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Abstract
Douady and Hubbard (Ann Sci Ec Norm Suppl 4 18(2):287–343,
1985) introduced the notion of polynomiallike maps. They used it to identify
homeomorphic copies M of the Mandelbrot set inside the Mandelbrot
set M. These copies can be primitive (with a root cusp) or satellite (without
a root cusp). They conjectured that the primitive copies are quasiconformally
homeomorphic to M, and that the satellite ones are quasiconformally
homeomorphic to M outside any small neighbourhood of the root. These conjectures are now Theorems due to Lyubich (Ann Math 149:319–420, 1999).
The satellite copies Mp/q are clearly not qc homeomorphic to M. But are
they mutually qc homeomorphic? Or even qc homeomorphic to half of the
logistic Mandelbrot set? In this paper we prove that, in general, the induced
Douady–Hubbard homeomorphism is not the restriction of a qc homeomorphism:
For any two satellite copies M and M the induced Douady–Hubbard
homeomorphism is not qc if the root multipliers λ = e2πip
/q and λ = e2πip/q have q = q.
1985) introduced the notion of polynomiallike maps. They used it to identify
homeomorphic copies M of the Mandelbrot set inside the Mandelbrot
set M. These copies can be primitive (with a root cusp) or satellite (without
a root cusp). They conjectured that the primitive copies are quasiconformally
homeomorphic to M, and that the satellite ones are quasiconformally
homeomorphic to M outside any small neighbourhood of the root. These conjectures are now Theorems due to Lyubich (Ann Math 149:319–420, 1999).
The satellite copies Mp/q are clearly not qc homeomorphic to M. But are
they mutually qc homeomorphic? Or even qc homeomorphic to half of the
logistic Mandelbrot set? In this paper we prove that, in general, the induced
Douady–Hubbard homeomorphism is not the restriction of a qc homeomorphism:
For any two satellite copies M and M the induced Douady–Hubbard
homeomorphism is not qc if the root multipliers λ = e2πip
/q and λ = e2πip/q have q = q.
Translated title of the contribution  Om quasikonform (in) kompatibilitet af satelit kopier af Mandelbrotmængden I: 

Original language  English 
Journal  Inventiones Mathematicae 
Volume  210 
Issue number  2 
Pages (fromto)  615644 
Number of pages  30 
ISSN  00209910 
DOIs  
Publication status  Published  2017 
Projects
 1 Finished

Holomorpic Dynamics and Orthogonal Polynomials
Petersen, C. L., Petersen, H. L., Henriksen, C. & Christiansen, J. S.
01/11/2015 → 31/10/2018
Project: Research