The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets

1998/03/01 by Mitsuhiro Shishikura · 2 citations
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Cellular Automata and Applications #advanced mathematical theories

paper · doi:10.2307/121009

openalex publication_date 1998/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

It is shown that the boundary of the Mandelbrot set M has Hausdorff dimension two and that for a generic c ∈ ∂M , the Julia set of z → z 2 + c also has Hausdorff dimension two.The proof is based on the study of the bifurcation of parabolic periodic points.

Citations

• Absolutely continuous invariant measures for one-parameter families of one-dimensional maps
• Repellers for real analytic maps

Cited by

• Dynamics of multicritical circle maps
• Mandelbrot set [wikipedia]