Geometric Aspects of Low Dimensional Dynamics

这项研究计划集中在复杂和真实的低维动力学的几个相互关联的几何主题上。它包括黎曼球面和区间上的多项式动力学，实Henon族和复Henon族中的动力学，来自统计物理的射影空间有理自同态的一些特殊族中的动力学，以及相关的Riemann曲面和双曲层的几何。这些主题中的大多数都被重整化的思想统一起来，重整化是渗透到动力学对象的小尺度结构的强大工具，旨在对这些对象进行完整的分类。该项目将导致对动力系统的小尺度结构的更深层次的洞察，这一主题为理解自然界中许多最基本的过程提供了关键，这些过程是科学和工程研究的基本对象。它将包括培训高素质的研究生和博士后，他们将以不同的方式和地点应用他们的技能：在学术界和工业中，在创造更广泛的真实和复杂动力学和统计物理各分支专家之间的互动方面，在促进动力系统领域内的有效交流方面。

This research program focuses on several interconnected geometric themes of complex and real low-dimensional dynamics. It includes polynomial dynamics on the Riemann sphere and the interval, dynamics in the real and complex Henon family, dynamics in some special families of rational endomorphisms of projective space coming from statistical physics, and the geometry of associated Riemann surfaces and hyperbolic laminations. Most of these themes are unified by the idea of renormalization as a powerful tool for penetrating into the small-scale structure of dynamical objects, aimed towards a complete classification of these objects.The project will result in deeper insights into small-scale structure of dynamical systems, a subject that furnishes a key to the understanding of many of the most basic processes in nature, the fundamental objects of study in science and engineering. It will include the training of highly qualified graduate students and postdocs, who will apply their skills in diverse ways and venues: in academia and industry, in the creation broader interactions between experts in various branches of real and complex dynamics and statistical physics, and in promoting efficient communication within the field of dynamical systems.