Extensions of homeomorphisms between limbs of the Mandelbrot set

Branner, Bodil; Fagella, Núria

doi:10.1090/S1088-4173-01-00069-8

Extensions of homeomorphisms between limbs of the Mandelbrot set
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by Bodil Branner and Núria Fagella

Conform. Geom. Dyn. 5 (2001), 100-139

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

Published electronically: October 18, 2001

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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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