Julia sets appear quasiconformally in the Mandelbrot set

TL;DR

This paper demonstrates that Julia sets can be quasiconformally embedded within the Mandelbrot set near certain parameters, revealing nested structures and density of specific parameter types, thus answering a longstanding question.

Contribution

It proves the quasiconformal appearance of Julia sets in the Mandelbrot set near parabolic and Misiurewicz parameters, including nested structures and density results.

Findings

Quasiconformal images of Julia sets appear near specific parameters in the Mandelbrot set.

Nested structures (

A dense set of semihyperbolic parameters exists with Hausdorff dimension 2 on the boundary of 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.