The Patterson-Sullivan construction and global leaf geometry for Anosov flows
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909975498981376 |
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| author | Butler, Clark |
| author_facet | Butler, Clark |
| contents | We give a new construction of the measure of maximal entropy for transitive Anosov flows through a method analogous to the construction of Patterson-Sullivan measures in negative curvature. In order to carry out our procedure we prove several new results concerning the global geometry of the leaves of the center-unstable foliation of an Anosov flow. We show that the universal covers of the center-unstable leaves are Gromov hyperbolic in the induced Riemannian metric and their relative Gromov boundaries canonically identify with the unstable leaves within in such a way that the Hamenstädt metrics on these leaves correspond to visual metrics on the relative Gromov boundary. These center-unstable leaves are then uniformized according to a technique inspired by methods of Bonk-Heinonen-Koskela which, in addition to its utility in the construction itself, also leads to rich analytic properties for these uniformized leaves such as supporting a Poincaré inequality. As a corollary we obtain that the fundamental group of a closed Riemannian manifold with Anosov geodesic flow must be Gromov hyperbolic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21308 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Patterson-Sullivan construction and global leaf geometry for Anosov flows Butler, Clark Dynamical Systems Metric Geometry We give a new construction of the measure of maximal entropy for transitive Anosov flows through a method analogous to the construction of Patterson-Sullivan measures in negative curvature. In order to carry out our procedure we prove several new results concerning the global geometry of the leaves of the center-unstable foliation of an Anosov flow. We show that the universal covers of the center-unstable leaves are Gromov hyperbolic in the induced Riemannian metric and their relative Gromov boundaries canonically identify with the unstable leaves within in such a way that the Hamenstädt metrics on these leaves correspond to visual metrics on the relative Gromov boundary. These center-unstable leaves are then uniformized according to a technique inspired by methods of Bonk-Heinonen-Koskela which, in addition to its utility in the construction itself, also leads to rich analytic properties for these uniformized leaves such as supporting a Poincaré inequality. As a corollary we obtain that the fundamental group of a closed Riemannian manifold with Anosov geodesic flow must be Gromov hyperbolic. |
| title | The Patterson-Sullivan construction and global leaf geometry for Anosov flows |
| topic | Dynamical Systems Metric Geometry |
| url | https://arxiv.org/abs/2512.21308 |