Minimality and unique ergodicity of Veech 1969 type interval exchange transformations

We give conditions for minimality of \({\mathbb {Z}}/N{\mathbb {Z}}\) $\mathbb{Z}/N\mathbb{Z}$ extensions of a rotation of angle \(\alpha \) $\alpha$ with one marked point, solving the problem for any prime N: for \(N=2\) $N=2$ , these correspond to the Veech 1969 examples, for which a necessary and sufficient condition was not known yet. We provide also a word combinatorial criterion of minimality valid for general interval exchange transformations, which applies to \({\mathbb {Z}}/N{\mathbb {Z}}\) $\mathbb{Z}/N\mathbb{Z}$ extensions of any interval exchange transformation with any number of marked points. Then we give a condition for unique ergodicity of these extensions when the initial interval exchange transformation is linearly recurrent and there are one or two marked points.