Interactions between dynamics, topology and combinatorics in low dimensions

混沌动力系统和底层空间的拓扑性质之间有着丰富的相互作用的历史。当动力系统是一个流时，这些关系可以追溯到所谓的双曲流或Anosov流的经典工作。在三维中，双曲流的现代表现是伪Anosov流。这些流动已成为重要的理论双曲三流形由于工作，例如，弗里德，Calegari，克里斯蒂，芬利和莫舍。一些这样的深层次关系涉及到表面的动力学，流的周期轨道的同伦性质，以及底层流形的大尺度几何。在最近的结果中，伪Anosov流是一个显着的对应关系，伪Anosov流转向三角形。转向三角剖分是由Agol引入的一种纯组合结构。除了动力学之外，三角剖分还阐明了底层空间的各种拓扑性质。我们将进一步探索伪Anosov流和三维流形理论之间的关系。一个主要的动机将是与转向三角测量的对应关系。用转向三角形交换流动，使我们能够用组合学和计算来研究流动。一些问题激励我们的工作是决定当一个任意流承认转向三角形，确定当两个流是等价的，枚举所有的流量，并了解如何修改流修改其三角形。这些问题将需要新的技术和对当前机制的详细了解。

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.