Topological entropy of nonautonomous dynamical systems on uniform spaces

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.