Cascades in the dynamics of measured foliations
[Cascades dans la dynamique des feuilletages mesurés]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 1, pp. 1-39.

Nous étudions le comportement des feuilletages mesurés harmoniques sur les surfaces de Riemann compactes. Quand les périodes relatives varient, on peut observer des cascades dans la dynamique d'un tel feuilletage. Dans le cas du genre 2, on montre que le lieu de bifurcation résultant d'une telle variation est un sous-ensemble dénombrable et fermé de , qui se plonge dans ωω.

This paper studies the behavior of harmonic measured foliations on compact Riemann surfaces. Cascades in the dynamics of such a foliation can occur as its relative periods are varied. We show that in the case of genus 2, the bifurcation locus arising from such a variation is a closed, countable set of  that embeds in ωω.

Publié le :
DOI : 10.24033/asens.2237
Classification : 30F30.
Keywords: Riemann surfaces, Abelian differentials, measured foliations, periods.
Mot clés : Surfaces de Riemann, différentielles abéliennes, feuilletages mesurés, périodes.
@article{ASENS_2015__48_1_1_0,
     author = {McMullen, Curtis T.},
     title = {Cascades in the dynamics  of measured foliations},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1--39},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
     number = {1},
     year = {2015},
     doi = {10.24033/asens.2237},
     mrnumber = {3335837},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2237/}
}
TY  - JOUR
AU  - McMullen, Curtis T.
TI  - Cascades in the dynamics  of measured foliations
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2015
SP  - 1
EP  - 39
VL  - 48
IS  - 1
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2237/
DO  - 10.24033/asens.2237
LA  - en
ID  - ASENS_2015__48_1_1_0
ER  - 
%0 Journal Article
%A McMullen, Curtis T.
%T Cascades in the dynamics  of measured foliations
%J Annales scientifiques de l'École Normale Supérieure
%D 2015
%P 1-39
%V 48
%N 1
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2237/
%R 10.24033/asens.2237
%G en
%F ASENS_2015__48_1_1_0
McMullen, Curtis T. Cascades in the dynamics  of measured foliations. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 1, pp. 1-39. doi : 10.24033/asens.2237. http://www.numdam.org/articles/10.24033/asens.2237/

Arnoux, P.; Bernat, J.; Bressaud, X. Geometrical models for substitutions, Exp. Math., Volume 20 (2011), pp. 97-127 (ISSN: 1058-6458) | DOI | MR | Zbl

Arnoux, P., Ergodic theory (Sem., Les Plans-sur-Bex, 1980) (French) (Monograph. Enseign. Math.), Volume 29, Univ. Genève, Geneva, 1981, pp. 5-38 | MR | Zbl

Arnoux, P. Un exemple de semi-conjugaison entre un échange d'intervalles et une translation sur le tore, Bull. Soc. Math. France, Volume 116 (1988), pp. 489-500 (ISSN: 0037-9484) | DOI | Numdam | MR | Zbl

Arnoux, P.; Schmidt, T. A. Veech surfaces with nonperiodic directions in the trace field, J. Mod. Dyn., Volume 3 (2009), pp. 611-629 (ISSN: 1930-5311) | DOI | MR | Zbl

Arnoux, P.; Yoccoz, J.-C. Construction de difféomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris Sér. I Math., Volume 292 (1981), pp. 75-78 (ISSN: 0151-0509) | MR | Zbl

Boshernitzan, M. D. Rank two interval exchange transformations, Ergodic Theory Dynam. Systems, Volume 8 (1988), pp. 379-394 (ISSN: 0143-3857) | DOI | MR | Zbl

Borevich, A. I.; Shafarevich, I. R., Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, 20, Academic Press, 1966, 435 pages | MR | Zbl

Calabi, E., Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 101-117 | MR | Zbl

Dynnikov, I. A. Interval identification systems and plane sections of 3-periodic surfaces, Proc. Steklov Inst. Math., Volume 263 (2008), pp. 65-77 (ISSN: 0371-9685) | DOI | MR | Zbl

Fathi, A. Some compact invariant sets for hyperbolic linear automorphisms of torii [tori], Ergodic Theory Dynam. Systems, Volume 8 (1988), pp. 191-204 (ISSN: 0143-3857) | DOI | MR | Zbl

Fathi, A.; Laudenbach, F.; Poénaru, V. Travaux de Thurston sur les surfaces, Astérisque, Volume 66–67 (1979) | Numdam | MR | Zbl

Franks, J., Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 61-93 | MR | Zbl

Fried, D. The geometry of cross sections to flows, Topology, Volume 21 (1982), pp. 353-371 (ISSN: 0040-9383) | DOI | MR | Zbl

Griffiths, P.; Harris, J., Wiley-Interscience, 1978, 813 pages (ISBN: 0-471-32792-1) | MR | Zbl

Hubert, P.; Lanneau, E.; Möller, M. The Arnoux-Yoccoz Teichmüller disc, Geom. Funct. Anal., Volume 18 (2009), pp. 1988-2016 (ISSN: 1016-443X) | DOI | MR | Zbl

Kenyon, R. The construction of self-similar tilings, Geom. Funct. Anal., Volume 6 (1996), pp. 471-488 (ISSN: 1016-443X) | DOI | MR | Zbl

Kenyon, R.; Smillie, J. Billiards on rational-angled triangles, Comment. Math. Helv., Volume 75 (2000), pp. 65-108 (ISSN: 0010-2571) | DOI | MR | Zbl

Kenyon, R.; Solomyak, B. On the characterization of expansion maps for self-affine tilings, Discrete Comput. Geom., Volume 43 (2010), pp. 577-593 (ISSN: 0179-5376) | DOI | MR | Zbl

Kontsevich, M.; Zorich, A. Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. math., Volume 153 (2003), pp. 631-678 (ISSN: 0020-9910) | DOI | MR | Zbl

Leutbecher, A. Über die Heckeschen Gruppen G(λ). II, Math. Ann., Volume 211 (1974), pp. 63-86 (ISSN: 0025-5831) | DOI | MR | Zbl

Levitt, G. Foliations and laminations on hyperbolic surfaces, Topology, Volume 22 (1983), pp. 119-135 (ISSN: 0040-9383) | DOI | MR | Zbl

Lowenstein, J. H.; Poggiaspalla, G.; Vivaldi, F. Interval exchange transformations over algebraic number fields: the cubic Arnoux-Yoccoz model, Dyn. Syst., Volume 22 (2007), pp. 73-106 (ISSN: 1468-9367) | DOI | MR | Zbl

Masur, H. Two boundaries of Teichmüller space, Duke Math. J., Volume 49 (1982), pp. 183-190 http://projecteuclid.org/euclid.dmj/1077315079 (ISSN: 0012-7094) | DOI | MR | Zbl

McMullen, C. T. Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., Volume 16 (2003), pp. 857-885 (ISSN: 0894-0347) | DOI | MR | Zbl

McMullen, C. T. Teichmüller geodesics of infinite complexity, Acta Math., Volume 191 (2003), pp. 191-223 (ISSN: 0001-5962) | DOI | MR | Zbl

McMullen, C. T. Teichmüller curves in genus two: the decagon and beyond, J. reine angew. Math., Volume 582 (2005), pp. 173-199 (ISSN: 0075-4102) | DOI | MR | Zbl

McMullen, C. T. Prym varieties and Teichmüller curves, Duke Math. J., Volume 133 (2006), pp. 569-590 (ISSN: 0012-7094) | DOI | MR | Zbl

McMullen, C. T. Dynamics of SL2() over moduli space in genus two, Ann. of Math., Volume 165 (2007), pp. 397-456 (ISSN: 0003-486X) | DOI | 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

McMullen, C. T. Diophantine and ergodic foliations on surfaces, J. Topol., Volume 6 (2013), pp. 349-360 (ISSN: 1753-8416) | DOI | MR | Zbl

McMullen, C. T. Navigating moduli space with complex twists, J. Eur. Math. Soc. (JEMS), Volume 15 (2013), pp. 1223-1243 (ISSN: 1435-9855) | DOI | MR | Zbl

McMullen, C. T. Moduli spaces of isoperiodic forms on Riemann surfaces, Duke Math. J., Volume 163 (2014), pp. 2271-2323 (ISSN: 0012-7094) | DOI | MR | Zbl

Möller, M. Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc., Volume 19 (2006), pp. 327-344 (ISSN: 0894-0347) | DOI | MR | Zbl

Möller, M., Handbook of Teichmüller theory. Vol. II (IRMA Lect. Math. Theor. Phys.), Volume 13, Eur. Math. Soc., Zürich, 2009, pp. 369-387 | DOI | MR | Zbl

Mazurkiewicz, S.; Sierpiński, W. Contribution à la topologie des ensembles dénombrables, Fund. Math., Volume 1 (1920), pp. 17-27 | DOI | JFM

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

Novikov, S. P. The Hamiltonian formalism and a multivalued analogue of Morse theory, Russian Math. Surveys, Volume 37 (1982), pp. 1-56 (ISSN: 0042-1316) | DOI | MR | Zbl

Schwartzman, S. Asymptotic cycles, Ann. of Math., Volume 66 (1957), pp. 270-284 (ISSN: 0003-486X) | DOI | MR | Zbl

Strebel, K., Ergebn. Math. Grenzg., 5, Springer, Berlin, 1984, 184 pages (ISBN: 3-540-13035-7) | DOI | MR | Zbl

Smillie, J.; Ulcigrai, C., Dynamical numbers—interplay between dynamical systems and number theory (Contemp. Math.), Volume 532, Amer. Math. Soc., Providence, RI, 2010, pp. 29-65 | DOI | MR | Zbl

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. On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc., Volume 19 (1988), pp. 417-431 (ISSN: 0273-0979) | DOI | MR | Zbl

Veech, W. A. Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod2 , Trans. Amer. Math. Soc., Volume 140 (1969), pp. 1-33 (ISSN: 0002-9947) | MR | Zbl

Veech, W. A. The metric theory of interval exchange transformations. III. The Sah-Arnoux-Fathi invariant, Amer. J. Math., Volume 106 (1984), pp. 1389-1422 (ISSN: 0002-9327) | DOI | MR | Zbl

Veech, W. A. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. math., Volume 97 (1989), pp. 553-583 (ISSN: 0020-9910) | DOI | MR | Zbl

Zorich, A., Pseudoperiodic topology (Amer. Math. Soc. Transl. Ser. 2), Volume 197, Amer. Math. Soc., Providence, RI, 1999, pp. 135-178 | MR | Zbl

Cité par Sources :