Research in Dynamical Systems

该提案概述了一个探索低维动力学不同方面的计划：全纯动力学的组合和几何问题，它与一维真实动力学的相互作用（特别是在刚性和重整化理论中），与三维双曲几何和理论分支的相互作用，二维复杂和真实动力学的解析和几何方面。动力系统理论研究物理、生物、工程等各种系统的演化，以及这些系统的简化数学模型。在过去的25年里，人们已经认识到，即使是一维或二维模型也能显示出丰富而重要的现象。人们还认识到，复变分析方法为处理这些系统提供了强有力的工具。该提案旨在进一步探索这一动态分支。

The proposal outlines a program of exploring different aspects of lowdimensional dynamics:combinatorial and geometric issues of holomorphic dynamics,its interplay with one-dimensional real dynamics (particularly in the rigidity and renormalization theories),interaction with three dimensional hyperbolic geometry and the theory offoliations, analytic and geometric aspects of two dimensional complex andreal dynamics.The dynamical systems theory studies the evolution of various systemsarising in physics, biology, engineering etc., as well as simplifiedmathematical models for those systems. It has been realized in the past 25years that even one- or two- dimensional models can exhibit rich andimportant phenomena. It has been also realized that the methods of complexanalysis provide powerful tools for dealing with these systems. Theproposal intends to explore further this branch of dynamics.