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
Full-text PDF Free Access
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.
- Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
- Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089
- Gamaliel Blé, External arguments and invariant measures for the quadratic family, Discrete Contin. Dyn. Syst. 11 (2004), no. 2-3, 241–260. MR 2083418, DOI https://doi.org/10.3934/dcds.2004.11.241
- Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383
- E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 0231999
- 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 [DH]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]Ep A. Epstein, Counterexamples to the quadratic mating conjecture, Manuscript 1998. [F]F P. Fatou, Mémoire sur les équations fonctionnelles, Bull. S.M.F 47 (1919), 161–271; 48 (1920), 33–94 and 208–314.
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
- O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas; Die Grundlehren der mathematischen Wissenschaften, Band 126. MR 0344463
- 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
- John Milnor, Pasting together Julia sets: a worked out example of mating, Experiment. Math. 13 (2004), no. 1, 55–92. MR 2065568
- John Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993), no. 1, 37–83. With an appendix by the author and Lei Tan. MR 1246482
- 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
- Carsten Lunde Petersen, The Herman-Światek theorems with applications, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 211–225. MR 1765090
- 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), no. 1, 1–52. MR 2051390, DOI https://doi.org/10.4007/annals.2004.159.1 [Re]Re M. Rees, Realization of matings of polynomials as rational maps of degree two, Manuscript 1986. [Sh]Sh M. Shishikura, On a theorem of M. Rees for matings of polynomials, Preprint IHES, 1990.
- Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122 (1985), no. 3, 401–418. MR 819553, DOI https://doi.org/10.2307/1971308 [TL]TL T. Lei, Accouplements de polynômes complexes, Ph.D. Thesis, Université Paris-Sud 1987.
- 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
- Michael Yampolsky and Saeed Zakeri, Mating Siegel quadratic polynomials, J. Amer. Math. Soc. 14 (2001), no. 1, 25–78. MR 1800348, DOI https://doi.org/10.1090/S0894-0347-00-00348-9
- 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
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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
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.