Billiards in regular 2 n ‐gons and the self‐dual induction

We build a coding of the trajectories of billiards in regular 2 n ‐gons, similar but different from the one in J. Smillie and C. Ulcigrai [in ‘Beyond Sturmian sequences: coding linear trajectories in the regular octagon’, Proc. Lond. Math. Soc. (3) 102 (2011) 291–340], by applying the self‐dual induction [S. Ferenczi and L.Q. Zamboni, ‘Structure of K ‐interval exchange transformations: induction, trajectories, and distance theorems’, J. Anal. Math. 112 (2010) 289–328] to the underlying one‐parameter family of n ‐interval exchange transformations. This allows us to show that, in that family, for n =3 non‐periodicity is enough to guarantee weak mixing, and in some cases minimal self‐joinings, and for every n we can build examples of n ‐interval exchange transformations with weak mixing, which are the first known explicitly for n >6.