This project involves the extension of ergodic theoretic results about interval exchange transformations (IETs) to cases of interval exchanges involving infinitely many intervals. The primary question of interest is to classify the locally finite ergodic invariant measures of an infinite IET. Interest in the infinite case is motivated by connections between infinite IETs and more classical topics. Progress on these classes of examples will make significant contributions to the field of infinite ergodic theory, and to the subject of dynamical systems. Recently, the work of the principal investigator and others in the field have shown the existence of parallels between the theory of infinite IETs, and the much more well developed theory of horocyclic flows on non-compact hyperbolic surfaces. Motivated and directed by these connections, known results about infinite IETs will be extended and new results will be produced.
Dynamical systems is the study of any system which updates according to a fixed rule. For example, a finite number of particles behaving according to the rules of Newtonian mechanics is a dynamical system. This project concerns systems which are said to have low complexity, meaning that similar states for the system take a very long time to become dissimilar. These systems are of interest because they arise naturally from geometric, algebraic and physical questions. Despite the relevance of these systems, questions involving low complexity systems often can not be directly addressed by approaches prevalent in the theory of dynamical systems, because, for instance, these systems are not hyperbolic. This is particularly true for questions about the long term statistical behavior of low complexity systems. This project will extend the collection of tools which do apply to low complexity systems, and further extend the classes of systems to which these tools apply. Progress on these questions is of importance to the development and applicability of the theory of dynamical systems, which is one of the primary mathematical approaches available to understand the physical world. The PI will be working with undergraduates on related research during the summer.
