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+{"text":"\\section{CHIRAL SYMMETRY}\n\\label{s1}\n\nThe outer components of nuclear forces are dominated by pion-exchanges\nand involve just a few basic subamplitudes, describing pion interactions\nwith either nucleons or other pions.\nThe simplest process $N \\!\\rar\\! \\pi N$, corresponding to the emission or \nabsorption of a single pion by a nucleon, is rather well understood and \ngives rise to the one-pion exchange  $N\\!N$ potential ($OPEP$).\nThe scattering reaction $\\pi N \\!\\rar\\! \\pi N$ comes next and determines both \nthe very important two-pion exchange term in the $N\\!N$ force \nand the leading three-body interaction, as shown in Fig.\\ref{F1}.\n\n\\vspace{-4mm}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.7\\columnwidth,angle=0]{Robilotta_Osaka-01.eps}\n\\vspace{-3mm}\n\\caption{Free $\\pi N$ amplitude (a)\nand two-pion exchange two-body (b) and three-body (c) potentials.} \n\\label{F1}\n\\end{center}\n\\end{figure}\n\n\\vspace{-5mm}\n\nThe theoretical understanding of the $\\pi N$ amplitude proved\nto be very challenging and a suitable description was only produced \nby means of chiral symmetry.\nThis framework provides a  natural explanation for the observed \nsmallness of $\\pi N$ scattering lengths and plays a fundamental\nrole in Nuclear Physics.\nNowadays, the use of chiral symmetry in low-energy pion-interactions\nis justified by $QCD$.\n\nThe small masses of the quarks $u$ and $d$, treated as perturbations \nin a chiral symmetric lagrangian, give rise to a well defined chiral \nperturbation theory (ChPT).\nHadronic amplitudes are then expanded in terms of a typical \nscale $q$, set by either pion four-momenta or nucleon three-momenta, \nsuch that $q\\ll 1$ GeV.\nThis procedure is rigorous and many results have the status of {\\em theorems}.\nIn general, these theorems are written as power series \nin the scale $q$ and involve both {\\em leading order terms} and \n{\\em chiral corrections}.\nThe former can usually be derived from tree diagrams, whereas the\nlatter require the inclusion of pion loops and are the main object \nof ChPT. \nAt each order, predictions for a given process must be unique and the \ninclusion of corrections cannot change already existing leading terms.\n\nThe relationship between chiral expansions of the $\\pi N$ amplitude and of \ntwo-pion exchange $(TPE)$ nuclear forces is discussed in the sequence.\nFor the $\\pi N$ amplitude, tree diagrams yield ${\\cal{O}}(q, q^2)$ terms \nand corrections up to ${\\cal{O}}(q^4)$ have already been evaluated, by means of both \ncovariant\\cite{BL}(CF) and heavy baryon\\cite{FM}(HBF) formalisms. \nIn the case of the $N\\!N$ potential, the leading term is ${\\cal{O}}(q^0)$ and\ngiven by the $OPEP$.\nThe tree-level $\\pi N$ amplitude yields $TPE$ contributions at ${\\cal{O}}(q^2,q^3)$\nand corrections at ${\\cal{O}}(q^4)$ are available, based on both \nHBF\\cite{HB} and CF\\cite{HR,HRR}.\nTree-level $\\pi N$ results also determine the leading ${\\cal{O}}(q^3)$ three-body\nforce and partial corrections at ${\\cal{O}}(q^4)$ begin to be derived \n\\cite{IR07,E3NP}.\nAs this discussion suggests, ${\\cal{O}}(q^4)$ corrections to both\ntwo- and three-nucleon forces require just the ${\\cal{O}}(q^3)$\n$\\pi N$ amplitude.\n\nThe full empirical content of the $\\pi N$ amplitude cannot be predicted\nby chiral symmetry alone. \nExperimental information at low energies is usually encoded into the \nsubthreshold coefficients introduced by H\\\"ohler and collaborators\\cite{H83}\nwhich can, if needed, be translated into the low-energy contants (LECs)\nof chiral lagrangians. \nTherefore, in order to construct a ${\\cal{O}}(q^3)$ $\\pi N$ amplitude,\none uses chiral symmetry supplemented by \nsubthreshold information, as indicated in Fig. \\ref{F2}.\nThe first two diagrams correspond to the nucleon pole, \nwhereas the other ones represent a smooth background.\nThe third graph reproduces the Weinberg-Tomozawa contact interaction,\nthe fourth one summarizes LEC contributions and the last two describe \nmedium range pion-cloud effects.\n\n\\vspace{-5mm}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.8\\columnwidth,angle=0]{Robilotta_Osaka-02.eps}\n\\vspace{-4mm}\n\\caption{Representation of the $\\pi N$ amplitude at ${\\cal{O}}(q^3)$.} \n\\label{F2}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{TWO-BODY POTENTIAL}\n\\label{s2}\n\nWith the purpose of discussing the problem of predicted \n$\\times$ observed chiral hierarchies, in this section we review \nbriefly results obtained by our goup\\cite{HR,HRR} for the \n$TPE$-$N\\!N$ potential at ${\\cal{O}}(q^4)$.\nThis component is determined by the three families of diagrams\nshown in Fig. \\ref{F3}. \nFamily $I$ begins at ${\\cal{O}}(q^2)$ and implements the minimal realization of \nchiral symmetry\\cite{RR94},\nwhereas family $I\\!I$ depends on $\\pi\\p$ correlations and is ${\\cal{O}}(q^4)$.\nThey involve only the constants $g_A$ and $ f_\\pi$\nand all dependence on the LECs is concentrated in family $I\\!I\\!I$,\nwhich begins at ${\\cal{O}}(q^3)$.\n\n\\vspace{-2mm}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.7\\columnwidth,angle=0]{Robilotta_Osaka-03.eps}\n\\caption{Dynamical structure of the two-pion exchange potential.} \n\\label{F3}\n\\end{center}      \n\\end{figure}\n\n\\vspace{-5mm}\n\nAs far as chiral orders of magnitude are concerned, on finds that \nthe various components of the force begin as follows\\cite{HRR}:\n${\\cal{O}}(q^2)\\rightarrow V_{SS}^+,  V_T^+, V_C^-$ and \n${\\cal{O}}(q^3)\\rightarrow V_C^+, V_{LS}^+, V_{LS}^-, V_{SS}^-, V_T^-$,\nwhere the superscripts $(+)$ and $(-)$ refer to terms proportional \nto either the identity or $\\mbox{\\boldmath $\\tau$}^{(1)}\\!\\cdot\\! \\mbox{\\boldmath $\\tau$}^{(2)}$ in isospin space.\nAn interesting feature of these results is that the role played by \nfamily $I\\!I$ is completely irrelevant.\nOn the other hand, family $I$ dominates almost completely the components\n$V_{LS}^+$, $V_T^+$, $V_{SS}^+$ and $V_C^-$, \nwhereas family $I\\!I\\!I$ does the same for $V_C^+$, $V_T^-$and $V_{SS}^-$.\n\n\\vspace{-4mm}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.49\\columnwidth,angle=0]{Robilotta_Osaka-04a.eps}\n\\includegraphics[width=0.49\\columnwidth,angle=0]{Robilotta_Osaka-04b.eps}\n\\vspace{-3mm}\n\\caption{$OPEP$ and $TPEP$ contributions to \nspin-spin (left) and tensor (right) NUisovector components.} \n\\label{F4}\n\\end{center}      \n\\end{figure}\n\n\\vspace{-5mm}\n\nThe relationship between the $OPEP[={\\cal{O}}(q^0)]$ and $TPEP[={\\cal{O}}(q^3)]$ \ncontributions to the $V_{SS}^-$ and $V_T^-$ profile functions is \nshown in Fig. \\ref{F4}, where it is possible to see that the chiral\nhierarchy is respected.\n\n\nIn Fig. \\ref{F5}, the two central components $V_C^-[={\\cal{O}}(q^2)]$ and \n$V_C^+[={\\cal{O}}(q^3)]$ are displayed side by side and two features \nare to be noted.\nThe first one concerns the favorable comparison with the empirical\nArgonne\\cite{Arg} potentials in both cases.\nThe second one is that $|V_C^+| \\sim 10 \\, |V_C^-|$ in  regions of physical\ninterest, defying strongly the predicted chiral hierarchy.\nThis problem will be further discussed in the sequence.\n\n\\vspace{-3mm}\n\n\\begin{figure}[h]\n\\begin{center}\n\\hspace*{-4mm}\n\\includegraphics[width=0.50\\columnwidth,angle=0]{Robilotta_Osaka-05a.eps}\n\\includegraphics[width=0.50\\columnwidth,angle=0]{Robilotta_Osaka-05b.eps}\n\\vspace{-2mm}\n\\caption{Isospin odd (left) and even (right) central components of the \ntwo-pion exchange potential.} \n\\label{F5}\n\\end{center}      \n\\end{figure}\n\n\\vspace{-4mm}\n\nViolations of the chiral hierarchy are also present in the \n{\\em drift potential}\\cite{Rdrift}, which corresponds to kinematical\ncorrections due to the fact that the two-body center of mass is allowed to \ndrift inside a larger system. \nIn terms of Jacobi coordinates, it is represented by the operator\n\\begin{eqnarray}\nV(r)^\\pm  =  \\left. V(r)^\\pm \\right] _{cm} + V_D^\\pm \\, \\Omega_{D}\n\\;\\;\\;\\;\\;\\;\\;\\; \\leftrightarrow \\;\\;\\;\\;\\;\\;\\;\\;\n\\Omega_D = \\frac{1}{4\\sqrt{3}}\\, (\\mbox{\\boldmath $\\sigma$}^{(1)}\\!-\\! \\mbox{\\boldmath $\\sigma$}^{(2)}) \\!\\cdot\\! \\mbox{\\boldmath $r$} \\!\\times \\! \\;,\n(-i \\mbox{\\boldmath $\\nabla$}^{^{\\!\\!\\!\\!\\!\\!\\!\\!^\\leftrightarrow}}_\\rho)\\;.\n\\nonumber\n\\end{eqnarray}\n\n\nThe profile function $V_D^+$ together with $V_{LS}^+$,  are \ndisplayed in Fig. \\ref{F6}.\nDrift corrections begin at ${\\cal{O}}(q^4)$ and, in principle, should be smaller \nthan the spin-orbit terms, which begin at ${\\cal{O}}(q^3)$.\nHowever, in this channel, the hierarchy is again not respected.\n\n\\vspace{20mm}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.7\\columnwidth,angle=0]{Robilotta_Osaka-06.EPS}\n\\vspace{-35mm}\n\\caption{Isospin even drift (full and dotted lines)\nand spin-orbit (dashed line) potentials.} \n\\label{F6}\n\\end{center}      \n\\end{figure}\n\n\n\n\n\\section{THREE-BODY POTENTIAL}\n\\label{s3}\n\nThe leading term in the three-nucleon potential, known as $TPE$-$3NP$,\nhas long range and corresponds to the process shown in fig.1c,\nin which a pion is emitted by one of the nucleons, scattered by a second one, \nand absorbed by the last nucleon.\nIs this case, the intermediate $\\pi N$ amplitude, which is ${\\cal{O}}(q)$ for \nfree pions, becomes ${\\cal{O}}(q^2)$ and the three-body force begins at \n${\\cal{O}}(q^3)$.\nThe first modern version of this component of the force was produced by \nFujita and Miyazawa\\cite{F-M}, its chiral structure has bee much debated\nsince the seventies\\cite{3NP} and, nowadays, a sort o consensus has been \nreached about its form\\cite{Rtokyo}. \nThe leading $TPE$-$3NP$ has a generic structure given by\n\\begin{eqnarray}\n&& V_L(123) = -\\,\\frac{\\mu}{(4\\pi)^2}  \n\\left\\{  \\delta_{ab} \\left[  a \\, \\mu - b \\, \\mu^3 \\, \\mbox{\\boldmath $\\nabla$}_{12} \\!\\cdot\\! \\mbox{\\boldmath $\\nabla$}_{23} \\right]  \n+ d\\, \\mu^3 \\; i\\,\\epsilon_{bac} \\tau_c^{(2)}\\;\ni \\, \\mbox{\\boldmath $\\sigma$}^{(2)} \\!\\cdot\\! \\mbox{\\boldmath $\\nabla$}_{12} \\times \\mbox{\\boldmath $\\nabla$}_{23}\\right\\} \n\\nonumber\\\\\n&& \\;\\;\\;\\;\\; \\times \n\\left[  (g_A \\, \\mu\/2 \\,f_\\pi)\\;\\tau_a^{(1)} \\;\\mbox{\\boldmath $\\sigma$}^{(1)} \\!\\cdot\\! \\mbox{\\boldmath $\\nabla$}_{12} \\right] \\;\n\\left[  (g_A \\, \\mu\/2 \\,f_\\pi)\\;\\tau_b^{(3)} \\;\\mbox{\\boldmath $\\sigma$}^{(3)} \\!\\cdot\\! \\mbox{\\boldmath $\\nabla$}_{23} \\right] \\;\nY(x_{12}) \\; Y(x_{23}) \\;,\n\\nonumber\n\\end{eqnarray}\n\n\\noindent\nwhere $\\mu$ is the pion mass and $a$, $b$ and $d$ are strength parameters,\ndetermined by either LECs or subhtreshold coefficients. \n\nThe evaluation of ${\\cal{O}}(q^4)$ corrections requires the inclusion of single\nloop effects and is associated with a large number of \ndiagrams, which are being calculated by Epelbaum and \ncollaborators\\cite{E3NP}.\nIn order to produce a feeling for the structure of these corrections,\nwe discuss a particular set of processes belonging to the \n$TPE$-$3NP$ class, considered recently\\cite{IR07}.\nFull results involve expressions which are too long and cumbersome\nto be displayed here.\nHowever, their main qualitative features can be summarized in the\nstructure  \n$V(123)=V_L(123)+ [V_{\\delta L}(123) + \\delta V(123)]$, \nwhere $V_L$ is the leading term shown above and the factors within \nsquare brackets are ChPT corrections.\nThe function $V_{\\delta L}$ can be obtained directly from $V_L$, by replacing \n$(a,b,c) \\rightarrow (\\delta a,\\delta b, \\delta c)$, where \nthe $\\delta s$ indicate changes smaller than $10\\%$.\nThis part of the ChPT correction corresponds just to shifts in the \nparameters of the leading component.\nThe term $\\delta V(123)$, on the other hand, represents effects associated \nwith new mathematical functions involving both non-local operators \nand complicated propagators containg loop integrals,\nin place of the Yukawa functions.\nThe strengths  of these new functions are determined by a new set of \nparameters $e_i$, which are also typically about $10\\%$ of the \nleading ones.\n\nIn summary, ChPT gives rise both to small changes in already existing \ncoefficients and to the appearance of many new mathematical structures.\nThe latter are the most interesting ones, since they may be instrumental\nin explaining effects such as the $A_y$ puzzle.\n\n\n\n\\section{THE CHIRAL PICTURE}\n\\label{s4}\n\nChiral symmetry has already been applied to about 20 components \nof nuclear forces, allowing a comprehensive picture to be assessed.\nAccording to ChPT, the various effects begin to appear at different\norders and the predicted hierarchy is displayed in the table below.\n\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular} {|c|ccc|}\n\\hline\nbeginning\t  \t& TWO-BODY & TWO-BODY\t& THREE-BODY \t   \\\\ \n\t\t\t\t& $OPEP$   & $TPEP$\t\t& $TPEP$\t\t   \\\\ \\hline \n${\\cal{O}}(q^0)$\t\t& $V_T^-, V_{SS}^-$\t&&    \t  \t   \\\\[1mm] \\hline\n${\\cal{O}}(q^2)$\t\t& $V_D^-$ & $V_C^-; V_T^+, V_{SS}^+$ &\t   \\\\[1mm]\\hline\n${\\cal{O}}(q^3)$\t\t&& $V_{LS}^-, V_T^-, V_{SS}^-; V_C^+, V_{LS}^+$ \n& $d; a, b$ \t\\\\[1mm] \\hline\n${\\cal{O}}(q^4)$\t\t&& $V_D^-; V_Q^+, V_D^+$ & $ e_i $ \\\\[1mm] \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nIn Ref.\\refcite{HRR}, the relative importance of $O(q^2)$, $O(q^3)$ \nand $O(q^4)$ terms in each component of the $TPEP$-$N\\!NP$ has been studied.\nIn general, convergence at distances of physical interest is \nsatisfactory, except for $V_C^+$, where the ratio between \n${\\cal{O}}(q^4)$ and ${\\cal{O}}(q^3)$ contributions \nis larger than $0.5$ for distances smaller than $2.5$ fm.\n\nAs far as the relative sizes of the various dynamical effects are\nconcerned, one finds strong violations of the predicted hierarchy\nwhen one compares $V_C^+$ with $V_C^-$ and $V_D^+$ with $V_{LS}^+$,\nas discussed above. \nIt is interesting to note that, in both cases,\nthe unexpected enhancements occur in the isoscalar sector. \nThe numerical explanation for this behavior is that some of the LECs \nused in the calculation are large and\ngenerated dynamically  by delta intermediate states.\nHowever, it is also possible that perturbation theory may not apply \nto isoscalar interactions at intermediate distances.\nThis aspect of the problem is explored in the next section.\n\n\n\n\n\n\\section{SCALAR FORM FACTOR}\n\\label{s5}\n\n\n\nThe structure of $V_C^+$ was  scrutinized in Ref.\\refcite{HRR}\nand  found to be heavily dominated by a term of the form\n\\begin{eqnarray}\nV_C^+(r) \\sim -\\, (4\/f_\\pi^2)\\;\\left[  (c_3 - 2c_1) - c_3 \\; \\mbox{\\boldmath $\\nabla$}^2\/2 \\right]  \\;\n\\tilde{\\sigma}_{N_N}(r)\\;,\n\\nonumber\n\\end{eqnarray}\nwhere the $c_i$ are LECs and $\\tilde{\\sigma}_{N_N}$ is the leading contribution \nfrom the pion cloud to the nucleon scalar form factor.\nThis close relationship between $\\tilde{\\sigma}_{N_N}$ and $V_C^+$ indicates \nthat the study of the former can shed light into the properties of the \nlatter.\n\nThe nucleon scalar form factor is defined as \n\\begin{eqnarray}\n\\langle  N(p') | \\!-\\! {\\cal{L}}_{sb}\\, | N(p) \\rangle  = \\sigma_N(t) \\; \\bar u(p')\\; u(p) \\;,\n\\nonumber\n\\end{eqnarray}\n\n\\noindent\nwhere ${\\cal{L}}_{sb}$ is the symmetry breaking lagrangian. \nIt has already been expanded\\cite{BL} up to ${\\cal{O}}(q^4)$ and receives its leading \n${\\cal{O}}(q^2)$ contribution from a tree diagram associated with the LEC $c_1$.\nCorrections at ${\\cal{O}}(q^3)$ and ${\\cal{O}}(q^4)$ are produced by two triangle diagrams, \ninvolving nucleon and delta intermediate states.\nIn configuration space\\cite{sigma}, the scalar form factor is denoted\nby $\\tilde{\\sigma}$ and one writes \n\\begin{eqnarray}\n\\tilde{\\sigma}_N(\\mbox{\\boldmath $r$}) = - 4\\, c_1\\, \\mu^2\\, \\delta^3(\\mbox{\\boldmath $r$}) + \n\\tilde{\\sigma}_{N_N}(r) + \\tilde{\\sigma}_{N_\\Delta}(r) \\;,\n\\nonumber\n\\end{eqnarray}\n\n\\noindent\nwhere $\\tilde{\\sigma}_{N_N}$ and $\\tilde{\\sigma}_{N_\\Delta}$ are the finite-range \ntriangle contributions. \n\n\\begin{figure}[t]\n\\begin{center}\n\\hspace*{-4mm}\n\\includegraphics[width=0.35\\columnwidth,angle=-90]{Robilotta_Osaka-07a.ps}\n\\includegraphics[width=0.35\\columnwidth,angle=-90]{Robilotta_Osaka-07b.ps}\n\\caption{Ratios $\\tilde{\\sigma}_N(r)\/(\\mu^2 f_\\pi^2)=(1-\\cos\\theta)$ (left)\nand $\\tilde{\\sigma}_{N_\\Delta}(r)\/\\tilde{\\sigma}_{N_N}(r)$ (right) as functions of \nthe distance $r$.}\n\\label{F7}\n\\vspace{-3mm}\n\\end{center}      \n\\end{figure}\n\nThe symmetry breaking lagrangian can be expressed in terms of the chiral angle \n$\\theta$ as ${\\cal{L}}_{sb} = f_\\pi^2 \\, \\mu^2 \\,(\\cos\\theta-1)$.\nThe ratio $\\tilde{\\sigma}_N(r)\/(\\mu^2 f_\\pi^2)=(1-\\cos\\theta)$ describes the \ndensity of the $q\\bar{q}$ condensate around the nucleon and is \ndisplayed in Fig.~\\ref{F7}.\nOne notes that it vanishes at large distances and increases monotonically \nas one approaches the center.\nThis means that the function $\\tilde{\\sigma}_N(r)$ becomes meaningless beyond \na critical radius $R$, corresponding to $\\theta = \\pi\/2$,\nsince the physical interpretation of the quark condensate \nrequires the condition $q\\bar{q}>0$.\nIn Ref. \\refcite{sigma}, the condensate was assumed to no longer exist in \nthe region $r<R$ and the $\\pi N$ sigma-term was evaluated using the expression\n\\begin{eqnarray}\n\\sigma_N = \\frac{4}{3} \\pi R^3 \\;f_\\pi^2 \\mu^2\n+ 4\\pi \\int_{R}^\\infty dr\\;r^2\\;\\tilde{\\sigma}_N(\\mbox{\\boldmath $r$})\\;.\n\\nonumber\n\\end{eqnarray}\n\nThis procedure yields 43 MeV$<\\sigma_N<\\;$49 MeV, depending on the value \nadopted for the $\\pi N\\Delta$ coupling constant, in agreement  \nwith the empirical value 45$\\pm 8$ MeV.\nThis picture of the nucleon scalar form factor is sound \nand can be used to gain insight about $V_C^+$.\n\nInspecting Fig. \\ref{F7} (right), one learns that the hierarchy predicted \nby ChPT is subverted for distances smaller than 1.5 fm, since the \n${\\cal{O}}(q^4)$ delta becomes more important than the ${\\cal{O}}(q^3)$ nucleon.\nOn the other hand, the good prediction obtained for the nucleon $\\sigma$-term \n(and also for the $\\Delta$ $\\sigma$-term\\cite{sigma}) \nindicates that the functions $\\tilde{\\sigma}_{N_N}(r)$ and $\\tilde{\\sigma}_{N_\\Delta}(r)$\ncan be trusted up to the critical radius $R\\sim$~0.6~fm.\nJust outside this radius, the chiral angle is close to $\\pi\/2$,\nindicating that the pion cloud is non-perturbative in that region.\nThis picture is supported by Fig. \\ref{F5} (right) since, \nat least up to 1 fm, the prediction for $V_C^+$ agrees well with \nthe Argonne phenomenological potentials.\nThis leads to our main conclusion, namely that the range of validity of \ncalculations based on nucleon and delta intermediate states is wider that \nthat predicted by ChPT.\n\n\n\\section*{Acknowledgments}\n\nIt was a great pleasure participating in the Chiral 2007 meeting\nand I would like to thank the organizers for the very nice conference,\nfor the warm and friendly hospitality, and for supporting my \nstay in Osaka.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nDark matter makes up $80$\\% of the total mass budget in the Universe,\nbut its exact nature remains doggedly elusive. Without direct\ndetection of a single dark matter particle we have to infer its\nproperties from its macroscopic distribution, and any observable that\nmight hint at its underlying particulate nature is seized upon with\nvigour. A plethora of observations, such as large scale\nstructures, lensing and scaling relations, and early fluctuations on\nthe Cosmic Microwave Background, agree surprisingly well with\npredictions of a rather simple cosmological model with a cosmological\nconstant, denoted as $\\Lambda$, and a collisionless cold dark matter\n(CDM) component~\\citep[e.g.][]{Ade:2013zuv}. This {$\\Lambda$CDM} ~paradigm has\nbecome the standard and most widely accepted model today for the\nobserved Universe.\n\n{$\\Lambda$CDM}\\ makes testable predictions at galactic scales that can be\ncontrasted with observations. Among the most fundamental of them is\nfor the structure of dark matter halos, with a density profile\nparameterised by a Navarro-Frenk-White (NFW)\nprofile~\\citep{Navarro_1997}, which at the inner radii scales as a power\nlaw $r^{-\\alpha}$ with $\\alpha\\approx1$. Observations, however, of the\nrotation curves of spiral galaxies, including dwarf and low surface\nbrightness galaxies, often exhibit an inner circular velocity of stars and\ngas that increases more mildly than expected from a CDM\nhalo~\\citep[e.g.\n][]{Flores:1994gz,Moore:1994yx,Persic:1995ru,Kuzio_2008,Oh_2015},\nindicating an inner density profile shallower than the NFW cusp, i.e.,\n$\\alpha<1$. This discrepancy can be further generalised as the mass\ndeficit problem: the CDM halo contains too much dark matter mass in\nthe inner regions than inferred from\nobservations~\\citep{MBK_2011,Ferrero_2012, Papastergis_2015, Papastergis2016, Schneider_2016}.\n\nA more intriguing observation is that the galactic rotation curves exhibit a large diversity.\nIndividual fits to galaxy rotation curves \nspan a spectrum from \ncores $\\alpha\\approx 0$ \\citep[e.g.][]{Cote2000,deBlok_2008,deNaray:2009xj} to cusps \\citep[e.g.\n][]{vandenbosch2000,Swaters_2003, Spekkens2005}, and in the case\nof cored profiles, the central densities can differ by factor of $10$\nfor galaxies inhabiting similar halos~\\citep{deNaray:2009xj}.\nRecently, \\cite{Oman_2015} quantified this diversity in rotation\ncurves by comparing ${V_{2\\,\\rm kpc}}$ for a fixed ${V_{\\rm max}}$, where ${V_{2\\,\\rm kpc}}$ is the\ncircular velocity at $2~{\\rm kpc}$ and ${V_{\\rm max}}$ is the peak circular\nvelocity. For ${V_{\\rm max}}$ in the range of $50\\textup{--}250\\;{\\rm km\\, s^{-1}}$,\nthe scatter in ${V_{2\\,\\rm kpc}}$ is a factor of $3\\textup{-}4$ and consequently\nthe mechanism invoked to generate cored profiles must also accommodate\nthis large variations in rotation curve shapes.\nNotably mass modelling is subject to significant uncertainties,\nespecially at the scale of low mass dwarfs where pressure support\neffects, triaxiality, inclination, hidden bars and other\nirregularities hamper the utility of\ncircular velocity profiles for mass modelling\n\\citep{Hayashi2004,Rhee2004,Pineda_2016,Read2016},\nmaking it hard to assess the precise spread in the rotation curves. In this paper, we take the result reported in Oman et al. as our reference.\n\nThe distribution of rotation curves is clearly too heterogeneous to be \nindexed with a single parameter. For example, several different galaxy formation\nmodels seem able to reproduce the Tully-Fisher \\citep{TullyFisher} relation within\n{$\\Lambda$CDM}\\ \\citep{McCarthy_2012,Brook_2016,\n  DiCintio_2016, Sales_2016, Ferrero_2016, Santos-Santos2016,\n  Katz:2016hyb}  provided\nthe stellar feedback populates dark matter halos with the right\nstellar mass and size \\citep[though see also][]{Pace:2016oim},\nyet despite this global consensus the circular velocity profiles\npredicted by each model differ in detail, with some cases leading to the\nformation of cores \\citep[e.g.][]{Governato_2010,\n  Pontzen_2012, DiCintio_2014,Read:2015sta, Wetzel_2016} whereas other\nsimulations preserve the inner dark matter cusps\n\\citep[e.g.][]{Sawala:2014baa, Vogelsberger_2014,Schaller2015}. The\ndisagreement may arise from the inclusion of different physical\nprocesses, different numerical implementation or may even vary with\nstar formation histories~\\citep[e.g.][]{Onorbe2015, Tollet2016}.\n\nIn particular, the feedback model applied by \\citet{Oman_2015} appears\nunable to explain the scatter in observations within the $\\Lambda$CDM\nframework. On the other hand, \\citet{Brook2015b} finds good agreement\nbetween simulations and observations when looking at the scatter in\nthe ratio between the circular velocity at $1$ kpc and ${V_{\\rm max}}$.\nSimilarly \\citet{Read2016} has also found consistency between baryon\ninduced cores within $\\Lambda$CDM and the shape of observed rotation\ncurves. It is unclear, however, that this baryonic solution would hold if the\nsize of the cores measured in observations is $\\gtrsim 1$~kpc\n\\citep[see for instance][]{Tollet2016}. In fact, this may represent a\nserious limitation of this solution. For example IC~2574\nhas an inferred cored inner halo which extends for $8 \\; \\rm kpc$,\nwell beyond the radius of the stars, with a stellar half-mass radius\nof $5\\; \\rm kpc$. This type of object is a challenge to the hypothesis\nof feedback generated cores since, by construction, the extent of the\ncores in such scenarios is limited to the region where stars form and deposit their\nenergy~\\citep{DiCintio_2014}.\n\nAn alternative solution is to consider cores formed out of\nself-interacting dark matter (SIDM), which we explore in this paper.\nThe SIDM model retains important features of the CDM model\nincluding the distribution of halo concentrations as a function of mass  \\citep{Rocha:2012jg}. \nSIDM differs, however, in that scattering between dark matter particles leads to heat transfer and the generation of dense cores in the\ninner regions of halos ~\\citep{Spergel:1999mh,Firmani:2000ce}.\nRecent\nhigh-resolution N-body simulations have shown that the\nself-interaction cross-section per unit mass $\\som\\sim1~{\\rm cm^2\\, g^{-1}}$\nis required to have core densities that are preferred by galaxy\nobservations\n\\citep{Vogelsberger_2012,Zavala:2012us,Rocha:2012jg,Elbert:2014bma}.\n\\cite{Kaplinghat_2016} finds $\\som\\approx1\\textup{--}3~{\\rm cm^2\\, g^{-1}}$\nby directly fitting the rotation curves of dwarf galaxies. On the\nother hand, there are various constraints on $\\som$, including merging\nclusters~\\citep{Randall_2008,Kahlhoefer:2013dca,Robertson_2016,Kim:2016ujt},\nshapes of elliptical galaxies and galaxy\nclusters~\\citep{MiraldaEscude_2002,Feng:2009hw,Peter_2013}, core sizes of\nclusters~\\citep{Yoshida_2000,Rocha:2012jg,Kaplinghat_2016,Elbert_2016},\nand survival of dwarf halos from evaporation~\\citep{Gnedin_2001}.\n\nAmong these constraints the strongest limit is $\\som \\lesssim 0.1\\;{\\rm cm^2\\, g^{-1}}$ in\ngalaxy clusters~\\citep{Yoshida_2000,Kaplinghat_2016,Elbert_2016},\nwhere the relative dark matter velocity $v\\sim1500~{\\rm km~s^{-1}}$.\nAs such, the self-interaction cross-section should have a mild velocity\ndependence, which can be naturally realised in a class of hidden\nsector dark matter models with a light force\nmediator~\\citep[e.g.][]{Feng:2009mn,Feng:2009hw,Buckley:2009in,Loeb:2010gj,Tulin:2013teo,Tulin:2012wi,Boddy:2014yra,Boddy:2016bbu}.\nA velocity-dependent SIDM model, based a Yukawa potential, has been\nimplemented in zoom-in N-body\nsimulations~\\citep{Vogelsberger_2012,Vogelsberger:2015gpr}, and \nfurther generalised in the ETHOS (Effective Theory of\nStructure Formation) framework, mapping\nunderlying particle physics parameters to astrophysical\nobservables~\\citep{Vogelsberger:2015gpr,Cyr-Racine:2015ihg}. \n\nThis approach has several distinct features that help explain the\nrotation curve diversity. Firstly, the\nscatter in the halo concentration leads to variations in core density\ndirectly~\\citep{Kaplinghat_2016}. Secondly, the self-interactions\nthermalise the inner halo with its central dark matter density\n\\emph{dependent} upon the baryonic extent. In dark matter dominated\ngalaxies, a dense core forms whose density is determined by the\nself-interaction cross-section and the halo mass. With a cored profile, the features in\nthe baryon distribution are more prominently reflected in the rotation\ncurves. In contrast, in galaxies where the baryon component dominates\nthe potential, the thermalisation process lead to a denser central\ncore with sizes influenced by the baryonic scale\nradius~\\citep{Kaplinghat_2014}. Analytical calculations to address\nthe diversity problem have been carried out by \\citet{Kamada2016} using isothermal\napproximations to the effects of SIDM with a baryonic potential in\norder to construct fits for a diverse range of individual spiral\ngalaxies.\n\nIn this work we explore the combined effects of SIDM and baryonic\npotentials by performing a series of numerical experiments.\nOur focus is on global trends but we include a\npair of individual fits to test our results on the strongest outliers\nfrom observations. We sample a realistic range of concentrations and\nhalo masses taken from cosmological simulations together with observed\ntrends in the mass-size relation of galaxies. Our numerical approach\nand the analytical one presented in~\\citet{Kamada2016} complement each\nother in addressing the diversity problem in the SIDM model. The\nstructure of this paper is as follows. In Section~\\ref{sec:sims} we\nintroduce in detail our simulations. Sections.~\\ref{sec:V2_Vmax_dm} and\n\\ref{sec:V2_Vmax_bar} explore the diversity problem for a large sample\nof halos with ${V_{\\rm max}}=[30$-$250]$~km~s$^{-1}$ with the latter including the effect\nof baryons. In Sec.~\\ref{sec:obs_curve_sims} we\nconfirm that we can find rotation curves that are reasonable matches\nto a pair of extreme (one cusp-like and one core-like) observed\nrotation curves. In Sec.~\\ref{sec:discussion} we summarise and\nconclude.\n\n\n\n\\section{Numerical simulations}\\label{sec:sims}\n\nWe use a combination of N-body simulations and analytical tools to\nstudy the mass profile of galaxies within the SIDM framework including\nthe effects driven by their baryons. The main focus of this work is on\ndisk dominated galaxies, which we model with a disk\ncomponent embedded within a massive and more extended dark matter halo\n(i.e. no bulge). For the N-body simulations we must produce\ndiscretised initial conditions which we describe for the halo in\nSection~\\ref{sec:bar_ic} in addition to the baryonic potential in\n\\ref{sec:diskpot}. We evolve these initial conditions with and without\nthe effect of a baryonic disk and also with and without\nself-interaction terms using the simulation code described in\nSection~\\ref{sec:nbody}.\n\n\n\\subsection{N-body code}\\label{sec:nbody}\nWe use the {\\sc Arepo} code \\citet{Springel_2010} with the modifications of \n\\citet{Vogelsberger_2012} and \\citet{Vogelsberger:2015gpr} to account for dark matter self-interactions.\nThis methodology uses a Monte-Carlo approach\nto model the scattering of dark matter via the probabilistic\nscattering of the macroscopic dark matter particles in the simulation,\nwhere the density distribution due to an individual particle is\nsmoothed over some kernel which extends over the $k$-nearest\nneighbours. Such an explicit method requires a time step limit $\\Delta\nt_{\\rm sidm}$ such that only $\\lesssim 1$ scattering can occur per\nparticle per timestep. Notably in the cusp region of an NFW halo (i.e.\n$\\rho \\sim r^{-1}$ as $r\\to 0$) as one moves towards the centre one\nfinds that $\\Delta t_{\\rm sidm} \\propto \\frac{1}{\\rho \\sigma} \\sim\nr^{\\frac{1}{2}}$ (with $\\sigma$ the velocity dispersion),\n i.e the centre of the halo is so dynamically cold\nand dense that the mean-free path is below the particle separation. To\navoid such small time steps one additionally needs to limit them to a\nsmall fraction of the acceleration time step.\n\n\nWe have modified this code to include a baryonic potential of a\nMiyamoto-Nagai (hereafter MN) disk (Section~\\ref{sec:diskpot}) and\nused a constant self-interaction cross-section for scattering, which\nis a good approximation in the dark matter low velocity limit, e.g.,\n$v\\lesssim 400 \\; {\\rm km\\, s^{-1}}$ as shown in Fig.~1 of \\citet{Kaplinghat_2016}.\n\nIn our calculations we assume a Hubble constant\n$H_0 = 70\\; {\\rm km\\, s^{-1}}\\,{\\rm Mpc}^{-1}$.  Our simulation period is fixed at\n$10$ Gyr, slightly shorter than the Hubble time\n($H_0^{-1} \\approx 13.96$~Gyr) in order to account, in an approximate\nway, for the assembly time of such galactic disks. It should be\nemphasised that isolated simulations with a static disk potential is\nwell-motivated in the SIDM model; for $\\som\\gtrsim1~{\\rm cm^2\/g}$,\ndark matter self-interactions occur multiple times in the inner halo\nover the age of galaxies, which drive thermalisation of the inner halo\nin the presence of the stellar disk, and as such the final inner SIDM\nhalo profile is more robust to changes in the formation history (given\na known baryon distribution) than that of CDM.\n\n\\subsection{Isolated halo initial conditions}\\label{sec:bar_ic}\n\nFor each of our galaxies, we generate a large set of compound galaxies\nconsisting of an exponential disk embedded in a dark matter halo using\nthe code {\\sc MakeDisk} \\citep{Hernquist_1993, Springel_2005}. The\nsystems are set up in near equilibrium by requiring a joint solution\nto the combined phase space of both disk and halo component.  In\nparticular, the density profile of the halo is set up with a\n\\citet{Hernquist_1990} density profile:\n\\begin{equation}\n\\rho_{\\rm dm}(r) = \\frac{M_{\\rm dm} r_{\\rm hq}}{2\\pi r \\left(r+r_{\\rm hq}\\right)^3} \\, ,\n\\end{equation}\nwith $r_{\\rm hq}$ the scale radius of the halo. \\citet{Hernquist_1990} profiles\nshow a $r^{-1}$ slope in the inner density profile and $r^{-4}$ in the\noutskirts, in close agreement with the inner\/outer slopes of $-1$ and\n$-3$, respectively, of the cosmologically inspired NFW\nprofiles \\citep{Navarro_1997}. The Hernquist profile has the extra\nadvantage of having an analytic expression for its distribution\nfunction, which facilitates the process of setting up the initial\nconditions.\n\n\nHernquist halos can be ``scaled'' to match a given NFW\nprofile quite closely and we therefore report our results in terms of\nvirial mass\\footnote{Virial quantities are defined at the virial\n  radius $r_{200}$, where the enclosed density is $200$ times the critical\n  density of the Universe. } $M_{200}$ and concentration parameter $c$\ncorresponding to NFW halos. Finding the equivalent \nhalo (by matching the dark matter mass within $r_{200}$ and the \nasymptotic density profile as $r\\to0$) gives the relation between parameters \n\\begin{eqnarray}\nM_{\\rm dm} &=& M_{200} - {M_{\\rm d}} \\\\\nr_{\\rm hq} &=& \\left(\\frac{ G M_{200}}{100 H_0^2}\\right)^{1\/3} c^{-1} \\sqrt{2 \\ln (1 + c) - \\frac{2c}{1+c}} \\label{eq:rhq}\n\\end{eqnarray}\ndependent upon the baryonic component ${M_{\\rm d}}$ and the redshift\nzero Hubble constant $H_0$.\n\nThe matching of the internal regions of our Hernquist profile to the\ndesired NFW is less accurate as we move close to the scale radius of\nthe NFW $r_{\\rm s}$, with the density of a Hernquist halo falling off more\nsteeply beyond that point. In terms of the velocity dispersion\nprofile, the Hernquist halo has a peak velocity dispersion shell that \nis more compact and this velocity dispersion is lower than the NFW value.\nThis can affect the heat exchange, which stops once the\nisothermal core has been established. As a consequence of the lower\nvelocity dispersion peak, Hernquist profiles can be more prone to the\nso called ``core collapse'' phenomena, or the runaway collapse in the\ninner regions that ultimately reverses the process of mass scattering\naway from the core and condenses instead more mass in the inner\nregions. This process would be exacerbated by the presence of baryons\nand as $\\som$ increases.\n\nWe have extensively tested this scenario and we include some comparisons\nin Appendix~\\ref{sec:NFW_vs_HQ}. We find \nthat it makes a negligible difference in\nmost of our simulations except for extreme cases with concentrated\nhalos, however as we shall see later it is the \\emph{low} concentration\nhalos where most of the interesting discrepancy between CDM and SIDM predictions lie, i.e. \nthese halos are expected to host the low central (dark matter) \ndensity galaxies that are most challenging to CDM. As such our approach of\nusing Hernquist halos is conservative, as the effect will be small and any\ndeviations will cause us to underestimate our conclusions.\n\nFor the simulations showed in Sec.~\\ref{sec:results}, we set up our\nhalos using $n_p=10^5$ particles to model the dark matter\ndistribution, which results in a spread for the mass per particle\n${\\rm m_p}=8.97\\times 10^4$--$2.66\\times 10^7 \\; \\rm M_\\odot$ according to the mass of the simulated halo.\nWe use a gravitational softening length $\\epsilon=50$~pc, \nand we have tested for numerical convergence using a\nhalo with $M_{200}=1.1\\times 10^{11} \\; \\rm M_\\odot$ and increasing\/decreasing the\nparticle number by a factor $10$ with respect to our \nfiducial runs and find agreement within $10\\%$ in the\n(dark matter only) circular velocity profiles (and by proxy mass) for all radii outside $1.0$~kpc. \n\n\\subsection{Disk potential}\\label{sec:diskpot}\nThe baryonic component of our galaxies is modelled by a fixed disk\npotential, which is not only computationally cheaper, but also allows\nus to maintain full control on the exact distribution of the baryons.\nThis is particularly important for dwarf galaxies which undergo hundreds\nof orbits over a Hubble time and are thus particularly sensitive to \n(unphysical) discretisation-induced instabilities. \nWe therefore strip the disk particles created\nin our initial conditions, and use instead an extra component in the\ngravitational force that accounts for the removed disk particles. \nThis approach is not fully self-consistent since it\nignores the back-reaction of the halo onto the baryons, but it provides\na useful tool to explore the evolution of halo potentials in SIDM in a\nset of controlled experiments where the {\\it final} baryonic\ndistribution is known. \n\nWe implemented a static potential in {\\sc Arepo} following a\n\\citet{Miyamoto_Nagai_1975} disk:\n\\begin{equation}\n\\Phi_{\\rm MN}(R,z) = \\frac{-G{M_{\\rm d}}}{\\sqrt{R^2+\\left({R_{\\rm d}}+\\sqrt{{z_{\\rm d}}^2+z^2}\\right)^2}} \\, ,\n\\end{equation}\nfor a disk of mass ${M_{\\rm d}}$ and scale radius ${R_{\\rm d}}$ and we keep the disk relatively\nthick in all cases by setting ${z_{\\rm d}}\/{R_{\\rm d}}=0.3$ (appropriate to model\ndwarf irregular galaxies). Note that the half-mass radius of the flat \n(${z_{\\rm d}}=0$) MN disk is $\\sqrt{3}{R_{\\rm d}}$ which is very close \nto the ratio of half-mass to scale radius of an exponential disk of \n$1.678$ and so unless ones disk is extremely thick, the scale radius \nof an exponential fit will be within a few percent\\footnote{The\n  differences are much more pronounced in the tails of the\n  distribution, indeed one cannot choose to correspond a flat\n  exponential to a MN density profile via for example the\n  mass weighted root-mean square radii since for the latter that is a\n  divergent quantity.}. From here on we will use ${R_{\\rm d}}$ interchangeably\nto refer to either the scale length of our baryonic disks in simulations\n(the MN scale radius) or the exponential disks associated with observed\nlate-type dwarf galaxies.\n\nNotice that the initial conditions are created assuming an exponential\nprofile while the fixed potential uses an MN model for the\ndisk. We have tested that this small inconsistency does not affect our\nresults.  Due to the diffusive nature of SIDM the resulting deviations\ndue to non-equilibria are of the order of a few percent and quickly\ndamped. Alternative approaches include that of \\citet{Elbert_2016}\nwhere the MN disk is `grown' adiabatically (compared to the orbital\ntimescale) from an initially halo-only setup. We have tested the extreme case \nof inserting the fixed potential instantaneously and find that after 10~Gyr \nevolution in SIDM at 2~kpc this makes a relative difference in the ${V_{2\\,\\rm kpc}}$ \nof $-0.2\\;{\\rm km\\, s^{-1}}$ or $0.6\\%$. Our method, being closer to\nequilibrium, is expected to be lower than these bounds. \n\nWe run the simulations for $10$ Gyr and compute the circular velocity\nprofiles. For the CDM cases this is in the most part a straightforward\nnumerical test since the initial conditions are created in equilibrium\nand we do not see significant evolution of the mass distribution with\ntime. For the most extended disks (${R_{\\rm d}}=6\\; \\rm kpc$), however, the\ncentre-of-mass is not well `tied' to the centre of the (shallow)\nbaryonic potential, and the centre-of-mass can `drift' on the order of\na kpc over 10~Gyr (i.e. an average of $0.1\\;{\\rm km\\, s^{-1}}$) from the centre of the\nbaryonic disk potential, making the assessment of the mass\ndistribution within 2~kpc problematic.  This evolution is expected due\nto discretisation of ICs and asymmetry in tree-based gravitational\nmethods, and is relatively hard to suppress even with a tight timestep\ncriteria. This process can of course also occur to the SIDM halos,\nalthough it is somewhat suppressed since their profiles have less\ncompact centres.  Since evolving CDM halos in extreme cases is\nsomewhat tangential to the purpose of this paper, however, we have\ntested the CDM evolution on the more compact (${R_{\\rm d}}=0.5$ and\n$1.5\\; \\rm kpc$) halos, whose ${V_{2\\,\\rm kpc}}$ evolve by an average of $-7.7\\%$\nover $10$~Gyr. For the ${R_{\\rm d}}=6\\; \\rm kpc$ CDM halos we will use the\ninitial conditions, as we believe those to be more accurate.\n\n\n\n\\section{Results}\n\\label{sec:results}\n\n\n\\subsection{Circular velocity profiles in dark matter only halos}\n\\label{sec:V2_Vmax_dm}\n\nTo illustrate the effect of dark matter self-interactions on the density profile of a dwarf halo without the influence of baryons, we use the analytical Jeans method proposed in~\\citep{Kaplinghat_2014, Kaplinghat_2016} and tested against simulations in~\\citep{Rocha:2012jg,Elbert:2014bma}. We have further checked that its accuracy is within $10\\textup{--}20\\%$ compared to the results from our N-body code~\\citep{Vogelsberger_2012}. The details of the formalism can be found in Appendix~\\ref{sec:jeans}.\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{ccccccc}\n\\hline\n$V_{200}$ & $M_{200}$ &  concentration & $r_{\\rm s}$ (kpc)  & $\\rho_{\\rm s}$ ($10^6 \\; \\rm M_\\odot \\; kpc^{-3}$)$^\\star$ & $\\rm {M_{\\rm d}}$ & ${\\rm med.} \\; {R_{\\rm d}}$\\\\\n(${\\rm km\\, s^{-1}}$) & ($\\rm M_\\odot$) & ($-2\\sigma$,median,$+2\\sigma$) & $-2\\sigma$,med.,$+2\\sigma$ & $-2\\sigma$,med.,$+2\\sigma$ & ($\\rm M_\\odot $) & ($\\rm kpc^{\\star \\star}$)\\\\\n\\hline\n$15$ & $1.12\\times 10^{9}$ & $9.99$, $15.60$, $24.38$ & $2.15$, $1.37$, $0.88$ & $6.07$, $18.42$, $57.80$ & $\\dagger$ & $\\dagger$ \\\\\n$30$ & $8.97\\times 10^{9}$ & $8.32$, $12.99$, $20.30$ & $5.15$, $3.30$, $2.11$ & $3.89$, $11.63$, $36.04$ & $  1.71\\times 10^{8}$ & ($1.93$) \\\\\n$50$ & $4.15\\times 10^{10}$ & $7.27$, $11.36$, $17.74$ & $9.83$, $6.29$, $4.03$ & $2.82$, $8.32$, $25.52$ & $  8.84\\times 10^{8}$ & ($2.53$) \\\\\n$70$ & $1.14\\times 10^{11}$ & $6.65$, $10.39$, $16.23$ & $15.04$, $9.62$, $6.16$ & $2.29$, $6.69$, $20.36$ & $  2.79\\times 10^{9}$ & ($3.03$) \\\\\n$100$ & $3.32\\times 10^{11}$ & $6.05$, $9.46$, $14.78$ & $23.60$, $15.11$, $9.67$ & $1.84$, $5.31$, $16.05$ & $  1.09\\times 10^{10}$ & ($3.60$) \\\\\n$150$ & $1.12\\times 10^{12}$ & $5.44$, $8.50$, $13.28$ & $39.41$, $25.22$, $16.14$ & $1.43$, $4.10$, $12.27$ & $  3.72\\times 10^{10}$ & ($4.09$) \\\\\n$200$ & $2.66\\times 10^{12}$ & $5.04$, $7.88$, $12.30$ & $56.69$, $36.28$, $23.22$ & $1.20$, $3.42$, $10.15$ & $  6.22\\times 10^{10}$ & ($4.53$) \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the halos used in our diversity analysis. $V_{200}$ and $M_{200}$, concentrations with $\\pm 2 \\sigma$ relative to the median relation from \\citet{Ludlow_2014}, and in NFW parameters we give the scale radius $r_{\\rm s}$ and density $\\rho_{\\rm s}$ for this range of concentrations. For the simulations where we include a baryonic disk, $\\rm M_d$ refers to the total baryonic disk mass.\\newline\n  ${}^\\star$ NFW density $\\rho_{\\rm s}$ is computed in the dark matter-only halo case. For the simulations with baryonic disks we fix $M_{200}$ but add a baryonic component, and thus $\\rho_{\\rm s}$ is multiplied by the dark fraction $M_{\\rm dm}\/M_{200}$.\\newline\n  ${}^{\\star\\star}$ We quote median baryonic disk radii only for reference, since we attempt to span the distribution with $0.5$-$6\\;\\rm kpc$ as described in the main text.\\newline\n${}^\\dagger$ This halo was simulated in DM-only.}\n\\label{table:bar_param}\n\\end{table*}\n\n\nOur suite of halos aims to span the `dwarf' galaxy distribution with 7 halos of $V_{200}$ from $15$-$200~{\\rm km\\, s^{-1}}$, the upper limit being near Milky Way-like (where velocity dependent corrections may play some role) down to dwarfs so baryon-poor that we expect negligible deviations from the DM-only results. To encompass the diversity in concentrations we sample $\\pm 2\\sigma$ deviations from the cosmological\nmass-concentration relation presented in \\citet{Ludlow_2014}, whose median\nis well approximated by the linear fit\n\\begin{equation}\\label{eq:linear_ludlow}\nc = 10.51 \\times \\left(\\frac{M_{200}}{10^{11} \\; \\rm M_\\odot}\\right)^{-0.088} \\, ,\n\\end{equation}\nand at each halo mass the distribution is approximately log-normal where $1\\sigma$ corresponds to a ratio of $\\approx 1.25$. The corresponding halo mass ($M_{200}$) and NFW $r_{\\rm s}$ and $\\rho_{\\rm s}$ are given for the \n$70\\; {\\rm km\\, s^{-1}}\\, \\rm Mpc^{-1}$ cosmology in Table~\\ref{table:bar_param}.\n\n\n\\begin{figure*}\n\\includegraphics[width=\\columnwidth]{figures\/dens_vs_rad_70kms.pdf}\n\\includegraphics[width=\\columnwidth]{figures\/V2_Vmax_sig2_sig5_sig10_jeans_sigma_regions.pdf}\n\\caption{\\emph{Left panel:} core formation in a dark matter only halo (NFW) with different\n  $\\som$. Dark matter density profile evolution for SIDM over $10$~Gyr\n  in \\emph{blue}, \\emph{magenta} and \\emph{green} for $\\som=2$, $5$\n  and $10\\; {\\rm cm^2\\, g^{-1}}$ respectively, with CDM ($\\som=0\\; {\\rm cm^2\\, g^{-1}}$) in\n  \\emph{black dashed}. \\emph{Vertical dotted lines} indicate $r_1$ (the radius of unit scattering). \\emph{Right panel:} effects of SIDM on the ${V_{2\\,\\rm kpc}}$ vs ${V_{\\rm max}}$\n  relation compared to CDM for dark matter only\n  halos. \\emph{Black line} is the CDM case, \\emph{blue} the\n  $\\som=2\\; {\\rm cm^2\\, g^{-1}}$ where the shaded regions denote $\\pm 1 \\, \\sigma$\n  scatter (due to halo concentrations; we have interpolated the values from the $\\pm 2\\sigma$ runs in Table~\\ref{table:bar_param}). \\emph{Magenta} and\n  \\emph{green} denote $\\som=5\\; {\\rm cm^2\\, g^{-1}}$ and $10\\; {\\rm cm^2\\, g^{-1}}$\n  respectively. \\emph{Green points} indicate observed values collated in \\citet{Oman_2015}~Table~1, from which we also include the mass deficits, $5\\times 10^8$ and $10^9${$\\rm M_\\odot$} and their effect on ${V_{2\\,\\rm kpc}}$ in \n\\emph{blue shaded regions}.}\n\\label{fig:dm_only}\n\\end{figure*}\n\n\nThe left panel\nof Fig.~\\ref{fig:dm_only} shows the density profiles of a halo with\n$V_{200}=70 \\; {\\rm km\\, s^{-1}}$\n($M_{200}=1.1\\times 10^{11} \\; \\rm M_\\odot$) and a concentration\n$c=6.65$ after $10$~Gyr evolution with CDM ($\\som=0\\; {\\rm cm^2\\, g^{-1}}$, black\ndashed line) vs. SIDM with $\\som=2$, $5$ and $10\\; {\\rm cm^2\\, g^{-1}}$ (see solid\nlines according to labels). \nThe radius of unit scattering, $r_1$ (see also Appendix~\\ref{sec:jeans}) outside of which the halo is undisturbed is marked for the three cases. This radius grows with cross-section, and in the interior we see all\nthree cross-sections produce cores for this halo that extend beyond $2$~kpc.\n\n\nFig.~\\ref{fig:dm_only} (right panel) shows the circular velocity of the SIDM halo profiles evaluated at $2~{\\rm kpc}$ as a function of $V_{\\rm max}$. For the comparison, we also plot the corresponding CDM one with the\nmass-concentration relation from~\\citet{Ludlow_2014} and the range spanned by a compilation of the observed rotation curves in galaxies taken from~\\citet{Oman_2015}. It is clear that ${V_{2\\,\\rm kpc}}$ increases steadily with\n$V_{\\rm max}$ in the case with CDM, although the relation stays below the $1:1$ line. In contrast, SIDM predicts a much shallower relation. ${V_{2\\,\\rm kpc}}$ is almost independent of ${V_{\\rm max}}$ in the range explored, and the median ${V_{2\\,\\rm kpc}}$\nvalue at a given ${V_{\\rm max}}$ depends mildly on cross-section.  For low mass halos (low circular velocity),\n${V_{\\rm max}}$ tends to converge to near ${V_{2\\,\\rm kpc}}$, since the objects are \nsmall and the core size becomes much less than $2~{\\rm kpc}$ and ${V_{\\rm max}}$ is measured near $2\\, \\rm kpc$. It is interesting to note this is a common feature in CDM as well as SIDM halos. Shaded regions indicate\n$\\pm 1 \\sigma$ in concentration (we interpolate from the $\\pm 2 \\sigma$).\n\nThe range of ${V_{2\\,\\rm kpc}}$ and ${V_{\\rm max}}$ statistics of the galaxy rotation curve distribution is shown in Fig.~\\ref{fig:dm_only} \\citep[taken from][]{Oman_2015}.\nThe scatter found in the data is significantly\nlarger than derived purely from that of concentration variations (in either CDM or SIDM). \nA large fraction of the observed ${V_{2\\,\\rm kpc}}$ scatter\nbelow the relation expected for CDM, suggesting that such objects have\na lower dark matter density in the inner regions than predicted by the\nmodel. Instead, self-interactions seem to have the opposite\nbehaviour, providing a better  match to these low density objects but\npredicting too low a ${V_{2\\,\\rm kpc}}$ to explain most of the data. These\ncalculations, however, ignore the contribution of the baryons, and as\nwe argue below, the mass and radial extent of the gas and stars in\ngalaxies may significantly change this prediction to bring SIDM in\ncloser agreement with observations. \n\n\n\\begin{figure*}\n  \\includegraphics[width=2\\columnwidth]{figures\/4panel_V200_70_cc-2.pdf}\n  \\caption{\\emph{Left three panels:} Dark matter contribution to the circular velocity profile \n    for a dwarf galaxy ($V_{200}=70\\; {\\rm km\\, s^{-1}}$, $c=-2\\sigma$)\n    with a fixed baryonic disk component with scale radii\n    $0.5$, $1.5$ and $6\\; \\rm kpc$ from left to right respectively,\n    simulated for 10~Gyr with a cross-section $\\som=2 \\;{\\rm cm^2\\, g^{-1}}$.\n    \\emph{Red lines}\n    indicate the baryonic contribution, \\emph{grey} the\n    dark matter (SIDM), and \\emph{thick solid black} the total. For\n    comparison, the \\emph{thick dashed black} lines indicate the CDM\n    total. \\emph{Right panel} combines the three profiles, with the CDM\n    comparisons in \\emph{grey dashed lines}.}\n\\label{fig:vel_vs_rd}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=\\columnwidth]{figures\/V2_Vmax_sig2_cc-2_0_2.pdf}\n\\includegraphics[width=\\columnwidth]{figures\/V2_Vmax_sig2_likelihood_regions_shen_2sigma.pdf}\n\\caption{N-body simulations in the presence of galactic disks. \\emph{Left panel} shows\n  the individual simulations with disk scale lengths ${R_{\\rm d}}=0.5$, $1.5$ and 6~kpc in \\emph{stars}, \\emph{squares} and \\emph{triangles} respectively, where each is repeated for the three different\n  $-2\\sigma$,  median, and $+2\\sigma$ concentrations and SIDM in \\emph{solid magenta} and CDM in \\emph{open blue symbols}. \\emph{Dashed black line} indicates the CDM relation without baryons, and \\emph{solid black} the corresponding SIDM (with $\\pm 1\\sigma$ shading). \n\\emph{Right panel} shows 2-$\\sigma$ likelihood contours about the median ${R_{\\rm d}}$ and concentration (\\emph{solid lines}) for CDM in the presence of a baryonic disk in \\emph{blue} and\n  SIDM in \\emph{magenta}, inferred from the fits in\n  Eqns.~(\\ref{eq:Shen_size})~and~(\\ref{eq:Shen_size_scatter}), (see also main text). \\emph{Green points} indicate observed values collated in Oman 2015 Table~1.}\n\\label{fig:SIDM_bar_distrib}\n\\end{figure*}\n\n\\subsection{Circular velocity profiles in the presence of baryons}\\label{sec:V2_Vmax_bar}\n\nFor collisionless dark matter halos, the presence of baryons may\nchange the shape  of the {\\it total} circular velocity profile by\nadding the contribution from the gas and the stars. This contribution\nmay or may not change the overall profile, depending on the relative\nfraction of baryonic mass and also the spatial distribution of these\nbaryons\\footnote{Notice that baryons could\nadditionally cause the dark halo to contract \\citep{Blumenthal_1986}, but given our\ninitial condition set-up, the halos will be created already in equilibrium\nwith the baryonic potential}. If gas and stars are very centrally\nconcentrated, they will dominate the contribution to the circular\nvelocity modifying the profile to rise more steeply than the original\nNFW halo. On the other hand, if the baryons are fairly extended, their\ncontribution could be sub-dominant to the dark matter, in which case\nthe total velocity profile including baryons does not deviate\nsignificantly from that of the original NFW halo. \n\nFig.~\\ref{fig:vel_vs_rd} (leftmost three panels) illustrates this\npoint using the N-body simulations set up described in\nSec.~\\ref{sec:sims}. For a fixed halo with $V_{200}=70\\; {\\rm km\\, s^{-1}}$, \n${M_{\\rm d}} = 2.8\\times 10^9$~{$\\rm M_\\odot$}\\ (according to abundance matching\nrelations from \\citet{Behroozi_2013}) and a concentration $-2\\sigma$ below the\nmedian.\nWe plot the equivalent  `circular velocity' inferred for a spherically\naveraged mass distribution\n\\begin{equation}\\label{eq:vcirc}\nV_{\\rm cir}^2 = G M(r)\/r = V_{\\rm cir, dm}^2 + V_{\\rm cir, bar}^2\n\\end{equation}\nwhich is an approximation for non-spherical distributions\n(for a pure thin MN disk without a halo these \n deviations are at maximum $\\approx 15\\%$).\nWe assume that\nthe baryons distribute in a MN disk with scale length\nequivalent ${R_{\\rm d}} = 0.5$, $1.5$ and $6\\; \\rm kpc$ (left to right). The\nlong-dashed lines indicate the final profiles including the baryons\nfor CDM, which show different shapes according to ${R_{\\rm d}}$, with the\ndifferences tracing back to the contribution of the baryons in each\ncase.\n\n\nAs expected, a more concentrated baryonic distribution (small ${R_{\\rm d}}$)\nshows a steep raise of the circular velocity curve, whereas larger\n${R_{\\rm d}}$ values grow more gently. The case with ${R_{\\rm d}}=0.5$~kpc results in\na declining circular velocity profile which might not be the typical\ngalaxy included in studies of rotation curves (probably representing a\nbulge dominated object instead). However, we include this case for\nillustration of the possible extremes and how CDM and SIDM would react to\nthe presence of such a compact galaxy. \n\n\n\nThe same exercise of considering a fixed halo and baryonic content\nbut changing the spatial distribution can be done assuming the SIDM\nscenario. The three left panels in Fig.~\\ref{fig:vel_vs_rd} show the\nexpected total profiles assuming a self-interaction cross-section\n$\\som=2 \\; {\\rm cm^2\\, g^{-1}}$ (see solid black lines). The dark matter component still\nshows a cored profile due to the self-interactions (see thin grey\nline) but this is obscured in the \\emph{total} velocity profile. In fact,\nthe inclusion of baryons creates a fair amount of {\\it diversity} in\nthe shapes of the circular velocity profiles for CDM and SIDM (see\nright panel in the same figure). For example, if we measure ${V_{2\\,\\rm kpc}}$ in\nthese halos (vertical red shaded region), CDM covers a range $\\approx\n45-83$ ${\\rm km\\, s^{-1}}$ which is comparable to that spanned by SIDM, ${V_{2\\,\\rm kpc}}\n\\approx 30-75$ ${\\rm km\\, s^{-1}}$. Encouragingly, the effect of baryons in the case\nwith self-interactions creates a wide range of (total) rotation curves but\nalso maintains a lower central density (in the cases where baryons are\nextended), allowing better accommodation of the low ${V_{2\\,\\rm kpc}}$ values\nthat are occur in observations (see right panel\nFig.~\\ref{fig:dm_only}) and for which CDM alone has no explanation.\n\n\n\n\n\nWe generalise this by simulating all halos introduced in\nSec.~\\ref{sec:V2_Vmax_dm} but including a baryonic disk. The mass of the\ndisk is the sum of a stellar and gaseous component, with the first set\nby the abundance matching relation between $M_*$ and $M_{200}$\nintroduced in \\citet{Behroozi_2013} and the gas mass computed from the\n$M_{\\rm gas}$-$M_\\star$ relations of \\citet{Huang_2012}. To account\nfor variations in halo concentration $c$, we sample three different halos\nof median and $\\pm 2\\sigma$ extremes according\nto the mass-concentration relation from\n\\citet{Ludlow_2014}. Table~\\ref{table:bar_param} summarises the main\nproperties of all our runs. Each halo is then set up in equilibrium\nwith their baryonic MN disk of scale lengths\n${R_{\\rm d}}=0.5,\\; 1.5,\\; \\rm and \\; 6.0$~kpc. We fix the scale\nheight of the disks to ${z_{\\rm d}} = 0.3{R_{\\rm d}}$, a slightly hotter\nstructure than for Milky-Way-like galaxies but that provides a reasonable\ndescription of lower mass galaxies, the main target of this study.\n\n\nWe run our simulations for $10$ Gyr assuming $i)$ collisionless cold dark\nmatter (CDM) and $ii)$ a fiducial self-interaction cross-section\n$\\som = 2\\;{\\rm cm^2\\, g^{-1}}$ for the SIDM case.  We investigate the relation\nbetween $V_{\\rm max}$ and ${V_{2\\,\\rm kpc}}$ of our halos in CDM and SIDM in the\nleft panel of Fig.~\\ref{fig:SIDM_bar_distrib}. Symbol shapes (square,\nstar, triangle) indicate the different scale lengths sampled whereas\nopen-blue and filled-magenta differentiate between CDM and SIDM\nruns. A quick inspection suggests that self-interactions are able to\ngenerate a wider range of ${V_{2\\,\\rm kpc}}$ at fixed $V_{\\rm max}$ which is in\nbetter agreement with observed values from the compilation in\n\\citet{Oman_2015} (green dots).  The diversity in both cases, CDM and\nSIDM, arises from different contribution by the baryons.  Since SIDM\ncores imply a lower dark matter density in the inner regions, the\ncontribution of baryons is \\emph{more} important that in the\ncollisionless case, creating a larger variety in ${V_{2\\,\\rm kpc}}$. Notice that\nfor the more compact disks (${R_{\\rm d}}=0.5$ and $1.5$ kpc) the\npredictions from CDM and SIDM are not that different, and in both\ncases ${V_{2\\,\\rm kpc}}$ can be very close to ${V_{\\rm max}}$, which suggests these\nrotation curves are relatively flat. One should also note that\nin a cosmological context the central density would be even further\npromoted by the adiabatic contraction introduced by baryonic collapse\n\\citep[e.g.][]{Elbert_2016}, which occurs even in the models with\nbursty feedback \\citep{Tollet2016}.\nIn the case with non-dominant\nbaryonic components (triangles) the points stay close to the dark\nmatter-only case (thin solid lines), which is significantly lower in\nthe case of self-interactions, as discussed in\nSec.~\\ref{sec:V2_Vmax_dm}. \n\nThe lower envelope of points in either scenario corresponds to the\nmost extended disks populating the $-2\\sigma$ outliers in halo\nconcentration. A few of the observed points in the left panel of\nFig.~\\ref{fig:SIDM_bar_distrib} still remain unexplained\n(${V_{\\rm max}} \\approx 70 \\; {\\rm km\\, s^{-1}}$ and ${V_{2\\,\\rm kpc}} \\approx 20\\; {\\rm km\\, s^{-1}}$) showing\ncentral densities even lower than can be obtained in SIDM with\n$\\som=2\\;{\\rm cm^2\\, g^{-1}}$. This could be suggesting the need for a larger cross\nsection for self-interactions or more extreme outliers on the\nmass-concentration relation. We should also bear in mind the\npossibility of observational errors as discussed in Sec.~\\ref{sec:intro}.\n\nAlthough the exercise of fixing ${R_{\\rm d}}$ is more clear and intuitive,\nin practise we know that galaxy size will scale with stellar mass, and\nthe average behaviour in ${V_{\\rm max}}$-${V_{2\\,\\rm kpc}}$ plane will depend on the\naverage ${R_{\\rm d}}$ of the population at fixed ${V_{\\rm max}}$.  In the right\npanel of Fig.~\\ref{fig:SIDM_bar_distrib} we take this into account as\nfollows.  The first step is to characterise ${R_{\\rm d}}$ as a function of\n${V_{\\rm max}}$. Because of the tight relation between halo mass and stellar\nmass, this is equivalent to finding a relation between ${R_{\\rm d}}$ and\n$M_*$ or, in our case, ${M_{\\rm d}}$. We compute the stellar size of the\ndisk and then estimate the baryonic one by a simple correction by gas\nfraction. In detail, for our stellar radii we use a fit to the\nlate-type SDSS galaxies of \\citet{Shen_2003}:\n\n\\begin{equation}\\label{eq:Shen_size}\n  R_{\\star} = 0.1  \\, \\left( \\frac{M_\\star}{1 \\; M_\\odot } \\right)^{0.14} \\left( 1 + \\frac{M_\\star}{3.98 \\times 10^{10} \\; M_\\odot} \\right)^{0.25} \\; \\rm kpc\n\\end{equation}\n\n\\noindent\nwith scatter\n\\begin{equation}\\label{eq:Shen_size_scatter}\n  \\sigma_{{\\rm ln} R} = 0.34 + \\frac{0.13}{1 + \\left( \\frac{M_\\star}{3.98 \\times 10^{10} \\; M_\\odot} \\right)^2}\n\\end{equation}\n\n\\noindent\nalong with an estimate of the baryonic size as: ${R_{\\rm d}} = (1+f_{\\rm\n  gas}) R_\\star$, where $f_{\\rm bar}$ is estimated from $M_*$ and\nobservations by \\citet{Huang_2012} as explained before. \nThis procedure then provides a median and $\\pm\n2\\sigma$ disk radii for each halo mass. We then take these dispersions\nin ${R_{\\rm d}}$ and interpolate between our simulations to find the\nappropriate ${V_{\\rm max}}$ and ${V_{2\\,\\rm kpc}}$, finally combining this dispersion in\nquadrature with the dispersion due to the $M_{200}$-concentration\nrelation to infer the median and $\\pm2 \\sigma$ likelihood regions as a\nfunction of halo mass.\n\nThese likelihood regions are shown in the right panel of\nFig.~\\ref{fig:SIDM_bar_distrib}. The expected median in each scenario\nis notably shallower than the points in the left panel, which is\nmainly due to the average disk size rising as ${V_{\\rm max}}$\nincreases. Indeed, as we saw in the left panel of the same figure, at\nfixed ${V_{\\rm max}}$ a more extended baryonic disk will have lower\n${V_{2\\,\\rm kpc}}$. For comparison we include the effects of the mass deficits of \n$5\\times 10^8$ and $10^9${$\\rm M_\\odot$} within 2~kpc.\nSeveral observed points still scatter above and below the median\nrelations for both CDM and SIDM hinting at the need of more extreme outliers\n(more than $2 \\sigma$) to explain such objects. \n\nWe have probably overstated the extremity of these outliers since we \nhave at zero-th order ignored correlations between halo and disk properties \nthat would alleviate the tension. \nFor instance, if more compact disks systematically\npopulate more concentrated halos whereas extended disks live in low\nconcentration halos, the shaded areas could be larger. We have also not \nmade an attempt to account for observational biases in these\ncalculations. Regardless of the absolute value, however, we emphasise that at\nfixed assumptions, SIDM seems to always provide a larger\nrange of diversity than CDM (with scatter at least $150\\%$ of the latter).\n\nAn interesting corollary of the analysis above is that, fixing the\nrelation ${V_{\\rm max}}$-${V_{2\\,\\rm kpc}}$, a given halo in CDM and SIDM would be\npredicted to have different contribution\nfrom the baryons at $2$~kpc, since in SIDM the baryonic matter will have to\n``compensate'' for the lower density of the cored dark matter halo. It\nis therefore important to compare the dark matter fractions predicted\nin both models to check that they are consistent with observed\ngalaxies. Fig.~\\ref{fig:DM_2kpc} shows the dark matter\nfractions $f_{\\rm dm}=M_{\\rm dm}\/M_{\\rm tot}$ measured at $2$~kpc for\nour CDM and SIDM halos (open and filled symbols, respectively)\ncompared to a subsample of galaxies taken from the rotation curves\npresented in \\citep{Oh_2015} and \\citet{Kuzio_2008}.  For the former\nwe use their disk-halo decomposition from the stellar, HI and\nkinematic data, and for \\citet{Kuzio_2008} we use their `popsynth'\nmass modelling rather than the extremal fits. In a couple of cases the\nlast-measured point does not reach 2~kpc and for these we used the\ndark matter fraction at the last measured point. \n\nThere is a significant overlap between the collisionless and the \nself-interacting scenarios. Also, spatial resolution coupled to\nuncertainties in the mass-to-light ratios for the observed objects\nmakes $f_{\\rm dm}$ probably not a good enough indicator to distinguish\nbetween these alternative models. Nonetheless, it is reassuring that\nthe same SIDM objects that reproduce better the observations in the\n${V_{\\rm max}}$-${V_{2\\,\\rm kpc}}$ plane, seem to have dark matter fractions that are\nconsistent with current observations.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/Vmax_V2ratio_dm_plus_CDM.pdf\n\\caption{Dark matter contribution to the rotation curve at 2~kpc for SIDM\n  simulations and observed dwarfs (see main text). \\emph{Filled magenta squares}\n  are the SIDM simulations, in comparison to the CDM in \\emph{open blue squares}.\n  \\emph{Greyscale symbols} refer to observed\n  galaxies from \\citep{Oh_2015} (\\emph{dark circles}),\n  \\citet{Kuzio_2008} (\\emph{light circles}), \\citet{deBlok_2001} \n  (\\emph{light squares}) and \\citet{deBlok_2008} (\\emph{dark squares}).}\n\\label{fig:DM_2kpc}\n\\end{figure}\n\n\n\\subsection{Two extreme examples: IC~2574 and UGC~5721}\\label{sec:obs_curve_sims}\n\nIn Sec.~\\ref{sec:V2_Vmax_bar}, we presented a detailed statistical\nanalysis of a sample of objects simulated within the SIDM\nparadigm. This, however, left unexplained one of the main motivations\nof this work, which is the existence of extreme outliers which are not\npossible to accommodate within the standard CDM scenario. We turn then\nour attention to two particular examples highlighted in\n\\citet{Oman_2015}, UGC~5721 and  IC~2574. These two galaxies have\nsimilar circular velocity measured at the outermost point of the\nrotation curve $\\approx 80$ ${\\rm km\\, s^{-1}}$, which probably indicate that they\npopulate dark matter halos with similar mass, but the {\\it shapes} of\nthe inner regions are remarkably different: whereas UGC~5721 is\nconsistent with a `cuspy' NFW profile, IC~2574 has a very extended core.\nThis pair is an interesting case\nof the aforementioned diversity and we can use it as benchmark for our\nSIDM scenario.\n\nIn Figs.~\\ref{fig:UGC5721} and \\ref{fig:ICgal} we show the data together\nwith the best fit CDM (dashed line) and SIDM (solid black) curves. \nFor UGC~5721 (NGC3724) we use the rotation curve data\nfrom \\citet{Swaters_2003} and stellar and gas densities from Andrew\nPace (priv. communication, for comparison one can see\nFig.~1 in \\citealp{Swaters_2010} which has a slightly higher\nmass-to-light ratio).\n                     \n                     \nFor IC~2574 we use the rotation curve, gas and stellar densities from\n\\citet{Oh_2011}. We multiply the stellar velocity contribution by\n$0.5$ (mass-to-light ratio lower by a factor $4$), so the stellar\ncontribution lies between that in \\citet{Sanders_2002} and\n\\citet{Oh_2011}. For both cases our SIDM halo is run using a cross\nsection for self-interactions of $\\som = 3\\,{\\rm cm^2\\, g^{-1}}$\nfollowing \\citet{Kamada2016}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lccc}\n\\hline\n\n\nProperty & Symbol & IC~2574 & UGC~5721 \\\\\n\\hline\nDisk scale length & ${R_{\\rm d}}$ & $3.0$ & $0.50$ kpc \\\\\nDisk scale height & ${z_{\\rm d}}$ & $0.9$ & $0.15$ kpc \\\\\nStellar mass & $M_\\star$  & $3.00 \\times 10^8$ & $4.48\\times 10^8$ {$\\rm M_\\odot$}\\\\\nGas mass$^\\dagger$ & $M_{\\rm gas}$ & $2.72\\times10^9$  & $8.47\\times10^8 $ {$\\rm M_\\odot$}\\\\\nHalo mass & $M_{200}$ & $9.78\\times10^{10}$ & $5.15\\times 10^{10}$ {$\\rm M_\\odot$} \\\\\nConcentration & $c$ & $6.2$ ($-2.3\\sigma$) & $13.6$ ($+0.9\\sigma$) \\\\\nDM cross-section &$ \\som $ & $3$ & $3$ cm$^2$ g$^{-1}$ \\\\\nNFW radius & $r_{\\rm s}$ & $15.4$ & $5.65\\;\\rm kpc$ \\\\\nNFW density & $\\rho_{\\rm s}$ & $1.9\\times10^{-3}$ & $0.013 \\, \\rm M_\\odot\\, pc^{-3}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the SIDM halos and baryonic potentials for the two galaxies in Fig.~\\ref{fig:UGC5721} and \\ref{fig:ICgal}. \\newline \n$\\dagger$ The gaseous component as added in post-processing due to its disturbed nature, see also main text in Sec.~\\ref{sec:obs_curve_sims}. \\newline \n}\n\\label{table:gal_params_fits}\n\\end{table}\n\nOur fitting process relied upon making an initial guess for the \nhalo mass using ${V_{\\rm max}}$ and sampling the concentration with $\\pm 2.5 \\sigma$ of the \nmass-concentration relation in Eqn.~(\\ref{eq:linear_ludlow}). \nThe combined stellar plus gaseous disks in these dwarfs are not\nwell represented by a single MN disk (especially their gas profiles\nwhich exhibit a much more extended, nearly constant surface density), \nhowever since these differences are primarily in the outer regions where we\nexpect the effects of self-interactions to be sub-dominant, we chose\nto approximate only the \\emph{stellar} distribution with an MN disk \nand apply the gas contribution to the rotation curve in a \npost-processing step (i.e. add in quadrature as in Eqn.~\\ref{eq:vcirc}),\nand after several iterations with the disk component we find the \nparameters indicated in Table~\\ref{table:gal_params_fits}.\nAs noted in Appendix~\\ref{sec:NFW_vs_HQ},\nthe initial conditions are constructed with Hernquist profiles \nconverted from the corresponding NFW parameters, \nbut if one is using a real NFW then in the higher\nconcentration case using a lower $r_{\\rm s}$ would give a\nslightly better match.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/UGC5721.pdf}\n\\caption{Observations for UGC~5721 along with SIDM and CDM simulations. \\emph{Red points} indicate the stellar contribution, \\emph{cyan} the gas and \\emph{grey squares with error bars} the observed rotation curve. The SIDM simulation (with parameters from Table~\\ref{table:gal_params_fits}) has the baryonic mass contribution indicated by the \\emph{green line}, which is a composition of the interpolated gas distribution along with an MN fit to the stellar disk. The halo contribution is given in the \\emph{dotted blue line} and the total curve in \\emph{solid black}. The comparison curve for CDM is given in the \\emph{dashed black line}.}\n\\label{fig:UGC5721}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures\/IC2574.pdf}\n\\caption{Observations for IC~2574 along with SIDM and CDM simulations. \\emph{Red points} indicate the stellar contribution, \\emph{cyan} the gas and \\emph{grey squares with error bars} the observed rotation curve. The SIDM simulation (with parameters from Table~\\ref{table:gal_params_fits}) has the baryonic mass contribution indicated by the \\emph{green line}, which is a composition of the interpolated gas distribution along with an MN fit to the stellar disk. The halo contribution is given in the \\emph{dotted blue line} and the total curve in \\emph{solid black}. The comparison curve for CDM is given in the \\emph{dashed black line.}}\n\\label{fig:ICgal}\n\\end{figure}\n\nFig.~\\ref{fig:UGC5721} shows that the SIDM fit for UGC~5721 including all components has an\noverall ``cuspy'' profile (see black solid line), which is \\emph{steeper} than the circular\nvelocity expected for the corresponding NFW halo without \nself-interactions (dashed line). For this fit we need a concentrated\nhalo at $+1\\sigma$ above the median mass-concentration relation, which\ntogether with the observed compact stellar disk (scale radius of\n$0.5$~kpc) combine to create a steeply raising velocity curve. \nThis is an excellent example of the importance of baryons in the shape\nof the rotation curves, even for dwarf galaxies. \nThe $\\chi^2\/\\rm dof$ of the SIDM fit is $3.04$ ($4.44$ below the initial, \nwhich is of course not the best individual CDM halo fit). \nThe visual impression of a superior fit is likely due to \nThis $\\chi^2$ is largely driven by the $\\lesssim 0.3$~kpc data with\ntight error bars over a non-monotonic feature, contributing to \na visually impression of a superior fit.\nClearly a more accurate fit to such features would require additional parameters (particularly for the stellar components), however this is outside the scope of this paper.\nWithout the presence of baryons, UGC~5721 is a challenge for SIDM\ndue to the development of an extended low density core at the centre. In our simulations, however, the\nSIDM thermal transfer in cooperation with the baryon dominance in the\ncentral $1.5$~kpc causes the halo to shrink even further than the\npure NFW model. \nThe compact stellar component excavates a potential well sufficient that\nthermalisation of the dark component leads to an almost-NFW final\nconfiguration \\citep[see also ][]{Kamada2016}.\n\n\nOn the other hand, Fig.~\\ref{fig:ICgal} presents the rotation curve\nfit for IC~2574 with an extended and slowly rising core in its\ncentre. The gas has a disturbed profile, but the baryons as a whole\ncontribute little to the total density except within the innermost\n$0.5$~kpc, and make a modest contribution at $r> 6$~kpc. For such a\ndark matter dominated galaxy to deviate so far from a cusp-like\nprofile is clearly a challenge for vanilla CDM \\citep{BlaisOuellette_2001, Sanders_2002,\n  Swaters_2003, Swaters_2010}. Notice that any baryonic solution to\nthis problem within CDM -- for instance assuming that supernova\nfeedback can create a core-- would not work for such object, since the\ncore extends well beyond where the stars are (see discussion in\n\\citealt{Oman_2015}). \nThe $\\chi^2\/\\rm dof$ of the SIDM fit is $1.57$, ($1.39$ below the initial\nconditions), which reflects the introduction of the core\nand quantifies the capacity of SIDM as a solution to this conundrum.\n\n\nWhilst SIDM is relatively efficient in producing cores, this process\nbecomes increasingly sedate as the core size approaches the scale\nradius of the halo (in this case $15.4$~kpc) and so we have had to\nchoose a relatively large scale radius (i.e. low concentration)\nhalo. The ${V_{2\\,\\rm kpc}}$ of our CDM halo is around $35\\; {\\rm km\\, s^{-1}}$ whilst in SIDM\nit is around $27\\; {\\rm km\\, s^{-1}}$, bringing it in better agreement with the\nobserved rotation curve ${V_{2\\,\\rm kpc}} \\approx 23 \\; {\\rm km\\, s^{-1}}$. \n\nWe note that for IC~2574 both the low concentration and self-interactions are required to achieve a good fit. As such one way to test our model is to perform a statistical study of extreme outliers like IC~2574. Since we assume the SIDM model inherits the halo mass-concentration relation of the CDM one, such a study also provides a direct test for {$\\Lambda$CDM}\\ on galactic scales.\n\n\nWe have shown that the SIDM model in the presence of baryons can provide good fits to two extreme outliers. We demonstrate that with SIDM the halo concentration, the baryon concentration and the back-reaction of baryons on the density profile play roles in explaining the rotation curves of these two extremes. SIDM halo profiles are flexible enough to accommodate the diverse rotation curves of spiral galaxies. \n\n\n\\section{Summary and Conclusions}\\label{sec:discussion}\n\n\nThe SIDM model modifies the CDM paradigm by inclusion of non-gravitational interactions between dark matter particles in the halo evolution, and these self-interaction effects are most significant in the regions of halos with the high densities and velocity dispersions, i.e. within the scale radius of the halo. In the outer halo and at large scales SIDM is almost collisionless and retains the successes of {$\\Lambda$CDM}.\nIn the inner regions of halos the dark matter is redistributed into a central core, whose density and size depend on the microscopic particle interaction cross-section, the halo mass and even the baryon concentration. The mass profiles inferred from astronomical observations of the rotation curves of spiral galaxies thus offer an avenue to both constrain the self-interaction cross-section of dark matter and falsify or support specific self-interaction models.\n\n\nObservationally, the shapes of the rotation curves of galaxies exhibit a\nwide diversity. We follow \\citet{Oman_2015} by quantifying the range in the rotation curves using the relation\nbetween the total circular velocity measured within $2~{\\rm kpc}$ and at the peak, ${V_{2\\,\\rm kpc}}\\textup{--}{V_{\\rm max}}$. For a given ${V_{\\rm max}}$, the spread in ${V_{2\\,\\rm kpc}}$ can be a factor of four, which is difficult to reconcile in the prevailing {$\\Lambda$CDM}~paradigm~\\citep{Oman_2015}, where dark matter is assumed to be collisionless over the cosmological timescale. This problem stems from the hierarchical structure formation in {$\\Lambda$CDM}\\ which produces a self-similar halo density profile, which is essentially parameterised by one parameter, the halo mass (or concentration). After we determine the halo mass by ${V_{\\rm max}}$, the halo circular velocity is completely fixed at all radii, up to the scatter, including the inner density cusp that is in contrast to the shallow (cored) one preferred by many dwarf galaxies. In addition, the enclosed mass of a CDM halo cusp tends to overwhelm the baryonic contribution and consequently also the scatter in ${V_{2\\,\\rm kpc}}$ caused by the spread in the baryon concentration.\nThis inflexibility in CDM may be alleviated by introducing particular prescriptions of baryonic feedback processes that can dynamically heat the dark matter \\citep[and others]{Navarro_1996,Pontzen_2012}, however SIDM offers an attractive alternative solution.\n\nIn this paper we have used controlled N-body simulations as a numerical experiment to test the SIDM model's ability to address the diversity problem. We use the {\\sc Arepo} code to run isolated simulations of a live halo in the presence of a static baryonic disk in order to assess the gravitational impact of gas and stars on the halo evolution. This approach is suited to the SIDM model as dark matter self-interactions thermalise the inner halo and the final (inner) density profile is largely insensitive to the formation history. We sample a wide range of halo masses spanning about three orders of magnitude and vary the halo concentration within $\\pm2\\sigma$ from the its mean value. For each halo mass, we use abundance matching relations to choose the disk mass and we take three disk scale radii to span the baryon distributions. This sampling of the model parameter domain enables us to make concrete predictions of the rotation curves in the SIDM model.\n\n\nOur main conclusion (see Fig.~\\ref{fig:SIDM_bar_distrib}) is that the SIDM mechanism accommodates galaxies with a wide range of  ${V_{2\\,\\rm kpc}}$ over the domain of ${V_{\\rm max}}$ from $30$--$250\\;{\\rm km\\, s^{-1}}$. The spread in core densities is due in part to the variance in concentration (within $\\pm2\\sigma$ from the mean), however this alone is insufficient to account for the large relative scatter of the observed values. Including self-interactions allows lower ${V_{2\\,\\rm kpc}}$, which requires low central densities in \\emph{both} baryons and dark matter. High values of ${V_{2\\,\\rm kpc}}$ are still achieved with compact baryonic disks where thermalisation into the deep baryonic potentials can induce NFW-like densities. \nAs a result the predicted range of ${V_{2\\,\\rm kpc}}$ for a fixed ${V_{\\rm max}}$ in SIDM is $50\\%$ larger than the CDM one, leading to a better agreement with the observed rotation curve distribution.\n\nSome observed galaxies in the range ${V_{\\rm max}}=60$--$80\\; {\\rm km\\, s^{-1}}$ and ${V_{2\\,\\rm kpc}}=20$--$30\\;{\\rm km\\, s^{-1}}$ are still challenging.\nThese cases might require \nassumptions about the structure of the copula of the combination of \nhalo concentrations and baryonic disk sizes, in particular that very\nextended disks populate the most under-concentrated halos to reproduce the very low ${V_{2\\,\\rm kpc}}$, which is not an \nunrealistic assumption to first order. \nNevertheless, the SIDM prediction is an improvement over CDM in all cases.\n\n\n\nTo further test our results, we have performed simulations to provide individual fits for two of the most extreme\noutliers highlighted in~\\citet{Oman_2015}, UGC~5721 and IC~2574, cusp- and core-\ndominated galaxies respectively, both with similar ${V_{\\rm max}} \\approx\n80 \\; \\rm {\\rm km\\, s^{-1}}$ (see Figs.~\\ref{fig:UGC5721} and~~\\ref{fig:ICgal}). Picking extreme $+0.9\\sigma$ and $-2.3\n\\sigma$ concentrations from the halo concentration-mass\nrelation, together with a compact ${R_{\\rm d}} = 0.5~{\\rm kpc}$ and\nextended ${R_{\\rm d}} = 3.0~{\\rm kpc}$ baryonic distribution respectively, we find good fits in the SIDM paradigm. Interestingly, we find that the SIDM density profile can be as dense as the NFW one in a baryon-concentrated galaxy such as UGC~5721, because SIDM particles follow an isothermal distribution within the deep baryon potential. Our simulation result confirms the theoretical expectation first predicted in~\\cite{Kaplinghat_2014}. \nAt the other extreme, IC~2574 with an inferred central dark matter dominated core is more straightforwardly tractable with a model that allows dark matter thermalisation, and is more challenging to the collisionless CDM model.\n\nThe implication of these elements is that with the inclusion of realistic baryonic components, SIDM models can produce quantitatively superior fits to rotation curves, both at the level of individual rotation curves and over the statistical distribution of a quantitative measure of their shapes. As one would expect, fitting the most extreme outliers requires an interplay between the combinations of halo concentration, the baryon distribution, and the influence of baryons on the SIDM halo profiles. The results from our numerical experiments fit well with the previous theoretical calculations of \\citet{Kamada2016}. \n\nA number of avenues for future research are apparent. On the observational side, more detailed observations for mass modelling would allow tighter constraints on the dark matter distribution and consequently the interaction cross-section. \nOrthogonally, larger sample sizes would allow the ${V_{2\\,\\rm kpc}}$--${V_{\\rm max}}$ distribution to better discriminate between CDM and SIDM. \nOn the computational side, we have ignored the effects of the hierarchical assembly of dark matter halos. Whilst this is not expected to play a large role in the inner halo, it will nevertheless introduce additional scatter as a function of assembly time. Finally, all these will likely show strong correlations with the baryonic statistics, and investigations of galaxy formation in SIDM using full hydrodynamical cosmological simulations with the baryonic feedback processes are already being explored \\citep[e.g.][]{Vogelsberger_2014,Fry_2015}.\n\n\n\\section*{Acknowledgements}\nThe authors would like to thank Andrew Pace for sharing his collated data set for the rotation curves, \nVolker Springel for making {\\sc Arepo} available for this work and\nJames Bullock, Simon White, Oliver Elbert, and Manoj Kaplinghat for useful discussions. This work was supported by the U.S. Department of Energy under Grant  No. de-sc0008541 (HBY). HBY acknowledges support from the Hellman Fellows Fund. MV acknowledges support through an MIT RSC award and the support of the Alfred P. Sloan Foundation. \n\n\n\\bibliographystyle{mnras}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAstrophysical jets emanate from sources spanning a huge range in energy output \nor length scale - among them young stellar objects (YSO), \nstellar mass compact objects as X-ray binaries or $\\mu$-quasars, \nor the powerhouses of some active galactic nuclei (AGN) which host a super-massive\nblack hole.\nIn particular for radio-loud quasars, \nfor which synchrotron emission dominates the radio spectrum,\nrelativistic jets are a generic feature. \nDue to the omnipresent angular momentum conservation, mass accretion to all of\nthese objects features a disk structure around the central mass.\nIt is commonly believed that jets are launched as disk winds, \nwhich are further accelerated and collimated by magnetic forces\n(see \\citet{1982MNRAS.199..883B, 1983ApJ...274..677P, 1986A&A...162...32C, 1997SvPhU..40..659B, \n2003ApJ...596.1240H, 2007prpl.conf..277P}.\nRelativistic jets may gain further energy by interaction with the\nblack hole magnetosphere \n\\citep{1977MNRAS.179..433B, 1997MNRAS.292..887G, 2005MNRAS.359..801K}.\n\nThe magnetohydrodynamic (MHD) self-collimation of non-relativistic jets has been \nproven in general by \ntime-dependent simulations \\citep{1995ApJ...439L..39U,1997ApJ...482..712O} and \nhave been investigated in further detail considering additional physical effects \nas magnetic diffusivity by \n\\cite{2002A&A...395.1045F}, a variation in \\cite{1999MNRAS.309..233O}, \nnon-axisymmetric instabilities in the launching region \\citep{2003ApJ...582..292O},\nor a variation in the mass flow profile or the magnetic field geometries\n\\citep{2006ApJ...651..272F, 2006MNRAS.365.1131P},\nor the influence of a central magnetic field \\citep{2009ApJ...692..346F,2008A&A...477..521M}.\n\nIn the case of relativistic jets the efficiency of MHD self-collimation is under debate. \nThe main reason is the existence of electric fields which are negligible for non-relativistic\nMHD and which are commonly thought to have a net de-collimating effect on the jet.\nEssentially, \\cite{1991ApJ...377..462C} have demonstrated the current carrying \nrelativistic jet can be highly collimated.\nHowever, the actual structure of these jets still remains unclear - mainly due to \nthe need for simplifying assumptions to solve the corresponding set of MHD equations.\n\nSo far, a variety of theoretical models have been developed for the case of \n{\\em self-similar} jets\n\\citep{1992ApJ...394..459L, 1994ApJ...432..508C, 1995ApJ...446...67C,\n      2003ApJ...596.1104V,2006A&A...447..797M},\nalthough it seems clear that relativity does not obey self-similarity.  \nFully 2.5D theoretical solutions for the internal magnetic jet structure could be\nobtained by neglecting matter inertia\n\\citep{1997A&A...319.1025F, 2001A&A...365..631F}.\nThese force-free solutions for the field structure can in principle be coupled\nto the dynamical wind solution along the field lines\n\\citep{1996A&A...313..591F, 2001A&A...369..308F, 2004ApJ...608..378F}.\n\\cite{1997A&A...319.1025F} obtained solutions for the internal jet force balance \nin Kerr metric with an asymptotically cylindrical jet emerging from a disk-like\nstructure around the central rotating black hole. \nThe shape of the collimating jet boundary was obtained as result of the internal \nforce equilibrium, in particular considering the regularity condition along the \njet outer light surface.\n\nTime-dependent simulations of relativistic MHD jet formation have been performed\nconsidering a general relativistic metric, including also the evolution of \nthe underlying accretion disk.\nEarly - seminal - simulations did last for a few inner disk rotations \nonly \\citep{1998ApJ...495L..63K, 1999ApJ...522..727K}, which is\nsufficient time to demonstrate the launching of an outflow, \nbut hardly sufficient in order to investigate the long-term dynamical evolution of\nthe emerging jet.\n\nMore recent simulations were able to follow several 100 disk rotations and show \nthe formation of a so-called funnel flow origination in the shear layer between \nthe horizon and the inner disk radius\n\\citep{2005ApJ...620..878D, 2007MNRAS.375..513M, 2008MNRAS.388..551T, 2009MNRAS.394L.126M}.\nThese simulations indicate a highly time-variable mass ejections of rather low degree \nof collimation. However, the funnel flow achieves Lorentz factors of up to 50.\nGeneral relativistic MHD simulations are also able to determine the interrelation \nbetween jet formation and the Blandford-Znajek mechanism \n\\citep{2005ApJ...630L...5M, 2007MNRAS.377L..49K}.\n\nUltra-relativistic MHD simulations of accelerating and collimating\njets have been presented by \\citet{2007MNRAS.380...51K},\nspanning over a huge range of length scale and providing jets of \nlarge Lorentz factor $\\Gamma \\sim 10$.\nTheir simulations, however, did not start from the very base of the \njet - the accretion disk, but at some fiducial boundary above the equatorial\nplane.\nSince the jet has been launched already with super-escape speed,\ngravity has not been considered. \nThe jet flow has been confined within a rigid wall of predefined shape \nwhich naturally affects the opening angle of the MHD jet nozzle and thus\njet collimation and acceleration.\n\nThe focus of our present paper is \ni) to concentrate on the formation and acceleration of a relativistic MHD jet right from\nthe launching area the accretion disk surface,\nii) to investigate the (self-) collimation of relativistic MHD jets under the influence\nof de-collimating electric forces and an \"open\" boundary condition for the outflow \niii) to consider gravity as en essential gradient to provide a realistic disk boundary condition in\nequilibrium,\niv) to run long-term simulations lasting more than 1000 inner disk rotations until the\njet reaches steady state,\nv) to concentrate on MHD disk jets as disks are the natural origin for the mass load for\nAGN jets.\n\nThe outline of this paper is as follows: In section \\ref{sec:concepts} we discuss the \nconcepts of ideal special relativistic magnetohydrodynamics in the\nperspective of jet formation.  \nSection \\ref{sec:model setup} is devoted to the initial- and boundary conditions of the\nnumerical simulations, whose results are shown in section \\ref{sec:results}. \nWe conclude in section \\ref{sec:conclusions}.\n\n\\section{Concepts of relativistic MHD jets}\\label{sec:concepts}\n \nIt is well known that relativistic jets must be strongly magnetized \n\\citep{1969ApJ...158..727M, 1986A&A...162...32C, 1993ApJ...415..118L}.\nThis simply reflects the fact that the lower the mass flux, the more electro-magnetic energy (Poynting flux) \ncan be transferred into high kinetic energy per unit mass.\nRelativistic MHD is also limited for very strong magnetization as then\nthe MHD assumption can be violated since a sufficiently large amount of electric charges is lacking which are needed to drive the electric current system. Such a situation might arrise in the ultra-relativistic regime of pulsar-winds but is unlikely for the disk-winds investigated here.  \n\nThis paper deals with the time-dependent formation of relativistic MHD jets \nby using the special relativistic MHD module of the PLUTO code provided by \\cite{2007ApJS..170..228M}\nand applying a Newtonian description of gravity. \n\\subsection{Relativistic MHD equations}\n\nThe relativistic MHD module of PLUTO solves the system of special relativistic conservation laws.\nIn a covariant formulation, \nthe equations follow naturally as a set of hyperbolic equations.\nIt is solved for energy and momentum conservation\n\\begin{equation}\n\\partial_{\\alpha} T^{\\alpha \\beta} = 0 \\label{eq:energy}\n\\end{equation}\nof an ideal magnetized fluid \n\\begin{equation}\nT^{\\alpha \\beta} = (\\rho h + b^2) u^\\alpha u^\\beta + \n\\left(p + \\frac{1}{2} b^2\\right) g^{\\alpha \\beta} -b^\\alpha b^\\beta\n\\end{equation}\nwith the specific plasma enthalpy \n\\begin{equation}\nh=\\frac{\\gamma}{\\gamma-1}\\frac{p}{\\rho}+1,\n\\end{equation}\nthe isotropic gas pressure $p$, density $\\rho$ (both in the local rest-frame), four-velocity $(u^\\alpha)=(\\Gamma,\\Gamma \\mathbf{\\beta})^T$, velocity $\\mathbf{\\beta}=\\mathbf{v}\/c$, Lorentz factor $\\Gamma=(1-\\beta^2)^{-0.5}$ and the magnetic field pseudo vector\n\\begin{equation}\nb^\\alpha = -\\frac{1}{2} \\epsilon^{\\alpha \\beta \\gamma \\delta} u_{\\beta} F_{\\gamma \\delta}.\n\\end{equation}\n\nAssuming infinite conductivity, thus vanishing electric fields in the \nrest-frame of the plasma $F^{\\alpha \\beta} u_{\\beta}= 0$, the \nhomogenous Maxwell equation\n\\begin{equation}\n \\partial_{\\alpha}\\ ^{*}\\!F^{\\alpha \\beta} = 0 \\label{eq:hommaxwell}\n\\end{equation}\ncan be written solely in terms of the magnetic four-vector $b^\\alpha$\n\\begin{equation}\n^{*}\\!F^{\\alpha \\beta} \\equiv \\frac{1}{2} \\epsilon^{\\alpha \\beta \\gamma \\delta} F_{\\gamma \\delta} \n                     =   b^\\alpha u^\\beta - b^\\beta u^\\alpha\\label{eq:hommhd}\n\\end{equation}\nand for the field vector components it \nfollows\\footnote{following the convention that Greek indices run \nfrom 0 to 4 whereas Latin indices go from 1 to 3}\n\\begin{eqnarray}\nB^{i} &=& ^{*}\\!F^{i 0} = b^{i} u^0 - b^{0}u^{i}\\\\\nE^{i} &=& \\epsilon^{ijk}b^j u^k \\label{eq:mhd-cond}.\n\\end{eqnarray}\nEquation \\ref{eq:mhd-cond} represents the ideal MHD condition \n$\\mathbf{E} = - \\mathbf{\\beta} \\times \\mathbf{B}$ \nand is the reason why all electric fields can be eliminated from the equations.  \nThe magnetic four-vector turns out as $b^0 = B^{i} u^{i};\\ \\  b^{i} = (B^{i}+b^0 u^{i})\/u^0$.  \nThe conservation of the Faraday tensor given by Eq.\\,\\ref{eq:hommhd} results in the \nnon-relativistic (ideal) induction equation\nand the solenoidal condition $\\mathbf{\\nabla} \\cdot \\mathbf{B} = 0$.  \nMass conservation is guaranteed by the continuity equation \n\\begin{equation}\n\\partial_{\\alpha} (\\rho u^\\alpha) = 0. \\label{eq:continuity}\n\\end{equation}\nWe apply a polytropic equation of state for the gas with the polytropic\nindex $\\gamma =5\/3$.\n\n\\subsection{Gravity in special relativity}\nThe outcome of MHD simulations is mainly determined by its boundary conditions and therefore requires great care in describing the proper physical state of \ninterest.  \nSince in our simulations the jet is considered to be launched as a wind from a \nrotating disk,\nit is essential to take into account a proper disk model as boundary condition.\nFor the disk boundary we choose a (sub-) Keplerian rotation profile and a\nhydrostatic pressure gradient (see section \\ref{sec:injection}).\nThis choice implies a hot, ``puffed up'' corona with a sound speed \n$c_{s}^2\\simeq 2\/3\\ v_{\\phi}^2$.\nThis is the main reason why we have added gravity to our special relativistic\ntreatment.\nThe direct impact of gravity on the jet dynamics is marginal as the outflow\nis accelerated to super escape speed quickly.\n\nA fully self-consistent relativistic treatment of gravity would imply\na general relativistic approach.\nHowever, we are not interested in the region very close to the horizon \nof the central object, but in the long-term dynamics and evolution of a disk wind into\na relativistic jet.\nWe can therefore neglect general relativistic effects in our simulation \ndomain.\n\nWe apply a softened gravitational potential\n\\begin{equation}\n\\phi = -\\frac{GM}{R+r_{S}}\n\\end{equation}\nwith a softening length of $r_{S}=1\/3$ that may be related to the to the Schwarzschild \nradius of a non-rotating black hole\\footnote{The capital $R=\\sqrt{r^2+z^2}$ denotes the spherical radius throughout this work.}.\n\\red\nThe corresponding acceleration reads\n\\begin{equation}\n\\mathbf{a} = -\\nabla\\phi = -\\frac{GM}{(R+r_{S})^2 R}\\mathbf{r}\n\\end{equation}\nand hence instead of solving equation \\ref{eq:energy}, we solve \n\\begin{equation}\n\\partial_{\\alpha}T^{\\alpha \\beta} = f^\\beta\n\\end{equation}\nwith the four force density \n$\n\t(f^\\beta) = \\Gamma \\rho (\\mathbf{a \\cdot v},\\mathbf{a})^T\n$\nas a local source term on the right-hand side.  \nThis is incorporated in PLUTO as a \"body force\" ($\\Gamma \\mathbf{a}$) using the infrastructure of the code.  \n\\black\n\nOmission of softening would lead to numerical errors \n(due to the unresolved steep gradients in the potential close to the origin), \npiling up to produce artificial acceleration along the spine of the jet close to \nthe axis.\nSoftening is clearly a compromise avoiding the singularity (by limiting the \nrequired resolution) on little cost of realism.  \nAnother choice could be the well-known pseudo-potential by \\cite{1980A&A....88...23P}\nwhich has just the negative softening $\\phi_{PW}=-GM\/(R-r_{S})$.  \nFor the cylindrical geometry of our choice, the singularity would become even more problematic,\ncomplicating the setup a great deal.  \n\n\n\\subsection{Relations in axisymmetric MHD}\nThe region of jet formation may be fairly well approximated in {\\em axisymmetry}.\nIn fact, non-axisymmetric distortions may actually hinder the formation of powerful jets as probably \ndemonstrated by the existence of a variety of strongly magnetized, rapidly rotating accretion disk \nsystems which, however, do not exhibit jets (e.g. cataclysmic variables or most pulsars). \n\nUnder the assumed symmetry in a cylindrical coordinate system, \nthe magnetic field vector can be written as\n\\begin{equation}\n\t\\mathbf{B} = \\mathbf{B}_{p} + B_{\\phi} \\mathbf{e_{\\phi}},\n\\end{equation}\nwhere $B_{\\phi}$ can now be an arbitrary function of $r$ and $z$, as the solenoidal condition\ntranslates to $\\mathbf{\\nabla\\cdot B_{p}}=0$. \nThe stream function \n$\\Psi(r,z)= (1\/2\\pi) \\int \\mathbf{dS \\cdot B_{p}} = rA_{\\phi}$ \nmeasures the magnetic flux through the surface area $S$ \nand follows from the toroidal component of the vector potential,\n\\begin{equation}\n\\mathbf{B_{p}}= \\mathbf{\\nabla}\\times \\mathbf{A_{\\phi}} \n              = \\mathbf{\\nabla}\\times \\frac{\\Psi\\ \\mathbf{e_{\\phi}}}{r} \n              = \\frac{1}{r}\\mathbf{\\nabla}\\Psi \\times\\ \\mathbf{e_{\\phi}}\n\\end{equation}\n\nFor the electric field, the ideal MHD condition $\\mathbf{E = B \\times \\beta}$ gives \n\\begin{equation}\n\\mathbf{E} = \\frac{r\\Omega^{F}}{c} B_{p} \\mathbf{n} = \\frac{r}{r_{\\rm L}} B_{p}\\mathbf{n}\\label{eq:electric-field}\n\\end{equation}\nin terms of the so called angular velocity of the field line \n$\\Omega^F = \\left(v_{\\phi} - v_{p}B_{\\phi}\/B_{p}\\right)\/r$, \nor the so-called light cylinder radius \\footnote{This is the radius at which the hypothetical angular \nvelocity of a field line supercedes the speed of light} of a field line $r_{\\rm L} \\equiv c\/\\Omega^F$.  \nThe direction of the electric field is given by \n$\\mathbf{n = B_{p}}\/B_{p}\\times \\mathbf{e_{\\phi}}$ and is perpendicular to the magnetic\nflux surface $\\Psi(r,z)$.  \nThe poloidal Poynting flux $\\mathbf{\\mathcal{S}} = (c\/4\\pi)\\mathbf{E\\times B_{\\phi}}$ \nsimplifies to \n\\begin{equation}\n\t\\mathbf{\\mathcal{S}} = -\\frac{c}{4\\pi} \\frac{r}{r_{\\rm L}} B_{\\phi}\\mathbf{B_{p}} \n                             = -r\\Omega^{F} \\frac{B_{\\phi}\\mathbf{B_{p}}}{4\\pi}.\n\\end{equation}\n\n\\subsubsection{Perpendicular and parallel force-balance}\\label{sec:force-balance}\nThe processes leading to flow collimation can be identified directly \nfrom the (steady-state) trans-field force-balance equation \n\\citep{1991ApJ...377..462C, 1993A&A...270...71A}.\nHere, we adopt the notation of the latter paper when investigating the \ncollimation behavior in the quasi steady-state time domain of our \nsimulations.\nThe curvature \n$\\kappa \\equiv \\mathbf{n}\\left( \\mathbf{B_{p}} \\cdot \\nabla \\right) \\mathbf{B_{p}} \/B_{p}^2$\nof a flux surface $\\Psi(r,z)$ results from the summation of perpendicular forces,\n\\begin{equation}\n\\begin{split}\n\\kappa \\frac{B_{p}^2}{4\\pi}\\left(1-M^2-\\frac{r^2\\Omega^{F^2}}{c^2}\\right) =  \\\\\n+ \\left(1-\\frac{r^2\\Omega^{F^2}}{c^2}\\right)\\nabla_{\\bot}\\frac{B_{p}^2}{8\\pi} \n+ \\nabla_{\\bot}\\frac{B_{\\phi}^2}{8\\pi}+\\nabla_{\\bot}p \\\\\n+ \\left(\\frac{B_{\\phi}^2}{4\\pi r}-\\frac{\\rho h u_{\\phi}^2}{r}\\right)\\nabla_{\\bot}r\n- \\frac{B_{p}^2\\Omega^{F}}{4\\pi c^2}\\nabla_{\\bot}\\left(r^2\\Omega^F\\right) \\\\\n+ \\Gamma \\rho \\nabla_{\\bot}\\phi, \n\\end{split}\n\\label{eq:trans-field}\n\\end{equation}\nwhere we have added the collimating component of the gravitational force. \nFor the ease of use in section \\ref{sec:trans-field}, we label the terms as \n$(F_{\\rm curv}, F_{\\rm pbp}, F_{\\rm pbphi}, F_{\\rm p}, F_{\\rm pinch}, \nF_{\\rm cf}, F_{\\rm el}, F_{\\rm grav})$ \nin the order of their appearance in Eq.\\ref{eq:trans-field}.\nThe (poloidal) Alfv\\'en Mach number $M$ is relativistically defined as \n\\begin{equation}\n\tM^2 = \\frac{4\\pi \\rho h u_{p}^2}{B_{p}^2}.\n\\end{equation}\nThe gradient $\\nabla_{\\perp}\\equiv\\mathbf{n}\\cdot \\nabla$ \nis projected perpendicular to the magnetic flux surfaces $\\Psi$, and thus along the \n(inward pointing) electric field.  \nThe light surface of a magnetosphere is located where \n$r_{\\rm L}(\\Psi) = r_{\\rm L}(r,z) \\equiv c \/ \\Omega^F(\\Psi)$\n\\footnote{Thus, the light surface consists of the points \nof intersection between the field lines with their corresponding light cylinder}, it hence depends on the flux-geometry as well as on the rotation profile $\\Omega^F$.  \nEach flux surface \/ magnetic field line crosses the light surface at most once (see\nalso the discussion in \\citet{1997A&A...323..999F})\nSome field lines $\\Psi(r,z)$ never cross the light surface, indicating an asymptotic radius\n$r_{\\infty}(\\Psi) < r_{\\rm L,\\Psi}$. For these field lines relativistic effects due\nto rotation (electric fields) are less important.\nFor others, the asymptotic radius is $r_{\\infty}(\\Psi) > r_{\\rm L, \\Psi}$.\nThe light surface constitutes a critical point of the stationary axisymmetric wind equation only\nin the mass-less limit in which it is identical to the modified poloidal Alfv\\'en surface \n$M_{A}^2 = 1-(r_{A}\/r_{L})^2$ \\citep{1986A&A...156..137C, 1986A&A...162...32C}. \nHowever, it is essential to note that at the light surface the dynamical behavior of \nthe poloidal magnetic pressure term changes - the force changes sign.\nThis leads to the existence of three dynamically different regimes in the asymptotic\n(collimated) region of a relativistic jet (see Fig.\\,\\ref{fig:lightcylinder}).\nIn region I, for all field lines $r_{\\infty}(\\Psi) < r_{\\rm L, \\Psi}$ corresponding\nto $(1-(r\/r_{\\rm L})^2)>0$, and, thus, a de-collimating magnetic pressure term.\nField lines in region II do cross their light cylinder, and, since $r> r_{\\rm L}$,\nthe magnetic pressure term acts as collimating for $r > r_{\\rm L}$.\nField lines in Region III  never reach their light cylinder,\nand here the magnetic pressure term is de-collimating again.\nThe slope of the outer part of the light surface critically depends on the dynamics\nand the magnetic field structure of the outflow in the very inner part.\n\nSimilarly, the forces due to the electric field $E = r\/r_{\\rm L} B_{\\rm p}$ \n(second last term in\nEq.\\,\\ref{eq:trans-field}) scale with the relative position to the light surface,\nhence they are important in region II of the jet formation region only.\n\nEquation \\ref{eq:trans-field} together with Fig.\\,\\ref{fig:lightcylinder} once\nmore demonstrates the need to resolve the whole acceleration and collimation\nregion of a jet in radial and vertical direction. \nOnly when the light surface is taken into account self-consistently, the proper\nforce-balance is applied along and across the flow. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=6cm]{f1}\n\\caption{The different dynamical regimes in special relativistic\ndisk-winds. Region I and III stay sub-relativistic, while region II\nis relativistic, i.e. electric forces are not negligible\n(see Sect.\\,\\ref{sec:force-balance} for a discussion).\n\\label{fig:lightcylinder}}\n\\end{figure}\n\nSimilarly one can derive the parallel-field force equation, it becomes:\n\\begin{equation}\n\\begin{split}\n\t\\frac{B_{p}^2}{4\\pi}\\nabla_{||}M^2 = \n\t\\kappa_{||} \\frac{B_{p}^2}{4\\pi}\\left(1-M^2\\right) \\\\\n\t-\\nabla_{||}\\left(p+\\frac{B_{p}^2}{8\\pi}+\\frac{B_{\\phi}^2}{8\\pi}\\right) \\\\\n\t- \\left(\\frac{B_{\\phi}^2}{4\\pi r}-\\frac{\\rho h u_{\\phi}^2}{r}\\right)\\nabla_{||}r \\\\\n\t- \\Gamma \\rho \\nabla_{||} \\phi\n\t\\label{eq:parallel-field}\n\\end{split}\n\\end{equation}\nwith the necessary definitions $\\nabla_{||}\\equiv\\mathbf{B_{p}}\/B_{p}\\nabla$ and \n$\\kappa_{||}B_{p}^2\\equiv \n\\mathbf{B_{p}}\/B_{p}\n\\left(\n\\mathbf{B_{p}}\n\\cdot \n\\nabla\n\\right)\n\\mathbf{B_{p}}\n$.\nWe see how the change in the Mach-number is mediated by the interplay of tension-, pressure-,\npinch-, centrifugal- and gravitational- acceleration.   Electric fields (pointing in the\nperpendicular direction) don't contribute and the equation reduces to the Newtonian case:\nIn steady-state, there is no electric acceleration!\n\n\\red\nA number of self-similar approaches to the relativistic\n jet formation have beed published (eg. \\cite{2003ApJ...596.1104V}).\n While the self-similar ansatz is a powerfull and highly\n successfull tool to solve the non-relativistic MHD problem\n (starting with the Blandford-Payne solution), we believe\n that using self-similarity for relativistic MHD jets is\n problematic.\n \nWe note that neither the light surface nor the relativistic Alfv\\'en\nsurface obeys a self-similar structure.\nIt is well known that forcing self-similarity into the relativistic \nMHD equations constrains\nthe rotation law for the magnetosphere $\\Omega^F(r)  \\propto r^{-1}$.\n(see also discussion in \\citet{1992ApJ...394..459L,1993ApJ...415..118L}).\nThis is a major difference to the non-relativistic self-similar\napproach.\nWe further note that also the scaling for the electric\nfield depends on the radial position of the light surface \n(see Eq.~\\ref{eq:electric-field}).\nThis is, however, of uttermost importance for the structure of relativistic\nmagnetospheres as the electric field forces play a leading role in the\ntrans-field force-balance (equation \\ref{eq:trans-field}).\nSimilar arguments hold for the inner light surfaces around Kerr \nblack holes or the geometry of the black hole ergosphere. \n\nWe therefore believe that a steady-state self-similar relativistic MHD \napproach is intrisically inconsistent with the relativistic \ncharacteristic of the flow.\n\\black\n\\subsubsection{field line constants}\\label{sec:flconst}\nStationary axisymmetric MHD flows conserve the following five quantities along\nthe magnetic flux-function $\\Psi$.\nFrom the iso-rotation law together with the ideal MHD condition follows\nthe rest mass energy-flux per magnetic induction, \n\\begin{equation}\nk = k(\\Psi) \\equiv \\frac{\\rho u_{p}}{B_{p}}\\label{eq:kf}\n\\end{equation}\nand the iso-rotation parameter \n\\begin{equation}\n\\Omega^F = \\Omega^F(\\Psi) = \\frac{1}{r}\\left(v_{\\phi}-v_{p}\\frac{B_{\\phi}}{B_{p}}\\right)\\label{eq:omegaf}\n\\end{equation}\n(often interpreted as angular velocity of the field lines).  \nIn absence of shocks the (pseudo-) entropy \n\\begin{equation}\nQ=\\frac{p}{\\rho^\\gamma} = Q(\\Psi) \\label{eq:Qf}\n\\end{equation}\nis conserved as well as the angular momentum flux\n\\begin{equation}\nl=-\\frac{I}{2\\pi k c} + r u_{\\phi} = l(\\Psi)\\label{eq:lf}\n\\end{equation}\nand the flux ratio of total energy to rest-mass energy,\n\\begin{equation}\n\\mu = \\frac{\\mathcal{S+K+M+T+G}}{\\mathcal{M}} = \\mu(\\Psi) \\label{eq:mu}\n\\end{equation}\nwhere we identify the individual terms as\n(purely) kinetic energy flux\n$\\mathcal{K} \\equiv (\\Gamma-1) \\rho\\ u_{p}$,\nrest-mass energy flux\n$\\mathcal{M} \\equiv \\rho\\ u_{p}$,\nthermal energy flux\n$\\mathcal{T} \\equiv \\Gamma \\frac{\\gamma}{\\gamma-1} p\\ u_{p}$,\nand gravitational energy flux\n$\\mathcal{G} \\equiv \\rho\\ \\phi\\ u_{p}$,\nrespectively.  \nThe cold, asymptotic limit of (\\ref{eq:mu}) is particularly of interest, it reads\n\\begin{equation}\n\\mu = \\Gamma(\\sigma+1)\n\\end{equation}\nwhere $\\sigma=\\mathcal{S\/(K+M)}$ is the customarily defined magnetization parameter\n- the ratio of Poynting to kinetic flux. \nThis simple relation provides a theoretical maximum for the Lorentz-factor \n$\\Gamma^{*}=\\mu$, when the entire electromagnetic energy is converted into kinetic \nenergy. \n\nThe essential point in the quest for relativistic jets is to find a highly energetic \ndisk solution with values of $\\mu$ beyond the anticipated Lorentz-factor.  \nPrevious studies obtaining highly relativistic jets by \\cite{2007MNRAS.380...51K} do\nnot start from a realistic disk solution but inject the jet material with \nan artificial rotation profile, \na high injection speed $v_{\\rm p} > 0.5 c $, and \na density low enough to obtain a high energy flux $12\\leq \\mu_{\\rm max}\\leq 18$.   \nIn the present study we are aiming to improve on the problems just mentioned \nby applying a physical boundary condition as a Keplerian disk-corona in equilibrium.\n\n\n\\red\n\\subsection{Accretion disk coronae}\\label{sec:coronae}\nIt is our ambition to connect the wind solutions to the ambiance of a realistic accretion disk.  \nIn our simulations, the flow originates in the high entropy atmosphere called a corona.\nOptically thin coronae are an integral part in models of the X-ray features of AGN\n\\citep[e.g.][]{1993ARA&A..31..717M}\nand $\\mu$-Quasars \\citep[e.g.][]{2002MNRAS.332..856N,2003NewAR..47..491M}.\n\nWhile Compton cooling can provide the observed spectra, the heating mechanism is not easily found.  \nJust as in the case of the sun, the coronal heat cannot directly be transfered from the colder photosphere\/accretion disk (according to the second law of thermodynamics) and the nature of vertical energy transport is an active field of research.  \nExternal irradiation of flared disks by the central object (or central disk) is certainly present in a multitude of objects \\citep[see][for a review]{2008ChJAS...8..302C} but might not be the primary energy source.  \nAmong the most promising mechanisms we should highlight magnetic reconnection heating as proposed by \\cite{1991ApJ...380L..51H}.  \n\n\nBetween the mid-plane and the coronal point of injection in our simulations, ideal MHD can not provide a realistic picture.  \nIn order for an accretion disk to {\\em work}, a torque of viscous or magnetic origin has to be exerted onto the material. \nAdditionally, the flux-freezing constraint of ideal MHD must be relaxed since it would lead to an accumulating magnetic pressure that ultimately stops the accretion process.  \nStudies modelling both the accretion motion and the super Alfv\\'enic jet based on a stationary self-similar approach (e.g. \\cite{1993ApJ...410..218W, 1993A&A...276..625F, 1995ApJ...444..848L}) rely on global-scale magnetic fields and ad-hoc assumptions on the viscosity and (anisotropic) magnetic diffusion.  \nIn order for these models to be stable against strong magnetic compression on the one side and the magneto rotational instability (MRI; see \\cite{1998RvMP...70....1B}) on the other, \\cite{1995A&A...295..807F} require equipartition for the thermal and magnetic pressure.\n\\cite{2000A&A...361.1178C} demonstrated the importance of heating for the mass-loading or the jet. \nTime-dependend numerical simulations of these magnetized accretion ejection structures (MAES) were presented by \n\\cite{2002ApJ...581..988C, 2004ApJ...601...90C, 2007A&A...469..811Z} \nand adopt a fixed in time anomalous resistivity profile in order to connect the two dynamical regimes.  \n\nAlthough the stationarity of the aforementioned simulations is possibly hampered by numerical diffusion and low resolution effects, MAES with global-scale fields are to date the most successful models to create collimated outflows.  \n\nIt is now widely believed that the source of viscosity and resistivity in weakly magnetized disks is the turbulence seeded by the MRI.  \nLocal, stratified shearing box simulations by \\cite{2000ApJ...534..398M} suggest a quenching of the MRI in the strongly magnetized coronal region, as the  magnetic scale height exceeds the thermal one.  This is in contrast to the equipartition fields proposed for the MAES.  \nMagnetic buoyancy of the large-scale fields eventually created by a turbulent dynamo could then provide the coronal heating.  \nTypically, the MRI results in toroidally dominated coronal fields with $B_{\\phi}^2 > 10 B_{p}^2$ and opening the field lines towards a topology favorable for wind acceleration remains a challenge.  In the context of the solar corona this process is discussed by \\cite{2003ApJ...599.1404W}.  \n \nBy restraining parts of the accretion disk structure, \\cite{2003A&A...398..825V} could achieve outflows in time-dependend simulations of disk-corona structures where the magnetic field is sustained by a mean-field dynamo.  \nAnalytic models of this turbulent disk-corona-outflow connection are however in still its infancy (see the discussion by \\cite{2004ApJ...616..669K} and attempts by \\cite{2009ApJ...704L.113B}).\n\n\n\\black\n\\section{Model setup for the MHD simulations}\\label{sec:model setup}\n\\red\nWith the aforementioned considerations we choose the following model for our investigation.  \nA global-scale poloidal field favorable of wind acceleration is adopted.  Whether it is advected by the accretion flow or created by an underlying dynamo is not of our concern.  \nThe jet base resembles a corona in the sense that it is hot (electron  temperature $\\sim10^9 \\rm K$), has no mechanism of cooling, is non-turbulent (no viscosity) and highly ionized (infinite conductivity).  We choose a Keplerian rotation profile for the field lines.  \nThe flow starts with sub-escape velocity and we investigate sub-magnetosonic injection where mass loading is determined by the internal dynamics as well as mass-fluxes imposed by the boundary condition.  \nWe perform axisymmetric special relativistic MHD simulations of jet formation for a set of different magnetic field geometries and field strengths.  In the following we discuss the numerical realization of our problem.\n\\black\n\\subsection{Boundary conditions}\n\nGiven the 2.5 dimensional nature of the problem, three geometrical boundaries have to be \nprescribed. \nThese are the inlet boundary along $z=0$ from which material is injected into the domain \n({\\em inflow})\nand the two outer boundaries at $r=r_{\\rm end}$ and $z=z_{\\rm end}$ where we expect \nmaterial to leave the computational domain ({\\em outflow}). \nThe boundary condition along $r=0$ ($\\rm R_{beg}$) follows from cylindrical symmetry.  \nFigure \\ref{fig:setup} gives an overview of the different regions.  \n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=7cm]{f2}\n\\caption{Sketch of the different regimes of our grid and boundary.  \nIn both directions we set 20 equidistant cells in $[0,1]$.\nThen follows a stretched grid until we add five equidistant cells one unit-radius before the outflow boundary.  For $r\\in[0,1]$, the hydrostatic corona is fixed to minimize the influence of the central region on the disk-wind.  \n\\label{fig:setup}}\n\\end{figure}\n\n\\subsubsection{Injection boundary ($Z_{beg}$)}\\label{sec:injection}\nPursuing the aim to follow the acceleration of an disk wind from as close to the accretion \ndisk as possible, we start with a sub slow-magnetosonic wind.\nWe are hence free to choose four constraining boundary conditions without overdetermining\nthe system (see \\cite{Bogovalov1997} and Appendix \\ref{sec:MHD-inj} for more details).  \n\nOur choice is to fix the toroidal electric field component $E_{\\phi}=0$. \nThis suppresses the evolution of the bounding poloidal magnetic field and is realized\nby requiring $\\mathbf{v_p || B_p}$.  \nThe field line iso-rotation is kept constant in time and follows a Keplerian rotation law\n$\\Omega^F\\propto r^{-1.5}$.\nUnless specified otherwise, the boundary condition starts out initially in a force-free state with zero toroidal field $\\Omega^F = v_{\\phi}(r)\/r$ corresponding to a disk in hydrodynamic equilibrium.  \nThis is an essential ingredient as - within stationary ideal MHD - $\\Omega^{F}$ just \nequals the mid-plane angular velocity of the material $\\omega(r)$. \nRelaxation of the infinite conductivity constraint would, however, lead to an inequality \n$\\Omega^F(r) \\le \\omega(r)$ owing to the diffusion of magnetic field. \nFor these reasons $\\Omega^F$ should closely follow the expected disk rotational profile \nand should be limited by the maximal velocity in the mid-plane, \ntypically at the inner edge of the disk located at $r=1$.  \n\nA radial force-equilibrium along the whole boundary is enforced by balancing the centrifugal and \npressure support against gravity via the sub-Keplerianity of the\nrotation $\\sqrt{\\chi} = v_\\phi(r=1)\/v_K$ \n\\red\nthat also determines the inlet density, \n\\black\nwhere $v_K$ is the circular velocity that alone sustains against gravity at $r=1$.\n\\red\nA more convenient parametrization is in terms of the\n\trelative temperature \t\n\t$\\epsilon \\equiv c_s^2\/v_\\phi^2 = (\\gamma-1)(1-\\chi)\/\\chi$.\n\\black\nWe investigate two cases - a hot corona with $\\epsilon=2\/3$ and a version with $\\epsilon=1\/6$ \n\\red\n($\\chi=0.5$ and $\\chi=0.8$)\n\\black.\n\nIf we interpret $r=1$ as the innermost stable circular orbit (ISCO) around a black hole, $v_{\\rm K}$ is a measure of the black hole spin.  In the case of a Schwarzschild black hole it is $v_{K}\\simeq0.6c$ while we choose the scaling velocity $v_{\\phi}(r=1) = 0.5 c$ for convenience.  \n\nThe inner disk-edge is numerically difficult to model because of the transition to the inflow of the disk-wind and the steep gradients in gravity.  Within $r<1$, the so-called plunge region, a physical solution would allow for (radial and vertical) accretion onto the central object. \nSince the dynamics in this area would then require a general relativistic treatment  which we cannot provide in this context, we simply minimize the dynamical effect of this region by freezing the hydrostatic solution initially present in the domain.  \nOther authors have assumed a thin funnel flow along the axis (e.g. \\cite{1999ApJ...526..631K}) or added an internal sink-cell (like \\cite{2002ApJ...581..988C}) to circumvent this problem \nThe transition between the ``inner corona'' and disk-wind is smoothed via the Fermi step function \n$F(x) = (1+e^{(1-x)\/0.1})^{-1}$.  \nRotational support is thus turned on by the setting \n\\begin{eqnarray}\n\\rho_{\\rm d}(r,z) &=& \\frac{1}{1-\\chi F^2(R)} (R+r_{S})^{1\/(1-\\gamma)}\\label{eq:rhodisk},\\\\\nv_{\\phi}(r) &=& \\sqrt{\\chi} v_{K} F(R) R^{-0.5}.\n\\end{eqnarray}\nDensity given by equation \\ref{eq:rhodisk} and the coronal pressure $p$ constitute the third and fourth fixed in time conditions.  \n\nWe emphasize that is is not possible to specify both injection velocity and density-profile and thus the mass-flux for sub-(magneto)sonic flows, \nas this is determined by the sonic point.  \nTherefore we match the vertical velocity $v_{z}$ to the domain via $\\partial_{z} v_{z} = 0$, \nwhile the radial component follows from the $E_{\\phi}=0$ condition. \nWe limit the injection speed by the local slow magnetosonic speed in the case when \nthe velocity just above the boundary becomes trans-sonic.\nThis provides the fifth constraint needed in that case.  \n\nWith the induction of a toroidal magnetic field component in the jet, the rotational velocity \nneeds to be adjusted in order to satisfy Eq.\\,\\ref{eq:omegaf},\n\\begin{equation}\nv_{\\phi} = r\\Omega^F + \\frac{v_{p}}{B_{p}}B_{\\phi}\n\\end{equation}\nas we apply the condition $\\partial_{z}B_{\\phi} = \\partial_{z} B_{r} = 0$ \n(while $B_{z}$ then follows from $\\mathbf{\\nabla\\cdot B = 0}$).  \n\nBy letting the jet solution alone determine Poynting- and mass- flux, \nwe loose control over the energy flux parameter $\\mu$ and the limiting asymptotic \nLorentz factor $\\Gamma^*$.  \nIt will rather be a consequence of the MHD under the constraints we have given, while we have used our freedom to provide a boundary most closely resembling a realistic hot disk corona.\nA graphical summary of the disk wind boundary conditions is shown in Fig.\\,\\ref{fig:disk}. \n\nIn order to extend the parameter space towards higher $\\mu$, we also investigate\ncases where we have over-determined the boundary conditions by specifying the mass-flux \nthrough\n\\begin{equation}\nv_{z}(r,0) = v_{\\rm inj} v_{\\phi}(r,0)\n\\end{equation}\nsimilar to e.g. \\citet{1997ApJ...482..712O, 2002A&A...395.1045F}.\nAnother parameter run adopts a $1\/r$ profile for the toroidal \nmagnetic field $B_{\\phi}(r) = - \\eta F(r) \/ r $ in order \nto specify the Poynting flux with an additional parameter $\\eta$.  \nThese runs \n\\red\nand the influence of the over-determination \n\\black\nis discussed separately in section \\ref{sec:extended}.  \n \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\columnwidth]{f3}\n\\caption{\nProfiles of the fixed in time variables for the inlet in hydrodynamic equilibrium.  Here we give constraints on $\\Omega^F,\\rho,p,E_{\\phi}$ ($E_{\\phi}=0$ not shown).  \nParameters are $v_{\\rm K}=0.5,\\epsilon=2\/3$. \nThe thin dotted line is the Fermi step function used to smooth those variables experiencing a sharp transition at the inner disk radius $r=1$.  \n\\label{fig:disk}}\n\\end{figure}\n\n\\subsubsection{Outflow boundaries ($R_{end},Z_{end}$)}\nThe standard outflow boundary conditions for many numerical codes are zero-gradient \nconditions, which are usually sufficient as the plasma velocities are constrained to be outward-pointing.  \n\n\\red\nIn the case of sub fast-magnetosonic outflows, this strategy is unfortunately insufficient as the flow inside of the domain will depend on the flow beyond the boundary via the incoming characteristics.  \nJust as for the inlet boundary, the now missing information has to be supplied by constraints that describe best the physical conditions downstream of the boundary.  \nIn the case of an outflow, the conditions leading to an untampered flow are however impossible to know a priori.  \nA way to circumvent this unphysical feedback is to avoid any causal contact by moving the boundary far away such that the characteristics will not enter the domain of interest within the simulated time.  \n\\black\n\n\\red\nWhen considering a boundary outside of causal contact ``very far away'', we estimate for Alfv\\'en waves to travel over $10^3$ scale radii within the anticipated simulation time.  The computational effort of such huge grids does not allow a large parameter study at the current time and we must leave this option for future endeavours.  \n\\black\n\n\n\nIn the absence of a substantially better solution, zero-gradients are  used for the \nprimitive variables except for magnetic fields for which this simple approach leads \nto artificial electric currents implying an inward-pointed Lorentz-force.  \nEspecially for low plasma-$\\beta$ this may result in a devastating artificial\ncollimation - preventing any steady state to establish and artificially collimating \nthe outflow increasingly thin with time. \n\n\\cite{1999ApJ...516..221U} have performed a systematic study comparing different \napproaches for outflow conditions including a (toroidal) force-free condition \n$\\mathbf{j_{p}||B_{p} = 0}$ and a more sophisticated version including an additional numerical factor that needs to be determined a posteriori.  \nFor the outflow conditions in our simulations we instead recover the magnetic field \ncomponents by imposing constraints on the poloidal \n($j_{r}=-\\partial_{z} B_{\\phi}$, $j_{z}=r^{-1}\\partial_{r}rB_{\\phi}$) and\ntoroidal $(j_{\\phi}=\\partial_{z}B_{r}-\\partial_{r}B_{z})$ electric  currents.  \nFor the toroidal magnetic field (poloidal electric current) we radially extrapolate \nthe expected $1\/r$ law of a marginal $j_{z}$ at the radial end ($\\rm R_{end})$.\nUsing $\\partial_{z}B_{\\phi}=0$ allows to specify $j_{r}$ at the upper end of the domain ($\\rm Z_{end})$.  \nConcerning the poloidal magnetic field components we implement a current-free \nboundary condition by enforcing $j_{\\phi} = 0$.\nThis is a novel approach designed to minimize spurious effects of collimation.\n\\red\nWe convinced oursevels that boundary effects have only a marginal effect on the solution by varying the grid-size and geometry.  \nFor a detailed discussion and comparison of various outflow conditions we refer to appendix \\ref{sec:zerocurrent}.  \n\\black  \n \nWe note that, as a further complication, a fully relativistic version for a force-free\nor force-balance boundary conditions would also need to take into account electric \nforces. \nWe have estimated the impact of such an upgrade and found that due to the geometry\nof our outflow (in particular the location of the light surface)\nit would play a minor role and is thus not worth the effort to implement.\n\n\\subsection{Initial conditions}\nAs initial state we prescribe a force-free coronal magnetic field,\n$F^{\\alpha \\beta}j_{\\beta} = 0$, together with a gas distribution in hydrostatic\nequilibrium.\nBoth is essential in order to avoid artificial relaxation processes\ncaused by a non-equilibrium initial condition.\nWe apply a polytropic equation of state $p=K\\rho^\\gamma$ with a ``classical''\npolytropic index of $\\gamma=5\/3$ since our flows are always cold when compared\nto the rest-mass.  \nTo further strengthen this choice, we performed a comparison simulation with the \n\\cite{1948PhRv...74..328T} equation of state as described by \\cite{Mignone2005} \nwhich produced an identical jet once the hot shock has passed through.  \nThe constant $K$ is determined by the radial force-balance of the inlet.  \n\nFor the initial magnetic field configuration we apply two different geometries.\n\\red\n\tField configuration A\n\\black\nis a potential field of hourglass shape as applied by \n\\citet{1997ApJ...482..712O} and \\citet{2002A&A...395.1045F} with the magnetic field components\n\\begin{eqnarray}\nB_{r} &=& \\frac{1}{r} \n\t\\left[1-\\frac{(z+z_{d})}{\\left(r^2+(z+z_{\\rm d})^2\\right)^{1\/2}}\\right]\\\\\nB_{z} &=& \\frac{1}{\\left(r^2+(z+z_{\\rm d})^2\\right)^{1\/2}},\n\\end{eqnarray}\ncorresponding to a vector potential \n\\begin{equation}\nA_{\\phi} =\\frac{1}{r} \\left[\\sqrt{r^2+(z+z_{d})^2}-(z+z_{\\rm d})\\right]\\label{eq:aphi-op}\n\\end{equation}\nin cylindrical coordinates with $B_{r} = -\\partial_{z} A_{\\phi}$ and\n$B_{z} = r^{-1}\\partial_{r} \\ r A_{\\phi}$.\nThe dimensionless disk thickness $z_{\\rm d}$ with $(z_{\\rm d}+z)>0$ is introduced to\navoid kinks in the field distribution for $z<0$ (the ghost zones) and we choose $z_{d}=1$ for convenience.  \n\nOur other option for the initial magnetic field \n\\red\n(configuration B)\n\\black\nis the ``split monopole'' \n\\citep{1987PASJ...39..821S} with the magnetic field components  \n\\begin{eqnarray}\nB_{r} &=& \\frac{r}{\\left(r^2+(z+z_{d})^2\\right)^{3\/2}}\\\\\nB_{z} &=& \\frac{z+z_{d}}{\\left(r^2+(z+z_{d})^2\\right)^{3\/2}}.\n\\end{eqnarray}\nIn the split-monopole setup \nthe parameter $z_{\\rm d}$ defines the offset of the \nfiducial center of the monopole from the grid origin, and adjusts \nthe initial angle of the field lines with respect to the disk-surface.\nWe either adopt an angle of $\\theta=85^\\circ$ or $\\theta=77^\\circ$ for the field line passing through $r=1$.\nSimilar to Eq. \\ref{eq:aphi-op}, the split monopole field can be described by a vector potential\n\\begin{equation}\nA_{\\phi} = \\frac{1}{r}\n\\left[1 - \\frac{z+z_{\\rm d}} {\\left(r^2+(z+z_{\\rm d})^2\\right)^{1\/2}}\\right].\n\\end{equation}\n\nThe fields are scaled to satisfy the the choice of the plasma-$\\beta$\n\\begin{equation}\n\\beta \\equiv \\left.\\frac{B_{p}^2}{8\\pi p}\\right|_{r=1,z=0}\n\\end{equation} \nat the inner disk radius.  \n\n\\red\n\tIt should be kept in mind that plasma-$\\beta$ largely varies along the disk boundary.  In configuration A, the profile $\\beta(r)$ monotonically decreases until for large radii it is $\\beta(r)\\propto r^{-0.5}$ leading to a magnetically dominated outer corona.  \n\tIn the split-monopole, $\\beta(r)$ decreases first to a minimum value (at $r^*(\\theta=77^{\\circ})\\approx5$ and $r^*(\\theta=85^{\\circ})\\approx15$) and increases for large radii according to $\\beta(r)\\propto r^{1.5}$ leading to thermal dominance.  \n\\black\n\nIn summary, for our injection boundary condition we are left with the following five dynamical parameters,\n\\begin{equation}\n(v_{\\rm K},\\beta,\\epsilon, v_{\\rm inj},\\eta),\n\\end{equation}\nwhere strictly speaking we are only allowed to choose the first three when launching \nsub-slow. \nAn overview of the simulations performed in this parameterization is shown in \nTab.\\,\\ref{tab_all}.  \n\n\\subsection{Numerical grid and physical scaling}\n\nWe use a numerical grid of $512\\times1024$ cells applying cylindrical coordinates.\nOnward from the inner region $(r<1,z<1)$, which is resolved with $20\\times20$ \nequidistant cells, we apply a stretched grid with the element size increasing \nby a factor of $\\lesssim 1.005$. \nThis leads to a domain size of $(r\\times z)=(100\\times200)r_{\\rm i}$ \ncorresponding to $(300\\times600)\\ r_{\\rm s}$ if $r_{\\rm i}=3r_{\\rm s}$ (see sketch in figure \\ref{fig:setup}).  \nStaggered magnetic fields treated via constrained transport \\citep{1999JCoPh.149..270B} are used to ensure $\\mathbf{\\nabla\\cdot B = 0}$.\n\nBecause of the constraints imposed on the cell aspect ratio by the zero-current \nboundary (appendix \\ref{sec:zerocurrent}), \nwe set the last five grid cells to be equally spaced with maximal aspect \nratios $<3\/1$.  \n\n\\red\nThe dimensionless nature of our simulations allows for various astrophysical interpretations.  We provide a physical scaling of simulation variables (marked with a prime) in the following paragraph.  \n\nSince velocities are given in terms of the speed of light ($c'=1$), relativistic simulations are in need of only two additional scales.    \nThe simulation variables are connected to their physical counterparts via\n\\begin{eqnarray}\nv&=&v'c;\\ \\ l=l'l_{0}; \\ \\ t=t't_{0}=t'l_{0}\/v_{0}; \\ \\ \\rho = \\rho ' \\rho_{0} \\\\\np &=& p'p_{0} = p' \\rho_{0}c^2; \\ \\ B = B'B_{0} = B'\\sqrt{4\\pi\\rho_{0}c^2}.\n\\end{eqnarray}\nIf we assume a Schwarzschild black hole as central body, we may set the spatial scale $l_{0}=6r_{g}$, equating the inner disk radius with the ISCO.  Then it becomes\n\\begin{eqnarray}\nv_{0} &=& 3\\times 10^{10} {\\rm cm}\\ {\\rm s}^{-1}\\\\\nl_{0} &=& 9\\times 10^{5}{\\rm cm} \n                       \\left(\\frac{\\rm M_{\\bullet}}{\\rm M_{\\odot}}\\right)\\\\\nt_{0} &=& 3\\times 10^{-5} {\\rm s} \n                       \\left(\\frac{\\rm M_{\\bullet}}{\\rm M_{\\odot}}\\right).\n\\end{eqnarray}\nAssuming a physical outflow mass-loss rate in terms of the Eddington limited accretion rate\n$\\dot{M}=0.01\\dot{M}_{\\rm edd}$ \nwe can provide a scale for the density by comparison to the mass loss rate of the simulation $\\dot{M}'$\n\\begin{equation}\n\\rho_{0} = 6\\times 10^{-7}\\frac{1}{\\dot{M}'}\\left(\\frac{M_{\\bullet}}{M_{\\odot}}\\right)^{-1}\\rm g\\ cm^{-3}\n\\end{equation}\nwhere we applied a radiative efficiency of $\\eta^*=0.1$.  \nThe scaling of pressure and magnetic fields then follows as \n\n\\begin{eqnarray}\np_{0} &=& 5\\times 10^{14}\\frac{1}{\\dot{M}'}\\left(\\frac{M_{\\bullet}}{M_{\\odot}}\\right)^{-1}\\rm g\\ cm^{-1}\\ s^{-2} \\\\\nB_{0} &=& 8\\times 10^7\\left.\\dot{M}'\\right.^{-0.5}\\left(\\frac{M_{\\bullet}}{M_{\\odot}}\\right)^{-0.5} \\rm Gauss . \n\\end{eqnarray}\n\nUnder these considerations, the only remaining scaling parameter is the mass of the compact object $M_{\\bullet}$.  \nNeglecting additional physical processes as radiation pressure or radiative \ncooling leaves us with a scale-free model that can be applied to any disk-wind\nlaunched jet around compact objects.  \nTable \\ref{tab:scaling} provides a fiducial scaling for a microquasar with $M_{\\bullet}=10 M_{\\odot}$ and for an AGN with $M_{\\bullet}=10^8 M_{\\odot}$.  \n\\black\nThe scale-free nature becomes obvious if we recall the rest-frame temperatures\nfor an ideal gas,\n$\n        T = \\frac{p'\\ c^2}{\\rho' k_{\\rm B}}\\langle \\mu \\rangle \\rm m_{\\rm p}\n$\n\\citep{1989cup..book.....A},\nwith the mean molecular weight $\\langle \\mu\\rangle$,\n    the proton mass ${\\rm m_{\\rm p}}$,\nand the Boltzmann constant $k_{\\rm B}$.\nFor ionized hydrogen $\\langle\\mu\\rangle=0.5$ one would find temperatures of\n$T = (p'\/\\rho')\\ 5.45\\times10^{12} \\rm K$,\nwhile for an electron-positron plasma ($\\mu\\simeq1\/2000$) the temperatures are \nlower by three orders of magnitude.\nThese ultra-high scaling temperatures are only reached in the very inner corona between\nthe central object and the inner disk radius. \nAs we do not intend to follow the dynamics here, this does not really pose a problem. \nIn principle, once $p'\/\\rho'\\gtrsim 1$, the\nequation of state transcends towards $\\gamma = 4\/3$ according to a Synge-gas.\nIn the jet, thermal pressure quickly looses importance and the temperatures \nare significantly lower.\n\n\n\n\\begin{deluxetable}{lllllll}\n\\tablewidth{\\columnwidth}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Fiducial scaling\\label{tab:scaling}\n}\n\\tablehead{\n\\multirow{2}*{ $\\frac{M_{\\bullet}}{M_{\\odot}}$  }&\n$l\/l'$ & \n$t\/t'$ &\n$\\rho\/\\rho'$& \n$p\/p'$ &\n$B\/B'$ \\\\\n& \n$\\rm [cm]$ &\n$\\rm [s]$ &\n$\\rm [g\\, cm^{-3}]$ &\n$\\rm [g\\, s^{-2}\\, cm^{-1}]$ &\n[Gauss]\n}\n\\startdata\n$10^8$ & \n$9\\times10^{13}$&\n$3\\times 10^{3}$ &\n$1.8\\times10^{-16}$  &\n$1.5\\times10^{5}$  &\n$1.4\\times 10^3$\\\\\n$10$ &\n$9\\times10^{6}$ &\n$3\\times10^{-4}$&\n$1.8\\times10^{-9}$ &\n$1.5\\times10^{12} $&\n$4.4 \\times 10^{6}$\n\\enddata\n\\tablecomments{Scaling for simulation WA05 ($\\dot{M}'=32.67$) assuming a physical mass loss rate of $\\dot{M}=1\\%\\dot{M}_{edd}$ with an efficiency of $\\eta*=0.1$.  }\n\\end{deluxetable}\n\n\n\\section{Results and discussion}\\label{sec:results}\n\nWe now present the results of our numerical simulations considering the formation\nof relativistic MHD jets from accretion disks.\nEach simulation consumed approximately 48 hours on 16 processors.  \nThe overall goal is to test whether the paradigm of MHD self-collimation\nof non-relativistic jets established from numerical simulations\n\\cite{1995ApJ...439L..39U, 1997ApJ...482..712O, 1999ApJ...526..631K, 2002A&A...395.1045F}\nalso holds in the relativistic case.\n\n\n\\begin{deluxetable*}{ccccccl||crrrrr}\n\\tablecaption{Parameter summary of our disk-wind simulations. \\label{tab_all}\n}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{\\textwidth}\n\\tablehead{\n \\colhead{ID}                      & \n \\colhead{Top}               &\n \\colhead{$\\beta$}                     & \n \\colhead{$\\epsilon$}       & \n \\colhead{$v_{\\rm inj} $}        & \n \\colhead{$\\eta$}         &\n \\colhead{Remarks}      &\n \\colhead{$\\Gamma_{\\rm max}$} &\n \\colhead{$\\mu_{\\rm max}$} &\n  \\colhead{$\\xi$} &\n \\colhead{$v_{p,\\rm max}$} &\n \\colhead{$r_{\\rm jet}$} &\n \\colhead{$\\dot{M}$}\n }\n\\startdata\n\n\n\n\n\n\n\n\nWA01 & A & 0.2 & 2\/3 & var. & var. & - & \n              1.23&         1.33 &        18.84 &       0.54 &        20.26 &        23.77\n\\\\\n\nWA02 & A & 1 & 2\/3 & var. & var. & - & \n       1.27 &         1.37 &        11.47 &       0.58 &        21.86 &        26.62\n\\\\\nWA03 & A & 2 & 2\/3 & var. & var. & - & \n\n1.26&         1.34 &        10.69 &       0.57 & \n       22.21 &        24.15\n\\\\\nWB01 & B & 1 & 2\/3 & var. & var. & $\\theta=77^\\circ$ & \n       1.33&         1.41 &        8.27 &       0.64 &        24.19 &        16.48\n\\\\\nWB02 & B & 1 & 2\/3 & var. & var. & $\\theta=85^\\circ$ & \n1.29&         1.33 &        8.19 &       0.60 & \n       21.27 &        15.66\n\\\\\nWA04 & A & 0.2 & 1\/6 & var. & var. & - & \n       1.27&         1.38 &        13.40 &       0.58 &        21.14 &        36.08\n\\\\\nWA05 & A & 1 & 1\/6 & var. & var. & - &\n1.25&         1.33 &        9.84 &       0.57 &         22.77 &        32.67\n\\\\\nWA06 & A & 2 & 1\/6 & var. & var. & - &\n1.25&         1.32 &        9.34 &       0.57 & \n       22.92 &        29.27\n\n\n\n\n\n\n\n\\enddata\n\\tablecomments{Columns are from left to right: simulation ID; \n initial magnetic field distribution (Top): potential field (A) or \n     split monopole (B);\n plasma-$\\beta$; accretion disk temperature parameter $\\epsilon$; \n     injection speed $v_{\\rm inj}$ as a fraction of $v_{\\phi}(r)$;\n     toroidal disk-field scaling $\\eta$;  specific remarks; to the right we show values of the steady state, the maximum Lorentz factor $\\Gamma_{\\rm max}$ collimation degree $\\xi$, maximal poloidal velocity $v_{p,\\rm max}$, jet radius $r_{\\rm jet}$ and the total mass flux out of the domain $\\dot{M}$. \n}\n\\end{deluxetable*}\n\n\n\n\\subsection{Overall evolution of the outflow}\n\nThe initial evolution of the disk corona is governed by the propagation of \ntoroidal Alfv\\'en waves launched due to the rotation of the field line \nfoot-points.  \nThe initial force-free magnetic field structure is adapted to a new dynamic\nequilibrium according to a rotating wind magnetosphere.\n\nA wind is launched from the disk boundary and is continuously accelerated \ndriving a shock front through the initial hydrostatic corona and sweeping\nthis material out of the computational domain (Fig.\\,\\ref{fig:simul}).\nThe disk wind evolves into a collimated outflow of super-magnetosonic speed.\nAlong the symmetry axis the hydrostatic initial condition is very well \npreserved.\nOnce the bow shock has passed through the domain, the jet mass flux declines\nto a value which is solely governed by the internal outflow dynamics\nand the injection boundary conditions. \nSimilarly, the post-shock magnetic field distribution follows as well\nfrom the internal outflow dynamics and has in principle little in common with \nthe initial setup.  \nCertain combinations of boundary conditions for mass flux and magnetic field\nwill result in a quasi-stationary\\footnote{We denote the dynamical state as\n{\\em quasi} stationary as due to Keplerian disk rotation the disk outflow\na large radii has evolved for a considerably lower number of disk rotations.\nTherefore, a slight change in the dynamical state of the outer outflow\ncan be expected after another few outer, thus 1000s of inner rotations.}\nstate of the outflow evolution \n(see next section).\nFrom this point onwards we can start our investigations of collimation and\nacceleration.\nIn this paper we concentrate on analysis when the flow has reached a\nquasi-steady state.\nWe usually terminate our simulations after 500 inner disk-rotations $P$, \nwhile a quasi-steady state is established over most of the domain after \nabout 200 rotations.\n\nFigure \\ref{fig:simul} shows the time evolution for two exemplary \nsimulations with an initial hourglass-shaped potential field distribution\n(case A) and a split-monopole field distribution (case B),\neach for the parameter choice $(\\beta,v_{K},\\epsilon) = (1,0.5,2\/3)$.\n\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{f4}\n\n\n\n\n\n\\caption{\nFormation of a relativistic MHD jet.\nShown is the Lorentz factor (color gradient) in terms of $log_{10}(\\Gamma-1)$ \nat the time of 25, 250 and 500 (left to right) inner disk rotations. \n{\\em Top:} Field distribution case A, hourglass-shape potential field \n(run WA02).\n{\\em Bottom:} Split monopole initial field distribution case B (run WB01). \nShown are poloidal magnetic field lines (solid white lines);\nthe critical MHD surfaces (solid black lines);\nthe light surface $r\\cdot \\Omega = c$ (solid black line). \nFor time $T\/P=250$ electric current flow lines are added (solid green line).\nTime step $T\/P=500$ also show the initial field lines for comparison \n(dashed white lines).\n\\label{fig:simul}}\n\\end{figure*}\nThe figure shows the \nLorentz factor, the poloidal magnetic field lines, poloidal electric current flow lines and the critical MHD surfaces.\nIn addition the light surface\nis drawn.\n\nPhenomenologically, the solutions form a magnetic nozzle with, depending on the disk flux distribution, considerable difference in the width, but comparable final opening angles of the fast component.  A broader initial field distribution (case B) also results in broader and faster winds where the material originating from the inner disk is more effectively thinned out.  \nIn analogy to hydrodynamic nozzles, the flow reaches the slow-magnetosonic speed directly above the throat.  \nCollimation happens mainly before the fast-magnetosonic surface is reached. Afterwards, the opening angle of a given field line is approximately conserved.  \n\nOf particular interest is the electric current distribution\n(shown for the time step $T\/P = 250$).\nThe electric current distribution \nis a consequence of the dynamical\nevolution of the outflow and therefore a direct outcome of the\ndisk boundary magnetic flux profile and the Keplerian field line rotation.\n\nIn general, the electric current leaves the outer disk to return within the fast component of the outflow.  \nIt is expected to enter the inner disk and then flow radially outwards \nclosing with the outgoing current.\nSuch butterfly-shaped circuits are expected in Keplerian disks while\nthe $j_{r}$ plays a leading role \nin the disk-jet feedback \\citep{1997A&A...319..340F}.\nA positive radial electric current in the disk-corona supports accretion \nby braking the disk material due to its magnetic torque $j_{r}\\times B_{z}$ \nsimilar to a Barlow-wheel\\footnote{\nHowever, this region is not resolved \nwithin our numerical domain, as it is located below our injection boundary\nas part of the underlying non-ideal MHD accretion disk. \nSee \\cite{2002ApJ...581..988C, 2007A&A...469..811Z} for non-relativistic\nsimulations of the disk-jet interaction}.\n\nThe inclination between the poloidal current vector and the magnetic field line\nindicates the direction of (de-) collimating magnetic forces acting on the flow.\nWhen the inclination becomes less than $90^{\\circ}$, the Lorentz force\n$j_{p}\\times B_{\\phi}$ changes from collimation to de-collimation.  \nThis can be clearly seen in the snapshots at $T\/P=250$ of the case A simulation where actual\nfield lines are indicated in white and initial field lines in red.\nIn the actual field distribution, the field lines are somewhat pushed away from\nthe surface $\\mathbf{j_{p} \\perp B_{p}}$ (this is also where magnetic acceleration is most effective).\nFor case B this happens  \nbeyond the light surface.\nAs a result, both electric and magnetic forces deflect the flow towards \nthe disk boundary which leads to a highly unstable layer just above \nthe outer disk.\nWe will provide an in-depth analysis including all forces acting on the flow\nin section \\ref{sec:trans-field}.\n\nThe locations of the characteristic surfaces are signatures for the MHD flow.\nDepending on the initial magnetic flux distribution (case A,B) and the \nmass flux profile (see also \\citet{2006ApJ...651..272F}), \nthis location may vary a great deal. \nIn our case B simulations, we generally observe surfaces which leave the domain\nin radial direction (parallel to the disk surface).\nFor the case A simulations these surfaces tend to \"collimate\" leaving the domain\nin vertical direction. \nThe latter implies a two-layered structure of the jet - a central super-fast magnetosonic jet surrounded by a \nsub-Alfv\\'enic outflow.\nThis is an interesting aspect for observational modeling \nand for stability analysis of sheath-spine jets \n\\citep{2005MNRAS.356..859P, 2007ApJ...662..835M, 2007ApJ...668L..27K, \n       2007ApJ...664...26H, 2009MNRAS.397.1486B}.\nThe broad wind launched from the outer regions of the disk has much lower velocities, decreasing continously with increasing launching-radius. For example, the terminal velocity of the flow originating from $r_{\\rm fp}>32$ of the case A simulations drops below $0.2c$, consistent with the X-ray absorption features observed in a mounting number of AGN\n\\citep{2006AN....327.1012C, 2009A&ARv..17...47T}.  \nIn principle, our dynamical models can provide basic ingredients (e.g. flow geometries and velocity gradients) for the modeling of spectral line profiles of disk winds \\citep{1995MNRAS.273..225K, 2008MNRAS.388..611S}.  \n\n\nFollowing  \\cite{2006ApJ...651..272F}, we may define an average collimation \ndegree $\\xi$ of the outflow measured as the fraction of vertical and radial \nmass flux through equal-area surfaces at a certain height (here at $z=z_{m}$),\n\\begin{equation}\t\n\t\\xi = \\frac{\n\t\\int_{0}^{r_{m}}r \\Gamma v_{z} \\rho |_{z_{m}}\\ dr\n\t}{\n\t\\int_{z_{m}-r_{m}\/2}^{z_{m}}r_{m}\\Gamma v_{r}\\rho |_{r_{m}}\\ dz\n\t}.\n\\end{equation}\nSimilarly, we define a mass flux weighted jet radius,\n\\begin{equation}\n\tr_{jet} = \\frac{\n\t\t\\int_{0}^{r_{m}}r \\Gamma v_{z} \\rho |_{z_{m}} \\ dr\n\t}{\n\t\\int_{0}^{r_{m}} \\Gamma v_{z} \\rho |_{z_{m}}\\ dr\n\t}.\n\\end{equation}\nThe corresponding values for $\\xi$ and $r_{\\rm jet}$ \nderived for $z_{m}=200$ at the upper end of \nthe domain and for time $t\/P=500$ are given in Tab.\\,\\ref{tab_all}\nalong with the maximum Lorentz factor $\\Gamma_{\\rm max}$, the maximum\npoloidal velocity $v_{p,max}$, and the total mass flux $\\dot{M}$.  \nFigure \\ref{fig:diag} shows the time evolution of these quantities in the top panel.  In general, we observe that the collimation degree $\\xi$ is the most sensitive tracer for secular trends among the observables mentioned.  \nIn the lower panel, we show the evolution of jet-power in the individual energy channels leaving the computational domain (radial and vertical).  \n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\columnwidth]{f5}\n\\caption{\n\\red\nTime evolution of characteristic quantities.  \\textit{Top panel:} After an initial adjustment till $t\/P\\simeq200$, mass flux $\\dot{M}$, jet radius $r_{jet}$, collimation degree $\\xi$, maximal Lorentz-factor $\\Gamma_{\\rm max}$  and poloidal velocity  $v_{p, \\rm max}$ cease to evolve. \n\\textit{Lower panel}: Power in the individual energy chanels out of  the domain. Thermal power ($T$)\npeaks when the shock reaches the upper boundary and is negligible otherwise.\nThe total outgoing power (labeled accordingly) is dominated by rest-mass ($M$) and Poynting flux ($S$). Also shown are the gravitational ($G$) and (purely) kinetic ($K$) contributions (simulation run WA02).  \n\\black\n\\label{fig:diag}}\n\\end{figure}\nAfter the re-configuration of the initial stationary-state to the dynamical solution, the partitioning of energies is completed at around $t\/P=100$.  Thermal energy-flux peaks when the hot bow-shock passes through the upper boundary.  Far away from the central object, gravitational and thermal energy-flux are negligible\nThe integrated energy-flux is dominated by rest-mass, reflecting the fact that only the inner component reaches significant Lorentz-factors.  \nThe balance between Poynting and kinetic flux is of particular interest. Figure \\ref{fig:diag} shows merely the end result of the spatial conversion history with the remaining electromagnetic energy $\\mathcal{S}$ above the purely kinetic part $\\mathcal{K}$. \nMore detailed insight into how this is established is provided in the following section using an individual field line. \n\\subsection{Stationary state analysis}\nSimulations starting from an initial field distribution A evolve into\na quasi-stationary flow solution after about 200 inner disk rotations.\nFigure \\ref{fig:stationary_flow} shows our reference simulation WA04\nat time $t\/P=250$, including an enlarged subgrid of the innermost area\nof the domain.\n\nSteady state solutions are helpful to understand the flow structure\nfor a number of reasons. \nFirstly, by using MHD conservation laws, the conserved quantities \n(see Sect.\\,\\ref{sec:flconst})\nallow to identify the momentum and {\\em energy channels} of the flow during \nacceleration and collimation.\nSecondly, by using the force-balance equations \n\\ref{eq:trans-field},\\ref{eq:parallel-field} we may identify the\nleading forces on the material along the outflow.\nThirdly, the cross-check for conserved quantities provides another test \nfor the quality of our setup and the numerical approach.\nA secondary indicator of stationarity is the alignment of poloidal velocities\nwith the poloidal magnetic field lines, $E_{\\phi} = 0$.\nFigure \\ref{fig:stationary_flow} shows corresponding velocity vectors\nconfirming this picture.\n\nThis is confirmed by checking in detail the complete set of integrals of motion\nof the MHD-flow $k,\\Omega^F,Q,l,\\mu$ as defined in equations $\\ref{eq:kf}-\\ref{eq:mu}$. \nFigure \\ref{fig:const} shows the relative deviation of these quantities\nfrom their average value along a given field line after $t\/P=500$.  \nThe integrals are conserved within $1\\%$-accuracy already right above the \ninjection boundary - clearly demonstrating the quality the choice \nof our numerical setup, in particular the injection boundary conditions\ncarefully constructed from an equilibrium of Keplerian rotation and \ngas pressure.\n\nDue to the differential rotation law, the number of Keplerian rotations \n$t\/P(r)$ scales with radius as $t\/P(r) = (t\/P) r^{-3\/2}$, implying\nthat at the end of our simulations ($t\/P=500$), \nwe have performed roughly one rotation at $r=64$ and half a rotation \nat $r=100$. \nNonetheless the integrals of motion for the field line $r_{\\rm fp}=64$ are \nconserved within $0.1\\%$.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\columnwidth]{f6}\n\\caption{Field line constants (conserved quantities).\nShown is the deviation from the average value along the field line\nrooted at $r_{\\rm fp}=2$ (simulation run WA02).\n\\label{fig:const}}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8cm]{f7a}\n\\includegraphics[width=8cm]{f7b}\n\\caption{Logarithmic (rest-frame) density of the stationary \nflow (simulation run WA04).\nShown are poloidal magnetic field lines (solid white), \nelectric current flow lines (solid black), \ncharacteristic MHD surfaces (various dot-dashed green), \nsurface of escape velocity (dotted green), \nlight surface (solid green).  \nArrows in the top plot indicate the velocity field.\nThe bottom figure is an enlarged picture of the central region \nindicating the three regimes defined by the light surface.  \n\\label{fig:stationary_flow}}\n\\end{figure}\n\n\\subsubsection{Collimating and accelerating forces}\\label{sec:trans-field}\nIn this section we identify the forces responsible for jet acceleration\nand collimation applying the steady-state parallel and transversal\nforce-equilibrium Eqs.\\,\\ref{eq:trans-field},\\ref{eq:parallel-field}.\n\nFig.\\,\\ref{fig:f-hot} compares these forces for a number of reference \nsimulations (WB01, WA02, WA05)\nalong a field line rooted at $r_{\\rm fp}=2$.\nAs check for consistency, we also show the gradient of \nthe Mach number $a \\equiv B_{p}^2\/4\\pi\\nabla_{||}M^2$ which \njust coincides with the summation \nof the parallel forces, indicating a steady state\n(see yellow solid and black dashed line).\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{f9a}\n\\includegraphics[width=0.49\\textwidth]{f9b}\n\\includegraphics[width=0.49\\textwidth]{f9c}\n\\includegraphics[width=0.49\\textwidth]{f9d}\n\\includegraphics[width=0.49\\textwidth]{f9e}\n\\includegraphics[width=0.49\\textwidth]{f9f}\n\\caption{Accelerating (left column) and collimating (right column) \nforces along the field line rooted at $r_{\\rm fp}=2$ for the three\nreference simulations (from top to bottom):\n{\\it WB01} (split monopole), \n{\\it WA02} (hourglass potential field, hot case), and \n{\\it WA05} (hourglass potential field, cool case).\nShown are the contributions from gravity, gas pressure gradient, \ncentrifugal force, poloidal magnetic field pressure gradient, \npoloidal field tension and pressure\ngradient, toroidal magnetic field pressure gradient and tension, and\nforces due to the electric field.\nVertical lines indicate the critical surfaces - the Alfv\\'en point along\nthis field line, denoted by 'A', the light cylinder radius denoted by 'lc',\nand the fast magnetosonic point denoted by 'F'.\nIn this logarithmic representation a change of sign in the force\ndirection is indicated by the singularities along the graphs.\n\\label{fig:f-hot}}\n\\end{figure*}\n\nIn general, the outflow starts with sonic speed and is first launched by thermal pressure in the hot disk corona, respectively \nthe centrifugal force in the colder version.\nUntil the Alfv\\'en point, the Lorentz-force of the poloidal electric current \n($F_{\\rm pbphi}+F_{\\rm pinch}$) is the main magnetic driver.\nUltimately the poloidal tension ($F_{\\rm curv}$) keeps the\nacceleration up even above the fast surface.\n\nConcerning the transverse force, we reproduce the expected sign-change \nof the curvature (tension) force (first collimating until the Alfv\\'en \nsurface, de-collimating beyond) and the poloidal pressure force \n(de-collimating until the light cylinder, collimating beyond).\nFor the cross-field balance, we observe the following three regimes: \n\nJust on top of the inlet, the main de-collimating forces besides poloidal magnetic pressure are thermal pressure in the hot case (WB02, WA02) and centrifugal support in the colder case (WA05).  Gravity is here the strongest force towards the origin and the situation just reflects the radial force-equilibrium we have applied for the inlet boundary.  This is the hydrodynamic regime.  \n\nAt the Alfv\\'en point, the residual of the pinch- and toroidal pressure-force ($\\mathbf{j_{p}}\\times B_{\\phi}$) is the main collimator, balanced by the centrifugal term.  Thermal pressure quickly looses importance.  \nThis is the magneto-hydrodynamic regime.  \n\nIn the asymptotic region beyond the light-cylinder, de-collimation by electric forces overcomes the centrifugal force and is balanced by the poloidal magnetic pressure that changes its sign at the light-cylinder (best seen in WB01).  \nThis is the relativistic regime.  \n\nTo get a global impression on the relative importance of the individual\nforces we show a radial cut throughout the asymptotic jet in figure\n\\ref{fig:force-balance}.\nThe strongest forces arise across the inner asymptotic light \nsurface which separates field lines of high angular velocity from \nthose in the non-rotating corona along the axis.\nHere the electric de-collimation is essential.\nThe $B_{\\phi}(r)$ profile is curled up from the inner disk radius along \nthe outflow - resulting in a magnetic pressure gradient that works \nin unison with the toroidal field pinch force until at some radius the \ntoroidal field surpasses its maximum and decreases (negative gradient).\n\nThe strong gradients in toroidal field and rotation induce a current sheet\nand give rise to an electric charge.\nThe space charge $\\rho_{\\rm e}=(1\/4)\\pi\\nabla\\cdot \\mathbf{E}$ is positive\nclose to the axis and changes its sign at a critical line as defined\nby \\cite{1969ApJ...157..869G}.\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\columnwidth]{f10}\n\\caption{\nTrans-field force cut at $z=200$, inner ($lc_{i}$) and outer ($lc_{o}$) light-cylinder.  The differentially rotating field lines are fastest at the inner disk-radius, resulting in the inner light-cylinder - here electric de-collimation is important.  The $B_{\\phi}(r)$ profile is curled up from the inner disk radius onwards and results in a magnetic pressure gradient that works in unison with the pinch force until the toroidal field surpasses its maximum. Within $r<3$, we omit the curves for thermal and poloidal pressure. These terms fluctuate around $\\pm10^{-5}$ while balancing each other.  \n (run WA02)\n\\label{fig:force-balance}}\n\\end{figure}\n\n\n\\subsubsection{Energy conversion}\nIn the simulations where injection is sub-magnetoslow, the energy flux is \nnot a free parameter, but is consistently determined by the simulation of the disk-wind. \nIt is hence of interest how the partitioning and conversion is realized.    \nFrom the values of $\\mu_{\\rm max}$ given in Tab.\\,\\ref{tab_all} it is obvious \nthat our disk-corona supports only mildly relativistic flows \nbelow $\\Gamma=1.5$ (section \\ref{sec:flconst}).  \n\nIn Fig.\\,\\ref{fig:vp} (bottom left panel) we show the efficiency $\\sigma$ of Poynting flux to kinetic \nflux conversion along the field line with $r_{\\rm fp}=2$ in the fast component of the jet.  \n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{f8a}\n\\includegraphics[width=0.49\\textwidth]{f8b}\n\\caption{ \nDynamical quantities as a function of the distance along the \nfield line rooted at $r_{\\rm fp}=2$ for the model WA02. \nVertical lines indicate the slow-magnetosonic, Alfv\\'en, light surface and fast magnetosonic transitions (from left to right).\n\\textit{Left:} \nIsorotation parameter $\\Omega^F$, poloidal and toroidal velocity are shown in the top panel. \nLorentz factor $\\Gamma$, energy conversion efficiency $\\sigma$, \nand normalized total energy flux $\\mu$ in the bottom panel.  The flow is below equipartition already at the base of the jet.  \n\\textit{Right:}\nComplete energy flux ratios.  In the asymptotic region, thermal and gravitational fluxes are obviously negligible. \n\\label{fig:vp}}\n\\end{figure*}\nHere, $\\sigma$ is below equipartition already at the inlet and it further \ndecreases as $\\Gamma$ approaches $\\mu$.  \nIn Fig.~\\ref{fig:vp} (top left) the toroidal velocity shows that the flow decouples \nfrom co-rotation with the magnetic field at the Alfv\\'en point.\nBeyond the Alfv\\'en point, angular momentum is then carried predominantly\nby the magnetic field.  \nThe poloidal velocity increases from low injection value (sonic velocity) to \n$\\sim0.5c$. \nFurther acceleration can not be expected as the bulk of the energy is already \nin kinetic form.  \nThe right panel of Figure \\ref{fig:vp} shows the individual energy channels compared to the\nrest-mass flux for the same field line. \nAt the base of the jet, the strong poloidal electric currents (a strong toroidal field)\ngive rise to an outflow with $\\mathcal{K}<\\mathcal{T}<\\mathcal{-G}<\\mathcal{S}<\\mathcal{M}$, predominantly transporting energy via rest-mass and Poynting-flux.  \n\nThe kinetic energy flux surpasses the thermal flux at the Alfv\\'en point and \nfurther overcomes the gravitational binding energy term shortly thereafter.  This is not surprising, since the escape-surface can be close to the Alfv\\'en surface at least for the inner field lines (see also Fig. \\ref{fig:stationary_flow}).\n\nOnly then, the cold limit $\\mu = \\Gamma(\\sigma+1)$ is applicable - it is certainly \nvalid in the asymptotical outflow where thermal and gravitational energy fluxes \nare negligible.\n\n\n\\subsection{Dependence on the launching environment}\nFor the simulations described up to now, we have performed in addition several \nparameter runs in order to investigate how the resulting jet dynamics\ndepends on the (prescribed) launching conditions - the disk corona\n(see Tab.~\\ref{tab_all}). \nWe now focus on the impact of the plasma $\\beta$ and the disk temperature \nparameter $\\epsilon$.  \n\nIn general, a low $\\beta$ (a stronger magnetic field) we find that the outflow tends\nto collimate more, as indicated by the higher average collimation degree $\\xi$ \nand a lower momentum weighted jet radius $r_{\\rm jet}$.  \nThis in principle decreases the MHD acceleration efficiency which critically \ndepends on the divergence of flux surfaces.  \nIt is straightforward to define the mass flux-weighted (half-) opening angle of \noutflow,\n\\begin{equation}\n\t\\theta_{\\dot{M}} = \\rm atan\\ \\xi^{-1},\n\\end{equation}\nwhich translates to angles of $3^{\\circ}<\\theta_{\\dot{M}}<7^\\circ$ for the \noutflows under consideration.\n\nThe impact of the magnetic field strength on the amount of mass flux is not clearly visible, \nas the two simulations with $\\epsilon=1\/6, 2\/3$ show a different trend. \nAs the $\\epsilon$-parameter is simply a proxy for the disk corona density, \nit will affect the collimation in the following manner: \nA higher inflow density lowers the Alfv\\'en surface towards the disk\nsurface which in turn broadens the current topology \nand therefore widens the flow.  This is also the trend that we observe in the indicators $\\xi$ and $r_{\\rm jet}$.  \nGiven that the injection speed calculated iteratively from the outflow\nsimulation approaches the slow-magnetosonic speed, \nwe expect the mass flux to scale as $\\dot{M}\\propto \\sqrt{p \\rho}$.  \nIn fact, this is approximately realized since we have \n$\\dot{M}(\\epsilon=1\/6)\/\\dot{M}(\\epsilon=2\/3) =\\sqrt{2.5}\\simeq 1.6$.  \n\nThe change of the initial split-monopole inclination $\\theta$ has little effect\non the overall jet collimation angle.\nIn particular, we observe an opposite trend as\nthe wider initial field with $\\theta=77^\\circ$ ends up slightly more\ncollimated than the one with $\\theta=85^\\circ$.  \nClearly a wider initial field leads to a larger jet radius $r_{\\rm jet}$.  \nHere, we like to stress the point that for the final steady state solutions\nin our simulations the initial field structure is important only insofar\nas it also prescribes the poloidal magnetic field profile along the outflow\nlaunching boundary. \nThe field structure is completely changed from the initial steady structure\nto a new dynamic equilibrium.\nThus it makes no sense to compare the collimation of the initial field\nwith the collimation of the outflow field distribution. \n\n\n\\par \nHaving pointed out the crucial role of the \n\\red\nvertical energy flux from the disk surface \n\\black\n$\\mu$ and the closely\nrelated quantity $\\sigma = \\mathcal{S\/(K+M)}$, \nwe now study the two most promising handles in increasing $\\mu$.  \nThat is (i) a decrease in mass flux $\\mathcal{M}$ and \n        (ii) an increase in Poynting-flux $\\mathcal{S}$.  \nWe first focus on (i) and describe (ii) thereafter.  \n\n\\subsubsection{Towards low mass loading}\\label{sec:light-winds}\n\\red\nA way to obtain high-speed jets seems to be a lower mass\nload injected into a similarly-strong magnetic flux.\nAccording to the well-known Michel-scaling \\citep{1969ApJ...158..727M}, \nthe asymptotic outflow velocity depends on the mass flux \n$u_{\\infty}\\propto \\dot{M}^{-1\/3}$.  \nWe investigate this interrelation running another set of simulations \nwhere we change the mass flux by prescribing a low injection velocity\n(instead of lowering the density of the injected gas,\n see Tab.~\\ref{tab:extended1})\n\n\\begin{deluxetable*}{ccccccl||crrrrr}\n\\tablecaption{Effect of mass-loading\n\\label{tab:extended1}}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{\\textwidth}\n\\tablehead{\n \\colhead{ID}                      & \n \\colhead{Top}               &\n \\colhead{$\\beta$}                     & \n \\colhead{$\\epsilon$}       & \n \\colhead{$v_{\\rm inj} $}        & \n \\colhead{$\\eta$}         &\n \\colhead{Remarks}      &\n \\colhead{$\\Gamma_{\\rm max}$} &\n \\colhead{$\\mu_{\\rm max}$} &\n  \\colhead{$\\xi$} &\n \\colhead{$v_{p,\\rm max}$} &\n \\colhead{$r_{\\rm jet}$} &\n \\colhead{$\\dot{M}$}\n }\n\\startdata\nI001 & A & 0.2  & 2\/3  &  0.01  &  var.  & - &  \n1.66&         2.01 &        25.61 &       0.75 & \n       21.11 &        10.63\n\\\\\n\nI01 & A & 0.2  & 2\/3  &  0.1  &  var.  & - &  \n       1.52&         1.76 &        25.58 &       0.71 & \n       20.49 &        13.78\n\\\\\n\nI02 & A & 0.2  & 2\/3  &  0.2  &  var.  & - &  \n1.38&         1.55 &        26.02 &       0.65 & \n       19.34 &        17.53\n\\\\\n\nI04 & A & 0.2  & 2\/3  &  0.4  &  var.  & - &  \n1.31&         1.45 &        25.30 &       0.60 &        17.07 &        25.84\n\\\\\nI08 & A & 0.2  & 2\/3  &  0.8  &  var.  & - &  \n1.16&         1.24 &        40.05 &       0.47 &        14.74 &        51.30\n\\\\\nI12 & A & 0.2  & 2\/3  &  1.2  &  var.  & - &  \n       1.21&         1.21 &        33.77 &       0.50 &        14.10 &        81.88\n      \n\\enddata\n\\tablecomments{As in table \\ref{tab_all}, but for the extended parameter study with a given $v_{\\rm inj}$.}\n\\end{deluxetable*}\n\n\nHowever, we observe that for low injection speeds $v_{z}<0.4\\ v_{\\phi}$ \nthe resulting (numerical) mass flux does not follow the expected linear \nrelation $\\dot{M} \\propto v_{\\rm inj}$.\nInstead, for lower and lower injection speed it approaches some\nseemingly unphysical offset value (Fig.~\\ref{fig:diag-comp}).\nThis value is unphysical in the sense that it departs from the \nprescribed value at the boundary condition \n$d\\dot{M}_{\\rm inj}\/dr = 2\\pi r \\rho_{\\rm inj} v_{\\rm inj}$.\nThis difficulty arises because the injected flow is sub-magnetosonic\nand thus overdetermined by simultaneously assigning a profile in $\\rho$ and $v_{z}$.\n\n\nAlthough this is in principle problematic, it is not necessarely fatal \nfor the investigation of low mass flux outflows.\nThe mass flux is governed by the magneto-slow point and is thus re-arranged\nalong the flow.\nIndeed we find that above the magneto-slow surface the field line constants \nare very well conserved. \nThis indicates that the flow dynamics transcends at the magneto-slow surface \nfrom a seemingly unphysical state into a MHD flow that satisfies the critical \nconditions at the magnetosonic surfaces.\n\nThis technique of self-adjusting the sub-slow mass inflow, however, turned out\nto be limited towards lower mass fluxes.\nThe resulting mass fluxes are unfortunately, still too high and do not allow \nsubstantially higher magnetisation\n(e.g. $\\mu_{\\rm max}=2.01$ for simulation I001 compared to $\\mu_{\\rm max}=1.33$ in run WA01).  \nFigure \\ref{fig:diag-comp} shows the trends concerning collimation $\\xi$, jet radius $r_{\\rm jet}$,\nand integral mass flux $\\dot{M}$ in the asymptotic flow. \n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\columnwidth]{f11}\n\\caption{Jet collimation and dynamics against injection speed parameter $v_{\\rm inj}$, The horizontal line indicates the mass flux when $v_{\\rm inj}$ is not specified (WA01).  \nFor $v_{\\rm inj}<0.4$, the mass flux approaches an obviously unphysical offset value.\nAt higher $v_{\\rm inj}$, the mass flux follows the expected linear dependence on the injection parameter (thin solid line).  \n\\label{fig:diag-comp}}\n\\end{figure}\nIncreasing the mass flux enhances collimation while $\\mu_{\\rm max}$ decreases accordingly.  \nFor high injection speed, we observe that the wind originating from the very inner disk evolves\ninto a thin ballistic flow layer that has little in common with the jets we are interested in.  \n\\black\n\\subsubsection{Poynting dominated flows}\\label{sec:extended}\nGiven the limitations mentioned above, we prescribe {\\it a priori} \nthe limiting energy flux parameter $\\mu$ by constraining both the mass flux and \nthe Poynting flux along the injection boundary - with the hope of thus providing \na sufficiently energetic disk wind. \n\nTo achieve this, we adopted a fixed-in-time toroidal magnetic field distribution,\n$B_{\\phi}\\propto -\\eta\/r$ which necessarily changes \n$\\Omega^F(r)$\\footnote{\nIn case of a central black hole causality requires that $r\\Omega^F(r)< 0.6$ \nwhich corresponds to the ISCO velocity for the Schwarzschild case with $r_{ISCO}=1$ \nin our scaling.}.\nThe toroidal field distribution following a $1\/r$ profile corresponds to $j_{z}=0$ \nand has a profound physical motivation as it anticipates the radial currents expected\nin the disk-corona.  \n\nTable \\ref{tab:extended2} summarizes the simulation runs performed within this setup. \nThese simulations have a considerably stronger toroidal magnetic field at the injection\npoint.\nWe apply up to $B_{\\phi}\\sim 4 B_{p}$.\n\\begin{deluxetable*}{ccccccl||crrrrr}\n\\tablecaption{Poynting dominated flows\n\\label{tab:extended2}}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{\\textwidth}\n\\tablehead{\n \\colhead{ID}                      & \n \\colhead{Top}               &\n \\colhead{$\\beta$}                     & \n \\colhead{$\\epsilon$}       & \n \\colhead{$v_{\\rm inj} $}        & \n \\colhead{$\\eta$}         &\n \\colhead{Remarks}      &\n \\colhead{$\\Gamma_{\\rm max}$} &\n \\colhead{$\\mu_{\\rm max}$} &\n  \\colhead{$\\xi$} &\n \\colhead{$v_{p,\\rm max}$} &\n \\colhead{$r_{\\rm jet}$} &\n \\colhead{$\\dot{M}$}\n }\n\\startdata\n\nM02 & A & 0.2  & 2\/3  &  0.1  &  2  & - &\n       2.18&         3.29 &        27.55 &       0.86 &        23.71 &        22.48\n\\\\\n\nM04 & A & 0.2  & 2\/3  &  0.1  &  4  & - & \n       3.51&         7.46 &        8.96 &       0.94 &        29.00 &        48.85\n\\\\\n\nM08 & A & 0.2  & 2\/3  &  0.1  &  8  & - &\n       6.11&         25.61 &        6.8377 &       0.97 &        35.89 &        80.40\n\\enddata\n\\tablecomments{As in table \\ref{tab_all}, but for the extended parameter study with a given $v_{\\rm inj}$ and $\\eta$.}\n\\end{deluxetable*}\n\nAn exemplary process enhancing large-scale toroidal fields in the disk corona \ncould be the MRI-driven dynamo under current investigation by many authors \n\\citep[e.g.][]{2000ApJ...534..398M, 2003A&A...398..825V}.\nThe jet eventually evolving from these disks is not propelled by the\n\\citet{1982MNRAS.199..883B} mechanism, but driven by the toroidal magnetic \npressure \\citep{1996LNP...471...74C, 2003ApJ...599.1238D, 2004ApJ...605..307K}. \nThese so called Tower jets have initially been proposed by \\cite{1996MNRAS.279..389L} and directly extract Poynting-flux from the dist rather than first converting rotational energy into the twisted magnetosphere that is present at the Alfv\\'en surface.\\footnote{See \\cite{2007Ap&SS.307...11K} for a review. \nNote also that these jets have successfully been reproduced by \\cite{2005MNRAS.361...97L} in laboratory experiments with purely radial\ncurrent distributions at the base.}\n\nThe simulations described here are of a mixed type, since they combine large-scale open field lines with a toroidal field emerging from a radial current.  \nFor an example simulation of this type, we show the conversion of energy in \nFig.~ \\ref{fig:vp-ext2} similar to Fig.~\\ref{fig:vp} for the low-energy case.  \nBy design, the injected Poynting flux surpasses the rest mass flux with $\\sigma\\simeq5$ \nwithin the fast outflow component.  \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\columnwidth]{f12}\n\\caption{As in Fig.~\\ref{fig:vp}, however calculated for simulation run M04.  The injected flow is strongly magnetized and remains Poynting-dominated when leaving the computational domain. \n\\label{fig:vp-ext2}}\n\\end{figure}\n\nThe outflow, which is initially Poynting flux dominated, does not reach \nequipartition at the fast magnetosonic surface $r=r_{\\rm F}$, where we merely find $\\Gamma\\sim\\mu^{1\/3}$ following \\cite{1969ApJ...158..727M,1998MNRAS.299..341B}.  \nIn the asymptotic region $r\\gg r_{F}$, the length scales for additional flow acceleration \nand collimation would increase exponentially with $\\Gamma\\propto (\\mu \\ln r)^{1\/3}$ \n(e.g. \\cite{1994PASJ...46..123T}), which is clearly beyond the reach of our numerical \nmethod.  \n\n\nOur simulations indicate that, given sufficient Poynting flux, bulk Lorentz factors derived\nfrom AGN jet observations can be obtained within several hundred Schwarzschild radii, \n$\\Gamma \\simeq 6$ for model M08. \nSince acceleration has proven to be most effective around the Alfv\\'en point \nwhich is expected to be very close to the central object ($r_{\\rm A}<r_{\\rm lc}$), \nwe expect this conclusion to remain valid also for higher $\\mu$ and thus higher \nterminal $\\Gamma$ flows.  \nHowever, we note that when increasing $\\sigma$, the energy conversion efficiency decreases, \na situation commonly denoted as $\\sigma$-problem in pulsar winds \n\\citep{1974MNRAS.167....1R, 1984ApJ...283..694K}.\nSeveral authors have recently addressed this issue with partly controversial results\n\\citep{2009MNRAS.394.1182K, 2009ApJ...699.1789T, 2009ApJ...698.1570L},\nso that after the successful acceleration towards relativistic speeds, Poynting-flux could still remain and one is tempted to ask: Is there a $\\sigma$-problem for AGN-jets? \nThe simulations presented here are not fit to answer this question satisfactory.\nHowever, we certainly know that the mildly relativistic disk winds presented earlier\ndo not suffer from this, as they are launched already in sub-equipartition.  \n\n\\section{Summary}\\label{sec:conclusions}\nWe have presented ideal MHD simulations of the formation of special relativistic disk \nwinds using the PLUTO 3.0 code.  \nOn the technical side, the key points are: \\\\\n\\textit{i)} \nThe inclusion of (Newtonian) gravity allows us to specify an astrophysically sensible boundary\ncondition of a hydrodynamically stable disk corona.\nWe can thus consistently follow the acceleration from initially sub-escape velocity winds.  \\\\\n\\textit{ii)} \nMuch dedication has been put in the development and testing of a novel realization for the\noutflow boundary that enables us to simulate for hundreds of inner disk rotations while \nminimizing spurious collimation due to artificial boundary currents.  \nOur detailed study of jet collimation is possible only through this effort.  \n\nAs a general result we obtain well collimated jets with a mass flux weighted half-opening\nangle of $3 - 7^\\circ$ and mildly relativistic velocities depending on the launching\nconditions for the outflow.\nThe flow collimation happens mainly in the classical (non-relativistic) regime before the \nlight surface.   \nA major result of our simulations is that we - for the first time - self-consistently \ncalculated the shape of that light surface.\nThe light surface determines the \"relativistic\" charater of the flow. Material which\ntraverses the light surface experiences the full relativistic effects.  \n\nWe can identify three dynamically distinct regions in terms of flow collimation. \\\\\n\\textit{i)} \nIn the hydrodynamic regime upstream of the Alfv\\'en surface, \ngravity balances thermal and magnetic pressure, respectively the centrifugal force in the colder case. \\\\\n\\textit{ii)} \nIn the magneto-hydrodynamic regime following the the Alfv\\'en surface downstream,\nthe residuals of magnetic pinch and the toroidal magnetic pressure gradient balances\nthe centrifugal force.  \\\\\n\\textit{iii)} \nIn the relativistic regime located downstrem of the light surface, the poloidal magnetic \npressure gradients now impose a collimating force against electric field de-collimation.\nElectric forces ultimately overcome the classical magneto-centrifugal contribution.  \n\nA steep rotation profile of the field line as given by a Keplerian disk\nresults in a light surface geometry which steepens for large radii.\nDepending on the magnetic field profile, the light surface may even\ncollimate along the flow for large radii.\nIn such a case the relativistic core inside the light surface is naturally confined \nby a non-relativistic wind.\nThe ability of both the relativistic jets and the non-relativistic disk winds to collimate may provide confining agents for an axial ultra-relativistic funnel \nwhich could probably launched by the Blandford-Znajek process.  \n\nThe relatively slow winds found to arrise at large distances of around 100 Schwarzschild radii may be observed as X-ray absorption winds in radio-quiet AGN.  \n\nIn the case of Blandford-Payne disk winds ($B_{\\phi}=0$ initially), the \noutflow is kinetic energy-dominated with the ratio of electromagnetic energy flux \nto kinetic energy flux $\\sigma<1$ already at the jet base and with this ratio \nfurther de-creasing downstream the outflow.  \nThese disk winds start out at sonic speed and reach only mildly relativistic speeds \nup to Lorentz factors $\\Gamma < 1.5$.\nWe have also investigated cases where the jet Poynting flux is increased by\ndirectly injecting an additional toroidal magnetic field from the disk boundary.\nIn this case we achieve terminal Lorentz-factors up to $\\Gamma_{\\rm max}\\simeq 6$\nwhile the super fast-magnetosonic jet remains Poynting-flux dominated.  \n\n\\begin{acknowledgements}\nWe thank Andrea Mignone and the PLUTO team for the possibility to use the\nPLUTO code and for his support with the relativistic module.  \nNumerical simulations were performed on the PIA cluster of the Max Planck Institute for Astronomy (Heidelberg) located at the Rechen-Zentrum in Garching.\nO.P. likes to thank Bhargav Vaidya for his commitment in numerous discussions.  \nWe are pleased to acknowledge clarifying conversations with Max Camenzind.\n\\end{acknowledgements}\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\n\tWe study $\\cat(0)$ groups and the boundaries of the spaces on which they act. A group is $\\cat(0)$ if it acts geometrically (properly discontinuously, cocompactly, and by isometries) on some $\\cat(0)$ space. The motivating question for this article is the following:\n\n\n\n\\begin{question} \\label{question: pconn}\n\tIf $G$ acts geometrically on a 1-ended $\\cat(0)$ space $X$, when is the visual boundary, $\\partial X$, path connected?\n\\end{question}\n\nThere are two main reasons to investigate this property for boundaries of $\\cat(0)$ spaces. The first comes from the theory of hyperbolic groups. All 1-ended hyperbolic groups have locally connected boundaries  \\cite{BestvinaMess,Swarup,BowCutPoints}. Some natural examples of non-hyperbolic $\\cat(0)$ groups, such as $F_2\\times \\mathbb{Z}$, show that this phenomenon does not occur in general. However, a boundary which is connected and locally connected is also globally path connected. In this sense, path connectivity is the `next best thing' one can hope for in the boundary of a $\\cat(0)$ group. At the same time, there are known examples of $\\cat(0)$ groups which act on spaces with non-path connected visual boundaries (see Example \\ref{example: Croke Kleiner} for one). Therefore, we want to find conditions on $G$ (or on $X$) which determine when $\\partial X$ is path connected.\n\nAnother reason to be interested in path connectivity is due to its relationship with semistability. A proper, 1-ended $\\cat(0)$ space $X$ is \\textit{semistable at infinity} if any two geodesic rays are properly homotopic. Being semistable at infinity is a quasi-isometry invarient, so showing a group is semistable at infinity comes down to finding an appropriate space on which this group acts. Geoghegan  conjectured that all $\\cat(0)$ groups are semistable at infinity (in fact, he has conjectured that all finitely presented groups are semistable at infinity). Geoghegan also shows that given a $\\cat(0)$ space $X$ if $\\partial X$ is path connected, then $X$ is semistable at infinity \\cite{GeoghPconnSemistable}. Piecing all of this together, knowing that $G$ acts geometrically on a $\\cat(0)$ space with a path connected visual boundary shows that $G$ is semistable at infinity.\n\nThe majority of this paper is dedicated to answering Question \\ref{question: pconn} for a class of $\\cat(0)$ groups which exhibit more hyperbolicity than others: $\\cat(0)$ groups with isolated flats.  These groups are hyperbolic relative to a collection of flat stabilizers and therefore share many properties with hyperbolic groups. Hruska and Ruane have found necessary and sufficient conditions for a  1-ended $\\cat(0)$ group with isolated flats to have a locally connected boundary \\cite{HR17}. There are many examples which do not have locally connected boundaries, see Section \\ref{section: examples} for a few. We therefore study the general case. In this paper, we prove the following:\n\n\\begin{customthm}{\\ref{thm main}}Let $G$ be a 1-ended $\\cat(0)$ group with isolated flats acting geometrically on $X$. Then $\\partial X$ is path connected.\n\\end{customthm} \n\nAs an immediate corollary (via the results in \\cite{GeoghPconnSemistable}) we have the following:\n\n\\begin{corollary}\n\tLet $G$ be a 1-ended $\\cat(0)$ group with isolated flats. Then $G$ is semistable at infinity.\n\\end{corollary}\nOther authors have some results which overlap with this corollary. For example, Mihalik and Swenson  show that many 1-ended relatively hyperbolic groups are semistable at infinity \\cite{MihalikSwensonSemistable}. However, they do not cover all cases. They assume that their relatively hyperbolic groups have a trivial maximal peripheral splitting and many $\\cat(0)$ groups with isolated flats have non-trivial maximal peripheral splittings. Hruska and Ruane  show that most groups which are hyperbolic relative to polycyclic groups are semistable at infinity \\cite{HruskaRuaneSemistable}. They assume that these groups have no non-central elements of order 2. We require no assumptions on the $\\cat(0)$ groups with isolated flats to conclude it is semistable at infinity.\n\nAlong the way to proving Theorem \\ref{thm main} we introduce a combination theorem for $\\cat(0)$ spaces and their boundaries:\n\n\\begin{customthm}{\\ref{combo}}\n\n\t\tLet $X$ be a $\\cat(0)$ space and $\\mathcal{B}$ a block decomposition. Suppose the following conditions hold:\n\t\n\t\\begin{enumerate}\n\t\t\\item for each $B\\in \\mathcal{B}$, $\\partial B$ is path connected,\n\t\t\\item for each $W\\in \\mathcal{W}$, $\\partial W$ is non-empty, and\n\t\t\\item all irrational rays are lonely.\n\t\\end{enumerate}\n\tThen $\\partial X$ is path connected.\n\\end{customthm}\n\nFor exact definitions of the terms in Theorem \\ref{combo}, see Section \\ref{section block decomp}. The example to keep in mind is when we have a $\\cat(0)$ group $G$ which is an amalgamated product $G=A\\ast_C B$. In many cases of interest, there is a natural `tree-of-spaces' $X$ on which $G$ acts geometrically and then Theorem \\ref{combo} essentially says if (1) $A$ and $B$ act on spaces $X_A$ and $X_B$ with path connected boundaries, (2) $\\partial X$ is connected,  and (3) a technical condition related to how the boundaries of $X_A$ and $X_B$ interact in $X$, then $\\partial X$ is path connected. Amalgamating $\\cat(0)$ groups is the primary method for forming new $\\cat(0)$ groups. In order to fully understand what types of spaces can appear as the boundary of a $\\cat(0)$ group, we need to understand properties of the boundaries of amalgams.\n\n\\begin{Outline}\n\tIn Section \\ref{section: examples}, we describe three key examples that we return to throughout the paper. Sections \\ref{section visual boundary}, \\ref{rel hyp section}, and \\ref{section:isolated flats} provide the necessary background information on the visual boundary, relatively hyperbolic groups, and $\\cat(0)$ groups with isolated flats. We formally define a block decompositions and prove Theorem \\ref{combo} in Section \\ref{section block decomp}. \n\t\n\tThe remaining sections build up to the proof of Theorem \\ref{thm main}. In Section \\ref{section:fixing a space}, for a given 1-ended $\\cat(0)$ group $G$ with isolated flats, we use the group's maximal peripheral splitting (introduced by Bowditch in \\cite{Bow01}), Hruska-Ruane's convex splitting theorem \\cite{HR17}, and Bridson-Haefliger's Equivariant Gluing Theorem \\cite{BH99} to fix a space $X$ on which $G$ acts geometrically. Hruska-Kleiner's work \\cite{HK05} shows that if $G$ is $\\cat(0)$ with isolated flats, then the boundary is well-defined. Therefore fixing a space on which $G$ acts does not affect the boundary. We then focus on a collection of subsets $Y_v\\subset X$ which cover $X$. These subsets are defined at the end of Section \\ref{section:fixing a space}. In Sections \\ref{section:decomp}, \\ref{section:proper and cocompact}, and \\ref{section:local conn at flats} we show that $\\partial Y_v$ is path connected. Applying  Theorem \\ref{combo} finishes the proof of Theorem \\ref{thm main}. \n\t\n\tOf potentially independent interest is the work in Section \\ref{section:proper and cocompact}. Here, we establish a connection between neighborhoods of points in $\\overline{F}$, the closure of a flat, and neighborhoods in  $\\partial Y_v \\setminus F$. This work was done by Haulmark in the locally connected setting \\cite{Ha17} and we expand the results to our setting.\n\t\n\\end{Outline}\n\n\\begin{ack}\n\tI would like to thank Kim Ruane for her help and guidance throughout this project. I would also like to thank Genevieve Walsh, Mike Mihalik, and Rob Kropholler for many interesting discussions and for feedback on my drafts.\n\\end{ack}\n\n\n\n\\section{Key Examples} \\label{section: examples}\n\nIn this section, we outline three key examples that capture the difficulty in this work. Two are surface group amalgams and the third is a right-angled Coxeter group which is an amalgamation of groups which are virtually surface groups. All three examples are $\\cat(0)$ groups with isolated flats and the boundaries of the spaces on which they act are not locally connected.\n\n\n\\begin{figure}[h]\n\t\\includegraphics[width=.4\\textwidth]{s2-glued-torus}\n\t\\centering\n\t\\caption{A torus and a genus 2 surface with the identification $x=y$.}\n\t\\label{figure: s2 torus}\n\\end{figure}\n\n\n\n\n\n\\begin{example} \\label{example: s2 torus}\n\tLet $\\tilde{X_1}$ be the surface amalgam seen in Figure \\ref{figure: s2 torus}, $G_1$ its fundamental group, and $X_1$ its universal cover. This group is $\\cat(0)$ with isolated flats, with the flats coming from the torus. $X_1$ consists of Euclidean and hyperbolic planes fitting together in a tree-like way. The boundary, $\\partial X_1$, is made up of circles coming from these two types of planes along with the boundary points of rays which pass through infinitely many such planes. It is clear that the union of the circles is path connected since adjacent planes share a pair of boundary points. Because of the isolated flats, if two based rays pass through the same infinite collection of planes, then the rays are asymptotic. Applying Theorem \\ref{combo}, we can conclude that $\\partial X_1$ is path connected.\n\\end{example}\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=.4\\textwidth]{punctured-tori.png}\n\t\\caption{Two tori, each with a boundary component, with the identification $[a,b]^2=[c,d]^2$}\n\t\\label{figure:tori with boundary}\n\t\n\\end{figure}\n\n\\begin{example}\\label{example: punctured tori}\n\tLet $\\tilde{X_2}$ be 2 tori with boundary components as identified in Figure \\ref{figure:tori with boundary} and let $G_2$ be the fundamental group. By identifying $([a,b])^2$ with $([c,d])^2$, we create a Klein bottle group in $G$ generated by $[a,b]$ and $[c,d]$. This subgroup (and its conjugates) are the peripheral subgroups for the relatively hyperbolic structure on $G_2$. Since the Klein bottle group is virtually $\\mathbb{Z}^2$, this makes $G_2$ $\\cat(0)$ with isolated flats and with maximal peripheral splitting seen in Figure \\ref{fig: max periph splitting G2}.\n\t\n\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\n\t\\node[below] at (0,0) (A) {$K=\\langle[a,b],[c,d]\\rangle$};\n\t\\node[below] at (-4,0) (B) {$F_2=\\langle a,b\\rangle$};\n\t\\node[below] at (4,0) (C) {$F_2=\\langle c,d \\rangle$};\n\t\\node[above] at (2,0) {$\\mathbb{Z}=\\langle [c,d] \\rangle$};\n\t\\node[above] at (-2,0) {$\\mathbb{Z}=\\langle [a,b]\\rangle $};\n\n\t\n\t\\draw[fill] (0,0) circle [radius=0.05];\n\t\\draw[fill] (4,0) circle [radius=0.05];\n\t\\draw[fill] (-4,0) circle [radius=0.05];\n\t\\draw (-4,0) -- (0,0) -- (4,0);\n\t\\end{tikzpicture}\n\t\\caption{The maximal peripheral splitting of $G_2$}\n\t\\label{fig: max periph splitting G2}\n\\end{figure}\nA $\\cat(0)$ space $X_2$ on which $G_2$ acts geometrically can be constructed using Bridson and Haefliger's Equivariant Gluing Theorem \\cite{BH99}. This construction captures the isolated flats explicitly since $X_2$ consists of Euclidean planes coming from the Klein bottle group and truncated hyperbolic planes coming from the free groups. Once again, these spaces fit together in a tree-like way and the boundary $\\partial X_2$ is made up of boundary points from the truncated hyperbolic planes, the Euclidean planes, and the rays which pass through infinitely many such spaces.\n\nTo see that $\\partial X_2$ is path connected, consider a truncated hyperbolic plane union every flat which shares a boundary point with it. Call this space $Y$. This is equivalent to taking a $F_2$-vertex $v$ of the Bass-Serre tree for the splitting and letting $Y$ be the of the vertex space for $v$ union the vertex spaces for each of the vertices of the Bass-Serre tree adjacent to $v$. It is clear that $X_2$ is the union of all such $Y$. To see that $\\partial Y$ is path connected, notice that it is a Cantor set coming from the truncated hyperbolic planes union a collection of circles coming from the Euclidean planes. These circle `fill in' the gaps between points of the Cantor set, essentially replacing the missing intervals with circles. Therefore $\\partial Y$ is a circle with a collection of extra arcs attached and is path connected. \n\nSince $\\partial Y_v$ is path connected for each $v$ in the Bass-Serre tree and $X$ is the union of all such $Y_v$, then the only points to check are those which come from rays traveling through infinitely many truncated hyperbolic and Euclidean plans. Once again, the isolated flats condition covers this and we can apply Theorem \\ref{combo} to conclude $\\partial X_2$ is path connected.\n\\end{example}\n\n\\begin{figure}[h]\n\t\\centering\n\t\\begin{tikzpicture}\n\t\\newdimen\\r\n\t\\r=1cm\n\t\\draw (0:\\r)--(60:\\r)--(120:\\r)--(180:\\r)--(240:\\r)--(300:\\r)--(360:\\r);\n\t\\draw (330:1.7)--(300:\\r)--(0:0)--(60:\\r)--(30:1.7)--(0:2)--(330:1.7);\n\t\n\t\\draw[fill] (0:\\r) circle [radius=0.05];\n\t\\draw[fill] (60:\\r) circle [radius=0.05];\n\t\\draw[fill] (120:\\r) circle [radius=0.05];\n\t\\draw[fill] (180:\\r) circle [radius=0.05];\n\t\\draw[fill] (240:\\r) circle [radius=0.05];\n\t\\draw[fill] (300:\\r) circle [radius=0.05];\n\t\\draw[fill] (360:\\r) circle [radius=0.05];\n\t\\draw[fill] (0:0) circle [radius=0.05];\n\t\\draw[fill] (330:1.7) circle [radius=0.05];\n\t\\draw[fill] (30:1.7) circle [radius=0.05];\n\t\\draw[fill] (0:2) circle [radius=0.05];\n\t\n\t\n\t\n\t\\end{tikzpicture}\n\t\\caption{A Right-angled Coxeter group consisting of two hexagons sharing a pair of vertices.}\n\t\\label{figure:RACG}\n\\end{figure}\n\n\\begin{example}\\label{example: RACG}\nLet $G_3$ be the Right-angled Coxeter group in Figure \\ref{figure:RACG} and let $X$ be the Davis complex for $G_3$. This group is $\\cat(0)$ with isolated flats, with the flats coming from the square in the graph. Let $H$ be the RACG whose defining graph is a hexagon, then $G_3$ has the splitting seen in Figure \\ref{figure: $G_3$ splitting}.\n\n\\begin{figure}[h]\n\t\\centering\n\t\n\t\\begin{tikzpicture}\n\t\\node[below] at (-2,0) {$H$};\n\t\\node[below] at (0,0) {$(\\mathbb{Z}_2\\ast \\mathbb{Z}_2)^2$};\n\t\\node[below] at (2,0) {$H$};\n\t\\node[above] at (-1,0) {$\\mathbb{Z}_2\\ast \\mathbb{Z}_2$};\n\t\\node[above] at (1,0) {$\\mathbb{Z}_2\\ast \\mathbb{Z}_2$};\n\t\n\t\\draw[fill] (0,0) circle [radius=0.05];\n\t\\draw[fill] (2,0) circle [radius=0.05];\n\t\\draw[fill] (-2,0) circle [radius=0.05];\n\t\n\t\\draw (-2,0)--(0,0)--(2,0);\n\t\\end{tikzpicture}\n\t\n\t\\caption{A splitting of $G_3$.}\n\t\\label{figure: $G_3$ splitting}\n\\end{figure}\n\n$H$ is virtually a surface group, therefore $X$ consists of spaces quasi-isometric to $\\H^2$, which we will call $\\tilde{H}$, and Euclidean planes. Applying the same reasoning as in Example \\ref{example: s2 torus}, we can conclude that $\\partial X$ is path connected. This example is important for two reasons. First, the way we split the group is vital for applying Theorem \\ref{combo}. If we split the group in the `obvious' way:\n\\[H\\ast_{\\mathbb{Z}_2\\ast \\mathbb{Z}_2} H\\]\nthen the virtual $\\mathbb{Z}^2$ is not captured in the splitting (i.e. this is not a peripheral splitting). Because of this, it is no longer true that if two rays go through the same sequence of $\\tilde{H}$s, then the rays are asymptotic. Second, this is the type of group for which Hruska-Ruane's results on semistability do not apply \\cite{HruskaRuaneSemistable}. However, it is known that RACGs  are semistable at infinity \\cite{MihalikArtinCoxeterSemistable}.\n\n\n\n\\end{example}\n\nWe will refer to all three of these examples again in Section \\ref{section:isolated flats} to describe the isolated flats structure on each group, and again in Section \\ref{section block decomp} to describe the block decompositions of each space.\n\t\n\\section{The Visual Boundary} \\label{section visual boundary}\nWe refer the reader to \\cite{BH99} for an introduction to the theory of $\\cat(0)$ spaces and groups. Throughout this section, $X$ is a proper $\\cat(0)$ space. \n\nWe describe the topology on $\\overline{X}=X\\cup \\partial_{x_0} X$ and then restrict this topology to the visual boundary.\n\n\\begin{defn}[$\\overline{X}$ and the cone topology]\n\t\n\tFix a basepoint $x_0\\in X$. Let $\\partial_{x_0} X$ be the set of geodesic rays $c\\colon [0,\\infty)\\to X$ based at $x_0$. Let $\\overline{B}(x_0,r)$ be the closed ball of radius $r$ and $\\pi_r$ be the projection map to this ball. Define sets of the following form:\n\t\n\t\\[ U(c,r,D):= \\{ x\\in \\overline{X} : d(x,x_0)>r, d(\\pi_r(x),c(r))<D\\}\\]\n\twhere $c$ is a geodesic segment or ray based at $x_0$, $r,D>0$. A neighborhood basis for the \\textit{cone topology} on $\\overline{X}$ consists of sets of the form $U(c,r,D)$ along with metric balls in $X$.\n\t\n\tThe cone topology can be restricted to $\\partial_{x_0} X$ by taking sets of the following form to be a neighborhood basis:\n\t\n\t\\[ U(c,r,D)= \\{ c'\\in \\partial_{x_0} X: d(c(r),c'(r))<D\\}\\]\n\twith $r,D>0$.\n\t\t\\end{defn}\n\t\n\tDue to the follow result in \\cite{BH99}, the topology on $\\partial_{x_0} X$ is independent of choice of basepoint. Therefore we can denote the visual boundary as $\\partial X$ without ambiguity. In practice, however, it often useful to fix a basepoint and work with $\\partial_{x_0} X$.\n\t\n\t\\begin{prop}[{\\cite[II.8]{BH99}}]\n\t\tFor any choice of $x_0$ and $x_1$, $\\partial_{x_0} X$ and $\\partial_{x_1} X$ are homeomorphic.\n\t\\end{prop}\n\n\nWe use a metric on $\\partial X$ which was first introduced by Osajda. We use this metric in Section \\ref{section:decomp}.\n\n\\begin{defn}[Metric on $\\partial X$]\n\tFix $x_0\\in X$ and $D>0$. The metric $d_D$ on $\\partial_{x_0} X$ is defined as follows: given two rays $c,c'\\in \\partial_{x_0} X$, $d_D(c,c')=\\dfrac{1}{r}$ where $r\\in [0,\\infty)$ such that $d(c(r),c'(r))=D$. Since $d(c(t),c'(t))$ increases monotonically with $t$, there is a unique $r$ satisfying $d(c(r),c'(r))=D$.\n\t\n\tThis metric is compatible with the cone topology \\cite{OsajdaSwic}. Furthermore, the spaces $(\\partial X, d_D)$ and $(\\partial X, d_{D'})$ are isometric, so the metric is independent of choice of $D$.\n\\end{defn}\n\n\t\n\\section{Relatively Hyperbolic Group and Peripheral Splittings} \\label{rel hyp section}\n\nIn this section, we define relatively hyperbolic groups, the Bowditch boundary, and state a theorem of Tran which relates the Bowditch boundary to the $\\cat(0)$ boundary. We then introduce the maximal peripheral splitting of a relatively hyperbolic group. We will use the definition of relative hyperbolicity from \\cite{Yaman}. This definition uses an action of a group on a compact, metrizable topological space $M$, which will ultimately be the Bowditch boundary.\n\n\\begin{defn}[Convergence action]\nLet $M$ be a compact metrizable space and $G$ a finitely generated group. An action of $G$ on $M$ is a \\textit{convergence action} if it is properly discontinuous on distinct triples of $M$.\n\\end{defn}\n\n\\begin{defn}[Geometrically finite]\nA convergence group action of $G$ on $M$ is \\textit{geometrically finite} if every point of $M$ is either a bounded parabolic point or a conical limit point. \n\\end{defn}\n\n\n\\begin{defn}[Relatively hyperbolic]\n\tLet the action of $G$ on $M$ be a geometrically finite convergence action and let $\\mathbb{P}$ be a collection of subgroups of $G$. Then $G$ is \\textit{hyperbolic relative to $\\mathbb{P}$} if $\\mathbb{P}$ is exactly the collection of maximal parabolic subgroups. \n\\end{defn}\n\n\\begin{defn}[Bowditch boundary]\n\tThe space $M$ is uniquely determined by the pair $(G,\\mathbb{P})$ and is the \\textit{Bowditch boundary}, denoted $\\partial(G,\\mathbb{P})$.\n\\end{defn}\n\nSince we do not deal with the Bowditch boundary directly, we will not define all the terms above. See \\cite{Yaman} for precise definitions. We need the Bowditch boundary in order to apply Theorem \\ref{Tran Theorem} and to use a handful results of Bowditch which are summarized in Theorem \\ref{thm: peripheral splitting}. Both theorems can be found below.\n\nThere are many other ways to define a relatively hyperbolic group, see for example \\cite{FarbBoundedCoset, GrovesManningCusped}. Each of these ultimately has $G$ acting on a $\\delta$-hyperbolic metric space and takes the boundary of this space to be the Bowditch boundary. This is equivalent to our definition of the Bowditch boundary. Furthermore, all of these definitions of relatively hyperbolic are equivalent.\n\n\nSuppose $G$ is a $\\cat(0)$ group acting geometrically on a $\\cat(0)$ space $X$ and $(G,\\mathbb{P})$ is also relatively hyperbolic. We have so far introduced two different boundaries for $G$: the $\\cat(0)$ boundary and the Bowditch boundary. Work of Tran tells us how to relate these two boundaries:\n\n\n\\begin{thm}[\\cite{Tran13}] \\label{Tran Theorem}\n\tLet $(G,\\mathbb{P})$ be a relatively hyperbolic group acting geometrically on a $\\delta$-hyperbolic or $\\cat(0)$ space $X$. The quotient space formed from $\\partial X$ by collapsing the limit sets of each $P\\in \\mathbb{P}$ to a point is $G$-equivariantly homeomorphic to the Bowditch boundary $\\partial (G,\\mathbb{P})$.\n\\end{thm}\n\nWe will now introduce the maximal peripheral splitting of a relatively hyperbolic group. This plays a vital role in establishing the block decomposition of the relatively hyperbolic groups we study.\n\n\\begin{defn}[Peripheral Splitting]\n\tLet $(G,\\mathbb{P})$ be a relatively hyperbolic group. A \\textit{peripheral splitting} of $(G,\\mathbb{P})$ is a splitting of $G$ as a finite bipartite graph of groups $\\mathcal{G}$ whose vertices have two type: peripheral and component. Furthermore, the collection of subgroups of $G$ which are conjugate to a peripheral vertex group is the same as $\\mathbb{P}$, and $\\mathcal{G}$ does not contain a vertex of degree 1 which is contained in the adjacent peripheral vertex group.\n\\end{defn}\n\nA splitting of this form is called \\textit{trivial} if one of the vertex groups is equal to $G$ and \\textit{nontrivial} otherwise. A splitting $\\mathcal{H}$ is a \\textit{refinement} of another splitting $\\mathcal{G}$ if $\\mathcal{G}$ can be obtained from $\\mathcal{H}$ by a sequence of foldings over edges which preserves the peripheral\/component coloring. A peripheral splitting is \\textit{maximal} if it is not a refinement of any other splitting. \n\n\nWe combine a number of results of Bowditch about peripheral splittings into one large theorem:\n\\begin{thm}[Peripheral Splittings Results] \\label{thm: peripheral splitting}\n\tLet $(G,\\mathbb{P})$ be a 1-ended relatively hyperbolic group. Then the following hold:\n\t\n\t\\begin{enumerate}\n\n\n\t\t\\item if each $P\\in \\mathbb{P}$ is finitely presented, 1- or 2-ended, and does not contain an infinite torsion subgroup, then $\\partial (G,\\mathbb{P})$ is connected \\cite[10.1]{Bow12} and locally connected \\cite[1.5]{Bow01},\n\t\t\n\t\t\\item if $G_v$ is a component vertex group of a peripheral splitting, then $G_v$ is hyperbolic relative to $\\mathbb{P}_v:= \\{ P\\cap G_v: P\\in \\mathbb{P}\\}$ \\cite[1.3]{Bow01},\n\t\t\n\n\t\\item if $G_v$ is a component vertex of a maximal peripheral splitting and both $\\mathbb{P}$ and $\\mathbb{P}_v$ consist of finitely generated, 1- or 2-ended groups with no infinite torsion subgroups, the $\\partial (G_v,\\mathbb{P}_v)$ is connected and locally connected \\cite[1.3 and 1.5]{Bow01}.\n\t\t\n\t\n\t\t\n\n\t\t\n\n\t\t\n\t\t\\item if $G_v$ is a component vertex of a maximal peripheral splitting and both $\\mathbb{P}$ and $\\mathbb{P}_v$ consist of finitely generated, 1- or 2-ended groups with no infinite torsion subgroups, then $\\partial (G_v,\\mathbb{P}_v)$ contains no global cut points \\cite[1.4]{Bow01}.\n\t\\end{enumerate}\n\\end{thm}\n\nIn the setting we are dealing with ($\\cat(0)$ groups with isolated flats) each component vertex is virtually $\\mathbb{Z}^n$ for $n\\geq 2$, and so 1-ended with no infinite torsion subgroups. Furthermore, if $G_v$ is a component vertex, $\\mathbb{P}_v$ consists of 1- or 2-ended groups.  We use the first point of this theorem in order to accurately restate a result of Hruska-Ruane in Section \\ref{section:decomp}. The second point allows us to make sense of the last two. The third point is used at the end of Section \\ref{section:decomp} and the final point is used in a proposition stated in Section \\ref{section:local conn at flats}.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Isolated Flats} \\label{section:isolated flats}\n\nHere we define $\\cat(0)$ isolated flats and outline some results about their large scale geometry. \n\n\\begin{defn}[Flat]\n\tA subset $F\\subset X$ is a \\textit{flat} if it is isometric to $\\mathbb{E}^n$ for some $n\\geq 2$.\n\\end{defn}\n\nThere are many equivalent definitions of isolated flats. Hruska and Kleiner prove the equivalence of these definitions in \\cite{HK05}. We will take the following as our definition:\n\n\n\\begin{defn}[$\\cat(0)$ space with isolated flats]\n\tLet $X$ be a $\\cat(0)$ space admitting a geometric group action by $G$. Let $\\mathcal{F}$ be a $G$-invariant collection of flats. Then $X$ has \\textit{isolated flats} if the following two properties hold:\n\t\n\t\\begin{enumerate}\n\t\t\\item Maximality: There is  $D<\\infty$ such that for any flat $F\\subset X$, there is a flat $F'\\in \\mathcal{F}$ with $F\\subset N_D(F')$. \n\t\t\\item Isolated: For any $r>0$, there is a $\\rho=\\rho(r)>0$ such that for any distinct $F,F'\\in \\mathcal{F}$,\t\n\t\t$$\\text{diam}(N_r(F)\\cap N_r(F'))<\\rho$$\t\t\n\t\\end{enumerate}\n\\end{defn}\n\nWe say a group $G$ is \\textit{$\\cat(0)$ with isolated flats} if it acts geometrically on a $\\cat(0)$ space with isolated flats.\n\n\n\\begin{example}~\n\t\n\t\n\t\\begin{enumerate}\n\t\t\\item Return to Example \\ref{example: s2 torus}. The universal cover of the surface group amalgam in Figure \\ref{figure: s2 torus} is a $\\cat(0)$ space with isolated flats. The fundamental group acts geometrically on the universal cover. The flats, coming from the universal cover of the torus, are isolated in the sense that there is a hyperbolic plane separating them.\n\t\t\n\t\t\\item Consider Example \\ref{example: punctured tori}. The universal cover of the two tori with boundary components with identifications given in Figure \\ref{figure:tori with boundary} is a $\\cat(0)$ space with isolated flats. The $\\mathbb{bR}^2$'s coming from the Klein bottle are separated by truncated $\\H^2$'s.\n\t\t\n\t\t\\item From Example \\ref{example: RACG}, the Davis complex of this RACG is $\\cat(0)$ with isolated flats admitting a geometric group action by the RACG. Here, the flat comes from the square in the defining graph. As with the example above, pairs of flats are separated by (something quasi-isometric to) hyperbolic planes.\n\t\t\n\t\t\\item Let $K$ be the figure 8-knot and $M=S^3\\setminus K$ and let $X$ be the universal cover of $M$. Then $X$ is isometric to truncated $\\H^3$ and is a $\\cat(0)$ space with isolated flats, with the lifts of the torus-boundary of the manifold making up the flats.\n\t\\end{enumerate}\n\\end{example}\n\n\n\nOne of the major theorems from Hruska and Kleiner's work is that there are a number of equivalent characterizations of $\\cat(0)$ spaces with isolated flats. \n\n\\begin{thm}[{\\cite[1.2.1]{HK05}}]\n\tLet $X$ be a $\\cat(0)$ space admitting a geometric group action by $G$. Then the following are equivalent\n\t\n\t\\begin{enumerate}\n\t\t\\item $X$ has isolated flats.\n\t\t\\item Each component of the Tits boundary $\\partial_T X$ is either an isolated point or a standard Euclidean sphere.\n\t\t\\item $X$ is relatively hyperbolic with respect to a family of flats $\\mathcal{F}$.\n\t\t\\item $G$ is relatively hyperbolic with respect to a collection of virtually abelian subgroups of rank at least 2.\n\t\\end{enumerate}\n\\end{thm}\n\nOne key fact we use to establish that $\\cat(0)$ groups with isolated flats satisfy condition (3) in Theorem \\ref{combo} is a quasiconvexity result from Hruska-Ruane. Recall the definition of quasiconvex:\n\n\\begin{defn}[Quasiconvex]\n\tLet $X$ be a geodesic metric space. A subset $C$ is \\textit{$\\kappa$-quasiconvex} if for any pair of points $x,y\\in C$, any geodesic between $x$ and $y$ is in $N_\\kappa(C)$. \n\t\n\tWe say a subset is \\textit{quasiconvex} if it is $\\kappa$-quasiconvex for some $\\kappa$. \n\\end{defn}\n\n\\begin{thm} [{\\cite[5.6]{HR17}}] \\label{Quasi-convex Constant}\n\t\n\tLet $X$ be a $\\cat(0)$ space with isolated flats with respect to $\\mathcal{F}$. There is a constant $\\kappa>0$ such that the following holds:\n\t\\begin{enumerate}\n\t\t\\item Given two flats $F_1,F_2\\in \\mathcal{F}$ with $c$ the shortest length geodesic from $F_1$ to $F_2$, then $F_1\\cup F_2\\cup c$ is $\\kappa$-quasiconvex in $X$.\n\t\t\\item Given a point $p$ and a flat $F\\in \\mathcal{F}$ with $c$ the shortest path from $p$ to $F$, then $c\\cup F$ is $\\kappa$-quasiconvex in $X$. In particular, if $c'$ is any geodesic joining a point of $c$ to a point of $F$, then $c'$ intersects the $\\kappa$-neighborhood of the endpoint in the flat of $c$.\n\t\\end{enumerate}\n\\end{thm}\n\n\n\\section{Block Decompositions} \\label{section block decomp}\n\nThis section is devoted to defining a block decomposition of a space, providing some motivating examples, and concludes with the proof of Theorem \\ref{combo}. A block decomposition was first introduced in \\cite{Moo10} as a generalization of the results from \\cite{CK00}. In both papers, the authors focus on spaces which admit geometric group actions by groups of the form $G=A\\ast_C B$. The spaces on which these groups act can be viewed as the union of closed, convex pieces, called blocks, and the intersection of these blocks follows the subgroup intersection described in the Bass-Serre tree for the splitting. While the definition of a block decomposition is independent of a group action, this paper's motivating examples of such a decomposition come from the splitting of a group.\n\nWe will use the following definition from \\cite{Moo10}:\n\n\\begin{defn} \\label{block decomposition}\n\tA \\textit{block decomposition} of a CAT(0) space $X$ is a collection $\\mathcal{B}$ of closed, convex subsets (called blocks) such that the following conditions hold:\n\t\n\t\\begin{enumerate}\n\t\t\\item \t$X = \\bigcup \\limits_{B\\in \\mathcal{B}} B$,\n\t\t\\item each block intersects at least two other blocks,\n\t\t\\item Parity condition: every block has a $(+)$ or $(-)$ parity such that two blocks intersect only if they have opposite parity,\n\t\t\\item $\\epsilon$-condition: there is an $\\epsilon>0$ such that two blocks intersect iff their $\\epsilon$-neighborhoods intersect.\n\t\\end{enumerate}\n\t\n\\end{defn}\n\nWe will mostly use the facts that each $B$ is closed and convex and that $X=\\bigcup\\limits_{B\\in \\mathcal{B}} B$. Conditions (2)-(4) are there to prevent any degenerate cases (e.g. splitting $X$ into only two pieces, decomposing a plane as a union of lines). \n\n\\begin{defn}\n\tA \\textit{wall} is a non-trivial intersection of blocks and we denote the collection of walls $\\mathcal{W}$. \n\\end{defn}\n\nSince each block is convex, note that each wall is convex as well. The nerve, denoted $\\Nerve(\\mathcal{B})$, is a graph which records block intersections. There is a vertex for each block and an edge if two blocks (non-trivially) intersect. We will use the following notation: If $v$  and $w$ are vertices of $\\Nerve(\\mathcal{B})$, then $B_v, B_w\\in \\mathcal{B}$ are the blocks corresponding to $v$ and $w$, respectively. Given an edge $e$ with vertices $v$ and $w$, then $B_v$ and $B_w$ intersect in a wall by definition and we denote this wall $W_e$.\n\nFollowing results from \\cite{CK00} and \\cite{Moo10}, we get that the nerve is always a tree.\n\\begin{lem}\n\t$\\Nerve(\\mathcal{B})$ is a tree.\n\\end{lem}\n\nSince $\\Nerve(\\mathcal{B})$ is a tree, we will denote it $\\mathcal{T}_\\mathcal{B}$ or $\\mathcal{T}$ when $\\mathcal{B}$ is understood. Here are some examples of block decompositions:\n\n\\begin{figure}[h]\n\t\\includegraphics[width=.4\\textwidth]{torus-complex}\n\t\\centering\n\t\\caption{Three tori with the curve $a$ identified with $b$ and the curve $c$ identified with $d$.}\n\t\\label{fig:torus complex}\n\\end{figure}\n\n\n\n\n\\begin{example} \\label{example: Croke Kleiner}\n\t\n\t\n\t\n\tIn \\cite{CK00}, the authors describe what they call a torus complex, see Figure \\ref{fig:torus complex}. Let $G$ be the fundamental group of the torus complex and $X$ the universal cover. The space $X$ is $\\cat(0)$ and admits a block decomposition coming from a the following splitting of $G$:\n\t$$G = \\left( F_2\\times \\mathbb{Z} \\right) \\ast_{\\mathbb{Z}^2} \\left( F_2\\times \\mathbb{Z} \\right)$$\n\tThe fundamental group of two tori with a pair of curves identified is $F_2\\times \\mathbb{Z}$ which acts geometrically on $T_4\\times \\mathbb{bR}$, where $T_4$ is the valence 4 tree. The blocks come from the lifts of the middle torus and either the left or right torus, meaning each $B\\in \\mathcal{B}$ is isometric to $T_4\\times \\mathbb{bR}$. The walls are the lifts of the middle torus and are all isometric to $\\mathbb{bR}^2$. \n\t\n\t Croke and Kleiner \\cite{CK00} use $\\mathcal{B}$ to show that by changing the angle of intersection between the curves $b$ and $c$ on the middle torus, $G$ can act geometrically on spaces $X_1$ and $X_2$ which have non-homeomorphic boundaries. This answered a question of Gromov. Later, in \\cite{CK02}, Croke and Kleiner vastly extend their findings on this specific example to a class of groups they call `admissible.'\n\\end{example}\n\n\\begin{example} We return to Example \\ref{example: s2 torus}. The universal cover of the surface amalgam from this example is $\\cat(0)$ and has a block decomposition coming from the splitting of the fundamental group. Here, if $G_1$ is the fundamental group, then $G_1$ splits as\n\t$$G_1 = \\pi_1(S_2)\\ast_\\mathbb{Z} \\mathbb{Z}^2$$\n\twhere $S_2$ is the surface with genus 2. The blocks of this decomposition come from the lifts of the two surfaces. Therefore there are two types of blocks: those isometric to $\\H^2$ and those isometric to $\\mathbb{bR}^2$. The walls, however, are all isometric to $\\mathbb{bR}$. Unlike the torus complex, this group is $\\cat(0)$ with isolated flats. \n\\end{example}\n\n\\begin{example}\n\tConsider Example \\ref{example: punctured tori} once again. Let $G_2$ be the fundamental group of this surface amalgam and $X_2$ its universal cover. Recall that this group admits the maximal peripheral splitting as seen in Figure \\ref{fig: max periph splitting G2}. $X_2$ admits a block structure coming from this splitting. The vertex groups which are free act geometrically on truncated hyperbolic space and the vertex group which is the Klein bottle group acts geometrically on a Euclidean plane. These truncated hyperbolic planes and Euclidean planes in $X_2$ make up the block structure. That is, each block is isometric to either a Euclidean plane or a truncated hyperbolic plane.\n\\end{example}\n\n\\begin{example}\n\tReturn now to the RACG in Example \\ref{example: RACG}. Let $G_3$ be the fundamental group and $X_3$ the universal cover of its Davis complex. The block structure on $X_3$ once again comes from a splitting of $G_3$ which we saw in Figure \\ref{figure: $G_3$ splitting}. The blocks of $X_3$ come from the subspaces on which the vertex groups act geometrically. The hexagon subgroup $H$ acts geometrically on a space quasi-isometric to $\\H^2$ while $(\\mathbb{Z}_2\\ast \\mathbb{Z}_2)^2$ acts geometrically on $\\mathbb{bR}^2$. These two types of spaces form the blocks of $X_3$. \n\\end{example}\n\n\n\\begin{example}\n\tMore generally, if $X$ is a $\\cat(0)$ space coming from the construction in the Equivariant Gluing Theorem (\\cite{BH99} Theorem II.11.18), then $X$ admits a block decomposition coming from the splitting of the group $G=G_1\\ast_H G_2$. There is a natural map $p\\colon  X\\to \\mathcal{T}$, where $\\mathcal{T}$ is the Bass-Serre tree for the splitting. If $v$ is a vertex of $\\mathcal{T}$ then the blocks are each of the form  $p^{-1}(N_{1\/2}(v))$. We will be using this in Section \\ref{section:fixing a space}.\n\\end{example}\n\nWith these examples in mind, we can say something about the way block boundaries intersect. We say a wall $W_e\\in \\mathcal{W}$ \\textit{separates} blocks $B_v$ and $B_u$ if any path from $B_v$ to $B_u$ must pass through $W_e$. This is equivalent to $e$ being an edge on any path from $v$ to $u$ in $\\mathcal{T}$. \n\n\\begin{lem} \\label{Block int}\n\tFor two blocks $B_u,B_v\\in \\mathcal{B}$ corresponding to vertices $u,v\\in \\mathcal{T}$, then one of the following holds:\n\t\n\t\\begin{enumerate}\n\t\t\\item $\\partial B_u \\cap \\partial B_v = \\emptyset$\n\t\t\\item The vertices $u$ and $v$ are adjacent and $\\partial B_u \\cap \\partial B_v = \\partial W$ where $W=B_u \\cap B_v$.\n\t\t\\item The vertices $u$ and $v$ are not adjacent in $\\mathcal{T}$ but $\\partial B_u \\cap \\partial B_v$ is non-empty. Then $\\partial B_u \\cap \\partial B_v \\subset \\partial W$ where $W$ is \\textit{any} wall between $B_u$ and $B_v$.\n\t\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n\tWe may assume that $\\partial B_u \\cap B_v \\neq \\emptyset$, so we are in either case (2) or case (3). If $u$ and $v$ are adjacent, then case (2) follows immediately. Suppose then that $\\alpha\\in \\partial B_u \\cap \\partial B_v$ and that $u$ and $v$ are not adjacent in $\\mathcal{T}$. Let $W$ be any wall separating $B_u$ and $B_v$. \n\t\n\tPick basepoints $x_0\\in B_u$ and $x_1\\in B_v$ and let $y$ be a point in $W$ along the geodesic segment from $x_0$ to $x_1$. Let $c_0$ and $c_1$ be rays representing $\\alpha$ based at $x_0$ and $x_1$, respectively. For each $n\\in \\mathbb{N}$, let $y_n$ be a point in $W$ along the geodesic segment from $c_0(n)$ to $c_1(n)$. Since $W$ separates the blocks, these $y_n$ exist. Since $c_0$ and $c_1$ are asymptotic, there is a constant $K$ such that $d(c_0(t),c_1(t))\\leq K$ for all $t$. Therefore $y_n$ is $K$-close to both $c_0$ and $c_1$. The limit of the geodesic segments from $y$ to $y_n$, therefore, is asymptotic to $c_0$ and $c_1$, so is a representative of $\\alpha$. Since $W$ is convex, then $\\alpha\\in \\partial W$.\n\\end{proof}\n\n\n\n\\begin{remark}\n\tThe following should be noted from the proof above: it is possible for $\\alpha\\in \\partial B$ but a ray representing $\\alpha$ is disjoint from $B$ entirely. This happens above with the block $B_u$ and the ray $c_1$.\n\\end{remark}\n\nWe use the following terminology and the definition below from \\cite{CK00}. A ray $c$ \\textit{enters} a block $B$ if for some time $t$, $c(t)\\in B$ and $c(t)$ is not in any other block (and therefore is not in a wall). Similarly, a ray $c$ \\textit{exits} a block $B$ if for some $t$, $c(t)\\in B$ and not in any wall and for some $t'>t$ $c(t')\\notin B$. \n\n\\begin{defn}\n\tLet $x_0\\notin W$ for all $W\\in \\mathcal{W}$. The \\textit{itinerary} of a ray $c$ based at $x_0$, denoted $\\Itin_{x_0}(c)$, is the sequence of blocks $c$ enters. If $x_0$ is understood, then the itinerary will be denoted $\\Itin(c)$. Abusing notation, if the basepoint is understood and $c(\\infty)=\\alpha\\in \\partial X$, then $\\Itin(\\alpha)$ is used to denote the itinerary of the ray representing $\\alpha$. \n\t\n\\end{defn}\n\nIn order for this definition to be well defined, we need the following lemma:\n\n\\begin{lem}\n\tIf $\\mathcal{B}$ is a block decomposition for $X$, then there are points of $X$ which are not in any walls. In fact, for each $B\\in \\mathcal{B}$, there are points of $B$ which do not lie in any walls.\n\\end{lem}\n\n\\begin{proof}\n\tLet $B\\in \\mathcal{B}$ and let $W$ be a wall of $B$ such that $B\\cap B_0 = W$. Define $D=\\inf_{W'\\subset B} d(W,W')$.\n\t\n\tIf $D>0$, then all points in $N_D(W)\\setminus W \\cap B$ are points of $B$ which do not lie in any walls. Since $D>0$, then this set is non-empty.\n\t\n\tAssume, for contradiction, that $D=0$. Then pick $\\delta< \\epsilon$ and consider $N_\\delta(W)$. Since $D=0$, there is some wall $W'\\subset B$ which non-trivially intersects $N_\\delta(W)$. Let $B\\cap B'=W'$. Since $\\delta<\\epsilon$, $N_\\delta(W)\\cap W'$ means that in fact $N_\\epsilon(B_0)\\cap W'\\neq 0$. Since $W'\\subset B'$ and applying the $\\epsilon$-condition on the block decomposition, $B_0\\cap B'\\neq \\emptyset$. But then $B, B', B_0$ all pairwise intersect, contradicting the parity condition. Therefore $D>0$.\n\\end{proof}\t\n\n\\begin{remark} It is important to note that it is possible for $\\alpha\\in \\partial B$ but for $B\\notin \\Itin(\\alpha)$. This can happen as in the proof of Lemma \\ref{Block int}, where $c_0$ represent $\\alpha$, $\\alpha\\in \\partial B_v$, but $\\Itin_{x_0}(c_0)=\\Itin_{x_0}(\\alpha)=\\{B_u\\}$. In other words, this can happen when $\\alpha$ lies in the boundary of multiple blocks.\n\\end{remark}\n\n\nThe lemma below follows exactly in the same way as \\cite{CK00} (Lemma 2), which uses the fact that each $B$ is convex and and $B$'s topological frontier is covered by the collection of blocks corresponding to the link in $\\mathcal{T}$ of the vertex corresponding to $B$.\n\n\\begin{lem} [{\\cite[2]{CK00}}] \\label{Itinerary to Nerve} \n\tLet $X$ be a $\\cat(0)$ space with a block decomposition $\\mathcal{B}$ and corresponding nerve $\\mathcal{T}$. The itinerary for any $\\alpha\\in \\partial X$ consists of blocks which correspond to vertices on a geodesic segment or ray in $\\mathcal{T}$.  \n\\end{lem}\n\n\n\n\nThis lemma gives rise to the following definition:\n\n\\begin{defn}\n\t\n\t\n\t\n\tFix a basepoint $x_0$ which is not in any wall and let $c$ be a geodesic ray based at $x_0$ representing the point $\\alpha\\in \\partial X$. The point $\\alpha$ is \\textit{rational} if $\\Itin(c)$ is finite. The point $\\alpha$ is \\textit{irrational} if $\\Itin(c)$ is infinite.\n\\end{defn}\n\nIn order for this definition to be useful, being rational or irrational must be independent of representative for $\\alpha$.\n\n\\begin{lem}\n\tIf $c$ represents $\\alpha$ and $c$ is rational, then all representatives of $\\alpha$ are rational. \n\\end{lem}\n\n\\begin{proof}\n\tLet $\\alpha \\in \\partial X$ and let $c\\colon [0,\\infty)\\to X$ be a ray representing $\\alpha$ based at $x_0$ be a finite itinerary and let $B$ be the final block of this itinerary. So $\\alpha\\in \\partial B$. Let $c'\\colon [0,\\infty)\\to X$ be a ray representing $\\alpha$ based at $x_1$, and suppose, for contradiction, that $\\Itin_{x_1}(c')$ is infinite. By Lemma \\ref{Itinerary to Nerve}, $\\Itin(c')$ represents a geodesic ray in $\\mathcal{T}$. \n\t\n\tFirst, we show that if $d(v_B,v_{B'})\\geq 2n$ for $v_B,v_{B'}\\in \\mathcal{T}$, then $d(B,B')\\geq 2n\\epsilon$ in $X$. Let $B=B_1,B_2,...,B_{2n}=B'$ denote the $2n$-blocks between $B$ and $B'$. To see $d(B,B')\\geq 2\\epsilon$, notice if $d(v_B,v_{B'})=2$, then by the $\\epsilon$-condition, $d(B,B')\\geq 2\\epsilon$ since the $\\epsilon$-neighborhoods of the blocks do not intersect. So let $x\\in B$ and $x'\\in B'$ and let $[x,x']$ be the geodesic segment between them. For each $0\\leq i \\leq n$, there are $z_i\\in B_{2i}$ such that $d(z_i,z_{i+2})\\geq 2\\epsilon$. Therefore $d(x,x')\\geq 2n\\epsilon$. \n\t\n\tNow, for any $K>0$, there exists some $n$ such that $2n\\epsilon>K$. Furthermore, since $\\Itin(c')$ represent a geodesic ray in $\\mathcal{T}$, there is a block $B'$ such that $d(v_B,v_{B'})\\geq 2n$. Thus $c'$ is not in the $K$-neighborhood of $B$ for any $K$. But $c$ and $c'$ are asymptotic, a contradiction. Thus $\\Itin(c')$ must be finite.\n\t\n\\end{proof}\n\n\\begin{corollary}\n\tIf $\\Itin(c)$ is infinite for some ray $c$, then all rays representing $c(\\infty)$ have infinite itineraries.\n\\end{corollary}\n\n\\begin{proof}\n\tSuppose $c'$ represents $c(\\infty)$ and is finite. Then by the argument above, all rays representing $c'(\\infty)$ have finite itineraries, which cannot happen since $\\Itin(c)$ is infinite.\n\\end{proof}\nLet $\\partial_R X$ and $\\partial_I X$ denote the rational and irrational boundary points of $\\partial X$, respectively. Then we have:\n\\begin{enumerate}\n\t\\item $\\partial_R X = \\bigcup\\limits_{B\\in \\mathcal{B}} \\partial B$\n\t\\item $\\partial_I X = \\partial X \\setminus \\partial_R X$\n\\end{enumerate}\n\nOnce a basepoint is fixed we can use Lemma \\ref{Itinerary to Nerve} to create a map \\linebreak $\\eta\\colon  \\partial_I X \\to \\partial \\mathcal{T}$, mapping a point $\\alpha$ to the end of $\\mathcal{T}$ representing its itinerary at that basepoint. This map is surjective but not necessarily injective. The points in $\\eta ^{-1}(\\alpha)$ are the points of the $\\partial X$ with the same itinerary as $\\alpha$. We will care when $\\alpha$ is the only boundary point with its itinerary.\n\n\\begin{defn}\n\tIf $\\alpha$ is irrational and is the only point in $\\partial X$ with its itinerary, then we say $\\alpha$ is \\textit{lonely}. We will denote the set of lonely rays $\\partial_{I_L} X$.\n\\end{defn}\n\n\n\n\\begin{thm}\\label{combo}\n\tLet $X$ be a $\\cat(0)$ space and $\\mathcal{B}$ a block decomposition. Suppose the following conditions holds:\n\t\n\t\\begin{enumerate}\n\t\t\\item for each $B\\in \\mathcal{B}$, $\\partial B$ is path connected,\n\t\t\\item for each $W\\in \\mathcal{W}$, $\\partial W$ is non-empty,\n\t\t\\item all irrational rays are lonely\n\t\\end{enumerate}\n\tthen $\\partial X$ is path connected.\n\\end{thm}\n\n\\begin{remark}\n\tIt should be noted that we only assume $\\partial W$ is non-empty. We do not need that $\\partial W$ is path connected. The walls having non-empty boundary is equivalent to $\\partial X$ being connected.\n\\end{remark}\n\nGoing forward, we shall refer to condition (3) as the \\textit{lonely condition}.\n\nBefore proving Theorem \\ref{combo}, it should be noted that conditions (1) and (2), without (3), are insufficient for guaranteeing $\\partial X$ being path connected. Returning to Example \\ref{example: Croke Kleiner}, the block decomposition satisfies conditions (1) and (2), but $\\partial X$ is not path connected. In \\cite{CMT}, the authors explicitly construct a pair of distinct irrational rays with the same itinerary which are in a different path component from the union of the block boundaries. In that example, there is something akin to the Topologist's Sine Curve in the boundary, which obstructs path connectivity. Theorem \\ref{combo} shows that the lonely condition is sufficient for avoiding this pathology.\n\nWe will prove Theorem \\ref{combo} in two parts. First we will show that $\\partial_R X$ is path connected. We then show if $\\partial_R X$ is path connected, then $\\partial_R X \\cup \\partial_{I_L} X$ is path connected as well. By the lonely condition, $\\partial_{I_L} X = \\partial_I X$, concluding the proof.\n\n\\begin{proposition}\n\tSuppose for each $B\\in \\mathcal{B}$, $\\partial B$ is path connected and for each $W\\in \\mathcal{W}$, $\\partial W$ is non-empty. Then $\\partial_R X$ is path connected.\n\\end{proposition}\n\n\\begin{proof}\n\tFix $x_0\\in X\\setminus \\mathcal{W}$.\tLet $x_0\\in B$ and $B'$ be an adjacent block. We show that $\\partial B \\cup \\partial B'$ is path connected then proceed by induction. Since block boundaries are path connected, it suffices to show that there is a path from a point in $\\partial B$ to a point in $\\partial B'$. Let $\\alpha\\in \\partial B$ and $\\alpha' \\in \\partial B'$. Since $B$ and $B'$ are adjacent, there is a $W\\in \\mathcal{W}$ such that $B\\cap B' = W$. Let $\\beta\\in \\partial W$, which exists since $\\partial W$ is non-empty. Since $\\partial W = \\partial B \\cap \\partial B'$ and each block has path connected boundary, there are paths from $\\alpha$ to $\\beta$ and from $\\beta$ to $\\alpha'$ in $\\partial B \\cup \\partial B'$. The concatenation of these paths is a path from $\\alpha$ to $\\alpha'$.\n\t\n\tLet $\\Itin(\\alpha')= \\{B_1, B_2, ..., B_n\\}$ with $B_1=B$. The union of the boundary of adjacent blocks is path connected, so $\\partial B_{n-1} \\cup \\partial B_n$ is path connected. By the induction hypothesis, $\\partial B_1 \\cup \\partial B_2 \\cup ... \\cup \\partial B_{n-1}$ is path connected, and therefore $\\partial B_1 \\cup \\partial B_2 \\cup ... \\cup \\partial B_n$ is path connected, which concludes the proof. \n\\end{proof}\n\nBefore proving the next proposition, we need the following lemma from \\cite{CK02}:\n\n\\begin{lem} \\label{Croke Kleiner Converging Rays}\n\tLet $X$ be a locally compact CAT(0) space with a closed, convex subset $Y$. Let $x_0\\in X$ and $\\alpha_i\\in \\partial X$ with $\\alpha_i\\to \\alpha$ and the rays $r_i$ based at $x_0$ representing $\\alpha_i$ and $r$ the ray based at $x_0$ representing $\\alpha$. If $r_i$ intersects $Y$ for all $i$, then either $r$ intersects $Y$ or $\\alpha \\in \\partial Y$.\n\\end{lem}\n\n\\begin{proposition} \\label{lonely plus rational is pconn}\n\tIf $\\partial_R X$ is path connected, then $\\partial_R X \\cup \\partial_{I_L} X$ is path connected.\n\\end{proposition}\n\n\\begin{proof}\n\tFix $x_0\\in X\\setminus \\mathcal{W}$ and let $\\alpha \\in \\partial_{I_L} X$ with $\\Itin(\\alpha) = \\{B_1, B_2, ...\\}$. Let $c\\colon [0,\\infty)$ be a path in $\\partial_R X$ such that $c([n-1,n])\\subset B_n$ for all $n\\in \\mathbb{N}$. Our goal is to extend $c$ to $[0,\\infty]$ such that $c(\\infty)=\\alpha$. \n\t\n\tLet $\\{t_n\\}\\subset [0,\\infty)$ be a monotone sequence tending to infinity. \n\t\n\t\n\tSince $\\partial X$ is compact then, by passing to a subsequence if necessary, $\\{c(t_n)\\}$ converges to some $\\alpha'$. It suffices to show that $\\alpha = \\alpha '$, which we will do by showing $\\Itin(\\alpha) = \\Itin(\\alpha')$.\n\t\n\tFor each $n$, $\\Itin(c(t_n))=\\{B_1, B_2, ..., B_{k_n}\\}$ where $k_n = \\lceil t_n \\rceil$. Let $r_n$ be the ray based at $x_0$ representing $c(t_n)$ and let $r$ be the ray based at $x_0$ representing $\\alpha'$. By our construction, $r_n\\to r$.\n\t\n\tFix a block $B_m$ in $\\Itin(\\alpha)$. There is an $N$ such that for all $n>N$, $B_m\\in \\Itin(c(t_n))$ and so $r_n\\cap B_m \\neq \\emptyset$. By Lemma \\ref{Croke Kleiner Converging Rays}, either $r\\cap B_m\\neq \\emptyset$ or $r\\in \\partial B_m$. If the former, then $B_m\\in \\Itin(\\alpha')$. If the latter is true, pick points $x_n(m)\\in r_n \\cap B_m$ such that $d(x_0, x_n(m))\\to \\infty$. Such a sequence exists since $r\\in \\partial B_m$ and $r_n\\to r$. Then, after possibly passing to a subsequence, the geodesic segments from $x_0$ to $x_n(m)$ converge to a ray $\\rho_m$ with $\\rho_m(\\infty) = \\alpha'$. For each $i<m$, $B_i$ separates $B_m$ from $x_0$, so $B_i\\in \\Itin(\\alpha')$.\n\t\n\tBut if this is true for some $m$, then it is true for all $M>m$ and so $\\alpha \\in \\partial B_M$ for all $M>m$. Consider the ray $\\rho_M$ by the same construction as the previous paragraph. Using the same argument, $B_i\\in \\Itin(\\rho_M(\\infty)) = \\Itin(\\alpha')$ for all $i<M$. Since $m<M$, this means $B_m\\in \\Itin(\\alpha')$. Therefore $\\alpha'$ and $\\alpha$ have the same itinerary. Since $\\alpha$ is lonely, $\\alpha = \\alpha'$.\n\\end{proof}\n\n\\begin{remark}\n\tProving path connectivity of the boundary does not require each block to have a path connected boundary. The proposition only assumes that $\\partial_R X$ is path connected. This distinction will come up later when we show all 1-ended $\\cat(0)$ groups with isolated flats have path connected visual boundaries.\n\\end{remark}\n\n\n\\begin{remark}\n\tWhile the lonely condition is sufficient, it is not necessary for path connectivity. This comes up for the following group\n\t\\[G=F_2\\times \\mathbb{Z} = \\mathbb{Z}^2\\ast_\\mathbb{Z} \\mathbb{Z}^2\\]\n\t$G$ acts the universal cover of two tori with a curve on each identified. The block decomposition comes from viewing $G$ as the amalgamated product of two $\\mathbb{Z}^2$s over a $\\mathbb{Z}$. The boundary is path connected since it is a suspension of a Cantor set, but irrational rays are not lonely.\n\\end{remark}\n\n\n\n\n\\section{Fixing a Space}\\label{section:fixing a space}\n\nNormally when dealing with $\\cat(0)$ groups and their boundaries, one needs to be particularly careful about the space they are working with. This is because Croke and Kleiner \\cite{CK00} showed  that, unlike in the hyperbolic setting, $\\cat(0)$ groups do not have a well defined visual boundary. When it comes to $\\cat(0)$ groups with isolated flats, however, work of Hruska and Kleiner \\cite{HK05} shows that the visual boundary of these groups is well defined . In this section, we will fix the space on which a 1-ended $\\cat(0)$ group with isolated flats will act on and define the block decomposition for this space.\n\nThroughout this section, let $G$ be a $\\cat(0)$ group which is hyperbolic relative to $\\mathbb{P}$, the collection of flat-stabilizers so that $(G,\\mathbb{P})$ is the relatively hyperbolic pair. Let $G$ act geometrically on some $\\cat(0)$ space $X$. \n\nWe want more control over the space on which $G$ acts, so we use the peripheral splitting theorem of Bowditch introduced in Section \\ref{rel hyp section}, a convex splitting theorem of Hruska and Ruane \\cite{HR17}, and a combination theorem of Bridson and Haefliger \\cite{BH99} to fix a different space which admits a geometric group action by $G$. We quote the two remaining theorems below. But first, a definition.\n\n\n\\begin{defn}\n\tLet $G$ be a CAT(0) group acting geometrically on a $\\cat(0)$ space $X$. A subgroup $H\\leq G$ is \\textit{convex} if $H$ stabilizes a closed, convex subspace $Y$ of $X$, and $H$ acts cocompactly on $Y$. Therefore $H$ act geometrically on $Y$, a $\\cat(0)$ space.\n\\end{defn}\n\n\\begin{thm}[Convex Splitting Theorem, {\\cite[1.3]{HR17}}]\n\tLet $G$ act geometrically on a $\\cat(0)$ space $X$. Suppose $G$ splits as the fundamental group of a graph of groups $\\mathcal{G}$ such that each edge group of $\\mathcal{G}$ is convex. Then each vertex group is convex as well. In particular, each vertex group is a $\\cat(0)$ group.\n\\end{thm}\n\n\nLet $\\mathcal{G}$ be the maximal peripheral splitting for our group $G$. Since $G$ has isolated flats, all peripheral subgroups in the graph of groups $\\mathcal{G}$ are virtually abelian and therefore so are all edge groups. By the Flat Torus Theorem \\cite{BH99}, all virtually abelian subgroup of a $\\cat(0)$ group are convex and therefore we can apply the Convex Splitting Theorem to the maximal peripheral splitting of $G$.\n\nLet $\\mathcal{C}$ be the collection of component vertices and $\\P$ the set of peripheral vertices of $\\mathcal{G}$. For each $v\\in \\mathcal{C}$ with vertex group $G_v$, let $X_v$ be the convex subspace of $X$ on which $G_v$ acts geometrically. For $w\\in \\P$, the vertex group $G_w$ is virtually abelian and therefore acts geometrically on some Euclidean space $F_w$. \n\nEach edge $e$ of $\\mathcal{G}$ is incident to a peripheral vertex $w$ and a component vertex $v$. Let $F_e$ be the flat subspace of $F_w$ on which $P_e$ acts geometrically. Each edge also comes equipped with a monomorphism $\\phi_e\\colon P_e\\to G_v$. The Flat Torus Theorem gives a $\\phi_e$-equivariant isometric embedding of $F_e$ into $X_v$. \n\nWith all of this set up, we can use the following theorem to build our desired $X$:\n\n\\begin{thm}[Equivariant Gluing Theorem, {\\cite[II.11.18]{BH99}}]\\label{thm:equivariant gluing}\n\tLet $\\Gamma_0$, $\\Gamma_1$, and $H$ be groups acting geometrically on complete CAT(0) spaces $X_0, X_1$, and $Y$, respectively. Suppose for $j=0,1$, there exists monomorphisms $\\phi_j\\colon H \\hookrightarrow X_j$ and $\\phi_j$-equivariant isometric embeddings $f_j\\colon  Y\\to X_j$, then $\\Gamma=\\Gamma_0\\ast_H \\Gamma_1$ acts geometrically on a CAT(0) space $X$.\n\\end{thm}\n\nThe space uses the Bass-Serre tree $\\mathcal{T}$ for the splitting $\\mathcal{G}$ of $G$. For each component vertex $v$ of $\\mathcal{T}$, take a copy of $X_v$ and for each peripheral vertex $w$ of $\\mathcal{T}$, take a copy of $P_w$. We shall refer to these spaces as \\textit{vertex spaces}. Then for each edge $e$ incident to $v$ and $w$, take a copy of $F_e\\times [0,1]$, which we shall refer to as \\textit{edge spaces}. Identify the $F_e\\times \\{0\\}$ part with the image of $F_e$ in $X_v$ and $F_e\\times \\{1\\}$ is identified with the image of $F_e$ in $F_w$. Following the Bass-Serre tree, all the copies of the different vertex and edge spaces are glued up to form $X$. \n\nWith this $X$ fixed, we can also describe the block decomposition of $X$. For each vertex space $X_v$ or $P_w$, take the closed $\\frac{1}{2}$-tubular neighborhood to be a block $B$. Then each block intersects in $F_e\\times \\{\\frac{1}{2}\\}$. This defines a block decomposition of $X$.\n\n\\begin{remark}\n\tThroughout the remaining sections, when referring to a $\\cat(0)$ group with isolated flats acting geometrically on $X$, we shall be referring to the $X$ from this construction. \n\\end{remark}\n\nOur goal for the next few sections is to show that $\\partial X$ is path connected. In Lemma \\ref{IFP blocks are lonely}, we will show that with this block decomposition, all irrational rays are lonely. Therefore we just need to show that $\\partial_R X$ is path connected.\n\nIn general, each component vertex is not 1-ended. When this happens, $\\partial X_v$ is cannot be path connected and therefore showing $\\partial_R X$ is path connected requires more work. Therefore we introduce $Y_v$:\n\n\\begin{defn}\\label{defn Y}\n\tFor each component vertex $v$, let $Star(v)$ be the star of $v$ in $\\mathcal{T}$. There is a continuous map $p\\colon X\\to \\mathcal{T}$ which maps vertex spaces to vertices and edge spaces to edges in the obvious way. Let $Y_v= p ^{-1} ( Star(v))$.\n\t\n\t Put another way, let $\\mathcal{E}_v$ be the collection of edges incident to $v$ and $\\P_v$ be the collection of vertices of $\\mathcal{T}$ adjacent to $v$. Since $\\mathcal{T}$ is the Bass-Serre tree for the maximal peripheral splitting, each $w\\in \\P_v$ is a peripheral vertex. Then $Y_v$ is\n\t\n\t\\[Y_v := X_v \\cup \\left(\\bigcup_{e\\in \\mathcal{E}_v} F_e\\times[0,1]\\right) \\cup \\left(\\bigcup_{w\\in \\P_w} F_w \\right) \\]\n\tPut a third way, $Y_v$ consists of $X_v$ and every flat of $X$ which shares a boundary point with $Y_v$.\n\t\n\tUltimately, we want to show that $\\partial Y_v$ is path connected for each $v\\in \\mathcal{C}$. When dealing with a generic $Y_v$, we will drop the subscript and denote it $Y$.\n\\end{defn}\n\nThe motivating example for needing to this is Example \\ref{example: punctured tori}. The maximal peripheral splitting of this group is in Figure \\ref{fig: max periph splitting G2}. Since the component vertices are free groups, they have Cantor sets for boundaries. Additionally, there is no obvious intermediate peripheral splitting which has component vertices with path connected visual boundaries. Therefore, one needs to look at one of the Cantor sets along with all the circles attached to it. We will show this space is path connected.\n\n\n\n\\section{Decomposition Spaces}\\label{section:decomp}\nIn this section we will repeat some results from \\cite{HR17} about decompositions of spaces and their relationship to CAT(0) groups with isolated flats. In particular, we establish that the map $\\pi\\colon \\partial X \\to \\partial (G,\\mathbb{P})$ from Theorem \\ref{Tran Theorem} can be viewed through the lens of decomposition theory. We then use this to establish a decomposition of $\\partial Y$ and ultimately conclude that boundary points in $Y$ which are not in the boundary of a flat are locally connected. \n\nMuch of the results here follow from elementary decomposition theory, which can be found in \\cite{Dav86}. For a topological space $M$, a \\textit{decomposition}, $\\mathcal{D}$, is a partition of $M$. The decomposition map $\\pi\\colon M\\to M\/\\mathcal{D}$ is the quotient map where each $d\\in \\mathcal{D}$ is collapsed to a point and the resulting space is given the quotient topology. In this way, decompositions of $M$ are equivalent to quotients of $M$. Any decomposition can be made into a quotient map and any quotient map can be made into a decomposition by taking point pre-images as elements of the decomposition.\n\n\n\\begin{defn}[Upper semicontinuous decomposition]\n\tA decomposition $\\mathcal{D}$ of $M$ is \\textit{upper semicontinuous} if each $d\\in \\mathcal{D}$ is compact and for each $d\\in \\mathcal{D}$ and each open $U\\subset M$ containing $d$, there is an open $V\\subset M$ containing $d$ such that every $d\\in \\mathcal{D}$ which intersects $V$ is contained in $U$. A quotient is \\textit{upper semicontinuous} if the associated decomposition is upper semicontinuous.\n\\end{defn}\n\n\\begin{prop}[{\\cite[I.1.1]{Dav86}}]\n\tLet $\\mathcal{D}$ be a decomposition of a space $M$ with each $d\\in \\mathcal{D}$ compact. The following are equivalent:\n\t\\begin{enumerate}\n\t\t\\item $\\mathcal{D}$ is upper semicontinuous.\n\t\t\\item For an open set $U\\subset M$, let $U^*$ be the union of $d\\in \\mathcal{D}$ such that $d\\subset U$. Then $U^*$ is open.\n\t\t\\item The decomposition map $\\pi\\colon  M\\to M\/\\mathcal{D}$ is closed.\n\t\\end{enumerate}\n\t\n\\end{prop}\n\\begin{comment}\n\\begin{proof} \n\t\n\t$(1)\\Rightarrow (2)$: Let $U$ open in $M$. If there are no $d\\in \\mathcal{D}$ such that $d\\subset U$, then $U^*$ is empty and therefore open. For each $d\\subset U$, let $V_d$ be the open subset of $U$ from the definition of upper semicontinuous. Let $V=\\cup_{d\\subset U}V_d$, then $V$ is open and we claim that $V=U^*$. The inclusion $U^*\\subset V$ is obvious. Suppose $x\\in V$, then $x\\in V_d$ for some $d\\subset U$. Since $\\mathcal{D}$ is a decomposition, $x\\in d'$ for some $d'\\in \\mathcal{D}$. Since $d'\\cap V_d\\neq \\emptyset$, then $d'\\subset  V_d \\subset U$ and therefore $d'\\subset U^*$. This shows the other inclusion.\n\t\n\t$(2)\\Rightarrow (3)$: Let $C\\subset M$ be closed and let $U$ be $C$'s complement in $M$. To show $\\pi\\colon M\\to M\/\\mathcal{D}$ is closed, we must show that $\\pi ^{-1} (\\pi(C))$ is closed. Call this set $C'$, then we have \n\t$$C'= \\{x\\in M: \\pi(x)\\in \\pi(C)\\} = \\cup \\{ d\\in \\mathcal{D}: \\pi(d) \\in \\pi(C) \\}$$\n\tLet $U'= M\\setminus C'$. We claim $U^*=U'$. First, $U^*\\subset U'$ since if $d\\subset U$, then $\\pi(d)\\notin \\pi(C)$ and therefore $d\\cap C'=\\emptyset$. For the other inclusion, suppose $x\\notin C'$, then $x\\in d$ for some $d\\in \\mathcal{D}$ such that $\\pi(d)\\notin \\pi(C)$. Since $\\pi(d)\\notin \\pi(C)$, then $d\\cap C = \\emptyset$, so $d\\subset U$ and thus $d\\subset U^*$. This shows the other inclusion.\n\t\n\t$(3)\\Rightarrow (1)$: Let $d\\in \\mathcal{D}$ and $U$ open in $M$ such that $d\\subset U$. Let $C=M\\setminus U$. Since $\\pi$ is a closed map, then $\\pi(C)$ is closed and therefore $\\pi^{-1}(\\pi(C))$ is closed as well. Let $U'=M\\setminus \\pi^{-1}(\\pi(C))$. Then $U'\\subset U$ and if $d\\cap U' \\neq \\emptyset$, then $d\\subset U$. If $d\\subset U$, then $\\pi(d)\\notin \\pi(C)$ and therefore $d\\subset U'$. Suppose $d'\\cap U' \\neq \\emptyset$, then $\\pi(d') \\notin \\pi(C)$, so $d'\\subset U$. \n\\end{proof}\n\\end{comment}\n\\begin{prop}[{\\cite[I.2.1 and I.2.2]{Dav86}}]\n\tIf $\\mathcal{D}$ is an upper semicontinuous decomposition of a Hausdorff space, then $M\/\\mathcal{D}$ is Hausdorff. If $\\mathcal{D}$ is an upper semicontinuous decomposition of a metric space, then $M\/\\mathcal{D}$ is metrizable.\n\\end{prop}\n\nBoundaries of proper CAT(0) spaces are metrizable, and therefore so are their quotients coming from upper semicontinuous decompositions.\n\n\\begin{prop}[{\\cite[I.3.1]{Dav86}}]\n\tIf $\\mathcal{D}$ is an upper semicontinuous decomposition of $M$, then $\\pi\\colon M\\to M\/\\mathcal{D}$ is a proper map.\n\\end{prop}\n\n\\begin{defn}[Monotone decomposition]\n\tA decomposition $\\mathcal{D}$ of $M$ is \\textit{monotone} if each $d\\in \\mathcal{D}$ is compact and connected.\n\\end{defn}\n\n\\begin{prop} [{\\cite[I.4.1]{Dav86}}] \\label{Montone if and only if connected}\n\tLet $\\mathcal{D}$ be an upper semicontinuous decomposition of a space $M$. Then $\\mathcal{D}$ is monotone if and only if $\\pi ^{-1}(C)$ is connected whenever $C$ is a connected subset of $M\/\\mathcal{D}$.\n\\end{prop}\n\\begin{comment}\n\\begin{proof}\n\t($\\Rightarrow$) Let $C$ be a connected subset of $M\/\\mathcal{D}$ and suppose, for contradiction, that $\\pi^{-1}(C)$ is disconnected. Then there are disjoint closed sets $U$ and $V$ such that $\\pi^{-1}(C)\\subset U \\cup V$. Since each $d\\in \\mathcal{D}$ is connected, if $d\\subset \\pi^{-1}(C)$, then $d\\subset U$ or $d\\subset V$. Furthermore, if $x\\in \\pi^{-1}(C)$ and $x\\in d$ for some $d\\in \\mathcal{D}$, then $d\\subset \\pi^{-1}(C)$. But then $\\pi(U)\\cup \\pi(V)$ are disjoint, cover $C$ and are closed since the decomposition induces a closed map. This is a contradiction, so $\\pi^{-1}(C)$ is connected.\n\t\n\t($\\Leftarrow$) Singletons are connected in $M\/\\mathcal{D}$, therefore $\\pi^{-1}(x)$ is connected for each $x\\in M\/\\mathcal{D}$ and $\\pi^{-1}(x)=d$ for some $d\\in \\mathcal{D}$ by definition of the decomposition map. Thus each $d\\in \\mathcal{D}$ is connected so the decomposition is monotone.\n\\end{proof}\n\\end{comment}\n\n\\begin{prop}[{\\cite[8.5]{HR17}}] \\label{Singleton points are locally connected}\n\tLet $\\mathcal{D}$ be an upper semicontinuous monotone decomposition of $M$. Let $\\{x\\}$ be a singleton member of $\\mathcal{D}$. If $M\/\\mathcal{D}$ is locally connected at $\\pi(x)$, then $M$ is locally connected at $x$.\n\\end{prop}\n\\begin{comment}\n\\begin{proof}\n\tLet $U$ be a neighborhood of $x$ in $M$. Then $U^*$ is an open neighborhood of $x$ contained in $U$. Since $M\/\\mathcal{D}$ is locally connected at $x$, then $\\pi(U^*)$ contains a connected neighborhood $V$ containing $\\pi(x)$. By Proposition \\ref{Montone if and only if connected} $\\pi^{-1}(V)$ is connected and $\\pi^{-1}(V)\\subset U^*\\subset U$. \n\\end{proof}\n\\end{comment}\nThe following definition allows us to construct subspace decompositions:\n\n\\begin{defn}[Subspace Decomposition]\n\tLet $\\mathcal{D}$ be a decomposition of a Hausdorff space $M$. Let $W\\subset M$ such that if $d \\cap W$ is non-empty for any $d\\in \\mathcal{D}$, then $d\\subset W$. The \\textit{induced subspace decomposition} of $W$ is the decomposition consisting of all members of $\\mathcal{D}$ which are contained in $W$. If $\\mathcal{D}$ is upper semi-continuous, then the induced subspace decomposition is as well.\n\\end{defn}\nThis definition will become vital for understanding the decomposition of $\\partial Y\\subset \\partial X$.\n\n\\begin{defn}[Null Family]\n\tA collection of subsets $\\mathcal{A}$ of a metric space is a \\textit{null family} if for each $\\varepsilon>0$, only finitely many $A\\in \\mathcal{A}$ have diameter greater than $\\varepsilon$.\n\\end{defn}\n\n\\begin{prop}[Saturation condition, {\\cite[8.9]{HR17}}] \\label{Null Family Saturation}\n\tLet $\\mathcal{A}$ be a null family of compact sets in a metric space $M$. Suppose $q\\in M$ and $q$ is not in any of the members of $\\mathcal{A}$. Then each neighborhood $U$ of $q$ contains a smaller neighborhood $V$ of $q$ such that for each $A\\in \\mathcal{A}$, if $A\\cap V \\neq \\emptyset$, then $A\\subset U$.\n\\end{prop}\n\\begin{comment}\n\\begin{proof}\n\tLet $U$ a neighborhood of $q$ and suppose $B(q,\\varepsilon)\\subset U$. Then choose $\\delta>0$ such that $\\delta<\\varepsilon\/2$ and such that $d(q,A)>\\delta$ for each of the finitely many $A$ with diameter greater than $\\varepsilon\/2$. If $V=B(q,\\delta)$, then the result follows. \n\\end{proof}\n\\end{comment}\n\n\\begin{remark}\\label{remark: null family condition}\n\t\tIn view of how the metric works on $\\partial X$, the notion of a null family can be reformulated topologically as follows: Fix $x_0$ as the basepoint for the cone topology. A collection $\\mathcal{A}$ of subspaces is a null family in $\\partial X$ if there is some $D>0$ such that for each $r<\\infty$, only finitely many members of $\\mathcal{A}$ are not contained in a set of the form $U(\\cdot, r, D)$. Under the metric $d_D$ on $\\partial_{x_0} X$, only those members of $\\mathcal{A}$ which lie in $U(\\cdot, r, D)$ have diameter at most $1\/r$.\n\t\n\tA similar condition can be put on the cone topology of $\\overline{X}$. We do not need to deal with this directly, however. Work of Bestvina says that $\\overline{X}$ is metrizable and this metric is compatible with the cone topology, which is sufficient for our needs \\cite{Bes96}.\n\\end{remark}\n\n\n\\begin{proposition} [{\\cite[8.12]{HR17}}] \\label{Family of Spheres}\n\tLet $X$ be a CAT(0) space with isolated flats with respect to the family $\\mathbb{P}$. Let $\\mathcal{A}$ be the collection of spheres $\\{ \\partial F: F\\in \\mathbb{P} \\}$. Then $\\mathcal{A}$ is a null family of disjoint compact subsets of $\\partial X$. \n\\end{proposition}\n\n\n\n\\begin{prop}[{\\cite[I.2.3]{Dav86}}]\n\tLet $\\mathcal{D}$ be a decomposition of a metric space $M$ such that each $d\\in \\mathcal{D}$ is compact. Let $\\mathcal{A}$ be the collection of $d\\in \\mathcal{D}$ such that $d$ is not a singleton. Then if $\\mathcal{A}$ is a null family, $\\mathcal{D}$ is an upper semicontinuous decomposition. \n\\end{prop}\n\\begin{comment}\n\\begin{proof}\n\tLet $U$ be an open set in $M$ and $d\\subset U$. For each $x\\in d$, let $\\varepsilon_x$ be such that $B(x,\\varepsilon_x)\\subset U$. For each $x$, let $0<\\delta_x<\\varepsilon_x\/2$ such that $d(x,A)>\\delta$ for each $A\\in \\mathcal{A}$ with diameter at least $\\varepsilon_x\/2$. Then let $V=\\cup_{x\\in d} B(x,\\delta_x)$. It is clear that $V\\subset U$ by our choices of $\\delta_x$. Suppose $y\\in d'\\cap V$ for some $d'\\in \\mathcal{D}$. If $d'=\\{y\\}$, then $d'\\subset U$. If not, then $y\\in B(x,\\delta_x)$ for some $x\\in d$. By our choice of $\\delta_x$, it must be that the diameter of $d'$ is smaller than $\\varepsilon_x\/2$, so $d'\\subset B(x,\\varepsilon)\\subset U$. Therefore $\\mathcal{D}$ defines an upper semicontinuous decomposition of $M$.\n\\end{proof}\n\\end{comment}\n\\begin{corollary}[{\\cite[8.13]{HR17}}]\n\tLet $G$ be a CAT(0) group with isolated flats relative to $\\mathbb{P}$ acting geometrically on a CAT(0) space $X$. Let $\\mathcal{A}$ be the collection of spheres which bound maximal dimensional flats in $\\partial X$. Then the quotient map $\\partial X \\to \\partial X\/ \\mathcal{A} \\cong \\partial (G,\\mathbb{P})$ given in Theorem \\ref{Tran Theorem} is upper semicontinuous and monotone.\n\\end{corollary}\n\n\\begin{corollary}[{\\cite[8.14]{HR17}}]   \\label{Non-flat points are l.c.}\n\tLet $G$ be 1-ended and CAT(0) with isolated flats relative to $\\mathbb{P}$. Then $\\partial X$ is locally connected at each $x$ which is not in the boundary of a flat. \n\\end{corollary}\n\n\\begin{proof}\n\tFrom Theorem \\ref{thm: peripheral splitting}, $\\partial(G,\\mathbb{P})$ is locally connected. Since the decomposition is upper semicontinuous and monotone, the singletons in the decomposition are locally connected as well. In this decomposition, the singletons are exactly the points which are not in the boundary of any flat.\n\\end{proof}\n\n\nRecall the definition of $Y$(Definition \\ref{defn Y}) at the end of Section \\ref{section:fixing a space}: Let $p\\colon X\\to \\mathcal{T}$ be the projection map onto the Bass-Serre tree $\\mathcal{T}$. Then for a component vertex $v\\in \\mathcal{T}$, $Y_v=p^{-1}(Star(v))$. Since we will being working with generic $Y_v$, we shall just denote it as $Y$.\n\\begin{thm} \\label{monotone usc decomp of Y}\n\tThe restriction of $\\pi\\colon \\partial X \\to \\partial (G,\\mathbb{P})$ to $\\partial Y$ defines a monotone upper semicontinuous decomposition and the image of $\\partial Y$ is $\\partial (G_v, \\mathbb{P}_v)$, the Bowditch boundary for the corresponding component vertex. Here \\linebreak $\\mathbb{P}_v : = \\{ P\\cap G_v: P\\in \\mathbb{P}\\}$.\n\\end{thm}\n\n\\begin{proof}\n\tFirst, see that restricting $\\pi$ to $\\partial Y$ is a subspace decomposition for if $d\\cap \\partial Y$, then $d\\subset \\partial Y$. This is clear since for each $d\\in \\mathcal{D}$, either $d$ is a singleton or $d$ is the boundary of a maximal dimensional flat and in either of those cases, $d\\cap \\partial Y$ means $d\\subset \\partial Y$. Additionally this subspace decomposition is monotone. Therefore we have an upper semicontinuous decomposition $\\pi\\colon  \\partial Y \\to \\pi(\\partial Y)$. \n\t\n\tNext, consider the relatively hyperbolic pair $(G_v,\\mathbb{P}_v)$. By the arguments above, the map $\\pi'\\colon  \\partial X_v \\to \\partial (G_v, \\mathbb{P}_v)$ is an upper semicontinuous decomposition. Since the peripheral structure of $G_v$ is inherited from $G$, then $\\pi(\\partial X_v) = \\pi'(\\partial X_v)$.\n\t\n\tLastly, we need to show that $\\pi(\\partial Y) = \\pi (X_v)$. To see this, note that if $d\\subset \\partial Y$, then either $d$ is a singleton or $d$ is the boundary of a flat. In the former case, then $d\\in \\partial X_v$. In the latter case, $d\\cap \\partial X_v \\neq \\emptyset$. The map $\\pi$ collapses all points of $d$ to a single point, therefore $\\pi(d) = \\pi (d\\cap \\partial X_v)$, and therefore $\\pi(\\partial Y) = \\partial (G_v,\\mathbb{P}_v)$.\n\t\n\t\n\t\n\\end{proof}\n\n\\begin{corollary}\\label{Y non-flat points lconn}\n\tIf $x\\in \\partial Y$ and $x\\notin \\partial F$ for any flat $F$, then $\\partial Y$ is locally connected at $x$.\n\\end{corollary}\n\n\\begin{proof}\n\tThis follows from the fact that we have a monotone, upper semicontinuous decomposition to a locally connected space (Theorem \\ref{thm: peripheral splitting}) and that $x$ is a singleton in this decomposition.\n\\end{proof}\n\n\n\n\\section{Proper and Cocompact Action} \\label{section:proper and cocompact}\n\nOur goal of this section is to show that the stabilizer of a flat of $X_v$ also acts properly and cocompactly on $\\partial Y$ minus the higher-dimensional flat. Recall $X_v$ is the vertex-space for a component vertex $v$ of the maximal peripheral splitting of $G$. Since we need to distinguish between flats of $X_v$ and flats of $Y$, we need to establish some notation. Throughout this section, fix a flat $F\\subset X_v$ and let $F^*$ be the maximal dimensional flat of $X$ which contains $F$. By our construction of $Y$, $F^*\\subset Y$ but $F^*$ is not necessarily contained in $X_v$. Let $P$ and $P^*$ stabilize $F$ and $F^*$ respectively. Lastly, define $\\Omega:= \\partial Y \\setminus \\partial F^*$.\n\nThe ideas from this section come from \\cite{Ha17}, which proves the results in the case where $\\partial X$ is locally connected. In that setting, $Y$ and $X_v$ are the same and $F=F^*$.\n\nThe stabilizer $P$ acts properly and cocompactly on $F$. With some work, one can show that $P$ also acts properly and cocompactly on $\\Omega$. Furthermore, the fundamental domains for the action of $P$ on both can be picked in a way which allows us to relate the two spaces. The following lemma of Bowditch  establishes the proper and cocompact action of $P$ on $\\Omega$ \\cite{Bow12}:\n\n\\begin{lem}\\label{Bow1}\n\tLet $P$ be a group acting on topological spaces $C$ and $D$. Define the action of $P$ on $C\\times D$ as the diagonal action and let $\\mathbb{R} \\subset C\\times D$. If $\\mathbb{R}$ is $P$-invariant and the projection maps $\\pi_C$ and $\\pi_D$ from $\\mathbb{R}$ to $C$ and $D$ are both proper and surjective, then the following are equivalent:\n\t\\begin{enumerate}[(a)]\n\t\t\\item $P$ acts properly and cocompactly on $C$\n\t\t\\item $P$ acts properly and cocompactly on $D$\n\t\t\\item $P$ acts properly and cocompactly on $\\mathbb{R}$\n\t\\end{enumerate}\n\\end{lem}\n\nDefine $\\bot(F)$ to be the set of geodesic rays orthogonal to $F$. Recall a geodesic ray $r\\colon [0,\\infty) \\to X$ is \\textit{orthogonal} to a convex set $C\\subset X$ if $r(0)\\in C$ and for every $t>0$ and any $y\\in C$, the Alexandrov angle, $\\angle_{r(0)}(r(t),y)$, is greater than or equal to $\\pi\/2$. \n\nWe will apply this lemma where $F$ and $\\Omega$ are $C$ and $D$ respectively.  Let $\\mathbb{R}:= \\{ (x,q): \\text{there exists } q\\in \\bot(F) \\text{ with } d(x,q(0))\\leq A\\}$, where $A$ is the diameter of the fundamental domain for the action of $P$ on $F$. Throughout, we will assume our basepoint for the topology on $\\partial Y$ is in $F$. Since $P$ acts properly and cocompactly on $F$, we need to show the $P$-invariance of $\\mathbb{R}$ and the projection maps to $F$ and $\\Omega$ are both proper and surjective. \n\n\n\\begin{lem}\n\t$\\mathbb{R}$ is $P$-invariant.\n\\end{lem}\n\n\\begin{proof}\n\t$P$ acts by isometries on $Y$, so if $(x,q)\\in \\mathbb{R}$ and $p\\in P$, then \\linebreak $d(p\\cdot x, p\\cdot q(0))\\leq A$. It is clear that $p\\cdot q$ is a ray, but we need to show that it is orthogonal to $F$.\n\t\n\tFor all $t$, $\\pi_F(q(t))=q(0)$, so $q(0)$ is the closest point of $F$ to each point along the ray. Assume, for contradiction, that $\\pi_F( p\\cdot q(t)) = y$ and \\linebreak $y\\neq p\\cdot q(0)$. Then\t\n\t$$d(p\\cdot q(t), y) < d(p\\cdot q(t), p\\cdot q(0))$$\n\tThis is a strict inequality because of the uniquness of projection points. Applying $p^{-1}$ to all the points gives us:\n\t$$d(q(t),p^{-1} \\cdot  y) < d(q(t),q(0))$$\n\tSince $p^{-1}\\in P$ and $P$ fixes $F$, then $p^{-1} y \\in F$. But $q(0)$ is the closest point of $F$ to $q(t)$, so $y=p\\cdot q(0)$ and therefore $p\\cdot q\\in \\bot(F)$.\n\\end{proof}\n\n\\begin{lem}\\label{lemma: orthogonal rays for all bnd points}\n\tLet $\\alpha\\in \\Omega$. If $r$ is a ray representing $\\alpha$, then there is a ray $q\\in \\bot(F)$ asymptotic to $r$.\n\\end{lem}\n\nTo prove this lemma, we make use of the following lemma from \\cite{Ha17}, which follows closely to Lemma I.5.31 in \\cite{BH99}.\n\n\\begin{lem} \\label{isometric embeddings converge}\n\tIf $(Y,\\rho)$ is a separable metric space, $(X,d)$ a proper metric space, $y_0\\in Y$, and $K$ a compact subset of $X$, then any sequence of isometric embeddings, $c_n\\colon Y \\to X$, with $c_n(y_0)\\in K$ has a subsequence which converges point-wise to an isometric embedding $c\\colon Y\\to X$.\n\\end{lem}\n\nThroughout the remaining of the section, let $[x,y]$ denote the geodesic segment connecting points $x,y\\in X$. \n\n\\begin{proof}[Proof of Lemma \\ref{lemma: orthogonal rays for all bnd points}]\n\tFix a basepoint $x_0$, let  $x_n:=r(n)$ and let $y_n:= \\pi_F(x_n)$ for all $n\\in \\mathbb{N}$. First, assume for contradiction that $y_n$ is unbounded. Then $y_n\\to \\eta \\in \\partial F$. By \\cite{HK09}, there is a constant $M>0$ such that for all $n$ $d(x_0,[x_n,y_n])<M$, where $[x_n,y_n]$ is the geodesic segment in $Y$ from $x_n$ to $y_n$. Let $m$ be the point of this geodesic closest to $x_0$. Since the geodesic $[x_n,y_n]$ is orthogonal to $F$ and $x_0,y_n\\in F$, then the largest side of the triangle with vertices $x_0,y_n,m$ is the edge $[x_0,m]$. Therefore $d(x_0,y_n)<M$, contradicting $y_n$ tending to infinity.\n\t\n\tFor each $n\\in \\mathbb{N}$, let $f_n\\colon [0,n]\\to Y$ be the geodesic $[y_n,x_n]$, which is orthogonal to $F$. For each $k$ and all $n\\geq k$, consider the sequence $(f_n|_{[0,k]})$. By Lemma \\ref{isometric embeddings converge}, these converge to a map $q_k\\colon [0,k]\\to Y$. Notice for each $\\ell < k$, $q_\\ell = q_k|_{[0,\\ell]}$, since the tails of the sequences defining $q_\\ell$ and $q_k$ are identical. Therefore, $q\\colon [0,\\infty)\\to Y$ is a geodesic ray which is the limit of the sequence $(q_n)$. \n\t\n\tIt suffices to show that $q\\in \\bot(F)$ and $q$ is asymptotic to $r$. For the former, fix $y\\in F$ and notice that since $[y_n,x_n]$ is a geodesic segment orthogonal to $F$,  $\\angle_{y_n}(y,x_n) \\geq \\pi\/2$ for all $n\\in \\mathbb{N}$. Then, applying Proposition II.3.3(1) in \\cite{BH99}, see that the map $(x,y,p)\\to \\angle_p(x,y)$ is upper semicontinuous. Therefore the limiting segments $q_n$ are orthogonal to $F$ so $q\\in \\bot(F)$. To see that $q$ and $r$ are asymptotic, notice that $[y_n,x_n]\\subset B_M(r)$, and so $q$ is $M$-close to $r$, thus $q$ and $r$ are asymptotic. \n\t\n\t\n\t\n\\end{proof}\n\n\\begin{corollary}\n\tThe projection maps $\\pi_F$ and $\\pi_\\Omega$ are surjective. \n\\end{corollary}\n\n\\begin{proof}\n\tThe lemma above shows that $\\pi_\\Omega$ is surjective. Using the $P$-action and the fact that $A$ is the diameter of the fundamental domain, then $\\pi_F$ is surjective as well.\n\\end{proof}\n\nPart of showing the projection maps are proper requires an understanding of what happens to the boundary points when a sequence of rays in $\\bot(F)$ converge. \n\n\n\\begin{lem}\n\tIf $(r_n)$ is a sequence of rays orthogonal to $F$ which converge to an element $r\\in \\bot(F)$, then $r_n(\\infty)$ converges to $r(\\infty)$ in the cone topology.\n\\end{lem}\n\n\\begin{proof}\n\tFix $x_0\\in F$ a basepoint for the cone topology on $\\partial Y$. For each $r_n$, let $c_n$ be the ray based at $x_0$ asymptotic to $r_n$. Since $r_n\\to r$, then there is a constant $D$ such that $d(r_n(t),c_n(t))\\leq D$ for all $t$. Applying Lemma \\ref{isometric embeddings converge}, the rays $c_n$ converge to a ray $c$ which is asymptotic to $r$. \n\t\n\tFix $\\epsilon>0$ and let $s>0$. $U(c,s,\\epsilon)$ is a basic neighborhood of $c$ in $\\partial Y$. Since $c_n\\to c$ pointwise, then there is an $N\\in \\mathbb{N}$ such that $d(c_m(s),c(s))<\\epsilon$ for all $m>N$ and therefore $c_n(\\infty)\\to c(\\infty)$.\n\\end{proof}\n\n\nWhat remains is to show that the projection maps are both proper. Recall from Theorem \\ref{Quasi-convex Constant} there is a universal constant $\\kappa>0$ which makes certain subsets of a $\\cat(0)$ space with isolated flats $\\kappa$-quasiconvex. Haulmark extends this result to include geodesic rays which are orthogonal to some flat $F$.\n\n\\begin{lem}[{\\cite[3.7]{Ha17}}] \\label{Haulmark F-q quasiconvex}\n\tLet $F\\in \\mathcal{F}$ and $q\\in\\bot(F)$, then there exists a constant $\\kappa$ such that $q\\cup F$ is $\\kappa$-quasiconvex in $X$.\n\\end{lem}\n\n\\begin{remark}\n\tThe constants $\\kappa$ in Theorem \\ref{Quasi-convex Constant} and Lemma \\ref{Haulmark F-q quasiconvex} above are the same $\\kappa$.\n\\end{remark}\n\n\\begin{corollary}\n\tLet $F^*$ a flat of $X$, $F$ a flat subset of $F^*$ and $q\\in \\bot(F)$, then $q\\cup F$ is $\\kappa$-quasiconvex in $Y$.\n\\end{corollary}\n\n\\begin{proof}\n\tIn $Y$, $\\bot(F)=\\bot(F^*)$ since $F$ separates $F^*$ from the rest of $Y$. $Y$ inherits the metric from $X$, so $q\\cup F^*$ is quasiconvex in $Y$. Any geodesic in $Y$ from points of $F$ to points of $q$ are $\\kappa$-close to $F^*$ or $q$. If they are $\\kappa$-close to $F^*$, then they are $\\kappa$-close to $F$ because $F$ separates $F^*$ from $Y$. Lastly, since $Y$ is convex, all geodesics from between points of $q\\cup F^*$ are in $Y$ to begin with. Thus, $q\\cup F$ is $\\kappa$-quasiconvex.\n\\end{proof}\n\nThe two lemmas below relate how close together a ray $q\\in \\bot(F)$ and $c$ can be, where $c$ is the ray based at $x_0$ asymptotic to $r$.\n\n\\begin{lem}\n\tThere is a constant $M=M(\\kappa)>0$ such that for any ray $q\\in \\bot(F)$, $d(q(0),c(t))< M$ where $c$ is the ray asymptotic to $q$ based at $x_0$ and $t=d(x_0,q(0))$.\n\\end{lem}\n\n\\begin{proof}\n\tLet $\\beta\\colon [0,t]\\to F$ be the geodesic from $x_0$ to $q(0)$. Using the quasiconvexity result above, there is a $\\kappa>0$ such that $c$ is contained in the $\\kappa$-neighborhood of $q\\cup F$. Therefore, there are points $s\\in[0,\\infty)$, $x\\in q$, $y\\in F$ such that $d(c(s),x)\\leq \\kappa$ and $d(c(s),y)\\leq \\kappa$. Using the triangle inequality, $d(x,y)\\leq 2\\kappa$. Since $x\\in q$, $q$ is orthogonal $F$ and $y\\in F$, then $d(x,q(0))\\leq 2\\kappa$ as well and so $d(c(s),q(0))\\leq 3\\kappa$. \n\t\n\tThe triangle $\\triangle(x_0,c(s),q(0))$ has edges with side lengths $t$, $s,$ and at most $3\\kappa$, so $t\\in [s-3\\kappa,s+3\\kappa]$. Thus $|t-s|\\leq 3\\kappa$ and so \\[d(c(t),q(0)) \\leq d(c(t),c(s))+ d(c(s),q(0)) \\leq 3\\kappa+3\\kappa.\\] Setting $M=6\\kappa$ is the desired constant.\n\\end{proof}\n\nThe next lemma relates rays in $\\bot(F)$ and their geodesic representatives, based on how close the former ray's starting point is to a ray in $F$.\n\n\\begin{lem} \\label{delta(epsilon,M)}\n\tFix $\\epsilon>0$ and let $M=M(\\kappa)$ from above. Then there is a constant $\\delta=\\delta(\\epsilon,M)$ such that for any $n>0$ and $\\eta \\in \\partial F$ the following holds: if $q\\in \\bot(F)$ and $q(0)\\in U(\\eta, n, \\epsilon)$, then $c(n)\\in U(\\eta,n,\\delta)$, where $c$ is the ray asymptotic to $q$.\n\\end{lem}\n\n\\begin{proof}\n\tLet $\\beta\\colon  [0,t]\\to F$ be the geodesic from $x_0$ to $q(0)$. The lemma above shows that if $t=n$, then $d(c(t),q(0))\\leq M$ so $d(c(t),\\eta(n))\\leq M+\\epsilon$.\n\t\n\tIf $n<t$, notice that $d(c(t),\\eta(n)) \\leq d(c(t),\\beta(t))+ d(\\beta(t),\\eta(n)) \\leq M+\\epsilon$. Consider the triangle $\\triangle(x_0,c(t),\\eta(n))$. This has side lengths $t$, $n$, and at most $M+\\epsilon$, so $t-n\\leq M+\\epsilon$. With the triangle inequality we have:\t\n\t$$d(c(n),\\eta(n)) \\leq d(c(t),c(n)) + d(c(t), \\eta(n)) \\leq M+\\epsilon + M + \\epsilon$$\n\tLastly, if $t<n$, then $d(\\beta(t),\\eta(n))\\leq \\epsilon$ and $d(\\beta(t),c(t)) \\leq M$ so \\linebreak $d(\\eta(n),c(t))\\leq M+\\epsilon$. Consider the triangle $\\triangle(x_0,\\eta(n),\\beta(t)$, which has sidelengths $t$, $n$, and at most $\\epsilon$. Therefore $n-t< \\epsilon$. Applying the triangle inequalty we get:\t\n\t$$d(c(n),\\eta(n)) \\leq d(c(t),c(n))+ d(c(t),\\eta(n)) \\leq \\epsilon + M+\\epsilon$$\n\tTherefore, letting $\\delta = 2(M+\\epsilon)$ covers all three cases.\n\\end{proof}\n\nWith these two lemmas, we can now prove that the projection maps are proper:\n\n\\begin{prop}\n\tThe projection map $\\pi_\\Omega\\colon \\mathbb{R} \\to \\Omega$ is a proper map.\n\\end{prop}\n\n\\begin{proof}\n\tLet $K\\subset \\Omega$ be a compact set and consider \n\t\\[\\pi_\\Omega^{-1}(K) = \\{ (x,q): q\\in \\bot(F),~ q(\\infty)\\in K,~ d(x,q(0))\\leq A\\}\\subset \\mathbb{R}\\]\n\tLet $C$ be the following set\n\t\\[ C := \\{ x\\in F: \\exists q\\in \\bot(F), ~q(\\infty)\\in K,~ d(q(0),x)\\leq A\\} \\]\n\tWe claim $C$ is compact and we will show this by showing that it is closed and bounded. \n\t\n\tFirst, suppose it is unbounded. Then there is a sequence $x_n$ tending to infinity and so there is a sequence of rays $q_n$ with $q_n(0)$ tending to a point $\\eta\\in \\partial F$. Let $c_n$ be the geodesic ray based at $x_0$ asymptotic to $q_n$. Fix $\\epsilon>0$ and consider $U(\\eta,n,\\epsilon)$. Such a neighborhood contains all but finitely many of $q_i(0)$. By Lemma \\ref{delta(epsilon,M)}, $U(\\eta,n,\\delta)$ contains all but finitely many $c_i$ and therefore $c_i(\\infty)\\to \\eta$. But $c_i(\\infty)\\in K$ and $K$ is compact, so $\\eta\\in K$, a contradiction. Therefore $C$ is bounded.\n\t\n\tNext, let $x_i\\to x$ be a sequence of points in $C$. For each $x_i$, there is a a $q_i\\in \\bot(F)$ with $d(q_i(0),x_i)\\leq A$. So for sufficiently large $i$, $q_i(0)\\in \\overline{B_A(x)}$. Applying Lemma \\ref{isometric embeddings converge}, $q_i$ converge to $q\\in \\bot(F)$ pointwise. Therefore $q(0)\\in \\overline{B_A(x)}$. Since $q_i(\\infty)\\in K$ for all $i$ and $K$ is compact, then $q(\\infty)\\in K$ and so $q(0)\\in C$ and so is $x$.\n\t\n\tNow let $(x_n,q_n)$ be a sequence in $\\pi_\\Omega^{-1}(K)$. Since $q_n(0)\\in C$ and $C$ is compact, by Lemma \\ref{isometric embeddings converge}, after possibly passing to a subsequence, $q_n\\to q$ pointwise. Similarly, $x_n\\in C$, so after passing to a subsequence, $x_n$ converge as well and so $(x_n,q_n)$ can be made to converge to $(x,q)$.\n\t\n\tLastly, to see that $q\\in \\bot(F)$, take any $y\\in F$. Let $p_n=q_n(0)$ and $x_n=q_n(1)$. Since each $q_n$ is orthogonal to $F$, $\\angle_{p_n}(x_,y)\\geq \\pi\/2$. The map $(x,y,p)\\to \\angle_p(x,y)$ is upper semicontinuous (see \\cite{BH99} Proposition II.3.3(1))), so $q\\in \\bot(F)$. This concludes the proof.\n\t\n\\end{proof}\n\\begin{prop}\n\tThe projection map $\\pi_F\\colon \\mathbb{R} \\to F$ is a proper map.\n\\end{prop}\n\n\n\\begin{proof}\n\tLet $C\\subset F$ be compact. Then we have\n\t\\[ \\pi_F^{-1} (C) = \\{ (x,q): q\\in \\bot(F), ~x\\in C, ~d(q(0),x)\\leq A\\}\\]\n\tLet $(x_,q_n)$ be a sequence in $\\pi_F ^{-1}(C)$. Since $C$ is compact, so is $\\overline{B_A(C)}$ and $q_i(0)\\in \\overline{B_A(C)}$ for each $i$. By Lemma \\ref{isometric embeddings converge}, after possibly passing to a subsequence, $q_n\\to q$ pointwise. Furthermore, $x_n\\to x$ since $x_n\\in C$. Therefore, $(x_n,q_n)\\to (x,q)$.\n\t\n\t\tTo see that $q\\in \\bot(F)$, take any $y\\in F$. Let $p_n=q_n(0)$ and $x_n=q_n(1)$. Since each $q_n$ is orthogonal to $F$, $\\angle_{p_n}(x_,y)\\geq \\pi\/2$. The map $(x,y,p)\\to \\angle_p(x,y)$ is upper semicontinuous (see \\cite{BH99} Proposition II.3.3(1))), so $q\\in \\bot(F)$ and thus $(x,q)\\in \\pi_F^{-1}(C)$ making this set compact.\n\\end{proof}\nCombining everything we get the following theorem:\n\n\\begin{thm}\n\tLet $P$ be the stabilizer of a flat $F$ of $X_v$. Then $P$ acts properly and cocompactly on $\\Omega:= \\partial Y \\setminus \\partial F^*$, where $F^*$ is the maximal dimensional flat in $Y$ which contains $F$.\n\\end{thm}\n\nMore importantly, the work above gives the following association between compact sets of $F$ and those of $\\Omega$:\n\n\\begin{corollary}\n\t\\label{Compact Set Association}\n\tGiven a compact set $K\\subset \\Omega$, there exists a compact set $C\\subset F \\subset F^*$ associated to $K$  such that for each $\\eta \\in K$, there is a ray $c\\colon [0,\\infty) \\to Y$ such that $c\\in \\bot(F)=\\bot(F^*)$, $c(0)\\in C$ and $c(\\infty)=\\eta$. Furthermore, the set $C$ can be chosen $P$-equivariantly in the sense that if $C$ is associated to $K$, then $pC$ is associated to $pK$.\n\\end{corollary}\n\n\\begin{proof}\n\tThis follows immediately from the proofs of the projection maps being proper. For any $K\\subset \\Omega$, take $C$ to be $\\pi_F(\\pi_\\Omega^{-1}(K))$, which is compact as seen above and this association is $P$-equivariant since $\\mathbb{R}$ is.\n\\end{proof}\n\nThis corollary is vital for relating neighborhoods of points in $F$ to neighborhoods of points in $\\Omega$, which will ultimately allow us to show that $\\partial Y$ is locally connected.\n\n\\section{Local Connectivity in Flats} \\label{section:local conn at flats}\nThe goal of this section to show that $\\partial Y$ is locally connected at each of the remaining points. Once that is done, we can combine everything we have done to prove Theorem \\ref{thm main}. The decomposition theory gives that for each $\\xi\\in \\partial Y$, if $\\xi\\notin \\partial F^*$ for any flat $F^*$ of $Y$, then $\\partial Y$ is locally connected at $\\xi$. Additionally, if $\\beta\\in \\partial F^*$ and $\\beta \\notin \\partial X_v$, then $\\partial Y$ is locally connected at $\\beta$ since $\\partial F^*$ is a sphere. Therefore all that is left to show is that if $\\alpha \\in \\partial X_v \\cap \\partial F^*= \\partial F$, then $\\partial Y$ is locally connected at $\\alpha$.\n\nInstead of showing local connectivity, we show that if $\\alpha\\in \\partial F$, then $\\partial Y$ is weakly locally connected at $\\alpha$. The distinction between locally connected and weakly locally connected is toward the end of the section, Definition \\ref{def:wlc}.\n\nMany of the ideas in this section come from \\cite{HR17}. The authors work in the setting where $G$ is a $\\cat(0)$ group with isolated flats with a trivial maximal peripheral splitting and ultimately conclude that $\\partial G$ is  locally connected at every point. We show here that the ideas work in our setting as well.\n\nThe approach is to use the association established in Corollary \\ref{Compact Set Association} to relate neighborhoods in $\\overline{F}$ to neighborhoods of $\\overline{\\Omega}$. Recall $F$ is the subflat of $F^*$ such that $\\partial F \\subset \\partial X_v$. Before establishing this connection, we need to understand $\\overline{\\Omega}$.\n\n\\begin{lem}\n\t$\\overline{\\Omega}=\\Omega \\cup \\partial F$.\n\\end{lem}\n\n\\begin{proof}\n\tFirst, we will show that $\\Omega \\cup \\partial F$ is closed. We know that $\\partial F^* $ and $\\partial F$ are both closed in $\\partial Y$. Therefore $\\partial F^* \\setminus \\partial F$ is open. We can also see that \n\t$$\\partial Y \\setminus (\\Omega \\cup \\partial F) = \\partial F^* \\setminus \\partial F$$\n\tTherefore $\\Omega \\cup \\partial F$ is closed since its complement in $\\partial Y$ is open and so $\\overline{\\Omega}\\subset \\Omega \\cup \\partial F$.\n\t\n\tNow suppose $\\alpha\\in \\partial F$, then we want to show that $\\alpha\\in \\overline{\\Omega}$. We will do this by creating a sequence of points in $\\Omega$ which limit to $\\alpha$. Fix $x_0\\in F$ and let $c\\colon [0,\\infty)\\to Y$ be the ray representing $\\alpha$ and let $C$ be a fundamental domain for the action of $P$ on $F$. Using the $P$-action, let $p_i\\cdot C$ cover the ray $c$. For each $p_i$, let $q_i\\in \\bot(F)$ such that $q_i(0)\\in p_i\\cdot C$. Let $r_i$ be the geodesic ray based at $x_0$ asymptotic to $q_i$.\n\t\n\tFix $\\epsilon>0$ and notice that since the translates of $C$ cover $c$, then for all but finitely many $i$, $q_i(0)\\in U(c,n,\\epsilon)$. Applying Lemma \\ref{delta(epsilon,M)}, we have  \\[r_i(n)\\in U(c,n,\\epsilon+\\delta)\\]for all but finitely many $i$. This means that $r_i(\\infty)\\in U(c,n,\\epsilon+\\delta)$ for all but finitely $i$ and thus $r_i(\\infty)\\to \\alpha$ as $n\\to \\infty$, as desired.\t\t\n\t\n\t\n\\end{proof}\n\nNext we restate a result of Haulmark's to emphasize the connection between rays orthogonal to the flat $F$ and their corresponding endpoints:\n\n\\begin{lem} [\\cite{Ha17}] \\label{Haulmark Quasi Convex}\n\tLet $\\kappa$ be the CAT(0) with isolated flats constant as in Theorem \\ref{Quasi-convex Constant}. Suppose $c'\\in \\bot(F)$ and $c$ is some ray contained in $F$. If $c'(0)\\in U(c,r,D)$ for some constants $r,D$, then $c'(\\infty)\\in U(c,r,D+\\kappa)$. Conversely, if $c'(\\infty)\\in U(c,r,D)$, then $c'(0)\\in U(c,r,D+\\kappa)$.\n\\end{lem}\n\n\nWe make use of the following result of Bestvina:\n\n\\begin{prop}[\\cite{Bes96}]\n\tLet $H$ be a group acting geometrically on a $\\cat(0)$ space $Z$. Let $C\\subset Z$ be a compact set. The $H$-translates of $C$ form a null family in the compact space $\\overline{Z}$.\n\\end{prop}\n\n\n\\begin{prop}\n\tIf $K\\subset \\Omega$ is compect, then the collection of $P$-translates of $K$ is a null family in $\\overline{\\Omega}$, where $P$ is the stabilizer of the flat $F\\subset X_v$.\n\\end{prop}\n\n\\begin{proof}\t\n\tLet $K\\subset \\Omega$ be compact and let $C\\subset F$ be the compact set associated to $K$ from Corollary \\ref{Compact Set Association}. Fix a positive constant $D>0$ and let $\\kappa$ be the constant from Theorem \\ref{Quasi-convex Constant}. From the Remark \\ref{remark: null family condition}, it suffices to show that for each $r<\\infty$, only finitely many $P$-translates of $K$ are not contain in any set of the form $U(\\cdot, r, D)$. \n\t\n\tUsing the Bestvina result above, $P^*$-translates of $C$ form a null family in $\\overline{F^*}$. Since $P\\subset P^*$ and $\\overline{F}\\subset \\overline{F^*}$, then the $P$-translates of $C$ form a null family in $\\overline{F}$. So only finitely many of the $P$-translates of $C$ are not in set of the form $U(\\cdot, r,D+\\kappa)$. For any $p\\in P$, if $pK$ lies in a set of the form $U(\\cdot, r, D)$, then $pC$ lies in a set of the form $U(\\cdot, r, D+\\kappa)$ by Lemma \\ref{Haulmark Quasi Convex}. Therefore the $P$-translates of $K$ form a null family in $\\overline{\\Omega}$.\n\\end{proof}\n\n\n\nThe association between compact sets of $F$ and those of $\\Omega$ established in Corollary \\ref{Compact Set Association} can be improved even further: the compact sets chosen can be connected fundamental domains for the action of $P$. To see this is connected, first observe the following:\n\n\\begin{prop} \\label{Connected Fundamental Domain}\n\tLet $(G,\\mathbb{P})$ be a CAT(0) group with isolated flats and peripheral structure $\\mathbb{P}$. Let $(G_v,\\mathbb{P}_v)$ be a component vertex of $G$'s maximal peripheral splitting with the peripheral structure coming from $G$. Let \\linebreak $\\rho \\in \\partial(G_v, \\mathbb{P}_v)$ be a parabolic point stabilized by $P$. Then there exists a connected, compact fundamental domain $C$ for the action of $P$ on $\\partial (G_v, \\mathbb{P}_v) \\setminus \\{\\rho\\}$. \n\\end{prop}\n\n\\begin{proof}\n\t\n\tSince $(G_v,\\mathbb{P}_v)$ is a component vertex of a 1-ended CAT(0) group with isolated flats, we know that $\\partial (G_v, \\mathbb{P}_v)$ is connected and locally connected and therefore path connected. Notice that $\\pi(F)=\\pi(F^*)=\\rho$. We also know that since $G_v$ is a component vertex of a maximal peripheral splitting, then $G_v$ does not split further relative to $\\mathbb{P}_v$. Therefore $\\rho$ is not a global cut point of $\\partial (G_v, \\mathbb{P}_v)$. These facts follow from Theorem \\ref{thm: peripheral splitting}.\n\t\n\tSince $\\rho$ is not a global cut point and $\\partial (G_v, \\mathbb{P}_v)$ is path connected, then $\\partial (G_v, \\mathbb{P}_v)\\setminus \\{\\rho\\}$ is path connected as well. We know that $P$ acts cocompactly on $\\partial (G_v, \\mathbb{P}_v)\\setminus \\{\\rho\\}$. Let $C_0$ be a compact fundamental domain for this action. Our goal is to show that $C_0$ is contained in a connected fundamental domain $C$. \n\t\n\tLet $d= d(C_0, \\rho)$ in $\\partial (G_v, \\mathbb{P}_v)$. We can cover $C_0$ with finitely many open connected sets of diameter at most $d\/2$. The union of the closures of these sets form a compact set $C_1\\subset \\partial (G_v, \\mathbb{P}_v)$ containing $C_0$ and having only finitely many connected components. By our choice of $d$, $C_1 \\subset \\partial (G_v, \\mathbb{P}_v) \\setminus \\{\\rho\\}$. We can then join the finitely many connected components of $C_1$ with finitely many paths in $\\partial (G_v, \\mathbb{P}_v) \\setminus \\{\\rho\\}$, giving us a compact connected fundamental domain $C$.\n\t\n\\end{proof}\n\nUsing this, we can find a connected fundamental domain for $P$'s action on $\\Omega$.\n\n\\begin{prop}\\label{Fundamental Domain on Omega}\n\tThere is a compact, connected fundamental domain $K\\subset \\Omega$ for the action of $P$ on $\\Omega$. Furthermore, if $S$ is any finite generating set for $P$, then $K$ can be chosen such that $sK\\cap K \\neq \\emptyset$ for each $s\\in S$. \n\\end{prop}\n\n\\begin{proof}\n\tWe will use the decomposition map $\\pi\\colon  \\partial Y \\to \\partial (G_v,\\mathbb{P}_v)$ and then pull back the compact, connected fundamental domain for $P$'s action on $\\partial(G_v,\\mathbb{P}_v) \\setminus \\{\\rho \\}$ to get a fundamental domain on $\\Omega$.\n\t\n\tLet $C_0$ be a connected fundamental domain for the action of $P$ on \\linebreak $\\partial (G_v, \\mathbb{P}_v) \\setminus \\{\\rho\\}$ as in Proposition \\ref{Connected Fundamental Domain}. Without loss of generality, we may increase the size of $C_0$ such that $C_0\\cap sC_0 \\neq \\emptyset$ for all $s\\in S$. We can then repeat the process from Proposition \\ref{Connected Fundamental Domain} to construct a connected $C$.\n\t\n\tSince the map $\\pi\\colon \\partial Y \\to \\partial (G_v, \\mathbb{P}_v)$ defines an upper semicontinuous monotone decomposition, then we know that $\\pi ^{-1}(C)$ is compact and connected. \tThe map $\\pi$ is $G$-equivariant, and therefore since $C$ is a fundamental domain, $K$ is as well. Since $C$ satisfies the desired intersection property, then $K$ does as well.\n\\end{proof}\n\nUsing the association between subsets of $F$ and those of $\\Omega$, we can establish a way to determine when subsets of $\\Omega$ are connected.\n\n\\begin{prop} \\label{K and C}\n\tThere exists compact, connected fundamental domains $K$ and $C$ for the action of $P$ on $\\Omega$ and $F$, respectively, such that the following property holds: Let $\\P$ be any subset of $P$. Then:\n\t\n\t\\begin{center}\n\t\tIf $\\bigcup\\limits_{p\\in \\P} pC$ is connected in $F$, then $\\bigcup\\limits_{p\\in \\P} pK$ is connected in $\\Omega$.\n\t\\end{center}\n\\end{prop}\n\n\\begin{proof}\n\tChoose a compact, connected fundamental domain $C$ for the action of $P$ on $F$. Let $S$ be the set of $p\\in P$ such that $pC\\cap C\\neq \\emptyset$. Then $S$ is a finite generating set for $P$. Use Proposition \\ref{Fundamental Domain on Omega} to pick $K$ such that $sK\\cap K \\neq \\emptyset$ for all $s\\in S$. This implies the following, which implies the desired conclusion:\t\n\t\\begin{center}\n\t\tIf $C\\cap pC \\neq \\emptyset$, then $K\\cap pK \\neq \\emptyset$\n\t\\end{center}\n\\end{proof}\n\nWe will now fix the compact, connected fundamental domains $K$ and $C$ for $P$'s action on $\\Omega$ and $F$, respectively. Using Corollary \\ref{Compact Set Association}, let $c'\\colon [0,\\infty)\\to Y$ be a geodesic ray which meets $F$ orthogonally. Using the $P$-action, we may assume that $c'(0)\\in C$ and $c'(\\infty)\\in K$. We will call $c'(0)=q_0$ and use $q_0$ as the basepoint for the cone topology.\n\nLet $\\alpha\\in \\partial F$. Our method is two-fold. First,  associate connected neighborhoods of $\\alpha$ in $\\overline{F^*}$ to neighborhoods of $\\alpha$ in $\\overline{\\Omega}$, which by Proposition \\ref{K and C} will be connected. Then, given a neighborhood of $\\alpha$ in $\\overline{\\Omega}$, find a connected neighborhood of $\\alpha$ in $\\overline{F^*}$ to associate inside of the neighborhood in $\\overline{\\Omega}$. \n\nTo make this possible, we need to find suitably nice neighborhoods in $\\overline{F^*}$. \n\n\n\\begin{defn}\n\tLet $Z$ be a $\\cat(0)$ space and $\\xi\\in \\partial Z$. A neighborhood $\\overline{N}$ of $\\xi$ in $\\overline{Z}$ is \\textit{clean} if the following conditions hold:\n\t\n\t\\begin{enumerate}\n\t\t\\item $\\overline{N}$ is connected.\n\t\t\\item $N=\\overline{N}\\cap Z$ is connected, and\n\t\t\\item Every point of $\\Lambda:=\\overline{N}\\cap \\partial Z$ is a limit point of $N$.\n\t\\end{enumerate}\n\\end{defn}\n\nHruska and Ruane prove  (\\cite{HR17}, Proposition 3.3) that for any $\\xi\\in \\partial Z$, $\\xi$ has a local base of clean connected neighborhoods. We show below that we can pick this neighborhood basis to be clean and connected both in $\\overline{F}$ and $\\overline{F^*}$. \n\n\\begin{lem}\n\tThere is a local base of neighborhoods of $\\alpha$ in $\\overline{F^*}$ which are clean and connected in $\\overline{F}$ and in $\\overline{F^*}$. \n\\end{lem}\n\n\\begin{proof}\n\tFix $q_0$ as the basepoint for the cone topology and let $c\\colon [0,\\infty)\\to F^*$ be the ray based at $q_0$ representing $\\alpha$. It is shown in \\cite{HR17} that basic open sets, $U(c,r,D)$, are clean and connected in $\\overline{F^*}$.  \n\t\n\tFirst note that since $c(0)\\in F$, $c(\\infty)\\in \\partial F$ and $F$ is convex in $F^*$, then $U(c,r,D)\\cap \\overline{F}$ is non-empty. Denote $N_F= F\\cap U(c,r,D)$. \n\t\n\tA point $p\\in F$ is in $N_F$ if and only if the geodesic segment from $c(0)$ to $p$ intersects $B_D(c(r))$. Let $\\gamma\\colon [0,t]\\to F$ be the geodesic segment $[c(0),p]$, then we can choose $s$ so that $d(\\gamma(s),c(r))<D$. It follows that the entire geodesic segment $[\\gamma(s),\\gamma(t)]\\subset U(c,r,D)$, so every point of $N_F$ is contained in a connected subset of $N_F$ which intersects the connected ball $B_D(c(r))$ and thus $N_F$ is connected. All of $U(c,r,D)$ is contained in the closure of $N_F\\subset \\overline{F}$, and therefore is connected as well. Since $U(c,r,D)$ is contained in the closure of $N$, the limit set requirement is satisfied as well.\n\t\n\\end{proof}\n\nFor a given clean, connected neighborhood $\\overline{N}=N\\cup \\Lambda$, let \\linebreak $\\P:=\\{p\\in P: pC\\cap N\\neq \\emptyset\\}$. Since $q_0\\in F$, $\\P$ is non-empty and by Proposition \\ref{K and C}, $\\bigcup\\limits_{p\\in \\P} pC$ is connected. \n\n\\begin{defn}\n\tLet $\\alpha\\in \\partial F$ and $\\overline{N}$ be a clean, connected neighborhood with $\\overline{N}\\cap F = N$ and $\\overline{N}\\cap \\partial F = \\Lambda'$. Let $\\P := \\{p\\in P: pC\\cap N\\neq \\emptyset\\}$. Then the \\textit{neighborhood in $\\overline{\\Omega}$ of $\\alpha$ associated to $\\overline{N}$} is $\\overline{Z}= \\left( \\bigcup\\limits_{p\\in \\P} pK\\right) \\cup \\Lambda'$. \n\\end{defn} \n\nWe now need to show three things: $\\overline{Z}$ is a neighborhood of $\\alpha$, it is connected, and when $\\overline{N}$ can be chosen to be `small,' then $\\overline{Z}$ is `small' as well.\n\n\\begin{lem} \\label{Z is a nieghborhood}\n\tThe set $\\overline{Z}$ associated to $\\overline{N}$ is a neighborhood of $\\alpha$ and is connected.\n\\end{lem}\n\n\\begin{proof}\n\tFirst, we show that $\\overline{Z}$ is a neighborhood of $\\alpha$. Let $c\\colon [0,\\infty)\\to F^*$ be the ray based at $q_0$ representing $\\alpha$ and fix some $D>0$. Since $\\overline{N}$ is a neighborhood of $\\alpha$, we can choose $R$ large enough so that $U(c,R,D+\\kappa)\\cap \\overline{F^*}$ is contained in $\\overline{N}$, and therefore $U(c,R,D)\\cap \\overline{F^*}$ is contained in $\\overline{N}$ as well. The $P$-translates of $K$ form a null family, so there is a neighborhood $V$ of $\\alpha$ with $V\\subset U(c,R,D)$ such that if $pK\\cap V \\neq \\emptyset$, then $pK\\subset U(c,R,D)$.\n\t\n\tNow it suffices to show that $V\\subset \\overline{Z}$. For $\\eta\\in V$, either $\\eta\\in \\Omega$ or $\\eta\\in \\partial F$. In the former case, $\\eta\\in pK$ for some $p\\in P$ and so $pK\\subset U(c,R,D)$. In particular, $p\\cdot q_\\infty\\in U(c,R,D)$ and by Lemma \\ref{Haulmark Quasi Convex}, $p\\cdot q_0\\in U(c,R,D+\\kappa)$. Since $q_0\\in C$ and $U(c,R,D+\\kappa)\\cap \\overline{F^*}\\subset \\overline{N}$, then $pC\\cap N \\neq \\emptyset$. Therefore, $\\eta\\in \\overline{Z}$ by the construction.\n\t\n\tIn the latter case, $\\eta\\in \\partial F\\cap V$, so\n\t\n\t$$\\eta \\in U(c,R,D)\\cap \\partial F \\subset U(c,R,D+\\kappa)\\cap \\partial F \\subset \\overline{N}\\cap \\partial F = \\Lambda'\\subset \\overline{Z}$$\n\tTherefore, $V\\subset \\overline{Z}$ and so it is a neighborhood of $\\alpha$ in $\\overline{\\Omega}$.\n\t\n\tNow we must show that $\\overline{Z}$ is connected. By Proposition \\ref{K and C}, $\\overline{Z}\\cap \\Omega$ is connected. Therefore it suffices to show that every point of $\\Lambda'$ is a limit point of $\\overline{Z}\\cap \\Omega$. Let $\\xi\\in \\Lambda'$. Since $\\overline{N}$ is clean, $\\xi$ is the limit of a sequence $x_i\\in N$. Each $x_i\\in p_iC$ for some $p_i\\in P$. Fix $\\epsilon>0$ and note that since $x_i\\to \\xi$, $x_i\\in U(\\xi,R,\\epsilon)$ for all but finitely many $i$. Let $A$ be the diameter of $C$, then $p_i\\cdot q_0\\in U(\\xi,R,\\epsilon+A)$ for all but finitely many $i$, and so by Lemma \\ref{delta(epsilon,M)}, $p_i\\cdot q(R)\\in U(\\xi,R,\\epsilon+A+\\delta)$ for all but finitely many $i$. Therefore, $p_i\\cdot q(\\infty)\\in U(\\xi,R,\\epsilon+A+\\delta)$ and thus $p_i\\cdot q(\\infty)$ converge to $\\xi$. Since $p_i\\cdot C \\cap N \\neq \\emptyset$, then $p_i\\cdot K\\cap \\overline{Z} \\neq \\emptyset$ and so $\\xi$ is the limit of points in $\\overline{Z}\\cap \\Omega$ as desired.\n\\end{proof}\n\n\\begin{lem} \\label{Z small}\n\tThe neighborhood $\\overline{Z}$ associated to $\\overline{N}$ can be made arbitrarily small by choosing $\\overline{N}$ to be a sufficiently small neighborhood of $\\alpha$ in $\\overline{F^*}$\n\\end{lem}\n\n\\begin{proof}\n\tLet $U$ be a neighborhood of $\\alpha$ in $\\overline{\\Omega}$. Since the $P$-translates of $K$ form a null family. There is a neighborhood $V\\subset U$ such that if $pK\\cap V$ is non-empty, then $pK\\subset U$. Since $V$ is a neighborhood of $\\alpha$, there are constants $R>0$ and $D>0$ such that $U(c,R,D+\\kappa)\\subset V$ and $U(c,R,D)\\cap \\partial F\\subset V$. Then, by Lemma \\ref{Haulmark Quasi Convex}, $U(c,R,D)$ is a neighborhood of $\\alpha$ in $\\overline{F^*}$ such that if $p\\cdot q_0\\in U(c,R,D)$, then $p\\cdot q_\\infty \\in U(c,R,D+\\kappa)$ and so $pK\\subset U$. Furthermore, since the $P$-translates of $C$ form a null family, there is a $W\\subset U(c,R,D)$ such that if $pC\\cap W$ is non-empty, then $pC\\subset U(c,R,D)$. Finally, there is a clean, connected neighborhood $\\overline{N}$ within $W$.\n\t\n\tIt remains to show that the associated neighborhood $\\overline{Z}$ is contained in $U$. Let $\\eta\\in \\overline{Z}$. By our choice of $U(c,R,D)$, if $\\eta\\in \\partial F$, then $\\eta \\in V$ and therefore in $U$. If $\\eta \\in \\Omega$, then $\\eta\\in pK$ for some $p$, so $pK\\cap \\overline{Z}$ is non-empty and thus $pC\\cap N$ is non-empty. This means that $pC\\cap W\\neq \\emptyset$ so $pC\\subset U(c,R,D)$ and thus $pK$ intersects $V$ non-trivially. By the choice of $V$, $pK\\subset U$ and so $\\eta\\in U$ as desired. Thus $\\overline{Z}$ is contained in $U$.\n\t\n\t\n\t\n\\end{proof}\n\nRecall the definitions of weakly locally connected and locally connected:\n\n\\begin{defn}[Weakly locally connected]\\label{def:wlc}\n\tLet $M$ be a topological space. $M$ is \\textit{weakly locally connected} at $m\\in M$ if for every open set $V$, there  is a connected (and not necessarily open) set $N\\subset V$ with $m$ in the interior of $N$.\n\t\n\tWe say a space $M$ is weakly locally connected if it is weakly locally connected at every point.\n\\end{defn}\n\n\n\\begin{defn}[Locally connected]\n\tLet $M$ be a topological space. $M$ is \\textit{locally connected} at $m\\in M$ if for every open set $V$ containing $m$, there is a connected open set $U$ with $m\\in U \\subset V$. \n\t\n\tWe say $M$ is locally connected if it is locally connected at every point.\n\\end{defn}\n\nIn Lemma \\ref{Z is a nieghborhood} and Lemma \\ref{Z small}, we do not know if $\\overline{Z}$ is open or not, so we cannot prove local connectivity in $\\partial Y$. Instead, we show the following:\n\\begin{prop}\\label{Y is wlc}\n\tIf $\\alpha\\in \\partial Y \\cap \\partial F$, then $\\partial Y$ is weakly locally connected at $\\alpha$.\n\\end{prop}\n\n\\begin{proof}\n\tLet $U$ be a neighborhood of $\\alpha$ in $\\partial Y$. Since $\\partial Y = \\overline{\\Omega} \\cup \\partial F^*$, then $U=U_1\\cup U_2$, with $U_1$ a neighborhood of $\\alpha$ in $\\overline{\\Omega}$ and $U_2$ a neighborhood of $\\alpha$ in $\\partial F^*$. Combining Lemma \\ref{Z is a nieghborhood} and Lemma \\ref{Z small}, there is a $V_1\\subset U_1$ which is connected and contains $\\alpha$. Since $\\partial F^*$ is a sphere, it is locally connected, so there is some $V_2\\subset U_2$ which is connected and contains $\\alpha$. Lastly, since $V_1\\cap V_2$ is non-empty, then their union is connected as well. Thus $V_1\\cup V_2\\subset U$ is a connected neighborhood of $\\alpha$.\n\\end{proof}\n\nCombining all of this work leads to the following theorem:\n\n\\begin{thm}\\label{Y is pconn}\n\tLet $(G,\\mathbb{P})$ be a $\\cat(0)$ group with isolated flats acting geometrically on $X$, let $G_v$ be a component vertex in its maximal peripheral splitting, acting geometrically on $X_v\\subset X$ and let $Y$ be as in Definition \\ref{defn Y}. Then $\\partial Y$ is path connected. \n\\end{thm}\n\n\\begin{proof}\n\t\n\tIt suffices to show that $\\partial Y$ is both connected and weakly locally connected at every point. By Theorem 27.16 in \\cite{Wil70}, a space which is weakly locally connected at every point is locally connected at every point. Also by Theorem 31.2 in \\cite{Wil70}, a space which is compact, connected, locally connected and metrizable is locally path connected, which implies globally path connected (by \\cite{Wil70}, Theorem 27.5). Since boundaries of proper metric spaces are compact and metrizable, all we need to show is that $\\partial Y$ is connected and weakly locally connected at each point. \n\t\n\tConnectedness follows from Theorem \\ref{monotone usc decomp of Y}. Since the map $\\pi\\colon \\partial Y \\to \\partial (G_v,\\mathbb{P}_v)$ is monotone and $\\partial(G_v,\\mathbb{P}_v)$ is connected, then $\\partial Y$ is connected as well.\n\t\n\tFor weakly locally connected, there are three types of points in $\\partial Y$ to consider: points in $\\partial X_v$ and not a flat, points just in a flat, and points in both $\\partial X_v$ and $\\partial F$ for some flat $F$. By Corollary \\ref{Y non-flat points lconn}, if $x\\in \\partial Y$ and $x\\notin \\partial F$ for any flat $F$, then $\\partial Y$ is locally connected at $x$ and therefore weakly locally connected at $x$. If $\\xi\\in \\partial F^*$ for some flat $F^*$ and $\\xi \\notin \\partial X_v$, then $\\partial Y$ is locally connected at $\\xi$ and so weakly locally connected at $\\xi$. This is because sufficiently small neighborhoods of $\\xi$ only contain points in $\\partial F^*$, which is locally connected. Lastly, if $\\alpha\\in \\partial X_v \\cap \\partial F$ for some flat $F$, then by Proposition \\ref{Y is wlc}, $\\partial Y$ is weakly locally connected at $\\alpha$.\n\\end{proof}\n\nWe are almost ready to combine all this work to conclude that $\\cat(0)$ groups with isolated flats have path connected visual boundaries. But first we need to show that these groups satisfy the lonely condition.\n\n\\begin{lem} \\label{IFP blocks are lonely}\n\tSuppose $G$ is a 1-ended $\\cat(0)$ group with isolated flats acting geometrically on a $\\cat(0)$ space $X$. Let $\\mathcal{B}$ be the block decomposition coming from the maximal peripheral splitting of $\\mathcal{G}$, then all irrational rays are lonely.\n\\end{lem}\n\n\n\\begin{proof}\n\tFor any flat $F$, let $\\gamma_F\\colon [0,b]\\to X$ be the geodesic segment from $x_0$ to the closest point of $F$. Since $F$ is convex, a closest point exists and is unique. By Theorem \\ref{Quasi-convex Constant}, $\\gamma_F\\cup F$ is $\\kappa$-quaisconvex. In particular, every geodesic starting at $x_0$ which intersects $F$ must be the in $\\kappa$-neighborhood of $\\gamma_F(b)$. \n\t\n\tLet $c,c'$ be rays in $X$ based at $x_0$ with the same, infinite itinerary. Since these rays have the same itinerary, they pass through the same sequence of flats, and must pass through infinitely many flats. But then the rays are $2\\kappa$-close infinitely often. By the convexity of the distance metric, the rays must be $2\\kappa$-close always, and therefore $c$ and $c'$ are asymptotic.\n\t\n\\end{proof}\n\n\n\\begin{thm} \\label{thm main} \n\tAll 1-ended $\\cat(0)$ groups with isolated flats have path connected visual boundaries.\n\\end{thm}\n\n\\begin{proof}\n\tLet $(G,\\mathbb{P})$ be a $\\cat(0)$ group with isolated flats. It suffices to show there is a space $X$ on which $G$ acts which has path connected visual boundary. Let $\\mathcal{G}$ be the maximal peripheral splitting for $(G,\\mathbb{P})$, $X$ be the tree-of-spaces on which $G$ acts geometrically, and $\\mathcal{B}$ the block decomposition coming from this peripheral splitting.\n\t\n\tEach peripheral vertex group in the splitting is virtually $\\mathbb{Z}^n$ for some $n$, so has path connected visual boundary. If each component vertex has path connected visual boundary as well, then by Theorem \\ref{combo} $\\partial X$ is path connected. By Lemma \\ref{IFP blocks are lonely}, the irrational rays are all lonely, so we can apply the theorem.\n\t\n\tIf there is a peripheral vertex which does not have path connected visual boundary, then by Theorem \\ref{Y is pconn}, $\\partial_R X$ is path connected. This is true because\n\t$$\\partial_R X = \\bigcup_{v\\in \\mathcal{C}} \\partial Y_v$$\n\twhere $\\mathcal{C}$ is the collection of component vertices in the tree $\\mathcal{T}$ for the splitting $\\mathcal{G}$. Therefore, we can apply Proposition \\ref{lonely plus rational is pconn} to conclude that $\\partial X$ is path connected.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{alpha\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper, we study the stability of the absolutely \ncontinuous spectrum of one-dimensional Stark operators\nunder various classes of perturbations. Stark Schr\\\"odinger operators describe behavior of the charged particle in the constant electric field. The absolutely continuous spectrum is a manifestation of the fact that the particle described by the operator propagates to infinity at a rather fast rate (see, e.g. \\cite{AS}). It is therefore interesting to describe the classes of perturbations which preserve the absolutely continuous spectrum of the Stark operators. In the first part\nof this work, we study perturbations of Stark operators by decaying potetnials.\nThis part is inspired by the recent work of Naboko and Pushnitski \\cite{NaPu}.\nThe general picture that we prove is very similar to the \ncase of perturbations of free Schr\\\"odinger operators \\cite{Kis1}. \nIn accordance with physical intuition, however, the absolutely continuous spectrum is stable under stronger perturbations than in the free case. \nIf in the free case the short range potentials preserving purely absolutely continuous spectrum of the free operator are given by condition (on the power scale)\n$|q(x)| \\leq C(1+|x|)^{-1-\\epsilon},$ in the Stark operator case the corresponding condition reads $|q(x)| \\leq C(1+|x|)^{-\\frac{1}{2}-\\epsilon}.$ If $\\epsilon$ is allowed to be zero in the above bounds, imbedded eigenvalues may occur in both cases (see, e.g. \\cite{NaPu}, \\cite{WN}). Moreover, in both cases if we allow potential to decay slower by an arbitrary function growing to infinity, very rich singular spectrum, such as a dense set of  eigenvalues, may occur (see \\cite{Na} for the free case and \\cite{NaPu} for the Stark case for precise formulation and proofs of these results). The first part of this work draws the paprallel further, showing that the absolutely continuous spectrum of Stark operators is preserved under perturbations satisfying  $|q(x)| \\leq C(1+|x|)^{-\\frac{1}{3} -\\epsilon},$ in particular even in the regimes where a dense \nset of eignevalues occurs; hence in such cases these eigenvalues are genuinely imbedded. Similar results for the free case were proven in \\cite{Kis1}, \\cite{Kis2}. Our main strategy of the proof here is similar to that in \\cite{Kis1} and \\cite{Kis2}: we study the asymptotics of the generalized eigenfunctions and then apply Gilbert-Pearson theory \\cite{GiPe} to derive spectral consequences.\n\nWhile the main new tool we introduce in our treatment of Stark operators is the same as in the free case, namely  the a.e. convergence of the Fourier-type integral operators, there are some major differences. First of all, the spectral parameter enters the final equations that we study in a different way and this makes analysis more complicated. Secondly, we employ a different method to analyze the asymptotics. Instead of Harris-Lutz asymptotic method we study appropriate Pr\\\"ufer transform variables, simplifying the overall consideration. \n\n In the second part of the work we \ndiscuss perturbations by potentials having some additional smoothness properties, but without decay. It turns out that for Stark operators the effects of decay or of additional smoothness of potential on the spectral properties are somewhat similar. \nIt was known for a long time that if a potential perturbing Stark operator has two bounded derivatives the spectrum remains purely absolutely continuous (actually, \ncertain growth of derivatives is also allowed, see Section 3 for details or Walter \\cite{Wa} for the original result). We note that the results similar to Walter's on the preservation on absolutely continuous spectrum were also obtained in \\cite{Ben} by applying different type of technique (Mourre method instead of studying asymptotics of solutions).\n  On the other hand, if the perturbing potential is a sequence of derivatives of $\\delta$ functions in integer points on $R$\nwith certain couplings,   the spectrum may turn pure point \\cite{AEL}, \\cite{Ex}.\n In some sense, the $\\delta'$ interaction is the most singular and least\n``differentiable\" among all available natural perturbations of one-dimensional Schr\\\"odinger operators \\cite{KiSi}. Hence we have very different spectral properties on the very opposite sides of the smoothness scale.  This work closes the part of the gap.  We improve the well-known results of Walter\n\\cite{Wa}  concerning the minimal smoothness required for the preservation of the absolutely continuous spectrum and show that in fact  existence and minimal smoothness of the first derivative is sufficient to imply absolute continuity of the spectrum.      \n\n\n\\section{Decaying perturbations}\n\nConsider a self-adjoint operator $H_{q}$ defined by the differential \nexpression \n\\[ H_{q}u= -u''-xu+q(x)u \\]\non the $L^{2}(-\\infty,\\infty).$  Let us introduce some notation. For the function $f \\in L^{2}$ we denote by $\\Phi f$ its Fourier transform:\n\\[ \\Phi f(k)= L^{2}-\\lim_{N\\rightarrow \\infty} \\int\\limits_{-N}^{N} \\exp(ikx)f(x)\\,dx. \\] \nFor locally integrable function $g$ we denote by $M^{+}g$ the function\n\\[ M^{+}g(x) = \\sup_{1>h>0} \\frac{1}{2h} \\int\\limits_{0}^{h} |g(x+t)+g(x-t)|\\,dt. \\]\nWe denote by ${\\cal M}^{+}f$ the set where $M^{+}f$ is finite. By the general results on the maximal functions (see, e.g. \\cite{Rud}), it is easy to conclude that the complement of the set ${\\cal M}^{+}$ has Lebesgue measure zero. \nWe will prove the following theorem: \\\\\n\\bf Theorem 1.1 \\it Suppose that the potential $q(x)$ satisfies $|q(x)| \\leq C(1+x)^{-\\frac{1}{3}-\\epsilon},$ $\\epsilon >0.$\nThen the absolutely continuous spectrum of multiplicity one fills the whole real axis. Imbedded singular spectrum may occur, but only on the complement of the set \n\\[ S= {\\cal M}^{+}\\left(\\Phi \\left(\\exp \\left(ix^{3}-i\\int_{0}^{x^{3}}q(ct^{\\frac{2}{3}})ct^{-\\frac{2}{3}}\\,dt\\right)q(c x^{2})x^{\\frac{1}{6}}\\right)\\right), \\]\nwhere $c= (\\frac{3}{2})^{\\frac{2}{3}}.$\n\n\\rm We will prove Theorem 1.1 by studying the asymptotics of the  solutions of the equation \n\\begin{equation}\n -u'' -xu+q(x)u=\\lambda u .\n\\end{equation}\nWe will then apply the subordinacy theory developed by Gilbert and Pearson \\cite{GiPe} to derive spectral consequences. For future reference, we summarize here one of the results of the subordinacy theory for Schr\\\"odinger operators defined on the whole axis  due to Gilbert \\cite{Gi} that we will use. \n\nLet us call the solution $u_{1}$ of the equation (1) subordinate on the right if for any different solution $u_{2}$ the limit \n\\[ \\lim_{N \\rightarrow +\\infty} \\frac{\\|u_{1}\\|_{L^{2}(0,N)}}{\\|u_{2}\\|_{L^{2}(0,N)}} \\]\nis equal to zero.  Subordinacy on the left is defined similarly. The set of the energies at which there exists a  solution subordinate both on the right and on the left constitutes the essential support of the absolutely continuous part of the spectral measure  of Schr\\\"odinger operator. Moreover, if for every $\\lambda$ from this set there exists a solution subordinate either on the right or on the left, the absolutely continuous spectrum has multiplicity one. \n\nSince in our case the total potential grows to plus infinity  on the negative  semi-axis, it is clear that for every $\\lambda$ there is a solution subordinate on the left and hence we need only to study the behavior of solutions on the positive semi-axis.\n\nIn studying Schr\\\"odinger operators with  sufficiently smooth potentials going to $-\\infty$ as $x \\rightarrow \\infty$ one of the useful tools is Liouville transform (see, e.g.\n\\cite{Har}). Suppose that  the equation is given by \n\\[ -y''+(v(x)+q(x))y= \\lambda y, \\]\nwhere  $v$ is two times differentiable and goes to $-\\infty$ as $x \\rightarrow \\infty$ sufficiently fast (specifically, $\\int_{1}^{\\xi} dx\/|v(x)|^{\\frac{1}{2}}$ diverges as $\\xi \\rightarrow \\infty$) and $q$ is a perturbation. The Lioville transformation is given\nby\n\\[  \\xi(x) = \\int\\limits_{0}^{x} \\sqrt{|v(s)|} \\,ds, \\,\\,\\,\\,\\,\n\\rho(\\xi) = |v(x(\\xi))|^{\\frac{1}{4}}y(x(\\xi)). \\]\nThe function $\\rho$ satisfies the equation\n\\begin{equation}\n-\\rho''+ Q(\\xi) \\rho = \\rho,\n\\end{equation}\nwhere\n\\[ Q(\\xi)= \\frac{5 |v'(x(\\xi))|^{2}}{16 |v(x(\\xi))|^{3}} - \\frac{v''(x(\\xi))}{4|v(x(\\xi))|^{2}}+\\frac{q(x(\\xi))-\\lambda}{|v(x(\\xi))|}. \\]\nIn some cases this new equation does not contain infinite potential and \nis simpler to deal with.\n\nIn our case $v(x)=-x$ and we  bring the equation (1) to a convenient form using the following explicit Liouville transformation (see also \\cite{NaPu}):\n$\\xi = \\frac{2}{3}x^{\\frac{3}{2}},$ $\\omega (\\xi) = x(\\xi)^{\\frac{1}{4}}u(x(\\xi)).$ Then the function $\\omega$ \nsolves the  Schr\\\"odinger equation\n\\begin{equation}\n -\\omega'' + \\left( \\frac{5}{36\\xi^{2}} + \\frac{-\\lambda + q(c\\xi^{\\frac{2}{3}})}{c\\xi^{\\frac{2}{3}}}\\right)\\omega = \\omega \n\\end{equation}\n(we remind that $c=(\\frac{3}{2})^{\\frac{2}{3}}$).\nWe introduce the short-hand notation\n\\[ b(\\xi)= \\frac{5}{36\\xi^{2}}-\\frac{cq(\\xi^{\\frac{2}{3}})}{c\\xi^{\\frac{2}{3}}} \\]\nand $V(\\xi, \\lambda)$ for the total potential in the equation (3).\nLet us further apply Pr\\\"ufer transformation to the equation for $\\omega,$ setting for each $\\lambda$\n\\[ \\omega(\\xi, \\lambda) = R(\\xi, \\lambda) \\sin (\\theta(\\xi, \\lambda)) \\]\n\\[ \\omega'(\\xi, \\lambda) = R(\\xi, \\lambda) \\cos (\\theta(\\xi, \\lambda)). \\]\nThe equations for $R$ and $\\theta$ are as follows: \n\\begin{equation}\n (\\log R(\\xi, \\lambda))' = \\frac{1}{2}V(\\xi, \\lambda)\\sin (2\\theta(\\xi, \\lambda)) \n\\end{equation}\n\\begin{equation}\n\\theta'(\\xi, \\lambda) = 1- \\frac{1}{2}V(\\xi, \\lambda)(1-\\cos (2\\theta(\\xi, \\lambda))). \n\\end{equation}\nOur goal now is to study the asymptotics of  solutions of (3).\nIn particular, we would like to establish convergence  of the integral\n\\begin{equation}\n \\int\\limits_{0}^{N} \\left( \\frac{5}{36\\xi^{2}} + \\frac{-\\lambda + q(c\\xi^{\\frac{2}{3}})}{c\\xi^{\\frac{2}{3}}}\\right)\\sin (2\\theta(\\xi, \\lambda))\\, d \\xi \n\\end{equation}\nas $N \\rightarrow \\infty$ for a.e.$\\lambda.$ This goal is motivated by the following  \\\\\n\\bf Lemma 1.2. \\it Suppose that for some value of $\\lambda$ the integral (6) converges. Then for this value of $\\lambda,$ there is no subordinate solution of the original equation (1). Moreover, if the integral (6) converges for a.e.~$\\lambda,$ the absolutely continuous part of the spectral measure of the operator $H_{q}$ fills the whole real axis.  \\\\\n\\bf Proof. \\rm If for a given value of $\\lambda$ the integral (6) converges,\nit follows from  (4) and (5) that all solutions of the equation (3) are bounded  and moreover there are two linearly independent solutions with the following asymptotics as $\\xi \\rightarrow +\\infty:$\n\\[ \\omega_{1}(\\xi, \\lambda) = \\sin (\\xi + r_{1}(\\xi, \\lambda))(1+o(1)), \\]\n\\[ \\omega_{2}(\\xi, \\lambda)= \\cos(\\xi + r_{2}(\\xi, \\lambda))(1+o(1)), \\]\nwhere $r_{i}'(\\xi, \\lambda) \\leq C(1+ \\xi)^{-\\frac{2}{3}}.$\nGoing back to the original equation (1), we infer that for a.e. $\\lambda$\nthis equation has only solutions $u_{\\alpha}(x, \\lambda)$ (where $\\alpha$ is some paprametrization of solutions, say by boundary condition at zero) with the following asymptotical behavior as $x \\rightarrow \\infty:$\n\\[ u_{\\alpha}( x, \\lambda) = x^{-\\frac{1}{4}}\\sin (\\frac{2}{3}x^{\\frac{3}{2}}+ f_{\\alpha}(x, \\lambda))(1+o(1)), \\]\nwhere $|f_{\\alpha}'(x, \\lambda)|\\leq C(1+x)^{-\\frac{1}{2}}$ uniformly over $\\alpha.$ \nFor any $u_{\\alpha}$ we then find\n\\[ \\int\\limits_{0}^{N} |u_{\\alpha}(x)|^{2}\\,dx = \\int\\limits_{0}^{N}\nx^{-\\frac{1}{2}}(\\sin(\\frac{2}{3}x^{\\frac{3}{2}}+ f_{\\alpha}(x))^{2}\\, dx \\left( 1+o(1)\\right) = \\]\n\\[ = \\left( N^{\\frac{1}{2}} + \\frac{1}{2}\\int\\limits_{0}^{N}x^{-\\frac{1}{2}}\n\\cos (\\frac{4}{3}x^{\\frac{3}{2}}+2 f_{\\alpha}(x, \\lambda))\\, dx \\right)\\left(1+o(1)\\right)= N^{\\frac{1}{2}}\\left( 1+o(1)\\right). \\]\nThe last equality follows from integrating by parts the integral in the previous expression. Hence, we obtain that for a.e.$\\lambda$ there are no subordinate solutions: as $N \\rightarrow \\infty,$ the $L^{2}$ norm grows at the same rate for all solutions. The last claim of the lemma follows  by direct  application of subordinacy theory. $\\Box$ \\\\\n\\it Remark. \\rm 1. Note that in particular we obtained that for a.e. $\\lambda$ all solutions of the equation (1) are bounded (and even power-decaying). However the results deriving absolute continuity  of the spectrum from the boundedness of solutions (see, e.g., \\cite{Sto}, \\cite{Si2}) are not applicable here because the potential goes to $-\\infty$ on $R^{+}.$ Instead, one has to apply directly Gilbert-Pearson theory. \\\\\n2. Note that while for a.e.$\\lambda$ there is no subordinate solution of equation (1) and hence the absolutely continuous spectrum fills the whole axis, there may be a complementary set of measure zero where the singular part of the spectral measure might be supported. As examples show \\cite{NaPu}, the imbedded spectrum may even be dense. The a.e. convergence of Fourier type integral operators  allows to prove a.e. convergence of the integral (6) while permitting for exceptional measure zero set where convergence may fail. \\\\\n\nWe begin analyzing the integral (6) with the following simple \\\\\n\\bf Lemma 1.3. \\it Suppose that the function $h(\\xi)$ satisfies \n\\[ h(\\xi)= \\xi + g(\\xi), \\]\nwhere $|g'(\\xi)|\\leq C\\xi^{-\\frac{2}{3}}.$ \nThen the integrals\n\\[ \\int\\limits_{1}^{N} \\xi^{-\\frac{2}{3}} \\exp (\\pm i h(\\xi))\\, d\\xi \\]\nconverge and , moreover, \n\\[ \\int\\limits_{N}^{\\infty}\\xi^{-\\frac{2}{3}} \\exp (\\pm i h(\\xi))\\, d\\xi= O(N^{-\\frac{1}{3}}). \\]\n\\bf Proof. \\rm \n\\[ \\int\\limits_{1}^{N} \\xi^{-\\frac{2}{3}} \\exp (\\pm i h(\\xi))\\, d\\xi = N^{-\\frac{2}{3}}\\int\\limits_{1}^{N} \\exp (\\pm i h(\\xi))\\,d\\xi +\\]\n\\[ + \\frac{2}{3}\\int\\limits_{1}^{N}\\xi^{-\\frac{5}{3}}\\int\\limits_{1}^{\\xi}\n\\exp (\\pm i h(\\eta))\\, d\\eta d\\xi. \\]\n Pick $\\xi_{0}$ so that \nfor $\\xi>\\xi_{0}$ $|g'(\\xi)|<\\frac{1}{2}$ ( we can do it by the assumption of Lemma). Then \n\\[ \\int\\limits_{\\xi_{0}}^{N} \\exp(\\pm i h(\\xi))\\,d\\xi=\n\\int\\limits_{h(\\xi_{0})}^{h(\\xi)}\\exp(\\pm i h)\\frac{1}{1+g'(\\xi(h))}\\, dh, \\]\nwhere $|g'(\\xi(h)))| \\leq C_{1}h^{-\\frac{2}{3}}.$\nExpanding the fraction in the last integral, it we find that \n\\[ \\int\\limits_{\\xi_{0}}^{N} \\exp(\\pm  i h(\\xi))\\,d\\xi=O(N^{\\frac{1}{3}}). \\]\nHence we see that the integral in question converges.\nIt is also straightforward to establish the last estimate of the lemma. \n$\\Box$\n\nFrom Lemma 1.3 and (5) it follows that the integrals \n\\[ \\int\\limits_{0}^{N} \\xi^{-\\frac{2}{3}} \\exp(\\pm 2i \\theta(\\xi, \\lambda)) \\,d\\xi \\]\nconverge for every value of $\\lambda.$\nTherefore, we see  that to establish convergence  of the integral (6)\nit suffices to study  the integral\n\\begin{equation}\n \\int\\limits_{1}^{N} \\left( \\frac{5}{36\\xi^{2}}-q(c\\xi^{\\frac{2}{3}})(c\\xi)^{-\\frac{2}{3}}\\right)\\sin(2\\theta(\\xi, \\lambda)\\, d\\xi). \n\\end{equation}\n(Of course, for the fast decaying term  the convergence issue is trivial, but  we incorporate it into the expression for future computational convenience.) \n\n We will need two  lemmata on the a.e. convergence of Fourier-type integrals. The first lemma we need is exactly Lemma 1.3 from \\cite{Kis1}.\nWe refer to that paper for a proof. \\\\\n\\bf Lemma 1.4. \\it Consider the function $f \\in L^{2}(R).$ Then for every $\\lambda_{0} \\in {\\cal M}^{+}(\\Phi(f))$ we have\n\\[ \\int\\limits_{-N}^{N} f(x) \\exp(i\\lambda_{0}x)\\,dx = O(\\log N). \\]\n\n\\rm The second lemma is a variation of \nLemma 1.4 from \\cite{Kis1}. We provide here the statement and a sketch of the proof for the sake of completness. \\\\\n\\bf Lemma 1.5. \\it Suppose that a function $f(x)$ satisfies $|f(x)| \\leq \nC(1+|x|)^{-\\alpha},$ where $\\alpha > \\frac{1}{2}.$ Then for a.e. $\\lambda$ the integral \n\\[ \\int\\limits_{0}^{N} \\exp(i\\lambda x)f(x)\\, dx \\]\nconverges as $x \\rightarrow \\infty$ and moreover for every $\\delta >0$ \n\\begin{equation}\n \\int\\limits_{N}^{\\infty} \\exp(i\\lambda x)f(x)\\, dx = o(N^{-\\alpha+\\frac{1}{2}+\\delta}) \n\\end{equation}\nfor a.e. $\\lambda.$ Explicitely, we can say that (8) holds for every $\\lambda \\in {\\cal M}^{+}(\\Phi (f(x)x^{\\alpha-\\frac{1}{2}-\\epsilon}))$ where $\\epsilon<\\delta.$ \\\\\n\\bf Proof. \\rm   Consider an apriori estimate \n\\[ \\int\\limits_{N}^{\\infty} \\exp (i\\lambda x) x^{\\alpha-\\frac{1}{2}-\\epsilon}f(x)x^{-\\alpha+\\frac{1}{2}+\\epsilon}\\,dx= \nN^{-\\alpha+\\frac{1}{2}+\\epsilon}\\int\\limits_{0}^{N}\\exp(i\\lambda x)\n(f(x)x^{\\alpha-\\frac{1}{2}-\\epsilon})\\,dx+ \\]\n\\[+(\\frac{1}{2}-\\alpha+\\epsilon)\\int\\limits_{N}^{\\infty}x^{-\\alpha-\\frac{1}{2}+\\epsilon}\\left(\\int\\limits_{0}^{x}\\exp(i \\lambda y)(f(y)y^{\\alpha-\\frac{1}{2}-\\epsilon}\\,dy \\right)\\,dx. \\]\nBy Lemma 1.4, for every $\\lambda \\in {\\cal M}^{+}(\\Phi (f(x)x^{\\alpha-\\frac{1}{2}-\\epsilon}))$ we have an estimate\n\\[ \\int\\limits_{0}^{N}\\exp(i \\lambda y)f(y)y^{\\alpha-\\frac{1}{2}-\\epsilon}\\,dy =O(\\log N) \\]\n Therefore for such $\\lambda$\nwe obtain \n\\[ \\int\\limits_{N}^{\\infty}\\exp(i \\lambda x)f(x)\\,dx = O(N^{-\\alpha+\\frac{1}{2}+\\epsilon}\\log N) = o(N^{-\\alpha +\\frac{1}{2} +\\delta}) \\]\nsince $\\delta>\\epsilon.$ The lemma is proven. $\\Box$ \\\\\n\\it Remark. \\rm We note that by Zygmund's theorem \\cite{Zyg} the Fourier integral converges a.e. for every $f \\in L^{p},$ $1 \\leq p<2.$ One can avoid using Lemma 1.4 by applying directly this theorem in the proof of Lemma 1.5. However this approach is less direct and also does not provide any explicit information about the convergence set. \\\\\n\n\\noindent \\bf Corollary 1.6. \\it Suppose that the function $f(\\xi)$ satisfies $|f(\\xi)| \\leq C(1+|\\xi|)^{-\\alpha-\\epsilon},$ where $\\alpha \\geq \\frac{5}{6}$ and $\\epsilon$ is an arbitrarily small positive number. Then for a.e.$\\lambda$ the integral \n\\[ \\int\\limits_{0}^{N} f(\\xi) \\exp(i\\lambda \\xi^{\\frac{1}{3}})\\, d\\xi \\]\nconverges as $N \\rightarrow \\infty.$ Moreover, for every $\\lambda \\in {\\cal M}^{+}(\\Phi (f(\\xi^{3})\\xi^{3\\alpha-\\frac{1}{2}}))$ we have\n\\[ \\left| \\int\\limits_{N}^{\\infty} f(\\xi)\\exp(i\\lambda \\xi^{\\frac{1}{3}})\\,d\\xi \\right| \\leq CN^{-\\alpha +\\frac{5}{6}}. \\]\n\\bf Proof. \\rm \nThe change of variables $y=\\xi^{\\frac{1}{3}}$ transforms the integral to the form\n\\[ \\int\\limits_{0}^{N^{\\frac{1}{3}}} f(y^{3})y^{2}\\exp (6i\\lambda y)\\, dy. \\]\nNote that \n\\[ |f(y^{3})y^{2}| \\leq C(1+|y|)^{-3\\alpha-3\\epsilon+2} \\]\nby assumption. Applying Lemma 1.5 (and remembering to take into account  the lower limit) we obtain the statements of the Corollary. $\\Box$\n\nWe prove the final lemma we need for the proof of the theorem. \nLet us introduce some short-hand notation:\n\\[ \\sigma (x, \\lambda)= 2\\theta(x, \\lambda)- 2x + \\int\\limits_{0}^{x}\n\\left( q(\\xi^{\\frac{2}{3}})\\xi^{-\\frac{2}{3}}-\\lambda\\xi^{-\\frac{2}{3}}\\right)\\,d\\xi+\\lambda \\int\\limits_{0}^{x}\\xi^{-\\frac{2}{3}}\\cos 2\\theta(\\xi, \\lambda)\\,d\\xi; \\]\n\\[ \\gamma(x, \\lambda) = 2\\theta(x, \\lambda) - \\sigma(x, \\lambda). \\] \nFrom the equation (5) we find\n\\begin{equation}\n\\gamma'(x, \\lambda)= b(x, \\lambda) \\cos (\\gamma(x, \\lambda)+\\sigma(x, \\lambda)). \n\\end{equation}\n\\bf Lemma 1.7. \\it Suppose that the potential $q(x)$ satisfies $|q(x)| \\leq C(1+|x|)^{-\\frac{1}{3}-\\epsilon}.$ Then the integrals \n\\[ \\int\\limits_{0}^{N} b(\\xi) \\exp (\\pm \\sigma (\\xi, \\lambda))\\,d\\xi \\]\nconverge as $N \\rightarrow \\infty$ for a.e. $\\lambda$ and moreover\nfor a.e. $\\lambda$ \n\\begin{equation}\n \\int\\limits_{N}^{\\infty} b(\\xi) \\exp (\\pm \\sigma (\\xi, \\lambda))\\,d\\xi = O(N^{-\\frac{1}{18}}). \n\\end{equation}\n\\bf Proof. \\rm We will consider the case of plus sign; the other case is analogous. By Lemma 1.3 and the definitions of $b(\\xi)$ and $\\sigma(\\xi, \\lambda),$ it is clear that it suffices to investigate the convergence of the following integral\n\\begin{equation} \\int\\limits_{0}^{N} q(c\\xi^{\\frac{2}{3}})\\xi^{-\\frac{2}{3}} \\exp\\left(\n2i\\xi -i\\int_{0}^{\\xi}q(\\eta^{\\frac{2}{3}})\\eta^{-\\frac{2}{3}}\\,d\\eta+ 6\\lambda i\\xi^{\\frac{1}{3}} +ic(\\lambda)+i\\xi^{-\\frac{1}{3}}h(\\lambda, \\xi)\\right), \n\\end{equation}\nwhere the function $h(\\xi, \\lambda)$ is bounded for every  $\\lambda.$ \nWe used Lemma 1.3 to   replace \n\\[ \\lambda \\int\\limits_{0}^{\\xi} \\eta^{-\\frac{2}{3}}\\cos (2\\theta (\\eta, \\lambda))\\, d\\eta \\]\nby \\[ c(\\lambda) +  \\xi^{-\\frac{1}{3}}h(\\lambda, \\xi) \\]\nin the exponent in (11).\nLet us denote \n\\begin{equation}\n \\tilde{q}(\\xi)=q(c\\xi^{\\frac{2}{3}})\\xi^{-\\frac{2}{3}}\\exp(2i\\xi-i\\int_{0}^{\\xi}q(c\\eta^{\\frac{2}{3}})c\\eta^{-\\frac{2}{3}}\\,d\\eta).\n\\end{equation}\nWe can rewrite the integral (11) as \n\\[ \\int\\limits_{0}^{N} \\tilde{q}(\\xi) \\exp (6i\\lambda \\xi^{\\frac{1}{3}} -\ni\\xi^{-\\frac{1}{3}}h(\\lambda, \\xi))\\, d\\xi \\]\n(henceforth we omit the constant factor $\\exp (ic(\\lambda))$).\nNote that as $\\xi \\rightarrow \\infty,$ we have  \n\\[ \\exp(i \\xi^{-\\frac{1}{3}} h(\\lambda, \\xi))= 1+ \\xi^{-\\frac{1}{3}}\\tilde{h}(\\lambda,\\xi), \\]\nwhere $\\tilde{h}$ is bounded  in $\\xi$ for every $\\lambda.$\nHence we can write the integral (11) as a sum of two integrals\n\\[ \\int\\limits_{0}^{N}\\tilde{q}(\\xi) \\exp (6i\\lambda \\xi^{\\frac{1}{3}})\\,d\\xi + \\int\\limits_{0}^{N}\\tilde{q}(\\xi)\\xi^{-\\frac{1}{3}}\\tilde{h}(\\xi, \\lambda) \\exp (6i\\lambda, \\xi)\\,d\\xi. \\]\nNote that by assumption on the decay of $q$ and (12) we have \n\\begin{equation}\n \\tilde{q}(\\xi) \\leq C\\xi^{-\\frac{8}{9}-\\frac{2}{3}\\epsilon}. \n\\end{equation}\nTherefore, the  second integral converges for every $\\lambda$ because the integrand is absolutely integrable. It is also easy to estimate the rate of convergence \nfor the second integral:\n\\[ \\left|\\int\\limits_{N}^{\\infty}\\tilde{q}(\\xi)\\xi^{-\\frac{1}{3}}\\tilde{h}(\\xi, \\lambda) \\exp (6i\\lambda \\xi)\\,d \\xi \\right| \\leq C(\\lambda)N^{-\\frac{2}{9}} \\]\nfor every $\\lambda$ because of the assumption on decay of $q.$\nTo study the behavior of the first integral, we note that because of the estimate (13) we can apply Corollary 1.6.\nWe obtain that for a.e. $\\lambda$ the first integral converges, and, moreover, for a.e. $\\lambda$ (explicitely, for every $\\lambda \\in {\\cal M}^{+}(\\Phi (\\tilde{q}(\\xi^{3})\\xi^{\\frac{13}{6}})$) the estimate on the rate of convergence is given by\n\\[ \\int\\limits_{N}^{\\infty}\\tilde{q}(\\xi) \\exp (6i\\lambda \\xi^{\\frac{1}{3}})\\,d\\xi \\leq C(\\lambda)N^{-\\frac{1}{18}}. \\]\nCombining the two estimates, we see that the lemma is proven.\n $\\Box$\n\nWe are now in a position to complete the proof of the theorem. The main idea is to take an advantage of certain symmetry of the equations (4), (5) and estimate (6). \\\\\n\\bf Proof of Theorem 1.1. \\rm  As we already remarked, to complete the proof we only have to study the a.e. convergence of the integral (7). Using (9) we rewrite this integral as\n\\[ \\int\\limits_{1}^{\\infty} b(\\xi) \\sin(\\gamma (\\xi, \\lambda)+\\sigma(\\xi, \\lambda))\\,d\\xi. \\]\nWe will integrate by parts two times, differentiating the terms which contain $\\gamma(\\xi, \\lambda).$ We omit the arguments of $\\gamma$ and $\\sigma$ in the following computations.\n\\[ \\int\\limits_{1}^{N} b(\\xi)\\sin (\\gamma +\\sigma)\\,d \\xi  = \\int\\limits_{1}^{N} b(\\xi)(\\sin (\\gamma )\\cos(\\sigma )+ \\sin (\\sigma)\\cos (\\gamma))\\, d\\xi = \\]\n\\[ = -\\left( \\left. \\cos (\\gamma) \\int\\limits_{\\xi}^{\\infty} b(\\eta)\n\\sin(\\sigma)\\, d\\eta + \\sin (\\gamma) \\int\\limits_{\\xi}^{\\infty} b(\\eta) \\cos (\\sigma)\\,d \\eta \\right) \\right|^{N}_{1}+ \\]\n\\[ + \\int\\limits_{1}^{N} b(\\xi)\\cos(\\gamma+\\sigma)\\left(\\cos (\\gamma )\\int\\limits_{\\xi}^{\\infty} b(\\eta)\\cos(\\sigma)\\,d \\eta-\\sin (\\gamma)\\int\\limits_{\\xi}^{\\infty} b(\\eta)\\sin(\\sigma)\\,d \\eta \\right)\\,d\\xi. \\]\nWe remark that by Lemma 1.7 the conditional integrals are well-defined for a.e. $\\lambda.$ Given that, we see that for a.e. $\\lambda$ the off-diagonal terms stay finite. We rearrange the integral term and write it as follows:\n\\[ \\int\\limits_{1}^{N} (\\sin(\\gamma))^{2} \\left( b(\\xi) \\sin (\\sigma)\\int\\limits_{\\xi}^{\\infty} b(\\eta)\\sin(\\sigma)\\,d \\eta \\right)  d \\xi - \\]\n\\[- \\int\\limits_{1}^{N} (\\sin(\\gamma)\\cos (\\gamma)) \\left( b(\\xi) \\cos (\\sigma)\\int\\limits_{\\xi}^{\\infty} b(\\eta)\\sin(\\sigma)\\,d \\eta +  b(\\xi) \\sin (\\sigma)\\int\\limits_{\\xi}^{\\infty} b(\\eta)\\cos(\\sigma)\\,d \\eta \\right) d \\xi  + \\]\n\\[ + \\int\\limits_{1}^{N} \\cos(\\gamma)^{2} \\left( b(\\xi) \\cos (\\sigma)\\int\\limits_{\\xi}^{\\infty} b(\\eta)\\cos(\\sigma)\\,d \\eta  \\right) d \\xi . \\]\nIntegrating by parts (terms in brackets being integrated), we again  obtain off-diagonal terms which are a.e.$\\lambda$ finite and the following integral term:\n\\[ \\int\\limits_{1}^{N} \\sin (2\\gamma) b(\\xi) \\cos (\\gamma+\\sigma) \\left( \\left( \\int\\limits_{\\xi}^{\\infty}b(\\eta)\\sin (\\sigma)\\,d \\eta \\right)^{2} - \\left( \\int\\limits_{\\xi}^{\\infty}b(\\eta)\\cos (\\sigma)\\,d \\eta \\right)^{2}\\right)\\, d \\xi - \\]\n\\[ - \\int\\limits_{1}^{N} \\cos (2\\gamma) b(\\xi) \\cos (\\gamma+\\sigma)  \\left( \\int\\limits_{\\xi}^{\\infty}b(\\eta)\\sin (\\sigma)\\,d \\eta \\right)\\left(\\int\\limits_{\\xi}^{\\infty}b(\\eta)\\cos (\\sigma)\\,d \\eta \\right) \\, d \\xi. \\]\nHence, the integral (7) and therefore (6) is convergent if \n\\[ b(\\xi) \\left( \\int\\limits_{\\xi}^{\\infty} b(\\eta) \\exp (\\pm \\sigma(\\eta, \\lambda))\\, d \\eta \\right)^{2} \\in L^{1}(0, \\infty). \\]\nBecause of the assumption on the decay of $q$ definition of $b(\\xi)$ and Lemma 1.7, we find that for a.e. $\\lambda \\in R$ the expression above is bounded above by \n\\[ C(\\lambda) \\xi^{-\\frac{2}{3}}\\xi^{-\\frac{2}{9}-\\frac{\\epsilon}{2}}\\xi^{-\\frac{1}{9}}\\leq C(\\lambda) \\xi^{-1-\\frac{\\epsilon}{2}} \\]\nand hence is absolutely integrable for a.e. $\\lambda.$  This proves the convergence of the integral (6) for a.e. $\\lambda.$ Since from the proof of Lemma 1.7  follows that the estimate (10) holds for every $\\lambda \\in {\\cal M}^{+}(\\Phi (\\tilde{q}(\\xi^{3}))\\xi^{-\\frac{13}{6}})),$ we also obtain an explicit set to which the singular part of the spectral measure gives zero weight, as stated in the  theorem. $\\Box$\n\n\\section{The case of sufficiently smooth potential}\n\nIn this section, we take up the study of an alternative kind of conditions implying the absolute continuity of the spectrum of Stark operators. \nWe begin with the remark that if the perturbation $q(x)$ is twice differetiable, $|q'(x)| \\leq C(1+|x|)$ and $|q''(x)| \\leq C(1+|x|)^{\\alpha},$ $\\alpha<\\frac{1}{2},$ we can apply the Liouville transformation directly to the \nwhole potential $-x+q(x)$, obtaining the equation (2) with absolutely integrable \npotential $Q.$ Then by Lemma 1.2, for every $\\lambda$ there is no subordinate solution of the original equation and hence the spectrum is purely absolutely continuous. The situation is more subtle if $q$ does not have two derivatives and hence we cannot apply Liouville transformation directly to the whole potential $-x+q(x).$\nOur goal is to show the following improvement of results proven in  \\cite{Wa}, \\cite{Ben}: \\\\\n\\bf Theorem 2.1. \\it Suppose that the perturbation $q(x)$ is bounded and differentiable, and the derivative $q'(x)$ is bounded and H\\\"older continuous with any exponent $\\alpha >0,$ i.e.\n\\[ \\sup_{x \\in R^{+}} \\left(\\sup_{y} \\frac{|q'(x)-q'(y)|}{|x-y|^{\\alpha}}\\right)<C. \\]\nThen the absolutely continuous part of the spectral measure $\\rho_{\\rm{ac}}^{H_{q}}$ associated with the Stark operator $H_{q}$ fills the whole real axis. \\\\\n\\bf Proof. \\rm \nThe strategy of the proof will be the same as for decaying case. We begin by analyzing equations (4), (5):\n\\[ (\\log R)'(\\xi, \\lambda)= \\frac{1}{2} V(\\xi)\\sin (2\\theta (\\xi, \\lambda)) \\]\n\\[ \\theta'(\\xi, \\lambda) = 1 -\\frac{1}{2}V(\\xi)+\\frac{1}{2}V(\\xi)\\cos(2\\theta(\\xi, \\lambda)). \\]\nWe remind that \n\\[ V(\\xi) =  \\frac{5}{36\\xi^{2}}+\\frac{\\lambda - q(c\\xi^{\\frac{2}{3}})}{c\\xi^{\\frac{2}{3}}}  \\]\nTo treat the phase equation, we introduce quantities similar to $\\gamma$ and $\\sigma$ in the previous proof:\n\\[ \\tilde{\\sigma}(\\xi, \\lambda)= 2 \\xi - \\int\\limits_{0}^{\\xi} V(\\eta)\\,d \\eta \\]\n\\[ \\tilde{\\gamma}(\\xi, \\lambda) = 2\\theta (\\xi, \\lambda) - \\tilde{\\sigma}(\\xi, \\lambda). \\]\n The computation completely repeating that in the proof of Theorem 1.1\nallows us to conclude that $R(\\xi, \\lambda)$ is bounded for a.e. $\\lambda$\nif the expression \n\\begin{equation}\n V(\\xi,\\lambda )\\left[ \\int\\limits_{\\xi}^{\\infty} V(\\eta, \\lambda) \\exp \\left( \\pm \\left( 2i\\eta -\\int\\limits_{0}^{\\eta} V(t, \\lambda)\\, dt \\right)\\right)\\, d\\eta \\right]^{2} \n\\end{equation}\nis well-defined and absolutely integrable for a.e. $\\lambda.$\nLet us now consider this control expression more carefully. \nConsider the  integral \n\\[ w_{\\pm}(\\xi, \\lambda)=\\int\\limits_{\\xi}^{\\infty} V(\\eta, \\lambda) \\exp \\left( \\pm \\left( 2i\\eta -\\int\\limits_{0}^{\\eta} V(t, \\lambda)\\, dt \\right)\\right)\\, d\\eta = \\]\n\\[ = -\\int\\limits_{\\xi}^{\\infty} \\frac{q(c\\eta^{\\frac{2}{3}})}{c\\eta^{-\\frac{2}{3}}} \\exp \\left( \\pm \\left( 2i\\eta -\\int\\limits_{0}^{\\eta} V(t, \\lambda)\\, dt \\right)\\right)\\, d\\eta \\,\\,+ \\] \\[ + \\int\\limits_{\\xi}^{\\infty} \\left(\\frac{5}{36\\eta^{2}} +\\frac{\\lambda}{c\\eta^{\\frac{2}{3}}} \\right) \\exp \\left( \\pm \\left( 2i\\eta -\\int\\limits_{0}^{\\eta} V(t, \\lambda)\\, dt \\right)\\right)\\, d\\eta. \\]\nWe see that the  second integral on the right-hand side converges for all $\\lambda$ and is $O(\\xi^{-\\frac{1}{3}})$ by  Lemma 1.3 \nHence it remains to study the first integral. We rewrite this integral as follows:\n\\begin{equation}\n \\int\\limits_{\\xi}^{\\infty} q(c\\eta^{\\frac{2}{3}}) \\exp \\left( \\pm i \\left(  -\\frac{5}{36\\eta}+ \\int\\limits_{0}^{\\eta}q(ct^{\\frac{2}{3}})ct^{-\\frac{2}{3}}\\,dt \\right)\\right) c\\eta^{-\\frac{2}{3}}\\exp \\left( \\pm \\left(\n2i\\eta + 6i\\lambda \\eta^{\\frac{1}{3}} \\right)\\right)\\, d\\eta. \n\\end{equation}\nLet us introduce a short-hand notation $g(\\eta)$ for the function \n\\[ \\exp \\left( \\pm i \\left(  -\\frac{5}{36\\eta}+ \\int\\limits_{0}^{\\eta}q(ct^{\\frac{2}{3}})ct^{-\\frac{2}{3}}\\,dt \\right)\\right).\\]\nNote that $g$ is continously differentiable and $|g(\\eta)|+|g'(\\eta)|$ is bounded.\nIn (15) we differentiate by parts, differentiating the first two terms and integrating next two. Note that \n\\[ \\int\\limits_{\\xi}^{\\infty} \\eta^{-\\frac{2}{3}} \\exp \\left(\\pm \\left(\n2i\\eta +6i\\lambda \\eta^{\\frac{1}{3}} \\right)\\right) \\, d\\eta = \\frac{1}{2}\\xi^{-\\frac{2}{3}}\\exp \\left( \\pm 2i\\xi \\pm 6i\\lambda \\xi^{\\frac{1}{3}} \\right) +\nO(\\xi^{-\\frac{5}{3}}). \\]\nGiven this observation, we obtain that the integral (15) is equal to\n\\[ O(\\xi^{-\\frac{2}{3}}) + \\left( \\frac{c}{6}\\int\\limits_{\\xi}^{\\infty} \n\\left(2\\eta^{-\\frac{1}{3}}q'(c\\eta^{\\frac{2}{3}}) \\pm i q(c\\eta^{\\frac{2}{3}})\n\\eta^{-\\frac{2}{3}} \\mp i \\frac{5}{36c\\eta^{2}} \\right) \\right. \\times \\]\n\\[ \\times \\left. g(\\eta)\\exp(\\pm 2i \\eta \\pm 6i \\lambda \\eta^{\\frac{1}{3}})\n\\eta^{-\\frac{2}{3}}\\left( 1+ O(\\eta^{-1})\\right) \\right) = \\]\n\\[ = O(\\xi^{-\\frac{1}{3}}) + \\frac{c}{6} \\int\\limits_{\\xi}^{\\infty}q'(c\\eta^{\\frac{2}{3}})\\eta^{-1}g(\\eta)\\exp \\left( \\pm 2i\\eta \\pm 6i\\lambda \\eta^{\\frac{1}{3}} \\right)\\, d \\eta. \\]\nBy assumption, $q'$ is is H\\\"older continuous uniformly over $R$ with \nexponent $\\alpha.$ Hence for every $h>0,$\n\\begin{equation}\n q'(c(\\eta + h)^{\\frac{2}{3}}) - q'(c\\eta^{\\frac{2}{3}}) = C( \\eta, h)\\eta^{-\\frac{\\alpha}{3}}h^{\\alpha}, \n\\end{equation}\nwhere $C(\\eta, h)$ is bounded uniformly for all $\\eta,$ $0<h<1.$ In the rest of the proof we will keep notation $C(\\eta, h)$ for different functions satisfying the same property. \nLet us integrate by parts the last integral ``$\\alpha$\" times. That is, let $h>0$ be fixed, such that $\\exp(ih) -1 \\neq 0.$ Then \n\\[ \\int\\limits_{\\xi}^{\\infty} q'(\\eta^{\\frac{2}{3}})g(\\eta)\\eta^{-1} \\exp (\\pm 6 i\\lambda \\eta^{\\frac{1}{3}})\\exp(\\pm 2i\\eta) \\left( \\exp(\\pm 2ih)-1 \\right)\\, d \\eta= \\]\n\\[ O(\\xi^{-1}) + \\int\\limits_{\\xi}^{\\infty} \\left( q'((\\eta+h)^{\\frac{2}{3}})(\\eta+h)^{-1}g(\\eta+h)\\exp( \\pm 6i\\lambda (\\eta+h)^{\\frac{1}{3}})- \\right. \\] \\[- \\left. q'(\\eta^{\\frac{2}{3}})\\eta^{-1}g(\\eta) \\exp(\\pm 6i\\lambda\\eta^{\\frac{1}{3}}) \\right) \\exp( \\pm 2i\\eta)\\, d \\eta. \\]\nNote that \n\\[ \\exp(\\pm 6i\\lambda(\\eta+h)^{\\frac{2}{3}})-\\exp(\\pm 6i\\lambda\\eta^{\\frac{2}{3}}) = \\eta^{-\\frac{2}{3}}C_{\\lambda}(\\eta, h). \\]\nHence we can rewrite the last integral as\n\\[ O(\\xi^{-\\frac{2}{3}}) + \\int\\limits_{\\xi}^{\\infty} \\left((q'((\\eta+h)^{\\frac{2}{3}})(\\eta+h)^{-1}g(\\eta+h)- q'(\\eta^{\\frac{2}{3}})\\eta^{-1}g(\\eta)\\right)\\exp(\\pm 2i\\eta \\pm 6i\\lambda \\eta^{\\frac{1}{3}})\\, d \\eta.  \\]\nSince we have\n\\begin{equation}\n|(\\eta+h)^{-1}g(\\eta+h)-\\eta^{-1}g(\\eta)| \\leq C\\eta^{-\\frac{2}{3}},\n\\end{equation}\na simple computation taking into account (16) and (17) shows that the last expression is equal to \n\\[  O(\\xi ^{-\\frac{2}{3}})+\\int\\limits_{\\xi}^{\\infty} C(\\eta, h)\\eta^{-1-\\frac{\\alpha}{3}}\\exp(\\pm 2i\\eta \\pm 6i\\lambda \\eta^{\\frac{1}{3}})\\, d \\eta.  \\]\n By Corollary 1.6, we obtain that the last expression converges for a.e. $\\lambda$ and is bounded by $C\\xi^{-\\frac{1}{3}(\\frac{1}{2}+\\alpha)+\\epsilon}$ for every $\\epsilon >0.$\nTherefore we have completed an estimation of the integral $w_{\\pm}(\\xi, \\lambda)$ and obtained that for a.e. $\\lambda$\n\\[ w_{\\pm}(\\xi, \\lambda) \\leq C(\\lambda) (\\xi^{-\\frac{1}{3}+\\epsilon}+\n\\xi^{\\frac{1}{3}(\\frac{1}{2}+\\alpha)+\\epsilon}). \\]\nTaking into account the fact that $|V(\\xi)| \\leq C\\xi^{-\\frac{2}{3}},$ this implies that for a.e. $\\lambda$ the control expression (14)\nis bounded by $C(\\lambda)\\xi^{-1-\\frac{2\\alpha}{3}+\\epsilon}$\nand hence is absolutely integrable. This allows us to conclude the boundedness of solutions of the equation (3) for a.e. $\\lambda.$ Applying Lemma 1.2, we find that for a.e. $\\lambda$ there is no subordinate solution of the original equation (1). This completes the proof of Theorem 2.1. $\\Box$ \\\\\n\n\\section*{Acknowledgment} I am very grateful to Professor  for stimulating discussions and to Professor Stolz for turning author's attention to the problem.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}