|Diagonal changes for every interval exchange transformation |
Auteur(s): Ferenczi Sébastien
(Article) Publié: Geometriae Dedicata, vol. p. (2015)
Ref HAL: hal-01263098_v1
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We give a geometric version of the induction algorithms defined in  and generalizing the self-dual induction of . For all interval exchanges, whatever the permutation and the disposition of the discontinuities, we define diagonal changes which generalize those of : 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  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.