Common fixed points of commuting holomorphic maps
In 1964, \textit{A. L. Shields} [Proc. Am. Math. Soc. 15, 703-706 (1964; Zbl 0129.291)] proved that every family of commuting holomorphic maps of the unit disk \(\Delta\) of \({\mathbb{C}}\) into itself continuous up to the boundary admits a fixed point in \({\bar \Delta}\). Later, \textit{D. J. Eustice} [Mich. Math. J. 19, 347-352 (1972; Zbl 0254.32008)] extended Shield's theorem to holomorphic maps of \(\Delta ^ 2=\Delta \times \Delta \subset {\mathbb{C}}^ 2\), and \textit{T. J. Suffridge} [Mich. Math. J. 21(1974), 309-314 (1975; Zbl 0333.47026)] to holomorphic maps of the unit ball of \({\mathbb{C}}^ n\). In this paper, Shields' theorem is generalized to bounded strongly convex domains of \({\mathbb{C}}^ n\) with \(C^ 3\) boundary. The main tools used in the proof are the iteration theory in strongly convex domains developed by the author [Math. Z. 198, 225-238 (1988; Zbl 0628.32035)] and the theory of complex geodesics for the Kobayashi distance in convex domains, essentially due to \textit{L. Lempert} [Bull. Soc. Math. France 109, 427-474 (1981; Zbl 0492.32025)]. In particular, the first section of the paper is devoted to prove several facts regarding the uniqueness of complex geodesics passing through given points of the closure of the domain.
scientific article; zbMATH DE number 4054039
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