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.