Complex dynamics via tropical moduli spaces

动力学是研究系统随时间演化的学科。一个显著的特征是，一个在非常简单的规则下演化的系统可以表现出极其复杂的长期行为。这通常表现在分形的无处不在-无限复杂的形状。复动力学是研究全纯映射--例如（z-->z^2-1）--在重复应用下的行为。当研究一个动力系统时，主要的问题是：长期行为如何依赖于初始条件？动力系统的“Julia集”是划分具有不同长期行为的初始条件的边界。在上面的例子中，以及更一般地，Julia集是一个分形。很自然地，不仅要研究孤立的一个映射的动力学行为，而且要研究映射族中动力学行为的变化。例如，我们可以考虑所有形式为（z-->z^2+constant）的动力系统，而不是只考虑（z-->z^2-1）。复动力学领域经历了一个转变与Douady和哈伯德的曼德尔布罗特集的探索，它生活在动力系统的空间（z-->z^2+常数）。曼德尔布罗特集的边界是一个分形，它划分了具有不同长期行为的动力系统。一个映射的动力学行为是由临界点的长期行为反映出来的，临界点是指导数为零的点。例如，（z-->z^2+常数）的Julia集是连通的当且仅当临界点“0”有界轨道。后临界有限（PCF）映射是每个临界点最终落在周期性循环中的映射。它们的动力学行为可以组合编码，理解PCF映射对于更普遍地理解复杂动力学至关重要。PCF映射在有理映射族中具有非常特殊的分布，例如它们在Mandelbrot集的边界上是稠密的。PCF地图还提供了一个迷人的一个变量和多个变量之间的动态联系。Thurston通过构造一个称为“Thurston拉回映射”的动力系统，证明了PCF映射的一个相应刚性结果，该动力系统的不动点是单变量PCF映射。瑟斯顿的拉回映射作用于高维Teichmuller空间：理解它们在“无穷大附近”的动力学行为对于理解PCF映射以及理解有理映射的退化至关重要。通过科赫的工作，瑟斯顿的拉回映射有代数几何的“阴影”，称为赫尔维茨对应。这提供了一个新的机会，使用组合代数几何的模空间的工具，以解决复杂的动力学问题。热带几何是研究代数几何中的退化：它是一个非常适合用来研究赫维茨对应和瑟斯顿拉回映射的动力学的框架。但是，它尚未应用于此设置。在这项研究中，我们将使用热带赫维茨对应的动力学，以统一的对象是活跃的研究课题，但在不同的领域。我们将联系动力学程度（代数动力学在许多变量）与瑟斯顿障碍（Teichmuller-theoric对象）。这将链接的全局代数动力学赫维茨对应的局部动力学无穷大附近的瑟斯顿的拉回地图。我们将把哈伯德树（一个变量的有理动力学）与容许覆盖（组合代数几何）和热带曲线联系起来。

Dynamics is the study of systems evolving with time. One striking feature is that a system evolving under a very simple rule can exhibit extremely complicated long-term behaviour. This often manifests through the ubiquitousness of fractals --- infinitely complicated shapes. Complex dynamics is the study of the behavior of holomorphic maps --- for example (z-->z^2-1) --- under repeated application. When studying a dynamical system, the main question is: how does long term behaviour depend on initial condition? The "Julia set" of a dynamical system is the boundary demarcating initial conditions with different long-term behaviour. In the above example, as well as more generally, the Julia set is a fractal. It is natural to investigate not just the dynamical behavior of one map in isolation, but also the variation of dynamical behavior within families of maps. For example, instead of looking just at (z-->z^2-1), one could consider all dynamical systems of the form (z-->z^2+constant). The field of complex dynamics underwent a transformation with Douady and Hubbard's exploration of the Mandelbrot set, which lives inside the space of dynamical systems (z-->z^2+constant). The boundary of the Mandelbrot set is a fractal that demarcates dynamical systems with qualitatively different long-term behaviour. The dynamical behaviour of a map is reflected by the long-term behaviour of critical points --- points where the derivative is zero. For example, the Julia set of (z-->z^2+constant) is connected if and only only if the critical point "0" has bounded orbit. Post-critically finite (PCF) maps are maps for which every critical point eventually lands in a periodic cycle. Their dynamical behaviour can be encoded combinatorially, and understanding PCF maps is crucial for understanding complex dynamics more generally. PCF maps have a very special distribution in families of rational maps, for example they are dense in the boundary of the Mandelbrot set. PCF maps also provide a fascinating link between dynamics in one variable and in many variables. Thurston proved a consequential rigidity result for PCF maps by constructing dynamical systems called "Thurston pullback maps", whose fixed points are PCF maps in one variable. Thurston's pullback maps act on high-dimensional Teichmuller spaces: understanding their dynamical behavior "near infinity" is of crucial importance for understanding PCF maps, as well as for understanding degenerations of rational maps. By work of Koch, Thurston's pullback maps have algebro-geometric "shadows" called Hurwitz correspondences. This provides a new opportunity to use tools from combinatorial algebraic geometry of moduli spaces in order to address questions in complex dynamics. Tropical geometry is the study of degenerations in algebraic geometry: it is a very well-suited framework to use to study the dynamics of Hurwitz correspondences and Thurston's pullback map. However, it has not yet been applied to this setting. In this research, we will use the dynamics of tropical Hurwitz correspondences in order to unify objects that are active topics of research, but in different fields. We will link dynamical degrees (algebraic dynamics in many variables) with Thurston obstructions (Teichmuller-theoric objects). This will link the global algebraic dynamics of Hurwitz correspondences to the local dynamics near infinity of Thurston's pullback map. We will link Hubbard trees (rational dynamics in one variable) with admissible covers (combinatorial algebraic geometry) and tropical curves.