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Disques de Siegel et anneaux de Herman. (Siegel disks and Herman rings). (French) Zbl 0638.58023
Sémin. Bourbaki, 39ème année, Vol. 1986/87, Exp. No. 677, Astérisque 152/153, 151-172 (1987).
[For the entire collection see Zbl 0627.00006.]
This is a survey of the modern state of the linearization problem of one- dimensional complex analytic maps (defined in the neighbourhood of a fixed point or an invariant circle). The exposition is given from the geometrical viewpoint based on geometrical function theory and methods of quasi-conformal deformations. The paper consists of three sections. (1) Periodic points and components of \({\bar {\mathbb{C}}}\setminus J(f)\). (2) Shishikura’s inequalities. (3) Numbers and functions.
In the first section Fatou-Julia-Sullivan’s classification of periodic points and components of \({\bar {\mathbb{C}}}\setminus J(f)\) for rational map \(f: {\bar {\mathbb{C}}}\to {\bar {\mathbb{C}}}\) is described (here J(f) is the Julia set of f). In the second section Shishikura’s precise estimates of the number of non-repelling cycles of f are explained. The last section acquaints the reader with the recent results due to M. Herman, J.-C. Yoccoz and others. Let us formulate two of them.
Let \({\mathcal S}\) be the set of rotation numbers \(\alpha\in {\mathbb{R}}\) such that all analytic maps \(f: z\mapsto e^{2\pi i\alpha}z+O(z^ 2)\) are analytically linearizable (i.e. there exists a univalent map \(\phi\) : \(U\to {\mathbb{C}}\) in a neighbourhood of the origin such that \(\phi (fz)=e^{2\pi i\alpha}\phi (z))\). Proposition 3. (Yoccoz, 1987). If \(P_{\alpha}: z\mapsto e^{2\pi i\alpha}z+z^ 2\) is linearizable then \(\alpha\in {\mathcal S}.\)
The maximal domain \({\mathcal D}\) where f is linearizable is called a Siegel disk. There was a problem (related to one of Fatou’s results) that the boundary \(\partial D\) contains a critical point of f. Proposition 7 (Herman, 1986). There exists \(\alpha\in {\mathbb{R}}\) such that \(P_{\alpha}: z\mapsto e^{2\pi i\alpha}z+z^ 2\) has a Siegel disk \({\mathcal D}\) whose boundary does not contain a critical point (and \(\partial D\) is a quasi- circle).
Reviewer: M.Lyubich

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
30C99 Geometric function theory
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