Holomorphic Self-maps of the Disk Intertwining Two Linear Fractional Maps

We characterize (in almost all cases) the holomorphic self-maps of the unit disk that intertwine two given linear fractional self-maps of the disk. The proofs are based on iteration and a detailed analysis of the solutions of Schroeder’s and Abel’s equations. In particular, we characterize the maps that commute with a given linear fractionalmap (in the cases that are not already known) and, as an application, determine all “roots” of such maps in the sense of iteration (if any). This yields a short proof of a recent theorem on the embedding of a linear fractional transformation into a semigroup of holomorphic self-maps of the disk.