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arXiv:math/9201282math.DS

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

Mitsuhiro Shishikura

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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.

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Subjects
math.DSDynamical Systems

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  1. Recursion and Fractals 1 comment

    • >There is a common misconception, repeated here, that fractals are defined by their self-similarity. This is not a misconception. Self-similarity at all length scales is indeed a defining factor of fractals. From that fact it _does_ follow that _most_ fractals have a non-integer…

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