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Annals of Mathematics, II. Series Vol. 149, No. 2, pp. 319420 (1999) 

FeigenbaumCoulletTresser universality and Milnor's hairiness conjectureMikhail LyubichReview from Zentralblatt MATH: In this long paper, which forces the reader into a position of a student, the author presents the following results: Let $M_0$ be the Mandelbrot set and $N$ the full family of Mandelbrot copies $M\subset M_0$ different from $M_0$. Let $QG$ be the space of quadraticlike germs. For $M\in N$ there is a subset $T_M\subset QG$ (the renormalization strip) and a renormalization operator $R_M:T_M\to QG$. Given a family $L\subset N$ one may consider the operator $R_L:\bigcup_{M\in L} T_M\to QG$. If the family is finite the operator is called of bounded type, if all $M\in L$ are centered on the real line the operator has real combinatorics. Hyperbolicity Theorem. If there exists a renormalization operator $R=R_L$ of real bounded type defined on the union of renormalization strips then there is a compact $R$invariant set $A$ (the renormalization horseshoe) such that: 1) The restriction $R_{A}$ is topologically conjugate to a shift transformation on finite alphabet bisequences and is uniformly hyperbolic. 2) Any stable leaf $W^s(f)$, $f\in A$ coincides with the hybrid class of $f$ and has codimension 1. 3) Any unstable leaf $W^u(f)$ is an analytic curve transversal to all real hybrid classes except the cusp one (for $c=0.25$). Hairiness Theorem. Let $c\in [2,0.25]$ be the Feigenbaum parameter value. Then the rescalings of Mandelbrot set near $c$ converge in the Hausdorff metric on compact plane to the whole complex plane. SelfSimilarity Theorem. Let $M$ be a real Mandelbrot copy and $\sigma:M\to M_0$ be the homeomorphism of $M$ onto the whole $M_0$. Then $\sigma$ has a unique fixed point $c$. Moreover $\sigma$ is $C^{1+\alpha}$conformal at $c$, with the derivative at $c$ equal to the Feigenbaum universal scaling constant $\lambda_M>1$. Universality Theorem. Let $S=\{f_\mu\}$ be a real analytic oneparameter family of quadraticlike maps transversally intersecting the hybrid class $H_c$ at $\mu_*$. Then for all sufficiently big $n$, $S$ has a unique intersection point $\mu_n$ near $\mu_*$ with the hybrid class $H_{c_n}$ and $\mu_n\mu_*\approx a\lambda_M^{n}$, in particular $c_nc\approx b\lambda_M^{n}$. Here $c_n$ are super attracting points of periods $p_M^n$, which are obtained from the center of $M$. HD Theorem. For any finite real family $L$ containing at least two elements the set of parameters of infinite renormalizable maps of type $L$ is a Cantor set with dimension strictly between 0 and 1. QC Theorem. Any primitive (i.e. not attached to a hyperbolic component) copy $M$ is quasiconformally equivalent to $M_0$. Reviewed by T.Nowicki Keywords: renormalization; hairiness; universality; quadratic maps; quasiconformal equivalence; Mandelbrot copies; hyperbolicity; Feigenbaum parameter; Hausdorff metric Classification (MSC2000): 37E20 37D05 37F25 37F30 37E45 Full text of the article:
Electronic fulltext finalized on: 8 Sep 2001. This page was last modified: 21 Jan 2002.
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