Iteration and the solution of functional equations for functions analytic in the unit disk

Abstract

This paper considers the classical functional equations of Schroeder f o p = Xf, and Abel f o p = f + 1, and related problems of fractional iteration where 9p is an analytic mapping of the open unit disk into itself. The main theorem states that under very general conditions there is a linear fractional transformation 4) and a function a analytic in the disk such that ) a = oa p and that, with suitable normalization, 4) and a are unique. In particular, the hypotheses are satisfied if 4p is a probability generating function that does not have a double zero at 0. This intertwining relates solutions of functional equations for T to solutions of the corresponding equations for (D. For example, it follows that if 4p has no fixed points in the open disk, then the solution space of f o (p = Xf is infinite dimensional for every nonzero X. Although the discrete semigroup of iterates of T usually cannot be embedded in a continuous semigroup of analytic functions mapping the disk into itself, we find that for each z in the disk, all sufficiently large fractional iterates of p can be defined at z. This enables us to find a function meromorphic in the disk that deserves to be called the infinitesimal generator of the semigroup of iterates of p. If the iterates of (p can be embedded in a continuous semigroup, we show that the semigroup must come from the corresponding semigroup for 4), and thus be real analytic in t. The proof of the main theorem is not based on the well known limit technique introduced by Koenigs (1884) but rather on the construction of a Riemann surface on which an extension of 9p is a bijection. Much work is devoted to relating characteristics of (p to the particular linear fractional transformation constructed in the theorem.