Fixed points and circle maps
The main purpose of the paper is to study the heuristic relation between holomorphic dynamics near an indifferent fixed point and the dynamics of analytic circle homeomorphisms. In this paper, the author develops a powerful geometric method. Using this method, the author makes many substantial contributions to this subject. For example, the author obtains a dictionary between the two problems, gives a new natural proof of Naishul's theorem, proves Dulac-Moussu's conjecture, and generalizes the snail lemma. The author also proves the nonexistence of periodic points on the boundary of Siegel disks of the first type. The main ingredients of the method are the following theorems: Theorem 1. Let \(f(z)= e^{2\pi i\alpha}z+ O(z^2)\), \(\alpha\in\mathbb{R}\), be a local holomorphic diffeomorphism. Let \(U\) be a Jordan neighborhood of the indifferent fixed point \(0\). Assume that \(f\) as well as \(f^{-1}\) are defined and univalent on a neighborhood of the closure of \(U\). Then there exists a set \(K\) such that (i) \(K\) is compact, connected and full; (ii) \(0\in K\subset\overline U\); (iii) \(K\cap\partial U\neq\emptyset\); (iv) \(f(K)= K\) and \(f^{-1}(K)= K\). Moreover, if \(f\) is not of finite order, \(f\) is linearizable at \(0\) if and only if \(0\in\text{int}(K)\). The compact set \(K\) in the theorem is called a Siegel compactum. When the fixed point is indifferent irrational and \(K\) is not contained in the closure of a linearization domain, then \(K\) is called a hedgehog. Theorem 2. Let \(\overline{\mathbb{C}}\) be the Riemann sphere, \(\mathbb{D}\) the unit disk and \(\mathbb{S}^1=\partial\mathbb{D}\). Assume that the hypotheses in Theorem 1 are satisfied and \(K\) is the Siegel compactum given by Theorem 1. Let \(h:\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}\to \overline{\mathbb{C}}\setminus K\), \(h(\infty)= \infty\), be a conformal representation of the exterior of \(K\). Then the map \(g= h^{-1}\circ f\circ h\) extends to an analytic circle diffeomorphism of \(\mathbb{S}^1\) with rotation number \(\rho(g)= \alpha\).
scientific article; zbMATH DE number 1172065