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Topological flows for hyperbolic groups

Part of: Dynamical systems with hyperbolic behavior Special aspects of infinite or finite groups

Published online by Cambridge University Press:  30 October 2020

RYOKICHI TANAKA Open the ORCID record for RYOKICHI TANAKA [Opens in a new window]
RYOKICHI TANAKA*
Affiliation:
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
*
e-mail: ryokichi.tanaka@tohoku.ac.jp
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Abstract

We show that for every non-elementary hyperbolic group the Bowen–Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion.

Keywords

hyperbolic groupgeodesic currentPatterson–Sullivan measureCannon’s automatic structureHausdorff dimension

MSC classification

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 37D35: Thermodynamic formalism, variational principles, equilibrium states 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3474 - 3520
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

Bridgeman, M., Canary, R. and Sambarino, A.. An introduction to pressure metrics for higher Teichmüller spaces. Ergod. Th. & Dynam. Sys. 38 (2017), 2001–2035.CrossRefGoogle Scholar
Bader, U. and Furman, A.. Some ergodic properties of metrics on hyperbolic groups. Preprint, 2017, arXiv:1707.02020v2.Google Scholar
Blachère, S., Haissinsky, P. and Mathieu, P.. Harmonic measures versus quasiconformal measures for hyperbolic groups. Ann. Sci. Éc. Norm. Supér. (4) 44(4) (2011), 683–721.CrossRefGoogle Scholar
Bourdon, M.. Actions quasi-convexes d’un groupe hyperbolique, flot géodésique. PhD Thesis, Université de Paris-Sud, 1993.Google Scholar
Bourdon, M.. Structure conforme au bord et flot géodésique d’un CAT $\left(-1\right)$ -espace. Enseign. Math. (2) 41(1–2) 63–102, 1995.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, 2nd revised edn (Lecture Notes in Mathematics, 470). Springer, Berlin, 2008.CrossRefGoogle Scholar
Bowen, R. and Ruelle, D.. The ergodic theory of Axion $A$ flows. Invent. Math. 29(3) (1975), 181–202.CrossRefGoogle Scholar
Bonk, M. and Schramm, O.. Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10(2) (2000), 266–306.10.1007/s000390050009CrossRefGoogle Scholar
Calegari, D.. The ergodic theory of hyperbolic groups. Geometry and Topology Down Under (Contemporary Mathematics, 597). American Mathematical Society, Providence, RI, 2013, pp. 15–52.CrossRefGoogle Scholar
Cannon, J. W.. The combinatorial structure of cocompact discrete hyperbolic groups. Geom. Dedicata 16(2) (1984), 123–148.CrossRefGoogle Scholar
Coulon, R., Dougall, R., Schapira, B. and Tapie, S.. Twisted Patterson-Sullivan measures and applications to amenability and coverings. Preprint, 2018, arXiv:1809.10881.Google Scholar
Calegari, D. and Fujiwara, K.. Combable functions, quasimorphisms, and the central limit theorem. Ergod. Th. & Dynam. Sys. 30(5) (2010), 1343–1369.CrossRefGoogle Scholar
Constantine, D., Lafont, J.-F. and Thompson, D. J.. Strong symbolic dynamics for geodesic flows on CAT $\!\left(-1\right)$ spaces and other metric Anosov flows. J. Éc. Polytech. Math. 7 (2020), 201–231.Google Scholar
Coornaert, M.. Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov. Pacific J. Math. 159(2) (1993), 241–270.CrossRefGoogle Scholar
Fisher, T. and Hasselblatt, B.. Hyperbolic Flows (Lecture Notes in Mathematical Sciences, 16). The University of Tokyo, Tokyo, 2018.Google Scholar
Furman, A.. Coarse-geometric perspective on negatively curved manifolds and groups. Rigidity in Dynamics and Geometry (Cambridge, 2000). Springer, Berlin, 2002, pp. 149–166.CrossRefGoogle Scholar
Garncarek, L.. Boundary representations of hyperbolic groups. Preprint, 2016, arXiv:1404.0903v2.Google Scholar
Ghys, E. and de la Harpe, P.. Sur les groupes hyperboliques d’après Mikhael Gromov (Progress in Mathematics, 83). Birkhäuser Boston, Inc., Boston, MA, 1990.CrossRefGoogle Scholar
Gouëzel, S.. Local limit theorem for symmetric random walks in Gromov-hyperbolic groups. J. Amer. Math. Soc. 27(3) (2014), 893–928.CrossRefGoogle Scholar
Gromov, M.. Hyperbolic groups. Essays in Group Theory (Mathematical Sciences Research Institute Publications, 8). Springer, New York, NY, 1987, pp. 75–263.CrossRefGoogle Scholar
Gekhtman, I., Taylor, S. J. and Tiozzo, G.. Counting loxodromics for hyperbolic actions. J. Topol. 11(2) (2018), 379–419.10.1112/topo.12053CrossRefGoogle Scholar
Heinonen, J.. Lectures on analysis on metric spaces. Universitext. Springer, New York, 2001.Google Scholar
Izumi, M., Neshveyev, S. and Okayasu, R.. The ratio set of the harmonic measure of a random walk on a hyperbolic group. Israel J. Math. 163 (2008), 285–316.CrossRefGoogle Scholar
Kaimanovich, V. A.. Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds. Ann. Inst. H. Poincaré Phys. Théor. 53(4) (1990), 361–393.Google Scholar
Kaimanovich, V. A.. Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces. J. Reine Angew. Math. 455 (1994), 57–103.Google Scholar
Kaimanovich, V. A., Kapovich, I. and Schupp, P.. The subadditive ergodic theorem and generic stretching factors for free group automorphisms. Israel J. Math. 157 (2007), 1–46.CrossRefGoogle Scholar
Kapovich, I. and Nagnibeda, T.. Geometric entropy of geodesic currents on free groups. Dynamical Numbers—Interplay Between Dynamical Systems and Number Theory (Contemporary Mathematics, 532). American Mathematical Society, Providence, RI, 2010, pp. 149–175.CrossRefGoogle Scholar
Knieper, G.. Volume growth, entropy and the geodesic stretch. Math. Res. Lett. 2(1) (1995), 39–58.CrossRefGoogle Scholar
Ledrappier, F.. Structure au bord des variétés à courbure négative. Séminaire de Théorie Spectrale et Géométrie. Vol. 13. Université Grenoble I, Saint-Martin-d’Hères, 1994–995, pp. 97–122.Google Scholar
Mineyev, I.. Flows and joins of metric spaces. Geom. Topol. 9 (2005), 403–482.CrossRefGoogle Scholar
Nica, B.. Proper isometric actions of hyperbolic groups on ${L}^p$ -spaces. Compos. Math. 149(5) (2013), 773–792.CrossRefGoogle Scholar
Nica, B. and Špakula, J.. Strong hyperbolicity. Groups Geom. Dyn. 10(3) (2016), 951–964.CrossRefGoogle Scholar
Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 268 (1990), 187–188.Google Scholar
Paulin, F., Pollicott, M. and Schapira, B.. Equilibrium states in negative curvature. Astérisque 373 (2015), viii+281 pp.Google Scholar
Tanaka, R.. Hausdorff spectrum of harmonic measure. Ergod. Th. & Dynam. Sys. 37(1) (2017), 277–307.CrossRefGoogle Scholar