Geometry and dynamics of systems with a hyperbolic flavor

双曲线系统的几何和动力学

• 批准号: RGPIN-2017-04592
• 负责人: Barthelme, Thomas
• 金额: $ 2.04万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758905
• 关键词: Geometry; dynamics; systems; hyperbolic; flavor

The goal of this proposal is to promote research and the training of HQPs in domains of theoretical mathematics at the intersection of dynamical systems, geometry and low dimensional topology. My research program focuses on two distinct subfamilies of questions. The first involves the understanding and topological classification of certain types of dynamical systems in manifolds of low dimension. The second is concerned with a range of questions regarding the interactions between the dynamical properties of systems of geometrical origin (the geodesic flow of a Finsler or Riemannian manifold being the foremost), and geometric or spectral data coming from the metric.Both of these research domains have a long and fruitful history and my proposed research will answer some naturally arising new questions as well as revisit older open problems with new tools and ideas. This research will lead to a better understanding of the relationships between some dynamical systems and the geometrical or topological aspects of the ambient space. The tools used in my work are also applicable in many different domains. Both Finsler geometry and dynamical systems with some hyperbolic behaviour can be found, and used, for instance, in Physics, Biology, or Numerical Analysis.A very general question, which has been studied in various forms for many years, is the following: Given a specific type of dynamical system, what properties must a manifold possess in order to support such a system? And, given one such manifold, we want to obtain a classification of all the "different" dynamical systems of the original type that exists on the manifold. In this proposal, I aim to make progress on that question for two related types of dynamical systems -- Anosov flows and partially hyperbolic diffeomorphisms -- on 3- or 5-manifolds, as, for now at least, the low dimensional case is the only one where we can hope to obtain significant results. The existence of such dynamical systems, which are two stereotypical examples of systems with chaotic behaviour, have already been shown to be intrinsically linked to some fine topological properties of the manifolds supporting them. My research will lead to a deeper understanding of these relationships.The second topic highlighted in this proposal is concerned with Finsler or Riemannian metrics with hyperbolic behaviour (defined here as either negatively curved metrics, or more generally, metrics with Anosov geodesic flow) and revolves around the generic question of how "similar" two metrics have to be to have the same unmarked length spectrum (i.e., the length of their closed geodesics, counted with multiplicity, are the same). This type of question (think of the famous "Can one hear the shape of a drum?") has previously led to some significant achievements and my work will further our knowledge of the relationships between some dynamical and geometrical data.

本提案的目标是在动力系统、几何和低维拓扑交叉的理论数学领域促进hqp的研究和训练。我的研究计划主要集中在两个不同的亚家族的问题。第一部分涉及对低维流形中某些类型动力系统的理解和拓扑分类。第二个是关于几何起源系统的动力学特性（芬斯勒流形或黎曼流形的测地线流是最重要的）与来自度量的几何或光谱数据之间相互作用的一系列问题。这两个研究领域都有着悠久而富有成果的历史，我提出的研究将回答一些自然产生的新问题，并使用新的工具和想法重新审视旧的开放问题。这项研究将有助于更好地理解一些动力系统与周围空间的几何或拓扑方面之间的关系。我工作中使用的工具也适用于许多不同的领域。芬斯勒几何和具有某些双曲行为的动力系统都可以在物理学、生物学或数值分析中找到并使用。一个非常普遍的问题，已经以各种形式研究了许多年，它是这样的：给定一种特定类型的动力系统，流形必须具有什么性质才能支持这样的系统？并且，给定一个这样的流形，我们想要得到流形上存在的所有原始类型的“不同”动力系统的分类。在本提案中，我的目标是在3-或5-流形上的两种相关类型的动力系统——阿诺索夫流和部分双曲微分同态——的问题上取得进展，因为，至少现在，低维情况是我们唯一希望获得显著结果的情况。这类动力系统的存在，是具有混沌行为的系统的两个典型例子，已经被证明与支持它们的流形的一些精细拓扑性质有内在联系。我的研究将导致对这些关系的更深层次的理解。本提案中强调的第二个主题涉及具有双曲行为的芬斯勒或黎曼度量（这里定义为负弯曲度量，或者更一般地说，具有Anosov测地线流的度量），并围绕着两个度量如何“相似”才能具有相同的未标记长度谱（即，它们的封闭测地线的长度，用多重计数，是相同的）的一般问题。这类问题（想想著名的“一个人能听到鼓的形状吗？”）已经取得了一些重要的成就，我的工作将进一步加深我们对一些动力学和几何数据之间关系的认识。