Topological entropy zero and asymptotic pairs
The topological entropy is a measure of the complexity of topological dynamics. A basic question in this regard is whether the topological entropy is zero or positive. In the setting of measure-preserving dynamical systems, a system is called deterministic if its (Kolmogorov-Sinai) entropy is zero. Determinism is equivalent to three conditions that are formulated in terms of Pinsker sigma-algebras and factors/extensions. It was H. Furstenberg who called a topological dynamical system ``deterministic'' when its topological entropy is zero. This being the case, the authors of this paper define three more notions of determinism analogous to the above three characterizations of deterministic, measure-preserving systems. All of them are expressed in terms of asymptotic pairs and factors/extensions. Given a continuous map \(T\) on a compact metric space \((X,d)\), a pair of distinct points \(x,y\) in \(X\) is asymptotic if \(d(T^{n}x, T^{n}y)\) converges to zero when \(n\) goes to infinity. These results simplify the characterization of topological determinism and provide an analogy between measure-theoretical and topological dynamics. However, the analogy does not extend to the notion of Pinsker factor, as discussed at the end of the paper.
scientific article; zbMATH DE number 6105475