Rigidity of generalized Veech 1969/Sataev 1975 extensions of rotations

TL;DR

This paper investigates the rigidity properties of generalized extensions of rotations with marked points, revealing conditions under which these transformations are rigid or not, and providing new examples of non-rigid interval exchange transformations.

Contribution

It extends previous work by analyzing rigidity in generalized Veech and Sataev extensions, identifying conditions based on Ostrowski expansions, and constructing new non-rigid examples.

Findings

Rigidity when the rotation angle has unbounded partial quotients.

Non-rigidity when the coding is linearly recurrent.

First examples of non linearly recurrent, non rigid interval exchange transformations.

Abstract

We look at $d$-point extensions of a rotation of angle $\alpha$ with $r$ marked points, generalizing the examples of Veech 1969 and Sataev 1975, together with the square-tiled interval exchange transformations of \cite{fh2}. We study the property of rigidity, in function of the Ostrowski expansions of the marked points by $\alpha$: we prove that $T$ is rigid when $\alpha$ has unbounded partial quotients, and that $T$ is not rigid when the natural coding of the underlying rotation with marked points is linearly recurrent. But there remains an interesting grey zone between these two cases, in which we have only partial results on the rigidity question; they allow us to build the first examples of non linearly recurrent and non rigid interval exchange transformations.