On the characterization of a Riemann surface by its semigroup of endomorphisms

Abstract

Suppose D 1 {D_1} and D 2 {D_2} be Riemann surfaces which have bounded nonconstant holomorphic functions. Denote by E ( D i ) E({D_i}) , i = 1 , 2 i = 1,2 , the semigroups of all holomorphic endomorphisms. If ϕ : E ( D 1 ) → E ( D 2 ) \phi :E({D_1}) \to E({D_2}) is an isomorphism of semigroups then there exists a conformal or anticonformal isomorphism ψ : D 1 → D 2 \psi :{D_1} \to {D_2} such that ϕ \phi is the conjugation by ψ \psi . Also the semigroup of injective endomorphisms as well as some parabolic surfaces are considered.