Interactions between dynamics, topology and combinatorics in low dimensions

低维动力学、拓扑学和组合学之间的相互作用

• 批准号: 2749484
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2749484
• 关键词: Interactions; between; dynamics; topology; combinatorics

There is a history of rich interplay between chaotic dynamical systems and the topological properties of the underlying space. When the dynamical system is a flow, these relationships date back to classical work on so-called hyperbolic or Anosov flows.In dimension three, a modern rendition of a hyperbolic flow is a pseudo-Anosov flow. These flows have become important in the theory of hyperbolic three-manifolds due to the work, for example, of Fried, Calegari, Christy, Fenley and Mosher. Some such deep relationships involve the dynamics of surface diffeomorphisms, homotopic properties of periodic orbits of the flow, and the large-scale geometry of the underlying manifold. Among recent results on pseudo-Anosov flows is a remarkable correspondence between pseudo-Anosov flows veering triangulations. Veering triangulations are a purely combinatorial structure introduced by Agol. In addition to dynamics, the triangulations illuminate various topological properties of the underlying space.We will further explore the relationships between pseudo-Anosov flows and the theory of three-manifolds. A primary motivation will be the correspondence with veering triangulations. Trading a flow for a veering triangulation allows us to study the flow with combinatorics and computation. Some problems motivating our work are deciding when an arbitrary flow admits a veering triangulation, determining when two flows are equivalent, enumerating all flows, and understanding how modifying the flow modifies its triangulation. Such problems will require novel techniques and a detailed understanding of the current machinery.

在混沌动力系统和底层空间的拓扑特性之间有着丰富的相互作用的历史。当动力系统是流动时，这些关系可以追溯到所谓的双曲流或阿诺索夫流的经典研究。在三维空间中，双曲流的现代表现形式是伪阿诺索夫流。由于Fried、Calegari、Christy、Fenley和Mosher等人的工作，这些流动在双曲三流形理论中变得非常重要。一些这样的深层关系涉及表面微分同态的动力学，流动周期轨道的同伦性质，以及底层流形的大尺度几何。在伪阿诺索夫流的最新结果中，伪阿诺索夫流转向三角测量之间具有显著的对应关系。转向三角网是Agol引入的一种纯组合结构。除了动力学，三角剖分阐明了底层空间的各种拓扑性质。我们将进一步探讨伪阿诺索夫流与三流形理论之间的关系。一个主要动机将是与转向三角测量相对应。将流转换为转向三角测量使我们能够使用组合学和计算来研究流。激励我们工作的一些问题是决定一个任意流何时允许转向三角测量，确定两个流何时是等效的，枚举所有流，以及理解如何修改流修改其三角测量。这样的问题需要新的技术和对现有机制的详细了解。