Affine mappings of translation surfaces: shrinking targets and Diophantine properties

TL;DR

The paper studies Diophantine approximation properties of orbits under the affine group of a translation surface with lattice Veech group. It constructs an induced $SL_2({\mathbb R})$-action on a bundle $Y$ and embeds it as an affine-invariant suborbifold in a moduli space of marked translation surfaces, enabling the use of a spectral gap result (Avila–Gouëzel) and a quantitative mean ergodic theorem. A shrinking-target framework is developed to relate target-avoidance in the ambient $SL_2({\mathbb R})$-action to Diophantine approximation by affine maps on the surface, yielding a lower bound $\alpha_{\omega} \ge \beta_{\omega} > 0$ for the Diophantine exponent, with optimality in tempered cases (when $\beta_{\omega}=1$). The work clarifies when the exponent is governed by the spectral gap and discusses potential non-tempered behavior via complementary series, contributing to both translation-surface dynamics and Diophantine approximation via homogeneous dynamics.

Abstract

Let $(X,ω)$ be a translation surface whose Veech group $Γ$ is a lattice. We prove that the generic orbit of the group of affine homeomorphisms of $(X,ω)$ can be used to approximate each point of $X$ with Diophantine precision. The proof utilizes an induced $SL_2(\mathbb{R})$-action on a fiber bundle $Y$ whose base is $SL_2(\mathbb{R})/Γ$ and whose fiber is $X$. We observe that this bundle embeds as an $SL_2(\mathbb{R})$-orbit closure in the moduli space of once marked translation surfaces, and hence we may invoke the spectral gap results of Avila and Gouëzel and a quantitative mean ergodic theorem for the $SL_2(\mathbb{R})$-action on the mean-zero, square-integrable functions on $Y$.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Proposition 2.1
• proof
• Lemma 2.2: Borel-Cantelli argument
• proof
• Lemma 2.3
• proof
• Proposition 2.4