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- Diagonal changes for every interval exchange transformation doi link

Auteur(s): Ferenczi Sébastien

(Article) Publié: Geometriae Dedicata, vol. p. (2015)


Ref HAL: hal-01263098_v1
DOI: 10.1007/s10711-014-0031-y
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Résumé:

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.