Variational equalities of entropy in nonuniformly hyperbolic systems
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- by Chao Liang, Gang Liao, Wenxiang Sun and Xueting Tian PDF
- Trans. Amer. Math. Soc. 369 (2017), 3127-3156 Request permission
Abstract:
In this paper we prove that for a nonuniformly hyperbolic system $(f,\widetilde {\Lambda })$ and for every nonempty, compact and connected subset $K$ with the same hyperbolic rate in the space $\mathcal {M}_{inv}(\widetilde {\Lambda },f)$ of invariant measures on $\widetilde {\Lambda }$, the metric entropy and the topological entropy of basin $G_K$ are related by the variational equality \[ \inf \{h_\mu (f)\mid \mu \in K\}=h_{\mathrm {top}}(f,G_K).\] In particular, for every invariant (usually nonergodic) measure $\mu \!\in \! \mathcal {M}_{inv}(\widetilde {\Lambda },f)$, we have \[ h_\mu (f)=h_{\mathrm {top}}(f,G_{\mu }).\] We also verify that $\mathcal {M}_{inv}(\widetilde {\Lambda },f)$ contains an open domain in the space of ergodic measures for diffeomorphisms with some hyperbolicity. As an application, the historical behavior is shown to occur robustly with a full positive entropy for diffeomorphisms beyond uniform hyperbolicity.References
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Additional Information
- Chao Liang
- Affiliation: Department of Applied Mathematics, The Central University of Finance and Economics, Beijing 100081, People’s Republic of China
- Email: chaol@cufe.edu.cn
- Gang Liao
- Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
- Address at time of publication: School of Mathematical Sciences, Soochow University, Suzhou 215006, People’s Republic of China
- MR Author ID: 906104
- Email: liaogang@math.pku.edu.cn, lg@suda.edu.cn
- Wenxiang Sun
- Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
- MR Author ID: 315192
- Email: sunwx@math.pku.edu.cn
- Xueting Tian
- Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
- Email: xuetingtian@fudan.edu.cn
- Received by editor(s): September 3, 2013
- Received by editor(s) in revised form: April 23, 2015
- Published electronically: August 22, 2016
- Additional Notes: The first author was supported by NNSFC(#11471344) and Beijing Higher Education Young Elite Teacher Project (YETP0986)
The second author is the corresponding author
The third author was supported by NNSFC (#11231001)
The fourth author was supported by NNSFC (#11301088) - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 3127-3156
- MSC (2010): Primary 37B40, 37D25, 37C40
- DOI: https://doi.org/10.1090/tran/6780
- MathSciNet review: 3605966