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
DOI: https://doi.org/10.1090/S0894-0347-00-00348-9
Published electronically: October 2, 2000
MathSciNet review: 1800348
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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?)

  • Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. MR 115006, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/S0764-4442%2898%2980088-8
  • [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
  • [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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1007/BF02392976
  • [Sh]Shishikura M. Shishikura, On a theorem of M. Rees for matings of polynomials, London Math. Soc. Lecture Note Ser., 274. [Si]Siegel C. L. Siegel, Iteration of analytic functions, Ann. of Math., 43 (1942) 607-612. [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
  • Lei Tan, Matings of quadratic polynomials, Ergodic Theory Dynam. Systems 12 (1992), no. 3, 589–620. MR 1182664, DOI https://doi.org/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 https://doi.org/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: yampol@ihes.fr, yampol@math.toronto.edu

Saeed Zakeri
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: zakeri@math.upenn.edu

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