区间映射的迭代根与嵌入流及相关的函数方程问题

A flow is regarded as a 1-time mapping, which contains the dynamical system of the mapping under iteration. To the opposite, for a given continuous mapping it is not. easy to find a flow to embed, that is the problem of embedding flow. Embedding flow is a bridge of connecting concrete and continuous dynamical systems. Because of the lack of embedding flows for a lot of continuous mappings, many properties only belong to concrete dynamical system. Although lots of continuous mappings have no embedding flows, they may embed to flows asymptotically, this manner is used to investigate the bifurcation for discrete dynamical systems by the continuous ones. Therefore, which kind of continuous mappings can embed to a flow while which ones can embed asymptotically, which ones are local while which ones are global, remains very important problems. In this project, we study the weak problem of embedding flow for continuous mappings on interval, called the iterative roots and asymptotic iterative roots, and we study from continuous mappings on interval. Using the eigenvalues of a given mapping, the methods of linearization and normal form to simplify and extend such mappings piece by piece, we give the existence conditions for iterative roots and asymptotic iterative roots. Then we further study the topological conjugacies and the relation of entropy between those roots, which show the changes and complexity of their dynamical systems.

一个流可以定义一个单位时间映射，从而可以包含该映射迭代的动力系统; 相反，一个连续映射未必能找到能够嵌入的流，这就是嵌入流问题。嵌入流问题是沟通离散动力系统和连续动力系统的桥梁。正是因为很多连续映射没有可嵌入的流，因此离散动力系统具有很多连续动力系统不具备的性质。尽管很多连续映射没有可嵌入的流，但却可能渐近嵌入到某个流，这个性质已经被用来借助连续动力系统分岔理论的便利研究离散动力系统的分岔。因此，哪些连续映射可嵌入流，哪些连续映射可渐近嵌入流，哪些嵌入或渐近嵌入是局部的，哪些是全局的，仍然是十分重要的问题。本项目从区间连续映射入手，研究上述问题的弱问题，即区间连续映射迭代根与渐近迭代根问题。我们利用映射的特征值以及相应的线性化、正规化方法对映射分段简化、逐段延拓，给出迭代根与渐近迭代根的存在条件，进而研究迭代根之间的共轭等价关系以及其熵的关系，展示其动力学性质的变化及复杂性.