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.
Highlights
With its origins going back several centuries, analysis of discrete dynamical systems has become an increasingly central methodology for many mathematical problems related to a wide range of applications in modern science and engineering
It was observed that the metric entropy is related to the exponential growth of distinguishable orbits, which in turn has a certain communication interpretation following from the fact that information theory models can be reformulated as Bernoulli schemes
The Rokhlin– Krengel formula for metric entropy of ergodic systems played a key role in our treatment of both topological and Kolmogorov–Sinai entropy
Summary
With its origins going back several centuries, analysis of discrete dynamical systems has become an increasingly central methodology for many mathematical problems related to a wide range of applications in modern science and engineering. Owing to Lemma 2, either λ(B) = 1 or λ(B) = 0, which proves the ergodicity of the Boole mapping (39) with respect to the same invariant Lebesgue measure λ on R and completes the proof. It is worth mentioning here the well-known result [1,13,47,48,70,71] that the doubling map (18) is isomorphic to the 1-dimensional Boole-type transformation f : R x → (x − 1/x)/2 ∈ R,.
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