On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points

Fei Liu; Xiaokai Liu; Fang Wang

doi:10.3934/dcds.2021057

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2021, Volume 41, Issue 10: 4791-4804. Doi: 10.3934/dcds.2021057

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$ p $

-Laplacian problem with indefinite weight

On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points

1.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China
2.
Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China
3.
School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

^* Corresponding author: Fang Wang
^* Corresponding author: Fang Wang

Received: August 31, 2020

Early access: March 2021

Published: October 2021

F. Liu is partially supported by Natural Science Foundation of Shandong Province under Grant No. ZR2020MA017, and NSFC under Grant Nos. 11301305, 11571207. F. Wang is partially supported by NSFC under Grant No. 11871045 and the State Scholarship Fund from China Scholarship Council (CSC). The research is also partially supported by key research project of the Academy for Multidisciplinary Studies, Capital Normal University

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Abstract

If $ (M,g) $ is a smooth compact rank $ 1 $ Riemannian manifold without focal points, it is shown that the measure $ \mu_{\max} $ of maximal entropy for the geodesic flow is unique. In this article, we study the statistic properties and prove that this unique measure $ \mu_{\max} $ is mixing. Stronger conclusion that the geodesic flow on the unit tangent bundle $ SM $ with respect to $ \mu_{\max} $ is Bernoulli is acquired provided $ M $ is a compact surface with genus greater than one and no focal points.

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Mathematics Subject Classification: Primary: 37A25, 37D40; Secondary: 53C22.

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