Pointwise convergence on the boundary in the Denjoy-Wolff Theorem
  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. 
Pietro Poggi-Corradini