Affine mappings of translation surfaces: shrinking targets and Diophantine properties
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910002889883648 |
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| author | Judge, Chris Southerland, Josh |
| author_facet | Judge, Chris Southerland, Josh |
| contents | 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 $Y$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_02541 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Affine mappings of translation surfaces: shrinking targets and Diophantine properties Judge, Chris Southerland, Josh Dynamical Systems 37D40 (Primary), 32G15, 11K60 (Secondary) 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 $Y$. |
| title | Affine mappings of translation surfaces: shrinking targets and Diophantine properties |
| topic | Dynamical Systems 37D40 (Primary), 32G15, 11K60 (Secondary) |
| url | https://arxiv.org/abs/2601.02541 |