On Definitions of Chaos in Discrete Dynamical System

By a topological dynamical system (TDS for short) (X,f),we mean a compact metric space X together with a surjective continuous map f:X rarr X.The chaoticity of a TDS is a centrum topic of the research since the introducing the term of chaos in 1975 by Li and York, known as Li-York chaos today, and notions of sensitivity is the kernel in the definition of chaos in the Ruelle-Takens chaos and Schweizer-Smital chaos (also called distributional chaos). Now chaos has been one of the main research subjects concerned by scientists in all fields. So it is very meaningful to discuss the relations between the various notions of chaos. In this paper, using the method of construction and combining with the symbolic space, we study the relations between distributional chaos in a sequence, distributional chaos and Ruelle-Takens chaos, and prove that they are not equivalent to each other in the compact system (X,f).