|
Extensions of homeomorphisms between limbs of the Mandelbrot set
Author(s):
Bodil
Branner;
Núria
Fagella
Journal:
Conform. Geom. Dyn.
5
(2001),
100-139.
MSC (2000):
Primary 37F45;
Secondary 37F10
Posted:
October 18, 2001
Retrieve article in:
PDF
This article is available free of charge
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:
-
- [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
Similar Articles:
Retrieve articles in Conformal Geometry and Dynamics
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:
10.1090/S1088-4173-01-00069-8
PII:
S 1088-4173(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
Posted:
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
Copyright of article:
Copyright
2001,
American Mathematical Society
|
Comments: webmaster@ams.org