Univalent Wandering Domains in the Eremenko-Lyubich Class

Theorem 1.1. There exists an entire transcendental function f ∈B and a wandering Fatou component U of f such that f|fn(U) is univalent for all n ≥0. Our example uses the Folding Theorem in [Bis15] (see Section 2) and is in fact a careful modification of Bishop’s original construction. Very roughly speaking Bishop’s function behaves like (z −zn)mn for some mn →∞, on some subsequence of wandering components. We replace these maps by (z −zn)mn +δn ·(z −zn) on subsets of the same components, which ar

Theorem 1.1. Acknowledgements. We are indebted to David Mart´ı-Pete and Mitsuhiro Shishikura for their careful reading of and numerous comments on a previous version of this paper. We are grateful to Chris Bishop for his comments on a preliminary version of the paper. We would also like to thank Mikhail Lyubich and Lasse Rempe-Gillen for helpful discussions. We finally thank the Universitat de Barcelona and the Institut de Matem`atiques de la UB for their hospitality during the visits that led to

Theorem 2.1. Let T be an unbounded connected graph and let τ be a conformal map defined on each complementary domain C \ T as above. Assume that: (i) No two D-components of C \ T share a common edge. (ii) T is bipartite with uniformly bounded geometry. (iii) The map τ on a D-component with 2n edges maps the vertices to the 2nth roots of unity. (iv) On R-components the τ-sizes of all edges are uniformly bounded from below. Then there is an r0 > 0, a transcendental entire f, and a K-quasiconformal

Theorem 2.2 ([Pom75a, Section 1.3]). Let F be a univalent function on the disk D(a, r) for some a ∈C and r > 0. Then (a) F(D(a, r)) ⊃D F(a), 1 4|F ′(a)|r
. (b) For all z ∈D(a, r). r2|z −a||F ′(a)| (r + |z −a|)2 ⩽|F(z) −F(a)| ⩽r2|z −a||F ′(a)| (r −|z −a|)2 . (c) For all z ∈D(a, r), 1 − z−a

Theorem 2.3 below is termed the Measurable Riemann Mapping Theorem. For a proof, history, and references we refer to Chapter 4 of [Hub06]. It is used to produce the quasi- conformal mapping φ of Theorem 2.1. Theorem 2.4 below will be used in normal family arguments to deduce the dilatation of a limit of a sequence of quasiconformal mappings. An exposition of Theorem 2.4 is given in Section IV.5.6 of [LV73].

Theorem 2.3. If µ ∈L∞(C) with ||µ||∞< 1, there exists a quasiconformal mapping φ : C →C so that φz/φz = µ a.e.. Moreover, given any other quasiconformal Φ : C →C with φz/φz = Φz/Φz a.e., there exists a conformal ψ : C →C so that Φ = ψ ◦φ.

Theorem 2.4. ([Ber57]) Let φn : C →C be a sequence of K-quasiconformal mappings converging to a quasiconformal mapping φ : C →C with complex dilatation µ uniformly on compact subsets of C. If the complex dilatations µn(z) of φn tend to a limit µ∞(z) almost everywhere, then µ∞(z) = µ(z) almost everywhere. As explained above, the key idea behind Theorem 1.1 is to obtain the desired entire function f as the composition of a quasiregular map σ ◦η as given by Theorem 2.1, and a quasiconformal map φ g

Lemma 3.1. Let ψ(z) := zm + δzη(z) for z ∈D with r := 1 −(4δ)/m. There exist m0 ∈N, and δ0 > 0 such that if m > m0 and δ < δ0, then r > (δ/m)1/(m−1) and ||ψz ψz ||L∞(D) < 4/5.

Lemma 3.2. There exists a constant k0 < 1 independent of w ∈D(0, 3/4) such that ||(ρw)z (ρw)z ||L∞(D) < k0. For the proof of Lemma 3.2, see Section 3 of [FGJ15]. In the following sections, we will consider the following quasiregular self-map of the unit disc: (3.3) ρw ◦ψδ,m : D →D.

Proposition 3.3. Let s < 1, and let µw,δ,m := (ρw ◦ψδ,m)z/(ρw ◦ψδ,m)z. There exists m0 ∈N (depending on s) and δ0, k0 (independent of s), such that if m ≥m0, δ < δ0, and w ∈D(0, 3/4), one has ||µw,δ,m||L∞< k0 and supp(µw,δ,m) ⊂{z ∈D : |z| > s}.

Theorem 4.1. There exist m0 ∈N, δ0 > 0, and k0 < 1 such that if m(n) > m0, 0 ≤δ(n) < δ0, and w(n) ∈D(0, 3/4) for all n ∈N, then, for any λ > 1, g as in (4.1) may be extended to a quasiregular map g : C →C such that ||gz/gz||L∞(C) < k0. The function g : C →C satisfies g(−z) = g(z), g(z) = g(z) for all z ∈C, and the singular set of g consists only of the critical values ±1 and

Proposition 4.4. There exist λ0 ∈R, m0 ∈NN such that if δ, w are permissible, λ > λ0, and m ≥m0, then there exist constants a1, a0, a−1 ∈C such that (4.3) φ(z) = a1z + a0 + a−1 z + O  1 |z|2  as z →∞, where φ is any quasiconformal mapping as in Theorem 2.3 such that g ◦φ−1 is holomorphic.

Proposition 4.6. For any C > 0, ε > 0, R ≥1, there exist λ0 ∈R, m0 ∈NN, such that if λ > λ0, m ≥m0 and the parameters δ, w are permissible, then there exists a quasiconformal mapping φ : C →C satisfying (4.5) such that g ◦φ−1 is holomorphic and: (4.6) |φ(z) −z| < C |z| for |z| > R, and (4.7) |φ(z) −z| < ε for all z ∈C.

Proposition 4.11. Let n ≥1, and suppose that g′(x) ≥2 for x ≥1/32 and that λ, δ, m, w are permissible. Assume furthermore that supp(gz) ∩S+ ⊂{z ∈S+ : dist(z, ∂S+) < 1/16}. Then  (1/8)2(1/4−2ε0) (3/8)2(1/4) n ·Qn k=1(g−1)′(gk(1/2)+2ε0)≤|(f−n)′(gn(1/2))|≤  (5/8)2(1/4+2ε0) (3/8)2(1/4)λ n · 1

Corollary 4.14. There exists n′ ∈N such that if δ, m, and w are permissible, then f −n(Dpn) ⊂D(1/2, 1/8) ⊂D(0, 3/4) for all n > n′.

Proposition 5.1. There exist a subsequence (nk)∞ k=1 of natural numbers, a sequence (Cnk)∞ k=1 of positive real numbers, and a permissible parameter m ∈NN such that: for any choice of permissible w, δ one has:

Proposition 5.3. There exist permissible w, δ such that f := g ◦φ−1 has a wandering component containing φ(D− p1).

Proposition 6.1. Let n ≥0, and consider the function f := g ◦φ−1 as given in Proposition 5.3. The map f is univalent on the Fatou component containing the domain f n(φ(D− p1)).