Fractional iteration in the disk algebra: prime ends and composition operators

In this paper we characterize the semigroups of analytic functions in the unit disk which lead to semigroups of operators in the disk algebra. These characterizations involve analytic as well as geometric aspects of the iterates and they are strongly related to the classical theorem of Carathéodory about local connection and boundary behaviour of univalent functions. 1. Introduction and Statement of the Results Let H(D) be the Fréchet space of all analytic functions in the disk endowed with the topology of uniform convergence on compact subsets of D. A (one-parameter) semigroup of H(D) or, a semigroup of analytic functions, is any continuous homomorphism Φ : t 7→ Φ(t )= ϕt from the additive semigroup of nonnegative real numbers R+ into the composition semigroup of all functions ϕ ∈ H(D) with ϕ(D) ⊂ D.T hat is,Φ satisfies the