Dynamics of interval exchange transformations and flows on flat surfaces

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This project seeks to better understand the dynamics of interval exchange transformations, straight line flow on flat surfaces and billiard flow in rational polygons. These represent low complexity, zero entropy systems that have a high complexity, positive entropy renormalization map on them. Additionally, Eskin-Mirzakhani-Mohammadi recently showed that if one looks at the systems arising from the whole flat surface simultaneously then the geometry shows greater structure than one might naively expect. This project seeks to connect these related and different phenomena to prove results about the straight line flow on these surfaces. Ergodic theory is interested in describing the long term behavior of orbits of dynamical systems. The questions this project is interested come from statistical mechanics, number theory and ergodic theory. Here are some sample questions: if there are two point masses on a flat surface and one is sent out in a random direction, how quickly does it get close to the other? If one fires the point masses in different directions will the trajectories be related? When is it possible for them to be related? This project hopes to attack these problems through tools coming from geometry, homogeneous dynamics and ergodic theory. The hope is to better understand what typical behavior is and what untypical behavior can occur.