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arXiv · 2601.02541

Affine mappings of translation surfaces: shrinking targets and Diophantine properties

Chris Judge·Josh Southerland
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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 $Y$.

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Chris Judge, Josh Southerland. 2026-01-05. Affine mappings of translation surfaces: shrinking targets and Diophantine properties. https://arxiv.org/abs/2601.02541

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