Phase Transitions in Smooth Dynamical Systems

开展遍历理论和动力系统的研究、教育和推广项目，分析非紧或非均匀双曲系统的随机性质。在PI和其他人先前工作的基础上，该提案确定了与此类系统相关的某些现象与统计物理学中的关键现象（“相变”）之间的类比。PI提出了一个详细的程序来利用这个类比系统地探索非一致双曲或非紧性的遍历理论效应。该项目的智力价值在于它使用了一种新颖的方法来研究一类重要的动力系统：应用中的大多数动力系统不是均匀双曲的，或者只是在其相空间的非紧化部分上是双曲的。更广泛的影响是针对三个群体的组成部分：本科生、研究生和专家。宾夕法尼亚州立大学数学系有一个现有的项目，旨在将研究注入到早期的学习过程中：数学高等研究学期（MASS）项目。PI打算在这个项目中开设一门主题课程，目的是吸引本科生参加数学研究生课程。研究生部分包括培养博士生，PI将努力吸引博士生。专家组成部分是撰写一篇研究专著的计划，内容涉及PI在与提案相关的主题上的最新结果。建议的专著的标题是“可数马尔可夫位移的热力学形式”。

Sarig conduct a project of research, education and outreach in ergodic theory and dynamical systems, analyzing the stochastic nature of non-compact or non-uniformly hyperbolic systems. Building on previous work of the PI and others, the proposal identifies an analogy between certain phenomena associated with such systems and critical phenomena in statistical physics (`phase transitions'). The PI proposes a detailed program for utilizing this analogy to explore the ergodic theoretic effects of non-uniform hyperbolicity or non-compactness systematically. The intellectual merit of the project is that it uses an original approach to study an important class of dynamical systems: Most of the dynamical systems in applications are not uniformly hyperbolic, or are hyperbolic only on a non-compact part of their phase space. The broader impact is a component aimed at three groups: undergraduate students, graduate students, and experts. The mathematics department in the Pennsylvania State University has an existing program aimed at injecting research into the learning process at an early stage: the Mathematics Advanced Studies Semesters (MASS) program. The PI intends to develop a topic course to be taught within this program, with the purpose of attracting undergraduates to join a graduate program in mathematics. The graduate component consists of training Ph.D. students, which the PI will endeavor to attract. The expert component is a plan to write a research monograph, concerning the recent results of the PI on topics related to the proposal. The title of the suggested monograph is `The thermodynamic formalism of countable Markov shifts'.