# Uniformly convergent iterates for a function analytic on the complex unit disk

I have asked this question to several leading mathematicians in dynamical systems, and they all told me to ask someone else, until I was directed back to asking my advisor, with whom I first posed the question!...

I feel like this is probably in a textbook somewhere that I haven't read.

The Denjoy-Wolff theorem guarantees that, except for elliptic automorphisms, any analytic function on the open unit disk will have its iterates converge to a single point in the closed unit disk (the Denjoy Wolff point), uniformly so on compact subsets of the open unit disk.

Is there a complete characterization of functions from the open unit disk into itself whose iterates converge to their Denjoy-Wolff point uniformly on the -entire- open unit disk at once? An example would be f(z) = z/2 + 1/2, whose iterates converge to 1 uniformly on all of D.

I'm hoping that there is a characterization somewhere of all such analytic functions.

Thanks!

• When the D-W point is in the open unit disk $D$, the necessary and sufficient condition is that $\overline{f^{n}(D)}\subset D$ for some $n$... which is admittedly not very explicit. The boundary case is more interesting though, and I've no clue there. This user might be the right person to ask, but he's not on Math.SE. – user147263 Feb 3 '15 at 18:29
• Thank you! I knew that if f is continuous on the boundary, then f_{n}(Dbar) would be sufficient. Do you know the proof of that version being both necessary and sufficient? – Derek Thompson Feb 4 '15 at 19:47