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By A.M. Fink

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13 this is the case if and only if p ∈ F (g). ITERATION OF INNER FUNCTIONS 15 By the reflection principle {z : z ∈ V } is also a fundamental set for g on G and Ψ(z) := Φ(z) is holomorphic such that, for each z ∈ G, Ψ(g(z)) = Φ(g(z)) = Φ(g(z)) = T (Φ(z)) = S(Ψ(z)), where S(w) := T (w), for each w ∈ Ω. Thus we have that g|G ∼ T and g|G ∼ S. Assume that there exists σ ∈ {−1, 1} such that (Ω, T ) = (−iH, id−iH +iσ). Then S = id−iH − iσ and hence g|G ∼ id−iH + i and g|G ∼ id−iH − i. 4. Thus we conclude that T = id−iH + i and T = id−iH − i.

6 we conclude that there exists ζ ∈ ∂D \ Q such that ∞ ∈ C(φ; ζ). Thus φ(D \ Q) is unbounded. This is a contradiction to φ(D \ Q) ⊂ G. ✷ This leads to the following result. 12. Let f be a transcendental entire function. Suppose that there exists an unbounded component of F(f ). Let G ⊂ C be a bounded Jordan domain such that G ∩ J (f ) = ∅. Then ∂G ∩ F(f ) has infinitely many components. Proof. 11 we can assume that there is no completely invariant component of any iterate of f . This implies that there are infinitely many unbounded components of the Fatou set of f .

3. 1 does not carry over to those Baker domains where f |D ∼ idH ± 1. 3 holds if and only if V is a B¨ottcher domain of F . In this case we have to distinguish the cases where F |V is a proper self-map of V or not. Both cases are possible. 5 (Bergweiler). The function F (z) := 21 z 2 exp(2 − z) has a B¨ottcher domain V around 0 such that F |V is a proper self-map of V and ∂V is a Jordan curve. Hence D := exp−1 (V ) is a Baker domain of the function f (z) := 2 − log 2 + 2z − exp(z) such that f |D ∼ λidH , for some λ > 1, f |D is an automorphism of D, and ∂D is a Jordan curve.

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