diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzijvo" "b/data_all_eng_slimpj/shuffled/split2/finalzzijvo"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzijvo"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nThe test set problem is NP-hard. The polynomial time approximation\nalgorithms using in practice includes \"greedy\" heuristics\nimplemented by set cover criterion or by information\ncriterion\\cite{ms}. Test set can not be approximated within\n$(1-\\varepsilon)\\ln n$ for any $\\varepsilon>0$ unless $NP\\subseteq\nDTIME(n^{\\log\\log n})$\\cite{bh,bdk}. Recently, the authors of\n\\cite{bdk} designed a new information type greedy algorithm,\ninformation content heuristic (ICH for short), and proved its\nperformance guarantee $\\ln n+1$, which almost matches the\ninapproximability results.\n\nThe setcover greedy algorithm (SGA for short) is a natural\napproximation algorithm for test set. In practice, its average\nperformance is virtually the same as information type greedy\nalgorithms\\cite{ms,dk}. The performance guarantee $2\\ln n$ of SGA is\nobtained by transforming the test set problem as a set cover\nproblem. The authors of \\cite{bh} give the tight performance\nguarantee $11\/8$ of SGA on instances with the size of tests no\ngreater than 2.\n\nOblivious rounding, a derandomization technique to obtain simple\ngreedy algorithm for set cover problems by conditional probabilities\nwas introduced in \\cite{y}. Young observed the number of elements\nuncovered is an \"potential function\" and the approximation algorithm\nonly need to drive down the potential function at each step, thus he\nshowed another proof of the well-known performance guarantee $\\ln\nn+1$.\n\nIn this paper, the author presents a tighter analysis of SGA. The author uses\nthe potential function technique of \\cite{y} to improve the performance\nguarantee $2\\ln n$ which derives from set cover problem to $1.1354\\ln n$, and\nconstruct instances to give a nontrivial lower bound $1.0004609\\ln n$ of the\nperformance guarantee. The latter result confirms the fact ICH is slightly\nbetter than SGA in worst case. In this analysis, the author refers to the tight\nanalysis of the greedy algorithm for set cover problem in \\cite{s}.\n\nIn Section 2, the author shows the two main theorems, and some definitions,\nnotations and facts are given. In Section 3, the author analyzes\ndifferentiation distribution of item pairs and uses the potential function\nmethod to prove the improved performance guarantee. In Section 4, the author\nshows the nontrivial lower bound by constructing instances of test set with\narbitrary large size. Section 5 is some discussions.\n\n\\section{Overview}\nThe input of test set problem consists of $S$, a set of items (called\nuniverse), and $\\mathcal T$, a collection of subsets (called tests) of $S$. An\nitem pair $\\{i,j\\}$ is a subset of $S$ containing two different items of $S$. A\ntest $T$ differentiates item pair $\\{i,j\\}$ if $|T\\cap \\{i,j\\}|=1$. $\\mathcal\nT$ is a test set of $S$, i.e. any item pair of $S$ is differentiated by one\ntest in $\\mathcal T$. The objective is to find $\\mathcal T'\\subseteq \\mathcal\nT$ with minimum cardinality which is also a test set of $S$. We use $\\mathcal\nT^*$ to represent the optimal test set. Denote $n=|S|$, and $m^*=|\\mathcal\nT^*|$. In this paper, we assume $m^*\\ge 2$.\n\nAmong an instance of test set problem, there are $n\\choose 2$ different item\npairs. Let $i,j$ be two different items, and $S_{1},S_{2}$ be two disjoint\nsubsets of $S$. If $i,j\\in S_{1}$, we say $\\{i,j\\}$ is an item pair inside\n$S_{1}$, and if $i\\in S_{1}$ and $i\\in S_{2}$, we say $\\{i,j\\}$ is an item pair\nbetween $S_{1}$ and $S_{2}$.\n\nLet $S'$ be a subset of $S$, $\\mathcal T$ be a collection of tests of $S$, we\nsay $\\mathcal T$ is a test set of $S'$ iff any item pair inside $S'$ is\ndifferentiated by one test in $\\mathcal T$. Notice tests in $\\mathcal T$ may\ncontains items that are not in $S'$. Clearly, if $\\mathcal T$ is a test set of\n$S$, $\\mathcal T$ is a test set of $S'$.\n\nWe use $\\{i,j\\}\\perp T$ to represent that $T$ differentiates $\\{i,j\\}$ and\n$\\{i,j\\}\\parallel T$ to represent that $T$ does not differentiate $\\{i,j\\}$. We\nuse $\\{i,j\\}\\perp \\mathcal T$ to represent that at least one test in $\\mathcal\nT$ differentiates $\\{i,j\\}$, $\\{i,j\\}\\parallel \\mathcal T$ to represent that\nany test in $\\mathcal T$ does not differentiate $\\{i,j\\}$, and $\\perp\n(\\{i,j\\},\\mathcal T)$ to represent the number of tests in $\\mathcal T$ that\ndifferentiate $\\{i,j\\}$.\n\n{\\bf Fact 1.} \\it For three different items $i$, $j$ and $k$, if\n$\\{i,j\\}\\parallel \\mathcal T$ and $\\{i,k\\}\\parallel \\mathcal T$,\nthen $\\{j,k\\}\\parallel \\mathcal T$.\\rm\n\n{\\bf Fact 2.} \\it For three different items $i$, $j$ and $k$, and a test $T$,\nif $\\{i,j\\}\\perp T$ and $\\{i,k\\}\\perp T$, then $\\{j,k\\}\\parallel T$.\\rm\n\nGiven $\\mathcal T'\\subseteq \\mathcal T$, we define a binary relation\n$\\backsim_{\\mathcal T'}$ on $S$: for two item $i,j$,\n$i\\backsim_{\\mathcal T'}j$ iff $\\{i,j\\}\\parallel \\mathcal T'$. By\nFact 1, $\\backsim_{\\mathcal T'}$ is an equivalent relation. The\nequivalent classes containing $i$ is denoted as $[i]$.\n\n{\\bf Fact 3.} \\it If $\\mathcal T$ is a minimal test set, then\n$|\\mathcal T|\\le n-1$.\\rm\n\n{\\bf Fact 4.} \\it If $\\mathcal T$ is a test set, then $|\\mathcal\nT|\\ge \\log_{2} n$.\\rm\n\nTest set $\\mathcal T$ with $|\\mathcal T|=\\log_{2} n$ is\ncalled a compact test set. If $\\mathcal T$ is a compact test set,\nthen $|S|=2^q, q\\in Z^+$.\n\nIn set cover problem, we are given $U$, the universe, and $\\mathcal C$, a\ncollection of subsets of $U$. $\\mathcal C$ is a set cover of $U$, i.e.\n$\\bigcup_{c\\in \\mathcal C}=U$. The objective is to find $\\mathcal C'\\subseteq\n\\mathcal C$ with minimum cardinality which is also a set cover of $S$.\n\nThe greedy algorithm for set cover runs like that. In each\niteration, simply select a subset covering most uncovered elements,\nrepeat until all elements are covered, and return the set of\nselected subsets. Let $N$ be the size of the universe, and $M^*$ be\nthe size of the optimal set cover. The greedy algorithm for set\ncover has performance guarantee $\\ln N-\\ln\\ln N+\\Theta(1)$ by\n\\cite{s}.\n\nWe give two lemmas about the greedy algorithm for set cover. Lemma 1\nis a corollary of Lemma 2 in \\cite{s} and Lemma 2 is a corollary of\nLemma 1 and Lemma 4 in \\cite{s}.\n\n{\\bf Lemma 1.} \\it The size of set cover\nreturned by the greedy algorithm is at most $M^*(\\ln N-\\ln M^*+1)$.\\rm\n\n{\\bf Lemma 2.} \\it Given $N$ and $M^*$, there are instance of set\ncover problem such that the size of set cover returned by the greedy\nalgorithm is at least $(M^*-1)(\\ln N-\\ln M^*)$.\\rm\n\nTest set problem can be transformed to set cover problem in a\nnatural way. Let $(S,\\mathcal T)$ be an instance of test set, we\nconstruct an instance $(U,\\mathcal C)$ of set cover, where\n$U=\\{\\{i,j\\}|i,j\\in S,i\\ne j\\}$, and $\\mathcal C=\\{c(T)|T\\in\n\\mathcal T\\},c(T)=\\{\\{i,j\\}|i\\in T,j\\in S-T\\}.$\n\nClearly,  $\\mathcal T'$ is a test set of $S$ iff $\\mathcal\nC'=\\{c(T)|T\\in \\mathcal T'\\}$ is a set cover of $U$.\n\nSGA can be described as:\\\\\n\n\\indent{\\bf Input:} $S$,$\\mathcal T$;\\\\\n\\indent{\\bf Output:} a test set of $S$;\\\\\n\\indent$\\bar\\mathcal T\\leftarrow\\varnothing$;\\\\\n\\indent {\\bf while} $\\#(\\bar\\mathcal T)>0$ {\\bf do}\\\\\n\\indent\\indent select $T$ in $\\mathcal T-\\bar\\mathcal T$\nminimizing $\\#(\\bar\\mathcal T\\cup\\{T\\})$;\\\\\n\\indent\\indent $\\bar\\mathcal T\\leftarrow\\bar\\mathcal T\\cup\\{T\\}$;\\\\\n\\indent {\\bf endwhile}\\\\\n\\indent {\\bf return} $\\bar\\mathcal T$;\\\\\n\\rm\n\nIn SGA, we call $\\bar\\mathcal T$ the partial test set. The\ndifferentiation measure of $\\bar\\mathcal T$, $\\#(\\bar\\mathcal T)$,\nis defined as the number of item pairs not differentiated by\n$\\bar\\mathcal T$. The differentiation measure of $T$ w.r.t.\n$\\bar\\mathcal T$ is defined as $\\#(T,\\bar\\mathcal T)=\\#(\\bar\\mathcal\nT)-\\#(\\bar\\mathcal T\\cup \\{T\\})$.\n\nSGA is isomorphic to the greedy algorithm for set cover under the\nnatural transformation. Thus we immediately obtain the performance\nguarantee $2\\ln n$ of SGA. This paper shows a better performance\nguarantee and a nontrivial lower bound of performance guarantee. The\ntwo main theorems are:\n\n{\\bf Theorem 1.} \\it The performance guarantee of SGA can be\n$1.1354\\ln n$. \\rm\n\n{\\bf Theorem 2.} \\it There are arbitrarily large instances of test set problem\nsuch that the performance ratio of SGA on these instances is at least\n$1.0004609\\ln n$. \\rm\n\nIn this paper, denote $[n]:=\\{1,2,\\cdots,n\\}$. Denote $\\phi(x):=\\frac{1}{x}(\\ln\nx-1)$. The harmonious number is defined as $H_n:=\\sum_{i=1}^n{\\frac{1}{i}}$.\n\nTwo inequalities are listed here for convenience of proof in Section 3.\n\n{\\bf Fact 5.} \\it For any $0<x<1$, $(1-x)^{1\/x}<1\/e$. \\rm\n\n{\\bf Fact 6.} \\it For any $x>1$, $\\phi(x)\\le 1\/e^2=0.135\\cdots$. \\rm\n\n\\section{Improved Performance Guarantee}\n\n\\subsection{Differentiation Distribution}\nIn this subsection, the author analyzes the distribution of times for\nwhich item pairs are differentiated in instances of test set,\nespecially the relationship between the differentiation distribution\nand the size of the optimal test set.\n\n{\\bf Lemma 3.} \\it Given two disjoint subsets $S_1,S_2\\subseteq S$, and\n$\\mathcal T$, a set of tests of $S$, suppose $\\mathcal T$ is a test set of\n$S_1$ and a test set of $S_2$ , then at most $\\min(|S_1|,|S_2|)$ item pairs\nbetween $S_1$ and $S_2$ are not\ndifferentiated by any test in $\\mathcal T$.\\\\\n\\indent Proof. \\rm Suppose $|S_1|\\le |S_2|$. We claim for any item $i\\in S_1$,\nthere is at most one item $j$ in $S_2$ satisfying $\\{i,j\\}\\parallel \\mathcal\nT$. Otherwise there are two different items $j,k$ in $S_2$ such that\n$\\{i,j\\}\\parallel \\mathcal T$ and $\\{i,k\\}\\parallel \\mathcal T$, then by Fact 1\n, $\\{j,k\\}\\parallel \\mathcal T$, which contradicts $\\mathcal T$ is a test set\nof $S_2$. $\\Box$\n\n{\\bf Lemma 4.} \\it At most $n\\log_{2}n$ item pairs are\ndifferentiated by exactly one test in $\\mathcal T^*$.\\\\\n\\indent Proof. \\rm Let $B$ be the set of item pairs that are\ndifferentiated by exactly one test in $\\mathcal T^*$. We prove\n$|B|\\le n\\log_{2}n$ by induction. When $n=1$, $|B|=n\\log_{2}n$.\nSuppose the lemma holds for any $n\\le h-1$, we prove the lemma holds\nfor $n=h$.\n\nSelect $T\\in \\mathcal T^*$ such that $T\\neq \\varnothing$ and $T\\neq\nS$, then $|T|\\le h-1$, $|S-T|\\le h-1$. Since $\\mathcal T^*$ is a\ntest set of $T$, by induction hypothesis, at most $|T|\\log_{2}|T|$\nitem pairs inside $T$ are differentiated by exactly one test in\n$\\mathcal T^*$. Similarly, at most $|S-T|\\log_{2}|S-T|$ item pairs\ninside $S-T$ are differentiated by exactly one test in $\\mathcal\nT^*$.\n\nBy Lemma 3, at most $\\min(|T|,|S-T|)$ item pairs between $T$ and\n$S-T$ are not differentiated by any test in $\\mathcal T^*-\\{T\\}$.\nTherefore at most $\\min(|T|,|S-T|)$ item pairs between $T$ and\n$S-T$ are differentiated by exactly one test in $\\mathcal T^*$.\n\nW.l.o.g, suppose $|T|\\le |S-T|$, then\n\\begin{eqnarray*}\n&|B|&\\le|T|\\log_{2}|T|+|S-T|\\log_{2}|S-T|+|T|\\\\\n&&=|T|\\log_{2}(2|T|)+|S-T|\\log_{2}|S-T|\\\\\n&&\\le|T|\\log_{2}|S|+|S-T|\\log_{2}|S|\\\\\n&&=|S|\\log_{2}|S|.\n\\end{eqnarray*}\n$\\Box$\n\n{\\bf Lemma 5.} \\it Given $S''\\subseteq S'\\subseteq S$, and $\\mathcal T$, a set\nof tests of $S'$, suppose $\\mathcal T$ is a test set of $S''$ and a test set of\n$S'-S''$ , then at most $|S'|\\log_2{|S'|}$ item pairs between $S''$ and\n$S'-S''$ are differentiated by exactly one test in $\\mathcal T$.\\\\\n\n\\indent Proof. \\rm Let $B$ be the set of item pairs between $S''$ and $S'-S''$\nwhich are differentiated by exactly one test in $\\mathcal T$. We prove that\n$|B|\\le |S'|\\log_2{|S|'}$ by induction. When $|S|=1$ and $|S|=2$, the lemma\nholds. Suppose the lemma holds for any $|S|\\le h-1$, $h\\ge 3$, we prove the\nlemma holds for $|S|=h$.\n\nSelect $T\\in \\mathcal T$ such that $T\\neq \\varnothing$ and $T\\neq S'$,\n then $|T|\\le h-1$, $|S'-T|\\le h-1$ (see Figure 1).\nSince $\\mathcal T-\\{T\\}$ is a test set of $S''\\cap T$ and a test\nset of $(S'-S'')\\cap T$, by induction hypothesis, at most\n$|T|\\log_2{|T|}$ item pairs between $S''\\cap T$ and $(S'-S'')\\cap T$\nare differentiated by exactly one test in $\\mathcal T$. Similarly,\nat most $|S'-T|\\log_{2}|S'-T|$ item pairs between $S''\\cap (S'-T)$\nand $(S'-S'')\\cap (S'-T)$\n are differentiated by exactly one test in $\\mathcal T$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth,bb=140 290 440 560]{lemma.eps}\n\\caption{illustration of Lemma 5}\n\\end{center}\n\\end{figure}\n\nSince $\\mathcal T-\\{T\\}$ is a test set of $S''\\cap T$ and a test set\nof $(S'-S'')\\cap (S'-T)$, by Lemma 3, at most $\\min(|S''\\cap\nT|,|(S'-S'')\\cap (S'-T)|)$ item pairs between $S''\\cap T$ and\n$(S'-S'')\\cap (S'-T)$ are not differentiated by any test in\n$\\mathcal T-\\{T\\}$. Hence at most $\\min(|S''\\cap T|,|(S'-S'')\\cap\n(S'-T)|)$ item pairs between $S''\\cap T$ and $(S'-S'')\\cap (S'-T)$\nare differentiated by exactly one test in $\\mathcal T$. Similarly,\nat most $\\min(|(S'-S'')\\cap T|,|S''\\cap (S'-T)|)$ item pairs between\n$(S'-S'')\\cap T$ and $S''\\cap (S'-T)$ are differentiated by exactly\none test in $\\mathcal T$\n\nClearly,\n\\begin{eqnarray*}\n&|T|\\ge&\\min(|S''\\cap T|,|(S'-S'')\\cap (S'-T)|)\\\\\n&&+\\min(|(S'-S'')\\cap T|,|S''\\cap (S'-T)|).\n\\end{eqnarray*}\n\nW.l.o.g, suppose $|T|\\le |S'-T|$, then\n\\begin{eqnarray*}\n&|B|&\\le|T|\\log_{2}|T|+|S'-T|\\log_{2}|S'-T|+|T|\\\\\n&&=|T|\\log_{2}(2|T|)+|S'-T|\\log_{2}|S'-T|\\\\\n&&\\le|T|\\log_{2}|S'|+|S'-T|\\log_{2}|S'|\\\\\n&&=|S'|\\log_{2}|S'|.\n\\end{eqnarray*}\n$\\Box$\n\n{\\bf Lemma 6.} \\it At most $n\\log_2{n}{m^*}^{t-1}$ item pairs are\ndifferentiated by exactly $t$ test in $\\mathcal T^*$, where $t\\ge\n2$.\\\\\n\\indent Proof. \\rm Let $B_{t}$ be the set of item pairs that are\ndifferentiated by exactly $t$ test in $\\mathcal T^*$. For any\ncombination $\\pi$ of $t-1$ tests in $\\mathcal T^*$, let $B_{\\pi}$\nbe the subset of $B_{t}$ such that each item pair in $B_{\\pi}$ is\ndifferentiated by any test in $\\pi$.\n\nLet $\\backsim_{\\pi}$ be the equivalent relation induced by\n $\\pi$. For any equivalent class $[i]$, there is exactly one\n equivalent class $[j]$, such that each item pair between $[i]$\n and $[j]$ is differentiated by any test in $\\pi$ (Fact 2).\n\nSince $\\mathcal T^*-\\pi$ is a test set of $[i]$ and a test set of\n$[j]$, by Lemma 5, at most $(|[i]\\cup[j]|)\\log_2{|[i]\\cup[j]|}$ item\npairs between $[i]$ and $[j]$ are differentiated by exactly one test\nin $\\mathcal T^*-\\pi$. In another word, at most\n$(|[i]\\cup[j]|)\\log_2{|[i]\\cup[j]|}$ item pairs between $[i]$ and\n$[j]$ are differentiated by exactly $t$ tests in $\\mathcal T^*$.\nHence\n\\begin{eqnarray*}\n&|B_{\\pi}|&\\le\\sum_{[i],[j]}{|[i]\\cup[j]|\\log_2{|[i]\\cup[j]|}}\\le n\\log_2{n}.\n\\end{eqnarray*}\n\nTherefore,\n\\begin{eqnarray*}\n&|B_{t}|&\\le\\sum_{\\pi}{|B_{\\pi}|}\\le{m^*\\choose\n{t-1}}n\\log_2{n}\\le n\\log_2{n}{m^*}^{t-1}.\n\\end{eqnarray*}\n$\\Box$\n\n{\\bf Lemma 7.} \\it At most $2n\\log_2{n}{m^*}^{t-1}$ item pairs\nare differentiated by at most $t$ test in $\\mathcal T^*$, where\n$t\\ge 2$.\\\\\n\\indent Proof. \\rm Let $B$ be the set of item pairs that are\ndifferentiated by at most $t$ test in $\\mathcal T^*$, and $B_{t}$ be\nthe set of item pairs that are differentiated by exactly $t$ test in\n$\\mathcal T^*$. By Lemma 6,\n\\begin{eqnarray*}\n&|B|&=|B_{1}|+|B_{2}|+\\cdots+|B_{t}|\\\\\n&&\\le n\\log_2{n}(1+m^*+\\cdots+{m^*}^{t-1})\\\\\n&&\\le 2n\\log_2{n}{m^*}^{t-1}.\n\\end{eqnarray*}\ngu $\\Box$\n\n\\subsection{Proof of Theorem 1}\nIn this subsection, the author uses the potential function technique\nto derive improved performance guarantee of SGA for test set. Our\nproof is based on the trick to \"balance\" the potential function by\nappending a negative term to the differentiation measure.\n\nLet $I=\\lceil \\ln\\frac{n-1}{4\\log_2{n}}\/\\ln m^*\\rceil$, then\n$2n\\log_2{n}{m^*}^{I-1}<{n\\choose 2}\\le 2n\\log_2{n}{m^*}^I$. Let\n$\\#_{0}=1$, $\\#_{1}=n\\log_2{n}$, $\\#_{t}=2n\\log_2{n}{m^*}^{t-1},2\\le\nt\\le I$, and $\\#_{I+1}=n(n-1)\/2$. Let $k_{t}=\\frac{m^*}{t}\\ln\n\\frac{t\\#_{t}}{\\#_{t-1}}$, $2\\le t\\le I+1$.\n\nDenote by $p$ the probability distribution on tests in $\\mathcal\nT^*$ drawing one test uniformly from $\\mathcal T^*$. For any $T\\in\n\\mathcal T^*$ , the probability of drawing $T$ is\n$p(T)=\\frac{1}{m^*}$.\n\nWe divide a run of the algorithm into $I+1$ phases. For $I+1\\ge t\\ge\n1$, Phase $t$ begins when $\\#(\\bar\\mathcal T)\\ge\\#_{t-1}$ and lasts\nuntil $\\#(\\bar\\mathcal T)<\\#_{t-1}$. Phase $t$ is blank if when\nPhase $t+1$ ends, $\\#(\\bar\\mathcal T)<\\#_{t-1}$.\n\nLet the set of selected tests in Phase $t$ is $\\mathcal T_{t}$, the\npartial test set when Phase $t$ ends is $\\bar\\mathcal T_{t}$, and\nthe returned test set is $\\mathcal T'$. Then $\\bar\\mathcal\nT_{t}=\\cup_{t\\le s\\le I+1}\\mathcal T_{s}$, $1\\le t\\le I+1$, and\n$\\mathcal T'=\\bar\\mathcal T_{2}\\cup\\mathcal T_{1}$. Set\n$\\bar\\mathcal T_{I+2}=\\varnothing$. If Phase $t$ is not blank, let\nthe last selected test in Phase $t$ is $T'_{t}$.\n\nIn Phase $t$, $I+1\\ge t\\ge 2$, define the potential function as\n$$f(\\bar\\mathcal T)=(\\#(\\bar\\mathcal T)-\\textstyle\\frac{t-1}{t}\\displaystyle\\#_{t-1})(1-\n\\frac{t}{m^*})^{k_{t}-|\\bar\\mathcal T-\\bar\\mathcal T_{t+1}|}.$$\n\nBy the definition of $\\bar\\mathcal T_{t+1}$ and Fact 5,\n$$f(\\bar\\mathcal T_{t+1})<(\\#_{t}-\\textstyle\\frac{t-1}{t}\\displaystyle\\#_{t-1})(1-\\frac{t}{m^*})^{k_{t}}\n<\\frac{\\#_{t-1}}{t}.$$\n\nBy the definition of $f(\\bar\\mathcal T)$ and the facts $p(T)\\ge 0$\nand $\\sum_{T\\in \\mathcal T^*}{p(T)}=1$,\n\\begin{eqnarray*}\n&&\\min_{T\\in \\mathcal T}{f(\\bar\\mathcal T\\cup \\{T\\})}\\\\\n&&\\le\\min_{T\\in \\mathcal T^*}{f(\\bar\\mathcal T\\cup \\{T\\})}\\\\\n&&\\le\\sum_{T\\in \\mathcal T^*}({p(T)}f(\\bar\\mathcal T\\cup\n\\{T\\}))\\\\\n&&=(\\#(\\bar\\mathcal\nT)-\\textstyle\\frac{t-1}{t}\\displaystyle\\#_{t-1}-\\sum_{T\\in \\mathcal\nT^*}{(p(T)\\#(T,\\bar\\mathcal\nT))})(1-\\frac{t}{m^*})^{k_{t}-|\\bar\\mathcal T-\\bar\\mathcal\nT_{t+1}|-1}\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n&&\\sum_{T\\in \\mathcal\nT^*}{(p(T)\\#(T,\\bar\\mathcal T))}\\\\\n&&=\\sum_{\\{i,j\\}\\parallel \\bar\\mathcal T}{\\sum_{T\\in\n\\mathcal T^*:\\{i,j\\}\\perp T}{p(T)}}\\\\\n&&\\ge\\sum_{\\{i,j\\}\\parallel \\bar\\mathcal T}{\\frac{t}{m^*}}\n-\\sum_{\\{i,j\\}\\parallel \\bar\\mathcal T:\\perp(\\{i,j\\},\\mathcal T^*)\\le t-1}{\\frac{t-1}{m^*}}\\\\\n&&\\ge(\\#(\\bar\\mathcal\nT)-\\textstyle\\frac{t-1}{t}\\displaystyle\\#_{t-1})\\frac{t}{m^*}\n\\end{eqnarray*}\nby Lemma 4 and Lemma 7.\n\nTherefore,\n$$\\min_{T\\in \\mathcal T}{f(\\bar\\mathcal T\\cup\\{T\\})}\\le(\\#(\\bar\\mathcal\nT)-\\textstyle\\frac{t-1}{t}\\displaystyle\\#_{t-1})(1-\\frac{t}{m^*})^{k_{t}-|\\bar\\mathcal\nT-\\bar\\mathcal T_{t+1}|}=f(\\bar\\mathcal T).$$\n\nDuring Phase $t$, the algorithm selects $T$ in $\\mathcal T$ to\nminimize $f(\\bar\\mathcal T\\cup\\{T\\}$). Therefore, $f(\\bar\\mathcal\nT_{t}-\\{T'_{t}\\})\\le f(\\bar\\mathcal T_{t+1})<\\frac{\\#_{t-1}}{t}$.\n\nOn the other hand, $\\#(\\bar\\mathcal T_{t}-\\{T'_{t}\\})\\ge \\#_{t-1}$\nby definition of Phase $t$. Hence\n$$f(\\bar\\mathcal T_{t}-\\{T'_{t}\\})=\\frac{\\#_{t-1}}{t}(1-\\frac{t}{m^*})^{k_{t}-|\\mathcal\nT_{t}-\\{T'_{t}\\}|}.$$\n\nTherefore, $(1-\\frac{t}{m^*})^{k_{t}-|\\mathcal\nT_{t}-\\{T'_{t}\\}|}<1$, $|\\mathcal T_{t}-\\{T'_{t}\\}|<k_{t}$, and\n$|\\mathcal T_{t}|<k_{t}+1$.\n\nTo sum up,\n\\begin{eqnarray*}\n&|\\bar\\mathcal T_{2}|&<\\sum_{2\\le t\\le I+1}{k_{t}}+I\\\\\n&&=m^*(\\sum_{2\\le t\\le I+1}{\\frac{1}{t}\\ln\\frac\n{\\#_{t}}{\\#_{t-1}}}+\\sum_{2\\le t\\le I+1}{\\frac{\\ln t}{t}})+I.\n\\end{eqnarray*}\n\nWhen all Phase $t$, $I+1\\ge t\\ge 2$, end, we obtain an instance of\nset cover $(U,\\mathcal C)$, where $U=\\{\\{i,j\\}|\\{i,j\\}\\parallel\n\\bar\\mathcal T_{2}\\}$ and $\\mathcal C=\\{c(T)|c(T)\\cap U\\ne\n\\varnothing\\}$. Clearly, $|U|<\\#_{1}$. Let $M^*$ be the size of the\noptimal set cover of this instance. Then $|M^*|\\le m^*$.\n\nConsider the following two cases: (a)$|M^*|\\le \\frac{m^*}{2}$;\n(b)$|M^*|>\\frac{m^*}{2}$.\n\nIn case (a),\n$$|\\mathcal T_{1}|\\le M^*(\\ln\\#_{1}+1)\\le m^*(\\textstyle\\frac12\\displaystyle+o(1))\\ln n,$$\n\n\\noindent and\n\\begin{eqnarray*}\n&|\\bar\\mathcal T_{2}|&\\le m^*(\\sum_{2\\le t\\le I+1}{\\frac12\\ln\\frac\n{\\#_{t}}{\\#_{t-1}}}+\\frac12 \\ln^2 (I+2))+I\\\\\n&&=m^*(\\textstyle\\frac12\\displaystyle+o(1))\\ln n.\n\\end{eqnarray*}\n\nHence\n$$|\\mathcal T'|=m^*(1+o(1))\\ln n.$$\n\nIn case (b), by Lemma 1,\n\\begin{eqnarray*}\n&|\\mathcal T_{1}|&\\le M^*(\\ln\\#_{1}-\\ln M^*+1)=m^*((1+o(1))\\ln n-\\ln\nm^*),\n\\end{eqnarray*}\n\n\\noindent and\n\\begin{eqnarray*}\n&|\\bar\\mathcal T_{2}|&< m^*(H_{I+1}\\ln m^*+\\frac12\\ln 2+\\frac12 \\ln^2 (I+2))+I\\\\\n&&\\le m^*(\\ln \\frac{\\ln n}{\\ln m^*}\\ln m^*+o(1)\\ln n).\n\\end{eqnarray*}\n\nNotice $\\frac{\\ln n}{\\ln m^*}>1$, by Fact 3.\n\nBy Fact 6,\n$$|\\mathcal T'|\\le m^*(1+\\phi(\\frac{\\ln n}{\\ln m^*})+o(1))\\ln n\n\\le m^*(1.13533\\cdots+o(1))\\ln n.$$\n\n\\section{Lower Bound}\nIn this section, we discuss a variation of test set problem. Given disjoint\nsets $S^{1},\\cdots,S^{r}$ and $\\mathcal T$, set of subsets of the universe\n$S=S^{1}\\cup\\cdots\\cup S^{r}$, we seek $\\mathcal T'\\subseteq \\mathcal T$  with\nminimum cardinality which is a test set of any $S^{p}$ for $1\\le p \\le r$.\nDenote the instance by $(S^p;\\mathcal T)$.\n\nIn our construction, $r=2N-J!2^{q}$, let $2^{k-1}<N^2\\le 2^{k}$, we could use\nthe copy-split trick similar to that used in \\cite{bh} to make $2^{k}$ copies\nof $S^{1},\\cdots,S^{r}$, split all copies by $k$ tests and split all $S^{p}$\nfor each copy by $r-1$ tests. Thus the splitting overhead for each copy could\nbe ignored.\n\nSuppose $\\hat\\mathcal T$ is a compact test set of $[2^q]$. For example, we can\nlet $\\hat\\mathcal T=\\{T_{k}|k\\in[q]\\}$, where $T_{k}$ contains integer $x$\nbetween $1$ and $2^q$ such that the $k$-th bit of $x$'s binary representation\nis $1$.\n\n\\subsection{Atom Instances}\nFirstly, we give the level-$t$ atom instances $(S_t^{y};\\mathcal T)$. The\nuniverse $S_t$ includes integral points in $(t+1)$-dimension Euclid space.\\\\\n\n{\\bf Construction of Atom Instances.} $S_t=\\bigcup_{y}{S_t^{y}}$.\n$S_t^{y}=\\{(x_{1},\\cdots,x_{t},y)|x_{i}\\in [2^q]\\}$, $1\\le y\\le 2^{q-2}$.\n$\\mathcal T_t=\\mathcal T^*_t\\cup \\mathcal T'_t$. $\\mathcal T^*_t=\\mathcal\nT^*_{t,1}\\cup\\cdots\\cup\\mathcal T^*_{t,t}$. $\\mathcal\nT^*_{t,i}=\\{T^*_{t,i;j}|j\\in[2^{q}]\\}$, $1\\le i\\le t$. $\\mathcal T'_t=\\mathcal\nT'_{t,1}\\cup\\cdots\\cup\\mathcal T'_{t,t}$, $\\mathcal\nT'_{t,i}=\\{T'_{t,i;j,k}|j\\in[2^{q-2}],k\\in[q]\\}$, $1\\le i\\le t$.\n\n$|\\mathcal T^*_t|=t 2^{q}$, and $|\\mathcal T'_{t}|=\\frac{q}{4}t 2^{q}$.\n$T^*_{t,i;j}$ contains points in $S_t^{y}$ with $x_{i}=j$. $T'_{t,i;j,k}$\ncontains points in $S_t^{j}$ with $x_{i}$ in the $k$-th test in $\\hat\\mathcal\nT$. We assign an order to tests $T'_{t,i;j,k}$ in $\\mathcal T'_t$, called {\\it\nnatural order}, as the\nlexical order of $(i,j,k)$.\\\\\n\nAn atom instance with $q=3$ and $t=2$ is shown in Figure 2.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\textwidth,bb=70 330 480 520]{atom.eps}\n\\caption{atom instance with $q=3$ and $t=2$}\n\\end{center}\n\\end{figure}\n\nWe claim SGA could return $\\mathcal T'_t$ according to their natural order in\nthe atom instance.\n\nAt the beginning of the algorithm, the differentiation measure of tests in\n$\\mathcal T'_t$ is $2^{2qt-2}$, and the differentiation measure of tests in\n$\\mathcal T_t^*$ is\n$$2^{q(t-1)}(2^{qt}-2^{q(t-1)})2^{q-2}=2^{2qt-2}(1-2^{-q}).$$\n\nThe algorithm could first select tests in $\\mathcal T'_{t,1}$ according to\ntheir natural order. After that, the differentiation measure of tests in\n$\\mathcal T'_t-\\mathcal T'_{t,1}$ decreases by a factor $2$, and the\ndifferentiation measure of tests in $\\mathcal T_t^*$ decreases by a factor at\nleast $2$. Hence, the algorithm could subsequently select tests in $\\mathcal\nT'_{t,2},\\cdots,\\mathcal T'_{t,t}$ according to their natural order.\n\n\\subsection{Level-$t$ Instances}\nSecondly, we construct a series of level-$t$ instances $(S_{t}^{y,z,w};\\mathcal\nT_{t})$, $1\\le t\\le J$ based on the atom instances. Tests in the atom instances\nare {\\it stretched} in $w$ dimension and {\\it cloned} in $z$ dimension. Let\n$M^*=J!2^q$, and $N=J!2^{q(J+1)}$. The universe $S_t$ includes $N$ integral\npoints in $(t+3)$-dimension Euclid space.\\\\\n\n{\\bf Construction of Level-$t$ Instances.} $S_t=\\bigcup_{y,z,w}{S_t^{y,z,w}}$.\\\\\n$S_t^{y,z,w}=\\{(x_{1},\\cdots,x_{t},y,z,w)|x_{i}\\in[2^q]\\}$, $1\\le y\\le\n2^{q-2},1\\le z\\le\\frac{J!}{t},1\\le w\\le t2^{q(J-t)+2}$. $\\mathcal T_t=\\mathcal\nT^*_t\\cup \\mathcal T'_t$. $\\mathcal T^*_t=\\mathcal\nT^*_{t,1}\\cup\\cdots\\cup\\mathcal T^*_{t,t}$. $\\mathcal\nT^*_{t,i}=\\{T^*_{t,i;j,l}|j\\in[2^{q}],l\\in[\\frac{J!}{t}]\\}$, $1\\le i\\le t$.\n$\\mathcal T'_t=\\mathcal T'_{t,1}\\cup\\cdots\\cup\\mathcal T'_{t,t}$, $\\mathcal\nT'_{t,i}=\\{T'_{t,i;j,k,l}|j\\in[2^{q-2}],k\\in[q],l\\in[\\frac{J!}{t}]\\}$, $1\\le\ni\\le t$.\n\n$|\\mathcal T^*_t|=M^*$, and $|\\mathcal T'_{t}|=\\frac{qM^*}{4}$. $T^*_{t,i;j,l}$\ncontains points in $S_t^{y,l,w}$ with $x_{i}=j$ for any $w$. $T'_{t,i;j,k,l}$\ncontains points in $S_t^{j,l,w}$ with $x_{i}$ in the $k$-th test in\n$\\hat\\mathcal T$ for any $w$. We assign an order to tests $T'_{t,i;j,k,l}$ in\n$\\mathcal T'_t$,\ncalled {\\it natural order}, as the lexical order of $(i,j,k,l)$.\\\\\n\nWe claim SGA could select all tests in $\\mathcal T'_{t,1}$ according to their\nnatural order in the first phase of the algorithm.\n\nAt the beginning of the algorithm, it is easy to prove that the differentiation\nmeasure of tests in $\\mathcal T'_{t,1}$ is $\\#^{begin}_t=2^{q(t-1)}N$, and the\ndifferentiation measure of tests in $\\mathcal T_t^*$ is\n$2^{q(t-1)}(1-2^{-q})N$.\n\nThe algorithm could select tests in $\\mathcal T'_{t,1}$ according to their\nnatural order while the differentiation measure of the selected test is kept\nequal to the differentiation measures of tests in $\\mathcal T'_{t,i}$ for $2\\le\ni\\le t$ and no less than the differentiation measure of any test in $\\mathcal\nT_t^*$.\n\nWhen the algorithm select the last test in $\\mathcal T'_{t,1}$, its\ndifferentiation measure is $\\#^{end}_t=2\\cdot 2^{q(t-2)}N$.\n\nThe algorithm could subsequently select tests in $\\mathcal\nT'_{t,2},\\cdots,\\mathcal T'_{t,t}$ according to their natural order, and\nreturns $\\mathcal T'_t$.\n\nWe explicitly show the three claims for proof of Section 4.4. \\\\\n\n{\\bf Claim 1.} \\it In each step of SGA, the differentiation measure of the\nselected test is no less than the differentiation measure of a subsequent test\naccording to their natural order.\\rm \\\\\n\n{\\bf Claim 2.} \\it In each step of SGA, the differentiation measure of the\nselected test in every step is no less than any test in $\\mathcal T_t^*$.\\rm\\\\\n\n{\\bf Claim 3.} \\it In the first phase of SGA, the differentiation measure of\nthe selected test in every step is at most $\\#^{begin}_t$ and at least\n$\\#^{end}_t$.\\rm\n\n\\subsection{Complete Instances}\nLet $(U,\\mathcal C)$ be the instance in Lemma 2, $U=\\{e_{1},\\cdots,e_{N}\\}$,\n$\\mathcal C^*$ be the optimal set cover, and $\\mathcal C'$ be the set cover\nreturned by the greedy algorithm. Construct an instance of test set\n$(S_0^{p};\\mathcal T_{0})$. $S_0=\\bigcup_{p}{S_0^p}$.\n$S_0^{p}=\\{e_{p},f_{p}\\}$, $1\\le p\\le N$. $\\mathcal T_{0}=\\mathcal\nT'_{0}\\cup\\mathcal T^*_{0}$. $\\mathcal T'_{0}=\\mathcal C'$. $\\mathcal\nT^*_{0}=\\mathcal C^*$.\n\nOn $(S_0^{p};\\mathcal T_{0})$, the algorithm could select all the tests in\n$\\mathcal T'_{0}$, the differentiation measure of selected tests ranges from\n$\\#^{begin}_0=N\/M^*$ to $\\#^{end}_0=1$ by the proof of Lemma 1 in \\cite{s}.\n\nConsequently, we construct a series of level-$t$ instances\n$(S_{t}^{y,z,w};\\mathcal T_{t})$, $1\\le t\\le J$. We modify tests in $\\mathcal\nT'_{t,1}$ by two operations: Enlargement and Merging. In the Enlargement\noperation, tests in $\\mathcal T'_{t,1}$ are enlarged by a factor $2$. In the\nMerging operation, tests in $\\mathcal T'_{t,i}$ for $i\\ge 2$ are merged to\ntests in $\\mathcal T'_{s,1}$ for $t-1\\ge\ns\\ge 1$. \\\\\n\n{\\bf Enlargement.} Let $\\mathcal T'_{t,1}=\\{T'_{t,1;j,k,l}|j\\in[2^{q-3}],k\\in[\nq],l\\in[\\frac{J!}{t}]\\}$ for $J\\ge t\\ge 1$. $T_{t,1;j,k,l}$ contains points in\n$S_t^{2j-1,l,w}$ and $S_t^{2j,l,w}$ with $x_{1}$ in the $k$-th test in\n$\\hat\\mathcal T$ for any $w$. As a result, $|\\mathcal\nT'_{t,1}|=\\frac{qM^*}{8t}$.\n\n{\\bf Merging.} By the decreasing order of $t$ for $J\\ge t\\ge 2$, merge tests in\n$\\mathcal T'_{t}-\\mathcal T'_{t,1}$ by their natural order one-by-one to tests\nin $\\mathcal T'_{s,1}$ for $t-1\\ge s\\ge 1$ by the decreasing order of $s$\n(primarily) and their natural order in $\\mathcal T'_{s,1}$ until tests in\n$\\mathcal T'_{t}-\\mathcal T'_{t,1}$ are exhausted.\\\\\n\nThe illustration of the Merging operations with $J=4$ is shown in Figure 3.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth,bb=160 310 460 510]{merging.eps}\n\\caption{Merging operation with $J=4$}\n\\end{center}\n\\end{figure}\n\nFor any $J\\ge t\\ge 2$, tests in $\\mathcal T'_{s,1}$ suffice in the Merging\noperation, provided that\n$$|\\mathcal T'_{t}-\\mathcal T'_{t,1}|=\\frac14(1-\\frac{1}{t})qM^*\n\\le\\frac{1}{8}H_{t-1}qM^*=|\\bigcup_{s=1}^{t-1}\\mathcal T'_{s,1}|.$$\n\nLet $\\mathcal T'=\\mathcal T'_{0}\\cup\\bigcup_{t=1}^{J}\\mathcal T'_{t,1}$ after\nthe two operations are performed. Note that after the Merging operation, tests\nin $\\mathcal T'_{t,1}$ for $J-1\\ge t\\ge 1$ will contain items from $S_{s}$ for\nall $J\\ge s\\ge t$. Let $\\mathcal T=\\mathcal T^*\\cup\\mathcal T'$.\n\nThe complete instance is $(S_0^{p},S_{t}^{y,z,w};\\mathcal T^*\\cup \\mathcal\nT')$. Let $S=\\bigcup_{t=0}^{J}{S_t}$, then $n=|S|=(J+1)N$. Suppose $\\mathcal\nT^*_{t}=\\{T^*_{t,1},\\cdots,T^*_{t,M^*}\\}$, $0\\le t\\le J$, let $\\mathcal\nT^*=\\{T^*_{0,1}\\cup\\cdots\\cup T^*_{J,1},\\cdots,T^*_{0,M^*}\\cup\\cdots\\cup\nT^*_{J,M^*}\\}$. Since $\\mathcal T^*_{0}$ is an optimal test set of $S_0$,\n$\\mathcal T^*$ is an optimal solution of the complete instance, and\n$m^*=|\\mathcal T^*|=M^*$.\n\n\\subsection{Proof of Theorem 2}\nWe intend to prove the performance ratio of SGA on this instance is at least\n$(1+\\frac{1}{J+1}(\\frac{H_{J}}{8\\ln 2}-1)-o(1))\\ln n$, for fixed $J$. When\n$J=391$, this performance ratio is at least $1.0004609\\ln n$. To accomplish the\nproof, we analyze the behavior of SGA on the complete instance. Indeed, the\nsequence of the selected tests projected onto $S_t$ is almost the same as the\nselected tests of SGA on the original level-$t$ instance for $J\\ge t\\ge 1$,\nexcept that the tests in $\\mathcal T'_{t,1}$ are enlarged.\n\nBefore the algorithm selects a test, let $\\#_t$ be the maximum differentiation\nmeasure of tests in $\\mathcal T'_{t,1}$ for $J\\ge t\\ge 1$, and $\\#^*$ to the\nmaximum differentiation measure of tests in $\\mathcal T^*$.\n\nLet $\\#_{t,s}$ be the number of item pairs inside of $S_s$ contributing to\n$\\#_t$ for $t\\le s\\le J$, $\\#_{t-1,s}$ be the number of item pairs inside $S_s$\ncontributing to $\\#_{t-1}$ for $t-1\\le s\\le J$,  $\\#_s^*$ be the number of item\npairs inside $S_s$ contributing to $\\#^*$ for $1\\le s\\le J$, , and $\\#_0^*$ be\nthe number of item pairs inside $S_0^{p}$ contributing to $\\#^*$. Then\n$\\#_t=\\sum_{s=t}^{J}{\\#_{t,s}}$, $\\#_{t-1}=\\sum_{s=t-1}^{J}{\\#_{t-1,s}}$, and\n$\\#^*=\\sum_{s=0}^{J}{\\#_s^*}$.\n\nNote that $\\#_{t,t}$ for $J\\ge t\\ge 1$ is twice of the corresponding number in\nthe original level-$t$ instance. Remember $\\#^{begin}_s=2^{q(s-1)}N$,\n$\\#^{end}_s=2\\cdot 2^{q(s-2)}N$, for $J\\ge s\\ge 1$, and\n$\\#^{begin}_0=\\frac{1}{J!}2^{-q}N$.\n\nBy Claim 1, $\\#_{t,s}\\ge \\#_{t-1,s}$ for $s>t$ and by Claim 3,\n$$\\#_{t,t}=2\\#_{t-1,t}\\ge\\#_{t-1,t}+\\#^{end}_t=\\#_{t-1,t}\n+2\\#^{begin}_{t-1}\\ge\\#_{t-1,t}+\\#_{t-1,t-1},$$\n\n\\noindent it follows that $\\#_t\\ge\\#_{t-1}$. Hence $\\#_t\\ge\\#_{s}$, for any\n$1\\le s< t$.\n\nBy Claim 2, $\\#_{t,s}\\ge \\#_s^*$ for $s>t$ and by Claim 3,\n$$\\#_{t,t}\\ge \\#_t^*+\\#^{end}_{t}= \\#_t^*+2\\#^{begin}_{t-1}\\ge\n\\#_t^*+\\sum_{s=0}^{t-1}\\#^{begin}_{s}\\ge \\#_t^*+\\sum_{s=0}^{t-1}{\\#^*_{s}},$$\n\n\\noindent it follows that $\\#_t\\ge\\#^*$.\n\nWe conclude the algorithm could select all tests in $\\mathcal T'_{t,1}$ in\ntheir natural order, for $J\\ge t\\ge 1$, and select all tests in $\\mathcal\nT'_{0}$, finally return $\\mathcal T'$.\n\nRemember $M^*=J!2^q$, $N=J!2^{q(J+1)}$, $m^{*}=M^{*}$, and $n=(J+1)N$. In the\ncondition $J$ is fixed, the size of returned solution is\n\\begin{eqnarray*}\n&|\\mathcal T'|&\\ge(M^*-1)(\\ln N-\\ln M^*)+\\frac{qM^*}{8}H_{J}\\\\\n&&= m^*(1+\\frac{1}{J+1}(\\frac{H_{J}}{8\\ln 2}-1)-o(1))\\ln n.\n\\end{eqnarray*}\n\\section{Discussion}\nThe author notes this is the first time to distinguish precisely the\nworst case performance guarantees of two types of \"greedy\nalgorithms\" implemented by set cover criterion and by information\ncriterion. In fact, the author definitely shows the pattern of\ninstances on which ICH performs better than SGA.\n\nIn a preceding paper\\cite{cl}, we proved the performance guarantee\nof SGA can be $(1.5+o(1))\\ln n$, and the proof can be extended to\nweighted case, where each test is assigned a positive weight, and\nthe objective is modified as to find a test set with minimum total\nweight.\n\nIn the minimum cost probe set problem\\cite{bc} of bioinformatics,\ntests are replaced with partitions of items. The objective is to\nfind a set of partitions with smallest cardinality to differentiate\nall item pairs. It is easily observed that the improved\nperformance guarantee in this paper is still applicable to this generalized case.\\\\\n\n\\noindent{\\bf Acknowledgements.} The author would like to thank Tao\nJiang and Tian Liu for their helpful comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec introduction}\n\\setcounter{equation}{0}\n\n\nLet $X$ be a proper non compact metric space and  $o$ be a point in $X$. Given a discrete group $\\Delta$ of isometries of $X$, consider the \\textit{orbital counting function}\n\n\n\n\\begin{center}\n$N_\\Delta(o,t):=\\#\\lbrace g\\in\\Delta:\\hspace{0,3cm} d_X(o,g\\cdot o)\\leq t \\rbrace$,\n\\end{center}\n\n\\noindent where $t\\geq 0$. The \\textit{orbital counting problem} consists on the study of the asymptotic behaviour of $N_\\Delta(o,t)$ as $t\\longrightarrow\\infty$.\n\n\n\nWhen $X=\\mathbb{R}^2$ and $\\Delta=\\mathbb{Z}^2$ this is known as the \\textit{Gauss circle problem} (see Phillips-Rudnick \\cite{PR}). For a negatively curved Hadamard manifold $X$ and $\\Delta$ co-compact, this problem was studied by Margulis in his PhD Thesis \\cite{Mar}: the author shows a purely exponential asymptotic for $N_\\Delta(o,t)$, the exponent being the topological entropy of the geodesic flow of the quotient space $\\Delta \\backslash X$. Many authors have generalized the work of Margulis to different contexts, see Roblin \\cite{Rob} and references therein for a fairly complete picture in the negatively curved setting. \n\n\n\nWhen $X$ is a (not necessarily Riemannian) symmetric space associated to a semisimple Lie group $G$ and $\\Delta <G$ is a lattice, these kind of problems were studied notably by Eskin-McMullen \\cite{EMcM} and Duke-Rudnick-Sarnak \\cite{DRS}. In the non-lattice case but restricted to Riemannian symmetric spaces, one also finds the work of Quint \\cite{Qui} and Sambarino \\cite{Sam2}. Quint deals with the case in which $\\Delta$ is a Schottky group (in the sense of Benoist \\cite{Ben4}). Sambarino treats more generally the case of Anosov subgroups (in the full flag variety of $G$) introduced by Labourie \\cite{Lab}.\n\n\n\n\n\nBefore stating precise results we discuss in an informal way the problems adressed by this paper. Fix $d:=p+q$ where $p\\geq 1$ and $q\\geq 2$, and let $\\langle\\cdot,\\cdot\\rangle_{p,q}$ be the bilinear symmetric form on $\\mathbb{R}^{d}$ defined by\n\n\\begin{center}\n$ \\langle (x_1,\\dots,x_d),(y_1,\\dots,y_d)\\rangle_{p,q}:= \\displaystyle\\sum_{i=1}^p x_iy_i - \\displaystyle\\sum_{i=p+1}^d x_{i}y_{i}$.\n\\end{center}\n\n\n\\noindent We denote by $G:=\\textnormal{PSO}(p,q)$ the group of projectivized matrices in $\\textnormal{SL}(d,\\mathbb{R})$ preserving $\\langle\\cdot,\\cdot\\rangle_{p,q}$ and by $X_G$ the \\textit{Riemannian symmetric space} of $G$, that is, the space of $q$-dimensional subspaces of $\\mathbb{R}^{d}$ on which the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ is negative definite. Let $d_{X_G}$ be the distance in $X_G$ induced by a $G$-invariant Riemannian metric. For closed subsets $A$ and $B$ of $X_G$, set\n\n\\begin{center}\n$d_{X_G}(A,B):=\\inf\\lbrace d_{X_G}(a,b):\\hspace{0,3cm} a\\in A, b\\in B\\rbrace$.\n\\end{center}\n\n\n\\noindent On the other hand, the \\textit{pseudo-Riemannian hyperbolic space of signature $(p,q-1)$} is the set\n\n\\begin{center}\n$\\mathbb{H}^{p,q-1}:=\\left\\lbrace o=[\\hat{o}]\\in\\mathbb{P}(\\mathbb{R}^d): \\hspace{0,3cm} \\langle \\hat{o},\\hat{o}\\rangle_{p,q}<0  \\right\\rbrace$,\n\\end{center}\n\n\n\\noindent endowed with a $G$-invariant pseudo-Riemannian metric coming from restriction of the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ to tangent spaces.\n\nLet $\\Delta$ be a discrete subgroup of $G$ and fix a point $o$ in $\\mathbb{H}^{p,q-1}$. In this paper we study counting problems in $X_G$ and in $\\mathbb{H}^{p,q-1}$.\n\n\n\\begin{itemize}\n\\item \\textbf{Counting in $X_G$:} Denote by\n\n\n\\begin{center}\n$S^o:=\\lbrace \\tau\\in X_G:\\hspace{0,3cm} o\\subset \\tau \\rbrace $.\n\\end{center}\n\n\n\n\n\\noindent It is a totally geodesic sub-manifold of $X_G$ isometric to the Riemannian symmetric space of $\\textnormal{PSO}(p,q-1)$. We define two counting functions in this setting. The first one is\n\n\n\\begin{center}\n$N_\\Delta(S^o,t):=\\#\\lbrace g\\in\\Delta:\\hspace{0,3cm} d_{X_G}(S^o,g \\cdot S^o)\\leq t\\rbrace$.\n\\end{center}\n\n\n \n\n\\noindent For the second one we pick a point $\\tau\\in S^o$ and define\n\n\\begin{center}\n$N_\\Delta(S^o,\\tau,t):=\\#\\lbrace g\\in\\Delta:\\hspace{0,3cm} d_{X_G}(\\tau,g\\cdot S^o)\\leq t\\rbrace$.\n\\end{center}\n\n\\item \\textbf{Counting in $\\mathbb{H}^{p,q-1}$:} We provide a geometric interpretation of the function $N_\\Delta(S^o,t)$ in $\\mathbb{H}^{p,q-1}$. It is the amount of \\textit{space-like} geodesic segments\\footnote{That is, geodesic segments which are tangent to positive vectors.} of length at most $t$, that connect $o$ with points of $\\Delta\\cdot o$. The function $N_\\Delta(S^o,\\tau,t)$ has a geometric interpretation in this setting as well, which is more involved, and that we postpone until Subsection \\ref{subsec counting hpq introd}.\n\n\\end{itemize}\n\n\n\\begin{rem}\nIf $q=1$ one has $\\mathbb{H}^{p}=X_G=\\mathbb{H}^{p,q-1}$ and $o=S^o=\\tau$. We have as well the equalities\n\\begin{center}\n$N_\\Delta(o,t)=N_\\Delta(S^o,t)=N_\\Delta(S^o,\\tau,t)$\n\\end{center} \n\n\\noindent and our results correspond to the classical and well-known counting theorems already quoted.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\nIn contrast with the counting function $N_\\Delta(o,t)$ described at the beginning, the functions $N_\\Delta(S^o,t)$ and $N_\\Delta(S^o,\\tau,t)$ could in general be equal to infinity for large values of $t$. Part of the results that we present here concern the study of conditions for the choice of $o$ (and $\\tau$) that guarantee that the new counting functions are real-valued for every $t\\geq 0$. Once this is established, one may ask if the \\textit{exponential growth rate}\n\n\n\\begin{center}\n$\\displaystyle\\limsup_{t\\longrightarrow\\infty}\\dfrac{\\log N_\\Delta(S^o,t)}{t}$\n\\end{center}\n\n\n\\noindent is positive, finite and independent on the choice of $o$ (and the analogue questions for $N_\\Delta(S^o,\\tau,t)$). A more subtle problem is to find an asymptotic for the functions $N_\\Delta(S^o,t)$ and $N_\\Delta(S^o,\\tau,t)$ as $t\\longrightarrow\\infty$. The main goal of this paper is to give an answer to this more subtle problem for an interesting class of subgroups $\\Delta$: images of word hyperbolic groups under \\textit{projective Anosov representations}. \n\n\n\n\n\n\n\n\\subsection{Main results in $X_G$} \\label{subsec counting XG introd}\n\n \nIn order to formally state our results we need to recall some basic facts concerning (projective) Anosov representations. Anosov representations are (a stable class of) faithful and discrete representations from word hyperbolic groups into semisimple Lie groups that share many geometrical and dynamical features with holonomies of convex co-compact hyperbolic manifolds. They were introduced by Labourie \\cite{Lab} in his study of the Hitchin component and further extended to arbitrary word hyperbolic groups by Guichard-Wienhard \\cite{GW}. After that, Anosov representations had been object of intensive research in the field of geometric structures on manifolds and their deformation spaces (see for instance the surveys of Kassel \\cite{Kas} or Wienhard \\cite{Wie} and references therein).\n\n\n\n\n\n\n\n\nLet $P_1^{p,q}$ be the stabilizer of an isotropic line in $\\mathbb{R}^d$, i.e. a line on which the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ equals zero. Then $P_1^{p,q}$ is a parabolic subgroup of $G$ and the quotient space $\\partial\\mathbb{H}^{p,q-1}:= G\/P_1^{p,q}$, called the \\textit{boundary} of $\\mathbb{H}^{p,q-1}$, identifies with the set of isotropic lines in $\\mathbb{R}^d$.\n\nFix a non elementary word hyperbolic group $\\Gamma$ and let $\\partial_\\infty\\Gamma$ be its Gromov boundary. Let $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation. By definition (see Section \\ref{sec anosov}) this means that there exists a continuous equivariant map\n\n\n\n\n\\begin{center}\n$\\xi:\\partial_\\infty\\Gamma\\longrightarrow \\partial\\mathbb{H}^{p,q-1}$\n\\end{center}\n\n\\noindent with the following properties:\n\n\\begin{itemize}\n\\item \\textbf{Transversality:} Let $\\cdot^{\\perp_{p,q}}$ denote the orthogonal complement with respect to the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$. Then the map $\\eta:=\\xi^{\\perp_{p,q}}$ satisfies $\\xi(x)\\oplus\\eta(y)=\\mathbb{R}^d$ for every $x\\neq y$ in $\\partial_\\infty\\Gamma$.\n\\item \\textbf{Uniform hyperbolicity:} Some flow associated to $\\rho$ satisfies a uniform contraction\/dilation property (see \\cite{Lab,GW}).\n\\end{itemize}\n\n\n\nWhen $\\rho$ is $P_1^{p,q}$-Anosov all infinite order elements in $\\rho(\\Gamma)$ are \\textit{proximal}. This means that they act on $\\mathbb{P}(\\mathbb{R}^d)$ with a unique attractive fixed line and a unique repelling hyperplane. The \\textit{limit set} of $\\rho$ is, by definition, the closure  of the set of attractive fixed lines of proximal elements in $\\rho(\\Gamma)$. It is denoted by $\\Lambda_{\\rho(\\Gamma)}$ and coincides with the image of $\\xi$.\n\n\n\nDefine\n\n\n\n\\begin{center}\n\n$\\pmb{\\Omega}_{\\rho}:=\\lbrace o=[\\hat{o}]\\in\\mathbb{H}^{p,q-1}: \\langle \\hat{o},\\hat{\\xi} \\rangle_{p,q}\\neq 0 \\textnormal{ for all }\\xi=[\\hat{\\xi}]\\in\\Lambda_{\\rho(\\Gamma)}\\rbrace$.\n\\end{center}\n\n\n\n\nIn the study of discrete groups of projective transformations, it is standard to consider sets similar to $\\pmb{\\Omega}_\\rho$ (see for instance Danciger-Gu\u00e9ritaud-Kassel \\cite{DGK1,DGK2} and references therein). Without any further assumption the set $\\pmb{\\Omega}_\\rho$ could be empty. An important class of Anosov representations for which $\\pmb{\\Omega}_\\rho$ is non empty is given by \\textit{$\\mathbb{H}^{p,q-1}$-convex co-compact subgroups} introduced in \\cite{DGK1}. However in our results we do not assume that $\\rho$ is $\\mathbb{H}^{p,q-1}$-convex co-compact, we only need that $\\pmb{\\Omega}_\\rho\\neq\\emptyset$ (see Example \\ref{ex omegarho no vacio}).\n\n\n\n\\begin{propsn}[Propositions \\ref{prop counting with nu is well defined} and \\ref{prop counting with lambdauno is well defined}]\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation, a point $o\\in\\pmb{\\Omega}_{\\rho}$ and $\\tau\\in S^o$. Then for every $t\\geq 0$ one has\\footnote{Even though finiteness of $N_{\\rho(\\Gamma)}(S^o,\\tau,t)$ follows directly from finiteness of $N_{\\rho(\\Gamma)}(S^o,t)$, in our proof we first show $N_{\\rho(\\Gamma)}(S^o,\\tau,t)<\\infty$ and use it to prove $N_{\\rho(\\Gamma)}(S^o,t)<\\infty$.}\n\n\\begin{center}\n$N_{\\rho(\\Gamma)}(S^o,\\tau,t)<\\infty$ and $N_{\\rho(\\Gamma)}(S^o,t)<\\infty$.\n\\end{center}\n\n\n\n\n\\end{propsn}\n\n\nThe main results of this paper in the Riemannian context are Theorems \\ref{teorema A} and \\ref{teorema B}. The notation $f(t) \\sim g(t)$ stands for\n\n\\begin{center}\n$\\displaystyle\\lim_{t\\longrightarrow\\infty}\\frac{f(t)}{g(t)}= 1$.\n\\end{center}\n\n\n\n\n\\begingroup\n\\setcounter{tmp}{\\value{teo}\n\\setcounter{teo}{0}\n\\renewcommand\\theteo{\\Alph{teo}\n\n\n\n\\begin{teo} \\label{teorema A}\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation and $o\\in\\pmb{\\Omega}_{\\rho}$. There exist positive constants $h=h_\\rho$ and $M=M_{\\rho,o}$ such that\n\n\n\n\\begin{center}\n$N_{\\rho(\\Gamma)}(S^o,t)\\sim \\dfrac{e^{ht}}{M}$.\n\\end{center}\n\n\n\\end{teo}\n\n\n\n\\begin{teo} \\label{teorema B}\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation, a point $o\\in\\pmb{\\Omega}_{\\rho}$ and $\\tau\\in S^o$. There exist positive constants $h=h_\\rho$ and $M'=M'_{\\rho,\\tau}$ such that\n\n\n\n\\begin{center}\n$N_{\\rho(\\Gamma)}(S^o,\\tau,t)\\sim \\dfrac{e^{ht}}{M'}$.\n\\end{center}\n\n\n\\end{teo}\n\n\n\n\\endgroup\n\n\n\n\nThe constant $h$ is the same in both Theorems \\ref{teorema A} and \\ref{teorema B} and it is independent on the choice of $o$ in $\\pmb{\\Omega}_\\rho$ (and $\\tau$ in $S^o$). It coincides with the topological entropy of the \\textit{geodesic flow} $\\phi^\\rho$ of $\\rho$, introduced by Bridgeman-Canary-Labourie-Sambarino \\cite{BCLS}, and can be computed as\n\n\\begin{center}\n$h=\\displaystyle\\limsup_{t\\longrightarrow\\infty}\\dfrac{\\log\\#\\lbrace[\\gamma]\\in[\\Gamma]:\\hspace{0,3cm}\\lambda_1(\\rho(\\gamma))\\leq t\\rbrace}{t}$.\n\\end{center}\n\n\\noindent Here $[\\gamma]$ denotes the conjugacy class of $\\gamma$ and $\\lambda_1(\\rho(\\gamma))$ denotes the logarithm of the spectral radius of $\\rho(\\gamma)$. The constants $M$ and $M'$ are related to the total mass of specific measures in the Bowen-Margulis measure class of $\\phi^\\rho$ (recall that the Bowen-Margulis measure class is the homothety class of measures maximizing entropy of $\\phi^\\rho$).\n\n\n\n\n\n\n\n\nSince the work of Margulis \\cite{Mar}, in order to obtain a counting result one usually studies the ergodic properties of a well chosen dynamical system. In order to find a dynamical system adapted to Theorem \\ref{teorema A} we introduce a decomposition of a specific subset of $G$, analogue to the Cartan Decomposition, but replacing \\textit{the} maximal compact subgroup of $G$ by $\\textnormal{PSO}(p,q-1)$ and \\textit{the} Cartan subspace by a smaller abelian subalgebra (Subsection \\ref{subsec HexpliebH}). For Theorem \\ref{teorema B} we use the more studied \\textit{polar decomposition} of $G$ (Subsection \\ref{subsec KexpliebH}).\n\n\n\\subsubsection{\\textnormal{\\textbf{Relation with the work of Oh-Shah}}} \\label{subsub ohshah}\n\n\n\n\n\n\nMotivated by the study of \\textit{Apollonian circle packings} on the Riemann sphere, Oh-Shah \\cite{OS} studied counting problems similar to ours. Indeed, let $p=1$ and $q=3$. Then $\\mathbb{H}^{1,2}$ identifies with the space of circles of the Riemann sphere or, equivalently, the space of totally geodesic isometric copies of $\\mathbb{H}^2$ inside $\\mathbb{H}^3$. In \\cite[Theorem 1.5]{OS} the cited authors prove that for a well-chosen $S^o\\cong \\mathbb{H}^2\\subset\\mathbb{H}^3$ and any point $\\tau\\in\\mathbb{H}^3$ one has\n\n\\begin{center}\n$\\#\\lbrace g\\in\\Delta:\\hspace{0,3cm} d_{\\mathbb{H}^3}(\\tau,g\\cdot S^o)\\leq t  \\rbrace\\sim M^{-1}e^{ht}$.\n\\end{center}\n\n\n\\noindent Hence Theorem \\ref{teorema B} can be interpreted as a higher rank generalization of this result. We note however that, for $p=1$ and $q=3$, our results only concern convex co-compact groups, while Oh-Shah's Theorem applies to a wider class of geometrically finite Kleinian subgroups. A slightly different counting theorem in $\\mathbb{H}^{1,2}$ was obtained by the cited authors in \\cite{OS3}. Effective versions of Oh-Shah's results (i.e. with an error term) have been obtained by Lee-Oh \\cite{LO} and Mohammadi-Oh \\cite{MO}. Our Theorem \\ref{teorema A} seems to be new even in this setting.\n\n\n\nThe approach by Oh-Shah is similar to the one of Eskin-McMullen \\cite{EMcM}: they study the equidistribution, with respect to certain measures, of the orthogonal translates of $S^o$ under the geodesic flow of $\\Delta\\backslash\\mathbb{H}^3$ (see Oh-Shah \\cite{OS2} for precisions). Here we use different techniques. We follow the approach by Sambarino \\cite{Sam} and construct a dynamical system on a compact space that contains the required geometric information.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Interpretation in $\\mathbb{H}^{p,q-1}$} \\label{subsec counting hpq introd}\n\nAnother part of our contributions concern geometric interpretations of Theorems \\ref{teorema A} and \\ref{teorema B} in $\\mathbb{H}^{p,q-1}$. We now state these interpretations.\n\n\nGeodesics in $\\mathbb{H}^{p,q-1}$ are intersections of projectivized $2$-dimensional subspaces of $\\mathbb{R}^d$ with $\\mathbb{H}^{p,q-1}$ and they are classified in three types, depending on the sign of the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ on its tangent vectors (see Subsection \\ref{subsub geod hpq}). We are mainly interested in \\textit{space-like geodesics}, i.e. geodesics associated to planes on which the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ has signature $(1,1)$. Let $o,o'\\in\\mathbb{H}^{p,q-1}$ be two points joined by a space-like geodesic and let $\\ell_{o,o'}$ be the length of this geodesic segment (see Subsection \\ref{subsub lengths of spacelike geod}). We denote by $\\mathscr{C}^>_o$ the set of points of $\\mathbb{H}^{p,q-1}$ that can be joined to $o$ by a space-like geodesic and we set\n\n\\begin{center}\n$\\mathscr{C}^>_{o,G}:=\\lbrace g\\in G:\\hspace{0,3cm} g\\cdot o\\in\\mathscr{C}^>_o\\rbrace$.\n\\end{center}\n\n\n\n\n\n\n\n\n\\begin{propsn}[Proposition \\ref{prop ell dXG y vertboverto}]\nLet $o\\in\\mathbb{H}^{p,q-1}$ and $g\\in \\mathscr{C}^>_{o,G}$. Then \n\n\\begin{center}\n$\\ell_{o,g\\cdot o}=d_{X_G}(S^o,g\\cdot S^o)$.\n\\end{center}\n\\end{propsn}\n\nIn Corollary \\ref{cor gammao in cowmayor} we prove that given a $P_1^{p,q}$-Anosov representation $\\rho:\\Gamma\\longrightarrow G$ and $o$ in $\\pmb{\\Omega}_\\rho$, then apart from possibly finitely many exceptions $\\gamma$ in $\\Gamma$ one has $\\rho(\\gamma)\\in \\mathscr{C}^>_{o,G}$. By Proposition \\ref{prop counting with lambdauno is well defined} we have\n\n\n\\begin{center}\n$\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\rho(\\gamma)\\in \\mathscr{C}^>_{o,G} \\textnormal{ and } \\ell_{o,\\rho(\\gamma)\\cdot o}\\leq t\\rbrace<\\infty$\n\\end{center}\n\n\n\\noindent for every positive $t$. Moreover, Theorem \\ref{teorema A} implies that this function is asymptotic to $M^{-1}e^{ht}$ as $t\\longrightarrow\\infty$.\n\n\n\n\nIn order to state the corresponding geometric interpretation of Theorem \\ref{teorema B} we follow Kassel-Kobayashi \\cite[p.151]{KK}. Let $o\\in\\mathbb{H}^{p,q-1}$ and $\\tau\\in S^o$. Then\n\n\\begin{center}\n$\\mathbb{H}^p_\\tau:=(o\\oplus\\tau^{\\perp_{p,q}})\\cap\\mathbb{H}^{p,q-1}$\n\\end{center}\n\n\n\\noindent is a space-like totally geodesic copy of $\\mathbb{H}^{p}$ passing through $o$. Let $K^\\tau$ be the (maximal compact) subgroup of $G$ stabilizing $\\tau$. As we shall see, for every $g$ in $G$ the point $g\\cdot o$ lies in the $K^\\tau$-orbit of a point $o_g$ in $\\mathbb{H}^{p}_\\tau$. The counterpart of Theorem \\ref{teorema B} in $\\mathbb{H}^{p,q-1}$ is provided by the following proposition.\n\n\n\n\\begin{propsn}[Proposition \\ref{prop interpretation of btau in symg}]\nFor every $g$ in $G$ one has\n\n\\begin{center}\n$\\ell_{o,o_g}=d_{X_G}(\\tau,g\\cdot S^o)$.\n\\end{center}\n\n\\end{propsn}\n\n\n\n\\subsubsection{\\textnormal{\\textbf{Relation with the work of Glorieux-Monclair and Kassel-Kobayashi}}}\\label{subsub GM and KK in introd}\n\n\nGlorieux-Monclair \\cite{GM} introduced an orbital counting function for $\\mathbb{H}^{p,q-1}$-convex co-compact representations that differs from\n\n\\begin{center}\n$t\\mapsto\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\rho(\\gamma)\\in \\mathscr{C}^>_{o,G} \\textnormal{ and } \\ell_{o,\\rho(\\gamma)\\cdot o}\\leq t\\rbrace$\n\\end{center}\n\n\n\\noindent by a constant. Indeed, they define an \\textit{$\\mathbb{H}^{p,q-1}$-distance}\n\n\\begin{center}\n$d_{\\mathbb{H}^{p,q-1}}(o,o'):=\\left\\{\\begin{array}{cc} \\ell_{o,o'} \\hspace{0,1cm}\\textrm{ if } o'\\in\\mathscr{C}^>_o \\\\ 0  \\hspace{0,6cm}\\textrm{ otherwise }\\end{array}\\right.$,\n\\end{center} \n\n\\noindent and show that it satisfies a version of the triangle inequality in the convex hull of the limit set of $\\rho$. This is used to prove that the exponential growth rate of the counting function\n\n\n\\begin{center}\n$t\\mapsto\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm}  d_{\\mathbb{H}^{p,q-1}}(o,\\rho(\\gamma)\\cdot o)\\leq t\\rbrace$\n\\end{center}\n\n\n\\noindent is independent on the choice of the basepoint $o$. The authors interpret this exponential rate as a \\textit{pseudo-Riemannian Hausdorff dimension} of the limit set of $\\rho$, with the purpose of finding upper bounds for this number (\\cite[Theorem 1.2]{GM}). A consequence of Theorem \\ref{teorema A} and Proposition \\ref{prop ell dXG y vertboverto} (see Remarks \\ref{rem crit exponent coindes with the one of GM} and \\ref{rem crit exponent coincides with the entropy}) is that this rate coincides with the topological entropy $h$ of $\\phi^\\rho$.\n\n\n\nOn the other hand, as we shall see in Section \\ref{sec generalized cartan} the number $\\ell_{o,o_g}$ is related to the \\textit{polar projection} of $g$ and therefore Theorem \\ref{teorema B} addresses the problems treated by Kassel-Kobayashi in \\cite[Section 4]{KK}. In \\cite{KK} the authors study the orbital counting function of Theorem \\ref{teorema B} for \\textit{sharp} subgroups of a real reductive symmetric space (see \\cite[Section 4]{KK}). Kassel-Kobayashi obtain some estimates on the growth of this function, but no precise asymptotic is established.\n\nThe method of \\cite{GM} is based on pseudo-Riemannian geometry: they construct analogues of Busemann functions, Gromov products and Patterson-Sullivan densities in $\\mathbb{H}^{p,q-1}$ using this viewpoint. Our approach is inspired by \\cite{KK} and has Lie-theoretic flavor: we study linear algebraic interpretations of the geometric quantities involved in the definition of the counting functions. This allows us to establish finiteness of these functions, to make a link between the different symmetric spaces and to apply Ledrappier's \\cite{Led} framework to our setting.\n\n\\subsection{Outline of the proof} \\label{subsec outline}\n\n\nThere are three major steps in the proof of Theorems \\ref{teorema A} and \\ref{teorema B}.\n\n\\subsubsection*{\\textnormal{\\textbf{First step}}} As we said, we interpret the geometric quantities involved in Theorems \\ref{teorema A} and \\ref{teorema B} as linear algebraic quantities. \n\n\nLet us be more precise. Fix $o \\in \\mathbb{H}^{p,q-1}$ and denote by $H^o$ the stabilizer in $G$ of this point. If we consider the symmetry of $\\mathbb{R}^d$ given by $J^o:=\\textnormal{id}_o\\oplus\\left(-\\textnormal{id}_{o^{\\perp_{p,q}}}\\right)$, we have that $H^o$ equals the fixed point set of the involution\n\\begin{center}\n$\\sigma^o:g\\mapsto J^ogJ^o$\n\\end{center}\n\n\n\\noindent of $G$ (see Subsection \\ref{subsubsec struc sym hpq}). This identifies the tangent space at $o$ of $\\mathbb{H}^{p,q-1}$ with the subspace of $\\mathfrak{so}(p,q)$ defined by $\\mathfrak{q}^o:=\\lbrace d\\sigma^o=-1\\rbrace$. In Propositions \\ref{prop ell dXG y vertboverto} and \\ref{prop linear alg interpr of bo} we prove that for every $g\\in \\mathscr{C}^>_{o,G}$ one has\n\\begin{equation}\\label{eq igualddad distancia con vap}\nd_{X_G}(S^o,g\\cdot S^o)=\\frac{1}{2}\\lambda_1(J^ogJ^og^{-1}).\n\\end{equation}\n\n\n\n\\noindent The main ingredient in the proof of equality (\\ref{eq igualddad distancia con vap}) is the following version of the classical Cartan Decomposition of $G$.\n\n\\begin{propsn}[Proposition \\ref{prop HBH}]\nLet $o\\in\\mathbb{H}^{p,q-1}$ and $\\mathfrak{b}^+\\subset\\mathfrak{q}^o$ be a ray such that $\\exp(\\mathfrak{b}^+)\\cdot o$ is space-like. Given $g\\in \\mathscr{C}^>_{o,G}$ there exists $h,h'\\in H^o$ and a unique $X\\in\\mathfrak{b}^+$ such that\n\n\n\\begin{center}\n$g=h\\exp(X)h'$.\n\\end{center}\n\n\\end{propsn} \n\nOn the other hand, the linear algebraic interpretation of the quantity $d_{X_G}(\\tau,g\\cdot S^o)$ is the following: the choice of $\\tau$ induces a norm $\\Vert\\cdot\\Vert_\\tau$ on $\\mathbb{R}^d$ invariant under the action of $K^\\tau$. We show in Propositions \\ref{prop interpretation of btau in symg} and \\ref{prop computing nu} that for every $g\\in G$ the following equality holds\n\\begin{equation}\\label{eq igualddad distancia con vasing}\nd_{X_G}(\\tau,g\\cdot S^o)=\\frac{1}{2}\\log\\Vert J^ogJ^og^{-1}\\Vert_\\tau.\n\\end{equation}\n\\noindent Once again the proof of this equality relies on a generalization of Cartan Decomposition (see Schlichtkrull \\cite[Chapter 7]{Sch}): every $g\\in G$ can be written as \n\n\\begin{center}\n$g=k\\exp(X)h$\n\\end{center}\n\n\\noindent for some $k\\in K^{\\tau}$, $h\\in H^o$ and a unique $X\\in\\mathfrak{b}^+$.\n\n\n\\subsubsection*{\\textnormal{\\textbf{Second step}}} In order to simplify the exposition we assume that $\\Gamma$ is torsion free. In this case every $\\gamma\\neq 1$ in $\\Gamma$ has a unique attractive (resp. repelling) fixed point in $\\partial_\\infty\\Gamma$, denoted by $\\gamma_+$ (resp. $\\gamma_-$). Consider $\\rho:\\Gamma\\longrightarrow G$ a $P_1^{p,q}$-Anosov representation. The key feature of choosing $o$ in $\\pmb{\\Omega}_\\rho$ is that it guarantees some \\textit{transversality condition} for the proximal matrices $J^o\\rho(\\gamma)J^o$ and $\\rho(\\gamma^{-1})$ and this allows to estimate the quantities (\\ref{eq igualddad distancia con vap}) and (\\ref{eq igualddad distancia con vasing}) in terms of the spectral radius of $\\rho(\\gamma)$.\n\n\nMore precisely, we will see in Proposition \\ref{prop fijos de Jo en borde} that \n\\begin{equation} \\label{eq omegarho en introduccion}\n\\pmb{\\Omega}_\\rho=\\lbrace o\\in\\mathbb{H}^{p,q-1}:\\hspace{0,3cm} J^o\\cdot\\xi(x)\\notin\\eta(x) \\textnormal{ for all } x\\in\\partial_\\infty\\Gamma \\rbrace.\n\\end{equation}\n\n\n\n\\noindent Fix $o\\in\\pmb{\\Omega}_\\rho$ and a distance $d$ in $\\mathbb{P}(\\mathbb{R}^d)$ induced by the choice of an inner product in $\\mathbb{R}^d$. By compactness of $\\partial_\\infty\\Gamma$ there exists a positive constant $r$ such that\n\n\n\\begin{center}\n$d(J^o\\cdot\\xi(x),\\eta(x))\\geq r$\n\\end{center}\n\n\\noindent holds for every $x\\in\\partial_\\infty\\Gamma$ (here $d(J^o\\cdot\\xi(x),\\eta(x))$ is the minimal distance between $J^o\\cdot\\xi(x)$ and the lines included in $\\eta(x)$). Further, if $\\gamma_+$ is uniformly far from $\\gamma_-$, with respect to some visual distance in $\\partial_\\infty\\Gamma$, then $\\xi(\\gamma_+)$ (resp. $\\xi(\\gamma_-)$) is uniformly far from $\\eta(\\gamma_-)$ (resp. $\\eta(\\gamma_+)$). In Lemma \\ref{lema jrhojrho prox dos} we combine all these facts with Benoist's work \\cite{Ben1} to conclude that  the product $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$ is proximal. Moreover, we obtain a comparison between the quantity (\\ref{eq igualddad distancia con vap}) (resp. (\\ref{eq igualddad distancia con vasing})) and\n\n\\begin{center}\n$\\lambda_1(\\rho(\\gamma))$\n\\end{center}\n\n\n\\noindent with very precise control on the error made in this comparison.\n\n\n\\subsubsection*{\\textnormal{\\textbf{Third step}}} \n\nWe apply Sambarino's outline \\cite{Sam} to our particular context\\footnote{The results in \\cite{Sam} are proved for fundamental groups of closed negatively curved manifolds. However, all the results obtained there remain valid when $\\Gamma$ is an arbitrary word hyperbolic group admitting an Anosov representation. This is explained in detail in Appendix \\ref{appendix distribution utaugamma y uogamma}.}. To a H\u00f6lder cocycle $c$ on $\\partial_\\infty\\Gamma$ the author associates a H\u00f6lder reparametrization $\\psi_t^c$ of the geodesic flow of $\\Gamma$. Recall that a \\textit{H\u00f6lder cocycle} is a map $c:\\Gamma\\times\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{R}$ satisfying\n\n\n\n\\begin{center}\n$c(\\gamma_0\\gamma_1,x)=c(\\gamma_0,\\gamma_1\\cdot x)+c(\\gamma_1,x)$\n\\end{center}\n\n\n\\noindent for every $\\gamma_0,\\gamma_1$ in $\\Gamma$ and $x\\in\\partial_\\infty\\Gamma$ and such that the map $c(\\gamma_0,\\cdot)$ is H\u00f6lder (with the same exponent for every $\\gamma_0$). The cocycle $c'$ is said to be \\textit{cohomologous} to $c$ if there exists a H\u00f6lder continuous function $U:\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{R}$ such that for every $\\gamma$ in $\\Gamma$ and $x$ in $\\partial_\\infty\\Gamma$ one has\n\n\n\n\\begin{center}\n$c(\\gamma,x)-c'(\\gamma,x)=U(\\gamma\\cdot x)-U(x)$.\n\\end{center}\n\n\\noindent In that case $\\psi_t^c$ is conjugate to $\\psi_t^{c'}$ (see \\cite[Section 3]{Sam}). By considering a Markov coding and applying Parry-Pollicott's Prime Orbit Theorem \\cite{PP}, Sambarino obtains an asymptotic for the number of periodic orbits of $\\psi_t^c$ of period less than or equal to $ t$ (see \\cite[Corollary 4.1]{Sam}). Obviously this is a purely dynamical result, i.e. changing $\\psi_t^c$ in its conjugacy class does not affect the asymptotics. \n\n\nHowever our problem is more subtle: one must find a particular cocycle, with some geometric meaning, and not just \\textit{any} cocycle in the given cohomology class. Indeed, the cocycles that we consider to prove Theorems \\ref{teorema A} and \\ref{teorema B} are cohomologous, but only the specific choices in such a cohomology class yield the respective results.\n\n\n\n\n\n\nLet us briefly sketch the proof of Theorem \\ref{teorema A} (Theorem \\ref{teorema B} is proved in a similar way). Fix $o\\in\\pmb{\\Omega}_\\rho$ and consider\n\n\\begin{center}\n$c_o:\\Gamma\\times\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} c_o(\\gamma,x):=\\dfrac{1}{2}\\log\\left\\vert\\dfrac{\\langle\\rho(\\gamma)\\cdot v_x,J^o\\rho(\\gamma)\\cdot v_x\\rangle_{p,q}}{\\langle v_x,J^o\\cdot v_x\\rangle_{p,q}}\\right\\vert$\n\\end{center}\n\n\\noindent where $v_{x}\\neq 0$ is any vector in $\\xi(x)$\\footnote{When $q=1$ this coincides with the \\textit{Busemann cocycle} of $\\mathbb{H}^{p}$, i.e. $c_o(\\gamma,x)=\\beta_{\\xi(x)}(\\rho(\\gamma^{-1})\\cdot o,o)$ where $\\beta_\\cdot(\\cdot,\\cdot):\\partial\\mathbb{H}^{p}\\times\\mathbb{H}^{p}\\times\\mathbb{H}^{p}\\longrightarrow\\mathbb{R}$ is the Busemann function.}. This is a well-defined function thanks to (\\ref{eq omegarho en introduccion}) and it is a H\u00f6lder cocycle.\n\n\n\nLet $\\partial_\\infty^{2}\\Gamma$ be the set of pairs of distinct points in $\\partial_\\infty\\Gamma$ and consider the action of $\\Gamma$ on $\\partial_\\infty^{2}\\Gamma\\times\\mathbb{R}$ given by\n\n\\begin{center}\n$\\gamma\\cdot (x,y,s):=(\\gamma \\cdot x,\\gamma\\cdot y, s-c_o(\\gamma,y))$.\n\\end{center}\n\n\n\n\n\n\n\\noindent We denote by $\\textnormal{U}_o\\Gamma$ the quotient space. The \\textit{translation flow} on $\\partial_\\infty^{2}\\Gamma\\times\\mathbb{R}$ given by\n\n\\begin{center}\n$\\psi_t(x,y,s):=(x,y,s-t)$\n\\end{center}\n\n\\noindent descends to a flow $\\psi_t=\\psi_t^o$ on $\\textnormal{U}_o\\Gamma$. As Sambarino shows in \\cite[Theorem 3.2(1)]{Sam} (see also Lemma \\ref{lema conj urhogamma y uogamma}) the flow $\\psi_t$ is conjugate to a H\u00f6lder reparametrization of the geodesic flow of $\\Gamma$ introduced by Gromov \\cite{Gro}. We will show (see Lemma \\ref{lema conj urhogamma y uogamma}) that periodic orbits of $\\psi_t$ are parametrized by conjugacy classes of \\textit{primitive} elements in $\\Gamma$, i.e. elements which cannot be written as a power of another element. If $\\gamma$ is primitive, the corresponding period is given by\n\n\\begin{center}\n$\\ell_{c_o}(\\gamma):=\\lambda_1(\\rho(\\gamma))$.\n\\end{center}\n\n\nWe show the following property concerning spectral radii in a projective Anosov representation.\n\n\n\n\\begin{propsn}[Proposition \\ref{prop geod flow is weak mixing}]\n\n\nLet $\\rho$ be a projective Anosov representation of $\\Gamma$. Then the set $\\lbrace\\lambda_1(\\rho(\\gamma))\\rbrace_{\\gamma\\in\\Gamma}$ spans a non discrete subgroup of $\\mathbb{R}$.\n\n\\end{propsn} \n\n\n\n\n\n\n\nDenote by $h$ the topological entropy of $\\psi_t$. The probability of maximal entropy of $\\psi_t$ can be constructed as follows: define the \\textit{Gromov product}\n\n\\begin{center}\n$[\\cdot,\\cdot]_o:\\partial_\\infty^{2}\\Gamma\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} [x,y]_o:=-\\dfrac{1}{2}\\log\\left\\vert  \\dfrac{\\langle v_x,J^o\\cdot v_x\\rangle_{p,q}\\langle v_y,J^o\\cdot v_y\\rangle_{p,q}}{\\langle v_x,v_y\\rangle_{p,q}\\langle v_y,v_x\\rangle_{p,q}}\\right\\vert$.\n\\end{center}\n\n\n\\noindent This function is well-defined thanks to (\\ref{eq omegarho en introduccion}) and transversality of $\\xi$ and $\\eta$. One can prove that\n\n\n\\begin{center}\n$[\\gamma \\cdot x,\\gamma\\cdot y]_o - [x,y]_o=-(c_o(\\gamma,x)+c_o(\\gamma,y))$\n\\end{center}\n\n\\noindent holds for every $\\gamma$ in $\\Gamma$ and $(x,y)\\in\\partial_\\infty^{2}\\Gamma$. Let $\\mu_o$ be a \\textit{Patterson-Sullivan probability} associated to $c_{o}$, that is, $\\mu_o$ is a probability on $\\partial_\\infty\\Gamma$ that satisfies\n\n\\begin{center}\n$\\dfrac{d\\gamma_*\\mu_o}{d\\mu_o}(x)=e^{-hc_{o}(\\gamma^{-1},x)}$\n\\end{center}\n\n\n\\noindent for every $\\gamma\\in\\Gamma$\\footnote{Recall that if $f:X\\longrightarrow Y$ is a map and $m$ is a measure on $X$ then $f_*(m)$ denotes the measure on $Y$ defined by $A\\mapsto m(f^{-1}(A))$.}. For the existence of such a probability see Subsection \\ref{subsub PS}. The measure\n\n\\begin{center}\n$e^{-h[\\cdot,\\cdot]_o}\\mu_o\\otimes\\mu_o\\otimes dt$\n\\end{center}\n\n\n\\noindent on $\\partial_\\infty^{2}\\Gamma\\times\\mathbb{R}$ is $\\Gamma$-invariant. It induces on the quotient $\\textnormal{U}_o\\Gamma$ the measure of maximal entropy of $\\psi_t$, which is unique up to scaling (see \\cite[Theorem 3.2(2)]{Sam} or Proposition \\ref{prop product is of maximal entropy}). \n\n\nDenote by $C_c^*(\\partial_\\infty^{2}\\Gamma)$ the dual of the space of compactly supported real continuous functions on $\\partial_\\infty^{2}\\Gamma$ equipped with the weak-star topology. For $x$ in $\\partial_\\infty\\Gamma$ let $\\delta_x$ be the Dirac mass at $x$. Inspired by the work of Roblin \\cite{Rob}, Sambarino \\cite[Proposition 4.3]{Sam} shows\n \n\\begin{center}\n$Me^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma, \\ell_{c_o}(\\gamma)\\leq t} \\delta_{\\gamma_-}\\otimes\\delta_{\\gamma_+}\\longrightarrow e^{-h[\\cdot,\\cdot]_o}\\mu_o\\otimes\\mu_o$\n\\end{center}\n\n\\noindent on $C_c^*(\\partial_\\infty^{2}\\Gamma)$ as $t\\longrightarrow\\infty$ (for a proof in our context see Proposition \\ref{prop distribution of periodic orbits}). The constant $M=M_{\\rho,o}>0$ equals the product of $h$ with the total mass of $e^{-h[\\cdot,\\cdot]_o}\\mu_o\\otimes\\mu_o\\otimes dt$ on the quotient space $\\textnormal{U}_o\\Gamma$.\n\n\nAs we show in Lemma \\ref{lema computing gromov on gammapm o}, the number $[\\gamma_-,\\gamma_+]_o$ is the precise error term in the comparison between $\\ell_{c_o}(\\gamma)$ and $\\frac{1}{2}\\lambda_1(J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1}))=d_{X_G}(S^o,\\rho(\\gamma)\\cdot S^o)$ provided by Benoist's Theorem \\ref{teo benoist}. This is the geometric step: we replace the period $\\ell_{c_o}(\\gamma)$ by the number $d_{X_G}(S^o,\\rho(\\gamma)\\cdot S^o)$ in the previous sum, using the Gromov product. \n\n\n\n\n\\begin{propsn}[Proposition \\ref{prop distribution on bg for length}]\nLet $\\Gamma$ be a torsion free word hyperbolic group, $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation and $o\\in\\pmb{\\Omega}_{\\rho}$. Then\n\n\n\\begin{center}\n$M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma, d_{X_G}(S^o,\\rho(\\gamma)\\cdot S^o) \\leq t} \\delta_{\\gamma_-}\\otimes\\delta_{\\gamma_+}\\longrightarrow \\mu_o\\otimes\\mu_o$\n\\end{center}\n\n\\noindent on $C^*(\\partial_\\infty\\Gamma\\times\\partial_\\infty\\Gamma)$ as $t\\longrightarrow\\infty$.\n\\end{propsn}\n\n\nThe proof of Proposition \\ref{prop distribution on bg for length} follows line by line the proof of \\cite[Theorem 6.5]{Sam}, which is again inspired by Roblin's work \\cite{Rob}.\n\nIt turns out that the previous proposition can be used to deduce Theorem \\ref{teorema A} in the general case, that is, if we admit torsion elements in $\\Gamma$.\n\n\n\\begin{propsn}[Proposition \\ref{prop distribution on bg for length with torsion}]\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation and $o\\in\\pmb{\\Omega}_{\\rho}$. Then\n\n\n\\begin{center}\n$M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma, d_{X_G}(S^o,\\rho(\\gamma)\\cdot S^o) \\leq t} \\delta_{\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}}}\\otimes\\delta_{\\rho(\\gamma)\\cdot o}\\longrightarrow \\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_o)$\n\\end{center}\n\n\\noindent on $C^*(\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d))$ as $t\\longrightarrow\\infty$.\n\\end{propsn}\n\n\n\n\n\n\n\n\n\\subsection{Organization of the paper}\n\n\n\nIn Section \\ref{sec symmetric spaces} we recall basic facts on the symmetric spaces $X_G$ and $\\mathbb{H}^{p,q-1}$. Of particular importance is Subsection \\ref{subsub endign of spacelike}, which is devoted to the study of end points of space-like geodesics passing through our preferred point $o\\in\\mathbb{H}^{p,q-1}$. We give several characterizations of this set that will allow us to understand $\\pmb{\\Omega}_\\rho$ in different ways, all of them used indistinctly in Sections \\ref{sec the set omegarho}, \\ref{section distrib wrt bo} and \\ref{section distrib wrt btau}. In Section \\ref{sec generalized cartan} we study the geometric quantities involved in Theorems \\ref{teorema A} and \\ref{teorema B}. Equalities (\\ref{eq igualddad distancia con vap}) and (\\ref{eq igualddad distancia con vasing}) are proven respectively in Subsections \\ref{subsec HexpliebH} and \\ref{subsec KexpliebH}. In Section \\ref{sec proximality} we recall Benoist's results on products of proximal matrices and Section \\ref{sec anosov} is devoted to reminders on Anosov representations. In Section \\ref{sec the set omegarho} we define the set $\\pmb{\\Omega}_\\rho$ and study the action of $\\Gamma$ on this set. We show in particular that the orbital counting functions involved in Theorems \\ref{teorema A} and \\ref{teorema B} are well-defined (Proposition \\ref{prop counting with lambdauno is well defined} and Proposition \\ref{prop counting with nu is well defined}). We also obtain some estimates for the spectral radius and operator norm of elements $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$ which are of major importance (c.f. Lemma \\ref{lema jrhojrho prox dos}). In Section \\ref{section distrib wrt bo} (resp. Section \\ref{section distrib wrt btau}) we prove Theorem \\ref{teorema A} (resp. Theorem \\ref{teorema B}). Finally, in Appendix \\ref{appendix distribution utaugamma y uogamma} we explain how to adapt the results of \\cite{Sam} to the context of arbitrary word hyperbolic groups admitting an Anosov representation.\n\n\n\n\n\n\\subsection*{Acknowledgements}\n\nThese problems were proposed to me by Rafael Potrie and Andr\u00e9s Sambarino. Without their guidance, their support and the (many) helpful discussions this work would not have been possible. I am extremely grateful for this.\n\nThe author also acknowledges Olivier Glorieux, Tal Horesh and Fanny Kassel for several enlightening discussions and comments.\n\n\nFinally, I would like to thank the referee of this article for careful reading and useful suggestions.\n\n\n\\section{Two symmetric spaces associated to $\\textnormal{PSO}(p,q)$} \\label{sec symmetric spaces}\n\\setcounter{equation}{0}\n\n\n\n\n\nFix two integers $p,q\\geq 1$ and let $d:=p+q$. We assume $d>2$. Denote by $\\mathbb{R}^{p,q}$ the vector space $\\mathbb{R} ^d$ endowed with the quadratic form\n\n\\begin{center}\n$ \\langle (x_1,\\dots,x_d),(y_1,\\dots,y_d)\\rangle_{p,q}:= \\displaystyle\\sum_{i=1}^p x_iy_i - \\displaystyle\\sum_{i=p+1}^d x_{i}y_{i}$.\n\\end{center}\n\n\n\n\n\\noindent From now on we denote by $G:=\\textnormal{PSO}(p,q)$ the subgroup of $\\textnormal{PSL}(d,\\mathbb{R})$ consisting on elements whose lifts to $\\textnormal{SL}(d,\\mathbb{R})$ preserve the form $ \\langle \\cdot,\\cdot\\rangle_{p,q}$.\n\n\nFor a subspace $\\pi$ of $\\mathbb{R}^d$ we denote by $\\pi^{\\perp_{p,q}}$ its orthogonal complement with respect to  $\\langle \\cdot,\\cdot\\rangle_{p,q}$, i.e. \n\n\\begin{center}\n$\\pi^{\\perp_{p,q}}:=\\lbrace x\\in\\mathbb{R}^d: \\hspace{0,3cm} \\langle x,y\\rangle_{p,q}=0 \\textnormal{ for all } y\\in \\pi\\rbrace$.\n\\end{center}\n\n\n\n\n\n\nLet $\\mathfrak{g}:=\\mathfrak{so}(p,q)$ be the Lie algebra of $G$. If $\\cdot^t$ denotes the \\textit{usual} transpose operator one has that $\\mathfrak{g}$ equals the set of matrices of the form\n\n\\begin{center}\n$\\left(\\begin{matrix}\nX_1 & X_2\\\\\nX_2^t & X_3\n\\end{matrix}\\right)$\n\\end{center}\n\n\n\\noindent where $X_1$ is of size $p\\times p$, $X_3$ is of size $q\\times q$ and both are skew-symmetric with respect to $\\cdot^t$. The \\textit{Killing form} of $G$ is the symmetric bilinear form $\\kappa$ on $\\mathfrak{g}$ defined by\n\n\\begin{center}\n$\\kappa(X,Y):=\\textnormal{tr}(\\textnormal{ad}_X\\circ\\textnormal{ad}_Y)$,\n\\end{center}\n\n\n\\noindent where $\\textnormal{ad}:\\mathfrak{g}\\longrightarrow\\textnormal{End}(\\mathfrak{g})$ is the adjoint representation.  It can be seen that the following equality holds:\n\n\n\n\n\n\n\n\\begin{center}\n$\\kappa(X,Y)=(d-2)\\textnormal{tr}(XY)$\n\\end{center}\n\n\n\n\n\n\\noindent  (see Helgason {\\cite[p.180 \\& p.189]{Hel}}).\n\n\n\n\n\\subsection{The Riemannian symmetric space $X_G$} \\label{subsec XG}\n\n\n\nA \\textit{Cartan involution} of $G$ is an involutive automorphism $\\tau:G\\longrightarrow G$ such that the bilinear form\n\n\\begin{center}\n$(X,Y)\\mapsto -\\kappa(X,d\\tau(Y))$\n\\end{center}\n\n\n\\noindent is positive definite. The fixed point set $K^\\tau$ of such an involution is a maximal compact subgroup of $G$ (see Knapp \\cite[Theorem 6.31]{Kna}). The \\textit{Riemannian symmetric space} of $G$ is the set consisting on Cartan involutions of $G$. It is denoted by $X_G$ and it is equipped with a natural action of $G$ which is transitive (c.f. \\cite[Corollary 6.19]{Kna}). The stabilizer of $\\tau$ is $K^\\tau$, thus\n\n\\begin{center}\n$G\/K^\\tau\\cong X_G$.\n\\end{center}\n\n\n\\begin{rem}\\label{rem Xg space of qplanes}\nThe space $X_G$ can be identified with the space of $q$-dimensional subspaces of $\\mathbb{R}^d$ on which the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ is negative definite. Explicitly, to a $q$-dimensional negative definite subspace $\\pi$ one associates the Cartan involution of $G$ determined by the inner product of $\\mathbb{R}^d$ which equals $-\\langle\\cdot,\\cdot\\rangle_{p,q}$ (resp. $\\langle\\cdot,\\cdot\\rangle_{p,q}$) on $\\pi$ (resp. $\\pi^{\\perp_{p,q}}$) and for which $\\pi$ and $\\pi^{\\perp_{p,q}}$ are orthogonal.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\nThe choice of a point $\\tau$ in $X_G$ determines a \\textit{Cartan decomposition}\n\n\n\n\\begin{center}\n$\\mathfrak{g}=\\mathfrak{p}^\\tau\\oplus\\mathfrak{k}^\\tau$\n\\end{center}\n\n\n\\noindent where $\\mathfrak{p}^{\\tau}:=\\lbrace d\\tau=-1\\rbrace$ and $\\mathfrak{k}^{\\tau}:=\\lbrace d\\tau=1\\rbrace$. The group $K^\\tau$ is tangent to $\\mathfrak{k}^\\tau$ and one has a $G$-equivariant identification \n\\begin{equation} \\label{eq liep es el tangente}\n\\mathfrak{p}^\\tau\\cong T_\\tau X_G\n\\end{equation}\n\n\\noindent given by $X\\mapsto \\left. \\frac{d}{dt}\\right\\vert_0 \\exp{(tX)}\\cdot\\tau$ (see \\cite[Theorem 3.3 of Ch. IV]{Hel}).\n\n\n\n\n\\begin{ex} \\label{ex explicit cartan involution}\n\nConsider the involution of $G$ defined by $\\tau(g):=(g^{-1})^t$. One sees that $\\tau\\in X_G$ and $\\mathfrak{p}^{\\tau}$ (resp. $\\mathfrak{k}^{\\tau}$) is the set of symmetric matrices (resp. skew-symmetric matrices) in $\\mathfrak{so}(p,q)$. Moreover $K^{\\tau}$ is the subgroup $\\textnormal{PS}(\\textnormal{O}(p)\\times \\textnormal{O}(q))$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{ex}\n\n\n\nThe Killing form $\\kappa$ is positive definite (resp. negative definite) on $\\mathfrak{p}^\\tau$ (resp. $\\mathfrak{k}^\\tau$). Thanks to (\\ref{eq liep es el tangente}) any positive multiple of $\\kappa$ induces a $G$-invariant Riemannian metric on $X_G$. It is well-known (see \\cite[Theorem 4.2 of Ch. IV]{Hel}) that $X_G$ equipped with any of these metrics is a symmetric space which is non-positively curved. \n\n\nWe already mentioned that in this paper we study counting problems not only in $X_G$ but also in $\\mathbb{H}^{p,q-1}$. In the next section we construct $\\mathbb{H}^{p,q-1}$, whose metric is induced by the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$. However, we will see that the Killing form induces as well a $G$-invariant metric on $\\mathbb{H}^{p,q-1}$. These two metrics differ by the scaling factor $(2(d-2))^{-1}$ (see Remark \\ref{rem form on qo} for further precisions). Since we want a simultaneous treatment of the geometry of the spaces $X_G$ and $\\mathbb{H}^{p,q-1}$, we fix the following normalization for the metric on $X_G$:\n\\begin{equation} \\label{eq distance in XG and killing}\nd_{X_G}(\\tau,\\exp(X)\\cdot\\tau):=\\left(\\dfrac{1}{2(d-2)}\\kappa(X,X)\\right)^{\\frac{1}{2}}\n\\end{equation}\n\n\n\n\\noindent for all $\\tau\\in X_G$ and all $X\\in\\mathfrak{p}^\\tau$.\n\n\n\n\n\n\\subsection{The pseudo-Riemannian hyperbolic space $\\mathbb{H}^{p,q-1}$} \\label{subsec hpq}\n\n\n\n\n\n\n\nLet \n\n\n\\begin{center}\n$\\hat{\\mathbb{H}}^{p,q-1}:=\\lbrace \\hat{o}\\in\\mathbb{R}^{p,q}: \\hspace{0,3cm} \\langle \\hat{o},\\hat{o}\\rangle_{p,q}=-1 \\rbrace$\n\\end{center}\n\n\\noindent endowed with the restriction of the form $\\langle \\cdot,\\cdot\\rangle_{p,q}$ to tangent spaces. This metric induces on\n\n\\begin{center}\n$\\mathbb{H}^{p,q-1}:=\\lbrace o=[\\hat{o}]\\in\\mathbb{P}(\\mathbb{R}^{p,q}): \\hspace{0,3cm} \\langle \\hat{o},\\hat{o}\\rangle_{p,q}<0 \\rbrace$\n\\end{center}\n\n\\noindent a pseudo-Riemannian structure invariant under the projective action of $G$. This space is called the \\textit{pseudo-Riemannian hyperbolic space of signature $(p,q-1)$}. The \\textit{boundary} of $\\mathbb{H}^{p,q-1}$ is the space of \\textit{isotropic lines} defined by\n\n\n\\begin{center}\n$\\partial\\mathbb{H}^{p,q-1}:=\\lbrace \\xi=[\\hat{\\xi}]\\in\\mathbb{P}(\\mathbb{R}^{p,q}): \\hspace{0,3cm} \\langle\\hat{\\xi},\\hat{\\xi}\\rangle_{p,q}=0 \\rbrace$.\n\\end{center}\n\n\n\\noindent It is also equipped with the natural (transitive) action of $G$. If we denote by $P_1^{p,q}$ the (parabolic) subgroup of $G$ stabilizing an isotropic line, then\n\n\\begin{center}\n$\\partial\\mathbb{H}^{p,q-1}\\cong G\/P_1^{p,q}$.\n\\end{center}\n\n\n\n\n\n\n\\subsubsection{\\textnormal{\\textbf{Structure of symmetric space}}} \\label{subsubsec struc sym hpq}\n\n\n\nThe action of $G$ on $\\mathbb{H}^{p,q-1}$ is transitive, hence $\\mathbb{H}^{p,q-1}\\cong G\/H^o$ where $H^o$ is the stabilizer in $G$ of the point $o\\in\\mathbb{H}^{p,q-1}$. For instance, when $o=[0,\\dots,0,1]\\in\\mathbb{H}^{p,q-1}$ one has\n\n\n\\begin{center}\n$H^o=\\left\\lbrace \\left[\\begin{matrix}\n\\hat{g} & 0\\\\\n0 & 1\n\\end{matrix}\\right]\\in G: \\hspace{0,3cm} \\hat{g}\\in \\textnormal{O}(p,q-1)  \\right\\rbrace$.\n\\end{center} \n\n\n\nFix any $o\\in\\mathbb{H}^{p,q-1}$. Since $o$ and $o^{\\perp_{p,q}}$ are transverse we can consider the matrix \n\n\\begin{center}\n$J^o:=\\textnormal{id}_o\\oplus\\left(-\\textnormal{id}_{o^{\\perp_{p,q}}}\\right)$. \n\\end{center}\n\n\\noindent It follows that $H^o=\\textnormal{Fix}(\\sigma^o)$ where $\\sigma^o$ is the involution of $G$ defined by\n\\begin{equation} \\label{eq involution}\n\\sigma^o(g):=J^ogJ^o.\n\\end{equation}\n\n\n\\noindent Thus $\\mathbb{H}^{p,q-1}\\cong G\/H^o$ is a symmetric space of $G$.\n\n\n\n\\begin{rem} \\label{rem form on qo}\nLet $o\\in\\mathbb{H}^{p,q-1}$ and $\\mathfrak{q}^o:=\\lbrace d\\sigma^o=-1 \\rbrace$. There exists a $G$-equivariant identification\n\n\n\\begin{center}\n$\\mathfrak{q}^o\\cong  T_o\\mathbb{H}^{p,q-1}$\n\\end{center}\n\n\n\n\n\\noindent given by $X\\mapsto \\left. \\frac{d}{dt}\\right\\vert_0 \\exp{(tX)}\\cdot o$. We denote by $\\langle\\cdot,\\cdot\\rangle$ the pull-back of the $(p,q-1)$-form on $T_o\\mathbb{H}^{p,q-1}$ under this map and, for $X\\in\\mathfrak{q}^o$, we set $\\vert X\\vert:=\\langle X,X\\rangle$\\footnote{This number can be positive, negative or zero for $X\\neq 0$ in $\\mathfrak{q}^o$.}. \n\n\nRecall that $\\kappa$ is the Killing form of $\\mathfrak{so}(p,q)$. From explicit computations (that we omit) one can conclude that the equality\n\\begin{equation} \\label{eq form on qo and killing}\n\\vert X\\vert=\\dfrac{1}{2(d-2)}\\kappa(X,X)\n\\end{equation}\n\\noindent holds for every $X\\in\\mathfrak{q}^o$. This justifies the choice of normalization made in Subsection \\ref{subsec XG}.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\n\n\\begin{rem} \\label{rem action of Ho en el tangente en SOpqmenos1}\n\n\nLet $o\\in\\mathbb{H}^{p,q-1}$. Then the action of the connected component of $H^o$ containing the identity is conjugate to the action of $\\textnormal{SO}(p,q-1)$ on $\\mathbb{R}^{p,q-1}$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\\subsubsection{\\textnormal{\\textbf{Geodesics of $\\mathbb{H}^{p,q-1}$}}}\\label{subsub geod hpq}\n\n\n\nGeodesics of $\\mathbb{H}^{p,q-1}$ are the intersections of straight lines of $\\mathbb{P}(\\mathbb{R}^{p,q})$ with $\\mathbb{H}^{p,q-1}$. They are divided in three types:\n\n\\begin{itemize}\n\\item \\textit{Space-like geodesics:} associated to 2-dimensional subspaces of $\\mathbb{R}^d$ on which $\\langle\\cdot, \\cdot\\rangle_{p,q}$ has signature $(1,1)$. They have positive speed and meet the boundary $\\partial\\mathbb{H}^{p,q-1}$ in two distinct points.\n\\item \\textit{Time-like geodesics:} associated to 2-dimensional subspaces of $\\mathbb{R}^d$ on which $\\langle\\cdot, \\cdot\\rangle_{p,q}$ has signature $(0,2)$. They have negative speed and do not meet the boundary (they are closed). \n\n\\item \\textit{Light-like geodesics:} associated to 2-dimensional subspaces of $\\mathbb{R}^d$ on which $\\langle\\cdot, \\cdot\\rangle_{p,q}$ has signature $(0,1)$, that is, is degenerate but has a negative eigenvalue. They have zero speed and meet the boundary in a single point.\n\\end{itemize}\n\n\n\n\n\\noindent For a point $o\\in\\mathbb{H}^{p,q-1}$ we denote by $\\mathscr{C}_o^{0}$ (resp. $\\mathscr{C}_o^{>}$) the set of points of $\\mathbb{H}^{p,q-1}$ that can be joined with $o$ by a light-like (resp. space-like) geodesic. Its closure in $\\mathbb{P}(\\mathbb{R}^{p,q})$ is denoted by $\\overline{\\mathscr{C}_o^0}$ (resp. $\\overline{\\mathscr{C}_o^>}$). \n\n\\subsubsection{\\textnormal{\\textbf{Light-cones}}}\\label{subsub light cones}\n\n\nThe following lemma is proved by Glorieux-Monclair in \\cite[Lemma 2.2]{GM}.\n\n\n\n\\begin{lema}\\label{lema geod between o and xi}\nLet $o\\in\\mathbb{H}^{p,q-1}$. Then $\\overline{\\mathscr{C}_o^0}\\cap\\partial\\mathbb{H}^{p,q-1}=o^{\\perp_{p,q}}\\cap\\partial\\mathbb{H}^{p,q-1}$.\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{lema}\n\n\n\n\\subsubsection{\\textnormal{\\textbf{Lenghts of space-like geodesics}}}\\label{subsub lengths of spacelike geod}\n\n\n\nFor a point $o'$ in $\\mathscr{C}_o^>$ we denote by $\\ell_{o,o'}$ the length of the geodesic segment connecting $o$ with $o'$. For instance the geodesic\n\\begin{equation} \\label{eq geod alpha}\ns\\mapsto [\\sinh(s),0\\dots,0,\\cosh(s)]\\in\\mathbb{H}^{p,q-1}\n\\end{equation}\n\n\n\\noindent is parametrized by arc-length.\n\n\n\n\n\n\\subsubsection{\\textnormal{\\textbf{Space-like copies of $\\mathbb{H}^p$}}} \\label{subsub space copies Hp}\nLet $\\pi$ be a $(p+1)$-dimensional subspace of $\\mathbb{R}^d$ of signature $(p,1)$. Then $\\mathbb{P}(\\pi)\\cap\\mathbb{H}^{p,q-1}$ identifies with\n\n\n\n\n\\begin{center}\n$\\lbrace o=[\\hat{o}]\\in\\mathbb{P}(\\mathbb{R}^{p,1}) \\hspace{0,3cm} \\langle \\hat{o},\\hat{o}\\rangle_{p,1}<0 \\rbrace$.\n\\end{center}\n\n\\noindent It follows that $\\mathbb{P}(\\pi)\\cap\\mathbb{H}^{p,q-1}$ is a totally geodesic isometric copy of $\\mathbb{H}^p$ inside $\\mathbb{H}^{p,q-1}$. Moreover this sub-manifold is space-like, in the sense that any of its tangent vectors has positive norm.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{\\textnormal{\\textbf{End points of space-like geodesics}}} \\label{subsub endign of spacelike}\n\n\n\nLet $o$ be a point in $\\mathbb{H}^{p,q-1}$. Note that $J^o$ preserves the form $\\langle\\cdot,\\cdot\\rangle_{p,q}$ and thus acts on $\\partial\\mathbb{H}^{p,q-1}$. Set\n\n\\begin{center}\n$\\mathcal{O}^o:=\\lbrace \\xi\\in\\partial\\mathbb{H}^{p,q-1}:\\hspace{0,3cm} J^o\\cdot \\xi\\neq \\xi\\rbrace$. \n\\end{center}\n\n\n\n\\begin{prop} \\label{prop fijos de Jo en borde}\nLet $o\\in\\mathbb{H}^{p,q-1}$. Then the following equalities hold:\n\n\n\n\n\\begin{equation*}\n\\begin{split}\n\\mathcal{O}^o & = \\lbrace \\xi\\in\\partial\\mathbb{H}^{p,q-1}: \\hspace{0,3cm} J^o\\cdot \\xi\\notin\\xi^{\\perp_{p,q}}\\rbrace\\\\ & = \\partial\\mathbb{H}^{p,q-1}\\setminus o^{\\perp_{p,q}} \\\\ & =\\partial\\mathbb{H}^{p,q-1}\\setminus \\overline{\\mathscr{C}_o^0}.\n\\end{split}\n\\end{equation*}\n\n\n\n\n\\end{prop}\n\n\nWe conclude that, unless $q=1$, the set $\\mathcal{O}^o$ is not the whole boundary of $\\mathbb{H}^{p,q-1}$. \n\n\n\\begin{proof}[Proof of Proposition \\ref{prop fijos de Jo en borde}]\n\n\nThe equality $\\partial\\mathbb{H}^{p,q-1}\\setminus o^{\\perp_{p,q}}=\\partial\\mathbb{H}^{p,q-1}\\setminus \\overline{\\mathscr{C}_o^0}$ is a consequence of Lemma \\ref{lema geod between o and xi}. The other equalities follow from definitions.\n\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Generalized Cartan decompositions} \\label{sec generalized cartan}\n\\setcounter{equation}{0}\n\n\n\n\nThe goal of this section is to define two generalized Cartan projections and to provide a link between them and Theorems \\ref{teorema A} and \\ref{teorema B}. The first one (Subsection \\ref{subsec KexpliebH}) is called the \\textit{polar projection} of $G$ and it is well-known. The second one (Subsection \\ref{subsec HexpliebH}) is new and can only be defined for elements in $G$ that satisfy some special property with respect to the choice of the basepoint $o$.\n\n\n\n\n\\subsection{Notations} \\label{subsec notations generalized cartan}\n\n\n\n\nThrough this section we fix a point $o\\in\\mathbb{H}^{p,q-1}$ and let $H^o=\\textnormal{Fix}(\\sigma^o)$ be its stabilizer in $G$ (c.f. Subsection \\ref{subsubsec struc sym hpq}). Let $\\mathfrak{h}^o$ be the Lie algebra of fixed points of $d\\sigma^o$ and $\\mathfrak{q}^o:=\\lbrace d\\sigma^o=-1\\rbrace$. One has the following decomposition of the Lie algebra $\\mathfrak{g}$ of $G$:\n\n\\begin{center}\n\n$\\mathfrak{g}=\\mathfrak{h}^o\\oplus\\mathfrak{q}^o$.\n\n\\end{center}\n\n\\noindent Moreover, this decomposition is orthogonal with respect to the Killing form of $\\mathfrak{g}$.\n\n\nLet $\\tau$ be a Cartan involution  commuting with $\\sigma^o$: such involutions always exist and two of them differ by conjugation by an element in $H^o$ (see Matsuki \\cite[Lemma 4]{Mat}). Let $K^{\\tau}:=\\textnormal{Fix}(\\tau)$, which is a maximal compact subgroup of $G$. Let $\\mathfrak{p}^\\tau$ and $\\mathfrak{k}^\\tau$ be the subspaces defined in Subsection \\ref{subsec XG}. As $\\sigma^o$ and $\\tau$ commute, the following holds:\n\n\n\\begin{center}\n$\\mathfrak{g}=(\\mathfrak{p}^{\\tau}\\cap\\mathfrak{q}^o) \\oplus(\\mathfrak{p}^{\\tau}\\cap\\mathfrak{h}^o)\\oplus(\\mathfrak{k}^{\\tau}\\cap\\mathfrak{q}^o)\\oplus(\\mathfrak{k}^{\\tau}\\cap\\mathfrak{h}^o)$.\n\\end{center}\n\n\n\n\\noindent Let $\\mathfrak{b}\\subset\\mathfrak{p}^{\\tau}\\cap\\mathfrak{q}^o$ be a (necessarily abelian) maximal subalgebra: two of them differ by conjugation by an element in $K^{\\tau}\\cap H^o$. We will consider closed Weyl chambers in $\\mathfrak{b}$ corresponding to positive systems of restricted roots of $\\mathfrak{b}$ in $\\mathfrak{g}^{\\sigma^o\\tau}:= (\\mathfrak{p}^{\\tau}\\cap\\mathfrak{q}^o)\\oplus (\\mathfrak{k}^{\\tau}\\cap\\mathfrak{h}^o)$. These closed Weyl chambers will be denoted by $\\mathfrak{b}^+$.\n\n\n\n\n\n\n\n\\begin{ex} \\label{ex explicit example}\nLet $o=[0,\\dots,0,1]$. Then $H^o$ is the upper left corner embedding of $\\textnormal{O}(p,q-1)$ in $G$ and the involution $\\sigma^o$ is obtained by conjugation by $J^o=\\textnormal{diag}(-1,\\dots,-1,1)$. One sees that $\\mathfrak{h}^o$ equals the upper left corner embedding of $\\mathfrak{so}(p,q-1)$ in $\\mathfrak{so}(p,q)$ and that\n\n\\begin{center}\n\n$\\mathfrak{q}^o=\\left\\lbrace\\left(\\begin{matrix}\n0 & 0 & Y_1\\\\\n0 & 0 & Y_2\\\\\nY_1^t & -Y_2^t & 0\n\\end{matrix}\\right): \\hspace{0,3cm} Y_1\\in \\textnormal{M}(p\\times 1,\\mathbb{R}), \\hspace{0,3cm} Y_2\\in \\textnormal{M}((q-1)\\times 1,\\mathbb{R})\\right\\rbrace$.\n\\end{center}\n\n\nLet $\\tau$ be the Cartan involution of Example \\ref{ex explicit cartan involution}. One observes that $\\tau$ commutes with $\\sigma^o$ and\n\n\n\n\n\\begin{center}\n\n$\\mathfrak{p}^{\\tau}\\cap\\mathfrak{q}^o=\\left\\lbrace X\\in\\mathfrak{q}^o: \\hspace{0,3cm} Y_2=0\\right\\rbrace$ \\hspace{0,5cm} $\\mathfrak{k}^{\\tau}\\cap\\mathfrak{q}^o=\\left\\lbrace X\\in\\mathfrak{q}^o: \\hspace{0,3cm} Y_1=0\\right\\rbrace$ .\n\\end{center}\n\n\n\n\n\\noindent Pick $\\mathfrak{b}$ to be the subset of $\\mathfrak{p}^{\\tau}\\cap\\mathfrak{q}^o$ of matrices with $Y_1$ of the form\n\n\\begin{center}\n$\\left(\\begin{matrix}\ns\\\\\n0\\\\\n\\vdots\\\\\n0\n\\end{matrix}\\right)$\n\\end{center}\n\n\n\\noindent for some $s\\in\\mathbb{R}$: this is a maximal subalgebra of $\\mathfrak{p}^{\\tau}\\cap\\mathfrak{q}^o$. A closed Weyl chamber $\\mathfrak{b}^+$ is defined by the inequality $ s\\geq 0$. \n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{ex}\n\n\n\n\n\n\n\nThe following remark will be used repeatedly in the sequel.\n\n\n\\begin{rem} \\label{rem J preserva norma}\nEven though $G$ does not act on $\\mathbb{R}^d$, it makes sense to ask if an element $g$ of $G$ preserves a norm on $\\mathbb{R}^d$ (this notion does not depend on the choice of a lift of $g$ to $\\textnormal{SL}(d,\\mathbb{R})$). Given a Cartan involution $\\tau$ commuting with $\\sigma^o$, let $\\Vert\\cdot\\Vert_\\tau$ be a norm on $\\mathbb{R}^d$ preserved by $K^\\tau$. We claim that this norm is preserved by $J^o$. Indeed, this is obvious for the choices of Example \\ref{ex explicit example} and follows in general by conjugating by an element $g$ in $G$ that takes $[0,\\dots,0,1]$ to the point $o$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\\subsection{The sub-manifold $S^o$}\\label{subsec So} Define\n\n\\begin{center}\n$S^o:=\\lbrace\\tau\\in X_G:\\hspace{0,3cm} \\tau\\sigma^o=\\sigma^o\\tau\\rbrace$.\n\\end{center}\n\n\n\n\n\\begin{rem}\\label{rem So space of hp trough o}\nRecall from Remark \\ref{rem Xg space of qplanes} that $X_G$ can be identified with the space of $q$-dimensional negative definite subspaces of $\\mathbb{R}^d$. Under this identification $S^o$ corresponds to the set of subspaces that contain the line $o$. By considering the $\\langle\\cdot,\\cdot\\rangle_{p,q}$-orthogonal complement we see that $S^o$ parametrizes the space of totally geodesic space-like copies of $\\mathbb{H}^{p}$ inside $\\mathbb{H}^{p,q-1}$ passing through $o$ (c.f. Subsection \\ref{subsub space copies Hp}).\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\nUsing the fact that two elements of $S^o$ differ by conjugation by an element in $H^o$ one observes that for any $\\tau\\in S^o$ the following holds\n\n\\begin{center}\n$S^o=H^o\\cdot\\tau$.\n\\end{center}  \n\n\\noindent Further, the group $H^o$ has several connected components but one can see that the connected component containing the identity acts transitively on $S^o$. Hence $S^o$ is connected and one can show that\n\n\\begin{center}\n$S^o=\\exp(\\mathfrak{p}^{\\tau}\\cap\\mathfrak{h}^o)\\cdot\\tau$. \n\\end{center}\n\n\n\\noindent It follows that $S^o$ is a totally geodesic sub-manifold of $X_G$ and $T_\\tau S^o\\cong \\mathfrak{p}^{\\tau}\\cap\\mathfrak{h}^o$ (see \\cite[Theorem 7.2 of Ch. IV]{Hel}). \n\n\n\n\n\n\n\n\\subsection{$K\\exp(\\mathfrak{b}^+)H$-decomposition}\\label{subsec KexpliebH}\n\n\n\nFor the rest of this section we fix a Cartan involution $\\tau\\in S^o$, a maximal subalgebra $\\mathfrak{b}\\subset\\mathfrak{p}^\\tau\\cap\\mathfrak{q}^o$ and a closed Weyl chamber $\\mathfrak{b}^+\\subset\\mathfrak{b}$. By Schlichtkrull \\cite[Proposition 7.1.3]{Sch} the following decomposition of $G $ holds:\n\\begin{equation}\\label{eq polar decomposition}\nG=K^{\\tau}\\exp(\\mathfrak{b}^+)H^o\n\\end{equation}\n\\noindent where the $\\exp(\\mathfrak{b}^+)$-component is uniquely determined and one can define\n\\begin{equation} \\label{eq def nu}\nb^\\tau:G\\longrightarrow\\mathfrak{b}^+\n\\end{equation}\n\n\n\\noindent by taking the $\\textnormal{log}$ of this component. This is a continuous map called the \\textit{polar projection} of $G$ associated to the choice of $\\tau$ and $\\mathfrak{b}^+$. It generalizes the usual Cartan projection of $G$.\n\n\\begin{rem} \\label{rem btau is proper}\n\nNote that $b^\\tau$ is not proper (unless $q=1$). However it descends to a map $\\mathbb{H}^{p,q-1}\\cong G\/H^o\\longrightarrow\\mathfrak{b}^+$ which, by definition, is proper.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\nWe now discuss geometric interpretations of the polar projection $b^\\tau$. The geometric interpretation in $\\mathbb{H}^{p,q-1}$ follows Kassel-Kobayashi \\cite[p.151]{KK}, while the geometric interpretation in $X_G$ is inspired by the work of Oh-Shah \\cite{OS} for the case $p=1$ and $q=3$.\n\nLet us begin with the interpretation in the pseudo-Riemannian setting. By Remark \\ref{rem So space of hp trough o}, the choice of $\\tau\\in S^o$ determines a totally geodesic space-like copy of the $p$-dimensional hyperbolic space, inside $\\mathbb{H}^{p,q-1}$ and passing through $o$. We denote this copy by $\\mathbb{H}^{p}_\\tau$. From explicit computations one can show that \n\n\\begin{center}\n$\\mathbb{H}^{p}_\\tau=\\exp(\\mathfrak{p}^\\tau\\cap\\mathfrak{q}^o)\\cdot o$.\n\\end{center}\n\n\n\\noindent In particular $\\mathbb{H}^{p}_\\tau$ contains the geodesic ray $\\exp(\\mathfrak{b}^+)\\cdot o$ starting from $o$. Equality (\\ref{eq polar decomposition}) tell us that for every $g$ in $G$ the point $g\\cdot o$ lies in the $K^\\tau$-orbit of $o_g:=\\exp(b^\\tau(g))\\cdot o$ (see Figure \\ref{fig interp geom polar}). The geometric interpretation of the polar projection is now clear: the number $\\vert b^\\tau(g)\\vert^{\\frac{1}{2}}$ equals the length of the geodesic segment connecting $o$ with $o_g$\\footnote{Recall that $\\vert\\cdot\\vert$ is the form on $\\mathfrak{q}^o$ defined in Remark \\ref{rem form on qo}.}.\n\n\n\n\n\n\n \n\n\n\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\scalebox{0.8}{%\n\\begin{overpic}[scale=1, width=1\\textwidth, tics=5]{fig1.pdf}\n\n \\put (30,79) { \\Large$g\\cdot o$}\n  \\put (87,92) { \\Large\\textcolor{blue}{$K^\\tau \\cdot o_g$}}\n    \\put (78,50) { \\Large\\textcolor{red}{$\\exp(\\mathfrak{b}^+)\\cdot o$}}\n        \\put (65,53) { \\Large$o_g$}\n        \\put (48,52) { \\Large$o$}\n        \\put (45,35) { \\Large$\\mathbb{H}^{p,q-1}$}\n        \\put (29,51) { \\Large$\\mathbb{H}^{p}_\\tau$}\n        \\put (5,35) { \\Large$\\partial\\mathbb{H}^{p,q-1}$}\n \\end{overpic}}\n \n\\hspace{0,3cm}\n\n\\begin{changemargin}{3cm}{3cm}    \n    \\caption{Geometric interpretation of polar projection in $\\mathbb{H}^{p,q-1}$.}\n  \\label{fig interp geom polar}\n\\end{changemargin}\n\n  \\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\nWe now turn our attention to the Riemannian symmetric space $X_G$.\n\n\n\\begin{prop} \\label{prop interpretation of btau in symg}\nFor every $g$ in $G$ one has \n\n\\begin{center}\n$\\vert b^\\tau(g)\\vert^{\\frac{1}{2}}=d_{X_G}(g^{-1}\\cdot \\tau,S^o)$.\n\\end{center}\n\\end{prop}\n\n\n\\begin{proof}\nThe function $g\\mapsto d_{X_G}(g^{-1}\\cdot\\tau,S^o)$ is $K^{\\tau}$-invariant on the left and $H^o$-invariant on the right, hence it suffices to check that the equality of the statement holds when $g=\\exp(X)$ for some $X\\in\\mathfrak{b}^+$.\n\n\nSince $X_G$ is non-positively curved, there exists a unique geodesic through $\\exp(-X)\\cdot \\tau$ which is orthogonal to $S^o=\\exp(\\mathfrak{p}^\\tau\\cap\\mathfrak{h}^o)\\cdot \\tau$. This geodesic is $\\exp(\\mathfrak{b})\\cdot \\tau$ and intersects $S^o$ in $\\tau$, hence\n\n\n\n\n\\begin{center}\n$d_{X_G}(\\exp(-X)\\cdot\\tau,S^o)=d_{X_G}(\\exp(-X)\\cdot\\tau,\\tau)$.\n\\end{center}\n\n\n\\noindent Thanks to Remark \\ref{rem form on qo} and (\\ref{eq distance in XG and killing}) the proof is complete.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\nWe finish this subsection with a linear algebraic interpretation of the polar projection. Let $\\Vert\\cdot\\Vert_{\\tau}$ be a norm on $\\mathbb{R}^d$ invariant under the action of $K^\\tau$.\n\n\\begin{prop} \\label{prop computing nu}\nFor every $g$ in $G$ one has\n\\begin{center}\n$ \\vert b^\\tau(g)\\vert^{\\frac{1}{2}}=\\frac{1}{2}\\log\\Vert J^o gJ^og^{-1}\\Vert_{\\tau} $.\n\\end{center}\n\n\n\\end{prop}\n\n\n\n\n\n\\begin{proof}\n\nWe prove the proposition for the particular choices of Example \\ref{ex explicit example}, the general case follows from this one by conjugating by appropriate elements of $G$.\n\n\nBy Remark \\ref{rem J preserva norma} the matrix $J^o$ preserves $\\Vert\\cdot\\Vert_\\tau$ thus  \n\n\\begin{center}\n$\\frac{1}{2}\\log\\Vert J^o gJ^og^{-1}\\Vert_{\\tau}=\\frac{1}{2}\\log\\Vert gJ^og^{-1}\\Vert_{\\tau}$.\n\\end{center}\n\n\n\\noindent The map $g\\mapsto\\frac{1}{2}\\log\\Vert  gJ^og^{-1}\\Vert_{\\tau}$ is $K^{\\tau}$-invariant on the left and $H^o$-invariant on the right, hence it remains to check that the equality of the statement holds on $\\exp(\\mathfrak{b}^+)$. Let $X\\in\\mathfrak{b}^+$, that is,\n\n\n\n\n\n\n\\begin{center}\n$X=\\left(\\begin{matrix}\n &   & &  & s\\\\\n  & &   & 0 & \\\\\n & & \\iddots &  & \\\\\n &  0  &  &  & \\\\\ns &  & & &   \n\\end{matrix}\\right)$\n\\end{center}\n\n\n\\noindent for some $s\\geq 0$. Since $X\\in\\mathfrak{q}^o$, one has $J^o\\exp(-X)=\\exp(X)J^o$ and thus\n\n\n\\begin{center}\n$\\vert X\\vert^\\frac{1}{2}=s=\\frac{1}{2}\\log\\Vert  \\exp(X)J^o\\exp(-X)\\Vert_{\\tau}$.\n\\end{center}\n\\end{proof}\n\n\n\n\n\n\n\\subsection{$H\\exp(\\mathfrak{b}^+)H$-decomposition}\\label{subsec HexpliebH}\n\n\nRecall from Subsection \\ref{subsub geod hpq} the definition of the set $\\mathscr{C}^>_o$ and define\n\n\n\\begin{center}\n$\\mathscr{C}_{o,G}^>:=\\lbrace g\\in G:\\hspace{0,3cm} g\\cdot o\\in \\mathscr{C}_{o}^>\\rbrace$. \n\\end{center}\n\n\n\n\n\\begin{prop}\\label{prop HBH}\nFor every $g$ in $\\mathscr{C}_{o,G}^>$ one can write\n\n\\begin{center}\n$g=h\\exp (X)h' $\n\\end{center}\n\n\n\\noindent for some $h,h'\\in H^o$ and a unique $X\\in\\mathfrak{b}^+$.\n\\end{prop}\n\n\nIt is clear that this decomposition of $g$ can only hold when $g\\in\\mathscr{C}_{o,G}^>$.\n\n\\begin{proof}[Proof of Proposition \\ref{prop HBH}]\n\nTake $h$ in $H^o$ such that $h^{-1}g\\cdot o\\in\\exp(\\mathfrak{b}^+)\\cdot o$. There exists then $X\\in\\mathfrak{b}^+$ and $h'\\in H^o$ such that $h^{-1}g=\\exp(X)h'$. Note that $X$ is unique since it is determined by the length of the geodesic segment connecting $o$ with $g\\cdot o$.\n\n\\end{proof}\n\n\nWe define the map\n\\begin{equation} \\label{eq def de bo}\nb^o:\\mathscr{C}_{o,G}^>\\longrightarrow\\mathfrak{b}^+: \\hspace{0,3cm} g\\mapsto b^o(g)\n\\end{equation}\n\n\\noindent where $g=h\\exp(b^o(g))h'$ for some $h,h'\\in H^o$. Note that $b^o$ descends to the quotient $\\mathscr{C}_{o}^>$ but this map is not proper (compare with Remark \\ref{rem btau is proper}).\n\n\n\n\n\n\n\n\n\\begin{prop}\\label{prop ell dXG y vertboverto}\nFor every $g$ in $\\mathscr{C}_{o,G}^>$ one has\n\n\\begin{center}\n$\\ell_{o,g\\cdot o}=\\vert b^o(g)\\vert^{\\frac{1}{2}}=d_{X_G}(S^o,g\\cdot S^o)$.\n\\end{center}\n\\end{prop}\n\n\n\\begin{proof}\n\n\nThe first equality was already discussed in the proof of Proposition \\ref{prop HBH}. For the second one write $g=h\\exp(b^o(g))h'$. Since $S^o=H^o\\cdot \\tau$ we have\n\n\\begin{center}\n$d_{X_G}(S^o,h\\exp(b^o(g))h'\\cdot S^o)=d_{X_G}(H^o\\cdot\\tau,\\exp(b^o(g))H^o\\cdot\\tau)$.\n\\end{center}\n\n\nSet $X:=b^o(g)$. If $X=0$ there is nothing to prove, so assume $X\\neq 0$. In that case $H^o\\cdot \\tau$ is disjoint from $\\exp(X)H^o\\cdot\\tau$: since the action of $\\mathfrak{b}$ on the geodesic $\\exp(\\mathfrak{b})\\cdot\\tau$ is free, this follows from the fact that $X_G$ is non-positively curved and the fact that $\\exp(\\mathfrak{b})\\cdot\\tau$ intersects orthogonally $H^o\\cdot\\tau$ (resp. $\\exp(X)H^o\\cdot\\tau$) in $\\tau$ (resp. $\\exp(X)\\cdot\\tau$).\n\n\n\\begin{cla} \\label{claim in prop interp geom of bo}\n\nTake $\\tau'\\in H^o\\cdot\\tau$ and $\\tau''\\in \\exp(X)H^o\\cdot\\tau$. Then the following holds:\n\\begin{center}\n$d_{X_G}(\\tau',\\tau'')\\geq d_{X_G}(\\tau,\\exp(X)\\cdot\\tau)$.\n\\end{center}\n\\end{cla}\n\n\n\\begin{proof}[Proof of Claim \\ref{claim in prop interp geom of bo}]\n\nLet $\\beta_1\\subset H^o\\cdot \\tau$ (resp. $\\beta_2\\subset \\exp(X)H^o\\cdot\\tau$) be the unit-speed geodesic connecting $\\beta_1(0)=\\tau$ (resp. $\\beta_2(0)=\\exp(X)\\cdot \\tau$) with $\\tau'$ (resp. $\\tau''$). Then $\\beta_1$ and $\\beta_2$ are disjoint and from the fact that $X_G$ is non-positively curved follows that the map\n\\begin{center}\n$(t,s)\\mapsto d_{X_G}(\\beta_1(t),\\beta_2(s))$\n\\end{center}\n\\noindent is smooth (see Petersen \\cite[p.129]{Pet}). Moreover, since $\\exp(\\mathfrak{b})\\cdot \\tau$ is orthogonal both to $H^o\\cdot\\tau$ and $\\exp(X)H^o\\cdot\\tau$ we conclude that the differential at $(0,0)$ of this map is zero. \n\nTake $t_0>0$ such that $\\beta_1(t_0)=\\tau'$ and a positive $a$ such that the geodesic $t\\mapsto\\beta_2(at)$ equals $\\tau''$ in $t_0$. By Busemann \\cite[Theorem 3.6]{Bus} the map\n\n\\begin{center}\n$t\\mapsto d_{X_G}(\\beta_1(t),\\beta_2(at))$\n\\end{center}\n\n\n\n\\noindent is convex. Since it has a critical point at $t=0$ the proof of the claim is finished.\n\n\n\\end{proof}\n\nThanks to Remark \\ref{rem form on qo} and (\\ref{eq distance in XG and killing}) the proof of Proposition \\ref{prop ell dXG y vertboverto} is now complete.\n\n\n\\end{proof}\n\n\n\n\n\n\n\nRecall that $\\lambda_1(g)$ denotes the logarithm of the spectral radius of $g\\in G$.\n\n\n\n\\begin{prop}\\label{prop linear alg interpr of bo}\nFor every $g$ in $\\mathscr{C}_{o,G}^>$ one has\n\n\\begin{center}\n$\\vert b^o(g)\\vert^{\\frac{1}{2}}=\\frac{1}{2}\\lambda_1(J^ogJ^og^{-1})$.\n\\end{center}\n\n\n\n\\end{prop}\n\n\\begin{proof}\n\nIt suffices to prove the proposition for the choices of $o$ and $\\mathfrak{b}^+$ of Example \\ref{ex explicit example}. Write $g=h\\exp(b^o(g))h' $ with\n\n\n\n\\begin{center}\n$b^o(g)=\\left(\\begin{matrix}\n &  & &   & s\\\\\n & & & 0 & \\\\\n & &  \\iddots &    & \\\\\n & 0 & & & \\\\\ns &  &   & &\n\\end{matrix}\\right)$\n\\end{center}\n\n\n\\noindent for some $s\\geq 0$. We have $\\vert b^o(g)\\vert^\\frac{1}{2}=s$. On the other hand, $J^o$ commutes with elements of $H^o$ and thus the number $\\frac{1}{2}\\lambda_1(J^ogJ^og^{-1})$ equals to\n\n\n\\begin{center}\n$\\frac{1}{2}\\lambda_1(J^oh\\exp(b^o(g))J^o\\exp(b^o(g))^{-1}h^{-1})=\\frac{1}{2}\\lambda_1(J^o\\exp(b^o(g))J^o\\exp(b^o(g))^{-1})$.\n\\end{center}\n\n\n\n\\noindent Since $b^o(g)\\in\\mathfrak{q}^o$ we have $J^o\\exp(b^o(g))^{-1}=\\exp(b^o(g))J^o$ and the proof is complete.\n\\end{proof}\n\n\n\n\\section{Proximality} \\label{sec proximality}\n\\setcounter{equation}{0}\n\n\n\nIn this section we recall basic facts on product of proximal matrices, the main one being Benoist's Theorem \\ref{teo benoist}. This results are well-known but we provide proofs for those which are not explicitly stated in the literature (the reader familiarized with these concepts may skip this section). Standard references are the works of Benoist \\cite{Ben3,Ben4,Ben1}. \n\n\n\\subsection{Notations and basic definitions}\\label{subsec notation definitions in sec proximality}\n\n\n\nA norm $\\Vert \\cdot \\Vert$ on $\\mathbb{R}^d$ will be fixed in the whole section. For $\\xi_{1},\\xi_2\\in\\mathbb{P}(\\mathbb{R}^d)$ define the distance\n\n\\begin{center}\n$d(\\xi_1,\\xi_2):=\\inf\\lbrace \\Vert v_{\\xi_1}-v_{\\xi_2}\\Vert: \\hspace{0,3cm} v_{\\xi_i}\\in \\xi_i \\textnormal{ and } \\Vert v_{\\xi_i}\\Vert =1 \\textnormal{ for all } i=1,2 \\rbrace$.\n\\end{center}\n\n\n\\noindent Let $\\mathsf{Gr}_{d-1}(\\mathbb{R}^d)$ be the Grassmannian of $(d-1)$-dimensional subspaces of $\\mathbb{R}^d$. There exists a $G$-equivariant identification $\\mathbb{P}((\\mathbb{R}^d)^*)\\longrightarrow \\mathsf{Gr}_{d-1}(\\mathbb{R}^d) $ given by \n\n\\begin{center}\n$\\theta\\mapsto\\ker\\theta$\n\\end{center}\n\n\n\\noindent where the action of $G$ on the left side is given by $g\\cdot\\theta:=\\theta\\circ g^{-1}$. This identification will be used from now on whenever convenient.\n\n\nFor $\\eta_1,\\eta_2\\in\\mathsf{Gr}_{d-1}(\\mathbb{R}^d)$ we let\n\n\\begin{center}\n$d(\\xi_1,\\eta_1):=\\min\\lbrace d(\\xi_1,\\xi):\\hspace{0,3cm} \\xi\\in\\mathbb{P}(\\eta_1)\\rbrace$\n\\end{center}\n\n\\noindent and we denote by $d^*(\\eta_1,\\eta_2)$ the distance on $\\mathbb{P}((\\mathbb{R}^d)^*)$ induced by the operator norm on $(\\mathbb{R}^d)^*$. Given a positive $\\varepsilon$ we set\n\n\\begin{center}\n\n$b_\\varepsilon(\\xi_1):=\\lbrace \\xi\\in\\mathbb{P}(\\mathbb{R}^d):\\hspace{0,3cm} d(\\xi_1,\\xi)<\\varepsilon\\rbrace$\n\\end{center}\n\n\n\\noindent and\n\n\n\\begin{center}\n\n$B_\\varepsilon(\\eta_1):=\\lbrace \\xi\\in\\mathbb{P}(\\mathbb{R}^d):\\hspace{0,3cm} d(\\xi,\\eta_1)\\geq\\varepsilon\\rbrace$.\n\\end{center}\n\n\nOn the other hand, let\n\n\\begin{center}\n$\\mathbb{P}^{(2)}:=\\lbrace (\\theta,v)\\in\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d): \\hspace{0,3cm} v\\notin\\ker\\theta \\rbrace$\n\\end{center}\n\n\n\\noindent and\n\n\n\\begin{center}\n$\\mathbb{P}^{(4)}:=\\lbrace (\\theta,v,\\phi,u)\\in\\mathbb{P}^{(2)}\\times\\mathbb{P}^{(2)}: \\hspace{0,3cm} v\\notin\\ker\\phi \\textnormal{ and } u\\notin\\ker\\theta \\rbrace$.\n\\end{center}\n\n\n\\noindent Observe that \n\\begin{equation} \\label{eq defi gcursiva}\n\\mathscr{G}_{\\Vert\\cdot\\Vert}=\\mathscr{G}:\\mathbb{P}^{(2)}\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} \\mathscr{G}( \\theta,v):=\\log\\dfrac{\\left\\vert\\theta(v)\\right\\vert}{\\Vert \\theta\\Vert\\Vert v\\Vert}\t\n \\end{equation} \n\\noindent is well-defined. Similarly the following map is well-defined\n\\begin{equation}\\label{eqdef crossratio}\n\\mathbb{B}:\\mathbb{P}^{(4)}\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} \\mathbb{B}(\\theta,v,\\phi,u):= \\log\\left\\vert\\dfrac{\\theta(u)}{\\theta(v)}\t\\dfrac{\\phi(v)}{ \\phi(u)}\\right\\vert\t\n \\end{equation} \n\n\n\\noindent and is called de \\textit{cross-ratio} of $(\\theta,v,\\phi,u)$\\footnote{Sometimes $e^{\\mathbb{B}}$ is called the cross-ratio.}. Both $\\mathscr{G}$ and $\\mathbb{B}$ are continuous.\n\n\n\n\n\n\\subsection{Product of proximal matrices}\\label{subsec product of proximal}\n\n\nGiven $g$ in $\\textnormal{End}(\\mathbb{R}^d)\\setminus\\lbrace 0\\rbrace$ we denote by\n\n\\begin{center}\n$\\lambda_1(g)\\geq\\dots\\geq\\lambda_d(g)$\n\\end{center}\n\n\\noindent the logarithms of the moduli of the eigenvalues of $g$, repeated with multiplicity (we use the convention $\\log 0=-\\infty$). The matrix $g$ is said to be \\textit{proximal} in $\\mathbb{P}(\\mathbb{R}^d)$ if $\\lambda_1(g)$ is simple. In that case we let $g_+$ (resp. $g_-$) to be the attractive fixed line (resp. repelling fixed hyperplane) of $g$ in $\\mathbb{P}(\\mathbb{R}^d)$. Note that if $g$ is non invertible then $g_-$ contains the kernel of $g$.\n\n\nWe now define a quantified version of proximality. The definition that we propose is (slightly) weaker than the one given by Benoist in \\cite{Ben3,Ben4,Ben1}. We provide proofs of the basic facts established in those works when necessary.\n\n\n\\begin{dfn} \\label{dfn of repsilon proximality}\nLet $0<\\varepsilon \\leq r$ and $g\\in\\textnormal{End}(\\mathbb{R}^d)\\setminus\\lbrace 0\\rbrace$ be a proximal matrix. The matrix $g$ is called \\textit{$(r,\\varepsilon)$-proximal} if $d(g_+,g_-)\\geq 2r$ and $g\\cdot B_\\varepsilon(g_-)\\subset b_\\varepsilon(g_+)$. \n\\end{dfn}\n\n\n\n\\begin{lema}[Benoist {\\cite[Corollaire 6.3]{Ben3}}]\\label{lema benoist proximal comp vasing y vap}\nLet $0<\\varepsilon\\leq r$. There exists a constant $ c_{r,\\varepsilon}>0$ such that for every $(r,\\varepsilon)$-proximal matrix $g$ one has\n\n\\begin{center}\n$\\log\\Vert g \\Vert - c_{r,\\varepsilon}\\leq \\lambda_1(g)\\leq \\log\\Vert g \\Vert$.\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{lema}\n\n\n\nThe following criterion of $(r,\\varepsilon)$-proximality will be very useful in the sequel.\n\n\n\\begin{lema}[Benoist {\\cite[Lemme 6.2]{Ben3}}]\\label{lema benoist lemma 1.2}\nLet $g$ be an element in $\\textnormal{End}(\\mathbb{R}^d)\\setminus\\lbrace 0\\rbrace$, $\\eta\\in\\mathsf{Gr}_{d-1}(\\mathbb{R}^d)$, $\\xi\\in\\mathbb{P}(\\mathbb{R}^d)$ and $0<\\varepsilon\\leq r$. If $d(\\xi,\\eta)\\geq 6r$ and $g\\cdot B_\\varepsilon(\\eta)\\subset b_\\varepsilon(\\xi)$ then $g$ is $(2r,2\\varepsilon)$-proximal with $d(g_+,\\xi)\\leq\\varepsilon$ and $d^*(g_-,\\eta)\\leq\\varepsilon$.\n\n\\end{lema}\n\n\n\\begin{proof}\nConsider the Hilbert distance on the convex set $B_\\varepsilon(\\eta)$ (see \\cite{Ben2}). The condition $g\\cdot B_\\varepsilon(\\eta)\\subset b_\\varepsilon(\\xi)$ implies that $g$ is contracting for this metric and thus has a unique fixed point in $B_\\varepsilon(\\eta)$, which belongs in fact to $b_\\varepsilon(\\xi)$. The proof now finishes as in \\cite[Lemme 6.2]{Ben3}.\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\\begin{cor}[Benoist {\\cite[Lemme 1.4]{Ben1}}]\\label{cor product of proximal transverse is proximal}\nLet $0<\\varepsilon\\leq r$. If $g_1$ and $g_2$ are $(r,\\varepsilon)$-proximal and satisfy \n\n\\begin{center}\n$d(g_{1_+},g_{2_-})\\geq 6r$ and $d(g_{2_+},g_{1_-})\\geq 6r$\n\\end{center}\n\n\n\\noindent then $g_1g_2$ is $(2r,2\\varepsilon)$-proximal. \n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{cor}\n\n\n\n\n\nLet $g_1$ and $g_2$ be two matrices as in Corollary \\ref{cor product of proximal transverse is proximal}. The goal now is to state a theorem (Theorem \\ref{teo benoist}) which provides a comparison between the spectral radius and operator norm of $g_1g_2$ in terms of the spectral radii of $g_1$ and $g_2$ and the maps $\\mathscr{G}$ and $\\mathbb{B}$.\n\n\n\n\\begin{lema}\\label{lemma atracrep of product}\n\nFix $r>0$ and $\\delta> 0$. For every $\\varepsilon$ small enough, the following property is satisfied: for every pair of $(r,\\varepsilon)$-proximal elements $g_1$ and $g_2$ such that\n\n\\begin{center}\n$d(g_{1_+},g_{2_-})\\geq 6r$ and $d(g_{2_+},g_{1_-})\\geq 6r$\n\\end{center}\n\n\n\\noindent one has\n\n\\begin{center}\n$\\vert \\mathscr{G}(g_{2_-},g_{1_+})-\\mathscr{G}((g_1g_2)_-,(g_1g_2)_+)\\vert<\\delta$.\n\\end{center}\n\n\n\\end{lema}\n\n\n\n\\begin{proof}\nFor every $0<\\varepsilon\\leq r$, consider the compact set $C_{r,\\varepsilon}$ of pairs $(g_1,g_2)$ of norm-one $(r,\\varepsilon)$-proximal matrices in $\\textnormal{End}(\\mathbb{R}^d)\\setminus\\lbrace 0\\rbrace$ satisfying \n\n\\begin{center}\n$d(g_{1_+},g_{2_-})\\geq 6r$ and $d(g_{2_+},g_{1_-})\\geq 6r$.\n\\end{center}\n\n\\noindent The function\n\\begin{center}\n$(g_1,g_2)\\mapsto \\vert \\mathscr{G}(g_{2_-},g_{1_+})-\\mathscr{G}((g_1g_2)_-,(g_1g_2)_+)\\vert$\n\\end{center}\n\n\n\\noindent is continuous and equals zero on $C_r:=\\displaystyle\\cap_{\\varepsilon >0}C_{r,\\varepsilon}\\subset \\textnormal{End}(\\mathbb{R}^d)\\setminus\\lbrace 0\\rbrace$.\n\n\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\\begin{teo}[Benoist {\\cite[Lemme 1.4]{Ben1}}] \\label{teo benoist}\n\n\nFix $r>0$ and $\\delta> 0$. Then for every $\\varepsilon$ small enough, the following properties are satisfied: for every pair of $(r,\\varepsilon)$-proximal elements $g_1$ and $g_2$ such that\n\n\\begin{center}\n$d(g_{1_+},g_{2_-})\\geq 6r$ and $d(g_{2_+},g_{1_-})\\geq 6r$\n\\end{center}\n\n\n\n\\noindent one has:\n\n\n\\begin{enumerate}\n\\item The number\n\n\n\\begin{center}\n\n\n$\\left\\vert \\lambda_1( g_1g_2)  - (\\lambda_1(g_1)+\\lambda_1(g_2))-\\mathbb{B}(g_{1_-},g_{1_+},g_{2_-},g_{2_+}) \\right\\vert$\n\n\\end{center}\n\n\n\\noindent is less than $\\delta$.\n\n\\item The number\n\n\\begin{center}\n$\\left\\vert \\log\\Vert g_1g_2 \\Vert - (\\lambda_1(g_1)+\\lambda_1(g_2))-\\mathbb{B}(g_{1_-},g_{1_+},g_{2_-},g_{2_+})+\\mathscr{G}(g_{2_-},g_{1_+}) \\right\\vert$\n\\end{center}\n\n\n\\noindent is less than $\\delta$.\n\n\\end{enumerate}\n\n\\end{teo}\n\n\n\n\\begin{proof}\n\n\\begin{enumerate}\n\\item[(1)] See \\cite[Lemme 1.4]{Ben1}.\n\n\\item[(2)] Let $\\varepsilon$ be as in (1). For every $g_1$ and $g_2$ as in the statement, Corollary \\ref{cor product of proximal transverse is proximal} implies that $g_1g_2$ is $(2r,2\\varepsilon)$-proximal. By \\cite[Lemma 5.6]{Sam} (and taking $\\varepsilon$ smaller if necessary) we have\n\n\n\\begin{center}\n$\\left\\vert \\log\\Vert g_1g_2 \\Vert - \\lambda_1(g_1g_2)+\\mathscr{G}((g_1g_2)_-,(g_1g_2)_+) \\right\\vert<\\delta$.\n\\end{center}\n\n\n\n\\noindent Lemma \\ref{lemma atracrep of product} finishes the proof.\n\n\n\\end{enumerate}\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Projective Anosov representations} \\label{sec anosov}\n\\setcounter{equation}{0}\n\n\nAnosov representations were introduced by Labourie \\cite{Lab} for surface groups and extended by Guichard-Wienhard \\cite{GW} to word hyperbolic groups. In this section we recall the definition of (projective) Anosov representations and some well-known facts concerning $(r,\\varepsilon)$-proximality of matrices in the image of such a representation. \n\n\n\n\n\n\n\n\n\\subsection{Singular values}\\label{subsec cartan dec in sec anosov}\n\n\nThe most useful characterization of Anosov representations for our purposes is the one given in terms of \\textit{singular values}. We begin by recalling this notion and we fix also some notations that we will use in the rest of the paper.\n\n\n\n\n\n\n\nLet $\\tau$ be a $q$-dimensional subspace of $\\mathbb{R}^d$ which is negative definite for $\\langle\\cdot,\\cdot\\rangle_{p,q}$. Consider $\\langle \\cdot,\\cdot\\rangle_\\tau$ to be the inner product of $\\mathbb{R}^d$ that coincides with $-\\langle\\cdot,\\cdot\\rangle_{p,q}$ (resp. $\\langle\\cdot,\\cdot\\rangle_{p,q}$) on $\\tau$ (resp. $\\tau^{\\perp_{p,q}}$) and for which $\\tau$ and $\\tau^{\\perp_{p,q}}$ are orthogonal. Given $g$ in $\\textnormal{PSL}(d,\\mathbb{R})$, we let $g^{*_\\tau}$ to be the adjoint operator with respect to $\\langle \\cdot,\\cdot\\rangle_\\tau$. Set\n\n\\begin{center}\n\n$a_1^\\tau(g)\\geq \\dots \\geq a_d^\\tau(g)$\n\\end{center}\n\n\\noindent to be the logarithms of the eigenvalues of $\\sqrt{g^{*_\\tau}g}$ repeated with multiplicity. These are called the $\\tau$-\\textit{singular values} of $g$. Geometrically, they represent the (logarithms of the) lengths of the semi axes of the ellipsoid which is the image by $g$ of the unit sphere\n\n\\begin{center}\n$\\mathbb{S}^{d-1}_\\tau:=\\lbrace x\\in\\mathbb{R}^d:\\hspace{0,3cm} \\langle x,x\\rangle_\\tau=1 \\rbrace$.\n\\end{center}\n\n\n\n\n\n\nLet $i=1,\\dots,d-1$. Given an element $g$ in $\\textnormal{PSL}(d,\\mathbb{R})$ such that $a^\\tau_i(g)>a^\\tau_{i+1}(g)$ we denote by $U_i(g)$ the $i$-dimensional subspace of $\\mathbb{R}^d$ spanned by the $i$ biggest axes of $g\\cdot \\mathbb{S}^{d-1}_\\tau$. We also set\n\n\n\\begin{center}\n$S_{d-i}(g):=U_{d-i}(g^{-1})$.\n\\end{center} \n\n\n\\begin{rem} \\label{rem complemento de Sdmenosuno va en Uuno}\nLet $\\varepsilon>0$. It follows from Singular Value Decomposition (see Horn-Johnson \\cite[Section 7.3 of Chapter 7]{HJ}), that there exists $L>0$ such that for every $g$ in $\\textnormal{PSL}(d,\\mathbb{R})$ satisfying $a^\\tau_1(g)-a^\\tau_2(g)>L$ one has\n\n\\begin{center}\n$g\\cdot B_\\varepsilon(S_{d-1}(g))\\subset b_\\varepsilon( U_1(g))$,\n\\end{center}\n\n\\noindent where $B_\\varepsilon(S_{d-1}(g))$ and $b_\\varepsilon( U_1(g))$ are defined as in Subsection \\ref{subsec notation definitions in sec proximality}.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\n\n\\subsection{The definition of projective Anosov representations}\\label{subsec deifnition anosov}\n\nA lot of work has been done in order to simplify the original definition of Anosov representations, here we follow mainly the work of Bochi-Potrie-Sambarino \\cite{BPS} (see also Guichard-Gu\u00e9ritaud-Kassel-Wienhard \\cite{GGKW} or Kapovich-Leeb-Porti \\cite{KLP1}).\n\n\n\nFix $\\tau$ as in the previous subsection and let $\\Gamma$ be a finitely generated group. Consider a finite symmetric generating set $S$ of $\\Gamma$ and take $\\vert\\cdot\\vert_\\Gamma$ to be the associated word length: for $\\gamma$ in $\\Gamma$, it is the minimum number required to write $\\gamma$ as a product of elements of $S$\\footnote{This number depends on the choice of $S$. However, the set $S$ will be fixed from now on hence we do not emphasize the dependence on this choice in the notation.}. Let $\\rho:\\Gamma\\longrightarrow \\textnormal{PSL}{(d,\\mathbb{R})}$ be a representation. We say that $\\rho$ is \\textit{projective Anosov} if there exist positive constants $C$ and $\\alpha$ such that for all $\\gamma\\in\\Gamma$ one has\n\\begin{equation}\\label{eq def anosov}\na^\\tau_1(\\rho(\\gamma))-a^\\tau_2(\\rho(\\gamma))\\geq \\alpha\\vert\\gamma\\vert_\\Gamma-C.\n\\end{equation}\nBy Kapovich-Leeb-Porti \\cite[Theorem 1.4]{KLP3} (see also \\cite[Section 3]{BPS}), condition (\\ref{eq def anosov}) implies that $\\Gamma$ is word hyperbolic\\footnote{We refer the reader to the book of Ghys-de la Harpe \\cite{GdlH} for definitions and standard facts on word hyperbolic groups.}. We assume in this paper that $\\Gamma$ is non elementary. Let $\\partial_\\infty\\Gamma$ be the Gromov boundary of $\\Gamma$ and $\\Gamma_{\\textnormal{H}}$ be the set of infinite order elements in $\\Gamma$. Every $\\gamma$ in $\\Gamma_{\\textnormal{H}}$ has exactly two fixed points in $\\partial_\\infty\\Gamma$: the attractive one denoted by $\\gamma_+$ and the repelling one denoted by $\\gamma_-$. The dynamics of $\\gamma$ on $\\partial_\\infty\\Gamma$ is of type \\textit{north-south}.\n\n\n\n\nFix $\\rho:\\Gamma\\longrightarrow\\textnormal{PSL}(d,\\mathbb{R})$ a projective Anosov representation. By \\cite{BPS,GGKW,KLP1} we know that there exist continuous equivariant maps\n\n\\begin{center}\n$\\xi:\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{P}(\\mathbb{R}^d)$ and $\\eta:\\partial_\\infty\\Gamma\\longrightarrow\\mathsf{Gr}_{d-1}(\\mathbb{R}^d)$\n\\end{center}\n\n\n\\noindent which are \\textit{transverse}, i.e. for every $x\\neq y$ in $\\partial_\\infty\\Gamma$ one has\n\\begin{equation} \\label{eq transv condition}\n\\xi(x)\\oplus\\eta(y)=\\mathbb{R}^d.\n\\end{equation}\n\n\n\n\n\\noindent One can see that condition (\\ref{eq def anosov}) implies that for every $\\gamma$ in $\\Gamma_{\\textnormal{H}}$ the matrix $\\rho(\\gamma)$ is proximal. Equivariance of $\\xi$ and $\\eta$ implies that\n\n\\begin{center}\n$\\xi(\\gamma_+)=\\rho(\\gamma)_+$ and $\\eta(\\gamma_+)=\\rho(\\gamma^{-1})_-$.\n\\end{center}\n\n\n\\noindent It follows that both $\\xi$ and $\\eta$ are homeomorphisms onto their images. In fact, these homeo\\-mor\\-phisms are H\u00f6lder (see Bridgeman-Canary-Labourie-Sambarino \\cite[Lemma 2.5]{BCLS}).\n\n\n\nWe denote by $\\Lambda_{\\rho(\\Gamma)}\\subset\\mathbb{P}(\\mathbb{R}^d)$ the image of $\\xi$, which is called the \\textit{limit set} of $\\rho(\\Gamma)$: it is the closure of the set of attractive fixed points in $\\mathbb{P}(\\mathbb{R}^d)$ of proximal elements in $\\rho(\\Gamma)$. The image of $\\eta$  is called the \\textit{dual limit set} of $\\rho(\\Gamma)$.\n\n\n\n\nHere is another characterization of the limit sets which is very useful. An explicit reference is \\cite[Theorem 5.3]{GGKW} (it can also be deduced from {\\cite[Subsection 3.4]{BPS}}). Let $d=d_\\tau$ (resp. $d^*=d^*_\\tau$) be the distance on $\\mathbb{P}(\\mathbb{R}^d)$ (resp. $\\mathbb{P}((\\mathbb{R}^d)^*)$) associated to $\\langle\\cdot,\\cdot\\rangle_\\tau$.\n\n\\begin{prop}\\label{prop limit with S and U and Uuno cerca gammamas}\nLet $\\rho:\\Gamma\\longrightarrow \\textnormal{PSL}(d,\\mathbb{R})$ be a projective Anosov representation. Then $\\xi(\\partial_\\infty\\Gamma)$ (resp. $\\eta(\\partial_\\infty\\Gamma)$) equals the set of accumulation points of sequences $\\lbrace U_1(\\rho(\\gamma_n))\\rbrace_n$ (resp. $\\lbrace S_{d-1}(\\rho(\\gamma_n))\\rbrace_n$) where $\\gamma_n\\longrightarrow\\infty$. Moreover, given a positive $\\varepsilon$ there exists $L>0$ such that for every $\\gamma$ in $\\Gamma_{\\textnormal{H}}$ with $\\vert\\gamma\\vert_\\Gamma>L$ one has\n\n\\begin{center}\n\n$d(U_1(\\rho(\\gamma)),\\rho(\\gamma)_+)<\\varepsilon$ and $d^*(S_{d-1}(\\rho(\\gamma)),\\rho(\\gamma)_-)<\\varepsilon$.\n\n\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{prop}\n\n\n\n\n\n\nWe are interested in projective Anosov representations whose image is contained in $G=\\textnormal{PSO}(p,q)$. The following remark is then important for our purposes.\n\n\\begin{rem}\nLet $\\rho:\\Gamma\\longrightarrow\\textnormal{PSL}(d,\\mathbb{R})$ be a projective Anosov representation. If $\\rho(\\Gamma)$ is contained in $G$ we say that $\\rho$ is \\textit{$P_1^{p,q}$-Anosov} (recall that $P_1^{p,q}$ denotes the (parabolic) subgroup of $G$ stabilizing an isotropic line). In this case, the image of $\\xi$ is contained in $\\partial\\mathbb{H}^{p,q-1}$ and the dual map $\\eta$ equals $\\xi^{\\perp_{p,q}}$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\n\n\\subsection{Proximality properties}\\label{subsec proximality anosov}\nThe following lemma will be useful in the next section.\n\n\\begin{lema}[c.f. {\\cite[Lemma 5.7]{Sam}}]\\label{lema sambarino lemma 5.7}\nLet $\\rho:\\Gamma\\longrightarrow \\textnormal{PSL}(d,\\mathbb{R})$ be a  projective Anosov representation and $0<\\varepsilon\\leq r$. Then\n\n\\begin{center}\n$\\#\\lbrace\\gamma\\in\\Gamma_{\\textnormal{H}}: \\hspace{0,3cm} d(\\rho(\\gamma)_+,\\rho(\\gamma)_-)\\geq 2r \\textnormal{ and } \\rho(\\gamma) \\textnormal{ is not } (r,\\varepsilon)\\textnormal{-proximal}\\rbrace<\\infty$.\n\\end{center}\n\n\n\\end{lema}\n\n\n\\begin{proof}\n\nConsider a sequence $\\gamma_n\\longrightarrow\\infty$ in $\\Gamma_{\\textnormal{H}}$ such that $d(\\rho(\\gamma_n)_+,\\rho(\\gamma_n)_-)\\geq 2r$ for all $n$. By Proposition \\ref{prop limit with S and U and Uuno cerca gammamas} for every $n$ big enough the following holds\n\n\\begin{center}\n$b_{\\frac{\\varepsilon}{2}}(U_1(\\rho(\\gamma_n)))\\subset b_{\\varepsilon}(\\rho(\\gamma_n)_+)$\n\\end{center}\n\n\\noindent and\n\n\\begin{center}\n$B_{\\varepsilon}(\\rho(\\gamma_n)_-)\\subset B_{\\frac{\\varepsilon}{2}}(S_{d-1}(\\rho(\\gamma_n)))$.\n\\end{center}\n\n\n\n\\noindent By Remark \\ref{rem complemento de Sdmenosuno va en Uuno} and (\\ref{eq def anosov}) the condition $\\rho(\\gamma_n)\\cdot B_{\\varepsilon}(\\rho(\\gamma_n)_-)\\subset b_{\\varepsilon}(\\rho(\\gamma_n)_+)$ is sa\\-tis\\-fied for sufficiently large $n$.\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\\section{The set $\\pmb{\\Omega}_\\rho$}\\label{sec the set omegarho}\n\\setcounter{equation}{0}\n\n\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation and define\n\n\n\n\n\n\n\\begin{center}\n$\\pmb{\\Omega}_{\\rho}:=\\lbrace o\\in \\mathbb{H}^{p,q-1}:\\hspace{0,3cm} J^o\\cdot \\xi(x)\\notin\\eta(x) \\textnormal{ for all } x\\in\\partial_\\infty\\Gamma\\rbrace$.\n\\end{center}\n\n\n\n\n\\noindent This section is structured as follows. In Subsection \\ref{subsec dynam on omegarho} we prove that the action of $\\Gamma$ on $\\pmb{\\Omega}_\\rho$ is properly discontinuous. Moreover, we show that if $o$ is a point in $\\pmb{\\Omega}_\\rho$ then the geodesic connecting $o$ with $\\rho(\\gamma)\\cdot o$ is space-like (apart from possibly finitely many exceptions $\\gamma\\in\\Gamma$). In Subsection \\ref{subsec proxim of jrhojrho} we study the matrices $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$ for a point $o$ in $\\pmb{\\Omega}_\\rho$: we apply to them Benoist's work on proximality. Finiteness of our counting functions is proved in Subsection \\ref{subsec orbital counting functions}. Finally, in Subsection \\ref{subsec weak triangle} we prove a proposition that will be needed in the proof of Proposition \\ref{prop distribution on bg for length}.\n\n\nBefore we start, let us discuss some examples for which $\\pmb{\\Omega}_\\rho$ is non empty. From Proposition \\ref{prop fijos de Jo en borde} we know that the following alternative description of $\\pmb{\\Omega}_\\rho$ holds\n\n\\begin{center}\n$\\pmb{\\Omega}_{\\rho}=\\lbrace o=[\\hat{o}]\\in \\mathbb{H}^{p,q-1}:\\hspace{0,3cm} \\langle \\hat{o},\\hat{\\xi}\\rangle_{p,q}\\neq 0 \\textnormal{ for all } \\xi=[\\hat{\\xi}]\\in\\Lambda_{\\rho(\\Gamma)}\\rbrace$.\n\\end{center}\n\n\n\\noindent We have the following important example.\n\n\n\n\\begin{ex}\\label{ex omegarho no vacio}\n\n\\item \n\n\\begin{itemize}\n\\item Let $\\Gamma$ be the fundamental group of a convex co-compact hyperbolic manifold of dimension $m\\geq 2$ and $\\iota_0:\\Gamma\\longrightarrow \\textnormal{SO}(m,1)$ be the holonomy representation. Fix $p\\geq m$ and $q\\geq 2$. Consider the embedding $\\mathbb{R}^{m,1}\\hookrightarrow \\mathbb{R}^{p,q}$ given by\n\n\\begin{center}\n$\\mathbb{R}^{m,1}\\cong \\textnormal{span}\\lbrace e_{p-m+1},\\dots,e_{p+1}\\rbrace$,\n\\end{center}\n\n\n\\noindent where $e_i$ is the vector of $\\mathbb{R}^d$ with all entries equal to zero except for the $i$-th entry which is equal to one. This induces a projection $j:\\textnormal{SO}(m,1)\\longrightarrow G$ and a representation $\\rho_0:\\Gamma\\longrightarrow G$ defined by\n\n\n\\begin{center}\n$\\rho_0:=j\\circ\\iota_0$.\n\\end{center}\n\n\n\\noindent Thus $\\rho_0$ is $P_1^{p,q}$-Anosov, because $\\iota_0$ is $P_1^{m,1}$-Anosov. The set $\\pmb{\\Omega}_{\\rho_0}$ is non empty: every point $o\\in\\mathbb{H}^{p,q-1}$ for which the subspace\n\n\\begin{center}\n$\\textnormal{span}\\lbrace o,e_{p+2},\\dots,e_d\\rbrace$\n\\end{center}\n\n\n\\noindent has signature $(0,q)$ belongs to $\\pmb{\\Omega}_{\\rho_0}$. Since the condition of being Anosov is open in the space of representations of $\\Gamma$ into $G$ and the limit map $\\xi$ varies continuously with the representation (see Guichard-Wienhard \\cite[Theorem 5.13]{GW}), we obtain that if $\\rho$ is a small deformation of $\\rho_0$ then $\\pmb{\\Omega}_\\rho$ is non empty.\n\n\n\n\\item The previous example generalizes to a large class of representations introduced by Danciger-Gu\u00e9ritaud-Kassel in \\cite{DGK1,DGK2} called \\textit{$\\mathbb{H}^{p,q-1}$-convex co-compact}\\footnote{These are inclusion representations induced by taking an infinite discrete subgroup $\\Gamma< G$ which preserves some properly convex non empty open set $\\Omega\\subset\\mathbb{P}(\\mathbb{R}^d)$ whose boundary is strictly convex and of class $C^1$. One requires that $\\Gamma$ preserves some \\textit{distinguished} non empty convex subset of $\\Omega$ on which the action is co-compact (see \\cite{DGK1,DGK2} for precisions).}. Let $\\Gamma <G$ be a $\\mathbb{H}^{p,q-1}$-convex co-compact group and $\\rho:\\Gamma\\longrightarrow G$ be the inclusion representation, which is $P_1^{p,q}$-Anosov as proved in \\cite[Theorem 1.25]{DGK2}. Let $\\Omega$ be a non empty $\\Gamma$-invariant properly convex open subset of $\\mathbb{H}^{p,q-1}$. By \\cite[Proposition 4.5]{DGK2}, $\\Omega$ is contained in $\\pmb{\\Omega}_\\rho$.\n\n\\item There exist examples of $P_1^{p,q}$-Anosov representations $\\rho$ whose image is not $\\mathbb{H}^{p,q-1}$-convex co-compact but satisfy $\\pmb{\\Omega}_\\rho\\neq\\emptyset$ (see \\cite[Examples 5.2 \\& 5.3]{DGK1}).\n\n\\end{itemize}\n\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{ex}\n\n\n\n\n\n\n\\subsection{Dynamics on $\\pmb{\\Omega}_\\rho$}\\label{subsec dynam on omegarho}\n\n\n\n\n\n\n\n\nObserve that $\\pmb{\\Omega}_\\rho$ is $\\Gamma$-invariant. The following proposition is well-known, we include a proof for completeness.\n\n\n\n\n\n\n\n\\begin{prop}\\label{prop action on omegarho is prop discont and limit set is limit set}\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation. Then the action of $\\Gamma$ on $\\pmb{\\Omega}_\\rho$ is properly discontinuous, that is, for every compact set $C\\subset\\pmb{\\Omega}_\\rho$ one has\n\n\\begin{center}\n$\\#\\left\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\rho(\\gamma)\\cdot C\\cap C\\neq \\emptyset\\right\\rbrace<\\infty$.\n\\end{center}\n\n\\noindent Moreover, for any point $o$ in $\\pmb{\\Omega}_\\rho$ the set of accumulation points of $\\rho(\\Gamma)\\cdot o$ in $\\mathbb{H}^{p,q-1}\\cup\\partial\\mathbb{H}^{p,q-1}$ coincides with the limit set $\\Lambda_{\\rho(\\Gamma)}$.\n\\end{prop}\n\n\n\\begin{proof}\n\n\n\n\n\n\n\nLet $C\\subset\\pmb{\\Omega}_\\rho$ be a compact set and fix a norm on $\\mathbb{R}^d$. By definition of $\\pmb{\\Omega}_\\rho$ we can take a positive $\\varepsilon$ such that\n\n\n\\begin{center}\n$C\\cap\\displaystyle\\bigcup_{x\\in\\partial_\\infty\\Gamma}b_\\varepsilon(\\xi(x))=\\emptyset$ and $C\\subset\\displaystyle\\bigcap_{x\\in\\partial_\\infty\\Gamma}B_\\varepsilon(\\eta(x))$.\n\\end{center}\n\n\n\n\n\nBy Proposition \\ref{prop limit with S and U and Uuno cerca gammamas}, Remark \\ref{rem complemento de Sdmenosuno va en Uuno} and (\\ref{eq def anosov}) we know that, apart from possibly finitely many exceptions $\\gamma$ in $\\Gamma$, the following holds:\n\n\n\\begin{center}\n$b_{\\frac{\\varepsilon}{2}}(U_1(\\rho(\\gamma)))\\subset \\displaystyle\\bigcup_{x\\in\\partial_\\infty\\Gamma}b_\\varepsilon(\\xi(x))$,\n\n\\vspace{0,3cm}\n\n$\\displaystyle\\bigcap_{x\\in\\partial_\\infty\\Gamma}B_\\varepsilon(\\eta(x))\\subset B_{\\frac{\\varepsilon}{2}}(S_{d-1}(\\rho(\\gamma)))$\n\n\n\\end{center}\n\n\n\\noindent and\n\n\\begin{center}\n\n$\\rho(\\gamma)\\cdot B_{\\frac{\\varepsilon}{2}}(S_{d-1}(\\rho(\\gamma))) \\subset b_{\\frac{\\varepsilon}{2}}(U_1(\\rho(\\gamma)))$.\n\n\\end{center}\n\n\\noindent For these $\\gamma$ we have then that $\\rho(\\gamma)\\cdot C$ is contained in the $\\varepsilon$-neighbourhood of $\\Lambda_{\\rho(\\Gamma)}$ and thus is disjoint from $C$.\n \n\n\n\nWe have shown that the action of $\\Gamma$ on $\\pmb{\\Omega}_\\rho$ is properly discontinuous and that for any point $o$ in $\\pmb{\\Omega}_\\rho$ the accumulation points of $\\rho(\\Gamma)\\cdot o$ belong to $\\Lambda_{\\rho(\\Gamma)}$. Conversely, the $\\Gamma$-orbit of any point in $\\Lambda_{\\rho(\\Gamma)}$ is dense in the limit set and now the proof is complete.\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\nLet $o\\in\\pmb{\\Omega}_\\rho$ and recall the notations introduced in Subsection \\ref{subsub geod hpq}. Given an open set $W\\subset\\partial\\mathbb{H}^{p,q-1}$ disjoint from $\\overline{\\mathscr{C}_o^0}\\cap\\partial\\mathbb{H}^{p,q-1}$ we denote by $\\mathscr{C}_{o}^{>_W}$ the subset of $\\mathscr{C}_o^{>}$ consisting of points $o'$ such that the (space-like) geodesic ray connecting $o$ with $o'$ has its end point in $W$.\n\n\nThe following corollary has been proved by Glorieux-Monclair \\cite{GM} for $\\mathbb{H}^{p,q-1}$-convex co-compact groups.\n\n\n\\begin{cor}\\label{cor gammao in cowmayor}\nLet $\\rho:\\Gamma\\longrightarrow G$ be a $P_1^{p,q}$-Anosov representation, a point $o\\in\\pmb{\\Omega}_{\\rho}$ and $W\\subset\\partial\\mathbb{H}^{p,q-1}$ an open set containing $\\Lambda_{\\rho(\\Gamma)}$ with closure disjoint from $\\overline{\\mathscr{C}_o^0}\\cap\\partial\\mathbb{H}^{p,q-1}$. Then apart from possibly finitely many exceptions $\\gamma$ in $\\Gamma$ one has $\\rho(\\gamma)\\cdot o \\in\\mathscr{C}_{o}^{>_W}$. In particular the geodesic joining $o$ with $\\rho(\\gamma)\\cdot o$ is space-like.\n\\end{cor}\n\n\n\n\\begin{proof}\n\nLet $C$ be the closure of $\\mathbb{H}^{p,q-1}\\setminus\\mathscr{C}_{o}^{>_W}$ in $\\mathbb{H}^{p,q-1}\\cup\\partial\\mathbb{H}^{p,q-1}$. Note that $C$ is compact and by Proposition \\ref{prop action on omegarho is prop discont and limit set is limit set} does not contain accumulation points of $\\rho(\\Gamma)\\cdot o$, hence $\\rho(\\Gamma)\\cdot o\\cap C$ is finite.  Since $\\gamma\\mapsto\\rho(\\gamma)\\cdot o$ is proper the proof is complete.\n\n\n\\end{proof}\n\n\\subsection{Proximality of $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$} \\label{subsec proxim of jrhojrho}\n\n\nFor the rest of the section we fix a $P_1^{p,q}$-Anosov representation $\\rho:\\Gamma\\longrightarrow G$, a point $o\\in\\pmb{\\Omega}_\\rho$ and a Cartan involution $\\tau\\in S^o$.\n\n\nThe next lemma is a direct consequence of Proposition \\ref{prop limit with S and U and Uuno cerca gammamas}, transversality condition (\\ref{eq transv condition}) and the definition of $\\pmb{\\Omega}_\\rho$. \n\n\n\n\\begin{lema}\\label{lema JUuno lejos de Sdmenos1}\nLet $d_\\tau$ be the distance on $\\mathbb{P}(\\mathbb{R}^d)$ induced by the norm $\\Vert\\cdot\\Vert_\\tau$. There exists a positive constant $D$ such that\n\n\n\\begin{center}\n$\\#\\lbrace \\gamma\\in\\Gamma:\\hspace{0,3cm} d_\\tau(J^o\\cdot U_1(\\rho(\\gamma)),S_{d-1}(\\rho(\\gamma^{-1})))< D\\rbrace <\\infty$.\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{lema}\n\n\n\n\n\n\\begin{lema} \\label{lema jrhojrho prox uno}\nThere exist $0<\\varepsilon\\leq r$ such that, apart from possibly finitely many exceptions $\\gamma\\in\\Gamma$, the matrix $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$ is $(r,\\varepsilon)$-proximal.\n\\end{lema}\n\n\n\n\\begin{proof}\n\n\n\nWe apply a \\textit{ping-pong} argument together with Lemma \\ref{lema benoist lemma 1.2}. By Lemma \\ref{lema JUuno lejos de Sdmenos1} we can take a positive constant $r$ and a finite subset $F\\subset\\Gamma$ such that for every $\\gamma\\in\\Gamma\\setminus F$ one has\n\\begin{equation}\\label{eq jrhojrho prox uno}\nd_\\tau (J^o\\cdot U_1(\\rho(\\gamma)),S_{d-1}(\\rho(\\gamma^{-1})))\\geq 6r.\n\\end{equation}\n\n\n\n\\noindent Take $0<\\varepsilon\\leq r$ such that for every $\\gamma\\in\\Gamma\\setminus F$ one has\n\n\\begin{center}\n$b_\\varepsilon(J^o\\cdot U_1(\\rho(\\gamma)))\\subset B_\\varepsilon(S_{d-1}(\\rho(\\gamma^{-1})))$.\n\\end{center}\n\n\n\\noindent By Remark \\ref{rem J preserva norma} the matrix $J^o$ preserves $d_\\tau$ thus\n\n\n\n\n\\begin{center}\n$J^o\\cdot b_\\varepsilon( U_1(\\rho(\\gamma)))\\subset B_\\varepsilon( S_{d-1}(\\rho(\\gamma^{-1})))$.\n\\end{center}\n\n\n\\noindent By taking $F$ larger if necessary we have that\n\n\\begin{center}\n$\\rho(\\gamma^{-1})\\cdot B_\\varepsilon(S_{d-1}(\\rho(\\gamma^{-1})))\\subset b_\\varepsilon(U_1(\\rho(\\gamma^{-1})))$\n\\end{center}\n\n\\noindent holds for every $\\gamma$ in $\\Gamma\\setminus F$. It follows that \n\n\\begin{center}$J^o\\rho(\\gamma^{-1})\\cdot B_\\varepsilon(S_{d-1}(\\rho(\\gamma^{-1})))\\subset B_\\varepsilon(S_{d-1}(\\rho(\\gamma)))$\n\\end{center}\n\n\\noindent and applying $\\rho(\\gamma)$ we obtain\n\n\\begin{center}\n$\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\cdot B_\\varepsilon(S_{d-1}(\\rho(\\gamma^{-1})))\\subset b_\\varepsilon(U_1(\\rho(\\gamma)))$.\n\\end{center}\n\n\\noindent Then\n\\begin{center}\n$J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\cdot B_\\varepsilon(S_{d-1}(\\rho(\\gamma^{-1})))\\subset b_\\varepsilon(J^o\\cdot  U_1(\\rho(\\gamma)))$.\n\\end{center}\n\n\\noindent By (\\ref{eq jrhojrho prox uno}) and Lemma \\ref{lema benoist lemma 1.2} the proof is finished.\n\n\\end{proof}\n\n\nThe following is a strengthening of Lemma \\ref{lema jrhojrho prox uno}. It provides a link between the generalized Cartan projections $b^o$ and $b^\\tau$ and the spectral radii of proximal elements in $\\rho(\\Gamma)$. For the remainder of the section we fix a maximal subalgebra $\\mathfrak{b}\\subset\\mathfrak{p}^\\tau\\cap\\mathfrak{q}^o$ and a closed Weyl chamber $\\mathfrak{b}^+$. \n\n\n\n\n\n\\begin{lema}\\label{lema jrhojrho prox dos}\n\nFix any $\\delta>0$ and  $A$ and $B$ two compact disjoint sets in $\\partial_\\infty\\Gamma$. Then there exist $0< \\varepsilon\\leq r$ such that, apart from possibly finitely many exceptions $\\gamma\\in \\Gamma_{\\textnormal{H}}$ with $\\gamma_-\\in A$ and $\\gamma_+\\in B$, the following holds:\n\n\\begin{enumerate}\n\\item The matrices $J^o\\rho(\\gamma)J^o$ and $\\rho(\\gamma^{-1})$ are $(r,\\varepsilon)$-proximal.\n\\item $d_\\tau(J^o\\cdot \\rho(\\gamma)_+,\\rho(\\gamma^{-1})_-)\\geq 6r$ and $d_\\tau(\\rho(\\gamma^{-1})_+,J^o\\cdot \\rho(\\gamma)_-)\\geq 6r$.\n\\item $d_\\tau((J^o\\rho(\\gamma)J^o)_+,\\rho(\\gamma^{-1})_-)\\geq 6r$ and $d_\\tau(\\rho(\\gamma^{-1})_+,(J^o\\rho(\\gamma)J^o)_-)\\geq 6r$.\n\\item The matrix $\\rho(\\gamma)$ belongs to $\\mathscr{C}_{o,G}^>$ and the number \n\n\\begin{center}\n$\\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}   - \\lambda_1(\\rho(\\gamma))$\n\\end{center}\n\n\\noindent is at distance at most $\\delta$ from\n\n\\begin{center}\n$\\frac{1}{2}\\mathbb{B}(J^o\\cdot \\rho(\\gamma)_-,J^o\\cdot \\rho(\\gamma)_+,\\rho(\\gamma^{-1})_-,\\rho(\\gamma^{-1})_+)$.\n\\end{center}\n\n\n\\item The number\n\n\\begin{center}\n\n$\\vert b^\\tau(\\rho(\\gamma))\\vert^{\\frac{1}{2}} - \\lambda_1(\\rho(\\gamma)) $\n\\end{center}\n\n\\noindent is at distance at most $\\delta$ from\n\n\\begin{center}\n$\\frac{1}{2} \\mathbb{B}(J^o\\cdot \\rho(\\gamma)_-,J^o\\cdot \\rho(\\gamma)_+,\\rho(\\gamma^{-1})_-,\\rho(\\gamma^{-1})_+)-\\frac{1}{2}\\mathscr{G}_\\tau(\\rho(\\gamma^{-1})_-,J^o\\cdot \\rho(\\gamma)_+)$.\n\\end{center}\n\n\n\\end{enumerate}\n\n\\end{lema}\n\n\n\n\n\\begin{proof}\n\nBy transversality condition (\\ref{eq transv condition}) there exists $r>0$ such that\n\\begin{equation}\\label{eq transvversalidad A y B distribucion nu}\nd_{\\tau}(\\xi(x),\\eta(y))\\geq 2r \\textnormal{ and } d_{\\tau}(\\xi(y),\\eta(x))\\geq 2r\n\\end{equation}\n\\noindent for all $(x,y)\\in A\\times B$. Further, since $o\\in\\pmb{\\Omega}_{\\rho}$ we may assume\n\\begin{equation} \\label{eq seisr en teo distribucion nu}\nd_{\\tau}(J^o\\cdot\\xi(x),\\eta(x))\\geq 6r\n\\end{equation}\n\n\n\\noindent for all $x\\in\\partial_\\infty\\Gamma$. Given these $r>0$ and $2\\delta>0$, we consider $\\varepsilon>0$ as in Benoist's Theorem \\ref{teo benoist}. \n\nBy Lemma \\ref{lema sambarino lemma 5.7} there exists a finite subset $F$ of $\\Gamma_{\\textnormal{H}}$ outside of which elements satisfying $d_{\\tau}(\\rho(\\gamma)_+,\\rho(\\gamma)_-)\\geq 2r$ are $(r,\\varepsilon)$-proximal. Thanks to (\\ref{eq transvversalidad A y B distribucion nu}), for all $\\gamma\\in\\Gamma_{\\textnormal{H}}\\setminus F$ with $\\gamma_-\\in A$ and $\\gamma_+\\in B$ one has that $\\rho(\\gamma^{\\pm 1})$ is $(r,\\varepsilon)$-proximal. Moreover, since $J^o=(J^o)^{-1}$ preserves $\\Vert\\cdot\\Vert_{\\tau}$ we have that $J^o\\rho(\\gamma)J^o$ is $(r,\\varepsilon)$-proximal with $(J^o\\rho(\\gamma)J^o)_{\\pm}=J^o\\cdot\\rho(\\gamma)_{\\pm}$. In fact, by (\\ref{eq seisr en teo distribucion nu}) we have\n\n\\begin{center}\n$d_{\\tau}(J^o\\cdot\\rho(\\gamma)_{+},\\rho(\\gamma^{-1})_{-})\\geq 6r \\textnormal{ and } d_{\\tau}(\\rho(\\gamma^{-1})_{+},J^o\\cdot\\rho(\\gamma)_{-})\\geq 6r$.\n\\end{center}\n\n\n\n\\noindent Thanks to Proposition \\ref{prop linear alg interpr of bo} (and Corollary \\ref{cor gammao in cowmayor}), Proposition \\ref{prop computing nu}, Theorem \\ref{teo benoist} and the fact that $\\lambda_1(\\rho(\\gamma^{-1}))$ equals $\\lambda_1(\\rho(\\gamma))$ for all $\\gamma$, the proof is finished.\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection{The orbital counting functions of Theorems \\ref{teorema A} and \\ref{teorema B}}\\label{subsec orbital counting functions}\n\n\n\n\n\n\n\n\\begin{prop}\\label{prop counting with nu is well defined}\n\nFor every $t\\geq 0$ one has\n\n\\begin{center}\n$\\#\\left\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\vert b^\\tau(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t\\right\\rbrace<\\infty$.\n\\end{center}\n\\end{prop}\n\n\n\n\\begin{proof}\n\n\nBy Remark \\ref{rem btau is proper} the map $b^\\tau$ descends to a proper map in $\\mathbb{H}^{p,q-1}\\cong G\/H^o$, that we still denote by $b^\\tau$. Hence\n\n\\begin{center}\n$C:=\\lbrace o'\\in\\mathbb{H}^{p,q-1}:\\hspace{0,3cm}  \\vert b^\\tau(o')\\vert\\leq t^2 \\rbrace$\n\\end{center}\n\n\n\n\n\n\n\\noindent is compact. By Proposition \\ref{prop action on omegarho is prop discont and limit set is limit set}, apart from possibly finitely many exceptions $\\gamma$ in $\\Gamma$, we have that $\\rho(\\gamma)\\cdot o$ does not belong to $C$.\n\n\n\\end{proof}\n\n\n\nThe next proposition follows from a combination of Propositions \\ref{prop linear alg interpr of bo} and \\ref{prop computing nu}, Lemmas \\ref{lema jrhojrho prox uno} and \\ref{lema benoist proximal comp vasing y vap}, and the previous proposition.\n\n\n\n\n\\begin{prop}\\label{prop counting with lambdauno is well defined}\n\nFor every $t\\geq 0$ one has\n\n\\begin{center}\n$\\#\\left\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\rho(\\gamma)\\in\\mathscr{C}_{o,G}^> \\textnormal{ and } \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t\\right\\rbrace<\\infty$.\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{prop}\n\n\\begin{rem}\\label{rem crit exponent coindes with the one of GM}\nAssume that $\\rho$ is $\\mathbb{H}^{p,q-1}$-convex co-compact and the basepoint $o$ belongs to the convex hull of the limit set of $\\rho$. By Corollary \\ref{cor gammao in cowmayor} and Proposition \\ref{prop ell dXG y vertboverto} we have that\n\n\\begin{center}\n$\\displaystyle\\limsup_{t\\longrightarrow\\infty}\\dfrac{\\log\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\rho(\\gamma)\\in\\mathscr{C}_{o,G}^> \\textnormal{ and } \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t\\rbrace}{t}$\n\\end{center}\n\n\\noindent coincides with\n\n\\begin{center}\n$\\displaystyle\\limsup_{t\\longrightarrow\\infty}\\dfrac{\\log\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} d_{\\mathbb{H}^{p,q-1}}(o,\\rho(\\gamma)\\cdot o)\\leq t\\rbrace}{t}$,\n\\end{center}\n\n\\noindent where $d_{\\mathbb{H}^{p,q-1}}$ is the $\\mathbb{H}^{p,q-1}$-distance introduced in \\cite{GM}.\n\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\\subsection{Weak triangle inequality}\\label{subsec weak triangle}\n\n\n\nThe following proposition is inspired by \\cite[Theorem 3.5]{GM}. \n\n\n\n\n\n\\begin{prop} \\label{prop traingle inequality}\nThere exists a constant $L>0$ such that for every $f\\in\\Gamma$ there exists $D_f>0$ with the following property:  for every $\\gamma\\in\\Gamma$  with $\\vert\\gamma\\vert_\\Gamma>L$ one has \n\n\n\\begin{center}\n$\\frac{1}{2}\\lambda_1(J^o\\rho(f)\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\rho(f^{-1}))\\leq D_f+\\frac{1}{2}\\lambda_1(J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1}))$.\n\\end{center}\n\n\n\n\\end{prop}\n\n\nWe can think about the content of Proposition \\ref{prop traingle inequality} as follows. Fix $f\\in\\Gamma$ such that $\\rho(f)\\in\\mathscr{C}_{o,G}^>$. By Corollary \\ref{cor gammao in cowmayor} for every $\\gamma$ with $\\vert\\gamma\\vert_\\Gamma$ large enough one has $\\rho(\\gamma)\\in\\mathscr{C}_{o,G}^>$ and $\\rho(f)\\rho(\\gamma)\\in\\mathscr{C}_{o,G}^>$. Thanks to Proposition \\ref{prop ell dXG y vertboverto} and Proposition \\ref{prop linear alg interpr of bo}, the inequality established in Proposition \\ref{prop traingle inequality} can be stated as\n\n\\begin{center}\n$\\ell_{o,\\rho(f)\\rho(\\gamma)\\cdot o}\\leq D_f+\\ell_{\\rho(f)\\cdot o,\\rho(f)\\rho(\\gamma)\\cdot o}$,\n\\end{center}\n\n\\noindent where the constant $D_f$ depends on the choice of $o$ and $f$ (and $\\rho$) but not on the choice of $\\gamma$. Even though the function $\\ell_{\\cdot,\\cdot}$ is not a distance, we can heuristically think about $D_f$ as the term that replaces $\\ell_{o,\\rho(f)\\cdot o}$ in the usual triangle inequality for distances.\n\n\n\n\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop traingle inequality}]\n\n\n\nTake $0<\\varepsilon\\leq r$ as in Lemma \\ref{lema jrhojrho prox uno}. Let $L>0$ such that for every $\\gamma$ in $\\Gamma$ with $\\vert \\gamma\\vert_\\Gamma >L$ the matrix $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$ is $(r,\\varepsilon)$-proximal. Fix $f\\in\\Gamma$ and let $\\gamma$ be a element in $\\Gamma$ with $\\vert \\gamma\\vert_\\Gamma >L$. We have\n\n\\begin{center}\n$\\frac{1}{2}\\lambda_1( J^o\\rho(f)\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\rho(f^{-1}))\\leq\\frac{1}{2}\\log\\Vert J^o\\rho(f)\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\rho(f^{-1})\\Vert_\\tau$.\n\\end{center}\n\n\\noindent By Remark \\ref{rem J preserva norma} the right side number equals $\\frac{1}{2}\\log\\Vert \\rho(f)\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\rho(f^{-1})\\Vert_\\tau$ which is less than or equal to\n\n\\begin{center}\n$  D_f'+\\frac{1}{2}\\log\\Vert J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})\\Vert_\\tau$\n\\end{center}\n\n\n\n\\noindent where $D_f':=\\frac{1}{2}\\log\\Vert \\rho(f)\\Vert_\\tau +\\frac{1}{2}\\log \\Vert\\rho(f^{-1})\\Vert_\\tau$. Since $J^o\\rho(\\gamma)J^o\\rho(\\gamma^{-1})$ is $(r,\\varepsilon)$-proximal, we conclude by applying Lemma \\ref{lema benoist proximal comp vasing y vap}.\n\n\\end{proof}\n\n\n\n\n\n\\section{Distribution of the orbit of $o$ with respect to $ b^o$}\\label{section distrib wrt bo}\n\\setcounter{equation}{0}\n\nIn this section we prove Theorem \\ref{teorema A}. The section is structured as follows: in Subsection \\ref{subsec cocycle co} we define a H\u00f6lder cocycle on $\\partial_\\infty\\Gamma$ and the corresponding flow. In Subsection \\ref{subsec dual and gromov co} we study the associated Gromov product. Theorem \\ref{teorema A} in the torsion free case (resp. general case) is proved in Subsection \\ref{subsec dist of fixed wrt bo} (resp. Subsection \\ref{subsec proof teo A}).\n\n\n\n\n\nFor the rest of the section we fix $\\rho:\\Gamma\\longrightarrow G$ a $P_1^{p,q}$-Anosov representation and a point $o$  in $\\pmb{\\Omega}_\\rho$.\n\n\n\\subsection{The cocycle $c_o$}\\label{subsec cocycle co}\n\nObserve that by definition of $\\pmb{\\Omega}_\\rho$ and equivariance of the curves $\\xi$ and $\\eta$ the following map is well-defined.\n\n\n\\begin{dfn} \\label{dfn cocycle o}\n\nLet\n\n\\begin{center}\n$c_o:\\Gamma\\times\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} c_o(\\gamma,x):=\\dfrac{1}{2}\\log\\left\\vert\\dfrac{\\theta_{x}\\left(\\rho(\\gamma^{-1})J^o\\rho(\\gamma)\\cdot v_x\\right)}{\\theta_{x}\\left(J^o\\cdot v_x\\right)}\\right\\vert$,\n\\end{center}\n\n\\noindent where $\\theta_{x}:\\mathbb{R}^d\\longrightarrow\\mathbb{R}$ is a non-zero linear functional whose kernel equals $\\eta(x)$ and $v_{x}\\neq 0$ belongs to $\\xi(x)$.\n\\end{dfn}\n\n\nA geometric interpretation of the map $c_o$ is provided by the following remark. This characterization will not be used in the sequel.\n\n\n\\begin{rem} \\label{rem co es la buseman de gm}\n\n\nOne can prove that for every $\\gamma\\in\\Gamma$ and $x\\in\\partial_\\infty\\Gamma$ one has\n\n\n\\begin{center}\n$c_{o}(\\gamma,x)=\\beta_{\\xi(x)}(\\rho(\\gamma^{-1})\\cdot o,o)$\n\\end{center}\n\n\n\\noindent where $\\beta_\\cdot(\\cdot,\\cdot)$ is the pseudo-Riemannian Busemann function defined by Glorieux-Monclair \\cite[Definition 3.8]{GM}.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\nRecall that a \\textit{H\u00f6lder cocycle} is a function $c:\\Gamma\\times\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{R}$ satisfying that for every $\\gamma_0,\\gamma_1$ in $\\Gamma$ and $x\\in\\partial_\\infty\\Gamma$ one has \n\n\\begin{center}\n$c(\\gamma_0\\gamma_1,x)=c(\\gamma_0,\\gamma_1\\cdot x)+c(\\gamma_1,x)$\n\\end{center}\n\n\\noindent and such that the map $c(\\gamma_0,\\cdot)$ is H\u00f6lder (with the same exponent for every $\\gamma_0$). The \\textit{period} of (an infinite order element) $\\gamma\\in \\Gamma_{\\textnormal{H}}$ is defined by $\\ell_{c}(\\gamma):=c(\\gamma,\\gamma_+)$.\n\n\n\n\\begin{lema}\\label{lema cocycle o and periods}\nThe map $c_o$ is a H\u00f6lder cocycle. The period of $\\gamma\\in\\Gamma_{\\textnormal{H}}$ is given by\n\n\\begin{center}\n$\\ell_{c_o}(\\gamma)=\\lambda_1(\\rho(\\gamma))>0$.\n\\end{center}\n\n\n\n\\end{lema}\n\n\n\n\n\\begin{proof} \n\n\n\nA direct computation shows that $c_o$ is a H\u00f6lder cocycle.\n\n\nOn the other hand let $\\gamma\\in\\Gamma_{\\textnormal{H}}$ and fix a particular choice of a linear functional $\\theta_{\\gamma_+}$. Since $\\lambda_1(\\rho(\\gamma))=\\lambda_1(\\rho(\\gamma^{-1}))$ one sees that $\\theta_{\\gamma_+}\\circ(\\pm\\rho(\\gamma^{-1}))$ coincides with $ e^{\\lambda_1(\\rho(\\gamma))}\\theta_{\\gamma_+}$ up to a sign (here $\\pm\\rho(\\gamma^{-1})$ denotes some lift of $\\rho(\\gamma^{-1})$ to $\\textnormal{SO}(p,q)$). The proof is now complete.\n\n\n\n\\end{proof}\n\n\nSet $\\partial_\\infty^{2}\\Gamma:=\\lbrace(x,y)\\in\\partial_\\infty\\Gamma\\times\\partial_\\infty\\Gamma:\\hspace{0,3cm} x\\neq y\\rbrace$ and consider the \\textit{translation flow} on $\\partial_\\infty^{2}\\Gamma\\times \\mathbb{R}$ defined by\n\\begin{equation}\\label{eq translation flow}\n\\psi_t(x,y,s):=(x,y,s-t).\n\\end{equation}\n\\noindent The group $\\Gamma$ acts on $ \\partial_\\infty^{2}\\Gamma\\times\\mathbb{R}$ by\n\\begin{equation} \\label{eq action via co}\n\\gamma\\cdot(x,y,s):=(\\gamma\\cdot x,\\gamma\\cdot y, s-c_o(\\gamma,y)).\n\\end{equation}\n\\noindent This action is proper and co-compact and we denote the quotient space by $\\textnormal{U}_o\\Gamma$. The flow $\\psi_t$ descends to a flow on $\\textnormal{U}_o\\Gamma$, still denoted $\\psi_t$, which is a H\u00f6lder reparametrization of the Gromov geodesic flow of $\\Gamma$ \\cite{Gro}. This is the analogue of Sambarino's Theorem \\cite[Theorem 3.2(1)]{Sam} (see also Lemma \\ref{lema conj urhogamma y uogamma}).\n\n\n\nWe say that an element $\\gamma$ in $\\Gamma$ is \\textit{primitive} if cannot be written as a positive power of another element in $\\Gamma$. Periodic orbits of $\\psi_t$ are in one-to-one correspondence with conjugacy classes of primitive elements in $\\Gamma$. If $[\\gamma]$ is such a conjugacy class, the period of the corresponding periodic orbit is \n\n\\begin{center}\n$\\ell_{c_o}(\\gamma)=\\lambda_1(\\rho(\\gamma))$\n\\end{center}\n\n\\noindent (see Fact \\ref{fact urhogamma is anosov} and Lemma \\ref{lema conj urhogamma y uogamma}). The topological entropy of $\\psi_t$ coincides with the \\textit{entropy} of $\\rho$ defined by Bridgeman-Canary-Labourie-Sambarino \\cite{BCLS}:\n\n\n\n\\vspace{0,2cm}\n\n\n\\begin{center}\n$h_\\textnormal{top}(\\psi_t)=h_\\rho:=\\displaystyle\\limsup_{t\\longrightarrow\\infty} \\dfrac{\\log\\#\\lbrace [\\gamma]\\in [\\Gamma]:\\hspace{0,3cm} \\gamma \\textnormal{ is primitive and }\\lambda_1(\\rho(\\gamma))\\leq t\\rbrace}{t}$.\n\n\\end{center} \n\n\n\\vspace{0,2cm}\n\n\n\\noindent It is positive and finite (c.f. Fact \\ref{fact entropy, equidistribution and counting for urhogamma}) and will be denoted by $h$ from now on.\n\n\n\n\n\n\n\\begin{rem}\nOne can prove that if we \\textit{push} all this construction by the limit map $\\xi:\\partial_\\infty\\Gamma\\longrightarrow\\Lambda_{\\rho(\\Gamma)}$, we recover the geodesic flow defined in \\cite[Subsection 6.1]{GM} for $\\mathbb{H}^{p,q-1}$-convex co-compact groups. This remark will not be used in the sequel.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\\subsection{Dual cocycle and Gromov product}\\label{subsec dual and gromov co}\n\n\nThanks to transversality condition (\\ref{eq transv condition}) and the fact that $o$ belongs to $\\pmb{\\Omega}_\\rho$ the following map is well-defined.\n\n\n\n\n\\begin{dfn}\nLet\n\n\n\\begin{center}\n$[\\cdot,\\cdot]_o:\\partial_\\infty^{2}\\Gamma\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} [x,y]_o:=-\\dfrac{1}{2}\\log\\left\\vert  \\dfrac{\\theta_{x}\\left(J^o\\cdot v_x\\right)\\theta_{y}\\left(J^o\\cdot v_y\\right)}{\\theta_{x}\\left(v_y\\right)\\theta_{y}\\left(v_x\\right)}\\right\\vert$,\n\\end{center}\n\n\\noindent where $\\theta_x$ (resp. $\\theta_y$) is a non-zero linear functional whose kernel is $\\eta(x)$ (resp. $\\eta(y)$) and $v_x$ (resp. $v_y$) is a non-zero vector in $\\xi(x)$ (resp. $\\xi(y)$).\n\\end{dfn}\n\n\\begin{rem}\\label{rem gromov o is the one of GM}\nThe map $[\\cdot,\\cdot]_o$ coincides, up to a sign, with the Gromov product introduced in \\cite[Subsection 3.5]{GM}. The authors give geometric interpretations of this function using pseudo-Riemannian geometry.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\n\n\\begin{rem}\\label{rem coco are dual}\nThe cocycle $c_o$ is dual to itself, i.e. $\\ell_{c_o}(\\gamma)=\\ell_{c_{o}}(\\gamma^{-1})$ for every $\\gamma\\in\\Gamma_{\\textnormal{H}}$. Indeed, this follows from Lemma \\ref{lema cocycle o and periods} and the fact that $\\lambda_1(g)=\\lambda_1(g^{-1})$ for all $g$ in $ G$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\n\n\nThe proof of the following lemma is a direct computation.\n\n\n\\begin{lema}\\label{lema gromov o}\nThe map $[\\cdot,\\cdot]_o$ is a Gromov product for the pair $\\lbrace c_o,c_o\\rbrace$, that is, for every $\\gamma\\in\\Gamma$ and every $(x,y)\\in\\partial_\\infty^{2}\\Gamma$ one has\n\n\n\\begin{center}\n$[\\gamma\\cdot x,\\gamma\\cdot y]_o-[x,y]_o=-(c_o(\\gamma,x)+c_o(\\gamma,y))$.\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{lema}\n\n\n\n\n\n\n\n\n\nThe following lemma will be very important in the proof of Theorem \\ref{teorema A}. It provides a geometric interpretation of the Gromov product different from the one given in Remark \\ref{rem gromov o is the one of GM}.\n\n\\begin{lema}\\label{lema computing gromov on gammapm o}\nLet $\\gamma$ be an element of $\\Gamma_{\\textnormal{H}}$. Then\n\n\\begin{center}\n$[\\gamma_-,\\gamma_+]_o=-\\frac{1}{2} \\mathbb{B}(J^o\\cdot\\rho(\\gamma)_-,J^o\\cdot\\rho(\\gamma)_+,\\rho(\\gamma^{-1})_-,\\rho(\\gamma^{-1})_+)$.\n\\end{center}\n\\end{lema}\n\n\n\\begin{proof}\n\n\nFrom Section \\ref{sec anosov} we know that $\\rho(\\gamma^{\\pm 1})$ is proximal and that the following holds:\n\n\\begin{center}\n$\\rho(\\gamma)_+=\\xi(\\gamma_{+})$, \\hspace{0,5cm} $\\rho(\\gamma^{-1})_+=\\xi(\\gamma_{-})$, \\hspace{0,5cm} $\\rho(\\gamma)_-=\\eta(\\gamma_-)$, \\hspace{0,5cm} $\\rho(\\gamma^{-1})_-=\\eta(\\gamma_{+})$.\n\n\\end{center}\n\n\n\n\n\n\\noindent Since $J^o=(J^o)^{-1}$, the matrix $J^o\\rho(\\gamma) J^o$ is proximal and one has the equalities\n\n\\begin{center}\n$(J^o\\rho(\\gamma) J^o)_{+}=J^o\\cdot \\xi(\\gamma_+)$ and $(J^o\\rho(\\gamma) J^o)_{-}=J^o\\cdot \\eta(\\gamma_-)$.\n\\end{center}\n\n\n\n\n\n\n\n\\noindent The proof finishes by a direct computation.\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Distribution of attractors and repellors with respect to $ b^o$} \\label{subsec dist of fixed wrt bo}\n\n\nRecall that $h=h_{\\textnormal{top}}(\\psi_t)$ and let $\\mu_{o}$ be a \\textit{Patterson-Sullivan probability} on $\\partial_\\infty\\Gamma$ associated to $c_{o}$, i.e. $\\mu_o$ satisfies\n\n\\begin{center}\n$\\dfrac{d\\gamma_*\\mu_o}{d\\mu_o}(x)=e^{-hc_{o}(\\gamma^{-1},x)}$\n\\end{center}\n\n\n\\noindent for every $\\gamma\\in\\Gamma$. Such a probability exists (see Subsection \\ref{subsub PS}). By Lemma \\ref{lema gromov o} the measure\n\\begin{equation}\ne^{-h[\\cdot,\\cdot]_o}\\mu_o\\otimes\\mu_o\\otimes dt\n\\end{equation}\n\\noindent on $\\partial_\\infty^{2}\\Gamma\\times\\mathbb{R}$ is $\\Gamma$-invariant and induces on the quotient $\\textnormal{U}_o\\Gamma$ a $\\psi_t$-invariant measure. By Sambarino \\cite[Theorem 3.2(2)]{Sam} this measure is, up to scaling, the probability of maximal entropy of $\\psi_t$ (see Proposition \\ref{prop product is of maximal entropy}).\n\n\n\nFor a metric space $X$ we denote by $C_c^*(X)$ the dual of the space of compactly supported continuous real functions on $X$ equipped with the weak-star topology. If $x$ is a point in $X$, let $\\delta_x\\in C_c^*(X)$ be the Dirac mass at $x$.\n\n\n\n\n\\begin{prop}[Sambarino {\\cite[Proposition 4.3]{Sam}}\\footnote{For a proof in our setting see Proposition \\ref{prop distribution of periodic orbits}.}]\\label{prop sambarino distribution wrt periods}\n\nThere exists a constant $M=M_{\\rho,o}>0$ such that\n\n\n\\begin{center}\n$Me^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}},\\ell_{c_o}(\\gamma)\\leq t} \\delta_{\\gamma_-}\\otimes\\delta_{\\gamma_+}\\longrightarrow e^{-h[\\cdot,\\cdot]_o}\\mu_o\\otimes\\mu_o$\n\\end{center}\n\n\n\\noindent as $t\\longrightarrow\\infty$ on $C_c^*(\\partial_\\infty^{2}\\Gamma)$.\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{prop}\n\n\n\n\n\nFrom Proposition \\ref{prop sambarino distribution wrt periods} we deduce Proposition \\ref{prop distribution on bg for length} which directly implies Theorem \\ref{teorema A} in the torsion free case. \n\n\nFix a point $\\tau\\in S^o$, a maximal subalgebra $\\mathfrak{b}\\subset\\mathfrak{p}^\\tau\\cap\\mathfrak{q}^o$ and a closed Weyl chamber $\\mathfrak{b}^+$ contained in $\\mathfrak{b}$.\n\n\\begin{prop} \\label{prop distribution on bg for length}\n\nThere exists a  constant $M=M_{\\rho,o}>0$ such that\n\n\\begin{center}\n$M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}}, \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t} \\delta_{\\gamma_-}\\otimes\\delta_{\\gamma_+}\\longrightarrow\\mu_o\\otimes\\mu_o$\n\\end{center}\n\n\n\\noindent as $t\\longrightarrow\\infty$ on $C^*(\\partial_\\infty\\Gamma\\times\\partial_\\infty\\Gamma)$.\n\n\\end{prop}\n\n\n\nRecall that the generalized Cartan projection $b^o$ is defined in the set $\\mathscr{C}_{o,G}^>$. The sum in Proposition \\ref{prop distribution on bg for length} is taken then over all elements $\\gamma\\in\\Gamma_{\\textnormal{H}}$ for which $\\rho(\\gamma)\\in\\mathscr{C}_{o,G}^>$ and $ \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t$. To make the formula more readable we do not emphasize the fact that $\\rho(\\gamma)$ must belong to $\\mathscr{C}_{o,G}^>$. On the other hand, by Corollary \\ref{cor gammao in cowmayor} this condition holds apart from finitely many exceptions $\\gamma\\in\\Gamma$.\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop distribution on bg for length}]\n\n\n\nSet \n\n\\begin{center}\n\n$\\theta_t:=M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}},  \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t} \\delta_{\\gamma_-}\\otimes\\delta_{\\gamma_+}$.\n\\end{center}\n\n\n\n\nWe first prove the statement outside the diagonal, that is, on subsets of $\\partial_\\infty^{2}\\Gamma$. Let $\\delta>0$ and $A,B\\subset\\partial_\\infty\\Gamma$ disjoint open sets. Consider an element $\\gamma\\in\\Gamma_{\\textnormal{H}}$ such that $\\gamma_-\\in A$ and $\\gamma_+\\in B$ and let $s:=[\\gamma_-,\\gamma_+]_{o}$. By taking $A$ and $B$ smaller we may assume \n\\begin{equation} \\label{eq aprox grom prod on dist thm nu}\n\\vert [x,y]_{o}-s\\vert<\\delta\n\\end{equation}\n\\noindent for all $(x,y)\\in A\\times B$.\n\n\n\n\nBy Lemma \\ref{lema jrhojrho prox dos}, apart from possibly finitely many exceptions $\\gamma\\in \\Gamma_{\\textnormal{H}}$ with $(\\gamma_-,\\gamma_+)\\in A\\times B$, the following holds:\n\n\n\n\n\\begin{center}\n$\\left\\vert  \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}-\\lambda_1(\\rho(\\gamma))  -\\frac{1}{2}\\mathbb{B}(J^o\\cdot\\rho(\\gamma)_-,J^o\\cdot \\rho(\\gamma)_+,\\rho(\\gamma^{-1})_-,\\rho(\\gamma^{-1})_+)\\right\\vert <\\delta$.\n\\end{center}\n\n\n\n\n\n\n\n\\noindent Applying Lemma \\ref{lema cocycle o and periods} and Lemma \\ref{lema computing gromov on gammapm o} we conclude that\n\n\\begin{center}\n$\\left\\vert   \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}-\\ell_{c_o}(\\gamma)  +[\\gamma_-,\\gamma_+]_o\\right\\vert <\\delta$.\n\\end{center}\n\n\n\n\n\n\\noindent By (\\ref{eq aprox grom prod on dist thm nu}) it follows that\n \n\n\\begin{center}\n$ \\ell_{c_{o}}(\\gamma)-s-2 \\delta< \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}<\\ell_{c_{o}}(\\gamma) -s+2 \\delta$\n\\end{center}\n\n\n\\noindent holds apart from finitely many exceptions $\\gamma\\in\\Gamma_{\\textnormal{H}}$ such that $\\gamma_-\\in A$ and $\\gamma_+\\in B$. From now on, the proof of the convergence\n\n\n\n\n\\begin{center}\n$\\theta_t(A\\times B)\\longrightarrow \\mu_o(A)\\mu_o(B)$\n\\end{center}\n\n\\noindent follows line by line the proof of \\cite[Theorem 6.5]{Sam}. \n\n\n\nIt remains to prove the convergence in the diagonal $\\lbrace(x,x): \\hspace{0,3cm} x\\in\\partial_\\infty\\Gamma\\rbrace$, but once again, the proof is the same as the one given in \\cite[Theorem 6.5]{Sam}. For completeness we briefly sketch it.\n\n\n\nSince $\\mu_o$ has no atoms (see Lemma \\ref{lema PS non atomic}), for every $\\gamma$ in $\\Gamma$ the diagonal has $(\\mu_o\\otimes \\gamma_*\\mu_o)$-measure equal to zero. We fix two elements $\\gamma_0,\\gamma_1\\in\\Gamma_{\\textnormal{H}}$ with no common fixed point in $\\partial_\\infty\\Gamma$ and let $\\varepsilon_0>0$. There exists a finite open covering $\\mathscr{U}$ of $\\partial_\\infty\\Gamma$ such that for $i=0,1$ one has\n\n\\begin{center}\n$\\displaystyle\\sum_{U\\in\\mathscr{U}}  \\mu_o(U) \\mu_o(\\gamma_i^{-1}\\cdot U)<\\varepsilon_0$.\n\\end{center}\n\n\n\\noindent We can assume that for every $U\\in\\mathscr{U}$ there exists $i\\in\\lbrace 0,1\\rbrace$ such that $\\gamma_i^{-1}\\cdot\\overline{U}$ is disjoint from $\\overline{U}$. There exists an open covering $\\mathscr{V}$ of $\\partial_\\infty\\Gamma$ with the following properties:\n\n\\begin{enumerate}\n\n\\item $\\displaystyle\\sum_{V\\in\\mathscr{V}}  \\mu_o(V) \\mu_o(\\gamma_i^{-1} \\cdot V)<\\varepsilon_0$ for $i=0,1$.\n\\item The closure of every element in $\\mathscr{U}$ is contained in a unique element of $\\mathscr{V}$ and if $\\gamma_i^{-1}\\cdot\\overline{U}$ is disjoint from $\\overline{U}$ the same holds for this element in $\\mathscr{V}$.\n\\item Suppose that $\\gamma_i^{-1}\\cdot\\overline{U}\\cap \\overline{U}=\\emptyset$ and let $V\\in\\mathscr{V}$ be the unique element such that $\\overline{U}\\subset V$. Then apart from finitely many exceptions $\\gamma$ such that $\\gamma_{\\pm}\\in U$ one has $(\\gamma_i^{-1}\\gamma)_-\\in V$ and $(\\gamma_i^{-1}\\gamma)_+\\in \\gamma_i^{-1}\\cdot V$.\n\n\\end{enumerate}\n\nSet $D:=\\displaystyle\\max_{i=0,1}\\lbrace D_{\\gamma_i^{-1}}\\rbrace$ where $D_{\\gamma_i^{-1}}$ is the constant given by Proposition \\ref{prop traingle inequality} and take $U\\in\\mathscr{U}$ as in (3). By Proposition \\ref{prop traingle inequality} we have\n\n\n\n\n\n\\begin{equation*}\n\\begin{split}\n\\theta_t(U\\times U) & \\leq Me^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}}, \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t+D} \\delta_{\\gamma_-}(V)\\delta_{\\gamma_+}(\\gamma_i^{-1}\\cdot V)\\\\ & +Me^{-ht}\\# F\n\\end{split}\n\\end{equation*}\n\n\n\n\n\\noindent where $F$ is a finite set independent of $t$. Since $V\\times\\gamma_i^{-1}\\cdot V$ is far from the diagonal the right side converges to\n\n\n\n\\begin{center}\n$e^D\\mu_o(V)\\mu_o(\\gamma_i^{-1}\\cdot V)$\n\\end{center}\n\n\n\\noindent as $t\\longrightarrow \\infty$. Adding up in $U\\in\\mathscr{U}$ we conclude\n\n\\begin{center}\n$\\displaystyle\\limsup_{t\\longrightarrow\\infty}\\displaystyle\\sum_{U\\in\\mathscr{U}}\\theta_t(U\\times U)\\leq 2e^D\\varepsilon_0$.\n\\end{center}\n\n\n\\noindent Hence $\\theta_t(\\lbrace(x,x):\\hspace{0,3cm} x\\in\\partial_\\infty\\Gamma\\rbrace)$ converges to zero and since the diagonal has measure zero for $\\mu_o\\otimes\\mu_o$ the proof is finished.\n\n\n\\end{proof}\n\n\n\n\\subsection{Proof of Theorem \\ref{teorema A}} \\label{subsec proof teo A}\n\n\n\nThe following is a corollary of Proposition \\ref{prop distribution on bg for length}.\n\n\n\n\\begin{cor}\\label{cor distr orbit o in gammah for bo}\nThere exists a constant $M=M_{\\rho,o}>0$ such that\n\n\n\\begin{center}\n$M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}},  \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t} \\delta_{\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}}}\\otimes\\delta_{\\rho(\\gamma)\\cdot o}\\longrightarrow \\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_o)$\n\\end{center}\n\n\\noindent on $C^*(\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d))$ as $t\\longrightarrow\\infty$.\n\n\n\\end{cor}\n\n\n\\begin{proof}\n\nSet\n\n\\begin{center}\n$\\nu_t^{\\textnormal{H}}:= M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}},  \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t} \\delta_{\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}}}\\otimes\\delta_{\\rho(\\gamma)\\cdot o}$\n\\end{center}\n\n\n\\noindent and take $\\theta_t$ the measure defined in the proof of Proposition \\ref{prop distribution on bg for length}. We know that \n\n\\begin{center}\n$(\\eta,\\xi)_*(\\theta_t)\\longrightarrow \\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_o)$.\n\\end{center}\n\n\n\\noindent Hence we only have to show the following convergence\n\\begin{equation}\\label{eq en cor orbit o para hiperbolicos}\n\\nu_t^{\\textnormal{H}}-(\\eta,\\xi)_*(\\theta_t)\\longrightarrow 0.\n\\end{equation}\n\n\n\nTake a small positive $\\delta$. By Proposition \\ref{prop limit with S and U and Uuno cerca gammamas} and the proof of Proposition \\ref{prop action on omegarho is prop discont and limit set is limit set} we know that, apart from finitely many exceptions $\\gamma$ in $\\Gamma_{\\textnormal{H}}$, one has\n\n\\begin{center}\n$d(\\rho(\\gamma)\\cdot o,\\rho(\\gamma)_+)<\\delta$ and $d(\\rho(\\gamma^{-1})\\cdot o,\\rho(\\gamma^{-1})_+)<\\delta$.\n\\end{center}\n\n\nBy taking $\\cdot^{\\perp_{p,q}}$ we can assume further that $d^*(\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}},\\rho(\\gamma)_-)<\\delta$. Now the proof of (\\ref{eq en cor orbit o para hiperbolicos}) follows from evaluation on continuous functions of $\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d)$.\n\n\n\\end{proof}\n\n\nWe now include torsion elements to the previous statement and finish the proof of Theorem \\ref{teorema A}.\n\n\n\\begin{prop}\\label{prop distribution on bg for length with torsion}\n\nThere exists a constant $M=M_{\\rho,o}>0$ such that\n\n\n\\begin{center}\n$M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma,  \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t} \\delta_{\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}}}\\otimes\\delta_{\\rho(\\gamma)\\cdot o}\\longrightarrow \\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_o)$\n\\end{center}\n\n\\noindent on $C^*(\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d))$ as $t\\longrightarrow\\infty$.\n\\end{prop}\n\n\n\n\n\\begin{proof}\n\n\nThe structure of the proof is the same as that of Proposition \\ref{prop distribution on bg for length}, that is, we first prove the statement outside the diagonal and deduce from that the statement on the diagonal. Here by \\textit{diagonal} we mean the set\n\n\\begin{center}\n$\\Delta:=\\lbrace (\\theta,v)\\in \\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d): \\hspace{0,3cm} \\theta(v)=0 \\rbrace$.\n\\end{center}\n\n\nLet \n\n\\begin{center}\n$\\nu_t:= M e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma,  \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t} \\delta_{\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}}}\\otimes\\delta_{\\rho(\\gamma)\\cdot o}$\n\\end{center}\n\n\n\\noindent and take $\\nu_t^{\\textnormal{H}}$ as in the proof of Corollary \\ref{cor distr orbit o in gammah for bo}.\n\n\nConsider first a continuous function $f$ on $\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d)$ whose support $\\textnormal{supp}(f)$ is disjoint from $\\Delta$.\n\n\n\\begin{cla} \\label{claim f suppor far from diagonal implies g in gh}\n\nThe following holds\n\n\\begin{center}\n$\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} (\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}},\\rho(\\gamma)\\cdot o)\\in \\textnormal{supp}(f) \\textnormal{ and } \\gamma\\notin\\Gamma_{\\textnormal{H}}\\rbrace<\\infty$.\n\\end{center}\n\n\\end{cla} \n\n\n\n\n\\begin{proof}[Proof of Claim \\ref{claim f suppor far from diagonal implies g in gh}]\n\n\n\n\nFix a positive $D$ such that for every $(\\theta,v)\\in \\textnormal{supp}(f)$ one has $d(\\theta,v)> D$. As we saw in the proof of Proposition \\ref{prop action on omegarho is prop discont and limit set is limit set}, the distances\n\n\\begin{center}\n$d(\\rho(\\gamma)\\cdot o,U_1(\\rho(\\gamma)))$ and $d^*(\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}},S_{d-1}(\\rho(\\gamma)))$\n\\end{center}\n\n\\noindent converge to zero as $\\gamma\\longrightarrow\\infty$. We conclude that, apart from possibly finitely many exceptions $\\gamma$ in $\\Gamma$ with $(\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}},\\rho(\\gamma)\\cdot o)\\in \\textnormal{supp}(f)$, one has\n\n\\begin{center}\n$d(U_1(\\rho(\\gamma)),S_{d-1}(\\rho(\\gamma)))>D$.\n\\end{center}\n\n\n\nNow apply (\\ref{eq def anosov}), Remark \\ref{rem complemento de Sdmenosuno va en Uuno} and Benoist's Lemma \\ref{lema benoist lemma 1.2} to conclude that for $\\vert\\gamma\\vert_\\Gamma$ large enough the matrix $\\rho(\\gamma)$ is proximal.\n\n\\end{proof}\n\n\n\nFrom Claim \\ref{claim f suppor far from diagonal implies g in gh} we conclude that \n\n\\begin{center}\n$\\displaystyle\\lim_{t\\longrightarrow\\infty}\\nu_t(f)=\\displaystyle\\lim_{t\\longrightarrow\\infty}\\nu_t^{\\textnormal{H}}(f)$\n\\end{center}\n\n\n\n\\noindent which by Corollary \\ref{cor distr orbit o in gammah for bo} equals $(\\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_o))(f)$.\n\n\nIt remains to prove the convergence on the diagonal. It suffices to prove that for every positive $\\varepsilon_0$ there exists an open covering $\\lbrace U^*\\times U\\rbrace$ of $\\Delta$ such that\n\n\\begin{center}\n$\\displaystyle\\limsup_{t\\longrightarrow\\infty}\\nu_t\\left(\\displaystyle\\bigcup (U^*\\times U)\\right)\\leq \\varepsilon_0$.\n\\end{center}\n\n\n\n\nThe proof is the same as in Proposition \\ref{prop distribution on bg for length}. Namely, take two elements $\\gamma_0,\\gamma_1$ in $\\Gamma_{\\textnormal{H}}$ with no common fixed point in $\\partial_\\infty\\Gamma$ and a coverings $\\mathscr{U}=\\lbrace U^*\\times U\\rbrace$ and $\\mathscr{V}=\\lbrace V^*\\times V\\rbrace$ of $\\Delta$ by open sets with the following properties:\n\n\n\n\\begin{enumerate}\n\\item For every $U^*\\times U$ in $\\mathscr{U}$ there exists $i=0,1$ such that $\\rho(\\gamma_i^{-1})\\cdot\\overline{U}$ is transverse to $\\overline{U^*}$.\n\\item $\\displaystyle\\sum_{V^*\\times V\\in\\mathscr{V}} (\\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_o))(V^*\\times\\rho(\\gamma_i^{-1})\\cdot V)<\\varepsilon_0$ for $i=0,1$.\n\\item The closure of every element in $\\mathscr{U}$ is contained in a unique element of $\\mathscr{V}$ and if $\\rho(\\gamma_i^{-1})\\cdot \\overline{U}$ is transverse to $\\overline{U^*}$ the same holds for this element in $\\mathscr{V}$.\n\\item Suppose that $\\rho(\\gamma_i^{-1})\\cdot \\overline{U}$ is transverse to $\\overline{U^*}$ and let $V^*\\times V\\in\\mathscr{V}$ be the unique element such that $\\overline{U}\\subset V$ and $\\overline{U^*}\\subset V^*$. Then, apart from possibly finitely many exceptions $\\gamma$ such that $(\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}},\\rho(\\gamma)\\cdot o)\\in U^*\\times U$, one has\n\n\\begin{center}\n$(\\rho((\\gamma_i^{-1}\\gamma)^{-1})\\cdot o^{\\perp_{p,q}},\\rho(\\gamma_i^{-1}\\gamma)\\cdot o)\\in V^*\\times \\rho(\\gamma_i^{-1})\\cdot V$.\n\\end{center}\n\\end{enumerate}\n\n\n\n\nProvided with this construction, the proof finishes in the same way as that of Proposition \\ref{prop distribution on bg for length}.\n\n\n\n\\end{proof}\n\n\n\\begin{rem}\\label{rem crit exponent coincides with the entropy}\nFrom Proposition \\ref{prop distribution on bg for length with torsion} we deduce that\n\n\\begin{center}\n$\\displaystyle\\lim_{t\\longrightarrow\\infty}\\dfrac{\\log\\#\\lbrace \\gamma\\in\\Gamma: \\hspace{0,3cm} \\rho(\\gamma)\\in\\mathscr{C}_{o,G}^> \\textnormal{ and } \\vert b^o(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t\\rbrace}{t}$\n\\end{center}\n\n\\noindent coincides with the entropy $h=h_\\rho$ of $\\rho$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\\section{Distribution of the orbit of $o$ with respect to $ b^\\tau$}\\label{section distrib wrt btau}\n\\setcounter{equation}{0}\n\n\nThe proof of Theorem \\ref{teorema B} follows the same lines of the proof of Theorem \\ref{teorema A}, we just have to pick a (slightly) different flow $\\psi_t$.\n\n\nFix a $P_1^{p,q}$-Anosov representation $\\rho:\\Gamma\\longrightarrow G$, a point $o$ in $\\pmb{\\Omega}_\\rho$ and $\\tau\\in S^o$.\n\n\n\n\\subsection{The cocycle $c_\\tau$}\\label{subsec cocycle ctau}\n\n\n\nLet $\\Vert\\cdot\\Vert_\\tau$ be the norm introduced in Subsection \\ref{subsec cartan dec in sec anosov}.\n\n\n\\begin{dfn} \\label{dfn cocycle tauo}\n\nLet\n\n\\begin{center}\n$c_{\\tau}:\\Gamma\\times\\partial_\\infty\\Gamma\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} c_{\\tau}(\\gamma,x):=\\dfrac{1}{2}\\log\\left(\\dfrac{\\Vert\\rho(\\gamma)\\cdot\\theta_{x}\\Vert_{\\tau}\\Vert\\rho(\\gamma)\\cdot v_{x}\\Vert_{\\tau}}{\\Vert\\theta_{x}\\Vert_{\\tau}\\Vert v_{x}\\Vert_{\\tau}}\\right)$\n\\end{center}\n\n\n\\noindent where $\\theta_{x}:\\mathbb{R}^d\\longrightarrow\\mathbb{R}$ is a non-zero linear functional whose kernel equals $\\eta(x)$ and $v_{x}\\neq 0$ belongs to $\\xi(x)$.\n\n\\end{dfn}\n\n\\begin{rem} \\label{rem ctau es el betauno}\nOne can prove that for every $\\gamma\\in\\Gamma$ and $x\\in\\partial_\\infty\\Gamma$ one has\n\n\n\\begin{center}\n$c_{\\tau}(\\gamma,x)=\\log\\dfrac{\\Vert\\rho(\\gamma)\\cdot v_{x}\\Vert_{\\tau}}{\\Vert v_{x}\\Vert_{\\tau}}$,\n\\end{center}\n\n\\noindent that is, $c_\\tau$ coincides with the map $\\beta_1(\\cdot,\\cdot)$ of \\cite[Section 5]{Sam}. This remark will not be used in the sequel.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\nThe following lemma holds by straightforward computations.\n\n\n\\begin{lema}\\label{lema cocycle tau and periods}\n\nThe function $c_{\\tau}$ is a H\u00f6lder cocycle. The period of $\\gamma$ in $\\Gamma_{\\textnormal{H}}$ is given by\n\n\\begin{center}  \n$\\ell_{c_{\\tau}}(\\gamma)=\\lambda_1(\\rho(\\gamma))>0$.\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{lema}\n\n\n\n\n\n\n\nThe quotient space of $\\partial_\\infty^{2}\\Gamma\\times \\mathbb{R}$ by the action of $\\Gamma$ induced by $c_\\tau$ will be denoted by $\\textnormal{U}_\\tau\\Gamma$. It is equipped with a flow that lifts to the translation flow (\\ref{eq translation flow}) on $\\partial_\\infty^{2}\\Gamma\\times\\mathbb{R}$.\n\n\n\n\n\n\n\\subsection{Dual cocycle and Gromov product}\\label{subsec dual and gromov ctau}\n\n\n\n\n\n\n\\begin{dfn}\n\nLet\n\\begin{center}\n$[\\cdot,\\cdot]_{\\tau}:\\partial_\\infty^{2}\\Gamma\\longrightarrow\\mathbb{R}: \\hspace{0,3cm} [x,y]_{\\tau}:=\\dfrac{1}{2}\\log\\left\\vert  \\dfrac{\\theta_{y}\\left(v_x\\right)\\theta_{x}\\left(v_y\\right)}{\\theta_{x}\\left(J^o\\cdot v_x\\right)\\Vert\\theta_{y}\\Vert_{\\tau}\\Vert v_y\\Vert_{\\tau}} \\right\\vert$.\n\\end{center}\n\\end{dfn}\n\n\n\\begin{rem} \\label{rem co dual to ctauo} \nRecall that $c_o$ is the cocycle defined in Section \\ref{section distrib wrt bo}. The cocycle $c_{\\tau}$ is dual to $c_o$, i.e. $\\ell_{c_o}(\\gamma)=\\ell_{c_{\\tau}}(\\gamma^{-1})$ for every $\\gamma\\in\\Gamma_{\\textnormal{H}}$.\n\\begin{flushright}\n$\\diamond$\n\\end{flushright}\n\\end{rem}\n\n\n\nThe proof of the following lemma is a direct computation.\n\n\\begin{lema}\nFor every $\\gamma\\in\\Gamma$ and every $(x,y)\\in\\partial_\\infty^{2}\\Gamma$ one has\n\n\n\\begin{center}\n$[\\gamma\\cdot x,\\gamma\\cdot y]_{\\tau}-[x,y]_{\\tau}=-(c_o(\\gamma,x)+c_{\\tau}(\\gamma,y))$.\n\\end{center}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{lema}\n\n\n\n\n\n\n\\begin{lema}\\label{lema computing gromov tauo on gammapm}\nLet $\\gamma$ be an element of $\\Gamma_{\\textnormal{H}}$. Then\n\n\\begin{center}\n$[\\gamma_-,\\gamma_+]_{\\tau}=-\\frac{1}{2}\\mathbb{B}(J^o\\cdot \\rho(\\gamma)_-,J^o\\cdot\\rho(\\gamma)_+,\\rho(\\gamma^{-1})_-,\\rho(\\gamma^{-1})_+)+\\frac{1}{2}\\mathscr{G}_{\\tau}(\\rho(\\gamma^{-1})_-,J^o\\cdot \\rho(\\gamma)_+)$.\n\\end{center}\n\\end{lema}\n\n\n\\begin{proof}\n\nRecall the definition of $[\\cdot,\\cdot]_o$ from Subsection \\ref{subsec dual and gromov co}. One has \n\n\n\\begin{center}\n\n$[\\gamma_-,\\gamma_+]_\\tau=[\\gamma_-,\\gamma_+]_o+\\dfrac{1}{2}\\log\\dfrac{\\left\\vert \\theta_{\\gamma_+}(J^o\\cdot v_{\\gamma_+}) \\right\\vert}{\\Vert \\theta_{\\gamma_+} \\Vert_\\tau\\Vert  v_{\\gamma_+}\\Vert_\\tau} $.\n\n\\end{center}\n\\noindent The proof then follows from Lemma \\ref{lema computing gromov on gammapm o} and Remark \\ref{rem J preserva norma}.\n\n\\end{proof}\n\n\n\n\n\\subsection{Distribution of attractors and repellors with respect to $ b^\\tau$} \\label{subsec dist of fixed wrt btau}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLet $\\mu_{\\tau}$ be a Patterson-Sullivan probability on $\\partial_\\infty\\Gamma$ associated to $c_{\\tau}$ and recall that $\\mu_o$ is the one associated to $c_o$. The analogue of Proposition \\ref{prop sambarino distribution wrt periods} is available for the flow on $\\textnormal{U}_\\tau\\Gamma$. The limit measure can be written in this case as\\footnote{For a proof, see Remark \\ref{rem BM for ctau and distribution}.}\n\n\\begin{center}\n$e^{-h[\\cdot,\\cdot]_\\tau}\\mu_o\\otimes\\mu_\\tau $.\n\\end{center}\n\n\n\nLet $\\mathfrak{b}^+$ be a closed Weyl chamber of a maximal subalgebra $\\mathfrak{b}\\subset\\mathfrak{p}^\\tau\\cap\\mathfrak{q}^o$.\n\n\n\\begin{prop} \\label{prop distribution on bg nu}\nThere exists a constant $M'=M'_{\\rho,\\tau}>0$ such that\n\n\n\\begin{center}\n$M' e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma_{\\textnormal{H}}, \\vert b^\\tau(\\rho(\\gamma))\\vert^{\\frac{1}{2}}\\leq t} \\delta_{\\gamma_-}\\otimes\\delta_{\\gamma_+}\\longrightarrow \\mu_o\\otimes\\mu_\\tau$\n\\end{center}\n\n\n\\noindent as $t\\longrightarrow\\infty$ on $C^*(\\partial_\\infty\\Gamma\\times\\partial_\\infty\\Gamma)$.\n\n\\end{prop}\n\n\n\n\n\n\\begin{proof}\n\n\nThe proof is the same that the one given in Proposition \\ref{prop distribution on bg for length} adapted to the pair $\\lbrace c_o,c_\\tau \\rbrace$ and the Gromov product $[\\cdot,\\cdot]_\\tau$: apply item (5) of Lemma \\ref{lema jrhojrho prox dos} and Lemma \\ref{lema computing gromov tauo on gammapm}.\n\n\n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{teorema B}} \\label{subsec proof teo B}\n\n\nThe following proposition, which implies Theorem \\ref{teorema B}, can be proved in the same way as Proposition \\ref{prop distribution on bg for length with torsion}.\n\n\n\\begin{prop}\\label{prop distribution on bg for btau with torsion}\nThere exists a constant $M'=M'_{\\rho,\\tau}>0$ such that\n\n\n\\begin{center}\n$M' e^{-ht}\\displaystyle\\sum_{\\gamma\\in\\Gamma, \\vert b^\\tau(\\rho(\\gamma))\\vert^{\\frac{1}{2}} \\leq t} \\delta_{\\rho(\\gamma^{-1})\\cdot o^{\\perp_{p,q}}}\\otimes\\delta_{\\rho(\\gamma)\\cdot o}\\longrightarrow \\eta_{*}(\\mu_o)\\otimes\\xi_*(\\mu_\\tau)$\n\\end{center}\n\n\\noindent on $C^*(\\mathbb{P}((\\mathbb{R}^d)^*)\\times\\mathbb{P}(\\mathbb{R}^d))$ as $t\\longrightarrow\\infty$.\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\end{prop}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nElectromagnetic metamaterials have been used to realize many exotic scattering responses over the last two decades. Effects including negative refractive index and cloaking have generated significant interest and have served to drive the community \\cite{smith2000composite, pendry2000negative, shelby2001experimental, smith2004metamaterials, schurig2006metamaterial}. A more applied, but still relevant metamaterials achievement, is that of graded designs \\cite{greegor2005simulation, smith2005gradient}. It was shown early on that unit-cells which form metamaterials may be designed with a spatial dependence across a surface and \/ or volume, and various lensing effects were shown that utilize spatial degrees of freedom. A principle of gradient metasurfaces is that their scattering properties change slowly as a function of spatial coordinate. Broadband metamaterial absorbers \\cite{landy2008perfect, avitzour2009wide}, and metamaterial spatial light modulators \\cite{shrekenhamer2013four}, also make use of dissimilar neighboring unit-cells, however there is no such requirement to make neighbors as alike as possible, and designs are simply cobbled together to achieve the desired response. This fact highlights a design feature of sub-wavelength metal-based metamaterials, i.e. their scattering response is primarily connected to their geometry and -- due to minimal neighbor interaction -- the unit-cell largely governs the electromagnetic properties of the array. Conventional optimization techniques built-in to modern day electromagnetic mode solvers are sufficient to achieve a desired response, and designs of differing units may be assembled with little change to the overall response.\n\nIn principle, all-dielectric metasurface (ADM) unit-cells may also be used to tesselate a surface in an arbitrary fashion similar to that achieved with metal-based metamaterials. However, ADMs resonators are only marginally sub-wavelength, and modes utilized are often not confined within their physical bound -- with an evanescent tail lying just outside their surface -- resulting in significant neighbor interaction. Further, surface modes related to the periodicity are not restricted to nearest neighbors, and may persist over several spatial periods. Despite the success of ADMs \\cite{sautter2015active, headland2015terahertz}, it is conceivable that still-richer electromagnetic scattering can be achieved if more complex geometries are employed. However, physical understanding for such metasurfaces is poor -- simple functional relationships, or even heuristic guidance regarding super unit-cell geometry and final electromagnetic properties is unavailable. The only contemporary means to estimate such metasurface properties, given a candidate geometry, is simulation or fabrication.  However, given the vast space of potential designs and the speed of conventional simulation and fabrication, it is completely infeasible to iterate over designs in order to achieve a desired response. \n\nA viable design alternative to numerical simulation for structured materials, including metamaterials \\cite{liu2018generative, ma2018deep, jiang2019global, nadell2019deep, campbell2019review}, photonic band gap \\cite{peurifoy2018nanophotonic, liu2018training, kiarashinejad2020knowledge, huang2020deep}, and plasmonics studies \\cite{inampudi2018neural, jiang2019free, wiecha2019deep}, is deep learning.  Deep neural networks (DNNs) have successfully learned a forward mapping $s=f(g)$ between a metasurface geometry $g$ and the resulting frequency dependent electromagnetic scattering $s$, where $f$ is an unknown complex (e.g., highly nonlinear) function. \\cite{gareth2013introduction}  A DNN -- once trained -- can effectively act as a high-speed simulator that may be used to find the electromagnetic scattering of a candidate geometry substantially faster (e.g., by six orders of magnitude \\cite{nadell2019deep}) than conventional simulation.\n\n\n\\section{Neural-Adjoint Method}\nWhile deep learning enables substantially faster evaluation of candidates, given the vast number of possibilities for many problems (e.g., approx. $10^{12}$ in our case), it is nonetheless difficult or impossible to iterate over all or most candidate designs. From a design perspective, what would be of greatest utility would be to instead specify a desired frequency dependent electromagnetic response $s$, and have a model or solver compute a specific approximate geometry $\\hat{g}$, which yields $s$ (here $\\hat{}$ denotes approximate). The solution used to search for such an ideal geometry may be cast as an inverse problem written as $\\hat{g}=\\hat{f}^{-1}(s)$. Hadamard described a solution method as well-posed if it met three criteria -- $H_1$: existence (at least one solution), $H_2$: uniqueness (only one solution), and $H_3$: stability (solution must depend continuously on $s$). A solution approach is then ill-posed if one of the $H_1-H_3$ conditions fails \\cite{mueller2012linear}. Past work has shown that finding a specific ADM consisting of distinct neighboring resonators in a super unit-cell geometry for a desired scattering state is an ill-posed non-linear inverse problem and, in particular, conditions $H_1$ and $H_2$ are not met \\cite{liu2018generative}.\n\n\\begin{figure}[t!]\n    \\centering\\includegraphics[width=0.5\\textwidth]{images\/fig1.pdf}\n    \\caption{(a) 3D illustration of an all-dielectric metasurface absorber. A schematic of the all-dielectric metasurfaces super-cell is shown -- top right. (b) Training phase of the neural-adjoint method which uses fixed geometry inputs and corresponding simulated spectrum. (c) The inference phase of the neural-adjoint method, in which the target spectrum drives the search of the optimal geometry. The geometry is shown in a green box here to indicate it is the only trainable parameter.}\n\\label{1}\n\\end{figure}\n\nHere we explore a recently-proposed deep inverse modeling approach, called the neural-adjoint (NA) method \\cite{ren2020benchmarking}, which outperformed many other methods for solving ill-posed inverse problems.  Here we adapt the NA method to solve a challenging problem involving the design of ADMs geometries with 14 free geometrical parameters; a much larger space than most recent work (e.g., $5-10$ free parameters in  \\cite{peurifoy2018nanophotonic, ma2018deep, nadell2019deep}).  Furthermore, existing inverse methods often involve complex training procedures, and ultimately produce sub-optimal solutions. In contrast, the NA only requires that we train a single conventional feed-forward neural network, and -- as we show -- appears to find close approximations to the globally optimal solution within just one minute, even for our complex ADM design problem.  Furthermore, we demonstrate how the NA method can be utilized to expand the design search space, essentially providing a form of active learning that is specifically tailored to solve inverse problems. We explore an example inverse problem, where a frequency dependent infrared absorptivity ($A(\\omega)=s$) is desired, which we would like to achieve using an ADM consisting of a square array of 2$\\times$2 resonators, each with an elliptical generatrix lying perpendicular to its directrix (line of length $h$) -- depicted in Fig. \\ref{1}. Although past experience and single unit-cell simulations guide us to choose approximate metasurface dimensions which yield resonances in the desired spectral range, our inverse problem is ill-posed, since we do not know if a solution exists, i.e. condition $H_1$ is not met. Further, we do not meet the condition of uniqueness, $H_2$, since many metasurface solutions (disparate geometries) may yield the same spectra -- at least within the accuracy of our chosen loss metric, and numerical precision. The proposed ADMs consists of a geometry space of fourteen parameters: [$h$, $p$, $r_{x_1}$--$r_{x_4}$, $r_{y_1}$--$r_{y_4}$, $\\theta_1$--$\\theta_4$]. As shown in Fig. \\ref{1}, $h$ is the height of all four elliptical resonators, $p$ is the periodicity of the super cell, $r_{x_1}$ -- $r_{x_4}$ and $r_{y_1}$ -- $r_{y_4}$ are the x-axis and y-axis radii of four elliptical resonators respectively, and $\\theta_1$ -- $\\theta_4$ are the rotational angles with respect to the center axis of each elliptical resonator. All geometry values are randomly sampled from the data grid shown in Table S1, which is included in the Supplemental. The numerical simulations use SiC for the proposed ADMs, and the Supplemental contains more details.\n\n\\begin{figure}[t!]\n    \\centering\\includegraphics[width=\\textwidth]{images\/fig2.pdf}\n    \\caption{Forward model mean-squared-error (MSE) between DNN predicted spectra and numerical simulations. The DNN result with (a) the best, (b) average, and (c) below average performance. (d) Histogram detailing the distribution of MSE for the entire validation set.}\n    \\label{2}\n\\end{figure}\n\n\n\\section{Results}\nThe DNN forward model is trained by pair of 14 geometrical parameters and 2000 frequency points from 100-500 THz. Results of the forward model trained on 60k $\\{g,s\\}$ pairs (approximately $5.75\\times10^{-8}$ of the entire geometry space) are shown in Fig. \\ref{2} (a)-(c), where red curves are numerical simulations, and blue curves are the DNN predictions. The absolute error between simulation and prediction is plotted as the gray curve on the right vertical axis of Fig. \\ref{2} (a)-(c). In Fig. \\ref{2} (d) we show the histogram of the mean-squared-error (MSE) for the entire validation set, and find that 95\\% have an MSE$\\leq2.49\\times10^{-3}$ (dashed gray line), and an average MSE of $1.2\\times10^{-3}$. Having an accurate trained forward model we next turn toward the inverse solution. The NA method \\cite{ren2020benchmarking} finds the optimal inverse solution by fixing all the weights and biases of $\\hat{f}$, and computing the forward model's gradient solely with respect to the input to the network (i.e., the geometry), starting from randomly chosen values, denoted $\\hat{g}_0$. It is important to highlight that $\\hat{f}$ is a closed-form differentiable expression, and thus calculation of $\\partial \\hat{f}\/\\partial g$ is trivial. Further, we can estimate the gradient of the input geometry with respect to a loss function $\\mathcal{L}$ that we are free to specify (e.g., mean squared error). Therefore, if $s$ is our desired spectrum, and $\\hat{g}_i$ is our current best estimate of the metasurface geometry, we can iteratively move along the loss surface to find a better solution using, \n\n\\begin{equation}\n    \\left. \\hat{g}_{i+1} = \\hat{g}_{i}+\\alpha\\frac{\\partial \\mathcal{L}(\\hat{f}(\\hat{g}_{i}),s)}{\\partial g} \\right\\vert_{g = \\hat{g}_{i}}\\\\\n    \\label{eq1}\n\\end{equation}\n\n\\noindent where $\\alpha$ is the learning rate. The $\\hat{g}_i$ can then be evaluated iteratively until some convergence criteria is satisfied (e.g., $\\mathcal{L}$ changes very little after each iteration).  Because this is a gradient-based procedure, it will only converge to a locally optimal solution.  As a result, the NA method prescribes that this search process be repeated $T$ times, each time starting from a different randomly chosen value of $\\hat{g}_{0}$. In practice we find that we may run greater than $10^4$ $T$ initializations in parallel with no cost in speed -- only limited by available memory. Therefore the NA method produces $T$ candidate designs, and we can choose the best design (or several designs, if desired) by passing each design back into $\\hat{f}$ and evaluating their similarity to the target scattering properties.\n\n\\begin{figure}[t]\n    \\centering\\includegraphics[width=\\textwidth]{images\/fig3.pdf}\n    \\caption{Neural adjoint inverse results for prediction of target spectra that exist within the geometry space. Example spectra from geometry predictions that have (a) the best, (b) average, and (c)-(d) below average performance compared to the target spectrum.}\n\\label{3}\n\\end{figure}\n\nNotably, while the user can specify many different choices for $\\mathcal{L}$, the NA method prescribes that a so-called \"boundary loss\", $\\mathcal{L}_{bnd}$ should be added to any user-chosen loss \\cite{ren2020benchmarking} and is given by: $\\mathcal{L}_{bnd} = ReLU(|\\hat{g}-\\mu_g|-\\frac{1}{2}R_g)$, where rectified linear unit (ReLU) is the activation function, $\\mu_g$ is the mean of the geometry training data, and $R_g$ is its range.  This boundary loss punishes the inference process with increasing loss if the geometry search process steps out of the space of the training data, where the forward model may produce inaccurate estimates of scattering parameters.  In our experiments we use the following total loss: $\\mathcal{L}= (s-\\hat{f}(\\hat{g}))^2 + \\mathcal{L}_{bnd}$. As an initial test of the NA inverse method, we feed in frequency dependent absorptivities $A(\\omega)$ where we know apriori that a solution $g$ exists, i.e. $s$ is a numerical simulation from which $\\hat{f}$ originates from. In Fig. \\ref{3} we show results of the NA inverse method and each sub-plot shows characteristic results of (a) the best results, (b) average results, and (c) and (d) below average results, all based on MSE.  In all of the these examples, the NA method identifies a close approximation to the correct solution.  This is impressive given the complexity of the spectra present in Fig. \\ref{3}. We suspect the small remaining errors in the predicted design are due largely to the limited precision of gradient descent as it nears solutions (i.e., minima) in the error space; due to the non-zero learning rate in Eq. \\ref{eq1}, it cannot converge to the exact minimum point.  We note however that learning rate can be gradually reduced during the search process, at the cost of additional computation time, until a solution of desired precision is obtained.\n\nWe next turn to the significantly more challenging task of applying the NA method to a spectra where we are unaware if a solution exists within the chosen geometrical parameter space, i.e. criterion $H_1$ for inverse problems is violated. We chose the frequency dependent external quantum efficiency (EQE) of gallium antimonide (GaSb) as $s$, shown as the gray curves of Fig. \\ref{4}. The metasurface will operate at elevated temperature, and thus we consider the so-called graybody spectra --  also termed the spectral exitance $M_{e,\\nu}(T)$ -- which is given by the blackbody radiation curve times the absorptivity. We keep the top 16,000 neural adjoint solutions (spectra) and determine $M_{e,\\nu}(T)$ for each of these at 100 temperatures between 1500 and 2500 k --  a total of $1.6\\times 10^6$ candidates. The shape of the EQE curve differs significantly from typical spectra we see in our geometry space (Fig. \\ref{3}). None-the-less we find a best solution resulting from the NA method at a temperature of $T=2100k$ that consists of a geometry of [$h=0.566$, $p=1.440$, $r_{x_1}=0.180$, $r_{x_2}=0.155$, $r_{x_3}=0.214$, $r_{x_4}=0.278$, $r_{y_1}=0.285$, $r_{y_2}=0.253$, $r_{y_3}=0.146$, $r_{y_4}=0.256$, $\\theta_1=-0.901^{\\circ}$, $\\theta_2=-20.677^{\\circ}$, $\\theta_3=-37.982^{\\circ}$, and $\\theta_4=39.046^{\\circ}$]. The spectral exitance resulting from this geometry -- calculated from $\\hat{f}$ -- is shown as the blue curve of Fig. \\ref{4} (b), and we find an MSE, compared to the EQE of GaSb, of 1.06$\\times$10$^{-2}$. We also apply a weighting function $W(\\nu)=1\\chi_{[100,275]}+0\\chi_{(275,300]}$ on the MSE forcing the NA method to focus on the region of interest for energy harvesting purposes. To verify our neural adjoint results, we numerically simulate the predicted geometry and plot the resulting $M_{e,\\nu}(T)$ in Fig. \\ref{5} (a) as the red curve -- again compared to the EQE of GaSb (gray curve). As can be seen, the simulated curve has many relatively sharp peaks that are not present in Fig. \\ref{4}(b). This is because, as noted earlier, $\\hat{f}$ does not perfectly match the numerical simulator, and this will introduce errors in the design process. Thus since the NA method relies on $\\hat{f}$ to search for designs, it is also limited by the accuracy of the forward model estimate. We also found that since $\\hat{f}$ is trained from geometries constrained to a grid, the discrepancy between NA solutions and numerical simulation arises because NA solutions are not confined to the grid, where our model is most accurate. None-the-less we find that our simulated $M_{e,\\nu}(T)$ achieves an MSE of  1.65$\\times10^{-2}$, as shown in Fig. \\ref{5} (a).\n\n\\begin{figure}[t]\n    \\centering\\includegraphics[width=\\textwidth]{images\/fig4.pdf}\n    \\caption{Neural adjoint inverse results for matching the EQE of GaSb. The target spectrum (gray), DNN prediction for $M_{e,\\nu}(T=2100K)$ (blue), for the expanded geometry space (a), and  the original geometry space (b) explored. Neural adjoint predictions versus $p$ and $h$ for (c) the expanded geometry space, and (d) the original space. Symbol colors indicate MSE, with values given in the colorbar.}\n\\label{4}\n\\end{figure}\n\nAnother major obstacle is that our design search space does not contain a geometry that can realize our target spectrum (i.e., Hadamard's criteria $H_1$).  This is suggested by the fact that our best NA solution, shown in Fig. \\ref{4}(a), still does not match our target spectrum.  However, we can use the NA output to identify where we should expand our search space so that it will include better designs. We can do this by visualizing the error of all inverse solutions returned by NA, and looking for trends e.g., we may find that all the best solutions are bunched up against some edge of our initial search space, suggesting that expanding along that dimension may yield better results. However, since we have a 14 dimensional design space, we are unable to easily visualize these data. To address this problem, we use the Uniform Manifold Approximation and Projection (UMAP) \\cite{mcinnes2018umap}, which is a type of dimensionality reduction method permitting us to visualize the distribution of our inverse solutions performance in 2D, so that we may more easily identify patterns. From this initial investigation with UMAP in Fig. S1 -- shown in Supplemental -- we find that our best NA inverse solutions are limited by height. Shown in Fig. \\ref{4} (d) are NA solutions as a function of height and periodicity color mapped by corresponding MSE values. It is evident that not only are our best solutions grouped at the maximum height allowed in our geometrical space, $h=0.6$ $\\mu$m, but also that the solutions improve as a function of height. Encouraged by these results, we expanded our original geometry space to include increased height values from 0.6 to 0.75 $\\mu$m, by simulating an additional 24k $\\{g,s\\}$ pairs. The NA model now trained on the expanded geometrical space indeed finds an improved solution, shown as the blue curve in Fig. \\ref{4} (a), where we realize an MSE that is reduced by a factor of 2.7. The simulated red curve in Fig. \\ref{5} (b) further validates the result that the MSE of numerical simulations  is also reduced -- here by a factor of 4.8. A plot of the 1000 best NA solutions in the expanded geometrical space shown in Fig. \\ref{4} (c), however, indicate that we may be able to make continued improvements, since we still have a gradient pushing for greater heights -- although the periodicity seems to be honing in on a value of 1.2 $\\mu$m.\n\n\\begin{figure}[t]\n   \\centering\\includegraphics[width=\\textwidth]{images\/fig5.pdf}\n    \\caption{Numerically simulated $M_{e,\\nu}(T=2100K)$ (red) of the optimal geometry predicted by the neural-adjoint method to match the EQE of GaSb (gray), for the original geometry space (a), and the expanded geometry space (b).}\n\\label{5}\n\\end{figure}\n\n\\section{Conclusion}\nWe have adapted the neural-adjoint inverse design method \\cite{ren2020benchmarking} to accurately predict the high-dimensional all-dielectric metasurface geometry needed to produce a targeted infrared absorptivity spectra. When the geometry needed to produce a desired spectrum lies outside of the bounds, the NA method appears to find the best possible solution within the permissible search space. Unlike other adjoint inverse approaches \\cite{lalau2013adjoint}, the NA method does not require any domain knowledge specific to the problem. In the event that the inverse solution does not exist in the parameter space explored, NA may be used to guide one to a better solution, through expansion of the design parameter search space. This may help to reduce the initial required number of numerical simulations, and to instead use NA guided simulation exploration. With its exceptional computational speed, high accuracy, and potential use in active learning that is explored here, the neural-adjoint method has an impressive future in not only ADMs thermal emitter but also any ADMs inverse problems. The NA method is not restricted to the case presented, but may be applied to many other systems including photonic band gap and plasmonics.\n\n\n\n\\section{Numerical Simulation}\nThe cylindrical resonators' geometry was previously demonstrated in the THz regime by\\cite{liu2017experimental, fan2017all}. To prove deep neural work capability with high-dimensional inputs, we increased the geometrical dimensionality by introducing the elliptical structures to previous cylindrical resonators, and each elliptical resonator undergoes a rotation angle ranging from -45 to 45 degrees. Furthermore, governing the fabrication practicality, we fixed all elliptical resonators to have the same height. To migrate from the THz regime to the infrared, materials and geometry sizes are scaled accordingly. We sized down our unit cell volume refers to the ratio of THz frequency and infrared frequency used in the legacy design and current design. Then an optimization on an adequate scale to finalize our geometry boundary listed in Table 1. We chose SiC for our simulations, considering its high melting point, high oxidation resistance, and reasonable absorption coefficient in near-infrared. We used experimentally measured relative permittivity data of SiC from 0 to 300 THz to fit the dispersive materials property of SiC in our simulations. To implement the unit cell boundary conditions in CST Microwave Studio we used for our simulation, we used finite frequency solvers to perform the numerical simulations, which also take considerations of the coupling effect between the four resonators within the unit cell. To minimize the time cost of the simulations, we lightly comprise the simulation accuracy to simulation speed. We tuned the Floquent mode to have one mode at both ports. The simulation mesh is tetrahedral, and we used a second-order solver with accuracy at $1e^-6$. The resulting spectrum of the simulation has 2001 data points within the range of 100-500 THz.\n\n\\begin{table}[htbp!]\n\\caption{Grid definition for the 14-dimensional input geometry parameters. h, p, and r are in units of microns. $\\theta$ is in unit of degrees.}\n\\centering\\begin{tabular}{ccccc}\n \\\\\n Step&h&p&$r_{x_n}$\/$r_{y_n}$&$\\theta_n$\\tnote{*}\\\\ \\hline\n 1&0.3&1&0.1&-45 \\\\\n 2&0.375&1.125&0.1125&-22.5 \\\\\n 3&0.45&1.25&0.125&0 \\\\\n 4&0.525&1.375&0.1375&22.5 \\\\\n 5&0.6&1.5&0.15&45 \\\\\n 6&-&-&0.1625&- \\\\\n 7&-&-&0.175&- \\\\\n 8&-&-&0.1875&- \\\\\n 9&-&-&0.25&- \\\\\n \\hline\n\\end{tabular}\n\\begin{tablenotes}\\footnotesize\n\\centering\\item[*] $n$ corresponds to the first to the fourth elliptical resonator in one super cell\n\\end{tablenotes}\n\\end{table}\n\nThe total possible number of geometrical combinations of our 2$\\times$2 metasurface is $8^9*6^5=1.04\\times 10^{12}$. It is impossible to use a conventional numerical simulation approach to exploit the entire geometry space to achieve the targeted spectrum. We find that the average simulation time per geometrical configuration per CPU is approximately three minutes. Thus it would take about 600 million years to finish exploring the entire geometry space with one CPU. The fast forward dictionary search (FFDS) inverse method was shown feasible for THz ADMs absorbers, where all 812 million possible geometries can be computed in a day \\cite{nadell2019deep}. To compute our entire geometry space with a size of over a trillion parameters would take FFDS over three and a half years. Thus the NA method is a good choice when the parameter space become too large for a FFDS approach. \n\n\\section{Deep Neural Network Architecture}\nWe built the entire network and neural-adjoint method using the PyTorch platform. The DNN used for the neural adjoint method consists of twelve fully connected linear layers, four 1D transpose convolutional layers for upsampling, and one final 1D convolutional layer for spectrum smoothing. The linear fully connected layers have the following structure[14, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 1000, 500\n], and all hidden layers except the last linear layer are batch normalized, activated by Leaky\\_Relu, followed by dropout layers with p = 0.05. The transpose convolutional layers have kernel size [16, 16, 33, 33] and filter size [4, 4, 4, 4]. The final convolutional layer has kernel size 1 and stride at 1. The data loader takes geometry inputs and the first 2000 data points of the absorptivity spectrum to generate the training and test datasets.  The post-processing truncates the predicted spectra's first and last fifty points to drop the convolutional layers bump at the edges.  We use L2 regularization, batch normalization, and the ADAM optimizer.\n\n\\section{NA Inverse Method}\nBecause NA a gradient-based procedure, it will only converge to a locally optimal solution. Because the NA method is amenable to parallelization on graphics processing units (standard hardware for deep learning), this entire process can be computed in very little time.  In our experiments, we can run the NA method with $T=1000$ on our desktop computer with an Nvidia 2080ti GPU and complete processing in under 1 minute.Since the NA method finds the globally optimal solution even in its worst-performing cases, our results suggest that the NA always (or nearly always) finds the globally optimal solutions, even for highly complex problems like ours.  Interestingly, this suggests that the main obstacle of custom design is no longer the inverse model, but rather the space over which we choose to search for designs i.e., the shapes we consider (cylinder, crosses, etc) and their parameter settings (e.g., radii, height).  Although powerful, to use deep learning methods we must necessarily define a range of these settings so that we can collect simulations to train our models, and this space limits where we can search for designs.  However, the design needed to realize our targeted scattering parameters may not exist in this initial search space.   As we show subsequently, the NA method can also be used to identify where this initial search space can be expanded so that it is most likely to contain the desired solution, providing a solution to this emergent obstacle to complex material design.    \n\n\\section{Data Augmentation}\nOur simulations' unit cell boundary conditions allow us to do four times data augmentation on our dataset because the infinite plane of unit cells consists of four different resonators' combinations that give almost identical spectra with fluctuations from CST software. However, we learned that the DNN could quickly learn the correlation between four different resonators' combinations. The forward model will know which input geometries share the same spectrum in high fidelity if the entire dataset is augmented before splitting into the training and validation dataset. Therefore, the forward model will give a false mean square error much lower than actual loss performance on an independent validation set. After applying data augmentation to the training and validation sets after the splits, we observe that the four times augmentation did not significantly improve accuracy. We believe that, with our 60000 simulations (24000 simulations after augmentation), the augmented data points are still too sparse to cover the entire geometry space defined by our geometry boundaries. \n\n\\section{Geometry Space Exploration Through UMAP}\nWe use Uniform Manifold Approximation and Projection (UMAP) to explore our solution geometry space and realize that angles have more random impacts on distributing the best NA solutions.  Thus, we plotted the UMAP with ten parameters, excluding rotational angles. The plotted UMAP demonstrates a clear trend that the MSE is decreasing in one direction. To confirm that the decreasing trend matches the increasing of resonators' height, we further marked the points from maximum and minimum height boundaries, respectively. The clustering of points towards the best MSE performance suggests that the NA method is finding the best local minima. \n\n\\begin{figure}[h!]\n    \\centering\\includegraphics[width=3.0in]{images\/figs1_umap.pdf}\n    \\caption{Uniform Manifold Approximation and Projection plotted with 10-dimensional geometry inputs indicates a strong correlation between the MSE performance and the increasing height.}\n\\label{5}\n\\end{figure}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{I. Introduction}\n\nCorrelation functions in disordered systems usually decay nonexponentially in time \\cite{angell}. The microscopic origin of this behavior is of importance for a detailed understanding of the dynamics in these systems. In principle, the stretched exponential form of the correlation functions can be explained by two extreme scenarios. Either different Debye-type relaxation rates superimpose to the total response, or the broadened form is an intrinsic property of the dynamics in glassy systems. These different scenarios have been denoted as dynamic heterogeneous and homogeneous \\cite{hethom}. Several, but not numerous, experimental techniques have been developed that allow the investigation of dynamic heterogeneity, like four-time NMR measurements \\cite{4timenmr}, a deep bleach experiment \\cite{deepbleach} and nonresonant hole burning (NHB) \\cite{NHB}. \n\nIn a theoretical investigation of NHB \\cite{gregor} it has been shown that indeed heterogeneous and homogeneous dynamics can be distinguished. Experimental realizations of NHB have been conducted on numerous systems like supercooled liquids \\cite{NHB,duvvuri,blochi}, relaxor materials \\cite{oli} and spin glasses \\cite{chamberlin}. In all these studies indications for dynamic heterogeneity surpass those for homogeneity, but also homogeneous behavior was found, e.g., in an amorphous ion conductor \\cite{richert}. Recently a mechanical variant of NHB has been developed \\cite{mckenna}.\n\n\nMainly accountable for the distinction between heterogeneous and homogeneous dynamics is the possibility to adress certain dynamics in the sample, i.e., the selection of specific dynamic subensembles. Frequency-selective behavior is obtained only if the measured response function is nonlinear in the applied field, which holds for the dynamic Kerr effect \\cite{kerreffect} as well as for NHB. In Kerr effect relaxation, frequency-selective behavior can be achieved if a driving AC field with varying frequency is applied.  We demonstrated frequency-selectivity in a Kerr effect experiment in the terahertz range in a recent publication \\cite{mykerr}. Treating the vibrational dynamics around the boson peak in a Brownian oscillator model we concluded that the frequency dependence of the vibrations' damping can be extracted experimentally. \n\nThe Kerr effect has to our knowledge not yet been used for an investigation of dynamic heterogeneities. In this article, we study the Kerr effect response in the range of slow reorientational motions in supercooled liquids. We propose the Kerr effect for a distinction between dynamic heterogeneous and homogeneous relaxation. The advantage of the Kerr effect compared to NHB lies in the fact that the Kerr effect response is purely nonlinear in the applied field. Therefore no separation of nonlinear contributions from the linear response background is required, and field strengths remarkably weaker than those usually applied in NHB ($100\\ kV\/cm$ \\cite{NHB,duvvuri,blochi}) may suffice. It should also be possible to study dynamics at different temperatures ranging from the glass transition temperature to high temperatures. Here, possible changes in a heterogeneous distribution with temperature \\cite{breitetemp,gammaverteilung} might be investigated.  \n\nThe paper is organized as follows. Section II outlines the principles of the Kerr effect, explains our suggested experiment and defines heterogeneous and homogeneous models. In Section III we give expressions for the Kerr effect response in the different approaches and we discuss the results. It turns out that indeed the distinction between dynamic homogeneity and heterogeneity is possible. The conclusions are given in Section IV.\n \n\n\\section*{II. Theory}\n\\subsection*{1. Dynamic Kerr effect}\nIf an anisotropically polarizable sample is exposed to a time-dependent electric field it becomes birefringent. This phenomenon is known as dynamic Kerr effect. In a theoretical description the coupling of external electric fields $E(t)$ to matter via the sample's permanent dipole moment $\\mu$ has to be taken into account as well as the coupling via the polarizability $\\alpha$. The structure of the Hamiltonian describing the interaction is thus of the form \\cite{hamop}\n\\be \\label{exthamilton}\n\\mathcal{H}_{ext}=-\\mu E(t)P_1(\\cos \\theta)-\\alpha E^2(t) P_2(\\cos \\theta)\n\\ee\nThe permanent dipole moment interaction is linear in $E$, while the polarizability interaction is of order $E^2$ since the induced dipole moment itself is $\\propto E$. The appearance of the Legendre polynomials $P_L(\\cos \\theta)$ is due to the tensorial nature of the dipole moments. Since the permanent dipole moment is a vector and the polarizability is a matrix, they transform like first and second rank Legendre polynomials, respectively. The scalar $\\alpha$ in Eq.(\\ref{exthamilton}) is to be identified with the difference between the polarizabilities parallel and perpendicular to the internal symmetry axis \\cite{deschardaeng}, and $\\theta$ is the angle between this axis and the applied field. \n\nThe time-dependent polarizability is the quantity of interest for a description of the dynamic Kerr effect. We therefore have to calculate the expectation value of the second Legendre polynomial\n\\be\\label{alpha}\n\\langle \\alpha P_2(\\cos \\theta (t) )\\rangle  =\\alpha \\langle P_2(t) \\rangle .\n\\ee\nThe brackets denote an expectation value over the whole sample. In this work we focus on reorientational motions. Here, the time dependence of the orientation (described by the angle $\\theta$) determines the time dependence of the polarizability. In the following, we will use the short hand notation on the right hand side of Eq.(\\ref{alpha}).\n\nThe expectation value $\\langle P_2(t) \\rangle$ is calculated in some dynamic model, where the applied external field that determines the time dependence is treated in perturbation theory. Even without specifying the model yet, it is clear that the linear response of order $E$ must vanish as long as isotropic systems are considered. This is because the linear response is always proportional to the first rank Legendre polynomial, see Eq.(\\ref{exthamilton}). Because of the orthogonality of the Legendre polynomials the linear response vanishes if the expectation value of the second rank Legendre polynomial, Eq.(\\ref{alpha}), is calculated. The Kerr effect response is thus of order $E^2$.\n \nIn some more detail we have two contributions to the expectation value $\\langle P_2(t) \\rangle$. The first one is proportional to the polarizability $\\alpha$ (formally calculated with the second term in the external Hamiltonian (\\ref{exthamilton}) in first order perturbation theory), the second one is quadratic in the permanent dipole moment $\\mu$ (second order perturbation theory with the first term in (\\ref{exthamilton})). We will denote these contributions in the following as $\\langle P_2^\\alpha(t) \\rangle$ and $\\langle P_2^{\\mu \\mu}(t) \\rangle$. \n\n\\subsection*{2. Experiment}\n\nWe propose an experiment as follows. First, a sinusoidal electric field is applied for an arbitrary number $N$ of (full) cycles to a sample in equilibrium.\n\\be\\label{feld}\nE(\\tau )=E \\sin(\\Omega_p \\tau) \\qquad \\tau <t_p=\\frac{2\\pi N}{\\Omega_p}\n\\ee\nThe most important experimental parameter is the pump frequency $\\Omega_p$. After the 'pump time' $t_p$ the field is switched off and the relaxation of the sample's polarizability back to thermal equilibrium is measured. In our terminology the time variable $t$ starts with $0$ at $t_p$. \n\nThe experiment is based on the same idea as NHB. If the dynamics is assumed to be heterogeneous (different relaxation rates), then it should be possible to address 'slow' or 'fast' dynamics separately by varying the pump frequency from 'low' to 'high'. The so-selected ensembles will then show 'slow' or 'fast' relaxation behavior afterwards. In a Fourier transformed representation of the relaxation, one would therefore expect an extremum position proportional to the applied field frequency. On the other hand, in a homogeneous system one would expect relaxation on a single time scale only, and different pump frequencies should not alter the time-dependence or extremum position in the Fourier transform. Thus, shifts in the extremum position can be considered as indications for dynamic heterogeneous relaxation while no shifts would support the homogeneous picture.\n\nThe aim of this paper is to demonstrate that the concept outlined above indeed holds in heterogeneous and homogeneous models. Because of the nonlinearity of the Kerr effect the contributions $\\langle P_2^\\alpha(t) \\rangle$ and $\\langle P_2^{\\mu \\mu}(t) \\rangle$ are not trivially related to the correlation function, and have thus to be explicitly calculated for a given field sequence like Eq.(\\ref{feld}). We specify the dynamic models used in the calculation of the response functions in the following section. \n\n\n\\subsection*{3. Dynamics in the rotational diffusion model}\n\nFor a description of the dynamics we use the well-known rotational diffusion model \\cite{kubo}. The rotational diffusion equation for the conditional probability $P_i(\\Omega,t|\\Omega ',t')$ to find the orientation $\\Omega$ at time $t$, assuming it was $\\Omega '$ at time $t'<t$ is\n\\be\\label{simplefopl}\n\\frac{\\partial}{\\partial t} P_i(\\Omega,t|\\Omega ',t')=\\hat \\Pi_i(\\Omega ,t) P_i(\\Omega,t|\\Omega ',t')\n\\ee\nwith the Fokker-Planck operator\n\\be\\label{rotdiff}\n\\hat \\Pi_i(\\Omega ,t) =D_i(t)\\left[ \\frac{1}{\\sin\\theta} \\partial_\\theta \\sin\\theta \\left( \\partial_\\theta +\\beta \\frac{\\partial {\\mathcal H}_{ext}}{\\partial \\theta}\\right) + \\frac{1}{\\sin^2\\theta} \\partial^2_\\phi \\right] .\n\\ee\nHere, $\\theta$ and $\\phi$ denote polar and azimuth angles with respect to the molecules internal symmetry axis. These angles constitute the orientation $\\Omega$. $D_i(t)$ is a (possibly time-dependent) rotational diffusion constant, $\\beta=(k_B T)^{-1}$ and ${\\mathcal H}_{ext}$ is the Hamiltonian for the interaction with the applied external field in Eq.(\\ref{exthamilton}). \n\nThe solution of Eq.(\\ref{rotdiff}) without driving field (${\\mathcal H}_{ext}=0$) is achieved via an expansion in Wigner rotation matrices $D^{L}_{MN}$ \\cite{wigner}. The ansatz\n\\begin{equation}\\label{ansatz}\nP_i(\\Omega,t|\\Omega ',t')=\\sum_{LMN} \\frac{2L+1}{8\\pi^2} D^{L}_{MN}(\\Omega) \\left[ D^{L}_{MN}(\\Omega ')\\right]^\\ast G^{(L)}_i(t,t')\n\\end{equation}\ninserted in Eq.(\\ref{simplefopl}) yields\n\\begin{equation}\\label{glvont}\nG^{(L)}_i(t,t')=\\exp\\left[ -L(L+1)\\int_{t'}^t d\\tau \\ D_i(\\tau)\\right]\n\\end{equation}\nand rotational correlation functions of the $L$-th Legendre polynomial are given by\n\\be\\label{corr}\n C^L_i(t,t') = \\langle  P_L(\\cos\\theta(t)) P_L(\\cos\\theta(t')) \\rangle_i =\\frac{1}{2L+1} G^{(L)}_i(t,t'). \n\\ee\nIn the presence of external fields perturbation theory is used to calculate the response functions. The corresponding perturbation operators result from the Fokker-Planck operator (\\ref{rotdiff}) including the external Hamiltonian in Eq.(\\ref{exthamilton}). Linear response functions obtained this way are related to the correlation function via the  fluctuation dissipation theorem \\cite{calabrese}.\n\n\n\n\n\n\n\n\\subsection*{Heterogeneous model}\nIn a heterogeneous scenario one assumes the coexistence of different, 'slow' and 'fast' environments related to small and large rotational diffusion constants $D_i$. If $D_i$ is time-independent, the propagator $P_i(\\Omega,t|\\Omega ',t')$ is time-translational invariant and the correlation function in case of a single rotational diffusion constant is given by\n\\be\\label{}\nC_i^L(t,t')=C_i^L(t-t')=\\frac{1}{2L+1} \\exp\\left[-L(L+1) D_i(t-t')\\right].\n\\ee\nDynamic heterogeneities are known to have finite life times of similar magnitude as the $\\alpha$-relaxation time \\cite{otptauschtaus,austauschsim}. Finite life times can be modelled via exchange models. Various approaches exist to include exchange in a model \\cite{xeigenwerte}. In a master equation ansatz \\cite{sillescu} one assumes that slow environments become fast and vice versa. The environments are related to rotational diffusion constants, and exchange between the environments leads to effectively time-dependent rotational diffusion constants. Time-translational invariance is preserved within this framework. We will denote the environments by $\\epsilon_i$ in the following, and we add a second index to quantities like the propagator. $P_{ij}(\\Omega, t | \\Omega ',t')=P_{ij}(\\Omega, t-t' |\\Omega ',0)$ thus is the probability to find the orientation $\\Omega$ in environment $\\epsilon_i$ at time $t$, assuming it was $\\Omega '$ in environment $\\epsilon_j$ at time $t'$. Transition rates $\\gamma_{ij}$ determine transitions from environment $\\epsilon_j$ to $\\epsilon_i$, meaning that the rotational diffusion constant changes its value from $D_j=D(\\epsilon_j)$ to $D_i=D(\\epsilon_i)$. Furthermore, the behavior of the orientation during a transition has to be specified. If we assume that each transition is associated with a rotational random jump, then an exchange model can be defined by \\cite{sillescu} \n\\be\\label{beckpfeif}\n\\frac{\\partial}{\\partial t} P_{ki}(\\Omega,t|\\Omega ',t')=\\left( \\hat \\Pi_k(\\Omega)-\\gamma_k \\right)  P_{ki}(\\Omega,t|\\Omega ',t') +  \\frac{1}{8\\pi^2} \\sum_{j \\not= k} \\gamma_{kj} \\int d\\Omega_0 P_{ji}(\\Omega_0,t|\\Omega ',t').\n\\ee\nHere, $\\hat \\Pi_k(\\Omega)$ is the Fokker-Planck operator (\\ref{rotdiff}) with $D_i(t)$ replaced by the time-independent $D_k$ and $\\gamma_k=\\sum_{j\\not= k}\\gamma_{jk}$ is the total rate out of state $\\epsilon_k$. \n\nThe solution of Eq.(\\ref{beckpfeif}) without external fields is obtained in a similar manner as the solution of Eq.(\\ref{simplefopl}). Here, the function $G^{(L)}_i(t,t')$ in Eq.(\\ref{ansatz}) is replaced $G^{(L)}_{ki}(t,t')$. For $L\\not= 0$ the elements of the matrix ${\\bf G}^{(L)}(t,t')$ are given by\n\\be\\label{c0gdiag}\nG^{(L\\not= 0)}_{ki}(t,t')=\\delta_{ki} \\exp\\left[ \\Lambda^{(L)}_i (t-t')\\right]\n\\ee\nwith the eigenvalues\n\\be\\label{xeigen}\n\\Lambda^{(L)}_i=-L(L+1)D_i-\\gamma_i.\n\\ee\nThe equilibrium probabilities $P_i^{eq}$ to find the system in environment $\\epsilon_i$ in the presence of exchange (finite $\\gamma_{ij}$) are obtained from the long-time limit of the propagator, $P_i^{eq}=\\lim_{(t-t') \\to \\infty} \\int d\\Omega  P_{ik}(\\Omega,t|\\Omega ',t')=G^{(0)}_{ik}(\\infty ,0)$, which guarantees detailed balance. Correlation functions in the heterogeneous model are then calculated as\n\\be\\label{hetcorr}\nC^L(t)=\\sum_i C^L_i(t) P^{eq}_i=\\frac{1}{2L+1}\\sum_i \\exp\\left[\\Lambda_i^{(L)}t\\right] P^{eq}_i .\n\\ee\nThe larger the exchange rates are, the more they dominate the eigenvalues in comparison to the $L$-dependent term in Eq.(\\ref{xeigen}). This also holds in other exchange models \\cite{xeigenwerte}. Large exchange rates thus lead to a weaker $L$-dependence of the eigenvalues, and the exponents in correlation functions (\\ref{hetcorr}) for different values of $L$ become more and more similar. The inclusion of exchange in the rotational diffusion model thus leads to very similar correlation times for different $L$, as usually found experimentally \\cite{willi}. The calculation of response functions is straightforward.\n\nExchange effects are neglected if $\\gamma_{ij}=0$ is chosen. Then the eigenvalues in Eq.(\\ref{hetcorr}) are given by $\\Lambda_i^{(L)}=-L(L+1)D_i$, and a distribution of rotational diffusion constants for a heterogeneous model can be arbitrarily chosen. If the relaxation times $\\tau_i \\propto D_i^{-1}$ are distributed according to the generalized Gamma distribution \\cite{gammaverteilung}, a stretched exponential form of the correlation function is obtained.\n\n\n \n\n\n\\subsection*{Homogeneous model}\nIn a homogeneous scenario one assumes that the relaxation rate (or rotational diffusion constant) depends on time. A decaying power law function\n\\be\\label{homd}\nD_i(t)=\\frac{\\beta_K}{2\\tau}  \\left( t\/\\tau \\right)^{\\beta_K-1} \n\\ee\ninserted into the correlation function (\\ref{corr}) using Eq.(\\ref{glvont}) yields \n\\be\\label{homcorr}\n\\langle P_1(t)P_1(t')\\rangle = \\frac{1}{3} \\exp\\left[ -(t'\/\\tau)^{\\beta_K}  \\right] \\exp\\left[ -(t\/\\tau)^{\\beta_K}  \\right] .\n\\ee\nThus, the special choice in Eq.(\\ref{homd}) reproduces a stretched exponential for the correlation function. Note that Eq.(\\ref{homcorr}) is not time-translational invariant, which is due to the time dependence of $D_i(t)$ \\cite{calabrese}. Due to the lack of time-translational invariance the system has an 'internal clock' that is hard to justify from a physical point of view. The consequences in our case will be discussed below. An alternative for modelling homogeneous dynamics are the so-called fractional Fokker-Planck approaches \\cite{fractional}. Here, time-translational invariance is maintained \\cite{fracttti}. \n\n\n\n\\section*{III. Results}\nIn this section we discuss the Kerr effect response to the alternating field in the various approaches considered in the previous section. After the orientational integrations have been carried out we find the expressions \n\\be\\label{alal}\n\\langle P_2^{\\alpha}(t) \\rangle=\\frac{6}{5} \\beta \\alpha E^2 \\sum_{i} \\int_0^{t_p} dt' \\sin^2(\\Omega_p t') D_i(t') G_{i}^{(2)}(t+t_p,t') P_i^{eq}\n\\ee\nfor the $\\alpha \\alpha$- contribution to the Kerr effect response and\n\\Be \\nonumber  \n\\langle P_2^{\\mu \\mu}(t) \\rangle=\\frac{4}{5}\\beta^2 \\mu^2 E^2 \\times \\\\ \\label{almumu}\n\\times \\sum_{i} \\int_0^{t_p} dt' \\int_0^{t'} dt'' \\sin(\\Omega_p t') \\sin(\\Omega_p t'') D_i(t') D_i(t'') G_{i}^{(2)}(t+t_p,t') G_{i}^{(1)}(t',t'') P_i^{eq} \n\\Ee\nfor the $\\alpha \\mu \\mu$-contribution. Eqns.(\\ref{alal},\\ref{almumu}) are valid for heterogeneous and homogeneous models. In the heterogeneous case the $D_i$ are time-independent and in the homogeneous case no summation over $i$ is required. In the presence of exchange $G_i^{(L)}(t_1,t_2)$ is to be identified with the diagonal elements of the matrix ${\\mathbf G}^{(L)}(t_1,t_2)$, see Eq.(\\ref{c0gdiag}).\n\nNote that Eq.(\\ref{alal}) is formally identical to the expression for the linear (dielectric) response, except for the fact that the external field of sine form enters quadratically and $G^{(2)}_i$ appears instead of $G^{(1)}_i$. As a consequence of this quasi-linearity the $\\alpha \\alpha$-contribution is -like the dielectric linear response- unable to distinguish between homogeneous and heterogeneous scenarios.\n\n\nLet us briefly discuss the simplest case where the system is described by a single, time-independent rotational diffusion constant. Here, $G^{(L)}_i(t_1,t_2)$ are exponential functions according to Eq.(\\ref{glvont}). We furthermore have to set $D_i=const$ and $\\sum_i P^{eq}_i=1$ in Eqns.(\\ref{alal},\\ref{almumu}). The time integrals can then be carried out analytically, leading to response functions of the form\n\\be\\label{respeinerallg}\n\\langle P_2^{\\Theta}(t) \\rangle=A^{\\Theta}(\\Omega_p, D_i,t_p)\\exp(-6 D_i t)\n\\ee\nwhere $\\Theta$ stands for $\\alpha$ or for $\\mu\\mu$. In the following discussion of the response functions we will show the imaginary part of Fourier-transformed response functions, denoted as $\\langle P_2^\\Theta(\\omega)\\rangle$. The symbol $\\omega_{max}$ will be used for the extremum positions in $\\langle P_2^\\Theta(\\omega)\\rangle$.\n\nBoth contributions (\\ref{respeinerallg}) with $\\Theta=\\alpha$ and $\\Theta=\\mu\\mu$ show the same time dependence, a monoexponential decay with the decay rate of the second Legendre polynomial. This is analogous to the step field-off-response discussed in Ref.\\cite{cole}. The maximum position in $\\langle P_2^\\Theta(\\omega)\\rangle$ therefore is located at frequency $\\omega_{max}=6D_i$ for both terms.\n\n\nThe amplitude functions $A^\\Theta(\\Omega_p,D_i,t_p)$ are given by \\cite{meinsenf}\n\\Be\\label{respeineralal}\nA^{\\alpha}(\\Omega_p,D_i,t_p) & = & \\frac{1}{30} \\beta \\alpha E^2   \\frac{\\Omega_p}{D_i} \\varepsilon ''(6D_i,2\\Omega_p) \\left[1-e^{-6 D_i t_p} \\right]   \\\\ \\nonumber\nA^{\\mu \\mu}(\\Omega_p,D_i,t_p) & = & \\frac{1}{30} \\beta^2 \\mu^2 E^2  \\varepsilon ''(2 D_i,\\Omega_p) \\times  \\\\\n\\label{respeineralmumu} & \\hspace{-0.6cm} \\times & \\hspace{-0.6cm} \\left( 3\\varepsilon ''(4D_i,\\Omega_p) \\left[e^{-6 D_i t_p}-e^{-2 D_i t_p} \\right] +\\frac{5}{3} \\varepsilon ''(6D_i,2\\Omega_p) \\left[ 1-e^{-6 D_i t_p} \\right] \\right)  \n\\Ee\nwith the dielectric loss defined as $\\varepsilon ''(x,y)=xy\/(x^2+y^2)$. In the context of frequency-selectivity the behavior of the amplitudes as a function of pump frequency is important. $A^{\\mu \\mu}(\\Omega_p,D_i,t_p)$ plotted versus pump frequency shows a peak because the amplitudes become large if the arguments in the dielectric loss functions $\\varepsilon ''(x,y)$ are equal, that means for $\\Omega_p\\approx 2D_i,3D_i,4D_i$. If for a moment we neglect the fact that the arguments of the dielectric loss functions have varying prefactors, then $E^2 \\varepsilon ''(D_i,\\Omega_p)^2$ represents the pump frequency-dependent energy absorbed by the system.\n\nNo frequency-selective behavior can be expected from the quasilinear $\\alpha \\alpha$-term, where the additional factor $\\Omega_p$ appearing in the nominator of $A^{\\alpha}(\\Omega_p,D_i,t_p)$ does not allow for a 'resonant' energy absorption. This is because the product $\\Omega_p \\varepsilon ''(6D_i,2\\Omega_p)$ is a strictly increasing function of $\\Omega_p$ in contrast to $\\varepsilon ''(6D_i,2\\Omega_p)$ which exhibits a peak at $\\Omega_p=3D_i$.\n\n\\subsection*{Heterogeneous model}\nIn a first discussion of Kerr effect responses in the heterogeneous model we neglect exchange effects. We choose the rotational diffusion constants distributed according to a generalized Gamma distribution with parameters such that the resulting (normalized) $(L=1)$-correlation has the form $\\exp\\left[ -(t\/\\tau)^{\\beta_K} \\right]$ with the KWW-exponent $\\beta_K=0.5$ and $\\tau =1$. The distribution of rotational diffusion constants is shown in the upper panel of Figure $1$. The lower panel shows the Kerr effect signal $\\langle P_2^{\\mu\\mu}(\\omega)\\rangle$ for different pump frequencies $\\Omega_p=0.01,0.1,1,10,100$ (indicated by letters A,B,C,D,E in the figure). One cycle has been applied to the system for all pump frequencies. \n\nIn order to understand the curves in Figure $1$ we refer to the previous section where we discussed the response of a single $D_i$. Here, the energy absorption was shown to have a maximum for $\\Omega_p \\approx 2D_i, 3D_i, 4D_i$. This means that in a distribution of $D_i$ most energy is absorbed if $D_i \\approx \\Omega_p\/2$ holds. Thus, a pump frequency dependent selection is possible. The $D_i$ corresponding to $\\Omega_p=10^{-2}\\dots 10^2$ are marked by the vertical arrows in the distribution function (upper panel of Fig.$1$). Under the assumption that the response is mostly governed by the selected $D_i$ we obtain a maximum position at $\\omega_{max}=6D_i$, thus at $\\omega_{max}\\approx 3 \\Omega_p$. Although the prefactor $3$ is not exactly recovered in the extremum positions in the lower panel of Fig.$1$, we indeed observe a proportionality of extremum position (indicated by dotted lines) and pump frequency for $\\Omega_p=100\\dots 0.1$ (E,D,C,B). These shifts of the extremum position are the hallmark of dynamic heterogeneity. For $\\Omega_p=0.01$ (A) the relative shift becomes weaker. Here, a selection of $D_i\\approx 0.01\/2$ does not determine the total response because these small $D_i$ are underrepresented in the distribution function, see the arrow (A) in the upper panel. In that case, rotational diffusion constants around $D_i \\approx 0.05$ (B in the upper panel) determine the extremum position. This also holds for still smaller pump frequencies. Here, the extremum position becomes independent of $\\Omega_p$ while the amplitudes of the signal strongly decrease. A similar behavior is observed at the high-frequency end of the distribution, which we do not show explicitly. The maximum position as a function of $\\Omega_p$ will be discussed in detail later compared to results in the homogeneous limit.\n\n\nIn order to discuss the width of the responses, we added a Debye function (dotted line near C in the lower panel of Figure $1$). The heterogeneous response (C) is somewhat broadened in comparison. The curve for $\\Omega_p=100$ (E) has a negative onset on the low-frequency side of the peak. Two remarks concerning this finding of a partially negative response function are appropriate. First, the negative sign of the response function is due to the first term in Eq.(\\ref{respeineralmumu}), which is proportional to the difference $\\left[\\exp(-6 D_i t_p)-\\exp(-2 D_i t_p) \\right]$. This is negative for finite $t_p$ in contrast to the other term $\\propto \\left[1-\\exp(-6 D_i t_p) \\right]$. A negative response function therefore is a switch-on effect, since the former contribution vanishes for large $t_p$. Indeed, if several cycles are applied, all response functions become positive in the whole frequency range. Second, even for finite $t_p$, the negative term can be considered as an artefact of the rotational diffusion model. This is because the rates $6D_i$, $2D_i$ in the exponents are the decay rates of the second and the first Legendre polynomial, which are known to be of similar magnitude in real systems as discussed in Section II. In any model where these rates are very similar (e.g. the exchange model discussed in Section II) or equal (e.g. isotropic rotational jumps instead of rotational diffusion) no negative sign of the response occurs for arbitrary $t_p$. \n\nFigure $2$ shows the contributions of the quasilinear term $\\langle P_2^\\alpha(\\omega)\\rangle$  with the same parameters as in Figure $1$. The extremum position is not proportional to pump frequency although it weakly depends on $\\Omega_p$. While the pump frequency spreads over five decades, the maximum position shifts only for less than one decade. The shift of the extremum positions is further diminished if several cycles are applied. This demonstrates the expected result that the quasilinear term is not able to detect dynamic heterogeneities. In the following discussion, we will therefore focus on the $\\alpha \\mu\\mu$-contribution. The quasilinear term can be viewed as a background contribution to the response function. As it is of quadratic order in the applied field like the $\\alpha \\mu \\mu$-contribution, no phase-cycling procedure allows the extraction of either of the contributions. However, the terms should be easily distinguishable since the quasilinear signal is at a more or less constant position. Furthermore, the problem with the quasilinear term is trivially minimized if systems with large permanent dipole moment $\\mu$ (compared to their polarizability $\\alpha$) are studied. In this case, $\\langle P_2^{\\alpha} (\\omega)\\rangle$ is much smaller than $\\langle P_2^{\\mu \\mu} (\\omega)\\rangle$.   \n\n\nWe now discuss the influence of exchange on the Kerr effect response in the framework of the exchange model outlined in Section II. For our calculations we assume that the exchange rates factor into terms that depend on initial and final environment only, $\\gamma_{ij}=\\rho(\\epsilon_i) X \\exp(\\beta \\epsilon_j)$. Here, $\\rho(\\epsilon_i)$ (probability to end in final state $\\epsilon_i$) is some normalized density of states and $X \\exp(\\beta \\epsilon_j)$ (probability to leave initial state $\\epsilon_j$) can be interpreted as thermally activated jumps out of environment $\\epsilon_j$. Here, we have identified the so-far not specified $\\epsilon_j$ with energies. The prefactor $X$ allows to adjust the 'strength' of exchange. We furthermore relate the rotational diffusion constants to $\\epsilon_i$ as $D_i=Y \\exp(\\beta \\epsilon_i)$.  The model is designed such that the total rate out of environment $\\epsilon_i$ is proportional to the appropriate rotational diffusion constant, $\\gamma_i\\approx (X\/Y) D_i$, meaning that slow environments exchange slowly and fast environments exchange fast. The ratio of $\\gamma_i$ and $D_i$, given by $(X\/Y)$, determines the influence of exchange on the decay of rotational correlation functions.\n\nFor the curves shown in Figure $3$ we chose $\\rho(\\epsilon_i)$ to be Gaussian with width $\\sigma=3$ and we have set $\\beta=1$. In order to obtain correlation times around unity we set $Y=100$. The upper panel of Figure $3$ shows normalized correlation functions for $L=1,2$ without exchange ($X\/Y=0$, dashed lines) and in the presence of exchange ($X\/Y=10$, solid lines). In the latter case the decay is strongly affected by exchange. All correlation functions shown can be well fitted with a function of KWW-type with a stretching exponent $\\beta_K \\approx 0.5\\ $. The ratios of correlation times $\\tau_L$ obtained from the fit are $\\tau_1\/\\tau_2 \\approx 3.0$ without exchange and $\\tau_1\/\\tau_2\\approx 1.3$ with exchange.\n\n\nThe nonlinear Kerr effect response $\\langle P_2^{\\mu\\mu}(\\omega)\\rangle$ in the presence of exchange is shown in the lower panel of Figure $3$. All parameters describing exchange are chosen as for the solid lines in the upper panel. In the limit without exchange, $(X\/Y)=0$, the model discussed in Figure $1$ is analytically recovered with a modified distribution of rotational diffusion constants. The resulting curves in that case are very similar to the responses in Fig.$1$. For that reason we show only the curves with exchange ($X\/Y=10$). The pump frequencies are chosen as $\\Omega_p=10^{-2} \\dots 10^{2}$ with one applied cycle for each frequency. The shifts of the extremum position obtained are very similar as in the heterogeneous case without exchange, cf. Fig.$1$. The peaks become slightly broader in the presence of exchange. No negative sign of the response occurs in any of the curves. \n\nWe therefore conclude that exchange does not affect the possibility to detect dynamic heterogeneities. It is important to stress that the Kerr effect experiment does not allow to conduct information regarding the lifetime of dynamic heterogeneities.\n\n\n\n\\subsection*{Homogeneous model}\nWe now discuss the responses in a homogeneous scenario, where the distribution of rotational diffusion constants is replaced by a single, but time-dependent rotational diffusion constant. The time dependence is chosen in a manner that the correlation function for $L=1$ is proportional to $\\exp\\left[ -(t\/\\tau)^{\\beta_K} \\right]$ with $\\beta_K=0.5$ and $\\tau=1$. The correlation function is thus the same as in the situation we discussed in Figure $1$. Inserting Eqns.(\\ref{glvont}),(\\ref{homd}) into (\\ref{almumu}) we find\n\\Be \n\\nonumber & & \\hspace{-1.6cm} \\langle P_2^{\\mu \\mu}(t) \\rangle=\\frac{1}{5} \\beta^2 \\mu^2 E^2 \\left(\\frac{\\beta_K}{\\tau}\\right)^2  \\exp\\left[ -3 \\left(\\frac{t+t_p}{\\tau} \\right)^{\\beta_K} \\right] \\times  \\\\ \\label{homoalmumu}\n& & \\hspace{-1.6cm} \\times \\int_0^{t_p} dt' \\int_0^{t'} dt'' \\sin(\\Omega_p t') \\sin(\\Omega_p t'') \\left(\\frac{t'}{\\tau} \\right)^{\\beta_K-1} \\left(\\frac{t''}{\\tau} \\right)^{\\beta_K-1}  \\exp\\left[ 2 \\left(\\frac{t'}{\\tau} \\right)^{\\beta_K}\\right]  \\exp\\left[ \\left(\\frac{t''}{\\tau} \\right)^{\\beta_K}\\right] .  \n\\Ee\nThe double integral does not depend on time $t$ and therefore represents only a prefactor to the response. The time-dependence is solely governed by the function $ \\exp \\left[ -3 \\left(\\frac{t+t_p}{\\tau} \\right)^{\\beta_K} \\right]$. Thus, the maximum position in $\\langle P_2^{\\mu\\mu}(\\omega)\\rangle$ is determined by the Fourier-transform of this function.\n\nFigure $4$ shows the homogeneous responses for the same pump frequencies as in Figure $1$. One cycle has been applied for each pump frequency, $t_p=2\\pi\/\\Omega_p$. It is this pump frequency-dependence of $t_p$ which leads to the shifts in the extremum positions in Figure $4$ (see the function above). Although the extrema are not at a constant position as usually expected in a homogeneous scenario, the shifts are less pronounced than in the heterogenous limit. The dashed curves corresponding to $\\Omega_p=1$ and $\\Omega_p=10$ are plotted negative. This is an example for the fact that signs of nonlinear responses are in general not easy to predict. The change of sign is not a systematic feature in our calculations.\n\n\n\n\\subsection*{Comparison of heterogeneous and homogeneous models}\n\nIn Figure $5$ we compare the maximum position in heterogeneous (upper straight line) and homogeneous (lower straight line) models, plotted versus pump frequency $\\Omega_p$ for one applied cycle. The distribution in the heterogeneous case is as in Fig.$1$, and the time-dependent rotational diffusion constant of the homogeneous model is as in Fig.$4$. In the homogeneous case a maximum position independent of pump frequency is obtained only for sufficiently large $\\Omega_p$. Clear shifts of the extremum position are observed in the heterogeneous limit. The shifting becomes weaker for $\\Omega_p < 0.1$. This is due to the shape of the chosen distribution function as elucidated in the discussion of Fig.$1$. \n \nFor a distinction between homogeneity and heterogeneity, the straight lines in Figure $5$ do not look too promising at first glance because the homogeneous extrema are not at a constant position. However, this finding should not be overemphasized. The observation of shifts in the homogeneous extremum positions relies directly on the lack of time-translational invariance of the specific homogeneous model used in the calculation. This leads to the shifted time $(t+t_p)$ in the function $\\exp \\left[ -3 \\left(\\frac{t+t_p}{\\tau} \\right)^{\\beta_K} \\right] $. The fact that the extrema are not at a constant position is only due to the appearance of the pump time $t_p$ in this function, which is related to the pump frequency via $t_p=2\\pi\/\\Omega_p$. It is furthermore easy to eliminate this effect. If $t_p$ is chosen independent of the pump frequency, then the homogeneous model obviously yields extremum positions completely independent of the pump frequency. This can be achieved by applying several cycles $N(\\Omega_p)\\propto \\Omega_p$ because of $t_p =2\\pi N(\\Omega_p)\/\\Omega_p$.\n  \nWe added some extremum positions for that case in Fig.$5$, denoted by symbols. We chose $t_p=2\\pi\/0.01$, meaning that one cycle is applied for the smallest pump frequency considered, $\\Omega_p=0.01$, and $10^4$ cycles are applied for the largest $\\Omega_p=100$. Then the homogeneous model extrema (open symbols) do not depend on $\\Omega_p$. The influence of several applied cycles on the extremum positions in the heterogeneous case is small. The extrema are located at somewhat smaller frequencies in comparison to the curve for one applied cycle. This is due to the pump time dependence of the function $A^{\\mu \\mu}(\\Omega_p,D_i,t_p)$. \n\nThe theoretical finding that the homogeneous ansatz leads to strictly constant extremum positions only if the pump time is constant for all applied frequencies also holds for NHB. If NHB is carried out as outlined in Ref.\\cite{mynhb} and the dynamics are treated in a rotational diffusion ansatz as in this work, we find in the homogeneous limit the same expression as Eq.(\\ref{homoalmumu}) up to prefactors and a modified time dependence. It is again the lack of time-translational invariance that leads to terms like $(t+t_p)^{\\beta_K}$ in the appearing exponential functions. We therefore suggest also in NHB to apply several cycles for higher pump frequencies. Also the heterogeneous model for NHB leads to very similar results as presented in the previous sections. Shifts of the extremum position are observed if the pump frequencies are varied in the range defined by the distribution of rotational diffusion constants. \n\n\\subsection*{Distribution width in the heterogeneous model}\n\nFinally, we discuss the influence of the width of a distribution of $D_i$ in a heterogeneous model. A linear behavior of the extremum position as a function of pump frequencies, $\\omega_{max} \\propto \\Omega_p^\\eta$ with $\\eta =1$, is obtained only if the distribution of rotational diffusion constants is sufficiently flat in the studied range. If the distribution is narrow, then usually exponents $\\eta <1$ are found.\n\nIn Figure $6$, extremum positions obtained from the heterogeneous model with a logarithmic Gaussian distribution of rotational diffusion constants $g(D_i)\\propto \\exp\\left(-[\\ln D_i - \\ln D_{\\infty}]^2\/[2\\sigma^2] \\right)$ are shown for three different distribution widths $\\sigma=1.8$ (diamonds), $\\sigma=1.2$ (squares) and $\\sigma=0.6$ (circles). Exchange effects are neglected. In each case, $D_{\\infty}$ was chosen such that the correlation time of the $(L=1)$-correlation function in a KWW-fit is $\\tau_1=1$. The corresponding KWW-exponents obtained from the fits of the correlation function are $\\beta_K=0.51$ (diamonds), $\\beta_K=0.67$ (squares) and $\\beta_K=0.88$ (circles). We applied the field for fixed pump time $t_p=2\\pi\/0.001$ independent of the pump frequency.\n\nHeterogeneities are observed via shifts of the extremum position. Constant behavior of $\\omega_{max}(\\Omega_p)$ is observed for small and large $\\Omega_p$. This is because, similar as in Figure $5$, the 'selected' $D_i$ in these cases are located beyond the main contributions to the distribution functions. In Fig.$5$ a constant behavior for large $\\Omega_p$ is not reached in the frequency range considered due to the tailing of the corresponding distribution of rotational diffusion constants, see the upper panel of Fig.$1$. We do not expect that a constant extremum position for large $\\Omega_p$ would be obtained experimentally. Here, other dynamics than isotropic reorientations will occur in real systems.\n\n\nOne can see that the rise of $\\omega_{max}$ with $\\Omega_p$ is the steeper the broader the distribution is, approaching the limit $\\eta =1$ for a very flat distribution. Acceptable power law fits for the shown curves are obtained in the window $\\Omega_p=0.1 \\dots 10$. Here, we find the exponents $\\eta \\approx 0.84$ (diamonds), $\\eta \\approx 0.63$ (squares) and $\\eta \\approx 0.2$ (circles). \n\n\\section*{IV. Conclusions}\n\nWe have calculated the Kerr after-effect response to an external oscillating electric field for reorientational dynamics in an isotropic system. Such an experiment is very similar to NHB, thus aiming at the question of the nature of the dynamics. The different assumptions of heterogeneous and homogeneous dynamics have been treated in a rotational diffusion model. We find that homogeneous and heterogeneous dynamics show different behaviour in the Kerr effect response function. \n\nWe propose the Kerr effect as an alternative to NHB. NHB and Kerr effect have in common that they are both nonlinear experiments. The advantage of the Kerr effect is that the signal has no linear response background. In an experiment, one would measure the reequilibration of the polarizability after the sample has been excited by a sinusoidal electric field. The frequency of the latter is then varied. If the time-dependent functions obtained in this way are plotted in a Fourier-transformed representation (imaginary part), then extremum positions as a function of the pump frequency shift under the assumption of heterogeneous dynamics, and they are constant in the homogeneous limit. The Kerr effect signal is thus able to distinguish between dynamic heterogeneity and homogeneity.\n\nThe interesting relation between dynamic and spatial heterogeneities is not resolved yet. We therefore emphasize that the Kerr effect experiment outlined in this article does not allow to obtain any information about spatial aspects of the heterogeneities.\n\nThe influence of exchange, and thus a finite lifetime of heterogeneities on the responses is small. Exchange in our model does not affect the possibility of frequency selective modification.\n\nIt turned out that the general trends of homogeneous and heterogeneous limits are obtained more clearly if several cycles of the pump field are applied. This is because switch-on effects are minimized for longer pump times. In addition, the extremum positions in our homogeneous model depend on the pump time $t_p=2\\pi N\/\\Omega_p$, and therefore on the pump frequency. In order to suppress this artificial dependence, we suggest to keep the pump time constant rather than the number of cycles applied if measurements with different $\\Omega_p$ are compared. If, e.g., $N$ cycles (where ideally $N>1$) are applied for pump frequency $\\Omega_p$, then $2N$ cycles should be applied if the pump frequency is $2\\Omega_p$.\n\n\nIf the width of a relaxation time distribution decreases then the shifts of the extremum position with pump frequency become weaker. This point might be of interest in the investigation of a sample at different temperatures. A steeper rise of the extremum position with pump frequency for lower sample temperatures would indicate an increasing width of the corresponding distribution function. A point we have not discussed explicitly is that of course also combinations of heterogeneous and homogeneous behavior may occur in real systems. If the weight of homogeneous and heterogenous character of the dynamics changes with temperature this should be observable in more or less pronounced shifts of the extrema.\n\n\n\n\n\\section*{Acknowledgement}\nThis work has been supported by the DFG under contract No. Di693\/1-2. \n\n\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is widely assumed that our understanding of gravity according to Einstein, together with effective local quantum field theories give an accurate description of IR physics. On the other-hand, many recent insights within the fields of holography and entropic gravity continue to corroborate the notion that spacetime and the laws of gravity should emerge from a UV-complete theory of quantum gravity\\cite{J95, M01, VR2010, susskind13, pad15, susskind17, felix17}\\nocite{smc} \\nocite{vl18}\n\\nocite{Heem}. It has become clear, particularly in the context of AdS\/CFT, that the emergence of the known laws of gravity is intimately connected with an entanglement area-law \\cite{ryu06, hubeny, myers13, swingle14, C16, C17}. Conversely, deviations from the area-law entanglement could lead to violations of gravitational physics. \n\nThat there could be novel physical ramifications of high-energy physics upon the laws that govern arbitrarily-large scales might seem very surprising, but just such a connection was recently proposed by E. Verlinde in the recent work \\cite{verl2016}. This recent work offers a novel hypothesis for the relation between the de Sitter entropy associated with the cosmological horizon, in terms of the dark energy, and its response due to the addition of matter. It is argued that the entropy carried by the dark energy obeys a volume-law, and that adding matter to a region of space generates competition between the volume-law and the area-law entropy associated with the matter. It is shown that there is a length-scale, which is related to the mass and the curvature scale, where the volume-law for the dark energy overwhelms the area-law scaling of the matter entropy. This leads to a violation of the holographic principle above this length-scale. It is this key role played by the violation of holography, at sufficiently large length-scales that is highlighted in the present work. \n\nIn \\cite{verl2016}, it was shown that the interplay between the de Sitter entropy, carried by the dark energy, and its local reduction due to the addition of a mass $M$, pertains to the following criteria,\n\n\\begin{equation}\n\\label{criteria}\n\\frac{M}{A(R)} \\lessgtr \\frac{a_0}{8 \\pi G},\n\\end{equation}\nwhere $a_0 = \\frac{ c^2 }{ L }$ is the MOND acceleration scale, $L$ is the Hubble length and $A(R)$ is the area of a sphere with radius $R$. This entropic condition coincides exactly with the empirical criteria for the scale upon which the phenomena attributed to dark matter become manifest in galactic rotation curves. This empirical observation is then given physical motivation by Verlinde's new hypothesis. Furthermore, it is shown that one can reproduce many key empirical findings in cosmology, such as Milgrom's fitting formula of MOND and the baryonic Tully-Fisher relation within this formalism \\cite{milgrom, btf}. There is also strong agreement between this framework and weak-lensing observations \\cite{brouwer16}.\n\nThe paper \\cite{verl2016} has been criticised on the basis of apparent inconsistencies in terms of associating an effective elastic description to the response governed by the removal (by the addition of matter) of a certain volume of the dark energy medium. Several of these features have since been addressed in the recent paper \\cite{sabine}, where a covariant proposal for the effective elastic response of the dark energy was considered, and the roles of the identifications between gravitational and elastic variables were clarified. Since the elastic description is intended to give an effective description, its negation would not necessarily rule out the underlying postulates of the emergent gravity framework. In the present work we make contact with the key results of \\cite{verl2016}, without using an effective elastic description.\n\nThe main proposal of the present paper is to use a straightforward modification of the setup described in \\cite{verl2010}, where Newtonian and Einstein gravity were derived from arguments involving so-called holographic screens. The modification entails a violation of holography above a critical length-scale, which thereby emulates a key feature of Verlinde's new theory. In the present paper, it is demonstrated that this modification is sufficient to describe the emergence of a dark gravity force, as well as a version of the baryonic Tully-Fisher relation. This work does not claim to go further than Verlinde's proposals, but instead the present goal is to provide a framework for the underlying ideas that is mathematically simpler, and which clarifies the key role played by the breakdown of holography in the emergence of dark gravity.\n\nThe rest of this paper is organised as follows. Section \\ref{review} contains a review of the recent proposal \\cite{verl2016} which identifies the core components that will be utilised in the derivation of emergent dark gravity, presented in section \\ref{main}. In section \\ref{main}, the main result is presented, where a modified thermodynamic setup based on arguments involving a holographic screen is constructed. It is demonstrated that the emergent entropic force associated to this system receives an additional contribution from the information in the bulk which is no-longer encoded holographically. This additional force has a $r^{-1}$ scaling-law required to describe flattening galactic rotation curves. In this setup it is also shown that the bulk and boundary energies obey a relation which is analogous to the baryonic Tully-Fisher relation, and that the bulk energy manifests as an apparent mass. By inputting the critical scale $r_c(M,L)$ identified in Verlinde's new theory to the setup, we find that the entropic force associated to the bulk then takes exactly the form of the force required to describe flat galactic rotation curves. It is also found that this value for the critical scale reproduces the baryonic Tully-Fisher relation up to a numerical factor. The key assumptions that are used in the present work are then reviewed. Finally, in section \\ref{relation} the present work is compared to the work of several recent papers with related goals.\n\n\n\\section{Emergent Gravity in de Sitter Space}\n\\label{review}\n\nIn \\cite{verl2016} a radical new explanation for the phenomena attributed to dark matter was offered in terms of emergent gravity in de Sitter space. In this section, we briefly review the key features of this work.\n\nIn \\cite{verl2016}, the static patch of de Sitter space was considered, with the metric,\n\\begin{equation}\n\\label{dsstatic}\nds^2 = -f(r) dt^2 + \\frac{1}{f(r)} dr^2 + d\\Omega_{d-2}^2,\n\\end{equation}\nwith $f(r) = 1 - \\frac{ r^2 }{ L^2 }$, where the cosmological horizon is at $r=L$. The Bekenstein-Hawking formula associates an entropy to de Sitter space which is determined by the area $A(L)$ of the cosmological horizon, as follows,\n\\begin{equation}\n\\label{dSent}\nS_{DE} = \\frac{ A(L) c^3 }{ 4 \\hbar G },\n\\end{equation}\nwhere, for reasons that will shortly be explained, the subscript ``DE\" denotes the dark energy. In \\cite{verl2016}, an interpretation for the entropy \\eqref{dSent} is proposed wherein the total entropy of de Sitter is associated to the dark energy which is distributed throughout the volume of de Sitter, leading to constant entropy density that obeys a volume-law. Accordingly, if we consider a spherical region of size $r$, the entropy associated to the dark energy within this region is proportional to the volume $V(r)$ of the sphere, so that we have\n\\begin{equation}\n\\label{SV}\nS_{DE}(r) = \\frac{ V(r) }{ V_0 },\n\\end{equation}\nwhere $V_0$ is the volume per unit of entropy of the dark energy. The condition that the entropy \\eqref{SV} coincides with the de Sitter entropy \\eqref{dSent} when $r=L$ then leads to the following formula for $V_0$,\n\\begin{equation}\n\\label{V0}\nV_0 = \\frac{ 4 G \\hbar L}{ d-1 }.\n\\end{equation}\nIt is then straightforward to show that formula \\eqref{dSent} can be re-written in the following way,\n\\begin{equation}\n\\label{SVA}\nS_{DE}(r) = \\frac{r}{L} \\frac{ A(r) c^3}{ 4 G \\hbar},\n\\end{equation}\nwhich makes it clear to see that when $r=L$, the above formula reproduces the formula for the total de Sitter entropy \\eqref{dSent}. In \\cite{verl2016}, the emergence of dark gravity is traced to the interplay that results from the local removal of a portion of the dark energy degrees of freedom due to the addition of matter. To ascertain this effect, one can turn on a matter source for a point mass M at the origin of the static patch \\eqref{dsstatic} by introducing the Newtonian potential with the following replacement,\n\\begin{equation}\n\\label{withpot}\nf(r) \\rightarrow 1 - \\frac{ r^2 }{ L^2 } + 2 \\phi(r),\n\\end{equation}\nwith,\n\\begin{equation}\n\\label{phi}\n\\phi(r) = \\frac{- GM }{r }.\n\\end{equation}\nIt is shown that the negative sign of $\\phi(r)$ leads to a reduction in the total de Sitter entropy when we add the mass $M$ to the origin of the static patch. This total reduction of the de Sitter entropy in fact corresponds to a local reduction of the entropy associated with the dark energy in the region surrounding the point mass. Accordingly one can consider the change in the growth of the area of a spherical region as a function of geodesic distance in the case with and without the matter to arrive at the following formula for reduction of the de Sitter entropy due to the addition of the mass $M$,\n\\begin{equation}\n\\label{SM}\nS_M(r) = \\frac{ 2 \\pi M c}{\\hbar} r.\n\\end{equation}\nA key result of \\cite{verl2016} is that, given a point mass M, there is a length-scale, depending on the mass and the curvature scale, where the de Sitter entropy \\eqref{SVA} exactly equals the amount of entropy \\eqref{SM} which is removed by the addition of a mass $M$. Using the formulas \\eqref{SVA} and \\eqref{SM} it is easy to calculate this scale to be the following,\n\\begin{equation}\n\\label{rcrit}\nr_c(M) = \\sqrt{\\frac{ G M L}{c^2}} = \\sqrt{\\frac{ G M}{ a_0}},\n\\end{equation}\nwhere the MOND acceleration scale $a_0 = c^2 \/ L$ \\cite{milgrom} has been identified. Below this critical length-scale, all of the de Sitter entropy is removed by the mass $M$; in this case there are only matter degrees of freedom in the bulk which are encoded in the degrees of freedom at the boundary. This corresponds to what is called the Newtonian regime. Conversely, for regions that are larger than this scale, the mass $M$ does not remove all of the de Sitter entropy. In this sub-Newtonian or ``dark gravity\" regime, there is therefore information associated to the dark energy in the bulk which is entangled with the bulk mass. This volume-law entanglement, which contains information about the bulk mass, then spoils the holographic encoding of the bulk mass. This scenario is depicted in figure \\ref{bigone}. As we will see, this is a key feature of these proposals that this work seeks to emulate in the thermodynamic setup presented in section \\ref{main}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\textwidth]\n{EGDS.jpg}\n\\caption{(Top) Cartoon illustrating the encoding of the information (shaded blue) pertaining to the bulk mass $M$ in de Sitter space. For regions much smaller than $r_c$, the bulk information pertaining to $M$ is redundantly encoded in the boundary bits $N_\\partial$ which obey an area-law. Conversely, for regions that are large compared to $r_c$, there are bulk degrees of freedom that scale extensively with the bulk and which are not encoded in the bits $N_\\partial$. (Bottom) Cartoon of a portion of a tensor network representation for the entanglement structure of de Sitter space. The bulk legs correspond to the green indices. The entanglement structure that builds the emergent geometry is encoded in the short-range correlations described by the grey, internal legs of the network. The state is endowed with a constant density of long-range entanglement by the single, large red tensor. The bulk indices participate in the long-range entanglement via their connection to the large-red tensor via the dark red legs. (Bottom Right) adding matter locally removes some of the long-range entanglement. We imagine that the long-range indices in the center (white) have been deleted. In this case, the bulk state in the yellow legs can be approximately reconstructed from the state on the portion of network bounded by the cut which passes through the blue legs of the network. For larger portions of the network, such as for the cut which passes through the purple legs, the state in the bulk legs within this cut cannot be reconstructed from the legs passing through the cut, leading to a breakdown of holography for regions that are sufficiently large.}\n\\label{bigone}\n\\end{figure}\n\nIn \\cite{verl2016}, a remarkable feature is observed when we consider the following criteria, associated to the length-scale above which the volume-law associated with the dark energy overwhelms the matter entropy \\eqref{SM}, that is where we have the following:\n\\begin{equation}\n\\label{entcrit}\nS_M(r) < S_{DE}(r).\n\\end{equation}\nUsing equations \\eqref{SVA} and \\eqref{SM}, we observe that the above criteria are equivalent to the criteria \\eqref{criteria}, as claimed in the introduction. \n\nSince the volume-law of the dark energy overwhelms the area-law scaling of the matter entropy in the dark gravity regime, then accordingly what emerges are not the known laws of gravity. Instead, in \\cite{verl2016}, this regime is understood by considering the effective, elastic response of the dark energy medium to its removal, from local inclusion regions, by the addition of matter. It is shown that this leads to the fitting formula of MOND and also the baryonic Tully-Fisher relation \\cite{milgrom, btf}. It is thereby argued that this framework provides an alternative explanation for the phenomena that are currently attributed to dark matter, which does not require dark matter to exist. This work will make contact with these two key results, but the effective elastic description described in \\cite{verl2016} will not be required.\n\n\\section{Emergent Dark Gravity from a (Non)Holographic Screen}\n\\label{main}\n\nThis section describes how to obtain the emergence of a dark gravitational force, which scales like $\\frac{1}{r}$ above a critical length-scale $r_c$, in a purely thermodynamic setting which is very similar to the setup considered in \\cite{verl2010}. The modification presented here is inspired by the arguments in \\cite{verl2016}, described in section \\ref{review}, where the interplay between the entropy associated to the dark energy and the entropy which is removed by the addition of matter leads to a violation of the holographic encoding of bulk beyond the critical length-scale $r_c$ as in \\eqref{rcrit}. Accordingly, the modification proposed here is the introduction of an arbitrary critical length-scale that controls the scale at which the holographic encoding of the bulk fails. \n\n\\subsection{A (Non)Holographic Screen}\n\\label{nhs}\n\nAt first we will assume precisely the setup described in section 3.2 of \\cite{verl2010}. Namely one is to imagine that there is a spherical region of space, known as a holographic screen, which separates the interior ``unemerged\" part of space from the exterior ``emerged\" part of space. A particular thermodynamic system is ascribed to this setup in which the ``unemerged\" part of space emerges. The emergence of space is imagined to arise due to a series of coarse-graining steps that push the holographic screen into the ``unemerged\" part of space, which leads to an overall reduction in the microscopic degrees of freedom associated with the holographic screen. In the emerged part of space, we imagine that there is a massive test particle of mass $m$ which is located at a small displacement $\\delta x$ from the holographic screen. The setup is contrived so that a change in the test particle's position contributes a change to the entropy of the screen, which leads to an entropic force acting on the test particle. The setup so-far described is depicted in figure \\ref{EDG} a). \n\nThe role of the holographic screen, within the earlier work, is to encode the information within the unemerged space. To make a clear connection between this setup and the AdS\/CFT literature, the unemerged part of space will be referred to, hereafter, as the bulk. The holographic screen then corresponds to the boundary of the bulk, but since this is an arbitrary boundary in space, and not a geometric boundary at spatial infinity as in AdS\/CFT, the boundary of the bulk (unemerged part of space) will be referred to as a screen. In the present setup, holography will be explicitly violated, so the term ``holographic screen\" will not be appropriate. The term non-holographic screen may then seem more appropriate, but for convenience this artificial boundary will simply be referred to as a screen.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]\n{edgfinal.jpg}\n\\caption{ a) the test particle of mass $m$ is located in the emerged part of space, and is displaced from the spherical screen of radius $r$ by $\\delta x \\ll 1$. b) for regions $r \\ll r_c$, the test particle contributes to a change in the entropy associated to the bits on the screen. c) for regions $r \\sim r_c$, the test particle contributes to a significant change in the entropy associated to the bulk.}\n\\label{EDG}\n\\end{figure}\n\nIn \\cite{verl2010} it is assumed that the information pertaining to an emergent mass $M$ in the bulk is encoded holographically in a quantity of bits $N_\\partial$ which is proportional to the area of the screen, in the following way,\n\\begin{equation}\n\\label{nbound}\nN_\\partial = \\frac{ A(r) c^3}{ 4 G \\hbar}.\n\\end{equation}\nThe main modification of \\cite{verl2010} presented in this paper is to assert that the holographic principle is violated in a way that depends on a critical length-scale $r_c$. We suppose that, in addition to the bits $N_\\partial$, there are a set of bits $N_\\Sigma$ that one also needs to have in order to be able to completely reconstruct the emergent mass $M$ in the bulk. We assert that the bits $N_\\Sigma$ pertaining to $M$ are not encoded holographically at the screen. Motivated by \\cite{verl2016} we assert that $N_\\Sigma$ should scale extensively with the \\emph{volume} of the bulk. We then associate the critical scale $r_c$ with the ratio of the numbers of bulk and screen bits as follows,\n\\begin{equation}\n\\label{ratio}\n\\frac{N_\\Sigma}{N_\\partial} = \\frac{r}{r_c},\n\\end{equation}\nwhich leads to the following form for $N_\\Sigma$,\n\\begin{equation}\n\\label{BV}\nN_\\Sigma = \\frac{r}{r_c} \\frac{ A(r) c^3}{ 4 G \\hbar}.\n\\end{equation}\nBased on the above observations we conclude that the role of the critical scale $r_c$ is to determine when the holographic, area-law scaling of the information associated to $M$ in the bulk is overwhelmed by the volume-law contribution of the bulk bits $N_\\Sigma$ that are not encoded holographically. Thus the critical scale $r_c$ that we have described plays a very similar role to the critical scale $r_c$ as in \\eqref{rcrit} identified in \\cite{verl2016}. In our setup, one could suppose that the scale $r_c$ that we are considering, should depend on the amount of information associated to the mass $M$ and the curvature scale $L$. This is indeed the case for the critical scale \\eqref{rcrit}. For now, this dependence will not be explicitly assumed.\n\nIf one imagines inserting a mass into the bulk, we can call the number of bits that characterise this matter $\\tilde{N}_\\Sigma$. In general this is different from $N_\\Sigma$, because when the holographic principle holds, we anticipate that the former are encoded redundantly at a screen, of size $r$, as a quantum secret-sharing scheme among the bits $N_\\partial(r)$ \\cite{harlow14, harlow15, harlow16}. In contrast, the bits $N_\\Sigma$ pertain to information about the bulk mass that is not encoded holographically, which do not therefore form a subset of the bits $N_\\partial$. The latter can be compared to the formula \\eqref{SVA}, so that the bits $N_\\Sigma$ here play the same role as the dark energy in \\cite{verl2016}. \n\n\\subsection{Thermodynamic Setup}\n\nNow we describe how to associate a thermodynamic system with the setup described in \\ref{nhs}. As per \\cite{verl2010}, we begin with the assumption that displacing the test particle which is outside the screen, but which is close to it, leads to a change to the entropy of the screen and the bulk. The change in the entropy of the total system will take the following form,\n\\begin{equation}\n\\label{composite}\n\\delta S = \\delta S_\\partial + \\delta S_\\Sigma ,\n\\end{equation}\nwhere $\\delta S_\\partial$ coincides with $\\delta S$ defined in section 3.2 of \\cite{verl2010}, which we identify as the change in the entropy of the screen. The quantity $\\delta S_\\Sigma$ is similarly defined as the contribution to the entropy of the bulk, which, as we have explained, is not encoded holographically. \n\nA change to the entropy as per \\eqref{composite} will lead to a change to the energy of the system $\\delta E = \\delta E_\\partial + \\delta E_\\Sigma$, where we have identified the contributions $\\delta E_\\partial$ and $\\delta E_\\Sigma$ which correspond to changes to the energy associated with the screen and bulk degrees of freedom respectively. Following \\cite{verl2010}, we assume that when the displacement $\\delta x$ of the test particle is small, the corresponding changes to the energies of the bulk and screen systems should be evenly divided over their respective degrees of freedom, so that the equipartition theorem holds for each system.\n\n\\begin{equation}\n\\label{equipartitions}\n\\delta E_\\partial = \\frac{1}{2} N_\\partial k_B T_\\partial \\hspace{1cm} \\delta E_\\Sigma = \\frac{1}{2} N_\\Sigma k_B T_\\Sigma ,\n\\end{equation}\nwhere $T_\\partial$ and $T_\\Sigma$ are the temperatures associated to the bulk and boundary systems. To proceed we further make the assumption that the bulk and boundary systems are in thermal equilibrium. In general this is something that we may expect to hold for situations where the emergent bulk is static. For now we will assume that this holds, so that we can identify $T_\\partial = T_\\Sigma = T$, where $T$ is the equilibrium temperature associated to the combined ensemble of the bulk and the screen. \n\nIf the first-law of thermodynamics holds for this system, then for near-equilibrium configurations the following relation must hold for the bulk and boundary subsystems,\n\\begin{equation}\n\\label{1stlaw}\n\\delta E_\\partial =T \\delta S_\\partial \\hspace{1cm} \\delta E_\\Sigma = T \\delta S_\\Sigma .\n\\end{equation}\nUsing equations \\eqref{equipartitions} and \\eqref{1stlaw}, we find that the changes to the entropy of the bulk and the screen obey the following relation,\n\\begin{equation}\n\\label{heidi}\n\\delta S_\\Sigma = \\frac{N_\\Sigma}{N_\\partial} \\delta S_\\partial = \\frac{r}{r_c} \\delta S_\\partial .\n\\end{equation}\nThe above relation has the intuitive interpretation that for scales $r \\ll r_c$, we have $\\delta S_\\Sigma \\sim 0$, and the significant contribution to the entropy of the system comes from the contribution to the screen, as depicted in figure \\ref{EDG} b). Conversely, at $r=r_c$, the test particle contributes the same amount of information to the bulk and boundary subsystems at $r=r_c$. Hence $r_c$ naturally described a scale where the contribution of a test particle to the bulk first becomes comparable to its corresponding contribution to the screen. For scales $r>r_c$, the corresponding contribution to the bulk overwhelms the contribution to the screen, as depicted in figure \\ref{EDG} c). Putting together equations \\eqref{heidi} and \\eqref{equipartitions} gives the following relation between the changes to the energies of the bulk and boundary subsystems,\n\\begin{equation}\n\\label{heidi2}\n\\delta E_\\Sigma = \\frac{N_\\Sigma}{N_\\partial} \\delta E_\\partial = \\frac{r}{r_c} \\delta E_\\partial .\n\\end{equation}\nTo obtain an emergent entropic force associated to this system, we express the change in energy given by the first law \\eqref{1stlaw} in terms of the work done by the force $F$ that displaces the test particle a distance $\\delta x$ \\emph{towards} the screen. In this way the entropic force must take the following form in terms of the change to the entropy of the system.\n\\begin{equation}\n\\label{ef}\nF = T \\Bigg( \\frac{\\partial S_\\Sigma}{\\partial x}+\\frac{\\partial S_\\partial}{\\partial x} \\Bigg) = T \\frac{\\partial S_\\partial}{\\partial x}\\Bigg( 1 + \\frac{r}{r_c} \\Bigg),\n\\end{equation}\nwhere equation \\eqref{heidi} was used in the last step. The term $\\frac{r}{r_c}$ includes a modification to the setup described in \\cite{verl2010} which is associated with the breakdown of holography that we have described. In particular, when $r \\ll r_c$ we can effectively set this term to zero, and in this case we would obtain the same result as \\cite{verl2010}, where the holographic principle is assumed to hold exactly.\n\n\\subsection{Emergent Dark Gravity}\n\\label{mainmain}\n\nIn \\cite{verl2010}, the form for the change $\\delta S_\\partial$ to the entropy of the screen can be motivated as follows. Following Bekenstein's derivation of the black hole entropy, one can propose that when the test particle is one Compton-wavelength from the screen, it should contribute a single bit of information to it \\cite{bek}. This contribution is assumed to be linear in $\\delta x$, at least approximately when $\\delta x$ is sufficiently small. This leads to the following form for the change of the boundary entropy,\n\\begin{equation}\n\\label{ds1}\n\\delta S_\\partial = \\frac{\\pi k_B}{2} \\frac{ m c }{\\hbar } \\delta x .\n\\end{equation}\nTo relate the quantities in this thermodynamic system to an emergent mass we suppose that the following relations hold,\n\\begin{equation}\n\\label{firstlaw}\n\\delta E_\\partial =M_B c^2 \\hspace{1cm} \\delta E_\\Sigma = M_D c^2 .\n\\end{equation}\nFor now $M_B$ and $M_D$ are just arbitrary constants with the units of mass. The relation \\eqref{heidi2} implies that these quantities obey the relation,\n\\begin{equation}\n\\label{BTF}\n\\frac{M_D}{M_B} = \\frac{r}{r_c} .\n\\end{equation}\nWe can now use either of the equations \\eqref{equipartitions} to determine the equilibrium temperature $T$ which is found to be the following,\n\\begin{equation}\n\\label{combtemp}\nT = \\frac{ 2 M_B G \\hbar}{c k_B \\pi r^2} .\n\\end{equation}\nNow we have all of the pieces to obtain the form for the entropic force that emerges from this thermodynamic system. In the case where $r \\ll r_c$, the bulk contribution to the entropy change can be neglected. In this limit we can neglect our modification to the setup described in \\cite{verl2010} and we accordingly obtain the same result contained therein, which is the emergence of Newton's gravitational force-law,\n\\begin{equation}\n\\label{Enewton}\nF = \\frac{ G m M_B }{ r^2 } .\n\\end{equation} \nEquation \\eqref{Enewton} justifies the interpretation of $M_B$ as describing an emergent mass in the bulk. For $r \\sim r_c$, the bulk contribution to the change in entropy of the screen becomes non-trivial, and using the formula \\eqref{combtemp} for the equilibrium temperature, together with the formulae \\eqref{heidi} and \\eqref{ds1} for the changes in the bulk and boundary entropy (respectively) and plugging these results into \\eqref{ef} for the entropic force, the entropic force takes the form,\n\\begin{equation}\n\\label{darkgravity}\nF = \\frac{ G m M_B }{ r^2 } + \\frac{G M_B m}{ r \\cdot r_c} .\n\\end{equation}\nThis force has the correct $\\frac{1}{r}$ scaling required to describe flat galactic rotation curves, where $M_B$ is the baryonic point mass located at the origin of the bulk. It is worth emphasising that this result essentially derived from the violation of holography as per equation \\eqref{ratio}, which eventually led to equation \\eqref{heidi}. In view of the derivation of Newtonian gravity in section 3.2 of \\cite{verl2010}, the additional factor of $r$ contained in \\eqref{heidi}, in this work, is what led to a force that scales like $r^{-1}$.\n\nPresently, the critical scale appearing in \\eqref{darkgravity} is arbitrary. On the other-hand, the critical length-scale \\eqref{rcrit} plays essentially the same role in \\cite{verl2016}. If we identify our critical scale with the scale \\eqref{rcrit}, we obtain the following emergent force,\n\\begin{equation}\n\\label{DEG}\nF= \\frac{G M_B m}{r^2} + \\sqrt{M_B G a_0} \\frac{m}{r},\n\\end{equation}\nwhich has precisely the form of Newtonian gravity with an additional dark gravity force which is observed at galactic scales above $r_c$ as per equation \\eqref{rcrit} \\cite{milgrom}. \n By inserting the relation \\eqref{BTF} into \\eqref{darkgravity} we see that the entropic force can be written in the following form,\n\\begin{equation}\n\\label{darkgravity2}\nF = \\frac{ G m }{ r^2 }  \\big( M_B + M_D(r) \\big) .\n\\end{equation}\nThis result serves to clarify the role of the quantity $M_D$, which describes the change in the bulk energy according to \\eqref{firstlaw}. In view of \\eqref{darkgravity2}, we conclude that the failure of holography for scales $r \\sim r_c$ leads to a contribution to the bulk energy (of the test particle to the screen) that manifests as an apparent, additional mass $M_D$ in the bulk. Notice that the scale $r_c$ drops out of equation \\eqref{darkgravity2}, but to ascertain the relation between the apparent mass $M_D$  and the mass $M_B$, we need to use the relation \\eqref{heidi2}, which does implicate $r_c$. This setup can therefore mimic the dark matter hypothesis, independently of the choice of $r_c$. We see that the identification between the mass $M_B$ and the apparent mass $M_D$ given by equation \\eqref{heidi2} has a similar role to the baryonic Tully-Fisher relation, which relates the baryonic mass distribution to the apparent distribution of dark matter \\cite{btf}. \n\nIf we consider the emergence of a spherically-symmetric mass distribution in the bulk which is entirely contained in a screen of size $r$. That is, suppose that there is an emergent mass profile $M_B(r)$ in the bulk, contained inside a screen of size $r$ and which corresponds to the change in energy $\\delta E_\\partial$ of the screen. Then if we then take the formula \\eqref{BTF}, with the critical scale $r_c$ identified by \\eqref{rcrit}, we recover a relation between the emergent mass profile $M(r)$ and the apparent mass profile $M_D(r)$, which, up to a numerical factor, has been shown to be equivalent to the baryonic Tully-Fisher relation \\cite{btf},\n\\begin{equation}\n\\label{BTFfinal}\nM_D(r)^2 = \\frac{a_0}{2 \\pi G} M_B(r),\n\\end{equation}\nwhich was shown to determine the apparent dark matter profile $M_D(r)$, given a profile $M_B(r)$ of (observed) baryonic matter in galaxies. Using this framework, we can therefore make contact with the main results obtained in \\cite{verl2016}. \n\n\\subsection{Review of Key Assumptions}\nThe key features of our main result, presented in \\ref{mainmain}, follow from the application of the following assumptions, whose role should be clarified,\n\\begin{enumerate}\n\\item{The bulk is not encoded holographically at the screen, and the non-holographic bulk degrees of freedom obey a volume-law, such that the failure to encode the bulk holographically is controlled by a length-scale $r_c$.}\n\\item{Changing the displacement of the test particle changes the entropy associated with the bulk and boundary subsystems.}\n\\item{The change in energy of the bulk and boundary subsystems obeys the equipartition theorem.}\n\\item{The bulk and boundary systems are in thermal equilibrium.}\n\\item{The first law of thermodynamics holds.}\n\\end{enumerate}\nAssumption 1 is our main assumption in this work, and it is the main modification to the setup described in \\cite{verl2010}. This is the modification that makes a connection to the new theory \\cite{verl2016}, where a violation of the holographic principle occurs for length-scales which are sufficiently large so that the inequality \\eqref{entcrit} holds. Assumption 2 is also made in \\cite{verl2010}, and this is the key feature that is contrived to produce an entropic force acting on the test particle. Assumption 3 is also made in \\cite{verl2010} in relation to what we have called the boundary system. That we should imagine this to hold for the bulk system seems like a natural extension, since when the particle is sufficiently close to the screen, we can imagine that the change in the energy of the bulk is evenly divided over the bulk bits. As in \\cite{verl2010}, we may not expect this to hold for displacements that are large compared to the Compton wavelength of the test particle, and we refer the reader to the former work for a justification of this. Assumption 4 is new here and it seems like a natural assumption to make when the emergent bulk and matter distribution is static, as was the case for the static patch of de Sitter that was considered in \\cite{verl2016}. In more general situations, where there are non-trivial dynamics, we may not expect assumption 4 to hold exactly. Assumption 5 was assumed in \\cite{verl2010}; the first law should evidently hold for any equilibrium thermodynamic system, such as the one that has been described here. The role of the first law is, as in the earlier work, that it allows one to relate the an entropic force, to the entropy gradient and the equilibrium temperature as per \\eqref{ef}.\n\nIn many cases, it may appear that the fundamental constants have been introduced in an ad-hoc fashion. This objection could also be raised about the work \\cite{verl2010}. Here, as in that work, the role of these contants is essentially to give quantities with the correct dimensions. Nevertheless, we see that $\\hbar$ drops out in the calculations of the dark gravity force \\eqref{DEG}, as it must in order to match with the idea that we are considering a Newtonian limit (that does not albeit lead to Newtonian gravity). So, as with the previous work, the constant $\\hbar$ remains arbitrary in this work.\n\n\\section{Comparison with Previous Work}\n\\label{relation}\n\nThe setup described in section \\ref{main} is strongly based on the setup described in \\cite{verl2010}. However, a key modification is included which is based on a central observation of the recent proposals \\cite{verl2016}, that the emergence of dark gravity is due to the breakdown of the holographic principle above a certain length-scale \\eqref{rcrit}, where the inequality \\eqref{entcrit} holds, as described in section \\ref{review}. Indeed, one of the aims of this work is to make a clear connection between the relatively simpler framework described in \\cite{verl2010} and the recent work \\cite{verl2016}, by extracting this key feature and implementing it as we have described in section \\ref{main}. The main results \\eqref{DEG} and \\eqref{BTFfinal} that we have derived for the emergent dark gravity force and the baryonic Tully-Fisher relation (respectively), rely on the identification for our critical scale as per \\eqref{rcrit}. We cannot motivate the use of this scale based on the efficacy of the empirical observations because in the present work, the role of the critical scale is to control the breakdown of holography, which does not immediately follow from the empirically observed behaviour above this scale. This effect is indeed currently widely attributed to the dark matter hypothesis, which is an entirely different proposal. Thus, the motivation for us to consider the scale \\eqref{rcrit}, in the present work, is tied to the proposals of \\cite{verl2016} which initially make contact with the observed criteria. In that work, this length-scale is motivated by according a particular interpretation to the de Sitter entropy, which it not considered here; it is instead shown that the ensuing violation of holography, with a volume-law scaling of the emergent matter degrees of freedom, essentially leads to the key features that match the observed galaxy-scale phenomena. To this end the effective elastic description that was described in \\cite{verl2016} was not required.\n\nThere have been several recent papers with related goals, which have considered how to derive the MOND fitting formula in terms of an entropic force which emerges from a thermodynamic setup involving holographic screens. These works include \\cite{Zhang}, where an argument analogous to \\cite{verl2010} is considered, but with the introduction of a modified inertia relation which accordingly produces a modified gravity force. In this work we do not assume that such a modified inertial relation holds. Rather, as we have said, the present setup is inspired by the observation of Verlinde in \\cite{verl2016} that the emergence of a dark gravity force is attributed to a breakdown of holography at a certain scale, which has been implemented explicitly in setion \\ref{main} of the present paper. There are also a pair of more recent papers \\cite{Abreu1,Abreu2} which have considered how to derive the MOND relation, again inspired by the holographic screen arguments \\cite{verl2010}, but the authors have made use of the Tsallis entropy to derive a modified gravity relation. Again, this differs significantly from the present approach, where Tsallis statistic have not been used; the key results presented here derive from a breakdown of holography which is not assumed in the previous papers \\cite{Abreu1,Abreu2}.\n\n\\section{Discussion \\& Outlook}\n\nIn this work, a clear connection has been established between the earlier work \\cite{verl2010} and the recent proposals \\cite{verl2016}, by implementing a breakdown of holography which is controlled by a critical length-scale via a straightforward modification to the earlier framework. Furthermore, when the value for the critical scale \\eqref{rcrit}, identified in the recent proposals, is adopted, the exact form of the dark gravity force, in addition to (up to a numerical factor) the baryonic Tully-Fisher relation, which are observed to produce flattening galactic rotation curves above precisely this scale, are obtained. This work then essentially clarifies the key role played by the breakdown of holography for sufficiently large length-scales (given an emergent mass $M$) in E. Verlinde's new hypothesis, whilst providing a mathematically simple framework with which one could explore this, and related ideas. \n\nIn figure \\ref{bigone}, a tensor network has been depicted which provides an analogy for the entanglement structure of an emergent de Sitter geometry as described in \\cite{verl2016}. The particular network represented here does not accurately depict the tensor network one might use to describe an emergent de Sitter geometry, however one could in principle obtain this via a discretisation of a constant time slice \\cite{B16, ev}, sewn with tensors that, as well as bulk and internal indices, each carry a small additional index which is contracted with the tensor that thereby endows the state with a constant density of long-range entanglement, whose role can therefore mimic the dark energy as per \\cite{verl2016}. Furthermore, this work offers the interpretation that the addition of a mass $M$ can be framed, in these terms, as a deletion of a portion of the long-range legs attached to a closed subregion of the network. This would be consistent with the idea that mass is associated with relative entropy in emergent gravity\\cite{verl2016}. This question is left open for future work.\n\n A particularly interesting open question for this work relates to the possibility of a finding covariant formulation, which could make contact with the recent work \\cite{sabine, wang}. Most of the components needed for this have been identified in the present work, but the formulation of a covariant model is left for future work. Another interesting and partially-related question  concerns a possible relation between the framework presented here and the effective elastic description offered in \\cite{verl2016} to describe the dark gravity regime.\n\nDespite the many successes of the dark matter paradigm, we have presently yet to observe the dark matter particle, and Verlinde's new theory offers the exciting possibility that understanding this regime may require a radical revision of our widely-held belief in the efficacy of GR and EFT on cosmological scales. In this work we hope to have provided a framework that will help to further our understanding of this new proposal and its underlying microscopic description.\n\n\n\\section{Acknowledgements}\nThe author would like to thank Henry Maxfield and Vaios Ziogas for helpful discussion and comments.\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}