Dynamical systems can be thought of as the study of objects in motion, e.g. planets, where the motion evolves over time according to a fixed set of rules. The evolution can be structured or quite complex and even chaotic. This research project studies a family of dynamical systems that are highly structured but whose long term behavior can often be understood via a chaotic system that renormalizes the dynamics. The analysis of these problems involves dynamical, geometric, and combinatorial arguments. The project also has a large component geared towards smoothing the progression of early career mathematicians. This includes research mentoring for undergraduate and graduate students, professional mentoring for graduate students, notes to introduce first year graduate students to ergodic theory while preparing them for qualifying exams in real analysis, a problem list of topics in the area, and a collection of key arguments in ergodic theory.
The goal of this project is to better understand the dynamics of interval exchange transformations, flows on translations surfaces, and Teichmueller geodesic flow. Interval exchange transformations and flows on translation surfaces are closely related objects. Interval exchange transformations arise as first return maps of flows on translation surfaces, and similarly a translation surface can be built from an interval exchange with additional suspension data. Questions of interest are generalizations of the isomorphism problem, non-unique ergodicity and its variants, the behavior of bounded sets, and the behavior of actions of matrices on the space of translation surfaces. These problems will be studied using renormalization dynamics -- Rauzy induction and Teichmueller geodesic flow -- as well as combinatorial constructions, especially in building examples with exotic properties.
