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

Minsky, Yair; Weiss, Barak

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

Une structure de translation sur une surface marquée $(S,\Sigma )$ donne lieu à deux feuilletages mesurés $ℱ$, $𝒢$ sur $S$ à singularités dans $\Sigma$ et, par intégration, à un couple de classes de cohomologie relative $\left[ℱ\right]$, $\left[𝒢\right]\in H^1(S,\Sigma ;ℝ)$. 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 $𝐛=\left[𝒢\right]$. 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 $\sigma \in {𝒮}_d$, l'ensemble ${ℝ}_{+}^d$ paramètre les échanges d'intervalles sur $d$ intervalles de permutation associée $\sigma$. Nous décrivons les droites $\ell$ de ${ℝ}_{+}^d$ dont presque tout point est uniquement ergodique. Nous démontrons aussi que si $\sigma$ est donnée par $\sigma \left(i\right)=d+1-i$, pour presque tout $s>0$, l'échange d'intervalles correspondant à $\sigma$ et à $(s,s^2,\cdots ,s^d)$ est uniquement ergodique. Une autre application est que lorsque $k=\left|\Sigma \right|\ge 2$, 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 $(S,\Sigma )$ sans connexion horizontale.

A translation structure on $(S,\Sigma )$ gives rise to two transverse measured foliations $ℱ,𝒢$ on $S$ with singularities in $\Sigma$, and by integration, to a pair of relative cohomology classes $\left[ℱ\right],\left[𝒢\right]\in H^1(S,\Sigma ;ℝ)$. Given a measured foliation $ℱ$, we characterize the set of cohomology classes $𝐛$ for which there is a measured foliation $𝒢$ as above with $𝐛=\left[𝒢\right]$. 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 $\sigma \in {𝒮}_d$, the space ${ℝ}_{+}^d$ parametrizes the interval exchanges on $d$ intervals with permutation $\sigma$. We describe lines $\ell$ in ${ℝ}_{+}^d$ such that almost every point in $\ell$ is uniquely ergodic. We also show that for $\sigma \left(i\right)=d+1-i$, for almost every $s>0$, the interval exchange transformation corresponding to $\sigma$ and $(s,s^2,...,s^d)$ is uniquely ergodic. As another application we show that when $k=\left|\Sigma \right|\ge 2,$ 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 $(S,\Sigma )$ without horizontal saddle connections. Here $B\subset SL(2,ℝ)$ is the subgroup of upper triangular matrices.

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

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