Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866910718949851136 |
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| author | Constantine, David Lafont, Jean-François Thompson, Daniel J. |
| author_facet | Constantine, David Lafont, Jean-François Thompson, Daniel J. |
| contents | 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, which verifies that the full power of symbolic dynamics is available in that setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1808_04395 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows Constantine, David Lafont, Jean-François Thompson, Daniel J. Dynamical Systems Metric Geometry 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, which verifies that the full power of symbolic dynamics is available in that setting. |
| title | Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows |
| topic | Dynamical Systems Metric Geometry |
| url | https://arxiv.org/abs/1808.04395 |