2. Speiser functions with few singular values

Let $f:\operatorname {\mathbb C}\to \operatorname {\widehat {\mathbb C}}$ be transcendental and meromorphic. We say that c is a critical value of f if c has a preimage with zero spherical derivative. With this definition, $\infty $ will be a critical value if there are any multiple poles. $a\in \operatorname {\widehat {\mathbb C}}$ is an asymptotic value of f if there exists a curve $\gamma $ tending to $\infty $ such that $f(\gamma )$ tends to a. As a simple example, $0$ and $\infty $ are asymptotic values of $e^{z}$ . A value s is called a singular value if it is either a critical or asymptotic value. See [Reference Bergweiler and EremenkoBE95] for a classification of singularities of the inverse of a meromorphic function. Singular values play a vital role in the dynamics of meromorphic functions; we refer to [Reference BergweilerBer93] for more details and explanations.

Meromorphic functions with two singular values. The following simple fact concerning meromorphic functions with two singular values is folklore and it is not easy to locate a reference. Therefore, the outline of a proof is presented for completeness. Recall that $\operatorname {\mathcal {S}}_{q}$ denotes the class of Speiser functions with exactly q singular values.

Proof Sketch of proof

Without loss of generality, we assume that the two singular values of f are $0$ and $\infty $ . Otherwise we consider $M_{1}\circ f$ , where $M_{1}$ is a Möbius transformation sending the two singular values of f to $0$ and $\infty $ . Then

$$ \begin{align*}f:\operatorname{\mathbb C}\setminus\{f^{-1}(0),\,f^{-1}(\infty)\}\rightarrow \operatorname{\widehat{\mathbb C}}\setminus\{0,\infty\}\end{align*} $$

is a covering map. Note that the fundamental group of $\operatorname {\widehat {\mathbb C}}\setminus \{0,\infty \}$ is isomorphic to $\mathbb {Z}$ . The transcendence of f then implies that the fundamental group of $\operatorname {\mathbb C}\setminus \{f^{-1}(0),\,f^{-1}(\infty )\}$ is trivial and thus $\operatorname {\mathbb C}\setminus \{f^{-1}(0),\,f^{-1}(\infty )\}$ is simply connected such that the above f is a universal covering to $\operatorname {\widehat {\mathbb C}}\setminus \{0,\infty \}$ . Since meromorphic functions are discrete maps, we have that $\operatorname {\mathbb C}\setminus \{f^{-1}(0),\,f^{-1}(\infty )\}=\operatorname {\mathbb C}$ . This implies that both singular values are actually omitted. Note that the exponential map $\exp $ is a holomorphic universal covering from $\operatorname {\mathbb C}$ to $ \operatorname {\widehat {\mathbb C}}\setminus \{0,\infty \}$ . Now it follows from the essential uniqueness of the universal covering spaces that there exists a holomorphic homeomorphism $\phi : \operatorname {\mathbb C}\to \operatorname {\mathbb C}$ such that $f=\exp \circ \phi $ . Since a holomorphic homeomorphism of the complex plane must be a linear map, we thus have

$$ \begin{align*}f(z)=e^{Az+B},\end{align*} $$

where $A(\neq 0)$ and B are complex constants. This completes the proof.

It follows immediately that if $f\in \operatorname {\mathcal {S}}_{2}$ , then the singular values of f are both asymptotic values. Moreover, if one of the asymptotic values is at $\infty $ , then f is of the form $\unicode{x3bb} e^{z}$ ; if both of them are finite, then f can be written as $M(e^{z})$ where M is Möbius, sending $0$ and $\infty $ to two finite points. This, together with the results obtained in [Reference Galazka and KotusGK18, Reference McMullenMcM87], shows that Theorem A holds; see also [Reference CuiCui21a].

Figure 1

Producing meromorphic functions in $\operatorname {\mathcal {S}}_{3}$ by using line complexes. Replacing the dashed line in the left picture by any one of the finite graphs on the right or any finite combination of these graphs will give functions with three singular values which are not topologically equivalent.

The next natural step would be to ask if a similar result stated in Theorem 2.1 will hold if one has more singular values. This need not be true. In fact, even for meromorphic functions with three singular values, one cannot expect a similar function-theoretic rigidity to that of Theorem 2.1. Recent work of Bishop [Reference BishopBis15] shows that there are uncountably many essentially different entire functions with two critical values. More precisely, the classes of topologically equivalent entire functions with two critical values are uncountable. See also [Reference CuiCui21a]. One can also use the so-called Maclane–Vinberg method to construct entire functions with two singular values; see [Reference Bergweiler, Fagella and Rempe-GillenBFRG15, Observation 5.2]. For general meromorphic functions with three singular values, one can resort to the theory of line complexes to construct such functions; see [Reference Goldberg and OstrovskiiGO08, Ch. 7]. Without giving a detailed account of line complexes, we mention that one can show that there exist infinitely many non-equivalent meromorphic functions in $\operatorname {\mathcal {S}}_{3}$ by modifying the left graph in Figure 1: Replacing the dashed line by one of finite graphs shown on the right of Figure 1 will produce meromorphic functions with three singular values. It is clear that there are infinitely many such functions, since one can consider any finite combination of these finite graphs which are then used to replace the dashed line. The meromorphic functions produced are those with rational Schwarzian derivatives. For a complete treatment of these functions, we refer to [Reference Goldberg and OstrovskiiGO08, Ch. 7] and [Reference ElfvingElf34].

3. The construction

For $\delta \in (0,2\pi ]$ , put

$$ \begin{align*}\operatorname{\mathbb C}_{\delta}=\{z=re^{i\theta}:\,r>0,\,0<\theta<\delta \}.\end{align*} $$

In particular, if $\delta =2\pi $ , then $\operatorname {\mathbb C}_{\delta }$ is the slit plane $\operatorname {\mathbb C}\setminus \mathbb {R}^{+}$ . Choosing the natural branch of the logarithm, for any $\alpha \in (0,2\pi ]$ , we set

$$ \begin{align*} h:\,\operatorname{\mathbb C}_{2\pi}&\to\operatorname{\mathbb C}_{\alpha},\\ z&\mapsto z^{\alpha/2\pi} \end{align*} $$

which then defines a conformal map.

To construct meromorphic functions in $\operatorname {\mathcal {S}}_{4}$ , we will consider the restriction of some carefully chosen Weierstraß elliptic function $\wp $ in the sector domain $\operatorname {\mathbb C}_{\alpha }$ . Then the function defined as $\wp \circ h$ will be meromorphic in $\operatorname {\mathbb C}_{2\pi }$ . We will then need to extend this function across the positive real axis in order to have a meromorphic function in the plane. However, the extension need not be continuous; in other words, for $x\in \mathbb {R}^{+}$ , $\wp \circ h (x)$ need not coincide when one approaches x respectively from the upper and lower half planes. This will be resolved by introducing a spiral map and then using a quasiconformal surgery. See Figure 2 for an illustration.

Figure 2

Illustration of the construction. We have put $\eta =\alpha \mu /(2\pi )$ . The map $\Phi $ is quasiconformal, which helps to remove the discontinuity arising on the logarithmic spiral $\Gamma $ .

From now on we fix one $\alpha $ and use the corresponding function h as defined above. Then in the sector $\operatorname {\mathbb C}_{\alpha }$ we consider a Weierstraß elliptic function $\wp $ with two periods $1$ and $\tau $ , where $\tau $ will satisfy the following condition:

(3.1) $$ \begin{align} \arg\tau= \begin{cases} ~\alpha&\text{if } \alpha<\pi,\\ ~\alpha-\pi&\text{if }\alpha>\pi. \end{cases} \end{align} $$

When $\alpha =\pi $ , the function defined as $\wp \circ h$ extends continuously across the positive real axis and thus gives a function meromorphic in the plane (which is actually $\wp (\sqrt {z})$ ). This is an easy case to deal with, so we will assume in what follows that $\alpha \neq \pi $ . The main point of condition (3.1) is to make sure that the two prime periods of the chosen Weierstraß $\wp $ -function lie on two boundaries of $\operatorname {\mathbb C}_{\alpha }$ . Now we put

$$ \begin{align*}e_{1}=\wp\bigg(\frac{1}{2}\bigg),\quad e_{2}=\wp\bigg(\frac{1+\tau}{2}\bigg),\quad e_{3}=\wp\bigg(\frac{\tau}{2}\bigg),\end{align*} $$

which are finite critical values of $\wp $ (with another critical value at $\infty $ ).

One can then see immediately that

$$ \begin{align*}g_{1}:=\wp\circ h:\,\operatorname{\mathbb C}_{2\pi}\to\operatorname{\mathbb C}\end{align*} $$

is well defined and meromorphic in the slit plane. However, $g_{1}$ may not be able to extend continuously across the positive real axis, as we mentioned above. One of the main objectives in the construction we will make is to circumvent this problem.

By choosing the natural branch of the power map, let

$$ \begin{align*} p: \operatorname{\mathbb C}_{2\pi} &\to \operatorname{\mathbb C}, \quad z\mapsto z^{1/\mu}. \end{align*} $$

It can readily be seen that if $\mu $ has real part equal to $1$ then the image of $\operatorname {\mathbb C}\setminus \mathbb {R}^{+} $ is an open set whose complement is a logarithmic spiral. Let us denote this spiral by $\Gamma $ . Now note that if $\{x_{n}\}_{n=0}^{\infty }$ and $\{y_{n}\}_{n=0}^{\infty }$ are two sequences of complex numbers approaching a point $z \in \mathbb {R}^{+}$ from different sides of the real axis, the limits of $p(x_{n})$ and $p(y_{n})$ as $n \rightarrow \infty $ may well be different. So the map p may not have a continuous extension to $\mathbb {R}^{+}$ . For our purposes, we put

$$ \begin{align*} \mu=1-i\,\frac{\log|\tau|}{\alpha}. \end{align*} $$

Denote by q the inverse of p; that is, $q(z)=z^{\mu }$ . Then the function

$$ \begin{align*} g_{2}: \operatorname{\mathbb C}\setminus\Gamma &\to\operatorname{\mathbb C}_{2\pi},\\ z&\mapsto g_{1}(q(z)) \end{align*} $$

is well defined and meromorphic in the complement of a logarithmic spiral $\Gamma $ . It may not extend continuously across $\Gamma $ , but by the choice of $\mu $ and $\tau $ , with a simple computation, we can extend $g_{2}$ continuously to a discrete set of $\Gamma $ whose points will be mapped by $h\circ q$ to poles of $\wp $ . We show below how to remove the discontinuities between this discrete set of points on $\Gamma $ by using a quasiconformal surgery. In short, we will construct a quasiconformal self-map $\Phi $ of $\operatorname {\mathbb C}_{\alpha }$ such that the new defined function $\wp \circ \Phi \circ h\circ q$ extends continuously across $\Gamma $ .

Let P denote the parallelogram formed by four vertices $0,\,1/2,\,(1+\tau )/2$ and $\tau /2$ . It follows from the basic properties of Weierstraß elliptic functions that $\delta :=\wp (\partial P)$ is a simple closed curve on $\operatorname {\widehat {\mathbb C}}$ , which passes through $\infty $ on both sides such that $\operatorname {\widehat {\mathbb C}}\setminus \delta $ consists of two domains A and B. Suppose without loss of generality that $A=\wp (P)$ . Note that all critical values of $\wp $ lie on $\delta $ . Now we can choose an analytic closed curve $\gamma $ in $\operatorname {\widehat {\mathbb C}}$ such that $\gamma \cap \delta =\{e_{2},\,\infty \}$ . To achieve this, we first choose an analytic curve $\gamma _{1}$ lying entirely in A with two endpoints being $e_{2}$ and $\infty $ ; similarly, an analytic curve $\gamma _{2}$ is chosen to lie completely in B with two endpoints $e_{2}$ and $\infty $ . Then $\gamma $ is defined as the union of $\gamma _{1}$ and $\gamma _{2}$ together with their common endpoints. See Figure 3 for an illustration. Now we consider suitable preimage of $\gamma $ under the function $\wp $ . More precisely, we have the following result; compare this with Proposition 3.1 of [Reference Aspenberg and CuiAC21]. Put

$$ \begin{align*}c=\dfrac{1+\tau}{2}.\end{align*} $$

Proposition 3.1. Let $\gamma $ be as above. Then there exist two piecewise analytic curves $\beta _{i}$ with $\wp (\beta _{i})=\gamma $ for all i, such that the following assertions hold.

Figure 3

The curve $\gamma $ is chosen such that it passes through the critical value $e_{2}$ and $\infty $ . Then we choose two suitable preimages $\beta _{1}$ and $\beta _{2}$ of $\gamma $ such that they are periodic and both start from c, which is a preimage of the critical value $e_{2}$ .

• • $\beta _{1}\cap \beta _{2}=\{c\}$ .

• • $\beta _{1}$ starts from the point c and is periodic with period $1$ , that is, $z\in \beta _{1}$ implies that $z+1\in \beta _{1}$ . Moreover, $\beta _{1}$ passes through poles at n of $\wp $ for all $n\geq 1$ .

• • $\beta _{2}$ starts from c and is periodic with period $\tau $ , that is, $z\in \beta _{2}$ implies that $z+\tau \in \beta _{2}$ . Moreover, $\beta _{2}$ passes through poles $n\tau $ for all $n\geq 1$ .

Proof. For convenience, we put $\gamma _{1}=\gamma \cap A$ and $\gamma _{2}=\gamma \cap B$ . We will use $W+c$ as the translation by a complex number c for any set $W\subset \operatorname {\mathbb C}$ ; in other words, $W+c=\{z+c: z\in W\}$ . Recall that P is the parallelogram formed by four points $0,\,1/2,\,(1+\tau )/2$ and $\tau /2$ . Now by elementary properties of Weierstraß elliptic functions we know that $\wp : P+\tfrac 12\to B$ is conformal. Therefore, $\gamma _{2}$ has a preimage denoted by $\beta ^{1}_{1}$ in $P+1/2$ which is an analytic curve connecting $(1+\tau )/2$ and $1$ . The latter property follows easily since $\gamma _{2}\subset B$ connects $e_{2}$ and $\infty $ . Similarly, since $\wp : P+1\to A$ is conformal we see immediately that $\gamma _{1}$ has a preimage $\beta ^{2}_{1}$ in $P+1$ which is an analytic curve connecting $1$ and $(3+\tau )/2$ . Now by periodicity of $\wp $ , the curve $\beta ^{1}_{1}\cup \beta ^{2}_{1}$ and its translations by $n\in \mathbb {N}$ will be mapped conformally onto $\beta $ . Now we define

$$ \begin{align*}\beta_{1}=\bigcup_{n\geq 0}(\beta^{1}_{1}\cup\beta^{2}_{1} + n).\end{align*} $$

It is clear that $\beta _{1}$ satisfies all the required properties.

In the same way we can define $\beta _{2}$ as claimed. We omit the details here.

We denote by V the domain contained in $\operatorname {\mathbb C}_{\alpha }$ and bounded by $\beta _{1}$ and $\beta _{2}$ . In what follows we concentrate on the construction of a quasiconformal mapping

$$ \begin{align*}\Phi: \operatorname{\mathbb C}_{\alpha}\to V\end{align*} $$

such that the function $\wp \circ \Phi \circ h\circ q$ extends continuously across the logarithmic spiral $\Gamma $ and thus gives a function continuous throughout the whole plane. See Figure 4 for the construction of $\Phi $ .

Figure 4

Sketch of the construction of the quasiconformal map $\Phi $ . $S^{\prime }_{1}$ and $S^{\prime }_{2}$ are the restrictions of $S_{1}$ and $S_{2}$ to the sector $\operatorname {\mathbb C}_{\alpha }$ , while $T^{\prime }_{1}$ and $T^{\prime }_{2}$ are the corresponding images of $S^{\prime }_{1}$ and $S^{\prime }_{2}$ under the maps $\phi _{1,1}$ and $\phi _{1,2}$ . The same applies to $\widehat {S}^{\prime }_{1}$ , $\widehat {S}^{\prime }_{2}$ and $\widehat {T}^{\prime }_{1},\,\widehat {T}^{\prime }_{2}$ .

By periodicity of Weierstraß elliptic functions and the construction of $\beta _{1}$ , we can define a periodic curve $\tilde {\beta }_{1}$ which is the extension of $\beta _{1}$ along the direction of the negative real axis. Now choose a real number $a>0$ such that $a>\max _{z\in \tilde {\beta }_{1}}\operatorname {Im}(z)$ . Define

$$ \begin{align*}T_{1}=\{z:\,\operatorname{Im}_{\substack{w\in\tilde{\beta}_{1}}, \,\substack{\operatorname{Re} w=\operatorname{Re} z}}(w)<\operatorname{Im}(z)<a\}.\end{align*} $$

Then by [Reference Aspenberg and CuiAC21, Lemma 3.1], there is a real number $a^{\prime }>0$ such that with

$$ \begin{align*}S_{1}=\{z:0<\operatorname{Im}(z)<a^{\prime}\}\end{align*} $$

there is a conformal map

$$ \begin{align*}\phi_{1,1}: S_{1}\to T_{1}\end{align*} $$

which fixes three boundary points $0$ and $\pm \infty $ and is periodic with period $1$ . In other words,

$$ \begin{align*}\phi_{1,1}(z+1)=\phi_{1,1}(z)+1\end{align*} $$

for any $z\in S_{1}$ . Moreover, $\phi _{1,1}$ extends to the boundary as a piecewise diffeomorphism.

Choose another real number b satisfying $b>\max \{a,\,a^{\prime }\}$ and put

$$ \begin{align*}S_{2}=\{z: a^{\prime}<\operatorname{Im}(z)<b\}\end{align*} $$

and

$$ \begin{align*}T_{2}=\{z: a<\operatorname{Im}(z)<b\}.\end{align*} $$

We now want to define a quasiconformal map from $S_{2}$ to $T_{2}$ which interpolates between the identity map on the upper boundary of $S_{2}$ and the extension of $\phi _{1,1}$ on the lower boundary of $S_{2}$ . For convenience, we denote by $\ell _{1}$ the horizontal curve $\{z: \operatorname {Im}(z)=a\}$ and by $\tilde {\ell }_{1}$ the curve $\{z: \operatorname {Im}(z)=a^{\prime }\}$ . We also put $\ell _{2}=\{z:\operatorname {Im}(z)=b\}$ . Suppose that $\chi _{1}$ is the boundary extension of $\phi _{1,1}$ to $\tilde {\ell }_{1}$ . Then it follows from the construction of $\phi _{1,1}$ , and the Schwarz reflection principle (and the fact that $\tilde {\ell }_{1}$ and $\ell _{1}$ are straight lines), that $\chi _{1}:\tilde {\ell }_{1}\to \ell _{1}$ is analytic. Moreover, $\chi _{1}(z+1)=\chi _{1}(z)+1$ for $z\in \mathbb {R}$ . By considering

$$ \begin{align*}\widetilde{\chi}_{1}(z)=\chi_{1}(z+ia^{\prime})-ia\end{align*} $$

we see that $\widetilde {\chi }_{1}: \mathbb {R}\to \mathbb {R}$ is increasing and analytic. We also define

$$ \begin{align*}\widetilde{\chi}_{2}(z)=z+i(a^{\prime}-a).\end{align*} $$

Note that $\widetilde {\chi }_{1}$ is obtained from $\chi _{1}$ and the identity map by moving the lower boundaries of $S_{2}$ and $T_{2}$ to the real axis. Suppose the new strips are $S^{\prime \prime }_{2}$ and $T^{\prime \prime }_{2}$ , respectively. Then

$$ \begin{align*}L(x+iy)=\bigg(1-\frac{y}{b-a^{\prime}}\bigg)\widetilde{\chi}_{1}(x)+\frac{y}{b-a^{\prime}}\widetilde{\chi}_{2}(x)+i\,\frac{b-a}{b-a^{\prime}}y\end{align*} $$

is the linear interpolation between $\widetilde {\chi }_{1}$ and $\widetilde {\chi }_{2}$ . This is actually a quasiconformal map, as can be seen by checking the Jacobian of L, which is

$$ \begin{align*}\frac{b-a}{b-a^{\prime}}\bigg(\bigg(1-\frac{y}{b-a^{\prime}}\bigg)\widetilde{\chi}^{\prime}_{1}(x)+\frac{y}{b-a^{\prime}}\widetilde{\chi}^{\prime}_{2}(x)\bigg).\end{align*} $$

It is strictly bigger than zero since both $\widetilde {\chi }_{1}$ and $\widetilde {\chi }_{2}$ are increasing. Together with the periodicity, this implies that L is quasiconformal. Now we define

$$ \begin{align*} \phi_{1,2}: S_{2} &\to T_{2},\\ z&\mapsto L(z-ia^{\prime})+ia. \end{align*} $$

This is a quasiconformal map.

Along the $\tau $ -direction, we use a similar idea to that above to obtain a quasiconformal map. More precisely, we first consider a periodic curve $\tilde {\beta }_{2}$ which is the natural extension of $\beta _{2}$ . With

$$ \begin{align*}\ell=\{z: \arg(z)=\arg(\tau)~\text{or}~\arg(\tau)-\pi\},\end{align*} $$

we define, for some real numbers $c^{\prime },\,c$ and d, where $c= (1+\tau )/2$ and $c^{\prime },d$ will be determined later,

$$ \begin{align*} \widetilde{\widehat{\ell}}_{1}&=\ell+c^{\prime}, \nonumber\\ \widehat{\ell}_{1}&=\ell+c, \nonumber \\ \widehat{\ell}_{2}&=\ell+d. \end{align*} $$

We also denote by $\widehat {S}_{1}$ the strip between $\ell $ and $\widetilde{\widehat{\ell }}_{1}$ , and by $\widehat {T}_{1}$ the domain bounded by $\tilde {\beta }_{2}$ and $\widehat {\ell }_{1}$ . Moreover, $\widehat {S}_{2}$ will be the strip bounded by $\widetilde{\widehat{\ell }}_{1}$ and $\widehat {\ell }_{2}$ , while $\widehat {T}_{2}$ is bounded by $\widehat {\ell }_{1}$ and $\widehat {\ell }_{2}$ .

Now $c^{\prime }$ and c are chosen (similarly to $a^{\prime }$ and a above) such that there exists a conformal map

$$ \begin{align*}\phi_{2,1}: \widehat{S}_{1}\to\widehat{T}_{1}\end{align*} $$

which is periodic with period $\tau $ and fixes three boundary points $0$ and $\pm \infty $ (here $\pm \infty $ are understood as infinity along two directions of $\ell $ ). In the same way as the map $\phi _{1,2}$ was defined, we can define a quasiconformal map

$$ \begin{align*}\phi_{2,2}: \widehat{S}_{2}\to\widehat{T}_{2}.\end{align*} $$

We omit the details here. For later purposes, we put

and

Moreover,

We now consider restrictions of the maps constructed above on the domain $\operatorname {\mathbb C}_{\alpha }$ . More precisely, we define

$$ \begin{align*} \begin{aligned} \phi_{1}: S &\longrightarrow T;~\,~ z\longmapsto \begin{cases} ~\phi_{1,1}(z)&\text{if } z\in S_{1},\\ ~\phi_{1,2}(z)&\text{if } z\in S_{2}. \end{cases} \end{aligned} \end{align*} $$

In a similar way, we have

$$ \begin{align*} \begin{aligned} \phi_{2}: \widehat{S} &\longrightarrow \widehat{T};~\,~z\longmapsto \begin{cases} ~\phi_{2,1}(z)&\text{if } z\in \widehat{S}_{1},\\ ~\phi_{2,2}(z)&\text{if } z\in \widehat{S}_{2}. \end{cases} \end{aligned} \end{align*} $$

Finally, we define

$$ \begin{align*} \begin{aligned} \phi_{3}: V_{\alpha} &\longrightarrow V_{\alpha},\\ z&\longmapsto z. \end{aligned} \end{align*} $$

Note that the half-strips S and $\widehat {S}$ overlap on a parallelogram

$$ \begin{align*} Q=S\cap \widehat{S} \end{align*} $$

which contains the origin on the boundary. Both $\phi _{1}$ and $\phi _{2}$ are defined on Q, but they do not necessarily coincide there. Away from Q, in $\operatorname {\mathbb C}_{\alpha }\setminus Q$ we have a well-defined quasiconformal map, which is $\phi _{1}$ in $S\setminus Q$ , $\phi _{2}$ in $\widehat {S}\setminus Q$ and the identity on $V_{\alpha }$ . However, this quasiconformal map still does not satisfy our requirement to remove the aforementioned discontinuity. To proceed, we need to change $\phi _{1}$ or $\phi _{2}$ further. We will change $\phi _{1}$ in what follows.

To this end, we first define suitable inverse branches of $\wp $ . Note that $\wp (\tilde {\beta }_{i})=\gamma $ . Let $\psi _{i}$ denote an inverse branch of $\wp $ from $\gamma $ into $\tilde {\beta }_{i}$ , for $i=1,2$ . Recall that $p: \operatorname {\mathbb C}_{2\pi } \mapsto \operatorname {\mathbb C} \setminus \Gamma $ , where $p(z) = z^{1/\mu }$ . Put $\tilde {p}(z) = p(z^{2\pi /\alpha })$ , which maps $\operatorname {\mathbb C}_{\alpha }$ onto $\operatorname {\mathbb C} \setminus \Gamma $ . We can extend $\tilde {p}$ to its boundary continuously. Moreover, from the definition of $\mu $ , we have that $x\in \mathbb {R}$ and $\tau x \in \ell = \{ z: \arg (z) = \arg (\tau ) \}$ are both mapped by $\tilde {p}$ onto the same point in $\Gamma $ . So, in a sense, the ‘transition function’ $t(x) = \tau x$ , for $x \in \mathbb {R}$ , identifies points on the lines bounding $\operatorname {\mathbb C}_{\alpha }$ which correspond to the same origin in $\Gamma $ . Now the desired ‘correction function’ is defined as

$$ \begin{align*} \begin{aligned} \kappa:\, \mathbb{R} &\longrightarrow\mathbb{R},\\ x&\longrightarrow \phi_{1}^{-1}\circ\psi_{1}\circ\wp\circ\phi_{2}(\tau x). \end{aligned} \end{align*} $$

Since $\tau x \in \ell $ , the above function $\kappa $ is well defined. Roughly speaking, the function $\kappa $ fixes the difference between $\phi _{1}$ and $\phi _{2}$ . With this function, we can now remove the discontinuity by considering a linear interpolation between $\kappa $ on the real axis and the identity on the horizontal line $\ell _{2}$ . More precisely, we define

$$ \begin{align*} \xi_{1}(x+iy)=\bigg(1-\frac{y}{b}\bigg)\kappa(x)+\frac{y}{b}\,x+iy \quad\text{for }0\leq y\leq b. \end{align*} $$

This map is quasiconformal, as one can check that the Jacobian of $\xi $ is non-zero almost everywhere. Moreover, we define

$$ \begin{align*} \xi_{2}(x+iy)=x+iy\quad\text{for }y\geq b. \end{align*} $$

In this way, we have just constructed a quasiconformal map of the upper half-plane by setting

$$ \begin{align*} \begin{aligned} \xi: \mathbb{H}^{+} &\longrightarrow \mathbb{H}^{+},\quad z\longmapsto \begin{cases} ~\xi_{1}(z)&\text{if } 0\leq\operatorname{Im}(z)\leq b,\\ ~\xi_{2}(z)&\text{if } \operatorname{Im}(z)\geq b. \end{cases} \end{aligned} \end{align*} $$

We still need to define a map on the parallelogram Q. Denote by $I_{1}$ the side of Q on the real axis (that is, $I_{1}=[0,d]$ ), and by $I_{2}$ the side on the line $\ell $ (that is, $I_{2}=\{re^{i\alpha }: 0\leq r\leq b/\sin (\alpha )\}$ ). The other two sides of Q are denoted by $I_{3}$ , which is parallel to $I_{1}$ , and $I_{4}$ , which is parallel to $I_{2}$ . We put $\hat {I}_{1}=\xi (I_{1})$ , $\hat {I}_{2}=I_{2}$ , $\hat {I}_{3}=I_{3}$ and $\hat {I}_{4}=\xi (I_{4})$ . Then the $\hat {I}_{i}$ , $i=1,\ldots ,4$ , form a quadrilateral, denoted by $\hat {Q}$ . We continue to define a new quadrilateral $\tilde {Q}$ formed by the $\tilde {I}_{i}$ , $i=1,\ldots ,4$ , where

$$ \begin{align*}\tilde{I}_{i}=\phi_{1}(\hat{I}_{i})\quad\text{for } i=1,4,\end{align*} $$

and

$$ \begin{align*}\tilde{I}_{i}=\phi_{2}(\hat{I}_{i})\quad\text{for }i=2,3.\end{align*} $$

Now we can define a boundary map between Q and $\tilde {Q}$ by using $\phi _{1}\circ \xi $ on $I_{1}\cup I_{4}$ and $\phi _{2}\circ \xi $ on $I_{2}\cup I_{3}$ . The boundary map extends to the interior of Q quasiconformally; see [Reference Branner and FagellaBF14, Lemma 2.24]. So we have a quasiconformal map

$$ \begin{align*}h: Q\to\tilde{Q}.\end{align*} $$

We can now define our promised map

$$ \begin{align*} \Phi :~ \operatorname{\mathbb C}_{\alpha}\,&\longrightarrow\, V,~\,~ z\longmapsto \begin{cases} ~\phi_{1}(\xi(z)) &\text{if }z\in S\setminus Q,\\ ~\phi_{2}(z) &\text{if }z\in \widehat{S}\setminus Q,\\ ~h(z) &\text{if }z\in Q,\\ ~z &\text{if }z\in V_{\alpha}. \end{cases} \end{align*} $$

One can then check that the map

(3.2) $$ \begin{align} G(z):=\wp\circ\Phi\circ h\circ q(z) \end{align} $$

extends continuously across the logarithmic spiral $\Gamma $ and thus gives us, by construction, a quasimeromorphic function of the plane.

So by the measurable Riemann mapping theorem (cf. [Reference AhlforsAhl06]) there exist a quasiconformal homeomorphism $\Psi $ and a meromorphic function f such that $G=f\circ \Psi $ . By our construction, G is quasiconformal only in $(h\circ q)^{-1}(W)$ , where $W=\operatorname {\mathbb C}_{\alpha }\setminus V_{\alpha }$ .

It follows from our construction that f has exactly four critical values and no asymptotic values. In other words, $f\in \operatorname {\mathcal {S}}_{4}$ . Moreover, f has only double poles.

To derive some asymptotic properties of f, we will need the well-known Teichmüller–Wittich–Belinskii theorem concerning conformality of a quasiconformal mapping at a point. We refer to [Reference AhlforsAhl06, Reference Lehto and VirtanenLV73] for background on quasiconformal mappings and also for this result. For our purpose, a stronger result is required. We first recall some relevant notions. Let $\varphi :\operatorname {\mathbb C}\to \operatorname {\mathbb C}$ be a quasiconformal mapping. Then the dilatation of $\varphi $ at a point z is

$$ \begin{align*}K_{\varphi}(z):=\frac{|\varphi_{z}|+|\varphi_{\bar{z}}|}{|\varphi_{z}|-|\varphi_{\bar{z}}|}=\frac{1+|\mu_{\varphi}|}{1-|\mu_{\varphi}|},\end{align*} $$

where $\mu _{\varphi }:=\varphi _{\bar {z}}/\varphi _{z}$ is the complex dilatation of $\varphi $ .

The above-mentioned result is stated as follows; see [Reference Aspenberg and CuiAC21, Lemma 2.2].

Note that W is the union of two half-strips, which means that the set $W\setminus \operatorname {\mathbb D}$ has finite logarithmic area, that is, the integration above holds. So $\Psi $ satisfies the conditions in Proposition 3.2. So we have, up to normalization,

(3.3) $$ \begin{align} \Psi(z)=z+o(z)\quad\text{as } z\to\infty. \end{align} $$

3.1. Distribution of poles of G

Now fix $R>0$ large. We will be interested in counting the number of poles in the closed disk $\overline {D}(0,R)$ . This will follow from the following area formula for a domain bounded by two logarithmic spirals. More precisely, let $\sigma $ be a (non-zero) complex number and $R>0$ . Put $\psi (z)=z^{\sigma }$ for $z\in \operatorname {\mathbb C}_{2\pi }$ with chosen principle branch. Let $\Gamma _{\beta }$ and $\Gamma _{\gamma }$ be the $\psi $ -images of two radial lines of arguments $\beta $ and $\gamma $ respectively, where $0\leq \beta <\gamma \leq 2\pi $ . See Figure 5. Denote by $A:=A_{\gamma -\beta }$ the bounded region bounded by $\Gamma _{\beta }$ , $\Gamma _{\gamma }$ and the circle $\{z:|z|=R\}$ , and by $B:=B_{\gamma -\beta }$ the preimage of A under $\psi $ . Then we have the following formula for the (Euclidean) area of A and B.

Figure 5

The area of the domain A bounded by two logarithmic spirals can be computed by using a logarithmic change of variable.

Proposition 3.3.

$$ \begin{align*} \operatorname{area} A&=\frac{(\gamma-\beta)|\sigma|^{2}}{2\operatorname{Re} \sigma}R^{2};\\[1pt] \operatorname{area} B&=\frac{\operatorname{Re} \sigma}{4\operatorname{Im}\sigma}(e^{2\gamma({\operatorname{Im}\sigma}/{\operatorname{Re}\sigma})} -e^{2\beta({\operatorname{Im}\sigma}/{\operatorname{Re}\sigma})})\cdot R^{{2}/{\operatorname{Re}(\sigma)}}. \end{align*} $$

Proof. Let C be the shaded domain in Figure 5, which is mapped by the exponential map to A. Then

$$ \begin{align*}\operatorname{area} A&=\iint_{C}|(e^{z})^{\prime}|^{2}|\,dz|^{2}\\[3pt] &=\int_{-\infty}^{\log R}e^{2x}\int_{({\beta-x\operatorname{Im}(1/\sigma)})/{\operatorname{Re}(1/\sigma)}}^{({\gamma-x\operatorname{Im}(1/\sigma)})/{\operatorname{Re}(1/\sigma)}}\,dy\,dx=\frac{(\gamma-\beta)|\sigma|^{2}}{2\operatorname{Re} \sigma}R^{2}.\end{align*} $$

Similarly, by using the map $e^{z/\sigma }$ we can obtain the area formula for B. We omit the details here.

We will also use in what follows some standard notions and notation from Nevanlinna theory [Reference Goldberg and OstrovskiiGO08, Reference HaymanHay64, Reference NevanlinnaNev70]. In particular, $n(r,f)$ denotes the number of poles of f in the disk $\overline {D}(0,r)$ , $m(r,f)$ the proximity function and $T(r,f)$ is the Nevanlinna characteristic of f. The order of a meromorphic function f is defined by

$$ \begin{align*}\rho(f)=\limsup_{r\to\infty}\frac{\log T(r,f)}{\log r}.\end{align*} $$

By using Proposition 3.3, we have the following estimate.

Proposition 3.4.

(3.4a) $$\begin{align} n(r,f)=\mathcal{O}(r^{({\alpha^{2}+(\log|\tau|)^{2}})/{\pi\alpha}}) , \end{align} $$

(3.4b) $$\begin{align} \rho(f)=\frac{\alpha^{2}+(\log|\tau|)^{2}}{\pi\alpha}. \end{align} $$

Proof. To estimate the number of poles of f in a certain disk, by (3.3) it suffices to estimate the poles for the map G for a sufficiently large disk. This can be obtained by comparing the area of the disk and the area of a preimage of a parallelogram for the Weierstrass elliptic function under the spiral function $h \circ q(z) = z^{\eta }$ , where $\eta = \mu \alpha /2\pi $ . Then it follows from Proposition 3.3 that

$$ \begin{align*}n(r,G)\sim C\,r^{{\alpha^{2}+(\log|\tau|)^{2}}/{\pi\alpha}},\end{align*} $$

for large r and for some positive constant C. This gives (3.4a ).

To obtain (3.4b ), we need a result of Teichmüller [Reference TeichmüllerTei37] which states that for a meromorphic function $f\in \operatorname {\mathcal {S}}$ , if $\infty $ is not an asymptotic value and the multiplicities of poles are bounded, then $m(r,f)$ is bounded. This means that the order of f can be estimated by using the integrated counting function $N(r,f)$ of f, where $N(r,f)=T(r,f)-m(r,f)$ . Then classical results on the comparison on the growth scale of $N(r,f)$ and $n(r,f)$ show that

$$ \begin{align*}\rho(f)=\limsup_{r\to\infty}\frac{\log n(r,f)}{\log r}.\end{align*} $$

Now (3.4b ) follows from our estimate of $n(r,f)$ .

In the above constructions, we have from the beginning fixed the parameters $\alpha $ and $\tau $ . If we now vary these parameters, we can achieve every finite order. To be more specific, we have the following proposition.

Proposition 3.5. For any given $\rho \in (0,\infty )$ , there exist $\alpha \in (0,2\pi ]$ and $\tau $ satisfying (3.1) such that

(3.5) $$ \begin{align} \rho=\frac{\alpha^{2}+(\log|\tau|)^{2}}{\pi\alpha}. \end{align} $$

Moreover, there is a meromorphic function $f\in \operatorname {\mathcal {S}}_{4}$ such that $\rho (f)=\rho $ .

We now put

(3.6) $$ \begin{align} \eta=\frac{\alpha\mu}{2\pi}=\frac{1}{2\pi}(\alpha-i\log|\tau|). \end{align} $$

Later on we will also need to estimate $|a^{\eta }|$ in terms of the modulus of a. Note that $\eta $ defined above is a complex number. We first prove the following result. For simplicity, we also use $\rho $ as given in (3.5).

Proposition 3.6. For $a\in \operatorname {\mathbb C}$ , we have

$$ \begin{align*}|a^{\eta} |=\mathcal{O}(|a|^{\rho/2}).\end{align*} $$

Proof. Let S be the standard strip $S = \{ z= x + i y : 0 < y < 2\pi \}$ and put $P = (1/\mu ) S = \{ z : \mu z \in S \}$ . First, recall that $\exp $ maps the oblique strip P onto $\operatorname {\mathbb C} \setminus \Gamma $ . It has a continuous extension to the boundary, by the definition of $\mu = 1-i \kappa $ (whose real part is equal to $1$ ). Here $\kappa =(\log |\tau |)/\alpha $ . To make it injective, let us extend $\exp $ only to the lower boundary of the strip P, so that, for instance, the preimage of $z=1$ under exp has its preimage at the origin, that is, the argument of this preimage is $0$ . Then the argument of an arbitrary complex number $a \in \mathbb {C}$ is the imaginary part of the ‘spiral branch’ $\log _{P}(a)$ of a; that is, $\log _{p}$ is this particular inverse of $\exp $ mapping $\operatorname {\mathbb C} \setminus \Gamma $ onto P.

By definition $a^{\mu } = e^{\mu \log a} = e^{(1-i\kappa ) (\ln r + i \theta (r))}$ , where $\theta =\theta (r)$ now depends on $r:=|a|$ in such a way that, if $\tilde {\theta } \in [0,2\pi )$ is the natural argument for a, then $\theta (r) = \tilde {\theta } + 2\pi k $ , where k is the unique integer such that

$$ \begin{align*} \tilde{\theta} + 2\pi k \in [\kappa \log r, \kappa \log r + 2\pi). \end{align*} $$

Hence there is some $\delta \in [0,2\pi )$ such that

$$ \begin{align*} k = \frac{\kappa \log r - \tilde{\theta} + \delta}{2\pi}. \end{align*} $$

Replacing the expression for k in $\theta (r)$ , and with $\kappa = (\log |\tau |)/\alpha $ , we have

$$ \begin{align*} |a^{\eta}|= |a^{\mu \alpha/2\pi}| = r^{\alpha/2\pi} e^{\theta(r) \log|\tau|/2\pi} = r^{({\alpha^{2} + (\log|\tau|)^{2}})/{2\pi \alpha}} e^{{\delta \log |\tau|}/{2 \pi}}.\\[-38pt] \end{align*} $$

The strategy below follows in a similar way to [Reference Aspenberg and CuiAC21, §3]. We first estimate asymptotic behaviours of the constructed function f near its poles, which will be useful in estimating the Hausdorff dimension of escaping sets later on. Then we show that by varying parameters $\alpha $ and $\tau $ we obtain functions in $\operatorname {\mathcal {S}}_{4}$ which are actually equivalent. This is crucial for completing the proof of Theorem 1.2.

3.2. Local behaviours near poles

We first consider the local behaviours of the quasimeromorphic map G near its poles. In the following, $A\sim B$ means that A and B have comparable modulus. Let $z_{0}$ be a pole of G. By (3.2), $\Phi (h(q(z_{0})))$ is a pole of the function $\wp $ . Put $\zeta =\Phi (h(q(z)))$ and $\zeta _{0}=\Phi (h(q(z_{0})))$ . Since $\zeta _{0}$ is a pole of $\wp $ , we see that there exists a constant C such that

(3.7) $$ \begin{align} \wp(\zeta)\sim\bigg(\frac{C}{\zeta-\zeta_{0}}\bigg)^{2}\quad\text{as } \zeta\to\zeta_{0}. \end{align} $$

By the construction of $\Phi $ , we have that

$$ \begin{align*}\zeta-\zeta_{0}\sim C^{\prime}( h(q(z))-h(q(z_{0})))=C^{\prime}( z^{\alpha\mu/(2\pi)}-z_{0}^{\alpha\mu/(2\pi)} )\quad\text{as } z\to z_{0},\end{align*} $$

where $C^{\prime }$ is some constant. This, together with (3.2) and (3.7), shows that

(3.8) $$ \begin{align} G(z)\sim\bigg(\frac{C^{\prime\prime}}{z^{\alpha\mu/(2\pi)}-z_{0}^{\alpha\mu/(2\pi)}}\bigg)^{2}\quad\text{as } z\to z_{0}, \end{align} $$

where $C^{\prime \prime }$ depends only on C and $C^{\prime }$ . Note that $z_{0}$ is a double pole of G. We may thus assume that

(3.9) $$ \begin{align} G(z)\sim\bigg(\frac{a(z)}{z-z_{0}}\bigg)^{2}\quad\text{as } z\to z_{0} \end{align} $$

for some function a which is holomorphic near in some neighbourhood of $z_{0}$ and, moreover, $a(z_{0})\neq 0$ . By comparing (3.8) and (3.9), we see that

$$ \begin{align*}a(z)\sim C^{\prime\prime}\frac{z-z_{0}}{z^{\alpha\mu/(2\pi)}-z_{0}^{\alpha\mu/(2\pi)}}\quad\text{as } z\to z_{0}.\end{align*} $$

Recall that $\eta $ is defined in (3.6). By taking limit and using L’Hospital’s rule, we obtain

$$ \begin{align*}a(z_{0})\sim C^{\prime\prime\prime} z_{0}^{1-\alpha\mu/(2\pi)}=C^{\prime\prime\prime} z_{0}^{1-\eta}.\end{align*} $$

Denote $w=\Phi (z)$ and $w_{0}=\Phi (z_{0})$ . Then $w_{0}$ is a double pole of f. Assume that

$$ \begin{align*}f(w)\sim\bigg( \frac{b(w)}{w-w_{0}} \bigg)^{2}\quad\text{as } w\to w_{0}.\end{align*} $$

Here $b(w)$ is a function holomorphic in $w_{0}$ and $b(w_{0})\neq 0$ . So with (3.9) we see that

$$ \begin{align*}b(w)\sim a(z)\frac{w-w_{0}}{z-z_{0}}\end{align*} $$

as $z\to z_{0}$ . Note that (3.3) implies that $w\to w_{0}$ as $z\to z_{0}$ . Again, (3.3) will also imply that

$$ \begin{align*}b(w_{0})\sim a(z_{0})\lim_{z\to z_{0}}\frac{w-w_{0}}{z-z_{0}}=a(z_{0}).\end{align*} $$

It follows that near a pole $w_{0}$ of f, we have the following asymptotic relation:

(3.10) $$ \begin{align} f(w)=\bigg( \frac{C_{1} w_{0}^{1-\eta}}{w-w_{0}} \bigg)^{2}\quad\text{as } w\to w_{0}, \end{align} $$

where $C_{1}$ is some constant.

3.5. Infinite order

As for infinite-order functions in $\operatorname {\mathcal {S}}_{4}$ , consider the function

$$ \begin{align*}f(z)=\wp(2\pi\cosh(z)),\end{align*} $$

where $\wp $ is a Weierstrass elliptic function with two periods $2\pi $ and $\tau $ such that $\tau $ is not a real multiple of $2\pi $ . It follows by construction that f has four critical values which are exactly the critical values of $\wp $ and has no asymptotic values. That $\rho (f)=\infty $ follows directly from [Reference Edrei and FuchsEF64, Corollary 1.2]. This can also be obtained by checking the counting functions of poles.

4. Estimate of the dimension

We will need to estimate the Hausdorff dimension of the escaping sets for the Speiser functions in $\operatorname {\mathcal {S}}_{4}$ constructed in the previous section, which then finishes the proof of Theorem 1.1. This will follow from the following estimate.

Before we prove this, we state the following direct consequence.

Therefore, Theorem 1.1 is proved except for the case where $d=2$ , which will be discussed in the final part of this section. By combining this with Proposition 3.7, Theorem 1.2 follows.

The rest of the proof will be devoted to the proof of the above Proposition 4.1. We only give a sketch of the proof, as the estimate of the Hausdorff dimension for escaping sets for our constructed functions will use the same idea as in [Reference Aspenberg and CuiAC21]: the upper bound follows from a result of Bergweiler and Kotus [Reference Bergweiler and KotusBK12], while the lower bound uses the spherical version of a well-known result of McMullen [Reference McMullenMcM87].

Proof of Proposition 4.1

For any given $\rho \in [0,\infty )$ , there exist $\alpha $ and $\mu $ satisfying

$$ \begin{align*}\rho=\frac{\alpha^{2}+(\log|\tau|)^{2}}{\pi\alpha}.\end{align*} $$

The constructions in §3 then gives us a Speiser function f in $\operatorname {\mathcal {S}}_{4}$ whose order is $\rho $ . This function f has only critical values and no asymptotic values. Moreover, all poles have the multiplicity $2$ .

Upper bound. It is clear that the above function f satisfies the conditions of [Reference Bergweiler and KotusBK12, Theorem 1.1]. So we have

$$ \begin{align*} \operatorname{dim}\operatorname{\mathcal{I}}(f)\leq \frac{2\rho}{1+\rho}. \end{align*} $$

Lower bound. Suppose that $a_{j}$ are the poles of f, where $\cdots \leq |a_{j}|\leq |a_{j+1}|\leq \cdots $ . Then it follows from (3.10) and Proposition 3.6 that

$$ \begin{align*}f(z)\sim \bigg(\frac{b_{j}}{z-a_{j}}\bigg)^{2}\quad\text{as } z\to z_{j},\end{align*} $$

where

$$ \begin{align*}|b_{j}|\sim |a_{j}^{1-\eta}|\sim |a_{j}|^{1-\rho/2}.\end{align*} $$

Since f is Speiser, we may take a large $R_{0}>0$ such that $D(0,R_{0})$ contains all singular values of f. Now with $B(R)=\operatorname {\widehat {\mathbb C}}\setminus \overline {D}(0,R)$ and $R>R_{0}$ , each component of $f^{-1}(B(R))$ is bounded, simply connected and contains one pole of f. Let $U_{j}$ be the component containing the pole $a_{j}$ . By using Koebe’s distortion and one-quarter theorem, we may obtain

(4.1) $$ \begin{align} D\bigg(a_{j},\frac{|b_{j}|}{4\sqrt{R}} \bigg)\subset U_{j}\subset D\bigg(a_{j},\frac{2|b_{j}|}{\sqrt{R}} \bigg). \end{align} $$

See [Reference Aspenberg and CuiAC21, §4] for more details. Moreover, if $g_{j}$ is an inverse branch of f from some domain $\Omega $ to $U_{j}$ , where $\Omega \subset B(R)$ , then

(4.2) $$ \begin{align} |g^{\prime}_{j}(z)|\leq B_{1}\,\frac{|b_{j}|}{|z|^{3/2}}\quad\text{for } z\in \Omega. \end{align} $$

Here $B_{1}>0$ is some constant.

We denote by $\operatorname {diam} (E)$ the diameter of the set E in the plane and by $\operatorname {diam}_{\chi } (E)$ the spherical diameter of E. The above estimates (4.1) and (4.2) will give us good control over the sizes of the pullbacks of $U_{k}$ under f for large k. More precisely, for sufficiently large k, we have

$$ \begin{align*}\operatorname{diam} g_{j}(U_{k})\leq\sup_{z\in U_{k}}|g^{\prime}_{j}(z)|\operatorname{diam} U_{k} \leq B_{1}\frac{|b_{j}|}{|a_{k}|^{3/2}}\frac{|b_{k}|}{\sqrt{R}},\end{align*} $$

and if the indices $j_{1},\,\ldots ,\,j_{\ell }$ are chosen such that $U_{j_{k}}$ is contained in $B(R)$ , where $k=1,\ldots ,\ell $ , we obtain, in terms of the spherical metric,

(4.3) $$ \begin{align} \operatorname{diam}_{\chi}(g_{j_{1}}\circ g_{j_{2}}\circ\cdots\circ g_{j_{\ell-1}})(U_{j_{\ell}})\leq B_{1}^{\ell-1}\,\frac{32}{\sqrt{R}}\,\prod_{k=1}^{\ell}\,\frac{|b_{j_{k}}|}{|a_{j_{k}}|^{3/2}}. \end{align} $$

We now consider the set

$$ \begin{align*}\operatorname{\mathcal{I}}_{R}(f)=\{z\in\operatorname{\mathcal{I}}(f): f^{n}(z)\in B(R)\text{ for all }n\in\mathbb{N}\}.\end{align*} $$

In other words, we are considering those escaping points whose iterates always stay in $B(R)$ . Apparently, this is a subset of $\operatorname {\mathcal {I}}(f)$ . Let $E_{l}$ be the collection of all components V of $f^{-l}(B(R))$ for which $f^{k}(V)\subset B(R)$ holds for $0\leq k\leq l-1$ . We will estimate the sizes of components of $E_{l}$ by using (4.3). For such a component V, by definition there exist $j_{1},\,\ldots ,\,j_{l-1}$ such that

$$ \begin{align*}f^{k}(V)\subset U_{j_{k+1}}\quad\text{for } k=0,1,\ldots,l-1.\end{align*} $$

So, using (4.3), one can have, for some constants $B_{2}$ and $B_{3}$ ,

$$ \begin{align*} \operatorname{diam}_{\chi}(V)\leq B_{1}^{\ell-1}\,\frac{32}{\sqrt{R}}\,\prod_{k=1}^{\ell}\,\frac{|b_{j_{k}}|}{|a_{j_{k}}|^{3/2}}=B_{1}^{\ell-1}\,\frac{32}{\sqrt{R}}\,\prod_{k=1}^{\ell}\,\frac{|B_{2}|^{\ell}}{|a_{j_{k}}|^{1/2+\rho/2}}\leq \bigg(\frac{B_{3}}{|R|^{1/2+\rho/2}}\bigg)^{\ell}. \end{align*} $$

Put

(4.4) $$ \begin{align} d_{\ell}=\bigg(\frac{|B_{3}|}{|R|^{1/2+\rho/2}}\bigg)^{\ell}. \end{align} $$

In addition to the term $d_{\ell }$ defined above, McMullen’s lower-bound estimate for the Hausdorff dimension also involves a lower-bound estimate for the density of $\overline {E}_{l+1}$ in V. Here $\overline {E}_{\ell }$ represents the union of all elements of $E_{\ell }$ . We will also define . For this purpose, we consider an annulus $A(s):=\{z: s<|z|<2s\}$ which is contained in $B(R)$ , that is, $s> R$ . Then the number of $U_{j}$ contained in $A(s)$ is $B_{4}(n(2s,f)-n(s,f))=B_{5} s^{\rho }$ , where $B_{4}$ and $B_{5}$ are positive numbers. So we have

$$ \begin{align*}\operatorname{diam} U_{j}\geq\frac{|b_{j}|}{2\sqrt{R}}=B_{6}\frac{|a_{j}|^{1-\rho/2}}{2\sqrt{R}}\geq \frac{B_{7}}{s^{\rho/2-1/2}}.\end{align*} $$

where $B_{6}, B_{7}$ are constants. Therefore,

$$ \begin{align*}\operatorname{dens}(\overline{E}_{1}, A(s))=\frac{\operatorname{area}(\overline{E}_{1} \cap A(s))}{\operatorname{area} A(s)}\geq \frac{B_{5} s^{\rho}\,\pi(B_{7}/s^{\rho/2-1/2})^{2}}{3\pi s^{2}}=B_{8}\, s,\end{align*} $$

where $B_{8}>0$ is a constant. By repeating the argument used in [Reference Aspenberg and CuiAC21, §4], which we do not repeat here, we have the following estimate, for some constant $B_{9}>0$ :

(4.5) $$ \begin{align} \operatorname{dens}_{\chi}(\overline{E}_{\ell+1}, V)\geq \frac{B_{9}}{R}=:\Delta_{\ell}.\end{align} $$

Now we can apply McMullen’s result by using (4.4) and (4.5) to obtain

$$ \begin{align*}\operatorname{dim} E\geq 2-\limsup_{\ell\to\infty}\frac{\sum_{j=1}^{\ell+1}|\log \Delta_{j}|}{|\log d_{\ell}|}\geq 2-\frac{\log B_{9} - \log R}{\log B_{3} - (({1}/{2})+({\rho}/{2}))\log R}.\end{align*} $$

With $R\to \infty $ , we have $\operatorname {dim} E\geq {2\rho }/({1+\rho })$ . The next step is to use this estimate to give the estimate for the Hausdorff dimension of the escaping set by taking a sequence $(R_{k})$ which tends to infinity increasingly and consider those points whose k-iterate lies in $B(R_{k})$ . This goes in the same way as in the aforementioned reference and so we omit the details. We conclude directly that

$$ \begin{align*}\operatorname{dim}\operatorname{\mathcal{I}}(f)\geq \frac{2\rho}{1+\rho}.\end{align*} $$

Combined with the upper bound discussed above, we have thus finished the proof.

To complete the proof of Theorem 1.1, we still need to find a function in $\operatorname {\mathcal {S}}_{4}$ with a full-dimensional escaping set. For this purpose, we put

$$ \begin{align*}f_{1}(z)=\wp_{1}(2\pi\cosh(z))\end{align*} $$

This function belongs to $\operatorname {\mathcal {S}}_{4}$ , as mentioned before.

Proposition 4.2.

$$ \begin{align*}\operatorname{dim}\operatorname{\mathcal{I}}(f_{1})=2.\end{align*} $$

The proof of this result is obtained in the same manner as in [Reference Aspenberg and CuiAC21, §4.3]. We leave the details to the interested reader.