Computational Methods and Function Theory 1 (2001), No. 1, 41--49
Copyright Heldermann Verlag 2001
Walter Bergweiler
bergweiler@math.uni-kiel.de, Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany.
Let $U$ be a Baker domain of a transcendental entire function $f$. Denote by $\lambda_U$ the hyperbolic metric in $U$ and, for $w\in U$ and $n\in\N$, define $\rho_n(w)=\lambda_U(f^{n+1}(w),f^{n}(w))$ and $\rho(w)=\lim_{n\to\infty} \rho_n(w)$. Here $f^n$ denotes the $n$--th iterate of $f$. It is shown that if the set of singularities of $f^{-1}$ that are contained in $U$ is bounded, then $$ \rho_{n}(w)= \frac1{2n}+a\frac{\log n}{n^2}+\landauO\left(\frac1{n^2}\right) $$ for some $a\in\R$ if $\rho(w)=0$ and $$ \rho_{n}(w)= \rho(w)+ \frac{b}{n^3}+\landauO\left(\frac1{n^4}\right) $$ for some $b\geq 0$ if $\rho(w)>0$, but $\inf_{w\in U}\rho(w)=0$. The result is applied to certain entire functions of finite order.
Keywords: Iteration, entire function, inner function, Julia set, Baker domain, singularity, residue fixed point index, Denjoy-Wolff point, hyperbolic metric.
MSC 2000: Primary 37F10; Secondary 30D05, 37F45.
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