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Local unstable entropies of partially hyperbolic diffeomorphisms

Part of: Dynamical systems with hyperbolic behavior Measure-theoretic ergodic theory

Published online by Cambridge University Press:  26 February 2019

WEISHENG WU
WEISHENG WU*
Affiliation:
Department of Applied Mathematics, College of Science, China Agricultural University, Beijing, 100083, PR China email wuweisheng@cau.edu.cn
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Abstract

Consider a $C^{1}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$. Following the ideas in establishing the local variational principle for topological dynamical systems, we introduce the notions of local unstable metric entropies (and local unstable topological entropy) relative to a Borel cover ${\mathcal{U}}$ of $M$. It is shown that they coincide with the unstable metric entropy (and unstable topological entropy, respectively), when ${\mathcal{U}}$ is an open cover with small diameter. We also define the unstable tail entropy in the sense of Bowen and the unstable topological conditional entropy in the sense of Misiurewicz, and demonstrate that both of them vanish. Some generalizations of these results to the case of unstable pressure are also investigated.

Keywords

partial hyperbolicitylocal variational principlelocal unstable entropylocal unstable pressure

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 37D35: Thermodynamic formalism, variational principles, equilibrium states 28D20: Entropy and other invariants
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 8 , August 2020 , pp. 2274 - 2304
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Copyright
© Cambridge University Press, 2019

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