Entropy and ergodicity of boole-type transformations

Denis Blackmore, Alexander A. Balinsky, Radoslaw Kycia, Anatolij K. Prykarpatski

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Article number1405
JournalEntropy
Volume23
Issue number11
DOIs
StatePublished - Nov 2021

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Mathematical Physics
  • Physics and Astronomy (miscellaneous)
  • Electrical and Electronic Engineering

Keywords

  • Bernoulli type transformations
  • Boole-type transformations
  • Discrete transformations
  • Entropy
  • Ergodicity
  • Fibered multidimensional mappings
  • Induced transformations
  • Invariant measure

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