3.1 Proof of Theorem 1.1(a): the contracting case

What we prove in this subsection is actually stronger than what Theorem 1.1(a) claims: we will show that, if U contains a non-empty open set V such that

$$ \begin{align*} d_{U_n}(f^n(z), f^n(w))\to 0\text{ as }n\to +\infty \quad \text{for every }z, w\in V, \end{align*} $$

then the same is true for every $z, w\in U$ . In order to do this, we will use results of Benini et al relating to non-autonomous dynamics in the unit disc (more specifically, [Reference Benini, Evdoridou, Fagella, Rippon and Stallard7, §2]). As preparation for that, we prove the following lemma, which tells us how to associate the dynamics of a wandering domain to a composition of inner functions. Recall that an inner function is a holomorphic function $g:\mathbb {D}\to \mathbb {D}$ such that the radial limit

$$ \begin{align*} g(e^{i\theta}) := \lim_{r\nearrow 1} g(re^{i\theta}) \end{align*} $$

exists and satisfies $|g(e^{i\theta })| = 1$ for Lebesgue almost every $\theta \in [0, 2\pi )$ .

Proof. The existence, analyticity, and uniqueness of the functions $g_n$ are guaranteed by Lemma 2.3. It remains to show that they are inner functions.

Since $\varphi _n(\mathbb {D}) = U_n$ and U is a wandering domain, it is clear that $f\circ \varphi _n(\mathbb {D})$ is not dense in $\widehat {\mathbb {C}}$ (indeed, it omits every other $U_m$ , $m\neq n + 1$ ); thus, we can (if necessary) compose $f\circ \varphi _n$ with a Möbius transformation $m_n$ to obtain a bounded map $m_n\circ f\circ \varphi _n:\mathbb {D}\to \mathbb {C}$ . By Fatou’s theorem [Reference Hayman18, Lemma 6.9], the radial limit $f\circ \varphi _n(e^{i\theta })$ exists and is finite for $\theta $ outside a set $E_n\subset [0, 2\pi )$ of Lebesgue measure zero. Furthermore, also by Fatou’s theorem, there exists a set $E_n'$ of measure zero such that $g_n(e^{i\theta })$ exists for $\theta \in [0, 2\pi )\setminus E_n'$ . Now, for every $n\geq 0$ , the set $F_n := E_n\cup E_n'$ has measure zero, and for $\theta \in [0,2\pi )\setminus F_n$ we have

$$ \begin{align*} \lim_{r\nearrow 1} \varphi_{n+1}\circ g_n(re^{i\theta}) = \lim_{r\nearrow 1} f\circ\varphi_n(re^{i\theta}), \end{align*} $$

where both limits exist and are finite. Since $\varphi _n$ , being a covering map, has no asymptotic values in $U_n$ and f is continuous, the right-hand side converges to a point in $\partial U_{n+1}$ . Thus, on the left-hand side, we must have $|g_n(re^{i\theta })|\to 1$ as $r\nearrow 1$ by continuity of $\varphi _{n+1}$ and the open mapping theorem. It follows that the $g_n$ are inner functions.

An immediate consequence of Lemma 3.2 is that, defining $G_n = g_n\circ g_{n-1}\circ \cdots \circ g_0$ , we have

(6) $$ \begin{align} \varphi_n\circ G_{n-1} = f^n\circ\varphi_0 \quad\text{ for every }n\geq 1. \end{align} $$

However, since the domains $U_n$ are multiply connected, the limiting behaviours of $G_n$ and $(f^n)|_U$ relative to the hyperbolic metric are not necessarily the same, except in one particular case.

Proof. Let $w\in \mathbb {D}$ ; since $G_n(w)\to 0$ as $n\to +\infty $ , we know that $d_{\mathbb {D}}(0, G_n(w))\to 0$ . By (6), we have

$$ \begin{align*} d_{U_n}(f^n(z_0), f^n\circ\varphi_0(w)) = d_{U_n}(\varphi_n(0), \varphi_n\circ G_{n-1}(w))\quad\text{for }w\in\mathbb{D}, \end{align*} $$

and applying the Schwarz–Pick lemma to the right-hand side yields

$$ \begin{align*} d_{U_n}(f^n(z_0), f^n\circ\varphi_0(w)) \leq d_{\mathbb{D}}(0, G_{n-1}(w)), \end{align*} $$

which goes to zero as claimed.

For a sequence $(G_n)_{n\in {\mathbb {N}}}$ as above, [Reference Benini, Evdoridou, Fagella, Rippon and Stallard7, Theorem 2.1] tells us that $d_{\mathbb {D}}(0, G_n(w))\to 0$ for $w\in \mathbb {D}$ if and only if

$$ \begin{align*} \sum_{n\geq 0} (1 - |g_n'(0)|) = +\infty, \end{align*} $$

or equivalently, if none of the $g_n'(0)$ equal zero, $G_n'(0) \to 0$ . Since $f^n$ lifts to $G_n$ , the local conformal invariance of hyperbolic distortion (recall (3)) implies that this is equivalent to saying that a sufficient condition for U to be a contracting wandering domain is

$$ \begin{align*} \lim_{n\to +\infty} \|Df^n(z_0)\|_U^{U_n} = 0. \end{align*} $$

Now our course of action is to choose a $z_0\in V$ , the contracting neighbourhood inside U, and show that the condition above holds. To this end, we prove the following hyperbolic version of Landau’s theorem. Unlike in Landau’s original theorem, the function here is not normalized (if it were, it would be a hyperbolic isometry!), and thus there is dependence on the derivative of f.

Lemma 3.4. Let $f:\Omega _1\to \Omega _2$ be a holomorphic map between hyperbolic domains, let $r> 0$ , and let $z_0\in \Omega _1$ . If $0 < \|Df(z_0)\|_{\Omega _1}^{\Omega _2} < 1$ , then $f(B_{\Omega _1}(z_0, r))$ contains a hyperbolic disc of radius

$$ \begin{align*} 2{L}r^*\|Df(z_0)\|_{\Omega_1}^{\Omega_2}, \end{align*} $$

where $r^* = \tanh (r/2)$ and $L\in (0.5, 0.544)$ is Landau’s constant.

Proof. First, we take uniformizing maps $\varphi :\mathbb {D}\to \Omega _1$ and $\psi :\mathbb {D}\to \Omega _2$ with $\varphi (0) = z_0$ and $\psi (0) = f(z_0)$ . Then, by Lemma 2.3, f admits a lift $F:\mathbb {D}\to \mathbb {D}$ (that is, F satisfies $\psi \circ F = f\circ \varphi $ ), which can be chosen to satisfy $F(0) = 0$ , and by (3) we also have $|F'(0)| = \|Df(z_0)\|_{\Omega _1}^{\Omega _2}$ . The ball $B_{\mathbb {D}}(0, r)$ is contained (by the Schwarz–Pick lemma) in $\varphi ^{-1}(B_{\Omega _1}(z_0, r))$ , and its image under F contains (by Landau’s theorem [Reference Landau24]; see [Reference Minda27] for a discussion of the bounds given here) a disc of Euclidean radius $Lr^*|F'(0)| = Lr^*\|Df(z_0)\|_{\Omega _1}^{\Omega _2}$ , where we have set $r^* = \tanh (r/2)$ so that $d_{\mathbb {D}}(0, r^*) = r$ . Since we do not know where the centre $w_0$ of this disc is, we cannot accurately calculate its hyperbolic size. Nevertheless, we do know that, for any w on its boundary,

$$ \begin{align*} d_{\mathbb{D}}(w_0, w) = \int_\gamma \rho_{\mathbb{D}}(s)\,|ds|\geq 2\int_\gamma\, |ds| \geq 2|w - w_0| = 2Lr^*\|Df(z_0)\|_{\Omega_1}^{\Omega_2}, \end{align*} $$

where $\gamma \subset \mathbb {D}$ is a hyperbolic geodesic connecting $w_0$ to w. Hence, $F(B_{\mathbb {D}}(0, r))$ contains a disc of hyperbolic radius $2Lr^*\|Df(z_0)\|_{\Omega _1}^{\Omega _2}$ .

In order to transfer this knowledge from $\mathbb {D}$ to $\Omega _2$ , we will use tools from [Reference Beardon, Carne, Minda and Ng3] (see also [Reference Keen and Lakic21, Ch. 10]). More specifically, for a subdomain D of $\Omega _2$ , let $R(D, \Omega _2)$ denote the hyperbolic radius of the largest hyperbolic disc contained in D; this is the hyperbolic Bloch constant, and Beardon et al proved in [Reference Beardon, Carne, Minda and Ng3, Lemma 4.2] that it is invariant under unbranched covering maps. More specifically, let $B^*$ denote the component of $\psi ^{-1}(f(B_{\Omega _1}(z_0, r)))$ containing the origin. Then, since $f\circ \varphi = \psi \circ F$ , $F(0) = 0$ , and $B^*$ is connected, we have ${B^*\supset F(B_{\mathbb {D}}(0, r))}$ , and thus

$$ \begin{align*} R(f(B_{\Omega_1}(z_0, r)), \Omega_2) = R(B^*, \mathbb{D}) \geq R(F(B_{\mathbb{D}}(0, r)), \mathbb{D}). \end{align*} $$

Since we know that $F(B_{\mathbb {D}}(0, r))$ contains a disc of hyperbolic radius $2Lr^*\|Df(z_0)\|_{\Omega _1}^{\Omega _2}$ , we have

$$ \begin{align*} R(F(B_{\mathbb{D}}(0, r)), \mathbb{D}) \geq 2Lr^*\|Df(z_0)\|_{\Omega_1}^{\Omega_2} \end{align*} $$

and we are done.

With these results in hand, we can finally prove what we intended. Notice that, by the triangle inequality, the hypotheses of Theorem 1.1(a) imply those of Theorem 3.5.

Theorem 3.5. Let U be a wandering domain of a meromorphic function f. If U contains a non-empty open set V such that

$$ \begin{align*} d_{U_n}(f^n(z), f^n(w))\to 0 \end{align*} $$

for every z and w in V, then U is contracting.

Proof. Take any $z_0\in V$ , and take uniformizing maps ${\varphi _n:\mathbb {D}\to U_n}$ such that $\varphi _n(0) = f^n(z_0)$ . Let $g_n$ be the lifts of f given by Lemma 3.2. We want to show that

(7) $$ \begin{align} \sum_{n\geq 0} (1 - |g_n'(0)|) = +\infty, \end{align} $$

whence our theorem will follow by Lemma 3.3 and [Reference Benini, Evdoridou, Fagella, Rippon and Stallard7, Theorem 2.1]. We can assume that none of the $g_n'(0)$ are zero; indeed, if there are infinitely many such $g_n$ , then (7) is trivially true, and if there are only finitely many such $g_n$ we pass to $U_N$ , $f^N(z_0)$ , and $f^N(V)$ for some sufficiently large N. Thus (recall that $G_n = g_n\circ \cdots \circ g_0$ ), as anticipated on page 8, (7) is equivalent to $|G_n'(0)|\to 0$ or, by (3), to $\|Df^n(z_0)\|_U^{U_n}\to 0$ .

Assume now that this is not the case; notice that, again by the Schwarz–Pick lemma, the sequence $(\|Df^n(z_0)\|_U^{U_n})_{n\in {\mathbb {N}}}$ is decreasing, and so if it gets arbitrarily close to zero on a subsequence then it is in fact tending to zero. In other words, there must exist some constant $c> 0$ such that $\|Df^n(z_0)\|_U^{U_n}> c$ for all $n\geq 0$ . Choose some $r> 0$ such that $K := \overline {B_U(z_0, r)}\subset V$ ; then, by Lemma 3.1, we have $\mathrm {diam}_{U_n}(f^n(K))\to 0$ , while by Lemma 3.4 $f^n(K)$ always contains a hyperbolic ball of radius $2Lc\cdot \tanh (r/2)$ . This is clearly a contradiction; we are done.

3.2 Proof of Theorem 1.1(b): the semi-contracting case

In this subsection, we want to show that if there exist some point $z_0\in U$ and an open neighbourhood V of $z_0$ such that $u_n(z)\searrow u(z)> 0$ without ever reaching it for $z\in V$ outside of a discrete set ( $u_n$ and u as defined at the beginning of §3, with $z_0$ as base point), then the same holds for every $z\in U$ except those for which $f^n(z) = f^n(z_0)$ for some $n\in {\mathbb {N}}$ .

We will divide the proof into two cases. We prove first that no point $z\in U$ can be contracting relative to $z_0$ , and then that no point can be eventually isometric relative to $z_0$ .

For the first case, we will revisit the function u; in particular, we notice that it is ‘f-invariant’ in the sense that, if $u^*:U_1\to [0, +\infty )$ is defined taking as a base point $f(z_0)$ , then $u^*(f(z)) = u(z)$ . By an abuse of notation, we will keep referring to the functions defined on $U_n$ with base points $f^n(z_0)$ as u and hope it will not lead to confusion. Now what we want to show is that $u(z) = 0$ if and only if $f^n(z) = f^n(z_0)$ for some n.

The ‘if’ is trivial. For the ‘only if’ part, let $z{\in U}$ be such that $f^n(z)\neq f^n(z_0)$ for all n, and assume that $u(z) = 0$ . Then, since U is semi-contracting in V, we can take a closed curve $\gamma \subset V$ that surrounds $z_0$ and avoids the zeros of u, which are discrete in V by hypothesis; by Lemma 3.1, $u|_\gamma $ achieves a minimum $c> 0$ . Additionally, V is in the Fatou set, meaning that $(f^n)|_V$ is holomorphic for every $n\in {\mathbb {N}}$ , and so by the argument principle $f^n(\gamma )$ surrounds $B_{U_n}(f^n(z_0), c)$ for all $n\in {\mathbb {N}}$ . Therefore, since we have that $d_{U_n}(f^n(z), f^n(z_0))\to 0$ , there must be $N_1\in {\mathbb {N}}$ such that $f^{N_1}(\gamma )$ surrounds $f^{N_1}(z)$ , and thus (again by the argument principle) there exists $w\in V$ surrounded by $\gamma $ such that $f^{N_1}(w) = f^{N_1}(z)$ . Now, by f-invariance of u, we have $u(w) = u(z) = 0$ , and so by the definition of V there exists $N_2\in {\mathbb {N}}$ for which $f^{N_2}(w) = f^{N_2}(z_0)$ . Finally, we see that $f^{N_1+N_2}(z) = f^{N_1+N_2}(z_0)$ , which is a contradiction since we assumed that $f^n(z)\neq f^n(z_0)$ for all $n\in {\mathbb {N}}$ . It follows that $u(z)> 0$ .

Let us now exclude the possibility of eventually isometric points. We show that if there exists $z\in U$ such that

(8) $$ \begin{align} d_{U_n}(f^n(z), f^n(z_0)) = c(z, z_0)> 0 \quad\text{ for }n\geq N, \end{align} $$

then $f:U_n\to U_{n+1}$ is a local hyperbolic isometry (equivalently, an unbranched covering map) for $n\geq N$ . Indeed, let $\gamma \subset U_n$ be a hyperbolic geodesic joining $f^n(z)$ to $f^n(z_0)$ . Then, by the Schwarz–Pick lemma,

$$ \begin{align*} \ell_{U_{n+1}}(f\circ\gamma) \leq \ell_{U_n}(\gamma), \end{align*} $$

while by the definition of the hyperbolic distance and (8) we have

$$ \begin{align*} \ell_{U_{n+1}}(f\circ\gamma) \geq d_{U_{n+1}}(f^{n+1}(z), f^{n+1}(z_0)) = d_{U_n}(f^n(z), f^n(z_0)) = \ell_{U_n}(\gamma). \end{align*} $$

Hence, $\ell _{U_{n+1}}(f\circ \gamma ) = \ell _{U_n}(\gamma )$ , or, equivalently,

$$ \begin{align*} \int_{\gamma} \rho_{U_n}(s)\,|ds| = \int_{f\circ\gamma} \rho_{U_{n+1}}(s')\,|ds'|. \end{align*} $$

By a change of variables, this becomes

$$ \begin{align*} \int_\gamma \rho_{U_n}(s)\,|ds| = \int_\gamma \rho_{U_{n+1}}(f(s))|f'(s)|\,|ds|, \end{align*} $$

and since $0\leq \rho _{U_{n+1}}(f(z))|f'(s)|\leq \rho _{U_n}(s)$ by the Schwarz–Pick lemma, we are forced to conclude that $\rho _{U_{n+1}}(f(s))|f'(s)| = \rho _{U_n}(s)$ for every $s\in \gamma $ . From the equality case of the Schwarz–Pick lemma, we deduce that $f:U_n\to U_{n+1}$ is a local hyperbolic isometry.

From now on, we will exchange U for $U_N$ , $z_0$ for $f^N(z_0)$ , and V for $f^N(V)$ for some sufficiently large N, and work as if f maps one wandering domain onto the next one locally isometrically for every $n\in {\mathbb {N}}$ . We can do this because of the ‘f-invariance’ of u: if $V\subset U$ is semi-contracting relative to $z_0\in V$ , then $f^N(V)\subset U_N$ is semi-contracting relative to $f^N(z_0)\in f^N(V)$ .

We will see that V being semi-contracting relative to $z_0\in V$ is incompatible with $f:U_n\to U_{n+1}$ being a local hyperbolic isometry. Indeed, let w be any point in V such that $f^n(w)\neq f^n(z_0)$ for all n and $\overline {B_U(z_0, d_U(z_0, w))}\subset V$ . Since V is semi-contracting, $u_n(w) = d_{U_n}(f^n(w), f^n(z_0))$ forms a non-increasing sequence that is also not eventually constant, that is, there is no N such that $u_n(w)$ is constant for $n\geq N$ . However, as $f:U_n\to U_{n+1}$ is a local hyperbolic isometry for all $n\geq 0$ , we can produce a sequence $(w_n)_{n\in {\mathbb {N}}}$ in $U\setminus \{w\}$ such that $f^n(w) = f^n(w_n)$ and $d_{U_n}(f^n(w), f^n(z_0)) = d_U(w_n, z_0)$ for all $n\in {\mathbb {N}}$ as follows.

Taking distance-minimizing geodesics $\gamma _n\subset U_n$ joining $f^n(z_0)$ to $f^n(w)$ , we apply the path lifting property to $f^n:U\to U_n$ (see [Reference Donaldson14, Proposition 10] or [Reference Hatcher17, Proposition 1.30]) to obtain a curve $\tilde {\gamma }_n\subset U$ joining $z_0$ to a point $w_n\in U$ such that $f^n(w_n) = f^n(w)$ . Since $f^n:U\to U_n$ is a local hyperbolic isometry, it preserves curve length, and thus

$$ \begin{align*} d_{U_n}(f^n(z_0), f^n(w)) = \ell_{U_n}(\gamma_n) = \ell_U(\tilde{\gamma}_n) \geq d_U(z_0, w). \end{align*} $$

On the other hand, by the Schwarz–Pick lemma,

$$ \begin{align*} d_U(z_0, w_n) \geq d_{U_n}(f^n(z_0), f^n(w)), \end{align*} $$

and therefore $d_U(z_0, w_n) = d_{U_n}(f^n(z_0), f^n(w))$ for all $n\in {\mathbb {N}}$ . Notice, though, that $(d_{U_n}(f^n(z_0), f^n(w)))_{n\in {\mathbb {N}}}$ is, by hypothesis, a non-constant non-increasing sequence, so that $w_n\neq w$ for all large enough n.

By the same token, the sequence $(w_n)_{n\in {\mathbb {N}}}$ is confined to the annulus $\{z\in U : u(w) \leq d_U(z, z_0) \leq d_U(w, z_0)\}\subset V$ , and hence has an accumulation point $w^*\in V$ . Since u is f-invariant, every $w_n$ satisfies $u(w_n) = u(w)$ , and thus by the continuity of u we have $u(w^*) = u(w)$ . But it is also the case by the definition of the points $w_n$ that $u(w^*) = u(w) = \lim _{n\to +\infty } d_U(w_n, z_0)$ , and so by continuity of the hyperbolic distance we have

$$ \begin{align*} u(w^*) = d_U(w^*, z_0). \end{align*} $$

It follows from the definition of u that $w^*$ is an eventually isometric point relative to $z_0$ lying in V, which is a contradiction.

Together, these two arguments show that every point in U is semi-contracting relative to $z_0$ .

3.3 Proof of Theorem 1.1(c): the eventually isometric case

Here, we assume that U contains a non-empty open set V and a point $z_0\in V$ such that $d_{U_n}(f^n(z), f^n(z_0)) = c(z, z_0)> 0$ for every $z\in V$ (except for at most countably many points for which ${f^k(z) = f^k(z_0)}$ for some $k\in {\mathbb {N}}$ ) and all sufficiently large n (say, $n\geq N$ , with N locally uniform over compact subsets of V). We want to show that the same holds for every $w\in U$ relative to $z_0\in V$ . Perhaps surprisingly, this is the most delicate case we will deal with here; it needs certain machinery that we now take the time to introduce.

We consider the set $\mathcal {H}_2$ of hyperbolic surfaces, identifying any two surfaces that are isometric. In other words, $\mathcal {H}_2$ is a space of equivalence classes of hyperbolic surfaces. We can ‘refine’ it a little by considering marked hyperbolic surfaces: pairs $(S, p)$ where ${S\in \mathcal {H}_2}$ and $p\in S$ is a base point. The space of all such pairs, again up to isometry equivalence, is denoted $\mathcal {H}_2^*$ ; we will now imbue it with a topology.

Despite its (relatively) intuitive definition, this topology, called the geometric topology, offers little insight into its own properties. Fortunately, there is an equivalent way of defining it via Kleinian groups, originally due to Claude Chabauty, which we will not state here. The following result was proved using this alternative definition (see [Reference Benedetti and Petronio6, Theorem E.1.10], [Reference Canary, Epstein, Green, Canary, Epstein and Marden12, Corollary I.3.1.7], or [Reference Matsuzaki and Taniguchi25, Proposition 7.8]).

Notice that Lemma 3.6 makes no claim about the limit surface (or, to be more precise, the equivalence class of limit surfaces). The nature of the limit surface can, in fact, be extremely counter-intuitive, but since appreciating the intricacies of the geometric topology is not our aim here, Lemma 3.6 will suffice.

Convergence in the sense of Gromov can be though of as a ‘locally uniform convergence’ of the surfaces’ geometry around the base point. Thus, Lemma 3.6 is, in a certain sense, a ‘normal families’ criterion for $\mathcal {H}_2$ . We will also need more conventional results on normal families, such as the following restatement of the Arzelà–Ascoli theorem for Lipschitz functions; see [Reference Beardon and Minda5, Theorem 7.1].

With this in mind, let us begin our study of the eventually isometric case in Theorem 1.1.

Let U, f, $V\subset U$ and $z_0\in V$ be as in the statement of Theorem 1.1(c). We can exclude the possibility of contracting points in U (relative to $z_0$ ) by an argument similar to the one used in §3.2: the iterates of such a point $w\in U$ would eventually intersect $f^n(V)$ , and the f-invariance of u would imply that $u(w)> 0$ , contradicting our choice of w.

Now, suppose that there exists a point $w\in U\setminus V$ that is semi-contracting relative to $z_0$ . The same argument as in §3.2 shows that $f:U_n\to U_{n+1}$ is a local hyperbolic isometry for $n\geq N$ (N here being chosen according to some compact subset of V), but the rest of the argument does not carry through—there is nothing unexpected about finding an eventually isometric point relative to $z_0$ in V; think, for instance, of the ‘annulus model’ described in [Reference Ferreira15, §2]. Instead, we will proceed with a ‘normal families’ argument. As before, we swap U for $U_N$ , $z_0$ for $f^N(z_0)$ , and V for $f^N(V)$ while keeping the same notation.

First, we take a universal covering map $\varphi _0:\mathbb {D}\to U$ with $\varphi _0(0) = z_0$ , and build the functions $\psi _n:\mathbb {D}\to U_n$ given by $\psi _n(z) = f^n\circ \varphi _0(z)$ ; since $f^n$ is a local hyperbolic isometry, these are all universal covering maps with $\psi _n(0) = f^n(z_0)$ . Furthermore, the hypothesis that V is locally eventually isometric implies that, if $r_0$ is such that ${\overline {B_U(z_0, r_0)}\subset V}$ is an embedded disc (and N was chosen accordingly), then $f^n$ maps $B_U(z_0, r_0)$ isometrically onto $f^n(B_U(z_0, r_0))$ for all n. In particular, the sets $f^n(B_U(z_0, r_0))$ will all be embedded discs of hyperbolic radius $r_0$ . Therefore, the injectivity radii at $f^n(z_0)$ of the hyperbolic surfaces $U_n$ are uniformly bounded below by $r_0$ . Thus, by Lemma 3.6, the sequence of marked hyperbolic surfaces $((U_n, f^n(z_0)))_{n\in {\mathbb {N}}}$ admits a subsequence $((U_{n_k}, f^{n_k}(z_0)))_{k\in {\mathbb {N}}}$ converging in the sense of Gromov to a marked hyperbolic surface $(U^*, z^*)$ (say).

It follows from the definition, then, that we can take some $r> 10d_U(z_0, w)$ (where $w\in U\setminus V$ is assumed to be semi-contracting relative to $z_0$ ) and find diffeomorphisms $\phi _k:\overline {B_{U^*}(z^*, r)}\to \phi _k(\overline {B_{U^*}(z^*, r)})\subset U_{n_k}$ such that $\phi _k(z^*) = f^{n_k}(z_0)$ and each $\phi _k$ is $K_k$ -bi-Lipschitz relative to the hyperbolic metrics on $U^*$ and $U_{n_k}$ , with $K_k\to 1$ as $k\to +\infty $ . Since $K_k\to 1$ , we can assume without loss of generality that $\sup _k K_k < 10$ , so that every $\phi _k^{-1}$ is defined on a hyperbolic ball of radius $d_U(z_0, w)$ around $f^{n_k}(z_0)$ . In particular, the functions

$$ \begin{align*} \widehat{\psi}_k = \phi_k^{-1}\circ \psi_{n_k} : B_{\mathbb{D}}(0, d_U(z_0, w)) \to U^* \end{align*} $$

are all well defined and map $0$ to $z^*$ . We want to show that $\{\widehat {\psi }_k\}_{k\in {\mathbb {N}}}$ is a normal family in the sense of precompactness relative to locally uniform convergence (see [Reference Beardon and Minda5], as well as [Reference Branner and Fagella11, Theorem 1.21] and [Reference Hinkkanen and Martin19] for an account of this interpretation of normality) with analytic limit functions; to this end, we make two claims.

Proof. For z and $z'$ in $B_w = B_{\mathbb {D}}(0, d_U(z_0, w))$ , we have, using the bi-Lipschitz constant of $\phi _k$ and the Schwarz–Pick lemma respectively,

$$ \begin{align*} d_{U^*}(\phi_k^{-1}\circ\psi_{n_k}(z), \phi_k^{-1}\circ\psi_{n_k}(z')) &\leq K_kd_{U_{n_k}}(\psi_{n_k}(z), \psi_{n_k}(z'))\\ &\leq K_kd_{\mathbb{D}}(z, z') \leq K_kd_{B_w}(z, z'). \end{align*} $$

Since $K_k\to 1$ , we have that $K := \sup _k K_k < 10$ is a uniform Lipschitz constant for $\phi _k^{-1}\circ \psi _{n_k} = \widehat {\psi }_k$ .

Now, since $\{\widehat {\psi }_k(0) : k\in {\mathbb {N}}\} = \{z^*\}$ is clearly a relatively compact subset of $U^*$ , Lemma 3.7 tells us that $(\widehat {\psi }_k)_k$ is a normal family, and so admits a subsequence $(\widehat {\psi }_{k_m})_m$ converging locally uniformly to a limit function

$$ \begin{align*} \psi^*:B_{\mathbb{D}}(0, d_U(z_0, w))\to U^*. \end{align*} $$

Since $K_k\to 1$ and the limit of K-quasiregular functions is K-quasiregular, it follows by Claim 3.8 that $\psi ^*$ is $1$ -quasiregular (see, for example, [Reference Bojarski, Gutlyanskii, Martio and Ryanazov10, Theorem 4.2], [Reference Hinkkanen and Martin19], [Reference Astala, Iwaniec and Martin2, Corollary 5.5.7], or [Reference Rickman28, Theorem VI.8.6]) and hence analytic by the quasiregular version of Weyl’s lemma (see [Reference Branner and Fagella11, Proposition 1.37]).

We want to show that $\psi ^*$ is not constant; to that end, take a point $z'\in B_{\mathbb {D}}(0, d_U(z_0, w))$ such that $\varphi _0(z')\in V\setminus \{z_0\}$ . Then, by the definition of the maps $\widehat {\psi }_n$ and the bi-Lipschitz property of $\phi _k$ ,

$$ \begin{align*} d_{U^*}(z^*, \widehat{\psi}_{k_m}(z'))\geq \frac{d_{U_n}(\psi_{k_m}(0), \psi_{k_m}(z'))}{K_{k_m}} \geq \frac{d_{U_n}(\psi_{k_m}(0), \psi_{k_m}(z'))}{10}; \end{align*} $$

since $\psi _n = f^n\circ \varphi _0$ and $f^n$ is isometric when restricted to V, we have that $d_{U_n}(\psi _{k_m}(0), \psi _{k_m}(z')) = d_U(z_0, \varphi _0(z'))> 0$ for all m. Thus, by making $m\to +\infty $ , we see that $d_{U^*}(\psi ^*(0), \psi ^*(z'))> 0$ , and so $\psi ^*$ is indeed non-constant.

We are now in position to make a case against the existence of w, the semi-contracting point relative to $z_0$ . The fact that $f:U_n\to U_{n+1}$ are local hyperbolic isometries implies that, in order for $d_{U_n}(f^n(w), f^n(z_0))$ to decrease infinitely many times, we must (as in §3.2) be able to find a sequence $w_n\in U\setminus \{w\}$ such that $f^n(w_k) = f^n(w)$ for $1\leq k\leq n$ and $d_U(w_n, z_0) = d_{U_n}(f^n(w), f^n(z_0)) < d_U(z_0, w)$ . By the completeness of the hyperbolic metric in U, the sequence $w_n$ admits an accumulation point ${w^*\in B_U(z_0, d_U(z_0, w))}$ , and this lifts to a sequence $\tilde {w}_n\in B_{\mathbb {D}}(0, d_U(z_0, w))$ with an accumulation point ${\tilde {w}^*\in B_{\mathbb {D}}(0, d_U(z_0, w))}$ . However, repeated application of the triangle inequality yields

$$ \begin{align*} d_{U^*}(\psi^*(\tilde{w}_{k_m}), \psi^*(\tilde{w}^*)) &\leq d_{U^*}(\psi^*(\tilde{w}_{k_m}), \widehat{\psi}_{k_l}(\tilde{w}_{k_m}))\\ &\quad+ d_{U^*}(\widehat{\psi}_{k_l}(\tilde{w}_{k_m}), \widehat{\psi}_{k_l}(\tilde{w}_{k_l})) + d_{U^*}(\widehat{\psi}_{k_l}(\tilde{w}_{k_l}), \psi^*(\tilde{w}^*)), \end{align*} $$

where $m\in {\mathbb {N}}$ is fixed and $l\geq m$ . By the definitions of $\tilde {w}_n$ and $\widehat {\psi }_k$ , we have that $\widehat {\psi }_{k_l}(\tilde {w}_{k_l}) = \widehat {\psi }_{k_l}(\tilde {w}_{k_m}$ ), so that the middle term in the sum above vanishes. As for the other two, taking the limit $l\to +\infty $ makes them arbitrarily small, since $\widehat {\psi }_{k_l}$ converges to $\psi ^*$ locally uniformly. It follows that $\psi ^*(\tilde {w}_{k_m}) = \psi ^*(\tilde {w}^*)$ for all $m\in {\mathbb {N}}$ . This is a contradiction, since $\psi ^*$ is analytic and as such $(\psi ^*)^{-1}(p)$ is discrete for any $p\in U^*$ .

We have completed the proof of case (c), and hence finished the proof of Theorem 1.1.

4.1 The patient

Benini et al constructed the first known examples of semi-contracting wandering domains using approximation theory (the one we are interested in is [Reference Benini, Evdoridou, Fagella, Rippon and Stallard7, Example 2(a)]). As such, we can describe its properties mostly in an asymptotic fashion; but by controlling the rate of convergence, this will suffice for our ends.

To describe the example, we shall need the following sets and functions. Let ${T_n:\mathbb {C}\to \mathbb {C}}$ and $b_n:\mathbb {D}\to \mathbb {D}$ be defined as

$$ \begin{align*} T_n(z) = z + 4n \quad \text{and }\quad b_n(z) = z\cdot\frac{z + a_n}{1 + a_nz}, \end{align*} $$

where $a_n$ is a real sequence satisfying $0 < a_n < 1$ and $a_n\nearrow 1$ fast enough that

$$ \begin{align*} \unicode{x3bb} := \prod_{n = 1}^{+\infty} a_n> 0; \end{align*} $$

as our construction progresses, we will impose further restrictions on the speed with which $a_n\nearrow 1$ . Define also the discs $\Delta _n' := B(4n, r_n)$ and $\Delta _n := B(4n, R_n)$ , where $0 < r_n < 1 < R_n$ , both sequences $(r_n)_{n\in {\mathbb {N}}}$ and $(R_n)_{n\in {\mathbb {N}}}$ tend to $1$ as $n\to +\infty $ , and the rate of convergence is relatively ‘free’—that is, $1 - r_n$ and $R_n - 1$ have upper bounds depending only on $1 - a_n$ .

With all that in place, their example consists of an entire function f with a bounded simply connected wandering domain U such that, for all $n\geq 0$ , the following properties hold.

1. (A) $\overline {\Delta _n'}\subset U_n\subset \Delta _n$ , so, in particular, $U_n$ ‘is asymptotically a disc’ (recall that $U_n$ is the Fatou component containing $f^n(U)$ ).

2. (B) $|f(z) - T_{n+1}\circ b_{n+1}\circ T_n^{-1}(z)| < \epsilon _{n+1}$ for $z\in \overline {\Delta _n}$ , where $\epsilon _n < (R_n - r_n)/4$ .

3. (C) $f^n(0) = 4n$ .

4. (D) $\deg f|_{U_n} = \deg b_{n+1} = 2$ and $f|_{U_n}$ satisfies

   $$ \begin{align*} |f(z) - z - 4|\to 0 \quad \text{ locally uniformly for }z\in U_n\text{ as }n\to +\infty. \end{align*} $$

   In particular, the critical points $z_n$ of $f|_{U_n}$ satisfy

   $$ \begin{align*} \mathrm{dist}(z_n, \partial U_n)\to 0\quad\text{ as }n\to+\infty, \end{align*} $$

   where $\mathrm {dist}$ denotes Euclidean distance.

It was shown in [Reference Benini, Evdoridou, Fagella, Rippon and Stallard7, Example 2] that the wandering domain U is semi-contracting; let us take a closer look at it.

Let $B_n(z) := b_n\circ b_{n-1}\circ \cdots \circ b_1(z)$ for $n\in {\mathbb {N}}$ . By Montel’s theorem, $B_n$ admits a subsequence converging locally uniformly to some $B:\mathbb {D}\to \overline {\mathbb {D}}$ ; since $b_n(0) = 0$ for all $n\in {\mathbb {N}}$ , we have $B(0) = 0$ , and thus $B:\mathbb {D}\to \mathbb {D}$ . Furthermore, by the Weierstrass convergence theorem and the assumptions on $a_n$ ,

$$ \begin{align*} B'(0) = \lim_{n\to+\infty} B_n'(0) = \prod_{n=1}^{+\infty} a_n = \unicode{x3bb}> 0, \end{align*} $$

and so B is non-constant. Additionally, since $B_n'(0) = \prod _{k=1}^n a_k> 0$ and, by Schwarz’s lemma applied to the sequence $(B_n)_{n\in {\mathbb {N}}}$ ,

(9) $$ \begin{align} |B_n(z)|\searrow |B(z)| \quad \text{for all }z\in\mathbb{D}, \end{align} $$

the limit function B is unique.

Next, we recall [Reference Benini, Evdoridou, Fagella, Rippon and Stallard7, Corollary 2.4].

Lemma 4.1. Let $h:\mathbb {D}\to \mathbb {D}$ be a holomorphic function with $h(0) = 0$ and $|h'(0)| = \mu $ . Then, for all $w\in \mathbb {D}$ ,

$$ \begin{align*} h_1(|w|) := |w|\cdot\frac{\mu - |w|}{1 - \mu|w|} \leq |h(w)| \leq |w|\cdot\frac{\mu + |w|}{1 + \mu|w|} =: h_2(|w|). \end{align*} $$

Elementary calculus shows that the function $h_1:[0, 1]\to \mathbb {R}$ is a concave function with maximum

$$ \begin{align*} \frac{2\mu^2 + (2 - \mu^2)\sqrt{1 - \mu^2} - 2}{\mu^2\sqrt{1 - \mu^2}}. \end{align*} $$

Thus, if we choose $c\in (0, 1)$ such that

$$ \begin{align*} c < \frac{2\unicode{x3bb}^2 + (2 - \unicode{x3bb}^2)\sqrt{1 - \unicode{x3bb}^2} - 2}{\unicode{x3bb}^2\sqrt{1 - \unicode{x3bb}^2}}, \end{align*} $$

then there exists a round annulus $A\subset {\mathbb {D}}$ centred at $0$ such that $|B(z)|> c$ for $z\in A$ . Now recall (9); in particular, since $(|B_n(z)|)_{n\in {\mathbb {N}}}$ is a non-increasing sequence for all $z\in \mathbb {D}$ and $B_n(z) = b_n\circ b_{n-1}\circ \cdots \circ b_1(z)$ , we also have $|b_n(z)|> c$ for $z\in A$ for all $n\in {\mathbb {N}}$ .

Now take some positive $c' < c$ ; applying Rouche’s theorem to condition (B) satisfied by f, we conclude that, for all sufficiently large n, there exists a topological annulus $A_n'\subset \overline {\Delta _n'}$ such that $|f(z) - 4(n+1)|> c'$ for all $z\in A_n'$ , and $A_n'$ surrounds the disc $\{z\in \mathbb {C} : |z - 4n| \leq c'\}$ .

4.2 The surgery

At each $U_n$ , we want to cut out a small disc and replace f by an appropriately rescaled version of the Joukowski map $z\mapsto z + z^{-1}$ inside of it, giving us a pole in each domain. After that, we must join f to the Joukowski map through quasiconformal interpolation in an appropriate annulus $A_n\subset U_n$ such that $\overline {A_n}\subset U_n$ . Since we want the resulting quasiregular map $g_0$ to be quasiconformally conjugate to a meromorphic one, there are two conditions we require to hold.

1. (1) Ensure that the dilatations $K_n$ of the interpolating map in $A_n$ (respectively) satisfy

   $$ \begin{align*} \prod_{n=1}^{+\infty} K_n < +\infty; \end{align*} $$

2. (2) For any $z\in A_n$ , its orbit under $g_0$ does not intersect $A_m$ for $m < n$ .

The two main results we will use in this surgery are both due to Kisaka and Shishikura. First, we have a way to interpolate quasiconformally in round annuli [Reference Kisaka, Shishikura, Rippon and Stallard22, Lemma 6.2].

Lemma 4.2. Let $k\in {\mathbb {N}}$ , $0 < R_1 < R_2$ , and $\varphi _j$ be analytic on a neighbourhood of ${C_j := \{|z| = R_j\}}$ such that $\varphi _j|_{C_j}$ winds around the origin k times ( $j = 1, 2$ ). If there exist positive constants $\delta _0$ and $\delta _1$ such that

(10) $$ \begin{align} \bigg|\log\bigg(\frac{\varphi_2(R_2e^{i\theta})}{R_2^k}\frac{R_1^k}{\varphi_1(R_1e^{i\theta})}\bigg)\bigg|\leq \delta_0 \end{align} $$

and

(11) $$ \begin{align} \bigg|z\frac{d}{dz}\bigg(\log\frac{\varphi_j(z)}{z^k}\bigg)\bigg| \leq \delta_1,\quad z = R_je^{i\theta},\ (j = 1, 2), \end{align} $$

for every $\theta \in [0, 2\pi )$ , and if $\delta _0$ and $\delta _1$ satisfy

(12) $$ \begin{align} C = 1 - \frac{1}{k}\bigg(\frac{\delta_0}{\log(R_2/R_1)} + \delta_1\bigg)> 0, \end{align} $$

then there exists a quasiregular map

$$ \begin{align*} H : \{z : R_1 \leq |z|\leq R_2\}\to\mathbb{C}^* \end{align*} $$

without critical points that interpolates between $\varphi _1$ and $\varphi _2$ and has dilatation constant

(13) $$ \begin{align} K_H \leq \frac{1}{C}. \end{align} $$

Second, we need a sufficient condition for us to conjugate the resulting quasiregular map to a meromorphic one [Reference Kisaka, Shishikura, Rippon and Stallard22, Theorem 3.1]. Although it was originally stated only for entire maps, it is easy to see how to deal with the presence of poles: this lemma is a particular case of the necessary and sufficient conditions given by Sullivan’s straightening theorem, which is flexible enough for transcendental meromorphic maps (see [Reference Branner and Fagella11, §§5.2 and 5.3]).

We have our patient and our tools; let us begin the surgery. We refer to Figure 1 for a sketch of some of the sets and functions here and their relations to each other. First, we take the points $z_n := 4n\in U_n$ , $n\geq N$ , which form an orbit under f; N here is large enough that the annuli $A_n'$ described at the end of §4.1 exist for $n\geq N$ . We take some $r> 0$ such that, for $n\geq N$ , the circle $C_n = \{z : |z_n - z| = r\}$ (shown by the blue inner disc in Figure 1) is surrounded by $A_n'$ . Inside the discs $\{z : |z_n - z| \leq r\}$ , we will remove f and transplant appropriately translated versions of the rescaled Joukowski map

$$ \begin{align*} J_n(z) = \frac{\mu_n r^2}{\mu_n^2r^2 - 1}\bigg(\mu_nz + \frac{1}{\mu_nz}\bigg), \end{align*} $$

where the $\mu _n> 1/r$ are parameters that will give us great control over the dilatation of the interpolating maps. The strange scaling constant here requires explaining; its role is to guarantee that $J_n(C_n)$ surrounds $C_{n+1}$ , which will help us enforce condition (2). It can be calculated as follows: as explained in [Reference Ahlfors1, pp. 94–95], the function $z\mapsto z + 1/z$ maps a circle of radius $\rho> 1$ injectively onto an ellipse with major semi-axis $\rho + 1/\rho $ and minor semi-axis $\rho - 1/\rho $ . Denoting the (as yet unknown) scaling constant by $\unicode{x3bb} $ , the condition ‘ $J_n(C_n)$ surrounds $C_{n+1}$ ’ then becomes

$$ \begin{align*} \unicode{x3bb}\bigg(\mu_nr - \frac{1}{\mu_n r}\bigg) = r, \end{align*} $$

and solving for $\unicode{x3bb} $ yields precisely

$$ \begin{align*} \unicode{x3bb} = \frac{\mu_nr^2}{\mu_n^2r^2 - 1}. \end{align*} $$

Figure 1 In the blue inner discs bounded by $C_n$ , the original map f was replaced by appropriately translated versions of $\gamma _n$ . To connect this with f, we interpolate on the red outer annuli $A_n$ using the quasiconformal maps  $\phi _n$ .

Next, since $f|_{U_n}$ converges locally uniformly to the translation $z\mapsto z + 4$ (in the sense outlined in property (D)), we can take some $r'> r$ with $C_n' := \{z : |z_n - z| = r'\}$ (red in Figure 1) also surrounded by $A_n'$ , and such that $|f(C_n') - z_{n+1}|> r + \delta $ for all sufficiently large n. Now, if n is (again) large enough, the only pre-image of $z_{n+1}$ under f surrounded by $C_n'$ is $z_n$ itself, and hence $\mathrm {ind}(f\circ C_n', z_{n+1}) = 1$ (the notation $\mathrm {ind}(\gamma , \alpha )$ denotes the winding number of the curve $\gamma $ around the point $\alpha \in \mathbb {C}$ ). We are ready to start interpolating the maps $T_{n+1}\circ J_n\circ T_n^{-1}$ on $C_n$ and f on $C_n'$ . For the sake of convenience, we pass to the unit disc, interpolating instead the functions $J_n$ and $T_{n+1}^{-1}\circ f\circ T_n$ on $\{z\in \mathbb {C} : |z| = r\}$ and $\{z\in \mathbb {C} : |z| = r'\}$ (respectively).

Needless to say, we are applying Lemma 4.2 with $k = 1$ . We would like first to draw attention to condition (11) in the case $j = 1$ , that is, for the function $J_n$ . The left-hand quantity takes the form

$$ \begin{align*} \bigg|\frac{2}{\mu_n^2r^2e^{2i\theta} + 1}\bigg|, \end{align*} $$

which has the upper bound (achieved for $\theta = \pi /2$ )

$$ \begin{align*} \frac{2}{\mu_n^2r^2 - 1}. \end{align*} $$

Clearly, taking $\mu _n\to +\infty $ gives us excellent control over how small this term is; it is through this ‘trick’ that we will control the dilatation introduced by the Joukowski map.

For the case $j = 2$ , we have a function of the form $f(z) = b_{n+1}(z) + \epsilon _n(z)$ , where $\epsilon _n\to 0$ uniformly as $n\to +\infty $ and $b_n$ are the Blaschke products described in §4.1. The exact form of the left-hand side of condition (11) is

(14) $$ \begin{align} \bigg|z\frac{d}{dz}\bigg(\log\frac{b_{n+1}(z) +\epsilon_n(z)}{z}\bigg)\bigg|, \end{align} $$

and by controlling how fast $a_n\nearrow 1$ and applying Cauchy’s integral formula to $\epsilon _n$ , we can ensure that $\epsilon _n$ and its derivative tend to zero as fast as necessary to control the size of the constants in Lemma 4.2. Thus, up to a small error term tending to zero arbitrarily fast, (14) takes the form

$$ \begin{align*} \bigg|z\frac{b_{n+1}'(z)}{b_{n+1}(z)} - 1\bigg| = \bigg|\frac{(1 +a_{n+1})z +a_{n+1}^2z^2 +a_{n+1}}{(1 +a_{n+1}^2)z + a_{n+1}z^2 + a_{n+1}} - 1\bigg|, \end{align*} $$

which again tends to zero arbitrarily fast by choosing an appropriate sequence $a_n\nearrow 1$ . Finally, the left-hand side of condition (10) takes the form

$$ \begin{align*} \bigg|\log\bigg(\frac{b_{n+1}(r'e^{i\theta}) + \epsilon_n(r'e^{i\theta})}{r'e^{i\theta}}\frac{re^{i\theta}}{J_n(re^{i\theta})}\bigg)\bigg| \end{align*} $$

for $\theta \in [0, 2\pi )$ , and we can use the triangle inequality to bound it by

$$ \begin{align*} \bigg|\log\frac{b_{n+1}(r'e^{i\theta}) + \epsilon_n(r'e^{i\theta})}{r'e^{i\theta}}\bigg| +\bigg|\log\frac{re^{i\theta}}{J_n(re^{i\theta})}\bigg|. \end{align*} $$

Ignoring the error term again, this becomes

(15) $$ \begin{align} \bigg|\log\frac{r'e^{i\theta} + a_{n+1}}{1 + a_{n+1}r'e^{i\theta}}\bigg| + \bigg|\log\frac{re^{i\theta}}{({\mu_n^2r^3e^{i\theta}}/(\mu_n^2r^2 - 1)) - (r/(e^{i\theta}(\mu_n^2r^2 - 1)))}\bigg|. \end{align} $$

Thus, we see that, by making appropriate, independent choices of $a_n\nearrow 1$ and $\mu _n\to +\infty $ , we can make (15) go to zero arbitrarily fast as $n\to +\infty $ .

It follows that we can arrange conditions (10) and (11) to hold in each annulus ${A_n := \{z\in \mathbb {C} : r < |z - z_n| < r'\}}$ with constants $\delta _{n,0}$ and $\delta _{n,1}$ that are as small as we need them to be. Thus, by invoking Lemma 4.2, we obtain a sequence of quasiconformal maps $\phi _n$ that interpolate between $J_n$ and f on the annuli $A_n$ with dilatation as close to $1$ as we please, (say) $K_n < 1 + 1/n^2$ , by (12) and (13). It follows that the $K_n$ can be made to satisfy

$$ \begin{align*} K_\infty := \prod_{n=1}^{+\infty} K_n < +\infty. \end{align*} $$

We now define the map

$$ \begin{align*} g_0(z) := \begin{cases} J_n(z), & z\in \mathrm{int}(C_n), n\geq N, \\ \phi_n(z), & z\in \overline{A_n}, n\geq N, \\ f(z) & \text{elsewhere}, \end{cases} \end{align*} $$

and claim that it satisfies the hypotheses of Lemma 4.3. Indeed, for any $z\in A_n$ , the fact that $J_n(C_n)$ surrounds $C_{n+1}$ while $C_{n+1}'$ surrounds $f(C_n')$ implies that $g_0(z)\in A_{n+1}$ , and so the $g_0$ -orbit of every $z\in \mathbb {C}$ meets each $A_n$ at most once, guaranteeing hypothesis (i). Hypotheses (ii)–(iv) are also satisfied by the construction of the $J_n$ and our choices of $a_n$ and $\mu _n$ , and so we can apply Lemma 4.3 and obtain a $K_{\infty }$ -quasiconformal map $\psi $ such that

$$ \begin{align*} g(z) := \psi\circ g_0\circ\psi^{-1}(z) \end{align*} $$

is a transcendental meromorphic function.

We claim that:

1. (i) the new map g has a wandering domain V;

2. (ii) V is semi-contracting; and

3. (iii) V is infinitely connected.

The first claim will follow from the fact that $\psi $ conjugates $g_0$ to g, and that we left ‘enough’ of f intact in each $U_n$ . More specifically, recall the round annulus A from §4.1; it satisfies $|B(z)|> c$ for $z\in A$ . By Rouche’s theorem, we find a topological annulus $A'\subset U_N$ (where, again, N is large enough that the annuli $A_n'$ exist for $n\geq N$ ) such that, for $z\in A'$ , $f^n(z)\in U_{n+N}$ does not intersect the discs $\{z\in \mathbb {C} : |z - 4(n + N)| < c'\}\subset U_{n+N}$ , where $c'$ was also defined in §4.1. In particular, for $z\in A'$ , the f-orbit of z is not affected by the surgery, and therefore is conjugated by $\psi $ to its g-orbit; it follows that $\psi (A')$ is contained in a Fatou component V of g. Furthermore, since g has a pole at $\psi (4n)$ for $n\geq N$ , V is at least doubly connected.

We now show that V is semi-contracting. We begin by showing that V (and each $V_n$ , the Fatou component of g containing $g^n(V)$ ) is contained in $\psi (U_N)$ (respectively, $\psi (U_{n+N})$ ). This, in turn, will follow from the fact that we did not modify f outside of a small disc properly contained within each $U_n$ . Indeed, if w is a point on $\partial U_N$ , it belongs to the Julia set of f, and is therefore approached by a sequence $(w_n)_{n\in {\mathbb {N}}}$ of repelling periodic points of f (see, for instance, [Reference Bergweiler8, Theorem 4]). Since the $w_n$ are periodic points in the Julia set, their orbits do not intersect the discs $\{z\in \mathbb {C} : |z - 4(n + N)| < c'\}\subset U_{n+N}$ for any $n\in {\mathbb {N}}$ , and so $g_0$ agrees with f on their f-orbits. It follows that the conjugacy $\psi $ takes these f-orbits to corresponding g-periodic orbits $\psi (w_n)$ , which we can show to be repelling by the local topological dynamics as follows.

Take one of the $w_n$ , of minimal period $k_n\geq 1$ (say), and apply Koenig’s linearization theorem [Reference Milnor26, Theorem 8.2] to find a neighbourhood W of $w_n$ and a biholomorphic map $\phi :W\cup f^{k_n}(W)\to \phi (W\cup f^{k_n}(W))\subset \mathbb {C}$ such that $\phi $ conjugates $f^{k_n}|_W$ to multiplication by $\alpha _n := (f^{k_n})'(w_n)$ , which satisfies $|\alpha _n|> 1$ since $w_n$ is repelling. This allows us to find (if necessary) a smaller neighbourhood $W'\subset W$ such that $f^m(W')\cap \{z\in \mathbb {C} : |z - 4(n + N)| < c'\} = \emptyset $ for all $n\in {\mathbb {N}}$ and $1\leq m \leq k_n$ , so that $\psi (f^m(z)) = g^m(\psi (z))$ for $z\in W'$ and $1\leq m\leq k_n$ . From this, we may conclude that $\psi (w_n)$ is repelling for g: for all ${z\in \psi (W')\setminus \{w_n\}}$ , there must exist $M\in {\mathbb {N}}$ such that $g^{Mk_n}(z)$ is not in $\psi (W')$ (see, for instance, [Reference Milnor26, p. 84]). It follows that $\psi (w)$ , being accumulated by repelling periodic points, belongs to $J(g)$ ; a similar argument applies to each $V_n$ , $n\in {\mathbb {N}}$ .

Thus, for any points $z, w\in V$ , we have by the Schwarz–Pick lemma and the fact that $V_n\subset \psi (U_{n+N})$ that

$$ \begin{align*} d_{V_n}(g^n(z), g^n(w)) \geq d_{\psi(U_{n+N})}(g^n(z), g^n(w)); \end{align*} $$

if $\psi $ were a conformal map, our work would be done—but it is not, and it does not preserve the hyperbolic metric. Instead, for a domain $D\subset \mathbb {C}$ and points $z, w\in D$ , let us define

$$ \begin{align*} k_D(z, w) := \inf_\gamma \int_\gamma \frac{1}{d(s, \partial D)}\,|ds|, \end{align*} $$

where the infimum runs over every rectifiable arc $\gamma \subset D$ joining z and w. This is the quasihyperbolic metric; if D is simply connected, standard estimates for the hyperbolic metric [Reference Carleson and Gamelin13, p. 13] show that

$$ \begin{align*} \frac{k_D(z, w)}{2} \leq d_D(z, w) \leq 2k_D(z, w). \end{align*} $$

Additionally, since $\psi $ is $(K_\infty )^{-1}$ -Hölder continuous [Reference Branner and Fagella11, p. 31], we know (see [Reference Gehring and Osgood16, Theorem 3]) that there exists a constant $C> 0$ depending only on $K_\infty $ such that

$$ \begin{align*} k_{U_{n+N}}(z', w') \leq C\cdot\max\{k_{\psi(U_{n+N})}(z, w), k_{\psi(U_{n+N})}(z, w)^{1/K_\infty}\}, \end{align*} $$

where $z' = \psi ^{-1}(z)$ and $w' = \psi ^{-1}(w)$ are points in $U_{n+N}$ . For points $z'\in A'$ , the construction of g implies that

$$ \begin{align*} \psi(f^n(z')) = g^n(\psi(z)) \quad\text{ for all }n, \end{align*} $$

and so taking $z'$ and $w'$ in $A'$ yields

$$ \begin{align*} &k_{U_{n+N}}(f^n(z'), f^n(w'))\\ &\quad\leq C\cdot\max\{k_{\psi(U_{n+N})}(g^n(z), g^n(w)), k_{\psi(U_{n+N})}(g^n(z), g^n(w))^{1/K_\infty}\}. \end{align*} $$

The left-hand side of this expression is bounded below by $d_{U_{n+N}}(f^n(z'), f^n(w'))/2$ , which, since U is semi-contracting, is in turn bounded below by $c(z', w')/2> 0$ . Also, since the exponent on the right-hand side is independent of n, we can get a uniform, positive lower bound $c'(z, w)$ on $k_{\psi (U_{n+N})}(g^n(z), g^n(w))$ regardless of which term is the maximum. Combining everything, we see that

$$ \begin{align*} \frac{c'(z, w)}{2} \leq d_{\psi(U_{n+N})}(g^n(z), g^n(w)) \leq d_{V_n}(g^n(z), g^n(w)) \quad \text{for }z\text{ and }w\text{ in }\psi(A'), \end{align*} $$

and thus V contains a non-empty open subset that is semi-contracting relative to any $z_0$ in this subset—we claim that $(d_{V_n}(g^n(z), g^n(w)))_{n\in {\mathbb {N}}}$ is not a constant sequence, since each $g|_{V_n}$ ‘inherits’ a critical point from f. Indeed, $f|_{U_n}$ has a single critical point $z_n^*$ for every n, and since $z_n^*$ approaches $\partial U_n$ as $n\to +\infty $ (in the sense outlined in property (D)), we have $z_n^*\in U_n\setminus \{z\in \mathbb {C} : |z - z_n| \leq r'\}$ for all sufficiently large n. Hence, the critical point is not affected by the surgery for sufficiently large n, and $\psi (z_n^*)\in V_n$ is a critical point of g for all large n.

We have only shown that V is semi-contracting on the topological annulus $\psi (A')$ . For the remainder of V, we choose some $z_0$ in $\psi (A')$ and apply Theorem 1.1.

To prove that V is infinitely connected, notice that (since $V_n\subset \psi (U_{n+N})$ ) all $V_n$ are bounded, and so $\deg g|_{V_n}$ is always finite. By applying [Reference Ferreira15, Theorem 1.1], we deduce that V must be infinitely connected. This concludes the proof of Theorem 1.2.