Hyperbolic Dynamics in Physical Systems and Ergodic Theory

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This project concerns mathematical research in the fields of dynamical systems and ergodic theory as well as applications to mathematical physics. The research focuses on hyperbolic dynamical systems, where a small perturbation causes exponential uncertainty in both negative and positive time. Such systems appear in many real-world dynamics, for example in interacting particle systems. Much is known for hyperbolic systems of particles with low degrees of freedom and with hard ball interactions by models of mathematical billiards. An important goal of this research is to investigate the case of high degrees of freedom, which is more relevant for physical systems - ergodic theory provides an abstract point of view on dynamical systems by studying time and space averages instead of the local geometry. Another research goal is to extend the understanding of infinite ergodic theory and partial chaos by hyperbolic dynamics examples. Moreover, a summer school will be offered to advanced high school or first year undergraduate students and several research projects will involve both undergraduate and Ph.D. students.The research is comprised of three main parts. The first part concerns the study high dimensional billiards and deterministic walks. Topics that are well understood in two-dimensional billiards (complexity bounds, long flight times) will be investigated in higher dimensions. The second part is devoted to hyperbolic dynamics in physical systems. Based on earlier results by the investigator on smaller building blocks, larger systems of mass and energy transport will be studied. Both these parts make significant advances towards better mathematical models of high dimensional real-life dynamical systems. The third part is to study the role of hyperbolic dynamics in infinite ergodic theory: a field which is more suitable for some large physical systems than traditional finite ergodic theory. New examples of partial chaos will also be constructed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.