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The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. (English) Zbl 0922.58047

It is well known that there are two important sets associated to dynamics of the complex quadratic polynomials \(P_c(z)=z^2+c\): the Julia set \(J_c\) which is the closure of repelling periodic points of \(P_c\), and the Mandelbrot set \(M\) that is the subset of parameters \(c\in{\mathbb C}\) for which the set \(K_c=\{z\in {\mathbb C}\mid P_c^n(z)\not\to \infty\) as \(n\to\infty\}\) is connected. The boundary of the Mandelbrot set is characterized as: \(\partial M=\{c\in {\mathbb C}\mid P_c\) is not \(J\)-stable}.
In this paper the author establishes results concerning the Hausdorff dimension, \(\dim_H\), of these sets. Namely: The Hausdorff dimension of the boundary of the Mandelbrot set, \(\dim_H(\partial M)=2\), and for any open subset \(U\) intersecting \(\partial M\), \(\dim_H(U\cap \partial M)=2\). This result comes to explain the complexity of the Mandelbrot set.
The result concerning the dimension of Julia sets states that there exists a residual subset \(\mathcal R\) of \(\partial M\) such that if \(c\in {\mathcal R}\), then \(\dim_H(J_c)=2\), and there exists a residual set \({\mathcal R}'\) of the unit circle \({\mathbf S}^1\) such that if \(P_c\) has a periodic point with multiplier \(\exp(2\pi i \alpha)\), \(\alpha\in{\mathcal R}'\), then \(\dim_H(J_c)=2\).
In order to prove these results the author investigates much more aspects concerning Hausdorff dimension and hyperbolic dimension of some subsets in dynamical plane and parameter space of a family of rational maps.
It was observed in computer experiments that after a small perturbation of a parabolic periodic point, the Julia set may inflate. The author shows that such an inflated part of the Julia set can have Hausdorff dimension close to 2. The main tool in the study of such a bifurcation is the theory of Ecalle cylinders. Using this theory, the author introduces a new renormalization procedure associated with parabolic fixed points.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
28A80 Fractals
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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