### Canonical conjugations at fixed points other than the Denjoy-Wolff point.

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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...

Let $\Omega $ be a bounded simply connected domain in the complex plane, $\u2102$. Let $N$ be a neighborhood of $\partial \Omega $, let $p$ be fixed, $1\<p\<\infty ,$ and let $\widehat{u}$ be a positive weak solution to the $p$ Laplace equation in $\Omega \cap N.$ Assume that $\widehat{u}$ has zero boundary values on $\partial \Omega $ in the Sobolev sense and extend $\widehat{u}$ to $N\setminus \Omega $ by putting $\widehat{u}\equiv 0$ on $N\setminus \Omega .$ Then there exists a positive finite Borel measure $\widehat{\mu}$ on $\u2102$ with support contained in $\partial \Omega $ and such that $$\begin{array}{c}\hfill \int |\nabla \widehat{u}{|}^{p-2}\phantom{\rule{0.166667em}{0ex}}\langle \nabla \widehat{u},\nabla \phi \rangle \phantom{\rule{0.166667em}{0ex}}dA=-\int \phi \phantom{\rule{0.166667em}{0ex}}d\widehat{\mu}\end{array}$$ whenever $\phi \in {C}_{0}^{\infty}\left(N\right).$ If $p=2$ and if $\widehat{u}$ is the Green function for $\Omega $ with pole at $x\in \Omega \setminus \overline{N}$ then the measure $\widehat{\mu}$ coincides...

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