分形分析在多项式动力系统中的应用

Julia sets of polynomials are one of main objects studied in complex dynamics. Locally connected Julia sets are a very important and rich class of fractals. This project deals with analysis on Julia sets. Using the ideas of analysis on fractals some problems in complex dynamics are considered. Since Julia sets are more complicated than fractals studied in the past (such as self-similar sets,postcritically finite sets) and their formations are also completely different, analysis on Julia sets becomes more difficult. By means of the graph approximation method, Hilbert projective metric and some properties of locally connected Julia sets we hope to construct Dirichlet forms on the Julia set of a polynomial, to calculate the corresponding Laplacian spectrum, and to discuss the relationship between the spectrum and polynomial. Further we will discuss mating of two polynomials with locally connected Julia sets using theory of complex dynamics and effective resistance metrics on Julia sets. Therefore this project makes analysis on fractals development to Julia sets and solves some of problems in complex dynamics. From the point of view of combination of fractal theory and complex dynamics, these works are to achieve innovation and to form characteristics.

多项式的Julia 集是复动力系统的主要研究对象之一，而局部连通的Julia集又是相当丰富和重要的一类分形集. 本项目希望对这类Julia集进行分析研究, 用分形分析理论讨论多项式的动力学问题. 由于Julia集较以往所研究的分形集(如自相似集、后临界有限集)更复杂,生成方式也完全不同, 故其上的分析研究更有难度.本项目希望借助图逼近方法、Hilbert投影度量和局部连通Julia集的性质，在其上构造Dirichlet型，计算相应的Laplace的谱，讨论谱与多项式之间的关系；也希望利用复动力系统理论、Julia集上的有效阻抗度量，讨论两个具有局部连通的Julia集的多项式的粘合. 因而，本项目研究除了将分形分析理论发展到重要的Julia集, 还将解决复动力系统中的一些问题, 从复动力系统与分形论结合的角度对上述问题进行研究，达到创新，形成特色.

我们研究了粒子运动在具有关联记忆非更新连续时间的随机游动外力场作用下的反常扩散，得到Fokker-Planck型动力学方程稳态解具有Boltzmann-Gibbs形式，等待时间和从属的渐近行为具有拉伸的高斯分布；应用动力学方法研究了临界有限的自相似分形集上的自相似能量的存在性，得到了在一定条件下，自相似分形集上的自相似能量是存在的；研究了具有Siegel盘的多项式的Douady猜测，借助于拟共形手术对三次多项式构造了具有结构稳定性的三次多项式族。这些研究将对于分形集(包括Julia集)的更深刻的认识有非常重要的作用。