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arXiv · 2512.21308

The Patterson-Sullivan construction and global leaf geometry for Anosov flows

Clark Butler
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Abstract

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 Hamenst\"adt 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 Poincar\'e inequality. As a corollary we obtain that the fundamental group of a closed Riemannian manifold with Anosov geodesic flow must be Gromov hyperbolic.

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Clark Butler. 2025-12-24. The Patterson-Sullivan construction and global leaf geometry for Anosov flows. https://arxiv.org/abs/2512.21308

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