1 Introduction

Transcendental Hénon maps are automorphisms of \({{\mathbb {C}}}^2\) with constant Jacobian of the form

$$\begin{aligned} F(z,w):=(f(z)-\delta w,z) \text { with}\, f:{{\mathbb {C}}}\rightarrow {{\mathbb {C}}}\, \text {entire transcendental}. \end{aligned}$$

In analogy with classical complex Hénon maps, for which f is assumed to be a polynomial (see e.g. [4,5,6, 15,16,17]), the dynamical investigation of transcendental Hénon maps can rely on tools and knowledge from one dimensional complex dynamics, which is better understood than its higher dimensional counterpart. They have been introduced in [13]. General properties of transcendental Hénon maps were established in [1,2,3] and examples with interesting dynamical features were presented.

Let \({{\mathbb {P}}}^2\) be the complex projective space obtained by compactifying \( {{\mathbb {C}}}^2\) by adding the line at infinity \(\ell _\infty \). We define the Fatou set of F as the set of points in \({{\mathbb {C}}}^2\) near which the iterates form a normal family with respect to the complex structure induced by \({{\mathbb {P}}}^2\) (compare with [1, section 1]). A Fatou component is a connected component of the Fatou set. Given a Fatou component \(\Omega \) we call a function \(h:\Omega \rightarrow {{\mathbb {P}}}^2\) a limit function for \(\Omega \) if there exists a subsequence \(n_k\) such that \(F^{n_k}\rightarrow h\) uniformly on compact subsets of \(\Omega \). The image \(h(\Omega ) \) of a limit function h is called a limit set (for \(\Omega \)). By Lemma 4.3 and 2.4 in [1], each limit set is either contained in \({{\mathbb {C}}}^2\) or contained in \(\ell _\infty \).

In this paper we investigate escaping Fatou components, that is Fatou components for which all limit sets lie in the line at infinity.

More precisely we construct a transcendental Hénon map with a cycle of escaping Fatou components satisfying the following properties. Let \(\mathbb {H}\) denote the right half plane, \(-\mathbb {H}\) denote the left half plane.

Theorem 1.1

Let

$$\begin{aligned} F(z,w):=(e^{-z^2}+e^{\pi i }\delta w,z),\quad \delta >2. \end{aligned}$$

Then F has a cycle of four Fatou components \(\Omega ^{ab}\) with \(a,b\in \{+,-\}\), each of which is biholomorphic to \(\mathbb {H}\times \mathbb {H}\). There are exactly two limit functions \(h_1,h_2\), both of rank 1, such that

$$\begin{aligned} h_1(\Omega ^{aa})=h_2(\Omega ^{a(-a)})=\mathbb {H}\text { and } h_1(\Omega ^{a(-a)})=h_2(\Omega ^{aa})=-\mathbb {H}\text { for all}\, a. \end{aligned}$$

Moreover, F is conjugate to its linear part on every \(\Omega ^{ab}\).

Notice that \(\delta \) is the Jacobian of F, hence the latter is expansive. Its role is not relevant, as long as \(\delta >1\). For convenience we take \(\delta >2\).

The main points of interest of this result are that the limit functions have rank one, that each Fatou component has two disjoint limit sets (compare [18] for restrictions on the presence of several limit sets), and that the limit sets \(\mathbb {H}, -\mathbb {H}\) are hyperbolic.

For general automorphisms of \({{\mathbb {C}}}^2\) there are very few examples of limit functions of rank 1 [9, 18], and for polynomial Hénon maps, it is not even known whether rank 1 limit functions can exist; in fact, their existence has been excluded provided the Jacobian is small enough [20]. On the other hand, they are abundant for holomorphic endomorphisms of \({{\mathbb {C}}}^2\) [10, Theorem 4]. For transcendental Hénon maps, rank 1 limit functions seem to appear naturally for escaping Fatou components [8]. To our knowledge there were no previous examples of hyperbolic limit sets for automorphisms of \({{\mathbb {C}}}^2\). One possible reason for the natural appearance of these phenomena might be that F is not defined on \(\ell _\infty \), hence there is no natural dynamics on limit sets contained there.

One can see F as a special case of maps of the form

$$\begin{aligned} F(z,w):=\left( e^{-z^k}+e^{\frac{2\pi i }{k}}\delta w,z\right) ,\quad \delta >2,\ k\in {{\mathbb {N}}}\end{aligned}$$

Analogous results hold for such maps, and are proven in [7] with similar techniques.

2 Proof of Theorem 1.1

From now on let F be as in Theorem 1.1,

$$\begin{aligned} F(z,w)=(e^{-z^2}-\delta w,z) \quad \text {with }\, \delta >2. \end{aligned}$$
(2.1)

Throughout the paper, given a point \(P=(z_0,w_0)\in {{\mathbb {C}}}^2\) and \(n\in {{\mathbb {N}}}\) we denote its iterates by \(F^n(P)=: (z_n,w_n)\).

2.1 Computing limit functions

In this section we give an explicit expression for the iterates of F and their formal limit. A direct computation (compare [8]) shows that

$$\begin{aligned} F^{2n}(z_0,w_0)&= (-\delta )^n\left( z_0+\sum _{j=1}^n(-\delta )^{-j}f(z_{2j-1}),w_0+\sum _{j=1}^n(-\delta )^{-j}f(z_{2j-2})\right) \\ F^{2n+1}(z_0,w_0)&= (-\delta )^n\left( -\delta \left( w_0+\sum _{j=1}^{n+1}(-\delta )^{-j}f(z_{2j-2})\right) ,z_0+\sum _{j=1}^n(-\delta )^{-j}f(z_{2j-1})\right) . \end{aligned}$$

For \(n\in {{\mathbb {N}}}\) define the following holomorphic functions from \({{\mathbb {C}}}^2\) to \(\hat{{{\mathbb {C}}}}\)

$$\begin{aligned} \Delta _1^n(z_0,w_0)&:= \sum _{j=1}^n (-\delta )^{-j}f(z_{2j-1}) \\ \Delta _2^n(z_0,w_0)&:= \sum _{j=1}^n (-\delta )^{-j}f(z_{2j-2}) \end{aligned}$$

With this notation the iterates of F take the form

$$\begin{aligned} F^{2n}(z_0,w_0)&= (-\delta )^n\left( (z_0+\Delta _1^n(z_0,w_0),w_0+\Delta _2^n(z_0,w_0)\right) \end{aligned}$$
(2.2)
$$\begin{aligned} F^{2n+1}(z_0,w_0)&=(-\delta )^n \left( -\delta w_0-\delta \Delta _2^{n+1}(z_0,w_0),z_0+\Delta _1^n(z_0,w_0)\right) . \end{aligned}$$
(2.3)

Let

$$\begin{aligned} \Delta _1(z,w)&=\Delta _1^\infty (z,w):=\lim _{n\rightarrow \infty } \Delta _1^n(z,w)\\ \Delta _2(z,w)&=\Delta _2^\infty (z,w):=\lim _{n\rightarrow \infty } \Delta _2^n(z,w)\\ \Delta (z,w)&=\max \left( \left| \Delta _1(z,w) \right| , \left| \Delta _2(z,w)\right| \right) . \end{aligned}$$

Notice that \(\Delta _1,\Delta _2\) are holomorphic functions to \(\hat{{{\mathbb {C}}}}\) on open sets on which they are well defined.

We can deduce the following formal limits.

$$\begin{aligned} h_1(z,w)&:=\lim _{n\rightarrow \infty }\frac{z_{2n}}{w_{2n}}=\frac{z+\Delta _1(z,w)}{w+\Delta _2(z,w)} \end{aligned}$$
(2.4)
$$\begin{aligned} h_2(z,w)&:= \lim _{n\rightarrow \infty }\frac{z_{2n+1}}{w_{2n+1}}=\frac{-\delta ( w+\Delta _2(z,w))}{z+\Delta _1(z,w)}=-\frac{\delta }{h_1(z,w)}. \end{aligned}$$
(2.5)

We have that \(h_1,h_2\) are holomorphic functions to \(\hat{{{\mathbb {C}}}}\) on open sets on which \(\Delta _1\) and \(\Delta _2\) are holomorphic functions to \(\hat{{{\mathbb {C}}}}\). We will show in Proposition 2.7 that \(h_1 \ne h_2\).

2.2 Existence of Fatou components and rank of the limit functions

In this section we construct a forward invariant open set W on which the even and the odd iterates converge, from which we deduce the existence of Fatou components. We then show that the limit functions have rank 1 on such Fatou components.

For \(A\subseteq {{\mathbb {C}}}^2\) and \(a,b \in \{ +,- \}\) define

$$\begin{aligned} A^{ab}:=A\cap \{(z,w)\in {{\mathbb {C}}}^2\,a{\text {Re}}(z)>0,\ b{\text {Re}}(w)>0 \}. \end{aligned}$$
(2.6)

If \(A\cap (\{{\text {Re}}z=0\}\cup \{{\text {Re}}w=0\})=\varnothing \) then \(A=\bigcup _{a,b \in \{+,-\}}A^{ab}.\)

We start by defining a set on which we have control on the dynamics. Let

$$\begin{aligned} \mathcal {S}&:=\{z\in {{\mathbb {C}}}:\,|{\text {Im}}(z)|<|{\text {Re}}(z)|\} \subset {{\mathbb {C}}}\\ S&:=\mathcal {S} \times \mathcal {S}\subset {{\mathbb {C}}}^2 \end{aligned}$$

A sketch of \(\mathcal {S}\) can be found in Fig. 1.

Lemma 2.1

Let \(z\in \mathcal {S}\), then \(|f(z)|=|e^{-z^2}|< 1.\)

Proof

If \(z\in \mathcal {S}\), then \(|{\text {arg}}(z)|< \frac{\pi }{4}\) and hence \({\text {Re}}(z^2)>0\) from which we have \(|e^{-z^2}|=e^{-{\text {Re}}z^2}<1\). \(\square \)

Lemma 2.2

(Orbits contained in S) For any \(P=(z_0,w_0)\in S^{ab}\) such that \(F(P)\in S\) and \(|{\text {Re}}w_0|>\frac{1}{\delta }\) we have that \(F(P)\in S^{(-b)a}\).

From now on assume that \(F^n(P)\in S \) for all \(n\in {{\mathbb {N}}}\). Then

$$\begin{aligned} F^{2n}(z_0,w_0)&\rightarrow h_1(z_0,w_0)\\ F^{2n+1}(z_0,w_0)&\rightarrow h_2(z_0,w_0). \end{aligned}$$

Fix \(\lambda >0\) and assume also that \(|{\text {Re}}z_0|,|{\text {Re}}w_0|>\frac{1+\lambda }{\delta -1}\). Then

$$\begin{aligned} |{\text {Re}}z_{2n-1}|&=|{\text {Re}}w_{2n}|> |{\text {Re}}w_0|+n\lambda \end{aligned}$$
(2.7)
$$\begin{aligned} |{\text {Re}}z_{2n}|&=|{\text {Re}}w_{2n+1}|>|{\text {Re}}z_0|+n\lambda . \end{aligned}$$
(2.8)

Proof

By hypothesis, \(F(P)\in S\) hence \(F(P)\in S^{\tilde{a}\tilde{b}}\) for some \(\tilde{a},\tilde{b} \in \{+,-\}\). Since \({\text {Re}}w_1={\text {Re}}z_0\) we have that \({\tilde{b}}=a\). Moreover \({\text {Re}}z_1=-\delta {\text {Re}}w_0+{\text {Re}}(e^{-z_0^2}) \) and since \(P\in S\), \(|{\text {Re}}(e^{-z_0^2})|<1\) by Lemma 2.1. Hence the sign of \({\text {Re}}z_1\) is opposite to the sign of \({\text {Re}}w_0\) provided \(|{\text {Re}}w_0|>\frac{1}{\delta }\), and \({\tilde{a}}=-b\) as required.

Assume from now on that \(F^n(P)\in S\) for all \(n\in {{\mathbb {N}}}\). It follows that \(z_n\in \mathcal {S}\) for all \(n\in {{\mathbb {N}}}\) and hence by Lemma 2.1\(|f(z_n)|< 1\) for all \(n\in {{\mathbb {N}}}\). Since \(\delta >2\) this implies

$$\begin{aligned} \Delta (z_0,w_0)<\sum _{j=1}^\infty \delta ^{-j}<1 \text { whenever}\, F^n(z_0,w_0)\in S\, \text { for all}\, n\in {{\mathbb {N}}}, \end{aligned}$$
(2.9)

which implies convergence of the even and odd iterates of F according to the expression in (2.2), (2.3).

We now prove (2.7), (2.8). Using the expression of F and since \(P\in S \), by Lemma 2.1 we have \(|{\text {Re}}z_1|\ge \delta |{\text {Re}}w_0|-|e^{-z_0^2}|\ge \delta |{\text {Re}}w_0|-1\) which is larger than \(|{\text {Re}}w_0|+\lambda \) if \(|{\text {Re}}w_0|>\frac{1+\lambda }{\delta -1}\). It follows that

$$\begin{aligned} |{\text {Re}}z_1|&> |{\text {Re}}w_0|+\lambda \end{aligned}$$
(2.10)
$$\begin{aligned} |{\text {Re}}z_2|&>|{\text {Re}}w_1|+\lambda =|{\text {Re}}z_0|+\lambda , \end{aligned}$$
(2.11)

where the claim for \(z_{2}\) follows because \(w_1=z_0\). The more general formula follows by induction, using that \(F^n(P)\in S \) for all \(n\in {{\mathbb {N}}}\). \(\square \)

Corollary 2.3

Let \(A\subset S\) be forward invariant. If \(P=(z_0,w_0)\in A\) such that \(|{\text {Re}}w_0|>\frac{1}{\delta }\) then Lemma 2.2 holds for \(\lambda =1\), in particular, if \(P\in A^{ab}\) then \(F(P)\in A^{(-b)a}\).

For \(R>0\) and \(0<k<1\) define the sets (see Fig. 1.)

$$\begin{aligned} \mathcal {W}_{k,R}&:=\{z\in {{\mathbb {C}}}: |{\text {Im}}z|<k|{\text {Re}}z|, \ |{\text {Re}}z|>R\}\subset {{\mathbb {C}}}\\ W_{k,R_1,R_2}&:=\mathcal {W}_{k,R_1} \times \mathcal {W}_{k,R_2}\subset {{\mathbb {C}}}^2. \end{aligned}$$

Observe that \(\mathcal {W}_{k,R} \subset \mathcal {S}\) and that \(\mathcal {W}_{1,0}=\mathcal {S}\).

Lemma 2.4

Let \(n\in {{\mathbb {N}}}\), and let \((z_0,w_0)\in W_{k,R_1,R_2}\). Let \(0<k<\tilde{k} < 1\). If \(R_2> \frac{2}{\delta (\tilde{k}-k)}\) then

$$\begin{aligned} \left| \frac{{\text {Im}}z_1}{{\text {Re}}z_1} \right|< \tilde{k}\quad \text {and}\quad \left| \frac{{\text {Im}}w_1}{{\text {Re}}w_1} \right| < {k}. \end{aligned}$$

Proof

Let \((z_0,w_0)\in W_{k,R_1,R_2}\). The claim for \(w_1\) is immediate because \(w_1=z_0\). Using the expression of F, the triangular inequality, the estimate in Lemma 2.1 and the fact that \(|{\text {Im}}w_0|<k |{\text {Re}}w_0|\) we have

$$\begin{aligned} \left| \frac{{\text {Im}}z_1}{{\text {Re}}z_1}\right|&<\frac{\delta |{\text {Im}}w_0|+1}{\delta |{\text {Re}}w_0|-1}<\frac{k \delta |{\text {Re}}w_0 |+1}{\delta |{\text {Re}}w_0| -1}. \end{aligned}$$

Setting the resulting expression to be less than \({\tilde{k}}\) we get \(|{\text {Re}}w_0|>\frac{1+\tilde{k}}{\delta ({\tilde{k}}-k)}\). Since \(\tilde{k}< 1\), it is enough to take \(|{\text {Re}}w_0|>\frac{2}{\delta ({\tilde{k}}-k)}\) as required. \(\square \)

Let \(k_n:=1-\frac{1}{n+2}\) and \(R_n:=(\frac{\delta }{2})^{\frac{n}{2}} R_0\) for \(R_0>2\) sufficiently large depending only on \(\delta \) (see (2.12)). Let \(R_{-1}=R_0\) and set

$$\begin{aligned} W_n:={\left\{ \begin{array}{ll} W_{k_n,R_n,R_{n-1}}^{++}\quad \text { if}\, n=0\mod 4\\ W_{k_n,R_n,R_{n-1}}^{-+}\quad \text { if}\,n=1\mod 4\\ W_{k_n,R_n,R_{n-1}}^{--}\quad \text { if}\,n=2\mod 4\\ W_{k_n,R_n,R_{n-1}}^{+-}\quad \text { if}\,n=3\mod 4 \end{array}\right. }. \end{aligned}$$

and define

$$\begin{aligned} W:=\bigcup _{n\in {{\mathbb {N}}}} W_n. \end{aligned}$$
Fig. 1
figure 1

A sketch in \({{\mathbb {C}}}\) of the set \({{\mathcal {S}}}\) and three of the sets \(\mathcal {W}_{k_n,R_n}\). The set W is obtained by taking appropriate products of (parts of) the sets \(\mathcal {W}_{k_n,R_n}\)

Full size image

Proposition 2.5

(Invariance of W) The set W is open and \(W\subset S\). For any \(n\in {{\mathbb {N}}}\) we have that \(F(W_n)\subset W_{n+1}\), hence W is forward invariant. The set W consists of four connected components \(W^{ab}\) with \(a,b\in \{+,-\}\) and \(F(W^{ab})\subset W^{(-b) a}\).

Proof

The fact that W is open and \(W\subset S\) follows from the definition. Fix \(n\in {{\mathbb {N}}}\). Let \((z_0, w_0)\in W_n\) and let \((z_1,w_1)\) be its image. Since \(w_1=z_0\), the signs of \({\text {Re}}w_1,{\text {Re}}z_0\) are the same, and we have that \(|{\text {Re}}w_1|=|{\text {Re}}z_0|>R_n\) and that

$$\begin{aligned} \left| \frac{{\text {Im}}w_1}{{\text {Re}}w_1}\right| =\left| \frac{{\text {Im}}z_0}{{\text {Re}}z_0}\right|<k_{n}<k_{n+1}. \end{aligned}$$

Hence to show that \(F(W_n)\subset W_{n+1}\) it is enough to see that \(|{\text {Re}}z_1|>R_{n+1}\) and that

$$\begin{aligned} \left| \frac{{\text {Im}}z_1}{{\text {Re}}z_1}\right| <k_{n+1}. \end{aligned}$$

Let \(\lambda _n:=R_{n+1}-R_{n-1}\). Since \(P\in S\), by (2.10) we have that

$$\begin{aligned} |{\text {Re}}z_1|>|{\text {Re}}w_0|+\lambda _n > R_{n-1}+\lambda _n=R_{n+1} \end{aligned}$$

provided \(R_{n-1}>\frac{1+\lambda _n}{\delta -1}\). Substituting the expression for \(\lambda _n\) we get \(R_{n+1}<\delta R_{n-1}-1\). Substituting the expression for \(R_{n+1}\) and \(R_{n-1}\) we get

$$\begin{aligned} \delta ^{\frac{n+1}{2}}R_0>2^{\frac{n+1}{2}} \end{aligned}$$

which is satisfied because \(\delta >2\), provided \(R_0\ge 1\). This gives \(|{\text {Re}}z_1|>R_{n+1}.\)

We now prove \(\left| \frac{{\text {Im}}z_1}{{\text {Re}}z_1}\right| <k_{n+1}.\) By Lemma 2.4, it is enough to check that \(R_{n-1}>\frac{2}{\delta (k_{n+1}-k_n)}=\frac{2(n+2)(n+3)}{\delta }\), that is

$$\begin{aligned} R_0>2^{\frac{n+1}{2}}\delta ^{-\frac{n+1}{2}}(n+2)(n+3)\quad \text { for all}\, n\in {{\mathbb {N}}}. \end{aligned}$$
(2.12)

Since the function on the right hand side is bounded in n for any \(\delta >2\) (in fact, it tends to 0 as \(n\rightarrow \infty \)), such \(R_0\) exists and depends only on \(\delta \).

Finally, for any \((z,w)\in W\) we have \({\text {Re}}z,{\text {Re}}w\ne 0\), so the sets \(W^{ab}\) are well defined. By construction, \(W_n\cap W_{n+4}\ne \varnothing \) so each \(W^{ab}\) is connected. It follows that W consists of 4 connected components \(W^{ab}\). Since W is forward invariant and contained in S, the orbits of points in W are contained in S hence Corollary 2.3 applies. \(\square \)

Proposition 2.6

(Existence of Fatou components) On each \( W^{ab}\) we have that

$$\begin{aligned} F^{2n}\rightarrow h_1,\ F^{2n+1}\rightarrow h_2 \text { uniformly on compact subsets of}\, W^{ab}. \end{aligned}$$

It follows that each \(W^{ab}\) is contained in a Fatou component that we denote by \(\Omega ^{ab}\).

Proof

Since \(W\subset S\) and is forward invariant by Proposition 2.5, (2.9) holds hence \(F^{2n}\) and \(F^{2n+1}\) converge uniformly on W to \(h_1,h_2\) respectively, hence W is contained in the Fatou set. Since each \(W^{ab} \) is open and connected it is contained in a unique Fatou component that we denote by \(\Omega ^{ab}\). \(\square \)

We will see in Proposition 2.18 that in fact the components \(\Omega ^{ab}\) are all distinct and that the notation \(\Omega ^{ab}\) matches the definition of \(A^{ab}\) given in Sect. 2.1 for a general set A.

Proposition 2.7

Both \(h_1\) and \(h_2\) have (generic) rank 1 on W, and \(h_1 \ne h_2\).

Proof

Recall that \(\Delta (z,w)<1\) on W by (2.9). It follows that by the explicit expression of \(F^{2n}, F^{2n+1}\), the iterates of any point in W converge to the line at infinity. So \(h_i(W)\subset \ell _\infty \), and \(h_1\), \(h_2\) either have generic rank 1 or are constants. Suppose by contradiction that \(h_1=c\) is constant. If \(|c|\ne \infty \), then one has:

$$\begin{aligned} |z_0|-\Delta (z_0,w_0)\le \left| z_0+\Delta _1(z_0,w_0)\right| =|c|\left| w_0+\Delta _2(z_0,w_0))\right| \le |c||w_0|+|c|\Delta (z_0,w_0), \end{aligned}$$

hence

$$\begin{aligned} |z_0|\le |c||w_0|+(|c|+1), \end{aligned}$$

contradicting the fact that \((z_0,w_0)\) could be any point in W, which is unbounded in the z direction for any choice of w. If \(c=\infty \), we have \(|w_0| \le 1\), again a contradiction. It follows that \(h_1 \ne h_2\). Indeed, \(h_1 \cdot h_2=-\delta \) is constant, if we had \(h_1=h_2\) it would follow that \(h_1^2\) (and hence \(h_1\)) would be constant as well, contradicting the argument above. \(\square \)

2.3 Construction of an absorbing set

Let \(\Omega ^{ab}\) with \(a,b \in \{+,-\}\) be the Fatou components defined in Proposition 2.6 and let

$$\begin{aligned} \Omega :=\bigcup _{ab}\Omega ^{ab}. \end{aligned}$$

Since each \(\Omega ^{ab}\) is connected, \(\Omega \) consists of at most 4 Fatou components. This section is devoted to find an absorbing set \(W_I\) for \(\Omega \) under F. Its existence will be used in Sect. 2.5 to show that the Fatou components \(\Omega ^{ab}\) are all distinct and to describe both their limit sets and their geometric structure. We use an argument based on harmonic functions used also in [1, 8, 14].

Definition 2.8

(Absorbing sets) A set A is absorbing for an open set \(\Omega \supset A \) under a map F if for any compact \(K\subset \Omega \) there exists \(N>0\) such that

$$\begin{aligned} F^{n}(K)\subset A\quad \text { for all }\, n\ge N. \end{aligned}$$

If A is absorbing for \(\Omega \), then \(\Omega =\bigcup _{n}F^{-n}(A)\).

Fix \(C\ge 1\) and let

$$\begin{aligned} I=I(C):=\{z\in {{\mathbb {C}}}\,\,|{\text {Im}}z|^2 < |{\text {Re}}z|^2-C^2\}=\{ z\in {{\mathbb {C}}}:\,{\text {Re}}(z^2) > C^2 \} \subset \mathcal {S}. \end{aligned}$$

Notice that if \(z\in I\), then \(|{\text {Re}}z|> C\).

Define

$$\begin{aligned} W_I=W_{I}(C):=\Omega \cap \{(z,w) \in {{\mathbb {C}}}^2: F^n(z,w)\in I\times I \text { for all}\, n\ge 0\}. \end{aligned}$$

Proposition 2.9

We have that \(W_I^{ab}\ne \varnothing \) for all \(a,b\in \{+,-\}\). For every \(a,b\in \{+,-\}\),

$$\begin{aligned} F(W_I^{ab})\subset W_I^{(-b)a}. \end{aligned}$$

The sets \(W_I^{++}\cup W_I^{--}\), \(W_I^{-+}\cup W_I^{+-}\) are both forward invariant under \(F^2\). Moreover \(F^{2n}\) and \(F^{2n+1}\) are convergent on \(W_I\).

Proof

Each \(W_I^{ab}\) contains the set \(\{(z,w)\in {{\mathbb {C}}}^2: a{\text {Re}}z> M,\ b{\text {Re}}w> M,\ {\text {Im}}z={\text {Im}}w=0\}\) for M sufficiently large. The set \(W_I\subset S\) is forward invariant hence Corollary 2.3 applies. Convergence of even and odd iterates follows by (2.9). \(\square \)

It will turn out that \(W_I\) is open as well (Proposition 2.16).

The rest of this section is devoted to proving the following proposition.

Proposition 2.10

The set \(W_I\) is absorbing for \(\Omega \) under F, that is,

$$\begin{aligned} \Omega =\bigcup _n F^{-n} (W_{I})=:{{\mathcal {A}}}_I. \end{aligned}$$

Let

$$\begin{aligned} \mathcal {X}:=\{ (z,w)\in \Omega :\,h_1(z,w)=0,\infty \}. \end{aligned}$$

Since \({\mathcal {X}}\) is an analytic set, being the union of the 0-set and the \(\infty \)-set of a meromorphic function, it is locally a finite union of 1-complex-dimensional varieties (see [11]).

Let K be a compact subset of \(\Omega {\setminus }\mathcal {X}\), hence \(h_i(P)\ne 0,\infty \) for all \(P\in K\), and \(i=1,2\). Define

$$\begin{aligned} M:=\max _{K, i }|h_i|<\infty . \end{aligned}$$
(2.13)

Note that \(M>1\) because \(h_2=-\frac{\delta }{h_1}\) and \(\delta >1\). By Corollary 2.3 in [8] if \(\varepsilon >0\) is sufficiently small there exists a constant c such that for every \((z_0,w_0)\in K\)

$$\begin{aligned} |z_n|&\le c(M+\varepsilon )^n. \end{aligned}$$
(2.14)
$$\begin{aligned} |w_n|=|z_{n-1}|&\le c(M+\varepsilon )^{n-1}. \end{aligned}$$
(2.15)

The proof of Proposition 2.10 relies on the following technical lemma. Recall that for \(P=(z_0,w_0)\), we write \(F^n(P)=(z_n,w_n)\).

Lemma 2.11

Define the sequence of harmonic functions \(u_n\) from \(\Omega \) to \({{\mathbb {R}}}\) as

$$\begin{aligned} u_n(z_0,w_0):=\frac{-{\text {Re}}( z_n^2)}{n}. \end{aligned}$$
(2.16)

Then

  1. (1)

    Let \(K\subset \Omega \) compact. Then there exists \(M=M(K) \) and \(N\in {{\mathbb {N}}}\) such that \(u_n\le \log M\) on K for \(n>N\);

  2. (2)

    \(u_n\rightarrow -\infty \) uniformly on compact subsets of W;

  3. (3)

    If \(P\in \Omega {\setminus } {{\mathcal {A}}}_I\), for every \(\varepsilon >0\) there is a subsequence \(n_k\rightarrow \infty \) such that \(u_{n_k}(P)\ge -\varepsilon \).

Lemma 2.12

Let \(z\in {{\mathbb {C}}}\), \(k<1\). If

$$\begin{aligned} \left| \frac{{\text {Im}}z}{{\text {Re}}z}\right|&\le k<1 \text { then } \end{aligned}$$
(2.17)
$$\begin{aligned} \left| \frac{{\text {Im}}z^2}{{\text {Re}}z^2}\right|&\le \frac{2k}{1-k^2}. \end{aligned}$$
(2.18)

Proof

Let \(z=re^{i\theta }\) satisfying (2.17); then \(|\tan \theta |\le k<1\). Hence since \(z^2=r^2e^{2i\theta }\),

$$\begin{aligned} \left| \frac{{\text {Im}}z^2}{{\text {Re}}z^2}\right| =|\tan (2\theta )|=\left| \frac{2\tan \theta }{1-\tan ^2\theta }\right| \le \frac{2k}{1-k^2}. \end{aligned}$$

\(\square \)

The following fact is certainly known, however we give a proof in the Appendix. Given a set A, let \(\mathring{A}\) denote its interior.

Proposition 2.13

Let L be a compact set and H be an analytic subset of dimension one of \({{\mathbb {C}}}^2\). For any compact K s.t. \(K\subset \mathring{L}\) there exists \(\eta =\eta (K,L,H)\) such that for any u harmonic defined in a neighborhood of L and such that

$$\begin{aligned} u\le \alpha <\infty \text { on}\, L{\setminus }( \eta \text {-neighborhood of }\, H) \end{aligned}$$

we have

$$\begin{aligned} u\le \alpha \text { on}\, K \end{aligned}$$

Proof of Lemma 2.11

  1. (1)

    Let K be a compact subset of \(\Omega \). Let \(\eta \) as obtained by applying Proposition 2.13 to a slightly larger compact set \(L\subset \Omega \) and to the analytic set \({\mathcal {X}}\). Let \(U_\eta ({\mathcal {X}})\) be an \(\eta \)-neighborhood of \({\mathcal {X}}\). In view of Proposition 2.13 it is enough to prove that there exists \(N\in {{\mathbb {N}}}\) such that \(u_n\le \log M\) for \(n>N\) and for some M on the set

    $$\begin{aligned} {\tilde{K}}:=K{\setminus } U_\eta ({\mathcal {X}}) \end{aligned}$$

    which is a compact subset of \(\Omega {\setminus } {\mathcal {X}}\). Hence it is enough to prove the claim for any K compact subset of \(\Omega {\setminus } {\mathcal {X}}\).

    Fix \(\varepsilon >0\) sufficiently small and let M, c be as in (2.14) and (2.15) for K. Suppose that there exists a subsequence \((n_j)\) and points \((z,w)=(z(j), w(j))\in K\) such that

    $$\begin{aligned} -\frac{{\text {Re}}({z}_{n_j}^2)}{n_j}>\beta \end{aligned}$$

    for some \(\beta \). We will show that \(\beta \le M\).

    Using (2.14) and (2.15) we have that

    $$\begin{aligned} c(M+\varepsilon )^{n_j+1}&\ge |z_{n_{j+1}}|=|e^{-z_{n_j}^2}-\delta w_{n_j}| \ge |e^{-z_{n_j}^2}|-\delta |w_{n_j}| \\&\ge e^{-{\text {Re}}(z_{n_j}^2)}-\delta c(M+\varepsilon )^{n_j-1} \ge e^{\beta n_j}-\delta c (M+\varepsilon )^{n_{j}-1}. \end{aligned}$$

    Hence, using \(M>1\) and \(\varepsilon >0\) sufficiently small,

    $$\begin{aligned} e^{\beta n_j}\le \delta c (M+\varepsilon )^{n_{j}-1}+ c(M+\varepsilon )^{n_j+1}\le c(\delta +1)(M+\varepsilon )^{n_j+1}. \end{aligned}$$

    Then

    $$\begin{aligned} \beta \le \frac{\log \big (c(\delta +1)\big )}{n_j}+\frac{n_j+1}{n_j}\,\, \log (M+\varepsilon )\rightarrow \log M \end{aligned}$$

    as \(n_j\longrightarrow \infty \) and \(\varepsilon \longrightarrow 0\).

  2. (2)

    It is enough to show that \(u_n(z_0, w_0)\rightarrow -\infty \) for any point \((z_0,w_0)\in W\) and it will follow for any compact subset of W. Since W is forward invariant, \(F^n(z_0,w_0)\subset W\subset S\) for all \(n\in {{\mathbb {N}}}\) and \(\Delta (z_0,w_0)<1\) by (2.9). Using the explicit expression for iterates of F given by (2.2), (2.3) we have

    $$\begin{aligned} |z_n^2|=|z_n|^2=\left\{ \begin{array}{ll} \delta ^{n}|z_0+\Delta _1^{n/2}(z_0,w_0)|^{2}\ge \delta ^{n}|z_0-1|^2 &{}\quad \text { if n even};\\ \delta ^{(n+1)}|w_0+\Delta _2^{(n+1)/2}(z_0,w_0)|^2\ge \delta ^{n+1}|w_0-1|^2 &{}\quad \text { if n odd}. \end{array}\right. \end{aligned}$$

    In both cases, since \(|z_0|, |w_0|>R_0>2\) we obtain \(|z_n^2|\ge \delta ^{n} \). Since \(W=\bigcup _j W_j\) as defined in Sect. 2.2, \((z_0,w_0)\in W_j\) for some j, hence by Proposition 2.5,

    $$\begin{aligned} F^n(z_0,w_0)\in W_{j+n} \quad \text {for all }\, n\in {{\mathbb {N}}}, \end{aligned}$$

    hence \(\left| \frac{{\text {Im}}z_n}{{\text {Re}}z_n}\right| \le k_{j+n}<1\) and by Lemma 2.12 we obtain

    $$\begin{aligned} \left| \frac{{\text {Im}}z_n^2}{{\text {Re}}z_n^2}\right| \le \frac{2k_{j+n}}{1-k_{j+n}^2}=:\alpha _n\sim n \quad \text {as }\, n\rightarrow \infty , \end{aligned}$$

    where the estimate \(\alpha _n\sim n\) as \(n\rightarrow \infty \) is computed using the explicit expression for \(k_{j+n}\). It follows that

    $$\begin{aligned} \delta ^n\le |z_n^2|=\sqrt{({\text {Re}}(z_n^2))^2+({\text {Im}}(z_n^2))^2}\le {\text {Re}}(z_n^2)\sqrt{1+\alpha _n^2} \end{aligned}$$

    hence \({\text {Re}}(z_n^2)\ge \frac{\delta ^{n}}{\sqrt{1+\alpha _n^2}}\sim \frac{\delta ^{n}}{n}\ge \delta ^{n/2}\) for n large. Finally

    $$\begin{aligned} u_{n}(z_0,w_0)=-\frac{{\text {Re}}(z_n^2)}{n}\le -\frac{\delta ^{n/2}}{n}\rightarrow -\infty \text { as}\, n\rightarrow \infty \end{aligned}$$
  3. (3)

    Suppose by contradiction that there exists \(P=(z_0,w_0)\in \Omega {\setminus }{{\mathcal {A}}}_I\), \(\varepsilon >0\) and \(N\in \mathbb {N}\) such that

    $$\begin{aligned} u_n(z_0,w_0)=\frac{-{\text {Re}}z_n^2}{n} < - \varepsilon \quad \text {for all}\, n\ge N. \end{aligned}$$

    Hence there exists \(N'>N\) depending on \(\varepsilon , C\) (where C is the constant used to define \(W_I\)) such that

    $$\begin{aligned} {\text {Re}}(z_n^2)> \varepsilon n> C^2 \text { for all}\, n\ge N'. \end{aligned}$$

    Since \(w_n=z_{n-1}\) and since \(P\in \Omega \) for hypothesis, we have that \(F^n(P)\in I\times I\) for all \(n\ge N'\) hence \(P\in F^{-n}(W_I)\subset {{\mathcal {A}}}_I\), a contradiction.

This concludes the proof. \(\square \)

Lemma 2.14

(Good holomorphic disks) Let \(P\in \Omega \), W as before. Then there exists \(\varphi :\overline{{{\mathbb {D}}}}\rightarrow \Omega \) holomorphic in a neighborhood of \(\overline{{{\mathbb {D}}}}\) such that

  • \(\varphi (0)=P\)

  • \(\varphi ({{\mathbb {D}}})\Subset \Omega \) and \(\partial \varphi ({{\mathbb {D}}})\) is analytic

  • The one-dimensional Lebesgue measure of \(\partial \varphi ({{\mathbb {D}}})\cap W\) is greater than 0.

Proof

Since W is open it is enough to have \(\varphi ({{\mathbb {D}}})\cap W\ne \varnothing \) to ensure that the one-dimensional Lebesgue measure of \(\partial \varphi ({{\mathbb {D}}})\cap W\) is greater than 0. Let \(a,b\in \{+,-\}\) such that \(P\in \Omega ^{ab}\). Since \(W^{ab}\ne \varnothing \) for all \(a,b\in \{+,-\}\) there exists \(Q\in W^{ab}\). Since \(\Omega ^{ab} \) is connected and open there exists a simple real analytic curve passing through P and Q in \(\Omega ^{ab}\). Complexifying this curve we obtain a holomorphic disc passing through P that we can write as \(\varphi ({{\mathbb {D}}})\) for some \(\varphi \) holomorphic defined in a neighborhood of \(\overline{{{\mathbb {D}}}}\). Up to precomposing \(\varphi \) with a Moebius transformation we can assume that \(P=\varphi (0)\). \(\square \)

In our proof, we are going to use the mean value property for the harmonic functions \(u_N\).

Lemma 2.15

(Mean value property for holomorphic disks) Let \({{\mathbb {D}}}\subset {{\mathbb {C}}}\) be the open unit disk and \(\varphi :\overline{{{\mathbb {D}}}}\rightarrow \Omega \) be a holomorphic map. Let u be harmonic on the holomorphic open disk \(D=\varphi ({{{\mathbb {D}}}})\) and continuous up to the boundary of D. Let \(P_0:=\varphi (0)\). Then

$$\begin{aligned} u(P_0)=\frac{1}{2\pi }\int _{\partial {{\mathbb {D}}}}u(\zeta )|\varphi '(\zeta )|^{-1}d\zeta \end{aligned}$$

Proof

Consider the function \(u\circ \varphi :\overline{{{\mathbb {D}}}}\rightarrow {{\mathbb {R}}}\). First, note that it is harmonic on \({{\mathbb {D}}}\) and continuous up to the boundary. Indeed if \(u:D\rightarrow {{\mathbb {R}}}\) is \(\mathcal {C}^2\)-smooth, then we can explicitly compute its Laplacian

$$\begin{aligned} \nabla ^2(u\circ \varphi )=\nabla ^2(u)|\varphi '|^2=0\, \end{aligned}$$

while if u is not \(\mathcal {C}^2\)-smooth, the result follows by approximating u with harmonic smooth functions.

Hence for \(u\circ \varphi \) the classical Mean Value Property holds. By computing \(u(P_0)\) we get

$$\begin{aligned} u(P_0)\,=\,u(\varphi (0))\,=\,\frac{1}{2\pi }\int _{\partial {{\mathbb {D}}}}u(\varphi (\eta ))\,d\eta \,=\,\frac{1}{2\pi }\int _{\partial {{\mathbb {D}}}}u(\zeta )\,|\varphi '(\zeta )|^{-1}\,d\zeta . \end{aligned}$$
(2.19)

\(\square \)

Proof of Proposition 2.10

Let \(P\in \Omega {\setminus } {{\mathcal {A}}}_I\) and \(D:=\varphi ({{\mathbb {D}}}) \) where \(\varphi \) is given by Lemma 2.14. Let \(\mu \) be the pushforward under \(\varphi \) of the one-dimensional Lebesgue measure on \(\partial {{\mathbb {D}}}\). Let K be a compact subset of W such that \(\mu (K\cap \partial D)>0\).

Let \(\mu _{\text {good}}=\mu (\partial D\cap K)>0\) and \(\mu _{\text {bad}}=\mu (\partial D\cap (\Omega {\setminus } K))\). Since \(\Omega \) contains D, \(\partial D=(\partial D\cap K )\cup (\partial D\cap (\Omega {\setminus } K))\), and since K is compact and \(\Omega \) is open, the sets in question are measurable.

By Lemma 2.11 for any given \({{\mathcal {M}}}>0\) there exists N such that \(u_N\le -{{\mathcal {M}}}\) on K, \(u_N(P)\ge -\varepsilon \) for some \(\varepsilon >0\) since \(P\in \Omega {\setminus } {{\mathcal {A}}}_I\), and \(u_N\le \log M\) on \(\overline{D}\) (with \(M=M(\overline{D})\)). By the Mean value property (2.19) for \(u_N\) we have

$$\begin{aligned} -\varepsilon \le u_N(P)&=\frac{1}{2\pi }\int _{\partial D} u_N(\zeta )|\varphi '(\zeta )|d\zeta =\frac{1}{2 \pi } \int _{\partial D\cap K} u_N(\zeta )|\varphi '(\zeta )|d\zeta \\&\quad +\frac{1}{2\pi }\int _{ \partial D\cap (\Omega {\setminus } K) } u_N(\zeta )|\varphi '(\zeta )|d\zeta \le \\&\le \frac{1}{2\pi } \left( -{{\mathcal {M}}}\mu _{\text {good}}+ \log M\mu _{\text {bad}}\right) \cdot \sup _{\partial {{\mathbb {D}}}}|\varphi '|^{-1}. \end{aligned}$$

Since \({{\mathcal {M}}}\) is arbitrarily large, this gives a contradiction. \(\square \)

Proposition 2.16

The set \(W_I\) is open.

Proof

Let \(P \in W_I\). We want to find \(V\subset W_I\) neighborhood of P. Since \(W_I\subset \Omega \cap (I\times I)\) which is open there is a neighborhood U of P which is compactly contained in \(\Omega \cap (I\times I)\). Since \(W_I\) is absorbing for \(\Omega \) under F there exists \(N>0\) such that

$$\begin{aligned} F^n(\overline{U})\subset W_I\quad \text { for all}\, n\ge N. \end{aligned}$$
(2.20)

As usual let us define \(P_j:=F^j(P)\); by definition of \(W_I\), \(P_j\subset I\times I\) for all \(j\ge 0\), which is an open set. Hence for each \(j\ge 0\) there is a neighborhood \(U_j\subset I\times I\) of \(P_j\). So up to making the \(U_j\) smaller, we can assume that \(U_j\subset F^j(U)\).

Let

$$\begin{aligned} V:=\bigcap _{j=0}^N F^{-j}(U_j)\subset U. \end{aligned}$$

The set V is open since it is a finite intersection of open sets. We only need to check that \(V\subset W_I\), or equivalently, that \(F^j(V)\subset I\times I \) for all \(j\ge 0\). For \(j\le N-1\), this is true by definition, since \(F^j(V)\subset U_j\subset I\times I\). For \(j\ge N\), this is true by (2.20). Since \(P\in V\) by construction, V is a neighborhood of P in \(W_I\) as required. \(\square \)

2.4 Geometric structure of \(\Omega \)

In this section we show that \(\Omega \) is the union of four disjoint Fatou components \(\Omega ^{ab}\), \(a,b\in \{+,-\}\), each of which is biholomorphic to \(\mathbb {H}\times \mathbb {H}\).

We first show conjugacy of F to its linear part on \(\Omega \), and estimate the distance between the conjugacy and the identity map.

Proposition 2.17

(Conjugacy) F is conjugate to the linear map \(L(z,w)=(-\delta w,z)\) on \(\Omega \) via an injective holomorphic map \(\varphi \). If P is such that \(F^n(P)\in S\) for all \(n\in {{\mathbb {N}}}\), then \(\Vert (\varphi - Id) (P)\Vert <\sqrt{2}\). Finally, \(\varphi (\Omega )\subset S\).

Proof

We first show that F is conjugate to L on \(W_I\).

For \(n\in {{\mathbb {N}}}\) let \(\varphi _n:{{\mathbb {C}}}^2\rightarrow {{\mathbb {C}}}^2\) be the automorphisms defined as

$$\begin{aligned} \varphi _n:=L^{-n}\circ F^n. \end{aligned}$$

If we show that the \(\varphi _n\) converge to a map \(\varphi \) uniformly on \(W_I\) we obtain that \(\varphi \) satisfies the functional equation \(\varphi =L^{-1}\circ \varphi \circ F\) and hence is a conjugacy between F and L.

Computing \(L^{-n}\) and using the explicit expressions for the iterates of F we obtain

$$\begin{aligned} \varphi _{2n}(z,w)&=\left( z+\Delta _1^n(z,w),w+\Delta _2^n(z,w)\right) , \end{aligned}$$
(2.21)
$$\begin{aligned} \varphi _{2n+1}(z,w)&=\left( z+\Delta _1^n(z,w),w+\Delta _2^{n+1}(z,w)\right) . \end{aligned}$$
(2.22)

Both have the same formal limit

$$\begin{aligned} \varphi (z,w)=\left( z+\Delta _1(z,w),w+\Delta _2(z,w)\right) . \end{aligned}$$

If \(P=(z,w) \in W_I\), then \(F^n(P)=(z_n,w_n)\subset I\times I\subset S\) for all j, hence, by (2.9), we have that \(\Delta (z,w)<1\); in particular, \(\Delta _1(z,w)\) and \(\Delta _2(z,w)\) are convergent. Hence \(\varphi \) is a holomorphic map from \(W_I\) to \(\varphi (W_I)\) (\(W_I\) is open by Proposition 2.16). Moreover, for any point (z, w) whose orbit is contained in S,

$$\begin{aligned}{} & {} \left\| (\varphi - {\text {Id}}) (z,w)\right\| =\left\| \left( \Delta _1(z,w),\Delta _2(z,w)\right) \right\| \nonumber \\{} & {} <\sqrt{2}\Delta (z,w)<\sqrt{2}. \end{aligned}$$
(2.23)

It follows that \(\varphi \) is open because \(W_I\) is an unbounded set, hence if \(\varphi \) had rank 0 or 1, \(\left\| (\varphi - {\text {Id}})\right\| \) could not be bounded on \(W_I\). Hence the map \(\varphi \) is injective by Hurwitz Theorem (see [19, Exercise 3 on page 310]) because the maps \(\varphi _n\) are injective and their limit has rank 2. It follows that \(\varphi \) is a biholomorphism between \(W_I\) and \(\varphi (W_I)\).

To extend \(\varphi \) to all of \(\Omega \) recall that \(W_I\) is absorbing for \(\Omega \). So if \(P\in \Omega \), we have that \(F^k(P)\in W_I\) for some \(k\in {{\mathbb {N}}}\), hence we can define \(\varphi (P)= L^{-k}\circ \varphi \circ F^k(P)\). Since F is an automorphism, \(\varphi \) extends as a biholomorphism to \(\Omega \).

It remains to show that \(\varphi (\Omega )\subseteq S\). By (2.23) we have that \(\varphi (W_I)\) is contained in a \(\sqrt{2}\) neighborhood U of \(W_I\). Suppose by contradiction that there exists \(Q=(z,w)\in \varphi (W_I){\setminus } S\). Since \(W_I\) is forward invariant under F and \(\varphi \) is a conjugacy we have that \(\varphi (W_I)\) is forward invariant under L. Up to considering L(Q) if necessary, and since \(\theta \) is such that \(r e^{i\theta }\notin S\), we can assume that \(z=re^{i \theta }\notin {{\mathcal {S}}}\). By forward invariance \(L^{2n}(Q)=((-\delta ^{n}) r e^{i\theta },(-\delta )^{n}w) \in \varphi (W_I)\).

Since \((-\delta )^{n} r \) tends to infinity, the distance of \(L^{2n}(Q)\) from the boundary of S tends to infinity, hence so does the distance of \(L^{2n}(Q)\) from \(W_I\subset S\), contradicting \(\varphi (W_I)\subset U\). Hence \(\varphi (W_I)\subset S\). Since \(W_I \) is an absorbing set for \(\Omega \) under F, \(\varphi \circ F=L\circ \varphi \), and \(\varphi (W_I)\) is completely invariant under L, we have that

$$\begin{aligned} \varphi (\Omega )= \varphi \left( \bigcup _{n\ge 0} F^{-n}(W_I)\right) =\bigcup _{n\ge 0} L^{-n}(\varphi (W_I))\subset \varphi (W_I) \subset S. \end{aligned}$$
(2.24)

\(\square \)

We are now able to understand the geometric structure of \(\Omega \).

Proposition 2.18

(Geometry of \(\Omega \)) \(\Omega \) consists of four distinct connected components, each of which is biholomorphic to \(\mathbb {H}\times \mathbb {H}\), and which form a cycle of period 4.

We recall the following simple topological lemma. Here \(\partial \) denotes the topological boundary.

Lemma 2.19

Let \(A,B\subset {{\mathbb {C}}}^n\) be open, A connected. If \(A\cap B\ne \varnothing \) and \(\partial B\cap A=\varnothing \) then \(A\subseteq B\).

Proof

Since \(A\cap \partial B=\varnothing \) we can write

$$\begin{aligned} A=(A\cap B)\cup (A{\setminus } \overline{B}). \end{aligned}$$

Both \(A\cap B\) and \(A{\setminus } \overline{B}\) are open and \(A\cap B\ne \varnothing \) by assumption, so since A is connected, \(A{\setminus } \overline{B}=\varnothing \). \(\square \)

Recall also that if a set A is invariant under a map F, by continuity of the latter we have \(F(\overline{A})\subset \overline{A}\). The following lemma is also known.

Lemma 2.20

Let \(\Omega _1,\Omega _2\) be two Fatou components for an automorphism F of \({{\mathbb {C}}}^2\). Then if \(F(\Omega _1)\cap \Omega _2\ne \varnothing \), \(F(\Omega _1)=\Omega _2\).

Proof

We have that \(F(\Omega _1)\subset \Omega _2\), indeed otherwise, \(F(\Omega _1)\) would intersect the boundary of \(\Omega _2\) which is contained in the forward Julia set, and this is impossible because the Fatou set is completely invariant. On the other hand suppose for a contradiction that there is \(P\in \Omega _2{\setminus } F(\Omega _1)\). Then since \(F(\Omega _1)\cap \Omega _2\ne \varnothing \) and both \(\Omega _1, F(\Omega _1)\) are connected there exists \(Q\in \Omega _2\cap F(\partial \Omega _1)\), which is impossible because \(\partial \Omega _1\) is contained in the forward Julia set which is forward invariant. \(\square \)

Observe that we could not simply use the same argument applied to \(F^{-1}\), since the Fatou components for F and \(F^{-1}\) are, in general, different sets.

Proof of Proposition 2.18

We prove the claim by showing that \(\Omega \) is biholomorphic to S. Since S has four connected components \(S^{ab}\) each of which is biholomorphic to \(\mathbb {H}\times \mathbb {H}\), the same holds for \(\Omega \). Since by definition \(\Omega =\bigcup _{a,b\in \{+,-\}}\Omega ^{ab}\) and each \(\Omega ^{ab}\) is connected, these are exactly the connected components of \(\Omega \).

Recall the definition of the set \(W\subset S\) from Sect. 2.2 and recall that it is forward invariant and contained in S. Hence (2.23) holds. Also recall that by (2.24) \(\varphi (\Omega )\subset S\).

Let U be a \(2\sqrt{2}\)-neighborhood of \(\partial W\). Fix \(a,b\in \{+,-\}\). We want to apply Lemma 2.19 to the sets \(A=W^{ab}{\setminus }\overline{ U}\) and \(B=\varphi (W^{ab})\). So we need to show that

  • \(A\cap B=(W^{ab}{\setminus }\overline{ U})\cap \varphi (W^{ab}) \ne \varnothing \)

  • \(\partial B\cap A =\partial \varphi (W^{ab})\cap ( W^{ab}{\setminus }\overline{ U})=\varnothing \).

The second item is true because \(\partial (\varphi (W^{ab}))\subset \varphi (\partial W^{ab})\Subset U\) by (2.23). So we now show that \((W^{ab}{\setminus }\overline{ U})\cap \varphi (W^{ab}) \ne \varnothing \). Let \(P\in W^{ab}\) such that the ball of radius \(\sqrt{2}\) centered at P is contained in \(W^{ab}{\setminus }\overline{ U}\). This is possible because this set contains arbitrarily large balls. By (2.23), \(\Vert P-\varphi (P)\Vert <\sqrt{2}\) hence \(\varphi (P)\in W^{ab}{\setminus }\overline{ U} \).

Hence, applying Lemma 2.19 we obtain that for each \(a,b\in \{+,-\}\) we have \(\varphi (W^{ab})\supset (W^{ab}{\setminus } U)\) hence

$$\begin{aligned} \varphi (\Omega )\supset \varphi (W)\supset (W{\setminus } U). \end{aligned}$$

We now show that this implies that \(\varphi (\Omega )\supset S\). Notice that S can be written as

$$\begin{aligned} S=\left\{ (r_1 e^{i\theta _1},r_2 e^{i\theta _2})\in {{\mathbb {C}}}^2: r_1,r_2>0,\,\text { and for each}\, i=1,2\, \text {either}\, |\theta _i|<\frac{\pi }{4}\,\text { or}\, |\theta _i-\pi |<\frac{\pi }{4} \right\} . \end{aligned}$$

Fix \(\alpha <\frac{\pi }{4}\). By definition of W there exists \(R=R(\alpha )\) such that \(W{\setminus } U\) contains the set

$$\begin{aligned} W{\setminus } U\supset X_{\alpha ,R}:=\{(r_1 e^{i\theta _1}, r_2 e^{i\theta _2}):r_1,r_2>R\,\text { and for each}\, i=1,2\,\text { either} |\theta _i|<\alpha \, \text {or}\, |\theta _i-\pi |<\alpha \}. \end{aligned}$$

Hence \(\varphi (\Omega )\supset \varphi (W)\supset (W{\setminus } U)\supset X_{\alpha ,R}\).

By the explicit form of L, \(\bigcup _{j\ge 0}L^{-j}X_{\alpha ,R}=X_{\alpha ,0} \). Hence by backward invariance of \(\varphi (\Omega )\) under L we have that

$$\begin{aligned} \varphi (\Omega )\supset \bigcup _{j\ge 0}L^{-j}X_{\alpha ,R}=X_{\alpha ,0}\quad \text { for every}\, \alpha <\frac{\pi }{4}. \end{aligned}$$

It follows that

$$\begin{aligned} \varphi (\Omega )\supset \bigcup _{\alpha <\frac{\pi }{4}} X_{\alpha ,0} =S. \end{aligned}$$

Hence \(\varphi (\Omega )=S\).

It remains to show that the Fatou components \(\Omega ^{ab}\) with \(a,b\in \{+,-\}\) form a cycle of period four, more precisely, that

$$\begin{aligned} F(\Omega ^{ab})=\Omega ^{(-b)a} \quad \text { for all }\, a,b\in \{+,-\}. \end{aligned}$$
(2.25)

By definition \(\Omega ^{ab}\supset W^{ab}\) and by Lemma 2.2, \(F(W^{ab})\cap W^{(-b)a}\ne \varnothing \). Hence \(F(\Omega ^{ab})\cap \Omega ^{(-b)a}\ne \varnothing \). By Lemma 2.20, \(F(\Omega ^{ab})= \Omega ^{(-b)a}\). \(\square \)

2.5 Limit sets on \(\Omega ^{ab}\)

Let \(\mathbb {H}\) and \(-\mathbb {H}\) denote the right and left half plane respectively. In this section we show the following.

Proposition 2.21

(Limit set for \(\Omega \)) We have that

$$\begin{aligned}&h_1(\Omega ^{ab})={\mathbb {H}}&\text { and }{} & {} h_2(\Omega ^{ab})=-{\mathbb {H}}&\text { if}\, a=b \\&h_1(\Omega ^{ab})=-{\mathbb {H}}&\text { and }{} & {} h_2(\Omega ^{ab})={\mathbb {H}}&\text { if}\, a\ne b. \end{aligned}$$

Let W be as defined in Sect. 2.2 and \(W_I\) as defined in Sect. 2.3. Since both are forward invariant and contained in S we have that, for any \(a,b\in \{+,-\}\), \(F(W^{ab})\subset W^{(-b)a}\) and \(F(W_I^{ab})\subset W_I^{(-b)a}\). Compare with Lemma 2.2 and Corollary 2.3.

We first study the image of \(W_I^{ab}\) under \(h_1,h_2\).

Lemma 2.22

$$\begin{aligned}&h_1(W_I^{ab})\subset {\mathbb {H}}&\text { and }{} & {} h_2(W_I^{ab})\subset {-\mathbb {H}}&\text { if}\, a=b \\&h_1(W_I^{ab})\subset {-\mathbb {H}}&\text { and }{} & {} h_2(W_I^{ab})\subset {\mathbb {H}}&\text { if}\, a\ne b \end{aligned}$$

Proof

Recall that \(h_1(z_0,w_0)=\lim _{n\rightarrow \infty }\frac{z_{2n}}{w_{2n}}\). Let \(I_+:=\left( -\frac{\pi }{4},\frac{\pi }{4}\right) \) and \(I_-:=\left( \frac{3}{4}\pi ,\frac{5}{4}\pi \right) \).

For \(a,b\in \{+,-\}\) and \((z,w)\in W_I^{ab}\), then \({\text {arg}}(z)\in I_a\) and \({\text {arg}}( w)\in I_b\). Hence \({\text {arg}}\left( \frac{z}{w}\right) \in \left( -\frac{\pi }{2},\frac{\pi }{2} \right) \) if \(a=b\), and \({\text {arg}}\left( \frac{z}{w}\right) \in \left( \frac{\pi }{2},\frac{3}{2}\pi \right) \) if \(a\ne b\). Since \(F^2(W_I^{++}\cup W_I^{--})\subset W_I^{++}\cup W_I^{--}\), If \((z,w)\in W_I^{++}\cup W_I^{--}\) then all of its even iterates \((z_{2n}, w_{2n})\in W_I^{++}\cup W_I^{--}\), hence by taking the limit \(h_1 (W_I^{++}), h_1(W_I^{--})\subset \overline{\mathbb {H}}\). Similarly if \((z,w)\in W_I^{+-}\cup W_I^{-+}\) then all of its even iterates \((z_{2n}, w_{2n})\in W_I^{+-}\cup W_I^{-+}\), hence \(h_1 (W_I^{+-}), h_1(W_I^{-+})\subset \overline{-\mathbb {H}}\). The analogous results for \(h_2\) hold by observing that \(h_2=\frac{-\delta }{h_1}\). Since \(W_I\) is open by Proposition 2.16, its image under a holomorphic map of maximal rank is open, hence we can replace \(\overline{\mathbb {H}},\overline{-\mathbb {H}}\) by \(\mathbb {H},-\mathbb {H}\). \(\square \)

Lemma 2.23

$$\begin{aligned}&\mathbb {H}\subset h_1(W^{ab})&\text { and }{} & {} -\mathbb {H}\subset h_2(W^{ab})&\text { if}\, a=b \\ -&\mathbb {H}\subset h_1(W^{ab})&\text { and }{} & {} \mathbb {H}\subset h_2(W^{ab})&\text { if}\, a\ne b. \end{aligned}$$

Before proving Lemma 2.23 let us see how Lemmas 2.22 and 2.23 imply Proposition 2.21.

Proof of Proposition 2.21

We prove the claims for \(h_1\); for \(h_2=\frac{-\delta }{h_1}\), it follows by symmetry.

Clearly \(h_1(\Omega ^{ab})\supset h_1(W^{ab})\) for any \(a,b\in \{+,-\}\) since \(\Omega ^{ab}\supset W^{ab}\). So in view of Lemma 2.23, \(h_1(\Omega ^{ab})\supset -\mathbb {H}\) or \(h_1(\Omega ^{ab})\supset \mathbb {H}\) depending on whether \(a=b\).

We now consider limit sets for \(\Omega ^{++}\) and \(\Omega ^{--}\); the other cases are analogous. By (2.25),

$$\begin{aligned} F^{2}(\Omega ^{++})\subset \Omega ^{--} \quad \text {and}\quad F^{2}(\Omega ^{--})\subset \Omega ^{++}. \end{aligned}$$

It follows that for any \(n>0\),

$$\begin{aligned} F^{2n}(\Omega ^{++}\cup \Omega ^{--})\subset \Omega ^{++}\cup \Omega ^{--}. \end{aligned}$$

In view of this, and since \(W_I\) is absorbing for \(\Omega \), we have that for any \(P\in (\Omega ^{++}\cup \Omega ^{--} )\)

$$\begin{aligned} F^{2n}(P)\subset W_I\cap (\Omega ^{++}\cup \Omega ^{--})=W_I^{++}\cup W_I^{--} \text { for any}\, n\,\text { large enough}. \end{aligned}$$

Hence \(h_1(P)\in h_1(W_I^{++}\cup W_I^{--} )\subset {\mathbb {H}}\) for every \(P\in (\Omega ^{++}\cup \Omega ^{--} )\), hence \(h_1(\Omega ^{++}\cup \Omega ^{--} )\subset {\mathbb {H}}\). It follows that \(h_1(\Omega ^{++})=h_1(\Omega ^{--})= \mathbb {H}\). \(\square \)

We devote the rest of this section to proving Lemma 2.23. We first give a version of Rouché’s Theorem which relies on one of the many versions of Rouché’s Theorem existing in one variable (compare with Theorem 3.4 in [8]; we will use it with the spherical instead of the Euclidean metric). This is certainly known to experts in the field but we are not aware of a reference. In this section \(\partial \) denotes the topological boundary, and \({\text {dist}}_{{\text {spher}}}\) denotes the spherical distance.

Theorem 2.24

(Rouché’s Theorem in \({{\mathbb {C}}}^2\)) Let \(B\subset {{\mathbb {C}}}^2\) be a polydisk, F, G be holomorphic maps defined in a neighborhood of \(\overline{B}\) which take values in \(\hat{{{\mathbb {C}}}}\). Let \(c\in G(B)\), let \(\varepsilon = {\text {dist}}_{{\text {spher}}}(c, G(\partial B))>0\) and assume

$$\begin{aligned} {\text {dist}}_{{\text {spher}}}( F,G)<\varepsilon \quad \text {on }\, \partial B. \end{aligned}$$

Then \(c\in F(B)\).

Notice that the assumptions imply that F, G have generic rank 1: They cannot have rank 2 because the target is \(\hat{{{\mathbb {C}}}}\), and G cannot be constant otherwise there could not be \(c\in G(B)\) with positive distance from \(G(\partial B)\). One can check that F cannot be constant either.

Proof

Let D be a horizontal disk passing through a point \(P_c\in G^{-1}(c)\cap B\), such that \(\partial D\subset \partial B\). Let \(g:=G|_{D}\), \(f:=F_{D}\). They are holomorphic in a slightly larger horizontal disk. Notice that \({\text {dist}}_{{\text {spher}}}(g,f)<\varepsilon \) on \(\partial D\), and that \({\text {dist}}_{{\text {spher}}}(G(\partial D),c)\ge \varepsilon \) because \(\partial D\subset \partial B\). By Rouché’s Theorem in one variable, \(c\in f(D)\subset F(B)\) as required. \(\square \)

Remark 2.25

Unless \(P_c\) is an isolated point in \(G^{-1}(c)\) we obtain a curve of points in \(F^{-1}(c)\). Indeed, the proof gives a point in \(F^{-1}(c)\) for any Euclidean disk passing through points in \(G^{-1}(c)\), for example, a family of disks passing through \(P_c\) along different complex directions. The points obtained for \(F^{-1}(c)\) are distinct unless they always coincide with \(P_c\). On the other hand, if \(P_c\) is an isolated point in \(G^{-1}(c)\) then \(P_c\in F^{-1}(c)\) is also isolated in \(F^{-1}(c)\). Indeed otherwise we could reverse the role of F and G and obtain one point in \(G^{-1}(c)\) for any Euclidean disk passing through any point in \(F^{-1}(c)\) and obtain a curve of points in \(G^{-1}(c)\). The proof as it is works when B is any \({{\mathbb {C}}}\)-convex set instead of a polydisk, and can certainly be generalized further.

Proof of Lemma 2.23

We show \(\mathbb {H}\subset h_1(W_I^{++})\). The other cases are analogous. Recall that orbits of points in W are contained in S, hence (2.9) holds. Since

$$\begin{aligned} \frac{z_{2n}}{w_{2n}}=\frac{z_0+\Delta _1^n(z_0,w_0)}{w_0+\Delta _2^n(z_0,w_0)}, \end{aligned}$$

dividing the numerator and the denominator by \(w_0\) and using \(\frac{1}{1+x}=1+\sum _{j=1}^\infty ( -x)^j\) for \(|x|<1\) we have that

$$\begin{aligned} \frac{z_{2n}}{w_{2n}}-\frac{z_{0}}{w_{0}}&=\left( \frac{z_0}{w_0}+\frac{\Delta _1^n(z_0,w_0)}{w_0}\right) \left( 1+\sum _{j=1}^\infty \left( \frac{-\Delta _2^n(z_0,w_0)}{w_0} \right) ^{j}\right) -\frac{z_{0}}{w_{0}} \end{aligned}$$
(2.26)
$$\begin{aligned}&=\frac{\Delta _1^n(z_0,w_0)}{w_0} +\left( \frac{z_0}{w_0}+ \frac{\Delta _1^n(z_0,w_0)}{w_0}\right) \sum _{j=1}^\infty \left( \frac{-\Delta _2^n(z_0,w_0)}{w_0} \right) ^{j} \forall n\ge 0. \end{aligned}$$
(2.27)

This expression makes sense for \(|x|=\left| \frac{-\Delta _2^n(z_0,w_0)}{w_0} \right| <1\), hence, in view of (2.9), for \(|w_0|>1\). Recall also that \(|\sum _{j=1}^\infty x^{j}|=\frac{|x|}{1-x}\le 2|x|\) if \(|x|<\frac{1}{2}\). Let \(K\subset {\hat{{{\mathbb {C}}}}}\) be a compact set and suppose that \(\frac{z_0}{w_0}\) takes values in K. By (2.26) and using (2.9), for any \(\varepsilon >0\) there exists \(M=M(K,\varepsilon )\) such that

$$\begin{aligned} \left| \frac{z_{2n}}{w_{2n}}-\frac{z_{0}}{w_{0}}\right| <\varepsilon \text { for }\, |w_0|>M\,\text { and }\,\frac{z_0}{w_0}\in K. \end{aligned}$$
(2.28)

Consider the function \(G(z,w):=\frac{z}{w}\) Observe that

$$\begin{aligned} G^{-1}(re^{i\theta })=\left\{ (r_1e^{i\theta _1},r_2 e^{i\theta _2})\in {{\mathbb {C}}}^2: \frac{r_1}{r_2}=r, \theta =\theta _1-\theta _2 \right\} . \end{aligned}$$

Let \(c\in \mathbb {H}\). By the shape of W we have that \(G(W^{++})=\mathbb {H}\), that \(\varepsilon :=\frac{1}{2}{\text {dist}}_{{\text {spher}}}(c, G(\partial W ))>0\), and that we can choose \(Q=(z_0,w_0)\in W^{++} \in G^{-1}(c)\) such that \(|w_0|\) is arbitrarily large. By taking a limit in n in equation (2.26) and on a sufficiently small polydisk centered at Q we can ensure that \({\text {dist}}_{{\text {spher}}}(h_1,G)<\varepsilon \), hence the claim follows by Rouché’s Theorem. \(\square \)

The main Theorem is a direct consequence of Propositions 2.6, 2.17,  2.18, 2.21.