Semigroups of Holomorphic Maps of a Riemann Surface into itself which are Homomorphs of the Set of Positive Reals Considered Additively

Abstract

Maps F: S × P ↦ S, where S is a Riemann surface and P is the set of positive reals, are studied under the assumption that f _t : p ↦ F(p,t),p ∈ S, is holomorphic and f _s+t = f _s ○ f _t , s, t ∈ P. Apart from cases where the fundamental group of S is abelian, the f _t are conformal automorphisms of S and F is trivial if it is continuous. When S is the open unit disk and F is continuous, the latter admits a simple geometric characterization with the aid of the notion of a region convex in a given direction thanks to work of Grunsky (J.M.A.A. 34 (1971), 685–701). The case, S a plane region and F continuous, was studied independently by Berkson and Porta (Mich. Math. J. 25 (1978), 101–115) with the view to investigating semigroups of composition operators on Hardy classes. Their work does not include the trivialization of F when the fundamental group of S is non-abelian nor the connection with the theory of regions convex in a given direction.