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

Sébastien Ferenczi (1)

1 IML - Institut de mathématiques de Luminy

We build a coding of the trajectories of billiards in regular 2n-gons, similar but different from the one in [16], by applying the self-dual induction [9] 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. In [16], see also [15], John Smillie and Corinna Ulcigrai develop a rich and original theory of billiards in the regular octagons, and more generally of billiards in the regular 2n-gons, first studied by Veech [17]: their aim is to build explicitly the symbolic trajectories, which generalize the famous Sturmian sequences (see for example [1] among a huge literature), and they achieve it through a new renormalization process which generalizes the usual continued fraction algorithm. In the present shorter paper, we show that similar results, with new consequences, can be obtained by using an existing, though recent, theory, the self-dual induction on interval exchange transformations. As in [16], we define a trajectory of a billiard in a regular 2n-gon as a path which starts in the interior of the polygon, and moves with constant velocity until it hits the boundary, then it re-enters the polygon at the corresponding point of the parallel side, and continues travelling with the same velocity; we label each pair of parallel sides with a letter of the alphabet (A 1 , ...A n), and read the labels of the pairs of parallel sides crossed by the trajectory as time increases; studying these trajectories is known to be equivalent to studying the trajectories of a one-parameter family of n-interval exchange transformations, and to this family we apply a slightly modified version of the self-dual induction defined in [9]. Now, the self-dual induction is in general not easy to manipulate, as its states are a family of graphs, and its typical itineraries, or paths in the so-called graph of graphs, are quite complicated to describe; but in our main Theorem 7 below, we show that for any non-periodic n-interval exchange in this particular family, after at most 2n − 2 steps our self-dual induction goes back, up to small modifications, to the initial state of another member of the family. This gives us a renormalization process, which differs from the one in [16] essentially because it is applied to lengths of intervals instead of angles, and allows us to compute the whole itinerary of the original interval exchange transformation under the self-dual induction in function of a single sequence of integers between 1 and 2n − 1, which act as the partial quotients of a continued fraction algorithm applied to initial lengths of subintervals.

Domaines: Mathématiques [math]

Sébastien Ferenczi. Billiards in regular 2n-gons and the self-dual induction. Journal of the London Mathematical Society, 2013, 87 (3), pp.766-784. ⟨10.1112/jlms/jds075⟩. ⟨hal-01263794⟩