Dynamical systems without ergodic theory

Wednesday, 1st March 2023

Abstract

The study of the distribution of geodesics on flat surfaces has traditionally relied on results in ergodic theory such as Birkhoff's ergodic theorem. However, while this approach leads to very strong statements over an infinite time range, it does not seem possible to make corresponding statements over finite time ranges. By relying on number theory, combinatorics, geometry and linear algebra instead, we have developed a method which leads to very strong time-quantitative statements concerning uniformity and density in cases where the slope of the geodesic satisfies certain diophantine restrictions. In this talk, we describe some of these developments. (Joint work with Jozsef Beck, Michael Donders and Yuxuan Yang)