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.

动力学是对随着时间发展的系统发展的研究。一个惊人的特征是，在非常简单的规则下发展的系统可以表现出极其复杂的长期行为。这通常通过分形的无处不在 - 无限复杂的形状。复杂的动力学是对全体形态图的行为的研究 - 例如（z-> z^2-1） - 在重复应用下。在研究动态系统时，主要问题是：长期行为如何取决于初始条件？动态系统的“朱莉娅集”是边界以不同的长期行为来划定初始条件。在上面的示例中，更普遍地，朱莉娅集合是分形的。不仅要孤立地研究一个地图的动力学行为，而且是地图家族家族中动态行为的变化是很自然的。例如，人们可以考虑形式的所有动态系统（z-> z^2+常数），而不是仅仅看（z-> z^2-1）。复杂动力学领域通过Douady和Hubbard对Mandelbrot集的探索进行了转换，该集合生活在动态系统的空间内（z-> z^2+常数）。 Mandelbrot集的边界是一种分形，它以质量不同的长期行为来划定动态系统。地图的动态行为反映了临界点的长期行为 - 衍生物为零的点。例如，仅当临界点“ 0”具有界限时，才连接（Z-> Z^2+常数）的Julia集。批判性有限的（PCF）地图是地图，每个临界点最终都会在周期性周期中降落。它们的动力学行为可以组合编码，理解PCF图对于更普遍地理解复杂的动态至关重要。 PCF地图在理性地图的家族中具有非常特殊的分布，例如它们在Mandelbrot集的边界中密集。 PCF地图还提供了一个变量和许多变量中的动力学之间的引人入胜的联系。 Thurston通过构造称为“ Thurston Realback Maps”的动态系统，证明了PCF地图的结果刚度结果，其固定点是一个变量中的PCF地图。 Thurston的回溯图作用于高维teichmuller空间：了解其动态行为“近乎无限”对于理解PCF地图以及理解理性地图的退化至关重要。根据科赫（Koch）的工作，瑟斯顿（Thurston）的回调图具有称为hurwitz对应的代数几何“阴影”。这为使用模量空间的组合代数几何形状的工具提供了一个新的机会，以解决复杂动力学中的问题。热带几何形状是对代数几何形状中的退化的研究：它是一个非常适合研究Hurwitz对应关系和Thurston的回溯图的动力学的框架。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，以统一积极的研究主题但在不同领域的对象。我们将将动力学学位（许多变量的代数动力学）与瑟斯顿障碍物（Teichmuller-theocor对象）联系起来。这将将Hurwitz对应的全球代数动力学联系到Thurston urback映射无穷大的局部动力学。我们将将哈伯德树（一个变量中的理性动力学）与可允许的覆盖物（组合代数几何形状）和热带曲线联系起来。