On quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I

Luna Lomonaco, Carsten Lunde Petersen

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

Douady and Hubbard (Ann Sci Ec Norm Suppl 4 18(2):287–343,
1985) introduced the notion of polynomial-like 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 q-c homeomorphic to M. But are
they mutually q-c homeomorphic? Or even q-c 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 q-c homeomorphism:
For any two satellite copies M and M the induced Douady–Hubbard
homeomorphism is not q-c if the root multipliers λ = e2πip
/q and λ = e2πip/q have q = q.
Translated title of the contributionOm quasikonform (in) kompatibilitet af satelit kopier af Mandelbrotmængden I:
Original languageEnglish
JournalInventiones Mathematicae
Volume210
Issue number2
Pages (from-to)615-644
Number of pages30
ISSN0020-9910
DOIs
Publication statusPublished - 2017

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