几类广义周期点在拓扑空间的推广及其相关性质

【作者】 张云；

【导师】 朱培勇；

【作者基本信息】 电子科技大学 ， 应用数学， 2008， 硕士

【摘要】 连续自映射的回归点、非游荡点、链回归点等广义周期点是拓扑动力系统的重要研究内容之一。近三十年来,国内外众多学者对此都非常感兴趣并且一直积极投入,他们在实线段甚至度量空间上对这些广义周期点进行了许多深入地讨论,得到了许多重要的研究成果.然而,随着现代动力系统的研究不断向高维空间和抽象空间发展,自然产生如下两个问题:(1)如何将拓扑动力系统中各种广义周期点及其相关理论推广到抽象的拓扑空间?(2)推广后的广义周期点与广义周期点集有何特有的重要性质?本文主要就上述两问题进行讨论,得到了如下一些结果:首先,对实线段上连续自映射的周期点、回归点和非游荡点进行了推广。在一般拓扑空间中,获得了连续自映射的周期点集、回归点集和非游荡点集的一些性质,并进一步证明了这些性质的正确性。其次,推广了实线段上连续自映射的不稳定流形。在一般拓扑空间中,获得了连续自映射的不稳定流形的一些性质并且证明了这些性质的正确性,同时也讨论了不稳定流形与周期点、ω-极限点、非游荡点和同宿点之间的关系。最后主要是将实线段上连续自映射的链回归点和ω-极限点推广到度量空间(特殊的拓扑空间)中,在一般度量空间或者紧度量空间中,获得了链回归点和ω-极限点的一些性质,并且证明了这些性质的正确性。上述结果,丰富和推广了拓扑动力系统中广义周期点的基本理论。在一定程度上为动力系统理论的抽象化发展以及混沌数学理论的拓扑推广奠定了一定的理论基础。更多还原

【Abstract】 Generalized periodic point of continuous self-mapping, such as recurrent point, nonwandering point, chain recurrent point, is one of the most important researches of topological dynamical system. For almost 30 years, scholars at home and abroad are very interested in this area and have been actively involved in. They discussed the generalized periodic points on real segment, even metric space in depth, and got a lot of important research results. However, with modern dynamic system keeps the constant development to high-dimensional space and abstract space, some questions turn up: How to generalize the results of generalized periodic point on topological dynamical system to abstract topological space? Are there any important properties of generalized periodic point after the promotion?This paper deals with the above questions and obtains some results.Firstly, the concepts of periodic points, recurrent points and nonwandering points on real segment are generalized. In general topological space, some properties of periodic points, recurrent points and nonwandering points of continuous self-map are obtained; moreover, the correctness of these properties has been verified.Secondly, the concepts of unstable manifold on real segment are generalized. In general topological space, some properties of unstable manifold of continuous self-map are obtained; moreover, the correctness of these properties has been verified. At last, the paper describes the relationship among unstable manifold, periodic points,ω- limiting points, nonwandering points and homoclinic points.At last, the concepts of chain recurrent points andω- limiting points on real segment are generalized to metric space. Metric space is a special topological space. It is shown some properties of the chain recurrent points andω-limiting points in metric space. So, the corresponding results of closed interval have been improved.To a certain extent, these results enrich and promote the basic theory of periodic point in topological dynamical system. They lay the theoretical foundation of abstract development in dynamical systems theory and topology promotion in chaos mathematical theory.更多还原