Chaos Caused By The Expanding Maps

In this paper we mainly discuss the chaotic properties on the compact metric space and symbolic space, then reach some important results as follows:1. Let ( X , d1 )(Y , d 2)be compact metric spaces without isolated point and h :X→Ybe a semi-conjugacy between f and g , where f :X→Xg :Y→Y are continuous maps. We investigated the expanding maps with transitivity. And we gave a sufficient condition about expanding maps are chaotic in the sense of Wiggins and Kato.2. Let (Σ,ρ)be a one-sided symbolic space(with two symbols) andσbe the shift onΣ,and (Σ,σ)be the topological dynamical system(TDS for short).We gave a sufficient condition for the extension of (Σ,σ)和( X , f)being distributively chaotic in a sequence.And if f : X→X is transitive and minimal, then there is a factor map is distributively chaotic in a sequence.