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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).

37B99 Topological dynamics
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
28A80 Fractals
Full Text: DOI
[1] [B] Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. Am. Soc.11, 85–141 (1984) · Zbl 0558.58017 · doi:10.1090/S0273-0979-1984-15240-6
[2] [CE] Campanino, M., Epstein, H.: On the existence of Feigenbaum’s fixed point. Commun. Math. Phys.79, 226–302 (1981) · Zbl 0474.58013 · doi:10.1007/BF01942063
[3] [CEL] Collet, P., Eckmann, J.-P., Lanford III, O.E.: Universal properties of maps on an interval. Commun. Math. Phys.76, 211–254 (1980) · Zbl 0455.58024 · doi:10.1007/BF02193555
[4] [D] Douady, A.: Système dynamiques holomorphes, exposé no. 599, séminaire Bourbaki 1982/83. Astérisque105–106, 39–63 (1983)
[5] [DH1] Douady, A., Hubbard, J.H.: Etude dynamique des polynômes complexes. Part I. Publication mathématique d’Orsay, 84-02, 1984
[6] [DH2] Douady, A., Hubbard, J.H.: Etude dynamique des polynômes complexes. Part II. Publication mathématique d’Orsay, Orsay, 85–102, 1985
[7] [DH3] Douady, A., Hubbard, J.H.: On the dynamics of polynomial-like mappings. Ann. Scient. Éc. Norm. Sup., 4e série, t.18, 287–343 (1985)
[8] [EE] Eckmann, J.-P., Epstein, H.: Scaling of Mandelbrot sets generated by critical point preperiodicity. Commun. Math. Phys.101, 283–289 (1985) · Zbl 0587.30027 · doi:10.1007/BF01218762
[9] [EEW] Eckmann, J.-P., Epstein, H., Wittwer, P.: Fixed points of Feigenbaum’s type for the equationf p (\(\lambda\)x)f(x). Commun. Math. Phys.93, 495–516 (1984) · Zbl 0565.58031 · doi:10.1007/BF01212292
[10] [EW] Eckmann, J.-P., Wittwer, P.: A complete proof of the Feigenbaum conjectures. J. Stat. Phys.46, Nos. 3/4, 455–475 (1987) · Zbl 0683.46052 · doi:10.1007/BF01013368
[11] [F1] Feigenbaum, M.: The universal metric properties of nonlinear transformations. J. Stat. Phys.21, 669–706 (1979) · Zbl 0515.58028 · doi:10.1007/BF01107909
[12] [F2] Feigenbaum, M.: Universal behavior in nonlinear systems. Los Alamos Sci.1, 4–27 (1980)
[13] [GSK] Gol’berg, A.I., Sinai, Ya.G., Khanin, K.M.: Universal properties of bifurcations of period three. Russ. Math. Surv.38, 187–188 (1983) · Zbl 0542.30039 · doi:10.1070/RM1983v038n01ABEH003398
[14] [GT] Großmann, S., Thomae, S.: Invariant distributions and stationary correlation functions of one-dimensional discrete processes. Z. Naturforsch.32a, 1353–1363 (1977)
[15] [L] Lanford III, O.E.: A computer-assisted proof of the Feigenbaum conjectures. Bull. (new series) Am. Soc., Vol.6, Number 3, 427–434 (1982) · Zbl 0487.58017 · doi:10.1090/S0273-0979-1982-15008-X
[16] [M] Milnor, J.: Self-similarity and hairiness in the Mandelbrot set. In Computers in Geometry and Topology. Martin, C. (ed.). Tangora. New York, Basel: Dekker, pp. 211–257 1989 · Zbl 0676.58036
[17] [PR] Peitgen, H.-O., Richter, P.H.: The beauty of fractals. Berlin, Heidelberg, New York: Springer 1986 · Zbl 0601.58003
[18] [PS] Peitgen, H.-O., Sauper, D. (eds.): The science of fractal images. Berlin, Heidelberg, New York: Springer 1988
[19] [PJS] Peitgen, H.-O., Jürgens, H., Sauper, D.: The Mandelbrot set: a paradigm for experimental mathematics, ECM/87-Educational Computing in Mathematics, Banchoff, T.F. et al. (eds.), Elsevier Science Publishers BV. (North-Holland), 1988
[20] [T1] Tan Lei: Ordre du contact des composantes hyperboliques deM, in [DH2], exposé no. XV
[21] [T2] Tan Lei: Ressemblence entre l’ensemble de Mandelbrot et l’ensemble de Julia au voisinage d’un point de Misiurewicz, in [DH2], exposé no. XXIII
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