Topology Of The Chaotic Properties Of Dynamical Systems In Several Parts

The relation and the dynamical properties of several partial chaos proper-ties in topological dynamical systems are systematically explored, the paper is organized as follows:In chapter 1, the background, status and applications of chaos theory, tran-sitivity, weakly mixing, sensitivity and hyperspace are recommended, then, the ideas of investigating problems are introduced and the main content of this paper is presented.In chapter 2 and 3, motivated by ideas of F. Blandchard and W. Huang, the concepts of transitive subsets, partial transitive property, initial value-sensitive subsets and partial sensitive-dependence on initial value are introduced. Firstly, fundamental properties of transitive subsets as a kind of partial properties are investigated, and we give three equivalent conditions of transitive subsets, two necessary and sufficient conditions for a regular closed subset A to be weakly mixing are established, the main result of the chapter 2 is the equivalence among partial transitivity of the induced hyperspace topological dynamical system, par-tial weak-mixing of the introduced hyperspace dynamical system, and the partial weak-mixing of the underlying dynamical system under the condition that A is a regular closed subset. Finally, the relation between transitive subsets and initial value-sensitive subsets is explored, and we proved that if A is a transi-tive and infinite subset of (X, d,f) and P(f) is dense in A, then A is an initial value-sensitive set of (X, d,f).In chapter 4, the concept of Devaney chaos subset is introduced, Under the condition that A is a weakly mixing subset of (X, d,f), it was proved that if A is a subset of Devaney chaos of (X,d,f), then (2X,dH,2f) is partial Devaney chaos property, Otherwise, if 2A is a subset of Devaney Chaos of (2X,dH,2f), then A is a weakly mixing subset of (X, d,f).