[Sur l'équation cohomologique pour les échanges d'intervalles]
On présente une classe explicite d'échanges d'intervalles T, de mesure pleine, pour laquelle l'équation cohomologique Ψ−Ψ∘T=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 Ψ−Ψ∘T=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.
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@article{CRMATH_2003__336_11_941_0,
author = {Marmi, Stefano and Moussa, Pierre and Yoccoz, Jean-Christophe},
title = {On the cohomological equation for interval exchange maps},
journal = {Comptes Rendus. Math\'ematique},
pages = {941--948},
publisher = {Elsevier},
volume = {336},
number = {11},
year = {2003},
doi = {10.1016/S1631-073X(03)00222-X},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00222-X/}
}
TY - JOUR AU - Marmi, Stefano AU - Moussa, Pierre AU - Yoccoz, Jean-Christophe TI - On the cohomological equation for interval exchange maps JO - Comptes Rendus. Mathématique PY - 2003 SP - 941 EP - 948 VL - 336 IS - 11 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(03)00222-X/ DO - 10.1016/S1631-073X(03)00222-X LA - en ID - CRMATH_2003__336_11_941_0 ER -
%0 Journal Article %A Marmi, Stefano %A Moussa, Pierre %A Yoccoz, Jean-Christophe %T On the cohomological equation for interval exchange maps %J Comptes Rendus. Mathématique %D 2003 %P 941-948 %V 336 %N 11 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(03)00222-X/ %R 10.1016/S1631-073X(03)00222-X %G en %F CRMATH_2003__336_11_941_0
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://www.numdam.org/articles/10.1016/S1631-073X(03)00222-X/
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