Thermodynamic Formalism and Dynamical Systems Arising from Geometry

热力学形式主义和几何产生的动力系统

• 批准号: 1101576
• 负责人: Daniel Thompson
• 金额: $ 8.88万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-15 至 2012-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101576&HistoricalAwards=false
• 关键词: Thermodynamic; Formalism; Dynamical; Systems; Arising

The key aims of this research are to further develop the theory of thermodynamic formalism and to apply powerful techniques from ergodic theory and dynamical systems to examples of special geometric interest. Thermodynamic formalism is a powerful and versatile tool in the study of the global statistical properties of dynamical systems, originally motivated by ideas from statistical mechanics. Key goals of the project are (1) to study the evolution of dynamical invariants of negatively curved manifolds under Ricci flow, (2) to develop thermodynamic techniques to study the beta-transformation and related maps, (3) to develop thermodynamic formalism for the Teichmueller flow. Since the 1930?s, a key motivation in the development of the ergodic theory of dynamical systems has been the study of problems in geometry. This project belongs to that rich tradition, and covers a selection of problems where state of the art techniques will produce new results at the intersection of dynamical systems and geometry. The investigation of the effect of Ricci flow on dynamical invariants of negative curvature manifolds combines the powerful techniques of smooth dynamical systems with state of the art innovations from the Ricci flow literature. The beta-transformation, which has been studied extensively since 1957, arises naturally in number theory. New results are now possible due to a recent breakthrough co-authored by the Principal Investigator. Teichmueller theory is an area of intensive current research at the intersection of geometry, topology, number theory and dynamics. There is great scope for the development of thermodynamic formalism for systems arising in this context, and the results will be useful for a variety of geometric and statistical applications. The project will both advance the theory of dynamical systems and build connections between different branches of mathematics (dynamics, PDE, number theory, Teichmueller theory). The focus of this research is on deriving fundamental pure results, so there is significant potential that the tools developed here will yield future applications in dynamics, geometry and beyond. In addition, the project has a number of educational benefits. The Principle Investigator will (1) disseminate the research through publications and talks, (2) integrate research with teaching by delivering mini-courses and seminars for graduate students and advanced undergraduates, (3) work with Penn State Outreach to promote learning and participation at the K-12 level, with particular emphasis on underrepresented groups.

这项研究的关键目的是进一步发展热力学形式主义的理论，并将来自Ergodic理论和动力学系统的强大技术应用于特殊几何兴趣的示例。热力学形式主义是研究动态系统的全球统计特性的强大而多才多艺的工具，最初是由统计力学的思想促进的。该项目的关键目标是（1）研究RICCI流下的负弯曲流形动力不变的演变，（2）开发热力学技术来研究β-变换及相关图，（3）以开发用于为特性的形式而发展的热力学形式。 Teichmueller流。自1930年以来，动力学系统的厄运理论发展的一个关键动机一直是对几何问题的研究。该项目属于这种丰富的传统，并涵盖了一些问题，在这些问题中，最先进的技术将在动态系统和几何形状的交集中产生新的结果。 RICCI流对负曲率歧管动力不变的影响的研究结合了平滑动力学系统的强大技术与RICCI流量文献的最新创新状态。自1957年以来已经进行了广泛研究的β-转化，在数量理论中自然而然地出现。由于主要研究人员的最新突破，现在可以新的结果。 Teichmueller理论是在几何学，拓扑，数理论和动力学的交集中进行深入研究的一个领域。在这种情况下为系统产生的系统的热力学形式主义的开发范围很大，结果将对多种几何和统计应用有用。该项目既将推进动力学系统的理论，又将建立数学不同分支（动态，PDE，数字理论，Teichmueller理论）之间的联系。这项研究的重点在于得出基本的纯效果，因此这里开发的工具具有很大的潜力，将在动态，几何以及其他方面产生未来的应用。此外，该项目具有许多教育益处。原则研究者将（1）通过出版和谈判传播研究，（2）通过为研究生和高级本科生提供迷你教堂和研讨会，将研究与教学整合，（3）与宾夕法尼亚州立大学的宣传合作，以促进学习和参与K-12水平特别强调代表性不足的群体。