Phase Transitions in Smooth Dynamical Systems

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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'.