Thermodynamic Formalism and Homological Characteristics of Anosov Flows

阿诺索夫流的热力学形式和同调特性

• 批准号: 2105821
• 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: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2105821
• 关键词: Thermodynamic; Formalism; Homological; Characteristics; Anosov

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.

该项目是在千古理论之间的接口：动态系统理论的分支，该理论使用数学分析来理解统计属性和低维拓扑。该项目的目的是使用热力学形式主义的技术来研究受同源限制的周期性轨道的分布，并找到相关数量的渐近结果，例如周期性轨道的链接数量。该项目的起点是Contreras的工作，他们使用Bowen的等均分配理论评估了3-Sphere中双曲线流的周期性轨道的渐近平均值。人们可以利用现在通过热力学形式主义而存在的更一般的等分分布理论，从而将Contreras的工作推广到加权平均和任意3个manifolds，这似乎是合理的。之后，人们希望能够在平均链接数字和流体动力螺旋之间建立联系。由于Arnold和Vogel的工作，已经知道螺旋性具有类似于这些平均链接数字的形式。这将是该项目的主要目的。关于周期性轨道如何分布在同源类课程中的许多先前的结果。该项目中的任何进一步工作都将尝试将这些结果延续到受损坏的同源性课程的背景下，在这种情况下，它不是歧管的第一个同源组，而是研究了定期轨道补充的第一个同源性小组。在几何学，拓扑和数学分析的领域中，并且正在完全扭动数学科学主题。