Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



Extensions of homeomorphisms between limbs of the Mandelbrot set

Authors: Bodil Branner and Núria Fagella
Journal: Conform. Geom. Dyn. 5 (2001), 100-139
MSC (2000): Primary 37F45; Secondary 37F10
Published electronically: October 18, 2001
MathSciNet review: 1872159
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using holomorphic surgery techniques, we construct a homeomorphism between a neighborhood of any limb without root point of the Mandelbrot set and a neighborhood of any other of equal denominator, in such a way that the limbs are mapped to each other. On the limbs, the homeomorphism coincides with that constructed in ``Homeomorphisms between limbs of the Mandelbrot set'' (J. Geom. Anal. 9 (1999), 327-390) which proves - without assuming local connectivity of the Mandelbrot set - that these maps are compatible with the embedding of the limbs in the plane. Outside the limbs, the constructed extension is quasi-conformal.

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

  • [A] L. Ahlfors, Lectures on quasi-conformal mappings, Wadsworth & Brooks/Cole mathematics series. MR 88b:30030
  • [AB] L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Annals of Math. 72 (1960), 385-404. MR 22:5813
  • [At] P. Atela, Bifurcations of Dynamic Rays in Complex Polynomials of Degree Two, Erg. Th. and Dyn. Sys. 12 (1991), 401-423. MR 94d:58128
  • [Bl] P. Blanchard, Complex Analytic Dynamics on the Riemann Sphere, Bull. Amer. Math. Soc. 11 (1984), 85-141. MR 85h:58001
  • [Br] B. Branner, The Mandelbrot Set, Proc. Symp. Applied Math. (1989), 75-105. CMP 21:16
  • [BD] B. Branner and A. Douady, Surgery on Complex Polynomials, Proc. of the Symposium on Dyn. Syst., Mexico, 1986, Lecture Notes in Math., 1345 11-72, Springer. MR 90e:58114
  • [BF] B. Branner and N. Fagella, Homeomorphisms between Limbs of the Mandelbrot Set, J. Geom. Anal. 9 (1999), 327-390. MR 2001j:37082
  • [D] A. Douady, Chirurgie sur les Applications Holomorphes, Proc. Int. Congr. Math., Berkeley, 1986, 724-738. MR 89k:58139
  • [DH1] A. Douady and J. H. Hubbard, Etude dynamique des polynomes complexes, I, II, Publ. Math. Orsay (1984, 1985). MR 87f:58072a; MR 87f:58072b
  • [DH2] A. Douady and J. H. Hubbard, On the Dynamics of Polynomial-like Mappings, Ann. Scient., Ec. Norm. Sup. $4^e$series, 18 (1985) 287-343. MR 87f:58083
  • [Mc] C. T. McMullen, Complex Dynamics and Renormalization, Annals of Mathematics Studies, 135, Princeton University Press, 1994. MR 96b:58097
  • [Mi] J. Milnor, Dynamics in One Complex Variable. Introductory Lectures. Vieweg, 1999. CMP 2000:03
  • [MSS] R. Mañe, P. Sad and D. Sullivan, On the Dynamics of Rational Maps, Ann. Scie. École Norm. Sup. (4) 16 (1983), 193-217. MR 85j:58089
  • [S] M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. Ec. Norm. Sup. 20 (1987), 1-29. MR 88i:58099

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 37F45, 37F10

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

Additional Information

Bodil Branner
Affiliation: Department of Mathematics, Technical University of Denmark, Building 303, DK-2800 Kongens Lyngby, Denmark

Núria Fagella
Affiliation: Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

Keywords: Holomorphic dynamics, complex polynomials, Julia sets, the Mandelbrot set, quasi-conformal mappings, surgery.
Received by editor(s): September 4, 2000
Received by editor(s) in revised form: May 10, 2001
Published electronically: October 18, 2001
Additional Notes: Partially supported by SNF Grant No. 9701387
Partially supported by DGICYT Grant No. PB96-1153, BFM2000-0805-C02-01 and CIRIT 2000SGR-27
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society