Complex dynamics via tropical moduli spaces

通过热带模空间的复杂动力学

• 批准号: EP/X026612/1
• 负责人: Rohini Ramadas
• 金额: $ 28.41万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX026612%2F1
• 关键词: Complex; dynamics; via; tropical; moduli

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.

动力学是对随时间演化的系统的研究。一个显着的特征是，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。这通常通过无处不在的分形（无限复杂的形状）来体现。复杂动力学是研究全纯映射的行为——例如（z-->z^2-1）——在重复应用下。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“朱莉娅集”是划分初始条件与不同长期行为的边界。在上面的例子中，以及更一般的情况下，朱莉娅集是一个分形。很自然地，不仅要研究孤立的一张地图的动态行为，还要研究地图族内动态行为的变化。例如，我们可以考虑 (z-->z^2+constant) 形式的所有动力系统，而不是只考虑 (z-->z^2-1)。随着 Douady 和 Hubbard 对曼德尔布罗特集的探索，复杂动力学领域经历了一场转变，曼德尔布罗特集存在于动力系统的空间内（z-->z^2+常数）。曼德尔布罗特集的边界是一个分形，它划分了具有不同性质的长期行为的动力系统。地图的动态行为由临界点（导数为零的点）的长期行为反映。例如，当且仅当临界点“0”具有有界轨道时，(z-->z^2+constant) 的 Julia 集才连通。后临界有限 (PCF) 映射是每个临界点最终落在周期循环中的映射。它们的动力学行为可以组合编码，并且理解 PCF 图对于更广泛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 集的边界上是密集的。 PCF 图还提供了一个变量和多个变量的动态之间的迷人联系。 Thurston 通过构造称为“Thurston 回拉图”的动力系统，证明了 PCF 映射的相应刚性结果，其固定点是一个变量中的 PCF 映射。瑟斯顿的回拉图作用于高维 Teichmuller 空间：理解它们“接近无穷大”的动态行为对于理解 PCF 图以及理解有理图的退化至关重要。通过科赫的工作，瑟斯顿的回拉图具有代数几何“阴影”，称为赫尔维茨对应。这提供了使用模空间组合代数几何工具来解决复杂动力学问题的新机会。热带几何是对代数几何退化的研究：它是一个非常适合用来研究赫维茨对应和瑟斯顿回拉图动力学的框架。但是，它尚未应用于此设置。在这项研究中，我们将利用热带赫尔维茨对应的动态来统一不同领域的活跃研究主题的对象。我们将把动力学度（许多变量中的代数动力学）与瑟斯顿障碍（Teichmuller 理论对象）联系起来。这将把赫尔维茨对应的全局代数动力学与瑟斯顿回拉图的接近无穷大的局部动力学联系起来。我们将哈伯德树（一个变量的有理动力学）与可接受的覆盖（组合代数几何）和热带曲线联系起来。