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 $\partial U \cap I(f)^c$ has harmonic measure zero relative to U. More precisely, if $\phi$ is a conformal map of the open unit disc $\mathbb{D}$ onto U, then for all $\zeta \in \partial \mathbb{D}$ apart from a set of capacity zero the angular limit $\phi(\zeta)$ exists and lies in $\partial U \cap I(f)$ . The first statement of Theorem 1.1 follows from the second because the set of angular limits of the conformal map $\phi$ forms the set of accessible boundary points of U, which has full harmonic measure in U, and any set of boundary points of $\partial \mathbb{D}$ of linear measure zero (in particular, of capacity zero) gives rise to a set of accessible boundary points of U of harmonic measure zero; see [9, page 206]. The paper [2] gives several examples of univalent Baker domains: some with connected boundaries and some with disconnected boundaries; some for which (1.1) holds and some for which it does not hold. Baker and Domínguez [1, 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 $\partial U \cap I(f)^c$ has harmonic measure zero relative to U. It remains an intriguing open question whether $\partial U \cap I(f) \neq \emptyset$ 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 $\partial U$ , with respect to harmonic measure in U, are escaping. We prove Theorem 1.1 using a new result on the boundary behaviour of conformal maps, which we state and prove in Section 3. Using Theorem 1.1 together with [15, Theorem 5.1], we can give a new sufficient condition for I(f) to be connected, and so satisfy Eremenko's conjecture in a particularly strong way.

Theorem 1.2. Let f be a transcendental entire function and let E be a set such that $E \subset I(f)$ and $J(f) \subset \overline{E}$ . 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) (see [12]) and I'(f) (see [5]). Each of these sets contains at least three points and is backwards invariant, so its closure contains J(f) by Montel's theorem. Thus we obtain the following corollary of Theorem 1.2.

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 unbounded [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, Theorem 5.1] in a different way.

Theorem 1.3. Let f be a transcendental entire function. Either I(f) is connected 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.

Theorem 2.1. Suppose that $f : \mathbb{D} \to \mathbb{C}$ is a conformal map. Then for all $\zeta \in \partial \mathbb{D}$ apart from a set of capacity 0 the angular limit $f(\zeta)$ exists and is finite. Theorem 2.1 is a classical result of Beurling [10, Theorem 9.19]. Throughout the paper we use the notation $f(\zeta)$ , where $\zeta \in \partial \mathbb{D}$ , for the angular limit at $\zeta$ 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 $f(\mathbb{D})$ that can only be reached along relatively long paths in $f(\mathbb{D})$ form a small subset of $\partial f(\mathbb{D})$ in some sense.

Theorem 2.2. Suppose that $f: \mathbb{D} \to \mathbb{C}$ is a conformal map, $V \subset f(\mathbb{D})$ is open, $E \subset \partial \mathbb{D}$ is a Borel set, and $\alpha \in (0,1]$ . If - $\operatorname{dist}(f(0), V) \ge \alpha |f'(0)|,$ - $\Lambda(f(C) \cap V) \geq \beta > 0$ , for all curves C in $\mathbb{D}$ that connect 0 to E, then $$\Lambda(E) \le 2\pi \operatorname{cap} E < \frac{15}{\sqrt{\alpha}} \exp\left(-\frac{\pi\beta^2}{\operatorname{area} V}\right).$$ 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 $E_n$ , $n \ge 1$ , be Borel subsets of $\mathbb{C}$ . - (a) If $E_1 \subset E_2$ , then $\operatorname{cap} E_1 \leq \operatorname{cap} E_2$ . - (b) If $\phi(z) = az + b$ , then $\operatorname{cap} \phi(E) = |a| \operatorname{cap} E$ . - (c) If $\phi$ is a Lipschitz map with constant M > 0, then $\operatorname{cap} \phi(E) \leq M \operatorname{cap} E$ . (d) If $E = \bigcup_{n=1}^{\infty} E_n$ and diam $E \leq d$ , then $$\frac{1}{\log(d/\operatorname{cap} E)} \le \sum_{n=1}^{\infty} \frac{1}{\log(d/\operatorname{cap} E_n)}.$$ - (e) The union of countably many sets of capacity zero has capacity zero. - (f) If $\phi$ is a Möbius transformation and E has capacity zero, then $\phi(E)$ has capacity zero. Finally, we need Bagemihl's ambiguous point theorem; see [10, Corollary 2.20]. Let f be a complex-valued function defined in $\mathbb{D}$ . For $\zeta \in \partial \mathbb{D}$ and a path $\gamma \subset \mathbb{D}$ , we define the cluster set of f at $\zeta$ along $\gamma$ as follows: $$C_{\gamma}(f,\zeta) = \{w : \lim_{n \to \infty} f(z_n) = w, \text{ for some sequence } z_n \in \gamma \text{ such that } \lim_{n \to \infty} z_n = \zeta\}.$$ A point $\zeta \in \partial \mathbb{D}$ is said to be an ambiguous point of f if there exist two paths $\gamma$ and $\gamma'$ in $\mathbb{D}$ each ending at $\zeta$ such that $C_{\gamma}(f,\zeta) \cap C_{\gamma'}(f,\zeta) = \emptyset$ . Bagemihl's theorem is as follows — note that there are almost no hypotheses here about the function f; it need not be continuous even.

Theorem 2.4. Let f be a complex-valued function with domain $\mathbb{D}$ . Then f has at most countably many ambiguous points. In fact we shall use the obvious adaptation of Theorem 2.4 from $\mathbb{D}$ to the upper half-plane $\mathbb{H} = \{z : \Im z > 0\}$ .

Theorem 3.1. Let $\phi : \mathbb{H} \to \mathbb{C}$ be a conformal map, let $w_0 \in \mathbb{C} \setminus \phi(\mathbb{H})$ , and let $\lambda > 1$ and $\varepsilon > 0$ . Also, for $n \geq 0$ , put $$I_n = [\lambda^{n-1/2}, \lambda^{n+1/2}]$$ and $E_n = \{t \in I_n : |\phi(t) - w_0| \ge \varepsilon\},$ $$J_n = [n - \frac{1}{2}, n + \frac{1}{2}]$$ and $F_n = \{t \in J_n : |\phi(t) - w_0| \ge \varepsilon\}.$ (a) If $\phi(\lambda^n i) \to w_0$ as $n \to \infty$ , then (3.1) $$\sum_{n=0}^{\infty} \frac{1}{\log(\lambda^n/\operatorname{cap} E_n)} < \infty.$$ (b) If $\phi(n+i) \to w_0$ as $n \to \infty$ , then (3.2) $$\sum_{n=0}^{\infty} \frac{1}{\log(1/\operatorname{cap} F_n)} < \infty.$$