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

TL;DR

This paper constructs the measure of maximal entropy for transitive Anosov flows using a Patterson-Sullivan-like approach, revealing new geometric and analytic properties of the flow's leaves and implications for the fundamental group.

Contribution

It introduces a novel construction method for the measure of maximal entropy in Anosov flows and establishes the Gromov hyperbolicity of the universal covers of their center-unstable leaves.

Findings

Universal covers of center-unstable leaves are Gromov hyperbolic.

Relative Gromov boundaries of leaves identify with unstable leaves.

Uniformized leaves support a Poincaré inequality.

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.