On Generic Properties of Nonautonomous Dynamical Systems

Abstract

We consider nonautonomous dynamical systems consisting of sequences of continuous surjective maps of a compact metric space [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text] denote the space of systems [Formula: see text], which are uniformly convergent, or equicontinuous, or pointwise convergent (to a continuous map), respectively. We show that for [Formula: see text], the generic system in any of the spaces has infinite topological entropy, while, if [Formula: see text] is the Cantor set, the generic system in any of the spaces has zero topological entropy. This shows, among others, that the general results of the above type for arbitrary compact space [Formula: see text] are difficult if not impossible.