Entropy and Ergodicity of Boole-Type Transformations

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Authors

BLACKMORE Denis BALINSKY Alexander A KYCIA Radoslaw Antoni PRYKARPATSKI Anatolij K

Year of publication 2021
Type Article in Periodical
Magazine / Source Entropy
MU Faculty or unit

Faculty of Science

Citation
Web https://doi.org/10.3390/e23111405
Doi http://dx.doi.org/10.3390/e23111405
Keywords discrete transformations; invariant measure; ergodicity; entropy; Bernoulli type transformations; Boole-type transformations; fibered multidimensional mappings; induced 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.
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