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]
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.

Reçu le : 2019-04-04
Accepté le : 2019-11-18
Publié le : 2020-01-15
DOI : https://doi.org/10.5802/jep.115
Classification : 37D40,  37D35,  51F99
Mots clés: Flots géodésiques, espaces CAT(-1), flots métriques d’Anosov, dynamique symbolique, représentations projectives d’Anosov
@article{JEP_2020__7__201_0,
     author = {David Constantine and Jean-Fran\c cois Lafont and Daniel J. Thompson},
     title = {Strong symbolic dynamics for geodesic flows on CAT$(-1)$ spaces and other metric Anosov flows},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     pages = {201-231},
     doi = {10.5802/jep.115},
     zbl = {07152735},
     mrnumber = {4054334},
     language = {en},
     url = {jep.centre-mersenne.org/item/JEP_2020__7__201_0/}
}
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/item/JEP_2020__7__201_0/

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