Escaping Fatou components with disjoint hyperbolic limit sets

We construct automorphisms of $\mathbb{C}^2$ with a cycle of escaping Fatou components, on which there are exactly two limit functions, both of rank 1. On each such Fatou component, the limit sets for these limit functions are two disjoint hyperbolic subsets of the line at infinity.


Introduction
Transcendental Hénon maps are automorphisms of C 2 with constant Jacobian of the form F (z, w) := (f (z) − δw, z) with f : C → C entire transcendental.
In analogy with classical complex Hénon maps, for which f is assumed to be a polynomial, 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 [Duj04].General properties of transcendental Hénon maps were established in [ABFP19], [ABFP23], [ABFP21] and examples with interesting dynamical features were presented.
Let P 2 be the complex projective space obtained by compactifying C 2 by adding the line at infinity ℓ ∞ .We define the Fatou set of F as the set of points in C 2 near which the iterates form a normal family with respect to the complex structure induced by P 2 (compare with [ABFP19], section 1).A Fatou component is a connected component of the Fatou set.Given a Fatou component Ω we call a function h : Ω → P 2 a limit function for Ω if there exists a subsequence n k such that F n k → h uniformly on compact subsets of Ω.The image h(Ω) of a limit function h is called a limit set (for Ω).By Lemma 4.3 and 2.4 in [ABFP19], each limit sets is either contained in C 2 or contained in ℓ ∞ .
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 H denote the right half plane, −H denote the left half plane.
Then F has a cycle of four Fatou components Ω ab with a, b ∈ {+, −}, each of which is biholomorphic to H × H.There are exactly two limit functions h 1 , h 2 , both of rank 1, such that h 1 (Ω aa ) = h 2 (Ω a(−a) ) = H and h 1 (Ω a(−a) ) = h 2 (Ω aa ) = −H for all a.
Moreover, F is conjugate to its linear part on every Ω ab .
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 [JL04] for restrictions on the presence of several limit sets), and that the limit sets H, −H are hyperbolic.
For general automorphisms of C 2 there are very few examples of limit functions of rank 1 ( [JL04], [BTBP21]), 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 ( [LP14]).On the other hand, they are abundant for holomorphic endomorphisms of C 2 ([BTFP15], Theorem 4).For transcendental Hénon maps, rank 1 limit functions seem to appear naturally for escaping Fatou components ( [BSZ23]).To our knowledge there were no previous examples of hyperbolic limit sets for automorphisms of C 2 .One possible reason for the natural appearance of these phenomena might be that F is not defined on ℓ ∞ , hence there is no natural dynamics on limit sets contained there.
One can see F as a special case of maps of the form Analogous results hold for such maps, and are proven in [Belpt] with similar techniques.
2 Proof of Theorem 1.1 From now on let F be as in Theorem 1.1, Throughout the paper, given a point P = (z 0 , w 0 ) ∈ C 2 and n ∈ N we denote its iterates by F n (P ) =: (z n , w n ).

Computing limit functions
In this section we give an explicit expression for the iterates of F and their formal limit.A direct computation (compare [BSZ23]) shows that For n ∈ N define the following holomorphic functions from With this notation the iterates of F take the form Notice that ∆ 1 , ∆ 2 are holomorphic functions to Ĉ on open sets on which they are well defined.
We can deduce the following formal limits. . (2.5) We have that h 1 , h 2 are holomorphic functions to Ĉ on open sets on which ∆ 1 and ∆ 2 are holomorphic functions to Ĉ.We will show in Proposition 2.7 that h 1 = h 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 ⊆ C 2 and a, b ∈ {+, −} define We start by defining a set on which we have control on the dynamics.Let Lemma 2.2 (Orbits contained in S).For any P = (z 0 , w 0 ) ∈ S ab such that F (P ) ∈ S and | Re w 0 | > 1 δ we have that F (P ) ∈ S (−b)a .From now on assume that F n (P ) ∈ S for all n ∈ N. Then 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 ∈ S, by Lemma 2.1 we have where the claim for z 2 follows because w 1 = z 0 .The more general formula follows by induction, using that F n (P ) ∈ S for all n ∈ N.
For R > 0 and 0 < k < 1 define the sets Observe that W k,R ⊂ S and that W 1,0 = S.
Lemma 2.4.Let n ∈ N, and let 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 Setting the resulting expression to be less than k we get n 2 R 0 for R 0 > 2 sufficiently large depending only on δ (see (2.12)).Let R −1 = R 0 and set Proof.The fact that W is open and W ⊂ S follows from the definition.Fix n ∈ N. Let (z 0 , w 0 ) ∈ W n and let (z 1 , w 1 ) be its image.Since w 1 = z 0 , the signs of Re w 1 , Re z 0 are the same, and we have that Hence to show that Let λ n := R n+1 − R n−1 .Since P ∈ S, by (2.10) we have that (2.12) Since the function on the right hand side is bounded in n for any δ > 2 (in fact, it tends to 0 as n → ∞), such R 0 exists and depends only on δ.
Finally, the set W consists of 4 connected components W ab by construction, since for any (z, w) ∈ W we have Re z, Re w = 0 and each W ab is connected.Since W ⊂ S and we now know that W is forward invariant, the orbits of points in W are contained in S hence Corollary 2.3 applies.
Proposition 2.6 (Existence of Fatou components).On each W ab we have that It follows that each W ab is contained in a Fatou component that we denote by Ω ab .Proof.Since W ⊂ 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 Ω ab .We will see in Proposition 2.18 that in fact the components Ω ab are all distinct and that the notation Ω ab matches the definition of A ab given in Section 2.1 for a general set A.
Proposition 2.7.Both h 1 and h 2 have (generic) rank 1 on W , and Proof.Recall that ∆(z, w) < 1 on W by (2.9).Since h i (W ) ⊂ ℓ ∞ , h 1 and h 2 either have generic rank 1 or are constants.Suppose by contradiction that h 1 = c is constant.If |c| = ∞, then one has: 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.
(and hence h 1 ) would be constant as well, contradicting the argument above.

Construction of an absorbing set
Let Ω ab with a, b ∈ {+, −} be the Fatou components defined in Proposition 2.6 and let Since each Ω ab is connected, Ω consists of at most 4 Fatou components.This section is devoted to find an absorbing set W I for Ω under F .Its existence will be used in Section 2.5 to show that the Fatou components Ω 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 [For04], [ABFP19], [BSZ23].
Definition 2.8 (Absorbing sets).A set A is absorbing for an open set Ω ⊃ A under a map F if for any compact K ⊂ Ω there exists N > 0 such that Fix C ≥ 1 and let Proposition 2.9.We have that W The rest of this section is devoted to proving the following proposition.
Proposition 2.10.The set W I is absorbing for Ω under F , that is, Since X is an analytic set, being the union of the 0-set and the ∞-set of a meromorphic function, it is locally a finite union of 1-complex-dimensional varieties (see [Chi89]).
Let K be a compact subset of Ω \ X , hence h i (P ) = 0, ∞ for all P ∈ K, and i = 1, 2. Define M := max if ε > 0 is sufficiently small there exists a constant c such that for every (z 0 , w 0 ) ∈ K (2.14) 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 Ω to R as (2.16) (2.18) The following fact is certainly known, however we give a proof in the Appendix.Given a set A, let Å denote its interior.
Proposition 2.13.Let L be a compact set and H be an analytic subset of dimension one of C 2 .For any compact K s.t.K ⊂ L there exists η = η(K, L, H) such that for any u harmonic defined in a neighborhood of L and such that 1. Let K be a compact subset of Ω.Let η as obtained by applying Proposition 2.13 to a slightly larger compact set L ⊂ Ω and to the analytic set X .Let U η (X ) be an ηneighborhood of X .In view of Proposition 2.13 it is enough to prove that there exists N ∈ N such that u n ≤ log M for n > N and for some M on the set which is a compact subset of Ω \ X .Hence it is enough to prove the claim for any K compact subset of Ω \ X .
Fix ε > 0 suffciently 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)) ∈ K such that for some β.We will show that β ≤ M .
Using (2.14) and (2.15) we have that Hence, using M > 1 and ε > 0 sufficiently small, 2. It is enough to show that u n (z 0 , w 0 ) → −∞ for any point (z 0 , w 0 ) ∈ W and it will follow for any compact subset of W . Since W is forward invariant, F n (z 0 , w 0 ) ⊂ W ⊂ S for all n ∈ N and ∆(z 0 , w 0 ) < 1 by (2.9).Using the explicit expression for iterates of F given by (2.2), (2.3) we have In both cases, since |z 0 |, |w 0 | > R 0 > 2 we obtain |z 2 n | ≥ δ n .Since W = j W j as defined in Section 2.2, (z 0 , w 0 ) ∈ W j for some j, hence by Proposition 2.5, Re zn ≤ k j+n < 1 and by Lemma 2.12 we obtain where the estimate α n ∼ n as n → ∞ is computed using the explicit expression for k j+n .It follows that 3. Suppose by contradiction that there exists P = (z 0 , w 0 ) ∈ Ω \ A I , ε > 0 and N ∈ N such that Hence there exists N ′ > N depending on ε, C (where C is the constant used to define W I ) such that Re(z 2 n ) > εn > C 2 for all n ≥ N ′ .
Since w n = z n−1 and since P ∈ Ω for hypothesis, we have that F n (P ) ∈ I × I for all n ≥ N ′ hence P ∈ F −n (W I ) ⊂ A I , a contradiction.
Lemma 2.14 (Good holomorphic disks).Let P ∈ Ω, W as before.Then there exists ϕ : Since Ω ab is connected and open there exists a simple real analytic curve passing through P and Q in Ω ab .Complexifying this curve we obtain a holomorphic disc passing through P that we can write as ϕ(D) for some ϕ holomorphic defined in a neighborhood of D. Up to precomposing ϕ with a Moebius transformation we can assume that P = ϕ(0).
In our proof, we are going to use the mean value property for the harmonic functions u N .
while if u is not C 2 -smooth, the result follows by approximating u with harmonic smooth functions.
Hence for u • ϕ the classical Mean Value Property holds.By computing u(P 0 ) we get Proof of Proposition 2.10.Let P ∈ Ω \ A I and D := ϕ(D) where ϕ is given by Lemma 2.14.Let µ be the pushforward under ϕ of the one-dimensional Lebesgue measure on ∂D.Let K be a compact subset of W such that µ(K ∩ ∂D) > 0.
By Lemma 2.11 for any given M > 0 there exists N such that u N ≤ −M on K, u N (P ) ≥ −ε for some ε > 0 since P ∈ Ω \ A I , and u N ≤ log M on D (with M = M (D)).By the Mean value property (2.19) for u N we have Since M is arbitrarily large, this gives a contradiction.
Proposition 2.16.The set W I is open.
Proof.Let P ∈ W I .We want to find V ⊂ W I neighborhood of P .Since which is open there is a neighborhood U of P which is compactly contained in Ω ∩ (I × I).Since W I is absorbing for Ω under F there exists N > 0 such that (2.20) As usual let us define P j := F j (P ); by definition of W I , P j ⊂ I × I for all j ≥ 0, which is an open set.Hence for each j ≥ 0 there is a neighborhood U j ⊂ I × I of P j .So up to making the U j smaller, we can assume that U j ⊂ F j (U ). Let The set V is open since it is a finite intersection of open sets.We only need to check that V ⊂ W I , or equivalently, that F j (V ) ⊂ I × I for all j ≥ 0. For j ≤ N − 1, this is true by definition, since F j (V ) ⊂ U j ⊂ I × I.For j ≥ N , this is true by (2.20).Since P ∈ V by construction, V is a neighborhood of P in W I as required.

Geometric structure of Ω
In this section we show that Ω is the union of four disjoint Fatou components Ω ab , a, b ∈ {+, −}, each of which is biholomorphic to H × H.
We first show conjugacy of F to its linear part on Ω, and estimate the distance between the conjugacy and the identity map.
Proof.We first show that F is conjugate to L on W I .
For n ∈ N let ϕ n : C 2 → C 2 be the automorphisms defined as If we show that the ϕ n converge to a map ϕ : C 2 → C 2 uniformly on W I we obtain that ϕ satisfy the functional equation ϕ = L −1 • ϕ • 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 (2.22)Both have the same formal limit ϕ(z, w) = z + ∆ 1 (z, w), w + ∆ 2 (z, w) .
If P = (z, w) ∈ W I , then F n (P ) = (z n , w n ) ⊂ I ×I ⊂ S for all j, hence, by (2.9), we have that ∆(z, w) < 1; in particular, ∆ 1 (z, w) and ∆ 2 (z, w) are convergent.Hence ϕ is a holomorphic map from W I to ϕ(W I ) (W I is open by Proposition 2.16).Moreover, It follows that ϕ is open because W I is an unbounded set, hence if ϕ had rank 0 or 1, (ϕ − Id) could not be bounded on W I .Hence the map ϕ is injective by Hurwitz Theorem (see [Kra01], Exercise 3 on page 310) because the maps ϕ n are injective and their limit has rank 2. It follows that ϕ is a biholomorphism between W I and ϕ(W I ).
To extend ϕ to all of Ω recall that W I is absorbing for Ω.So if P ∈ Ω, we have that F k (P ) ∈ W I for some k ∈ N, hence we can define ϕ(P ) = L −k • ϕ • F k (P ).Since F is an automorphism, ϕ extends as a biholomorphism to Ω.
It remains to show that ϕ(Ω) ⊆ S. By (2.23) we have that ϕ(W I ) is contained in a √ 2 neighborhood U of W I .Suppose by contradiction that there exists Q = (z, w) ∈ ϕ(W I ) \ S. Since W I is forward invariant under F and ϕ is a conjugacy we have that ϕ(W I ) is forward invariant under L. Up to considering L(Q) if necessary, and since θ is such that re iθ / ∈ S,we can assume that z = re iθ / ∈ S. By forward invariance Since (−δ) 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 ⊂ S, contradicting ϕ(W I ) ⊂ U .Hence ϕ(W I ) ⊂ S. Since W I is an absorbing set for Ω under F , ϕ • F = L • ϕ, and ϕ(W I ) is completely invariant under L, we have that (2.24) We are now able to understand the geometric structure of Ω.
Proposition 2.18.Ω consists of four distinct connected components, each of which is biholomorphic to H × H, and which form a cycle of period 4.
We recall the following simple topological lemma.Here ∂ denotes the topological boundary.
Recall also that if a set A is invariant under a map F , by continuity of the latter we have F (A) ⊂ A. The following lemma is also known.
Lemma 2.20.Let Ω 1 , Ω 2 be two Fatou components for an automorphism F of C 2 .Then if Proof.We have that F (Ω 1 ) ⊂ Ω 2 , indeed otherwise, F (Ω 1 ) would intersect the boundary of Ω 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 ∈ Ω 2 \ F (Ω 1 ).Then since F (Ω 1 ) ∩ Ω 2 = ∅ and both Ω 1 , F (Ω 1 ) are connected there exists Q ∈ Ω 2 ∩ F (∂Ω 1 ), which is impossible because ∂Ω 1 is contained in the forward Julia set which is forward invariant.
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 Ω is biholomorphic to S. Since S has four connected components S ab each of which is biholomorphic to H × H, the same holds for Ω.Since by definition Ω had at most four connected components Ω ab with a, b ∈ {+, −}, these are exactly the connected components of Ω.
Recall the definition of the set W ⊂ S from Section 2.2 and recall that it is forward invariant.We first show that ϕ(W ab ) ∩ S ab = ∅ for every a, b ∈ {+, −}; (2.25) Indeed, by construction every W ab contains points (z, w) with |z|, |w| > M > √ 2. For such points, by (2.23) we have that ϕ(z, w) − (z, w) < √ 2, but since W ab ⊂ S ab by Proposition 2.17, the Euclidean distance of (z, w) from S ã, b is larger than M if (a, b) = (ã, b) (indeed, if you take a point in one sector, no point in a different sector is closer than the origin).
We now show that ϕ(Ω) = S.By (2.24) we know that ϕ(Ω) ⊂ S. Notice that S can be written as S = {(r 1 e iθ 1 , r 2 e iθ 2 ) ∈ C 2 : r 1 , r 2 > 0, and for each i = 1, 2 either Hence by backward invariance of ϕ(Ω) under L we have that It remains to show that the Fatou components Ω ab with a, b ∈ {+, −} form a cycle of period four, more precisely, that (2.26) By definition Ω ab ⊃ W ab and by Lemma 2.2,

Limit sets on Ω ab
Let H and −H denote the right and left half plane respectively.In this section we show that Proposition 2.21 (Limit set for Ω).We have that Let W be as defined in Section 2.2 and W I as defined in Section 2.
Before proving Lemma 2.23 let us see how Lemmas 2.22 and 2.23 imply Proposition 2.21.
We now consider limit sets for Ω ++ and Ω −− ; the other cases are analogous.By (2.26), It follows that for any n > 0, In view of this, and since W I is absorbing for Ω, we have that for any P ∈ (Ω ++ ∪ Ω −− ) for any n large enough. Hence 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 [BSZ23]; 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 ∂ denotes the topological boundary, and dist spher denotes the spherical distance.Notice that the assumptions imply that F, G have generic rank 1: They cannot have rank 2 because the target is Ĉ, and G cannot be constant otherwise there could not be c ∈ G(B) with positive distance from G(∂B).One can check that F cannot be constant either.
Proof.Let D be a horizontal disk passing through a point P c ∈ G −1 (c) ∩ B, such that ∂D ⊂ ∂B.Let g := G| D , f := F D .They are holomorphic in a slightly larger horizontal disk.Notice that dist spher (g, f ) < ε on ∂D, and that dist spher (G(∂D), c) ≥ ε because ∂D ⊂ ∂B.By Rouché's Theorem in one variable, c ∈ f (D) ⊂ F (B) as required.
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 ∈ 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 C-convex set instead of a polydisk, and can certainly be generalized further.
Proof of Lemma 2.23.We show H ⊂ 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 , dividing the numerator and the denominator by w 0 and using 1 1+x = ∞ j=0 (−x) j for |x| < 1 we have that Let K ⊂ Ĉ be a compact set and suppose that z 0 w 0 takes values in K.By (2.27) and using (2.9), for any ε > 0 there exists (2.28) Consider the function G(z, w) := z w Observe that Let c ∈ H.By the shape of W we have that G(W ++ ) = H, that ε := 1 2 dist spher (c, G(∂W )) > 0, and that we can choose Q = (z 0 , w 0 ) ∈ W ++ ∈ G −1 (c) such that |w 0 | is arbitrarily large.By taking a limit in n in equation (2.27) and on a sufficiently small polydisk centered at Q we can ensure that dist spher (h 1 , G) < ε, hence the claim follows by Rouché's Theorem.
The main Theorem is a direct consequence of Propositions 2.6, 2.17, 2.18, 2.21.

Appendix: Proof of Proposition 2.13
We split the proof of Proposition 2.13 over several lemmas.Definition 3.1.Let E ⊂ C n .A vector v ∈ C n is called tangent to E at a point P ∈ E if there exist a sequence of points P J ∈ E and real numbers t j > 0 such that P i → P and t j (P j − P ) → v as j → ∞.The set of all such tangent vectors is the tangent cone to E at P .The tangent cone is indeed a cone in C n = T P C n .If the set E is a C 1 -smooth manifold, the tangent cone coincides with the tangent space.
For complex analytic sets of dimension one, the following is a well known fact.For a proof, see [Chi89], Corollary on page 80. Lemma 3.2.Let H ⊂ C n be an analytic set of dimension one.For all x ∈ H the tangent cone of H at x consists of a finite union of complex lines (whose number is not greater than the number of irreducible components of H at x). Definition 3.3.Let B ⊂ C n be a polidisc.The torus T with same center and same poliradius as B is called its Šilov boundary.We will denote it by ∂ S B.
The Šilov boundary is a very general notion, for Banach algebras, but we will not need it here in all generality.For details, we refer to [DSSST06], from page 325.Lemma 3.5.Let H ⊂ C 2 be an analytic set of dimension one.Then for every P ∈ H there exists an arbitrarily small torus T P centered in P such that T P ∩ H = ∅.
Proof.Let P ∈ H. Consider the tangent cone C P of H at P .By Lemma 3.2, C P is a finite set of directions α 1 , . . ., α k ∈ Ĉ.Up to a rotation, we can suppose all directions to be in C. Up to choosing η > 0 small enough, we can ensure that the polidisk B η of poliradius η centered in P intersects only one connected component of H.Moreover, by the definition of tangent cone, we can choose a small neighbourhood K ⊂ C of all α j such that H ∩ U η ⊂ ∪ α∈K (P + (z, αz)) .
We can suppose K to be small enough that there is 0 < b < 1 such that K ∩ {β ∈ C | |β| = b} = ∅.Proof of Proposition 2.13.Let K and L be compacts sets as in the statement.For each P ∈ H ∩ K, by Lemma 3.5 there exists a torus T P ⊂ L centered in P such that T P ∩ H = ∅.Each torus T P is the Šilov boundary of a polidisk B P centered in P .Since {B P } P ∈H∩K is a covering of H ∩ K, by compactness we can extract a finite covering {B 1 , . . ., B k }.
There is a η-neighbourhood U η of H such that U η ∩ K ⊂ ∪B j .If the harmonic function u satisfies u ≤ α on L \ U η then it satisfies the same estimate on all tori T j , and by Lemma 3.4 the same estimate holds on all B j .Hence u ≤ α on K.
.8) Proof.By hypothesis, F (P ) ∈ S hence F (P ) ∈ S ãb for some ã, b ∈ {+, −}.Since Re w 1 = Re z 0 we have that b = a.Moreover Re z 1 = −δ Re w 0 + Re(e −z 2 0 ) and since P ∈ S, | Re(e −z 2 0 )| < 1 by Lemma 2.1.Hence the sign of Re z 1 is opposite to the sign of Re w 0 provided | Re w 0 | > 1 δ , and ã = −b as required.Assume from now on that F n (P ) ∈ S for all n ∈ N. It follows that z n ∈ S for all n ∈ N and hence by Lemma 2.1 |f Proposition 2.5 (Invariance of W ). The set W is open and W ⊂ S. For any n ∈ N we have that F (W n ) ⊂ W n+1 , hence W is forward invariant.The set W consists of four connected components W ab with a, b ∈ {+, −} and F (W ab ) ⊂ W (−b)a .
ab I = ∅ for all a, b ∈ {+, −}.For every a, b ∈ {+, −}, F (W ab I ) ⊂ W (−b)a I .The sets W ++ I ∪ W −− I , W −+ I ∪ W +− I are both forward invariant under F 2 .Moreover F 2n and F 2n+1 are convergent on W I .Proof.Each W ab I contains the set {(z, w) ∈ C 2 : a Re z > M, b Re w > M, Im z = Im w = 0} for M sufficiently large.The set W I ⊂ S is forward invariant hence Corollary 2.3 applies.Convergence of even and odd iterates follow by (2.9.)It will turn out that W I is open as well (Proposition 2.16).

Lemma 2. 15 (
Mean value property for holomorphic disks).Let D ⊂ C be the open unit disk and ϕ : D → Ω be a holomorphic map.Let u be harmonic on the holomorphic open disk D = ϕ(D) and continuous up to the boundary of D. Let P 0 := ϕ(0).Then u(P 0 ) = 1 2π ∂D u(ζ)|ϕ ′ (ζ)| −1 dζ Proof.Consider the function u • ϕ : D → R. First, note that it is harmonic on D and continuous up to the boundary.Indeed if u :
Lemma 3.4.Let B be a polydisk, ∂ S B be its Šilov boundary, and u : U → R be a harmonic function defined on a neighbourhood U of B. Thenmax B u = max ∂ S B u.Recall that ∂ denotes the topological boundary.Proof.For every P ∈ ∂B \ ∂ S B there is a horizontal Euclidean disc D through P which is contained in ∂B whose boundary is in ∂ S B. Being u harmonic in a neighbourhood of B, u is harmonic on such a closed disc, hence its value at P is less or equal to the maximum of u at its boundary ∂D ⊂ ∂ S B. Hence max ∂B u = max ∂ S B u .If P ∈ B, we can find a disc through P with boundary in ∂B and repeat the argument, getting the conclusion max B