Asymptotic Pairs,Stable Sets And Chaos In Positive Entropy Systems

We consider positive entropy G-systems for certain countable, discrete, in?nite left-orderable amenable groups G. By undertaking local analysis, the existence of asymptotic pairs and chaotic sets will be studied in connecting with the stable sets.Examples are given for the case of integer lattice groups, the Heizenberg group, and the groups of integral unipotent upper triangular matrices.The thesis is organized as follows:In Chapter 1, we brie?y describe the development and the main contents of dynamical systems(in particular, topological dynamics and ergodic theory). We also have a brief review of the stability and the chaos theory, and gradually introduce the background and main contents in our text.In Chapter 2, we begin with a brief introduction of topological dynamics and ergodic theory. Next, for some dynamical systems for amenable group, we introduce some classical de?nition of topological entropy and metric entropy as well as their basic properties. Finally, we brie?y review the ergodic decomposition and the variational principle of entropy.In Chapter 3, we investigate dynamical systems under the action of a countable,discrete, in?nite amenable group with algebraic past. For such dynamical systems,we give the Pinsker formula of entropy, introduce the notion of Pinsker σ-algebra and discuss its basic properties.In Chapter 4, our main purpose is to prove Theorem 1 and Theorem 2. We?rstly review the de?nition of asymptoticity and Li-Yorke pairs for discrete dynamical systems, and expand these conceptions to some more general groups. Next, we use local analysis method to study dynamical systems for a class of amenable groups with algebra past. We show that these systems always have a relative ”big” stable set in the positive position, and the stable set re?ect some chaotic phenomenas(including Li-Yorke chaotic set of Cantor type) in the opposite position.In Chapter 5, we will construct counter-examples in a large class of amenable groups, in order to explain the existence of dynamical system with positive entropy which has no proper asymptotic pairs. The counter-example give us some guidance to study the stability and chaos for general dynamical systems. Speci?cally, in the?rst section, we construct a skew-product dynamical system for Z which has no proper Z-asymptotic pairs. Then, in the second section, we extend the construction to these more general groups with a subgroup Z.