Cutting sequences in Veech surfaces

TL;DR

This paper characterizes cutting sequences on Veech surfaces like the regular hexagon and Bouw-M"oller surfaces, extending known results from the square and octagon cases and providing new symbolic codings and dictionaries.

Contribution

It introduces a unified approach to characterize cutting sequences across various Veech surfaces, including hexagons and Bouw-M"oller surfaces, adapting procedures used for octagons and squares.

Findings

Characterization of cutting sequences in a regular hexagon.

Development of a dictionary linking hexagon and torus codings.

Extension of methods to Bouw-M"oller surfaces.

Abstract

A cutting sequence is a symbolic coding of a linear trajectory on a translation surface corresponding to the sequence of sides hit in a polygonal representation of the surface. We characterize cutting sequences in a regular hexagon with opposite sides identified by translations exploiting the same procedure used by Smillie and Ulcigrai for the regular octagon. In the case of the square, cutting sequences are the well known Sturmian sequences. We remark the differences between the procedure used in the case of the square and the one used in the cases of the regular hexagon and regular octagon. We also show how to adapt the latter to work also in the case of the square, giving a new characterization for this case. We also show how to create a dictionary to pass from the symbolic coding with respect to the hexagon to the symbolic coding with respect to the parallelogram representing it in the space of flat tori. Finally we consider the Bouw-M\"oller surfaces $\mathscr M(3,4)$ and $\mathscr M(4,3)$ and we use their semi-regular polygons representations to prove our main result, which is a theorem analogous to the central step used to characterize cutting sequences in the previous cases.