Complexity,Rigidity And Relative Uniform Positive Entropy In Dynamical Systems

In this thesis,we study the complexity and rigidity of Z-systems,the directional complexity and rigidity of Zq-measure preserving systems(q≥2),and the dynamics and topological connection of factor map between G-systems and induced factor map,where G is a countably infinite discrete amenable group.The thesis is divided into five chapters as follows:In Chapter 1,we recall a brief history and main topics of topological dynamical sytems and ergodic theory.And the motivation and results of our study are given.In Chapter 2,we recall some preliminaries and known results of topological dynamical system and ergodic theory,which will be used in the rest of the thesis.In Chapter 3,for Z-actions,we study measure rigidity,measure complexity and formulas of entropy.For a given Z-system(X,T)and a T-invariant Borel probability measure ρ,we introduce two metrics:the max metric dn,q and the mean metric dn,q.By these two metrics,we give an equivalence of rigid measure-preserving systems,i.e.,the following statements are equivalent:(1)the measure-preserving system(X,BX,ρ,T)is measure rigid;(2)the invariant measure ρ has bounded complexity corresponding to dn,q;(3)the invariant measure ρ has bounded complexity corresponding to dn,q.Additionally,we obtain formulas for the measure-theoretic entropy of an ergodic Zmeasure preserving system(resp.the topological entropy of a Z-system)in these two metrics dn,q and d,q respectively.In Chapter 4,for a given Zq measure-preserving system and a non-zero real direction,we study measure rigidity,directional measure complexity,and the ergodic decomposition formula of the measure-theoretical directional entropy.We introduce directional complexity for Zq measure-preserving dynamical systems via a collection of new metrics along non-zero directions in Rq.It turns out that a Zq measure-preserving dynamical system is measure rigid if and only if the invariant measure has bounded directional complexity.We also obtain ergodic decomposition formula for the measuretheoretical directional entropy of a Zq measure-preserving dynamical system.In Chapter 5,let G be a countably infinite discrete amenable group,we consider the dynamical properties and topological connection of the factor map between G-systems and induced factor map.Note that a G-system(X,G)naturally induces a G-system(M(X),G),where M(X)denotes the space of Borel probability measures on a compact metric space X endowed with the weak*-topology.A factor map π:(X,G)→(Y,G)between two G-systems induces a factor map π:(M(X),G)→(M(Y),G).By building the relative connection between original systems and induced systems,we prove that π is open if and only if π is open.When Y is fully supported,it is shown that πhas relative uniformly positive entropy if and only if π has relative uniformly positive entropy.