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...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-10-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/23/11/1405 |
| _version_ | 1797510470208323584 |
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| 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 |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-03-10T05:31:56Z |
| format | Article |
| id | doaj.art-e17f33b2f9a34537b60e9e99022ff415 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T05:31:56Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-e17f33b2f9a34537b60e9e99022ff4152023-11-22T23:14:30ZengMDPI AGEntropy1099-43002021-10-012311140510.3390/e23111405Entropy and Ergodicity of Boole-Type TransformationsDenis Blackmore0Alexander A. Balinsky1Radoslaw Kycia2Anatolij K. Prykarpatski3Department of Mathematical Sciences and CAMS, New Jersey Institute of Technology, Newark, NJ 07102, USAMathematics Institute at the Cardiff University, Cardiff CF24 4AG, UKFaculty of Physics, Mathematics and Computer Science, Cracow University of Technology, 31-155 Kraków, PolandFaculty of Physics, Mathematics and Computer Science, Cracow University of Technology, 31-155 Kraków, PolandWe 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.https://www.mdpi.com/1099-4300/23/11/1405discrete transformationsinvariant measureergodicityentropyBernoulli type transformationsBoole-type transformations |
| spellingShingle | Denis Blackmore Alexander A. Balinsky Radoslaw Kycia Anatolij K. Prykarpatski Entropy and Ergodicity of Boole-Type Transformations Entropy discrete transformations invariant measure ergodicity entropy Bernoulli type transformations Boole-type transformations |
| title | 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_short | Entropy and Ergodicity of Boole-Type Transformations |
| title_sort | entropy and ergodicity of boole type transformations |
| topic | discrete transformations invariant measure ergodicity entropy Bernoulli type transformations Boole-type transformations |
| url | https://www.mdpi.com/1099-4300/23/11/1405 |
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