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Entropy and ergodicity of Boole-type transformations

Blackmore, Denis, Balinsky, Alexander A. ORCID: https://orcid.org/0000-0002-8151-4462, Kycia, Radoslaw and Prykarpatski, Anatolij K. 2021. Entropy and ergodicity of Boole-type transformations. Entropy 23 (11) , 1405. 10.3390/e23111405

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Abstract

We review some analytic, measure-theoretic and topological techniques for studying ergodicity and entropy of discrete dynamical systems, with a focus on Boole-type transformations and their generalizations. In particular, we present a new proof of the ergodicity of the 1-dimensional Boole map and prove that a certain 2-dimensional generalization is also ergodic. Moreover, we compute and demonstrate the equivalence of metric and topological entropies of the 1-dimensional Boole map employing “compactified”representations and well-known formulas. Several examples are included to illustrate the results. We also introduce new multidimensional Boole-type transformations invariant with respect to higher dimensional Lebesgue measures and investigate their ergodicity and metric and topological entropies.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Additional Information: This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/
Publisher: MDPI
ISSN: 1099-4300
Date of First Compliant Deposit: 2 November 2021
Date of Acceptance: 20 October 2021
Last Modified: 10 Nov 2022 09:58
URI: https://orca.cardiff.ac.uk/id/eprint/145260

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