Computational Methods and Function Theory

, Volume 1, Issue 1, pp 41–49 | Cite as

Singularities in Baker Domains

Article

Abstract

Let U be a Baker domain of a transcendental entire function f. Denote by λU the hyperbolic metric in U and, for wU and n ∈ ℕ, define ρn(w) = λ U(f n+1}(w),f n(w)) and ρ(w) = lim n→∞ρn(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
$$\rho_n (w) = {1 \over 2n} + a {{\rm log\ n} \over n^{2}} + {\cal O} \bigg({1 \over n ^{2}}\bigg)$$
for some a ∈ ℝ if ρ(w) = 0 and
$$\rho_{n}(w) = \rho(w)+ {b \over n^{3}}+ {\cal O} \bigg({1 \over n^{4}}\bigg)$$
for some b ≥ 0 if ρ(w) > 0, but inf wU ρ(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 

2000 MSC

37F10 30D05 37F45 

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Copyright information

© Heldermann Verlag 2001

Authors and Affiliations

  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany

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