A Transversal for horocycle flow on H(2)

TL;DR

This paper constructs a Poincaré section for horocycle flow on genus 2 translation surfaces with a singular cone point, using zippered rectangle coordinates, and analyzes return times for these surfaces.

Contribution

It introduces a parametrization of a Poincaré section for the horocycle flow on a specific moduli space using zippered rectangle coordinates.

Findings

Bounded return times for certain surfaces with large parameter sums

Explicit parametrization of the Poincaré section

Insights into the dynamics of horocycle flow on genus 2 surfaces

Abstract

Using zippered rectangle coordinates we parametrize a Poincar\'e section for horocycle flow on the space of genus 2 translation surfaces with one singular cone point of angle $6\pi$. In addition, we bound the return time under horocycle flow to this Poincar\'e section by examining a subset of surfaces where a certain sum of parameters is large.