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In this paper we prove that a commuting family of continuous self-maps of a bounded convex domain in C(n) which are holomorphic in the interior has a common fixed point. The proof makes use of three basic ingredients: iteration theory of holomorphic maps, a precise description of the structure of the boundary of a convex domain, and a similar result for commuting families of self-maps of a hyperbolic domain of a compact Riemann surface

Common Fixed Points in Hyperbolic Riemann Surfaces and Convex Domains