Diagonal changes for every interval exchange transformation

Abstract

We give a geometric version of the induction algorithms defined in Ferenczi (Ann. Inst. Fourier (Grenoble), http://iml.univ-mrs.fr/~ferenczi/fie.pdf) and generalizing the self-dual induction of Ferenczi and Zamboni (J Anal Math 112, 289–328, 2010). For all interval exchanges, whatever the permutation and the disposition of the discontinuities, we define diagonal changes which generalize those of Dlecroix and Ulcigrai (Geom Dedic): they are exchange of unions of triangles on a set of triangulated polygons, which may be glued to create 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 Dlecroix and Ulcigrai (Geom Dedic) 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.