Topologically Transitive and Mixing Properties of Set-Valued Dynamical Systems

Abstract

We introduce and study two properties of dynamical systems: topologically transitive and topologically mixing under the set-valued setting. We prove some implications of these two properties for set-valued functions and generalize some results from a single-valued case to a set-valued case. We also show that both properties of set-valued dynamical systems are equivalence for any compact intervals.

Transitive points in CR-dynamical systems

“…Next, we give an example of a non-transitive CR-dynamical system (X,G) such that trans 2 (G) ∅. This is an example that proves that the statement of [SS,Proposition 8,page 2] is incorrect.…”

“…Also, several papers on the topic of dynamical systems with (upper semicontinuous) set-valued functions have appeared recently, see [BEGK,CP,LP,LYY,LWZ,KN,KW,MRT,R,SS,SS2], where more references may be found. However, there is not much known of such dynamical systems and therefore, there are many properties of such set-valued dynamical systems that are yet to be studied.…”

“…At the end of the paper, two examples are given proving that the statement of [SS,Theorem 9,page 3], saying that any dynamical system with an upper semicontinuous set-valued function is transitive if and only if there is a point with a dense orbit, is incorrect.…”

Some Properties on Sensitivity, Transitivity and Mixing of Set-Valued Dynamical Systems

“…It is obvious that the orbits are not uniquely determined in set-valued case as shown by Example 2.3 in [19]. Note that the product set-valued function F × F : X × X → 2 X×X is defined by (F × F )(x, x ) = {(y, y ) ∈ X × X : y ∈ F (x) and y ∈ F (x )} for all x, x ∈ X.…”

“…Metzger et al [12] proposed topological stability for set-valued maps and extended several results from classical single-valued cases into set-valued cases. As a continuation work from [19], the aim of this paper is to introduce the concept of sensitivity for set-valued dynamical systems and investigate more properties on the transitivity and mixing of set-valued dynamical systems. We define sensitivity under setvalued setting and show that topologically mixing set-valued function implies sensitive.…”