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A note on preimage entropy

Part of: Dynamical systems with hyperbolic behavior Ergodic theory

Published online by Cambridge University Press:  19 June 2023

TAO WANG
TAO WANG*
Affiliation:
MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China
*
e-mail: twang@hunnu.edu.cn
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Abstract

Cheng and Newhouse (Ergod. Th. & Dynam. Sys. 25 (2005), 1091–1113) proved a variational principle for topological preimage entropy $h_{\mathrm {pre}}(f)$:

$$ \begin{align*} h_{\mathrm{pre}}(f)=\sup_{\mu\in\mathcal{M}(X,f)}h_{\mathrm{pre},\mu}(f). \end{align*} $$

Unfortunately, we show in this note that this variational principle is not true.

Keywords

preimage entropyvariational principle

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states 37A35: Entropy and other invariants, isomorphism, classification
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 5 , May 2024 , pp. 1468 - 1472
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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