|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
Slope gap distributions
of Veech surfaces
Luis Kumanduri, Anthony Sanchez and Jane Wang |
|
Algebraic & Geometric Topology 24 (2024) 951–980
|
 |
Abstract |
|
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 nonanalyticity 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 nonanalyticity. We do so by taking an explicit parametrization of a Poincaré section to the horocycle flow on associated to an arbitrary Veech surface , 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 a Veech surface always has quadratic decay. |
Keywords
gap distributions, slope gaps, saddle connections, holonomy
vectors, translation surfaces, Veech surfaces
|
Mathematical Subject Classification
Primary: 32G15, 37D40
Secondary: 14H55
|
References |
Publication
Received: 5 December 2021
Revised: 5 August 2022
Accepted: 3 September 2022
Published: 12 April 2024
|
Authors |
| Luis Kumanduri | |
| Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States |
|
| Anthony Sanchez | |
| Department of Mathematics University of California San Diego La Jolla, CA United States |
|
| Jane Wang | |
| Department of Mathematics and
Statistics University of Maine Orono, ME United States |
|
| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
Open Access made possible by participating institutions via Subscribe to Open.
