Holomorphic self-maps of the disk intertwining two linear fractional maps

Details Holomorphic self-maps of the disk intertwining two linear fractional maps Holomorphic self-maps of the disk intertwining two linear fractional maps

2008-06-13

arxiv.org/abs/0806.2276v2

Mathematics

Dynamical Systems

This is a revised version

Scientific paper

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.

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