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 Carath\'eodory about local connection and boundary behaviour of univalent functions.
"Fractional iteration in the disk algebra: prime ends and composition operators." Rev. Mat. Iberoamericana 21 (3) 911 - 928, December, 2005.