Thermodynamic Formalism and Dynamical Systems Arising from Geometry

本研究的主要目的是进一步发展热力学形式主义理论，并将遍历理论和动力系统的强大技术应用于特殊几何兴趣的例子。热力学形式论是研究动力系统的全局统计性质的一种强大而通用的工具，它最初受到统计力学思想的启发。该项目的主要目标是：(1)研究负弯曲流形在Ricci流下的动力学不变量的演化；(2)开发研究β变换和相关映射的热力学技术；(3)建立Teichmueller流的热力学形式。从1930年代开始？在动力学系统遍历理论的发展中，一个关键的动机是对几何问题的研究。这个项目属于这一丰富的传统，涵盖了一系列问题，其中最先进的技术将在动力系统和几何的交叉点产生新的结果。Ricci流对负曲率流形动力学不变量的影响的研究结合了光滑动力系统的强大技术和Ricci流文献的最新创新。自1957年以来被广泛研究的β变换自然出现在数论中。由于最近由首席研究员共同撰写的突破，新的结果现在可能成为可能。Teichmueller理论是目前在几何学、拓扑学、数论和动力学交叉领域进行深入研究的一个领域。在这种情况下产生的系统的热力学形式主义有很大的发展空间，其结果将对各种几何和统计应用有用。该项目将推进动力系统理论，并建立数学不同分支（动力学、偏微分方程、数论、泰奇穆勒理论）之间的联系。这项研究的重点是得出基本的纯结果，因此这里开发的工具有很大的潜力，未来将在动力学、几何等领域得到应用。此外，该项目还有许多教育方面的好处。首席研究员将(1)通过出版物和演讲传播研究成果，(2)通过为研究生和高级本科生提供迷你课程和研讨会，将研究与教学结合起来，(3)与宾夕法尼亚州立大学外展中心合作，促进K-12阶段的学习和参与，特别强调代表性不足的群体。

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.