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Variational equalities of entropy in nonuniformly hyperbolic systems


Authors: Chao Liang, Gang Liao, Wenxiang Sun and Xueting Tian
Journal: Trans. Amer. Math. Soc. 369 (2017), 3127-3156
MSC (2010): Primary 37B40, 37D25, 37C40
DOI: https://doi.org/10.1090/tran/6780
Published electronically: August 22, 2016
MathSciNet review: 3605966
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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

$\displaystyle \inf \{h_\mu (f)\mid \mu \in K\}=h_{\mathrm {top}}(f,G_K).$

In particular, for every invariant (usually nonergodic) measure $ \mu \!\in \! \mathcal {M}_{inv}(\widetilde {\Lambda },f)$, we have

$\displaystyle h_\mu (f)=h_{\mathrm {top}}(f,G_{\mu }).$

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.

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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
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
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

DOI: https://doi.org/10.1090/tran/6780
Keywords: Metric entropy, periodic points, weak shadowing property
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)
Article copyright: © Copyright 2016 American Mathematical Society