核心分解及其应用

For a compact set K in the sphere, like the Julia set of a rational function, we consider two families of monotone decompositions of K; in one of those families we require that the corresponding hyperspace be a Peano space, in the other the hyperspace is assumed to be semi-Peano. We will show that the finest elements of those two families both exist and respectively call them "the core decomposition with Peano hyperspace" and "the core decomposition with semi-Peano hyperspace". Among others we have two types of motivations; the first one is from continuum theory connected with dynamical systems and the other from quite recent results concerning dynamics of rational functions and the structure of Mandelbrot set. Our investigation takes a topological viewpoint and tries to analyze the elements of those core decompositions, each is a continuum in the plane; in particular, it is of some interest to compare those planar continua with the cluster sets of certain univalent functions. The results we aim at are interesting and have applications to the study of dynamical systems, such as the sub-system formed by a rational function restricted to its Julia set and discussions on topological aspects of the Mandelbrot set.

针对球面上的紧子集，例如有理函数的Julia集，考虑两个单调分解族，第一个要求对应的超空间在商拓扑下是Peano空间，第二个要求超空间是半Peano空间。我们证明，这两个分解族各存在一个最细的分解，分别叫做：“以Peano空间为超空间的核心分解”、“以半Peano空间为超空间的核心分解”。这两类核心分解的讨论，有两种动机，其一来源于跟动力系统结合在一起的连续统理论的研究，其二来源于有理函数动力系统以及Mandelbrot集结构的近期研究。我们主要从拓扑动力系统的角度来分析这些核心分解的分解元的构造特征，与某些单页函数的凝聚集的结构相比较。所得到的结果，在动力系统的研究中有重要的应用，例如，有理函数在Julia集上的子系统的研究、Mandelbrot集的某些拓扑性质的详细刻画。

基于近期复多项式和有理函数动力学的研究进展，建立平面紧集的原子结构理论，用于平面拓扑、复分析、复动力系统的研究。主要成果包括：得到平面紧集原子结构的存在性和不变性；给出经典Torhorst定理的量化版本；在合理前提下得到圆域上共形同胚连续延拓到边界的充分必要条件；提出多项式核熵的最一般定义，推广了经典核熵概念，联系到双可达维数，分析了二次多项式核熵函数的单调性与间断点的典型性质。