On the Myrberg Limit Sets and Bowen-Margulis-Sullivan Measures for Visibility Manifolds without Conjugate Points

TL;DR

This paper explores the relationship between Myrberg limit sets and ergodic properties of geodesic flows on visibility manifolds without conjugate points, extending classical hyperbolic results to a broader setting.

Contribution

It establishes the equivalence between Patterson-Sullivan measure positivity of Myrberg limit sets and geodesic flow conservativity on these manifolds, generalizing prior hyperbolic results.

Findings

Positivity of Patterson-Sullivan measure is equivalent to flow conservativity.

Myrberg limit set has full Patterson-Sullivan measure within the conical limit set.

Results extend classical hyperbolic manifold theories to manifolds without conjugate points.

Abstract

In this paper, we clarify the strong relationship between Myrberg type dynamics and the ergodic properties of the geodesic flows on (not necessarily compact) uniform visibility manifolds without conjugate points. We prove that the positivity of the Patterson-Sullivan measure of the Myrberg limit set is equivalent to the conservativity of the geodesic flow with respect to the Bowen-Margulis-Sullivan measure. Moreover we show that the Myrberg limit set is a full Patterson-Sullivan measure subset of the conical limit set. These results extend the classical works of P. Tukia and B. Stratmann from hyperbolic manifolds to the manifolds without conjugate points.