The Study Of Complexity And Recurrence In Topological Dynamical System

In this thesis,we mainly study the complexity and recurrence of the dynamical system.The thesis is organized as follows:In chapter 1,we briefly recall the development courses and main objectives of the topological dynamical system and ergodic theory.We also introduce the research background and main results of our study.In Chapter 2,we briefly introduce some basic definitions and properties of topolog-ical dynamical system and ergodic theory.We also discuss some concepts and propo-sitions of the thesis.In Chapter 3,we give some equivalent characterizations of equicontinuous systems in the mean sense,we extend some results which are valid under minimal conditions to the general case.We show that the dynamical system is mean equicontinuous if and only if it is equicontinuous in the mean if and only if it is Banach mean equicontinuous.Meanwhile,we also give the mean equicontinuous structure relation of the dynamical system,thus the mechanism of the maximal mean equicontinuous factor of the dynam-ical system is clarified.In chapter 4,we mainly describe the structure of null systems.Some results have been obtained under minimal conditions,we now focus on general topological null sys-tems.We prove that for null system,if it is distal,then it is equicontinuous;if it has a closed proximal relation,then it is mean equicontinuous.As a direct application,it follows that a null dynamical system with dense minimal points is also mean equicon-tinuous.In chapter 5,we study the problem of topological multiple recurrence of weakly mixing minimal systems for integer-valued generalized polynomials.We show that:Let(X,T)be a weakly mixing minimal system,and p1,…,pd be non-degenerate integer-valued generalized polynomials.Then there exists dense Gδ subset X0 of X such that for any x ∈ X0,{(Tp1(n)x,…,Tpd(n)x):n∈Z}is dense in Xd,which extending Huang,Shao and Ye’s work[47]on the validity of ratio-nal coefficients integer-valued polynomials to integer-valued generalized polynomials.In chapter 6,we mainly study the multiplicative combinatorial properties of the sets of return times of the Zl-system.We prove that for a minimal Zl-system(X,T1,...,Tl),there exists a residual subset X0 of X,such that for any x∈X0 and any non-empty open subset U of X,the return time set N(x,U)contains arbitrarily long geometric progressions(in Nl),extending Glasscock,Koutsogiannis and Richter’s work[38]to Zl-system.