On ergodic properties of geodesic flows on uniform visibility manifolds without conjugate points

TL;DR

This paper investigates the ergodic properties of geodesic flows on uniform visibility manifolds without conjugate points, establishing conditions for ergodicity, uniqueness of measures of maximal entropy, and related geometric and dynamical properties.

Contribution

It develops a Patterson-Sullivan construction for the measure of maximal entropy and explores ergodic, hyperbolic, and rigidity properties under various geometric assumptions.

Findings

Geodesic flow is ergodic under certain conditions.

Unique measure of maximal entropy exists with Bernoulli property.

Counting and volume asymptotics are derived for closed geodesics and Riemannian balls.

Abstract

In this paper, we conduct a comprehensive study on ergodic properties of the geodesic flow on a $C^\infty$ uniform visibility manifold $M$ without conjugate points. If $M$ is a closed surface of genus at least two without conjugate points, and with continuous Green bundles and bounded asymptote, we study the geometric properties of singular geodesics and show that if all singular geodesics are closed, then there are at most finitely many isolated singular closed geodesics and finitely many generalized strips. In particular, the geodesic flow is ergodic with respect to Liouville measure under the above assumption. Let $(M,g)$ be a closed uniform visibility manifold without conjugate points and $X$ its universal cover. Under the entropy gap assumption, the geodesic flow has a unique measure of maximal entropy (MME for short) by \cite[Theorem 1.2]{MR}. We develop a Patterson-Sullivan construction of this unique MME and show that it has local product structure, is fully supported, and has the Bernoulli property. If we assume further that $M$ has continuous Green bundles and the geodesic flow has a hyperbolic periodic point, using the nonuniform hyperbolic structure on an open dense subset and the symbolic approach developed in \cite{LP}, we show that for any H\"{o}lder continuous function, the equilibrium state is unique under the pressure gap condition. Under the same conditions above, we apply the mixing properties of the MME to count the number of free-homotopy classes containing a closed geodesic, as well as the volume asymptotics of Riemannian balls in the universal cover. We then obtain some rigidity results involving the Margulis function. Finally, for a uniform visibility manifold $M=X/\Gamma$ (not necessarily compact) without conjugate points, we show that if $\Gamma$ is non-elementary and contains an expansive isometry, then the Hopf-Tsuji-Sullivan dichotomy holds.