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

Minsky, Yair; Weiss, Barak

doi:10.24033/asens.2214

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.

Résumé
Abstract

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.

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.

Publié le : 2014-09-24

MR Zbl | 1 citation dans Numdam

DOI : 10.24033/asens.2214

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.

@article{ASENS_2014__47_2_245_0,
     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://www.numdam.org/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://www.numdam.org/articles/10.24033/asens.2214/

Bibliographie
Cité par

Bonahon, F. Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form, Ann. Fac. Sci. Toulouse Math., Volume 5 (1996), pp. 233-297 (ISSN: 0240-2955) | DOI | Numdam | MR | Zbl

Cheung, Y.; Masur, H. A divergent Teichmüller geodesic with uniquely ergodic vertical foliation, Israel J. Math., Volume 152 (2006), pp. 1-15 (ISSN: 0021-2172) | DOI | MR | Zbl

Calta, K.; Wortman, K. On unipotent flows in $ℋ (1, 1)$ , Ergodic Theory Dynam. Systems, Volume 30 (2010), pp. 379-398 (ISSN: 0143-3857) | DOI | MR | Zbl

Eskin, A.; Marklof, J.; Witte Morris, D. Unipotent flows on the space of branched covers of Veech surfaces, Ergodic Theory Dynam. Systems, Volume 26 (2006), pp. 129-162 (ISSN: 0143-3857) | DOI | MR | Zbl

Kleinbock, D., Smooth ergodic theory and its applications (Seattle, WA, 1999) (Proc. Sympos. Pure Math.), Volume 69, Amer. Math. Soc., 2001, pp. 639-660 | DOI | MR | Zbl

Kerckhoff, S.; Masur, H.; Smillie, J. Ergodicity of billiard flows and quadratic differentials, Ann. of Math., Volume 124 (1986), pp. 293-311 (ISSN: 0003-486X) | DOI | MR | Zbl

Kleinbock, D.; Weiss, B. Bounded geodesics in moduli space, Int. Math. Res. Not., Volume 2004 (2004), pp. 1551-1560 (ISSN: 1073-7928) | DOI | MR | Zbl

Kleinbock, D.; Weiss, B. Badly approximable vectors on fractals, Israel J. Math., Volume 149 (2005), pp. 137-170 (ISSN: 0021-2172) | DOI | MR | Zbl

Masur, H. Interval exchange transformations and measured foliations, Ann. of Math., Volume 115 (1982), pp. 169-200 (ISSN: 0003-486X) | DOI | MR | Zbl

Masur, H. Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J., Volume 66 (1992), pp. 387-442 (ISSN: 0012-7094) | DOI | MR | Zbl

Mattila, P., Cambridge Studies in Advanced Math., 44, Cambridge Univ. Press, 1995, 343 pages (ISBN: 0-521-46576-1; 0-521-65595-1) | MR | Zbl

McMullen, C. T. Foliations of Hilbert modular surfaces, Amer. J. Math., Volume 129 (2007), pp. 183-215 (ISSN: 0002-9327) | DOI | MR | Zbl

Masur, H.; Smillie, J. Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math., Volume 134 (1991), pp. 455-543 (ISSN: 0003-486X) | DOI | MR | Zbl

Masur, H.; Tabachnikov, S., Handbook of dynamical systems, Vol. 1A, North-Holland, 2002, pp. 1015-1089 | DOI | MR | Zbl

Minsky, Y.; Weiss, B. Nondivergence of horocyclic flows on moduli space, J. reine angew. Math., Volume 552 (2002), pp. 131-177 (ISSN: 0075-4102) | DOI | MR | Zbl

Penner, R. C.; Harer, J. L., Annals of Math. Studies, 125, Princeton Univ. Press, 1992, 216 pages (ISBN: 0-691-08764-4; 0-691-02531-2) | MR | Zbl

Schwartzman, S. Asymptotic cycles, Volume 3 (2008) no. 12, 5463 pages ( http://www.scholarpedia.org/article/Asymptotic_cycles )

Sullivan, D. Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., Volume 36 (1976), pp. 225-255 (ISSN: 0020-9910) | DOI | MR | Zbl

Thurston, W. P. Minimal stretch maps between hyperbolic surfaces (preprint arXiv:math.GT/9801039 ) | MR

Veech, W. A. Interval exchange transformations, J. Analyse Math., Volume 33 (1978), pp. 222-272 (ISSN: 0021-7670) | DOI | MR | Zbl

Veech, W. A. Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., Volume 115 (1982), pp. 201-242 (ISSN: 0003-486X) | DOI | MR | Zbl

Veech, W. A. Measures supported on the set of uniquely ergodic directions of an arbitrary holomorphic 1-form, Ergodic Theory Dynam. Systems, Volume 19 (1999), pp. 1093-1109 (ISSN: 0143-3857) | DOI | MR | Zbl

Vorobets, Y. B. Plane structures and billiards in rational polygons: the Veech alternative, Russian Math. Surveys, Volume 51 (1996), pp. 779-817 | DOI | MR | Zbl

Zemljakov, A. N.; Katok, A. B. Topological transitivity of billiards in polygons, Mat. Zametki, Volume 18 (1975), pp. 291-300 (ISSN: 0025-567X) | MR | Zbl

Zorich, A., Frontiers in number theory, physics, and geometry. I (Cartier, P.; Julia, B.; Moussa, P.; Vanhove, P., eds.), Springer, 2006, pp. 437-583 | DOI | MR | Zbl

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