  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.
