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

Abstract

Let $(M,ω)$ be a translation surface such that every leaf of its horizontal foliation is either closed, or joins two zeros of $ω$. Then, $M$ decomposes as a union of horizontal Euclidean cylinders. The $\textit{twist torus}$ of $(M,ω)$, denoted $\mathbb{T}(ω)$, consists of all translation surfaces obtained from $(M,ω)$ 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 $g_t\cdot \mathbb{T}(ω)$ on moduli spaces of translation surfaces in cases where $(M,ω)$ is a $\textit{Veech surface}$. We provide sufficient criteria for these tori to become dense within the conjectured limiting locus $\mathcal{M} :=\overline{\mathrm{SL}_2(\mathbb{R})\cdot \mathbb{T}(ω)}$ as $t\rightarrow \infty$. 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 $g_t\cdot \mathbb{T}(ω)$ as $t\rightarrow\infty$. 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.