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

2023/04/29 by Liu, Fei, Liu, Xiaokai, Wang, Fang · 1 citation
#37D25 (Secondary) #37D40 (Primary) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2305.00301

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 Poincaré series diverges at the critical exponent, the conical limit set has full Patterson-Sullivan measure, etc.

Cited by

• On the Dirichlet Problem at Infinity and Poisson Boundary for Certain Manifolds without Conjugate Points

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