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.

摘要。主要研究员（PI）旨在研究全纯族中的复杂动力系统。 这些问题集中在家庭内的稳定性，分叉和退化的概念。更具体地说，PI将追求（1）关于有理映射的模空间的紧化的问题，其中它仍然开放以构建捕获动力学退化信息的自然边界，并精确描述哪些稳定族在模空间中有界;（2）高维动力系统族的稳定性刻画，其中一维动力学的方法不能推广，（3）对单变量多项式族和与之相关的动态树空间的研究，它为所有次数的多项式提供了一个连续的组合模型，并应模拟多项式模空间中逃逸轨迹的结构;以及（4）与所有维度的多项式族有关的超限直径的性质，其中讨论涉及分析和算术方法的组合。更一般的概述：围绕动力系统的扰动的长期影响有许多公开的问题。 PI研究一个变量的有理函数的迭代，这是具有非平凡动态的不可逆系统的最简单示例之一。 在这个项目中，她的目标是回答一个全球性的问题：什么是结构的稳定制度在一个复杂的分析家庭的合理功能？或者，什么样的退化会发生，它们告诉我们关于分叉点的什么？ 研究这些特殊的家庭的动机来自有趣的连接与双曲几何，代数和算术几何，和潜在的理论。