Geometric Structures in Holomorphic Dynamics and Teichmuller Theory

这将是一个广泛的研究全纯动力学几何方面、Teichmuller理论、层积和相关领域的项目。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彻底理解复二次族的动力学。正规化猜想和普适性猜想涉及动力系统相域和参数域的基本刚性特征。正则猜想或随机猜想将给出单峰映射动力学的完整测度理论图景。Ehrenpreis猜想断言任意两个紧致黎曼曲面都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理论和层积理论之间的相互作用，以及真实动力学和复杂动力学在一变量和两变量之间的相互作用。动力系统理论研究由微分方程或单个映射的迭代描述的各种系统的演化。它在天体力学、统计物理学、流体动力学、生物学和其他自然科学分支中有许多应用。全纯动力学是动力系统理论中处理复杂解析映射迭代的部分。它已被证明是理解实际低维动力学的重要模型的有力工具。在全纯动力学、几何分析、双曲几何和叶状空间理论之间有许多相互联系。全纯动力学也产生了美丽的分形物体，比如Julia集合和Mandelbrot集合，它们复杂的结构几十年来一直吸引着科学家。所有这些结构及其应用将由石溪动力学小组进一步探索。

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.