Thermodynamic Formalism and Homological Characteristics of Anosov Flows

这个项目是关于遍历理论和低维拓扑之间的接口：动力系统理论的一个分支，使用数学分析来理解统计特性。这个项目的目的是利用热力学形式的技巧来研究同调约束下流动的周期轨道的分布，并找到相关参数的渐近结果，如周期轨道的连接数。该项目的起点是Contrera的工作，他利用Bowen的等分布理论计算了三维球面双曲流周期轨道的渐近平均连接数。人们似乎可以利用现在通过热力学形式主义而存在的更一般的均匀分布理论，将Contrera的工作推广到加权平均和任意的3-流形。在此之后，人们希望能够在平均连接数和流体动力学螺旋度之间建立联系；由于Arnold和Vogel的工作，螺旋度已经被认为具有类似于这些平均连接数的形式。这将是该项目的主要目标。关于周期轨道如何在同调类中分布，已经有许多已有的结果。这个项目中的任何进一步工作都将尝试将这些结果推广到分支同调类的背景下，在分支同调类中，人们不是看流形的第一同调群，而是看周期轨道补集的第一同调群。研究领域是几何、拓扑和数学分析，完全扭曲了数学科学的主题。

This project is on the interface between ergodic theory: the branch of the theory of dynamical systems that uses mathematical analysis to understand statistical properties, and low-dimensional topology. The aim of this project is to use techniques from thermodynamic formalism to study the distribution of periodic orbits of flows subject to homological constraints, and find asymptotic results for related quantities, such as the linking numbers of periodic orbits.A starting point for the project is the work of Contreras, who used the equidistribution theory of Bowen to evaluate asymptotic average linking numbers of periodic orbits of hyperbolic flows in the 3-sphere. It seems plausible that one could utilise the more general equidistribution theory that now exists through thermodynamic formalism, to generalise Contreras' work to weighted averages and arbitrary 3-manifolds. Following this, one would hope to be able to make a connection between average linking numbers and hydrodynamical helicity; the helicity is already known to have a form similar to these average linking numbers, due to the work of Arnold and Vogel. This will be the main aim of the project.There are many pre-existing results on how periodic orbits are distributed in homology classes. Any further work in this project will be an attempt to carry these results over to the context of ramified homology classes, where instead of the manifold's first homology group, one looks at the first homology group of the complement of a periodic orbit.The research is in the areas of geometry, topology and Mathematical analysis and is wholly writhing the Mathematical Sciences themer.