Time-Restricted Sensitivity And Entropy For Topological Dynamical Systems

In this thesis,we study the correlation of two concepts in topological dynamical systems:the time-restricted sensitivity and the entropy.The theoretical basis is the Brin-Katok local entropy formula and the variational principle of entropy.The thesis is divided into four chapters as follows:In chapter 1,combined with the development process of topological dynamical systems and ergodic theory,we explain the background of the research theme and introduce the main conclusions of this thesis.In chapter 2,we briefly introduce the basic concepts of topological dynamical systems and ergodic theory at first.Then,we mainly introduce the conditional measure,conditional entropy and the related knowledge of the topological dynamical system with infinity countable discrete amenable group action.In Chapter 3,according to the relative Brin-Katok local entropy formula,we construct the inherent relationship between relative time-restricted sensitivity and conditional entropy.Firstly,we introduce the defination of relative time-restricted asymptotic rate.Secondly,the properties of relative time-restricted asymptotic rate are analyzed.At last,we give a specific characterization of the relative time-restricted sensitivity.Then the quantitative relationship between this asymptotic rate and conditional entropy is acquired through the relative Brin-Katok local entropy formula.In Chapter 4,for infinity countable discrete amenable group actions,we give the concepts of the measure-theoretic restricted sensitivity and the topological restricted sensitivity.And their relationship with the local entropy and the topological entropy are explored,respectively.Specifically,for the measure-theoretic restricted sensitivity,we use the method similar to that in Chapter 3 to find the relationship between the asymptotic rate of measure-theoretic restricted sensitivity and the local entropy by using the local entropy formula of the infinity countable discrete amenable group actions.We use the variational principle of entropy to find the corresponding relationship between the topological restricted asymptotic rate and the topological entropy.