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

DOI:
https://doi.org/10.1090/S1088-4173-01-00069-8

Published electronically:
October 18, 2001

MathSciNet review:
1872159

Full-text PDF Free Access

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.

**[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.**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**

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

Email:
B.Branner@mat.dtu.dk

**Núria Fagella**

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

Email:
fagella@maia.ub.es

DOI:
https://doi.org/10.1090/S1088-4173-01-00069-8

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