Fixed points and boundary behaviour of the Koenigs function

Abstract

We analyze the relationship between the fixed points of different iterates of an analytic self-map of the unit disk. We show that, in general, a boundary fixed point of such a function is not a fixed point of its iterates. However, in the context of fractional iteration, all the iterates have the same fixed points. We also present results, in terms of the Koenigs function, of self-maps whose behaviour are not so extreme as above.