Complete solution of Siegel’s linearisation problem of an application that is holomorphic in the neighborhood of a fixed point (after J.-C. Yoccoz). (Solution complète au problème de Siegel de linéarisation d’une application holomorphe au voisinage d’un point fixe [d’après J.-C. Yoccoz].) (French) Zbl 0791.30019
Séminaire Bourbaki, Vol. 1991/92. Exposés 745-759 (avec table par noms d’auteurs de 1948/49 à 1991/92). Paris: Société Mathématique de France, Astérisque. 206, 273-310 (Exp. No.753) (1992).
The study of holomorphic dynamics near fixed points naturally leads to the following problem: Let $f$ be a function holomorphic on a neighbourhood of the origin with the power series $f(z)=az\sp n+$ higher terms and $n>0$. Does there exists a conformal conjugation between $f$ and $z \mapsto az\sp n$?
The present paper is an excellent survey on classical and recent results concerning this problem. First, the author recalls classical results dealing with the cases $n \ne 1$ and $n=1 \ne \vert a \vert$. In the remaining case one usually writes $a=\exp (2 \pi t)$ with $t \in \bbfR$. It is well-known, that $t \in \bbfQ$ implies the answer to the question stated above is “no”. Hence we may assume $t \in \bbfR \backslash \bbfQ$. Then the above problem becomes the so-called “Zentrumproblem”. After recalling partially results due to Cremer, Siegel and Brjuno the author explains the final solution recently given by Yoccoz. Then he points out analogies to the question on linearization of circle diffeomorphisms and presents several theorems due to Yoccoz concerning the linearization of analytic self mappings of the circle.
Next, the author scetches Yoccoz’ solution of the “Zentrumproblem” who introduces a new geometric method in the problem of small divisors. At the end, he deals with the modifications of this approach neeeded for the problem concerning circle diffeomorphisms and other applications. Especially, he establishes his results on the existence of periodic points accumulating at the origin. For the entire collection see [Zbl 0772.00016
|30D05||Functional equations in the complex domain, iteration and composition of analytic functions|
|39B12||Iterative and composite functional equations|