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On preimage entropy, folding entropy and stable entropy

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  14 January 2020

WEISHENG WU  and
YUJUN ZHU
WEISHENG WU
Affiliation:
Department of Applied Mathematics, Science College, China Agricultural University, Beijing100083, P.R. China email wuweisheng@cau.edu.cn
YUJUN ZHU
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen361005, P.R. China email yjzhu@xmu.edu.cn
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Abstract

For non-invertible dynamical systems, we investigate how ‘non-invertible’ a system is and how the ‘non-invertibility’ contributes to the entropy from different viewpoints. For a continuous map on a compact metric space, we propose a notion of pointwise metric preimage entropy for invariant measures. For systems with uniform separation of preimages, we establish a variational principle between this version of pointwise metric preimage entropy and pointwise topological entropies introduced by Hurley [On topological entropy of maps. Ergod. Th. & Dynam. Sys.15 (1995), 557–568], which answers a question considered by Cheng and Newhouse [Pre-image entropy. Ergod. Th. & Dynam. Sys.25 (2005), 1091–1113]. Under the same condition, the notion coincides with folding entropy introduced by Ruelle [Positivity of entropy production in nonequilibrium statistical mechanics. J. Stat. Phys.85(1–2) (1996), 1–23]. For a $C^{1}$-partially hyperbolic (non-invertible and non-degenerate) endomorphism on a closed manifold, we introduce notions of stable topological and metric entropies, and establish a variational principle relating them. For $C^{2}$ systems, the stable metric entropy is expressed in terms of folding entropy (namely, pointwise metric preimage entropy) and negative Lyapunov exponents. Preimage entropy could be regarded as a special type of stable entropy when each stable manifold consists of a single point. Moreover, we also consider the upper semi-continuity for both of pointwise metric preimage entropy and stable entropy and give a version of the Shannon–McMillan–Breiman theorem for them.

Keywords

preimage entropyfolding entropystable entropyShannon–Breiman–McMillan theoremvariational principle

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 37D30: Partially hyperbolic systems and dominated splittings
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 4 , April 2021 , pp. 1217 - 1249
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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