The Topological Sensitivity with respect to Furstenberg Families

Tengfei Wang,Jiandong Yin,Kai Jing

doi:10.1155/2020/7684072

Abstract

In this work, a dynamical system X,f means that X is a topological space and f:X⟶X is a continuous map. The aim of the article is to introduce the conceptions of topological sensitivity with respect to Furstenberg families, n-topological sensitivity, and multisensitivity and present some of their basic features and sufficient conditions for a dynamical system to possess some sensitivities. Actually, it is proved that every topologically ergodic but nonminimal system is syndetically sensitive and a weakly mixing system is n-thickly topologically sensitive and multisensitive under the assumption that X admits some separability.

Highlights

For a compact system (X, f) which means that f is a continuous self-map on a compact metric space (X, d), sensitive dependence on initial condition for (X, f)was firstly introduced by Ruelle [1] as if there exists δ > 0 such that for each x ∈ X and every open neighborhood Vx of x, there is a nonnegative integer n such that sup􏼈d(fn(x), fn(y)): y ∈ Vx􏼉 > δ
For the sake of dealing with the conception of topological sensitivity of dynamical systems in a unified way, we introduce the conceptions of topological sensitivity with respect to families of N0, n-topological sensitivity, and multisensitivity for dynamical systems and give some sufficient conditions for a dynamical system to possess distinct sensitivities
We introduce this notion for dynamical systems

Summary

Introduction

For a compact system (X, f) which means that f is a continuous self-map on a compact metric space (X, d), sensitive dependence on initial condition (sensitive for simplicity) for (X, f)was firstly introduced by Ruelle [1] as if there exists δ > 0 such that for each x ∈ X and every open neighborhood Vx of x, there is a nonnegative integer n such that sup􏼈d(fn(x), fn(y)): y ∈ Vx􏼉 > δ. One can write this in a slightly different way (see [2]) as follows. Some of the presented results improve or generalize the main results of [15] to a great extent

Preliminaries

Topological Sensitivity with respect to Families of Dynamical Systems

N-Topological Sensitivity of Dynamical Systems

Multitopological Sensitivity of Dynamical Systems