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

TL;DR

This paper investigates the distribution and density of expanding twist tori on moduli spaces of translation surfaces, especially Veech surfaces, providing criteria for their equidistribution and support without subsequence extraction.

Contribution

It offers new criteria for the density and measure distribution of expanding twist tori on moduli spaces of Veech surfaces, avoiding subsequence limitations.

Findings

Criteria for density of expanding tori in the moduli space.

Conditions ensuring uniform lower bounds on measure limits.

Examples of Veech surfaces satisfying the criteria.

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 $\textit{twist torus}$ 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 Teichm\"uller 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 $\textit{Veech surface}$. We provide sufficient criteria for these tori to become dense within the conjectured limiting locus $\mathcal{M} :=\overline{\mathrm{SL}_2(\mathbb{R})\cdot \mathbb{T}(\omega)}$ 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 $g_t\cdot \mathbb{T}(\omega)$ 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.