  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 careful analysis of the Denjoy-Wolff
points. In particular, we characterize the maps that commute with a given
linear fractional map (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 as a byproduct 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.
