Separation of horocycle orbits on moduli space in genus 2

TL;DR

This paper establishes a quantitative closing lemma for horocycle flows on genus 2 moduli space, linking small transversal separation to the existence of nearby pseudo-Anosov points, advancing understanding of orbit separation.

Contribution

It introduces a Margulis function to measure orbit separation and proves a new quantitative closeness result for horocycle flows on genus 2 surfaces.

Findings

Quantitative bounds on horocycle orbit separation.

Existence of nearby pseudo-Anosov points when orbits are close.

Connection to fractal dimension and Veech groups.

Abstract

We prove a quantitative closing lemma for the horocycle flow induced by the $\mathrm{SL}(2,\mathbb{R})$-action on the moduli space of Abelian differentials with a double-order zero on surfaces of genus 2. The proof proceeds via construction of a Margulis function measuring the discretized fractal dimension of separation of a horocycle orbit of a point from itself, in a direction transverse to the $\mathrm{SL}(2,\mathbb{R})$-orbit. From this, we deduce that small transversal separation guarantees the existence of a nearby point with a pseudo-Anosov in its Veech group. This is reminiscent of the initial dimension phases in Bourgain-Gamburd for random walks on compact groups, Bourgain-Lindenstrauss-Furman-Mozes for quantitative equidistribution in tori, and quantitative equidistribution of horocycle flow for a product of $\mathrm{SL}(2,\mathbb{R})$ with itself due to Lindenstrauss-Mohammadi-Wang, and multiple other works.