[Dynamique symbolique forte pour les flots géodésiques sur les espaces CAT et les flots métriques d’Anosov]
Nous montrons que le flot géodésique sur un espace compact localement ou, plus généralement, correspondant à une action convexe cocompacte d’un groupe non élémentaire sur un espace , 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 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
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
CC-BY 4.0
@article{JEP_2020__7__201_0,
author = {David Constantine and Jean-Fran\c{c}ois 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{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {201--231},
publisher = {\'Ecole polytechnique},
volume = {7},
year = {2020},
doi = {10.5802/jep.115},
zbl = {07152735},
mrnumber = {4054334},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.115/}
}
TY - JOUR AU - David Constantine AU - Jean-François Lafont AU - Daniel J. Thompson TI - Strong symbolic dynamics for geodesic flows on CAT$(-1)$ spaces and other metric Anosov flows JO - Journal de l’École polytechnique — Mathématiques PY - 2020 SP - 201 EP - 231 VL - 7 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.115/ DO - 10.5802/jep.115 LA - en ID - JEP_2020__7__201_0 ER -
%0 Journal Article %A David Constantine %A Jean-François Lafont %A Daniel J. Thompson %T Strong symbolic dynamics for geodesic flows on CAT$(-1)$ spaces and other metric Anosov flows %J Journal de l’École polytechnique — Mathématiques %D 2020 %P 201-231 %V 7 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.115/ %R 10.5802/jep.115 %G en %F 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/articles/10.5802/jep.115/
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