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Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk.

Pietro Poggi-Corradini — 2003

Revista Matemática Iberoamericana

A lot is known about the forward iterates of an analytic function which is bounded by 1 in modulus on the unit disk D. The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880's to the 1980's, have provided conjugations which yield very precise descriptions of the dynamics. Backward-iteration sequences are of a different nature because a point could have infinitely many preimages as well as none. However, if we insist in choosing preimages that are at a...

p Harmonic Measure in Simply Connected Domains

John L. LewisKaj NyströmPietro Poggi-Corradini — 2011

Annales de l’institut Fourier

Let Ω be a bounded simply connected domain in the complex plane, . Let N be a neighborhood of Ω , let p be fixed, 1 < p < , and let u ^ be a positive weak solution to the p Laplace equation in Ω N . Assume that u ^ has zero boundary values on Ω in the Sobolev sense and extend u ^ to N Ω by putting u ^ 0 on N Ω . Then there exists a positive finite Borel measure μ ^ on with support contained in Ω and such that | u ^ | p - 2 u ^ , φ d A = - φ d μ ^ whenever φ C 0 ( N ) . If p = 2 and if u ^ is the Green function for Ω with pole at x Ω N ¯ then the measure μ ^ coincides...

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