Dynamical Systems
On the cohomological equation for interval exchange maps
[Sur l'équation cohomologique pour les échanges d'intervalles]
Comptes Rendus. Mathématique, Tome 336 (2003) no. 11, pp. 941-948.

On présente une classe explicite d'échanges d'intervalles T, de mesure pleine, pour laquelle l'équation cohomologique ΨΨT=Φ admet une solution bornée Ψ, à condition que la donnée Φ appartienne à un sous-espace de codimension finie de l'espace des fonctions dont la dérivée sur chaque intervalle est de variation bornée.

Cette classe est définie par une condition diophantienne « de type Roth » exprimé dans une variante du développement en fraction continue de Rauzy–Veech–Zorich associé à T.

We exhibit an explicit full measure class of minimal interval exchange maps T for which the cohomological equation ΨΨT=Φ has a bounded solution Ψ provided that the datum Φ belongs to a finite codimension subspace of the space of functions having on each interval a derivative of bounded variation.

The class of interval exchange maps is characterized in terms of a diophantine condition of “Roth type” imposed to an acceleration of the Rauzy–Veech–Zorich continued fraction expansion associated to T.

Reçu le :
Accepté le :
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DOI : 10.1016/S1631-073X(03)00222-X
Marmi, Stefano 1, 2 ; Moussa, Pierre 3 ; Yoccoz, Jean-Christophe 4

1 Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, Loc. Rizzi, 33100 Udine, Italy
2 Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, Italy
3 Service de physique théorique, CEA/Saclay, 91191 Gif-Sur-Yvette, France
4 Collège de France, 3, rue d'Ulm, 75005 Paris, France
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Marmi, Stefano; Moussa, Pierre; Yoccoz, Jean-Christophe. On the cohomological equation for interval exchange maps. Comptes Rendus. Mathématique, Tome 336 (2003) no. 11, pp. 941-948. doi : 10.1016/S1631-073X(03)00222-X. http://archive.numdam.org/articles/10.1016/S1631-073X(03)00222-X/

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