Dynamics on Translation Surfaces and on Spaces of Translation Surfaces

The field of dynamical systems studies the way points in a closed system, such as planetary motion, travel within that system over time. Ergodic theory uses the notion of measure, which is a generalization of area for sets in the plane or volume for sets in space, to study dynamical systems. Over time, ergodic theory has developed connections to probability theory, number theory, physics and other areas.

Using tools from ergodic theory, this project plans to study dynamical systems that are connected to systems consisting of a point mass traveling in a frictionless polygon, where the point mass is assumed to obey the laws of elastic collision when hitting the side. They are also connected to other areas of mathematics including geometric topology and algebraic geometry. This project will also involve the training of graduate students in this area.

This project seeks to improve our understanding of the horocycle flows and "Real rel" flows on strata translation surfaces and translation flows on translation surfaces themselves. These systems are related, and the study of one informs the study of the other with tools and questions. The principal investigator will study the basic properties of these systems, including the behavior of points, the structure of the set of invariant measures and mixing properties.

Additionally, some questions motivated by geometric topology and geometric group theory will be considered.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.