Several Types Of Chaotic Set In Generalized Symbolic Dynamical Systems

In the dynamics, the research of chaos comes from the discovery of chaotic phenomenon. Li—Yorke provided exact definition of chaos for the first time in1975. According different determinant rules, people give different notions about chaos and study them. In the study of dynamical system, symbolic dynamical systems becomes the powerful tool of studying dynamical system. People have found various kinds of chaotic set in symbolic dynamical systems. In this paper we will study the chaos of generalized symbolic dynamical systems, we found a uncountable distributionally chaotic set, a transitive and invariant Li—Yorke chaotic set and a uncountable ω-chaotic set in S C E(Z+)\(∪∑(K).This paper is composed of four chapters. The first chapter introduces the research progress of generalized symbolic dynamical systems and prepare knowledge including some common definition of chaos, some notions and properties in symbolic dynamical systems and generalized symbolic dynamical systems.In the second chapter, we construct a uncountable distributionally chaotic set S in generalized symbolic dynamical systems, and the set S is outside of∪∑(K), that is S C E(Z+)\U∑(K)In the third chapter, we construct a uncountable Li-Yorke chaotic set in generalized symbolic dynamical systems (E(Z+),σ) and prove the set is transitive and invariant, and the Li-Yorke chaotic set D (？)∑(Z+)\U∑(K).In the fourth chapter, we construct a uncountable ω-chaotic set in generalized symbolic dynamical systems (E(Z+),σ), and prove the ω-chaotic set S’ C E(Z+)\U∑(K).