An interpolation theorem for slice-regular functions with application to very tame sets and slice Fatou–Bieberbach domains in ${\mathbb {H}}^2$

Abstract

We prove an interpolation theorem for slice-regular quaternionic functions. We define very tame sets in ${\mathbb {H}}^2$ to be the sets which can be mapped by compositions of automorphisms with volume 1 to the set ${{\mathcal {T}}}=\lbrace (2n-1,0), n \in {\mathbb {N}}\rbrace \cup \lbrace (2n + {\mathbb {S}},0), n \in {\mathbb {N}}\rbrace .$ We then show that any zero set of an entire slice-regular function of one variable embedded in ${\mathbb {H}}\times \lbrace 0 \rbrace \subset {\mathbb {H}}^2$ is very tame in ${\mathbb {H}}^2.$ A notion of slice Fatou–Bieberbach domain in ${\mathbb {H}}^2$ is introduced and, finally, a slice Fatou–Bieberbach domain in ${\mathbb {H}}^2$ avoiding ${{\mathcal {T}}}$ is constructed in the last section.

1 Introduction

It is known that zero sets of slice-regular functions consist of points and spherical zeroes, namely isolated two-dimensional spheres of the form $S(a,r)= a + {\mathbb {S}}r := \lbrace z, (z - a)^2 + r^2=0 \rbrace$ with $a \in {\mathbb {R}}, r > 0;$ the symbol ${\mathbb {S}}$ denotes the sphere of imaginary units in ${\mathbb {H}},$ namely ${\mathbb {S}}:= \lbrace I \in {\mathbb {H}}, I^2=-1 \rbrace .$ The sphere S(a, r) is the zero set of the quadratic polynomial $(q - a)^2 + r^2.$ Zero sets of entire slice-regular functions consist of a discrete set of points and a ‘discrete’ set of spherical zeroes $S(a_n,r_n), n \in M \subset {\mathbb {N}}$ in the sense that the set $a_n^2 + r_n^2, n \in M$ does not have accumulation points (see Theorem 3.21 in [3]). By the quaternionic version of the Weierstrass theorem proved in [5] we know, that any such a set, i.e., a union of a discrete set of points and a discrete set of isolated spheres as described above, is indeed a zero set of a slice-regular function. Since images of spheres $S(a_n,r_n)$ by slice-regular functions are no longer spheres, but rather generalized spheres of the form $S(q,\rho ) = q + {\mathbb {S}}\rho$ with $q,\rho \in {\mathbb {H}},$ the corresponding interpolation theorem for quaternionic functions is about assigning values to points and generalized spheres to spheres. We prove such a theorem in Sect. 2.

In Sect. 3 we generalize the complex-analytic notion of very tame discrete set to a quaternionic setting. In the complex-analytic case, a tame set in ${\mathbb {C}}^2$ is any set which can be mapped to a subset of the set ${\mathbb {N}}\times \lbrace 0 \rbrace \subset {\mathbb {C}}^2$ by an automorphism of ${\mathbb {C}}^2.$ If the automorphism can be chosen to have Jacobian equal to 1,  then the set is very tame (see [9]).

We have shown in [8] that the class of quaternionic automorphisms of ${\mathbb {H}}^2$ with volume (see Definition 5.1 in [8]) turns out to be a suitable counterpart in the quaternionic setting for holomorphic automorphisms of ${\mathbb {C}}^2.$ Quaternionic automorphisms with volume 1 are the quaternionic analogues for holomorphic automorphisms with Jacobian equal to 1 (for the definition of quaternionic Jacobian see Sect. 1.1).

With this in mind we define (Definition 3) a very tame set in ${\mathbb {H}}^2$ to be a union of a discrete set of points and a set of isolated spheres which can be injectively mapped to a subset of the set

$$\begin{aligned} {{\mathcal {T}}}=\lbrace (2n-1,0), n \in {\mathbb {N}}\rbrace \cup \lbrace (2n + {\mathbb {S}},0), n \in {\mathbb {N}}\rbrace \subset {\mathbb {H}}^2 \end{aligned}$$

by a diffeomorphism of ${\mathbb {H}}^2,$ which is a limit of a convergent sequence of quaternionic automorphisms with volume 1,  such that points are mapped to points and spheres to spheres.

Having the interpolation theorem at our disposal, we are then able to show that zero sets of slice-regular functions in ${\mathbb {H}}$ embedded in ${\mathbb {H}}^2$ are very tame sets in ${\mathbb {H}}^2.$ Moreover, the quaternionic automorphisms with volume 1 involved are shears (see Sect. 1.1).

In Sect. 4, we define quaternionic slice Fatou–Bieberbach domains and prove that there exists an example of a slice Fatou–Bieberbach domain in ${\mathbb {H}}^2$ which does not intersect ${{\mathcal {T}}}.$ The domain is obtained as a domain of convergence of compositions of shears which are slice-regular in the sense of [6].

1.1 Definitions, notation and basic tools

The present paper relies on previous work done by the authors, contained in [8], where the reader can find all the necessary definitions and theorems used in this paper. For reader’s convenience, we briefly recall here some of the definitions and notations.

By ${{\mathcal {S}}}{{\mathcal {R}}}(D)$ we refer to all slice-regular quaternionic functions on a (usually symmetric slice) domain $D\subseteq {\mathbb {H}}$ as defined in [3]. If a domain is a ball with radius R centered at 0,  i.e. $D = B^4(0,R) \subset {\mathbb {H}},$ then ${{\mathcal {S}}}{{\mathcal {R}}}(D)$ consists of all power series with coefficients on the right, i.e., a series of the form $\sum z^n a_n,$ converging in D. By ${{\mathcal {H}}}[z,w](\Omega ), \Omega \subset {\mathbb {H}}^2,$ we denote the set of all functions defined as quaternionic power series converging in $\Omega$ with no restriction on the position of the quaternionic coefficients. This space coincides with all real analytic power series in 8 real variables with values in ${\mathbb {H}},$ converging in $\Omega .$ If the coefficients of the series are only on the right hand-side, then the class is denoted by ${\mathcal H}_{\mathrm {rhs}}[z,w].$ The notation ${{\mathcal {H}}}^n[z,w](\Omega ),$ with $\Omega$ a domain in ${\mathbb {H}}^2,$ denotes the set of all mappings $\Psi (z,w) : \Omega \rightarrow {\mathbb {H}}^n$ with components in ${\mathcal H}[z,w](\Omega ).$ The mappings of the form $\Phi (z,w) = (z,w) + f(z - \mu w)(\mu ,1)$ or $\Psi (z,w) = (z,w) + g(w-\nu z)(1,\nu )$ with $\mu ,\nu \in {\mathbb {R}}$ and $f,g\in {{\mathcal {S}}}{{\mathcal {R}}}({\mathbb {H}})$ are called shears and are automorphisms with volume 1, see [8]. In particular any shear of the form $\Phi (z,w) = (z,w) + f(z)(0,1)$ will be called a vertical shear and, similarly, $\Psi (z,w) = (z,w) + g(w)(1,0)$ will be called a horizontal shear.

The (Lie) derivative of $f \in {{\mathcal {H}}}[z,w](\Omega )$ in the direction (h, k) at $(z,w)\in \Omega$ is defined as

$$\begin{aligned} \lim \limits _{{\mathbb {R}}\ni t \rightarrow 0}\dfrac{f(z + th,w+tk) -f(z,w)}{t}:={\mathcal {L}}_{(h,k)}f(z,w). \end{aligned}$$

In particular, we will use the notation

$$\begin{aligned} {\mathcal {L}}_{(h,0)}f(z,w)={\widehat{\partial }}_z\,\,f(z,w)[h],\qquad {\mathcal {L}}_{(0,k)}f(z,w)={\widehat{\partial }}_w\,\,f(z,w)[k] \end{aligned}$$

and call them, respectively, partial derivative of $f \in {\mathcal H}[z,w](\Omega )$ with respect to z (applied to h) and partial derivative of $f \in {{\mathcal {H}}}[z,w](\Omega )$ with respect to w (applied to k). The operators ${\widehat{\partial }}_z\,\,f(z,w)[h]$ and ${\widehat{\partial }}_w\,\,f(z,w)[h]$ are additive and real homogeneous in h. The Jacobian of the mapping $F =(f_1,f_2) \in {{\mathcal {H}}}^2[z,w](\Omega )$ applied to $(h_1,h_2)$ is defined to be

$$\begin{aligned} DF(z,w)[h_1,h_2]:= \begin{bmatrix} {\widehat{\partial }}_z\,\,f_1(z,w)[h_1] &\, & {\widehat{\partial }}_w\,\,f_1(z,w)[h_2] \\ {\widehat{\partial }}_z\,\,f_2(z,w)[h_1] &\, & {\widehat{\partial }}_w\,\,f_2(z,w)[h_2] \end{bmatrix}. \end{aligned}$$

Notice that if $F\in {{\mathcal {H}}}^2[z,w]$, then

$$\begin{aligned}F(z+h_1,w+h_2)-F(z,w)= DF(z,w)[h_1,h_2] \begin{bmatrix}1\\ 1\end{bmatrix}+ o(||(h_1,h_2)||) \end{aligned}$$

For the Jacobian, we use the matrix norm $| \cdot |_{\infty }$ defined by

$$\begin{aligned} \big |DF(z,w)\big |_{\infty } = \max \lbrace \big |DF(z,w)[h_1,h_2]\big |_{\infty }, |h_1|,|h_2| \le 1\rbrace , \end{aligned}$$

where

$$\begin{aligned}&\big |DF(z,w)[h_1,h_2]\big |_{\infty } = \max \lbrace \big |{\widehat{\partial }}_z\,\,f_1(z,w)[h_1] + {\widehat{\partial }}_w\,\,f_1(z,w)[h_2]\big |,\\&\qquad\qquad\qquad \big |{\widehat{\partial }}_z\,\,f_2(z,w)[h_1] + {\widehat{\partial }}_w\,\,f_2(z,w)[h_2]\big | \rbrace . \end{aligned}$$

The adapted chain rule holds, namely

$$\begin{aligned}D(F \circ G)(z,w)[h_1,h_2] = DF(G(z,w))\diamond DG(z,w)[h_1,h_2],\end{aligned}$$

where $\diamond$ is defined by

$$\begin{aligned}&D(F \circ G)(z,w)[h_1,h_2] \\&=\begin{bmatrix} {\widehat{\partial }}_z\,\,f_1(u,v)[a_1] + {\widehat{\partial }}_w\,\,f_1(u,v)[a_2] &\, &{\widehat{\partial }}_z\,\,f_1(u,v)[b_1] + {\widehat{\partial }}_w\,\,f_1(u,v)[b_2] \\ {\widehat{\partial }}_z\,\,f_2(u,v)[a_1] + {\widehat{\partial }}_w\,\,f_2(u,v)[a_2] &\, &{\widehat{\partial }}_z\,\,f_2(u,v)[b_1] + {\widehat{\partial }}_w\,\,f_2(u,v)[b_2] \end{bmatrix}, \end{aligned}$$

with

$$\begin{aligned} DF(u,v)= \begin{bmatrix} {\widehat{\partial }}_z\,\,f_1(u,v) &\, &{\widehat{\partial }}_w\,\,f_1(u,v) \\ {\widehat{\partial }}_z\,\,f_2(u,v) &\, &{\widehat{\partial }}_w\,\,f_2(u,v) \end{bmatrix} \hbox { and } DG(z,w)[h_1,h_2]= \begin{bmatrix} a_1 & b_1\\ a_2 & b_2 \end{bmatrix}. \end{aligned}$$

It is immediate that the norm $| \cdot |_{\infty }$ is definite, right absolutely homogeneous and sub-additive. It is also sub-multiplicative for the operator $\diamond ,$ i.e.

$$\begin{aligned}|D(F \circ G)(z,w)|_{\infty } \le |DF(G(z,w))|_{\infty }\cdot |DG(z,w)|_{\infty }.\end{aligned}$$

To show this, assume first that $|DG(z,w)[h_1,h_2]|_{\infty } = |a_1 + b_1|,$ define

$$\begin{aligned} k_1:=\frac{a_1 + b_1}{|a_1 + b_1|}, \; k_2:=\frac{a_2 + b_2}{|a_1 + b_1|}\end{aligned}$$

and notice that $|k_1|,|k_2| \le 1.$ To estimate the norm of the derivative we use additivity and real homogeneity of the operators ${\widehat{\partial }}_z,\,\,{\widehat{\partial }}_w$ in the variables h and k. We have

$$\begin{aligned} |D(F \circ G)(z,w)[h_1,h_2]|_{\infty }= & \max \{ \big |{\widehat{\partial }}_z\,\,f_1(u,v)[a_1] + {\widehat{\partial }}_w\,\,f_1(u,v)[a_2] + {\widehat{\partial }}_z\,\,f_1(u,v)[b_1] + {\widehat{\partial }}_w\,\,f_1(u,v)[b_2]\big |,\\&\big |{\widehat{\partial }}_z\,\,f_2(u,v)[a_1] + {\widehat{\partial }}_w\,\,f_2(u,v)[a_2] + {\widehat{\partial }}_z\,\,f_2(u,v)[b_1] + {\widehat{\partial }}_w\,\,f_2(u,v)[b_2]\big | \},\\= & \max \{ \big |{\widehat{\partial }}_z\,\,f_1(u,v)[a_1 + b_1] + {\widehat{\partial }}_w\,\,f_1(u,v)[a_2+b_2]\big |,\\&\big |{\widehat{\partial }}_z\,\,f_2(u,v)[a_1+ b_1] + {\widehat{\partial }}_w\,\,f_2(u,v)[a_2+b_2]\big |\} \\= & \max \{ \big |{\widehat{\partial }}_z\,\,f_1(u,v)[k_1] + {\widehat{\partial }}_w\,\,f_1(u,v)[k_2]\big |, \big |{\widehat{\partial }}_z\,\,f_2(u,v)[k_1] + {\widehat{\partial }}_w\,\,f_2(u,v)[k_2]\big |\} \cdot \\&\cdot |a_1 + b_1|\\\le & |DF(z,w)|_{\infty } |DG(z,w)[h_1,h_2]| \end{aligned}$$

and the same estimate $|D(F \circ G)(z,w)[h_1,h_2]|_{\infty } \le |DF(z,w)|_{\infty } |DG(z,w)[h_1,h_2]|$ holds also if $|DG(z,w)[h_1,h_2]|_{\infty } =|a_2 + b_2|.$ Taking the $\max$ over $|h_1|,|h_2| \le 1$ yields the desired estimate.

2 The interpolation theorem for slice-regular functions

As already mentioned in the Introduction, an interpolation theorem in the quaternionic setting should provide the existence of a quaternionic function which assigns prescribed values to points and prescribed generalized spheres to spheres. Let us begin by stating the formal definitions.

Definition 1

A generalized sphere in ${\mathbb {H}}$ is any set of the form $S(q,\rho ) = q + {\mathbb {S}}\rho ,$ with $q,\rho \in {\mathbb {H}}.$ If $q = a \in {\mathbb {R}}, \rho = r \in [0,\infty )$ we use the notation S(a, r) and we call them spheres. In particular, a sphere becomes a real number when $r=0$.

Remark 1

Zero sets of entire slice-regular functions may contain only spheres besides isolated quaternions (see e.g., [4]). In this framework we consider the zero set of a slice-regular function as a geometric object, i.e., as a union of points where the slice-regular function f vanishes. Algebraically, a point $z_k = a_k + Ib_k$ may be an isolated zero of multiplicity n in the sense that $f(z)=(z - z_k)^{*n} *f_1(z)$ and at the same time there may be a spherical zero $a_k + b_k S,$ which contains $z_k.$ We only consider the spherical zero in such cases for the purpose of the theorem.

Definition 2

Let the zero set Z of an entire slice-regular function be given. We define the decomposition of Z to be $Z = {\mathcal {S}} \cup {\mathcal {Z}} \cup {\mathcal {X}},$ where ${\mathcal {S}}$ is the discrete set of spheres, ${\mathcal {S}}:= \{S_j=S(a_j,r_j), j \in M_1 \subset {\mathbb {N}}, r_j > 0\},$ ${\mathcal {Z}}$ is the discrete set of non-real zeroes, ${\mathcal {Z}}:=\{z_k \in {\mathbb {H}}\setminus {\mathbb {R}}, k \in M_2 \subset {\mathbb {N}}\}$ and ${{\mathcal {X}}}$ is the discrete set of real zeroes, ${\mathcal {X}}:=\{x_l \in {\mathbb {R}}, l \in M_3 \subset {\mathbb {N}}\}.$

Theorem 1

(The Interpolation Theorem). Let the zero set Z of an entire slice-regular function be given with the decomposition $Z= {\mathcal {S}} \cup {\mathcal {Z}} \cup {\mathcal {X}}$ as in Definition 2. Assume that $z_k \not \in S(a_j,r_j)$ for any j, k. For any sequence of generalized spheres ${\check{S}}_j=S(q_j,\rho _j), j \in M_1$ and values $w_k, u_l \in {\mathbb {H}}, k \in M_2, l \in M_3,$ there exists an entire slice-regular function f such that $f(S(a_j,r_j)) = S(q_j,\rho _j),$ $f(z_k) = w_k$ and $f(x_l) = u_l$ for all $j \in M_1, k \in M_2$ and $l \in M_3.$

By the assumption on discreteness of the spheres $S_j$, the sequence $R_j^2 = a_j^2 + r_j^2$ has no accumulation point. We may assume that it is (not necessarily strictly) increasing. The main ingredient for the proof of the Theorem is the following

Lemma 1

Let the set Z and the sets $S_j, z_k$ and $x_l$ be as in Theorem 1. Choose $j_0 \in M_1, \varepsilon > 0$ and let $0< R < R_{j_0} = (a_{j_0}^2 + r_{j_0}^2)^{1/2}.$ For any generalized sphere ${\check{S}} = S(q,\rho )$ there exists a slice-regular function $f_{j_0}$ such that $f_{j_0}(S_{j_0}) = {\check{S}},$ $f_{j_0}$ vanishes on the set $Z \setminus S_{j_0}$ and $|f_{j_0}|_{B^4(0,R)} < \varepsilon .$ If all the generalized spheres are spheres, then the function $f_{j_0}$ can be chosen to be slice-preserving, i.e., such that it has a power series expansion with real coefficients.

Proof

For any point $z_k\in {\mathcal {Z}}$, we can consider the corresponding sphere ${\tilde{S}}_{k}\ni z_k$. By assumption, also the spheres ${\tilde{S}}_{k}$ are discrete; namely if $z_k = a_k + I_k r_k \in S(a_k, r_k) = {\tilde{S}}_k$, then ${\tilde{S}}_k \cap {S}_j = \emptyset$ for any $S_j\in {\mathcal {S}}$. By the quaternionic version of the Weierstrass theorem as in [5], there exists a slice-preserving function g with zeroes precisely on $S_j \ne S_{j_0}, {\tilde{S}}_k$ and $x_l$ so the coefficients of its power series are real. Therefore the image of the sphere $S_{j_0}$ is again a sphere, $g(S_{j_0}) = S(a,r),$ more precisely, $g(a_{j_0} + I r_{j_0} ) = a + I r$ with $a^2 + r^2 \ne 0.$ In order to satisfy also the condition $f(S_{j_0}) = {\check{S}},$ we are looking for a slice–regular function $\gamma (z) = z \alpha + \beta$ such that

$$\begin{aligned} g(a_{j_0} + I r_{j_0} )\gamma (a_{j_0} + I r_{j_0} ) = q + I\rho . \end{aligned}$$

(1)

Since $\gamma (a_{j_0} + Ir_{j_0} ) = a_{j_0}\alpha + \beta + Ir_{j_0} \alpha$, equation (1) implies

$$\begin{aligned} a_{j_0}\alpha + \beta +I r_{j_0} \alpha = \frac{1}{a^2 + r^2}(a - I r)(q + I\rho ) = \frac{1}{a^2 + r^2}(aq+r \rho + I(a \rho - r q)) \end{aligned}$$

and so we get

$$\begin{aligned} \alpha = \frac{a \rho - r q}{r_{j_0}(a^2 + r^2)}, \,\, \beta = \frac{aq+r \rho }{a^2 + r^2} - a_{j_0}\alpha . \end{aligned}$$

(2)

Hence $g\cdot \gamma$ is a slice–regular function which also satisfies the interpolation condition $(g \cdot \gamma )(S_{j_0})={\check{S}}$. For the statement of the theorem about approximation, observe that any given sphere $S_j = \lbrace a_j + r_j I, I \in {\mathbb {S}}\rbrace$ uniquely determines $\vartheta _j\in (0,1)$ such that $a_{j} + r_{j} I = R_j e^{I \pi \vartheta _j}$ for any point $a_{j} + r_{j} I \in S_j.$ If $\vartheta _j$ is a rational number, then for some $n \in {\mathbb {N}}$ we have $((a_{j} + r_{j} I)/R_j)^n= 1$ and so the requested estimate on the ball $B^4(0,R)$ is obtained if we multiply $g \cdot \gamma$ by $(z/R_{j_0})^{Nn}$ for N large enough. In other words, the function

$$\begin{aligned} f_{j_0}(z) = \left( \frac{z}{R_{j_0}}\right) ^{Nn} g(z)\,\gamma (z) \end{aligned}$$

has all the required properties. If $\vartheta _j$ is not rational, then the set of points $\{((a_{j} + r_{j} I)/R_j)^n, \;$ $n\in {\mathbb {N}}\}$ is dense in the unit circle $S_I^1$ of ${\mathbb {C}}_{I}$. Therefore, for some increasing sequence of natural numbers $\{n_k\}_{k\in {\mathbb {N}}}$, we have

$$\begin{aligned} \left( \frac{a_{j_0} + r_{j_0} I}{R_{j_0}}\right) ^{n_k} = c_{n_k} - d_{n_k} I \rightarrow \frac{ a_{j_0} - r_{j_0} I}{R_{j_0}}. \end{aligned}$$

(3)

Define $\alpha _{n_k}, \beta _{n_k} \in {\mathbb {R}}$ to be such that $c_{n_k} + d_{n_k} I = \alpha _{n_k} (a_{j_0} + r_{j_0} I) + \beta _{n_k}.$ Since by (3) we have $c_{n_k} + d_{n_k} I \approx (a_{j_0} + r_{j_0} I) /R_{j_0},$ it is clear that $\alpha _{n_k} \approx 1/R_{j_0}$ and $\beta _{n_k} \approx 0.$ Hence, we can find a sequence of functions $\psi _{n_k}(z) = \alpha _{n_k} z + \beta _{n_k}$ with $\alpha _{n_k}, \beta _{n_k} \in {\mathbb {R}},$ such that

$$\begin{aligned} \left( \frac{z}{R_{j_0}}\right) ^{n_k}\psi _{n_k}(z)=1 {\mathrm { \ for\ any}}\ z \in S_{j_0} \hbox { and } \psi _{n_k}(z) \rightarrow z/R_{j_0}. \end{aligned}$$

Furthermore, if $z\in B^4(0, R) \subset {\mathbb {H}}$, for any positive $\varepsilon$, there exists $k_0$ such that for any $k>k_0$ the estimate

$$\begin{aligned} \left| \left( \frac{z}{R_{j_0}}\right) ^{n_k}\psi _{n_k}(z)\,g(z)\,\gamma (z)\right| <\varepsilon \end{aligned}$$

holds. Then the function f defined by

$$\begin{aligned} f_{j_0}(z):= \left( \frac{z}{R_{j_0}}\right) ^{n_k}\psi _{n_k}(z)\,g(z)\,\gamma (z) \end{aligned}$$

for k sufficiently large has all the desired properties.

Notice that if $q,\rho \in {\mathbb {R}}$, then the numbers $\alpha ,\beta$ given by (2) are real, so $\gamma$ has real coefficients. Thus the last statement of the Lemma is also proven.$\square$

Lemma 2

Let the set Z and the sequences $S_j, z_k$ and $x_l$ be as in Theorem 1. Choose $k_0 \in M_2, \varepsilon > 0$ and let $0< R < R_{k_0} =|z_{k_0}|.$ For any $w \in {\mathbb {H}}$ there exists a slice-regular function $f_{k_0}$ such that $f_{k_0}(z_{k_0}) = w,$ $f_{k_0}$ vanishes on $Z \setminus \{z_{k_0}\}$ and $|f_{k_0}|_{B^4(0,R)} < \varepsilon$. The same holds with $x_{l_0}$ in place of $z_{k_0}.$

Proof

From [5] there exists an entire slice-regular function with zeroes precisely on $S_j, z_k \ne z_{k_0}$ and $x_l$ of the form

$$\begin{aligned} g(z) = S(z)\, X(z)\, Z(z), \end{aligned}$$

where S has precisely the union of spheres $S_j$ as zero set, X has the union of the real numbers $x_l$ as zero set and Z has the union of points $z_k \ne z_{k_0}$ as zero set. Define $\lambda = g(z_{k_0})$, $\lambda \ne 0$ by the assumptions above, and let $u = w^{-1} z_{k_0} w.$ Then for sufficiently large N the function

$$\begin{aligned} f_{k_0} (z) = z^N g(z)\, \lambda ^{-1} w u^{-N} \end{aligned}$$

fulfills all the requested conditions of the Lemma. $\square$

Proof of the Interpolation Theorem

We are looking for a function f of the form $f = F_1 + F_2 + F_3,$ where $F_1,\ F_2,\ F_3$ are slice–regular functions such that

• 1. $F_1(S_j)={\check{S}}_j$ for $j\in M_1$ and $F_1|_{{\mathcal {Z}} \cup {\mathcal {X}}}\equiv 0$

• 2. $F_2(z_k)=w_k$ for $k\in M_2$ and $F_2|_{{\mathcal {S}} \cup {\mathcal {X}}}\equiv 0$

• 3. $F_3(x_l)=u_l$ for $l\in M_3$ and $F_3|_{{\mathcal {S}} \cup {\mathcal {Z}}}\equiv 0$.

For the construction of $F_1$ observe that if the set $M_1$ is finite then we can simply define the function $F_1(z):= \sum _{j \in M_1} f_{j}$ with $f_j$ given by Lemma 1 since the sum is finite.

Now assume that the sequence of spheres is infinite and that the corresponding radii $R_j$ form an increasing sequence. If there are $n_j$ spheres with the same radius indexed from $j_n$ to $j_n + n_j -1,$ namely $R_{j_n} = \ldots = R_{j_n + n_j -1},$ we require that functions $f_j, j = j_n,\ldots j_n + n_j - 1$ satisfy the estimate $|f_j(z)| < 2^{-j}$ on $B(0, R_{j_n-1}).$ Then $F_1(z):= \sum _{j \in M_1} f_{j}$ converges, $F_1(S_j)={\check{S}}_j$ for $j\in M_1$ and $F_1$ vanishes in ${\mathcal {Z}} \cup {\mathcal {X}}$.

The constructions of $F_2$ and $F_3$ are similar and rely upon Lemma 2 in the place of Lemma 1.

Lemmas 1 and 2 together with Theorem 1 imply the

Corollary 1

If the set Z (as above) does not intersect the ball $B^4(0,r) \in {\mathbb {H}}$, then the interpolation function obtained in Theorem 1 can be chosen to be arbitrarily small on compact subsets of $B^4(0,r).$

3 Very tame sets in ${\mathbb {H}}^2$

There are different notions of tameness in the literature and we refer the interested reader to [1] for an updated historical review on this subject. We only recall here that the notion of very tameness in complex analysis for a discrete set refers to correspondence via volume-preserving automorphisms (that is automorphisms with Jacobian equal to 1) with an embedded copy of ${\mathbb {N}}$.

Here we give the following

Definition 3

The standard tame set is defined as

$$\begin{aligned} {\mathcal {T}}=\underbrace{\lbrace (2n-1, 0), n \in {\mathbb {N}}\rbrace }_{{\mathcal {T}}_p} \cup \underbrace{\left\{ (2n + {\mathbb {S}}, 0),\, n \in {\mathbb {N}}\right\} }_{{\mathcal {T}}_S}. \end{aligned}$$

A very tame set in ${\mathbb {H}}^2$ is any discrete set ${\mathfrak {T}}={\mathfrak {T}}_p\cup {\mathfrak {T}}_S$ of points (${\mathfrak {T}}_p$) and spheres (${\mathfrak {T}}_S$) which can be mapped injectively to a subset of ${\mathcal {T}}^{\prime} \subseteq \mathcal T$ by a diffeomorphism of ${\mathbb {H}}^2,$ which is a limit of a convergent sequence of quaternionic automorphisms with volume 1 (in the sense of Definition 5.1 in [8]) such that points are mapped to points and spheres are mapped to spheres, i.e.

$$\begin{aligned} {\mathfrak {T}}_p\cong {\mathcal {T}}_p\cap {\mathcal {T}}^{\prime}\ \mathrm {and}\ {\mathfrak {T}}_S\cong {\mathcal {T}}_S\cap {\mathcal {T}}^{\prime}.\end{aligned}$$

Remark 2

Recall that quaternionic automorphisms with volume 1 are generalizations of holomorphic automorphisms with Jacobian equal to 1; furthermore, any composition of holomorphic automorphisms with Jacobian equal to 1 is again a holomorphic automorphism with Jacobian equal to 1. In the quaternionic setting a composition of two automorphisms with volume 1 is not necessarily an automorphism with volume.

The result which motivates the definition of a very tame set in the quaternionic setting and gives a geometric interpretation of the link with the theory of slice-regular function is the following

Theorem 2

The zero set Z of any slice-regular function $f \in {{\mathcal {S}}}{{\mathcal {R}}}({\mathbb {H}}),\ f\not \equiv 0$ embedded in ${\mathbb {H}}\times \lbrace 0\rbrace$ is a very tame set in ${\mathbb {H}}^2$. More precisely, the set $Z \times \lbrace 0\rbrace$ can be mapped to ${{\mathcal {T}}}$ by a finite composition of slice-regular shears with only real coefficients in their power series expansions.

Proof

We follow the idea of the proof presented in the paper [9] by Rosay and Rudin where vertical and horizontal shears are applied in order to find the 1-1 correspondence of points in Z with the ones in ${\mathbb {N}}\times \{0\} \subset {\mathbb {C}}^2$. Geometrically, this is achieved by labelling points in $Z=\{z_j\ :\ j\in {\mathbb {N}}\}$ and using the vertical shear $\Phi _1(z,w)=(z,w+f_1(z))$ such that $\Phi _1(z_j,0)=(z_j,j)$ to lift them in ${\mathbb {C}}^2$ to different levels. Then, one considers the horizontal shear $\Psi (z,w)=(z+g(w), w)$ such that $\Psi (z_j,j)=(j,j)$ and then the verical shear $\Phi _2(z,w)=(z,w+f_2(z))$ such that $\Phi _2(j,j)=(j,0)$. Hence $\Phi _2\circ \Psi \circ \Phi _1$ is the desired composition of automorphisms.

We proceed similarly with the difference that we first find an automorphism which injectively maps the spheres of Z to the spheres of ${\mathcal {T}}$ and leave the points of Z fixed and, afterwards, find an automorphisms which injectively maps the points of Z to the points of ${\mathcal {T}}$ and keeps the spheres of ${\mathcal {T}}$ fixed.

We may (and will) also assume that the zero set Z contains infinitely many real and nonreal points and infinitely many spheres and that $(Z \times \{0\}) \cap {\mathcal {T}} = \emptyset ,$ otherwise we just translate the coordinate system a little bit in the real direction by a shear $\Psi _0(z,w)=(z + \varepsilon ,w),$ $\varepsilon \in {\mathbb {R}}$.

In the quaternionic setting, as a first step, we construct a composition of shears which maps spheres of Z to spheres of ${\mathcal {T}}$ and leave fixed the real isolated zeroes and the spheres ${\tilde{S}}_k$ generated by non-real isolated zeroes. The idea of keeping fixed any entire sphere generated by a non-real zero in Z allows us to consider shears in ${\mathcal {H}}_{\mathrm {rhs}}^2$ whose restrictions to ${\mathbb {C}}_I\times {\mathbb {C}}_I\simeq {\mathbb {C}}^2$ for any $I\in {\mathbb {S}}$ can be regarded as complex holomorphic automorphisms with real coefficients in their power series expansions.

All spheres $S_j = S(a_j,r_j)\in Z$ and have to be mapped according to the following diagram

$$\begin{aligned} (a_j + r_j I, 0) {\mathop {\longrightarrow }\limits ^{\Phi _1}} (a_j + r_jI, 2j(1+ I)) {\mathop {\longrightarrow }\limits ^{\Psi }} (2j + I, 2j (1 + I)) {\mathop {\longrightarrow }\limits ^{\Phi _2}} (2j + I,0). \end{aligned}$$

where $\Phi _1(z,w) =(z, w + f_1(z))$ is the first vertical shear, $\Psi (z,w)=(z + g(w),w)$ is the horizontal shear and finally the shear $\Phi _2(z,w)=(z, w + f_2(z))$ is the second vertical shear. The function $f_1$ in the first shear $\Phi _1$ has to be zero on all real points in Z and spheres ${\tilde{S}}_k$ generated by non-real zeroes $z_k\in Z$, (i.e $f_1({\tilde{S}}_k) = 0$ for any k and $f_1(x_l) = 0$ for any l) and, in addition, $f_1(a_j + r_jI) = 2j(1 + I)$ for all j;  these interpolation conditions for $f_1$ can be fulfilled because, by assumption, ${\mathcal {Z}} \cap {\mathcal {S}} = \emptyset .$ Next, the function g has to leave the zero level ${\mathbb {H}} \times \{0\}$ fixed and move the spheres in z-direction so that they have the correct projection to ${\mathbb {H}}\times \{0\}$. In other words, the conditions for g are

$$\begin{aligned} g(0) = 0, \quad g(2j(1+I)) = -(a_j + r_jI) + 2j + I=-a_j+2j+(-r_j+1)I\ \quad \forall j. \end{aligned}$$

At the end, using $\Phi _2$, we keep fixed the spheres ${\tilde{S}}_k$ and zeroes $x_l$ and map the spheres $\Psi \circ \Phi _1(S_j)$ to $(2j+{\mathbb {S}},0)\in {\mathcal {T}}_S$ by imposing $f_2(2j + I) =-2j(1+I)$ and $f_2({\tilde{S}}_k) = f_2(x_l) = 0, \forall k,l,$ which is possible because of the condition $(Z \times \{0\}) \cap {\mathcal {T}} = \emptyset .$

Notice that for this step the Interpolation Theorem is fundamental, since we have to assign spheres to given spheres, otherwise the map g could not be defined. Moreover, since spheres are mapped to spheres, the coefficients in the power series expansions of $f_1, f_2$ and g are real.

In the second step isolated zeroes are mapped to odd integers (we list all discrete zeroes in the sequence $y_k$ without distinguishing the real from the nonreal ones) and proceed as above by using new shears ${{\widetilde{\Phi }}}_1(z,w)=(z, w + \tilde{f_1}(z))$, ${{\widetilde{\Psi }}}=(z + {\tilde{g}}(w),w)$, ${{\widetilde{\Phi }}}_2=(z, w + \tilde{f_2}(z))$ according to the following diagram

$$\begin{aligned} (y_k,0) {\mathop {\longrightarrow }\limits ^{{{\widetilde{\Phi }}}_1}} (y_k,2k-1) {\mathop {\longrightarrow }\limits ^{{{\widetilde{\Psi }}}}} (2k - 1, 2k-1) {\mathop {\longrightarrow }\limits ^{{{\widetilde{\Phi }}}_2}} (2k - 1,0), k \in {\mathbb {N}}, \end{aligned}$$

where the interpolation conditions for the functions $\tilde{f_1}, \tilde{f_2}, {\tilde{g}}$ are $\tilde{f_1}(2j+{\mathbb {S}}) = 0,$ $\tilde{f_1}(y_k) = 2k-1$, ${\tilde{g}}(0) = 0,$ ${\tilde{g}}(2k-1) = 2k-1 - y_k,$ $\tilde{f_2}(2j+{\mathbb {S}}) = \tilde{f_2}(0) = 0,$ $\tilde{f_2}(2k-1) = -(2k-1)$ for any k, j. Notice that the interpolation conditions for ${\tilde{f}}_1$ do not contradict each other because of the assumption that $(Z \times \{0\}) \cap {\mathcal {T}} = \emptyset .$ Therefore ${{\widetilde{\Phi }}}_2\circ {{\widetilde{\Psi }}}\circ {{\widetilde{\Phi }}}_1\circ \Phi _2\circ \Psi \circ \Phi _1 \circ \Psi _0$ is the desired composition of shears which fulfills the correspondence of Z with ${\mathcal {T}}$. $\square$

Remark 3

Observe that shears in the proof of Theorem 2 are in fact slice-regular mappings.

4 Slice Fatou–Bieberbach domains in ${\mathbb {H}}^2$

In this section we give a definition of slice Fatou–Bieberbach domains in ${\mathbb {H}}^2$ and then construct an example of a slice Fatou–Bieberbach domain in ${\mathbb {H}}^2$ which avoids the (standard) tame set ${{\mathcal {T}}}.$ We recall that a Fatou–Bieberbach domain in ${\mathbb {C}}^n$ is a proper subset $\Omega \subset {\mathbb {C}}^n,$ biholomorphic to ${\mathbb {C}}^n.$ In the quaternionic setting the notion of analiticity is usually associated with the notion of slice-regulariwe can find a sequence ty. The class of slice-regular mappings is rather rigid, for example it is not preserved even by compositions with real linear transformations. This is the reason why in the next definition we require analiticity for the map $h: {\mathbb {H}}^2 \rightarrow \Omega ,$ i.e., $h \in {{\mathcal {H}}}^2[z,w]$ (see Sect. 1.1 for the definition) but in addition we also require that the intersection of $\Omega$ with complex subspaces ${\mathbb {C}}_I^2 = {\mathbb {C}}_I \times {\mathbb {C}}_I, I \in {\mathbb {S}},$ should coincide with the definition of a Fatou–Bieberbach domain in the complex setting.

Definition 4

A slice Fatou–Bieberbach domain in ${\mathbb {H}}^2$ is a proper subset $\Omega \subset {\mathbb {H}}^2$ such that there exists a real analytic diffeomorphism $h : {\mathbb {H}}^2 \rightarrow \Omega ,$ i.e., $h \in {{\mathcal {H}}}^2[z,w]$ and such that for every $I \in {\mathbb {S}}$ the restriction $h|_{{\mathbb {C}}^2_I}: {\mathbb {C}}^2_I \rightarrow \Omega \cap {{\mathbb {C}}_I^2}$ is a biholomorphic map; in other words, $\Omega \cap {{\mathbb {C}}_I^2}$ is a Fatou-Bieberbach domain in ${\mathbb {C}}_I^2$.

The main result of this section is the following

Theorem 3

There exists a slice Fatou–Biebaerbach domain in ${\mathbb {H}}^2$ avoiding ${{\mathcal {T}}}.$

We provide the construction of a slice Fatou–Bieberbach domain by adapting the standard procedure of ‘escaping points’ to the quaternionic setting and using shears as automorphisms with volume 1.

To this end we follow the approximation scheme presented in Lemma 2.1 of [2] and stated here as Lemma 5.

Before describing the approximation scheme in Lemma 5, we need to prove two technical lemmas. We use the following notation. Let $\Delta (r_1,r_2):= B^4(0,r_{1}) \times B^4(0,r_{2})$ be a biball, i.e., the product of two balls centered at the origin with radii $r_1, r_2 > 0.$ We denote by $\Delta =\Delta (1,1)$ the product of two unit balls; furthermore, if V is a subset of ${\mathbb {H}}^2$ and $\delta$ a positive real number, we define $V+\delta \Delta :=\bigcup \limits _{(p_1, \, p_2)\in V} B^4(p_1, \delta )\times B^4(p_2, \delta )$.

Lemma 3

Let U, V be two biballs, $U \Subset V$. Then there exists a constant c depending only on U and V such that for any shear $\Phi$ it holds that $|D (\Phi -\mathrm {id})|_{\infty ,U} < c\, |\Phi - \mathrm {id}|_V$.

Proof

Any shear can be written either the form $\Phi (z,w) = (z,w) + f(z - \mu w)(\mu ,1),$ ($\mu \in {\mathbb {R}},$ $f \in {{\mathcal {S}}}{{\mathcal {R}}}({\mathbb {H}}))$ or $\Phi (z,w) = (z,w) + (f(w),0), f \in {{\mathcal {S}}}{{\mathcal {R}}}({\mathbb {H}}).$

Consider $U = \Delta (\varrho _{1},\varrho _{2}),$ $V = \Delta (r_{1}, r_{2})$, with $\varrho _j<r_j$, $j=1,2$, and define $R_U= \varrho _{1} +|\mu | \varrho _{2},$ $R_V = r_{1}+|\mu |r_{2,}.$ Since $U \Subset V,$ then $R_U<R_V$. The Jacobian of a shear $\Phi (z,w) = (z,w) + f(z - \mu w)(\mu ,1)$ is

$$\begin{aligned} D\Phi (z,w)[h,k]= & \begin{bmatrix} h+\mu {\widehat{\partial }}_z\,\,f(z-\mu w)[h] &\, & \mu {\widehat{\partial }}_w\,\,f(z-\mu w)[k] \\ {\widehat{\partial }}_z\,\,f(z-\mu w)[h] &\, & k+{\widehat{\partial }}_w\,\,f(z-\mu w)[k] \end{bmatrix}\\= & \begin{bmatrix} h+\mu {\widehat{\partial }}_z\,\,f(z-\mu w)[h] &\,& -\mu ^2 {\widehat{\partial }}_z\,\,f(z-\mu w)[k] \\ {\widehat{\partial }}_z\,\,f(z-\mu w)[h] &\,& k - \mu {\widehat{\partial }}_z\,\,f(z-\mu w)[k] \end{bmatrix}, \end{aligned}$$

since for functions of the form $f(z - \mu w)$ with $\mu$ real the chain rule ${\widehat{\partial }}_w\,\,f(z-\mu w)=-\mu {\widehat{\partial }}_z\,\,f(z-\mu w)$ holds (see [8], equation (2.8)). Therefore, we have

$$\begin{aligned}&|D(\Phi - \mathrm {id})(z,w)|_{\infty } = \max \{|\mu ||{\widehat{\partial }}_z\,\,f(z-\mu w)[h]-\mu {\widehat{\partial }}_z\,\,f(z-\mu w)[k]|,\\&\qquad |{\widehat{\partial }}_z\,\,f(z-\mu w)[h]-\mu {\widehat{\partial }}_z\,\,f(z-\mu w)[k]|,\,|h|,|k| \le 1\}\\&\quad \le (1 + |\mu |)\max \{|{\widehat{\partial }}_z\,\,f(z-\mu w)[h]-\mu {\widehat{\partial }}_z\,\,f(z-\mu w)[k]|, |h|,|k| \le 1\}\\&\quad \le (1 + |\mu |)\max \{|{\widehat{\partial }}_z\,\,f(z-\mu w)[h]|+|\mu ||{\widehat{\partial }}_z\,\,f(z-\mu w)[k]|, |h|,|k| \le 1\}\\&\quad \le (1 + |\mu |)^2\max \{|{\widehat{\partial }}_z\,\,f(z-\mu w)[h]|,|h|\le 1\}. \end{aligned}$$

Assume that $|\Phi - \mathrm {id}| = \varepsilon$ on V. Then $|f(u)| \le \varepsilon /\sqrt{1 + \mu ^2} = {\tilde{\varepsilon }}$ for $|u| \le R_V$ and if $(z,w) \in U$ we have $|z- \mu w| \le R_U<R_V.$ Since f is by definition entire and slice-regular, the coefficients in its power series expansion $f(u) = \sum \nolimits _{m=0}^{\infty } u^m a_m$ can be estimated by Cauchy formula, $|a_m| \le {\tilde{\varepsilon }}/{R_V^m}.$ Furthermore, for $(z,w) \in U$, we have

$$\begin{aligned}|{\widehat{\partial }}_z(z- \mu w)^m[h]| \le m |h| |z - \mu w|^{m-1},\end{aligned}$$

so the following estimate holds

$$\begin{aligned} |{\widehat{\partial }}_z\,\,f(z-\mu w)[h]|&\le \frac{{\tilde{\varepsilon }} |h|}{R_V} \sum _{m=1}^{\infty } m \left| \frac{z - \mu w}{R_V}\right| ^{m-1} \\&\,\le \frac{{\tilde{\varepsilon }} |h|}{R_V} \sum _{m=1}^{\infty } m \left( \frac{{R}_{U}}{R_V}\right) ^{m-1} = \,\frac{{\tilde{\varepsilon }} |h| R_V}{(R_V - {R}_{U})^2}. \end{aligned}$$

Define $d := R_V(R_V - {R}_{U})^{-2}.$ The norm $| \cdot |_{\infty }$ of the matrix $D(\Phi (z,w) - (z,w))[h,k]$ for $|h|, |k| \le 1$ is estimated by

$$\begin{aligned} |D(\Phi (z,w) - (z,w))[h,k]|_{\infty } \le d {\tilde{\varepsilon }}(1 +|\mu |)^2 = d \varepsilon (1 +|\mu |)^2/\sqrt{1 + \mu ^2}. \end{aligned}$$

We claim that there exists a constant c depending only on U and V such that $d (1 +|\mu |)^2/\sqrt{1 + \mu ^2} \le c.$ Indeed, since the function

$$\begin{aligned} c(t) := d \frac{(1 + t)^2}{\sqrt{1 + t^2}} = \frac{(1+t)^2(r_{1} + t r_{2})}{\sqrt{1 + t^2}(r_{1} - \varrho _{1} + t(r_{2} - \varrho _{2}))^2}, \, t \ge 0, \end{aligned}$$

is continuous in t, positive and bounded, such a constant exists. Finally, if the shear is of the form $\Phi (z,w) = (z + f(w),w)$, then the constant is $c(\infty )=\dfrac{r_2}{(r_2-\varrho _2)^2}$. $\square$

$\square$

Let $\{U_n\}_{n \in {\mathbb {N}}}$ be an exhaustion of ${\mathbb {H}}^2$ by biballs such that $U_n \Subset U_{n+1}.$ For each n we define biballs $U_n'''\Subset U_n''\Subset U_n'$ by strictly increasing the radii of $U_n,$ so that we also have $U_n' \Subset U_{n+1},$ more precisely, we choose a (strictly decreasing) sequence of positive real numbers $\{\delta _n\}_{n \in {\mathbb {N}}}$ such that $U_n':=U_n + \delta _n \Delta \Subset U_n + 2 \delta _n \Delta \Subset U_{n+1}.$ Define $U_n'':=U_n+(2\delta _n/3) \Delta$ and $U_n''':=U_n+(\delta _n/3) \Delta .$ Then

$$\begin{aligned}U_n \Subset U_n'''\Subset U_n''\Subset U_n'\Subset U_{n+1}\end{aligned}$$

for every $n \in {\mathbb {N}}$.

Lemma 4

Let the sequences of sets $\{U_n\}_ {n \in {\mathbb {N}}}$ and $\{\delta _n\}_{n \in {\mathbb {N}}}$ be as above and let $\{\Phi _n\}_{ n \in {\mathbb {N}}}$ be a sequence of shears. Define $\Psi _{l,m} = \Phi _l \circ \ldots \circ \Phi _{m+1}$ for $l > m$ and $\Psi _{m,m} := \mathrm {id}.$ Assume that for every choice of $l > m \ge n$ the following inclusions hold:

$$\begin{aligned} \Psi _{l,m}(U_n) \Subset U_{n}''',\,\Psi _{l,m}(U_n''') \Subset U_{n}'',\,\Psi _{l,m}(U_n'') \Subset U_{n}',\,\Psi _{l,m}(U_n') \Subset U_{n+1}. \end{aligned}$$

(4)

Let a sequence $\{\varepsilon _n\}_{ n \in {\mathbb {N}}}$ of positive real numbers be given such that $|\Phi _n -\mathrm {id}|_{U_{n-1}'} < \varepsilon _{n-1}$ for every $n\in {\mathbb {N}}.$ Fix $k> 1$ and let $l > m \ge n \ge k.$ Then

$$\begin{aligned} |\Psi _{l,m} - \mathrm {id}|_{U_k''} < \sum _{j=m+1}^{l} \varepsilon _{j-1} \end{aligned}$$

(5)

and

$$\begin{aligned} |D(\Psi _{l,m}- \mathrm {id})|_{\infty ,\Psi _{m,n}(U_k''')} <\prod _{j=m+1}^l (1+ c_{j-1}\varepsilon _{j-1}) - 1 \end{aligned}$$

(6)

where $c_j$ are given by Lemma 3 with respect to sets $U=U_j''$ and $V = U_j'.$

Remark 4

By definition $\Psi _{l,m}\circ \Psi _{m,n}=\Psi _{l,n}$ and $\Psi _{m+1,m}=\Phi _{m+1}$.

Proof

For any choice of integers $l > m \ge n \ge k,$ the triangle inequality gives

$$\begin{aligned} |\Psi _{l,m} - \mathrm {id}|_{U_k''}= & |\Psi _{l,m} -\Psi _{l-1,m} +\Psi _{l-1,m} -\mathrm {id}|_{U_{k}''} \\\le & |\Phi _l\circ \Psi _{l-1,m} -\Psi _{l-1,m}|_{U_k''} +|\Psi _{l-1,m} -\mathrm {id}|_{U_k''}. \end{aligned}$$

Because $l > k,$ we have the inclusion $U_k' \subset U_{l-1}'$ and using (4) we estimate

$$\begin{aligned} |\Phi _{l}\circ \Psi _{l-1,m} -\Psi _{l-1,m}|_{U_k''} = |\Phi _l-\mathrm {id}|_{\Psi _{l-1,m}(U_{k}'')} \le |\Phi _l-\mathrm {id}|_{U_{k}'} \le |\Phi _l-\mathrm {id}|_{U_{l-1}'} \end{aligned}$$

and then by induction

$$\begin{aligned}|\Psi _{l,m} - \mathrm {id}|_{U_k''}\le \sum _{j=m+1}^l |\Psi _{j,m}-\Psi _{j-1,m}|_{ U_{k}'' }\le \sum _{j=m+1}^l |\Phi _j-\mathrm {id}|_{U_{j-1}'}<\sum _{j=m+1}^l \varepsilon _{j-1}. \end{aligned}$$

For the second estimate, first observe that $|\Phi _j -\mathrm {id}|_{U_{j-1}'} < \varepsilon _{j-1}$ implies $|D(\Phi _j -\mathrm {id})|_{\infty ,U_{j-1}''} < c_{j-1}\varepsilon _{j-1}$ by Lemma 3. Fix $m \ge n \ge k$ and proceed by induction on l. For $l = m+1$ we have $\Psi _{m+1,m} = \Phi _{m+1},$ so there is nothing to prove for the base case of induction. Assume that the claim holds true for $l-1 \ge m+1.$ By the chain rule for mappings in ${\mathcal {H}}^2[z,w]$ we have

$$\begin{aligned} D(\Phi _l \circ \Psi _{l-1,m})(z,w)[h,k] = D(\Phi _l)(\Psi _{l-1,m}(z,w)) \diamond D\Psi _{l-1,m}(z,w)[h,k]. \end{aligned}$$

Since the operations D and $\diamond$ are additive and the matrix norm $| \cdot |_{\infty }$ is sub-multiplicative with respect to $\diamond$ (see Sect. 1.1), we can write

$$\begin{aligned} D(\Psi _{l,m} -\mathrm {id})(z,w)[h,k]= & \,D(\Phi _l \circ \Psi _{l-1,m}- \Psi _{l-1,m})(z,w)[h,k] \\&+D(\Psi _{l-1,m} - \mathrm {id})(z,w)[h,k]\\= & \,D(\Phi _l- \mathrm {id})(\Psi _{l-1,m}(z,w)) \diamond D\Psi _{l-1,m}(z,w)[h,k] + \\&+D(\Psi _{l-1,m} - \mathrm {id})(z,w)[h,k] \end{aligned}$$

and, by induction hypothesis, we have

$$\begin{aligned} |D(\Psi _{l,m} -\mathrm {id})|_{\infty , \Psi _{m,n}(U_k''')} &= |D(\Phi _l - \mathrm {id})|_{\infty ,\Psi _{l-1,m}(\Psi _{m,n}(U_k'''))} |D\Psi _{l-1,m}|_{\infty , \Psi _{m,n}( U_k''')}\\&\quad+ |D(\Psi _{l-1,m}- \mathrm {id})|_{\infty ,\Psi _{m,n}(U_k''')} \\& \le \, |D(\Phi _l - \mathrm {id})|_{\infty ,\Psi _{l-1,n}(U_{k}''')} |D\Psi _{l-1,m}|_{\infty , \Psi _{m,n}(U_k''')}\\&\quad+|D(\Psi _{l-1,m}- \mathrm {id})|_{\infty ,\Psi _{m,n}(U_k''')} \\& < c_{l-1}\varepsilon _{l-1} \prod _{j = m+1}^{l-1}(1 + c_{j-1}\varepsilon _{j-1}) + \prod _{j = m+1}^{l-1}(1 + c_{j-1}\varepsilon _{j-1}) - 1 \\&= \,\prod _{j = m+1}^{l}(1 + c_{j-1}\varepsilon _{j-1}) -1, \end{aligned}$$

since the inclusions $\Psi _{l-1,n}(U_{k}''') \Subset U_{k}'' \Subset U_{l-1}''$ and the definition of constants $c_j$ imply $|D(\Phi _l - \mathrm {id})|_{\infty ,\Psi _{l-1,n}(U_{k}''')} \le |D(\Phi _l - \mathrm {id})|_{\infty ,U_{l-1}''} < c_{l-1}\varepsilon _{l-1}.$ Furthermore, by induction hypothesis, $|D\Psi _{l-1,m}|_{\infty , \Psi _{m,n}(U_k''')} \le 1+|D\Psi _{l-1,m}-\mathrm {id}|_{\infty , \Psi _{m,n}(U_k''')}<\prod _{j = m+1}^{l-1}(1 + c_{j-1}\varepsilon _{j-1}).$ $\square$

Lemma 5

Let the sequences $\{U_n\}_{n\in {\mathbb {N}}},\ \{\delta _n\}_{n\in {\mathbb {N}}}$ and $\{c_n \}_{n\in {\mathbb {N}}}$ be as in Lemma 4. Let $\lbrace \Phi _n \rbrace _{n \in {\mathbb {N}}}$ be a sequence of shears and let $\{\varepsilon _n\}_{n\in {\mathbb {N}}},\{\varepsilon _n'\}_{n\in {\mathbb {N}}},$ $\varepsilon _n,\varepsilon _n' \in (0,1)$ be strictly decreasing sequences converging to 0 such that $\varepsilon _n < \min \{\delta _{n}, \log (1 + \varepsilon _n')\}$ for any $n\in {\mathbb {N}}$ and such that for any $n\in {\mathbb {N}}$

• (i) $|\Phi _{n+1} - \mathrm {id}|_{U_n'} < 2^{-n} \min \lbrace 1,c_{n}^{-1} \rbrace \varepsilon _{n}/6,$

• (ii) if $\Phi \in {\mathcal {C}}^1(U_{n}''')$ satisfies $|\Phi - \mathrm {id}|_{U_n'''}< \varepsilon _n'$ and $|D(\Phi - \mathrm {id})|_{\infty , U_n'''}< \varepsilon _n'$, then $\Phi$ is one to one in ${U}_n$.

For each $n \in {\mathbb {N}}$ define $\Psi _n:= \Phi _n \circ \Phi _{n-1}\circ \ldots \circ \Phi _1.$ Let $\Omega _n:=\Psi _n^{-1}(U_n).$ Then $\Omega _n \Subset \Omega _{n+1}$ and the sequence $\{\Psi _n\}_{n\in {\mathbb {N}}}$ converges, uniformly on compacta of the open set $\Omega : = \cup _{n=1}^{\infty } \Omega _n$, to a map $\Psi \in {\mathcal {C}}^1(\Omega )$ which maps $\Omega$ bijectively onto ${\mathbb {H}}^2.$

Before the proof of the Lemma, we make a couple of remarks.

Remark 5

Observe that for any $U_n\Subset U_n'''$ one can always choose a positive $\varepsilon '_n$ such that if $\Phi \in {\mathcal {C}}^1(U_{n}''')$ approximates the identity in the sense of condition (ii), then it is one to one in ${U}_n.$

Remark 6

The condition (i) in Lemma 5 plays a similar role as condition (i) in Lemma 2.1 of [2]. It not only implies that for $m \ge n$ we have $|\Phi _{m+1}-\mathrm {id}|_{ U'_{n} } \le (2^{-m}\varepsilon _m/6)$ and hence $\Phi _{m+1}(U'_{n}) \subset (2^{-m}\varepsilon _m/6)\Delta \Subset U_{n+1} \subset U'_{m+1},$ but also the validity of inclusions (4).

Indeed, since $|\Phi _{m+2} - \mathrm {id}|_{U_{m+1}'} < 2^{-(m+1)}\varepsilon _{m+1}/6 ,$ we can estimate

$$\begin{aligned} |\Psi _{m+2,m} - \mathrm {id}|_{U_n'}< &\, |\Psi _{m+2,m} - \Psi _{m+1,m} +\Phi _{m+1}- \mathrm {id}|_{U_n'} \\< &\, |\Phi _{m+2} - \mathrm {id}|_{\Phi _{m+1}(U_n')} +|\Phi _{m+1}- \mathrm {id}|_{U_n'} \\< &\, |\Phi _{m+2} - \mathrm {id}|_{U_{m+1}'} +|\Phi _{m+1}- \mathrm {id}|_{U_m'} \\< &\, 2^{-(m+1)}\varepsilon _{m+1}/6+2^{-m}\varepsilon _m/6< \varepsilon _{m}/3 \le \varepsilon _n/3<\delta _{n}/3, \end{aligned}$$

so $\Psi _{m+2,m}(U'_{n}) \subset (\delta _{n}/3)\Delta \Subset U_{n+1}.$ Hence, inductively, for $l > m \ge n$ the estimate

$$\begin{aligned} |\Psi _{l,m} - \mathrm {id}|_{U_n'}< \left( \sum _{j = m+1}^{\infty } 2^{-(j-1)}\varepsilon _{j-1}/6\right)< \varepsilon _{m}/3<\delta _{n}/3 \end{aligned}$$

(7)

holds, which together with the definition of the sequence $\{\delta _n\}_{n \in {\mathbb {N}}}$ implies the validity of inclusions (4) for $l > m \ge n.$

Remark 7

Let us comment the changes in the formulation of Lemma 5 compared to Lemma 2.1 in [2]. In the complex setting the Cauchy formula for holomorphic functions guarantees that the sup-norm estimates on $U_n'$ imply $C^1$-estimates on ${U}_{n}'' \Subset U_n',$ which is not the case for an arbitrary automorphism $\Psi \in {\mathcal {H}}^2[z,w],$ since those are just real analytic. Such estimates, however, hold for mappings $\Psi - \mathrm {id},$ where $\Psi$ is a shear, as presented in Lemma 3.

For similar reasons we have to add $C^1$-estimates to point (ii) of the statement of Lemma 5. The new statement is then a classical result in real analysis: if we write $h=(h_0,h_1,h_2,h_3)$ and $z=(z_0,z_1,z_2,z_3)$ (with respect to the orthonormal basis (1, i, j, k) of ${\mathbb {H}}$) then, since the operator ${\widehat{\partial }}_z$ is additive and real linear in h,

$$\begin{aligned} {\widehat{\partial }}_z\,\,f(z,w)[h] = \sum _{l = 0}^3 h_l \frac{\partial }{\partial z_l}\,\,f(z,w), \end{aligned}$$

and so the condition $|D(\Phi (z,w) - (z,w))[h,k]| < \varepsilon$ for $|h|,|k| \le 1$ implies that $\Phi$ approximates the identity in $C^1$-sense as a real smooth mapping.

Remark 8

The map $\Psi ^{-1}$ is defined in ${\mathbb {H}}^2$ but we cannot argue that it is analytic, because in general it is not a slice-regular map in two variables and after composing slice-regular functions one cannot apply Cauchy estimates. So Theorem 3 is not a straightforward application of Lemma 5.

Proof of Lemma 5

With all the above lemmas the proof of Lemma 5 is now similar to the analogous one in [2].

To prove uniform estimates for the mappings and their derivatives on $\Omega ,$ choose a compact set $K \subset \Omega$ and $\varepsilon > 0.$ We are looking for $m_0 > 0$ so that for $l > m \ge m_0$ the estimates

$$\begin{aligned}|\Psi _l - \Psi _m|_{K}< \varepsilon \hbox { and } |D(\Psi _l - \Psi _m)|_{\infty ,K}< \varepsilon \end{aligned}$$

hold.

Define $\Psi _{l,m}$ as in Lemma 4. Notice that the assumptions of Lemma 4 are fulfilled, because by Remark 6, the inclusions (4) hold for all $l > m \ge n.$ In particular, $\Psi _{n+1,n}(U_n') \Subset U_{n+1},$ or, equivalently, $U_{n}' \Subset \Phi _{n+1}^{-1}(U_{n+1}).$

Define $\Omega _n':=\Psi _n^{-1}(U_n'),$ $\Omega _n'':=\Psi _n^{-1}(U_n'')$ and $\Omega _n''':=\Psi _n^{-1}(U_n''').$ The first three of the following inclusions

$$\begin{aligned}\Omega _n \Subset \Omega _n'''\Subset \Omega _n''\Subset \Omega _n'\Subset \Omega _{n+1}\end{aligned}$$

are obvious. The last inclusion $\Omega _n'\Subset \Omega _{n+1}$ is equivalent to

$$\begin{aligned} \Psi _{n+1}(\Omega _n') \Subset \Psi _{n+1}(\Omega _{n+1}) = U_{n+1}. \end{aligned}$$

(8)

Because $\Psi _{n+1}(\Omega _n') = \Phi _{n+1} (\Psi _{n}(\Omega _n')) = \Phi _{n+1}(U_n')$ the inclusion (8) is equivalent to $U_n' \Subset \Phi _{n+1}^{-1}(U_{n+1})$ which holds true as mentioned. As a consequence we have that $\{\Omega _n\}_{n\in {\mathbb {N}}}$ is an exhaustion of $\Omega =\cup _n \Omega _n$.

Notice that also

$$\begin{aligned} \Psi _m(\Omega _n''') = \Psi _{m,n}(U_n''') \Subset U_n'' \end{aligned}$$

(9)

holds for every $m \ge n.$

Fix $n_0\in {\mathbb {N}}$ such that $K\subset \Omega _{n_0}$. Hence it suffices to show that the sequence $\{\Psi _m\}_{m \in {\mathbb {N}}}$ converges uniformly in ${{\mathcal {C}}}^1$ sense on $\Omega _n$ for some $n\ge n_0$. Let $l > m \ge n\ge n_0.$ Because of the inclusions in (9) we have

$$\begin{aligned} |\Psi _l - \Psi _m|_{\Omega _n'''} = |\Psi _{l,m} - \mathrm {id}|_{\Psi _m(\Omega _n''')} \le |\Psi _{l,m} - \mathrm {id}|_{U_n''} \end{aligned}$$

(10)

and by Lemma 4 we estimate

$$\begin{aligned} |\Psi _l - \Psi _m|_{\Omega _n'''} \le \sum _{j = m}^{l-1}|\Phi _{j+1} - \mathrm {id}|_{U_n''} \le \sum _{j=n}^{\infty } 2^{-j} \varepsilon _{j}/6< 2^{-n} \varepsilon _{n}/3 < \varepsilon \end{aligned}$$

(11)

for large enough n.

The estimate for the derivative is a bit more delicate.

Since the norm $| \cdot |_{\infty }$ is sub-multiplicative and the derivative is additive, we have the following estimate

$$\begin{aligned}&|D(\Psi _l - \Psi _m)|_{\infty , \Omega _n'''} \le |D(\Psi _{l,m} - \mathrm {id})|_{\infty , \Psi _m(\Omega _n''')}\cdot |D\Psi _m|_{\infty , \Omega _n'''} \nonumber \\&=|D(\Psi _{l,m} - \mathrm {id})|_{\infty , \Psi _{m,n}(U_n''')}\cdot |D\Psi _m|_{\infty , \Omega _n'''} \end{aligned}$$

(12)

The assumptions of Lemma 4 are now fulfilled and so we can estimate the term $|D(\Phi _{l,m} - \mathrm {id})|_{\infty , \Psi _{m,n}(U_n''')}$ in (12) by inserting $2^{-j+1} \min \{1, c_{j-1}^{-1}\}\varepsilon _{j-1}/6$ instead of $\varepsilon _{j-1}$ in (6) and using $c_{j-1} \min \{1, c_{j-1}^{-1}\}\le 1$ to obtain

$$\begin{aligned} |D(\Psi _{l,m} - \mathrm {id})|_{\infty ,\Psi _{m,n} (U_{n}''')}&\le \prod _{j = m+1}^{\infty } (1 + 2^{-j+1}\varepsilon _{j-1}/6) - 1\nonumber \\&\le e^{\sum _{j=m+1}^{\infty } 2^{-j+1} \varepsilon _{j-1}/6} - 1\le e^{2^{-m}\varepsilon _{m}/3} - 1 \nonumber \\&<\varepsilon _m' \le \varepsilon _n' < \varepsilon \end{aligned}$$

(13)

for large enough $n\ge n_0$. We also observe that

$$\begin{aligned} 0< e^{2^{-n}\varepsilon _{n}/3} - 1<2 \hbox { for } n \ge 1 \end{aligned}$$

(14)

since $\varepsilon _{n} < 1.$

In order to estimate $|D\Psi _m|_{\infty ,\Omega _n'''}$ we first fix n so that the estimates (11) and (13) hold. The mapping $\Psi _n$ is entire, the set $\Omega _n''$ relatively compact, so $|D\Psi _{n}|_{\infty ,\Omega _n'''} < M$ for some positive constant M.

Taking into account that $\Psi _{n}(\Omega _n''') = U_n'''= \Psi _{n,n}(U_n''')$ we estimate

$$\begin{aligned} |D\Psi _m|_{\infty , \Omega _n'''}= &\, |D(\Psi _m - \Psi _{n})|_{\infty ,\Omega _n'''} + |D\Psi _{n}|_{\infty ,\Omega _n'''} \\\le &\, |D(\Psi _{m,n}-\mathrm {id})|_{\infty , \Psi _{n}(\Omega _n''')} |D\Psi _{n}|_{\infty ,\Omega _n'''}+ |D\Psi _{n}|_{\infty ,\Omega _n'''}\\\le &\, |D(\Psi _{m,n}-\mathrm {id})|_{\infty ,\Psi _{n,n}(U_n''')} |D\Psi _{n}|_{\infty ,\Omega _n'''+ |D\Psi _{n}|_{\infty ,\Omega _n'''}}\\\le &\, 2 M + M = 3M, \end{aligned}$$

since by (13) and (14) we have $|D(\Psi _{m,n}-\mathrm {id})|_{\infty , \Psi _{n,n}(U_n''')} < 2$ for all $m \ge n.$ Hence the derivative $D(\Psi _l - \Psi _m)$ is by (12) bounded by

$$\begin{aligned} |D(\Psi _l - \Psi _m)|_{\infty , \Omega _n'''}\le &\, |D(\Psi _{l,m} - \mathrm {id})|_{\infty , \Psi _{m,n}(U_n''')}\cdot |D\Psi _m|_{\infty , \Omega _n'''}\\\le &\, \varepsilon _m' 3M. \end{aligned}$$

Choose $m_0$ large enough so that $\varepsilon _{m_0}' < \varepsilon /(3M).$ Then we have

$$\begin{aligned} |D(\Psi _l - \Psi _m)|_{\infty , \Omega _n}\le |D(\Psi _l - \Psi _m)|_{\infty , \Omega _n'''}< \varepsilon \end{aligned}$$

as desired.

The sequence $\{\Psi _m\}_{m \in {\mathbb {N}}}$ thus converges to a map $\Psi$ defined in the open set $\Omega$ uniformly - in ${{\mathcal {C}}}^1$ sense - on compact sets. The limit map $\Psi : \Omega \rightarrow \Psi (\Omega )$ is at least ${\mathcal {C}}^1$ in $\Omega$.

On each $U_n'''$, the map $\Psi \circ \Psi _n^{-1}=\lim \limits _{m \rightarrow \infty } \Psi _{m,n}$ satisfies

$$\begin{aligned} |\Psi \circ \Psi _n^{-1} - \mathrm {id}|_{U_n'''} \le \lim _{m \rightarrow \infty }|\Psi _{m,n} - \mathrm {id}|_{U_n''} \le 2^{-n+1} \varepsilon _{n}/3 < \varepsilon _n' \end{aligned}$$

(15)

and

$$\begin{aligned} |D(\Psi \circ \Psi _n^{-1} - \mathrm {id})|_{\infty ,U_{n}'''}\le \lim _{m \rightarrow \infty }|D(\Psi _{m,n} - \mathrm {id})|_{\infty ,U_{n}'''} \le e^{2^{-n}{\varepsilon _{n}}/{3}} - 1 < \varepsilon _n' \end{aligned}$$

therefore by (ii) $\Psi \circ \Psi _n^{-1}$ is injective in each $U_n$, so the map $\Psi$ is injective in $\Omega _n.$ Therefore, it is injective in $\Omega$; furthermore, by Brouwer’s theorem on invariance of domains, the map $\Psi$ is also an open map.

To prove that $\Psi$ is surjective, we will actually show – as in [2] – that $\Psi (\Omega )$ contains all the sets $U_{n}''.$

Fix n such that $r_{n-1}:=d(\partial \,\,U_{n-1},0) > 1.$ We claim that $U_{n-1}'' \Subset \Psi ({\overline{\Omega }}_{n}).$ Define $G_n:=U_{n-1}''\cap \Psi ({\overline{\Omega }}_{n})$ which is a closed set in $U_{n-1}''$ because $\Psi ({\overline{\Omega }}_n)$ is compact (since $\overline{\Omega _n}$ is) and therefore closed. By construction $\Psi ({\overline{\Omega }}_{n}) = \Psi \circ \Psi _n^{-1}({\overline{U}}_n)$ and by (10)

$$\begin{aligned}|\Psi \circ \Psi _n^{-1} - \mathrm {id}|_{U_{n-1}''} \le 2^{-n+1}\varepsilon _{n}/3,\end{aligned}$$

so $\Psi \circ \Psi _n^{-1}(0) \in G_n$ because $|\Psi \circ \Psi _n^{-1}(0)|< \varepsilon _n< 1 < r_{n-1}.$ For $w \in G_n$ we have that $w = \Psi \circ \Psi _n^{-1}(z)$ for some $z \in {\overline{U}}_n$ and hence $|w - z|< 2^{-n+1}\varepsilon _{n}/3< \delta _{n}/3 < \delta _{n-1}/3,$ so $z \in U_{n-1}'' +\Delta \delta _{n-1}/3 \Subset U_{n-1}'$ and there is an open neighborhood $U \subset U_{n-1}'\subset U_n$ of z. Then $\Psi \circ \Psi _n^{-1}(U) \subset \Psi \circ \Psi _n^{-1}(U_n) = \Psi (\Omega _n)$ is an open set containing w,  since $\Psi \circ \Psi _n^{-1}$ is an open mapping. The set $\Psi \circ \Psi _n^{-1}(U) \cap U_{n-1}''$ is then an open neighborhood of w in $\Psi (\Omega _n) \cap U_{n-1}''\subset G_n$; in other words, the set $G_n$ is a nonempty open and closed subset of $U_{n-1}''$ and since the latter is connected, we conclude that $G_n = U_{n-1}''.$ $\square$

Remark 9

By the estimate (15), the sequence $\{\Psi \circ \Psi _m^{-1}\}_{m \in {\mathbb {N}}}$ converges uniformly on compact subsets of ${\mathbb {H}}^2$ to the identity mapping and hence also the sequence $\{ \Psi _m^{-1} = \Phi _1^{-1} \circ \ldots \circ \Phi _m^{-1}\}_{m \in {\mathbb {N}}}$ converges to $\Psi ^{-1}$ uniformly on compact subsets of ${\mathbb {H}}^2.$ Consider $\Phi _n(z,w) = (z,w) + f_n(z - \mu _n w)(\mu _n,1);$ we assume that each (slice-regular) function $f_n$ has a zero of multiplicity $M_n$ at the origin, with $M_n$ strictly increasing with n, and the power series representative (there are infinitely many of them, see [7]) for $f_n(z - \mu _n w)$ is the one obtained by formally plugging the term $z - \mu _n w$ into power series for $f_n,$ hence the series expansion for $f_n(z - \mu _n w)$ begins with the homogeneous polynomials of degree $M_n$ in (z, w). Then the power series expansions of $\{\Phi _1^{-1} \circ \ldots \circ \Phi _n^{-1}\}_{n\in {\mathbb {N}}},$ which are obtained inductively by formally plugging the power series for $\Phi _n^{-1}$ into the power series $\Phi _1^{-1} \circ \ldots \circ \Phi _{n-1}^{-1},$ converge to a formal power series expansion, since any fixed homogenous polynomial in the power series expansion changes only finitely many times, due to strictly increasing orders of vanishing of the functions $f_n$ at the origin. Because of the lack of Cauchy estimates we cannot say that the formal power series is convergent with limit $\Psi ^{-1}.$

Proof of Theorem 3

As a first step in the proof, we move the set ${{\mathcal {T}}} = \lbrace (2n-1, 0), n \in {\mathbb {N}}\rbrace \cup \lbrace (2n + {\mathbb {S}},0), n \in {\mathbb {N}}\rbrace$ to a more convenient position, namely to the set ${{{\mathcal {T}}^{\prime}}}\,=\lbrace (-n^2, -n^2), n \in {\mathbb {N}}\rbrace \cup \lbrace (n^2 I,n^2I), n \in {\mathbb {N}},\ \ I\in {\mathbb {S}}\rbrace$ by a composition of a vertical shear and a horizontal shear. The vertical one is defined as $\Phi _v(z,w) = (z, w + f(z))$ with f satisfying the interpolation conditions $f(2n-1)= -n^2, \,f(2n + I) = n^2I,$ and the horizontal one is defined as $\Phi _h(z,w) = (z + g(w),w)$ with g satisfying the interpolating conditions $g(-n^2) = -n^2 - 2n + 1, \, g(n^2I) = n^2I - 2n - I.$ Both f and g can be chosen to have real coefficients and therefore the shears $\Phi _h,\Phi _v$ both preserve ${\mathbb {C}}_I \times {\mathbb {C}}_I$ for all $I\in {\mathbb {S}}$.

Then, for any natural number m, we define the subset ${{\mathcal {T}}^{\prime}}_{m}\,\subset {{\mathcal {T}}^{\prime}}\,$ to be the set

$$\begin{aligned}{{\mathcal {T}}}_{m}\,'=\lbrace (-n^2, -n^2), n \ge m \rbrace \cup \lbrace (n^2 I,n^2I), n \ge m, \ I\in {\mathbb {S}}\rbrace .\end{aligned}$$

and construct two shears $\Phi _{v,m}=\mathrm {id}+ (0,-f_m)$ and $\Phi _{h,m}=\mathrm {id}+ (-g_m,0),$ where $f_m$ and $g_m$ are entire slice-regular functions with nonnegative coefficients in their power series expansions and such that for any $m\in {\mathbb {N}}$ the map $\Phi _{h,m} \circ \Phi _{v,m}$ has the property $\Phi _{h,m} \circ \Phi _{v,m}({{\mathcal {T}}}\,'_m) = {{\mathcal {T}}}\,'_{m+1}$ and $\Phi _{h,m}$ approximates the identity map on compact subsets of $m^2\Delta$ whereas $\Phi _{v,m}$ approximates the identity map on compact subsets of $(m+1)^2\Delta .$

To this end, for $n \ge m$, we require the following interpolation condition for $f_m$ and $g_m$:

$$\begin{aligned}&f_m(-n^2) = 2n+1, \,f_m(n^2 I) = -(2n+1) I, I \in {\mathbb {S}} \end{aligned}$$

(16)

$$\begin{aligned}&g_m(-(n+1)^2) = 2n+1,\, g_m((n+1)^2I) = -(2n+1)I,I \in {\mathbb {S}}, \end{aligned}$$

(17)

together with the property that $f_m$ is as small as we wish on compact subsets of $B^4(0,m^2)\subset {\mathbb {H}}$ and $g_m$ is as small as we wish on compact subsets of $B^4(0,(m+1)^2)\subset {\mathbb {H}},$ both with the order of vanishing at 0 as large as we wish.

We will actually look for such maps in the form $f_m:= f^P_m + f^S_m$ and $g_m:= g^P_m + g^S_m,$ where the maps with superscript P interpolate on points and leave the spheres fixed and the maps with superscript S interpolate on spheres and leave points fixed.

For each $n\in {\mathbb {N}}$, consider

$$\begin{aligned}&{\tilde{\phi }}^P_n(z):= \prod _{k \ne n}\left( 1 + \frac{z}{k^2}\right) ^2 \prod _{k = 1}^{\infty } \left( 1 + \frac{z^2}{k^4}\right) , \\ {\mathrm {and}}\\&\phi ^P_n(z):= \dfrac{{\tilde{\phi }}^P_n(z)}{{\tilde{\phi }}^P_n(-n^2)}. \end{aligned}$$

We have

$$\begin{aligned} \phi ^P_n(-n^2) = 1,\ \phi ^P_n(-k^2) = 0\ {\mathrm {if}}\ k \ne n,\ {\mathrm {and}}\ \phi ^P_n(m^2I) = 0, \forall m \in {\mathbb {N}}, \forall I \in {\mathbb {S}}. \end{aligned}$$

For a suitable choice of an increasing sequence $\{M_n\}_{n \in {\mathbb {N}}}$, the function

$$\begin{aligned} f_m^P(z):= \sum _{n = m}^{\infty } \left( \frac{z}{n^2}\right) ^{2M_n} \phi ^P_n(z) (2n+1) \end{aligned}$$

is well defined and satisfies the interpolation conditions $f_m^P(-n^2) = 2n+1$ if $n\ge m$ and $f^P_m(k^2I) = 0\ \forall k$. Furthermore the coefficients of the power series expansions of $f^P_m$ are nonnegative, $f^P_m$ is as small as we wish on compact subsets of $B^4(0, m^2)$ with order of vanishing at 0 at least $2M_m.$

In order to define $f_m^S$, we first consider

$$\begin{aligned} {\tilde{\phi }}(z):= \prod _{k=1}^{\infty }\left( 1 + \frac{z}{k^2}\right) , \; {\tilde{\phi }}_n(z):= \prod _{k \ne n} \left( 1 + \frac{z^2}{k^4}\right) ^2. \end{aligned}$$

We have

$$\begin{aligned} {\tilde{\phi }}(-k^2) = 0\,\; \forall k,\; {\tilde{\phi }}_n(k^2I) = 0,\; \forall k \ne n, \;\forall I \in {\mathbb {S}}. \end{aligned}$$

The value ${\tilde{\phi }}_n(n^2I)$ is positive and does not depend on $I\in {\mathbb {S}}$. Nonetheless ${\tilde{\phi }}(n^2I) = a + Ib\ne 0$ is not necessarily a real number. We are looking for a function of the form

$$\begin{aligned} {\tilde{\psi }}_n (z) = \left( \frac{z}{n^2}\right) ^{l+1}(\alpha z + \beta ),\;\; {\mathrm {with}\ }\alpha , \beta \ge 0, \;l \in \{0,1,2,3\}, \end{aligned}$$

such that

$$\begin{aligned} {\tilde{\psi }}_n(n^2I)(a + Ib) = -I. \end{aligned}$$

Choose $l \in \{0,1,2,3\}$ so that for $z_0 = -I^{l} (a + Ib)$ we have $\mathfrak {R}e(z_0) \ge 0$ and $\mathfrak {I}m(z_0) \le 0.$ Then $\alpha , \beta \in {\mathbb {R}}$ determined by the equality

$$\begin{aligned} \alpha n^2 I + \beta = \frac{{\bar{z}}_0}{|{\bar{z}}_0|^2} \end{aligned}$$

are nonnegative. Finally, define

$$\begin{aligned} \phi _n^S(z):= {\tilde{\psi }}_n(z){\tilde{\phi }}(z){\tilde{\phi }}_{n}(z)({\tilde{\phi }}_n(n^2I))^{-1} \end{aligned}$$

for some (and thus for all) $I \in {\mathbb {S}}.$ The coefficients in the power series expansion of $\phi _n^S$ are nonnegative as desired and, furthermore,

$$\begin{aligned}\phi _n^S(n^2I) = -I,\ \; {\phi }^S_n(k^2I) = 0,\; \forall k \ne n, \;\forall I \in {\mathbb {S}}\end{aligned}$$

and

$$\begin{aligned} {\phi }^S_n(-k^2) = 0, \ \forall k \in {\mathbb {N}}.\end{aligned}$$

For a suitable choice of an increasing sequence $\{M_n\}_{n \in {\mathbb {N}}}$, the function

$$\begin{aligned} f_m^S(z):= \sum _{n = m}^{\infty } \left( \frac{z}{n^2}\right) ^{4M_n }\phi ^S_n(z)(2n+1) \end{aligned}$$

is well defined with nonnegative coefficients in its power series expansion, as small as we wish on compact subsets of $B^4(0, m^2)$ with order of vanishing at 0 at least $4M_m$ and satisfies the interpolation conditions

$$\begin{aligned}f_m^S(n^2I) = -(2n+1)I,\ \; \forall n\ge m,\ f^S_m(-k^2) = 0,\ \forall k.\end{aligned}$$

Then the function $f_m:= f^P_m + f^S_m$ satisfies all the conditions required, namely $f_m$ is as small as we wish on compact subsets of $B^4(0, m^2)$ with order of vanishing at 0 as large as we wish, has nonnegative coefficients in its power series expansion and fulfills the interpolation conditions (16).

The shear $\Phi _{v,m}(z,w) = (z, w - f_m(z))$ maps $(-n^2, -n^2)$ to $(-n^2, -(n+1)^2)$ and $(n^2I, n^2I)$ to $(n^2I, (n+1)^2I)$ for $n \ge m, I \in {\mathbb {S}}$ and approximates the identity on compact subsets of $m^2\Delta$ as well as we wish. Also $\Phi _{v,m}^{-1}(z,w)= (z, w + f_m(z))$ approximates the identity on compact subsets of $m^2\Delta$ as well as $\Phi _{v,m}$ does and in addition it has nonnegative coefficients in its power series expansion.

In a similar manner we obtain the shear $\Phi _{h,m}(z,w) = (z-g_m(w), w )$ so that it maps $(-n^2, -(n+1)^2)$ to $(-(n+1)^2, -(n+1)^2),$ $(n^2I, (n+1)^2I)$ to $((n+1)^2I, (n+1)^2I)$ for $n \ge m, I \in {\mathbb {S}}$ and approximates the identity on compact subsets of $(m+1)^2\Delta$ as well as we wish.

To define the function $g_m^P,$ for each $n\in {\mathbb {N}}$, consider a suitable choice of an increasing sequence $\{M_n\}_{n \in {\mathbb {N}}}$, and put

$$\begin{aligned} g_m^P(z):= \sum _{n = m}^{\infty } \left( \frac{z}{(n+1)^2}\right) ^{2M_n} \phi ^P_{n+1}(z) (2n+1). \end{aligned}$$

The function $g_m^P$ is well defined, satisfies the interpolation conditions $g_m^P(-(n+1)^2) = 2n+1$, for $n\ge m$ and $g^P_m(k^2I) = 0 \ \forall k$; it has nonnegative coefficients in its power series expansion, is as small as we wish on compact subsets of $B(0, (m+1)^2)$ and with order of vanishing at 0 at least $2M_m.$

To define $g_m^S$, we recall that

$$\begin{aligned} {\tilde{\phi }}(-k^2) = 0\ \forall k, \; {\tilde{\phi }}_{n+1}(k^2I) = 0,\; \forall k \ne n+1, \;\forall I \in {\mathbb {S}}. \end{aligned}$$

Again, the value ${\tilde{\phi }}_{n+1}((n+1)^2I)$ is positive and does not depend on $I\in {\mathbb {S}}$ but ${\tilde{\phi }}((n+1)^2I) = a + Ib\ne 0$ is not necessarily a real number.

As in the previous case, we construct a function of the form

$$\begin{aligned} {\tilde{\psi }}_{n+1} (z) = \left( \frac{z}{(n+1)^2}\right) ^{l+1}(\alpha z + \beta ),\ \; {\mathrm {with}}\ \alpha , \beta \ge 0, \;l \in \{0,1,2,3\}, \end{aligned}$$

such that

$$\begin{aligned} {\tilde{\psi }}_{n+1}((n+1)^2I)(a + Ib) = -I. \end{aligned}$$

Let

$$\begin{aligned} \phi _{n+1}^S(z):= {\tilde{\psi }}_{n+1}(z)\,{\tilde{\phi }}(z)\,{\tilde{\phi }}_{n+1}(z)\,({\tilde{\phi }}_{n+1}((n+1)^2I))^{-1} \end{aligned}$$

for some (and thus for all) $I \in {\mathbb {S}}.$ The coefficients of the power series expansion of $\phi _n^S$ are nonnegative; furthermore,

$$\begin{aligned}\phi _{n+1}^S((n+1)^2I) = -I,\ \; {\phi }^S_{n+1}(k^2I) = 0,\; \forall k \ne n+1, \;\forall I \in {\mathbb {S}}\end{aligned}$$

and

$$\begin{aligned}{\phi }^S_{n+1}(-k^2) = 0 \, \forall k.\end{aligned}$$

For a suitable choice of a strictly increasing sequence $\{M_n\}_{n \in {\mathbb {N}}}$, the function

$$\begin{aligned} g_m^S(z):= \sum _{n = m}^{\infty } \left( \frac{z}{(n+1)^2}\right) ^{4M_n }\phi ^S_{n+1}(z)(2n+1) \end{aligned}$$

is well defined and satisfies the interpolation conditions

$$\begin{aligned}g_m^S((n+1)^2I) = -(2n+1)I,\; \ \forall n\ge m,\ g^S_m(-k^2) = 0,\ \forall k;\end{aligned}$$

it has nonnegative coefficients in its power series expansion, is as small as we wish on compact subsets of $B(0, (m+1)^2)$ and the order of vanishing at 0 is at least $4M_m.$

Then the function $g_m:= g^P_m + g^S_m$ satisfies all the conditions required, namely $g_m$ is as small as we wish on compact subsets of $B^4(0, (m+1)^2)$, has the order of vanishing at 0 as large as we wish and fulfills the interpolation conditions (17).

The shear $\Phi _{h,m}(z,w) = (z- g_m(w),w)$ maps $(-n^2, -(n+1)^2)$ to $(-(n+1)^2, -(n+1)^2)$ and $(n^2I, (n+1)^2I)$ to $((n+1)^2I, (n+1)^2I)$ for $n \ge m, I \in {\mathbb {S}}$ and approximates the identity on compact subsets of $(m+1)^2\Delta$ as well as we wish. Also $\Phi _{h,m}^{-1}(z,w)=(z+ g_m(w),w)$ approximates the identity on compact subsets of $(m+1)^2\Delta$ as well as $\Phi _{h,m}$ does and in addition it has nonnegative coefficients in its power series expansion.

We now define the sets $U_m$ and relabel the shears $\Phi _{v,m}$ and $\Phi _{h,m}$ accordingly. For $m \in {\mathbb {N}}$ define

$$\begin{aligned} \begin{array}{lll} U_{2m-1} := m^2\Delta ,&\,& \Phi _{2m-1}:= \Phi _{v,m},\\ U_{2m} := ((m+1)^2-1)\Delta , &\,& \Phi _{2m}:= \Phi _{h,m}. \end{array} \end{aligned}$$

Choose a strictly decreasing sequence of positive numbers $\{\delta _m\}_{m \in {\mathbb {N}}}$ such that $U_m':=U_m + \delta _m \Delta \Subset U_m + 2 \delta _m \Delta \Subset U_{m+1}.$ Define $U_m'':=U_m+(2\delta _m/3) \Delta$ and $U_m''':=U_m+(\delta _m/3) \Delta ,$ so

$$\begin{aligned}U_m \Subset U_m'''\Subset U_m''\Subset U_m'\Subset U_{m+1}\end{aligned}$$

for every $m \in {\mathbb {N}}.$

Let the sequence $\{c_m\}_{m \in {\mathbb {N}}}$ be given by Lemma 3 with respect to pairs of sets $U=U_{m}''$ and $V = U_{m}'.$

Choose a strictly decreasing sequence $\{\varepsilon _m'\}_{m \in {\mathbb {N}}}$ of positive numbers converging to 0 so that the condition (ii) of Lemma 5 holds (see also Remark 5).

Then choose a strictly decreasing sequence of positive real numbers $\{\varepsilon _m\}_{m \in {\mathbb {N}}}$ converging to 0 such that the condition $\varepsilon _m < \min \{\delta _{m}, \log (1 + \varepsilon _m')\}$ from Lemma 5 holds for any $m\in {\mathbb {N}}.$

Choose the sequence of shears $\{\Phi _m\}_{m \in {\mathbb {N}}}$ which approximate the identity so well, that the assumptions (i) and (ii) of Lemma 5 are fulfilled. Moreover, we require that the orders $M_m$ of vanishing at (0, 0) of $\Phi _m - \mathrm {id}$ form a strictly increasing sequence with $M_m > m,$ so $\Phi _m(0,0) = (0,0)$ for all $m \in {\mathbb {N}}.$

By Lemma 5, the sequence $\{\Psi _m = \Phi _m \circ \ldots \circ \Phi _1\}_{m\in {\mathbb {N}}}$ converges on compact subset of $\Omega$ to a diffeomorphism $\Psi :\Omega \rightarrow {\mathbb {H}}^2.$ Since for all $m \in {\mathbb {N}}$ and $I \in {\mathbb {S}}$ we have $|\Psi _m(-n^2, -n^2)| {\rightarrow } +\infty$ and $|\Psi _m(n^2I, n^2I)| {\rightarrow } +\infty ,$ as ${m \rightarrow +\infty },$ the set ${{{\mathcal {T}}^{\prime}}}\,$ is not contained in $\Omega .$

The limit map $\Psi$ as well as it inverse $\Psi ^{-1}$ are limits of mappings, which preserve complex subspaces ${\mathbb {C}}^2_I.$ Therefore, we have, for any fixed $I\in {\mathbb {S}}$ and a set $\Omega _I:= \Omega \cap {\mathbb {C}}_I^2 \ni (0,0),$ that

$$\begin{aligned}\Psi |_{\Omega _I}:\Omega _I \rightarrow {\mathbb {C}}_I^2 \hbox { and } \Psi ^{-1}|_{{\mathbb {C}}_I^2}: {\mathbb {C}}_I^2\rightarrow {\mathbb {C}}_I^2, \end{aligned}$$

so $\Psi |_{\Omega _I} = {\mathbb {C}}^2_I$ and $\Psi _{\Omega _I}^{-1} = (\Psi |_{{\mathbb {C}}_I^2})^{-1},$ since $\Psi$ is surjective. As the restrictions of $\Psi _m|_{\Omega _I}$ converge uniformly on compact subsets of $\Omega _I$, the limit mappings $\Psi |_{\Omega _I}$ are holomorphic, and hence $\Omega _I$ is a Fatou–Bieberbach domain for any $I\in {\mathbb {S}}$.

To conclude that $\Omega$ is a slice Fatou–Bieberbach domain, we have to show that $\Psi ^{-1}:{\mathbb {H}}^2\rightarrow \Omega$ is real analytic.

By construction, any shear $\Phi _k^{-1}$ is a diffeomorphism with nonnegative real coefficients in its power series expansion and the orders of vanishing at (0, 0) of $\Phi _k^{-1} - \mathrm {id}$ form a strictly increasing sequence, so by Remark 9 the formal power series expansion of $\Psi ^{-1}=\lim \limits _{m\rightarrow +\infty } \Psi _m^{-1}$ exists and has nonnegative coefficients. Then, according to [7], for any $m\in {\mathbb {N}}$ one can write¹

$$\begin{aligned}\Psi _m^{-1}(z,w)=\sum _{\alpha \in \Lambda } (z,w)^{\alpha }r_{\alpha ,m},\quad r_{\alpha ,m}\in {\mathbb {R}},\ r_{\alpha ,m}\ge 0, \end{aligned}$$

which converges absolutely and uniformly on compact subsets of ${\mathbb {H}}^2$. Put $\Lambda ^m = \cup _{d=0}^m \Lambda _d$ for $m \in {\mathbb {N}}.$ For any given $\alpha \in \Lambda$, the sequence $r_{\alpha ,m}, \, m \in {\mathbb {N}}$ is increasing and there exists $m_0\in {\mathbb {N}}$ such that, for $m\ge m_0$, $r_{\alpha ,m}=r_{\alpha ,m_0}=:r_{\alpha }.$ Even more, by assumption that the orders of vanishing of $\Psi _m^{-1} - \mathrm {id}$ form a strictly increasing sequence, we have $r_{\alpha ,m}=r_{\alpha ,m_0}=:r_{\alpha }$ for every $\alpha \in \Lambda ^{m_0}$ and $m \ge m_0.$ In particular, $r_{\alpha ,m}=r_{\alpha }$ for any $\alpha \in \Lambda ^{m}$.

The formal power series for $\Psi ^{-1}$ is

$$\begin{aligned}\sum _{\alpha \in \Lambda } (z,w)^{\alpha }r_{\alpha },\quad r_{\alpha }\in {\mathbb {R}},\ r_{\alpha }\ge 0, \end{aligned}$$

which converges absolutely and uniformly on compact subsets of ${\mathbb {C}}_I^2$ to $\Psi ^{-1}|_{{\mathbb {C}}_I^2}$ with the series expansion

$$\begin{aligned} \Psi ^{-1}|_{{\mathbb {C}}_I^2}=\sum _{p,q=0}^{\infty }z^pw^q \rho _{p,q}. \end{aligned}$$

In this case, the restriction to ${\mathbb {C}}_I^2$ of the formal power series for $\Psi ^{-1}$ is convergent and the variables commute, so we have

$$\begin{aligned}\rho _{p,q} = \sum _{\alpha \in \Lambda _{p+q}, | \alpha | = p} r_{\alpha }; \end{aligned}$$

since the coefficients $r_{\alpha }$ are nonnegative, we have $\rho _{p,q} \ge 0$ for all $p,q \in {\mathbb {N}}_0.$ For any index subset $\Lambda _1 \subset \Lambda$ and

$$\begin{aligned} S_{\Lambda _1} (z,w)= \sum _{\alpha \in \Lambda _1} (z,w)^{\alpha }r_{\alpha } \end{aligned}$$

the following estimate holds

$$\begin{aligned} |S_{\Lambda _1} (z,w)|\le \sum _{\alpha \in \Lambda _1} |(z,w)^{\alpha }|r_{\alpha } = S_{\Lambda _1}(|z|,|w|)\le \Psi ^{-1}|_{{\mathbb {C}}^2_I}(|z|,|w|) = \Psi ^{-1}(|z|,|w|). \end{aligned}$$

Fix a compact set $K \subset {\mathbb {H}}$ and $\varepsilon > 0.$ We claim that there exists $m_0\in {\mathbb {N}}$ so that if $m \ge m_0$ we have $|\Psi ^{-1}(z,w) - S_{\Lambda ^m}(z,w)| < \varepsilon$ on K. By uniform convergence on compacts, there exists $m_0$ so that if $m \ge m_0$ we have $|\Psi ^{-1}(z,w)-\Psi _m^{-1}(z,w)| < \varepsilon /2$ on K. Let $m_0$ be so large that $\sum \nolimits _{p+q\ge m_0}^{\infty }|z|^p|w|^q \rho _{p,q} < \varepsilon /2$ on K. Since for any $m$ we have $r_{\alpha } = r_{\alpha ,m}$ if $\alpha \in \Lambda ^{m}$, thus for $m \ge m_0$

$$\begin{aligned} |&\Psi _m^{-1}(z,w)- \sum _{\alpha \in \Lambda ^{m}} (z,w)^{\alpha }r_{\alpha }| = \big |\sum _{\alpha \in \Lambda } (z,w)^{\alpha }r_{\alpha ,m}-\sum _{\alpha \in \Lambda ^{m}} (z,w)^{\alpha }r_{\alpha } \big |\\& = \big | \sum _{\alpha \in (\Lambda \setminus \Lambda ^{m})} (z,w)^{\alpha }r_{\alpha ,m} -\sum _{\alpha \in \Lambda ^{m} } (z,w)^{\alpha }(r_{\alpha } - r_{\alpha ,m})\big | \\& = \big |\sum _{\alpha \in (\Lambda \setminus \Lambda ^{m})} (z,w)^{\alpha }r_{\alpha ,m}\big | \le \sum _{\alpha \in (\Lambda \setminus \Lambda ^{m})} (|z|,|w|)^{\alpha }r_{\alpha ,m} \\& \le \sum _{\alpha \in (\Lambda \setminus \Lambda ^{m})} (|z|,|w|)^{\alpha }r_{\alpha } \le \sum _{p+q> m_0}^{\infty }|z|^p|w|^q \rho _{p,q} < \varepsilon /2, \end{aligned}$$

so on K the estimate

$$\begin{aligned} |\Psi ^{-1}(z,w) - S_{\Lambda ^m}(z,w)| \le |\Psi ^{-1}(z,w)-\Psi _m^{-1}(z,w)| + |\Psi _m^{-1}(z,w)- { }\sum _{\alpha \in \Lambda ^m} (z,w)^{\alpha }r_{\alpha }| < \varepsilon \end{aligned}$$

holds. The formal series of $\Psi ^{-1}$ thus converges absolutely and uniformly on compact subsets of ${\mathbb {H}}^2$ to $\Psi ^{-1}$.

The mapping $F =\Phi _v^{-1} \circ \Phi _h^{-1} \circ \Psi ^{-1} \circ \Phi _h \circ \Phi _v$ is real analytic and the image $F({\mathbb {H}}^2)$ misses the set ${{\mathcal {T}}}.$ As the shears $\Phi _h,\Phi _v$ both have real coefficients and thus preserve ${\mathbb {C}}_I^2$ for all I,  the set $F({\mathbb {H}}^2)$ is a slice Fatou–Bieberbach domain. $\square$