Geodesic
Mating Siegel quadratic polynomials
Yampolsky, Michael1, 2 ; Zakeri, Saeed3
1Institut des Hautes Études Scientifiques, 35 route de Chartres, F-91440, Bures-sur-Yvette, France
2Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
3Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Journal of the American Mathematical Society, Tome 14 (2001) no. 1, pp. 25-78
Cet article a éte moissonné depuis la source American Mathematical Society

Voir la notice de l'article

Let $F$ be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers $\theta$ and $\nu$. Using a new degree $3$ Blaschke product model for the dynamics of $F$ and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that $F$ can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers $\theta$ and $\nu$.
DOI : 10.1090/S0894-0347-00-00348-9

Yampolsky, Michael  1 , 2   ; Zakeri, Saeed  3

1 Institut des Hautes Études Scientifiques, 35 route de Chartres, F-91440, Bures-sur-Yvette, France
2 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
3 Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395 
@article{10_1090_S0894_0347_00_00348_9,
     author = {Yampolsky, Michael and Zakeri, Saeed},
     title = {Mating {Siegel} quadratic polynomials},
     journal = {Journal of the American Mathematical Society},
     pages = {25--78},
     year = {2001},
     volume = {14},
     number = {1},
     doi = {10.1090/S0894-0347-00-00348-9},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-00-00348-9/}
}
TY  - JOUR
AU  - Yampolsky, Michael
AU  - Zakeri, Saeed
TI  - Mating Siegel quadratic polynomials
JO  - Journal of the American Mathematical Society
PY  - 2001
SP  - 25
EP  - 78
VL  - 14
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-00-00348-9/
DO  - 10.1090/S0894-0347-00-00348-9
ID  - 10_1090_S0894_0347_00_00348_9
ER  - 
%0 Journal Article
%A Yampolsky, Michael
%A Zakeri, Saeed
%T Mating Siegel quadratic polynomials
%J Journal of the American Mathematical Society
%D 2001
%P 25-78
%V 14
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-00-00348-9/
%R 10.1090/S0894-0347-00-00348-9
%F 10_1090_S0894_0347_00_00348_9
Yampolsky, Michael; Zakeri, Saeed. Mating Siegel quadratic polynomials. Journal of the American Mathematical Society, Tome 14 (2001) no. 1, pp. 25-78. doi: 10.1090/S0894-0347-00-00348-9

[1] Ahlfors, Lars, Bers, Lipman Riemann’s mapping theorem for variable metrics Ann. of Math. (2) 1960 385 404

[2] Atela, Pau Bifurcations of dynamic rays in complex polynomials of degree two Ergodic Theory Dynam. Systems 1992 401 423

[3] Bullett, Shaun, Sentenac, Pierrette Ordered orbits of the shift, square roots, and the devil’s staircase Math. Proc. Cambridge Philos. Soc. 1994 451 481

[4] Douady, Adrien Systèmes dynamiques holomorphes 1983 39 63

[5] Douady, Adrien Disques de Siegel et anneaux de Herman Astérisque 1987

[6] Douady, Adrien, Earle, Clifford J. Conformally natural extension of homeomorphisms of the circle Acta Math. 1986 23 48

[7] Douady, Adrien, Hubbard, John H. A proof of Thurston’s topological characterization of rational functions Acta Math. 1993 263 297

[8] Haïssinsky, Peter Chirurgie parabolique C. R. Acad. Sci. Paris Sér. I Math. 1998 195 198

[9] Mcmullen, Curtis T. Complex dynamics and renormalization 1994

[10] De Melo, Welington, Van Strien, Sebastian One-dimensional dynamics 1993

[11] Broer, Henk, Levi, Mark Geometrical aspects of stability theory for Hill’s equations Arch. Rational Mech. Anal. 1995 225 240

[12] Petersen, Carsten Lunde Local connectivity of some Julia sets containing a circle with an irrational rotation Acta Math. 1996 163 224

[13] Rees, Mary A partial description of parameter space of rational maps of degree two. I Acta Math. 1992 11 87

[14] Corliss, J. J. Upper limits to the real roots of a real algebraic equation Amer. Math. Monthly 1939 334 338

[15] Świątek, Grzegorz Rational rotation numbers for maps of the circle Comm. Math. Phys. 1988 109 128

[16] Tan, Lei Matings of quadratic polynomials Ergodic Theory Dynam. Systems 1992 589 620

[17] Tan, Lei, Yin, Yongcheng Local connectivity of the Julia set for geometrically finite rational maps Sci. China Ser. A 1996 39 47

[18] Yampolsky, Michael Complex bounds for renormalization of critical circle maps Ergodic Theory Dynam. Systems 1999 227 257

[19] Yoccoz, Jean-Christophe Il n’y a pas de contre-exemple de Denjoy analytique C. R. Acad. Sci. Paris Sér. I Math. 1984 141 144

Cité par Sources :