Slope gap distribution of the double heptagon and an algorithm for determining winning vectors

2025/08/10 by Assal, Fernando Al, Ali, Nada, Arengo, Uma +4
#14H55 (Primary) 37D40 #32G15 (Secondary) #37A17 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2508.19252

Abstract

In this paper, we study the distribution of renormalized gaps between slopes of saddle connections on translation surfaces. Specifically, we describe a procedure for finding the "winning holonomy vectors" as defined by Kumanduri-Sanchez-Wang in arXiv:2102.10069, which constitutes a key step in calculating the slope gap distribution for an arbitrary Veech surface. We then apply this method to explicitly compute the gap distribution for the regular double heptagon translation surface. This extends work of Athreya-Chaika-Lelievre in arXiv:1308.4203 on the gap distribution for the "golden L" translation surface, which is equivalent to the regular double pentagon surface.

Citations

• An Effective Slope Gap Distribution for Lattice Surfaces
• Gaps of saddle connection directions for some branched covers of tori
• The Boca-Cobeli-Zaharescu Map Analogue for the Hecke Triangle Groups G_q
• A Transversal for horocycle flow on H(2)
• Translation surfaces and their orbit closures: An introduction for a\n broad audience
• Poincaré sections for the horocycle flow in covers of SL(2,R)/SL(2,Z) and applications to Farey fraction statistics
• The gap distribution of slopes on the golden L
• A Poincaré section for horocycle flow on the space of lattices
• The distribution of gaps for saddle connection directions
• Siegel measures

Related

• The gap distribution of slopes on the golden L
• Gaps of saddle connection directions for some branched covers of tori
• An Effective Slope Gap Distribution for Lattice Surfaces
• The distribution of gaps for saddle connection directions
• Origami slope gaps and the Hall distribution