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SRB measures for pointwise hyperbolic systems on open regions

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Abstract

A diffeomorphism f : M → M is pointwise partially hyperbolic on an open invariant subset N ⊂ M if there is an invariant decomposition TNM = Eu ⊕ Ec ⊕ Es such that Dxf is strictly expanding on Eux and contracting on Eux at each x ∈ N. We show that under certain conditions f has unstable and stable manifolds, and admits a finite or an infinite u-Gibbs measure μ. If f is pointwise hyperbolic on N, then μ is a Sinai-Ruelle-Bowen (SRB) measure or an infinite SRB measure. As applications, we show that some almost Anosov diffeomorphisms and gentle perturbations of Katok’s map have the properties.

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Acknowledgements

The third author was supported by National Natural Science Foundation of China (Grant Nos. 11871120 and 11671093). The authors gratefully thank the referees for the helpful comments and suggestions which definitely helped to improve the quality of the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences & Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou, 215006, China

    Jianyu Chen

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Huyi Hu

  3. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Yunhua Zhou

Authors
  1. Jianyu Chen
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  2. Huyi Hu
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  3. Yunhua Zhou
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Correspondence to Huyi Hu.

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Dedicated to the Memory of Professor Shantao Liao

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Chen, J., Hu, H. & Zhou, Y. SRB measures for pointwise hyperbolic systems on open regions. Sci. China Math. 63, 1671–1720 (2020). https://doi.org/10.1007/s11425-020-1766-3

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  • DOI: https://doi.org/10.1007/s11425-020-1766-3

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