Hostname: page-component-7479d7b7d-fwgfc Total loading time: 0 Render date: 2024-07-09T19:51:21.287Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Cheng, W. C. and Newhouse, S.. Pre-image entropy. Ergod. Th. & Dynam. Sys. 25 (2005), 1091–1113.10.1017/S0143385704000240CrossRefGoogle Scholar
Downarowicz, T.. Entropy in Dynamical Systems (New Mathematical Monographs, 18). Cambridge University Press, Cambridge, 2011.10.1017/CBO9780511976155CrossRefGoogle Scholar
Fiebig, D., Fiebig, U. and Nitecki, Z.. Entropy and preimage sets. Ergod. Th. & Dynam. Sys. 23 (2003), 1785–1806.10.1017/S0143385703000221CrossRefGoogle Scholar
Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.10.1090/surv/101CrossRefGoogle Scholar
Hurley, M.. On topological entropy of maps. Ergod. Th. & Dynam. Sys. 15 (1995), 557–568.10.1017/S014338570000852XCrossRefGoogle Scholar
Langevin, R. and Przytycki, F.. Entropie de limage inverse dune application. Bull. Soc. Math. France 120 (1992), 237–250.10.24033/bsmf.2185CrossRefGoogle Scholar
Nitecki, Z. and Przytycki, F.. Preimage entropy for mappings. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), 1815–1843.10.1142/S0218127499001309CrossRefGoogle Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.10.1007/978-1-4612-5775-2CrossRefGoogle Scholar
Wu, W. and Zhu, Y.. On preimage entropy, folding entropy and stable entropy. Ergod. Th. & Dynam. Sys. 41 (2020), 1217–1249.10.1017/etds.2019.114CrossRefGoogle Scholar
Wu, W. and Zhu, Y.. Entropy via preimage structure. Adv. Math. 406 (2022), 108483.10.1016/j.aim.2022.108483CrossRefGoogle Scholar