Mathematical Sciences: Investigations in Conformal DynamicalSystems

两位主要研究人员将利用单侧Bernoulli移位理论来理解任意次多项式的Cantor Julia集上映射的动力学。利用这两个映射之间的共轭关系，它们将动力系统空间的拓扑与单侧移位的自同构群的结构联系起来。此外，穿孔环面的Teichmuller理论，黎曼球面上有理映射迭代产生的动力系统的分类，以及黎曼球面的分支覆盖的Catalan数都将是相关研究的主题。通过分析复杂平面上多项式映射的动力学，可以生成计算机生成的分形图。继续围绕平面反弹但从不落入周期吸引子的像素构成了具有奇异配置的不变分形图。这些计算机图像背后的数学需要几个现有理论的融合。首席调查员将继续研究这些相互关系。

The two principal investigators will use the theory of one- sided Bernoulli shifts to understand the dynamics of maps on Cantor Julia sets for polynomials of arbitrary degree. Using the conjugacy between these two maps, they will relate the topology of the space of dynamical systems to the structure of the automorphism group of the one-sided shift. In addition, Teichmuller theory for punctured tori, classifications of dynamical systems which arise from iteration of rational maps on the Riemann sphere, and Catalan numbers for branched coverings by Riemann spheres will be the subjects of related investigations. Computer-generated fractal images can be made by analyzing the dynamics of polynomial maps of the complex plane. Pixels which continue to bounce about the plane, but never fall into one of the periodic attractors, comprise invariant fractals with exotic configurations. The mathematics behind these computer images requires an amalgamation of several existing theories. The principal investigators will continue their studies of these interrelationships.