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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 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.
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.
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