Separation of horocycle orbits on moduli space in genus 2

We prove a quantitative closing lemma for the horocycle flow induced by the SL (2, 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 SL (2, 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 SL (2, R) with itself due to Lindenstrauss-Mohammadi-Wang, and multiple other works.