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On the Patterson-Sullivan measure for geodesic flows on rank 1 manifolds without focal points

  • * Corresponding author: Fang Wang

    * Corresponding author: Fang Wang 

The first author is partially supported by NSFC under Grant Nos.11301305 and 11571207. The second authoris partially supported by NSFC under Grant No.11571387 and by the State Scholarship Fund from China Scholarship Council (CSC). The third author is partially supported by NSFC under Grant Nos.11701559 and 11571387

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  • In this article, we consider the geodesic flow on a compact rank $ 1 $ Riemannian manifold $ M $ without focal points, whose universal cover is denoted by $ X $. On the ideal boundary $ X(\infty) $ of $ X $, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based on the the Patterson-Sullivan measure, we show that the geodesic flow on $ M $ has a unique invariant measure of maximal entropy. We also obtain the asymptotic growth rate of the volume of geodesic spheres in $ X $ and the growth rate of the number of closed geodesics on $ M $. These results generalize the work of Margulis and Knieper in the case of negative and nonpositive curvature respectively.

    Mathematics Subject Classification: Primary: 37D40; Secondary: 37B40.

    Citation:

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