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。

动力系统理论主要刻画事物随时间演变的规律，其中包括以微分方程描述的连续动力系统和以迭代为主的离散动力系统。嵌入流作为离散与连续系统的桥梁，一直是动力系统理论的一个重要研究课题。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。本项目从区间逐段单调连续映射入手，研究嵌入流的弱问题，即区间连续映射的迭代根问题。由于非单调高度是刻画区间映射复杂性的一个重要指标，我们在论文[On non-monotonicity height of piecewise monotone functions]得到映射高度为无穷的充分条件，并给出具体的算法，同时我们也证明了高度为无穷的映射在连续映射集的稠密性以及复合算子作用下高度的多样性；在上述结果的基础上，文[On piecewise monotone functions with height being infinity]进一步讨论高度无穷映射之间的共轭关系，分别得到拓扑共轭的充分与必要条件，同时我们也证明了高度为无穷的映射在复合下具有一定的对称性；最后，我们在文[Topological conjugacy for multifunctions with finitely many set-valued points and an application to iterative roots]中证明三类非连续映射拓扑共轭的充分与必要条件，并得到相应的迭代根结论。本项目的研究结果不仅致力于发展非定向映射与其迭代根的动力学性质，丰富迭代根的存在性与正规形理论，同时也将有助于嵌入流及相关函数方程问题的解决。