Pointwise Convergence on the Boundary in the Denjoy-wolff Theorem

If φ is an analytic selfmap of the disk (not an elliptic automorphism) the DenjoyWolff Theorem predicts the existence of a point p with |p| ≤ 1 such that the iterates φn 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 are 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 φ. We show that when |p| < 1(elliptic case), pointwise a.e. convergence holds if and only if φ is not an inner function. When |p| = 1 things are more delicate. We show that when φ is hyperbolic or non-zero-step parabolic, then pointwise a.e. convergence holds always. The last case, zero-step parabolic, remains open.