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We explicitly compute the limiting slope gap distribution for saddle connections on any 2n-gon for n greater than or equal to 3. Our calculations show that the slope gap distribution for a translation surface is not always unimodal, answering a question of Athreya. We also give linear lower and upper bounds for number of non-differentiability points as n grows. The latter result exhibits the first example of a non-trivial bound on an infinite family of translation surfaces and answers a question by Kumanduri-Sanchez-Wang.
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Slope gap distributions of Veech surfaces
The slope gap distribution of a translation surface is a measure of how random the directions of the saddle connections on the surface are. It is known that Veech surfaces, a highly symmetric type of translation surface, have gap distributions that are piecewise real analytic. Beyond that, however, very little is currently known about the general behavior of the slope gap distribution, including the number of points of non-analyticity or the tail. We show that the limiting gap distribution of slopes of saddle connections on a Veech translation surface is always piecewise real-analytic with finitely many points of non-analyticity. We do so by taking an explicit parameterization of a Poincaré section to the horocycle flow on SL(2,R)/SL(X,ω) associated to an arbitrary Veech surface SL(X,ω) and establishing a key finiteness result for the first return map under this flow. We use the finiteness result to show that the tail of the slope gap distribution of Veech surfaces always has quadratic decay.
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- Award ID(s):
- 2103136
- PAR ID:
- 10639733
- Publisher / Repository:
- MSP
- Date Published:
- Journal Name:
- Algebraic & Geometric Topology
- Volume:
- 24
- Issue:
- 2
- ISSN:
- 1472-2747
- Page Range / eLocation ID:
- 951 to 980
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.2140/agt.2024.24.951
