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,f0,∞) generated by a sequence of continuous self-maps f0,∞={fn}n=0∞ on a compact uniform space X. We prove that (X,f0,∞) and its k-th product system have the same entropy. We confirm that the entropy of (X,f0,∞) equals to that of its n-th compositions system, and also equals to the entropy of f0,∞ restricted to its non-wandering set if f0,∞ is equi-continuous. We prove that the entropy of (X,f0,∞) is less than or equal to that of its limit system (X,f) when f0,∞ 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 obtain the estimations of upper and lower bounds of entropy for an invariant subsystem of a coupled-expanding system associated with a transition matrix.