Central points of the double heptagon translation surface are not connection points

TL;DR

This paper investigates flow directions on double heptagon translation surfaces, identifying conditions for hyperbolic directions and showing that central points are not connection points, answering a question by Hubert and Schmidt.

Contribution

It provides a gcd-based criterion for hyperbolic directions and demonstrates that central points are not connection points on double heptagon surfaces.

Findings

Central points are not connection points.

A gcd algorithm determines hyperbolic directions.

Explicit points in the holonomy field are identified.

Abstract

We consider flow directions on the translation surfaces formed from double $(2n+1)$-gons, and give a sufficient condition in terms of a natural gcd algorithm for a direction to be hyperbolic in the sense that it is the fixed direction for some hyperbolic element of the Veech group of the surface. In particular, we give explicit points in the holonomy field of the double heptagon translation surface which are not so-called connection points. Among these are the central points of the heptagons, giving a negative answer to a question by P.Hubert and T.Schmidt.