Cutting sequences, regular polygons, and the Veech group

TL;DR

This paper explores the combinatorial properties of geodesic flow on regular polygons through cutting sequences and derivation, extending previous work on the octagon to more general polygons.

Contribution

It generalizes the analysis of cutting sequences and Veech groups from the regular octagon to other regular polygons, revealing broader structural properties.

Findings

Structural properties of the octagon generalize to other polygons

Development of a combinatorial derivation process for geodesic flows

Extension of Veech group analysis to regular polygons

Abstract

We describe the cutting sequences associated to geodesic flow on regular polygons, in terms of a combinatorial process called "derivation." This work is an extension of some of the ideas and results in Smillie and Ulcigrai's recent paper, where the analysis was made for the regular octagon. It turns out that the main structural properties of the octagon generalize in a natural way.