Thermodynamic Formalism and Dynamical Systems Arising from Geometry

本研究的主要目的是进一步发展的热力学形式主义的理论，并适用于强大的技术，从遍历理论和动力系统的特殊几何利益的例子。热力学形式主义是研究动力系统全局统计性质的一个强大而通用的工具，最初是由统计力学的思想激发的。该项目的主要目标是（1）研究Ricci流下负弯曲流形的动力学不变量的演化，（2）发展热力学技术来研究β变换和相关映射，（3）发展Teichmueller流的热力学形式。自1930年以来？年代，动力系统遍历理论发展的一个关键动力是几何问题的研究。这个项目属于丰富的传统，并涵盖了选择的问题，最先进的技术将产生新的结果，在动力系统和几何的交叉点。里奇流对负曲率流形的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里奇流文献的最新创新。自1957年以来一直被广泛研究的β变换，在数论中自然出现。由于主要研究者最近共同撰写的一项突破，现在有可能获得新的结果。Teichmueller理论是当前研究的一个领域，它是几何学、拓扑学、数论和动力学的交叉。有很大的发展范围内的系统产生的热力学形式主义在这方面，其结果将是有用的各种几何和统计应用。该项目将推进动力系统的理论，并建立数学的不同分支（动力学，PDE，数论，Teichmueller理论）之间的联系。这项研究的重点是获得基本的纯结果，因此这里开发的工具将在动力学，几何学等领域产生未来的应用。此外，该项目还具有一些教育效益。主要研究者将（1）通过出版物和讲座传播研究，（2）通过为研究生和高级本科生提供迷你课程和研讨会，将研究与教学相结合，（3）与宾夕法尼亚州立大学外展部合作，促进K-12水平的学习和参与，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.