On Recurrence And Sensitivity In Topological Dynamical System And Its Induced Spaces

In this thesis, we mainly study the properties of recurrence and sensitivity in topo-logical dynamical system and its two induced systems. The thesis is organized as fol-lows:In the Introduction, the origin and developments of the topological dynamical sys-tem and ergodic theory are recalled. Also some backgroud and main results of our study are presented.In Chapter 2, some basic definitions and properties of topological dynamical sys-tem and ergodic theory are recalled. And also some concepts and propositions which we use later.In Chapter 3, the finite and countable expansive systems, as generalizations of ex-pansive systems, and its applications are investigated. In particular, it is firstly proved that there exist (positively) essential (i.e. n-but not (n-1)-) expansive homeomor-phisms, solving completely a clarification question by Morales. Furthermore, an equiv-alent characterization for the existence of essential finite expansive homeomorphism on countable compact space are presented, and also some applications of the characteriza-tion are obtained.In Chapter 4, the mean forms of sensitivity are discussed. Mainly the relations between mean sensitivity and some other generalized forms of sensitivity and famous chaos are considered. In particular, it is shown that there exist a Devaney chaotic system which is neither mean sensitive nor mean equicontinuous, answering then negatively a question by Tu that if mean forms of Auslander-Yorke dichotomy theorem still hold for E-systems.In Chapter 5, the relations on properties of recurrence and sensitivity between hy-perspace and phase space are discussed. For the study of systems with strong recurrence property, it turns out that the hyperspace is a pointwise periodic system (resp. point-wise minimal system, weakly rigid system, M-system, E-system, weakly mixing sys-tem with dense distal points) if and only if the original system is a periodic system (resp. equicontinuous system, uniformly rigid system, M-system, E-system, weakly mixing system with dense distal sets). And for the study of mean sensitivity, it is particular-ly shown that there exists a Banach mean equicontinuous system but its hyperspace is sensitive.Besides, in Chapter 5 applications of hyperspace related to some open problems are also presented. It is worth emphasizing that every weakly mixing system with dense distal sets is disjoint from all minimal system, and meanwhile weakly mixing system with dense distal sets does not necessarily have dense distal points, which give then big and new progress on a Furstenberg’s question with disjointness.In Chapter 6, the relations on properties of recurrence and sensitivity between probability measure space and phase space are discussed. First for some system with strong recurrence property, it is similarly proved that the induced probability measure system is a pointwise periodic system (resp. uniformly rigid system, topologically exact system, P-system, E-system) if and only if the original system is a periodic system (re-sp. uniformly rigid system, topologically exact system, almost HY-system, E-system). Now consider the study of sensitivity property. On one hand rich results that are similar to hyperspace case are obtained. For example, let (X,T) be a topological dynamical system, (M(X).TM) be its induced probability measure system and F be a Fursten-berg family consisting of subsets of non-negative integers. Then the F-sensitivity of (M(X), TM) can be inherited into (X. T), and the converse is also true when F is addi-tionally a filter; (M(X),TM) is multi-sensitive if and only if (X.T) is multi-sensitive; there exist minimal sensitive or Li-Yorke sensitive systems (X, T) such that (M(X), TM) is not sensitive, and so on. On the other hand, some different conclusions are also de-rived. For example, if (X, T) is F-sensitive, then for any natural number n the subsys-tem (Mn(X), TM) of (M(X), TM) is also F-sensitive; if (M(X),TM) is Li-Yorke sensitive then so is (X,T); there exists a system (X,T) such that (K(X),TK) is sensitive but (M(X),TM) is not sensitive, and so on.