An official website of the United States government
Slope gap distribution of saddle connections on the 2n-gon
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.
more »
« less
- PAR ID:
- 10470913
- Publisher / Repository:
- American Institute of Mathematical Sciences
- Date Published:
- Journal Name:
- Discrete and Continuous Dynamical Systems
- Volume:
- 43
- Issue:
- 1
- ISSN:
- 1078-0947
- Page Range / eLocation ID:
- 1 to 56
- Subject(s) / Keyword(s):
- translation surfaces, saddle connections, gap distributions, horocycle flow, dynamical systems
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
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.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3934/dcds.2022141
