Ergodic Theory And Furstenberg Families

In this thesis we aim to establish a corresponding theory of Furstenberg's family for measurable dynamical systems (MDS) based on the similarities between ergodic theory and topological dynamics, especially on the characterization and classification of ergodic properties and transitive properties. The theory of family has been applied to study the complexities of topological dynamical systems (TDS) successfully in the recent years. We emphasize the similarities and dissimilarities of the application of the theory of family in MDS and TDS. The paper is organized as follows:In the introduction the origin and main contents of the topological dynamical system and ergodic theory are presented. We retrospect the history of their developments, and we mainly focus on the study of the recurrence properties, complexities and classifications of the systems. In the first chapter we introduce the basic notions and results of topological dynamics and ergodic theory, including some tools and manners in the study of dynamics.In the second chapter we use the return times set to characterize several mixing properties in MDS. Particularly, we show that a measure preserving transformation T is weakly mixing iff the return times set of T is a topological recurrence set iff the lower Banach density of the return times set of T is 1; T is mildly mixing iff the return times set of T is IP^*-set; T is intermixing iff the return times set of T is cofinite.In chapter 3 we introduce the basic notions and properties of Furstenberg families including the Ramsey property and the filters. We then give the definitions of some familiar families and show their relations with dynamics. Finally we summarize its applications in characterizing the transitive properties of TDS and the ergodic properties of MDS, from which one can find out the similarities.In chapter 4, for a family F and a MDS (X, B,μ, T), several notions of ergodicity related to F are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is F-convergency ergodic of order k if for any A₀,…,A_k of positive measure, 0 = e₀<…< e_k and e>0, {n∈Z₊:|μ(∩_i=0^k T^－ne_iA_i)－∏_i=0^kμ(A_i)|*-convergcncy ergodic of order 1: (2) T is strongly mixing; (3) T isΔ^*-convergency ergodic of order 2: whereΔ^* is the dual family of the family of difference sets. At the same time, it is shown that for any MDS (X,B,μ,T), any A∈B with positive measure and e>0, {n∈Z₊:μ(A∩T^－nA)>μ(A)²ï¼e}∈Δ^* which strengthens the well known Khintchine's result [75]. Moreover, using the results of this chapter and chapter 2 we answer some questions raised by V. Bergelson and T. Downarowicz [9].In the last chapter we introduce the notions of rank one system and the main method to construct such a system: cutting and stacking. Using the method wo recall a rank one system whicdi is first given by Chacon, and consider its ergodic properties. The system is intermixing [30]. We show that it is notΔ^*-uniformly positively ergodic and not weakly partially mixing. We also show for this system, there doesn't exist a subset J of N with lower Banach density 1 such that for any measurable sets A, B of the system satisfying lim_{J(?)n→∞}μ(A∩TⁿB)=μ(A)μ(B).