Recurrence And Chaotic Behavior Of Dynamical Systems

This doctoral dissertation is mainly divided into two parts.At the first part, we discuss on uniformly recurrent motions of topological semigroup actions. Let G (?) X be a topological action of a topological semigroup G on a compact metric space X. We show that for any given point x in X, the following two properties that both approximate to periodicity are equivalent to each other:● For any neighbourhood U of x, the return times set{g ∈ G:gx ∈ U} is syndetic of Furstenburg in G.● Given any ε> 0, there exists a finite subset K of G such that for each g in G, the ε-neighbourhood of the orbit-arc K[gx] contains the entire orbit G[x].This is a generalization of a classical theorem of Birkhoff for the case where G= R or Z. In addition, a counterexample is constructed to this statement, while X is merely a complete but not locally compact metric space.In the second part, we discuss Auslander-Yorke chaos and sensitivity of group actions on dendrites. First, we show that each sensitive group action on a dendrite contains an Auslander-Yorke chaotic subsystem. By this conclusion, we prove that each sensitive group action on a dendrite must have a ping-pong game, which implies that each sensitive finitely generated group action on a dendrite has positive geometric entropy, and each dendrite admits no sensitive nilpotent group actions. At last, we construct two examples:a sensitive but non-expansive slovable group action on a dendrite, and a sensitive group action on a ring domain without an Auslander-Yorke chaotic subsystem.