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Conformal Geometry and Dynamics

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Mating a Siegel disk with the Julia set of a real quadratic polynomial


Authors: G. Ble and R. Valdez
Journal: Conform. Geom. Dyn. 10 (2006), 257-284
MSC (2000): Primary 37F10; Secondary 37F45, 37F50
DOI: https://doi.org/10.1090/S1088-4173-06-00150-0
Published electronically: October 5, 2006
MathSciNet review: 2261051
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Abstract | References | Similar Articles | Additional Information

Abstract: In this work, we show that it is possible to construct the mating between a quadratic polynomial with a Siegel disk and a real quadratic polynomial possessing a postcritical orbit that is semi-conjugate to a rigid rotation with the same rotation number as the Siegel disk.


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

  • [Ah] L.V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Math. Studies, No. 10, D. Van Nostrand Co. Inc., New York, 1966. MR 0200442 (34:336)
  • [B] A.F. Beardon, Iteration of rational functions, Springer-Verlag, 1991. MR 1128089 (92j:30026)
  • [Bl] G. Ble, External arguments and invariant measures for the quadratic family, Disc. and Cont. Dyn. Sys. 11 (2004), 241-260. MR 2083418 (2005e:37097)
  • [CG] L. Carleson and T.W. Gamelin, Complex Dynamics, Springer-Verlag, 1993. MR 1230383 (94h:30033)
  • [CL] E. F. Collingwood and A.J. Lohwater, The theory of cluster sets, Cambridge at the University Press, 1966. MR 0231999 (38:325)
  • [D] A. Douady, Systèmes Dynamiques Holomorphes, Séminaire Bourbaki, 35é année # 599, Astérisque 105-106 (1983), 39-63. MR 0728980 (85h:58090)
  • [D2] A. Douady, Disques de Siegel et anneaux de Herman, Séminaire Bourbaki, 1986-87, exposé No.677, Astérisque 152-153, (1987), 151-172. MR 0936853 (89g:30049)
  • [DE] A. Douady and C.J. Earle, Conformally natural extension of homeomorphismes of the circle, Act. Math. 157 (1986), 25-48. MR 0857678 (87j:30041)
  • [DH] A. Douady and J.H. Hubbard, Étude dynamique des polynômes complexes I et II, Pub. Math. d'Orsay 84-02 and 85-02, (1984-85).
  • [Ep] A. Epstein, Counterexamples to the quadratic mating conjecture, Manuscript 1998.
  • [F] P. Fatou, Mémoire sur les équations fonctionnelles, Bull. S.M.F 47 (1919), 161-271; 48 (1920), 33-94 and 208-314.
  • [HW] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, Oxford University Press, 1979. MR 0568909 (81i:10002)
  • [LV] O. Lehto and J. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, 1973. MR 0344463 (49:9202)
  • [dMvS] W. de Melo and S. van Strien, One-dimensional dynamics, Springer Verlag, 1993. MR 1239171 (95a:58035)
  • [M1] J. Milnor, Pasting together Julia sets: A worked out example of mating, Experimen. Math. 13 (2004), 55-92. MR 2065568 (2005c:37087)
  • [M2] J. Milnor, Geometry and dynamics of quadratic rational maps, Experimen. Math. 2 (1993), 37-83. MR 1246482 (96b:58094)
  • [Pe] C.L. Petersen, Local connectivity of some Julia sets containing a circle with an irrational rotation, Act. Math. 177 (1996), 163-224. MR 1440932 (98h:58164)
  • [Pe1] C.L. Petersen, The Herman-Swiatek theorem, with applications, The Mandelbrot Set, theme and variations. London Mathematical Society, Lecture Note Series 274 (2000), 211-225. MR 1765090 (2001b:37061)
  • [PeZa] C.L. Petersen and S. Zakeri, On the Julia set of a typical quadratic polynomial with a Siegel disk, Ann. of Math. (2) 159 (2004), 1-52. MR 2051390 (2005c:37085)
  • [Re] M. Rees, Realization of matings of polynomials as rational maps of degree two, Manuscript 1986.
  • [Sh] M. Shishikura, On a theorem of M. Rees for matings of polynomials, Preprint IHES, 1990.
  • [S] D. Sullivan, Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. 122 (1985), 401-418. MR 0819553 (87i:58103)
  • [TL] T. Lei, Accouplements de polynômes complexes, Ph.D. Thesis, Université Paris-Sud 1987.
  • [Ya] M. Yampolsky, Complex bounds for renormalization of critical circle maps, Erg. Th. and Dyn. Sys. 19 (1999), 227-257. MR 1677153 (2000d:37053)
  • [YaZa] M. Yampolsky and S. Zakeri, Mating Siegel quadratic polynomials, J. Am. Math. Soc. 14, No. 1, (2001), 25-78. MR 1800348 (2001k:37064)
  • [Y] J.C. Yoccoz, Il n'y a pas de contre-example de Denjoy analytique, C. R. Acad. Sci. Paris 298 (1984), 141-144. MR 0741080 (85j:58134)

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Additional Information

G. Ble
Affiliation: División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, Km. 1 Carr. Cunduacán-Jalpa, C.P. 86690, Cunduacán, Tabasco, México
Email: gble@ujat.mx

R. Valdez
Affiliation: Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, col. Lomas de Chamilpa, C.P. 62210 Cuernavaca, Morelos, México
Email: rogelio@matcuer.unam.mx

DOI: https://doi.org/10.1090/S1088-4173-06-00150-0
Keywords: Holomorphic dynamics, rational map, mating, Julia set, Mandelbrot set
Received by editor(s): February 10, 2006
Published electronically: October 5, 2006
Additional Notes: The first author was supported by CONACYT, 42249
The second author was supported by PROMEP, UAEMOR-PTC-166
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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