Research On Several Problems In Entropy Theory Of Dynamical System

In this thesis,we mainly study two problems,one is the relationship of Bowen dimension entropy and measure-theoretic lower local entropy under the maximum mean metric,the other is the Katok entropy formula of the measure-theoretic r-entropy for amenable group actions.The following is the specific arrangement:In the chapter one,we introduce the backdrop and categorization of dynamical system and the evolution of entropy theory.In addition,we introduce some developments of entropy theory under the mean metric and dynamical system for amenable group actions as the background of this thesis.In the chapter two,we introduce some basic knowledge of the dynamical system,entropy theory and the dynamical system for amenable group actions.In the chapter three,we give the definitions of Bowen dimension entropy and measure-theoretic lower local entropy under the maximum mean metric,prove a covering lemma under the maximum mean metric,and then use this lemma to prove the Billingsley properties of Bowen dimension entropy and measure-theoretic lower local entropy under the maximum mean metric.In the chapter four,we give the definition of measure-theoretic r-entropy for amenable group actions,and the Katok entropy formula of measure-theoretic r-entropy for amenable group actions.In the chapter five,we make a concise summary of the results of this thesis,and make further foreground for the research issues.