The author will undertake several projects in the area of dynamical systems and ergodic theory. The first project concerns open systems from which orbits are allowed to escape. The project will study various mechanisms which facilitate or hinder escape in nonuniformly hyperbolic systems (such as billiards or Henon maps) and quantify the relation between escape rate and positive Lyapunov exponents. The second 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. The third project concerns the Ulam discretization problem for hyperbolic maps. This is an important tool used in the numerical study of dynamical systems. All three projects involve a detailed analysis of the spectral properties of the transfer operator associated with the corresponding 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 which 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 billiards 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. In addition, the method of Ulam discretization provides a practical way to approximate complex systems numerically. 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.
