A Continuous Map Of Topological Space Of Generalized Periodic Points And Chaotic State Study

In 1975, Li and Yorke considered an one-dimensional continuous map and obtained the well-known result, that is,"period 3 implies chaos"[7]. They first introduced a precise mathematic definition of chaos on a real interval. In 1987[8], Zhou Zuoling predigested the conditions of Li-Yorke's chaos and generalized the definition of Li-Yorke's chaos to a compact metric space. In 2002, Huang and Ye gave some equipollences of Li-Yorke's chaos and establish some criteria of chaos on compact metric spaces[13]. Therefore, it is naturally asked:Question: If can Li-Yorke's chaos be generalized from a metric space to a topological space?In this paper, we mainly discuss the above question and we have obtained the following results:1.We generalized some fundamental properties such as tanstivity, minimality and mixing property of topological dynamical systems base on compact metric spaces to countable spaces and sequentially compact spaces and establish equipollences of tanstivity, minimality and mixing property on these spaces.2.We study the generalized periodic points such as recurrent points, non-wonderi -ng points,ω-limit points and chain-recurrent points of continuous self-mappings on countable spaces and sequentially compact spaces, and we prove that: Let f be a continuous self-mapping on a sequentially space X , thenω-limit set of a point is a nonempty finite set iff it is a periodic orbit of f . If X is locally-connected and for every connected subset A in X , A ? A is finite, then every non-wandering point is a chain recurrent point. If f is homeomorphic on X , then the set of chain recurrent points of f is strongly invariant. For each natural number n , every chain recurrent point of f n is a chain recurrent point of f .3. Asymptotic pairs and proximal pairs are generalized toa first countable topological space.Therefore, Li-Yorke's chaos is naturally generalized to a topological space that is first countable.4.Based on the above generalizations, we prove that: Let X be a Baire space that is second countable and without isolate points, if ( X ,f) is a transitive system with a fixed point and the set of the asymptotic points is first category for any transitive points, then ( X ,f) is densely chaotic in the sense of Li-Yorke. This result can be used as criteria of Li-Yorke's chaos on topogical spaces. Notice that both the compact Hausdorff space and the complete metric space are Baire spaces. Thus, the existing results are extended an improved.Above all, we summarize the innovations and the subsequent works of our paper and give some advices to who want to work on this subject.