L_{[-n, 3n]^d}(X_{E_\\mathcal H})$ given by\n$$\\tilde a_{i_1, i_2, \\ldots i_d}= \\tilde x_{f(i_1), f(i_2), \\ldots, f(i_d)}.$$\nGiven $k, l \\in [-n, 3n]$ if $|k-l|=1$ then $|f(k)-f(l)|=1$. Thus $\\tilde a$ is a locally allowed pattern in $X_{E_\\mathcal H}$. Moreover since $f(-n)= f(3n)$ the pattern $\\tilde a$ is `periodic', meaning,\n$$\\tilde a_{i_1, i_2,\\ldots, i_{k-1}, -n, i_{k+1}, \\ldots, i_d }= \\tilde a_{i_1, i_2, \\ldots, i_{k-1}, 3n, i_{k+1}, \\ldots, i_d }$$\nfor all $i_1, i_2, \\ldots, i_d \\in [-n,3n]$. Also $\\tilde a|_{B_n}=\\tilde x|_{B_n}$. Then the configuration $\\tilde x^1$ obtained by tiling $\\mathbb{Z}^d$ with $\\tilde a|_{[-n,3n-1]^d}$, that is,\n$$\\tilde x^1_{\\vec i}= \\tilde a_{(i_1\\!\\!\\!\\mod 4n,\\ i_2\\!\\!\\!\\mod 4n,\\ \\ldots,\\ i_d\\!\\!\\!\\mod 4n)-(n, n, \\ldots, n)}\\text{ for all }\\vec i \\in \\mathbb{Z}^d$$\nis an element of $X_{E_\\mathcal H}$. Moreover $\\tilde x^1|_{B_n}= \\tilde a|_{B_n}= \\tilde x|_{B_n}$ and $Image(\\tilde x^1)= Image(\\tilde x|_{B_n})$. Since $diameter(B_n)=2dn$, $diameter(Image(\\tilde x^1))\\leq 2dn$. Let $\\tilde \\mathcal T= Image(\\tilde x^1)$. Then the fold-radius of $X_{\\tilde \\mathcal T}$ is less than or equal to $dn$. Let a stiff shift obtained by a sequence of config-folds starting at $X_{\\tilde \\mathcal T}$ be denoted by $Z'$. Since $\\tilde \\mathcal T$ folds into a graph consisting of a single edge, $Z^\\prime$ consists of two checkerboard patterns in the vertices of an edge in $\\tilde T$, say $\\tilde w_1$ and $\\tilde w_2$. Then by Proposition \\ref{prop: folding_ to _ stiffness_fixing_a_set}\n\\begin{eqnarray*}\nI_{X_{\\tilde\\mathcal T},n}(\\tilde x^1)|_{D_n}= \\tilde x^1|_{D_n}= \\tilde x|_{D_n}\\\\\nI_{X_{\\tilde\\mathcal T},n}(\\tilde x^1)|_{D_{(d+1)n-1}^c} \\in \\mathcal L_{D_{(d+1)n-1}^c}(Z^\\prime).\n\\end{eqnarray*}\n\\noindent We note that $I_{X_{\\tilde\\mathcal T},n}(\\tilde x^1)|_{\\partial D_{(d+1)n-1}}$ consists of a single symbol, say $\\tilde w_1$. Let $\\pi(\\tilde w_1)= w_1$. Then the configuration $x^\\prime=\\pi(I_{X_{\\tilde\\mathcal T},n}(\\tilde x^1)) \\in X_{\\mathcal H}$ satisfies\n\\begin{eqnarray*}\nx^\\prime|_{D_n}= x|_{D_n}\\text{ and}\\\\\nx^\\prime|_{\\partial D_{(d+1)n-1}}=w_1.\n\\end{eqnarray*}\u2022\n\\item \\textbf{Patching of an arbitrary pattern inside a configuration with non-maximal range.}\nWe will first prove that there exists a walk on $\\mathcal H$ from $w_1$ to $v_1$, $((w_1= u_1), u_2, \\ldots, (u_{3r+2}= v_1))$.\nSince the graph is not bipartite, it has a cycle $p_1$ such that $|p_1|\\leq r-1$ and is odd. Let $v^\\prime$ be a vertex in $p_1$. Then there exist walks $p_2$ and $p_3$ from $w_1$ to $v^\\prime$ and from $v^\\prime$ to $v_1$ respectively such that $|p_2|, |p_3|\\leq r-1$. Consider any vertex $w^\\prime\\sim_\\mathcal H v_1$. If\n$3r+1-|p_2|- |p_3|$ is even then the walk\n$$p_2\\star p_3 (\\star (v_1, w^\\prime, v_1))^{\\frac{3r+1-|p_2|- |p_3|}{2}}$$\nand if not, then the walk\n$$p_2\\star p_1 \\star p_3 (\\star (v_1, w^\\prime, v_1))^{\\frac{3r+1-|p_1|-|p_2|- |p_3|}{2}}$$\nis a walk of length $3r+1$ in $\\mathcal H$ from $w_1$ to $v_1$. This is the only place where we use the fact that $\\mathcal H$ is not bipartite. If it were bipartite, then we would require that $x^\\prime_{\\vec 0}$ and $y^\\prime_{\\vec 0}$ have to be in the same partite class to construct such a walk.\n\nGiven such a walk the configuration $z$ given by\n\\begin{eqnarray*}\nz|_{D_{(d+1)n}}&=& x^\\prime|_{D_{(d+1)n}}\\\\\nz|_{D^c_{(d+1)n +3r}}&= &y^\\prime|_{D^c_{(d+1)n +3r}}\\\\\nz|_{\\partial D_{(d+1)n+i-2}}&=& u_i\\text{ for all } 1\\leq i \\leq 3r+2\n\\end{eqnarray*}\n\\noindent is an element of $X_\\mathcal H$ for which $z|_{D_n}=x^\\prime|_{D_n}=x|_{D_n}$ and $z|_{D_{(d+1)n+3r+k}^c}=y^\\prime|_{D_{(d+1)n+3r+k}^c}=y|_{D_{(d+1)n+3r+k}^c}.$\n\\end{enumerate}\u2022\n\n\\end{proof}\n\nWe now return to the proof of Theorem \\ref{theorem: MRF fully supported }. Let $\\mu$ be an ergodic probability measure adapted to $X_\\mathcal H$ with positive entropy.\n\nSuppose $sl_{\\vec e_i}(x)= \\theta_i$ almost everywhere. By Lemma \\ref{lemma:slope 1 is frozen}, $\\theta_i<1$ for all $1\\leq i \\leq d$. Let $\\theta= \\max_i \\theta_i$ and $0<\\epsilon<\\frac{1}{4}\\left(1- \\theta\\right)$. Denote by $S^{d-1}$, the sphere of radius $1$ in $\\mathbb R^d$ for the $l^1$ norm. By Corollary \\ref{corollary: existence_of _slopes_in_reality} for all $\\vec{v}\\in S^{d-1}$\n$$\\lim_{n\\longrightarrow \\infty }\\frac{1}{n}h_x({\\vec{0}}, \\lfloor n \\vec{v}\\rfloor) \\leq \\theta$$\nalmost everywhere. Since $S^{d-1}$ is compact in $\\mathbb R^d$ we can choose a finite set $\\{\\vec{v}_1, \\vec{v}_2, \\ldots, \\vec{v}_t\\} \\subset S^{d-1}$ such that for all $\\vec{v}\\in S^{d-1}$ there exists some $1\\leq i\\leq t$ satisfying $\\|\\vec{v}_i -\\vec v\\|_1<\\epsilon$. By Egoroff's theorem \\cite{Follandreal1999} given $\\epsilon$ as above there exists $N_0\\in \\mathbb N$ such that for all $n\\geq N_0$ and $1\\leq i \\leq t$\n\\begin{equation}\n\\mu(\\{x\\in X_\\mathcal H\\:|\\:h_x({\\vec{0}}, \\lfloor n \\vec{v}_i\\rfloor)\\leq n\\theta + n\\epsilon\\ for\\ all\\ 1\\leq i\\leq t\\}) >1-\\epsilon.\\label{equation:uniform_continuity_of_heights}\n\\end{equation}\u2022\nLet $\\vec{v} \\in \\partial D_{n-1}$ and $1\\leq i_0\\leq t$ such that\n$\\|\\frac{1}{n}\\vec{v}-\\vec{v}_{i_0}\\|_1<\\epsilon$. If for some $x\\in X_\\mathcal H$ and $n \\in \\mathbb N$\n$$h_x({\\vec{0}}, \\lfloor n \\vec{v}_{i_0}\\rfloor)\\leq n\\theta + n\\epsilon$$\nthen\n$$h_x({\\vec{0}}, \\lfloor \\vec{v}\\rfloor)\\leq h_x({\\vec{0}}, \\lfloor n \\vec{v}_{i_0}\\rfloor) +\\lceil n \\epsilon \\rceil \\leq n\\theta + 2n\\epsilon+1.$$\nBy Inequality \\ref{equation:uniform_continuity_of_heights} we get\n$$\\mu\\left(\\{x\\in X_\\mathcal H\\:|\\: h_x\\left({\\vec{0}}, \\lfloor \\vec{v}\\rfloor\\right)\\leq n\\theta + 2n\\epsilon+1\\ for\\ all\\ \\vec{v}\\in \\partial D_{n-1}\\}\\right) >1-\\epsilon$$\nfor all $n\\geq N_0$. Therefore for all $n\\geq N_0$ there exists $x^{(n)}\\in supp(\\mu) $ such that\n$$Range_{\\partial D_{n-1}}\\left(x^{(n)}\\right)\\leq 2n\\theta + 4 n \\epsilon +2< 2n(1- \\epsilon)+2.$$\nLet $x \\in X_\\mathcal H$ and $n_0\\in \\mathbb N$. It is sufficient to prove that $\\mu([x]_{D_{n_0-1}})>0$. Suppose $r:=|\\mathcal H|$. Choose $k \\in \\mathbb N$ such that\n\\begin{eqnarray*}\nn_0(d+1)+3r+k+1&\\geq&N_0\\\\\n2\\left(n_0(d+1)+3r+k+1\\right)(1-\\epsilon)+2&\\leq&2k.\n\\end{eqnarray*}\u2022\nThen by Lemma \\ref{lemma:patching_various_parts} there exists $z\\in X_\\mathcal H$ such that either\n\\begin{equation*}\nz_{\\vec{j}}=\n\\begin{cases}\nx_{\\vec{j}} \\quad\\quad\\quad\\quad \\quad\\quad\\: &if \\ {\\vec{j}} \\in D_{n_0}\\\\\nx^{\\left(n_0(d+1)+3r+k+1\\right)}_{\\vec{j}} \\ &if \\ {\\vec{j}} \\in D_{n_0(d+1)+3r+k}^c\n\\end{cases}\n\\end{equation*}\u2022\nor\n\\begin{equation*}\nz_{\\vec{j}}=\n\\begin{cases}\nx_{{\\vec{j}} +\\vec e_1}\\quad\\quad\\quad\\quad \\quad\\quad \\:& if \\ {\\vec{j}} \\in D_{n_0}\\\\\nx^{\\left(n_0(d+1)+3r+k+1\\right)}_{\\vec{j}} \\ &if \\ {\\vec{j}} \\in D_{n_0(d+1)+3r+k}^c.\n\\end{cases}\n\\end{equation*}\u2022\nIn either case $(z, x^{\\left(n_0(d+1)+3r+k+1\\right)})\\in \\Delta_{X_\\mathcal H}$. Since $\\mu$ is adapted to $X_\\mathcal H$, $z\\in supp(\\mu)$. In the first case we get that $\\mu([x]_{D_{n_0-1}})=\\mu([z]_{D_{n_0-1}})>0$. In the second case we get that $$\\mu([x]_{D_{n_0-1}})=\\mu(\\sigma^{\\vec e_1}([x]_{D_{n_0-1}}))=\\mu([z]_{D_{(n_0-1)}-\\vec e_1})>0.$$\nThis completes the proof.\n\n\\end{proof}\n\nEvery shift space conjugate to an entropy minimal shift space is entropy minimal. However a shift space $X$ which is conjugate to $X_\\mathcal H$ for $\\mathcal H$ which is connected and four-cycle free need not even be a hom-shift. By following the proof carefully it is possible to extract a condition for entropy minimality which is conjugacy-invariant:\n\n\\begin{thm}\\label{theorem:conjugacy_invariant_entropy minimality condition}\nLet $X$ be a shift of finite type and $c$ a continuous sub-cocycle on $X$ with the property that $c(\\cdot, {\\vec{i}})\\leq \\|{\\vec{i}}\\|_1$ for all ${\\vec{i}} \\in \\mathbb{Z}^d$. Suppose every ergodic probability measure $\\mu$ adapted to $X$ satisfies:\n\\begin{enumerate}\n\\item\nIf $sl^c_{\\vec e_i}(x)=1$ almost everywhere for some $1\\leq i \\leq d$ then $h_\\mu< h_{top}(X)$.\n\\item\nIf $sl^c_{\\vec e_i}(x)<1$ almost everywhere for all $1\\leq i \\leq d$ then $supp(\\mu)=X$.\n\\end{enumerate}\nthen $X$ is entropy minimal.\n\\end{thm}\n\nHere is a sketch: By Proposition \\ref{proposition:entropyviamme} and Theorems \\ref{thm:equiGibbs}, \\ref{theorem: ergodic decomposition of markov random fields} it is sufficient to prove that every ergodic measure of maximal entropy is fully supported. If $X$ is a shift of finite type satisfying the hypothesis of Theorem \\ref{theorem:conjugacy_invariant_entropy minimality condition} then it is entropy minimal because every ergodic measure of maximal entropy of $X$ is an ergodic probability measure adapted to $X$; its entropy is either smaller than $h_{top}(X)$ or it is fully supported. To see why the condition is conjugacy invariant suppose that $f:X\\longrightarrow Y$ is a conjugacy and $c\\in C_Y$ satisfies the hypothesis of the theorem. Then by Proposition \\ref{proposition: sub-cocycles under conjugacy} it follows that ${f^\\star}(c)\\in C_X$ satisfies the hypothesis as well.\n\n\\begin{proof}[Proof of Theorem \\ref{theorem: pivot property for four cycle free}] By Proposition \\ref{proposition: pivot for disconnected} we can assume that $\\mathcal H$ is connected. Consider some $(x, y)\\in \\Delta_{X_\\mathcal H}$. By Corollary \\ref{corollary:covering_space_lifting_homoclinic} there exist $(\\tilde x, \\tilde y)\\in \\Delta_{X_{E_\\mathcal H}}$ such that $\\pi(\\tilde x)= x$ and $\\pi(\\tilde y)=y$. It is sufficient to prove that there is a chain of pivots from $\\tilde x$ to $\\tilde y$. We will proceed by induction on $\\sum_{{\\vec{i}} \\in \\mathbb{Z}^d} d_{E_\\mathcal H}(\\tilde x_{{\\vec{i}}}, \\tilde y_{\\vec{i}})$. The induction hypothesis (on $M$) is : If $\\sum_{{\\vec{i}} \\in \\mathbb{Z}^d} d_{E_\\mathcal H}(\\tilde x_{\\vec{i}}, \\tilde y_{\\vec{i}})= 2M$ then there exists a chain of pivots from $\\tilde x$ to $\\tilde y$.\n\nWe note that $d_{E_\\mathcal H}(\\tilde x_{\\vec{i}}, \\tilde y_{\\vec{i}})$ is even for all ${\\vec{i}} \\in \\mathbb{Z}^d$ since there exists ${\\vec{i}}^\\prime\\in \\mathbb{Z}^d$ such that $\\tilde x_{{\\vec{i}}^\\prime}= \\tilde y_{{\\vec{i}}^{\\prime}}$ and hence $\\tilde x_{\\vec{i}}$ and $\\tilde y_{\\vec{i}}$ are in the same partite class of $E_\\mathcal H$ for all ${\\vec{i}} \\in \\mathbb{Z}^d$.\n\nThe base case $(M=1)$ occurs exactly when $\\tilde x$ and $\\tilde y$ differ at a single site; there is nothing to prove in this case. Assume the hypothesis for some $M\\in \\mathbb N$.\n\nConsider $(\\tilde x, \\tilde y)\\in \\Delta_{X_{E_\\mathcal H}}$ such that\n$$\\sum_{{\\vec{i}} \\in \\mathbb{Z}^d} d_{E_\\mathcal H}(\\tilde x_{\\vec{i}}, \\tilde y_{\\vec{i}})=2M+2.$$\nLet\n$$B=\\{{\\vec{j}} \\in \\mathbb{Z}^d\\:|\\: \\tilde x_{\\vec{j}} \\neq \\tilde y_{\\vec{j}}\\}$$\nand a vertex $\\tilde v\\in E_\\mathcal H$. Without loss of generality we can assume that\n\\begin{equation}\n\\max_{{\\vec{i}} \\in B} d_{E_\\mathcal H}(\\tilde v, \\tilde x_{\\vec{i}})\\geq \\max_{{\\vec{i}} \\in B} d_{E_\\mathcal H}(\\tilde v, \\tilde y_{\\vec{i}}).\\label{equation:assumption_for_pivot}\n\\end{equation}\u2022\nConsider some ${\\vec{i}}_0 \\in B$ such that\n$$d_{E_\\mathcal H}(\\tilde v, \\tilde x_{{\\vec{i}}_0})= \\max_{{\\vec{i}} \\in B}d_{E_\\mathcal H}(\\tilde v, \\tilde x_{{\\vec{i}}}).$$\nConsider the shortest walks $(\\tilde v= \\tilde v_1, \\tilde v_2, \\ldots, \\tilde v_n=\\tilde x_{{\\vec{i}}_0})$ from $\\tilde v$ to $\\tilde x_{{\\vec{i}}_0}$ and $(\\tilde v= \\tilde v^\\prime_1, \\tilde v^\\prime_2, \\ldots, \\tilde v^\\prime_{n^\\prime}=\\tilde y_{{\\vec{i}}_0})$ from $\\tilde v$ to $\\tilde y_{{\\vec{i}}_0}$. By Assumption \\ref{equation:assumption_for_pivot}, $n^\\prime\\leq n$. Since these are the shortest walks on a tree, if $\\tilde v^\\prime_k=\\tilde v_{k^\\prime}$ for some $1\\leq k\\leq n^\\prime$ and $1\\leq k^{\\prime} \\leq n$ then $k =k^{\\prime}$ and $\\tilde v_l = \\tilde v_l^\\prime$ for $1\\leq l \\leq k$. Let\n$$k_0= \\max\\{1\\leq k\\leq n^\\prime\\:|\\: \\tilde v_k^\\prime= \\tilde v_k\\}.$$\nThen the shortest walk from $\\tilde x_{{\\vec{i}}_0}$ to $\\tilde y_{{\\vec{i}}_0}$ is given by $\\tilde x_{{\\vec{i}}_0}=\\tilde v_n, \\tilde v_{n-1}, \\tilde v_{n-2}, \\ldots, \\tilde v_{k_0}, \\tilde v^\\prime_{k_0+1}, \\ldots, \\tilde v^\\prime_{n^\\prime}= \\tilde y_{{\\vec{i}}_0}$.\n\nWe will prove for all $\\vec i \\sim \\vec i_{0}$, $\\tilde x_{\\vec i}= \\tilde v_{n-1}$. This is sufficient to complete the proof since then the configuration\n$$\\tilde x^{(1)}_{{\\vec{j}}}=\n\\begin{cases}\n\\tilde x_{\\vec{j}} \\quad\\ \\:if\\ {\\vec{j}} \\neq {\\vec{i}}_0\\\\\n\\tilde v_{n-2}\\ \\ if\\ {\\vec{j}} = {\\vec{i}}_0,\n\\end{cases}$$\n\\noindent is an element of $X_{E_\\mathcal H}$, $(\\tilde x,\\tilde x^{(1)})$ is a pivot and\n$$n+n^\\prime -2 k_0 -2=d_{E_\\mathcal H}{(\\tilde x^{(1)}_{{\\vec{i}}_0}, \\tilde y_{{\\vec{i}}_0})}< d_{E_\\mathcal H}(\\tilde x_{{\\vec{i}}_0}, \\tilde y_{{\\vec{i}}_0})=n+n^\\prime -2 k_0$$\n\\noindent giving us a pair $(\\tilde x^{(1)}, \\tilde y)$ such that\n$$\\sum_{{\\vec{i}} \\in \\mathbb{Z}^d} d_{E_\\mathcal H}(\\tilde x^{(1)}_{\\vec{i}}, \\tilde y_{\\vec{i}})=\\sum_{{\\vec{i}} \\in \\mathbb{Z}^d} d_{E_\\mathcal H}(\\tilde x_{\\vec{i}}, \\tilde y_{\\vec{i}})-2= 2M$$\n\nto which the induction hypothesis applies. There are two possible cases:\n\\begin{enumerate}\n\\item\n${\\vec{i}} \\in B$: Then $d_{E_\\mathcal H}(\\tilde v, \\tilde x_{\\vec i})=d_{E_\\mathcal H}(\\tilde v, \\tilde x_{\\vec i_0})-1$ and $\\tilde x_{\\vec{i}}\\sim_{E_\\mathcal H} \\tilde x_{\\vec i_0}$. Since $E_\\mathcal H$ is a tree, $\\tilde x_{\\vec{i}}= \\tilde v_{n-1}$.\n\n\n\\item ${\\vec{i}} \\notin B$: Then $\\tilde x_{\\vec{i}}= \\tilde y_{\\vec{i}}$ and we get that $d_{E_\\mathcal H}(\\tilde x_{{\\vec{i}}_0}, \\tilde y_{{\\vec{i}}_0})=2$. Since $\\tilde x_{\\vec{i}}\\sim_{E_\\mathcal H} \\tilde x_{{\\vec{i}}_0}$, the shortest walk joining $\\tilde v$ and $\\tilde x_{{\\vec{i}}}$ must either be $\\tilde v= \\tilde v_1, \\tilde v_2, \\ldots, \\tilde v_{n-1}= \\tilde x_{{\\vec{i}}}$ or $\\tilde v= \\tilde v_1, \\tilde v_2,\\ldots,\\tilde v_{n}= \\tilde x_{{\\vec{i}}_0}, \\tilde v_{n+1}= \\tilde x_{{\\vec{i}}}$. We want to prove that the former is true. Suppose not.\n\nSince $\\tilde y_{{\\vec{i}}_0}\\sim_{E_\\mathcal H} \\tilde x_{{\\vec{i}}}$ and ${\\vec{i}}_0\\in B$, the shortest walk from $\\tilde v$ to $\\tilde y_{{\\vec{i}}_0}$ is $\\tilde v= \\tilde v_1, \\tilde v_2,\\ldots,\\tilde v_{n}= \\tilde x_{{\\vec{i}}_0}, \\tilde v_{n+1}= \\tilde x_{{\\vec{i}}}, \\tilde v_{n+2}=\\tilde y_{{\\vec{i}}_0} $. This contradicts Assumption \\ref{equation:assumption_for_pivot} and completes the proof.\n\n\\end{enumerate}\n\n\\end{proof}\n\n\n\n\\section{Further Directions}\n\\subsection{Getting Rid of the Four-Cycle Free Condition}\n\nIn the context of the results in this paper, the four-cycle free condition seems a priori artificial; we feel that in many cases it is a mere artifact of the proof. To the author, getting rid of this condition is an important and interesting topic for future research. Here we will illustrate what goes wrong when we try to apply our proofs for the simplest possible example with four-cycles, that is, $C_4$.\n\n\nWe have shown (Example \\ref{Example: Folds to an edge}) that $X_{C_4}$ satisfies the hypothesis of Propositions \\ref{proposition: periodicfoldentropy} and \\ref{proposition: frozenfoldpivot} and thus it also satisfies the conclusions of Theorems \\ref{theorem: MRF fully supported } and \\ref{theorem: pivot property for four cycle free}. The proofs of Theorems \\ref{theorem: MRF fully supported } and \\ref{theorem: pivot property for four cycle free} however rely critically on the existence of lifts to the universal cover, that is, Proposition \\ref{proposition:covering_space_lifting}. However the conclusion of this proposition does not hold for $X_{C_4}$: The universal cover of $C_4$ is $\\mathbb{Z}$ and the corresponding covering map $\\pi: \\mathbb{Z} \\longrightarrow C_4$ is given by $\\pi(i)= i \\mod 4$. By the second remark following Theorem 4.1 in \\cite{chandgotia2013Markov} it follows that the induced map $\\pi: X_{\\mathbb{Z}}\\longrightarrow X_{C_4}$ is not surjective disproving the conclusion of Proposition \\ref{proposition:covering_space_lifting} for $X_{C_4}$.\n\n\\subsection{Identification of Hom-Shifts}\n\\noindent\\textbf{Question 1:} Given a shift space $X$, are there some nice decidable conditions which imply that $X$ is conjugate to a hom-shift?\n\nBeing conjugate to a hom-shift lays many restrictions on the shift space, for instance on its periodic configurations. Consider a conjugacy $f:X\\longrightarrow X_\\mathcal H$ where $\\mathcal H$ is a finite undirected graph. Let $Z\\subset X_\\mathcal H$ be the set of configurations invariant under $\\{\\sigma^{2\\vec e_i}\\}_{i=1}^d$. Then there is a bijection between $Z$ and $\\mathcal L_A(X_\\mathcal H)$ where $A$ is the rectangular shape\n$$A:=\\{\\sum_{i=1}^{d}\\delta_i \\vec e_i\\:|\\: \\delta_i\\in \\{0,1\\}\\}$$\nbecause every pattern in $\\mathcal L_A(X_\\mathcal H)$ extends to a unique configuration in $Z$. More generally given a graph $\\mathcal H$ it is not hard to compute the number of periodic configurations for a specific finite-index subgroup of $\\mathbb{Z}^d$. Moreover periodic points are dense in these shift spaces and there are algorithms to compute approximating upper and lower bounds of their entropy \\cite{symmtricfriedlan1997,louidor2010improved}. Hence the same then has to hold for the shift space $X$ as well. We are not familiar with nice decidable conditions which imply that a shift space is conjugate to a hom-shift.\n\n\\subsection{Hom-Shifts and Strong Irreducibility}\\label{subsection: homSI}\n\n\\noindent\\textbf{Question 2:} Which hom-shifts are strongly irreducible?\n\nWe know two such conditions:\n\\begin{enumerate}\n\\item\\cite{brightwell2000gibbs}\nIf $\\mathcal H$ is a finite graph which folds into $\\mathcal H^\\prime$ then $X_\\mathcal H$ is strongly irreducible if and only if $X_{\\mathcal H'}$ is strongly irreducible. This reduces the problem to graphs $\\mathcal H$ which are stiff. For instance if $\\mathcal H$ is dismantlable, then $X_\\mathcal H$ is strongly irreducible.\n\\item\\cite{Raimundo2014}\n$X_\\mathcal H$ is single site fillable. A shift space $X_{\\mathcal F}\\subset {\\mathcal A}^{\\mathbb{Z}^d}$ is said to be \\emph{single site fillable} if for all patterns $a\\in {\\mathcal A}^{\\partial\\{\\vec 0\\}}$ there exists a locally allowed pattern in $X_{\\mathcal F}$, $b\\in {\\mathcal A}^{D_1}$ such that\n$b|_{\\partial\\{\\vec 0\\}}=a$. In case $X_{\\mathcal F}= X_\\mathcal H$ for some graph $\\mathcal H$ then it is single site fillable if and only if given vertices $v_1, v_2, \\ldots, v_{2d}\\in \\mathcal H$ there exists a vertex $v\\in \\mathcal H$ adjacent to all of them.\n\\end{enumerate}\nIt follows that $X_{K_5}$ is single site fillable and hence strongly irreducible for $d=2$. In fact strong irreducibility has been proved in \\cite {Raimundo2014} for shifts of finite type with a weaker mixing condition called TSSM. This does not cover all possible examples. For instance it was proved in \\cite{Raimundo2014} that $X_{K_4}$ is strongly irreducible for $d=2$ even though it is not TSSM and $K_4$ is stiff. We do not know if it is possible to verify whether a given hom-shift is TSSM.\n\n\\subsection{Hom-Shifts and Entropy Minimality}\\textbf{Question 3:} Given a finite connected graph $\\mathcal H$ when is $X_\\mathcal H$ entropy minimal?\n\n\n\nWe have provided some examples in the paper:\n\\begin{enumerate}\n\\item\n$\\mathcal H$ can be folded to a single vertex with a loop or a single edge. (Proposition \\ref{proposition: periodicfoldentropy})\n\\item $\\mathcal H$ is four-cycle free. (Theorem \\ref{theorem:four cycle free entropy minimal})\n\\end{enumerate}\u2022\nAgain this does not provide the full picture. For instance $X_{K_4}$ is strongly irreducible when $d=2$ and hence entropy minimal even though $K_4$ is stiff and not four-cycle free.\nA possible approach might be via identifying the right sub-cocycle and Theorem \\ref{theorem:conjugacy_invariant_entropy minimality condition}.\n\n\\noindent\\textbf{Conjecture:} Let $d=2$ and $\\mathcal H$ be a finite connected graph. Then $X_\\mathcal H$ is entropy minimal.\n\n\n\\subsection{Hom-Shifts and the Pivot Property}\\label{subsection: Hom-shifts and the pivot property}\nWe have given a list of examples of graphs $\\mathcal H$ for which the shift space $X_\\mathcal H$ has the pivot property in Section \\ref{section: the pivot property}. In this paper we have provided two further sets of examples:\n\\begin{enumerate}\n\\item\n$\\mathcal H$ can be folded to a single vertex with a loop or a single edge. (Proposition \\ref{proposition: frozenfoldpivot})\n\\item\n$\\mathcal H$ is four-cycle free. (Theorem \\ref{theorem: pivot property for four cycle free})\n\\end{enumerate}\n\nWe saw in Section \\ref{section: the pivot property} that $X_{K_4}, X_{K_5}$ do not have the pivot property when $d=2$. However they do satisfy a weaker property which we will describe next.\n\nA shift space $X$ is said to have the \\emph{generalised pivot property} if there is an $r\\in \\mathbb N$ such that for all $(x,y)\\in \\Delta_X$ there exists a chain $x^1=x, x^2, x^3, \\ldots, y=x^n\\in X$ such that $x^i$ and $x^{i+1}$ differ at most on some translate of $D_r$.\n\nIt can be shown that any nearest neighbour shift of finite type $X\\subset {\\mathcal A}^\\mathbb{Z}$ has the generalised pivot property. In higher dimensions this is not true without any hypothesis; look for instance in Section 9 in \\cite{chandgotia2013Markov}. It is not hard to prove that any single site fillable nearest neighbour shift of finite type has the generalised pivot property. This can be generalised further: in \\cite{Raimundo2014} it is proven that every shift space satisfying TSSM has the generalised pivot property.\n\n\\noindent\\textbf{Question 4:} For which graphs $\\mathcal H$ does $X_\\mathcal H$ satisfy the pivot property? What about the generalised pivot property?\n\n\\section{Acknowledgments}\nI would like to thank my advisor, Prof. Brian Marcus for dedicated reading of a million versions of this paper, numerous suggestions, insightful discussions and many other things. The line of thought in this paper was begot in discussions with Prof. Tom Meyerovitch, his suggestions and remarks have been very valuable to me. I will also like to thank Prof. Ronnie Pavlov, Prof. Sam Lightwood, Prof. Michael Schraudner, Prof. Anthony Quas, Prof. Klaus Schmidt, Prof. Mahan Mj, Prof. Peter Winkler and Raimundo Brice\\~no for giving a patient ear to my ideas and many useful suggestions. Lastly, I will like to thank Prof. Jishnu Biswas; he had introduced me to universal covers, more generally to the wonderful world of algebraic topology. This research was partly funded by the Four-Year Fellowship at the University of British Columbia. Lastly I would like to thank the anonymous referee for giving many helpful comments and corrections largely improving the quality of the paper.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introduction}\nNeural network based approaches have become popular frameworks in many machine learning research fields, showing its advantages over traditional methods. In NLP tasks, two types of neural networks are widely used: Recurrent Neural Network (RNN) and Convolutional Neural Network (CNN). \n\nRNNs are powerful models in various NLP tasks, such as machine translation \\citep{cho-etal-2014}, sentiment classification \\citep{wang-and-tian-2016,liu-emnlp-2016,wang-etal-2016,zhang-etal-2016,liang-etal-2016}, reading comprehension \\citep{kadlec-etal-2016,dhingra-etal-2016,sordoni-etal-2016,cui-etal-2016,cui-etal-2017-aoa,yang-etal-2016}, etc. \nThe recurrent neural networks can flexibly model different lengths of sequences into a fixed representation. \nThere are two main implementations of RNN: Long Short-Term Memory (LSTM) \\citep{hochreiter-1997} and Gated Recurrent Unit (GRU) \\citep{cho-etal-2014}, which solve the gradient vanishing problems in vanilla RNNs. \n\nCompared to RNN, the CNN model also shows competitive performances in some tasks, such as text classification \\citep{kim-2014}, etc.\nHowever, different from RNN, CNN sets a pre-defined convolutional kernel to ``summarize'' a fixed window of adjacent elements into blended representations, showing its ability of modeling local context.\n\nAs both global and local information is important in most of NLP tasks \\citep{luong-etal-2015}, in this paper, we propose a novel recurrent unit, called Contextual Recurrent Unit (CRU). The proposed CRU model adopts advantages of RNN and CNN, where CNN is good at modeling local context, and RNN is superior in capturing long-term dependencies. We propose three variants of our CRU model: {\\em shallow fusion}, {\\em deep fusion} and {\\em deep-enhanced fusion}. \n\nTo verify the effectiveness of our CRU model, we utilize it into two different NLP tasks: sentiment classification and reading comprehension, where the former is sentence-level modeling, and the latter is document-level modeling. \nIn the sentiment classification task, we build a standard neural network and replace the recurrent unit by our CRU model.\nTo further demonstrate the effectiveness of our model, we also tested our CRU in reading comprehension tasks with a strengthened baseline system originated from Attention-over-Attention Reader (AoA Reader) \\citep{cui-etal-2017-aoa}.\nExperimental results on public datasets show that our CRU model could substantially outperform various systems by a large margin, and set up new state-of-the-art performances on related datasets.\nThe main contributions of our work are listed as follows.\n\\begin{itemize}[leftmargin=*]\n\t\\item We propose a novel neural recurrent unit called Contextual Recurrent Unit (CRU), which effectively incorporate the advantage of CNN and RNN. Different from previous works, our CRU model shows its excellent flexibility as GRU and provides better performance.\n\t\\item The CRU model is applied to both sentence-level and document-level modeling tasks and gives state-of-the-art performances.\n\t\\item The CRU could also give substantial improvements in cloze-style reading comprehension task when the baseline system is strengthened by incorporating additional features which will enrich the representations of unknown words and make the texts more readable to the machine.\n\\end{itemize}\n\n\\section{Related Works}\\label{related-work}\n\nGated recurrent unit (GRU) has been proposed in the scenario of neural machine translations \\citep{cho-etal-2014}. It has been shown that the GRU has comparable performance in some tasks compared to the LSTM. Another advantage of GRU is that it has a simpler neural architecture than LSTM, showing a much efficient computation.\n\nHowever, convolutional neural network (CNN) is not as popular as RNNs in NLP tasks, as the texts are formed temporally. But in some studies, CNN shows competitive performance to the RNN models, such as text classification \\citep{kim-2014}.\n\nVarious efforts have been made on combining CNN and RNN.\n\\citet{wang-etal-2016} proposed an architecture that combines CNN and GRU model with pre-trained word embeddings by word2vec. \n\\citet{liang-etal-2016} proposed to combine asymmetric convolution neural network with the bidirectional LSTM network. \n\\citet{zhang-etal-2016} presented Dependency Sensitive CNN, which hierarchically construct text by using LSTMs and extracting features with convolution operations subsequently. \n\\citet{cai-etal-2016} propose to make use of dependency relations information in the shortest dependency path (SDP) by combining CNN and two-channel LSTM units. \n\\citet{kim-etal-2016} build a neural network for dialogue topic tracking where the CNN used to account for semantics at individual utterance and RNN for modeling conversational contexts along multiple turns in history.\n\n\nThe difference between our CRU model and previous works can be concluded as follows.\n\\begin{itemize}[leftmargin=*]\n  \\item Our CRU model could adaptively control the amount of information that flows into different gates, which was not studied in previous works.\n  \\item Also, the CRU does not introduce a pooling operation, as opposed to other works, such as CNN-GRU \\citep{wang-etal-2016}. Our motivation is to provide flexibility as the original GRU, while the pooling operation breaks this law (the output length is changed), and it is unable to do exact word-level attention over the output. However, in our CRU model, the output length is the same as the input's and can be easily applied to various tasks where the GRU used to. \n  \\item We also observed that by only using CNN to conclude contextual information is not strong enough. So we incorporate the original word embeddings to form a \"word + context\" representation for enhancement.\n\\end{itemize}\n\n\n\\section{Our approach}\\label{cru}\nIn this section, we will give a detailed introduction to our CRU model.\nFirstly, we will give a brief introduction to GRU \\citep{cho-etal-2014} as preliminaries, and then three variants of our CRU model will be illustrated.\n\n\\subsection{Gated Recurrent Unit}\nGated Recurrent Unit (GRU) is a type of recurrent unit that models sequential data \\citep{cho-etal-2014}, which is similar to LSTM but is much simpler and computationally effective than the latter one. We will briefly introduce the formulation of GRU.\nGiven a sequence $x = \\{x_1, x_2, ..., x_n\\}$, GRU will process the data in the following ways. For simplicity, the bias term is omitted in the following equations.\n\\begin{gather}\nz_t = \\sigma(W_z x_t+U_z h_{t-1}) \\\\\nr_t = \\sigma(W_r x_t+U_r h_{t-1}) \\\\\n\\widetilde{h_t} = \\tanh(W x_t+U [r_t \\odot h_{t-1}]) \\\\\nh_t = z_t  h_{t-1} + (1-z_t) \\widetilde{h_t}\n\\end{gather}\n\nwhere $z_t$ is the update gate, $r_t$ is the reset gate, and non-linear function $\\sigma$ is often chosen as $sigmoid$ function.\nIn many NLP tasks, we often use a bi-directional GRU, which takes both forward and backward information into account.\n\n\n\\subsection{Contextual Recurrent Unit}\nBy only modeling word-level representation may have drawbacks in representing the word that has different meanings when the context varies. \nHere is an example that shows this problem.\n\n\\begin{quote}\n\\begin{scriptsize}\\begin{verbatim}\nThere are many fan mails in the mailbox. \nThere are many fan makers in the factory.\n\\end{verbatim}\\end{scriptsize}\n\\end{quote}\n\nAs we can see that, though two sentences share the same beginning before the word {\\em fan}, the meanings of the word {\\em fan} itself are totally different when we meet the following word {\\em mails} and {\\em makers}. The first {\\em fan} means ``a person that has strong interests in a person or thing\", and the second one means ``a machine with rotating blades for ventilation\".\nHowever, the embedding of word {\\em fan} does not discriminate according to the context. \nAlso, as two sentences have the same beginning, when we apply a recurrent operation (such as GRU) till the word {\\em fan}, the output of GRU does not change, though they have entirely different meanings when we see the following words.\n\nTo enrich the word representation with local contextual information and diminishing the word ambiguities, we propose a model as an extension to the GRU, called Contextual Recurrent Unit (CRU).\nIn this model, we take full advantage of the convolutional neural network and recurrent neural network, where the former is good at modeling local information, and the latter is capable of capturing long-term dependencies.\nMoreover, in the experiment part, we will also show that our bidirectional CRU could also significantly outperform the bidirectional GRU model.\n\nIn this paper, we propose three different types of CRU models: {\\em shallow fusion}, {\\em deep fusion} and {\\em deep-enhanced fusion},  from the most fundamental one to the most expressive one. \nWe will describe these models in detail in the following sections.\n\n\\subsubsection{Shallow Fusion}\nThe most simple one is to directly apply a CNN layer after the embedding layer to obtain blended contextual representations. Then a GRU layer is applied afterward. We call this model as {\\em shallow fusion}, because the CNN and RNN are applied linearly without changing inner architectures of both. \n\nFormally, when given a sequential data $x = \\{x_1, x_2, ..., x_n\\}$, a shallow fusion of CRU can be illustrated as follows.\n\\begin{gather} \ne_t = W_e \\cdot x_t ~~;~~ c_t = \\phi(\\widetilde{e_t}) \\\\\nh_t = GRU(h_{t-1}, c_t) \n\\end{gather}\n\nWe first transform word $x_t$ into word embeddings through an embedding matrix $W_e$.\nThen a convolutional operation $\\phi$ is applied to the context of $e_t$, denoted as $\\widetilde{e_t}$, to obtain contextual representations.\nFinally, the contextual representation $c_t$ is fed into GRU units.\n\nFollowing \\cite{kim-2014}, we apply embedding-wise convolution operation, which is commonly used in natural language processing tasks.\nLet $e_{i:j} \\in\\mathbb{R}^{\\mathcal \\\\j*d}$ denote the concatenation of $j-i+1$ consecutive $d$-dimensional word embeddings.\n\\begin{equation} e_{i:j} = concat[e_i, e_{i+1}, ..., e_j] \\end{equation}\n\nThe embedding-wise convolution is to apply a convolution filter {\\bf w} $\\in\\mathbb{R}^{\\mathcal \\\\k*d}$  to a window of $k$ word embeddings to generate a new feature, i.e., summarizing a local context of $k$ words. \nThis can be formulated as\n\\begin{equation} c_i = f({\\bf w} \\cdot e_{i:i+k-1} + b) \\end{equation}\nwhere $f$ is a non-linear function and $b$ is the bias.\n\nBy applying the convolutional filter to all possible windows in the sentence, a feature map $c$ will be generated.\nIn this paper, we apply a {\\em same-length} convolution (length of the sentence does not change), i.e. $c \\in\\mathbb{R}^{\\mathcal \\\\n*1}$.\nThen we apply $d$ filters with the same window size to obtain multiple feature maps.\nSo the final output of CNN has the shape of $C \\in\\mathbb{R}^{\\mathcal \\\\n*d}$, which is exactly the same size as $n$ word embeddings, which enables us to do exact word-level attention in various tasks.\n\n\\subsubsection{Deep Fusion}\nThe contextual information that flows into the update gate and reset gate of GRU is identical in shallow fusion.\nIn order to let the model adaptively control the amount of information that flows into these gates, we can embed CNN into GRU in a deep manner. We can rewrite the Equation 1 to 3 of GRU as follows.\n\\begin{gather}\nz_t = \\sigma(\\phi_z(\\widetilde{e_t})) + U_z h_{t-1}) \\\\\nr_t = \\sigma(\\phi_r(\\widetilde{e_t})) + U_r h_{t-1}) \\\\\n\\widetilde{h_t} = \\tanh(\\phi(\\widetilde{e_t}))+U [r_t \\odot h_{t-1}]) \n\\end{gather}\n\nwhere $\\phi_z, \\phi_r, \\phi$ are three different CNN layers, i.e., the weights are not shared.\nWhen the weights share across these CNNs, the deep fusion will be degraded to shallow fusion.\n\n\\subsubsection{Deep-Enhanced Fusion}\nIn shallow fusion and deep fusion, we used the convolutional operation to summarize the context.\nHowever, one drawback of them is that the original word embedding might be blurred by blending the words around it, i.e., applying the convolutional operation on its context. \n\nFor better modeling the original word and its context, we enhanced the deep fusion model with original word embedding information, with an intuition of ``enriching word representation with contextual information while preserving its basic meaning''.\nFigure \\ref{deep-e-example} illustrates our motivations.\n\n\\begin{figure}[tp]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{example2.pdf}\n  \\caption{\\label{deep-e-example} An intuitive illustration of variants of the CRU model. The gray scale represents the amount of information. a) original sentence; b) original representation of word ``shortcut''; c) applying convolutional filter (length=3); d) adding original word embedding; }\n\\end{figure}\n\nFormally, the Equation 9 to 11 can be further rewritten into\n\\begin{gather}\nz_t = \\sigma(W_z(\\phi_z(\\widetilde{e_t}) + e_t) + U_z h_{t-1} \\\\\nr_t = \\sigma(W_r(\\phi_r(\\widetilde{e_t}) + e_t) + U_r h_{t-1}) \\\\\n\\widetilde{h_t} = \\tanh(W(\\phi(\\widetilde{e_t}) + e_t)+U [r_t \\odot h_{t-1}])\n\\end{gather}\n\nwhere we add original word embedding $e_t$ after the CNN operation, to ``enhance'' the original word information while not losing the contextual information that has learned from CNNs.\n\n\n\\section{Applications}\\label{application}\nThe proposed CRU model is a general neural recurrent unit, so we could apply it to various NLP tasks.\nAs we wonder whether the CRU model could give improvements in both sentence-level modeling and document-level modeling tasks, in this paper, we applied the CRU model to two NLP tasks: sentiment classification and cloze-style reading comprehension.\nIn the sentiment classification task, we build a simple neural model and applied our CRU.\nIn the cloze-style reading comprehension task, we first present some modifications to a recent reading comprehension model, called AoA Reader \\cite{cui-etal-2017-aoa}, and then replace the GRU part by our CRU model to see if our model could give substantial improvements over strong baselines.\n\n\\subsection{Sentiment Classification}\\label{sentiment-classification}\nIn the sentiment classification task, we aim to classify movie reviews, where one movie review will be classified into the positive\/negative or subjective\/objective category.\nA general neural network architecture for this task is depicted in Figure \\ref{sc-arch}.\n\n\\begin{figure}[tp]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{sc-arch2.pdf}\n  \\caption{\\label{sc-arch} A general neural network architecture of sentiment classification task.}\n\\end{figure}\n\nFirst, the movie review is transformed into word embeddings.\nAnd then, a sequence modeling module is applied, in which we can adopt LSTM, GRU, or our CRU, to capture the inner relations of the text.\nIn this paper, we adopt bidirectional recurrent units for modeling sentences, and then the final hidden outputs are concatenated.\nAfter that, a fully connected layer will be added after sequence modeling.\nFinally, the binary decision is made through a single $sigmoid$ unit.\n\nAs shown, we employed a straightforward neural architecture to this task, as we purely want to compare our CRU model against other sequential models.\nThe detailed experimental result of sentiment classification will be given in the next section.\n\n\n\\subsection{Reading Comprehension}\\label{rc-task}\nBesides the sentiment classification task, we also tried our CRU model in cloze-style reading comprehension, which is a much complicated task. \nIn this paper, we strengthened the recent AoA Reader \\cite{cui-etal-2017-aoa} and applied our CRU model to see if we could obtain substantial improvements when the baseline is strengthened.\n\n\\subsubsection{Task Description}\nThe cloze-style reading comprehension is a fundamental task that explores relations between the document and the query.\nFormally, a general cloze-style query can be illustrated as a triple $\\langle {\\mathcal D}, {\\mathcal Q}, {\\mathcal A} \\rangle$, where $\\mathcal D$ is the document, $\\mathcal Q$ is the query and the answer $\\mathcal A$. \nNote that the answer is a {\\em single} word in the document, which requires us to exploit the relationship between the document and query.\n\n\n\\subsubsection{Modified AoA Reader}\nIn this section, we briefly introduce the original AoA Reader \\cite{cui-etal-2017-aoa}, and illustrate our modifications.\nWhen a cloze-style training triple $\\langle \\mathcal D, \\mathcal Q, \\mathcal A \\rangle$ is given, the Modified AoA Reader will be constructed in the following steps.\nFirst, the document and query will be transformed into continuous representations with the embedding layer and recurrent layer.\nThe recurrent layer can be the simple RNN, GRU, LSTM, or our CRU model.\n\nTo further strengthen the representation power, we show a simple modification in the embedding layer, where we found strong empirical results in performance.\nThe main idea is to utilize additional sparse features of the word and add (concatenate) these features to the word embeddings to enrich the word representations. The additional features have shown effective in various models \\cite{dhingra-etal-2016,pengli-etal-2016,yang-etal-2016}.\nIn this paper, we adopt two additional features in document word embeddings (no features applied to the query side).\n\n\\noindent{{$\\bullet$~~ \\bf Document word frequency}}: Calculate each document word frequency. This helps the model to pay more attention to the important (more mentioned) part of the document.\n\\begin{equation} freq(d) = \\frac{word\\_count(d)}{length(\\mathcal D)}, d\\in \\mathcal D \\end{equation}\n\n\\noindent{{$\\bullet$~~ \\bf Count of query word}}: Count the number of each document word appeared in the query. For example, if a document word appears three times in the query, then the feature value will be 3. We empirically find that instead of using binary features (appear=1, otherwise=0) \\cite{pengli-etal-2016}, indicating the count of the word provides more information, suggesting that the more a word occurs in the query, the less possible the answer it will be.\nWe replace the Equation 16 with the following formulation (query side is not changed), \n\\begin{equation}  \\small e(x) = concat[W_e \\cdot x, freq(x), CoQ(x)] , x\\in \\mathcal D \\end{equation}\nwhere $freq(x)$ and $CoQ(x)$ are the features that introduced above. \n\\begin{gather}\n{\\small \\overrightarrow{h_s(x)}} =  {\\small \\overrightarrow{RNN}(e(x)) ; \\overleftarrow{h_s(x)} = \\overleftarrow{RNN}(e(x))} \\\\\nh_s(x) = [\\overrightarrow{h_s(x)}; \\overleftarrow{h_s(x)}]\n\\end{gather}\n\nOther parts of the model remain the same as the original AoA Reader. For simplicity, we will omit this part, and the detailed illustrations can be found in \\citet{cui-etal-2017-aoa}.\n\n\\section{Experiments: Sentiment Classification}\\label{experiments-sc}\n\n\\subsection{Experimental Setups}\nIn the sentiment classification task, we tried our model on the following public datasets.\n\\begin{itemize}[leftmargin=*]\n  \\item {\\bf MR}\\footnote{\\url{http:\/\/www.cs.cornell.edu\/People\/pabo\/movie-review-data\/}} Movie reviews with one sentence each. Each review is classified into positive or negative \\cite{pang-and-lee-2005}.\n  \\item {\\bf IMDB}\\footnote{\\url{http:\/\/ai.stanford.edu\/~amaas\/data\/sentiment\/}} Movie reviews from IMDB website, where each movie review is labeled with binary classes, either positive or negative \\cite{maas-etal-2011}. Note that each movie review may contain several sentences.\n  \\item {\\bf SUBJ}$^1$ Movie review labeled with subjective or objective \\cite{pang-and-lee-2004}. \n\\end{itemize}\n\nThe statistics and hyper-parameter settings of these datasets are listed in Table \\ref{imdb-stats}.\n \n        \\begin{table}[htp]\n        \\begin{center}\n       \n        \\begin{tabular}{lccc}\n        \\toprule\n        & \\bf MR & \\bf IMDB & \\bf SUBJ \\\\\n        \\midrule\n        Train \\# & 10,662 & 25,000 & 10,000 \\\\\n       \n        Test \\# & 10-CV & 25,000 & 10-CV \\\\\n        \\midrule\n        Embed. size & 200 & 256 & 200 \\\\\n        Hidden size & 200 & 256 & 200 \\\\\n        Dropout & 0.3 & 0.3 & 0.4 \\\\\n        Pre-train Embed. & GloVe & - & GloVe \\\\\n        Initial LR & 0.0005 & 0.001 & 0.0005 \\\\\n        Vocab truncation & - & 50,000 & - \\\\\n        \\bottomrule\n        \\end{tabular}\n        \\end{center}\n        \\caption{\\label{imdb-stats} Statistics and hyper-parameter settings of MR, IMDB and SUBJ datasets. 10-CV represents 10-fold cross validation.}\n        \\end{table} \n\n        \nAs these datasets are quite small and overfit easily, we employed $l_2$-regularization of 0.0001 to the embedding layer in all datasets. \nAlso, we applied dropout \\cite{srivastava-etal-2014} to the output of the embedding layer and fully connected layer.\nThe fully connected layer has a dimension of 1024.\nIn the MR and SUBJ, the embedding layer is initialized with 200-dimensional GloVe embeddings (trained on 840B token) \\cite{pennington-etal-2014} and fine-tuned during the training process.\nIn the IMDB condition, the vocabulary is truncated by descending word frequency order.\nWe adopt batched training strategy of 32 samples with ADAM optimizer \\cite{kingma2014adam}, and clipped gradient to 5 \\cite{pascanu-etal-2013}.\nUnless indicated, the convolutional filter length is set to 3, and ReLU for the non-linear function of CNN in all experiments.\nWe use 10-fold cross-validation (CV) in the dataset that has no train\/valid\/test division.     \n\n\\subsection{Results}\\label{result-sc}\n\nThe experimental results are shown in Table \\ref{exp-class}.\nAs we mentioned before, all RNNs in these models are {\\bf bi-directional}, because we wonder if our bi-CRU could still give substantial improvements over bi-GRU which could capture both history and future information.\nAs we can see that, all variants of our CRU model could give substantial improvements over the traditional GRU model, where a maximum gain of 2.7\\%, 1.0\\%, and 1.9\\% can be observed in three datasets, respectively.\nWe also found that though we adopt a straightforward classification model, our CRU model could outperform the state-of-the-art systems by 0.6\\%, 0.7\\%, and 0.8\\% gains respectively, which demonstrate its effectiveness. \nBy employing more sophisticated architecture or introducing task-specific features, we think there is still much room for further improvements, which is beyond the scope of this paper.\n\n\n       \\begin{table}[t]\n        \\begin{center}\n        \\small\n        \\begin{tabular}{lccc}\n        \\toprule\n        \\bf System & \\bf MR & \\bf IMDB & \\bf SUBJ \\\\\n        \\midrule\n       \n        Multi-channel CNN & 81.1 & - & 93.2 \\\\\n       \n        HRL  & - & 90.9 & - \\\\\n        Multi-task arc-II  & - & {\\em 91.2} & {\\em 95.0} \\\\\n        CNN-GRU-wordvec  & 82.3 & - & - \\\\\n        DSCNN-Pretrain  & 82.2 & 90.7 & 93.9 \\\\\n        LR-Bi-LSTM & 82.1 & - & - \\\\\n        AC-BLSTM & 83.1& - & 94.2 \\\\\n        G-AC-BLSTM & {\\em 83.7} & -& 94.3 \\\\\n        \\midrule\\midrule\n        GRU & 81.0 & 90.9 & 93.9 \\\\\n        CRU (shallow fusion) & 82.1 & 91.3 & 95.0 \\\\\n        CRU (deep fusion) & 82.7 & 91.5 & 95.2 \\\\\n        CRU (deep-enhanced, filter=3) & {\\bf 83.7} & {\\bf 91.9} & {\\bf 95.8} \\\\\n        CRU (deep-enhanced, filter=5) & 83.2 & 91.7 & 95.2 \\\\\n        \\bottomrule\n        \\end{tabular}\n        \\end{center}\n        \\caption{\\label{exp-class} Results on MR, IMDB and SUBJ sentiment classification task. Best previous results are marked in italics, and overall best results are mark in bold face. {\\small {\\bf Multi-channel CNN} \\cite{kim-2014}: A CNN architecture with static and non-static word embeddings. {\\bf HRL} \\cite{wang-and-tian-2016}: A hybrid residual LSTM architecture. {\\bf Multi-task arc-II} \\cite{liu-emnlp-2016}: A deep architectures with shared local-global hybrid memory for multi-task learning. {\\bf CNN-GRU-word2vec} \\cite{wang-etal-2016}: An architecture that combines CNN and GRU model with pre-trained word embeddings by {\\em word2vec}. {\\bf DSCNN-Pretrain} \\cite{zhang-etal-2016}: Dependency sensitive convolutional neural networks with pretrained sequence autoencoders. {\\bf AC-BLSTM} \\cite{liang-etal-2016}: Asymmetric convolutional bidirectional LSTM networks. } }\n        \\end{table}\n        \nWhen comparing three variants of the CRU model, as we expected, the CRU with {\\em deep-enhanced fusion} performs best among them. This demonstrates that by incorporating contextual representations with original word embedding could enhance the representation power.\nAlso, we noticed that when we tried a larger window size of the convolutional filter, i.e., 5 in this experiment, does not give a rise in the performance. \nWe plot the trends of MR test set accuracy with the increasing convolutional filter length, as shown in Figure \\ref{mr-length}.\n\n    \\begin{figure}[t]\n      \\centering\n      \\includegraphics[width=0.45\\textwidth]{mr-length.pdf}\n      \\caption{\\label{mr-length} Trends of MR test set accuracy with the increasing convolutional filter length. }\n    \\end{figure}\n    \nAs we can see that, using a smaller convolutional filter does not provide much contextual information, thus giving a lower accuracy.\nOn the contrary, the larger filters generally outperform the lower ones, but not always.\nOne possible reason for this is that when the filter becomes larger, the amortized contextual information is less than a smaller filter, and make it harder for the model to learn the contextual information. However, we think the proper size of the convolutional filter may vary task by task. Some tasks that require long-span contextual information may benefit from a larger filter.\n    \nWe also compared our CRU model with related works that combine CNN and RNN \\cite{wang-etal-2016,zhang-etal-2016,liang-etal-2016}. \nFrom the results, we can see that our CRU model significantly outperforms previous works, which demonstrates that by employing {\\em deep fusion} and enhancing the contextual representations with original embeddings could substantially improve the power of word representations.\n    \nOn another aspect, we plot the trends of IMDB test set accuracy during the training process, as depicted in Figure \\ref{imdb-train}.\nAs we can see that, after iterating six epochs of training data, all variants of CRU models show faster convergence speed and smaller performance fluctuation than the traditional GRU model, which demonstrates that the proposed CRU model has better training stability.\n\n   \\begin{figure}[tb]\n      \\centering\n      \\includegraphics[width=0.45\\textwidth]{imdb-4.pdf}\n      \\caption{\\label{imdb-train} Trends of IMDB test set accuracy with the training time growing.}\n    \\end{figure}\n\n\\section{Experiments: Reading Comprehension}\\label{experiments-rc}        \n    \n    \\begin{table*}[ht]\n    \\begin{center}\n   \n    \\begin{tabular}{p{9cm} ccccc}\n    \\toprule\n    & \\multicolumn{2}{p{2cm}}{\\centering \\bf CBT NE} & \\multicolumn{2}{p{2cm}}{\\centering \\bf CBT CN} \\\\\n    & \\bf Valid & \\bf Test & \\bf Valid & \\bf Test\\\\\n    \\midrule\n    Human \\cite{hill-etal-2015} & - & {\\em 81.6} & - & {\\em 81.6} \\\\\n    MemNN \\cite{hill-etal-2015} & 70.4 & 66.6 & 64.2 & 63.0 \\\\ \n    AS Reader \\cite{kadlec-etal-2016} & 73.8 & 68.6 & 68.8 & 63.4 \\\\\n    GA Reader \\cite{dhingra-etal-2016} & 74.9 & 69.0 & 69.0 & 63.9 \\\\\n   \n    Iterative Attention \\cite{sordoni-etal-2016} & 75.2 & 68.6 & 72.1 & 69.2 \\\\\n    AoA Reader \\cite{cui-etal-2017-aoa} & 77.8 & 72.0 & 72.2 & 69.4 \\\\\n    NSE Adp. Com. \\cite{munkhdalai2016reasoning} & 78.2 & 73.2 & 74.2 & 71.4 \\\\\n    GA Reader + Fine-gating \\cite{yang-etal-2016} & 79.1 & {\\em 75.0} & 75.3 & 72.0  \\\\\n    AoA Reader + Re-ranking \\cite{cui-etal-2017-aoa} & {\\em 79.6} & 74.0 & {\\em 75.7} & {\\em 73.1} \\\\\n    \\midrule\n    M-AoA Reader (GRU) & 78.0 & 73.8 & 72.8 & 69.8 \\\\\n    M-AoA Reader (CRU) & 79.5 & 75.4 & 74.4 & 71.3 \\\\\n    M-AoA Reader (CRU) + Re-ranking & {\\bf 80.6} & {\\bf 76.1} & {\\bf 76.6} & {\\bf 74.5} \\\\\n    \\midrule\\midrule\n    AS Reader (Ensemble) & 74.5 & 70.6 & 71.1 & 68.9 \\\\\n    KnReader (Ensemble) & 78.0 & 73.3 & 72.2 & 70.6 \\\\\n    Iterative Attention (Ensemble) & 76.9 & 72.0 & 74.1 & 71.0 \\\\\n    AoA Reader (Ensemble) & 78.9 & 74.5 & 74.7 & 70.8 \\\\\n    AoA Reader (Ensemble + Re-ranking) & {\\em 80.3} & {\\em 75.7} & {\\em 77.0} & {\\em 74.1} \\\\\n    \\midrule\n    M-AoA Reader (CRU) (Ensemble) & 80.0 & 77.1 & 77.0 & 73.5 \\\\   \n    M-AoA Reader (CRU) (Ensemble + Re-ranking) & {\\bf 81.8} & {\\bf 77.5} & {\\bf 79.0} & {\\bf 76.8} \\\\        \n    \\bottomrule\n    \\end{tabular}\n    \\end{center}\n    \\caption{\\label{public-result} Results on the CBT NE and CN cloze-style reading comprehension datasets.\n     }\n    \\end{table*}\n\n\\subsection{Experimental Setups}   \nWe also tested our CRU model in the cloze-style reading comprehension task.\nWe carried out experiments on the public datasets: CBT NE\/CN \\cite{hill-etal-2015}.\nThe CRU model used in these experiments is the {\\em deep-enhanced} type with the convolutional filter length of 3.\nIn the re-ranking step, we also utilized three features: Global LM, Local LM, Word-class LM, as proposed by \\citet{cui-etal-2017-aoa}, and all LMs are 8-gram trained by SRILM toolkit \\cite{stolcke-2002}.\nFor other settings, such as hyperparameters, initializations, etc., we closely follow the experimental setups as \\citet{cui-etal-2017-aoa} to make the experiments more comparable.\n\n\n\\subsection{Results}\nThe overall experimental results are given in Table \\ref{public-result}. \nAs we can see that our proposed models can substantially outperform various state-of-the-art systems by a large margin.\n\n\\begin{itemize}[leftmargin=*]\n  \\item Overall, our final model (M-AoA Reader + CRU + Re-ranking) could give significant improvements over the previous state-of-the-art systems by 2.1\\% and 1.4\\% in test sets, while re-ranking and ensemble bring further improvements.\n  \\item When comparing M-AoA Reader to the original AoA Reader, 1.8\\% and 0.4\\% improvements can be observed, suggesting that by incorporating additional features into embedding can enrich the power of word representation. Incorporating more additional features in the word embeddings would have another boost in the results, but we leave this in future work.\n  \\item Replacing GRU with our CRU could significantly improve the performance, where 1.6\\% and 1.5\\% gains can be obtained when compared to M-AoA Reader. This demonstrates that incorporating contextual information when modeling the sentence could enrich the representations. Also, when modeling an unknown word, except for its randomly initialized word embedding, the contextual information could give a possible guess of the unknown word, making the text more readable to the neural networks.\n\\item The re-ranking strategy is an effective approach in this task. We observed that the gains in the common noun category are significantly greater than the named entity. One possible reason is that the language model is much beneficial to CN than NE, because it is much more likely to meet a new named entity that is not covered in the training data than the common noun.\n\\end{itemize}\n\n\\section{Qualitative Analysis}\\label{qualitative-analysis}        \nIn this section, we will give a qualitative analysis on our proposed CRU model in the sentiment classification task.\nWe focus on two categories of the movie reviews, which is quite harder for the model to judge the correct sentiment. The first one is the movie review that contains negation terms, such as ``not''. The second type is the one contains sentiment transition, such as ``clever but not compelling''. We manually select 50 samples of each category in the MR dataset, forming a total of 100 samples to see if our CRU model is superior in handling these movie reviews. The results are shown in Table \\ref{quality-result-table}. As we can see that, our CRU model is better at both categories of movie review classification, demonstrating its effectiveness.\n\n        \\begin{table}[h]\n        \\begin{center}\n       \n        \\begin{tabular}{lcc}\n        \\toprule\n        & \\bf GRU & \\bf CRU \\\\\n        \\midrule\n        Negation Term (50) & 37  & 42 \\\\\n        Sentiment Transition (50) & 34  & 40 \\\\\n        \\midrule\n        Total (100) & 71  & 82 \\\\\n        \\bottomrule\n        \\end{tabular}\n        \\end{center}\n        \\caption{\\label{quality-result-table} Number of correctly classified samples.}\n        \\end{table}\n\nAmong these samples, we select an intuitive example that the CRU successfully captures the true meaning of the sentence and gives the correct sentiment label. We segment a full movie review into three sentences, which is shown in Table \\ref{quality-table}.\n\n        \\begin{table}[htbp]\n        \\begin{center}\n        \\small\n        \\begin{tabular}{lcc}\n        \\toprule\n        \\bf Sentence & \\bf GRU & \\bf CRU \\\\\n        \\midrule\n        I like that Smith & POS & POS \\\\\n        \\midrule\n        I like that Smith, \\\\ he's {\\em not making fun of} these people, & POS & POS \\\\\n        \\midrule\n        I like that Smith, \\\\ he's {\\em not making fun of} these people, \\\\he's {\\em not laughing at} them. & {\\em NEG} & POS \\\\\n        \\bottomrule\n        \\end{tabular}\n        \\end{center}\n        \\caption{\\label{quality-table} Predictions of each level of the sentence.  }\n        \\end{table}\n\nRegarding the first and second sentence, both models give correct sentiment prediction. While introducing the third sentence, the GRU baseline model failed to recognize this review as a positive sentiment because there are many negation terms in the sentence. However, our CRU model could capture the local context {\\em during} the recurrent modeling the sentence, and the phrases such as ``not making fun'' and ``not laughing at'' could be correctly noted as positive sentiment which will correct the sentiment category of the full review, suggesting that our model is superior at modeling local context and gives much accurate meaning.\n\n\\section{Conclusion}\\label{conclusion}\nIn this paper, we proposed an effective recurrent model for modeling sequences, called Contextual Recurrent Units (CRU). \nWe inject the CNN into GRU, which aims to better model the local context information via CNN before recurrently modeling the sequence. \nWe have tested our CRU model on the cloze-style reading comprehension task and sentiment classification task.\nExperimental results show that our model could give substantial improvements over various state-of-the-art systems and set up new records on the respective public datasets. In the future, we plan to investigate convolutional filters that have dynamic lengths to adaptively capture the possible spans of its context.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}