Topological entropy of nonautonomous dynamical systems on uniform spaces

TL;DR

This paper investigates the properties, calculations, and estimations of topological entropy for nonautonomous dynamical systems on compact uniform spaces, establishing relations, bounds, and invariance under certain conditions.

Contribution

It provides new relations, bounds, and invariance results for topological entropy in nonautonomous systems on uniform spaces, extending classical concepts.

Findings

Entropy equals that on the non-wandering set under equi-continuity.

Entropy of the limit system bounds the nonautonomous system's entropy.

Same entropy for systems with finite-to-one equi-semiconjugacy.

Abstract

In this paper, we focus on some properties, calculations and estimations of topological entropy for a nonautonomous dynamical system $(X,f_{0,\infty})$ generated by a sequence of continuous self-maps $f_{0,\infty}=\{f_n\}_{n=0}^{\infty}$ on a compact uniform space $X$. We obtain the relations of topological entropy among $(X, f_{0,\infty})$, its $k$-th product system and its $n$-th iteration system. We confirm that the entropy of $(X, f_{0,\infty})$ equals to that of $f_{0,\infty}$ restricted to its non-wandering set provided that $f_{0,\infty}$ is equi-continuous. We prove that the entropy of $(X, f_{0,\infty})$ is less than or equal to that of its limit system $(X, f)$ when $f_{0,\infty}$ converges uniformly to $f$. We show that two topologically equi-semiconjugate systems have the same entropy if the equi-semiconjugacy is finite-to-one. Finally, we derive the estimations of upper and lower bounds of entropy for an invariant subsystem of a coupled-expanding system associated with a transition matrix.