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Strong symbolic dynamics for geodesic flows on CAT(-1) spaces and other metric Anosov flows
[Dynamique symbolique forte pour les flots géodésiques sur les espaces CAT(-1) et les flots métriques d’Anosov]
David Constantine1 ; Jean-François Lafont2 ; Daniel J. Thompson2
1 Wesleyan University, Mathematics and Computer Science Department Middletown, CT 06459
2 Department of Mathematics, Ohio State University Columbus, Ohio 43210
Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 201-231.

Nous montrons que le flot géodésique sur un espace compact localement CAT(-1) ou, plus généralement, correspondant à une action convexe cocompacte d’un groupe non élémentaire sur un espace CAT(-1), peut être codé par un flot symbolique irréductible de type fini avec une fonction toît höldérienne. Notre approche consiste à montrer que ces flots géodésiques sont des flots métriques d’Anosov, qui satisfont à une régularité höldérienne pour les temps de retour associés à une classe spéciale de sections géométriques transverses au flot. Nous obtenons un certain nombre de résultats sur la dynamique du flot par rapport aux mesures d’équilibre pour les potentiels höldériens. En particulier, nous démontrons que la mesure de Bowen-Margulis est Bernoulli, à l’exception du cas particulier où toutes les périodes d’orbites fermées sont des multiples entiers d’une constante commune. Nos techniques s’appliquent également au flot géodésique associé à une représentation projective d’Anosov [BCLS15], donnant accès à toute la puissance de la dynamique symbolique pour cette classe de flots.

We prove that the geodesic flow on a locally CAT(-1) metric space which is compact, or more generally convex cocompact with non-elementary fundamental group, can be coded by a suspension flow over an irreducible shift of finite type with Hölder roof function. This is achieved by showing that the geodesic flow is a metric Anosov flow, and obtaining Hölder regularity of return times for a special class of geometrically constructed local cross-sections to the flow. We obtain a number of strong results on the dynamics of the flow with respect to equilibrium measures for Hölder potentials. In particular, we prove that the Bowen-Margulis measure is Bernoulli except for the exceptional case that all closed orbit periods are integer multiples of a common constant. We show that our techniques also extend to the geodesic flow associated to a projective Anosov representation [BCLS15], which verifies that the full power of symbolic dynamics is available in that setting.

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DOI : 10.5802/jep.115
Classification : 37D40, 37D35, 51F99
Keywords: Geodesic flows, CAT$(-1)$ spaces, metric Anosov flows, symbolic dynamics, projective Anosov representations
Mot clés : Flots géodésiques, espaces CAT$(-1)$, flots métriques d’Anosov, dynamique symbolique, représentations projectives d’Anosov
David Constantine 1 ; Jean-François Lafont 2 ; Daniel J. Thompson 2

1 Wesleyan University, Mathematics and Computer Science Department Middletown, CT 06459
2 Department of Mathematics, Ohio State University Columbus, Ohio 43210
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Strong symbolic dynamics for geodesic flows on {CAT}$(-1)$ spaces and other metric {Anosov} flows},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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David Constantine; Jean-François Lafont; Daniel J. Thompson. Strong symbolic dynamics for geodesic flows on CAT$(-1)$ spaces and other metric Anosov flows. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 201-231. doi : 10.5802/jep.115. https://jep.centre-mersenne.org/articles/10.5802/jep.115/

[BAPP19] A. Broise-Alamichel, J. Parkkonen & F. Paulin - Equidistribution and counting under equilibrium states in negative curvature and trees, Progress in Math., vol. 329, Birkhauser, 2019 | MR | Zbl

[BCLS15] M. Bridgeman, R. Canary, F. Labourie & A. Sambarino - “The pressure metric for Anosov representations”, Geom. Funct. Anal. 25 (2015), p. 1089-1179 | DOI | MR | Zbl

[BCLS18] M. Bridgeman, R. Canary, F. Labourie & A. Sambarino - “Simple root flows for Hitchin representations”, Geom. Dedicata 192 (2018) no. 1, p. 57-86 | DOI | MR | Zbl

[BCS17] M. Bridgeman, R. Canary & A. Sambarino - “An introduction to pressure metrics for higher Teichmüller spaces”, Ergodic Theory Dynam. Systems (2017), p. 1-35 | Zbl

[BH99] M. R. Bridson & A. Haefliger - Metric spaces of non-positive curvature, Grundlehren Math. Wiss., vol. 319, Springer, Berlin, 1999 | MR | Zbl

[Bou95] M. Bourdon - “Structure conforme au bord et flot géodésique d’un CAT(-1)-espace”, Enseign. Math. 41 (1995) no. 1-2, p. 63-102 | MR | Zbl

[Bow72] R. Bowen - “Periodic orbits for hyperbolic flows”, Amer. J. Math. 94 (1972) no. 1, p. 1-30 | DOI | MR | Zbl

[Bow73] R. Bowen - “Symbolic dynamics for hyperbolic flows”, Amer. J. Math. 95 (1973) no. 2, p. 429-460 | DOI | MR | Zbl

[BR75] R. Bowen & D. Ruelle - “The ergodic theory of Axiom A flows”, Invent. Math. 29 (1975) no. 3, p. 181-202 | DOI | MR | Zbl

[BW72] R. Bowen & P. Walters - “Expansive one-parameter flows”, J. Differential Equations 12 (1972), p. 180-193 | DOI | MR | Zbl

[Can84] J. W. Cannon - “The combinatorial structure of cocompact discrete hyperbolic groups”, Geom. Dedicata 16 (1984), p. 123-148 | MR | Zbl

[Cha94] C. Champetier - “Petite simplification dans les groupes hyperboliques”, Ann. Fac. Sci. Toulouse Math. (6) 3 (1994) no. 2, p. 161-221 | DOI | Numdam | MR | Zbl

[CLT19] D. Constantine, J.-F. Lafont & D. Thompson - “The weak specification property for geodesic flows on CAT(-1) spaces”, 2019, to appear in Groups Geom. Dyn. | arXiv

[CP12] M. Coornaert & A. Papadopoulos - “Symbolic coding for the geodesic flow associated to a word hyperbolic group”, Manuscripta Math. 109 (2012), p. 465-492 | DOI | MR | Zbl

[DP84] M. Denker & W. Philipp - “Approximation by Brownian motion for Gibbs measures and flows under a function”, Ergodic Theory Dynam. Systems 4 (1984) no. 4, p. 541-552 | DOI | MR | Zbl

[DSU17] T. Das, D. Simmons & M. Urbański - Geometry and dynamics in Gromov hyperbolic metric spaces, Math. Surveys and Monographs, vol. 218, American Mathematical Society, Providence, RI, 2017 | MR | Zbl

[FH18] T. Fisher & B. Hasselblatt - Hyperbolic Flows, Lect. Notes in Math. Sciences, vol. 16, University of Tokyo, 2018, available at https://www.ms.u-tokyo.ac.jp/publication_e/lecturenote_e.html | Zbl

[GM10] K. Gelfert & A. E. Motter - “(Non)invariance of dynamical quantities for orbit equivalent flows”, Comm. Math. Phys. 300 (2010) no. 2, p. 411-433 | DOI | MR | Zbl

[Gro87] M. Gromov - “Hyperbolic groups”, in Essays in group theory (S. Gersten, ed.), MSRI Publications, vol. 8, Springer, 1987, p. 75-265 | DOI | MR

[KT19] T. Kucherenko & D. J. Thompson - “Measures of maximal entropy for suspension flows over the full shift”, 2019, to appear in Math. Z. | arXiv | DOI

[Min05] I. Mineyev - “Flows and joins of metric spaces”, Geom. Topol. 9 (2005), p. 403-482 | DOI | MR | Zbl

[MT04] I. Melbourne & A. Török - “Statistical limit theorems for suspension flows”, Israel J. Math. 144 (2004), p. 191-209 | DOI | MR | Zbl

[Pol87] M. Pollicott - “Symbolic dynamics for Smale flows”, Amer. J. Math. 109 (1987) no. 1, p. 183-200 | DOI | MR | Zbl

[PP90] W. Parry & M. Pollicott - Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, vol. 187-188, Société Mathématique de France, Paris, 1990 | Numdam | Zbl

[PS17] R. Potrie & A. Sambarino - “Eigenvalues and entropy of a Hitchin representation”, Invent. Math. 209 (2017) no. 3, p. 885-925 | DOI | MR | Zbl

[Put14] I. F. Putnam - A homology theory for Smale spaces, Mem. Amer. Math. Soc., vol. 232, no. 1094, American Mathematical Society, Providence, RI, 2014 | DOI | Zbl

[Rat73] M. Ratner - “The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature”, Israel J. Math. 16 (1973), p. 181-197 | DOI | MR | Zbl

[Rat74] M. Ratner - “Anosov flows with Gibbs measures are also Bernoullian”, Israel J. Math. 17 (1974), p. 380-391 | DOI | MR | Zbl

[Ric17] 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), p. 939-970 | DOI | MR | Zbl

[Rob03] T. Roblin - Ergodicité et équidistribution en courbure négative, vol. 95, Société Mathématique de France, Paris, 2003 | Numdam | MR | Zbl

[Rue76] D. Ruelle - “A measure associated with Axiom-A attractors”, Amer. J. Math. 98 (1976) no. 3, p. 619-654 | DOI | MR | Zbl

[Sam16] A. Sambarino - “On entropy, regularity and rigidity for convex representations of hyperbolic manifolds”, Math. Ann. 364 (2016) no. 1-2, p. 453-483 | DOI | MR | Zbl

[Tap11] S. Tapie - “A variation formula for the topological entropy of convex-cocompact manifolds”, Ergodic Theory Dynam. Systems 31 (2011), p. 1849-1864 | DOI | MR | Zbl

[Tro90] M. Troyanov - “Espaces à courbure négative et groupes hyperboliques” (E. Ghys & P. de la Harpe, eds.), Progress in Math., vol. 83, Birkhäuser, 1990, p. 47-66

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