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Singularities in Baker Domains

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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.

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Correspondence to Walter Bergweiler.

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Dedicated to the memory of Professor I. N. Baker

Supported by G.I.F., G-643-117.6/1999 and by INTAS-99-00089.

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Bergweiler, W. Singularities in Baker Domains. Comput. Methods Funct. Theory 1, 41–49 (2001).

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2000 MSC