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The boundary of the Mandelbrot set has Hausdorff dimension two. (English) Zbl 0813.58047
Camacho, C. (ed.) et al., Complex analytic methods in dynamical systems. Proceedings of the congress held at Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil, January 1992. Paris: Société Mathématique de France, Astérisque. 222, 389-405 (1994).
For \(P_ c = z^ 2 + c\) \((z,c \in \mathbb{C})\) the Julia set \(J_ c\) of \(P_ c\) and the Mandelbrot set \(M\) are defined as \(J_ c = \) the closure of repelling periodic points of \(P_ c\) and \(M = (c \in \mathbb{C}\mid J_ c\) is connected). The author already proved – concerning Hausdorff dimension – that \(H\text{-dim } \partial M = 2\). This is extended that for a generic \(c \in \partial M\), \(H\text{-dim } J_ c = 2\).
For the entire collection see [Zbl 0797.00019].
Reviewer: Y.Kozai (Tokyo)

MSC:
37F99 Dynamical systems over complex numbers
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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