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

TL;DR

This paper investigates the quasi-conformal compatibility of satellite copies of the Mandelbrot set, proving that the Douady-Hubbard homeomorphism between different satellite copies is generally not quasi-conformal when their root multipliers differ.

Contribution

It demonstrates that the Douady-Hubbard homeomorphism between satellite copies of the Mandelbrot set is not quasi-conformal if their root multipliers are different, clarifying the limitations of such mappings.

Findings

Douady-Hubbard homeomorphism is not q-c between satellite copies with different roots.

Satellite copies are not mutually q-c homeomorphic in general.

The homeomorphism is not q-c off neighborhoods of roots with different multipliers.

Abstract

In the paper 'On the dynamics of polynomial-like mappings' Douady and Hubbard introduced the notion of polynomial-like maps. They used it to identify homeomophic copies of the Mandelbrot set inside the Mandelbrot set. They conjectured that in case of primitive copies the homeomorphism between the homeomorphic copy of the Mandelbrot set and the Mandelbrot set is q.-c., and similarly in the satellite case, it is q.-c. off any small neighborhood of the root. These conjectures are now Theorems due to Lyubich. The satellite copies of the Mandelbrot set are clearly not q-c homeomorphic to the Mandelbrot set. 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 of the Mandelbrot set, the induced Douady-Hubbard homeomorphism is not q-c, if the root multipliers, which are primitive q and q' roots of unity, have q different from q'.