Entropy and drift for Gibbs measures on geometrically finite manifolds

Gekhtman, Ilya; Tiozzo, Giulio

doi:10.1090/tran/8036

Entropy and drift for Gibbs measures on geometrically finite manifolds
HTML articles powered by AMS MathViewer

by Ilya Gekhtman and Giulio Tiozzo PDF

Trans. Amer. Math. Soc. 373 (2020), 2949-2980 Request permission

Abstract:

We prove a generalization of the fundamental inequality of Guivarc’h relating entropy, drift and critical exponent to Gibbs measures on geometrically finite quotients of $CAT(-1)$ metric spaces. For random walks with finite superexponential moment, we show that the equality is achieved if and only if the Gibbs density is equivalent to the hitting measure. As a corollary, if the action is not convex cocompact, any hitting measure is singular to any Gibbs density.

References

André Avez, Entropie des groupes de type fini, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A1363–A1366 (French). MR 324741
Sébastien Blachère and Sara Brofferio, Internal diffusion limited aggregation on discrete groups having exponential growth, Probab. Theory Related Fields 137 (2007), no. 3-4, 323–343. MR 2278460, DOI 10.1007/s00440-006-0009-2
I. Benua and Zh. -F. Kén, Central limit theorem on hyperbolic groups, Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 1, 5–26 (Russian, with Russian summary); English transl., Izv. Math. 80 (2016), no. 1, 3–23. MR 3462675, DOI 10.4213/im8306
Sébastien Blachère, Peter Haïssinsky, and Pierre Mathieu, Asymptotic entropy and Green speed for random walks on countable groups, Ann. Probab. 36 (2008), no. 3, 1134–1152. MR 2408585, DOI 10.1214/07-AOP356
Sébastien Blachère, Peter Haïssinsky, and Pierre Mathieu, Harmonic measures versus quasiconformal measures for hyperbolic groups, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 4, 683–721 (English, with English and French summaries). MR 2919980, DOI 10.24033/asens.2153
Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181–202. MR 380889, DOI 10.1007/BF01389848
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Anne Broise-Alamichel, Jouni Parkkonen, and Frédéric Paulin, Equidistribution and counting under equilibrium states in negatively curved spaces and graphs of groups. Applications to non-Archimedean Diophantine approximation, in preparation, hal-01421211, 2016.
Chris Connell and Roman Muchnik, Harmonicity of Gibbs measures, Duke Math. J. 137 (2007), no. 3, 461–509. MR 2309151, DOI 10.1215/S0012-7094-07-13732-3
Chris Connell and Roman Muchnik, Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces, Geom. Funct. Anal. 17 (2007), no. 3, 707–769. MR 2346273, DOI 10.1007/s00039-007-0608-9
Michel Coornaert, Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993), no. 2, 241–270 (French, with French summary). MR 1214072, DOI 10.2140/pjm.1993.159.241
Thierry Coulhon, Ultracontractivity and Nash type inequalities, J. Funct. Anal. 141 (1996), no. 2, 510–539. MR 1418518, DOI 10.1006/jfan.1996.0140
Bertrand Deroin, Victor Kleptsyn, and Andrés Navas, On the question of ergodicity for minimal group actions on the circle, Mosc. Math. J. 9 (2009), no. 2, 263–303, back matter (English, with English and Russian summaries). MR 2568439, DOI 10.17323/1609-4514-2009-9-2-263-303
Cornelia Druţu and Mark Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005), no. 5, 959–1058. With an appendix by Denis Osin and Mark Sapir. MR 2153979, DOI 10.1016/j.top.2005.03.003
Matthieu Dussaule and Ilya Gekhtman, Entropy and drift for word metrics in relatively hyperbolic groups, arXiv:1811.10849, 2018.
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Alex Furman, Coarse-geometric perspective on negatively curved manifolds and groups, Rigidity in dynamics and geometry (Cambridge, 2000) Springer, Berlin, 2002, pp. 149–166. MR 1919399
Vaibhav Gadre, Joseph Maher, and Giulio Tiozzo, Word length statistics and Lyapunov exponents for Fuchsian groups with cusps, New York J. Math. 21 (2015), 511–531. MR 3368971
Ilya Gekhtman, Equidistribution of closed geodesics along random walk trajectories with respect to the harmonic invariant measure, arXiv:1711.04985, 2017.
Ilya Gekhtman, Victor Gerasimov, Leonid Potyagailo, and Wenyuan Yang, Martin boundary covers Floyd boundary, arXiv:1708.02133, 2017.
Victor Gerasimov and Leonid Potyagailo, Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 6, 2115–2137. MR 3120738, DOI 10.4171/JEMS/417
Sébastien Gouëzel, Frédéric Mathéus, and François Maucourant, Entropy and drift in word hyperbolic groups, Invent. Math. 211 (2018), no. 3, 1201–1255. MR 3763407, DOI 10.1007/s00222-018-0788-y
Y. Guivarc’h, Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire, Conference on Random Walks (Kleebach, 1979) Astérisque, vol. 74, Soc. Math. France, Paris, 1980, pp. 47–98, 3 (French, with English summary). MR 588157
Y. Guivarc’h and Y. Le Jan, Asymptotic winding of the geodesic flow on modular surfaces and continued fractions, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 1, 23–50. MR 1209912, DOI 10.24033/asens.1666
Peter Haïssinsky, Marches aléatoires sur les groupes hyperboliques, Géométrie ergodique, Monogr. Enseign. Math., vol. 43, Enseignement Math., Geneva, 2013, pp. 199–265 (French). MR 3220556
Ursula Hamenstädt, Harmonic measures for compact negatively curved manifolds, Acta Math. 178 (1997), no. 1, 39–107. MR 1448711, DOI 10.1007/BF02392709
Vadim A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. Reine Angew. Math. 455 (1994), 57–103. MR 1293874, DOI 10.1515/crll.1994.455.57
Vadim A. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math. (2) 152 (2000), no. 3, 659–692. MR 1815698, DOI 10.2307/2661351
F. Ledrappier, Harmonic measures and Bowen-Margulis measures, Israel J. Math. 71 (1990), no. 3, 275–287. MR 1088820, DOI 10.1007/BF02773746
François Ledrappier, Applications of dynamics to compact manifolds of negative curvature, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 1195–1202. MR 1404020
Grigoriy A. Margulis, On some aspects of the theory of Anosov systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows; Translated from the Russian by Valentina Vladimirovna Szulikowska. MR 2035655, DOI 10.1007/978-3-662-09070-1
Katsuhiko Matsuzaki, Yasuhiro Yabuki, and Johannes Jaerisch, Normalizer, divergence type and Patterson measure for discrete groups of the Gromov hyperbolic space, arXiv:1511.02664, 2015.
Denis V. Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006), no. 843, vi+100. MR 2182268, DOI 10.1090/memo/0843
William Parry and Mark Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990), 268 (English, with French summary). MR 1085356
S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3-4, 241–273. MR 450547, DOI 10.1007/BF02392046
Frédéric Paulin, Mark Pollicott, and Barbara Schapira, Equilibrium states in negative curvature, Astérisque 373 (2015), viii+281 (English, with English and French summaries). MR 3444431
Ja. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4(166), 21–64 (Russian). MR 0399421
Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202. MR 556586, DOI 10.1007/BF02684773
Matt Sunderland, Linear progress with exponential decay in weakly hyperbolic groups, arXiv:1710.05107, 2018.
Eric L. Swenson, Quasi-convex groups of isometries of negatively curved spaces, Topology Appl. 110 (2001), no. 1, 119–129. Geometric topology and geometric group theory (Milwaukee, WI, 1997). MR 1804703, DOI 10.1016/S0166-8641(99)00166-2
Ryokichi Tanaka, Dimension of harmonic measures in hyperbolic spaces, Ergodic Theory Dynam. Systems 39 (2019), no. 2, 474–499. MR 3893268, DOI 10.1017/etds.2017.23
A. M. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples, Uspekhi Mat. Nauk 55 (2000), no. 4(334), 59–128 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 4, 667–733. MR 1786730, DOI 10.1070/rm2000v055n04ABEH000314
Wolfgang Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, 2000. MR 1743100, DOI 10.1017/CBO9780511470967
Wenyuan Yang, Genericity of contracting elements in groups, To appear in Mathematische Annalen.