Siegel disks and Herman rings. (Disques de Siegel et anneaux de Herman.) (French) Zbl 0638.58023

Douady, Adrien

Sémin. Bourbaki, 39ème année, Vol. 1986/87, Exp. No. 677, Astérisque 152/153, 151-172 (1987).

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).
[For the entire collection see Zbl 0627.00006.]

Reviewer: M.Lyubich

MSC:

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

Keywords:

Siegel disk; one-dimensional complex analytic maps; geometrical function theory; quasi-conformal deformations; Periodic points; Julia set; non- repelling cycles; critical point

Citations:

Zbl 0627.00006