?-weakly Mixing Subsets And Mean Li-Yorke Chaos In Dynamical Systems With Positive Entropy

In this paper,based on some resent results of recurrence and multiple ergodic av-erage,we study the existence of ?-weakly mixing set and mean Li-Yorke chaos along integer polynomials for group actions with positive topological entropy.The thesis is divided into five chapters as follows:In Chapter 1,we firstly recall a brief history and main topics of topological dy-namical sytems and ergodic theory.And then the moivation and results of our study are given.In Chapter 2,some preliminaries and known results in topological dynamical sys-tem are reviewed for later use.In Chapter 3,for countable torsion-free discrete group actions,the relationships between entropy and the existence of ?-weakly mixing sets are investigated.More precisely,the notions and basic properties of ?-weakly mixing subsets for countable torsion-free discrete group actions are introduced.It is shown that for a finitely gen-erated torsion-free discrete nilpotent group action,positive topological entropy implies the existence of ?-weakly mixing subsets.However,by modifying the example intro-duced by Furstenberg,we construct a finitely generated torsion-free discrete solvable group action,which has positive topological entropy but without any ?-weakly mixing subsets.Moreover,we give an equivalent characterization of ?-weakly mixing sub-sets,which hleps us to define the asynchronous chaotic behavior of ?-weakly mixing subsets.By combing above results,it is shown that for a finitely generated torsion-free discrete nilpotent group action,positive toological entropy implies asynchronous chaos.In chapter 4,for Z-actions with positive topological entropy,we study the exis-tence of ?-weakly mixing subsets along some collections of sequences of integers.For collections of sequence of integers,we propose a mild condition,named Condition(**),and show that the Pinsker ?-algebra of any Z-measure preserving system is a character-istic ?-algebra averaging along collections of sequences of integers satisfying Condi-tion(**).In the meantime,we introduce the notion of ?-weakly mixing subsets along a collection of sequences of integers and prove that Z-dynamical systems with positive topological entropy have ?-weakly mixing subsets along collections of sequences of integers with "good" properties.As a consequence,we show that for Z-actions positive topological entropy implies multi-variant Li-Yorke chaos along polynomial times of the shift prime numbers.In Chapter 5,we consider the mean Li-Yorke chaos along non-constant integer polynomials in Z-dynamical systems.By analyzing the limiting behavior of ergodic averages of bounded functions along positive non-constant integer polynomials of sev-eral variables and disintegrating the given measur in an appropriate way,we prove that for Z-actions positive topological entropy implies mean Li-Yorke chaos along positive(or negative)non-constant polynomials on integers or primes.