Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Mating Siegel quadratic polynomials

Authors: Michael Yampolsky and Saeed Zakeri
Journal: J. Amer. Math. Soc. 14 (2001), 25-78
MSC (2000): Primary 37F10; Secondary 37F45, 37F50
Published electronically: October 2, 2000
MathSciNet review: 1800348
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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 $\nu$.

References [Enhancements On Off] (What's this?)

  • [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

Similar Articles

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

Retrieve articles in all journals with MSC (2000): 37F10, 37F45, 37F50

Additional Information

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

Saeed Zakeri
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395

Keywords: Holomorphic dynamics, rational map, Siegel disk, mating, Julia set
Received by editor(s): March 25, 1999
Received by editor(s) in revised form: June 9, 2000
Published electronically: October 2, 2000
Additional Notes: The first author was partially supported by NSF grant DMS-9804606
Article copyright: © Copyright 2000 American Mathematical Society