Diagonal changes for every interval exchange transformation

Sébastien Ferenczi (1)

1 IML - Institut de mathématiques de Luminy

We give a geometric version of the induction algorithms defined in [10] and generalizing the self-dual induction of [17]. For all interval exchanges, whatever the permutation and the disposition of the discontinuities, we define diagonal changes which generalize those of [7]: they are exchange of unions of triangles on a set of triangulated polygons, which may be glued to cre- ate a translation surface. There are many possible algorithms depending on decisions at each step, and when the decision is fixed each diagonal change is a natural extension of the corresponding induction, which extends the result shown in [7] in the particular case of the hyperelliptic Rauzy class. Furthermore, for that class, we can define decisions such that we get an algorithm of diagonal changes which is a natural extension of the underlying algorithm of self-dual induction, and we can thus compute an invariant measure for the normalized induction. The diagonal changes allow us also to realize the self-duality of the induction in the hyperelliptic class, and to prove this does not hold outside that class.

Domaines: Mathématiques [math]

Sébastien Ferenczi. Diagonal changes for every interval exchange transformation. Geometriae Dedicata, 2015, ⟨10.1007/s10711-014-0031-y⟩. ⟨hal-01263098⟩