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

Abstract:

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