HOMEOMORPHISMS BETWEEN LIMBS OF THE MANDELBROT SET

Bodil Branner and Nuria Fagella

Bodil Branner
The Technical University of Denmark
Building 303  
DK-2800 Lyngby 
Denmark
e-mail: branner@mat.dtu.dk

Nuria Fagella
Departament de Matematica Aplicada i Analisi
Universitat de Barcelona
Gran Via 585
08007 Barcelona
Spain
e-mail: fagella@maia.ub.es

ABSTRACT

Using a family of higher degree polynomials as a bridge, together with complex 
surgery techniques, we construct a homeomorphism between any two limbs of the 
Mandelbrot set of equal denominator. Induced by these homeomorphisms  and 
complex conjugation we obtain an involution between each limb and itself, 
whose fixed points form a topological arc.  All these maps have counterparts 
at the combinatorial level relating corresponding external arguments. Assuming 
local connectivity of the Mandelbrot set we may conclude that the constructed 
homeomorphisms between limbs are compatible with the embeddings of the limbs 
in the plane. As usual we plough in the dynamical planes and harvest in the 
parameter space.

REFERENCE

Journal of Geometric Analysis
To appear