Compact Composition Operators and Iteration

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Let $\Omega$ be a domain in the complex plane, $\phi$ an analytic map that maps $\Omega$ into itself, and $X$ an $F$-space of analytic functions in $\Omega$ that possesses certain mild regularity properties. (Some examples considered in the paper are Hardy spaces, Bergman spaces, and the space of all analytic functions in $\Omega$.) If composition with $\phi$ defines a compact operator on $X$ that has eigenvalues, then the iterates of $\phi$ converge to a constant $\lambda$ in the closure of $\Omega$. Furthermore, if $\lambda\in\partial\Omega$ and $\liminf_{\zeta\to\lambda}|\phi(\zeta)-\lambda|/|\zeta-\lambda| > 0$, then functions in $H^\infty(\Omega)$ and their derivatives have nice behavior near $\lambda$ in the sense that the functionals of evaluation at $\lambda$ for functions and their derivatives have weak${}^*$-continuous extensions to $H^\infty(\Omega)$. The following example is discussed in the above context. Start with an analytic map $\phi$ from a disk into itself that has an attractive fixed point $\lambda$ at the center. Form $\Omega$ by removing from the disk both $\lambda$ and a carefully chosen sequence of disjoint disks that converge to $\lambda$. Finally, let $X=H^\infty(\Omega)$. (Less)