平面紧子集的核心分解及其应用

The Julia sets of rational functions have very complex topological properties, which make it difficult to study their dynamical properties. The core decomposition theory of planar compacta provides a mathematical tool for constructing an appropriate factor system for them, and it can be used to study the topological properties of general planar compacta, such as Mandelbrot set. The existing theorems show that any planar compacta have a core decomposition with respect to be the property of Peano compacta, and this core decomposition is compatible with rational functions. Based on this research, the project will further explore and improve the value of core decomposition theory, and continue to explore the following questions: (1) Study the dynamical properties of the factor system of rational function systems; (2) Study the core decompositon of small Julia sets for polynomials of infinitely renormalization; (3) Exploring the local connectivity of Mandelbrot set from the perspective of core decomposition theory. These three questions are raised by combining the core decomposition theory and the hot topics in the dynamical system and topology research. They can enrich the study of topological properties of Julia sets , and we need to combine the ideas and methods from dynamic systems，complex analysis and continuum theory to solve them.

有理函数的Julia集拓扑性质非常复杂，导致了该动力系统的研究非常困难。平面紧子集的核心分解理论为构造该系统的一个合适的因子系统提供了数学工具，并且它可用于研究一般的平面紧子集，例如Mandelbrot集的拓扑性质。现有的结论表明，任何的平面紧子集都存在关于Peano紧集性质的核心分解，并且该分解与有理函数匹配良好。本项目将在这一研究的基础之上，进一步挖掘和完善核心分解理论的价值，继续深入探讨以下问题：(1)研究有理函数系统的因子系统的动力学性质；(2)研究无穷可重整多项式的小Julia集的核心分解问题；(3)从核心分解的观点出发，探索Mandelbrot集的局部连通性问题。这些问题是结合核心分解理论和动力系统以及拓扑学研究中的热点问题而提出，它能进一步丰富Julia集拓扑性质的研究，需要综合利用动力系统，复分析和连续统理论的思想和方法来解决。

核心分解理论是研究多项式Julia集及其动力系统的重要工具，本项目利用该工具进一步研究了它在多项式的核熵理论以及Mandelbrot集局部连通性中的应用。在本项目的资助下，项目主持人(1)推广了多项式的核熵的定义，研究了核熵在Mandelbrot集上的连续性和单调性，（2）比较了大小Julia集的核熵之间的关系，（3）研究了Mandelbrot集的核心分解，找到了更多的核心分解的单点集原子。