Holomorphic families of complex dynamical systems

抽象的。首席调查者(PI)致力于研究全纯族中的复杂动力系统。这些问题集中在家庭内部的稳定性、分叉和退化的概念上。更具体地说，PI将寻求(1)关于有理映射模空间的紧致性的问题，其中它仍然是开放的，以构造一个捕捉动力退化信息的自然边界，并精确地描述哪些稳定族在模空间中是有界的：(2)高维动力系统族的稳定性的刻画，其中来自一维动力学的技术不是泛化的，但利用多势理论已经有了最新的进展；(3)一元多项式族及其与动力学相关的树空间的研究，它为所有次数的多项式提供了一个连续的组合模型，并且应该在多项式的模空间中模拟逃逸轨迹的结构；以及(4)与所有维的多项式族有关的超限直径的性质，其中涉及分析和算术方法的结合。更广泛的概述：关于动力系统的扰动的长期影响，有许多悬而未决的问题。PI研究一元有理函数的迭代，这是具有非平凡动力学的不可逆系统的最简单的例子之一。在这个项目中，她的目标是回答全球性的问题：在一个复杂分析的有理函数家族中，稳定的制度是什么结构？或者，可以发生什么类型的退化，它们告诉我们关于分叉轨迹的什么？研究这些特殊家族的动机来自于与双曲几何、代数和算术几何以及位势理论的有趣联系。

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.