Topological Sequence Entropy Change Principle And Measure Brin-Katok Formula R- Entropy

This thesis investigates some problems about the topological se-quence entropy, measure-theoretic r-entropy and topological pressure in topological dynamical systems.In the first part of the thesis, we will introduce several types of topological sequence entropies of subsets, such as upper capacity topological sequence entropy, Bowen topological sequence entropy, weighted topological sequence entropy and packing topological se-quence entropy, and we define measure-theoretic sequence entropy by the idea of Brin and Katok. We establish the variational princi-ples for Bowen topological sequence entropy and packing topological sequence entropy of subsets.In the second part, inspired by the idea of Brin and Katok, we define the measure-theoretic r-entropy by a local view. And we show that the limit of measure-theoretic r-entropy is equal to the measure-theoretic entropy as r γâ†'0.In the third part, we introduce measure-theoretic pressure, topo-logical pressure [38], packing topological pressure and upper capacity topological pressure, and we get the product theorem for topological pressure [38].The paper is organized as follows:In chapter1, we mainly review development of the topological entropy and topological pressure, and we give the main results of this paper.In chapter2, we recall some classical notions in topological dy-namical systems and ergodic theory.In chapter3, we define measure-theoretic sequence entropy and we establish the variational principles for Bowen topological sequence entropy and packing topological sequence entropy of subsets.In chapter4, we define the measure-theoretic r-entropy by a local view and construct the Brin-Katok formula for measure-theoretic r- entropy.In chapter5, we introduce some types of topological pressures and get the product theorem for topological pressure [38].