Holomorphic dynamical systems. (Systèmes dynamiques holomorphes.) (French) Zbl 0532.30019

Douady, Adrien

Sémin. Bourbaki, 35e année, 1982/83, Exp. No. 599, Astérisque 105-106, 39-63 (1983).

This paper is a detailed survey of recent developments in the theory of rational maps of the Riemann sphere \(\Sigma\), containing outline proofs and numerous open questions. It begins with a brief account of the basic definitions and classical results: the classification of periodic points, the Julia set \(J\), etc.; and then passes to Sullivan’s theorem on the non- existence of wandering domains in \(\Sigma\)-\(J\), and recent work of Sullivan-Sad-Manẽ-Thurston on structural stability and hyperbolic maps.
The remainder of the paper largely concerns recent work of J. Hubbard and the author. The point of departure is a detailed study of quadratic polynomials \(f_c(z)=z^2+c\) for varying \(c\). If \(K_c=\{z\in\mathbb{C}: f^n_c(z)\to \infty \}\) then \(K_c\) is either connected or a Cantor set. The Mandelbrot set \(M\subseteq\mathbb{C}\) is the set of \(c\in\mathbb{C}\) for which the first alternative holds. Introducing the techniques of external and internal arguments, the Hubbard trees, the authors have made a detailed study of \(M\). \(M\) is connected and probably locally connected and has a kind of self-replicating structure first observed by Mandelbrot.
This structure is further analyzed by introducing the notion of functions of polynomial type, holomorphic functions which behave qualitatively like polynomials in some small disc. Many of the results on polynomials extend to this class. Two further techniques, modulation and coupling of polynomials allow one to remove one part of a dynamical system and replace it by another, perhaps forming the basis of a future theory of “surgery” for rational maps.
For the entire collection see [Zbl 0514.00009].

Reviewer: Caroline Series (Coventry)

MSC:

30D05	Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37Fxx	Dynamical systems over complex numbers
26C15	Real rational functions
30E99	Miscellaneous topics of analysis in the complex plane

Keywords:

rational map; Julia set; Mandelbrot set; Hubbard trees; coupling of polynomials; modulation of polynomials