Geometric Aspects of Low Dimensional Dynamics

这个研究计划的重点是复杂的和真实的低维动力学的几个相互关联的几何主题。它包括黎曼球面和区间上的多项式动力学，真实的和复Henon族中的动力学，来自统计物理的射影空间的某些特殊有理自同态族中的动力学，以及相关的黎曼曲面和双曲叠层的几何学。这些主题中的大多数都被重整化的思想所统一，重整化是一种强大的工具，可以渗透到动力学对象的小尺度结构中，旨在对这些对象进行完整的分类。该项目将导致对动力学系统的小尺度结构有更深入的了解，这是一个学科，它是理解自然界中许多最基本过程的关键，科学和工程学的基本研究对象。它将包括高素质的研究生和博士后的培训，他们将以不同的方式和场所应用他们的技能：在学术界和工业界，在真实的和复杂的动力学和统计物理学的各个分支专家之间建立更广泛的互动，并促进动力系统领域内的有效沟通。

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.