The Hopf-Tsuji-Sullivan Dichotomy on Visibility Manifolds Without Conjugate Points

TL;DR

This paper extends the Hopf-Tsuji-Sullivan dichotomy to visibility manifolds without conjugate points, linking ergodic properties of geodesic flows to geometric and measure-theoretic conditions.

Contribution

It establishes the dichotomy for a new class of manifolds and provides equivalent conditions for conservativity, connecting dynamics with geometric and measure-theoretic properties.

Findings

Geodesic flow is either conservative and ergodic or dissipative and non-ergodic.

Conservativity is equivalent to divergence of the Poincaré series at the critical exponent.

Full Patterson-Sullivan measure on the conical limit set indicates conservativity.

Abstract

In this article, we establish the Hopf-Tsuji-Sullivan dichotomy for geodesic flows on certain manifolds with no conjugate points: either the geodesic flow is conservative and ergodic, or it is completely dissipative and non-ergodic. We also show several equivalent conditions to the conservativity, such the Poincar\'e series diverges at the critical exponent, the conical limit set has full Patterson-Sullivan measure, etc.