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arXiv · 2305.00301

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

Fei Liu·Xiaokai Liu·Fang Wang
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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.

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Fei Liu, Xiaokai Liu, Fang Wang. 2023-04-29. The Hopf-Tsuji-Sullivan Dichotomy on Visibility Manifolds Without Conjugate Points. https://arxiv.org/abs/2305.00301

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