A Note on Nonautonomous Discrete Dynamical Systems

Abstract

A discrete nonautonomous dynamical system (a NDS for short) is a pair $(X,f_{1,\infty })$ where X is a topological space and $f_{1,\infty }$ is a sequence of continuous functions $\left( f_{1},f_{2},\ldots \right)$ from X to itself. The orbit of a point $x \in X$ is defined as the set $\mathscr {O}_{f_{1,\infty }}(x) := \{x, f_{1}^{1}(x), f_{1}^{2}(x), \ldots , f_{1}^{n}(x), \ldots \}$. NDS’ were introduced by S. Kolyada and L. Snoha in 1996 and they are related to several mathematical fields; among others the theory of difference equations. Notice that NDS’ generalize the usual notion of a discrete dynamical system: indeed, if suffices to take $f_{1,\infty }$ as a constant sequence. The aim of this note is twofold. First we analyze several definitions of a periodic point in the framework of NDS’. The interest of this notion comes from the fact that, in the realm of discrete dynamical systems, the third condition of the definition of Devaney’s chaos (sensitivity) follows from the first two (transitivity and the set of periodic points is dense). This result need not be true for NDS’ and the results in this context depend upon the definition of a periodic point we consider. Secondly, we present several results on transitivity. In contrast to the situation for discrete dynamical systems, there exists second countable, perfect metric NDS’ with the Baire property which have transitive points but they are not transitive. Among other things, we study the relationships between these two notions.