非单调映射迭代根的构造及其分类

项目名称： 非单调映射迭代根的构造及其分类

项目编号： No.11501471

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 刘鎏

作者单位： 西南交通大学

项目金额： 18万元

中文摘要： 嵌入流问题是动力系统研究的重要问题之一，它刻画了离散与连续动力系统之间的关系。迭代根问题是嵌入流问题的弱问题，这一问题的研究决定着迭代指数能否从整数次推广到有理数次。尽管对单调区间映射的迭代根问题已取得丰富结果，而非单调映射的迭代根问题是十分困难的，其根本原因是映射的定向性被破坏。本项目将研究逐段单调连续映射迭代根的构造及分类。刻画逐段单调映射的一个重要指标就是非单调性高度。高度=1时，映射存在特征区间，前人利用特征区间上的单调性通过延拓的办法给出迭代根的存在性。高度>1时，映射不再有特征区间，其迭代根的轨道更为复杂。本项目拟用迭代构造法寻找此情形下的迭代根。进而，用拓扑共轭方程来判断迭代根与其原映射动力学的保持性，对迭代根进行分类。对动力学性质不能保持的情形，拟研究迭代根与原映射拓扑动力学关系，包括传递性、混合性、拓扑熵等，并以多项式为例，用符号计算给出多项式及其根的高度和拓扑熵。

中文关键词： 迭代根；非单调性高度；特征区间；拓扑共轭；拓扑熵

英文摘要： The problem of embedding flows describes the relation between discrete and continuous dynamical system, which is one of important problems in dynamical system field. The problem of iterative roots, being a weak version of the problem of embedding flows, decides that whether the iterative order can be extended from integer to rational. Even though there are plentiful results for one-dimensional monotone mappings, it is still difficult problem to find iterative roots of non-monotone mappings, since the directionality of mappings is destroyed. In this project, we study the construction of iterative roots of non-monotonic mappings and their classification. An important index to describe piecewise monotone mappings is non-monotonicity height. When height is 1, the mapping has a characteristic interval such that the existence of iterative roots can be given by extension. When height is greater than 1, the orbit of iterative roots is more complicated since there is no characteristic interval. We will use the construction method to find iterative roots in our project. Furthermore, we will use topological conjugate equation to judge the retention of dynamic between piecewise monotone mappings and their iterative roots, and then give the classification of iterative roots. Finally, for an irretention case, we will study the relation of dynamic between piecewise monotone mappings and their iterative roots, including transitivity, mixing, topological entropy and so on. Moreover, as an example, we will use symbol computing method to give the height and topological entropy of polynomial and their iterative roots.

英文关键词： iterative root;non-monotonicity height;characteristic interval;topological conjugacy;topological entropy