The Complexity Of Dynamical Systems And Its Applications

In this thesis, we study the complexity of dynamical systems and the relationship between topological dynamics and combinatorial number theory. The thesis is orga-nized as follows.In the introduction, we briefly recall the history of the study of topological dynam-ics and ergodic theory, and some backgrounds of our study. Particularly, we summarize the recent progress in the study of chaotic phenomena and the connection between topological dynamics and combination number theory.In Chapter1, we introduce some basic notions and properties on topological dy-namics and ergodic theory which will be used in this thesis.In Chapter2, we introduce two useful tools in the study of topological dynamics: Furstenberg family and Ellis semigroup.In Chapter3, we recall the concept of chaos, and discuss the relationships among various kinds of chaos. The word "chaos" is used to describe the complexity of systems. Since there are different opinions on the complexity, various kinds of chaos appear in topological dynamics, such as Li-Yorke chaos, Devaney chaos, positive topological entropy and distributional chaos. A natural problem is that how to distinguish and find the relationships among various kinds of chaos. In this chapter, combining some recent work in this topic and some results of this thesis, we will discuss all those kinds of chaos in topological dynamics and their relationships.In Chapter4, we investigate the complexity of interval maps. In particular, we first point out the degree of multivariant chaos, by showing that every Li-Yorke chaotic interval map with zero topological entropy is Li-Yorke2-chaotic but not Li-Yorke3-chaotic. We also prove that for an interval map if it is topological null, then the maximal pattern entropy of every open cover is of polynomial order, answering a question by Huang and Ye when the space is the closed unit interval. Various chaotic properties and their relationships for interval maps are also discussed. For zero entropy interval maps, it is shown that the proximal relation is an equivalence relation and a pair is a sequence entropy pair if and only if it is f-nonseparable. We also get the structure of the set of f-nonseparable pairs and its relationship to Li-Yorke chaos. Moreover, some new equivalent conditions of positive topological entropy are obtained.In Chapter5, we will classify transitive systems by transitive points via Fursten-berg families, and show that a dynamical system is a weakly mixing E-system (resp. weakly mixing M-system, HY-system) if and only if it is{D-sets}-point transitive (resp.{central sets}-point transitive,{weakly thick sets}-point transitive). More specifically, assume that (X,T) is a topological dynamical system and (?) is a Furstenberg family (a collection of subsets of non-negative integers), a point x∈X is called an (?)-transitive point, if for every non-empty open subset U of X, the set{n∈Z+:T"x∈U}∈(?); the system (X,T) is called (?)-point transitive if there is some(?)-transitive point in X. Moreover, we show that every weakly mixing system is (?)ip-point transitive, while we construct an (?)ip-point transitive system which is not weakly mixing. We also apply the point transitivity to the study of disjointness and weak disjointness, and show that every transitive system with dense small periodic sets is disjoint from every totally minimal system and a system is Δ*{(?)wt)-transitive if and only if it is weakly disjoint from every P-system.In Chapter6, we will investigate the connection between topological dynamics and combinatorial number theory, and generalize a result of Furstenberg and Weiss that a kind of homogeneous system of linear diophantine equations can be solved in central sets. First, some basic properties of the Stone-Cech compactification J3N of the natural numbers N are discussed. And then we set up a general correspondence between the algebra properties of βN and the sets defined by dynamical properties, we also point out that the dynamical characterizations of quasi-central sets and D-sets, which were obtained several years before, are special cases of our result. We also investigate the set’s forcing that is the dynamical properties of a point along a subset of N. Moreover, we consider both addition and multiplication in N and βN. Particularly, we show that if F is a quasi-central set (resp. D-set, C-set) then for every n∈N both nF and n-1F are also quasi-central sets (resp. D-sets, C-sets). It is well known that C-sets are originally defined by the combinatorial method, we obtain a dynamical characterization of C-sets, and show that a kind of homogeneous system of linear diophantine equations can be solved in C-sets.