Some Dynamical Properties In Topological Dynamics

Dynamical properties such as recurrence and complexity are always important topics in topological dynamics.In this thesis,we characterizes dynamical behaviors of systems via some classic properties including equicontinuity,distality,almost periodicity,almost automorphicity and topological complexity.The thesis is divided into five chapters.In the first chapter,we give some preliminary notions and properties,and then introduce our main results.In the second chapter,we mainly investigate the relations among equicontinuity,uniform-ly almost periodicity,and regionally proximal relation in topological semiflows.Let(?,T,X)be a topological semiflows on a compact Hausdorff space X,where T is any discrete monoid with ?:T ×X?X being the phase map of(T,X)and each of its transition maps ?t being a surjection of X.We prove that(?,T,X)is equicontinuous if and only if it is uniformly almost periodic,and if and only if its regionally proximal relation is equal to the diagonal?x={(x,x):x ? X}.In the third chapter,we mainly concern about the lifting of locally almost periodicity of minimal semiflows.Based on the lifting,we use a different method from Sacker and Sell's to get the property of lifting of equicontinuity for semiflows.Moreover,in the case of a group action,we give answers to questions about lifting of almost automorphy for minimal flows posed by Sell,Shen and Yi[Math.Contemp.,215(1998),279-298].In the fourth chapter,we characterize the distality of a point through Furstenberg families of cental sets.Firstly,we show that a point is distal if and only if it is Fc-product recurrent(i.e.the pair formed by this point and a Fc-recurrent point in any system is recurrent in the product system),and thus get the equivalence of Finf-PR,Fps-PR and Fc-PR.Secondly,we show that a distal point is also a product Fc-recurrent point(i.e.the pair combined by this point and a Fc-recurrent point in any system is Fc-recurrent in the product system).These results generalize parts of Oprocha and Zhang's work[Adv.Math.244(2013),395-412].In the last chapter,we study the topological complexities of entropy zero systems for countable-infinite amenable group actions.Firstly,for a given F?lner sequence {Fn}n=0+?,we define respectively the entropy dimensions and the dimensions of generating sequences to characterize the sub-exponential growth of the topological complexity.Meanwhile,we investigate the relations among them.Secondly,we introduce the notion of a dimension set.Moreover,using it,we discuss the disjointness between the entropy zero systems which generalizes the results of Dou,Huang and Park[Trans.Amer.Math.Soc.363(2)(2011),659-680].