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Conformal Geometry and Dynamics

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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
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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

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