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

Abstract

The Mandelbrot set $\mathcal{M}$ contains infinitely many small copies of itself, each canonically homeomorphic to $\mathcal{M}$ via the Douady--Hubbard theory of polynomial-like maps. These copies come in two kinds: primitive copies, whose principal hyperbolic component carries a cusp at its root, and satellite copies, whose boundary is smooth at the root. Douady and Hubbard conjectured that the straightening maps of analytic families of polynomial-like maps are quasiregular, predicting that primitive copies are quasiconformally homeomorphic to $\mathcal{M}$ and de-rooted satellite copies to $\mathcal{M}\setminus \{1/4\}$ --- hence that satellite copies are mutually quasiconformally homeomorphic away from their roots. Lyubich proved the primitive case, and showed that satellite copies are quasiconformally homeomorphic to $\mathcal{M}$ outside every neighbourhood of the root. Whether the homeomorphisms between satellite copies are quasiconformal at the roots remained open. In a previous work we gave a negative answer: satellite copies ${\mathcal{M}}_{p/q}$ and ${\mathcal{M}}_{p^{\prime }/q^{\prime }}$ with $q\ne q^{\prime }$ are not quasiconformally homeomorphic, disproving the conjecture; and we conjectured that copies whose rotation numbers share the same denominator are quasiconformally homeomorphic. In the present paper we prove that they are. Together, these results yield a complete geometric classification: two satellite copies of $\mathcal{M}$ are quasiconformally equivalent if and only if their rotation numbers have the same denominator. This settles the quasiconformal geometry of the small copies of the Mandelbrot set.