Boundary behavior of the iterates of a self-map of the unit disk (Q450975)

The authors consider the dynamics of an arbitrary analytic self-map \(\varphi\) in the unit disk \(\mathbb D\). The central result in the area is the celebrated Denjoy-Wolff theorem. The iterates \((\varphi_n)\) converge to a certain point \(\tau\in \partial \mathbb D\) uniformly on compacta of \(\mathbb D\) if \(\varphi\) is different from an elliptic automorphism. They treat the parabolic case from a geometrical-dynamical point of view giving the following theorem. NEWLINENEWLINELet \(\varphi\) be a parabolic self-map of \(\mathbb D\) of zero hyperbolic step with Denjoy-Wolff point \(\tau\). If the associated Koenigs function \(\sigma\) has an angular limit almost everywhere on \(\partial \mathbb D\), then \((\varphi_n(\xi))\) converges to \(\tau\) for almost all \(\xi\in\partial\mathbb D\). NEWLINENEWLINEThey also provide some quantitative information about this convergence. Furthermore, it is showed that there is a proper boundary Denjoy-Wolff theorem for those parabolic self-maps of \(\mathbb D\) of zero hyperbolic step whose Koenigs function has an angular limit almost everywhere on \(\partial \mathbb D\).