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Similarity between the Mandelbrot set and Julia sets. (English) Zbl 0726.58026
For each \(c\in {\mathbb{C}}\), let \(f_ c(z)=z^ 2+c\), and let \(J_ c\) denote the Julia set of \(f_ c\). We say that c is eventually periodic under \(f_ c\), if there exist integers \(p,\ell >0\) such that \[ (*)\;f^ p(w)=w,\;w=f^{\ell}(c). \] In this case, c is called a Misiurewicz point, and such points are known to be dense in the boundary of the Mandelbrot set M. The author is concerned to formulate and prove rigorously the intuitive statement that M and \(J_ c\) are each self- similar at c and similar to each other there. Precisely because the statement is so appealing, and yet so complicated to make precise and to prove, it seems to the reviewer that this is an important paper.
A good brief description is the author’s own: ‘In this paper we will show that for every Misiurewicz point c, a powerful magnification of \(J_ c\) focussed at c and the same magnification of M focussed also at c will give very similar structures (up to a multiplication by a complex number), and the more powerful the magnification is, the more similar the structures are. We say that \(J_ c\) and M are asymptotically similar about c. Moreover, both \(J_ c\) and M are asymptotically self-similar about c in the sense that if we increase successively the magnifications of \(J_ c\) (respectively M) focussed at c by a carefully chosen factor, what we see are more and more the same’.
That helpful introduction makes it much easier to appreciate the formal precise definitions of the 3 types of local similarity mentioned by the author. These are formulated in terms of the Hausdorff metric on the space of compact non-empty subsets of \({\mathbb{C}}\). Local self-similarity of \(J_ c\) is then proved (when c is a Misiurewicz point) from Theorem 3.2, which applies more generally to sets which are completely invariant under a rational mapping f: \({\mathbb{C}}\to {\mathbb{C}}\). It depends on a classical lemma that says when f is locally conformally conjugate to its linear part, and (for \(J_ c)\) on the known fact that in (*), \(| (f^ p)'(w)| >1.\)
To prove the corresponding result for M is more difficult. Since the Julia sets form a family parametrized by M, the author formulates Proposition 4.1, which relates a family of closed sets \(X_{\lambda}\subseteq {\mathbb{C}}\) when \(\lambda\) runs through an open set \(\Delta\), to the subset \(M_ u\) of \(\Delta\) which consists of all \(\lambda\) for which \(X_{\lambda}\) meets the graph of a given continuous u: \(\Delta\to {\mathbb{C}}\). The Proposition gives conditions under which \(M_ u\) is locally similar to some \(X_{\lambda}\) (it takes nearly a page to state, and \({\mathbb{C}}\) is replaced by \({\mathbb{R}}^ n)\). The core of its proof concerns the question of showing when c in (*) depends analytically on c. After the application to M, there is a helpful Theorem 5.5, which reviews the various steps of the whole proof.
The paper contains several examples and illustrations, and concludes with a brief discussion of related problems and conjectures.
(The author’s preliminary definition of local asymptotic similarity has been developed independently, and from a more geometric point of view, by K. Wicks in his monograph ‘Fractals and Hyperspaces’, Springer Lecture Notes, Vol. 1492. Apart from choosing essentially the same definition, there is no overlap between the two works).

MSC:
37B99 Topological dynamics
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
28A80 Fractals
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