Hyperbolic components of McMullen maps
[Composantes hyperboliques des fractions de McMullen]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 3, pp. 703-737.

Dans cet article nous résolvons complètement une question posée par B. Devaney. Nous montrons que toutes les composantes hyperboliques sont des domaines de Jordan dans la famille de fractions rationnelles de type McMullen. De plus nous donnons une description précise de toutes les fractions du bord de la composante non bornée. Il en découle que les cusps sont denses dans le bord de la composante non bornée.

In this article, we completely settle a question raised by B. Devaney. We prove that all the hyperbolic components are Jordan domains in the family of rational maps of McMullen type. Moreover, we give a precise description of all the rational maps on the outer boundary. It follows that the cusps are dense on the outer boundary.

Publié le :
DOI : 10.24033/asens.2256
Classification : 37F45; 37F10, 37F15
Keywords: Parameter plane, McMullen map, Hyperbolic component, Jordan curve.
Mot clés : Plan de paramètres, application de McMullen, composante hyperbolique, courbe de Jordan. 
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     title = {Hyperbolic components of {McMullen} maps},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {703--737},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
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Qiu, Weiyuan; Roesch, Pascale; Wang, Xiaoguang; Yin, Yongcheng. Hyperbolic components of McMullen maps. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 3, pp. 703-737. doi : 10.24033/asens.2256. http://www.numdam.org/articles/10.24033/asens.2256/

Ahlfors, L. V.; Bers, L. Riemann's mapping theorem for variable metrics, Ann. of Math., Volume 72 (1960), pp. 385-404 (ISSN: 0003-486X) | DOI | MR | Zbl

Ahlfors, L. V., McGraw-Hill Book Co., New York, 1978, 331 pages (ISBN: 0-07-000657-1) | MR | Zbl

Borel, E. Les probabilités dénombrables et leurs applications arithmétiques, Rendiconti del Circolo Matematico di Palermo, Volume 27 (1909), pp. 247-271 | DOI | JFM

Carleson, L.; Gamelin, T. W., Universitext: Tracts in Mathematics, Springer, New York, 1993, 175 pages (ISBN: 0-387-97942-5) | DOI | MR | Zbl

Devaney, R. L. Structure of the McMullen domain in the parameter planes for rational maps, Fund. Math., Volume 185 (2005), pp. 267-285 (ISSN: 0016-2736) | DOI | MR | Zbl

Devaney, R. L. The McMullen domain: satellite Mandelbrot sets and Sierpiński holes, Conform. Geom. Dyn., Volume 11 (2007), pp. 164-190 (ISSN: 1088-4173) | DOI | MR | Zbl

Devaney, R. L. Intertwined internal rays in Julia sets of rational maps, Fund. Math., Volume 206 (2009), pp. 139-159 (ISSN: 0016-2736) | DOI | MR | Zbl

Devaney, R. L. Singular perturbations of complex polynomials, Bull. Amer. Math. Soc. (N.S.), Volume 50 (2013), pp. 391-429 (ISSN: 0273-0979) | DOI | MR | Zbl

Douady, A.; Hubbard, J. H. A proof of Thurston's topological characterization of rational functions, Acta Math., Volume 171 (1993), pp. 263-297 (ISSN: 0001-5962) | DOI | MR | Zbl

Complex dynamics. Twenty-five years after the appearance of the Mandelbrot set, Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held in Snowbird, UT, June 13–17, 2004 (Devaney, R. L.; Keen, L., eds.) (Contemporary Mathematics), Volume 396, Amer. Math. Soc., Providence, RI (2006) (ISBN: 0-8218-3625-0) | DOI | MR | Zbl

Devaney, R. L.; Look, D. M.; Uminsky, D. The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J., Volume 54 (2005), pp. 1621-1634 (ISSN: 0022-2518) | DOI | MR | Zbl

Devaney, R. L.; Pilgrim, K. M. Dynamic classification of escape time Sierpiński curve Julia sets, Fund. Math., Volume 202 (2009), pp. 181-198 (ISSN: 0016-2736) | DOI | MR | Zbl

Devaney, R. L.; Russell, E. Connectivity of Julia Sets for Singularly Perturbed rational Maps, Chaos, CNN, Memristors and beyond: A Festschrift for Leon Chua (Adamatzky, A. et al., eds.), World Scientific Publishing Co. (2013), pp. 239-245

Gardiner, F. P.; Jiang, Y.; Wang, Z., Geometry of Riemann surfaces (London Math. Soc. Lecture Note Ser.), Volume 368, Cambridge Univ. Press, Cambridge, 2010, pp. 156-193 | DOI | MR | Zbl

Haïssinsky, P.; Pilgrim, K. M. Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., Volume 28 (2012), pp. 1025-1034 (ISSN: 0213-2230) | DOI | MR | Zbl

Kozlovski, O.; Shen, W.; van Strien, S. Rigidity for real polynomials, Ann. of Math., Volume 165 (2007), pp. 749-841 (ISSN: 0003-486X) | DOI | MR | Zbl

Kozlovski, O.; van Strien, S. Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. Lond. Math. Soc., Volume 99 (2009), pp. 275-296 (ISSN: 0024-6115) | DOI | MR | Zbl

Lyubich, M. On the Lebesgue measure of the Julia set of a quadratic polynomial (preprint arXiv:math/9201285 )

Mañé, R. Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys., Volume 100 (1985), pp. 495-524 http://projecteuclid.org/euclid.cmp/1104114003 (ISSN: 0010-3616) | DOI | MR | Zbl

Mañé, R. Erratum: “Hyperbolicity, sinks and measure in one-dimensional dynamics” [Comm. Math. Phys. 100 (1985), 495–524], Comm. Math. Phys., Volume 112 (1987), pp. 721-724 http://projecteuclid.org/euclid.cmp/1104160062 (ISSN: 0010-3616) | DOI | MR | Zbl

McMullen, C. T., The Mandelbrot set, theme and variations (London Math. Soc. Lecture Note Ser.), Volume 274, Cambridge Univ. Press, Cambridge, 2000, pp. 1-17 | MR | Zbl

McMullen, C. T., Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) (Math. Sci. Res. Inst. Publ.), Volume 10, Springer, New York, 1988, pp. 31-60 | DOI | MR | Zbl

McMullen, C. T. Cusps are dense, Ann. of Math., Volume 133 (1991), pp. 217-247 (ISSN: 0003-486X) | DOI | MR | Zbl

McMullen, C. T., Annals of Math. Studies, 135, Princeton Univ. Press, Princeton, NJ, 1994, 214 pages (ISBN: 0-691-02982-2; 0-691-02981-4) | MR | Zbl

McMullen, C. T. Rational maps and Teichmüller space: analogies and open problems, Linear and Complex Analysis Problem Book (Havin, V. P.; Nikolskii, N. K., eds.) (Lecture Notes in Math.), Volume 1574 (1994), pp. 430-433

Milnor, J., Friedr. Vieweg & Sohn, Braunschweig, 2000, 257 pages (ISBN: 3-528-03130-1) | MR | Zbl

Mañé, R.; Sad, P.; Sullivan, D. On the dynamics of rational maps, Ann. Sci. École Norm. Sup., Volume 16 (1983), pp. 193-217 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Qiu, W.; Wang, X.; Yin, Y. Dynamics of McMullen maps, Adv. Math., Volume 229 (2012), pp. 2525-2577 (ISSN: 0001-8708) | DOI | MR | Zbl

Qiu, W.; Yin, Y. Proof of the Branner-Hubbard conjecture on Cantor Julia sets, Sci. China Ser. A, Volume 52 (2009), pp. 45-65 (ISSN: 1006-9283) | DOI | MR | Zbl

Roesch, P. Captures for the family Fa(z)=z2+a/z2 , Dynamics on the Riemann Sphere, EMS (2006) | DOI | Zbl

Roesch, P. Hyperbolic components of polynomials with a fixed critical point of maximal order, Ann. Sci. École Norm. Sup., Volume 40 (2007), pp. 901-949 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Roesch, P. Topologie locale des méthodes de Newton cubiques (1997) | MR

Slodkowski, Z. Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc., Volume 111 (1991), pp. 347-355 (ISSN: 0002-9939) | DOI | MR | Zbl

Steinmetz, N. On the dynamics of the McMullen family R(z)=zm+λ/zl , Conform. Geom. Dyn., Volume 10 (2006), pp. 159-183 (ISSN: 1088-4173) | DOI | MR | Zbl

Tan, L.; Yin, Y., Complex dynamics, A. K. Peters, Wellesley, MA, 2009, pp. 215-227 | MR | Zbl

Yin, Y.; Zhai, Y. No invariant line fields on Cantor Julia sets, Forum Math., Volume 22 (2010), pp. 75-94 (ISSN: 0933-7741) | DOI | MR | Zbl

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