Connection points on double regular polygons

TL;DR

This work analyzes connection points on the translation surface obtained from the double regular $n$-gon for odd $n\ge 7$, with a focus on higher-degree trace fields. It leverages Hecke groups and a modulo-two reduction to derive obstructions for periodic directions, constructing a large family of trace-field points that are not connection points (including central points) for $n\neq 9$, and conjecturing a complete non-connection classification for $n=7$. For prime $n$, the paper provides an explicit non-periodic separatrix through a central point, giving a constructive direction in the trace field that cannot extend to a saddle connection. Overall, the results advance understanding of non-quadratic Veech surfaces and the interaction between trace fields, modular obstructions, and geometric trajectories, with implications for billiards in rational polygons.

Abstract

We study connection points on the double regular $n$-gon translation surface, for $n \geq 7$ odd and its staircase model. For $n \neq 9$, we provide a large family of points with coordinates in the trace field that are not connection points. This family includes the central points, and for $n=7$ we conjecture that all the remaining points are connection points. Further, in the case where $n \geq 7$ is a prime number, we provide a constructive proof by exhibiting an explicit separatrix passing through a central point that does not extend to a saddle connection.

Figures (7)

• Figure 1: From the double regular heptagon to its staircase model.
• Figure 2: The staircase $S_{11}$ represented in the plane, with the bottom left corner of $R_5$ set as the origin.
• Figure 3: The region $Z_0$ in the proof of Proposition \ref{['prop:irrational_height']}.
• Figure 4: The point $P$ and a cylinder $C$ in which $P$ has irrational height. We consider alternative coordinates for elements in $C$ in which the expression of the twist along $C$ is simple. These coordinates coincide with the chosen coordinates for $S_n$ for points lying at the intersection of the cylinder $C$ with $R_j \cup R_{j-1} \cup R_{j+1}$. The line $L_C(P)$ of points having the same height in $C$ as $P$ and whose coordinates coincide with the coordinates for $S_n$ is represented in bold red.
• Figure 5: In the first step, we show that ${\mathcal{P}}_{(u,v)}$ is dense on $L_C(P)$ using the twist in a cylinder $C$ in which $P$ has irrational height. In the second step, we use a similar argument to show density of ${\mathcal{P}}_{(u,v)}$ in the strip of elements whose height in $C'$ is the same as an element of $L_C(P)$.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Conjecture 1.4
• Theorem 1.5
• Definition 2.1
• Theorem 2.2: Ve89
• Proposition 2.3
• Remark 2.4
• Lemma 2.5