PreprintarXiv licencearXiv
Asymptotics for the number of saddle connections on a slit torus
Abstract
We consider a flat torus with a marked line segment β and prove linear asymptotics for closed curves on the torus disjoint from the interior of the line segment. We also consider segments that begin and terminate at the end points of β whose interiors are disjoint from β and prove Llog L asymptotics. In all cases, after quotienting by the growth rate, we compute the constant explicitly, which differs depending on whether or not β lies in a periodic direction on the torus. In particular, when β lies along a non-periodic trajectory, we show the constant is $2/(|β| ζ(2))$. Moreover, we prove that as a function of the holonomy vector of β, the constants are continuous at the points where β lies in a non-periodic direction.
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BibTeX
@misc{aulicino2026asymptotics,
title = {{Asymptotics for the number of saddle connections on a slit torus}},
author = {David Aulicino and Howard Masur},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24429},
eprint = {2610.24429},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24429},
}
RIS
TY - UNPB
TI - Asymptotics for the number of saddle connections on a slit torus
AU - David Aulicino
AU - Howard Masur
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24429
UR - https://arxiv.org/abs/2610.24429
AB - We consider a flat torus with a marked line segment β and prove linear asymptotics for closed curves on the torus disjoint from the interior of the line segment. We also consider segments that begin and terminate at the end points of β whose interiors are disjoint from β and prove Llog L asymptotics. In all cases, after quotienting by the growth rate, we compute the constant explicitly, which differs depending on whether or not β lies in a periodic direction on the torus. In particular, when β lies along a non-periodic trajectory, we show the constant is $2/(|β| ζ(2))$. Moreover, we prove that as a function of the holonomy vector of β, the constants are continuous at the points where β lies in a non-periodic direction.
ER -