Comptes Rendus
Dynamical Systems
On the cohomological equation for interval exchange maps
Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 941-948.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00222-X

Stefano Marmi 1, 2; Pierre Moussa 3; Jean-Christophe Yoccoz 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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Stefano Marmi; Pierre Moussa; Jean-Christophe Yoccoz. On the cohomological equation for interval exchange maps. Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 941-948. doi : 10.1016/S1631-073X(03)00222-X. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00222-X/

[1] G. Forni 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] W.H. Gottschalk; G.A. Hedlund Topological dynamics, Amer. Math. Soc. Collog. Publ., Volume 36 (1955)

[3] M. Keane Interval exchange transformations, Math. Z., Volume 141 (1975), pp. 25-31

[4] M. Keane Non-ergodic interval exchange transformations, Israel J. Math., Volume 26 (1977), pp. 188-196

[5] H.B. Keynes; D. Newton A “minimal”, non-uniquely ergodic interval exchange transformation, Math. Z., Volume 148 (1976), pp. 101-105

[6] H. Masur Interval exchange transformations and measured foliations, Ann. of Math., Volume 115 (1982), pp. 169-200

[7] G. Rauzy Échanges d'intervalles et transformations induites, Acta Arit. (1979), pp. 315-328

[8] W. Veech Interval exchange transformations, J. Anal. Math., Volume 33 (1978), pp. 222-272

[9] W. Veech Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., Volume 115 (1982), pp. 201-242

[10] A. Zorich Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier, Volume 46 (1996) no. 2, pp. 325-370

[11] A. Zorich Deviation for interval exchange transformations, Ergodic Theory Dynamical Systems, Volume 17 (1997), pp. 1477-1499

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