Counting Saddle Connections on Hyperelliptic Translation Surfaces with a Slit
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917217727152128 |
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| author | Aulicino, David Masur, Howard Pan, Huiping Su, Weixu |
| author_facet | Aulicino, David Masur, Howard Pan, Huiping Su, Weixu |
| contents | We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that it satisfies an $L (\log L)^{d-2}$ growth rate, where $d$ is the complex dimension of the hyperelliptic stratum. The upper bound holds for all translation surfaces in the hyperelliptic stratum while the lower bound holds for almost every surface in the hyperelliptic stratum. The proof of the lower bound uses horocycle renormalization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15993 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting Saddle Connections on Hyperelliptic Translation Surfaces with a Slit Aulicino, David Masur, Howard Pan, Huiping Su, Weixu Dynamical Systems Geometric Topology We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that it satisfies an $L (\log L)^{d-2}$ growth rate, where $d$ is the complex dimension of the hyperelliptic stratum. The upper bound holds for all translation surfaces in the hyperelliptic stratum while the lower bound holds for almost every surface in the hyperelliptic stratum. The proof of the lower bound uses horocycle renormalization. |
| title | Counting Saddle Connections on Hyperelliptic Translation Surfaces with a Slit |
| topic | Dynamical Systems Geometric Topology |
| url | https://arxiv.org/abs/2601.15993 |