Geometric Aspects of Low Dimensional Dynamics

低维动力学的几何方面

• 批准号: 1007266
• 负责人: Mikhail Lyubich
• 金额: $ 13.2万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007266&HistoricalAwards=false
• 关键词: Geometric; Aspects; Low; Dimensional; Dynamics

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.

本研究计划的重点是复杂和真实低维动态的几个相互关联的几何主题。它包括黎曼球和区间上的多项式动力学，实和复Henon族的动力学，来自统计物理的射影空间的一些特殊有理自同态族的动力学，以及相关黎曼曲面和双曲层合的几何。这些主题中的大多数都统一于重整化的思想，重整化是一种深入动态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规模结构的更深入的了解，这一主题为理解自然界中许多最基本的过程提供了关键，这是科学和工程研究的基本对象。它将包括培养高素质的研究生和博士后，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，在真实和复杂动力学和统计物理的各个分支的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交流。