Boundaries of univalent Baker domains

Theorem 1.1. Let f be a transcendental entire function and let U be a univalent Baker domain of f. Then ∂U ∩I(f)c has harmonic measure zero relative to U. More precisely, if φ is a conformal map of the open unit disc D onto U, then for all ζ ∈∂D apart from a set of capacity zero the angular limit φ(ζ) exists and lies in ∂U ∩I(f). The first statement of Theorem 1.1 follows from the second because the set of angular limits of the conformal map φ forms the set of accessible boundary points of U, whi

Corollary 1.3] showed that the boundary of any non-univalent Baker domain is disconnected, and indeed has uncountably many components, so we have the following immediate corollary of Theorem 1.1.

Corollary 1.1. Let f be a transcendental entire function and let U be a Baker domain of f whose boundary is connected. Then ∂U ∩I(f)c has harmonic mea- sure zero relative to U. It remains an intriguing open question whether ∂U ∩I(f) ̸= ∅whenever U is a Baker domain. Note that in [15] we showed that if U is any wandering domain in I(f), then almost all points of ∂U, with respect to harmonic measure in U, are escaping. We prove Theorem 1.1 using a new result on the boundary behaviour of conformal

Theorem 1.2. Let f be a transcendental entire function and let E be a set such that E ⊂I(f) and J(f) ⊂E. Either I(f) is connected or it has infinitely many components that meet E; in particular, if E is connected, then I(f) is connected. Several subsets of I(f) have been studied, involving different rates of escape, including: the fast escaping set A(f) (see [6] and [14]), the slow escaping set L(f) and moderately slow escaping set M(f) (see [13]), the quite fast escaping set Q(f) (see [16]), Z(f)

Corollary 1.2. Let f be a transcendental entire function. If one of the sets A(f), L(f), M(f), Q(f), Z(f) or I′(f) is connected, then I(f) is connected. The fast escaping set A(f) has the property that all its components are un- bounded [14, Theorem 1.1]. Therefore, if we apply Theorem 1.2 in the case when the set E is A(f), then we obtain the following result, which seems to be the strongest general result so far on Eremenko’s conjecture. This result can also be deduced directly from [15, Theor

Theorem 1.3. Let f be a transcendental entire function. Either I(f) is con- nected or it has infinitely many unbounded components. Acknowledgement The authors are grateful to Walter Bergweiler and Dierk Schleicher for a discussion that led to the formulation of Theorem 1.3. 2. Background material We require several fundamental results from complex analysis, all of which can be found in [10], which we state here for the reader’s convenience. The first two results concern the boundary behaviour of a

Theorem 2.1. Suppose that f : D →C is a conformal map. Then for all ζ ∈∂D apart from a set of capacity 0 the angular limit f(ζ) exists and is finite.

Theorem 2.1 is a classical result of Beurling [10, Theorem 9.19]. Throughout the paper we use the notation f(ζ), where ζ ∈∂D, for the angular limit at ζ of the conformal map f, whenever this exists. The second result on conformal maps [10, Theorem 9.24] is a quantitative version of the fact that those boundary points of f(D) that can only be reached along relatively long paths in f(D) form a small subset of ∂f(D) in some sense.

Theorem 2.2. Suppose that f : D →C is a conformal map, V ⊂f(D) is open, E ⊂∂D is a Borel set, and α ∈(0, 1]. If • dist(f(0), V ) ≥α|f ′(0)|, • Λ(f(C) ∩V ) ≥β > 0, for all curves C in D that connect 0 to E, then Λ(E) ≤2π cap E < 15 √α exp  −πβ2 area V  . We also need various basic results on logarithmic capacity, which can be found in [10, pages 204, 208 and 209].

Theorem 2.3. Let E and En, n ≥1, be Borel subsets of C. (a) If E1 ⊂E2, then cap E1 ≤cap E2. (b) If φ(z) = az + b, then cap φ(E) = |a| cap E. (c) If φ is a Lipschitz map with constant M > 0, then cap φ(E) ≤M cap E.

Theorem 2.4. Let f be a complex-valued function with domain D. Then f has at most countably many ambiguous points. In fact we shall use the obvious adaptation of Theorem 2.4 from D to the upper half-plane H = {z : ℑz > 0}. 3. A result on conformal maps To prove Theorem 1.1 we require two results on the boundary behaviour of a conformal map, each of which states, roughly speaking, that if the map behaves in a certain way near a boundary point, then its boundary values behave in a similar way near

Theorem 2.2. For simplicity we state these results in the upper half-plane.

Theorem 3.1. Let φ : H →C be a conformal map, let w0 ∈C \ φ(H), and let λ > 1 and ε > 0. Also, for n ≥0, put In = [λn−1/2, λn+1/2] and En = {t ∈In : |φ(t) −w0| ≥ε}, Jn = [n −1 2, n + 1 2] and Fn = {t ∈Jn : |φ(t) −w0| ≥ε}. (a) If φ(λni) →w0 as n →∞, then (3.1) ∞ X n=0

Theorem 1.2.