Bowen Entropy And Weighted Average Dimension In The Dynamic System

The thesis is divided into three parts,mainly studying entropy and mean dimen-sion in dynamical systems.Firstly,we investigate the weighted entropy and mean dimension,generalizing Lindenstrauss and Weiss’s result to the weighted mean di-mension.Secondly,it is proved that the positive entropy implies the Li-Yorke mean chaos on the random dynamical system.Thirdly,we establish the Brin-Katok’s local entropy formula for fixed-point free flows by reparametrization balls and show the Bowen entropy of the whole compact space is equal to the topological entropy.The paper is organized as follows:In Chapter 1,we introduce the backgrounds of entropy,mean dimension in dy-namical systems.In Chapter 2,we define the weignted mean dimension,study some properties of it and proved weignted topological mean dimension is not greater than the weighted metric mean dimension.In Chapter 3,It is proved that positive entropy implies Li-Yorke mean chaos on the random dynamical system.In Chapter 4,we establish the Brin-Katok’s local entropy formula for fixed-point free flows and show the equivalence between topological entropy defined by reparametrization balls and classical topological entropy.