Entropy and Ergodicity of Boole-Type Transformations

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 pr...

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Auteurs principaux:	Denis Blackmore, Alexander A. Balinsky, Radoslaw Kycia, Anatolij K. Prykarpatski
Format:	article
Langue:	EN
Publié:	MDPI AG 2021
Sujets:	discrete transformations invariant measure ergodicity entropy Bernoulli type transformations Boole-type transformations Science Q Astrophysics QB460-466 Physics QC1-999
Accès en ligne:	https://doaj.org/article/e17f33b2f9a34537b60e9e99022ff415
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spelling	oai:doaj.org-article:e17f33b2f9a34537b60e9e99022ff4152021-11-25T17:29:23ZEntropy and Ergodicity of Boole-Type Transformations10.3390/e231114051099-4300https://doaj.org/article/e17f33b2f9a34537b60e9e99022ff4152021-10-01T00:00:00Zhttps://www.mdpi.com/1099-4300/23/11/1405https://doaj.org/toc/1099-4300We 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.Denis BlackmoreAlexander A. BalinskyRadoslaw KyciaAnatolij K. PrykarpatskiMDPI AGarticlediscrete transformationsinvariant measureergodicityentropyBernoulli type transformationsBoole-type transformationsScienceQAstrophysicsQB460-466PhysicsQC1-999ENEntropy, Vol 23, Iss 1405, p 1405 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	discrete transformations invariant measure ergodicity entropy Bernoulli type transformations Boole-type transformations Science Q Astrophysics QB460-466 Physics QC1-999
spellingShingle	discrete transformations invariant measure ergodicity entropy Bernoulli type transformations Boole-type transformations Science Q Astrophysics QB460-466 Physics QC1-999 Denis Blackmore Alexander A. Balinsky Radoslaw Kycia Anatolij K. Prykarpatski Entropy and Ergodicity of Boole-Type Transformations
description	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.
format	article
author	Denis Blackmore Alexander A. Balinsky Radoslaw Kycia Anatolij K. Prykarpatski
author_facet	Denis Blackmore Alexander A. Balinsky Radoslaw Kycia Anatolij K. Prykarpatski
author_sort	Denis Blackmore
title	Entropy and Ergodicity of Boole-Type Transformations
title_short	Entropy and Ergodicity of Boole-Type Transformations
title_full	Entropy and Ergodicity of Boole-Type Transformations
title_fullStr	Entropy and Ergodicity of Boole-Type Transformations
title_full_unstemmed	Entropy and Ergodicity of Boole-Type Transformations
title_sort	entropy and ergodicity of boole-type transformations
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/e17f33b2f9a34537b60e9e99022ff415
work_keys_str_mv	AT denisblackmore entropyandergodicityofbooletypetransformations AT alexanderabalinsky entropyandergodicityofbooletypetransformations AT radoslawkycia entropyandergodicityofbooletypetransformations AT anatolijkprykarpatski entropyandergodicityofbooletypetransformations
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Entropy and Ergodicity of Boole-Type Transformations

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