On qc compatibility of satellite copies of the Mandelbrot set: II

Abstract

The Mandelbrot set is a fractal which classifies the behaviour of complex quadratic polynomials. Although its remarkably simple definition: $\mathcal{M}:=\{c \in \mathbb{C}\,|\,Q_c(0)^n \nrightarrow \infty \mbox{ as } n\rightarrow \infty, \mbox{ where } Q_c(z)=z^2+c\}$, it is a central object in complex dynamics, and it has been charming and intriguing since it has first been defined and drawn. A fascinating fact is the presence of little copies of the Mandelbrot set in the Mandelbrot set itself (and in many other parameter planes). There exist two different kinds of little copies of the Mandelbrot set within the Mandelbrot set: the primitive copies, visually similar to the Mandelbrot set, with a cusp at the root of the principal hyperbolic component, and the satellite copies, whose principal hyperbolic component has no cusp, i.e. has smooth boundary across the root point. Lyubich proved that the primitive copies of $\mathcal{M}$ satisfy a stronger regularity condition: they are quasiconformally homeomorphic to $\mathcal{M}$. The satellite copies are homeomorphic to $\mathcal{M}$, but the homeomorphism is only quasiconformal outside neighbourhoods of the root. The question that remained open was: are the satellite copies mutually quasiconformally homeomorphic? In a previous work, we showed that the satellite copies with rotation numbers with different denominators are not quasiconformally homeomorphic. Here we complete the picture, by showing that, for any $q$, the satellite copies $\mathcal{M}_{p/q}$ and $\mathcal{M}_{p'/q}$ of the Mandelbrot set $\mathcal{M}$ with rotation numbers with the same denominator $q$ are quasiconformal homeomorphic.