A. Eremenko, ON THE CHARACTERIZATION OF A RIEMANN SURFACE BY ITS SEMIGROUP OF ENDOMORPHISMS, Transactions of the American Mathematical Society, 338(1), 1993, pp. 123-131
Suppose D1 and D2 be Riemann surfaces which have bounded nonconstant h
olomorphic functions. Denote by E(D(i)), i = 1 , 2, the semigroups of
all holomorphic endomorphisms. If phi: E(D1) --> E(D2) is an isomorphi
sm of semigroups then there exists a conformal or anticonformal isomor
phism psi: D1 --> D2 such that phi is the conjugation by psi. Also the
semigroup of injective endomorphisms as well as some parabolic surfac
es are considered.