Topological entropy of nonautonomous dynamical systems

TL;DR

This paper investigates the behavior of topological entropy in nonautonomous dynamical systems and their induced systems on probability measures, revealing conditions under which entropy vanishes, becomes infinite, or is not preserved.

Contribution

It establishes a link between the entropy of a nonautonomous system and its induced system on measures, and constructs examples where entropy is not preserved under extensions.

Findings

Vanishing entropy in the base system implies vanishing in the induced system.

Positive entropy in the base system leads to infinite entropy in the induced system.

Provides a counterexample to Bowen's inequality showing entropy not preserved under finite-to-one extensions.

Abstract

Let $\mathcal{M}(X)$ be the space of Borel probability measures on a compact metric space $X$ endowed with the weak$^\ast$-topology. In this paper, we prove that if the topological entropy of a nonautonomous dynamical system $(X,\{f_n\}_{n=1}^{+\infty})$ vanishes, then so does that of its induced system $(\mathcal{M}(X),\{f_n\}_{n=1}^{+\infty})$; moreover, once the topological entropy of $(X,\{f_n\}_{n=1}^{+\infty})$ is positive, that of its induced system $(\mathcal{M}(X),\{f_n\}_{n=1}^{+\infty})$ jumps to infinity. In contrast to Bowen's inequality, we construct a nonautonomous dynamical system whose topological entropy is not preserved under a finite-to-one extension.