Abstract
We construct automorphisms of \({{\mathbb {C}}}^2\) of constant Jacobian 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. In the literature there are currently very few examples of automorphisms of \({{\mathbb {C}}}^2\) with rank one limit sets on the boundary of Fatou components. To our knowledge, this is the first example in which such limit sets are hyperbolic, and moreover different limit sets of rank 1 coexist.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Transcendental Hénon maps are automorphisms of \({{\mathbb {C}}}^2\) with constant Jacobian of the form
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
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
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
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,
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
For \(n\in {{\mathbb {N}}}\) define the following holomorphic functions from \({{\mathbb {C}}}^2\) to \(\hat{{{\mathbb {C}}}}\)
With this notation the iterates of F take the form
Let
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.
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
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
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
Fix \(\lambda >0\) and assume also that \(|{\text {Re}}z_0|,|{\text {Re}}w_0|>\frac{1+\lambda }{\delta -1}\). Then
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
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
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.)
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
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
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
and define
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
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
Let \(\lambda _n:=R_{n+1}-R_{n-1}\). Since \(P\in S\), by (2.10) we have that
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
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
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
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:
hence
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
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
If A is absorbing for \(\Omega \), then \(\Omega =\bigcup _{n}F^{-n}(A)\).
Fix \(C\ge 1\) and let
Notice that if \(z\in I\), then \(|{\text {Re}}z|> C\).
Define
Proposition 2.9
We have that \(W_I^{ab}\ne \varnothing \) for all \(a,b\in \{+,-\}\). For every \(a,b\in \{+,-\}\),
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,
Let
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
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 \((z_0,w_0)\in K\)
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
Then
-
(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)
\(u_n\rightarrow -\infty \) uniformly on compact subsets of W;
-
(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
Proof
Let \(z=re^{i\theta }\) satisfying (2.17); then \(|\tan \theta |\le k<1\). Hence since \(z^2=r^2e^{2i\theta }\),
\(\square \)
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
we have
Proof of Lemma 2.11
-
(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)
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)
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
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
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
\(\square \)
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
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
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
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
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
Both have the same formal limit
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,
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
\(\square \)
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
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
We now show that this implies that \(\varphi (\Omega )\supset S\). Notice that S can be written as
Fix \(\alpha <\frac{\pi }{4}\). By definition of W there exists \(R=R(\alpha )\) such that \(W{\setminus } U\) contains the set
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
It follows that
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
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
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
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
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),
It follows that for any \(n>0\),
In view of this, and since \(W_I\) is absorbing for \(\Omega \), we have that for any \(P\in (\Omega ^{++}\cup \Omega ^{--} )\)
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
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
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
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
Consider the function \(G(z,w):=\frac{z}{w}\) Observe that
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.
References
Arosio, L., Benini, A.M., Fornæss, J.E., Peters, H.: Dynamics of transcendental Hénon maps. Math. Ann. 373(1–2), 853–894 (2019)
Arosio, L., Benini, A.M., Fornæss, J.E., Peters, H.: Dynamics of transcendental Hénon maps III: infinite entropy. J. Mod. Dyn. 17, 465–479 (2021)
Arosio, L., Benini, A.M., Fornæss, J.E., Peters, H.: Dynamics of transcendental Hénon maps-II. Math. Ann. 385(3–4), 975–999 (2023)
Bedford, E., Lyubich, M., Smillie, J.: Polynomial diffeomorphisms. IV: The measure of maximal entropy and laminar currents. Invent. Math. 112(1), 77–126 (1993)
Bedford, E., Smillie, J.: Polynomial diffeomorphisms of \({ C}^2\): currents, equilibrium measure and hyperbolicity. Invent. Math. 103(1), 69–99 (1991)
Bedford, E., Smillie, J.: Polynomial diffeomorphisms of \({\bf C}^2\). II. Stable manifolds and recurrence. J. Am. Math. Soc. 4(4), 657–679 (1991)
Beltrami, V.: Automorphisms of \(\mathbb{C}^2\) with cycles of escaping Fatou components with hyperbolic limit sets (2024). arXiv:2401.16903 [math.DS]
Benini, A.M., Saracco, A., Zedda, M.: Invariant escaping Fatou components with two rank-one limit functions for automorphisms of \(\mathbb{C} ^2\). Ergod. Theory Dyn. Syst. 43(2), 401–416 (2023)
Boc-Thaler, L., Bracci, F., Peters, H.: Automorphisms of \(\mathbb{C} ^2\) with parabolic cylinders. J. Geom. Anal. 31(4), 3498–3522 (2021). (MR4236533)
Boc-Thaler, L., Fornæss, J.E., Peters, H.: Fatou components with punctured limit sets. Ergod. Theory Dyn. Syst. 35(5), 1380–1393 (2015)
Chirka, E.M.: Tangent Cones and Intersection Theory, pp. 79–153. Springer Netherlands, Dordrecht (1989)
Della Sala, G., Saracco, A., Simioniuc, A., Tomassini, G.: Lectures on complex analysis and analytic geometry, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 3. Edizioni della Normale, Pisa (2006)
Dujardin, R.: Hénon-like mappings in \(\mathbb{C} ^2\). Am. J. Math. 126(2), 439–472 (2004)
Fornæss, J.E.: Short \({C^k}\), Complex analysis in several variables—Memorial Conference of Kiyoshi Oka’s Centennial Birthday. Adv. Stud. Pure Math. Math. Soc. Japan Tokyo 42(4), 95–108 (2004)
Fornæss, J.E., Sibony, N.: Classification of recurrent domains for some holomorphic maps. Math. Ann. 301(4), 813–820 (1995)
Fornæss, J.E., Sibony, N.: Fatou and Julia sets for entire mappings in \({{\mathbb{C} }}^k\). Math. Ann. 311(1), 27–40 (1998)
Hubbard, J.H., Oberste-Vorth, R.W.: Hénon mappings in the complex domain. I. The global topology of dynamical space. Inst. Hautes Études Sci. Publ. Math. 79, 5–46 (1994)
Jupiter, D., Lilov, K.: Invariant nonrecurrent Fatou components of automorphisms of \(\mathbb{C} ^2\). Far East J. Dyn. Syst. 6(1), 49–65 (2004)
Krantz, S.G.: Function Theory of Several Complex Variables. AMS Chelsea Publishing, Providence (2001). Reprint of the 1992 edition
Lyubich, M., Peters, H.: Classification of invariant Fatou components for dissipative Hénon maps. Geom. Funct. Anal. 24(3), 887–915 (2014)
Acknowledgements
This work was partially supported by the Indam Groups Gnampa and GNSAGA and by PRIN 2022 Real and Complex Manifolds: Geometry and Holomorphic Dynamics. A.S. thanks Franc Forstnerič for a useful discussion.
Funding
Open access funding provided by Università degli Studi di Parma within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Proof of Proposition 2.13
Appendix: Proof of Proposition 2.13
We split the proof of Proposition 2.13 over several lemmas.
Definition 3.1
Let \(E\subset {{\mathbb {C}}}^n\). A vector \(v\in {{\mathbb {C}}}^n\) is called tangent to E at a point \(P\in {\overline{E}}\) if there exist a sequence of points \(P_J \in E\) and real numbers \(t_j > 0\) such that \(P_i \rightarrow P\) and \(t_j(P_j - P) \rightarrow v\) as \(j\rightarrow \infty \). The set of all such tangent vectors is the tangent cone to E at P.
The tangent cone is indeed a cone in \({{\mathbb {C}}}^n=T_P{{\mathbb {C}}}^n\). If the set E is a \(\mathcal {C}^1\)-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 [11, Corollary on page 80].
Lemma 3.2
Let \(H\subset {{\mathbb {C}}}^n\) be an analytic set of dimension one. For all \(x\in H\) 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 \(B\subset {{\mathbb {C}}}^n\) be a polidisc. The torus \({{\mathbb {T}}}\) with same center and same poliradius as B is called its Šilov boundary. We will denote it by \(\partial _{S}B\).
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 [12, from page 325].
Lemma 3.4
Let B be a polydisk, \(\partial _{S}B\) be its Šilov boundary, and \(u:U\rightarrow {{\mathbb {R}}}\) be a harmonic function defined on a neighbourhood U of \(\overline{B}\). Then
Recall that \(\partial \) denotes the topological boundary.
Proof
For every \(P\in \partial B{\setminus } {\partial _{S}B}\) there is a horizontal Euclidean disc D through P which is contained in \(\partial B\) whose boundary is in \({\partial _{S}B}\). Being u harmonic in a neighbourhood of \({\overline{B}}\), 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 \(\partial D\subset \partial _{S} B\). Hence
If \(P\in B\), we can find a disc through P with boundary in \(\partial B\) and repeat the argument, getting the conclusion
\(\square \)
Lemma 3.5
Let \(H\subset {{\mathbb {C}}}^2\) be an analytic set of dimension one. Then for every \( P\in H\) there exists an arbitrarily small torus \({{\mathbb {T}}}_P\) centered in P such that \({{\mathbb {T}}}_P\cap H=\varnothing \).
Proof
Let \(P\in H\). Consider the tangent cone \(C_P\) of H at P. By Lemma 3.2, \(C_P\) is a finite set of directions \(\alpha _1,\ldots ,\alpha _k\in {\hat{{{\mathbb {C}}}}}\). Up to a rotation, we can suppose all directions to be in \({{\mathbb {C}}}\). Up to choosing \(\eta >0\) small enough, we can ensure that the polidisk \(B_\eta \) of poliradius \(\eta \) centered in P intersects only one connected component of H. Moreover, by the definition of tangent cone, we can choose a small neighbourhood \(K\subset {{\mathbb {C}}}\) of all \(\alpha _j\) such that
We can suppose K to be small enough that there is \(0<b<1\) such that \(K\cap \{\beta \in {{\mathbb {C}}}\ | \ |\beta |=b\}=\varnothing \).
For any \(0<\delta <\eta \), defining
we have that \({{\mathbb {T}}}_P\subset U_\delta \) and if \((z,w)\in {{\mathbb {T}}}_P\), \(\frac{w-w_P}{z-z_P}=\beta \) with \(|\beta |=b\). So
\(\square \)
Proof of Proposition 2.13
Let K and L be compact sets as in the statement. For each \(P\in H\cap K\), by Lemma 3.5 there exists a torus \({{\mathbb {T}}}_P\subset L\) centered in P such that \({{\mathbb {T}}}_P\cap H=\varnothing \). Each torus \({{\mathbb {T}}}_P\) is the Šilov boundary of a polidisk \(B_P\) centered in P. Since \(\{B_P\}_{P\in H\cap K}\) is a covering of \(H\cap K\), by compactness we can extract a finite covering \(\{B_1,\ldots ,B_k\}\).
There is a \(\eta \)-neighbourhood \(U_\eta \) of H such that \(U_\eta \cap K\subset \cup B_j\). If the harmonic function u satisfies \(u\le \alpha \) on \(L{\setminus } U_\eta \) then it satisfies the same estimate on all tori \({{\mathbb {T}}}_j\), and by Lemma 3.4 the same estimate holds on all \(B_j\). Hence \(u\le \alpha \) on K. \(\square \)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Beltrami, V., Benini, A.M. & Saracco, A. Escaping Fatou components with disjoint hyperbolic limit sets. Math. Z. 307, 37 (2024). https://doi.org/10.1007/s00209-024-03501-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00209-024-03501-z
Keywords
- Holomorphic dynamics
- Fatou set
- Henon maps
- Escaping Fatou components
- Baker domains
- Trascendental automorphisms of \(\mathbb {C}^2\)