Journal of the American Mathematical Society

Author(s): Michael Yampolsky; Saeed Zakeri
Journal: J. Amer. Math. Soc. 14 (2001), 25-78.
MSC (2000): Primary 37F10; Secondary 37F45, 37F50
Posted: October 2, 2000
Retrieve article in: PDF
This article is available free of charge

Let

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

Blaschke product model for the dynamics of

and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that

can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers $\theta$ and $\nu$ .

[AB]

L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Annals of Math., 72 (1960) 385-404. MR 22:5813

[A]

P. Atela, Bifurcations of dynamic rays in complex polynomials of degree two, Erg. Th. and Dyn. Sys., 12 (1991) 401-423. MR 94d:58128

[BS]

S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase, Math. Proc. Camb. Phil. Soc., 115 (1994) 451-481. MR 95j:58043

[dFdM]

E. de Faria and W. de Melo, Rigidity of critical circle mappings I, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 4, 339-392. CMP 2000:05

[Do1]

A. Douady, Algorithms for computing angles in the Mandelbrot set, in ``Chaotic Dynamics and Fractals,'' ed. Barnsley and Demko, Academic Press (1986) 155-168. CMP 19:01

[Do2]

A. Douady, Systèmes dynamiques holomorphes, Seminar Bourbaki, Astérisque, 105-106 (1983) 39-63. MR 85h:58090

[Do3]

A. Douady, Disques de Siegel et anneaux de Herman, Seminar Bourbaki, Astérisque, 152-153 (1987) 151-172. MR 89g:30049

[DE]

A. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math., 157 (1986) 23-48. MR 87j:30041

[DH]

A. Douady and J. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993) 263-297. MR 94j:58143

[Ep]

A. Epstein, Counterexamples to the quadratic mating conjecture, Manuscript in preparation.

[Ha]

P. Haïssinsky, Chirurgie parabolique, C. R. Acad. Sci. Paris, 327 (1998) 195-198. MR 99i:58127

[He]

M. Herman, Conjugaison quasisymetrique des homeomorphismes analytique des cercle a des rotations, Manuscript.

[Luo]

Jiaqi Luo, Combinatorics and holomorphic dynamics: Captures, matings, Newton's method, Thesis, Cornell University, 1995.

[Lyu]

M.Yu. Lyubich, The dynamics of rational transforms: The topological picture, Russian Math. Surveys 41 (1986) 43-117.

[Mc]

C. McMullen, Complex Dynamics and Renormalization, Annals of Math. Studies, vol. 135, 1994. MR 96b:58097

[dMvS]

W. de Melo, S. van Strien, One-dimensional dynamics, Springer-Verlag, 1993. MR 95a:58035

[Mi1]

J. Milnor, Dynamics in One Complex Variable: Introductory Lectures, Vieweg, 1999 (Available from the American Mathematical Society). CMP 2000:03

[Mi2]

J. Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math., 2 (1993) 37-83. MR 96h:58094

[Mi3]

J. Milnor, Periodic orbits, external rays, and the Mandelbrot set: An expository account, Asterisque 261 (2000). CMP 2000:12

[Mi4]

J. Milnor, Pasting together Julia sets - a worked out example of mating, to appear.

[Mo]

R.L. Moore, Concerning upper semi-continuous collection of continua, Trans. Amer. Math. Soc., 27 (1925) 416-428. CMP 95:18

[Pe]

C. Petersen, Local connectivity of some Julia sets containing a circle with an irrational rotation, Acta Math., 177 (1996) 163-224. MR 98h:58164

[Re1]

M. Rees, Realization of matings of polynomials as rational maps of degree two, Manuscript, 1986.

[Re2]

M. Rees, A partial description of parameter space of rational maps of degree two: part I, Acta Math., 168 (1992) 11-87. MR 93f:58205

[Sh]

M. Shishikura, On a theorem of M. Rees for matings of polynomials, London Math. Soc. Lecture Note Ser., 274. CMP 2000:14

[Si]

C. L. Siegel, Iteration of analytic functions, Ann. of Math., 43 (1942) 607-612. MR 4:76C

[ST]

M. Shishikura and L. Tan, A family of cubic rational maps and matings of cubic polynomials, Experiment. Math., 9 (2000) 29-53. CMP 2000:12

[Sw]

G. Swiatek, Rational rotation numbers for maps of the circle, Comm. Math. Phys., 119 (1988) 109-128. MR 90h:58077

[Tan]

L. Tan, Matings of quadratic polynomials, Erg. Th. and Dyn. Sys. 12 (1992) 589-620. MR 93h:58129

[TY]

L. Tan and Y. Yin, Local connectivity of the Julia set for geometrically finite rational maps,

Sci. China Ser. A 39 (1996) 39-47. MR 97g:58142

[Ya]

M. Yampolsky, Complex bounds for renormalization of critical circle maps, Erg. Th. and Dyn. Sys., 19 (1999) 227-257. MR 2000d:37053

[Yo1]

J.C. Yoccoz, Il n'y a pas de contre-example de Denjoy analytique, C. R. Acad. Sci. Paris Ser. I Math., 298 (1984) 141-144. MR 85j:58134

[Yo2]

J.C. Yoccoz, Petits Diviseurs en Dimension 1, Astérisque 231, 1995.

[Za1]

S. Zakeri, Biaccessibility in quadratic Julia sets I-II, to appear in Erg. Th. and Dyn. Sys.

[Za2]

S. Zakeri, Dynamics of cubic Siegel polynomials, Comm. Math. Phys., 206 (1999) 185-233. CMP 2000:07

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 37F10, 37F45, 37F50

Michael Yampolsky
Affiliation: Institut des Hautes Études Scientifiques, 35 route de Chartres, F-91440, Bures-sur-Yvette, France
Address at time of publication: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email: yampol@ihes.fr, yampol@math.toronto.edu

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN: 1088-6834(e) ISSN: 0894-0347(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS