FATOU COMPONENTS AND SINGULARITIES OF MEROMORPHIC

Theorem 1.1 ([Ber95, Theorem 3]). Let f be a transcendental entire function with an in- variant Baker domain U. If U ∩S(f) = ∅, then there exists a sequence pn ∈P(f) such that, as n →∞, (1) |pn| →∞, (2) |pn+1/pn| →1, (3) dist(pn, U) = o(|pn|). Here and in the sequel dist denotes the Euclidean distance. Additionally, as we mentioned above, Bergweiler provided an example (which inspired the example in Figure 1), where the postcritical set is located at a positive distance from the Baker domain, sh

Theorem 1.1 for meromorphic transcendental maps (and hence for Baker domains which are not necessarily simply connected) was proven by Mihaljevi´c-Brandt and Rempe-Gillen in [MBRG13, Theorem 1.5], with conclusion (2) replaced by sup n≥1

Lemma 3] and, additionally, has several applications concerning the relation between points in P(f) and wandering domains. The following consequence of Lemma 2.1, although stated for a general Fatou component, is most meaningful when U is a wandering or Baker domain. In particular, it contributes to an answer to Question 11 from [Ber93a]. Theorem B. Let f be a transcendental meromorphic map and U be a Fatou component of f. Denote by Un the Fatou component such that fn(U) ⊂Un. Then for every z ∈U

Lemma 2.1. Let f be a transcendental meromorphic map and U be a Fatou component of f. Denote by Un the Fatou component such that fn(U) ⊂Un. Then for every compact set K ⊂U and every ε > 0, M ≥1 there exists n0 > 0 such that for every n ≥n0, every z ∈K and every curve γ in C connecting fn(z) to a point w ∈∂Un with ℓ(γ) ≤M dist(fn(z), ∂Un) there exists a point p ∈D(γ, εℓ(γ)) ∩P(f).

Lemma 2.1 and the definition of N(k) we conclude that there is a point uj(k) ∈P(f) ∩D  γj(k), ℓ(γj(k)) k  . By the definition of γj(k), γj(k) ⊂D  0, rj(k) + rj(k) k  \ D 

Proposition 4.1. The map Nf(z) = z −tan(z), which is Newton’s method of the map f(z) = sin(z), has no wandering domains.

Proposition 4.2. The map Ng(z) = z + i + tan(z), which is the Newton’s method of the map g(z) = exp  − R z 0 du i+tan(u)  , has no wandering do- mains.

Proposition 4.4. If α, β ∈C, β ̸= 0 and infn≥0 dist(Nn h (u), J(Nh)) > 0, then Nh is topolog- ically hyperbolic.