The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets

TL;DR

This paper proves that the boundary of the Mandelbrot set and most Julia sets have Hausdorff dimension two, using bifurcation analysis of parabolic points, advancing understanding of fractal geometry in complex dynamics.

Contribution

It establishes that the boundary of the Mandelbrot set and generic Julia sets have Hausdorff dimension two, providing new insights into their fractal structure.

Findings

Boundary of Mandelbrot set has Hausdorff dimension two

Generic Julia sets have Hausdorff dimension two

Proof based on bifurcation of parabolic points

Abstract

It is shown that the boundary of the Mandelbrot set $M$ has Hausdorff dimension two and that for a generic $c \in \bM$, the Julia set of $z \mapsto z^2+c$ also has Hausdorff dimension two. The proof is based on the study of the bifurcation of parabolic periodic points.