Complex dynamics via tropical moduli spaces

动力学是研究系统随时间演变的学科。一个显著的特征是，在一个非常简单的规则下进化的系统可能会表现出极其复杂的长期行为。这通常通过无处不在的分形图表现出来-无限复杂的形状。复动力学是研究全纯映射在重复应用下的行为的学科，例如(z--&gt；z^2-1)。在研究动力系统时，主要问题是：长期行为如何取决于初始条件？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。在上面的示例中，以及更一般的情况下，Julia集是一个分形。不仅要单独研究一个映射的动力学行为，而且要研究映射族内的动力学行为的变化是很自然的。例如，不是只看(z--&gt；z^2-1)，而是可以考虑形式为(z--&gt；z^2+常量)的所有动力系统。随着Douady和Hubbard对Mandelbrot集的探索，复杂动力学领域经历了一次变革，Mandelbrot集位于动力系统(z--&gt；z^2+常量)的空间中。Mandelbrot集的边界是一个分形图，它用定性不同的长期行为来划分动力系统。映射的动力学行为通过临界点的长期行为来反映-导数为零的点。例如，(z--&gt；z^2+常量)的Julia集是连通的当且仅当临界点“0”有有界轨道。后临界有限(PCF)映射是每个临界点最终落在周期周期中的映射。它们的动力学行为可以组合编码，而理解PCF映射对于更广泛地理解复杂动力学是至关重要的。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图还提供了一个令人着迷的链接，在单变量和多变量的动力学之间。通过构造不动点为一元PCF映射的动力系统，瑟斯顿证明了PCF映射的一个相应刚性结果。瑟斯顿的拉回映射作用于高维Teichmuller空间：理解它们的动力学行为对于理解PCF映射以及理解有理映射的退化是至关重要的。根据科赫的研究，瑟斯顿的回撤地图具有被称为赫维茨对应的代数几何“阴影”。这为利用模空间的组合代数几何工具解决复杂动力学问题提供了新的机会。热带几何研究的是代数几何中的退化：它是一个非常适合用来研究赫维茨对应和瑟斯顿拉回图的动力学的框架。但是，它尚未应用于此设置。在这项研究中，我们将使用热带Hurwitz对应的动力学来统一作为活跃研究主题的不同领域的对象。我们将把动态度(多变量的代数动力学)与瑟斯顿障碍物(泰希穆勒理论对象)联系起来。这将把赫维茨对应的全局代数动力学与瑟斯顿拉回图的接近无穷远的局部动力学联系起来。我们将把Hubbard树(一元有理动力学)与可容许覆盖(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.