Dynamics of transcendental Hénon maps
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Abstract
The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist socalled Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental maps to consider. In this paper we find inspiration from the extensive work on the dynamics of complex Hénon maps. We introduce the family of transcendental Hénon maps, and study their dynamics, emphasizing the description of Fatou components. We prove that the classification of the recurrent invariant Fatou components is similar to that of polynomial Hénon maps, and we give examples of Baker domains and wandering domains.
Mathematics Subject Classification
32H50 37F50 37F101 Introduction
The main reason for considering transcendental Hénon maps and not arbitrary entire maps in \({\mathbb {C}}^2\) is that the space of entire maps is too large. Even the class of polynomials maps in two complex variables is often considered too diverse to study the dynamics of these maps all at the same time. On the other side, the family of polynomial automorphisms of \({\mathbb {C}}^2\) has received a large amount of attention. It portrays a wide variety of dynamical behavior, yet it turns out that this class of maps is homogeneous enough to describe its dynamical behavior in detail. A result of Friedland and Milnor [22] implies that any polynomial automorphism with nontrivial dynamical behavior is conjugate to a finite composition of polynomial Hénon maps. It turns out that finite compositions of polynomial Hénon maps behave in many regards similarly to single Hénon maps, and the family of Hénon maps is sufficiently rigid to allow a thorough study of its dynamical behavior.
Very little is known about the dynamics of holomorphic automorphisms of \({\mathbb {C}}^2\), although there have been results showing holomorphic automorphisms of \({\mathbb {C}}^2\) with interesting dynamical behavior, such as the construction of oscillating wandering domains by Sibony and the third named author [20], and a result of Vivas, Wold and the last author [28] showing that a generic volume preserving automorphisms of \({\mathbb {C}}^2\) has a hyperbolic fixed point with a stable manifold which is dense in \(\mathbb {C}^2\). Transcendental Hénon maps seems to be a natural class of holomorphic automorphisms of \({\mathbb {C}}^2\) with nontrivial dynamics, restrictive enough to allow for a clear description of its dynamics, but large enough to display interesting dynamical behaviour which does not appear in the polynomial Hénon case.
We classify in Sect. 4 the invariant recurrent components of the Fatou set of a transcendental Hénon map, that is, components which admits an orbit accumulating to an interior point. Invariant recurrent components have been described for polynomial Hénon maps in [5]; our classification holds not only for transcendental Hénon maps but also for the larger class of holomorphic automorphisms with constant Jacobian. Moreover, using the fact that f is a transcendental holomorphic function, we obtain in Sect. 3 results about periodic points and invariant algebraic curves. We show that the set \(\mathrm{Fix}(F^2)\) is discrete, and (if \(\delta \ne 1\)) that F admits infinitely many saddle points of period 1 or 2, which implies that the Julia set is not empty. We also show that there is no irreducible invariant algebraic curve (the same was proved by Bedford–Smillie for polynomial Hénon maps in [4]). The dynamical behavior can be restricted even further by considering transcendental Hénon maps whose map f has a given order of growth. For example, if the order of growth is smaller than \(\frac{1}{2}\), then \(\mathrm{Fix}(F^k)\) is discrete for all \(k\ge 1\).
We then give examples of Baker domains, escaping wandering domains, and oscillating wandering domains. Such Fatou components appear in transcendental dynamics in \(\mathbb {C}\), and for trivial reasons they cannot occur for polynomials. The existence of the filtration gives a similar obstruction for polynomial Hénon maps, but this filtration is lost when considering transcendental Hénon maps.
For a transcendental function a Baker domain is a periodic Fatou component on which the orbits converge locally uniformly to the point \(\infty \), which is an essential singularity [7]. We give an example in Sect. 5 of a transcendental Hénon map with a twodimensional analogue: a Fatou component on which the orbits converge to a point at the line at infinity \(\ell ^\infty \), which is (in an appropriate sense) an essential singularity. In one complex variable for any Baker domain there exists an absorbing domain, equivalent to a half plane \({\mathbb {H}}\), on which the dynamics is conjugate to an affine function, and the conjugacy extends as a semiconjugacy to the entire Baker domain. In our example the domain is equivalent to \({\mathbb {H}} \times {\mathbb {C}}\), and the dynamics is conjugate to an affine map.
The final part of the paper is devoted to wandering domains. Recall that wandering domains are known not to exist for onedimensional polynomials and rational maps [30], but they do arise for transcendental maps (see for example [7]). In higher dimensions it is known that wandering domains can occur for holomorphic automorphisms of \(\mathbb {C}^2\) [20] and for polynomial maps [1], but whether polynomial Hénon maps can have wandering domains remains an open question. We will consider two types of wandering Fatou components, each with known analogues in the onedimensional setting. We construct in Sect. 6 a wandering domain, biholomorphic to \({\mathbb {C}}^2\), which is escaping: all orbits converge to the point [1 : 1 : 0] at infinity. The construction is again very similar to that in one dimension. However, the proof that the domain and its forward images are actually different Fatou components is not the proof usually given in one dimension. Instead of finding explicit sets separating one component from another, we give an argument that uses exponential expansion near the boundary of each of the domains.
Finally, we construct in Sect. 7 a transcendental Hénon map F with a wandering domain \(\Omega \), biholomorphic to \(\mathbb {C}^2\), which is oscillating, that is it contains points whose orbits have both bounded subsequences and subsequences which converge to infinity. Up to a linear change of variable, the map F is the limit as \(k\rightarrow \infty \) of automorphisms of \(\mathbb {C}^2\) of the form \(F_k(z,w):=(f_k(z)+\frac{1}{2}w,\frac{1}{2}z)\), all having a hyperbolic fixed point at the origin. The family \((F_k)\) is constructed inductively using Runge approximation in one variable to obtain an entire function \(f_{k+1}\) which is sufficiently close to \(f_k\) on larger and larger disks, in such a way that the orbit of an open set \(U_0\subset \mathbb {C}^2\) approaches the origin coming in along the stable manifold of \(F_k\) and then goes outwards along the unstable manifold of \(F_k\), over and over for all \(k\in \mathbb {N}\).
Regarding the complex structure of those Fatou components, in both the Baker domain and the oscillating wandering domain case one encounters the same difficulty. Namely, in both cases one finds a suitable invariant domain A of the Fatou component on which it is possible to construct, using the dynamics of F, a biholomorphism to a model space (\(\mathbb {H}\times \mathbb {C}\) and \(\mathbb {C}^2\) respectively, where \(\mathbb {H}\) denotes the right halfplane). One then needs to prove that the domain A is in fact the whole Fatou component, and this is done by using the following plurisubharmonic method: If A is strictly smaller than \(\Omega \) then we can construct a plurisubharmonic function \(u:\Omega \rightarrow \mathbb {R}\cup \{\infty \}\) for which the submean value property is violated at points in \(\partial A\cap \Omega \). We note that a somewhat similar argument was given by the third author in [18], and we believe that this method can be applied in a variety of similar circumstances.
It is important to point out that for an entire map \(F:\mathbb {C}^2\rightarrow \mathbb {C}^2\) there are two natural definitions of the Fatou set, which correspond to compactifying \(\mathbb {C}^2\) either with the onepoint compactification \({\widehat{\mathbb {C}^2}}\), or with \(\mathbb {P}^2\). In one dimension the two Fatou sets coincide, and the same is true for polynomial Hénon maps, since by the existence of the filtration all forward orbits that converge to infinity converge to the same point on the line at infinity \(\ell ^\infty = {\mathbb {P}}^2 \setminus {\mathbb {C}}^2\). For a general entire selfmap of \(\mathbb {C}^2\) these two definitions can give two different Fatou sets (see Example 2.6). Notice that, if we compactify with \({\widehat{\mathbb {C}^2}}\), any open subset of \(\mathbb {C}^2\) on which the sequence of iterates \(F^n\) diverges uniformly on compact subsets would be in the Fatou set regardless of how the orbits go to infinity. This seems to be too weak a definition in two complex variables. We thus define the Fatou set compactifying \(\mathbb {C}^2\) with \(\mathbb {P}^2\) (which has the additional advantage of being a complex manifold). Section 2 is devoted to this argument.
2 The definition of the Fatou set
Let \(n\in \mathbb {N}, { n\ge 1}\) and let X be a complex manifold. There are (at least) two natural definitions of what it means for a family \(\mathcal {F}\subset \mathrm{Hol}(X,\mathbb {C}^n)\) to be normal. We denote by \({\widehat{\mathbb {C}^n}}\) the onepoint compactification of \(\mathbb {C}^n\), and with the symbol \(\infty \) we denote both the point at infinity and the constant map \(z\mapsto \infty \).
Definition 2.1
A family \(\mathcal {F}\subset \mathrm{Hol}(X,\mathbb {C}^n)\) is \(\mathbb {P}^n\)normal if for every sequence \((f_n)\in \mathcal {F}\) there exists a subsequence \((f_{n_k})\) converging uniformly on compact subsets to \(f\in \mathrm{Hol}(X,\mathbb {P}^n)\). In other words, \(\mathcal {F}\) is relatively compact in \(\mathrm{Hol}(X,\mathbb {P}^n)\).
A family \(\mathcal {F}\subset \mathrm{Hol}(X,\mathbb {C}^n)\) is \(\widehat{\mathbb {C}^n}\)normal if for every sequence \((f_n)\in \mathcal {F}\) which is not divergent on compact subsets there exists a subsequence \((f_{n_k})\) converging uniformly on compact subsets to \(f\in \mathrm{Hol}(X,\mathbb {C}^n)\). This is equivalent to \(\mathcal {F}\) being relatively compact in \(\mathrm{Hol}(X,\mathbb {C}^n)\cup \infty \subset C^0(X,\widehat{\mathbb {C}^n})\).
Remark 2.2
When \(n=1\) the two definitions are equivalent.
A family \(\mathcal {F}\subset \mathrm{Hol}(X,\mathbb {C}^n)\) is \(\mathbb {P}^n\)normal if and only if it is equicontinuous with respect to the FubiniStudy distance on \(\mathbb {P}^n\). This follows from the Ascoli–Arzelà theorem and from the fact that \(\mathrm{Hol}(X,\mathbb {P}^n)\) is closed in \(C^0(X,\mathbb {P}^n)\). One may think that, similarly, a family \(\mathcal {F}\subset \mathrm{Hol}(X,\mathbb {C}^n)\) is \(\widehat{\mathbb {C}^n}\)normal if and only if it is equicontinuous with respect to the spherical distance \(d_{\widehat{\mathbb {C}^n}}\) on \({\widehat{\mathbb {C}^n}}\), but this is not the case, as the following example shows.
Example 2.3
For \(n\ge 2\), the family \(\mathrm{Hol}(\mathbb {D},\mathbb {C}^n)\cup \infty \) is not closed in \(C^0(\mathbb {D},\widehat{\mathbb {C}^n})\). As a consequence, for a family \(\mathcal {F}\subset \mathrm{Hol}(\mathbb {D},\mathbb {C}^n)\), being relatively compact in \(C^0(\mathbb {D},\widehat{\mathbb {C}^n})\) is not equivalent to being \(\widehat{\mathbb {C}^n}\)normal.
Proof
Lemma 2.4
If a family \(\mathcal {F}\subset \mathrm{Hol}(X,\mathbb {C}^n)\) is \(\mathbb {P}^n\)normal, then it is \(\widehat{\mathbb {C}^n}\)normal.
Proof
Let \((f_n)\) be a sequence in \(\mathcal {F}\). Since \(\mathcal {F}\) is \(\mathbb {P}^n\)normal there exists a subsequence \((f_{n_k})\) converging uniformly on compact subsets to a map \(f\in \mathrm{Hol}(X,\mathbb {P}^n).\) If there is a point \(x\in X\) such that \(f(x)\in \ell ^\infty \), then \(f(X)\subset \ell ^\infty \). Indeed, it suffices to show that \(f^{1}(\ell ^\infty )\) is open, and this follows taking an affine chart around \(f(y)\in \ell ^\infty \) in such a way that \(\ell ^\infty =\{z_1=0\}\) and applying Hurwitz theorem to the sequence \(\pi _1\circ f_n\).
Thus, if the sequence \((f_n)\) is not diverging on compact subsets, the subsequence \((f_{n_k})\) converges uniformly on compact subsets to a map \(f\in \mathrm{Hol}(X,\mathbb {C}^n).\)\(\square \)
As a consequence of the previous discussion, for an entire map \(F:\mathbb {C}^n\rightarrow \mathbb {C}^n\) we have two possible definitions of the Fatou set.
Definition 2.5
A point \(z\in \mathbb {C}^n\) belongs to the \(\widehat{\mathbb {C}^n}\)Fatou set if the family of iterates \((F^n)\) is \(\widehat{\mathbb {C}^n}\)normal near z. A point \(z\in \mathbb {C}^n\) belongs to the \(\mathbb {P}^n\)Fatou set if the family of iterates \((F^n)\) is \(\mathbb {P}^n\)normal near z.
For a polynomial Hénon map, it follows from the existence of the invariant filtration that any forward orbit that converges to infinity must converge to the point \([1:0:0] \in \ell ^\infty \). Thus, the two definitions of Fatou set coincide. By Lemma 2.4 the \(\mathbb {P}^n\)Fatou set is always contained in the \(\widehat{\mathbb {C}^n}\)Fatou set. If \(n>1\) the inclusion may be strict as the following example shows.
Example 2.6
On the other hand, if the sequence \(N_j\) increases sufficiently fast, then for \(1 < z_0 \le 2\) we have that \(F^n(z_0, w_0) \rightarrow [0:1:0] \in \ell ^\infty \), again uniformly on compact subsets. It follows that \(\mathbb {D}\times {\mathbb {C}}\) is a \(\mathbb {P}^2\)Fatou component. Therefore in this example the single \(\widehat{\mathbb {C}^2}\)Fatou component contains infinitely many distinct \(\mathbb {P}^2\)Fatou components.
In what follows, we will only consider \(\mathbb {P}^2\)normality. We will call the \(\mathbb {P}^2\)Fatou set simply the Fatou set. The Julia set is the complement of the Fatou set.
3 Invariant subsets
3.1 Periodic points
If g is a transcendental function or a polynomial Hénon map, then, for each \(k \ge 1\), the set \(\mathrm{Fix}(g^k)\) is discrete. Clearly this statement is not satisfied for holomorphic automorphisms of \({\mathbb {C}}^2\). For example, one can consider any holomorphic conjugate of a rational rotation.
Lemma 3.1
If F is a transcendental Hénon map, then \(\mathrm{Fix}(F)\) and \(\mathrm{Fix}(F^2)\) are discrete.
Proof
The fixed points (z, w) of F satisfy \(z = w\) and thus \(z = f(z)\delta z\). Since f is not linear the set of solutions is discrete.
Without making further assumptions it is not clear to the authors that \(\mathrm{Fix}(F^k)\) is discrete when \(k\ge 3\). However, we can show discreteness when we assume that the function f has small order of growth.
Proposition 3.2
Let F be a transcendental Hénon maps such that f has order of growth strictly less than \(\frac{1}{2}\). Then \(\mathrm{Fix}(F^k)\) is discrete for all \(k\ge 1\).
Proof
Suppose for the purpose of a contradiction that the solution set in \({\mathbb {C}}^k\) of the system (2) is not discrete. Then there exists an unbounded connected component V. Let \(n \in {\mathbb {N}}\) be such that V intersects the polydisk \(D(0,r_n)^k\). Then V also intersects the boundary \(\partial D(0,r_n)^k\), say in a point \((z_0, \ldots , z_{k1})\). By the symmetry of the equations in (2) we may then well assume that \(z_0 = r_n\), and of course that \(z_j \le r_n\) for \(j= 1, \ldots , k1\).
By Wiman’s Theorem we may assume that \(g(z_0)\) is arbitrarily large, and in particular that \(f(z_0) > (1+\delta ) r_n\). But this contradicts the first equation in (2), completing the proof. \(\square \)
We now turn to the question whether transcendental Hénon maps always have periodic points.
Definition 3.3
For a selfmap h and for all \(n\ge 1\) we denote by \(\mathrm{Per}_n(h)\) the set of periodic points of h of minimal period n.
Recall that if f is an entire transcendental function, then \(\mathrm{Per}_2(f)\) has infinite cardinality by [6].
Proposition 3.4
If \(F(z,w)=(f(z)\delta w,z)\) is a transcendental Hénon map, then \( \mathrm{Fix}(F)\ne \varnothing \) unless \(f(z)z(\delta +1)=e^{h(z)}\) for some holomorphic function h(z). The set \(\mathrm{Per}_2(F)\) has infinite cardinality if \(\delta \ne 1\), and if \(\delta =1\) the set \(\mathrm{Per}_2(F)\) is not empty if and only if the set \(\{f=0\}\) contains at least two points.
Proof
Let \(Z:=\{fz(\delta +1)=0\}\subset \mathbb {C}\). Then \( \mathrm{Fix}(F)=\{(z,z)\in \mathbb {C}^2:z\in Z\}.\)
Remark 3.5
Notice that the set Z has finite cardinality if and only if \( f(z) z(\delta +1)= p(z) e^{h(z)}\), where p is a nonzero polynomial p and h is entire function. Thus in all other cases the sets \(\mathrm{Fix}(F)\) and \(\mathrm{Per}_2(F)\) have infinite cardinality.
If \(f:\mathbb {C}\rightarrow \mathbb {C}\) is an entire transcendental function, we have additional information on the multiplier of repelling periodic points of period \(n\ge 2\). Indeed we have the following theorem [8, Theorem 1.2]:
Theorem 3.6
Corollary 3.7
(Nonempty Julia set) If \(\delta \ne 1\), then there exist infinitely many saddle points in \({ \mathrm{Per}_2(F)}\), and thus the Julia set of F is nonempty.
Proof
3.2 Invariant algebraic curves
It follows from a result of Bedford–Smillie [4] that a polynomial Hénon map does not have any invariant algebraic curve. Indeed, given any algebraic curve, the normalized currents of integration on the pushforwards of this curve converge to the (1, 1) current \(\mu ^\), whose support does not lie on an algebraic curve.
This type of argument is not available for transcendental dynamics. Here we present a different argument.
Theorem 3.8
For the general case \(\{H = 0\}\), where we may now assume that we are not dealing with a graph, we will use the following two elementary estimates.
Lemma 3.9
Proof
As we have already shown that \(\{H = 0\}\) is not a graph, it follows that \(\{H = 0\}\) intersects all but finitely many lines \(\{z = c\}\). The result follows from the assumption that f is transcendental. \(\square \)
 (1)
\(H(z,w) = p(w)z^{N_1} + \sum _{k=0}^{N_11}\sum _{\ell =0}^{N_2}\alpha _{k,\ell }z^kw^\ell \).
 (2)
\(H(z,w) = q_0(z)+\sum _{\ell =1}^n q_\ell (z)w^\ell \).
Lemma 3.10
There exist d large enough so that if \(H(z,w)=0\) for z sufficiently large, then \(w<z^d.\)
Proof
If \(w>z^{d}\) for arbitrarily large z and d, then \(w^nq_n(z)\) dominates the other terms in the form (2), so H(z, w) cannot vanish. \(\square \)
Proof of Theorem 3.8
By Lemma 3.9 there exist \((z_j, w_j)\) with \(z_j \rightarrow \infty \), \(H(z_j,w_j)=0\) and \(f(z_j) > z_j^j\). Let \((z_j^\prime , w_j^\prime ) = F(z_j, w_j)\) so that \(z^\prime _j = f(z_j)\delta w_j\) and \(w^\prime _j=z_j\). Since \(\{H=0\}\) is invariant we have that \(H(z^\prime _j,w_j^\prime )=0\).
4 Classification of recurrent components
In this section we only assume that F is a holomorphic automorphism of \({\mathbb {C}}^2\) with constant Jacobian \(\delta \).
Definition 4.1
A point \(x\in \mathbb {C}^2\) is recurrent if its orbit \((F^n(x))\) accumulates at x itself. A periodic Fatou component \(\Omega \) is called recurrent if there exists a point \(z \in \Omega \) whose orbit \((F^n(z))\) accumulates at a point \(w \in \Omega \).
Since the class of holomorphic automorphism of \({\mathbb {C}}^2\) with constant Jacobian is closed under composition, by replacing F with an iterate we can restrict to the case where \(\Omega \) is invariant. For an invariant Fatou component \(\Omega \), a limit maph is a holomorphic function \(h:\Omega \rightarrow \mathbb {P}^2\) such that \(f^{n_k}\rightarrow h\) uniformly on compact sets of \(\Omega \) for some subsequence \(n_k\rightarrow \infty \).
Theorem 4.2

If \(\mathrm{dim}\,\Sigma =0\), then \(\Omega \) is the basin of an attracting fixed point, and is biholomorphically equivalent to \({\mathbb {C}}^2\).
 If \(\mathrm{dim}\,\Sigma =1\), either \(\Sigma \) is biholomorphic to a circular domain A, and there exists a biholomorphism from \(\Omega \) to \(A\times \mathbb {C}\) which conjugates the map F towhere \(\theta \) is irrational, or there exists \(j\in \mathbb {N}\) such that \(F^j_\Sigma =\mathrm{id}_\Sigma \), and there exists a biholomorphism from \(\Omega \) to \(\Sigma \times \mathbb {C}\) which conjugates the map \(F^j\) to$$\begin{aligned} (z,w)\mapsto \left( e^{i\theta }z, \frac{\delta }{e^{i\theta }} w\right) , \end{aligned}$$$$\begin{aligned} (z,w)\mapsto (z, \delta ^j w). \end{aligned}$$

\(\mathrm{dim}\,\Sigma =2\) if and only if \(\delta  = 1\). In this case there exists a sequence of iterates converging to the identity on \(\Omega \).
By a circular domain we mean either the disk, the punctured disk, an annulus, the complex plane or the punctured plane. For the polynomial Hénon maps case, see [5] and [19].
Let \((F^{n_k})\) be a convergent subsequence of iterates on \(\Omega \), with \(F^{n_k}(z) \rightarrow w\in \Omega \). We denote the limit of \((F^{n_k})\) by g.
Lemma 4.3
The image \(g(\Omega )\) is contained in \({\mathbb {C}}^2.\)
Proof
If there is a point \(x\in \Omega \) for which g(x) belongs to the line at infinity \(\ell ^\infty \), then \(g(\Omega )\subset \ell ^\infty \) (see e.g. the proof of Lemma 2.4), which gives a contradiction. \(\square \)
Definition 4.4
We define the maximal rank of g as \(\max _{p\in \Omega }\mathrm{rk}( d_pg)\).
4.1 Maximal rank 0
Lemma 4.5
Suppose that g has maximal rank 0. Then \(g(\Omega )\) is the single point w, which is an attracting fixed point.
Proof
Since the maximal rank is 0, the map g is constant and must therefore equal w. Since F and g commute, the point w must be fixed. Suppose that the differential \(d_wF\) has an eigenvalue of absolute value \(\ge 1.\) Then the same is true for all iterates \(F^{n_k}\). Hence they cannot converge to a constant map. So w must be an attracting fixed point. \(\square \)
It follows that \(\Omega \) is the attracting basin of the point w, and the entire sequence \(F^n\) converges to g. In this case the limit manifold \(\Sigma \) is the point \(\{w\}\).
4.2 Maximal rank 2
Theorem 4.6
Suppose that g has maximal rank 2. Then there exists a subsequence \((m_k)\) so that \(F^{m_k}\rightarrow \text{ Id }\) on \(\Omega .\)
Proof
Let x be a point of maximal rank 2. There exist an open neighborhood U of x and an open neighborhood \(V'\) of g(x) such that \(g:U\rightarrow V'\) is a biholomorphism. Denote \(h:=g^{1}\) defined on \(V'\). Let \(V\subset \subset V'\) be an open neighborhood of g(x). Since \(F^{n_k}\rightarrow g\) on U, we have that \(V\subset F^{n_k}(U)\) for large k and the maps \((F^{n_k})^{1}\) converge to h uniformly on compact subsets of V. In particular \(V\subset \Omega \). Replace \(n_k\) by a subsequence so that \(n_{k+1}n_k \nearrow \infty .\) We can then write \(F^{n_{k+1}n_k}=F^{n_{k+1}}\circ (F^{n_k})^{1}\) on V. If we set \(m_k:=n_{k+1}n_k\), then \(F^{m_k}\rightarrow \mathrm {Id}\) on V. Since we are in the Fatou set this implies that \(F^{m_k}\rightarrow \mathrm {Id}\) on \(\Omega .\)\(\square \)
It follows that every point \(p\in \Omega \) is recurrent and that F is volume preserving. The following fact is trivial but we recall it for convenience.
Lemma 4.7
Let \((G_n:\Omega \subset \mathbb {C}^2\rightarrow \mathbb {C}^2)\) be a sequence of injective holomorphic mappings which are volume preserving. If \(G_n\) converges to G uniformly on compact subsets, then G is holomorphic, injective and volume preserving.
Proof
The map G is holomorphic and \(dG_n{\mathop {\longrightarrow }\limits ^{n\rightarrow \infty }}dG\), and thus G is volume preserving. Thus by Hurwitz Theorem G is injective. \(\square \)
Corollary 4.8
Suppose that g has maximal rank 2. Then if h is any limit map of F on \(\Omega \), then either h is injective and \(h(\Omega )\subset \Omega \) or \(h(\Omega )\subset \ell ^\infty \).
Proof
Assume that \(h(\Omega )\subset \mathbb {C}^2\). Then by Lemma 4.7 the map h is holomorphic and injective. Arguing as in the proof of Theorem 4.6 we get that \(h(\Omega )\subset \Omega \). \(\square \)
Proposition 4.9
If g has maximal rank 2 then each orbit \((F^n(z))\) is contained in a compact subset of \(\Omega \).
Proof
Let \((K_j)\) be an exhaustion of \(\Omega \) by compact subsets such that \(K_j \subset {\mathop {K}\limits ^{\circ }}_{j+1} \) for all \(j\in \mathbb {N}\). By passing to a subsequence of the exhaustion if needed, we may assume that \(F(K_j) \subset \subset K_{j+1}\). Let \(p\in \Omega \). We can assume that \(p\in K_1\), and let \(r>0\) be such that \(B(p,r)\subset K_1\).
 (i)
Each \(F^{k_j}(p)\) lies in \(K_4 \setminus K_3\)
 (ii)
For \(k_j< n < l_j\) the points \(F^n(p)\) lie outside of \(K_3\).
 (iii)
Each \(F^{l_j}(p)\) lies outside \(K_{n_j}\),
 (iv)
Each point \(F^{m_j}(p)\) lies in \(K_1\),
 (v)
For \(k_j< n < m_j\) the points \(F^n(p)\) lie in \(K_{n_{j+1}2}\).
Corollary 4.10
The limit of any convergent subsequence \((F^{n_k})\) is an automorphism of \(\Omega \).
Proof
By Corollary 4.8 and Proposition 4.9 the limit h is a biholomorphic map \(h:\Omega \rightarrow h(\Omega )\subset \Omega \). Suppose that \(p\in \Omega \setminus h(\Omega ).\) Let K be a compact subset of \(\Omega .\) Then for all large enough k, \(p\in \Omega \setminus F^{n_k}(K).\) Hence \(F^{n_k}(p) \in \Omega \setminus K.\) Fix such k. By Theorem 4.6 there exists \(m>n_k\) so that \(F^m\) is close enough to the identity so that \(F^{mn_k}(p) \in \Omega \setminus K.\) This contradicts Proposition 4.9. \(\square \)
In this case the limit manifold \(\Sigma \) is the whole \(\Omega \).
Remark 4.11
It follows that the maximal rank of a limit map is independent of the chosen convergent subsequence.
4.3 Maximal rank 1
We now consider the case where the limit map g has maximal rank 1. By Remark 4.11 every other limit of a convergent subsequence on \(\Omega \) must also have maximal rank 1.
Recall that \((F^{n_k})\) is a convergent subsequence of iterates on \(\Omega \) such that \(F^{n_k}(z) \rightarrow w\in \Omega \). Replace \(n_k\) by a subsequence so that \(n_{k+1}n_k \nearrow \infty .\) Let \((m_k)\) be a subsequence of \((n_{k+1}n_k)\) so that \(F^{m_k}\) converges, uniformly on compact subsets of \(\Omega \). From now on we assume that g is the limit of the sequence \((m_k)\).
Remark 4.12
Notice that \(g(w)=w\). Actually, if follows by the construction that there exists an open neighborhood N of z in \(\Omega \) such that \(g(N)\subset \Omega \) and for all \(y\in g(N)\), \(g(y)=y\).
Lemma 4.13
The map F is strictly volume decreasing.
Proof
By assumption the Jacobian determinant \(\delta \) of F is constant. Since \(\Omega \) is recurrent we have \(\delta  \le 1\). Since \((F^{n_k})\) converges to the map \(g: \Omega \rightarrow {\mathbb {C}}^2\) of maximal rank 1, it follows that \(\delta  < 1\). \(\square \)
We write \(\Sigma :=g(\Omega )\). Notice that \(\Sigma \) is a subset of \(\overline{\Omega }\cap {\mathbb {C}}^2\), and that, since F and g commute, the map \(F_\Sigma :\Sigma \rightarrow \Sigma \) is bijective. We need a Lemma.
Lemma 4.14
Let U be an open set in \(\mathbb {C}^2\), and let \(h:U\rightarrow \mathbb {C}^2\) be a holomorphic map of maximal rank 1. Then for all \(w\in h(U)\), the fiber \(h^{1}(w)\) has no isolated points.
Proof
Assume by contradiction that \(q\in h^{1}(w)\) is isolated. If \(\epsilon \) is small enough, then \(h(\partial B(q,\epsilon ))\) is disjoint from w. Hence there exists a small ball \(B(w,\delta )\) which is disjoint from \(h(\partial B(q,\epsilon ))\), and if we restrict h to \(V:=h^{1}(B(w,\delta ))\cap B(q,\epsilon )\), then \(h:V\rightarrow B(w,\delta )\) is a proper holomorphic map. Let \(\zeta \in V\) be such that \(\mathrm{rk}_\zeta h=1\). Then the level set \(h^{1}(h(\zeta ))\) contains a closed analytic curve in V. Such curve is not relatively compact in V, and this contradicts properness. \(\square \)
It is not clear a priori that \(\Sigma \) is a complex submanifold, but we will show, following a classical normalization procedure, that there exists a smooth Riemann surface \({\widehat{\Sigma }}\) such that the selfmap F on \(\Sigma \) can be lifted to a holomorphic automorphism \({\widehat{F}}\) on \({\widehat{\Sigma }}\). Note that such normalization procedure was used in a similar context in [31].
Lemma 4.15
 (1)
\(\gamma _z(\Delta _z)\) is an irreducible local complex analytic curve which is smooth except possibly at \(\gamma _z(0)\) where it could have a cusp singularity,
 (2)
\(\gamma _z(\Delta _z)= g(U(z))\).
Proof
If g has rank 1 at z, the result follows immediately from the constant rank Theorem. So suppose that g has rank 0 at z, and let \(\Delta _z\subset \Omega \) be an affine disk through z on which g is not constant. By the Puiseux expansion of \(g: \Delta _z \rightarrow {\mathbb {C}}^2\), it follows that, up to taking a smaller \(\Delta _z\), \(g(\Delta _z)\) is an irreducible local complex analytic curve (with possibly an isolated cusp singularity at g(z)). Hence \(g(\Delta _z)\) is the zero set of a holomorphic function G defined in a open neighborhood V of g(z). Let U(z) be the connected component of \(g^{1}(V)\) containing z. We claim that \(G\circ g\) vanishes identically on U(z), which implies that \(g(U(z))=g(\Delta _z)\). If not, then \((G\circ g)^{1}(0)\) is a closed complex analytic curve in U(z) containing \(\Delta _z\). Pick a point \(q\in \Delta _z\) where locally \((G\circ g)^{1}(0)=\Delta _z\). Then \(g^{1}(g(q))\) is isolated at q since g is not constant on \(\Delta _z\), which gives a contradiction by Lemma 4.14.
Finally, again by the Puiseux expansion of \(g: \Delta _z \rightarrow {\mathbb {C}}^2\), there exists a holomorphic injective map \(\gamma _z:\Delta _z\rightarrow \mathbb {C}^2\) such that \(\gamma _z(\Delta _z)=g(\Delta _z)\). \(\square \)
Remark 4.16
For all \(z\in \Omega \), there exists a unique surjective holomorphic map \(h_z:U(z)\rightarrow \Delta _z\) such that \(g =\gamma _z\circ h_z\) on the neighborhood U(z). If g has rank 1 at z, then \(h_z_{\Delta _z}=\mathrm{id}\).
Consider the disjoint union \(\coprod _{z\in \Omega } \Delta _z\), and define an equivalence relation in the following way: \((x,z)\simeq (y, w)\) if and only if \(\gamma _z(x)=\gamma _w(y)\) and the images coincide locally near this point. Define \({\widehat{\Sigma }}\) as \({ \coprod _{z\in \Omega } \Delta _z/_{\simeq }}\), endowed with the quotient topology, and denote \(\pi _{\simeq }:\coprod _{z\in \Omega } \Delta _z\rightarrow {\widehat{\Sigma }}\) the projection to the quotient. It is easy to see that the map \(\pi _{\simeq }\) is open. For all \(z\in \Omega \), define a homeomorphism \(\pi _z:\Delta _z\rightarrow {\widehat{\Sigma }}\) as \( \pi _z(x):=[(x,z)]\).
Definition 4.17
We define a continuous map \(\gamma :{\widehat{\Sigma }}\rightarrow \mathbb {C}^2\) such that \(\gamma ({\widehat{\Sigma }})= \Sigma \) in the following way: \(\gamma ([(x,z)])=\gamma _z(x).\) Notice that this is well defined. The map \(g: \Omega \rightarrow \mathbb {C}^2\) can be lifted to a unique surjective continuous map \(\widehat{g}: \Omega \rightarrow \widehat{\Sigma }\) such that \(g=\gamma \circ {\widehat{g}}.\) Such map is defined on U(z) as \({\widehat{g}}:= \pi _z \circ h_z.\) Notice that if g has rank 1 at z then \({\widehat{g}}_{\Delta _z}=\pi _z\).
Lemma 4.18
The topological space \({\widehat{\Sigma }}\) is connected, second countable and Hausdorff.
Proof
Since \({\widehat{\Sigma }}=\hat{g}(\Omega )\), and \(\Omega \) is connected, it follows that \({\widehat{\Sigma }}\) is connected. Since \({\widehat{g}}\) is open, it follows also that \({\widehat{\Sigma }}\) is second countable. Let \([(x,z)]\ne [(y,w)]\in {\widehat{\Sigma }}\). Then we have two cases. Either \(\gamma _z(x)\ne \gamma _w(y)\), or \(\gamma _z(x)= \gamma _w(y)\) but the images do not coincide locally near this point. In both cases there exist a neighborhood \(U\subset \Delta _z\) of x and a neighborhood \(V\subset \Delta _w\) of y such that \(\pi _{\simeq }(U)\cap \pi _{\simeq }(V)=\varnothing .\)\(\square \)
Remark 4.19
With the complex structure just defined on \({\widehat{\Sigma }}\), the maps \(\gamma \) and \({\widehat{g}}\) are holomorphic.
Definition 4.20
Define \(R\subset {\widehat{\Sigma }}\) as the set of points \(\zeta \in {\widehat{\Sigma }}\) such that there exists \(z\in \Omega \) with \({\widehat{g}}(z)=\zeta \) and \(\mathrm{rk}_z{\widehat{g}}=1\).
Lemma 4.21
The set \({\widehat{\Sigma }}\setminus R\) is discrete.
Proof
Let \(w\in \Omega \) such that \(\mathrm{rk}_w{\widehat{g}}=0\). By the identity principle there exists a neighborhood V of w in \(\Delta _w\) such that \(\mathrm{rk}_z(h_w_{\Delta _w})=1\) for all \(z\in V\setminus \{w\}\). The result follows since \({\widehat{g}}= \pi _w\circ h_w\) on U(w), and \(\pi _w:\Delta _w\rightarrow \pi _w(\Delta _w)\) is a biholomorphism. \(\square \)
Lemma 4.22
Proof
Let \(\zeta \in R\), and let \(z\in \Omega \) such that \({\widehat{g}}(z)=\zeta \) and \(\mathrm{rk}_z{\widehat{g}}=1\). Define on \(\pi _z(\Delta _z)\) the map \({\widehat{F}}:= {\widehat{g}} \circ F \circ \pi _z^{1}\). This is welldefined and holomorphic away from the discrete closed set \({\widehat{\Sigma }}\setminus R\), and can be extended holomorphically to the whole \({\widehat{\Sigma }}\). \(\square \)
The inverse of \(\widehat{F}\) is given by \(\widehat{F^{1}}\), therefore \(\widehat{F}\) is an automorphism.
Lemma 4.23
The Riemann surface \({\widehat{\Sigma }}\) contains an open subset on which the sequence \((\widehat{F}^{m_k})\) converges to the identity.
Proof
Lemma 4.24
Either there exists a \(j \in {\mathbb {N}}\) for which \(\widehat{F}^j = \mathrm {Id}\), or \(\widehat{\Sigma }\) is biholomorphic to a circular domain, and the action of \(\widehat{F}\) is conjugate to an irrational rotation.
Recall that by a circular domain we mean either the disk, the punctured disk, an annulus, the complex plane or the punctured plane.
Proof
Assume that \(\widehat{F}^j \ne \mathrm {Id}\) for all \(j\le 1\). Since the holomorphic map \(\gamma :{\widehat{\Sigma }}\rightarrow \mathbb {C}^2\) is nonconstant, it follows that the Riemann surface \({\widehat{\Sigma }}\) is not compact. Thus if \({\widehat{\Sigma }}\) is not a hyperbolic Riemann surface, then it has to be biholomorphic either to \(\mathbb {C}\) or to \(\mathbb {C}^*\). In both cases, since \({\widehat{F}}\) is an automorphism, it is easy to see that Lemma 4.23 implies that \({\widehat{F}}\) is conjugate to an irrational rotation. If \({\widehat{\Sigma }}\) is hyperbolic, then the family \(({\widehat{F}}^n)\) is normal, and thus Lemma 4.23 implies that the sequence \(({\widehat{F}}^{m_k} )\) converges to the identity uniformly on compact subsets of \({\widehat{\Sigma }}\). Thus the automorphism group of \(\widehat{\Sigma }\) is nondiscrete. Hence (see e.g. [16, p. 294]) the Riemann surface \(\widehat{\Sigma }\) is biholomorphic to a circular domain and the action of \(\widehat{F}\) is conjugate to an irrational rotation. \(\square \)
Definition 4.25
Define the set \(I\subset {\widehat{\Sigma }}\times {\widehat{\Sigma }} \) as the set of pairs (x, y) such that \(x\ne y\) and \(\gamma (x)=\gamma (y)\). Define the set \(C\subset {\widehat{\Sigma }}\) as the set of x such that \(\gamma :{\widehat{\Sigma }}\rightarrow \mathbb {C}^2\) has rank 0 at x.
Since the map \(\pi _\simeq \) is open, it follows immediately that the set I is discrete in \({\widehat{\Sigma }}\times {\widehat{\Sigma }}\) and that the set C is discrete in \({\widehat{\Sigma }}\).
Our goal is to prove that the set \(\Sigma \) is a closed complex submanifold of \(\Omega \). We will first consider the case where \(\widehat{\Sigma }\) is biholomorphic to a circular domain and \(\widehat{F}\) is conjugate to an irrational rotation.
Lemma 4.26
If \({\widehat{F}}\) is conjugate to an irrational rotation then there is at most one element \(\zeta _0 \in C\). The set I is empty, and thus the map \(\gamma :{\widehat{\Sigma }}\rightarrow \Sigma \) is injective.
Proof
The set C is invariant by \({\widehat{F}}\). Since the action of \(\widehat{F}\) is conjugate to an irrational rotation, and since C is discrete it follows that C can only contain the center of rotation \(\zeta _0\) (if there is one).
Similarly, the set I is invariant by the map \((x,y)\mapsto ({\widehat{F}}(x),{\widehat{F}}(y)),\) but this contradicts the discreteness of I. \(\square \)
Lemma 4.27
The set \(C\subset {\widehat{\Sigma }}\) is empty.
Proof
If \({\widehat{\Sigma }}\) has no center of rotation, then there is nothing to prove. Suppose for the purpose of a contradiction that there is a center of rotation \(\zeta _0\in {\widehat{\Sigma }}\) and that \(\zeta _0\in C\).
Then \(\Sigma \) has a cusp at \(z:=\gamma (\zeta _0)\). Notice that \(F(z)=z\). Since \({\widehat{F}}\) acts on \(\widehat{\Sigma }\) as a rotation, it follows that the tangent direction of \(\Sigma \) to z is an eigenvector of \(d_zF\) with eigenvalue \(\lambda _1=1\). Since F is strictly volume decreasing, the other eigenvalue \(\lambda _2\) of \(d_zF\) satisfies \(0<\lambda _2<1\). Thus we obtain a forward invariant cone in \(T_z({\mathbb {C}}^2)\), centered at the line \(T_z(\Sigma )\). Extending this cone to a constant cone field in a neighborhood of z, it follows that we obtain stable manifolds through z and all nearby points in \(\Sigma \), giving a continuous Riemann surface foliation near the point z, see the reference [23] for background on normal hyperbolicity.
Since \({\widehat{F}}\) acts on \({\widehat{\Sigma }}\) as a rotation, the stable manifolds through different points in \(\Sigma \) must be distinct. However, this is not possible in a neighborhood of z. To see this, let h be a locally defined holomorphic function such that \(\Sigma \) equals the zero set of h near z. We may assume that h vanishes to higher order only at z. Now consider the restriction of h to the stable manifold through z. Then h has a multiple zero at z, hence by Rouché’s Theorem, the number of zeroes for nearby stable manifolds is also greater than one. But since nearby stable manifolds are transverse to \(\Sigma \), and h does not vanish to higher order in nearby points, it follows that the restriction of h to nearby stable manifolds must have multiple single zeroes. Hence these nearby stable manifolds intersect \(\Sigma \) in more than one point, giving a contradiction. \(\square \)
Let \({ \widehat{V}\subset \subset {\widehat{\Sigma }}}\) be open and invariant under the action of \(\widehat{F}\), and let V be its image in \({\mathbb {C}}^2\), which is an embedded complex submanifold of \(\mathbb {C}^2\).
Lemma 4.28
Proof
Lemma 4.29
The image \(\Sigma \) is contained in \(\Omega \).
Proof
Since F is \(C^1\), we can extend the invariant cone field to a neighborhood \(\mathcal {N}(V )\). Let z be a point whose forward orbit remains in \(\mathcal {N}(V )\), which holds in particular for all points in V. Then there exists a stable manifold \(W^s(z)\), transverse to V, and these stable manifolds fill up a neighborhood of V.
The forward iterates of F form a normal family on this neighborhood, which implies that V is contained in the Fatou set. \(\square \)
Corollary 4.30
The map \(g:\Omega \rightarrow \Sigma \) is a holomorphic retraction. In particular \(\Sigma \) is a closed smooth onedimensional embedded submanifold.
Proof
In the case where \(\widehat{F}^j\) equals the identity, it follows that \(F^j\) equals the identity on \(\Sigma \). The next lemma, together with the fact that the Jacobian \(\delta \) of F satisfies \(\delta <1\), shows that there cannot be cusps and selfintersections in this case.
Lemma 4.31
If a local biholomorphic map G of a neighborhood of the origin in \({\mathbb {C}}^2\) is the identity on a complex analytic curve \(\gamma \) with any kind of singularity at the origin, then \(d_0G\) is the identity.
Proof
The curve \(\gamma \) will be the zero set of a holomorphic function \(\phi \) which vanishes generically to first order on the curve, but vanishes to at least order 2 at the origin. Then we write \(G(z,w)=(z+A(z,w)\phi , w+B(z,w)\phi ),\) which gives the result. \(\square \)
We immediately get stable manifolds transverse to \(\Sigma \), which imply as above that \(g:\Omega \rightarrow \Sigma \) is a holomorphic retraction.
Lemma 4.32
The retraction g has constant rank 1 on \(\Omega \).
Proof
Remark 4.33
The stable manifolds of the points in \(\Sigma \) fill up a neighborhood of \(\Sigma \). Since all orbits in the Fatou component \(\Omega \) get close to \(\Sigma \), this implies that the stable manifolds of the points in \(\Sigma \) fill up the whole \(\Omega \). Thus for any limit map \(h:\Omega \rightarrow \mathbb {P}^2\) we have that \(h(\Omega )\subset \Sigma .\) Moreover, by Lemma 4.24, the restriction \(h_{\Sigma }\) is an automorphism of \(\Sigma \), and thus \(h(\Omega )=\Sigma \). Notice that for all \(z\in \Sigma \) the fiber \(g^{1}(z)\) coincides with the stable manifold \(W^s(z)\). Hence \(h=h_\Sigma \circ g.\)
Remark 4.34
In the view of the contents of this section if a Fatou component \(\Omega \) is recurrent then for every \(z\in \Omega \) the orbit is relatively compact in \(\Omega \).
Now we investigate the complex structure of \(\Omega \). Notice that \(\Sigma \), being an open Riemann surface, is Stein.
Proposition 4.35
If some iterate \(F^j_\Sigma \) is the identity, then there exists a biholomorphism \(\Psi :\Omega \rightarrow \Sigma \times \mathbb {C}\) which conjugates the map \(F^j\) to \((z,w)\mapsto (z, \delta ^j w).\)
Proof
Let L be the holomorphic line bundle on \(\Sigma \) given by \(\mathrm{Ker}\, dg\) restricted to \(\Sigma \). Since \(\Sigma \) is Stein, there exists a neighborhood U of \(\Sigma \) and an injective holomorphic map \(h:U\rightarrow L\) such that for all \(z\in \Sigma \) we have that \(h(z)=0_z\), and h maps the fiber \(g^{1}(z)\cap U\) biholomorphically into a neighborhood of \(0_z\) in the fiber \(L_z\) [21, Proposition 3.3.2]. We can assume that \(d_z h_{L_z}=\mathsf{id}_{L_z}\). Notice that the map \({ d_zF^j}_{L_z}:L_z\rightarrow L_z\) acts as multiplication by \(\delta ^j\). The sequence \(( dF_L^{nj} \circ h\circ F^{nj})\) is eventually defined on compact subsets of \(\Omega \) and converges uniformly on compact subsets to a biholomorphism \(\Psi :\Omega \rightarrow L\), conjugating \(F^j\) to \(dF^j_L:L\rightarrow L\). The line bundle L is holomorphically trivial since \(\Sigma \) is a Stein Riemann surface. \(\square \)
4.4 Transcendental Hénon maps
So far in this section we have not used that the maps we study here are of the form \(F(z,w) = (f(z)\delta w, z)\). Absent of any further assumptions we do not know how to use this special form to obtain a more precise description of recurrent Fatou components. We do however have the following consequence of Proposition 3.2 and Theorem 4.2.
Corollary 4.36
Let F be a transcendental Hénon map, and assume that f has order of growth strictly smaller than \(\frac{1}{2}\). Then any recurrent periodic Fatou component \(\Omega \) of rank 1 must be the attracting basin of a Riemann surface \(\Sigma \subset \Omega \) which is biholomorphic to a circular domain, and f acts on \(\Sigma \) as an irrational rotation.
5 Baker domain
For holomorphic automorphisms of \(\mathbb {C}^2\) the definition of essential singularity at infinity has to be adapted in order to be meaningful.
Definition 5.1
Let F be a holomorphic automorphism of \(\mathbb {C}^2\). We call a point \([p:q:0]\in \ell ^\infty \) an essential singularity at infinity if for all \([s:t:0] \in \ell ^\infty \) there exists a sequence \((z_n,w_n)\) of points in \(\mathbb {C}^2\) converging to [p : q : 0] whose images \(F(z_n,w_n)\) converge to [s : t : 0].
Remark 5.2
Remark 5.3
 (1)
\(z\mapsto \lambda z \in \mathrm{Aut}(\mathbb {H})\), where \(\lambda >1\),
 (2)
\( z\mapsto z\pm i \in \mathrm{Aut}(\mathbb {H})\),
 (3)
\(z\mapsto z+1 \in \mathrm{Aut}(\mathbb {C})\).
Lemma 5.4
Each domain \(R_{\alpha }\) is forward invariant.
Proof
We immediately obtain the following.
Corollary 5.5
Remark 5.6
Lemma 5.7
All the orbits in R converge uniformly on compact subsets to the point [1 : 1 : 0] on the line at infinity.
Proof
Definition 5.8
Lemma 5.9
Proof
Lemma 5.10
Proof
Lemma 5.11
On \(U\setminus A\), \( \limsup _{n\rightarrow +\infty } u_n = 0\).
Proof
Proposition 5.12
The set R is an absorbing domain.
Proof
Remark 5.13
Definition 5.14
We denote by \(\Omega \) the domain \(\{(z,w)\in \mathbb {C}^2: \mathrm {Re}\,({z} {w})>0\}\), which is biholomorphic to \(\mathbb {H}\times \mathbb {C}\).
Theorem 5.15
There exists a biholomorphism \(\psi :U\rightarrow \Omega \) which conjugates F to the map L. In particular U is biholomorphic to \(\mathbb {H}\times \mathbb {C}\).
Proof
Let \(n \ge N\) be such that \(\Vert \psi  \mathrm {Id}\Vert \le \frac{r}{2}\) on \({\overline{B}}((x_n, y_n),r)\). By Rouché’s Theorem in several complex variables it follows that \((x_n, y_n) \in \psi ({\overline{B}}((x_n, y_n),r)) \subset \psi (U)\). Since \(\Omega \) is completely invariant under L and U is completely invariant under F, it follow that \((x_0, y_0) \in \psi (U)\). \(\square \)
6 Escaping wandering domain
Definition 6.1
 (1)
is escaping if all orbits converge to the line at infinity,
 (2)
is oscillating if there exists an unbounded orbit and an orbit with a bounded subsequence,
 (3)
is orbitally bounded if every orbit is bounded.
For polynomials in \(\mathbb {C}\) it is known that wandering domains cannot exist [30]. For transcendental functions there are examples of escaping wandering domains. [7] uses for example the function \(f(z)=z+\lambda \sin (2\pi z)+1\) for suitable \(\lambda \). There are also examples of oscillating wandering domains [10, 15], and it is an open question whether orbitally bounded wandering domains can exist.
It follows from the existence of the filtration that a polynomial Hénon map does not admit any escaping or oscillating wandering domain. In the remainder of the paper we will give examples of both escaping and oscillating wandering domains for transcendental Hénon maps. The existence of orbitally bounded wandering domains is an open question for both polynomial and transcendental Hénon maps.
For simplicity of notation we consider the map f obtained by conjugating \(\tilde{f}\) with a real translation by \(\alpha \), so that the critical points \(c_{n,+}\) for \(\tilde{f} \) are mapped to critical points \(z_n=n\) for f. Thus, f will act on \({\mathbb {Z}}\) as a translation by 1, and f commutes with translation by 1.
Consider the function \(g(z) := f(z)  1\), which commutes with f and with the translation by 1. For the function g each point \(z_n\) is a superattracting fixed point. For all \(n\in \mathbb {Z}\) denote by \(B_n\) the immediate basin for g of the point \(z_n\). Clearly the basins \(B_n\) are disjoint, \(B_{n+1}=B_n+1\), and \(f(B_n)\subset B_{n+1}\). If one shows that each \(B_n\) is also a Fatou component for f, then clearly each \(B_n\) is an escaping wandering domain for f.
There are two classical ways in one dimensional complex dynamics to show that each \(z_n\) belongs to a different Fatou component for such a function f. One is by constructing curves in the Julia set that separate the points \(z_n\) [12, p. 183], and the other one is by showing that any two commuting maps which differ by a translation have the same Julia set [2, Lemma 4.5]. The first kind of argument is typically unavailable in higher dimensions. The proof we present is an ad hoc version of the second argument. We will define transcendental Hénon maps \(F,G:\mathbb {C}^2\rightarrow \mathbb {C}^2\) which behave similarly to f, g, then use G to show that the norms of the differentials \(dG^n\) and \(dF^n\) at points on the boundaries of the attracting basins for G grow exponentially in n, and finally use the latter information to imply that those boundaries must be contained in the Julia set for F. Since the attracting basins are disjoint for G, the corresponding sets are disjoint for F, giving a sequence of wandering domains.
Lemma 6.2
Proof
Theorem 6.3
Proof
Notice that \((T^k(Q))\) is an Forbit converging to the point [1 : 1 : 0] on the line at infinity. Since \(F^n=T^n\circ G^n\) for all \(n\in \mathbb {N}\), it follows that the Forbits of all points in A converge to [1 : 1 : 0]. Hence A is contained in a Fatou component \(\Omega \) of F. We show that \(\Omega \) equals A by proving that all boundary points of A are contained in the Julia set of F.
7 Oscillating wandering domain
We show in this section the existence of a transcendental Hénon map with an oscillating wandering domain biholomorphic to \(\mathbb {C}^2\).
Proposition 7.1
 (i)
\(\Vert F_{k}F_{k1}\Vert _{D(0,R_{k1})\times \mathbb {C}}\le \epsilon _{k}\) for all \(k\ge 1\);
 (ii)
\(F_{k}(P_n)=P_{n+1}\) for all \(0 \le n < n_{k}\);
 (iii)
\(F_k(B(P_n,\beta _n))\subset \subset B(P_{n+1},\beta _{n+1})\) for all \(0 \le n<n_k\), where \(\beta _n\le \frac{\theta _{\rho }}{2}\) for all \(0\le n\le n_k\);
 (iv)
\(z_n<R_{k} \theta _{\rho }\) for all \(0\le n< n_k\), and \(z_{n_{k}}> R_{k} +5 \theta _k\);
 (v)
\(P_{j}\in B(0,\frac{1}{k})\) for some j with \(n_{k1}\le j\le n_{k}\), for all \(k\ge 1\).
Before proving this proposition, let us show that it implies the existence of a wandering Fatou component. By (i), the maps \(F_k\) converge uniformly on compact subsets to a holomorphic automorphism \(F(z,w)=(f(z)+aw,az)\). By (ii), (iv) and (v) it follows that \((P_n)\) is an oscillating orbit for F, that is, it is unbounded and it admits a subsequence converging to the origin. Since by (iii) for all \(j\in \mathbb {N}\) the iterate images of \(B(P_j,\beta _j)\) have uniformly bounded Euclidean diameter (in fact, the diameter goes to zero), each ball \(B(P_j,\beta _j)\) is contained in the Fatou set of F.
Lemma 7.2
If \(i\ne j\), then \(P_i\) and \(P_j\) are in different Fatou components of F, and hence F has an oscillating wandering domain.
Proof
For all \(j\in \mathbb {N}\), denote \(\Omega _j\) the Fatou component containing \(B(P_j,\beta _j)\). Since \(\beta _n\rightarrow 0\) as \(n\rightarrow \infty \), by (iii) and by identity principle it follows that all limit functions on each \(\Omega _j\) are constants.
Assume \(j > i\) and let \(k = ji\). Let \(\mathcal {N}(0)\) be a neighborhood of the origin that contains no periodic points of order less than or equal to k. Since the orbit \((P_n)\) enters and leaves a compact subset of \(\mathcal {N}(0)\) infinitely often, there must be a subsequence \((n_j)\) for which \(P_{n_j}\) converges to a point \(z \in \mathcal {N}(0) \setminus \{0\}\). But then \(P_{n_j + k}\) converges to \(F^k(z) \ne z\), which implies that \(P_i\) and \(P_j\) cannot be contained in the same Fatou component. \(\square \)
We will prove Proposition 7.1 by induction on k. We start the induction by letting \(R_0:=1\), \(\theta _0:=1\), \(\beta _0:=\frac{1}{2}\) and \(P_0:=(z_0,w_0)\) with \(z_0 > 6\). We set \(F_0(z,w):= (bz+aw,az)\) and \(n_0:=0\).
Let us now suppose that (i)(v) hold for certain k, and let us proceed to define \(F_{k+1}\) and the points \((P_n)_{n_{k}< n\le n_{k+1}}\). This is done in two steps. The first step relies upon the classical Lambda Lemma 7.3:
Lemma 7.3
(Lambda Lemma, see e.g. [27]) Let G be a holomorphic automorphism of \({\mathbb {C}}^2\) with a saddle fixed point at the origin. Denote by \(W^s(0)\) and \(W^u(0)\) the stable and unstable manifolds respectively. Let \(p \in W^s(0)\setminus \{0\}\) and \(q \in W^u(0)\setminus \{0\}\), and let D(p) and D(q) be holomorphic disks through p and q, respectively transverse to \(W^s(0)\) and \(W^u(0)\). Let \(\epsilon >0\). Then there exists \(N\in \mathbb {N}\) and \(N_1>2N+1\), and a point \(x \in D(p)\) with \(G^{N_1}(x) \in D(q)\) such that \(\Vert G^n(x)  G^n(p)\Vert < \epsilon \) and \(\Vert G^{N_1n}(x)  G^{n}(q)\Vert < \epsilon \) for \(0\le n \le N\), and \(\Vert G^n(x)\Vert < \epsilon \) when \(N< n < N_1  N\).
Using the Lambda Lemma we find a finite \(F_k\)orbit \((Q_j)_{0\le j\le M}\), the new oscillation, which goes inward along the stable manifold of \(F_k\), reaching the ball \(B(0,\frac{1}{k+1})\), and then outwards along the unstable manifold of \(F_k\).
The second step of the proof relies upon another classical result:
Lemma 7.4
Using Runge approximation we find a map \(F_{k+1}\) connecting the previously constructed finite orbit \((P_n)_{0\le n\le n_{k}}\) with the new oscillation \((Q_j)_{0\le j\le M}\) via a finite orbit along which \(F_{k+1}\) is sufficiently contracting. The contraction neutralizes possible expansion along the new oscillation \((Q_j)_{0\le j\le M}\), and we refer to this connecting orbit as the contracting detour.
7.1 Finding the new oscillation
Lemma 7.5
Proof
Consider arbitrarily small disks \(D_1\) centered at \((u_0, v_0)\) and \(D_2\) centered at \((s_0, t_0)\) transverse to the stable respectively the unstable manifold. It follows from the Lambda Lemma 7.3 that there exists a finite \(F_k\)orbit \((Q_j):=(z_j',w_j')_{0\le j\le M}\) which intersects the ball \(B(0,\frac{1}{k+1})\) with \(Q_0\in D_1\) and \(Q_{M}\in D_2\). If all the perturbations are chosen small enough, then the sequence \((z'_j)_{0\le j\le M}\) avoids the disk \({\overline{D}}(z_{n_k},\theta _{k})\) and the point \(\frac{w'_0}{a}\). Setting \(\theta _{k+1}>0\) small enough completes the proof (Fig. 2). \(\square \)
7.2 Connecting the orbits via the contracting detour
Lemma 7.6
 (1)
\(z''_0=z_{n_k}, z''_{N}=\frac{w_0'}{a},\)
 (2)
\(z''_j>z_{n_k}+2\) for all \(1\le j\le N1\),
 (3)
\(z''_jz''_i>2\), for all \(0\le i\ne j\le N1\).
 (1)
h coincides with \(f_k\) on K,
 (2)
\(h_{{\overline{D}}(z''_j,\theta _k)}\) is constantly equal to \(z''_{j+1}aw''_j\) for all \(0\le j< N\),
 (3)
\(h_{{\overline{D}}(w'_0/a,\theta _{k+1})}\) is constantly equal to \( z_0'aw''_N,\)
 (4)
\(h_{{\overline{D}}(z'_{M},\theta _{k+1})}\) is constantly equal to some value \(A>R_{k+1}+5\theta _{k+1}.\)
 (1)
\(f_{k+1}(0)=h(0)=0, f_{k+1}'(0)=h'(0)=b,\)
 (2)
\(f_{k+1}(z_j)= h(z_j)\) for all \(0\le j< n_k\),
 (3)
\(f_{k+1}(z'_j)= h(z'_j)\) for all \(0\le j\le M\),
 (4)
\(f_{k+1}(z''_j)= h(z''_j)\) for all \(0\le j\le N\),
 (5)
\(\Vert f_{k+1}h\Vert _{K\cup H}<\epsilon _{k+1}.\)
 (1)
\(D(z''_j,\frac{\theta _k}{2})\subset \subset D(z''_j,\theta _k) \) for all \(0\le j< N\),
 (2)
\({\overline{D}}(w'_0/a,\frac{\theta _{k+1}}{2})\subset \subset {\overline{D}}(w'_0/a,\theta _{k+1})\),
 (3)
\({\overline{D}}(z'_{M},\frac{\theta _{k+1}}{2})\subset \subset {\overline{D}}(z'_{M},\theta _{k+1})\).
7.3 Complex structure
We will now prove that, by making the contracting detours sufficiently long, the oscillating wandering domains can be guaranteed to be biholomorphically equivalent to \({\mathbb {C}}^2\). We denote the Fatou component containing \(P_0\) by \(\Omega \).
Lemma 7.7
We can guarantee that the calibrated basin \(\Omega _{(P_n), (\beta _n)} \) is biholomorphic to \({\mathbb {C}}^2\).
Proof
While we cannot guarantee that equation (22) with \(a'^2<a''\) holds outside of the contracting detours, by making the contracting detours sufficiently long we can still ensure that \(\Phi _n\) converges uniformly on compact subsets to a biholomorphism \(\Phi :{\tilde{\Omega }}\rightarrow \mathbb {C}^2\), compare for example Theorem 1.3 of [29]. \(\square \)
Lemma 7.8
We can guarantee that the functions \(g_n\) converge to \(\log a\) on \(\Omega _{(P_n), (\beta _n)}\setminus \{P_0\}\).
Proof
Take sufficiently many contractions that are sufficiently close to multiplication by a. \(\square \)
Lemma 7.9
We can guarantee that \(g_{j_k} \rightarrow 0\) on \(\Omega \setminus \Omega _{(P_n), (\beta _n)}\), uniformly on compact subsets.
Proof
Proposition 7.10
With the previous choices, the wandering Fatou component \(\Omega \) equals the calibrated basin \(\Omega _{(P_n), (\theta _n)}\), and is thus biholomorphically equivalent to \({\mathbb {C}}^2\).
Proof
Suppose by contradiction that \(\Omega \ne \Omega _{(P_n), (\theta _n)}\). Let h be the uppersemicontinuous regularization of \(\lim _{k\rightarrow \infty } g_{n_k}.\) Clearly \(h\equiv \log a\) on \(\Omega _{(P_n), (\theta _n)}\) and \(h\equiv 0\) on \(\Omega \setminus \Omega _{(P_n), (\theta _n)}\). Then by [25, Prop 2.9.17] the function h is plurisubharmonic, and the submean value property at any \(\zeta \in \partial \Omega _{(P_n), (\theta _n)}\) gives a contradiction. \(\square \)
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