A Birkhoff theorem for Riemann surfaces

A classical theorem of Birkhoff asserts that there exists an entire function f such that the sequence of function (f(z + n))(n greater than or equal to 0) is dense in the space of entire functions. In this paper we give sufficient conditions on a Riemann surface R and on a given sequence (phi(n))(n greater than or equal to 0) of holomorphic self-mappings of R such that there exists a holomorphic function f on R such that (f circle phi(n))(n greater than or equal to 0) is dense in the space of holomorphic functions on R. The necessity of these conditions is examined. In particular, we characterize the Riemann surfaces R and the sequences (phi(n))(n greater than or equal to 0) of automorphisms of R for which there exists a holomorphic function f on R with the property that the sequence (f circle phi(n))(n greater than or equal to 0) is dense in the space of the holomorphic functions on R.