Phase Transitions in Smooth Dynamical Systems

SARIG在遍历理论和动力系统方面开展了一个研究、教育和推广项目，分析了非紧凑或非一致双曲系统的随机性质。在国际和平研究所和其他人以前工作的基础上，该提案确定了与这类系统有关的某些现象与统计物理中的关键现象(“相变”)之间的类比。PI提出了一个详细的程序，利用这个类比来系统地探索非均匀双曲性或非紧致性的遍历理论效应。该项目的智能优点在于，它使用了一种新颖的方法来研究一类重要的动力系统：大多数应用中的动力系统不是一致双曲的，或者只在其相空间的非紧部分上是双曲的。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，旨在将研究注入学习过程的早期阶段：数学高级研究学期(MASS)项目。国际数学学院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项目。研究生部分包括培养博士生，国际和平研究所将努力吸引这些学生。专家部分是编写一本研究专著的计划，涉及与提案有关的专题的国际和平倡议的最新结果。这本专著的标题是《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'.