## The Decoration Theorem of the Mandelbrot Set and Applications in Holomorphic Dynamics

- We prove the decoration theorem of the Mandelbrot set M: if a homeomorphic copy of M (coming from renormalization) is removed from M, then most of the remaining connected components (the decorations of the little copy) are small. We prove that for every hyperbolic component of the Mandelbrot set, any two limbs with equal denominators are homeomorphic so that the homeomorphism preserves periods of hyperbolic components. This settles a conjecture on the Mandelbrot set that goes back to 1994. We give a topological description of the space of quadratic rational maps with superattractive two-cycles: its "non-escape locus" M2 (the analog of the Mandelbrot set M) is locally connected, it is the continuous image of M under a canonical map, and it can be described as M (minus the 1/2-limb), mated with the lamination of the basilica. The latter statement is a refined version of a conjecture of Ben Wittner, which in its original version requires local connectivity of M to even be stated. Our methods of mating with a lamination also apply to dynamical matings of certain non-locally connected Julia sets. Using the theory of iterated monodromy groups we construct the line complex for a linearizer map.