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
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
DOI : 10.24033/asens.2237
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/
Geometrical models for substitutions, Exp. Math., Volume 20 (2011), pp. 97-127 (ISSN: 1058-6458) | DOI | MR | Zbl
, Ergodic theory (Sem., Les Plans-sur-Bex, 1980) (French) (Monograph. Enseign. Math.), Volume 29, Univ. Genève, Geneva, 1981, pp. 5-38 | MR | Zbl
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
Veech surfaces with nonperiodic directions in the trace field, J. Mod. Dyn., Volume 3 (2009), pp. 611-629 (ISSN: 1930-5311) | DOI | MR | Zbl
Construction de difféomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris Sér. I Math., Volume 292 (1981), pp. 75-78 (ISSN: 0151-0509) | MR | Zbl
Rank two interval exchange transformations, Ergodic Theory Dynam. Systems, Volume 8 (1988), pp. 379-394 (ISSN: 0143-3857) | DOI | MR | Zbl
, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, 20, Academic Press, 1966, 435 pages | MR | Zbl
, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 101-117 | MR | Zbl
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
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
Travaux de Thurston sur les surfaces, Astérisque, Volume 66–67 (1979) | Numdam | MR | Zbl
, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 61-93 | MR | Zbl
The geometry of cross sections to flows, Topology, Volume 21 (1982), pp. 353-371 (ISSN: 0040-9383) | DOI | MR | Zbl
, Wiley-Interscience, 1978, 813 pages (ISBN: 0-471-32792-1) |The Arnoux-Yoccoz Teichmüller disc, Geom. Funct. Anal., Volume 18 (2009), pp. 1988-2016 (ISSN: 1016-443X) | DOI | MR | Zbl
The construction of self-similar tilings, Geom. Funct. Anal., Volume 6 (1996), pp. 471-488 (ISSN: 1016-443X) | DOI | MR | Zbl
Billiards on rational-angled triangles, Comment. Math. Helv., Volume 75 (2000), pp. 65-108 (ISSN: 0010-2571) | DOI | MR | Zbl
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
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
Über die Heckeschen Gruppen
Foliations and laminations on hyperbolic surfaces, Topology, Volume 22 (1983), pp. 119-135 (ISSN: 0040-9383) | DOI | MR | Zbl
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
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
Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., Volume 16 (2003), pp. 857-885 (ISSN: 0894-0347) | DOI | MR | Zbl
Teichmüller geodesics of infinite complexity, Acta Math., Volume 191 (2003), pp. 191-223 (ISSN: 0001-5962) | DOI | MR | Zbl
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
Prym varieties and Teichmüller curves, Duke Math. J., Volume 133 (2006), pp. 569-590 (ISSN: 0012-7094) | DOI | MR | Zbl
Dynamics of
Foliations of Hilbert modular surfaces, Amer. J. Math., Volume 129 (2007), pp. 183-215 (ISSN: 0002-9327) | DOI | MR | Zbl
Diophantine and ergodic foliations on surfaces, J. Topol., Volume 6 (2013), pp. 349-360 (ISSN: 1753-8416) | DOI | MR | Zbl
Navigating moduli space with complex twists, J. Eur. Math. Soc. (JEMS), Volume 15 (2013), pp. 1223-1243 (ISSN: 1435-9855) | DOI | MR | Zbl
Moduli spaces of isoperiodic forms on Riemann surfaces, Duke Math. J., Volume 163 (2014), pp. 2271-2323 (ISSN: 0012-7094) | DOI | MR | Zbl
Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc., Volume 19 (2006), pp. 327-344 (ISSN: 0894-0347) | DOI | MR | Zbl
, Handbook of Teichmüller theory. Vol. II (IRMA Lect. Math. Theor. Phys.), Volume 13, Eur. Math. Soc., Zürich, 2009, pp. 369-387 | DOI | MR | Zbl
Contribution à la topologie des ensembles dénombrables, Fund. Math., Volume 1 (1920), pp. 17-27 | DOI | JFM
, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015-1089 | DOI | MR | Zbl
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
Asymptotic cycles, Ann. of Math., Volume 66 (1957), pp. 270-284 (ISSN: 0003-486X) | DOI | MR | Zbl
, Ergebn. Math. Grenzg., 5, Springer, Berlin, 1984, 184 pages (ISBN: 3-540-13035-7) | DOI | MR | Zbl
, 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
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
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
Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem
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
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
, Pseudoperiodic topology (Amer. Math. Soc. Transl. Ser. 2), Volume 197, Amer. Math. Soc., Providence, RI, 1999, pp. 135-178 | MR | Zbl
Cité par Sources :