On espère établir un isomorphisme entre le modèle supposé et e qu'il est ensé mo-
déliser en utilisant la théorie de renormalisation des fon tions presques paraboliques (voir
[1℄ Lars V.Ahlfors. Le tures onquasi onformalmappings,volume38of UniversityLe ture
Series. Ameri an Mathemati al So iety, Providen e, RI, se ond edition, 2006. With
supplemental hapters by C. J.Earle, I. Kra, M. Shishikuraand J.H. Hubbard.
[2℄ Mi hael Benedi ks and AnaRodrigues. Kneading sequen es for doublestandard maps.
to appear in Fundam. Mathem.
[3℄ Xavier Bu and Arnaud Chéritat. Upper bound for the size of quadrati Siegel disks.
Invent. Math., 156(1):124, 2004.
[4℄ Alexandre Dezotti. Conne tedness of the Arnold tongues for double standard maps.
Pro . Amer. Math. So ., 138(10):35693583, 2010.
[5℄ A. Douady and J. H. Hubbard. Étude dynamique des polyn mes omplexes. Partie I,
volume84ofPubli ationsMathématiquesd'Orsay[Mathemati alPubli ationsofOrsay℄.
Université de Paris-Sud, Département de Mathématiques,Orsay, 1984.
[6℄ A. Douady and J. H. Hubbard. Étude dynamique des polyn mes omplexes. Partie
II, volume 85 of Publi ations Mathématiques d'Orsay [Mathemati al Publi ations of
Orsay℄. Université de Paris-Sud, Département de Mathématiques, Orsay, 1985. With
the ollaborationof P. Lavaurs, TanLei and P. Sentena .
[7℄ Adrien Douady and John HamalHubbard. On the dynami s of polynomial-like map-
pings. Ann. S i. É ole Norm. Sup. (4), 18(2) :287343, 1985.
[8℄ Tomasz Downarowi z. Survey of odometers and Toeplitz ows. In Algebrai and to-
pologi al dynami s, volume 385 of Contemp. Math., pages 737. Amer. Math. So .,
Providen e, RI, 2005.
[9℄ Peter L. Duren. Univalent fun tions, volume 259 of Grundlehren der Mathematis hen
Wissens haften [Fundamental Prin iples of Mathemati al S ien es℄. Springer-Verlag,
New York, 1983.
[10℄ Adam Epstein. Transversality in holomorphi dynami s, manus ript, 2010.
[11℄ Adam Lawren e Epstein. In nitesimal thurston rigidity and the fatou-shishikura in-
equality,2010.
[12℄ A. È. Erëmenko and G. M. Levin. Estimation of the hara teristi exponents of a
polynomial. Teor. Funktsi Funktsional.Anal. i Prilozhen., (58) :3040(1993), 1992.
[13℄ InouHiroyukiandShishikuraMitsuhiro. Therenormalizationforparaboli xedpoints
J.-C.Yo oz. InTopologi almethods in modern mathemati s (StonyBrook, NY, 1991),
pages 467511.Publish orPerish, Houston,TX, 1993.
[15℄ John HamalHubbard. Tei hmüller theory and appli ations to geometry, topology, and
dynami s. Vol. 1. MatrixEditions,Itha a, NY, 2006. Tei hmüller theory, With ontri-
butions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David
Brown, Allen Hat her, Chris Hruska and Sudeb Mitra, With forewords by William
Thurston and Cliord Earle.
[16℄ Anatole Katok and Boris Hasselblatt. Introdu tion to the modern theory of dynami al
systems, volume 54 of En y lopedia of Mathemati s and its Appli ations. Cambridge
University Press, Cambridge,1995. Witha supplementary hapterby Katok and Leo-
nardo Mendoza.
[17℄ Genadi Levin. Dis onne ted Julia set and rotation sets. Ann. S i. É ole Norm. Sup.
(4), 29(1) :122, 1996.
[18℄ Genadi Levin. Multipliersof periodi orbitsin spa es of rationalmaps, i-ii,2008.
[19℄ GenadiLevin. Multipliersofperiodi orbitsofquadrati polynomialsandtheparameter
plane. Israel J. Math.,170 :285315, 2009.
[20℄ GenadiLevin. Rigidityand non lo al onne tivity of julia sets of some quadrati poly-
nomials,2009.
[21℄ Curtis T. M Mullen. Complex dynami s and renormalization,volume 135 of Annals of
Mathemati s Studies. Prin eton University Press, Prin eton, NJ, 1994.
[22℄ John Milnor. Periodi orbits, externals rays and the Mandelbrot set : an expository
a ount. Astérisque, (261) :xiii,277333, 2000. Géométrie omplexeet systèmes dyna-
miques (Orsay, 1995).
[23℄ John Milnor. Dynami s in one omplex variable, volume 160 ofAnnalsof Mathemati s
Studies. Prin eton University Press, Prin eton, NJ, third edition,2006.
[24℄ Mi haª Misiurewi z and Ana Rodrigues. Non-generi usps. to appear in the Transa -
tions of the AMS.
[25℄ Mi haª Misiurewi z and Ana Rodrigues. Double standard maps. Comm. Math. Phys.,
273(1) :3765,2007.
[26℄ Mi haªMisiurewi zandAnaRodrigues.Onthetipofthetongue.J.FixedPointTheory
Appl., 3(1) :131141, 2008.
[27℄ Zeev Nehari. Theellipti modularfun tion anda lass ofanalyti fun tions rst onsi-
dered by Hurwitz. Amer. J. Math.,69 :7086,1947.
[28℄ ZeevNehari. Conformalmapping. DoverPubli ationsIn .,New York,1975. Reprinting
of the 1952 edition.
[29℄ Ri ardo Pérez-Mar o. Topology of julia sets and hedgehogs. preprint, Universtié de
number of symmetries. Invent. Math., 119(1):67127, 1995.
[31℄ Ri ardo Pérez-Mar o. Hedgehogs dynami s. preprint, UCLA, 1996.
[32℄ Carsten LundePetersen and Pas ale Roes h. Carrotsfor dessert. 2010.
[33℄ P. Roes h. Some rational maps whose Julia sets are not lo ally onne ted. Conform.
Geom. Dyn.,10 :125135(ele troni ), 2006.
[34℄ P. Roes h. Frontière des domaines de stabilité, mémoire d'habilitation à diriger des
re her hes, 2009.
[35℄ Mitsuhiro Shishikura. On the quasi onformal surgery of rational fun tions. Ann. S i.
É ole Norm. Sup. (4), 20(1) :129, 1987.
[36℄ Mitsuhiro Shishikura. Bifur ation of paraboli xed points. In The Mandelbrot set,
theme and variations, volume 274 of London Math. So . Le ture Note Ser., pages 325
363. CambridgeUniv. Press, Cambridge,2000.
[37℄ Dan ErikKrarup Sørensen. In nitely renormalizablequadrati polynomials,with non-
lo ally onne ted Juliaset. J. Geom. Anal.,10(1) :169206, 2000.
[38℄ E. T. Whittaker and G. N. Watson. A ourse of modern analysis. Cambridge Mathe-
mati alLibrary.CambridgeUniversity Press, Cambridge,1996. An introdu tiontothe
general theory of in nite pro esses and of analyti fun tions; with an a ount of the
prin ipaltrans endental fun tions,Reprint ofthe fourth (1927) edition.
[39℄ SaeedZakeri. On Siegeldisksofa lassofentire maps. DukeMath.J., 152(3):481532,
Ini nite satellite renormalizables quadrati polynomials
Abstra t :
The onne tednessofthetonguesofthedoublestandardmapfamilyisshownbyquasi onformal deformation.
I determine the growth rate of the oe ient of the Laurent series of inverse of the Bött her map for quadrati polynomialswith es aping riti al point.
An inequality that yieldsa domainonwhi h thereis no riti alvalue of the multipliermap is derived fromthe theory of quadrati dierentials.
I give an alternate proof of Levin riterium for non lo al onne tedness of in nite satellite renormalizablequadrati Juliasets. A geometri modelof this situation isalsoinvestigated.
Les langues de Arnold de la famille standard double Explosion des y les dans la famille
z
2+ λ
Dire teur de thèse : Xavier Bu
Soutenue à l'université Paul Sabatier (Toulouse 3) le mardi 7 juin 2011
Résumé :
La onnexitédeslanguesdeArnolddelafamillestandarddoubleestdémontréepardéformation quasi onforme.
Je donne un équivalentpour les oe ients du développement en série de Laurent de l'inverse des oordonnées de Bött her pour lespolyn mes quadratiques dont lepoint ritique s'é happe.
Une généralisationd'une inégalité quisert àdéterminer un domaineà l'intérieurduquel il n'y a pas de valeur ritique de la fon tion multipli ateur est obtenue en utilisant les diérentielles quadratiques.
LestravauxdeLévine surune onditionde nonlo ale onnexitédeJuliain nimentsatellitere- normalisablessontrepris,suivisdel'étuded'unmodèlegéométriquedes renormalisationssatellites générantun modèletopologique hypothétique d'un ompa t invariantdans l'ensemblede Juliade es polyn mes.
Mots lé : dynamique holomorphe, appli ation de Bött her, di¯entielles quadratiques, fon tion mutlipli ateur, in niment renormalisable, renormalisation satellite, non lo ale onnexité, langues de Arnold, famillestandard double.
Dis ipline : mathématiques
Institut de mathématiques de Toulouse, Équipe Émile Pi ard
UniversitéPaulSabatier