Research On Entropy Of Dynamical Systems And It's Application

In this thesis, we study the so called local theory of entropy (including sequence entropy and complexity function) both in topological and measure-theoretical settings. We introduce new notions of localizations, which can be applied to the study of dynamical properties, structures and classifications of dynamical systems. We emphasize the similarities and the dissimilarities of entropy, sequence entropy and complexity function.The paper is organized as follows:In the first charpter, the origin, developments and main contents of the topological dynamical system and ergodic theory are presented, then the background, status and applications of dynamical entropy, especially of the localizations of dynamical entropy are introduced. In Charpter 2, preliminaries on topological dynamical system and ergodic theorey are recomendated.In Charpter 3, the notions of entropy sequences and entropy sets are introduced both in topological and measure-theoretical situations. It turns out that a subset is an entropy set if and only if each tuple (not on the diagonal) from the set is an entropy tuple, and that each maximal entropy set is closed. The systems with only one maximal entropy set are characterized. Moreover, it is proved that if a topological system has positive entropy, then there is a maximal entropy set with uncountably many points, and the topological entropy is the supremum of the entropies over all maximal entropy sets, which shows that in some sense the complexity of a system can be focused on it's entropy sets. At last, an example with a maximal entropy set containing only two points and a transitive but not topological K example with a unique maximal entropy set are constructed.In Charpter 4, notions and resultes on the localizations of sequence entropy are introduced. Then a new theorem involving disjointness via sequence entropy tuples is obtained, which can be used to show that weakly mixing systems are disjoint with the Morse minimal system. At last, some properties of sequence entropy sets are presented.In Charpter 5, null flows are investigated. Sequence entropy, sequence entropy pair and weakly mixing pair are introduced. We prove that a minimal null flow is an almost...