Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Mating Siegel quadratic polynomials
HTML articles powered by AMS MathViewer

by Michael Yampolsky and Saeed Zakeri PDF
J. Amer. Math. Soc. 14 (2001), 25-78 Request permission


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 $\nu$.
  • Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. MR 115006, DOI 10.2307/1970141
  • Pau Atela, Bifurcations of dynamic rays in complex polynomials of degree two, Ergodic Theory Dynam. Systems 12 (1992), no. 3, 401–423. MR 1182654, DOI 10.1017/S0143385700006854
  • Shaun Bullett and Pierrette Sentenac, Ordered orbits of the shift, square roots, and the devil’s staircase, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 3, 451–481 (English, with English and French summaries). MR 1269932, DOI 10.1017/S0305004100072236
  • [dFdM]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. [Do1]Douady3 A. Douady, Algorithms for computing angles in the Mandelbrot set, in “Chaotic Dynamics and Fractals,” ed. Barnsley and Demko, Academic Press (1986) 155-168.
  • Adrien Douady, Systèmes dynamiques holomorphes, Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp. 39–63 (French). MR 728980
  • Adrien Douady, Disques de Siegel et anneaux de Herman, Astérisque 152-153 (1987), 4, 151–172 (1988) (French). Séminaire Bourbaki, Vol. 1986/87. MR 936853
  • Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23–48. MR 857678, DOI 10.1007/BF02392590
  • Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263–297. MR 1251582, DOI 10.1007/BF02392534
  • [Ep]Epstein A. Epstein, Counterexamples to the quadratic mating conjecture, Manuscript in preparation.
  • Peter Haïssinsky, Chirurgie parabolique, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 2, 195–198 (French, with English and French summaries). MR 1645124, DOI 10.1016/S0764-4442(98)80088-8
  • [He]Herman2 M. Herman, Conjugaison quasisymetrique des homeomorphismes analytique des cercle a des rotations, Manuscript. [Luo]Luo Jiaqi Luo, Combinatorics and holomorphic dynamics: Captures, matings, Newton’s method, Thesis, Cornell University, 1995. [Lyu]L3 M.Yu. Lyubich, The dynamics of rational transforms: The topological picture, Russian Math. Surveys 41 (1986) 43-117.
  • Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
  • Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
  • [Mi1]Milnor1 J. Milnor, Dynamics in One Complex Variable: Introductory Lectures, Vieweg, 1999 (Available from the American Mathematical Society).
  • Henk Broer and Mark Levi, Geometrical aspects of stability theory for Hill’s equations, Arch. Rational Mech. Anal. 131 (1995), no. 3, 225–240. MR 1354696, DOI 10.1007/BF00382887
  • [Mi3]Milnor2 J. Milnor, Periodic orbits, external rays, and the Mandelbrot set: An expository account, Asterisque 261 (2000). [Mi4]Milnor4 J. Milnor, Pasting together Julia sets - a worked out example of mating, to appear. [Mo]Moore R.L. Moore, Concerning upper semi-continuous collection of continua, Trans. Amer. Math. Soc., 27 (1925) 416-428.
  • Carsten Lunde Petersen, Local connectivity of some Julia sets containing a circle with an irrational rotation, Acta Math. 177 (1996), no. 2, 163–224. MR 1440932, DOI 10.1007/BF02392621
  • [Re1]Rees1 M. Rees, Realization of matings of polynomials as rational maps of degree two, Manuscript, 1986.
  • Mary Rees, A partial description of parameter space of rational maps of degree two. I, Acta Math. 168 (1992), no. 1-2, 11–87. MR 1149864, DOI 10.1007/BF02392976
  • [Sh]Shishikura M. Shishikura, On a theorem of M. Rees for matings of polynomials, London Math. Soc. Lecture Note Ser., 274.
  • J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4, DOI 10.1080/00029890.1939.11998880
  • [ST]ST M. Shishikura and L. Tan, A family of cubic rational maps and matings of cubic polynomials, Experiment. Math., 9 (2000) 29-53.
  • Grzegorz Świątek, Rational rotation numbers for maps of the circle, Comm. Math. Phys. 119 (1988), no. 1, 109–128. MR 968483, DOI 10.1007/BF01218263
  • Lei Tan, Matings of quadratic polynomials, Ergodic Theory Dynam. Systems 12 (1992), no. 3, 589–620. MR 1182664, DOI 10.1017/S0143385700006957
  • Lei Tan and Yongcheng Yin, Local connectivity of the Julia set for geometrically finite rational maps, Sci. China Ser. A 39 (1996), no. 1, 39–47. MR 1397233
  • Michael Yampolsky, Complex bounds for renormalization of critical circle maps, Ergodic Theory Dynam. Systems 19 (1999), no. 1, 227–257. MR 1677153, DOI 10.1017/S0143385799120947
  • Jean-Christophe Yoccoz, Il n’y a pas de contre-exemple de Denjoy analytique, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 7, 141–144 (French, with English summary). MR 741080
  • [Yo2]Yoccoz2 J.C. Yoccoz, Petits Diviseurs en Dimension 1, Astérisque 231, 1995. [Za1]Zakeri1 S. Zakeri, Biaccessibility in quadratic Julia sets I-II, to appear in Erg. Th. and Dyn. Sys. [Za2]Zakeri2 S. Zakeri, Dynamics of cubic Siegel polynomials, Comm. Math. Phys., 206 (1999) 185-233.
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
  • Email:,
  • Saeed Zakeri
  • Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
  • Email:
  • 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
  • © Copyright 2000 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 14 (2001), 25-78
  • MSC (2000): Primary 37F10; Secondary 37F45, 37F50
  • DOI:
  • MathSciNet review: 1800348