Affine mappings of translation surfaces: shrinking targets and Diophantine properties

TL;DR

This paper demonstrates that for certain translation surfaces with lattice Veech groups, the affine group orbits can approximate points with Diophantine precision, leveraging ergodic theory and spectral gap results.

Contribution

It introduces a method to use affine group orbits on translation surfaces for Diophantine approximation, connecting dynamics with number theory.

Findings

Affine orbits approximate points with Diophantine precision.

Utilizes $SL_2(\mathbb{R})$-action and spectral gap results.

Embeds orbit closure in moduli space for analysis.

Abstract

Let $(X,\omega)$ be a translation surface whose Veech group $\Gamma$ is a lattice. We prove that the generic orbit of the group of affine homeomorphisms of $(X,\omega)$ 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 $SL_2(\mathbb{R})/\Gamma$ 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 Gou\"ezel and a quantitative mean ergodic theorem for the $SL_2(\mathbb{R})$-action on the mean-zero, square-integrable functions on $Y$.