Ergodicity and equidistribution in Hilbert geometry

Pierre-Louis Blayac; Feng Zhu

doi:10.3934/jmd.2023026

Article Contents

2023, Volume 19: 879-945. Doi: 10.3934/jmd.2023026

This volume Previous Article On length spectrum rigidity of dispersing billiard systems Next Article Limit distributions of expanding translates of shrinking submanifolds and non-improvability of Dirichlet's approximation theorem

Ergodicity and equidistribution in Hilbert geometry

Pierre-Louis Blayac^1, and
Feng Zhu^2,

1.
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48103, USA
2.
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr, Madison, WI 53706, USA

Received: November 05, 2021

Revised: August 11, 2023

Abstract / Introduction Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

We show that dynamical and counting results characteristic of negatively curved Riemannian geometry, or more generally CAT($ -1 $) or rank-one CAT(0) spaces, also hold for rank-one properly convex projective manifolds or orbifolds, equipped with their Hilbert metrics, admitting finite Sullivan measures built from appropriate conformal densities. In particular, this includes geometrically finite convex projective manifolds or orbifolds whose universal covers are strictly convex with $ C^1 $ boundary.

More specifically, with respect to the Sullivan measure, the Hilbert geodesic flow is strongly mixing, and orbits and primitive closed geodesics equidistribute, allowing us to asymptotically enumerate these objects.

Keywords:

Mathematics Subject Classification: Primary: 22D40, 20F69; Secondary: 37D40, 20F67, 22E40.

Citation:

Full Text(HTML)

Figure 1. The Hilbert distance d_Ω(x, y) is determined by the points a, x, y, b

Download: Full-size image PowerPoint slide

Figure 2. Points $ x,y\in\Omega $ in a horosphere centered at $ \xi\in \partial\Omega $. This picture is slight modification of [9, Fig. 2]

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	M. Babillot, On the mixing property for hyperbolic systems, Israel J. Math., 129 (2002), 61-76. doi: 10.1007/BF02773153.
[2]	M. Babillot, Points entiers et groupes discrets: de l'analyse aux systèmes dynamiques, in Rigidité, Groupe Fondamental et Dynamique, 1–119, with an appendix by Emmanuel Breuillard, Panor. Synthèses, 13, Soc. Math. France, Paris, 2002.
[3]	S. A. Ballas, J. Danciger and G.-S. Lee, Convex projective structures on nonhyperbolic three-manifolds, Geom. Topol., 22 (2018), 1593-1646. doi: 10.2140/gt.2018.22.1593.
[4]	Y. Benoist, Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal., 7 (1997), 1-47. doi: 10.1007/PL00001613.
[5]	Y. Benoist, Automorphismes des cônes convexes, Invent. Math., 141 (2000), 149-193. doi: 10.1007/PL00005789.
[6]	Y. Benoist, Convexes divisibles Ⅰ, in Algebraic Groups and Arithmetic, 3339–374, Tata Inst. Fund. Res., 2004.
[7]	Y. Benoist, Convexes divisibles Ⅲ, Ann. Sci. École Norm. Sup. (4), 38 (2005), 793–832. doi: 10.1016/j.ansens.2005.07.004.
[8]	Y. Benoist and H. Oh, Effective equidistribution of s-integral points on symmetric varieties, Annales de l'Institut Fourier, 62 (2012), 1889-1942. doi: 10.5802/aif.2738.
[9]	P.-L. Blayac, Topological mixing of the geodesic flow on convex projective manifolds, preprint, arXiv: 2009.05035, to appear in Ann. de l'Institut Fourier, 2020.
[10]	P.-L. Blayac, The boundary of rank-one divisible convex sets, preprint, arXiv: 2106.07581, 2021.
[11]	P.-L. Blayac, Dynamical Aspects of Convex Projective Geometry, PhD thesis, Université Paris-Saclay, 2021.
[12]	P.-L. Blayac, Patterson–Sullivan densities in convex projective geometry, preprint, arXiv: 2106.08089, 2021.
[13]	A. Borel, Linear algebraic groups, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), 3–19, Amer. Math. Soc., Providence, RI, 1966.
[14]	B. H. Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J., 77 (1995), 229-274. doi: 10.1215/S0012-7094-95-07709-6.
[15]	B. H. Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput., 22 (2012), 1250016, 66 pp. doi: 10.1142/S0218196712500166.
[16]	R. Bowen, The equidistribution of closed geodesics, Amer. J. Math., 94 (1972), 413-423. doi: 10.2307/2374628.
[17]	H. Bray, Ergodicity of Bowen-Margulis measure for the Benoist 3-manifolds, J. Mod. Dyn., 16 (2020), 305-329. doi: 10.3934/jmd.2020011.
[18]	H. Bray, Geodesic flow of nonstrictly convex Hilbert geometries, Ann. Inst. Fourier (Grenoble), 70 (2020), 1563-1593. doi: 10.5802/aif.3358.
[19]	M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9.
[20]	H. Busemann, The Geometry of Geodesics, Academic Press Inc., New York, 1955.
[21]	L. Carvajales, Counting problems for special-orthogonal Anosov representations, Ann. Inst. Fourier (Grenoble), 70 (2020), 1199-1257. doi: 10.5802/aif.3333.
[22]	L. Carvajales, Growth of quadratic forms under Anosov subgroups, Int. Math. Res. Not. IMRN, 2023 (2023), 785-854. doi: 10.1093/imrn/rnab181.
[23]	S. Choi, G.-S. Lee and L. Marquis, Convex projective generalized Dehn filling, Ann. Sci. Éc. Norm. Supér., 53 (2020), 217–266. doi: 10.24033/asens.2421.
[24]	D. Cooper, D. D. Long and S. Tillmann, On convex projective manifolds and cusps, Adv. Math., 277 (2015), 181-251. doi: 10.1016/j.aim.2015.02.009.
[25]	L. J. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and their Applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge, 1990.
[26]	M. Crampon, Entropies of strictly convex projective manifolds, J. Mod. Dyn., 3 (2009), 511-547. doi: 10.3934/jmd.2009.3.511.
[27]	M. Crampon, Dynamics and Entropies of Hilbert Metrics, Thèses, Université de Strasbourg; Ruhr-Universität Bochum, 2011.
[28]	M. Crampon and L. Marquis, Finitude géométrique en géométrie de Hilbert, Ann. Inst. Fourier (Grenoble), 64 (2014), 2299-2377. doi: 10.5802/aif.2914.
[29]	M. Crampon and L. Marquis, Le flot géodésique des quotients géométriquement finis des géométries de Hilbert, Pacific J. Math., 268 (2014), 313-369. doi: 10.2140/pjm.2014.268.313.
[30]	F. Dal'bo, Remarques sur le spectre des longueurs d'une surface et comptages, Bol. Soc. Brasil. Mat. (N.S.), 30 (1999), 199-221. doi: 10.1007/BF01235869.
[31]	F. Dal'bo, Topologie du feuilletage fortement stable, Ann. Inst. Fourier (Grenoble), 50 (2000), 981-993. doi: 10.5802/aif.1781.
[32]	F. Dal'bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel J. Math., 118 (2000), 109-124. doi: 10.1007/BF02803518.
[33]	J. Danciger, F. Guéritaud, F. Kassel, G.-S. Lee and L. Marquis, Convex compactness for Coxeter groups, preprint, arXiv: 2102.02757, 2021.,
[34]	J. Danciger, F. Guéritaud and F. Kassel, Geometry and topology of complete Lorentz spacetimes of constant curvature, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 1–56.
[35]	J. Danciger, F. Guéritaud and F. Kassel, Convex cocompact actions in real projective geometry, preprint, arXiv: 1704.08711, to appear in Ann. Sci. Éc. Norm. Supér., 2017.
[36]	S. Dey and M. Kapovich, Patterson–Sullivan theory for Anosov subgroups, Trans. Amer. Math. Soc., 375 (2022), 8687-8737. doi: 10.1090/tran/8713.
[37]	S. Edwards, M. Lee and H. Oh, Anosov groups: Local mixing, counting and equidistribution, Geom. Topol., 27 (2023), 513-573. doi: 10.2140/gt.2023.27.513.
[38]	D. Egloff, Uniform Finsler Hadamard manifolds, Ann. Inst. Henri Poincaré, Phys. Théor., 66 (1997), 323–357.
[39]	M. Gromov, Hyperbolic manifolds, groups and actions, in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, NY, 1978), 183–213, Ann. of Math. Stud., No. 97, Princeton Univ. Press, Princeton, N.J., 1981.
[40]	O. Guichard and A. Wienhard, Anosov representations: Domains of discontinuity and applications, Invent. Math., 190 (2012), 357-438. doi: 10.1007/s00222-012-0382-7.
[41]	G. C. Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol., 10 (2010), 1807-1856. doi: 10.2140/agt.2010.10.1807.
[42]	M. Islam, Rank-one Hilbert geometries, preprint, arXiv: 1912.13013, 2019.
[43]	M. Kapovich and B. Leeb, Relativizing characterizations of Anosov subgroups, I. (With an appendix by Gregory A. Soifer), Groups Geom. Dyn., 17 (2023), 1005-1071. doi: 10.4171/GGD/723.
[44]	F. Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math., 165 (2006), 51-114. doi: 10.1007/s00222-005-0487-3.
[45]	G. Link, Equidistribution and counting of orbit points for discrete rank one isometry groups of Hadamard spaces, Tunis. J. Math., 2 (2020), 791-839. doi: 10.2140/tunis.2020.2.791.
[46]	G. A. Margulis, Certain applications of ergodic theory to the investigation of manifolds of negative curvature, Funkcional. Anal. i Priložen, 3 (1969), 89-90.
[47]	L. Marquis, Espace des modules de certains polyèdres projectifs miroirs, Geom. Dedicata, 147 (2010), 47-86. doi: 10.1007/s10711-009-9438-2.
[48]	L. Marquis, Around groups in Hilbert geometry, in Handbook of Hilbert Geometry, 207–261, IRMA Lect. Math. Theor. Phys., vol. 22, Eur. Math. Soc., Zürich, 2014.
[49]	B. Nica, Linear groups - Malcev's theorem and Selberg's lemma, 2013.
[50]	D. V. Osin, Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc., 179 (2006), no. 843, ⅵ+100 pp. doi: 10.1090/memo/0843.
[51]	S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273. doi: 10.1007/BF02392046.
[52]	F. Paulin, M. Pollicott and B. Schapira, Equilibrium states in negative curvature, Astérisque, 2015, No. 373, ⅷ+281 pp.
[53]	M. Peigné, On the Patterson–Sullivan measure of some discrete group of isometries, Israel J. Math., 133 (2003), 77-88. doi: 10.1007/BF02773062.
[54]	J.-F. Quint, Mesures de Patterson-Sullivan en rang supérieur, Geom. Funct. Anal., 12 (2002), 776-809. doi: 10.1007/s00039-002-8266-4.
[55]	J.-F. Quint, Groupes de Schottky et comptage, Ann. Inst. Fourier (Grenoble), 55 (2005), 373-429. doi: 10.5802/aif.2102.
[56]	M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972.
[57]	R. Ricks, Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces, Ergodic Theory Dynam. Systems, 37 (2017), 939-970. doi: 10.1017/etds.2015.78.
[58]	T. Roblin, Ergodicité et équidistribution en courbure négative, Mémoires de la Société Mathématique de France, 95 (2003), 1–96. doi: 10.24033/msmf.408.
[59]	D. J. Rudolph, Ergodic behaviour of Sullivan's geometric measure on a geometrically finite hyperbolic manifold, Ergodic Theory Dynam. Systems, 2 (1982), 491-512. doi: 10.1017/S0143385700001735.
[60]	A. Sambarino, The orbital counting problem for hyperconvex representations, Ann. Inst. Fourier (Grenoble), 65 (2015), 1755-1797. doi: 10.5802/aif.2973.
[61]	A. Sambarino, Quantitative properties of convex representations, Comment. Math. Helv., 89 (2014), 443-488. doi: 10.4171/CMH/324.
[62]	B. Schapira and S. Tapie, Regularity of entropy, geodesic currents and entropy at infinity, Ann. Sci. Éc. Norm. Supér. (4), 54 (2021), 1–68. doi: 10.24033/asens.2455.
[63]	A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, in Contributions to Function Theory (Internat. Colloq. Function Theory, Bombay, 1960), 147–164, Tata Institute of Fundamental Research, Bombay, 1960.
[64]	D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. Inst. Hautes Études Sci., 50 (1979), 171-202.
[65]	P. Tukia, Conical limit points and uniform convergence groups, J. Reine Angew. Math., 501 (1998), 71-98. doi: 10.1515/crll.1998.081.
[66]	C. Walsh, The horofunction boundary of the Hilbert geometry, Adv. Geom., 8 (2008), 503-529. doi: 10.1515/ADVGEOM.2008.032.
[67]	C. Walsh, The horoboundary and isometry group of Thurston's Lipschitz metric, in Handbook of Teichmüller Theory. Vol. IV, volume 19 of IRMA Lect. Math. Theor. Phys., Eur. Math. Soc., Zürich, (2014), 327–353. doi: 10.4171/117-1/8.
[68]	C. Walsh, Gauge-reversing maps on cones, and Hilbert and Thompson isometries, Geom. Topol., 22 (2018), 55-104. doi: 10.2140/gt.2018.22.55.
[69]	A. Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. (Crelles Journal), 566 (2004), 41-89. doi: 10.1515/crll.2004.007.
[70]	F. Zhu, Ergodicity and equidistribution in strictly convex Hilbert geometry, preprint, arXiv: 2008.00328, 2020.
[71]	F. Zhu, Relatively dominated representations, Ann. Inst. Fourier (Grenoble), 71 (2021), 2169-2235. doi: 10.5802/aif.3449.
[72]	A. Zimmer, A higher rank rigidity theorem for convex real projective manifolds, Geom. Topol., 27 (2023), 2899–2936. doi: 10.2140/gt.2023.27.2899.
[73]	A. Zimmer, Projective Anosov representations, convex cocompact actions, and rigidity, J. Differential Geom., 119 (2021), 513-586. doi: 10.4310/jdg/1635368438.