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

Abstract

We build a coding of the trajectories of billiards in regular 2n-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.