Cohomology classes represented  by measured foliations, and Mahler's question for interval exchanges

Minsky, Yair; Weiss, Barak

doi:10.24033/asens.2214

Geodesic

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Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges
[Classes de cohomologie représentées par des feuilletages mesurés et question de Mahler pour les échanges d'intervalles]

Minsky, Yair ; Weiss, Barak

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 2, pp. 245-284.

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Abstract (VO)
Résumé

A translation structure on $(S, Σ)$ gives rise to two transverse measured foliations $ℱ, 𝒢$ on $S$ with singularities in $Σ$ , and by integration, to a pair of relative cohomology classes $[ℱ], [𝒢] \in H^{1} (S, Σ; ℝ)$ . Given a measured foliation $ℱ$ , we characterize the set of cohomology classes $𝐛$ for which there is a measured foliation $𝒢$ as above with $𝐛 = [𝒢]$ . This extends previous results of Thurston [19] and Sullivan [18].

We apply this to two problems: unique ergodicity of interval exchanges and flows on the moduli space of translation surfaces. For a fixed permutation $σ \in 𝒮_{d}$ , the space $ℝ_{+}^{d}$ parametrizes the interval exchanges on $d$ intervals with permutation $σ$ . We describe lines $ℓ$ in $ℝ_{+}^{d}$ such that almost every point in $ℓ$ is uniquely ergodic. We also show that for $σ (i) = d + 1 - i$ , for almost every $s > 0$ , the interval exchange transformation corresponding to $σ$ and $(s, s^{2}, ..., s^{d})$ is uniquely ergodic. As another application we show that when $k = | Σ | \geq 2,$ the operation of “moving the singularities horizontally” is globally well-defined. We prove that there is a well-defined action of the group $B ⋉ ℝ^{k - 1}$ on the set of translation surfaces of type $(S, Σ)$ without horizontal saddle connections. Here $B \subset SL (2, ℝ)$ is the subgroup of upper triangular matrices.

Une structure de translation sur une surface marquée $(S, Σ)$ donne lieu à deux feuilletages mesurés $ℱ$ , $𝒢$ sur $S$ à singularités dans $Σ$ et, par intégration, à un couple de classes de cohomologie relative $[ℱ]$ , $[𝒢] \in H^{1} (S, Σ; ℝ)$ . Étant donné un feuilletage mesuré $ℱ$ , nous caractérisons l'ensemble des classes de cohomologie $𝐛$ pour lesquelles il existe un feuilletage mesuré $𝒢$ comme ci-dessus tel que $𝐛 = [𝒢]$ . Cela généralise des résultats antérieurs de Thurston [19] et Sullivan [18].

Nous appliquons ce résultat à deux problèmes : l'unique ergodicité des échanges d'intervalles et les flots sur l'espace des modules des surfaces de translation. Étant donnée une permutation $σ \in 𝒮_{d}$ , l'ensemble $ℝ_{+}^{d}$ paramètre les échanges d'intervalles sur $d$ intervalles de permutation associée $σ$ . Nous décrivons les droites $ℓ$ de $ℝ_{+}^{d}$ dont presque tout point est uniquement ergodique. Nous démontrons aussi que si $σ$ est donnée par $σ (i) = d + 1 - i$ , pour presque tout $s > 0$ , l'échange d'intervalles correspondant à $σ$ et à $(s, s^{2}, \dots, s^{d})$ est uniquement ergodique. Une autre application est que lorsque $k = | Σ | \geq 2$ , l'opération consistant à « déplacer horizontalement les singularités » est bien définie. En notant $B$ le sous-groupe des matrices triangulaires supérieures de $SL (2, ℝ)$ , nous prouvons qu'il y a une action bien définie du groupe $B \times ℝ^{k - 1}$ sur l'ensemble des surfaces de translation de type $(S, Σ)$ sans connexion horizontale.

Publié le : 2014-05-27

MR Zbl

DOI : 10.24033/asens.2214

Classification : 37D40; 32G15 37F30 57M50.
Keywords: Cohomology classes, measured foliations, interval exchanges.
Mots-clés : Classes de cohomologie, feuilletages mesurés, la question de Mahler, échanges d'intervalles.

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     author = {Minsky, Yair and Weiss, Barak},
     title = {Cohomology classes represented  by measured foliations, and {Mahler's} question for interval exchanges},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {245--284},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 47},
     number = {2},
     year = {2014},
     doi = {10.24033/asens.2214},
     mrnumber = {3215923},
     zbl = {1346.37039},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.24033/asens.2214/}
}

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Minsky, Yair; Weiss, Barak. Cohomology classes represented  by measured foliations, and Mahler's question for interval exchanges. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 2, pp. 245-284. doi : 10.24033/asens.2214. http://geodesic.mathdoc.fr/articles/10.24033/asens.2214/

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