{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:29:41Z","timestamp":1760236181188,"version":"build-2065373602"},"reference-count":77,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,26]],"date-time":"2021-10-26T00:00:00Z","timestamp":1635206400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 \u201ccompactified\u201drepresentations 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.<\/jats:p>","DOI":"10.3390\/e23111405","type":"journal-article","created":{"date-parts":[[2021,10,26]],"date-time":"2021-10-26T23:49:52Z","timestamp":1635292192000},"page":"1405","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Entropy and Ergodicity of Boole-Type Transformations"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3524-9538","authenticated-orcid":false,"given":"Denis","family":"Blackmore","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences and CAMS, New Jersey Institute of Technology, Newark, NJ 07102, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8151-4462","authenticated-orcid":false,"given":"Alexander A.","family":"Balinsky","sequence":"additional","affiliation":[{"name":"Mathematics Institute at the Cardiff University, Cardiff CF24 4AG, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6390-4627","authenticated-orcid":false,"given":"Radoslaw","family":"Kycia","sequence":"additional","affiliation":[{"name":"Faculty of Physics, Mathematics and Computer Science, Cracow University of Technology, 31-155 Krak\u00f3w, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5124-5890","authenticated-orcid":false,"given":"Anatolij K.","family":"Prykarpatski","sequence":"additional","affiliation":[{"name":"Faculty of Physics, Mathematics and Computer Science, Cracow University of Technology, 31-155 Krak\u00f3w, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Aaronson, J. 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