Pointwise convergence on the boundary in the Denjoy-Wolff Theorem
| dc.creator | Poggi-Corradini, Pietro | |
| dc.date | 2004-07-08 | |
| dc.date.accessioned | 2026-07-07T05:10:05Z | |
| dc.date.available | 2026-07-07T05:10:05Z | |
| dc.description | If $ϕ$ 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 $ϕ_{n}$ 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 $ϕ$. 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 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. | |
| dc.description | 11 pages | |
| dc.identifier | https://arxiv.org/abs/math/0407133 | |
| dc.identifier | http://arxiv.org/abs/math/0407133 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/71819 | |
| dc.subject | Complex Variables | |
| dc.subject | 30D05;30C85 | |
| dc.title | Pointwise convergence on the boundary in the Denjoy-Wolff Theorem | |
| dc.type | text |