Julia sets appear quasiconformally in the Mandelbrot set

Abstract: In this paper we prove the following: Take any "small Mandelbrot set" and zoom in a neighborhood of a parabolic or Misiurewicz parameter in it, then we can see a quasiconformal image of a Cantor Julia set which is a perturbation of a parabolic or Misiurewicz Julia set. Furthermore, zoom in its middle part, then we can see a certain nested structure ("decoration") and finally another "smaller Mandelbrot set" appears. A similar nested structure exists in the Julia set for any parameter in the "smaller Mandelbrot set". We can also find images of a Julia sets by quasiconformal maps with dilatation arbitrarily close to 1. This answers a question by Adrian Douady. All the parameters belonging to these images are semihyperbolic and this leads to the fact that the set of semihyperbolic but non-Misiurewicz and non-hyperbolic parameters is dense with Hausdorff dimension 2 in the boundary of the Mandelbrot set.