Some Studies On Recurrence And Ergodicity

Recurrence and ergodicity are two important topics in topological dynamical sys-tems and ergodic theory.In this thesis,we mainly study distality,product recurrence,the mean ergodic theorems and multiple recurrence theorems.This thesis is divided into five chapters.In Chapter 1,the background and main results are stated.In Chapter 2,we generalize Furstenberg's realization theorem of IP-set from the positive integers to arbitrary infinite group and countable-infinite monoid satisfying suitable cancellation laws,and also generalize Galvin's theorem from the Stone-Cech compactification of a discrete semigroup to a compact Hausdorff right-topological semi-group with a dense topological center.At last we apply them to characterize distality,IP*-recurrence and product(IP-)recurrence of a flow with a compact Hausdorff phase space and phase semigroup.In Chapter 3,we mainly extend the classical mean ergodic theorem to a quasi-topological vector space over a general field of characteristic zero.We show that the mean ergodic theorem for arbitrary ergodic net in a quasi-topological vector space holds under an assumption that there exists a family of continuous additive homomorphisms from this space to a topological field which separates points and closed Q-subspaces.Moreover,we give a counterexample that without this assumption,the mean ergodic theorem fails.In Chapter 4,by using some "lifting" techniques,we prove that for measurable preserving set-valued maps,the Khintchine's recurrence theorem,and Furstenberg's multiple recurrence theorem still hold on some special "invariant" subsets with positive measure.And we also extend the multiple Birkhoff recurrence theorem from continuous transformations to upper semicontinuous set-valued maps with closed nonempty values.In Chapter 5,we discuss fiber bundle extensions(a generalization of cocycle ex-tensions),and prove that under some additional conditions,fiber bundle extensions can lift recurrence,uniformly recurrence and distality.At last we describe the intimate correspondence between fiber bundle extensions and coordinate cocycles.