Fixed points and uniqueness of complex geodesics
Let \(V\) be a bounded circular domain in \(\mathbb{C}^ n\) and let \(f\) be in the space of all holomorphic mappings from \(V\) to \(V\); if 0 and \(x\) are fixed points of \(f\), then the existence of a complex geodesic whose range contains 0 and \(x\) and is contained in the set fix\(f=\{Z\in V:f(Z)=Z\}\) is ensured in \textit{J.-P. Vigue}'s papers [Adv. Math. 52, 241-247 (1984; Zbl 0555.32015) and Trans. Am. Math. Soc. 289, 345-353 (1985; Zbl 0589.32043)]. The main result shows that the uniqueness of such a geodesic is equivalent to the condition of the intersection between fix\(f\) and the set which describes the behaviour of the boundary of \(V\) in the direction of \(x\).
scientific article; zbMATH DE number 60571
scientific article