Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture

Mikhail Lyubich

Review 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 quadratic-like 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 bi-sequences 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.

Self-Similarity 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 one-parameter family of quadratic-like 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 $|c_n-c|\approx b\lambda_M^{-n}$. 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 quasi-conformally 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

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