# Rigidity of square-tiled interval exchange transformations

## TL;DR

This paper investigates the rigidity of interval exchange transformations on square-tiled surfaces, establishing conditions related to the boundedness of partial quotients of tan θ and exploring specific classes like Veech triangles.

## Contribution

It provides a combinatorial characterization of rigidity for these transformations and identifies new classes of rigid flows and transformations in specific surface settings.

## Key findings

- Rigidity occurs if and only if tan θ has bounded partial quotients.
- When all vertices are singularities, T is not of rank one under bounded partial quotients.
- Uncountably many rigid flows and transformations are constructed for Veech triangle unfoldings.

## Abstract

We look at interval exchange transformations defined as first return maps on the set of diagonals of a flow of direction $\theta$ on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if tan $\theta$ has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and tan $\theta$ has bounded partial quotients, the square-tiled interval exchange transformation T is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an uncountable set of rigid directional flows and an uncountable set of rigid interval exchange transformations.