Topological Dynamics in Real and Complex Systems

该提案涉及低维动力学中的几个主题。PI计划研究层片的拓扑动力学；特别是他想进一步发展树木生长的动力学，并用它来研究Julia集的顶点的行为。他还想考虑一类非解析平面图，对于这些非解析型平面图，可以证明与众所周知的多项式结果类似的结果。他的其他研究方向涉及有理映射族的吸引点和进一步发展的旋转理论，该理论研究圈的共存取决于它们的特征，类似于一次圆映射的旋转数。动力系统理论为研究复杂系统提供了工具，例如复杂确定性过程随时间的演变。在通信理论、统计物理和化学动力学等其他科学中使用这些强大的工具有着丰富而杰出的传统。

The proposal deals with several topics in low-dimensionaldynamics. The PI plans to investigate the topological dynamicsof laminations; in particular he would like to further developthe dynamics of growing trees and use it to study the behaviorof the vertices of the Julia set. He also wants to consider classes of non-analytic plane maps for whom analogs of well-knownresults for polynomials can be proven. Other directions of hisresearch are related to attractos for classes of rational maps and to further developing the rotation theory which studies coexistence of cycles depending on their characteristics similarto rotation numbers for circle maps of degree one. The theory of dynamical systems gives tools to study complex systems, for example the evolution of a complicated deterministicprocess over time. There is a rich and distinguished tradition of using these powerful tools in other sciences like communication theory, statistical physics, and chemical kinetics.