On Nonautonomous Discrete Dynamical Systems

Abstract

We define and study expansiveness, shadowing, and topological stability for a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a metric space.

Various Non-autonomous Notions for Borel Measures

“…Such δ is called expansive constant for µ. Remark 3.2 (i) If the space is complete separable without isolated points, then every non-atomic Borel measure is expansive with respect to any expansive [14] time varying bi-measurable map. (ii) If X is compact, then expansiveness of measure does not depend on the choice of the metric.…”

“…(b) F is said to be topologically stable [14] if for every ǫ > 0 there is 0 < δ < 1 such that if G = {g n } n∈N is another time varying bi-measurable map on X with p(F, G) < δ, then there is a continuous map h : X → X such that d(h(x), x) < ǫ and d(F n (h(x)), G n (x)) < ǫ for all n ∈ Z. Theorem 4.2 Let F = {f n } n∈N be a time varying bi-measurable map with topological stability on a metric space (X, d). Then, every non-atomic Borel measure µ on X is topologically stable with respect to F .…”

“…Because of several clear motivation discussed in the introductory paper for non-autonomous systems, authors of [15] have generalized the classical spectral decomposition theorem for homeomorphisms on compact metric spaces. Most importantly in [14], authors have proved that a non-autonomous system with expansiveness and shadowing is topologically stable in the class of all time varying homeomorphisms. Our purpose is to extend this result to expansive measures with shadowing with respect to time varying homeomorphisms on relatively compact metric spaces.…”

A note on nonwandering set of a nonautonomous discrete dynamical system

“…For the time varying dynamical systems the terms orbit, periodicity and expansiveness are defined in [4]. We have studied expansiveness, shadowing and topological stability for a nonautonomous discrete system induced by a sequence of continuous maps on a compact metric space in [5] and for a nonautonomous discrete system induced by a sequence of homeomorphisms on a compact metric space in [6].…”

“…Definition 1.1 [6] Let (X, d) be a metric space and f n : X → X be a sequence of homeomorphisms, n = 0, 1, 2, .... For a point x 0 ∈ X, define a sequence as follows :…”