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

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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.

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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