Topological Mixing Of General Symbolic Dynamical Systems Subshift

Symbolic dynamical system is a powerful tool in the study of dynamical systems theories, the study of dynamical properties of the general symbolic dynamical system is a far unsolved problem. The topological mixing is a very complex property of dy-namical systems. A system with topological mixing property has many chaotic proper-ties in different senses. This paper mainly discusses the dynamical properties of shift map a on the space (Z+ ) of infinite sequences comprises sequencable infinite num-ber of symbols. It gives that ((Z+)) is topological mixing and emphasizes to dis-cuss the subshift determined by matrix A of order infinity, gives necessary and suffi-cient conditions for the subshift ( A(Z+), A) to be topological mixing. In addition,constructs a chaotic set in (Z+)\ U(K) in the sense of Li-Yorke.This paper consists of five parts.In the introduction, we briefly introduce the history of symbolic dynamics development, the research of the symbolic dynamical system of an finite number of sym-bols, the research of the general symbolic dynamical system, the research of chaotic properties of shift and subshift of symbolic dynamical system and lists main conclusions in the paper.In section 2, we introduce some basic notions and their relations, such as: Li -Yorke chaos, Devaney chaos, Xiong Jincheng chaos, topological mixing, and so on. Moreover, introduce some concepts: one - sided symbolic dynamical system, shift and subshift, etc.The section 3 and 4 are the main part of this paper, we prove necessary and sufficient conditions for the subshift (A. (Z+),A) to be topological mixing and con-struct a chaotic set in (Z+) \ U (K) in the sense of Li - Yorke.Finally, we point out some unsolved problems.