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.

摘要这将是一项广泛的研究计划，这些计划在几何方面，TeichMuller理论，层压层和相关领域的几何方面。该项目解决了这些领域中的中心问题：Mandelbrot集和Julia集的局部连接问题将有助于对复杂的二次家族的动态有透彻的了解。实质性正规化和普遍性猜测涉及动态系统的阶段和参数域的基本刚性特征。常规或随机的猜想将提供单峰地图动力学的完整量度理论图片。 Ehrenpreis的猜想断言，任何两个紧凑的Riemann表面几乎具有同构的紧凑型覆盖物。该项目将探索霍明型动力学，双曲线几何形状，teichmuller理论与层压理论之间的进一步相互作用，以及一个和两个变量中的真实和复杂动力学之间的相互作用。尼古拉系统理论研究研究了由微分方程或单映射的各种系统描述的各种系统的演变。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都有许多应用。 全体形态动力学是涉及复杂分析图的迭代的动力系统理论的一部分。事实证明，它是理解实际低维动力学重要模型的强大工具。圆锥形动力学，几何分析，双曲几何形状和叶状空间理论之间存在许多互连。全态动力学还会产生美丽的分形物体，例如朱莉娅（Julia）套装和曼德布罗（Mandelbrot）套装，其复杂的结构使科学家着迷了数十年。 Stony Brook Dynamics Group将进一步探讨所有这些结构及其应用。