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Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure

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Abstract

Given a non-conformal repeller \(\Lambda \) of a \(C^{1+\gamma }\) map, we study the Hausdorff dimension of the repeller and continuity of the sub-additive topological pressure for the sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use a substantially modified version of Katok’s approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the sub-additive topological pressure and obtain a sharp lower bound of the Hausdorff dimension of the repeller. The latter is given by the zero of the super-additive topological pressure.

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Acknowledgements

The authors would like to thank the referees for their careful reading and valuable comments which helped improve the paper substantially. The authors would like to thank Professor Dejun Feng and Wen Huang for their suggestions and comments. Ya. P. wants to thank Mittag-Leffler Institute and the Department of Mathematics of Weizmann Institute of Science where part of this work was done for their hospitality.

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Authors and Affiliations

  1. Department of Mathematics, East China Normal University, Shanghai, 200062, People’s Republic of China

    Yongluo Cao

  2. Department of Mathematics, Soochow University, Suzhou, 215006, Jiangsu, People’s Republic of China

    Yongluo Cao

  3. Center for Dynamical Systems and Differential Equation, Soochow University, Suzhou, 215006, Jiangsu, People’s Republic of China

    Yongluo Cao & Yun Zhao

  4. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Yakov Pesin

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

    Yun Zhao

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  1. Yongluo Cao
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  2. Yakov Pesin
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Correspondence to Yakov Pesin.

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Y. Cao: The first author is partially supported by NSFC (11771317, 11790274), Y. Pesin: author is partially supported by NSF grant DMS-1400027, Y. Zhao: author is partially supported by NSFC (11871361, 11790274).

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Cao, Y., Pesin, Y. & Zhao, Y. Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure. Geom. Funct. Anal. 29, 1325–1368 (2019). https://doi.org/10.1007/s00039-019-00510-7

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  • DOI: https://doi.org/10.1007/s00039-019-00510-7

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