The author will undertake several projects in the area of dynamical systems and ergodic theory. The first project concerns open systems, which are inspired by physical models in which mass or energy is allowed to escape. The project will study various mechanisms that facilitate or hinder escape in nonuniformly hyperbolic systems and use the relation between entropy and positive Lyapunov exponents to quantify the escape rate. The second project develops a powerful approach, the spectral decomposition of the transfer operator, to study the statistical properties of particle systems, which are an important class of models from mathematical physics. The third project investigates the behavior of dynamical systems which are comprised of (possibly infinitely many) smaller components linked together, with orbits or energy allowed to pass between components. When focused on one component at a time, such systems generalize the discussion of open systems in a natural way by allowing both entry and escape. All three projects involve a detailed analysis of the spectral properties of the transfer operator associated with the corresponding closed system without relying on restrictive Markovian assumptions on the dynamics.
Much research in dynamical systems has focused on closed systems in which the dynamics are self-contained. In many modeling situations, however, it is not possible to obtain such a global view so that it becomes necessary to study local systems that are influenced by other unknown systems, possibly on different scales. Such considerations motivate the study of the types of systems considered in this project: systems in which mass or energy may enter or exit through deterministic or random mechanisms. Many of these problems are motivated by models from mathematical physics. For example, open particle systems are used to model atom traps; extended and linked particle systems are used to create mechanical models of heat conduction in solids and to investigate metastability in molecular processes. The research will provide analytical tools to solve problems posed and approached formally in the physics literature. The project will both promote and be informed by this interdisciplinary dialogue. In addition, the project will support undergraduate research in mathematics. Using the highly visual nature and physical motivation of the problems described, the PI will recruit undergraduate students to work on projects related to these topics during the summers. Students will disseminate the results of their research at poster sessions and through publication in undergraduate or research journals, as appropriate. By stimulating interest in research careers in mathematics and creating a peer community supportive of that interest, the project will contribute to the important goal of integrating research and education.
