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Entropy and Ergodicity of Boole-Type Transformations.

Denis Blackmore¹, Alexander A Balinsky², Radoslaw Kycia³

¹Department of Mathematical Sciences and CAMS, New Jersey Institute of Technology, Newark, NJ 07102, USA.

Entropy (Basel, Switzerland)

|November 27, 2021

Summary

This summary is machine-generated.

This study explores ergodicity and entropy in discrete dynamical systems, focusing on Boole-type transformations. New proofs confirm ergodicity for 1D and 2D Boole maps, with equivalent metric and topological entropies demonstrated.

Keywords:

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

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Area of Science:

Mathematics
Dynamical Systems Theory
Measure Theory

Background:

Ergodicity and entropy are key concepts in understanding the long-term behavior of dynamical systems.
Boole-type transformations and their generalizations are a class of discrete dynamical systems with unique properties.

Purpose of the Study:

To review and apply analytic, measure-theoretic, and topological techniques to study ergodicity and entropy.
To present new proofs for the ergodicity of specific Boole-type transformations.
To investigate the relationship between metric and topological entropies for these systems.

Main Methods:

Analytic, measure-theoretic, and topological approaches.
Development of "compactified" representations.
Application of established formulas for entropy computation.

Main Results:

A novel proof for the ergodicity of the 1-dimensional Boole map.
Proof of ergodicity for a 2-dimensional generalization of the Boole map.
Demonstration of the equivalence between metric and topological entropies for the 1D Boole map.
Introduction of new multidimensional Boole-type transformations.

Conclusions:

The study provides rigorous mathematical proofs for the ergodicity of Boole-type transformations.
It establishes the equivalence of different entropy measures, enhancing theoretical understanding.
New multidimensional transformations are introduced for further research in dynamical systems.