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

  • * Corresponding author: Fang Wang

    * Corresponding author: Fang Wang

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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  • 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.

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


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