This project deals with ergodic theory and dynamical systems. The research will concentrate on three fundamental problems associated with various aspects of noncompactness in dynamical systems. The first problem considers measure rigidity phenomena for horocycle flows on infinite volume hyperbolic surfaces. The existing theory treats finite invariant measures, whereas the project involves a program to study the infinite, but locally finite, case. The second and third problems are concerned with the thermodynamic formalism for maps with infinite Markov partitions and for nonuniformly hyperbolic surface diffeomorphisms. Building on previous work of the principal investigator and others, the project will investigate an analogy between certain phenomena associated with such systems and critical phenomena in statistical physics (namely, phase transitions). The principal investigator will pursue a detailed program for utilizing this analogy to explore in a systematic way the ergodic theoretic effects of noncompactness or nonuniform hyperbolicity.
A dynamical system is a model that describes a system (think of a physical system) that can be in one of many possible states, together with a law that prescribes how the state of the system evolves in time. Such models are used frequently in mathematics, physics, biology, and engineering. Most of the mathematical research in dynamical systems hitherto has focused on systems whose collections of possible states are "small" in an appropriate sense (the precise mathematical term is "compact"). In contrast, this project studies dynamical systems with noncompact collections of states. It focuses on various dynamical phenomena that can appear only in the noncompact setting, most notably a new type of rigidity on the level of infinite invariant measures, as well as a certain collection of phenomena similar to critical phenomena that one encounters in physics. There are numerous potential applications of these ideas to other areas of mathematics, including geometry and mathematical physics. In particular, it is hoped that the results of the research will shed light on the theory of phase transitions in statistical physics.
