Geometric Structures in Holomorphic Dynamics and Teichmuller Theory

全纯动力学中的几何结构和 Teichmuller 理论

• 批准号: 0505652
• 负责人: Mikhail Lyubich
• 金额: --
• 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: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505652&HistoricalAwards=false
• 关键词: Geometric; Structures; Holomorphic; Dynamics; Teichmuller

ABSTRACTThis will be a broad program of research in geometric aspects of holomorphic dynamics, Teichmuller theory, laminations, and related areas. The project addresses central problems in these fields: The Local Connectivity Problem for the Mandelbrot set and for Julia sets would help to give a thorough understanding of dynamics for the complex quadratic family. TheRenormalization and Universality Conjectures concern fundamental rigidity features of the phase and parameter domains for dynamical systems. The Regular or Stochastic Conjecture would give a complete measure-theoretic picture of the dynamics of unimodal maps. The Ehrenpreis Conjecture asserts that any two compact Riemann surfaces have almost isomorphic compact coverings. The project would explore further interplay between holomorphic dynamics, hyperbolic geometry, Teichmuller theory, and the theory of laminations, as well as the interplay between real and complex dynamics in one and two variables.Dynamical systems theory studies evolution of various systems described by differential equations or by the iteration of a single map. It has numerous applications in celestial mechanics, statistical physics, fluid dynamics, biology, and other branches of natural science. Holomorphic dynamics is the part of dynamical systems theory that deals with iterates of complex analytic maps. It has proved to be a powerful tool in understanding important models of real low-dimensional dynamics. There are numerous interconnections between holomorphic dynamics, geometric analysis, hyperbolic geometry, and the theory of foliated spaces. Holomorphic dynamics also produces beautiful fractal objects, such as Julia sets and the Mandelbrot set, whose intricate structure has fascinated scientists for decades. All of this structure and its applications will be further explored by the Stony Brook dynamics group.

摘要这将是一个广泛的研究项目，涉及全纯动力学、Teichmuller 理论、叠层和相关领域的几何方面。该项目解决了这些领域的核心问题：Mandelbrot 集和 Julia 集的局部连通性问题将有助于全面了解复杂二次族的动力学。重整化和普遍性猜想涉及动力系统相域和参数域的基本刚性特征。正则或随机猜想将给出单峰映射动力学的完整测度理论图景。埃伦普雷斯猜想断言任何两个紧致黎曼曲面都具有几乎同构的紧致覆盖。该项目将进一步探索全纯动力学、双曲几何、Teichmuller 理论和叠片理论之间的相互作用，以及一变量和二变量中真实动力学和复杂动力学之间的相互作用。动力系统理论研究由微分方程描述的各种系统的演化或者通过单个映射的迭代。它在天体力学、统计物理学、流体动力学、生物学和自然科学的其他分支中有广泛的应用。 全纯动力学是动力系统理论的一部分，用于处理复杂解析图的迭代。事实证明，它是理解真实低维动力学重要模型的有力工具。全纯动力学、几何分析、双曲几何和叶状空间理论之间存在着许多相互联系。全纯动力学还产生美丽的分形对象，例如朱莉娅集和曼德尔布罗特集，其复杂的结构几十年来一直让科学家们着迷。所有这些结构及其应用将由石溪动力学小组进一步探索。