Chaos Of Iteration System

This thesis deals with chaos of iteration system.In chapter 2. we research topologically ergodic maps. In terms of sub-shifts of finite type determined by an irreducible matrix, affine maps of compacted connected metric abelian group and continuous maps of tree, the two concepts of topologically ergodic map and topologically transitive map are identical. Furthermore, examples show that the topologically ergodic map differs from the topologically mixing map and the topologically transitive map.In chapter 3, we study chaos for subshifts of finite type. We show that for any subshift of finite type determined by an irreducible and aperodic matrix, there is a finitely chaotic set with full Hausdorff dimension. Furthermore, we make a survey of relationship between the various sorts of chaos for subshifts of finite type.In chapter 4. we discuss the Hausdorff dimension of chaotic sets for interval self-maps. Some sufficient conditions that there is a chaotic set for interval self-map with positive Hausdorff dimension were obtained. Furthermore, we point out that for any interval Lipschitz map with positive topological entropy there is a chaotic set with positive Hausdorff dimension.In chapter 5. we study the chaos caused by sweeps out. A necessary and sufficient condition that there is a chaotic set with full measure was obtained. Furthermore, we study weak-mixing on sequence and the chaos caused by it.