Prime and composite Laurent polynomials

In the paper [J. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc. 23 (1922) 51-66] Ritt constructed the theory of functional decompositions of polynomials with complex coefficients. In particular, he described explicitly polynomial solutions of the functional equation f (p (z)) = g (q (z)). In this paper we study the equation above in the case where f, g, p, q are holomorphic functions on compact Riemann surfaces. We also construct a self-contained theory of functional decompositions of rational functions with at most two poles generalizing the Ritt theory. In particular, we give new proofs of the theorems of Ritt and of the theorem of Bilu and Tichy.