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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.16186 |
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| _version_ | 1866914000901505024 |
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| author | McAdam, Taylor Yu, Xiaoxing |
| author_facet | McAdam, Taylor Yu, Xiaoxing |
| contents | In this paper we review the theory of slope gap distributions of translation surfaces and summarize the state-of-the-art for calculating slope gap distributions of Veech surfaces. We then derive the slope gap distribution of a particular 10-tile origami by considering the origami's return times to a Poincaré section under the horocycle flow on the moduli space associated with the origami. We show that the resulting distribution is not a sum of scaled Hall distributions, unlike all previously published origami slope gap distributions. More generally, this demonstrates that the slope gap distribution of a branched covering of a translation surface cannot necessarily be represented as a sum of scaled copies of the slope gap distribution of the base surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16186 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Origami slope gaps and the Hall distribution McAdam, Taylor Yu, Xiaoxing Dynamical Systems Geometric Topology 37D40 (Primary) 14H55, 37A17, 32G15 (Secondary) In this paper we review the theory of slope gap distributions of translation surfaces and summarize the state-of-the-art for calculating slope gap distributions of Veech surfaces. We then derive the slope gap distribution of a particular 10-tile origami by considering the origami's return times to a Poincaré section under the horocycle flow on the moduli space associated with the origami. We show that the resulting distribution is not a sum of scaled Hall distributions, unlike all previously published origami slope gap distributions. More generally, this demonstrates that the slope gap distribution of a branched covering of a translation surface cannot necessarily be represented as a sum of scaled copies of the slope gap distribution of the base surface. |
| title | Origami slope gaps and the Hall distribution |
| topic | Dynamical Systems Geometric Topology 37D40 (Primary) 14H55, 37A17, 32G15 (Secondary) |
| url | https://arxiv.org/abs/2508.16186 |