A note on stronger forms of sensitivity for inverse limit dynamical systems

Hai Zhu,Lei Liu,Jian Wang

doi:10.1186/s13662-015-0425-7

Abstract

In this paper we study stronger forms of sensitivity for inverse limit dynamical system which is induced from dynamical system on a compact metric space. We give the implication of stronger forms of sensitivity between inverse limit dynamical systems and original systems. More precisely, the inverse limit system is syndetically sensitive (resp. cofinitely sensitive, ergodically sensitive, multi-sensitive) if and only if original system is syndetically sensitive (resp. cofinitely sensitive, ergodically sensitive, multi-sensitive). Also, we prove that the inverse limit system is syndetically transitive if and only if original system is syndetically transitive.

Highlights

Throughout this paper a topological dynamical system we mean a pair (X, f ), where X is a compact space and f : X → X is a surjective continuous map
It is well known that sensitive dependence on initial conditions characterizes the unpredictability of chaotic phenomenon
We prove that the inverse limit system is syndetically transitive if and only if original system is syndetically transitive

Summary

Introduction

Throughout this paper a topological dynamical system we mean a pair (X, f ), where X is a compact space and f : X → X is a surjective continuous map. A map f : X → X is syndetically transitive if Nf (U, V ) is syndetic, for any nonempty open sets U, V ⊂ X. ( ) f is syndetically sensitive if there exists a δ > such that for every nonempty open subset U ⊂ X, Nf (U, δ) is syndetic. ( ) f is ergodically sensitive if there exists a δ > such that for every nonempty open subset U ⊂ X, Nf (U, δ) has positive upper density. We shall prove that Nσf (U, V ) is a syndetic set for any nonempty open subsets U and V in lim←(X, f ). We shall prove that Nσf (U, δ) is syndetic for any nonempty open subset U in lim←(X, f ).

This shows that

Since σf is cofinitely sensitive with sensitive constant