Research On Some Problems In Entropy And Chaos Of Dynamic Systems For Group Actions

Chaos theory is a basic subject,which is used to study nonlinear systems.As an important condition for the definition of Devaney chaos,sensitivity has been widely concerned by experts and scholars.In recent years,the researches on sensitivity have been very mature and make a lot of progress.Since entropy and sensitivity are both important invariants to describe the complexity of dynamical systems,many scientists turn their attention to the relationship between entropy and various sensitivities of dynamical systems,and obtain many important results.Most of these researches are focus on the dynamical systems for Z-actions.In this thesis,we let the acting group of a topological dynamical system be Zd actions and introduce the definition of restricted sensitivity,and study its properties.The thesis is organized an follows:In the first part,some basic notions and theorems are introduced.In the second part,we introduce the definitions of restricted sensitivity and restricted pairwise sensitivity of measure-preserving dynamical system for Zd-actions,and show that restricted pairwise sensitivity is a stronger notion than restricted sensitivity when the metric is regular.Moreover,we obtain the relationship between restricted sensitivity,restricted pairwise sensitivity and entropy.In the third part,we introduce the definition of the restricted mean sensitivity of measure-preserving dynamical system for Z-actions,and obtain the relationship between it and entropy.