Classification Results of Orbit Closures of Translation Surfaces

Abstract

Translation surfaces are a type of flat surface that generalizes the dynamics on flat tori to higher genera. This has applications to billiards and the finite blocking problem. Studying dynamics on individual translation surfaces is often done by studying a different dynamical system on the moduli space of translation surfaces. This thesis covers three classification results of orbit closures in these moduli spaces. First, we use the transfer principle to classify periodic points on a certain family of Veech surfaces. The next result is classifying orbit closures in a product of two components of strata. This is done with an induction argument and investigating the boundary of orbit closures. Finally, we reprove the classification of rank 2 orbit closures in some genus 3 strata. The main contribution of this new proof is providing code that can automatically check certain conditions to significantly simplify the work needed for those proofs. This code would also be useful in classifying orbit closures in other strata.