Veech Surfaces and Expanding Twist Tori on Moduli Spaces of Abelian Differentials

Abstract

Let $(M,\omega )$ be a translation surface such that every leaf of its horizontal foliation is either closed, or joins two zeros of $\omega$. Then, $M$ decomposes as a union of horizontal Euclidean cylinders. The $\text{twist torus}$ of $(M,\omega )$, denoted $\mathbb{T}(\omega )$, consists of all translation surfaces obtained from $(M,\omega )$ by applying the horocycle flow independently to each of these cylinders. Let $g_t$ be the Teichmüller geodesic flow. We study the distribution of the expanding tori $g_t\cdot \mathbb{T}(\omega )$ on moduli spaces of translation surfaces in cases where $(M,\omega )$ is a $\text{Veech surface}$. We provide sufficient criteria for these tori to become dense within the conjectured limiting locus $\mathcal{M}:=\overline{{\mathrm{SL}}_2(\mathbb{R})\cdot \mathbb{T}(\omega )}$ as $t\to \infty$. We also provide criteria guaranteeing a uniform lower bound on the mass a given open set $U\subset \mathcal{M}$ must receive with respect to any weak-$\ast$ limit of the uniform measures on $g_t\cdot \mathbb{T}(\omega )$ as $t\to \infty$. In particular, all such limits must be fully supported in $\mathcal{M}$ in such cases. Finally, we exhibit infinite families of well-known examples of Veech surfaces satisfying each of these results. A key feature of our results in comparison to previous work is that they do not require passage to subsequences.

45 pages. Comments are welcome