非单调映射迭代根的构造及其分类

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.

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。

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。本项研究了严格逐段单调连续函数迭代根的存在性，拓扑共轭问题，多项式非单调高度的计算，以及与此相关的迭代方程问题：（i）针对严格逐段单调连续映射，我们讨论了不同高度下迭代根的存在性，部分解决了张景中和杨路在1983年提出的公开问题一和公开问题二；（ii）针对集值映射，我们讨论了具有上半连续且有有限多个集值点的映射性质，并给出其迭代根的存在性与根的具体构造方法；（iii）针对特殊的一类逐段单调映射--多项式映射，我们给出其非单调高度的算法；（iv）利用迭代根构造的思想，我们讨论了多项式型迭代方程解的存在性与具体算法，以及其解的稳定性；（v）利用多项式型迭代构造“k阶迭代均值”，并给出该均值的不变性。本项目对非单调映射迭代根的研究为今后判断迭代根与其原映射动力学的保持性，以及对迭代根进行分类等问题打下基础。同时，也为我们下一步研究与迭代根相关的共轭、嵌入流以及不变曲线等问题创造良好的条件。