Holomorphic families of complex dynamical systems

复杂动力系统的全纯族

• 批准号: 0813675
• 负责人: Laura DeMarco
• 金额: $ 4.92万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0813675&HistoricalAwards=false
• 关键词: Holomorphic; families; complex; dynamical; systems

ABSTRACT. The Principal Investigator (PI) aims to study complex dynamical systems in holomorphic families. The questions are centered on the notions of stability within families, bifurcations, and degenerations. More specifically, the PI will pursue (1) questions about the compactification of the moduli space of rational maps, where it remains open to construct a natural boundary which captures the information of dynamical degeneration, and to describe precisely which stable families are bounded in the moduli space; (2) characterizations of stability for families of higher-dimensional dynamical systems, where the techniques from one-dimensional dynamics do not generalize, but there has been recent progress using pluripotential theory; (3) an investigation of families of polynomials in one variable and the associated space of trees with dynamics, which provides a continuous combinatorial model for polynomials in all degrees, and should model the structure of the escape locus in the moduli space of polynomials; and (4) properties of the transfinite diameter in connection with families of polynomials in all dimensions, where the discussion involves a combination of analytic and arithmetic methods.A more general overview:There are many open questions around the long-term effects of perturbations of a dynamical system. The PI studies the iteration of rational functions of one variable, one of the simplest examples of a non-invertible system with non-trivial dynamics. In this project, she aims to answer questions of a global nature: what is the structure of the stable regime in a complex-analytic family of rational functions? Or, what type of degenerations can take place and what do they tell us about the bifurcation locus? The motivation for studying these particular families comes from interesting connections with hyperbolic geometry, algebraic and arithmetic geometry, and potential theory.

抽象的。首席研究员（PI）旨在研究全纯族中的复杂动力系统。 问题集中在家庭内部的稳定、分歧和堕落的概念上。更具体地说，PI 将追求（1）关于有理映射模空间的紧致化的问题，在该问题中，构建捕获动态退化信息的自然边界仍然是开放的，并准确描述哪些稳定族在有理映射中受到限制。模空间； （2）高维动力系统族的稳定性表征，其中一维动力学技术无法推广，但最近在使用多能理论方面取得了进展； (3) 研究一个变量中的多项式族以及与动力学相关的树空间，为所有次数的多项式提供连续组合模型，并且应该对多项式模空间中的逃逸轨迹的结构进行建模； （4）与所有维度的多项式族相关的超限直径的性质，其中的讨论涉及分析和算术方法的组合。更一般的概述：围绕扰动的长期影响存在许多悬而未决的问题一个动力系统。 PI 研究一个变量的有理函数的迭代，这是具有非平凡动力学的不可逆系统的最简单示例之一。 在这个项目中，她的目标是回答全球性的问题：理性函数的复杂分析族中稳定政权的结构是什么？或者，会发生什么类型的变性以及它们告诉我们有关分叉轨迹的什么信息？ 研究这些特殊族的动机来自于与双曲几何、代数和算术几何以及势论的有趣联系。