On iteration in planar domains (Q1363619)

Let \(G\) be a domain on the Riemann sphere and let \(f:G\rightarrow G\) be analytic. The purpose of this paper is to study the behavior of the iterates \(f^{n}\) of \(f\) in the case where \(G\) is hyperbolic (i.e., the complement of \(G\) has at least three points). The problem is well understood if \(f\) has a fixed point and thus only the case that \(f\) has no fixed points is considered here. The case that \(G\) is the unit disk \(D\) is also well understood. Let \(p:D\rightarrow G\) be a universal covering map. Then there exists an analytic function \(\widetilde f:D\rightarrow D\) such that \(p\circ \widetilde f=f\circ p\). The first result is that if \(z\in G\), then \(\lambda_{G}(f^{n}(z), f^{n+1}(z))\rightarrow 0\) if and only if \(\lambda_{D}(\widetilde f^{n}(0),\widetilde f^{n+1}(0))\rightarrow 0\). Here \(\lambda_{\Omega}(\cdot,\cdot)\) denotes the hyperbolic distance in a (hyperbolic) domain \(\Omega\). The hyperbolic distance in \(G\) may be replaced by the quasihyperbolic distance in \(G\) here, except in the case where \(f\) has an isolated boundary fixed point; that is, if there exists an isolated point \(a\) in the boundary of \(G\) such that \(f\) extends analytically to \(G\cup \{a\}\) by defining \(f(a)=a\). The main result of the paper says that if \(f:G\rightarrow G\) is as above, without isolated boundary fixed point, then there exists a domain \(H\subset {\mathbb{C}}\), an analytic function \(g:G\rightarrow H\) and a Möbius transformation \(\varphi:H\rightarrow H\) such that \(g\circ f=\varphi \circ g\). Moreover, \(H\) is hyperbolic if \(\lambda_{G}(f^{n}(z), f^{n+1}(z))\rightarrow 0\) for any \(z\in G\) and \(H={\mathbb{C}}\) otherwise. Further conclusions are obtained under additional hypotheses on \(f\) and \(G\).