The Research On Topological Entropies Of Free Semigroup Actions And Nonautonomous Dynamical Systems

In topological dynamical systems,topological entropy,as an important invariant of topological conjugacy,depicts the degree of chaos of a system.Since Bowen defined topological entropy for noncompact sets of autonomous discrete dynamical systems in a way which resembles Hausdorff dimension,Bowen topological entropy has been widely concerned by scholars.Furthermore,the topological entropies for noncompact sets of other dynamical systems are discussed.Compared with autonomous discrete dynamical systems,the research on topological entropies for noncompact sets of free semigroup actions and nonautonomous discrete dynamical systems is relatively less.At the same time,as a natural generalization of free semigroup actions and nonautonomous discrete dynamical systems,nonautonomous iterated function systems are more common in practical applications.The study of topological entropy of nonautonomous iterated function systems can reveal the common of dynamical characteristics of topological entropy between free semigroup actions and nonautonomous discrete dynamical systems.Based on the framework of C-P structure,this dissertation defines topological entropies for noncompact sets of free semigroup actions and nonautonomous discrete dynamical systems.Furthermore,we give the definition of measure entropy of nonautonomous iterated function systems based on measure entropies of free semigroup actions and nonautonomous discrete dynamical systems,and prove the partial variational principle between this kind of measure entropy and corresponding topological entropy of nonautonomous iterated function systems.The detailed research work is as follows:In Chapter 1,we summarize the development process and research significance of entropies in dynamical systems.The research status and progress of topological entropies and measure entropies of several important dynamical systems are introduced.The concepts,definitions and propositions related to the research work of this paper are briefly described.In Chapter 2,we study topological entropies of free semigroup actions in the sense of Bufetov for noncompact sets.By using C-P structure,we give the definitions of the topological entropy and upper(lower)capacity topological entropy of a free semigroup action,respectively.A sufficient condition for equality of topological entropy and upper capacity topological entropy for compact invariant sets is obtained.An equivalent definition of topological entropy and the definition of L+(L-)lower local entropy are given by Bowen balls.Furthermore,the upper(lower)bound of topological entropy of free semigroup actions for Borel subsets is estimated.Then,the relationship between the upper capacity topological entropies of a free semigroup action and a skew-product transformation is studied.Another equivalent definition of topological entropy is given by Bowen's approach.When the free semigroup action is generated by m generators of Lipschitz maps,we study the relation between topological entropy and Hausdorff dimension of any subset.In Chapter 3,we study Pesin topological entropies of nonautonomous discrete dynamical systems for noncompact sets.Pesin topological entropy and upper(lower)Pesin topological entropy of nonautonomous discrete dynamical systems are defined by C-P structure,respectively.We estimate the upper and lower bounds of Pesin topological entropy for Borel subsets by lower local entropy of nonautonomous discrete dynamical systems.Then,the dynamical properties of Pesin topological entropy are studied,including monotonicity,power rule and semiconjugate formula of Pesin topological entropy.When the nonautonomous discrete dynamical system is composed of Lipschitz or locally expanding maps,the relation between Pesin topological entropy and Hausdorff dimension,and the relation between upper(lower)Pesin topological entropy and upper(lower)box dimension of this kind dynamical systems are obtained.In Chapter 4,we study the measure entropy and topological entropy of nonautonomous iterated function systems.The properties of topological entropy of nonautonomous iterated function systems are discussed,and an example of calculating the topological entropy is given.Then,the relation between the topological entropies of nonautonomous iterated function systems and the corresponding skew product transformations is established.At the same time,a kind of measure entropy of nonautonomous iterated function systems is defined and its dynamical characteristics are studied.The partial variational principle between topological entropy and measure entropy is proved.