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SRB measures for attractors with continuous invariant splittings

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Abstract

We study the existence of SRB measures of \(C^2\) diffeomorphisms for attractors whose bundles admit Hölder continuous invariant (non-dominated) splittings. We prove the existence when one sub-bundle has the non-uniform expanding property on a set with positive Lebesgue measure and the other sub-bundle admits non-positive Lyapunov exponents on a total probability set.

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Notes

  1. The contradiction of the case \( \ell (Q)\mu _Q(Q)<1\) can be obtained similarly.

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Acknowledgements

We would like to thank Prof. J. Xia for useful discussions and suggestions. We also thank the anonymous referees who helped us to improve the presentation of this paper. Z. Mi would like to thank Northwestern University for their hospitality and the excellent research atmosphere, where this work was partially done. Z. Mi also would like to thank China Scholarship Council (CSC) for their financial support.

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Authors and Affiliations

  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, People’s Republic of China

    Zeya Mi, Yongluo Cao & Dawei Yang

Authors
  1. Zeya Mi
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  2. Yongluo Cao
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  3. Dawei Yang
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Corresponding author

Correspondence to Dawei Yang.

Additional information

D. Yang is the corresponding author. Y. Cao would like to thank the support of NSFC 11125103, D. Yang would like to thank the support of NSFC 11271152, NSFC11671288, Z. Mi would like to thank the support of NSFC 11671288. We are all partially supported by A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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Mi, Z., Cao, Y. & Yang, D. SRB measures for attractors with continuous invariant splittings. Math. Z. 288, 135–165 (2018). https://doi.org/10.1007/s00209-017-1883-2

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