Restrictive semigroups of holomorphic endomorphisms on Riemann surfaces

Abstract

Let R be a Riemann surface and X be any nonempty subset of R. E(R,X) denotes the semigroup, under composition, of all holomorpnic selfmaps of R which carry X into X and is referred to as a restrictive semigroup of holomorphic endomorphisms. Let R , R be Riemann surfaces which have bounded nonconstant holomorphic functions and X, Y be any subsets of R, R , respectively. If $phi$ : E($R_1$ , X) - E($R_2$ , Y) is an isomorphism of semigroups, then there exist a conformal (or anticonformal) isomorphism $Psi$ : X —> Y such that $Phi(f) = Psi o f o Psi^{-1}$ for every $f in E (R_1 ,X)$.