Li-Yorke Sensitivity of Set-Valued Discrete Systems

Heng Liu,Fengchun Lei,Lidong Wang

doi:10.1155/2013/260856

Heng Liu, Fengchun Lei + Show 1 more

Open Access

https://doi.org/10.1155/2013/260856

Copy DOI

Abstract

Consider the surjective, continuous mapf:X→Xand the continuous mapf¯of𝒦(X)induced byf, whereXis a compact metric space and𝒦(X)is the space of all nonempty compact subsets ofXendowed with the Hausdorff metric. In this paper, we give a short proof that iff¯is Li-Yoke sensitive, thenfis Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity offdoes not imply Li-Yorke sensitivity off¯.

Highlights

Throughout this paper a dynamical system (X, f) is a pair where X is a compact metric space with metric d and f : X → X is a surjective, continuous map.The idea of sensitivity from the work [1, 2] by Ruelle and Takens was applied to topological dynamics by Auslander and Yorke in [3] and popularized later by Devaney in [4]
A system (X, f) is called ε-sensitive if there exists a positive ε such that any x ∈ X is a limit of points y ∈ X satisfying the condition d(fn(x), fn(y)) > ε for some positive integer n
A system is Li-Yorke sensitive if there exists ε > 0 such that every x ∈ X is a limit of points y ∈ X such that the pair (x, y) is proximal but supn>N{d(fn(x), fn(y))} > ε for any N > 0, and the positive ε is said to be a Li-Yorke sensitive constant of the system

Summary

Introduction

Throughout this paper a dynamical system (X, f) is a pair where X is a compact metric space with metric d and f : X → X is a surjective, continuous map. According to Li and Yorke (see [5]), a subset S ⊂ X is a scrambled set (for f), if any different points x and y from S are proximal and not asymptotic; that is, lim inf d (fn (x) , fn (y)) = 0, n→∞. A system is Li-Yorke sensitive if there exists ε > 0 such that every x ∈ X is a limit of points y ∈ X such that the pair (x, y) is proximal but supn>N{d(fn(x), fn(y))} > ε for any N > 0, and the positive ε is said to be a Li-Yorke sensitive constant of the system. A dynamical system (X, f) is called spatiotemporal chaotic (see [6] or [7]) if every point is a limit point for points which are proximal to but not asymptotic to it. By suing the obtained results, we give positive answers to Sharma and Nagar’s question in [18]

The Denjoy Homeomorphism and an Interval Map

A Counter Example

Li-Yorke Sensitivity

Li-Yorke Sensitivity of Interval Maps