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

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 twist torus of $(M,ω)$, denoted T(ω), consists of all translation surfaces obtained from $(M,ω)$ by applying the horocycle flow independently to each of these cylinders. Let gₜ be the Teichmüller geodesic flow. We study the distribution of the expanding tori gₜ· T(ω) on moduli spaces of translation surfaces in cases where $(M,ω)$ is a Veech surface. We provide sufficient criteria for these tori to become dense within the conjectured limiting locus M :=SL₂(R)· T(ω)̄ as t→ ∞. We also provide criteria guaranteeing a uniform lower bound on the mass a given open set U must receive with respect to any weak- limit of the uniform measures on gₜ· T(ω) as t→∞. In particular, all such limits must be fully supported in 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.