拓扑动力系统的拓扑传递和拓扑弱混合的局部化

The transitive set and weakly mixing set are local concepts of topological transitivity and weak mixing, and provide a theoretical tool for studying local aspects of recurrence of topological dynamical systems.On the basis of preliminary results,it is the transitive set and the weakly mixing set that are mainly resear- ched, the relationships among transitive sets and weakly mixing sets and chaos are investigated, the conditions are given that a n-weakly mixing set is a n+1-weakly mixing set,the conditions are discussed that a transitive is a weakly mixing set in one dimension dynamical system and the properties of transitive sets and weakly mixing sets are investigated in induced set-valued discrete systems.Through deeply studying topological properties and topological structures of transitive sets and weakly mixing sets, the results are obtained that a system has a transitive set implies it is chaos, a n-weakly mixing set is a n+1-weakly mixing set and induce set-valued discrete system has a transitive set implies original system has a weakly mixing set. The research of this project will enrich the properties of transitive set and weakly mixing set and improve the dynamical behavior of set-valued discrete system.

传递集和弱混合集是拓扑传递和弱混合的局部化概念，为研究动力系统回复性局部方面的性质提供了理论依据。目前，对拓扑传递和弱混合的研究已有大量的成果，而对它们的局部化研究较少。本项目拟在前期成果的研究基础上，以传递集和弱混合集为主要研究目标，重点探索传递集和弱混合集与混沌的关系、n阶弱混合集是n+1阶弱混合集的条件、区间上连续映射传递集为弱混合集以及诱导的集值离散系统的传递集和弱混合集的性质等研究课题；通过深入研究传递集和弱混合集的拓扑性质和拓扑结构，以达到动力系统有传递集蕴含系统是混沌的，n阶弱混合集是n+1阶弱混合集以及诱导集值离散系统有传递集蕴含原系统有弱混合集等目的。此项目研究将丰富传递集和弱混合集的性质以及完善集值离散系统的动力学行为。

本项目在前期成果研究的基础上，以传递集和弱混合集为主要研究目标，重点探索了传递集和弱混合集与混沌的关系、n阶弱混合集是n+1阶弱混合集的条件、区间上连续映射传递集为弱混合集以及诱导的集值离散系统的传递集和弱混合集的性质等研究课题；通过深入研究传递集和弱混合集的拓扑性质和拓扑结构，得到动力系统有传递集蕴含系统是混沌的，n阶弱混合集是n+1阶弱混合集以及诱导集值离散系统有传递集蕴含原系统有弱混合集等目的。此项目的研究丰富了传递集和弱混合集的性质以及完善集值离散系统的动力学行为。