Topological Entropy Of Subsets For Random Dynamical Systems

The research of topological entropy and its variational principle has become the essential topic in ergodic theory of dynamical systems.In this thesis,we mainly study pseudosemimetric entropy for a homeomorphism on a compact metric space and topological entropy of subsets of continuous random dynamical systems.The thesis is organized as follows:In Chapter 1,we introduce the relevant knowledge of topological entropy,measure-theoretic entropy formulas,random dynamical systems,free semigroup actions and main results of this thesis.In Chapter 2,for a homeomorphism on a compact metric space,we consider the Bowen metric under pseudosemimetric,and define pseudosemimetric entropies by using spanning sets and separated sets respectively.We show that pseudosemimetric entropies are equivalent to topological entropy,and we obtain the corresponding Katok's entropy formula and Brin-Katok's entropy formula.In Chapter 3,under the actions of continuous random dynamical systems,we consider a kind of weighted probability measure model,and define the corresponding measure-theoretical upper,lower entropies and Bowen topological entropy of subsets,and show the relationships between them.We establish a variational principle between measure-theoretical lower entropy and Bowen topological entropy of subsets for random version,and give an application to free semigroup actions.