Entropy and Ergodicity of Boole-Type Transformations

J 2021

Entropy and Ergodicity of Boole-Type Transformations

BLACKMORE, Denis, Alexander A BALINSKY, Radoslaw Antoni KYCIA and Anatolij K PRYKARPATSKI

Basic information

Original name

Entropy and Ergodicity of Boole-Type Transformations

Authors

BLACKMORE, Denis, Alexander A BALINSKY, Radoslaw Antoni KYCIA and Anatolij K PRYKARPATSKI

Edition

Entropy, Basel (Schwitzerland), MDPI AG, POSTFACH, 2021, 1099-4300

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Country of publisher

Switzerland

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 2.738

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.3390/e23111405

UT WoS

000723932500001

Keywords in English

discrete transformations; invariant measure; ergodicity; entropy; Bernoulli type transformations; Boole-type transformations; fibered multidimensional mappings; induced transformations

Abstract

V originále

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.

Links

MUNI/A/1099/2022, interní kód MU

Name: Specifický výzkum v odborné a učitelské matematice 2023

Investor: Masaryk University

Citovat

BLACKMORE, Denis, Alexander A BALINSKY, Radoslaw Antoni KYCIA and Anatolij K PRYKARPATSKI. Entropy and Ergodicity of Boole-Type Transformations. Entropy. Basel (Schwitzerland): MDPI AG, POSTFACH, 2021, vol. 23, No 11. ISSN 1099-4300. Available from: https://dx.doi.org/10.3390/e23111405.

@article{2227819,
   author = {Blackmore, Denis and Balinsky, Alexander A and Kycia, Radoslaw Antoni and Prykarpatski, Anatolij K},
   article_location = {Basel (Schwitzerland)},
   article_number = {11},
   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},
   language = {eng},
   issn = {1099-4300},
   journal = {Entropy},
   title = {Entropy and Ergodicity of Boole-Type Transformations},
   url = {https://doi.org/10.3390/e23111405},
   volume = {23},
   year = {2021}
}

TY  - JOUR
ID  - 2227819
AU  - Blackmore, Denis - Balinsky, Alexander A - Kycia, Radoslaw Antoni - Prykarpatski, Anatolij K
PY  - 2021
TI  - Entropy and Ergodicity of Boole-Type Transformations
JF  - Entropy
VL  - 23
IS  - 11
PB  - MDPI AG, POSTFACH
SN  - 10994300
KW  - discrete transformations
KW  - invariant measure
KW  - ergodicity
KW  - entropy
KW  - Bernoulli type transformations
KW  - Boole-type transformations
KW  - fibered multidimensional mappings
KW  - induced transformations
UR  - https://doi.org/10.3390/e23111405
N2  - 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.
ER  -

BLACKMORE, Denis, Alexander A BALINSKY, Radoslaw Antoni KYCIA and Anatolij K PRYKARPATSKI. Entropy and Ergodicity of Boole-Type Transformations. \textit{Entropy}. Basel (Schwitzerland): MDPI AG, POSTFACH, 2021, vol.~23, No~11. ISSN~1099-4300. Available from: https://dx.doi.org/10.3390/e23111405.

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