[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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1016/S1631-073X(03)00222-X/
[1] Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math., Volume 146 (1997), pp. 295-344
[2] Topological dynamics, Amer. Math. Soc. Collog. Publ., Volume 36 (1955)
[3] Interval exchange transformations, Math. Z., Volume 141 (1975), pp. 25-31
[4] Non-ergodic interval exchange transformations, Israel J. Math., Volume 26 (1977), pp. 188-196
[5] A “minimal”, non-uniquely ergodic interval exchange transformation, Math. Z., Volume 148 (1976), pp. 101-105
[6] Interval exchange transformations and measured foliations, Ann. of Math., Volume 115 (1982), pp. 169-200
[7] Échanges d'intervalles et transformations induites, Acta Arit. (1979), pp. 315-328
[8] Interval exchange transformations, J. Anal. Math., Volume 33 (1978), pp. 222-272
[9] Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., Volume 115 (1982), pp. 201-242
[10] Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier, Volume 46 (1996) no. 2, pp. 325-370
[11] Deviation for interval exchange transformations, Ergodic Theory Dynamical Systems, Volume 17 (1997), pp. 1477-1499
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