Pointwise convergence on the boundary in the Denjoy-Wolff Theorem

Abstract

If $\phi$ is an analytic selfmap of the disk (not an elliptic automorphism) the Denjoy-Wolff Theorem predicts the existence of a point $p$ with $|p|\leq 1$ such that the iterates $\phi_{n}$ converge to $p$ uniformly on compact subsets of the disk. Since these iterates are bounded analytic functions, there is a subset of the unit circle of full linear measure where they all well-defined. We address the question of whether convergence to $p$ still holds almost everywhere on the unit circle. The answer depends on the location of $p$ and the dynamical properties of $\phi $. We show that when $|p|<1$(elliptic case), pointwise a.e. convergence holds if and only if $\phi$ is not an inner function. When $|p|=1$ things are more delicate. We show that when $\phi$ is hyperbolic or type I parabolic, then pointwise a.e. convergence holds always. The last case, type II parabolic remains open at this moment, but we conjecture the answer to be as in the elliptic case.