A class of Newton maps with Julia sets of Lebesgue measure zero

Abstract

Let $g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$ where p, q are polynomials and $c\in {\mathbb {C}}$, and let f be the function from Newton’s method for g. We show that under suitable assumptions on the zeros of $g''$ the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $f^n(z)$ converges to zeros of g almost everywhere in ${\mathbb {C}}$ if this is the case for each zero of $g''$ that is not a zero of g or $g'$. In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.

1 Introduction and results

For a rational function $f:\hat{{\mathbb {C}}}\rightarrow \hat{{\mathbb {C}}}$ that is not constant and not a Möbius transformation, or a transcendental meromorphic function $f:{\mathbb {C}}\rightarrow \hat{{\mathbb {C}}}$, let $f^n$ denote the nth iterate of f. The Fatou set, ${\mathcal {F}}(f)$, is the set of all z such that all iterates $f^n$ are defined and form a normal family in a neighbourhood of z. Its complement, ${\mathcal {J}}(f)$, is called the Julia set. For an introduction to the iteration theory of meromorphic functions, see, for example, [23] for rational functions and [2] for transcendental functions.

Let g be a non-constant meromorphic function. Newton’s root finding method for g consists of iterating the function

$$\begin{aligned} f(z)=z-\frac{g(z)}{g'(z)}. \end{aligned}$$

(1.1)

We also call f the Newton map corresponding to g. The zeros of g are precisely the attracting fixed points of f, and the simple zeros of g are even superattracting fixed points. Recall that, more generally, a periodic point $z_0$ of period p of a meromorphic function f is called attracting, indifferent or repelling depending on whether $\vert (f^p)'(z_0)\vert <1$, $\vert (f^p)'(z_0)\vert =1$ or $\vert (f^p)'(z_0)\vert >1$. The periodic point $z_0$ is called superattracting if $(f^p)'(z_0)=0$. An indifferent periodic point $z_0$ is called rationally indifferent if $(f^p)'(z_0)$ is a qth root of unity for some $q\in {\mathbb {N}}$, otherwise it is called irrationally indifferent.

We will investigate the Lebesgue measure of Julia sets of Newton maps corresponding to functions of the form

$$\begin{aligned} g(z)=\int _0^z p(t)e^{q(t)}\,dt+c \end{aligned}$$

(1.2)

where p is a polynomial with $p\not \equiv 0$, q is a non-constant polynomial and $c\in {\mathbb {C}}.$

In the following, we will assume that g is not of the form

$$\begin{aligned} g(z)=\tilde{p}(z)e^{\tilde{q}(z)} \end{aligned}$$

(1.3)

with polynomials $\tilde{p}$ and $\tilde{q}$. Then g has infinitely many zeros and f is transcendental. Newton’s method for functions of the form (1.3) has been studied by Haruta [9].

Let ${{\,\mathrm{dist}\,}}(\cdot ,\cdot )$ denote the Euclidean distance in ${\mathbb {C}}$. We will prove the following result.

Theorem 1.1

Let g be of the form (1.2) but not of the form (1.3),  and let f be the corresponding Newton map. Denote the zeros of $g''$ which are not zeros of g or $g'$ by $z_1,\ldots ,z_N.$ Suppose that for all $j\in \{1,\ldots ,N\},$ the point $z_j$ is attracted by a periodic cycle,  that is,  there exists a periodic cycle ${\mathcal {C}}$ of f such that $\lim _{n\rightarrow \infty }{{\,\mathrm{dist}\,}}(f^n(z_j),\,{\mathcal {C}})=0.$ Then the Lebesgue measure of ${\mathcal {J}}(f)$ is zero.

Jankowski [11, §3] proved that if f is the Newton map corresponding to a function g of the form $g(z)=r(z)e^{az}+b$ where r is a rational function and $a,b\in {\mathbb {C}}{\setminus }\{0\}$, and if for each of the zeros, $z_1,\ldots ,z_N$, of $g''$ that are not zeros of g or $g'$, the iterates $f^n(z_j)$ converge to a finite limit as $n\rightarrow \infty$, then the Julia set of f has Lebesgue measure zero. Note that if r is a polynomial, then g can be written in the form (1.2) with $q(t)=at$ and $p(t)=ar(t)+r'(t)$. Also, under the assumptions of Jankowski’s result, $f^n(z_j)$ is attracted by a cycle of period 1 for all $j\in \{1,\ldots ,N\}$. So Jankowski’s theorem for polynomial r is a special case of Theorem 1.1. The essential new difficulties we have to deal with in our proof come from the fact that we allow q to have degree greater than one.

Bergweiler [3, Theorem 3] also investigated Newton’s method for functions of the form (1.2). He proved the following result.

Theorem

(Bergweiler) Let g be of the form (1.2) but not of the form $e^{az+b}$ with $a,b\in {\mathbb {C}},$ and let f be the corresponding Newton map. Denote the zeros of $g''$ which are not zeros of g or $g'$ by $z_1,\ldots ,z_N$. If $f^n(z_j)$ converges to a finite limit for all $j\in \{1,\ldots ,N\},$ then $f^n(z)$ converges to zeros of g on an open dense subset of ${\mathbb {C}}$.

It is not difficult to see that under the assumptions of Bergweiler’s theorem, $f^n(z_j)$ converges to an attracting fixed point of f and hence a zero of g for all $j\in \{1,\ldots ,N\}$. So the theorem says that $f^n(z)$ converges to zeros of g on an open dense subset of ${\mathbb {C}}$, provided this is the case for each zero z of $g''$ that is not a zero of g or $g'$.

A component ${\mathcal {U}}$ of the Fatou set ${\mathcal {F}}(f)$ is called periodic if there is $p\in {\mathbb {N}}$ with $f^p({\mathcal {U}})\subset {\mathcal {U}}$, the component ${\mathcal {U}}$ is called preperiodic if there is $l\in {\mathbb {N}}$ such that $f^l({\mathcal {U}})$ is contained in a periodic Fatou component, and ${\mathcal {U}}$ is called a wandering domain if it is not (pre)periodic. It is known (see, e.g., [2, §4]) that if ${\mathcal {U}}$ is a periodic Fatou component of period p of f, then either $f^{np}\vert _{{\mathcal {U}}}$ converges to an attracting periodic point in ${\mathcal {U}}$ (immediate basin of attraction), $f^{np}\vert _{\mathcal {U}}$ converges to a rationally indifferent periodic point in $\partial {\mathcal {U}}$ (parabolic domain), $f^{np}\vert _{\mathcal {U}}$ converges to some $z_0\in \partial {\mathcal {U}}$ and $f^p(z_0)$ is not defined (Baker domain), or $f^{np}\vert _{\mathcal {U}}$ is conjugate to a rotation of a disk (Siegel disk) or an annulus (Herman ring).

Bergweiler’s theorem is proved by showing that under the given assumptions, f has neither wandering domains nor parabolic domains, Baker domains, Siegel disks or Herman rings.

The following corollary is a direct consequence of Theorem 1.1 and Bergweiler’s theorem.

Corollary 1.2

Let g be of the form (1.2) but not of the form (1.3),  and let f be the corresponding Newton map. Denote the zeros of $g''$ which are not zeros of g or $g'$ by $z_1,\ldots ,z_N$. If $f^n(z_j)$ converges to a finite limit for all $j\in \{1,\ldots ,N\},$ then $f^n(z)$ converges to zeros of g for almost all $z\in {\mathbb {C}}$.

For example, the assumptions of Corollary 1.2 and hence those of Theorem 1.1 are satisfied for $g(z)=\int _0^ze^{-t^2}\,dt+c$ with $-\sqrt{\pi }/2<c<\sqrt{\pi }/2$, see [3, §8]. Clearly, the conclusion of Corollary 1.2 cannot be true if there exists a cycle of period at least two in ${\mathcal {F}}(f)$.

In order to prove Theorem 1.1, we will first prove a general theorem giving conditions ensuring that the Julia set of a meromorphic function has Lebesgue measure zero. This may be of independent interest. For a meromorphic function f, we denote by ${{\,\mathrm{sing}\,}}(f^{-1})$ the set of singular values of f, that is, the set of critical and asymptotic values of f and limit points of those. For $n\ge 0$, let ${\mathcal {N}}_n=\{z:\,f^n(z)\text { is not defined}\}$. Let

$$\begin{aligned} {\mathcal {P}}(f):=\overline{\bigcup _{n=0}^\infty f^n({{\,\mathrm{sing}\,}}(f^{-1}){\setminus }{\mathcal {N}}_n)} \end{aligned}$$

denote the postsingular set of f. For $z_0\in {\mathbb {C}}$ and $r>0$, let ${\mathcal {D}}(z_0,r)$ denote the open disk centred at $z_0$ with radius r. Also, let ${{\,\mathrm{meas}\,}}(\cdot )$ denote Lebesgue measure, and for measurable ${\mathcal {A}},{\mathcal {B}}\subset {\mathbb {C}}$ with $0<{{\,\mathrm{meas}\,}}({\mathcal {B}})<\infty$, let

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {A}},{\mathcal {B}})=\frac{{{\,\mathrm{meas}\,}}({\mathcal {A}}\cap {\mathcal {B}})}{{{\,\mathrm{meas}\,}}({\mathcal {B}})} \end{aligned}$$

denote the density of ${\mathcal {A}}$ in ${\mathcal {B}}$.

Following [15], we call a measurable set ${\mathcal {A}}\subset {\mathbb {C}}$ thin at $\infty$ if there exist $R_0,\varepsilon _0>0$ such that for all $z\in {\mathbb {C}}$, we have

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {A}}, {\mathcal {D}}(z,R_0))<1-\varepsilon _0. \end{aligned}$$

Additionally, we introduce the concept that ${\mathcal {A}}$ is thin at $z_0\in {\mathbb {C}}$ if there exist $\delta _1, \varepsilon _1>0$ such that for all $z\in \overline{{\mathcal {D}}(z_0,\delta _1)}$, we have

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {A}}, {\mathcal {D}}(z, \vert z-z_0\vert ))<1-\varepsilon _1. \end{aligned}$$

(1.4)

We call ${\mathcal {A}}$ uniformly thin at ${\mathcal {B}}\subset {\mathbb {C}}$ if there are $\delta _1,\varepsilon _1>0$ such that (1.4) holds for all $z_0\in {\mathcal {B}}$.

Theorem 1.3

Let f be a meromorphic function that is not constant and not a Möbius transformation. Suppose that there exists $R_1>0$ such that

• (i) ${\mathcal {P}}(f)\cap {\mathcal {J}}(f)\cap \overline{{\mathcal {D}}(0,R_1)}$ is a finite set; 

• (ii) ${\mathcal {J}}(f)$ is thin at $\infty ;$

• (iii) ${\mathcal {J}}(f)$ is uniformly thin at $({\mathcal {P}}(f)\cap {\mathbb {C}}){\setminus }\overline{{\mathcal {D}}(0,R_1)}.$

Then the Lebesgue measure of ${\mathcal {J}}(f)$ is zero.

McMullen [15, Proposition 7.3] proved that if f is entire, ${\mathcal {P}}(f)$ is compact, ${\mathcal {P}}(f)\cap {\mathcal {J}}(f)=\emptyset$ and ${\mathcal {J}}(f)$ is thin at $\infty$, then ${{\,\mathrm{meas}\,}}({\mathcal {J}}(f))=0$. A meromorphic function f for which ${\mathcal {P}}(f)$ is compact and does not intersect ${\mathcal {J}}(f)$ is called hyperbolic. There are various results on iteration of hyperbolic meromorphic functions; see, for example, [4, 19, 20, 22, 25]. Stallard [21] extended McMullen’s result to entire functions f with possibly unbounded postsingular set such that ${{\,\mathrm{dist}\,}}({\mathcal {P}}(f),{\mathcal {J}}(f))>0$ and ${\mathcal {J}}(f)$ is thin at $\infty$. Meromorphic functions f with ${{\,\mathrm{dist}\,}}({\mathcal {P}}(f),{\mathcal {J}}(f))>0$ are sometimes called $\textit{topologically hyperbolic}$, and have also been considered in [1, 14].

Jankowski [11, 12] extended Stallard’s result by allowing that f is meromorphic and that there are certain exceptions to the condition ${{\,\mathrm{dist}\,}}({\mathcal {P}}(f),{\mathcal {J}}(f))>0$. A more general result was later obtained by Zheng [24, Theorem 5] who proved that if f is a meromorphic function such that the set ${\mathcal {P}}(f)\cap {\mathcal {J}}(f)$ is finite, there exists $R>0$ such that ${{\,\mathrm{dist}\,}}(({\mathcal {J}}(f)\cap {\mathbb {C}}){\setminus }{\mathcal {D}}(0,R), {\mathcal {P}}(f))>0$ and ${\mathcal {J}}(f)$ is thin at $\infty$, then ${{\,\mathrm{meas}\,}}({\mathcal {J}}(f))=0$.

The results by McMullen, Stallard, Jankowki and Zheng mentioned above are special cases of Theorem 1.3 since the condition that ${{\,\mathrm{dist}\,}}(({\mathcal {J}}(f)\cap {\mathbb {C}}){\setminus }{\mathcal {D}}(0,R),{\mathcal {P}}(f))>0$ implies that ${\mathcal {J}}(f)$ is uniformly thin at $({\mathcal {P}}(f)\cap {\mathbb {C}}){\setminus }\overline{{\mathcal {D}}(0,R')}$ for $R'>R$. Our theorem is the first of this kind to allow infinitely many postsingular values in the Julia set or an unbounded sequence of postsingular values whose distance to the Julia set tends to zero.

In general, the condition that ${\mathcal {J}}(f)$ is thin at $\infty$ cannot be dropped. If $\vert \alpha \vert$ is small, then the postsingular set of $f(z)=\sin (\alpha z)$ is a compact subset of ${\mathcal {F}}(f)$, and McMullen [15] showed that ${\mathcal {J}}(f)$ has positive measure. However, there are results where instead of assuming that ${\mathcal {J}}(f)$ is thin at $\infty$ other conditions are imposed, see [5, Theorem 8], [24, Theorems 3 and 4].

This article is structured as follows. In Sect. 2, we prove Theorem 1.3. In the remaining part of this paper, we prove Theorem 1.1. First, in Sect. 3, we introduce the change of variables $w=q(z)$. In Sect. 4, we give asymptotic representations of g and f. In Sect. 5, we introduce a class of subsets of ${\mathbb {C}}$ whose preimages under q are connected to the asymptotic behaviour of f. In Sect. 6, we investigate the postsingular set of f. In Sects. 7–10, we investigate the location and size of the set $q({\mathcal {F}}(f))$. Finally, in Sect. 11, we complete the proof of Theorem 1.1.

2 Julia sets of zero measure

In this section, we prove Theorem 1.3. The following lemma is an easy consequence of the well-known Koebe 1/4-theorem and Koebe distortion theorem (see, e.g., [18, Corollary 1.4, Theorem 1.6]).

Lemma 2.1

Let $z_0\in {\mathbb {C}}$ and $r>0,$ and let $f:{\mathcal {D}}(z_0,r)\rightarrow {\mathbb {C}}$ be holomorphic and injective. Then

$$\begin{aligned} f({\mathcal {D}}(z_0,r))\supset {\mathcal {D}}\left( f(z_0),\frac{1}{4}\vert f'(z_0) \vert r\right) . \end{aligned}$$

Moreover,  for $\rho \in (0,1),$

$$\begin{aligned} f({\mathcal {D}}(z_0,\rho r))\subset {\mathcal {D}}\left( f(z_0), \frac{\rho }{(1-\rho )^2}\vert f'(z_0)\vert r\right) \end{aligned}$$

and

$$\begin{aligned} \frac{\min _{z\in {\mathcal {D}}(z_0,\rho r)}\vert f'(z)\vert }{\max _{z\in {\mathcal {D}}(z_0,\rho r)}\vert f'(z)\vert } \ge \left( \frac{1-\rho }{1+\rho }\right) ^4. \end{aligned}$$

For ${\mathcal {A}}\subset {\mathbb {C}}$, denote the forward orbit of ${\mathcal {A}}$ by

$$\begin{aligned} {\mathcal {O}}^+({\mathcal {A}}):=\bigcup _{n=0}^\infty f^{n}({\mathcal {A}}{\setminus }{\mathcal {N}}_n) \end{aligned}$$

where ${\mathcal {N}}_n=\{z:\,f^n(z) \text { is not defined}\}$, and for ${\mathcal {B}}\subset \hat{{\mathbb {C}}}$, let

$$\begin{aligned} {\mathcal {O}}^-({\mathcal {B}}):=\bigcup _{n=1}^\infty f^{-n}({\mathcal {B}}) \end{aligned}$$

be the backward orbit of ${\mathcal {B}}$. For $z\in \hat{{\mathbb {C}}}$, write

$$\begin{aligned} {\mathcal {O}}^+(z):={\mathcal {O}}^+(\{z\}) \quad \text {and}\quad {\mathcal {O}}^-(z):={\mathcal {O}}^-(\{z\}). \end{aligned}$$

We call $z\in \hat{{\mathbb {C}}}$ an exceptional point of the meromorphic function f if ${\mathcal {O}}^-(z)$ is finite. It is not difficult to see that any meromorphic function that is not constant and not a Möbius transformation has at most two exceptional points.

Lemma 2.2

Let f be a meromorphic function that is not constant and not a Möbius transformation. If f is transcendental,  in addition suppose that ${\mathcal {O}}^-(\infty )$ is finite. Let ${\mathcal {K}}$ be a compact subset of ${\mathbb {C}}$ that contains no exceptional point of f, and let ${\mathcal {U}}\subset {\mathbb {C}}$ be open with ${\mathcal {U}}\cap {\mathcal {J}}(f)\ne \emptyset$. Then there is $n_0\in {\mathbb {N}}$ such that ${\mathcal {K}}\subset f^n({\mathcal {U}})$ for all $n\ge n_0$.

This is due to Fatou for rational [7, p.39] and entire functions [8, p.356]. His proof for entire functions also works for transcendental meromorphic functions where ${\mathcal {O}}^-(\infty )$ is finite. We also require the following result.

Lemma 2.3

Let f be a meromorphic function that is not constant and not a Möbius transformation,  and let ${\mathcal {C}}=\{z_0,f(z_0),\ldots ,f^p(z_0)=z_0\}$ be a periodic cycle of f. Suppose that $z\in {\mathcal {J}}(f)$ is attracted by ${\mathcal {C}}$. Then there exists $n\in {\mathbb {N}}$ such that $f^n(z)\in {\mathcal {C}}$.

Clearly, the hypotheses imply that ${\mathcal {C}}\subset {\mathcal {J}}(f)$. It is not difficult to see that the conclusion of Lemma 2.3 is true for repelling cycles. For rationally indifferent cycles, the result follows from the Leau flower theorem [16, §10], and for irrationally indifferent cycles, it was shown by Pérez Marco [17].

In Lemma 2.4, we give conditions ensuring that a point $z\in {\mathcal {J}}(f)$ is not a point of density of ${\mathcal {J}}(f)$. We will then use Lemma 2.4 and the Lebesgue density theorem to prove Theorem 1.3.

Lemma 2.4

Let f be a meromorphic function that is not constant and not a Möbius transformation,  and let $z\in {\mathcal {J}}(f){\setminus }{\mathcal {O}}^{-}({\mathcal {P}}(f)\cup \{\infty \})$. Suppose that there exist sequences $(n_k)$ of positive integers with $\lim _{k\rightarrow \infty }n_k=\infty$ and $(r_k)$ of positive real numbers satisfying the following conditions : 

• (i) ${\mathcal {D}}(f^{n_k}(z),r_k)\cap {\mathcal {P}}(f)=\emptyset$ for all $k\in {\mathbb {N}};$

• (ii) there is $\varepsilon >0$ such that ${{\,\mathrm{dens}\,}}({\mathcal {F}}(f), {\mathcal {D}}(f^{n_k}(z),r_k))\ge \varepsilon$ for all $k\in {\mathbb {N}}.$

Then z is not a point of density of ${\mathcal {J}}(f)$.

Proof

Let

$$\begin{aligned} \omega :=\sqrt{1-\frac{\varepsilon }{2}}. \end{aligned}$$

For $k\in {\mathbb {N}}$, let

$$\begin{aligned} z_k:=f^{n_k}(z),\quad {\mathcal {D}}_k:={\mathcal {D}}(z_k, r_k) \quad \text {and}\quad {\mathcal {D}}_k':={\mathcal {D}}\left( z_k,\omega r_k\right) . \end{aligned}$$

Since ${\mathcal {D}}_k\cap {\mathcal {P}}(f)=\emptyset$, there is a branch $\varphi _k$ of $f^{-n_k}$ defined in ${\mathcal {D}}_k$ with $\varphi _k(z_k)=z$. By Koebe’s theorems (see Lemma 2.1),

$$\begin{aligned} {\mathcal {D}}\left( z,\frac{\omega }{4}r_k\vert \varphi _k'(z_k)\vert \right) \subset \varphi _k({\mathcal {D}}_k')\subset {\mathcal {D}}\left( z,\frac{\omega }{(1-\omega )^2} r_k\vert \varphi _k'(z_k)\vert \right) . \end{aligned}$$

(2.1)

We claim that

$$\begin{aligned} \lim _{k\rightarrow \infty }\left| \varphi _k'(z_k)\right| r_k=0. \end{aligned}$$

(2.2)

If this was not true, there would be $\delta >0$ such that ${\mathcal {D}}(z,\delta )\subset \varphi _k({\mathcal {D}}_k')$ for infinitely many k, and hence $f^{n_k}({\mathcal {D}}(z,\delta ))\subset {\mathcal {D}}_k'$ for infinitely many k. If f is transcendental and ${\mathcal {O}}^-(\infty )$ is infinite, this is impossible because ${\mathcal {O}}^-(\infty )$ is dense in ${\mathcal {J}}(f)$. Suppose that f is rational or ${\mathcal {O}}^-(\infty )$ is finite. Fix $v\in {\mathcal {P}}(f)\cap {\mathbb {C}}$ and let ${\mathcal {K}}$ be of the form ${\mathcal {K}}=\{z:\,\vert z-v\vert =\rho \}$ where $\rho$ is chosen such that ${\mathcal {K}}$ does not contain any exceptional point of f. Then by Lemma 2.2, ${\mathcal {K}}\subset f^{n_k}({\mathcal {D}}(z,\delta ))\subset {\mathcal {D}}_k'\subset {\mathcal {D}}_k$ for all large k. But this implies $v\in {\mathcal {D}}_k$, contradicting (i). This proves (2.2).

We will now show that

$$\begin{aligned} \limsup _{r\rightarrow 0}\,{{\,\mathrm{dens}\,}}({\mathcal {F}}(f), {\mathcal {D}}(z,r))>0, \end{aligned}$$

that is, z is not a point of density of ${\mathcal {J}}(f)$. We have

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f), \varphi _k({\mathcal {D}}_k'))&\ge \left( \frac{\min _{\zeta \in {\mathcal {D}}_k'} \vert \varphi _k'(\zeta )\vert }{\max _{\zeta \in {\mathcal {D}}_k'}\vert \varphi _k' (\zeta )\vert }\right) ^2{{\,\mathrm{dens}\,}}({\mathcal {F}}(f), {\mathcal {D}}_k')\\&=\left( \frac{\min _{\zeta \in {\mathcal {D}}_k'}\vert \varphi _k'(\zeta )\vert }{\max _{\zeta \in {\mathcal {D}}_k'}\vert \varphi _k'(\zeta )\vert }\right) ^2 \frac{{{\,\mathrm{meas}\,}}({\mathcal {D}}_k'\cap {\mathcal {F}}(f))}{{{\,\mathrm{meas}\,}}{\mathcal {D}}_k'}\\&\ge \left( \frac{\min _{\zeta \in {\mathcal {D}}_k'}\vert \varphi _k'(\zeta )\vert }{\max _{\zeta \in {\mathcal {D}}_k'}\vert \varphi _k'(\zeta )\vert }\right) ^2\cdot \frac{{{\,\mathrm{meas}\,}}({\mathcal {D}}_k\cap {\mathcal {F}}(f)) -{{\,\mathrm{meas}\,}}({\mathcal {D}}_k{\setminus } {\mathcal {D}}_k')}{{{\,\mathrm{meas}\,}}{\mathcal {D}}_k}. \end{aligned}$$

Hence, by the Koebe distortion theorem and (ii),

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f),\varphi _k({\mathcal {D}}_k'))\ge \left( \frac{1-\omega }{1+\omega }\right) ^8 \cdot \left( \varepsilon -\frac{\pi r_k^2-\pi r_k^2\omega ^2}{\pi r_k^2}\right) =\left( \frac{1-\omega }{1+\omega }\right) ^8\cdot \frac{\varepsilon }{2}. \end{aligned}$$

(2.3)

By (2.3) and (2.1),

$$\begin{aligned}&{{\,\mathrm{dens}\,}}\left( {\mathcal {F}}(f), {\mathcal {D}}\left( z,\frac{\omega }{(1-\omega )^2} \vert \varphi _k'(z_k)\vert r_k\right) \right) \\&\quad \ge {{\,\mathrm{dens}\,}}({\mathcal {F}}(f), \varphi _k({\mathcal {D}}_k'))\cdot {{\,\mathrm{dens}\,}}\left( \varphi _k({\mathcal {D}}_k'), {\mathcal {D}}\left( z,\frac{\omega }{(1-\omega )^2} \vert \varphi _k'(z_k)\vert r_k\right) \right) \\&\quad \ge \left( \frac{1-\omega }{1+\omega }\right) ^8 \frac{\varepsilon }{2}\cdot \frac{1}{16}\left( 1-\omega \right) ^4. \end{aligned}$$

$\square$

Proof of Theorem 1.3

We first show that ${{\,\mathrm{meas}\,}}({\mathcal {P}}(f)\cap {\mathcal {J}}(f))$ is zero. In order to do so, write ${\mathcal {P}}(f)\cap {\mathcal {J}}(f)\cap {\mathbb {C}}={\mathcal {P}}_1\cup {\mathcal {P}}_2$ with ${\mathcal {P}}_1:={\mathcal {P}}(f)\cap {\mathcal {J}}(f)\cap \overline{{\mathcal {D}}(0,R_1)}$ and ${\mathcal {P}}_2:=({\mathcal {P}}(f)\cap {\mathcal {J}}(f)\cap {\mathbb {C}}){\setminus }\overline{{\mathcal {D}}(0,R_1)}$. Since ${\mathcal {P}}_1$ is a finite set, we only have to show that ${{\,\mathrm{meas}\,}}({\mathcal {P}}_2)=0$.

Since ${\mathcal {J}}(f)$ is uniformly thin at $({\mathcal {P}}(f)\cap {\mathbb {C}}){\setminus }\overline{{\mathcal {D}}(0,R_1)}$, there are $\delta _1,\varepsilon _1>0$ such that for all $v\in {\mathcal {P}}(f)\cap {\mathbb {C}}$ with $\vert v\vert > R_1$ and all $\zeta \in \overline{{\mathcal {D}}(v,\delta _1)}$, we have

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f),{\mathcal {D}}(\zeta ,\vert \zeta -v\vert ))>\varepsilon _1. \end{aligned}$$

(2.4)

Let $z\in {\mathcal {P}}_2$ and $r\in (0,2\delta _1)$. Then ${\mathcal {D}}(z+r/2,r/2)\subset {\mathcal {D}}(z,r)$ and ${{\,\mathrm{dens}\,}}({\mathcal {F}}(f),{\mathcal {D}}(z+r/2,r/2))>\varepsilon _1$. Thus,

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f),{\mathcal {D}}(z,r))&\ge {{\,\mathrm{dens}\,}}\left( {\mathcal {F}}(f),{\mathcal {D}}\left( z+\frac{r}{2},\frac{r}{2}\right) \right) \cdot {{\,\mathrm{dens}\,}}\left( {\mathcal {D}}\left( z+\frac{r}{2}, \frac{r}{2}\right) ,{\mathcal {D}}(z,r)\right) \\&>\frac{\varepsilon _1}{4}. \end{aligned}$$

Hence, z is not a point of density of ${\mathcal {J}}(f)$. By the Lebesgue density theorem (see, e.g., [13, Corollary 2.14]), the Lebesgue measure of ${\mathcal {P}}_2$ is zero. So ${{\,\mathrm{meas}\,}}({\mathcal {P}}(f)\cap {\mathcal {J}}(f)\cap {\mathbb {C}})=0$ and hence also ${{\,\mathrm{meas}\,}}({\mathcal {O}}^-({\mathcal {P}}(f)\cap {\mathcal {J}}(f)\cap {\mathbb {C}}))=0$. Since ${\mathcal {O}}^-(\infty )$ is countable, we obtain that ${{\,\mathrm{meas}\,}}({\mathcal {O}}^-(({\mathcal {P}}(f)\cap {\mathcal {J}}(f)) \cup \{\infty \}))=0$.

Next, we show that each $z\in {\mathcal {J}}(f){\setminus }{\mathcal {O}}^- ({\mathcal {P}}(f)\cup \{\infty \})$ satisfies

$$\begin{aligned} \limsup _{n\rightarrow \infty }{{\,\mathrm{dist}\,}}(f^n(z),{\mathcal {P}}_1)>0. \end{aligned}$$

(2.5)

In order to do so, suppose that $\lim _{n\rightarrow \infty }{{\,\mathrm{dist}\,}}(f^n(z),{\mathcal {P}}_1)=0$. We show that then $z\in {\mathcal {O}}^-({\mathcal {P}}_1)$, contradicting our assumption.

Because ${\mathcal {P}}_1$ is finite, there is a subsequence, $(f^{n_k}(z))$, that converges to some $w\in {\mathcal {P}}_1$. For all $j\in {\mathbb {N}}$, we have $f^j(w)=\lim _{k\rightarrow \infty }f^{n_k+j}(z)\in {\mathcal {P}}_1$. Thus, w is preperiodic, that is, $f^l(w)$ is periodic for some $l\in {\mathbb {N}}$. Assume without loss of generality that $l=0$, that is, there is $p\in {\mathbb {N}}$ with $f^p(w)=w.$

Let $\alpha >0$ such that the disks $\overline{{\mathcal {D}}(\zeta ,\alpha )}$ with $\zeta \in {\mathcal {P}}_1$ are pairwise disjoint, and let $\beta \in (0,\alpha )$ such that $f({\mathcal {D}}(f^j(w),\beta ))\subset {\mathcal {D}}(f^{j+1}(w),\alpha )$ for all $j\in \{0,\ldots ,p-1\}.$ Then by periodicity, this is true for all $j\ge 0$. For large k, we have ${{\,\mathrm{dist}\,}}(f^{n_k+j}(z),{\mathcal {P}}_1)<\beta$ for all $j\ge 0$ and $f^{n_k}(z)\in {\mathcal {D}}(w,\beta )$. Then $f^{n_k+1}(z)\in {\mathcal {D}}(f(w),\alpha )$. Since the disks $\overline{{\mathcal {D}}(\zeta , \alpha )}$ with $\zeta \in {\mathcal {P}}_1$ are disjoint, we have $\vert f^{n_k+1}(z)-\zeta \vert>\alpha >\beta$ for all $\zeta \in {\mathcal {P}}_1{\setminus }\{f(w)\}$. Thus, $\vert f^{n_k+1}(z)-f(w)\vert <\beta$. Inductively, we obtain that $f^{n_k+j}(z)\in {\mathcal {D}}(f^j(w), \beta )$ for all $j\in {\mathbb {N}}$. Thus, $f^n(z)$ is attracted by the cycle $\{w,f(w),\ldots ,f^{p-1}(w),f^p(w)=w\}$. By Lemma 2.3, z is eventually mapped to this cycle, so $z\in {\mathcal {O}}^-({\mathcal {P}}_1)$.

Now let $z\in {\mathcal {J}}(f){\setminus }{\mathcal {O}}^-({\mathcal {P}}(f)\cup \{\infty \})$. By (2.5), there exist a subsequence $(f^{n_k}(z))$ and $\eta >0$ such that

$$\begin{aligned} {{\,\mathrm{dist}\,}}(f^{n_k}(z), {\mathcal {P}}_1)>\eta \end{aligned}$$

(2.6)

for all $k\in {\mathbb {N}}$. We will show that z satisfies the assumptions of Lemma 2.4 and hence is not a point of density of ${\mathcal {J}}(f)$. Let

$$\begin{aligned} d_k:={{\,\mathrm{dist}\,}}(f^{n_k}(z), {\mathcal {P}}(f)), \end{aligned}$$

and let $a_k\in {\mathcal {P}}(f)$ with

$$\begin{aligned} \vert f^{n_k}(z)-a_k\vert =d_k. \end{aligned}$$

First suppose that the sequence $(d_k)$ is bounded, say $d_k\le \gamma$ for all k. We distinguish three cases.

1st case: $\vert a_k\vert \le R_1$ for infinitely many k. By passing to a subsequence if necessary, we can assume that $\vert a_k\vert \le R_1$ for all k. Then $(f^{n_k}(z))$ is bounded, and by again passing to a subsequence, we can assume that $f^{n_k}(z)$ converges to some $w\in {\mathcal {J}}(f)$. By (2.6), we have $w\notin {\mathcal {P}}_1$. If $\vert w\vert >R_1$, then for large k, we have

$$\begin{aligned} d_k=\vert f^{n_k}(z)-a_k\vert \ge \vert f^{n_k}(z)\vert -\vert a_k\vert \ge \vert f^{n_k}(z)\vert -R_1>\vert f^{n_k}(z)-w\vert . \end{aligned}$$

Thus, $w\notin {\mathcal {P}}(f)$, so $\nu :={{\,\mathrm{dist}\,}}(w, {\mathcal {P}}(f))>0$. For large k,

$$\begin{aligned} {\mathcal {D}}\left( w,\frac{\nu }{4}\right) \subset {\mathcal {D}}\left( f^{n_k}(z), \frac{\nu }{2}\right) \subset {\mathcal {D}}(w,\nu ). \end{aligned}$$

Thus, ${\mathcal {D}}(f^{n_k}(z),\nu /2)\cap {\mathcal {P}}(f)=\emptyset$, and

$$\begin{aligned}&{{\,\mathrm{dens}\,}}\left( {\mathcal {F}}(f), {\mathcal {D}}\left( f^{n_k}(z), \frac{\nu }{2}\right) \right) \\&\quad \ge {{\,\mathrm{dens}\,}}\left( {\mathcal {D}}\left( w,\frac{\nu }{4}\right) ,{\mathcal {D}}\left( f^{n_k}(z), \frac{\nu }{2}\right) \right) \cdot {{\,\mathrm{dens}\,}}\left( {\mathcal {F}}(f), {\mathcal {D}}\left( w,\frac{\nu }{4}\right) \right) \\&\quad =\frac{1}{4}{{\,\mathrm{dens}\,}}\left( {\mathcal {F}}(f), {\mathcal {D}}\left( w, \frac{\nu }{4}\right) \right) >0. \end{aligned}$$

By Lemma 2.4, z is not a point of density of ${\mathcal {J}}(f)$.

2nd case: $\vert a_k\vert > R_1$ and $d_k\le \delta _1$ for infinitely many k, without loss of generality for all k. Then by (2.4),

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f), {\mathcal {D}}(f^{n_k}(z), d_k))>\varepsilon _1. \end{aligned}$$

By Lemma 2.4, z is not a point of density of ${\mathcal {J}}(f)$.

3rd case: $\vert a_k\vert > R_1$ and $d_k>\delta _1$ for infinitely many k, without loss of generality for all k. Let

$$\begin{aligned} w_k:=a_k+\frac{\delta _1}{d_k}(f^{n_k}(z)-a_k). \end{aligned}$$

Then

$$\begin{aligned} \vert f^{n_k}(z)-w_k\vert =\left( 1-\frac{\delta _1}{d_k}\right) \vert f^{n_k}(z)-a_k\vert =d_k-\delta _1 \end{aligned}$$

and hence

$$\begin{aligned} {\mathcal {D}}(w_k,\delta _1)\subset {\mathcal {D}}(f^{n_k}(z), d_k). \end{aligned}$$

Also, $\vert w_k-a_k\vert =\delta _1$. By the hypotheses,

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f), {\mathcal {D}}(f^{n_k}(z),d_k))&\ge {{\,\mathrm{dens}\,}}({\mathcal {D}}(w_k,\delta _1), {\mathcal {D}}(f^{n_k}(z),d_k))\cdot {{\,\mathrm{dens}\,}}({\mathcal {F}}(f), {\mathcal {D}}(w_k,\delta _1))\\&\ge \frac{\delta _1^2}{d_k^2}\varepsilon _1\ge \frac{\delta _1^2}{\gamma ^2} \varepsilon _1. \end{aligned}$$

By Lemma 2.4, z is not a point of density of ${\mathcal {J}}(f)$.

Now suppose that the sequence $(d_k)$ is unbounded. Since ${\mathcal {J}}(f)$ is thin at $\infty$, there are $R_0,\varepsilon _0>0$ such that for all $v\in {\mathbb {C}}$, we have

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f),{\mathcal {D}}(v,R_0))>\varepsilon _0. \end{aligned}$$

By passing to a subsequence if necessary, we can assume that $d_k\ge R_0$ for all k. Then ${\mathcal {D}}(f^{n_k}(z),R_0)\cap {\mathcal {P}}(f)=\emptyset$ for all k. Also,

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f), {\mathcal {D}}(f^{n_k}(z),R_0))>\varepsilon _0. \end{aligned}$$

By Lemma 2.4, z is not a point of density of ${\mathcal {J}}(f)$.

Altogether, it follows that the set of density points of ${\mathcal {J}}(f)$ has Lebesgue measure zero. By the Lebesgue density theorem, the Lebesgue measure of ${\mathcal {J}}(f)$ is zero. $\square$

3 A change of variables

Throughout the remaining part of the paper, let g be defined by (1.2), that is,

$$\begin{aligned} g(z)=\int _0^zp(t)e^{q(t)}\,dt+c. \end{aligned}$$

Remark 3.1

Suppose that $q(t)=at^d+O(t^{d-1})$ as $t\rightarrow \infty$, where $a\in {\mathbb {C}}{\setminus }\{0\}$ and $d\ge 1$. Let $\alpha \in {\mathbb {C}}$ with $\alpha ^d=a$. Then $q(t/\alpha )=t^d+O(t^{d-1})$ as $t\rightarrow \infty$,

$$\begin{aligned} g(z/\alpha )=\int _0^{z/\alpha }p(t)e^{q(t)}\,dt+c=\int _0^z \frac{1}{\alpha }p\left( \frac{t}{\alpha }\right) e^{q\left( t/\alpha \right) }\,dt+c, \end{aligned}$$

and Newton’s method for $g(z/\alpha )$ is conjugate to Newton’s method for g via $z\mapsto \alpha z$. Thus, we can and will assume without loss of generality that $a=1$, that is,

$$\begin{aligned} q(t)=t^d+O(t^{d-1}) \end{aligned}$$

as $t\rightarrow \infty .$

Also, since the functions g and $b\cdot g$ for $b\in {\mathbb {C}}{\setminus }\{0\}$ have the same zeros and Newton’s methods for g and $b\cdot g$ coincide, we can and will assume without loss of generality that p has the form

$$\begin{aligned} p(t)=dt^m+O(t^{m-1}) \end{aligned}$$

as $t\rightarrow \infty$, where $d=\deg (q)$.

Let f be defined by (1.1), that is, f is the Newton map corresponding to g. In order to prove Theorem 1.1, it will be useful to consider the change of variables $w=q(z)$. Let $R>0$ such that all critical values of q are contained in ${\mathcal {D}}(0,R)$ and such that for $\vert z\vert \ge (1/2)R^{1/d}$, we have

$$\begin{aligned} \frac{1}{2^d}\vert z\vert ^d\le \vert q(z)\vert \le 2^d\vert z\vert ^d. \end{aligned}$$

(3.1)

Define

$$\begin{aligned} {\mathcal {G}}:={\mathbb {C}}{\setminus }(\overline{{\mathcal {D}}(0,R)}\cup [0,\infty )). \end{aligned}$$

Lemma 3.2

There exists $c>0$ such that the set $q^{-1}({\mathcal {G}})$ consists of d components,  ${\mathcal {S}}_1,\ldots , {\mathcal {S}}_d,$ satisfying

$$\begin{aligned} {\mathcal {S}}_j\subset \left\{ z:\,\vert z\vert >\frac{1}{2}R^{1/d}, \,\frac{2(j-1)\pi }{d}-\frac{c}{\vert z\vert }<\arg (z) <\frac{2j\pi }{d}+\frac{c}{\vert z\vert }\right\} \end{aligned}$$

and

$$\begin{aligned} {\mathcal {S}}_j\supset \left\{ z:\,\vert z\vert >2R^{1/d}, \,\frac{2(j-1)\pi }{d}+\frac{c}{\vert z\vert }<\arg (z)<\frac{2j\pi }{d}-\frac{c}{\vert z\vert }\right\} \end{aligned}$$

for all $j\in \{1,\ldots ,d\}$. Moreover,  q maps each ${\mathcal {S}}_j$ conformally onto ${\mathcal {G}}$.

Proof

Since ${\mathcal {G}}$ is simply connected and contains no critical values of q, its preimage $q^{-1}({\mathcal {G}})$ consists of d components, and q maps each of them conformally onto ${\mathcal {G}}$. By (3.1),

$$\begin{aligned} q\left( {\mathcal {D}}\left( 0,\frac{1}{2}R^{1/d}\right) \right) \subset {\mathcal {D}}(0,R) \end{aligned}$$

and

$$\begin{aligned} q\left( {\mathbb {C}}{\setminus } {\mathcal {D}}\left( 0,2R^{1/d}\right) \right) \subset {\mathbb {C}}{\setminus } {\mathcal {D}}(0,R). \end{aligned}$$

Also, for $z\in {\mathbb {C}}$, we have

$$\begin{aligned} \arg (q(z))&=\arg \left[ z^d\left( 1+{\mathcal {O}}\left( \frac{1}{z}\right) \right) \right] \equiv d\arg (z)+\arg \left( 1+{\mathcal {O}}\left( \frac{1}{z}\right) \right) \\&\equiv d\arg (z)+{\mathcal {O}}\left( \frac{1}{z}\right) \mod 2\pi \end{aligned}$$

as $z\rightarrow \infty$. Thus,

$$\begin{aligned} \arg (z)\equiv \frac{\arg (q(z))}{d}+{\mathcal {O}}\left( \frac{1}{z}\right) \mod \frac{2\pi }{d} \end{aligned}$$

as $z\rightarrow \infty$. Using that q is surjective, we obtain the desired conclusion. $\square$

For $j\in \{1,\ldots ,d\}$, let $\varphi _j$ be the branch of $q^{-1}$ defined in ${\mathcal {G}}$ with $\varphi _j({\mathcal {G}})={\mathcal {S}}_j$.

4 The asymptotics of g and f

In this section, we give asymptotic representations for $g(\varphi _j(w))$, g(z), $f(\varphi _j(w))$, f(z). Let

$$\begin{aligned} \lambda :=\frac{d-1-m}{d}. \end{aligned}$$

Then

$$\begin{aligned} \frac{p(z)}{q'(z)}=z^{-\lambda d} \left( 1+O\left( \frac{1}{z}\right) \right) \end{aligned}$$

(4.1)

as $z\rightarrow \infty$ and, for $j\in \{1,\ldots ,d\}$,

$$\begin{aligned} \left| \frac{p(\varphi _j(w))}{q'(\varphi _j(w))}\right| =\vert w\vert ^{-\lambda }\left( 1+O\left( \frac{1}{\vert w \vert ^{1/d}}\right) \right) \end{aligned}$$

(4.2)

as $w\rightarrow \infty$ in ${\mathcal {G}}$.

Lemma 4.1

Let $j\in \{1,\ldots ,d\}$. Then there exists $c_j\in {\mathbb {C}}$ such that

$$\begin{aligned} g(\varphi _j(w))=c_j+\frac{p(\varphi _j(w))}{q'(\varphi _j(w))} \left( 1+\frac{\lambda }{w}+O\left( \frac{1}{\vert w \vert ^{1+1/d}}\right) \right) e^w \end{aligned}$$

as $w\rightarrow \infty$ in ${\mathcal {G}}$.

In terms of $z=\varphi _j(w)$, Lemma 4.1 says the following.

Corollary 4.2

For $j\in \{1,\ldots ,d\},$ we have

$$\begin{aligned} g(z)=c_j+\frac{p(z)}{q'(z)}\left( 1+\frac{\lambda }{z^d} +O\left( \frac{1}{z^{d+1}}\right) \right) e^{q(z)} \end{aligned}$$

as $z\rightarrow \infty$ in ${\mathcal {S}}_j$.

Proof of Lemma 4.1

Let $x_0\in (-\infty ,-R)={\mathcal {G}}\cap (-\infty ,0]$ and $w\in {\mathcal {G}}$. Then

$$\begin{aligned} g(\varphi _j(w))&=\int _0^{\varphi _j(w)}p(t)e^{q(t)}\,dt+c\\&=\int _{\varphi _j(x_0)}^{\varphi _j(w)}p(t)e^{q(t)}\,dt +\int _0^{\varphi _j(x_0)}p(t)e^{q(t)}\,dt+c\\&=\int _{\varphi _j(x_0)}^{\varphi _j(w)}p(t)e^{q(t)}\,dt+g(\varphi _j(x_0))\\&=\int _{x_0}^w\varphi _j'(s)p(\varphi _j(s))e^s\,ds+g(\varphi _j(x_0)). \end{aligned}$$

Let

$$\begin{aligned} r(s):=\varphi _j'(s)p(\varphi _j(s))=\frac{p(\varphi _j(s))}{q'(\varphi _j(s))}. \end{aligned}$$

Repeated integration by parts yields

$$\begin{aligned} \int _{x_0}^wr(s)e^s\,ds=\left( r(s)-r'(s)+r''(s)\right) e^s\Big \vert _{x_0}^w-\int _{x_0}^wr'''(s)e^s\,ds. \end{aligned}$$

We have

$$\begin{aligned} r'(s)&=\varphi _j'(s)\frac{q'(\varphi _j(s))p'(\varphi _j(s))-q''(\varphi _j(s)) p(\varphi _j(s))}{q'(\varphi _j(s))^2}\\&=\left( \frac{1}{q'(\varphi _j(s))}\cdot \frac{p(\varphi _j(s))}{q(\varphi _j(s))}\right) \cdot \left( \frac{q(\varphi _j(s))}{p(\varphi _j(s))}\cdot \frac{q'(\varphi _j(s)) p'(\varphi _j(s))-q''(\varphi _j(s))p(\varphi _j(s))}{q'(\varphi _j(s))^2}\right) \\&=\frac{p(\varphi _j(s))}{q'(\varphi _j(s))s}\cdot \frac{q(\varphi _j(s))q' (\varphi _j(s))p'(\varphi _j(s))/p(\varphi _j(s))-q(\varphi _j(s))q'' (\varphi _j(s))}{q'(\varphi _j(s))^2}\\&=\frac{r(s)}{s}\cdot \frac{m-(d-1)}{d} \left( 1+O\left( \frac{1}{\vert s\vert ^{1/d}}\right) \right) \\&=-\frac{\lambda }{s}r(s)\left( 1+O\left( \frac{1}{\vert s \vert ^{1/d}}\right) \right) . \end{aligned}$$

Also, a computation shows that

$$\begin{aligned} r''(s)=r(s)O\left( \frac{1}{s^2}\right) \quad \text {and}\quad r'''(s)=r(s)O\left( \frac{1}{s^3}\right) \end{aligned}$$

as $s\rightarrow \infty$. With $h(x_0):=(r(x_0)-r'(x_0)+r''(x_0))e^{x_0}$, we obtain

$$\begin{aligned} \int _{x_0}^wr(s)e^s\,ds=r(w)e^w\left( 1+\frac{\lambda }{w} +O\left( \frac{1}{\vert w\vert ^{1+1/d}}\right) \right) -h(x_0)-\int _{x_0}^wr'''(s)e^s\,ds. \end{aligned}$$

We have

$$\begin{aligned} \int _{x_0}^wr'''(s)e^s\,ds=\int _{-\vert w\vert }^wr'''(s)e^s\,ds +\int _{-\infty }^{-\vert w\vert }r'''(s)e^s\,ds -\int _{-\infty }^{x_0}r'''(s)e^s\,ds. \end{aligned}$$

To estimate $\int _{-\vert w\vert }^wr'''(s)e^s\,ds$, let $\gamma$ be the part of the circle with centre 0 and radius $\vert w\vert$ that connects $-\vert w\vert$ and w in ${\mathcal {G}}$. Then ${{\,\mathrm{Re}\,}}s\le {{\,\mathrm{Re}\,}}w$ for $s\in \gamma$. We obtain

$$\begin{aligned} \left| \int _{-\vert w\vert }^wr'''(s)e^s\,ds\right|&\le {{\,\mathrm{length}\,}}(\gamma )\cdot \max _{s\in \gamma }\left| r'''(s) e^s\right| \le O(\vert w\vert )\vert r(w)\vert O \left( \frac{1}{\vert w\vert ^3}\right) e^{{{\,\mathrm{Re}\,}}w}\\&=\vert r(w)\vert O\left( \frac{1}{\vert w\vert ^2}\right) \vert e^w\vert . \end{aligned}$$

Let us now estimate $\int _{-\infty }^{-\vert w\vert }r'''(s)e^s\,ds$. By (4.2), we have $\vert r(s)\vert \sim \vert s\vert ^{-\lambda }$ as $\vert s\vert \rightarrow \infty$. First suppose that $\lambda \ge 0$. Using that $r'''(s)=r(s)O(1/s^3)$, we obtain

$$\begin{aligned} \left| \int _{-\infty }^{-\vert w\vert }r'''(s)e^s\,ds\right|&\le \vert r(w)\vert e^{-\vert w\vert }O \left( \frac{1}{\vert w\vert ^3}\right) \int _{-\infty }^{-\vert w\vert } e^{s+\vert w\vert }\,ds\\&\le \vert r(w)e^w\vert O\left( \frac{1}{\vert w\vert ^3}\right) \int _{-\infty }^0e^s\,ds=\vert r(w)e^w\vert O \left( \frac{1}{\vert w\vert ^3}\right) . \end{aligned}$$

Now suppose that $\lambda <0$. Then

$$\begin{aligned} \left| \int _{-\infty }^{-\vert w\vert }r'''(s)e^s\,ds\right| \le O\left( \frac{1}{\vert w\vert ^3}\right) \int _{-\infty }^{-\vert w\vert }\vert s\vert ^{-\lambda }e^s\,ds. \end{aligned}$$

Integration by parts yields

$$\begin{aligned} \int _{-\infty }^{-\vert w\vert }\vert s\vert ^{-\lambda } e^s\,ds=O(\vert w\vert ^{-\lambda }e^{-\vert w\vert }) \le O(\vert r(w)e^w\vert ) \end{aligned}$$

and hence

$$\begin{aligned} \int _{-\infty }^{-\vert w\vert }r'''(s)e^s\,ds=r(w)e^w O\left( \frac{1}{\vert w\vert ^3}\right) . \end{aligned}$$

Altogether, we obtain the desired conclusion with

$$\begin{aligned} c_j=g(\varphi _j(x_0))-h(x_0)+\int _{-\infty }^{x_0}r'''(s)e^s\,ds. \end{aligned}$$

$\square$

For the function f from Newton’s method for g, Lemma 4.1 yields the following.

Corollary 4.3

For $j\in \{1,\ldots ,d\},$ we have

$$\begin{aligned} f(\varphi _j(w))=\varphi _j(w)-\frac{1}{q'(\varphi _j(w))}\left( 1+\frac{\lambda }{w} +O\left( \frac{1}{\vert w\vert ^{1+1/d}}\right) \right) -\frac{c_je^{-w}}{p(\varphi _j(w))} \end{aligned}$$

as $w\rightarrow \infty$ in ${\mathcal {G}}$.

In terms of z, Corollary 4.3 says the following.

Corollary 4.4

For $j\in \{1,\ldots ,d\},$ we have

$$\begin{aligned} f(z)=z-\frac{1}{q'(z)}\left( 1+\frac{\lambda }{z^d} +O\left( \frac{1}{\vert z\vert ^{d+1}}\right) \right) -\frac{c_je^{-q(z)}}{p(z)} \end{aligned}$$

as $z\rightarrow \infty$ in ${\mathcal {S}}_j$.

5 Partitioning the plane

For a more detailed study of the behaviour of $f\circ \varphi _j$, we will divide the complex plane into several sets depending on how large $\vert e^{-w}\vert$ is compared to some power of $\vert w\vert$. More precisely, we consider sets whose boundary points satisfy

$$\begin{aligned} {{\,\mathrm{Re}\,}}w=\mu \log \vert w\vert -\log \alpha \end{aligned}$$

(5.1)

for certain $\mu \in {\mathbb {R}}$ and $\alpha >0$. Such sets were also considered by Jankowski [11]. In this section, we will show that given $\mu \in {\mathbb {R}}$, $\alpha >0$ and $y\in {\mathbb {R}}$ of sufficiently large modulus, there is a unique $x_y\in {\mathbb {R}}$ such that $w=x_y+iy$ satisfies (5.1). We also give a proof of several properties of the mapping $y\mapsto x_y$ which in part are also shown in [11, §3.3.4].

Lemma 5.1

Let $\mu \in {\mathbb {R}},$ $\alpha >0$ and $y\in {\mathbb {R}}$ with $\vert y\vert \ge 2\vert \mu \vert$. Then there exists a unique $x_y\in {\mathbb {R}}$ with

$$\begin{aligned} x_y =\mu \log \vert x_y+iy\vert -\log \alpha . \end{aligned}$$

(5.2)

If $x>x_y,$ then

$$\begin{aligned} x>\mu \log \vert x+iy\vert -\log \alpha . \end{aligned}$$

(5.3)

If $x<x_y,$ then

$$\begin{aligned} x<\mu \log \vert x+iy\vert -\log \alpha . \end{aligned}$$

(5.4)

Proof

Let $\varphi :{\mathbb {R}}\rightarrow {\mathbb {R}}$,

$$\begin{aligned} \varphi (x)=x-\mu \log \vert x+iy\vert =x-\frac{\mu }{2}\log (x^2+y^2). \end{aligned}$$

Then $\varphi (x)\rightarrow \infty$ as $x\rightarrow \infty$, and $\varphi (x)\rightarrow -\infty$ as $x\rightarrow -\infty$. Thus, $\varphi$ is surjective, so there exists $x_y$ satisfying (5.2).

Also,

$$\begin{aligned} \varphi '(x)=1-\frac{\mu x}{x^2+y^2}. \end{aligned}$$

Since

$$\begin{aligned} \frac{\vert \mu x\vert }{x^2+y^2}\le \vert \mu \vert \frac{\max \{\vert x\vert ,\vert y\vert \}}{\max \{x^2,y^2\}} =\frac{\vert \mu \vert }{\max \{\vert x\vert ,\vert y\vert \}} \le \frac{1}{2}, \end{aligned}$$

we have

$$\begin{aligned} \varphi '(x)\ge \frac{1}{2}. \end{aligned}$$

Thus, $\varphi$ is strictly increasing, which implies (5.3) and (5.4). In particular, $\varphi$ is injective, so $x_y$ is unique. $\square$

For $\mu \in {\mathbb {R}}$ and $\alpha >0$, let

$$\begin{aligned} \gamma _{\mu ,\alpha }:(-\infty ,-2\vert \mu \vert ] \cup [2\vert \mu \vert ,\infty )\rightarrow {\mathbb {R}},\,\gamma _{\mu ,\alpha }(y)=x_y. \end{aligned}$$

Lemma 5.2

Let $\mu \in {\mathbb {R}}$ and $\alpha >0$.

• (i) The function $\gamma _{\mu ,\alpha }$ is continuously differentiable.

• (ii) If $\mu >0,$ then $\lim _{\vert y\vert \rightarrow \infty }\gamma _{\mu ,\alpha }(y)=\infty .$ If $\mu <0,$ then $\lim _{\vert y\vert \rightarrow \infty }\gamma _{\mu ,\alpha }(y)=-\infty .$ For $\mu =0,$ $\gamma _{\mu ,\alpha }\equiv -\log \alpha .$

• (iii) $\vert \gamma _{\mu ,\alpha }'(y)\vert \le 2\vert \mu \vert /\vert y\vert$. In particular,  $\lim _{\vert y\vert \rightarrow \infty }\gamma _{\mu ,\alpha }'(y)=0$ uniformly in $\alpha$.

• (iv) For $\alpha>\beta >0,$ we have

  $$\begin{aligned} \frac{2}{3}\log \frac{\alpha }{\beta }\le \gamma _{\mu ,\beta }(y) -\gamma _{\mu ,\alpha }(y)\le 2\log \frac{\alpha }{\beta } \end{aligned}$$

  and

  $$\begin{aligned} \lim _{\vert y\vert \rightarrow \infty }(\gamma _{\mu ,\beta }(y) -\gamma _{\mu ,\alpha }(y))=\log \frac{\alpha }{\beta }. \end{aligned}$$

Proof

For $\mu =0$, the results are obvious. We will prove the lemma for $\mu >0$, the proof for $\mu <0$ is analogous. To prove (i)-(iii), note that the condition

$$\begin{aligned} x=\mu \log \vert x+iy\vert -\log \alpha \end{aligned}$$

is equivalent to

$$\begin{aligned} y^2=\alpha ^{2/\mu }e^{(2/\mu ) x}-x^2. \end{aligned}$$

The function

$$\begin{aligned} \psi (x)=\alpha ^{2/\mu }e^{(2/\mu ) x}-x^2 \end{aligned}$$

satisfies

$$\begin{aligned} \lim _{x\rightarrow -\infty }\psi (x)=-\infty \quad \text {and}\quad \lim _{x\rightarrow \infty }\psi (x)=\infty . \end{aligned}$$

(5.5)

Let $x_0:=\max \{x:\,\psi (x)=4\mu ^2\}$. Then $\psi (x)>4\mu ^2$ for $x>x_0$. Also,

$$\begin{aligned} \psi '(x)=\frac{2}{\mu }\alpha ^{2/\mu }e^{(2/\mu ) x} -2x=\frac{2}{\mu }(\psi (x)+x^2-\mu x). \end{aligned}$$

It is not difficult to see that $x^2-\mu x\ge -\mu ^2/4$ for all $x\in {\mathbb {R}}$. Thus,

$$\begin{aligned} \psi '(x)\ge \frac{2}{\mu }\left( \psi (x)-\frac{\mu ^2}{4}\right) >0 \end{aligned}$$

(5.6)

for $x>x_0$. In particular, $\psi :[x_0,\infty )\rightarrow [4\mu ^2,\infty )$ is bijective. This implies that

$$\begin{aligned} \gamma _{\mu ,\alpha }(y)=\psi ^{-1}(y^2) \end{aligned}$$

is a continuously differentiable function. By (5.5), (ii) is satisfied. Also, by (5.6) and since $y^2\ge 4\mu ^2$, we have

$$\begin{aligned} \vert \gamma _{\mu ,\alpha }'(y)\vert =\left| \frac{2y}{\psi '(\psi ^{-1}(y^2))}\right| \le \frac{2\vert y\vert }{(2/\mu )(\psi (\psi ^{-1}(y^2))-\mu ^2/4)} =\frac{\mu \vert y\vert }{y^2-\mu ^2/4}\le \frac{2\mu }{\vert y\vert }, \end{aligned}$$

that is, (iii) is satisfied. To prove (iv), let $y\in {\mathbb {R}}$ with $\vert y\vert \ge 2\mu$ be fixed, and let $\varphi$ be as in the proof of Lemma 5.1. Let $x_1:=\gamma _{\mu ,\alpha }(y)$ and $x_2:=\gamma _{\mu ,\beta }(y).$ Then by the mean value theorem,

$$\begin{aligned} \log \frac{\alpha }{\beta }=\varphi (x_2)-\varphi (x_1)=\varphi '(\xi )(x_2-x_1) \end{aligned}$$

for some $\xi \in [x_2,x_1]$. In the proof of Lemma 5.1, we have seen that $\varphi '(\xi )\ge 1/2$, and the same arguments show that $\varphi '(\xi )\le 3/2$. Also, $\varphi '(\xi )\rightarrow 1$ as $\vert y\vert \rightarrow \infty$. $\square$

For $\mu \in {\mathbb {R}},\,\alpha >0$ and $\nu \ge 2\vert \mu \vert$, define

$$\begin{aligned} {\mathcal {H}}(\mu ,\alpha ,\nu )&:=\{w:\,{{\,\mathrm{Re}\,}}w\ge \mu \log \vert w\vert -\log (\alpha ), \,\vert {{\,\mathrm{Im}\,}}w\vert \ge \nu \}\\&=\{x+iy:\,\vert y\vert \ge \nu ,\,x\ge \gamma _{\mu ,\alpha }(y)\}. \end{aligned}$$

Also, let

$$\begin{aligned} \Gamma (\mu ,\alpha )&:=\{w:\,\vert {{\,\mathrm{Im}\,}}w\vert \ge 2\vert \mu \vert , \,{{\,\mathrm{Re}\,}}w=\mu \log \vert w\vert -\log \alpha \}\\&=\{\gamma _{\mu ,\alpha }(y)+iy:\,\vert y\vert \ge 2\vert \mu \vert \}. \end{aligned}$$

Remark 5.3

Note that if $w\in \Gamma (\mu ,\alpha )$, then

$$\begin{aligned} \vert e^{-w}\vert =e^{-{{\,\mathrm{Re}\,}}w}=\alpha \vert w\vert ^{-\mu }; \end{aligned}$$

if $w\in {\mathcal {H}}(\mu ,\alpha ,\nu )$, then

$$\begin{aligned} \vert e^{-w}\vert \le \alpha \vert w\vert ^{-\mu }; \end{aligned}$$

and if $w\in {\mathbb {C}}{\setminus }{\mathcal {H}}(\mu ,\alpha ,\nu )$ with $\vert {{\,\mathrm{Im}\,}}w\vert \ge \nu$, then

$$\begin{aligned} \vert e^{-w}\vert >\alpha \vert w\vert ^{-\mu }. \end{aligned}$$

6 The singular values of f

Recall that

$$\begin{aligned} g(z)=\int _0^zp(t)e^{q(t)}\,dt+c \end{aligned}$$

where $p(t)=t^d+O(t^{d-1})$ and $q(t)=dt^m+O(t^{m-1})$ as $t\rightarrow \infty$, and f is the Newton map corresponding to g. Let us assume throughout the rest of the paper that g and f satisfy the assumptions of Theorem 1.1.

In this section, we determine the location of the singular values of f.

Lemma 6.1

[3, §7, p.238]The function f does not have finite asymptotic values.

So each singular value of f in ${\mathbb {C}}$ must be a critical value or a limit point of critical values. We have

$$\begin{aligned} f'(z)=\frac{g(z)g''(z)}{g'(z)^2}. \end{aligned}$$

Thus, the critical points of f are:

• 1. the zeros of g that are not zeros of $g'$. These are superattracting fixed points of f and form a discrete subset of ${\mathbb {C}}$.

• 2. the zeros of $g''$ that are not zeros of g or $g'$. There are only finitely many of these, $z_1,\ldots ,z_N$, and by assumption, each $z_j$ is attracted by a periodic cycle.

In particular, the set of critical values of f does not have limit points in ${\mathbb {C}}$. So every singular value of f in ${\mathbb {C}}$ is a critical value, and all but finitely many of them are superattracting fixed points.

Lemma 6.2

The set ${\mathcal {P}}(f)\cap {\mathcal {J}}(f)$ is finite.

Proof

Since the superattracting fixed points of f form a discrete subset of ${\mathbb {C}}$, the set ${\mathcal {P}}(f)\cap {\mathcal {J}}(f)$ is contained in the closure of ${\mathcal {O}}^+(\{z_1,\ldots ,z_N\})$. Each $z_j$ is attracted by a periodic cycle ${\mathcal {C}}$. In particular, ${\mathcal {O}}^+(z_j)$ is bounded and has only finitely many limit points. If $z_j\in {\mathcal {J}}(f)$, then Lemma 2.3 yields that $z_j$ is eventually mapped to ${\mathcal {C}}$, so the forward orbit of $z_j$ is finite.

By Corollary 4.2,

$$\begin{aligned} g(z)=c_j+\frac{p(z)}{q'(z)}\left( 1+\frac{\lambda }{z^d} +O\left( \frac{1}{z^{d+1}}\right) \right) e^{q(z)} \end{aligned}$$

as $z\rightarrow \infty$ in ${\mathcal {S}}_j$, for $j\in \{1,\ldots ,d\}$. We will see later that if $c_j\ne 0$, then g has infinitely many zeros in ${\mathcal {S}}_j$. It is easy to see that this cannot be the case for $c_j=0$. However, we will show now that under the assumptions of Theorem 1.1, the case $c_j=0$ does not occur.

Lemma 6.3

If $c_j=0$ for some $j\in \{1,\ldots ,d\},$ then f has a Baker domain.

Proof

If $c_j=0$, then Corollary 4.4 yields that

$$\begin{aligned} f(z)=z-\frac{1}{dz^{d-1}}+O\left( \frac{1}{z^d}\right) \end{aligned}$$

as $z\rightarrow \infty$ in ${\mathcal {S}}_j$. The claim now follows from [6, §8, §11] (see also [10, Theorem 2]). $\square$

Corollary 6.4

If the assumptions of Theorem 1.1 are satisfied,  then $c_j\ne 0$ for all $j\in \{1,\ldots ,d\}.$

Proof

A theorem by Bergweiler [3, Theorem 2] says that if g and f are defined by (1.2) and (1.1), then every cycle of Baker domains of f contains a singularity of $f^{-1}$. This cannot be true under the assumptions of Theorem 1.1. $\square$

We now investigate the location of the zeros of g. It turns out that the images under q of all but finitely many of them are close to the curves $\Gamma (\lambda ,1/\vert c_j\vert )$ defined in Sect. 5. More precisely, we have the following.

Lemma 6.5

For $j\in \{1,\ldots ,d\}$ and $k\in {\mathbb {Z}},$ let $v_{j,k}\in \Gamma (\lambda ,1/\vert c_j\vert )$ such that

$$\begin{aligned} {{\,\mathrm{Im}\,}}v_{j,k}={\left\{ \begin{array}{ll} \arg (-c_j)+\lambda (\pi /2+2\pi (j-1))+2k\pi &{}\text {if }k\ge 0\\ \arg (-c_j)+\lambda (-\pi /2+2\pi j)+2k\pi &{}\text {if }k<0. \end{array}\right. } \end{aligned}$$

If $z\in {\mathcal {S}}_j$ is a zero of g and $\vert z\vert$ is large,  then there exists $k\in {\mathbb {Z}}$ such that

$$\begin{aligned} q(z)=v_{j,k}+o(1) \end{aligned}$$

(6.1)

as $\vert z\vert \rightarrow \infty$. Vice versa,  if $j\in \{1,\ldots ,d\}$ and $\vert k\vert$ is large,  then g has a zero $z\in {\mathcal {S}}_j$ satisfying (6.1).

Proof

First suppose that $z\in {\mathcal {S}}_j$ is a zero of g. By Corollary 4.2 and (4.1),

$$\begin{aligned} g(z)=c_j+z^{-d\lambda }(1+o(1))e^{q(z)} \end{aligned}$$

as $z\rightarrow \infty$, and hence

$$\begin{aligned} e^{q(z)}=-c_jz^{d\lambda }(1+o(1)). \end{aligned}$$

(6.2)

Thus,

$$\begin{aligned} {{\,\mathrm{Re}\,}}q(z)&=\log \left| e^{q(z)}\right| =\log \vert c_j\vert +d\lambda \log \vert z\vert +o(1)\\&=\log \vert c_j\vert +\lambda \log \vert q(z)\vert +o(1)\\&=\lambda \log \vert q(z)\vert -\log \frac{1}{\vert c_j\vert }+o(1). \end{aligned}$$

In particular, ${{\,\mathrm{Re}\,}}q(z)=o(\vert q(z)\vert )$ as $z\rightarrow \infty$ and hence

$$\begin{aligned} \arg q(z)=\pm \frac{\pi }{2}+o(1). \end{aligned}$$

(6.3)

Let us now assume that ${{\,\mathrm{Im}\,}}q(z)>0$ and hence $\arg q(z)=\pi /2+o(1).$ The proof in the case where ${{\,\mathrm{Im}\,}}q(z)<0$ is analogous. By (6.2),

$$\begin{aligned} {{\,\mathrm{Im}\,}}q(z)\equiv \arg (-c_j)+d\lambda \arg (z)+o(1) \mod 2\pi . \end{aligned}$$

(6.4)

We have

$$\begin{aligned} \arg q(z) =\arg (z^d(1+o(1)))\equiv d\arg z+o(1)\mod 2\pi \end{aligned}$$

and hence

$$\begin{aligned} \arg z \equiv \frac{1}{d}\arg q(z)+o(1)\equiv \frac{\pi }{2d} +o(1)\mod \frac{2\pi }{d}. \end{aligned}$$

Since $z\in {\mathcal {S}}_j$, this implies

$$\begin{aligned} \arg z\equiv \frac{\pi }{2d}+\frac{2\pi (j-1)}{d}+o(1)\mod 2\pi . \end{aligned}$$

(6.5)

Inserting (6.5) into (6.4) yields

$$\begin{aligned} {{\,\mathrm{Im}\,}}q(z)\equiv \arg (-c_j)+\lambda \left( \frac{\pi }{2} +2\pi (j-1)\right) +o(1)\mod 2\pi . \end{aligned}$$

This completes the proof of the first part of Lemma 6.5.

Let us now prove the second part. As before, we will give the proof only for $k>0$, the proof for $k<0$ is analogous. Recall that $\varphi _j$ is the branch of $q^{-1}$ that maps ${\mathbb {C}}{\setminus } (\overline{{\mathcal {D}}(0,R)}\cup [0,\infty ))$ onto ${\mathcal {S}}_j$. For small $\varepsilon >0$, let ${\mathcal {G}}_{j,k}$ be the interior of the set of all

$$\begin{aligned} v\in {\mathcal {H}}\left( \lambda ,\,\frac{1+\varepsilon }{\vert c_j\vert }, \,2\vert \lambda \vert \right) {\setminus } {\mathcal {H}}\left( \lambda , \frac{1-\varepsilon }{\vert c_j\vert },\,2\vert \lambda \vert \right) \end{aligned}$$

satisfying

$$\begin{aligned} \vert {{\,\mathrm{Im}\,}}v-{{\,\mathrm{Im}\,}}v_{j,k}\vert <\pi . \end{aligned}$$

We will use the minimum principle to show that $g\circ \varphi _j$ has a zero in ${\mathcal {G}}_{j,k}$. For $v\in {\mathcal {G}}_{j,k}$, we have ${{\,\mathrm{Re}\,}}(v)=o(\vert v\vert )$, and hence

$$\begin{aligned} \arg (v)=\frac{\pi }{2}+o(1) \end{aligned}$$

as $\vert v\vert \rightarrow \infty$. Similar arguments as above and the definition of $v_{j,k}$ yield

$$\begin{aligned} \arg (\varphi _j(v))\equiv \frac{\pi }{2d}+\frac{2\pi (j-1)}{d}+o(1) \equiv \frac{\arg (-c_j)-{{\,\mathrm{Im}\,}}(v_{j,k})}{-d\lambda }+o(1) \mod \frac{2\pi }{d\lambda }.\qquad \end{aligned}$$

(6.6)

In particular, this is true for $v=v_{j,k}$. Also, since $v_{j,k}\in \Gamma (\lambda ,1/\vert c_j\vert )$, we have

$$\begin{aligned} e^{-{{\,\mathrm{Re}\,}}v_{j,k}} =\frac{1}{\vert c_j\vert }\vert v_{j,k} \vert ^{-\lambda }=\frac{1}{\vert c_j\vert }\vert \varphi _j (v_{j,k})\vert ^{-d\lambda }(1+o(1)), \end{aligned}$$

that is,

$$\begin{aligned} \vert \varphi _j(v_{j,k})\vert ^{-d\lambda }=\vert c_j\vert e^{-{{\,\mathrm{Re}\,}}v_{j,k}}(1+o(1)). \end{aligned}$$

Using Lemma 4.1 and (4.1), we obtain

$$\begin{aligned}&(g\circ \varphi _j)(v_{j,k})=c_j+\varphi _j(v_{j,k})^{-d\lambda }(1+o(1)) \exp (v_{j,k})\\&\quad =c_j+\vert \varphi _j(v_{j,k})\vert ^{-d\lambda } \exp (-id\lambda \arg (\varphi _j(v_{j,k})))(1+o(1))\exp (v_{j,k})\\&\quad =c_j+\vert c_j\vert \exp (-{{\,\mathrm{Re}\,}}v_{j,k})\exp (i(\arg (-c_j) -{{\,\mathrm{Im}\,}}v_{j,k}))(1+o(1))\exp (v_{j,k})\\&\quad =c_j-c_j(1+o(1))=o(1). \end{aligned}$$

Next, we will show that $g\circ \varphi _j$ is bounded below on $\partial {\mathcal {G}}_{j,k}$.

If $v\in \Gamma (\lambda ,(1+\varepsilon )/\vert c_j\vert )$, then

$$\begin{aligned} \vert (g\circ \varphi _j)(v)\vert&=\vert c_j+\varphi _j(v)^{-d\lambda }(1+o(1)) e^v\vert \ge \vert \vert c_j\vert -\vert v\vert ^{-\lambda }e^{{{\,\mathrm{Re}\,}}v} (1+o(1))\vert \\&=\left| \vert c_j\vert -\frac{\vert c_j\vert }{1+\varepsilon } (1+o(1))\right| =\left| \frac{\varepsilon \vert c_j\vert }{1+\varepsilon } -o(1)\right| \ge \frac{\varepsilon \vert c_j\vert }{2}, \end{aligned}$$

provided $\varepsilon$ is sufficiently small and k is sufficiently large. An analogous estimate yields that if $v\in \Gamma (\lambda ,(1-\varepsilon )/\vert c_j\vert )$, then

$$\begin{aligned} \vert (g\circ \varphi _j)(v)\vert \ge \left| \frac{\vert c_j\vert }{1-\varepsilon } (1+o(1))-\vert c_j\vert \right| \ge \varepsilon \vert c_j\vert , \end{aligned}$$

provided $\varepsilon$ is sufficiently small and k is sufficiently large. If ${{\,\mathrm{Im}\,}}(v)={{\,\mathrm{Im}\,}}(v_{j,k})\pm \pi$, then by (6.6),

$$\begin{aligned} \arg (\varphi _j(v)^{-d\lambda }e^v)\equiv \arg (-c_j)\pm \pi +o(1) \equiv \arg (c_j)+o(1)\mod 2\pi . \end{aligned}$$

Thus, for $v\in \overline{{\mathcal {G}}_{j,k}}$ with ${{\,\mathrm{Im}\,}}v={{\,\mathrm{Im}\,}}v_{j,k}\pm \pi$, we have

$$\begin{aligned} \vert (g\circ \varphi _j)(v)\vert&=\left| c_j+\varphi _j(v)^{-d\lambda } (1+o(1))e^v\right| \\&=\left| \vert c_j\vert \exp (i\arg (c_j)) +\vert v\vert ^{-\lambda }\vert e^v\vert \exp (i\arg c_j+o(1))(1+o(1))\right| \\&=\left| \vert c_j\vert +\vert v\vert ^{-\lambda }\vert e^v\vert (1+o(1))\right| \ge \vert c_j\vert . \end{aligned}$$

We obtain that if k is sufficiently large, then

$$\begin{aligned} \vert (g\circ \varphi _j)(v_{j,k})\vert =o(1)<\min _{v\in \partial {\mathcal {G}}_{j,k}} \vert v\vert . \end{aligned}$$

By the minimum principle, $g\circ \varphi _j$ has a zero $w\in {\mathcal {G}}_{j,k}$. The first part of the lemma yields that $z:=\varphi _j(w)$ satisfies (6.1). $\square$

Corollary 6.6

Let $j\in \{1,\ldots ,d\}$ and let $z\in {\mathcal {S}}_j$ be a zero of g. Then

$$\begin{aligned} \arg (z)={\left\{ \begin{array}{ll} \pi /(2d)+2\pi (j-1)/d+o(1)&{}\text {if }{{\,\mathrm{Im}\,}}q(z)>0\\ -\pi /(2d)+2\pi j/d+o(1) &{}\text {if }{{\,\mathrm{Im}\,}}q(z)<0 \end{array}\right. } \end{aligned}$$

as $\vert z\vert \rightarrow \infty .$ In particular, 

$$\begin{aligned} {{\,\mathrm{dist}\,}}(z,\partial {\mathcal {S}}_j)\ge \left( \frac{1}{d}+o(1)\right) \vert z\vert \end{aligned}$$

as $\vert z\vert \rightarrow \infty$.

Proof

The first part is stated in (6.5) in the case where ${{\,\mathrm{Im}\,}}q(z)>0$, and follows from (6.3) with similar arguments in the case where ${{\,\mathrm{Im}\,}}q(z)<0$. We obtain

$$\begin{aligned} {{\,\mathrm{dist}\,}}(z,\partial {\mathcal {S}}_j)=\sin \left( \frac{\pi }{2d}+o(1)\right) \vert z\vert \ge \left( \frac{1}{d}+o(1)\right) \vert z\vert . \end{aligned}$$

$\square$

7 The set $q({\mathcal {F}}(f))$: first part

For $j\in \{1,\ldots ,d\}$, let

$$\begin{aligned} {\mathcal {F}}_j:={\mathcal {F}}(f)\cap {\mathcal {S}}_j. \end{aligned}$$

In Sects. 7–9, we will investigate the location and size of $q({\mathcal {F}}_j)$ in three different subsets of ${\mathbb {C}}$, using the sets ${\mathcal {H}}(\mu , \alpha ,\nu )$ introduced in Sect. 5. The first subset is ${\mathcal {H}}(\lambda , 1/\vert c_j\vert , \nu )$, the second one is ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ){\setminus }{\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$ for small $\alpha _1>0$ and large $\beta _1>0$, and the third set is $\{w:\,\vert {{\,\mathrm{Im}\,}}w\vert \ge \nu \}{\setminus } {\mathcal {H}}(\lambda -1,\beta _2/\vert c_j\vert ,\nu )$ for large $\beta _2>0$. See Fig. 1 for an illustration of these sets.

Fig. 1

An illustration of the sets ${\mathcal {H}}(\lambda ,1/\vert c_j\vert ,\nu )$, ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert , \nu ){\setminus }{\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$ and $\{w:\,\vert {{\,\mathrm{Im}\,}}w\vert \ge \nu \}{\setminus }{\mathcal {H}}(\lambda -1, \beta _2/\vert c_j\vert ,\nu )$ in the case where $\lambda >0$

In this section, we investigate the location and size of $q({\mathcal {F}}_j)$ in ${\mathcal {H}}(\lambda , 1/\vert c_j\vert , \nu )$ for $j\in \{1,\ldots ,d\}$ and large $\nu >0$. Recall that the branch $\varphi _j$ of $q^{-1}$ maps ${\mathcal {H}}(\lambda , 1/\vert c_j\vert , \nu )$ to a subset of ${\mathcal {S}}_j$.

Lemma 7.1

Let $j\in \{1,\ldots ,d\}$. There exists $\nu _0>0$ such that

$$\begin{aligned} (f\circ \varphi _j)({\mathcal {H}}(\lambda ,1/\vert c_j\vert ,\nu _0))\subset {\mathcal {S}}_j. \end{aligned}$$

In particular,  if $(q\circ f\circ \varphi _j)^k(w)\in {\mathcal {H}}(\lambda ,1/\vert c_j\vert ,\nu _0)$ for all $k\in \{0,\ldots ,n-1\},$ then $(f^n\circ \varphi _j)(w)\in {\mathcal {S}}_j$ and $(q\circ f\circ \varphi _j)^n(w)=(q\circ f^n\circ \varphi _j)(w).$

Proof

Let $w\in {\mathcal {H}}(\lambda ,1/\vert c_j\vert ,\nu _0).$ By Corollary 4.3, (4.2) and Remark 5.3,

$$\begin{aligned}&\vert (f\circ \varphi _j)(w)-\varphi _j(w)\vert \\&\quad \le \frac{1}{\vert q'(\varphi _j(w))\vert } \left( 1+O\left( \frac{1}{\vert w\vert }\right) +\frac{\vert q'(\varphi _j(w))c_je^{-w}\vert }{\vert p(\varphi _j(w))\vert }\right) \\&\quad =\frac{1}{\vert q'(\varphi _j(w))\vert } \left( 1+O\left( \frac{1}{\vert w\vert }\right) +\vert w\vert ^{\lambda }\vert c_je^{-w}\vert \left( 1+O\left( \frac{1}{\vert w\vert ^{1/d}}\right) \right) \right) \\&\quad \le \frac{3}{\vert q'(\varphi _j(w))\vert } =3\vert \varphi _j'(w)\vert \end{aligned}$$

if $\vert w\vert$ is sufficiently large. For $\nu _0\ge 12+R$, with R as in Sect. 3, we obtain

$$\begin{aligned} f(\varphi _j(w))\in {\mathcal {D}}(\varphi _j(w),3\vert \varphi _j'(w)\vert )\subset {\mathcal {D}}\left( \varphi _j(w),\frac{\nu _0-R}{4}\vert \varphi _j'(w)\vert \right) . \end{aligned}$$

On the other hand, by Koebe’s 1/4-theorem,

$$\begin{aligned} {\mathcal {S}}_j\supset \varphi _j\left( {\mathcal {D}}\left( w,\nu _0-R\right) \right) \supset {\mathcal {D}}\left( \varphi _j(w),\frac{\nu _0-R}{4} \vert \varphi _j'(w)\vert \right) , \end{aligned}$$

whence the claim follows. $\square$

Next, we derive an asymptotic expression for

$$\begin{aligned} h_j(w)=(q\circ f \circ \varphi _j)(w) \end{aligned}$$

in ${\mathcal {H}}(\lambda ,2/\vert c_j\vert ,\nu _1)$ for large $\nu _1>0$.

Lemma 7.2

Let $j\in \{1,\ldots ,d\}.$ There exists $\nu _1>0$ such that

$$\begin{aligned} h_j(w)=w-1+\frac{2m+1-d}{2d}\cdot \frac{1}{w} +O\left( \frac{1}{\vert w\vert ^{1+1/d}}\right) -c_je^{-w}\varphi _j(w)^{d\lambda }\left( 1+O\left( \frac{1}{\vert w \vert ^{1/d}}\right) \right) \end{aligned}$$

(7.1)

as $w\rightarrow \infty$ in ${\mathcal {H}}(\lambda ,2/\vert c_j\vert ,\nu _1)$.

Remark 7.3

In fact, for any $\alpha >0$, there exists $\nu >0$ such that $h_j$ has an asymptotic expression of the form (7.1) in ${\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert , \nu )$. We will need that $\alpha >1$ so that ${\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu )\supset {\mathcal {H}}(\lambda ,1/\vert c_j\vert ,\nu )$.

Proof of Lemma 7.2

By Corollary 4.3, we have

$$\begin{aligned} f(\varphi _j(w))=\varphi _j(w)-\frac{\eta (w)}{q'(\varphi _j(w))} \end{aligned}$$

where

$$\begin{aligned} \eta (w)=1+\frac{\lambda }{w}+O\left( \frac{1}{\vert w \vert ^{1+1/d}}\right) +c_je^{-w}\frac{q'(\varphi _j(w))}{p(\varphi _j(w))} \end{aligned}$$

as $\vert w\vert \rightarrow \infty$. Note that $\eta$ is bounded in ${\mathcal {H}}(\lambda ,2/\vert c_j\vert , \nu _1)$. Taylor expansion of q around $\varphi _j(w)$ yields

$$\begin{aligned} h_j(w)&=q(f(\varphi _j(w)))=\sum _{k=0}^d\frac{1}{k!}q^{(k)} (\varphi _j(w))(f(\varphi _j(w))-\varphi _j(w))^k\nonumber \\&=\sum _{k=0}^d\frac{(-1)^k}{k!}\frac{q^{(k)}(\varphi _j(w))}{q' (\varphi _j(w))^k}\eta (w)^k\nonumber \\&=w-\eta (w)+\frac{1}{2}\frac{q''(\varphi _j(w))}{q'(\varphi _j(w))^2}\eta (w)^2 +\sum _{k=3}^d\frac{(-1)^k}{k!}O\left( \frac{1}{w^{k-1}}\right) \eta (w)^k\nonumber \\&=w-\eta (w)+\frac{1}{2}\frac{q''(\varphi _j(w))}{q'(\varphi _j(w))^2}\eta (w)^2 +O\left( \frac{1}{w^2}\right) \end{aligned}$$

(7.2)

as $w\rightarrow \infty$ in ${\mathcal {H}}(\lambda ,2/\vert c_j\vert ,\nu _1)$. Using that $\lambda =(d-1-m)/d$, we have

$$\begin{aligned}&-\eta (w)\nonumber \\&\quad =-1+\frac{m+1-d}{d}\cdot \frac{1}{w} +O\left( \frac{1}{\vert w\vert ^{1+1/d}}\right) -c_je^{-w}\varphi _j(w)^{d-1-m}\left( 1+O \left( \frac{1}{\vert w\vert ^{1/d}}\right) \right) . \end{aligned}$$

(7.3)

Moreover,

$$\begin{aligned} \frac{q''(\varphi _j(w))}{q'(\varphi _j(w))^2}=\frac{d-1}{d} \cdot \frac{1}{w}\left( 1+O\left( \frac{1}{\vert w \vert ^{1/d}}\right) \right) \end{aligned}$$

and

$$\begin{aligned} \eta (w)^2&=\left( 1+O\left( \frac{1}{w}\right) +c_je^{-w} \varphi _j(w)^{d\lambda }\left( 1+O\left( \frac{1}{\vert w \vert ^{1/d}}\right) \right) \right) ^2\\&=1+O\left( \frac{1}{w}\right) +c_je^{-w}\varphi _j (w)^{d\lambda }\cdot O(1). \end{aligned}$$

Hence,

$$\begin{aligned} \frac{1}{2}\frac{q''(\varphi _j(w))}{q'(\varphi _j(w))^2} \eta (w)^2=\frac{d-1}{2d}\frac{1}{w} +O\left( \frac{1}{\vert w\vert ^{1+1/d}}\right) +c_je^{-w}\varphi _j(w)^{d\lambda }O\left( \frac{1}{w}\right) . \end{aligned}$$

(7.4)

Combining (7.2), (7.3) and (7.4) yields the desired conclusion. $\square$

For the derivative of $h_j$, we obtain the following.

Lemma 7.4

Let $j\in \{1,\ldots ,d\}$. There exists $\nu _2>0$ such that

$$\begin{aligned} h_j'(w)=1+O\left( \frac{1}{\vert w\vert ^{1+1/d}}\right) +c_je^{-w}\varphi _j(w)^{d\lambda } \left( 1+O\left( \frac{1}{\vert w\vert ^{1/d}}\right) \right) \end{aligned}$$

as $w\rightarrow \infty$ in ${\mathcal {H}}(\lambda ,1/\vert c_j\vert , \nu _2)$.

Proof

Suppose $\nu _2\ge \nu _1+1$. By Lemma 7.2, there are holomorphic functions, $a_1, a_2$, satisfying $a_1(w)=O(1/\vert w\vert ^{1+1/d})$ and $a_2(w)=O(1/\vert w \vert ^{1/d})$ as $w\rightarrow \infty$ such that

$$\begin{aligned} h_j(w)=w-1+\frac{2m+1-d}{2d}\cdot \frac{1}{w}+a_1(w) -c_je^{-w}\varphi _j(w)^{d\lambda }\left( 1+a_2(w)\right) \end{aligned}$$

for $w\in {\mathcal {H}}(\lambda , 2/\vert c_j\vert , \nu _2-1)$. By Lemma 5.2 and Cauchy’s inequality, we have $a_1'(w)=O(1/\vert w\vert ^{1+1/d})$ and $a_2'(w)=O(1/\vert w\vert ^{1/d})$ as $w\rightarrow \infty$ in ${\mathcal {H}}(\lambda ,1/\vert c_j\vert , \nu _2)$. Also,

$$\begin{aligned} \frac{\text {d}}{\text {d}w}e^{-w}\varphi _j(w)^{d\lambda }&=-e^{-w}\varphi _j (w)^{d\lambda }\left( 1-\frac{d\lambda }{\varphi _j(w)}\varphi _j'(w)\right) \\&=-e^{-w}\varphi _j(w)^{d\lambda }\left( 1+O\left( \frac{1}{w}\right) \right) . \end{aligned}$$

Thus,

$$\begin{aligned} \frac{\text {d}}{\text {d}w} \left( c_je^{-w}\varphi _j(w)^{d\lambda } (1+a_2(w))\right) =-c_je^{-w}\varphi _j(w)^{d\lambda } \left( 1+O\left( \frac{1}{\vert w\vert ^{1/d}}\right) \right) . \end{aligned}$$

We obtain

$$\begin{aligned} h_j'(w)&=1-\frac{2m+1-d}{2d}\frac{1}{w^2}+a_1'(w) +c_je^{-w}\varphi _j(w)^{d\lambda } \left( 1+O\left( \frac{1}{\vert w\vert ^{1/d}}\right) \right) \\&=1+O\left( \frac{1}{\vert w\vert ^{1+1/d}}\right) +c_je^{-w}\varphi _j(w)^{d\lambda }\left( 1+O \left( \frac{1}{\vert w\vert ^{1/d}}\right) \right) \end{aligned}$$

as $w\rightarrow \infty$ in ${\mathcal {H}}(\lambda ,1/\vert c_j\vert ,\nu _2)$. $\square$

We will now proceed as follows. Recall that if $z_0\in {\mathcal {S}}_j$ is a superattracting fixed point of f, then $q(z_0)$ lies close to the curve $\Gamma (\lambda ,1/\vert c_j\vert )$. Also, every horizontal strip of width $2\pi +\varepsilon$ with $\varepsilon >0$ that is sufficiently far from the real axis contains such an image of a superattracting fixed point. We will show that if $z_0$ is a superattracting fixed point of f and ${\mathcal {A}}^*(z_0)$ is its immediate basin of attraction, then $q({\mathcal {A}}^*(z_0))$ contains a disk of a fixed radius around $q(z_0)$. We then consider preimages of this disk under iterates of $h_j=q\circ f\circ \varphi _j$. The function $h_j$ is not locally invertible at $q(z_0)$, but if $\alpha$ is slightly smaller than 1, then $h_j$ has a local inverse function, $\psi _j$, defined in ${\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu )$. If $\alpha$ is sufficiently close to 1, then ${\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu )$ intersects the disks contained in $q({\mathcal {A}}^*(z_0))$. We then show that the images of this intersection under $\psi _j$ have a certain size and are more or less evenly distributed in ${\mathcal {H}}(\lambda ,1/\vert c_j\vert ,\nu )$. The idea here is that if $w\in {\mathcal {H}}(\lambda ,1/\vert c_j\vert ,\nu )$ is sufficiently far from the boundary, then Lemma 7.2 yields $h_j(w)\approx w-1$, and hence $\psi _j(w)\approx w+1$. See Fig. 2 for an illustration of the abovementioned approach.

Lemma 7.5

If $z_0$ is a zero of g but not a zero of $g'$ and $\vert z_0\vert$ is sufficiently large,  then

$$\begin{aligned} {\mathcal {A}}^*(z_0)\supset {\mathcal {D}}\left( z_0,\frac{1}{3d\vert z_0\vert ^{d-1}}\right) . \end{aligned}$$

For the proof, we require the following theorem which essentially says that under suitable assumptions the solution of Böttcher’s functional equation in a neighbourhood of a superattracting fixed point extends to a conformal map defined in the entire immediate basin of attraction.

Fig. 2

The images of superattracting fixed points of f under q lie close to the dashed line. The white disks around them are contained in the images of the basins of attraction of the superattracting fixed points. To the right of the solid line, the inverse $\psi _j$ of $h_j$ is defined. The grey disks lie in the intersection of the images of the basins of attraction under q and the domain of definition of $\psi _j$, and their images under iteration of $\psi _j$ are contained in $q({\mathcal {F}}_j)$

Theorem 7.6

Let h be a meromorphic function,  and let $z_0$ be a superattracting fixed point of multiplicity k of h. Suppose that ${\mathcal {A}}^*(z_0)$ contains no critical point other than $z_0$ and no asymptotic value of h. Then there is a conformal map $\Phi :{\mathcal {D}}(0,1)\rightarrow {\mathcal {A}}^*(z_0)$ satisfying $\Phi (0)=z_0$ and

$$\begin{aligned} h(\Phi (z))=\Phi (z^k) \end{aligned}$$

for all $z\in {\mathcal {D}}(0,1)$.

A proof of this theorem can be found, for example, in [23, p. 65, Theorem 4]. There, the result is stated for rational functions, but the proof also works for meromorphic functions without asymptotic values in ${\mathcal {A}}^*(z_0)$.

Proof of Lemma 7.5

Let $z_0$ be a zero of g that is not a zero of $g'$, and assume that none of the finitely many zeros of $g''$ lies in ${\mathcal {A}}^*(z_0)$. Then $z_0$ is a superattracting fixed point of f, and there are no other critical points of f in ${\mathcal {A}}^*(z_0)$. Also,

$$\begin{aligned} f''(z)=\frac{g'(z)^2g''(z)+g(z)g'(z)g'''(z)-2g(z)g''(z)^2}{g'(z)^3}, \end{aligned}$$

and hence

$$\begin{aligned} f''(z_0)=\frac{g''(z_0)}{g'(z_0)}\ne 0. \end{aligned}$$

By Theorem 7.6, there is a conformal map $\Phi :{\mathcal {D}}(0,1)\rightarrow {\mathcal {A}}^*(z_0)$ satisfying $f(\Phi (z))=\Phi (z^2)$ and $\Phi (0)=z_0.$ Differentiating the equation $f(\Phi (z))=\Phi (z^2)$ twice yields

$$\begin{aligned} f''(\Phi (z))\Phi '(z)^2+f'(\Phi (z))\Phi ''(z)=2\Phi '(z^2)+4z^2\Phi ''(z^2). \end{aligned}$$

For $z=0$, we obtain

$$\begin{aligned} f''(z_0)\Phi '(0)^2=2\Phi '(0) \end{aligned}$$

and hence

$$\begin{aligned} \vert \Phi '(0)\vert =\frac{2}{\vert f''(z_0)\vert }. \end{aligned}$$

We have

$$\begin{aligned} f''(z_0)&=\frac{g''(z_0)}{g'(z_0)}=\frac{(p(z_0)q'(z_0) +p'(z_0))e^{q(z_0)}}{p(z_0)e^{q(z_0)}}\\&=q'(z_0)+\frac{p'(z_0)}{p(z_0)}=dz_0^{d-1} \left( 1+O\left( \frac{1}{z_0}\right) \right) . \end{aligned}$$

Hence, by Koebe’s 1/4-theorem,

$$\begin{aligned} {\mathcal {A}}^*(z_0)=\Phi ({\mathcal {D}}(0,1))\supset {\mathcal {D}}\left( z_0,\frac{1}{4}\vert \Phi '(0)\vert \right) ={\mathcal {D}}\left( z_0,\frac{1}{2\vert f''(z_0) \vert }\right) \supset {\mathcal {D}}\left( z_0, \frac{1}{3d\vert z_0\vert ^{d-1}}\right) \end{aligned}$$

if $\vert z_0\vert$ is sufficiently large. $\square$

Corollary 7.7

Let $z_0\in {\mathbb {C}}$ be a zero of g that is not a zero of $g'.$ If $\vert z_0\vert$ is sufficiently large,  then

$$\begin{aligned} q({\mathcal {A}}^*(z_0))\supset {\mathcal {D}}\left( q(z_0),\frac{1}{13}\right) . \end{aligned}$$

Proof

If $\vert z_0\vert$ is sufficiently large, then by Lemma 7.5,

$$\begin{aligned} {\mathcal {A}}^*(z_0)\supset {\mathcal {D}}\left( z_0, \frac{1}{3d\vert z_0\vert ^{d-1}}\right) , \end{aligned}$$

and q is injective in this disk. Koebe’s 1/4-theorem yields

$$\begin{aligned} q({\mathcal {A}}^*(z_0))\supset q\left( {\mathcal {D}}\left( z_0,\frac{1}{3d\vert z_0 \vert ^{d-1}}\right) \right) \supset {\mathcal {D}}\left( q(z_0), \frac{\vert q'(z_0)\vert }{12d\vert z_0\vert ^{d-1}}\right) . \end{aligned}$$

Since $q'(z)=dz^{d-1}(1+O(1/z))$ as $z\rightarrow \infty$, the claim follows. $\square$

The next lemma deals with preimages under $h_j$.

Lemma 7.8

Let $\alpha \in (0,1),\,\varepsilon \in (0,1-\alpha )$ and $j\in \{1,\ldots ,d\}$. There exists $\nu _3>0$ such that for each $w_0\in {\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu _3),$ there is a unique $w\in {\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu _3-1)$ with $h_j(w)=w_0.$ More precisely,  $w\in {\mathcal {D}}(w_0+1, \alpha +\varepsilon ).$

Proof

Let $w\in {\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu _3-1).$ By Lemma 7.2 and Remark 5.3,

$$\begin{aligned} \vert h_j(w)-(w-1)\vert&\le O\left( \frac{1}{\vert w\vert }\right) +\vert c_je^{-w}\vert \vert w\vert ^{\lambda } \left( 1+O\left( \frac{1}{\vert w\vert ^{1/d}}\right) \right) \\&\le O\left( \frac{1}{\vert w\vert }\right) +\alpha \left( 1+O \left( \frac{1}{\vert w\vert ^{1/d}}\right) \right) <\alpha +\varepsilon , \end{aligned}$$

provided $\nu _3$ and hence $\vert w\vert$ is sufficiently large. If $h_j(w)=w_0$, we obtain

$$\begin{aligned} \vert w-(w_0+1)\vert =\vert w_0-(w-1)\vert =\vert h_j(w)-(w-1)\vert <\alpha +\varepsilon , \end{aligned}$$

that is, $w\in {\mathcal {D}}(w_0+1,\alpha +\varepsilon )$. On the other hand, Lemma 5.2 yields that

$$\begin{aligned} \overline{{\mathcal {D}}(w_0+1,\alpha +\varepsilon )}\subset {\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu _3-1) \end{aligned}$$

if $\nu _3$ is sufficiently large. Thus, for $w\in \partial {\mathcal {D}}(w_0+1,\alpha +\varepsilon )$,

$$\begin{aligned} \vert (h_j(w)-w_0)-(w-1-w_0)\vert =\vert h_j(w)-(w-1)\vert <\alpha +\varepsilon =\vert w-1-w_0\vert . \end{aligned}$$

By Rouché’s theorem, there is a unique $w\in {\mathcal {D}}(w_0+1,\alpha +\varepsilon )$ satisfying $h_j(w)=w_0$. $\square$

By Lemma 7.8, there is a subset ${\mathcal {H}}_j\subset {\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu _3-1)$ such that $h_j$ maps ${\mathcal {H}}_j$ conformally onto ${\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu _3)$. Let $\psi _j:{\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu _3)\rightarrow {\mathcal {H}}_j$ be the corresponding inverse function. The next lemma yields that if $\vert {{\,\mathrm{Im}\,}}w\vert$ is sufficiently large, then all iterates $\psi _j^n(w)$ are defined and tend to $\infty$ as $n\rightarrow \infty$ in a horizontal strip whose width is bounded independent of w.

Lemma 7.9

Let $\alpha \in (0,1),$ $\varepsilon \in (0,1-\alpha )$ and $j\in \{1,\ldots ,d\}.$ Then there exist $\nu _4>0$ and $C>0$ such that $\psi _j^n(w)$ is defined for all $w\in {\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu _4)$ and all $n\in {\mathbb {N}},$ and satisfies

• (i) ${{\,\mathrm{Re}\,}}\psi _j^n(w)\ge {{\,\mathrm{Re}\,}}w+n(1-\alpha -\varepsilon );$

• (ii) $\vert \psi _j^n(w)\vert \ge \max \{n,\vert w\vert \}\cdot \dfrac{1-\alpha -\varepsilon }{4};$

• (iii) $\vert {{\,\mathrm{Im}\,}}\psi _j^n(w)-{{\,\mathrm{Im}\,}}w\vert \le C;$

• (iv) $e^{-{{\,\mathrm{Re}\,}}\psi _j^n(w)}\vert \psi _j^n(w)\vert ^{\lambda }=O(e^{-n(1-\alpha -\varepsilon )/2}).$

For the proof, we require the following lemma.

Lemma 7.10

For all $n_0\in {\mathbb {N}},$ we have

$$\begin{aligned} \sum _{n=n_0}^\infty \frac{1}{k^2}\le \frac{2}{n_0}. \end{aligned}$$

Proof

We have

$$\begin{aligned} \sum _{n=n_0}^\infty \frac{1}{k^2}\le \frac{1}{n_0^2} +\sum _{k=n_0+1}^\infty \int _{k-1}^k\frac{1}{t^2} \,dt =\frac{1}{n_0^2}+\int _{n_0}^\infty \frac{1}{t^2} \,dt=\frac{1}{n_0^2}+\frac{1}{n_0}\le \frac{2}{n_0}. \end{aligned}$$

$\square$

Proof of Lemma 7.9

Let

$$\begin{aligned} \delta :=1-\alpha -\varepsilon . \end{aligned}$$

First note that if $\psi _j^n(w)$ is defined, then Lemma 7.8 yields that $\psi _j^k(w)\in {\mathcal {D}}(\psi _j^{k-1}(w)+1, \alpha +\varepsilon )$ for all $k\in \{1,\ldots ,n\}$, and hence

$$\begin{aligned} {{\,\mathrm{Re}\,}}\psi _j^n(w)\ge {{\,\mathrm{Re}\,}}w+n\delta . \end{aligned}$$

So $\psi _j^n(w)$ satisfies (i). Also, if $n\le \vert w\vert /2$, then

$$\begin{aligned} \vert \psi _j^n(w)\vert \ge \vert w\vert -n(1+\alpha +\varepsilon ) \ge \vert w\vert -\frac{\vert w\vert }{2}(1+\alpha +\varepsilon ) =\vert w\vert \frac{\delta }{2}\ge n\delta . \end{aligned}$$

If $n>\vert w\vert /2$, then

$$\begin{aligned} \vert \psi _j^n(w)\vert \ge {{\,\mathrm{Re}\,}}\psi _j^n(w)\ge {{\,\mathrm{Re}\,}}w+n\delta \ge \lambda \log \vert w\vert -\log \frac{\alpha }{\vert c_j\vert }+n\delta \ge \frac{n\delta }{2}\ge \frac{\vert w\vert \delta }{4}, \end{aligned}$$

provided $\vert w\vert$ and hence also n is sufficiently large. In particular, $\psi _j^n(w)$ satisfies (ii).

Let

$$\begin{aligned} n_w:=\lfloor \vert w\vert \rfloor . \end{aligned}$$

We will show by induction that if $w\in {\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu _4)$ for sufficiently large $\nu _4>0$, then $\psi _j^n(w)$ is defined for all $n\in {\mathbb {N}}$ and

$$\begin{aligned} \vert {{\,\mathrm{Im}\,}}\psi _j^n(w)-{{\,\mathrm{Im}\,}}w\vert \le C' \left( \min \left\{ \frac{n}{\vert w\vert },1\right\} +n_w\sum _{k=n_w}^n\frac{1}{k^2}+\sum _{k=1}^n \frac{1}{k^{1+1/d}}+\sum _{k=1}^ne^{-k\delta /2}\right) \end{aligned}$$

(7.5)

where $C'$ does not depend on w or n. Note that by Lemma 7.10,

$$\begin{aligned}&C'\left( \min \left\{ \frac{n}{\vert w\vert },1\right\} +n_w\sum _{k=n_w}^n\frac{1}{k^2}+\sum _{k=1}^n\frac{1}{k^{1+1/d}} +\sum _{k=1}^ne^{-k\delta /2}\right) \\&\quad \le C'\left( 3+\sum _{k=1}^\infty \frac{1}{k^{1+1/d}} +\sum _{k=1}^\infty e^{-k\delta /2}\right) =:C<\infty . \end{aligned}$$

So (7.5) implies (iii). Clearly, (7.5) is true for $n=0$. Now suppose that (7.5) holds with n replaced by $n-1$. By Lemma 7.8, $\psi _j^n(w)$ is defined if and only if $\vert {{\,\mathrm{Im}\,}}\psi _j^{n-1}(w)\vert >\nu _3$. This is satisfied if $\vert {{\,\mathrm{Im}\,}}w\vert >\nu _3+C$. By Lemma 7.2,

$$\begin{aligned}&\vert {{\,\mathrm{Im}\,}}\psi _j^n(w)-{{\,\mathrm{Im}\,}}\psi _j^{n-1}(w)\vert =\vert {{\,\mathrm{Im}\,}}\psi _j^n(w)-{{\,\mathrm{Im}\,}}h_j(\psi _j^n(w))\vert \\&\quad \le \left| \frac{2m+1-d}{2d}{{\,\mathrm{Im}\,}}\left( \frac{1}{\psi _j^n(w)}\right) \right| +O\left( \frac{1}{\vert \psi _j^n(w)\vert ^{1+1/d}}\right) +2\vert c_j\vert e^{-{{\,\mathrm{Re}\,}}\psi _j^n(w)}\vert \psi _j^n(w)\vert ^{\lambda }, \end{aligned}$$

provided $\vert w\vert$ is sufficiently large. By (ii),

$$\begin{aligned} \frac{1}{\vert \psi _j^n(w)\vert ^{1+1/d}} =O\left( \frac{1}{n^{1+1/d}}\right) . \end{aligned}$$

If $n\le \vert w\vert$, then we estimate the first summand by

$$\begin{aligned} \left| {{\,\mathrm{Im}\,}}\left( \frac{1}{\psi _j^n(w)}\right) \right| \le \frac{1}{\vert \psi _j^n(w)\vert } =O\left( \frac{1}{\vert w\vert }\right) . \end{aligned}$$

If $n>\vert w\vert$, then by Lemma 7.8, (ii) and the induction hypothesis,

$$\begin{aligned} \left| {{\,\mathrm{Im}\,}}\left( \frac{1}{\psi _j^n(w)}\right) \right|&=\frac{\vert {{\,\mathrm{Im}\,}}\psi _j^n(w)\vert }{\vert \psi _j^n(w)\vert ^2} \le \frac{\vert {{\,\mathrm{Im}\,}}\psi _j^{n-1}(w)\vert +\alpha +\varepsilon }{\vert \psi _j^n(w)\vert ^2}\\&\le \frac{16(\vert {{\,\mathrm{Im}\,}}w\vert +C+\alpha +\varepsilon )}{\delta ^2n^2} \le \frac{17\vert w\vert }{\delta ^2n^2}=n_w\cdot O \left( \frac{1}{n^2}\right) , \end{aligned}$$

provided $\vert w\vert$ is sufficiently large.

Moreover, if $\lambda \ge 0$, then by (i), Lemma 7.8 and Remark 5.3,

$$\begin{aligned} \vert c_j\vert e^{-{{\,\mathrm{Re}\,}}\psi _j^n(w)}\vert \psi _j^n(w)\vert ^{\lambda }&\le \vert c_j\vert e^{-{{\,\mathrm{Re}\,}}w}(\vert w\vert +n(1+\alpha +\varepsilon ))^{\lambda } e^{-n\delta }\\&\le \alpha \vert w\vert ^{-\lambda }(\vert w\vert +n(1+\alpha +\varepsilon ))^{\lambda }e^{-n\delta }\\&=\alpha \left( 1+\frac{n}{\vert w\vert }(1+\alpha +\varepsilon )\right) ^{\lambda }e^{-n\delta }\\&\le \alpha (1+n(1+\alpha +\varepsilon ))^{\lambda } e^{-n\delta }=O(e^{-n\delta /2}), \end{aligned}$$

provided $\vert w\vert \ge 1$. If $\lambda <0$, then by (i), (ii) and Remark 5.3,

$$\begin{aligned} \vert c_j\vert e^{-{{\,\mathrm{Re}\,}}\psi _j^n(w)}\vert \psi _j^n(w)\vert ^{\lambda } \le \left( \frac{\delta }{4}\right) ^{\lambda }\vert w\vert ^{\lambda }\vert c_j\vert e^{-{{\,\mathrm{Re}\,}}w}e^{-n\delta } \le \left( \frac{\delta }{4}\right) ^{\lambda }\alpha e^{-n\delta }. \end{aligned}$$

In particular, (iv) is satisfied. Also, if $n\le \vert w\vert$, then

$$\begin{aligned} {{\,\mathrm{Im}\,}}\psi _j^n(w)-{{\,\mathrm{Im}\,}}\psi _j^{n-1}(w)=O\left( \frac{1}{\vert w\vert } \right) +O\left( \frac{1}{n^{1+1/d}}\right) +O\left( e^{-n\delta /2}\right) , \end{aligned}$$

and if $n>\vert w\vert$, then

$$\begin{aligned} {{\,\mathrm{Im}\,}}\psi _j^n(w)-{{\,\mathrm{Im}\,}}\psi _j^{n-1}(w) =n_wO\left( \frac{1}{n^2}\right) +O\left( \frac{1}{n^{1+1/d}}\right) +O(e^{-n\delta /2}). \end{aligned}$$

Thus, $\psi _j^n(w)$ satisfies (7.5), and hence also (iii). $\square$

We now estimate the derivative of $\psi _j^n$.

Lemma 7.11

Let $\alpha \in (0,1)$ and $j\in \{1,\ldots ,d\}$. There are $\nu _5>0$ and $B>0$ such that

$$\begin{aligned} \vert (\psi _j^n)'(w)\vert \ge B \end{aligned}$$

for all $w\in {\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert , \nu _5)$ and all $n\in {\mathbb {N}}$.

Proof

We have

$$\begin{aligned} (\psi _j^n)'=\prod _{k=0}^{n-1}\psi _j'\circ \psi _j^k =\frac{1}{\prod _{k=0}^{n-1}h_j'\circ \psi _j^{k+1}} =\frac{1}{\prod _{k=1}^nh_j'\circ \psi _j^k}. \end{aligned}$$

By Lemmas 7.4 and 7.9,

$$\begin{aligned}&\vert h_j'(\psi _j^{k}(w))\vert \\&\quad \le 1+O\left( \frac{1}{\vert \psi _j^{k}(w) \vert ^{1+1/d}}\right) +\vert c_j\vert e^{-{{\,\mathrm{Re}\,}}\psi _j^{k}(w)}\vert \psi _j^{k}(w)\vert ^{\lambda } \left( 1+O\left( \frac{1}{\vert \psi _j^{k}(w)\vert ^{1/d}}\right) \right) \\&\quad \le 1+O\left( \frac{1}{k^{1+1/d}}\right) +O\left( e^{-k(1-\alpha -\varepsilon )/2}\right) . \end{aligned}$$

Since the infinite product $\prod _{k=1}^\infty (1+O(1/k^{1+1/d})+O(e^{-k(1-\alpha -\varepsilon )/2}))$ converges, we obtain the desired conclusion. $\square$

Recall that ${\mathcal {F}}_j={\mathcal {F}}(f)\cap {\mathcal {S}}_j$.

Lemma 7.12

For $j\in \{1,\ldots ,d\}$ and $k\in {\mathbb {Z}},$ let $v_{j,k}$ be as in Lemma 6.5 and $w_{j,k}:=v_{j,k}+1/26$. There is $\vartheta >0$ such that if $\vert k\vert$ is sufficiently large,  then ${\mathcal {D}}(\psi _j^n(w_{j,k}),\vartheta )\subset q({\mathcal {F}}_j)$ for all $n\in {\mathbb {N}}$.

Remark 7.13

For sufficiently large $\vert k\vert$, the point $v_{j,k}$ is close to $q(z_0)$ for some attracting fixed point $z_0$ of f. The function $\psi _j$ is not defined in $q(z_0)$ and $v_{j,k}$. Therefore, we introduce the point $w_{j,k}$ which is in the domain of definition of $\psi _j$ for large $\vert k\vert$ and also lies in $q(A^*(z_0))$.

Proof of Lemma 7.12

By Lemma 6.5, there is a zero $z_0$ of g satisfying $q(z_0)=v_{j,k}+o(1)$. Thus, $w_{j,k}=q(z_0)+1/26+o(1)$. If $\vert k\vert$ is sufficiently large, we obtain

$$\begin{aligned} {\mathcal {D}}\left( w_{j,k},\frac{1}{27}\right) \subset {\mathcal {D}}\left( q(z_0) +\frac{1}{26},\frac{1}{26}\right) \subset {\mathcal {D}}\left( q(z_0), \frac{1}{13}\right) . \end{aligned}$$

By Corollary 7.7, this yields

$$\begin{aligned} {\mathcal {D}}\left( w_{j,k},\frac{1}{27}\right) \subset q({\mathcal {A}}^*(z_0)). \end{aligned}$$

Let $\nu >0$ be large and $\exp (-1/2(1/26-1/27))<\alpha <1.$ Then

$$\begin{aligned} 2\log \frac{1}{\alpha }<\frac{1}{26}-\frac{1}{27}. \end{aligned}$$

(7.6)

Since $v_{j,k}\in \Gamma (\lambda ,1/\vert c_j\vert )$, by (7.6) and Lemma 5.2(iv) and (iii), we get

$$\begin{aligned} {\mathcal {D}}\left( w_{j,k},\frac{1}{27}\right) \subset {\mathcal {H}}\left( \lambda ,\frac{\alpha }{\vert c_j\vert },\nu _5\right) \end{aligned}$$

if $\vert k\vert$ is sufficiently large. By Koebe’s 1/4-theorem and Lemma 7.11,

$$\begin{aligned} \psi _j^n\left( {\mathcal {D}}\left( w_{j,k},\frac{1}{27}\right) \right) \supset {\mathcal {D}}\left( \psi _j^n(w_{j,k}),\frac{\vert (\psi _j^n)' (w_{j,k})\vert }{4\cdot 27}\right) \supset {\mathcal {D}}\left( \psi _j^n (w_{j,k}),\frac{B}{4\cdot 27}\right) . \end{aligned}$$

With $\vartheta :=B/(4\cdot 27)$ we thus have

$$\begin{aligned} h_j^n({\mathcal {D}}(\psi _j^n(w_{j,k}),\vartheta ))\subset {\mathcal {D}}\left( w_{j,k},\frac{1}{27}\right) \subset q({\mathcal {F}}_j). \end{aligned}$$

Since by Lemma 7.1,

$$\begin{aligned} h_j^n(w)=(q\circ f\circ \varphi _j)^n(w)=(q\circ f^n\circ \varphi _j)(w) \end{aligned}$$

for $w\in {\mathcal {H}}(\lambda ,\alpha /\vert c_j\vert ,\nu _0)$, this implies

$$\begin{aligned} {\mathcal {D}}(\psi _j^n(w_{j,k}),\vartheta )\subset q({\mathcal {F}}_j), \end{aligned}$$

provided $\vert k\vert$ is sufficiently large. $\square$

The final result of this section says that $q({\mathcal {F}}_j)$ has positive density in rectangles of sufficiently large side lengths that are contained in ${\mathcal {H}}(\lambda ,1/\vert c_j\vert ,\nu )$.

Lemma 7.14

There are $D_0,\nu ,\eta _0>0$ such that for all $j\in \{1,\ldots ,d\}$ and any rectangle ${\mathcal {R}}\subset {\mathcal {H}}(\lambda ,1/\vert c_j\vert ,\nu )$ with sides parallel to the real and imaginary axis whose vertical and horizontal side lengths are both at least $D_0,$ we have

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), {\mathcal {R}})\ge \eta _0. \end{aligned}$$

Proof

First suppose that

$$\begin{aligned} {\mathcal {R}}=\{w:\,x_1\le {{\,\mathrm{Re}\,}}w\le x_2,\,y_1\le {{\,\mathrm{Im}\,}}\le y_2\} \end{aligned}$$

(7.7)

where

$$\begin{aligned} 2\pi +2(C+\vartheta )\le x_2-x_1, y_2-y_1\le 2(2\pi +2(C+\vartheta )), \end{aligned}$$

(7.8)

with C as in Lemma 7.9 and $\vartheta$ as in Lemma 7.12. Let $v_{j,k}$ be as in Lemma 6.5 and $w_{j,k}=v_{j,k}+1/26$. There is $k\in {\mathbb {Z}}$ such that $y_1+C+\vartheta \le {{\,\mathrm{Im}\,}}w_{j,k}={{\,\mathrm{Im}\,}}v_{j,k}\le y_2-C-\vartheta$. Also, by Lemma 7.8, there is $n\in {\mathbb {N}}$ such that $x_1+\vartheta<{{\,\mathrm{Re}\,}}\psi _j^n(w_{j,k})<x_2-\vartheta$. By Lemma 7.9, we have $y_1+\vartheta \le {{\,\mathrm{Im}\,}}\psi _j^n(w_{j,k})\le y_2-\vartheta$. Thus,

$$\begin{aligned} {\mathcal {D}}(\psi _j^n(w_{j,k}),\vartheta )\subset {\mathcal {R}}. \end{aligned}$$

Also, by Lemma 7.12,

$$\begin{aligned} {\mathcal {D}}(\psi _j^n(w_{j,k}),\vartheta )\subset q({\mathcal {F}}_j). \end{aligned}$$

We obtain

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), {\mathcal {R}})\ge \frac{{{\,\mathrm{meas}\,}}({\mathcal {D}}(\psi _j^n(w_{j,k}),\vartheta ))}{{{\,\mathrm{meas}\,}}{\mathcal {R}}}\ge \frac{\pi \vartheta ^2}{4(2\pi +2(C+\vartheta ))^2}=:\eta _0. \end{aligned}$$

Now, if ${\mathcal {R}}\subset {\mathcal {H}}(\lambda ,1/\vert c_j\vert ,\nu )$ is any rectangle whose horizontal and vertical side length both exceed $D_0:=2\pi +2(C+\vartheta )$, then ${\mathcal {R}}$ can be written as the union of rectangles of the form (7.7) that satisfy (7.8) and have pairwise disjoint interior, whence the claim follows. $\square$

8 The set $q({\mathcal {F}}(f))$: second part

In this section, we investigate the density of $q({\mathcal {F}}(f))$ in subsets of ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ){\setminus } {\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$ for small $\alpha _1>0$ and large $\beta _1>0$.

We first give an approximate expression for $h_j$ in ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ){\setminus } {\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$.

Lemma 8.1

Let $\varepsilon >0$ and $j\in \{1,\ldots ,d\}$. Then there are $\alpha _1,\beta _1,\nu >0$ such that for all $w\in {\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ){\setminus } {\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu ),$ we have

$$\begin{aligned} \left| \frac{h_j(w)-w}{-c_je^{-w}\varphi _j(w)^{d\lambda }} -1\right| <\varepsilon . \end{aligned}$$

Proof

Taylor expansion of q around $\varphi _j(w)$ yields

$$\begin{aligned} h_j(w)=q(f(\varphi _j(w)))=w+\sum _{k=1}^d\frac{q^{(k)} (\varphi _j(w))}{k!}(f(\varphi _j(w))-\varphi _j(w))^k. \end{aligned}$$

Thus,

$$\begin{aligned}&\frac{h_j(w)-w}{-c_je^{-w}\varphi _j(w)^{d\lambda }}-1\nonumber \\&\quad =\frac{q'(\varphi _j(w))(f(\varphi _j(w))-\varphi _j(w))}{-c_je^{-w}\varphi _j (w)^{d\lambda }}-1+\sum _{k=2}^d\frac{q^{(k)}(\varphi _j(w))}{k!} \frac{(f(\varphi _j(w))-\varphi _j(w))^k}{-c_je^{-w}\varphi _j(w)^{d\lambda }}. \end{aligned}$$

(8.1)

By Corollary 4.3,

$$\begin{aligned}&f(\varphi _j(w))\nonumber \\&\quad =\varphi _j(w)-\frac{1}{q'(\varphi _j(w))} \left( 1+O\left( \frac{1}{\vert w\vert }\right) +c_je^{-w}\varphi _j(w)^{d\lambda }\left( 1+O\left( \frac{1}{\vert w \vert ^{1/d}}\right) \right) \right) \end{aligned}$$

(8.2)

as $w\rightarrow \infty$. Hence,

$$\begin{aligned} \frac{q'(\varphi _j(w))(f(\varphi _j(w))-\varphi _j(w))}{-c_je^{-w} \varphi _j(w)^{d\lambda }}-1=\frac{1+O(1/\vert w\vert )}{c_je^{-w}\varphi _j(w)^{d\lambda }}+O\left( \frac{1}{\vert w \vert ^{1/d}}\right) . \end{aligned}$$

(8.3)

For $w\in {\mathbb {C}}{\setminus }{\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$ with $\vert {{\,\mathrm{Im}\,}}w\vert \ge \nu$, we have

$$\begin{aligned} \left| c_je^{-w}\varphi _j(w)^{d\lambda }\right| \ge \frac{\beta _1}{2}, \end{aligned}$$

provided $\nu$ is sufficiently large. Inserting this into (8.3) yields

$$\begin{aligned} \left| \frac{q'(\varphi _j(w))(f(\varphi _j(w))-\varphi _j(w))}{-c_je^{-w} \varphi _j(w)^{d\lambda }}-1\right| \le \frac{3}{\beta _1} +O\left( \frac{1}{\vert w\vert ^{1/d}}\right) <\frac{\varepsilon }{d} \end{aligned}$$

(8.4)

if $\beta _1$ and $\vert w\vert$ are sufficiently large.

Also, for $w\in {\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu )$, we have

$$\begin{aligned} \left| c_je^{-w}\varphi _j(w)^{d(\lambda -1)}\right| \le 2\alpha _1, \end{aligned}$$

provided $\nu$ is sufficiently large. By (8.2), this yields

$$\begin{aligned} \vert f(\varphi _j(w))-\varphi _j(w)\vert \le \frac{1}{\vert q'(\varphi _j(w))\vert } \left( 1+O\left( \frac{1}{\vert w\vert }\right) +3\alpha _1\varphi _j(w)^d\right) \le \frac{4}{d}\alpha _1\vert \varphi _j(w)\vert , \end{aligned}$$

(8.5)

provided $\vert w\vert$ is sufficiently large. If $k\ge 2$, then by (8.4) and (8.5), we have

$$\begin{aligned}&\left| \frac{q^{(k)}(\varphi _j(w))}{k!} \cdot \frac{(f(\varphi _j(w))-\varphi _j(w))^k}{-c_je^{-w} \varphi _j(w)^{d\lambda }}\right| \nonumber \\&\quad =\left| \frac{q'(\varphi _j(w))(f(\varphi _j(w)) -\varphi _j(w))}{-c_je^{-w}\varphi _j(w)^{d\lambda }} \right| \cdot \left| \frac{q^{(k)}(\varphi _j(w))}{k!q' (\varphi _j(w))}\right| \cdot \vert f(\varphi _j(w))-\varphi _j(w)\vert ^{k-1}\nonumber \\&\quad \le \left( 1+\frac{\varepsilon }{d}\right) \cdot \left| \frac{q^{(k)}(\varphi _j(w))}{k!q'(\varphi _j(w))}\right| \cdot \left( \frac{4}{d}\alpha _1\vert \varphi _j(w)\vert \right) ^{k-1}\nonumber \\&\quad \le \left( 1+\frac{\varepsilon }{d}\right) \left( {\begin{array}{c}d\\ k\end{array}}\right) \frac{2}{d} \vert \varphi _j(w)\vert ^{-k+1}\left( \frac{4}{d}\alpha _1\vert \varphi _j(w) \vert \right) ^{k-1}\nonumber \\&\quad =\left( 1+\frac{\varepsilon }{d}\right) \left( {\begin{array}{c}d\\ k\end{array}}\right) \frac{2\cdot 4^{k-1}}{d^k}\alpha _1^{k-1}<\frac{\varepsilon }{d} \end{aligned}$$

(8.6)

if $\vert w\vert$ is sufficiently large and $\alpha _1$ is sufficiently small. Inserting (8.4) and (8.6) into (8.1) yields the desired conclusion. $\square$

We will now proceed as follows. First, we show that $h_j$ maps the intersection of certain horizontal strips with ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ) {\setminus }{\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$ into ${\mathcal {H}}(\lambda ,1/c^*,\nu )$ where $c^*=\max _l\vert c_l\vert$. The idea is that if ${{\,\mathrm{Im}\,}}w$ lies in certain intervals, then the argument of $-c_je^{-w}\varphi _j(w)^{d\lambda }$ is small, and using that $h_j(w)\approx w-c_je^{-w}\varphi _j(w)^{d\lambda }$ by Lemma 8.1, one can deduce that ${{\,\mathrm{Re}\,}}h_j(w)$ is large. By Sect. 7, the set $q({\mathcal {F}}(f))$ has positive density in large bounded subsets of ${\mathcal {H}}(\lambda ,1/c^*,\nu )$. Together with the invariance of ${\mathcal {F}}(f)$ under f, we deduce that $q({\mathcal {F}}(f))$ has positive density in large bounded subsets of ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ){\setminus }{\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$.

The next lemma deals with the mapping behaviour of f in certain horizontal strips in ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ){\setminus }{\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$. For $j\in \{1,\ldots ,d\}$ and $n\in {\mathbb {Z}}$, let

$$\begin{aligned} y_{n}^j:={\left\{ \begin{array}{ll} \arg (-c_j)+\lambda (\pi /2+2\pi (j-1))+2n\pi &{}\text {if }n\ge 0\\ \arg (-c_j)+\lambda (-\pi /2+2\pi j)+2n\pi &{}\text {if }n<0. \end{array}\right. } \end{aligned}$$

Lemma 8.2

Let $\varepsilon \in (0,\pi /4)$. Then there are $\alpha _1, \beta _1, \nu >0$ such that the following holds. Let $j\in \{1,\ldots ,d\}$. Suppose that w lies in the closure of ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ){\setminus }{\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$ and there exists $n\in {\mathbb {Z}}$ with $\vert {{\,\mathrm{Im}\,}}w-y_n^j\vert \le \pi /4.$ Let $\beta \ge \beta _1$ such that $w\in \Gamma (\lambda ,\beta /\vert c_j\vert ),$ and let $\theta :={{\,\mathrm{Im}\,}}w-y_n^j$. Then, 

$$\begin{aligned}&\vert h_j(w)-w\vert \le (1+\varepsilon )\beta , \\&(1-\varepsilon )\beta \cos (\vert \theta \vert +\varepsilon ) \le {{\,\mathrm{Re}\,}}(h_j(w)-w)\le (1+\varepsilon )\beta \end{aligned}$$

and

$$\begin{aligned} (1-\varepsilon )\beta \sin (\vert \theta \vert -\varepsilon ) \le \vert {{\,\mathrm{Im}\,}}(h_j(w)-w)\vert \le (1+\varepsilon )\beta \sin (\vert \theta \vert +\varepsilon ). \end{aligned}$$

Proof

By Lemma 8.1,

$$\begin{aligned} \left| \frac{h_j(w)-w}{-c_je^{-w}\varphi _j(w)^{d\lambda }} -1\right| \le \frac{\varepsilon }{2}, \end{aligned}$$

(8.7)

provided $\alpha _1$ is sufficiently small and $\beta _1$ and $\nu$ are sufficiently large. Thus,

$$\begin{aligned} \left( 1-\frac{\varepsilon }{2}\right) \left| c_je^{-w}\varphi _j(w)^{d\lambda } \right| \le \vert h_j(w)-w\vert \le \left( 1+\frac{\varepsilon }{2}\right) \left| c_je^{-w}\varphi _j(w)^{d\lambda }\right| . \end{aligned}$$

Since $w\in \Gamma (\lambda ,\beta /\vert c_j\vert )$, this yields

$$\begin{aligned} (1-\varepsilon )\beta \le \vert h_j(w)-w\vert \le (1+\varepsilon )\beta \end{aligned}$$

if $\nu$ is sufficiently large. Also, by (8.7),

$$\begin{aligned} \left| \arg \left( \frac{h_j(w)-w}{-c_je^{-w}\varphi _j(w)^{d\lambda }} \right) \right| \le \arcsin \left( \frac{\varepsilon }{2}\right) \le \frac{\pi }{4}\varepsilon . \end{aligned}$$

(8.8)

We have

$$\begin{aligned} \arg w=\arg q(\varphi _j(w))\equiv \arg (\varphi _j(w)^d(1+o(1))) \equiv d\arg \varphi _j(w)+o(1)\mod 2\pi \end{aligned}$$

as $w\rightarrow \infty$. Since w lies in the closure of ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ) {\setminus }{\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$, we have

$$\begin{aligned} \arg w={{\,\mathrm{sgn}\,}}({{\,\mathrm{Im}\,}}(w))\frac{\pi }{2}+o(1)={{\,\mathrm{sgn}\,}}(n)\frac{\pi }{2}+o(1) \end{aligned}$$

if $\vert n\vert$ is sufficiently large. Hence,

$$\begin{aligned} \arg \varphi _j(w)\equiv {{\,\mathrm{sgn}\,}}(n)\frac{\pi }{2d}+o(1)\mod \frac{2\pi }{d}. \end{aligned}$$

Since $\varphi _j(w)\in {\mathcal {S}}_j$, we obtain

$$\begin{aligned} \arg \varphi _j(w)\equiv {\left\{ \begin{array}{ll} \pi /(2d)+2\pi (j-1)/d+o(1)&{}\text {if }n>0\\ -\pi /(2d)+2\pi j/d+o(1)&{}\text {if }n<0 \end{array}\right. }\mod 2\pi . \end{aligned}$$

Thus,

$$\begin{aligned} \arg \left( -c_je^{-w}\varphi _j(w)^{d\lambda }\right)&\equiv \arg (-c_j)-{{\,\mathrm{Im}\,}}(w)+d\lambda \arg \varphi _j(w)\\&\equiv -\theta -2n\pi +o(1)\equiv -\theta +o(1)\mod 2\pi . \end{aligned}$$

By (8.8), this implies

$$\begin{aligned} \vert \theta \vert -\varepsilon \le \vert \arg (h_j(w)-w)\vert \le \vert \theta \vert +\varepsilon \end{aligned}$$

if $\vert w\vert$ is sufficiently large. We obtain

$$\begin{aligned} {{\,\mathrm{Re}\,}}(h_j(w)-w)&\le \vert h_j(w)-w\vert \le (1+\varepsilon )\beta ,\\ {{\,\mathrm{Re}\,}}(h_j(w)-w)&=\vert h_j(w)-w\vert \cos (\arg (h_j(w)-w)) \ge (1-\varepsilon )\beta \cos (\vert \theta \vert +\varepsilon ),\\ \vert {{\,\mathrm{Im}\,}}(h_j(w)-w)\vert&=\vert h_j(w)-w\vert \cdot \vert \sin (\arg (h_j(w)-w))\vert \le (1+\varepsilon )\beta \sin (\vert \theta \vert +\varepsilon ),\\ \vert {{\,\mathrm{Im}\,}}(h_j(w)-w)\vert&=\vert h_j(w)-w\vert \cdot \vert \sin (\arg (h_j(w)-w))\vert \ge (1-\varepsilon ) \beta \sin (\vert \theta \vert -\varepsilon ). \end{aligned}$$

$\square$

Let

$$\begin{aligned} c^*:=\max _{1\le l\le d}\vert c_l\vert . \end{aligned}$$

(8.9)

The following lemma says that $h_j$ maps the intersection of ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ){\setminus } {\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$ with certain horizontal strips into ${\mathcal {H}}(\lambda ,1/c^*,\nu )$.

Lemma 8.3

There are $\alpha _1,\beta _1,\nu >0$ such that for all $j\in \{1,\ldots ,d\}, n\in {\mathbb {Z}}$ and all w in the closure of ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ) {\setminus }{\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$ with $\vert {{\,\mathrm{Im}\,}}w-y_n^j\vert \le \pi /4,$ we have $h_j(w)\in {\mathcal {H}}(\lambda ,1/c^*,\nu )$.

Proof

We have $w\in \Gamma (\lambda ,\beta /\vert c_j\vert )$ for some $\beta \ge \beta _1$. Since also $w\in {\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu )$, we have

$$\begin{aligned} \lambda \log \vert w\vert -\log \beta ={{\,\mathrm{Re}\,}}w\ge (\lambda -1) \log \vert w\vert -\log \alpha _1 \end{aligned}$$

and hence

$$\begin{aligned} \beta \le \alpha _1\vert w\vert . \end{aligned}$$

(8.10)

Let $\varepsilon \in (0,\pi /4)$, $\theta :={{\,\mathrm{Im}\,}}w-y_n^j$, and suppose that $\alpha _1,\beta _1,\nu$ are chosen such that the conclusion of Lemma 8.2 holds. Then

$$\begin{aligned} \vert {{\,\mathrm{Im}\,}}h_j(w)-{{\,\mathrm{Im}\,}}w\vert \le (1+\varepsilon )\beta \sin (\vert \theta \vert +\varepsilon )\le 2\beta . \end{aligned}$$

By (8.10) and since $\vert {{\,\mathrm{Im}\,}}w\vert =(1+o(1))\vert w\vert$ for w in the closure of ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ){\setminus }{\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$, this yields

$$\begin{aligned} \vert {{\,\mathrm{Im}\,}}h_j(w)\vert \ge \vert {{\,\mathrm{Im}\,}}w\vert -2\beta =(1+o(1)) \vert w\vert -2\beta \ge (1+o(1))\vert w\vert -2\alpha _1\vert w\vert >\nu , \end{aligned}$$

provided $\vert w\vert$ is sufficiently large and $\alpha _1$ is sufficiently small.

Also, by Lemma 8.2 and (8.10),

$$\begin{aligned} \vert h_j(w)-w\vert \le (1+\varepsilon )\beta \le (1+\varepsilon )\alpha _1\vert w\vert . \end{aligned}$$

If $\alpha _1$ is sufficiently small, we obtain

$$\begin{aligned} \frac{1}{2}\vert w\vert \le \vert h_j(w)\vert \le 2\vert w\vert \end{aligned}$$

and hence

$$\begin{aligned} \frac{1}{2}\vert h_j(w)\vert \le \vert w\vert \le 2\vert h_j(w)\vert . \end{aligned}$$

(8.11)

By Lemma 8.2 and (8.11),

$$\begin{aligned}&{{\,\mathrm{Re}\,}}h_j(w)\ge {{\,\mathrm{Re}\,}}w+(1-\varepsilon )\beta \cos \left( \frac{\pi }{4}+\varepsilon \right) \\&\quad =\lambda \log \vert w\vert -\log \beta +(1-\varepsilon )\beta \cos \left( \frac{\pi }{4} +\varepsilon \right) \\&\quad \ge \lambda \log \vert h_j(w)\vert -\vert \lambda \vert \log 2 -\log \beta +(1-\varepsilon )\beta \cos \left( \frac{\pi }{4}+\varepsilon \right) \\&\quad \ge \lambda \log \vert h_j(w)\vert -\log \frac{1}{c^*} \end{aligned}$$

if $\beta _1$ and hence $\beta$ is sufficiently large. Thus, $h_j(w)\in {\mathcal {H}}(\lambda ,1/c^*, \nu ).$ $\square$

Let us now define several sets. We start with subsets ${\mathcal {Q}}_{n,k}^j,\tilde{{\mathcal {Q}}}_{n,k}^j\subset {\mathbb {C}}{\setminus } {\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$ for $j\in \{1,\ldots ,d\}$, $k\in {\mathbb {N}}$ and $n\in {\mathbb {Z}}$.

Let $0<\theta _1<1/(6\pi )\arccos (5/6)$. For $j\in \{1,\ldots ,d\}$, $k\in {\mathbb {N}}$ and $n\in {\mathbb {Z}}$, let ${\mathcal {Q}}_{n,k}^j$ be the set of all

$$\begin{aligned} w\in {\mathcal {H}}\left( \lambda , \frac{2^{k+1}\beta _1}{\vert c_j\vert },\nu \right) {\setminus } {\mathcal {H}}\left( \lambda ,\frac{2^k\beta _1}{\vert c_j\vert },\nu \right) \end{aligned}$$

such that

$$\begin{aligned} \vert {{\,\mathrm{Im}\,}}w-y_{n}^j\vert \le \theta _1. \end{aligned}$$

Also, let $\tilde{{\mathcal {Q}}}_{n,k}^j$ be the set of all

$$\begin{aligned} w\in {\mathcal {H}}\left( \lambda ,\frac{2^{k+2}\beta _1}{\vert c_j\vert },\nu \right) {\setminus } {\mathcal {H}}\left( \lambda ,\frac{2^{k-1}\beta _1}{\vert c_j\vert },\nu \right) \end{aligned}$$

such that

$$\begin{aligned} \vert {{\,\mathrm{Im}\,}}w-y_{n}^j\vert \le 5\pi \theta _1. \end{aligned}$$

See Fig. 3 for an illustration of these sets. Note that ${\mathcal {Q}}_{n,k}^j\subset \tilde{{\mathcal {Q}}}_{n,k}^j$. If $\tilde{{\mathcal {Q}}}_{n,k}^j \subset {\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu )$, then by Lemma 8.3, we have $h_j(\tilde{Q}_{n,k}^j)\subset {\mathcal {H}}(\lambda ,1/c^*,\nu )$.

Fig. 3

An illustration of the sets $Q_{n,k}^j$ and $\tilde{Q}_{n,k}^j$

Moreover, let ${\mathcal {R}}_{n,k}^j$ be the rectangle containing all $v\in {\mathbb {C}}$ satisfying

$$\begin{aligned} \frac{3}{4}2^k\beta _1<{{\,\mathrm{Re}\,}}v-\lambda \log \vert n\vert <\frac{5}{2}2^{k}\beta _1 \end{aligned}$$

and

$$\begin{aligned} \vert {{\,\mathrm{Im}\,}}v-y_{n}^j\vert <3\cdot 2^{k}\beta _1\theta _1. \end{aligned}$$

Also, let $\tilde{{\mathcal {R}}}_{n,k}^j$ be the rectangle containing all $v\in {\mathbb {C}}$ satisfying

$$\begin{aligned} \frac{5}{8}2^{k}\beta _1<{{\,\mathrm{Re}\,}}v-\lambda \log \vert n\vert <3\cdot 2^{k}\beta _1 \end{aligned}$$

and

$$\begin{aligned} \vert {{\,\mathrm{Im}\,}}v-y_{n}^j\vert <4\cdot 2^{k}\beta _1\theta _1. \end{aligned}$$

Note that ${\mathcal {R}}_{n,k}^j\subset \tilde{{\mathcal {R}}}_{n,k}^j$.

Lemma 8.4

There are $\alpha _1, \beta _1, \nu , n_0>0$ such that the following holds. If $j\in \{1,\ldots ,d\},$ $n\in {\mathbb {Z}}$ with $\vert n\vert \ge n_0$ and $k\in {\mathbb {N}}$ are such that $\tilde{{\mathcal {Q}}}_{n,k}^j\subset {\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ),$ then $h_j({\mathcal {Q}}_{n,k}^j)\subset {\mathcal {R}}_{n,k}^j$ and $h_j(\tilde{{\mathcal {Q}}}_{n,k}^j)\supset \tilde{{\mathcal {R}}}_{n,k}^j.$

Proof

For $w\in {\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ){\setminus }{\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$, we have ${{\,\mathrm{Re}\,}}w=o(\vert w\vert )$ and hence $\vert w\vert =(1+o(1))\vert {{\,\mathrm{Im}\,}}w\vert$ as $w\rightarrow \infty$. If $\vert {{\,\mathrm{Im}\,}}w-y_n^j\vert \le 5\pi \theta _1$ and $\vert n\vert$ is sufficiently large, we obtain

$$\begin{aligned} \vert n\vert \le \vert w\vert \le e^2\vert n\vert . \end{aligned}$$

For $w\in \tilde{{\mathcal {Q}}}_{n,k}^j$, this implies that

$$\begin{aligned} {{\,\mathrm{Re}\,}}w\ge \lambda \log \vert w\vert -\log \frac{2^{k+2} \beta _1}{\vert c_j\vert }\ge \lambda \log \vert n\vert -2\vert \lambda \vert -\log \frac{2^{k+2}\beta _1}{\vert c_j\vert } \end{aligned}$$

(8.12)

and

$$\begin{aligned} {{\,\mathrm{Re}\,}}w\le \lambda \log \vert w\vert -\log \frac{2^{k-1} \beta _1}{\vert c_j\vert }\le \lambda \log \vert n\vert +2\vert \lambda \vert -\log \frac{2^{k-1}\beta _1}{\vert c_j\vert }. \end{aligned}$$

(8.13)

Let $\varepsilon >0$ be small and let $w\in {\mathcal {Q}}_{n,k}^j\subset \tilde{{\mathcal {Q}}}_{n,k}^j$. By Lemma 8.2 and (8.12), and since $0<\theta _1<1/(6\pi )\arccos (5/6)<(1/2)\arccos (5/6)$, we have

$$\begin{aligned} {{\,\mathrm{Re}\,}}h_j(w)&\ge {{\,\mathrm{Re}\,}}w+(1-\varepsilon )2^k\beta _1\cos (\theta _1+\varepsilon )\\&\ge {{\,\mathrm{Re}\,}}w+(1-\varepsilon )2^k\beta _1\cos (2\theta _1)\\&>\lambda \log \vert n\vert -2\vert \lambda \vert -\log \frac{2^{k+2}\beta _1}{\vert c_j\vert }+(1-\varepsilon )2^k \beta _1\cdot \frac{5}{6}\\&>\lambda \log \vert n\vert +\frac{3}{4}2^k\beta _1 \end{aligned}$$

if $\varepsilon$ is sufficiently small and $\beta _1$ is sufficiently large. Analogously,

$$\begin{aligned} {{\,\mathrm{Re}\,}}h_j(w)&\le {{\,\mathrm{Re}\,}}w+(1+\varepsilon ) 2^{k+1}\beta _1\\&\le \lambda \log \vert n\vert +2\vert \lambda \vert -\log \frac{2^{k-1}\beta _1}{\vert c_j\vert }+(1+\varepsilon )2^{k+1}\beta _1\\&<\lambda \log \vert n\vert +\frac{5}{2}2^{k}\beta _1 \end{aligned}$$

if $\varepsilon$ is sufficiently small and $\beta _1$ is sufficiently large. Moreover, by Lemma 8.2,

$$\begin{aligned} \vert {{\,\mathrm{Im}\,}}h_j(w)-y_{n}^j\vert&\le \vert {{\,\mathrm{Im}\,}}h_j(w) -{{\,\mathrm{Im}\,}}w\vert +\vert {{\,\mathrm{Im}\,}}w-y_n^j\vert \\&\le (1+\varepsilon )2^{k+1}\beta _1\sin (\theta _1+\varepsilon )+\vert {{\,\mathrm{Im}\,}}w -y_{n}^j\vert \\&\le (1+\varepsilon )2^{k+1}\beta _1(\theta _1+\varepsilon )+\theta _1\\&<3\cdot 2^{k}\beta _1\theta _1 \end{aligned}$$

if $\varepsilon$ is sufficiently small and $\beta _1$ is sufficiently large. Thus, $h_j({\mathcal {Q}}_{n,k}^j)\subset {\mathcal {R}}_{n,k}^j$.

In the following, we show that $h_j(\partial \tilde{{\mathcal {Q}}}_{n,k}^j)\cap \tilde{{\mathcal {R}}}_{n,k}^j=\emptyset$. Since we have already shown that $h_j(\tilde{{\mathcal {Q}}}_{n,k}^j)\cap \tilde{{\mathcal {R}}}_{n,k}^j\ne \emptyset$, this implies that $h_j(\tilde{{\mathcal {Q}}}_{n,k}^j)\supset \tilde{{\mathcal {R}}}_{n,k}^j.$

If $w\in \Gamma (\lambda ,2^{k-1}\beta _1/\vert c_j\vert )$ and $\beta _1$ is large, then by Lemma 8.2 and (8.13),

$$\begin{aligned} {{\,\mathrm{Re}\,}}h_j(w)&\le {{\,\mathrm{Re}\,}}w+(1+\varepsilon )2^{k-1}\beta _1\\&\le \lambda \log \vert n\vert +2\vert \lambda \vert -\log \frac{2^{k-1}\beta _1}{\vert c_j\vert } +(1+\varepsilon )2^{k-1}\beta _1\\&<\lambda \log \vert n\vert +\frac{5}{8}2^{k}\beta _1. \end{aligned}$$

If $w\in \Gamma (\lambda ,2^{k+2}\beta _1/\vert c_j\vert )$ and $\vert {{\,\mathrm{Im}\,}}w-y_{n}^j\vert \le 5\pi \theta _1$, then by Lemma 8.2 and (8.12), and since $0<\theta _1<1/(6\pi )\arccos (5/6)$, we have

$$\begin{aligned} {{\,\mathrm{Re}\,}}h_j(w)&\ge {{\,\mathrm{Re}\,}}w+(1-\varepsilon )2^{k+2} \beta _1\cos (5\pi \theta _1+\varepsilon )\\&\ge \lambda \log \vert n\vert -2\vert \lambda \vert -\log \frac{2^{k+2}\beta _1}{\vert c_j\vert } +(1-\varepsilon )2^{k+2}\beta _1\cos (6\pi \theta _1)\\&>\lambda \log \vert n\vert -2\vert \lambda \vert -\log \frac{2^{k+2}\beta _1}{\vert c_j\vert } +(1-\varepsilon )2^{k+2}\beta _1\cdot \frac{5}{6}\\&>\lambda \log \vert n\vert +3\cdot 2^{k}\beta _1, \end{aligned}$$

provided $\varepsilon$ is sufficiently small and $\beta _1$ is sufficiently large.

If $\vert {{\,\mathrm{Im}\,}}w-y_{n}^j\vert =5\pi \theta _1$ and $w\in {\mathcal {H}}(\lambda ,2^{k+2}\beta _1/\vert c_j\vert ,\nu ){\setminus } {\mathcal {H}}(\lambda ,2^{k-1}\beta _1/\vert c_j\vert ,\nu )$, then by Lemma 8.2,

$$\begin{aligned} \vert {{\,\mathrm{Im}\,}}h_j(w)-y_{n}^j\vert&\ge \vert {{\,\mathrm{Im}\,}}h_j(w) -{{\,\mathrm{Im}\,}}(w)\vert -\vert {{\,\mathrm{Im}\,}}w-y_{n}^j\vert \\&\ge (1-\varepsilon )2^{k-1}\beta _1\sin (5\pi \theta _1-\varepsilon )-5\pi \theta _1\\&\ge (1-\varepsilon )2^{k-1}\beta _1\frac{2}{\pi } (5\pi \theta _1-\varepsilon )-5\pi \theta _1\\&>4\cdot 2^{k}\beta _1\theta _1, \end{aligned}$$

provided $\varepsilon$ is sufficiently small and $\beta _1$ is sufficiently large. Thus, $h_j(\partial \tilde{{\mathcal {Q}}}_{n,k}^j)\subset {\mathbb {C}}{\setminus } \tilde{{\mathcal {R}}}_{n,k}^j$. $\square$

Next, we prove that the density of $q({\mathcal {F}}_j)$ in $\tilde{Q}_{n,k}^j$ is bounded below by a positive constant.

Lemma 8.5

There are $\alpha _1,\beta _1,\nu , \delta , n_1>0$ such that for all $j\in \{1,\ldots ,d\}$, $n\in {\mathbb {Z}}$ with $\vert n\vert \ge n_1$ and $k\in {\mathbb {N}}$ with $\tilde{{\mathcal {Q}}}_{n,k}^j\subset {\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ),$ we have

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), \tilde{{\mathcal {Q}}}_{n,k}^j)\ge \delta . \end{aligned}$$

Proof

By Sect. 6, in particular Lemma 6.5, the function $h_j=q\circ f\circ \varphi _j$ has no critical points in $\tilde{{\mathcal {Q}}}_{n,k}^j$ if $\nu$ and $\beta _1$ are sufficiently large. By Lemma 8.4, $h_j(\tilde{{\mathcal {Q}}}_{n,k}^j)\supset \tilde{{\mathcal {R}}}_{n,k}^j$ and $h_j({\mathcal {Q}}_{n,k}^j)\subset {\mathcal {R}}_{n,k}^j$. Let ${\mathcal {U}}$ be the component of $h_j^{-1}(\tilde{{\mathcal {R}}}_{n,k}^j)$ containing ${\mathcal {Q}}_{n,k}^j$. Then ${\mathcal {Q}}_{n,k}^j\subset {\mathcal {U}}\subset \tilde{{\mathcal {Q}}}_{n,k}^j$. Since $\tilde{{\mathcal {R}}}_{n,k}^j$ is simply connected, $h_j$ maps ${\mathcal {U}}$ conformally onto $\tilde{{\mathcal {R}}}_{n,k}^j$. Let $\psi :\,\tilde{{\mathcal {R}}}_{n,k}^j\rightarrow {\mathcal {U}}$ be the corresponding inverse function. By Lemma 8.3,

$$\begin{aligned} h_j(\tilde{{\mathcal {Q}}}_{n,k}^j)\subset {\mathcal {H}}\left( \lambda ,\frac{1}{c^*}, \nu \right) \subset {\mathbb {C}}{\setminus } (\overline{{\mathcal {D}}(0,R)}\cup [0,\infty )) =q({\mathcal {S}}_l) \end{aligned}$$

for all $l\in \{1,\ldots ,d\}$. Hence, there exists $l\in \{1,\ldots ,d\}$ such that $(f\circ \varphi _j)(\tilde{{\mathcal {Q}}}_{n,k}^j)\subset {\mathcal {S}}_l$. We have $\psi (q({\mathcal {F}}_l)\cap ~\tilde{{\mathcal {R}}}_{n,k}^j)=q({\mathcal {F}}_j)\cap {\mathcal {U}}$. By the Koebe distortion theorem, $\psi$ has bounded distortion in ${\mathcal {R}}_{n,k}^j$ independent of n, k and j. We obtain

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), \tilde{{\mathcal {Q}}}_{n,k}^j)&\ge {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), \psi ({\mathcal {R}}_{n,k}^j))\cdot {{\,\mathrm{dens}\,}}(\psi ({\mathcal {R}}_{n,k}^j), \tilde{{\mathcal {Q}}}_{n,k}^j)\\&={{\,\mathrm{dens}\,}}(\psi (q({\mathcal {F}}_l)\cap {\mathcal {R}}_{n,k}^j), \psi ({\mathcal {R}}_{n,k}^j)) \cdot {{\,\mathrm{dens}\,}}(\psi ({\mathcal {R}}_{n,k}^j), \tilde{{\mathcal {Q}}}_{n,k}^j)\\&\ge c{{\,\mathrm{dens}\,}}(q({\mathcal {F}}_l), {\mathcal {R}}_{n,k}^j)\cdot {{\,\mathrm{dens}\,}}({\mathcal {Q}}_{n,k}^j, \tilde{{\mathcal {Q}}}_{n,k}^j) \end{aligned}$$

for some $c>0$ independent of n, k and j. If $\beta _1$ is sufficiently large, then by Lemma 7.14,

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_l), {\mathcal {R}}_{n,k}^j)\ge \eta _0. \end{aligned}$$

Moreover, by Lemma 5.2,

$$\begin{aligned} {{\,\mathrm{meas}\,}}{\mathcal {Q}}_{n,k}^j\ge \frac{2}{3}\log 2\cdot 2\theta _1 \quad \text { and } {{\,\mathrm{meas}\,}}\tilde{{\mathcal {Q}}}_{n,k}^j\le 2\log 8\cdot 10\pi \theta _1. \end{aligned}$$

Hence,

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), \tilde{{\mathcal {Q}}}_{n,k}^j)\ge c\eta _0 \frac{\log 2}{15\pi \log 8} =:\delta . \end{aligned}$$

$\square$

The last lemma of this section says that there is a positive lower bound for the density of $q({\mathcal {F}}_j)$ in any sufficiently large rectangle contained in ${\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ) {\setminus }{\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu ).$

Lemma 8.6

There are $\alpha _1, \beta _1, \nu , D_1,\eta _1>0$ such that for all $j\in \{1,\ldots ,d\}$ and any rectangle ${\mathcal {R}}\subset {\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ){\setminus } {\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$ with sides parallel to the real and imaginary axis and side lengths at least $D_1,$ we have

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), {\mathcal {R}})\ge \eta _1. \end{aligned}$$

Proof

Suppose that

$$\begin{aligned}&D_1\ge 5\log 8,\\&D_1\ge y_{n_1}^l+2\pi +10\pi \theta _1 \text { and }D_1\ge \vert y_{-n_1}^l\vert +2\pi +10 \pi \theta _1\text { for all } l\in \{1,\ldots ,d\}, \end{aligned}$$

with $n_1$ as in Lemma 8.5. Let ${\mathcal {R}}\subset {\mathcal {H}}(\lambda -1,\alpha _1/\vert c_j\vert ,\nu ){\setminus } {\mathcal {H}}(\lambda ,\beta _1/\vert c_j\vert ,\nu )$ be a rectangle with sides parallel to the real and imaginary axis and side lengths at least $D_1$. By the definition of $\tilde{{\mathcal {Q}}}_{n,k}^j$ and Lemma 5.2, there are $k\in {\mathbb {N}}$ and $n\in {\mathbb {Z}}$ with $\vert n\vert \ge n_1$ such that

$$\begin{aligned} \tilde{{\mathcal {Q}}}_{n,k}^j\subset {\mathcal {R}}. \end{aligned}$$

If, in addition, the side lengths of ${\mathcal {R}}$ do not exceed $2D_1$, then by Lemmas 8.5 and 5.2,

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j),{\mathcal {R}})\ge {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), \tilde{{\mathcal {Q}}}_{n,k}^j) \cdot {{\,\mathrm{dens}\,}}(\tilde{{\mathcal {Q}}}_{n,k}^j,{\mathcal {R}})\ge \delta \frac{2/3\log 8 \cdot 10\pi \theta _1}{4D_1^2}. \end{aligned}$$

Since any general rectangle with side lengths at least $D_1$ can be written as the union of rectangles with side lengths between $D_1$ and $2D_1$ which are disjoint up to the boundary, the claim follows. $\square$

9 The set $q({\mathcal {F}}(f))$: third part

For $\nu >0$, let

$$\begin{aligned} {\mathcal {G}}_\nu :=\{w:\,\vert {{\,\mathrm{Im}\,}}w\vert \ge \nu \}. \end{aligned}$$

In this section, we investigate the density of $q({\mathcal {F}}(f))$ in subsets of ${\mathcal {G}}_\nu {\setminus }{\mathcal {H}}(\lambda -1,\beta _2/\vert c_j\vert ,\nu )$ for large $\beta _2>0$. First, we give an approximation for $h_j$ in ${\mathcal {G}}_\nu {\setminus }{\mathcal {H}}(\lambda -1,\beta _2/\vert c_j\vert ,\nu )$.

Lemma 9.1

Let $\varepsilon >0$ and $j\in \{1,\ldots ,d\}$. Then there are $\beta _2,\nu >0$ such that for all $w\in {\mathcal {G}}_\nu {\setminus } {\mathcal {H}}(\lambda -1,\beta _2/\vert c_j\vert ,\nu ),$ we have

$$\begin{aligned} \left| \frac{h_j(w)}{(-c_j/d)^de^{-dw}w^{-m}}-1\right| <\varepsilon . \end{aligned}$$

Proof

By Corollary 4.3,

$$\begin{aligned} f(\varphi _j(w))&=\varphi _j(w)-\frac{1}{q'(\varphi _j(w))} \left( 1+O\left( \frac{1}{\vert w\vert }\right) \right) -\frac{c_je^{-w}}{p(\varphi _j(w))}\nonumber \\&=O(\vert w\vert ^{1/d})-\frac{c_j}{d}e^{-w}\varphi _j(w)^{-m} \left( 1+O\left( \frac{1}{\vert w\vert ^{1/d}}\right) \right) \end{aligned}$$

(9.1)

as $w\rightarrow \infty$. Note that the $O(\cdot )$-terms do not depend on $\beta _2$. For $w\in {\mathcal {G}}_\nu {\setminus } {\mathcal {H}}(\lambda -1,\beta _2/\vert c_j\vert ,\nu )$, we have

$$\begin{aligned} \left| \frac{c_j}{d}e^{-w}\varphi _j(w)^{-m}\right| =\left| \frac{c_j}{d}e^{-w}\right| \cdot \vert w \vert ^{\lambda -1+1/d}(1+o(1))\ge \frac{\beta _2\vert w\vert ^{1/d}}{2d} \end{aligned}$$

(9.2)

if $\vert w\vert$ is sufficiently large. In particular,

$$\begin{aligned} \vert f(\varphi _j(w))\vert \ge \frac{\beta _2}{4d}\vert w\vert ^{1/d} \end{aligned}$$

if $\beta _2$ and $\vert w\vert$ are sufficiently large, and hence

$$\begin{aligned} h_j(w)=q(f(\varphi _j(w)))=f(\varphi _j(w))^d \left( 1+O\left( \frac{1}{\vert w\vert ^{1/d}}\right) \right) \end{aligned}$$

as $w\rightarrow \infty$ in ${\mathcal {G}}_\nu {\setminus } {\mathcal {H}}(\lambda -1,\beta _2/\vert c_j\vert ,\nu )$. Also, by (9.1) and (9.2),

$$\begin{aligned} \left| \frac{f(\varphi _j(w))}{(-c_j/d)e^{-w}\varphi _j(w)^{-m}} -1\right|&=\left| \frac{O(\vert w\vert ^{1/d})}{(-c_j/d) e^{-w}\varphi _j(w)^{-m}}+O\left( \frac{1}{\vert w\vert ^{1/d}} \right) \right| \\&\le \frac{2d}{\beta _2}O(1) +O\left( \frac{1}{\vert w\vert ^{1/d}}\right) , \end{aligned}$$

where the $O(\cdot )$-terms do not depend on $\beta _2$. Hence, we can achieve that

$$\begin{aligned} \left| \frac{f(\varphi _j(w))^d}{((-c_j/d)e^{-w} \varphi _j(w)^{-m})^d}-1\right| \le \frac{\varepsilon }{2} \end{aligned}$$

by taking $\beta _2$ and $\nu$ sufficiently large. Also,

$$\begin{aligned} \left( -\frac{c_j}{d}e^{-w}\varphi _j(w)^{-m}\right) ^d =\left( -\frac{c_j}{d}\right) ^de^{-dw}w^{-m} \left( 1+O\left( \frac{1}{\vert w\vert ^{1/d}}\right) \right) \end{aligned}$$

as $w\rightarrow \infty$, whence the claim follows. $\square$

We proceed similarly as in Sect. 8, that is, we show that $h_j$ maps certain subsets of ${\mathcal {G}}_\nu {\setminus }{\mathcal {H}}(\lambda -1,\beta _2/\vert c_j\vert ,\nu )$ into ${\mathcal {H}}(\lambda ,1/c^*,\nu )$. We then apply the results of Sect. 7 to show that $q({\mathcal {F}}(f))$ has positive density in large bounded subsets of ${\mathcal {G}}_\nu {\setminus }{\mathcal {H}}(\lambda -1,\beta _2/\vert c_j\vert ,\nu ).$

For $n\in {\mathbb {Z}},\,k\in {\mathbb {N}}$ and $j\in \{1,\ldots ,d\}$, let ${\mathcal {P}}_{n,k}^j$ be the set of all

$$\begin{aligned} w\in {\mathcal {H}}\left( \lambda -1,\frac{2^{k+2}\beta _2}{\vert c_j\vert },\nu \right) {\setminus } {\mathcal {H}}\left( \lambda -1,\frac{2^{k-1}\beta _2}{\vert c_j\vert }, \nu \right) \end{aligned}$$

satisfying

$$\begin{aligned} \frac{(2n-1)\pi }{d}\le {{\,\mathrm{Im}\,}}w\le \frac{2(n+1)\pi }{d}. \end{aligned}$$

There are $\theta _{n,k}^j\in [-\pi ,\pi )$ and $r_{n,k}^j>0$ such that for all $w\in {\mathcal {P}}_{n,k}^j$, we have

$$\begin{aligned} \vert w\vert =r_{n,k}^j(1+o(1))\quad \text {and } \arg (w)=\theta _{n,k}^j+o(1) \end{aligned}$$

as $\vert n\vert \rightarrow \infty$. Let $t_{n,k}^j\in [2n\pi /d,2(n+1)\pi /d)$ with

$$\begin{aligned} t_{n,k}^j\equiv \arg (-c_j)-\frac{m}{d}\theta _{n,k}^j\mod \frac{2\pi }{d}. \end{aligned}$$

Lemma 9.2

Let $\theta ^*\in (0,\pi /(4d))$. Then there are $\beta _2,\nu >0$ such that the following holds. Let $j\in \{1,\ldots ,d\},$ $k\in {\mathbb {N}}$ and $w\in {\mathcal {H}}(\lambda -1,2^{k+2}\beta _2/\vert c_j\vert ,\nu ) {\setminus }{\mathcal {H}}(\lambda -1,2^{k-1}\beta _2/\vert c_j\vert ,\nu )$ such that there exists $n\in {\mathbb {Z}}$ with $t_{n,k}^j-\pi /(4d)\le {{\,\mathrm{Im}\,}}w\le t_{n,k}^j-\theta ^*$. Let $\beta \in [2^{k-1}\beta _2,2^{k+2}\beta _2]$ such that $w\in \Gamma (\lambda -1,\beta /\vert c_j\vert )$ and let $\theta :=t_{n,k}^j-{{\,\mathrm{Im}\,}}w$. Then

$$\begin{aligned} \frac{3}{4}\left( \frac{\beta }{d}\right) ^dr_{n,k}^j \cos (2d\theta )<{{\,\mathrm{Re}\,}}h_j(w)<\frac{5}{4}\left( \frac{\beta }{d}\right) ^d r_{n,k}^j \end{aligned}$$

and

$$\begin{aligned} \frac{3}{4\pi }\left( \frac{\beta }{d}\right) ^dr_{n,k}^jd \theta<{{\,\mathrm{Im}\,}}h_j(w)<\frac{5}{2}\left( \frac{\beta }{d}\right) ^d r_{n,k}^jd\theta . \end{aligned}$$

Proof

Let $\varepsilon >0$ be small. By Lemma 9.1,

$$\begin{aligned} \left| \frac{h_j(w)}{(-c_j/d)^de^{-dw}w^{-m}}-1\right| <\varepsilon \end{aligned}$$

if $\beta _2$ and $\nu$ are sufficiently large. Thus,

$$\begin{aligned} \left( 1-\varepsilon \right) \left( \frac{\vert c_j\vert }{d}\right) ^d e^{-d{{\,\mathrm{Re}\,}}w}\vert w\vert ^{-m}\le \vert h_j(w)\vert \le \left( 1+\varepsilon \right) \left( \frac{\vert c_j\vert }{d}\right) ^d e^{-d{{\,\mathrm{Re}\,}}w}\vert w\vert ^{-m}. \end{aligned}$$

(9.3)

Since $w\in \Gamma (\lambda -1,\beta /\vert c_j\vert )$, we have

$$\begin{aligned} \vert w\vert ^{-1-m}e^{-d{{\,\mathrm{Re}\,}}w}=\vert w\vert ^{d(\lambda -1)} e^{-d{{\,\mathrm{Re}\,}}w}=\left( \frac{\beta }{\vert c_j\vert }\right) ^d. \end{aligned}$$

Thus,

$$\begin{aligned} \left( \frac{\vert c_j\vert }{d}\right) ^de^{-d{{\,\mathrm{Re}\,}}w}\vert w \vert ^{-m}=\left( \frac{\beta }{d}\right) ^d\vert w\vert =\left( \frac{\beta }{d}\right) ^dr_{n,k}^j(1+o(1)) \end{aligned}$$

as $\vert n\vert \rightarrow \infty$. Inserting the last equation into (9.3) yields

$$\begin{aligned} \frac{3}{4}\left( \frac{\beta }{d}\right) ^dr_{n,k}^j<\vert h_j(w) \vert <\frac{5}{4}\left( \frac{\beta }{d}\right) ^dr_{n,k}^j \end{aligned}$$

(9.4)

if $\varepsilon$ is sufficiently small and $\vert n\vert$ is sufficiently large. Also, by Lemma 9.1,

$$\begin{aligned} \left| \arg (h_j(w))-\arg \left( \left( -\frac{c_j}{d}\right) ^d e^{-dw}w^{-m}\right) \right| <\arcsin (\varepsilon )\le \frac{\pi }{2}\varepsilon . \end{aligned}$$

(9.5)

We have

$$\begin{aligned} \arg \left( \left( -\frac{c_j}{d}\right) ^de^{-dw}w^{-m}\right)&\equiv d\arg (-c_j)-d{{\,\mathrm{Im}\,}}w-m\arg w\\&\equiv d\arg (-c_j)-dt_{n,k}^j+d\theta -m\theta _{n,k}^j+o(1)\\&\equiv d\theta +o(1) \mod 2\pi \end{aligned}$$

as $\vert n\vert \rightarrow \infty$. By (9.5), this yields

$$\begin{aligned} \frac{d\theta }{2}<\arg h_j(w)<2d\theta \end{aligned}$$

(9.6)

if $\varepsilon$ is sufficiently small compared to $\theta ^*$. By (9.4), (9.6) and the fact that $(2/\pi )x\le \sin x\le x$ for $0\le x\le \pi /2$, we obtain

$$\begin{aligned} {{\,\mathrm{Re}\,}}h_j(w)&\le \vert h_j(w)\vert<\frac{5}{4} \left( \frac{\beta }{d}\right) ^dr_{n,k}^j,\\ {{\,\mathrm{Re}\,}}h_j(w)&=\vert h_j(w)\vert \cos (\arg h_j(w))>\frac{3}{4}\left( \frac{\beta }{d}\right) ^dr_{n,k}^j\cos (2d\theta ),\\ {{\,\mathrm{Im}\,}}h_j(w)&=\vert h_j(w)\vert \sin (\arg h_j(w))<\frac{5}{4} \left( \frac{\beta }{d}\right) ^dr_{n,k}^j\sin (2d\theta ) \le \frac{5}{2}\left( \frac{\beta }{d}\right) ^dr_{n,k}^jd\theta ,\\ {{\,\mathrm{Im}\,}}h_j(w)&=\vert h_j(w)\vert \sin (\arg h_j(w))>\frac{3}{4} \left( \frac{\beta }{d}\right) ^dr_{n,k}^j\sin \left( \frac{d\theta }{2}\right) \ge \frac{3}{4\pi } \left( \frac{\beta }{d}\right) ^dr_{n,k}^jd\theta . \end{aligned}$$

$\square$

Let us now define several sets. We start with subsets ${\mathcal {T}}_{n,k}^j,\tilde{T}_{n,k}^j\subset {\mathcal {G}}_\nu {\setminus } {\mathcal {H}}(\lambda -1,\beta _2/\vert c_j\vert ,\nu )$. Let

$$\begin{aligned} 0<\theta _2<\frac{1}{2\cdot 4^{d+1}d\pi }\arccos \left( \frac{11}{12}\right) . \end{aligned}$$

For $n\in {\mathbb {Z}}$, $k\in {\mathbb {N}}$ and $j\in \{1,\ldots ,d\}$, let ${\mathcal {T}}_{n,k}^j$ be the set of all

$$\begin{aligned} w\in {\mathcal {H}}\left( \lambda -1,\frac{2^{k+1}\beta _2}{\vert c_j\vert },\nu \right) {\setminus } {\mathcal {H}}\left( \lambda -1,\frac{2^k\beta _2}{\vert c_j\vert },\nu \right) \end{aligned}$$

satisfying

$$\begin{aligned} t_{n,k}^j-\theta _2\le {{\,\mathrm{Im}\,}}w\le t_{n,k}^j-\frac{\theta _2}{2}. \end{aligned}$$

Also, let $\tilde{{\mathcal {T}}}_{n,k}^j$ be the set of all

$$\begin{aligned} w\in {\mathcal {H}}\left( \lambda -1,\frac{2^{k+2}\beta _2}{\vert c_j\vert },\nu \right) {\setminus } {\mathcal {H}}\left( \lambda -1,\frac{2^{k-1}\beta _2}{\vert c_j\vert },\nu \right) \end{aligned}$$

satisfying

$$\begin{aligned} t_{n,k}^j-4^{d+1}\pi \theta _2\le {{\,\mathrm{Im}\,}}w\le t_{n,k}^j -\frac{1}{10\cdot 4^d\pi }\theta _2. \end{aligned}$$

Note that ${\mathcal {T}}_{n,k}^j\subset \tilde{{\mathcal {T}}}_{n,k}^j$. See Fig. 4 for an illustration of ${\mathcal {T}}_{n,k}^j$ and $\tilde{{\mathcal {T}}}_{n,k}^j$.

Fig. 4

An illustration of the sets ${\mathcal {T}}_{n,k}^j$ and $\tilde{{\mathcal {T}}}_{n,k}^j$

Moreover, let ${\mathcal {U}}_{n,k}^j$ be the rectangle containing all $v\in {\mathbb {C}}$ satisfying

$$\begin{aligned} \frac{11}{16}\left( \frac{2^{k}\beta _2}{d}\right) ^dr_{n,k}^j<{{\,\mathrm{Re}\,}}v<\frac{5}{4}\left( \frac{2^{k+1}\beta _2}{d}\right) ^dr_{n,k}^j \end{aligned}$$

and

$$\begin{aligned} \frac{3}{8\pi }\left( \frac{2^k\beta _2}{d}\right) ^dr_{n,k}^j d\theta _2<{{\,\mathrm{Im}\,}}v<\frac{5}{2}\left( \frac{2^{k+1}\beta _2}{d}\right) ^d r_{n,k}^jd\theta _2. \end{aligned}$$

Also, let $\tilde{{\mathcal {U}}}_{n,k}^j$ be the rectangle containing all $v\in {\mathbb {C}}$ satisfying

$$\begin{aligned} \frac{5}{8}\left( \frac{2^{k}\beta _2}{d}\right) ^dr_{n,k}^j<{{\,\mathrm{Re}\,}}v <\frac{11}{8}\left( \frac{2^{k+1}\beta _2}{d}\right) ^dr_{n,k}^j \end{aligned}$$

and

$$\begin{aligned} \frac{1}{4\pi }\left( \frac{2^{k}\beta _2}{d}\right) ^dr_{n,k}^j d\theta _2<{{\,\mathrm{Im}\,}}v<3\left( \frac{2^{k+1}\beta _2}{d}\right) ^d r_{n,k}^jd\theta _2. \end{aligned}$$

Note that ${\mathcal {U}}_{n,k}^j\subset \tilde{{\mathcal {U}}}_{n,k}^j$.

Lemma 9.3

There is $n_0\in {\mathbb {N}}$ such that for all $n\in {\mathbb {Z}}$ with $\vert n\vert \ge n_0,$ $k\in {\mathbb {N}}$ and $j\in \{1,\ldots ,d\},$ we have

$$\begin{aligned} \tilde{{\mathcal {U}}}_{n,k}^j\subset {\mathcal {H}}\left( \lambda ,\frac{1}{c^*},\nu \right) \end{aligned}$$

with $c^*=\max _l\vert c_l\vert$ as defined in (8.9).

Proof

Let $v\in \tilde{{\mathcal {U}}}_{n,k}^j$. Note that $r_{n,k}^j\rightarrow \infty$ as $\vert n\vert \rightarrow \infty$ uniformly in k. In particular,

$$\begin{aligned} {{\,\mathrm{Im}\,}}v>\frac{1}{4\pi }\left( \frac{2^{k}\beta _2}{d}\right) ^d r_{n,k}^jd\theta _2\ge \nu \end{aligned}$$

if $\vert n\vert$ is sufficiently large. Also,

$$\begin{aligned} {{\,\mathrm{Im}\,}}v<3\left( \frac{2^{k+1}\beta _2}{d}\right) ^dr_{n,k}^j d\theta _2=\frac{24}{5}2^dd\theta _2\cdot \frac{5}{8} \left( \frac{2^k\beta _2}{d}\right) ^dr_{n,k}^j <\frac{24}{5}2^dd\theta _2{{\,\mathrm{Re}\,}}v, \end{aligned}$$

and hence

$$\begin{aligned} \vert v\vert \le \vert {{\,\mathrm{Re}\,}}v\vert +\vert {{\,\mathrm{Im}\,}}v\vert <\left( 1+\frac{24}{5}2^dd\theta _2\right) {{\,\mathrm{Re}\,}}v. \end{aligned}$$

Thus,

$$\begin{aligned} {{\,\mathrm{Re}\,}}v\ge \frac{1}{1+(24/5)2^dd\theta _2}\vert v\vert \ge \lambda \log \vert v\vert -\log \frac{1}{c^*} \end{aligned}$$

if $\vert n\vert$ and hence $r_{n,k}^j$ and $\vert v\vert$ are sufficiently large. $\square$

Lemma 9.4

There are $\beta _2, \nu >0$ such that for all $j\in \{1,\ldots ,d\},$ $k\in {\mathbb {N}}$ and $n\in {\mathbb {Z}}$ with $\vert t_{n,k}^j\vert >\nu +4^{d+1}\pi \theta _2,$ we have

$$\begin{aligned} h_j({\mathcal {T}}_{n,k}^j)\subset {\mathcal {U}}_{n,k}^j\quad \text {and}\quad h_j (\tilde{{\mathcal {T}}}_{n,k}^j)\supset \tilde{{\mathcal {U}}}_{n,k}^j. \end{aligned}$$

Proof

First suppose that $w\in {\mathcal {T}}_{n,k}^j$. Then by Lemma 9.2 and the fact that $\theta _2<1/(2\cdot 4^{d+1}d\pi )\arccos (11/12)<1/(2d)\arccos (11/12),$ we have

$$\begin{aligned}&{{\,\mathrm{Re}\,}}h_j(w)>\frac{3}{4}\left( \frac{2^k\beta _2}{d}\right) ^d r_{n,k}^j\cos (2d\theta _2)>\frac{11}{16}\left( \frac{2^k \beta _2}{d}\right) ^dr_{n,k}^j,\\&{{\,\mathrm{Re}\,}}h_j(w)<\frac{5}{4}\left( \frac{2^{k+1}\beta _2}{d}\right) ^dr_{n,k}^j,\\&{{\,\mathrm{Im}\,}}h_j(w)>\frac{3}{8\pi }\left( \frac{2^k\beta _2}{d}\right) ^d r_{n,k}^jd \theta _2,\\&{{\,\mathrm{Im}\,}}h_j(w)<\frac{5}{2}\left( \frac{2^{k+1}\beta _2}{d}\right) ^d r_{n,k}^jd\theta _2. \end{aligned}$$

Hence, $h_j({\mathcal {T}}_{n,k}^j)\subset {\mathcal {U}}_{n,k}^j$.

Also, Lemma 9.2 yields the following. If $w\in \Gamma (\lambda -1,2^{k-1}\beta _2/\vert c_j\vert )$ with $t_{n,k}^j-4^{d+1}\pi \theta _2\le {{\,\mathrm{Im}\,}}w\le t_{n,k}^j-1/(10\cdot 4^d\pi )\theta _2$, then

$$\begin{aligned} {{\,\mathrm{Re}\,}}h_j(w)<\frac{5}{4}\left( \frac{2^{k-1}\beta _2}{d}\right) ^d r_{n,k}^j\le \frac{5}{8}\left( \frac{2^k\beta _2}{d}\right) ^dr_{n,k}^j. \end{aligned}$$

If $w\in \Gamma (\lambda -1,2^{k+2}\beta _2/\vert c_j\vert )$ with $t_{n,k}^j-4^{d+1}\pi \theta _2\le {{\,\mathrm{Im}\,}}w\le t_{n,k}^j-1/(10\cdot 4^d\pi )\theta _2$, then using that $\theta _2<1/(2\cdot 4^{d+1}d\pi )\arccos (11/12)$, we get

$$\begin{aligned} {{\,\mathrm{Re}\,}}h_j(w)>\frac{3}{4}\left( \frac{2^{k+2}\beta _2}{d}\right) ^d r_{n,k}^j\cos (2d4^{d+1}\pi \theta _2)>\frac{11}{8} \left( \frac{2^{k+1}\beta _2}{d}\right) ^dr_{n,k}^j. \end{aligned}$$

If $w\in {\mathcal {H}}(\lambda -1,2^{k+2}\beta _2/\vert c_j\vert ,\nu ){\setminus } {\mathcal {H}}(\lambda -1,2^{k-1}\beta _2/\vert c_j\vert ,\nu )$ and ${{\,\mathrm{Im}\,}}w = t_{n,k}^j-4^{d+1}\pi \theta _2$, then

$$\begin{aligned} {{\,\mathrm{Im}\,}}h_j(w)>3\left( \frac{2^{k+1}\beta _2}{d}\right) ^dr_{n,k}^jd \theta _2. \end{aligned}$$

If $w\in {\mathcal {H}}(\lambda -1,2^{k+2}\beta _2/\vert c_j\vert ,\nu ){\setminus } {\mathcal {H}}(\lambda -1,2^{k-1}\beta _2/\vert c_j\vert ,\nu )$ and ${{\,\mathrm{Im}\,}}w =t_{n,k}^j-1/(10\cdot 4^d\pi )\theta _2$, then

$$\begin{aligned} {{\,\mathrm{Im}\,}}h_j(w)<\frac{1}{4\pi }\left( \frac{2^{k}\beta _2}{d}\right) ^d r_{n,k}^jd\theta _2. \end{aligned}$$

Thus, $h_j(\partial \tilde{{\mathcal {T}}}_{n,k}^j)\cap \tilde{{\mathcal {U}}}_{n,k}^j =\emptyset$. Since ${\mathcal {T}}_{n,k}^j\subset \tilde{{\mathcal {T}}}_{n,k}^j$ and $h_j({\mathcal {T}}_{n,k}^j)\subset {\mathcal {U}}_{n,k}^j\subset \tilde{{\mathcal {U}}}_{n,k}^j$, we obtain that $h_j(\tilde{{\mathcal {T}}}_{n,k}^j)\supset \tilde{{\mathcal {U}}}_{n,k}^j.$ $\square$

Next, we show that the density of $q({\mathcal {F}}_j)$ in $\tilde{{\mathcal {T}}}_{n,k}^j$ is bounded below by a positive constant.

Lemma 9.5

There are $\delta >0$ and $n_1\in {\mathbb {N}}$ such that for all $j\in \{1,\ldots ,d\},$ $k\in {\mathbb {N}}$ and $n\in {\mathbb {Z}}$ with $\vert n\vert \ge n_1,$ we have

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), \tilde{{\mathcal {T}}}_{n,k}^j)\ge \delta . \end{aligned}$$

Proof

We only sketch the proof, since it is similar to the one of Lemma 8.5. By Lemma 9.3,

$$\begin{aligned} \tilde{{\mathcal {U}}}_{n,k}^j\subset {\mathcal {H}}\left( \lambda ,\frac{1}{c^*},\nu \right) . \end{aligned}$$

By Lemma 9.4, $h_j(\tilde{{\mathcal {T}}}_{n,k}^j)\supset \tilde{{\mathcal {U}}}_{n,k}^j$ and $h_j({\mathcal {T}}_{n,k}^j)\subset {\mathcal {U}}_{n,k}^j$. Let ${\mathcal {V}}\subset \tilde{{\mathcal {T}}}_{n,k}^j$ be the component of $h_j^{-1}(\tilde{{\mathcal {U}}}_{n,k}^j)$ containing ${\mathcal {T}}_{n,k}^j$. As in the proof of Lemma 8.5, we get that $f(\varphi _j({\mathcal {V}}))\subset {\mathcal {S}}_l$ for some $l\in \{1,\ldots ,d\}$, and that

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), \tilde{{\mathcal {T}}}_{n,k}^j)\ge c{{\,\mathrm{dens}\,}}(q({\mathcal {F}}_l), {\mathcal {U}}_{n,k}^j) \cdot {{\,\mathrm{dens}\,}}({\mathcal {T}}_{n,k}^j,\tilde{{\mathcal {T}}}_{n,k}^j) \end{aligned}$$

for some $c>0$ independent of n, k and j. If $\vert n\vert$ and hence $r_{n,k}^j$ is sufficiently large, then by Lemma 7.14,

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_l), {\mathcal {U}}_{n,k}^j)\ge \eta _0. \end{aligned}$$

Also, the density of ${\mathcal {T}}_{n,k}^j$ in $\tilde{{\mathcal {T}}}_{n,k}^j$ is bounded below independent of n, k and j, whence the claim follows. $\square$

The final result of this section says that the density of $q({\mathcal {F}}_j)$ in large rectangles in ${\mathcal {G}}_\nu {\setminus }{\mathcal {H}}(\lambda -1, \beta _2/\vert c_j\vert ,\nu )$ is bounded below.

Lemma 9.6

There are $\beta _2,\nu ,D_2,\eta _2>0$ such that for all $j\in \{1,\ldots ,d\}$ and any rectangle ${\mathcal {R}}\subset {\mathcal {G}}_\nu {\setminus } {\mathcal {H}}(\lambda -1,\beta _2/\vert c_j\vert ,\nu )$ with sides parallel to the real and imaginary axis and side lengths at least $D_2,$ we have

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j),{\mathcal {R}})\ge \eta _2. \end{aligned}$$

Proof

This is proved the same way as Lemma 8.6, using Lemma 9.5. $\square$

10 The set $q({\mathcal {F}}(f))$: conclusions

In this section, we combine the results of Sects. 7–9 to show that $q({\mathcal {F}}_j)$ has positive density in large bounded subsets of ${\mathbb {C}}$.

Lemma 10.1

There are $D,\eta _3>0$ such that for all $j\in \{1,\ldots ,d\}$ and any square ${\mathcal {R}}\subset {\mathbb {C}}$ with sides parallel to the real and imaginary axis and side lengths at least D,  we have

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j),{\mathcal {R}})\ge \eta _3. \end{aligned}$$

Proof

Let

$$\begin{aligned} {\mathcal {E}}_1:={\mathcal {H}}\left( \lambda ,\frac{\beta _1}{\vert c_j\vert },\nu \right) {\setminus } {\mathcal {H}}\left( \lambda ,\frac{1}{\vert c_j\vert },\nu \right) \end{aligned}$$

and

$$\begin{aligned} {\mathcal {E}}_2:={\mathcal {H}}\left( \lambda -1,\frac{\beta _2}{\vert c_j\vert },\nu \right) {\setminus } {\mathcal {H}}\left( \lambda -1,\frac{\alpha _1}{\vert c_j\vert },\nu \right) . \end{aligned}$$

Also, let $\gamma _1$ and $\gamma _2$ be the left boundary curves of ${\mathcal {E}}_1$ and ${\mathcal {E}}_2$, respectively, parametrised by $y={{\,\mathrm{Im}\,}}z$. Justified by Lemma 5.2, we suppose that $\nu$ is so large that

$$\begin{aligned} \vert \gamma _k'(y)\vert <\frac{1}{10} \text { for } \vert y\vert \ge \nu \text { and } k\in \{1,2\}. \end{aligned}$$

(10.1)

Using the notation of Lemmas 7.14, 8.6 and 9.6, suppose that

$$\begin{aligned} D>2\nu +5\max \{D_0,D_1,D_2\} \end{aligned}$$

(10.2)

and

$$\begin{aligned} D>20\max \left\{ \log \beta _1, \,\log \frac{\beta _2}{\alpha _1}\right\} . \end{aligned}$$

(10.3)

For ${\mathcal {S}}\subset {\mathbb {C}}$, let

$$\begin{aligned} {{\,\mathrm{diam}\,}}_x({\mathcal {S}}):=\sup \{\vert {{\,\mathrm{Re}\,}}(z-w)\vert :\,z,w\in {\mathcal {S}}\} \end{aligned}$$

and

$$\begin{aligned} {{\,\mathrm{diam}\,}}_y({\mathcal {S}}):=\sup \{\vert {{\,\mathrm{Im}\,}}(z-w)\vert :\,z,w\in {\mathcal {S}}\}. \end{aligned}$$

Define

$$\begin{aligned} {\mathcal {R}}_+:={\mathcal {R}}\cap \{z:\,{{\,\mathrm{Im}\,}}z\ge \nu \}, \quad {\mathcal {R}}_- :={\mathcal {R}}\cap \{z:\,{{\,\mathrm{Im}\,}}z\le -\nu \}, \end{aligned}$$

and let

$$\begin{aligned} {\mathcal {R}}_1:={\left\{ \begin{array}{ll} {\mathcal {R}}_+ &{}\text {if }{{\,\mathrm{diam}\,}}_y({\mathcal {R}}_+)\ge {{\,\mathrm{diam}\,}}_y({\mathcal {R}}_-)\\ {\mathcal {R}}_- &{}\text {otherwise.} \end{array}\right. } \end{aligned}$$

By (10.2), ${{\,\mathrm{diam}\,}}_y({\mathcal {R}}_1)>\max \{D_0,D_1,D_2\}.$

We now divide ${\mathcal {R}}_1$ into 5 rectangles, ${\mathcal {R}}_{1,1},\ldots ,{\mathcal {R}}_{1,5}$, with ${{\,\mathrm{diam}\,}}_y({\mathcal {R}}_{1,k})={{\,\mathrm{diam}\,}}_y({\mathcal {R}}_1)$ and ${{\,\mathrm{diam}\,}}_x({\mathcal {R}}_{1,k})=\frac{1}{5}{{\,\mathrm{diam}\,}}_x({\mathcal {R}}_1)$ for all $k\in \{1,\ldots ,5\}$ (see Fig. 5).

Fig. 5

The rectangle ${\mathcal {R}}_1$, bounded by the solid line, is divided into five smaller rectangles by the dashed lines

By (10.2), ${{\,\mathrm{diam}\,}}_x({\mathcal {R}}_{1,k})>\max \{D_0,D_1,D_2\}.$ By (10.1), Lemma 5.2, (10.3) and the fact that ${\mathcal {R}}$ is a square of side length at least D, we have

$$\begin{aligned} {{\,\mathrm{diam}\,}}_x ({\mathcal {E}}_l\cap {\mathcal {R}})&<\frac{1}{10}{{\,\mathrm{diam}\,}}_y({\mathcal {R}})+2\max \left\{ \log \beta _1,\log \frac{\beta _2}{\alpha _1}\right\} \\&<\frac{1}{10}{{\,\mathrm{diam}\,}}_y({\mathcal {R}})+\frac{1}{10}D\le \frac{1}{5}{{\,\mathrm{diam}\,}}_x({\mathcal {R}}) \end{aligned}$$

for $l\in \{1,2\}$. Thus, ${\mathcal {E}}_1$ and ${\mathcal {E}}_2$ each intersect at most two of the rectangles ${\mathcal {R}}_{1,k}$. Hence, there exists $l\in \{1,\ldots ,5\}$ such that ${\mathcal {R}}_{1,l}$ does not intersect ${\mathcal {E}}_1\cup {\mathcal {E}}_2$. This implies that ${\mathcal {R}}_{1,l}$ satisfies the hypothesis of one of Lemmas 7.14, 8.6 and 9.6. Hence,

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), {\mathcal {R}}_{1,l})\ge \min \{\eta _0,\eta _1,\eta _2\} \end{aligned}$$

and

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), {\mathcal {R}})&\ge {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j),{\mathcal {R}}_{1,l}) \cdot {{\,\mathrm{dens}\,}}({\mathcal {R}}_{1,l},{\mathcal {R}})\\&\ge \min \{\eta _0,\eta _1,\eta _2\}\cdot \frac{1}{10} \frac{{{\,\mathrm{diam}\,}}_x({\mathcal {R}})({{\,\mathrm{diam}\,}}_y({\mathcal {R}})-2\nu )}{({{\,\mathrm{diam}\,}}_x{\mathcal {R}})^2}\\&\ge \min \{\eta _0,\eta _1,\eta _2\}\cdot \frac{1}{10} \left( 1-\frac{2\nu }{D}\right) . \end{aligned}$$

$\square$

The following corollary is an immediate consequence of Lemma 10.1.

Corollary 10.2

There are $r_0,\eta >0$ such that for all $z\in {\mathbb {C}},$ all $r\ge r_0$ and all $j\in \{1,\ldots ,d\},$ we have

$$\begin{aligned} {{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j),{\mathcal {D}}(z,r))\ge \eta . \end{aligned}$$

Remark 10.3

Corollary 10.2 says that ${\mathbb {C}}{\setminus } q({\mathcal {F}}_j)$ is thin at $\infty$.

11 Proof of Theorem 1.1

Proof of Theorem 1.1

We will verify the assumptions of Theorem 1.3. By Lemma 6.2, the set ${\mathcal {P}}(f)\cap {\mathcal {J}}(f)$ is finite, so it remains to prove that there exists $R_1>0$ such that ${\mathcal {J}}(f)$ is uniformly thin at $({\mathcal {P}}(f)\cap {\mathbb {C}}){\setminus } \overline{{\mathcal {D}}(0,R_1)}$ and that ${\mathcal {J}}(f)$ is thin at $\infty$. In the previous sections we have studied the function obtained from f by the change of variables $w=q(z)$ introduced in Sect. 3, with the polynomial $q(z)=z^d+O(z^{d-1})$ from the definition of g. Now we use this to draw conclusions about f.

Let $r_1>0$ such that

• (a) $\vert q'(z)\vert \ge (d/2)\vert z\vert ^{d-1}$ for all $z\in {\mathbb {C}}$ with $\vert z\vert \ge r_1$;

• (b) each $z_0\in {\mathcal {P}}(f)$ with $\vert z_0\vert \ge r_1$ is a zero of g and hence a superattracting fixed point of f. Justified by Corollary 6.6, we also assume that there is $j\in \{1,\ldots ,d\}$ with $z_0\in {\mathcal {S}}_j$, and ${{\,\mathrm{dist}\,}}(z_0,\partial {\mathcal {S}}_j)\ge 3$. Moreover, suppose that the conclusion of Lemma 7.5 holds for $\vert z_0\vert \ge r_1$.

Let $r_0$ be the constant from Corollary 10.2. First, we will show that there exists $\eta _4>0$ such that for all $j\in \{1,\ldots ,d\}$, all $z\in {\mathcal {S}}_j$ with $\vert z\vert \ge r_1$ and all $r>8r_0/(d\vert z\vert ^{d-1})$ with ${\mathcal {D}}(z,2r)\subset {\mathcal {S}}_j$, we have

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f), {\mathcal {D}}(z,r))\ge \eta _4. \end{aligned}$$

(11.1)

Recall that q is injective in ${\mathcal {S}}_j$. By Koebe’s theorems,

$$\begin{aligned} {\mathcal {D}}\left( q(z),\frac{1}{4}\vert q'(z)\vert r\right) \subset q ({\mathcal {D}}(z,r))\subset {\mathcal {D}}(q(z),4\vert q'(z)\vert r). \end{aligned}$$

By (a) and the assumption on r, we have $(1/4)\vert q'(z)\vert r\ge r_0$. Hence, by Corollary 10.2,

$$\begin{aligned} {{\,\mathrm{dens}\,}}\left( q({\mathcal {F}}_j),{\mathcal {D}}\left( q(z),\frac{1}{4}\vert q'(z) \vert r\right) \right) \ge \eta . \end{aligned}$$

Thus,

$$\begin{aligned}&{{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j), q({\mathcal {D}}(z,r)))\\&\quad \ge {{\,\mathrm{dens}\,}}\left( {\mathcal {D}}\left( q(z),\frac{1}{4} \vert q'(z)\vert r\right) ,q({\mathcal {D}}(z,r))\right) \cdot {{\,\mathrm{dens}\,}}\left( q({\mathcal {F}}_j),{\mathcal {D}}\left( q(z), \frac{1}{4}\vert q'(z)\vert r\right) \right) \\&\quad \ge {{\,\mathrm{dens}\,}}\left( {\mathcal {D}}\left( q(z),\frac{1}{4}\vert q'(z) \vert r\right) ,{\mathcal {D}}(q(z),4\vert q'(z)\vert r)\right) \cdot \eta =\frac{1}{256}\eta . \end{aligned}$$

By the Koebe distortion theorem,

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f),{\mathcal {D}}(z,r))\ge \left( \frac{\min _{\zeta \in {\mathcal {D}}(z,r)}\vert q'(\zeta )\vert }{\max _{\zeta \in {\mathcal {D}}(z,r)} \vert q'(\zeta )\vert }\right) ^2{{\,\mathrm{dens}\,}}(q({\mathcal {F}}_j),q({\mathcal {D}}(z,r))) \ge \frac{1}{3^8\cdot 256}\eta . \end{aligned}$$

This implies (11.1) with $\eta _4=\eta /(3^8\cdot 256).$

Let us now prove that there exists $R_1>0$ such that ${\mathcal {J}}(f)$ is uniformly thin at $({\mathcal {P}}(f)\cap {\mathbb {C}}){\setminus }\overline{{\mathcal {D}}(0,R_1)}$. Let $\delta _1\in (0,1)$, $z_0\in {\mathcal {P}}(f)$ with $\vert z_0\vert > r_1+1$ and $z\in {\mathcal {D}}(z_0,\delta _1).$ By (b), ${\mathcal {D}}(z,2\delta _1)\subset {\mathcal {S}}_j$. Also, $\vert z\vert \ge r_1$. If $\vert z-z_0\vert \ge 8r_0/(d\vert z\vert ^{d-1})$, then by (11.1),

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f),{\mathcal {D}}(z,\vert z-z_0\vert ))\ge \eta _4. \end{aligned}$$

Now suppose that

$$\begin{aligned} \vert z-z_0\vert <\dfrac{8r_0}{d\vert z\vert ^{d-1}}. \end{aligned}$$

(11.2)

By Lemma 7.5, we have ${\mathcal {D}}(z_0,1/(3d\vert z_0\vert ^{d-1}))\subset {\mathcal {F}}(f)$. Hence,

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f),{\mathcal {D}}(z,\vert z-z_0\vert ))\ge {{\,\mathrm{dens}\,}}\left( {\mathcal {D}}\left( z_0,\dfrac{1}{3d\vert z_0\vert ^{d-1}}\right) , {\mathcal {D}}(z,\vert z-z_0\vert )\right) . \end{aligned}$$

The expression on the right hand side is bounded below independent of $z_0$ and $\vert z\vert$, provided (11.2) is satisfied. So ${\mathcal {J}}(f)$ is uniformly thin at $({\mathcal {P}}(f)\cap {\mathbb {C}}) {\setminus }\overline{{\mathcal {D}}(0,r_1+1)}$.

It remains to prove that ${\mathcal {J}}(f)$ is thin at $\infty$. Let R be as in Sect. 3 and let $r_2>\max \{2R^{1/d}, r_1\}$. If $r_2$ is sufficiently large, then Lemma 3.2 yields that $\bigcup _{j=1}^d\partial {\mathcal {S}}_j{\setminus }{\mathcal {D}}(0,r_2)$ is contained in d pairwise disjoint halfstrips, ${\mathcal {T}}_1,\ldots ,{\mathcal {T}}_d$, of width 1. We can assume that $r_2$ is so large that ${{\,\mathrm{dist}\,}}({\mathcal {T}}_k,{\mathcal {T}}_l)\ge 1$ for $k\ne l$. Then for $\vert z\vert \ge r_2+3$, the set ${\mathcal {D}}(z,3){\setminus }\bigcup _{j=1}^d{\mathcal {T}}_j$ contains a disk, ${\mathcal {D}}$, of radius 1/2. There is $j\in \{1,\ldots ,d\}$ with ${\mathcal {D}}\subset {\mathcal {S}}_j$. Let ${\mathcal {D}}'$ be the disk with the same center as ${\mathcal {D}}$ and radius 1/4. If $r_2$ is sufficiently large, then by (11.1), we have ${{\,\mathrm{dens}\,}}({\mathcal {F}}(f),{\mathcal {D}}')\ge \eta _4$, and hence

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f),{\mathcal {D}}(z,3))\ge {{\,\mathrm{dens}\,}}({\mathcal {F}}(f),{\mathcal {D}}')\cdot {{\,\mathrm{dens}\,}}({\mathcal {D}}', {\mathcal {D}}(z,3)) \ge \frac{\eta _4}{144}. \end{aligned}$$

We now consider the case that $\vert z\vert <r_2+3$. Let $\zeta _1,\ldots ,\zeta _n\in {\mathcal {D}}(0,r_2+3)$ such that

$$\begin{aligned} {\mathcal {D}}(0,r_2+3)\subset \bigcup _{k=1}^n{\mathcal {D}}(\zeta _k,1). \end{aligned}$$

Then

$$\begin{aligned} \eta _5:=\min _{1\le k\le n}{{\,\mathrm{dens}\,}}({\mathcal {F}}(f), {\mathcal {D}}(\zeta _k,1))>0. \end{aligned}$$

For $z\in {\mathcal {D}}(0,r_2+3)$, let $k\in \{1,\ldots ,n\}$ such that $z\in {\mathcal {D}}(\zeta _k,1)$. Then ${\mathcal {D}}(\zeta _k,1)\subset {\mathcal {D}}(z,3)$ and

$$\begin{aligned} {{\,\mathrm{dens}\,}}({\mathcal {F}}(f),{\mathcal {D}}(z,3))\ge \frac{1}{9}{{\,\mathrm{dens}\,}}({\mathcal {F}}(f),{\mathcal {D}}(\zeta _k,1)) \ge \frac{1}{9}\eta _5. \end{aligned}$$

Thus, ${\mathcal {J}}(f)$ is thin at $\infty$. Hence, Theorem 1.3 yields that ${\mathcal {J}}(f)$ has Lebesgue measure zero. $\square$