Parametrization of unstable manifolds and Fatou disks for parabolic skewproducts
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Abstract
We study the existence of Fatou components on parabolic skew product maps. We focus on skew products in which each coordinate has a fixed point that is parabolic. As in the geometrically attracting case, we prove that there exists maps F that have onedimensional disks that are mapped to a point in the Julia set of the restriction of F to an invariant onedimensional fiber. We first prove a linearization theorem for a onedimensional map, then for a parabolic skew product. Finally, we apply this result to construct the skew product map described above.
Keywords
Parabolic skewproduct maps Fatou coordinates Tangen to the identity maps1 Introduction
In this article, we investigate Fatou components for polynomial maps in two variables. More specifically, we fix our study in skewproduct polynomial maps \(F(t,z) = (h(t), f(t,z))\) with an invariant fiber, \(h(t_0) = t_0\).
In a recent joint paper with Peters [7], we studied the geometrically attracting case, that is \(0<h'(t_0) <1\) inspired by results of Lilov’s thesis. Lilov’s theorem [4] deals with the superattracting case, i.e., \(h'(t_0)=0\).
In this article, we focus on the parabolic case. That is, we study the dynamics close to the fixed fiber for the case when \(h'(t_0)=1\). To put our result into context, let us explain the results of [4] as well as [7].
Consider a skewproduct polynomial map \(F(t,z)= (h(t), f_t(z))\) such that \(h(t_0)=t_0\), with the condition that there exists an open attracting basin \(A \subset \mathbb C_t\) for \(t_0\), that is \(h(A)\subset A\), \(\lim _{n\rightarrow \infty }h^n(z)=t_0\) for all \(z \in A\), where \(t_0 \in \overline{A}\) (where \(\mathbb C_t= \{(0,t), t\in \mathbb C\}\)). Then the only options for \(h'(t_0)\) are the ones above, either the map h is superattracting, geometrically attracting or parabolic (up to an iterate). We want to investigate the dynamics of F on the set \(A\times \mathbb C\). Since \(f_{t_0}\) is a polynomial, by the nonwandering theorem of Sullivan [8], we know that all the onedimensional Fatou components of \(f_{t_0}\) are nonwandering (and in fact, preperiodic). Under our conditions on the existence of A as above, each onedimensional Fatou component of \(f_{t_0}\) is therefore contained, or in the boundary, of a twodimensional Fatou component of F. Lilov [4] proved that in the superattracting case all Fatou components in \(A\times \mathbb C\) are eventually mapped onto one of these fattened preperiodic Fatou components. As a consequence in the superattracting case, there are no wandering Fatou components in \(A\times \mathbb C\). Lilov proved this by proving a stronger result, namely:
Theorem
(Lilov [4]) Let \(t_1 \in A\), and let D be an open onedimensional disk lying in the \(t_1\)fiber. Then the forward orbit of D must intersect one of the fattened Fatou components of \(f_{t_0}\).
In a joint paper with Peters [7], we proved that this result does not hold in the geometrically attracting case. More precisely, we prove the following:
Theorem
(Peters [7]) There exist skewproduct polynomial maps of the form \(F(t, z) = (\alpha t, p(z) + q(t))\), where \(\alpha < 1\) and p and q are polynomials and a vertical holomorphic disk \(D \subset \{t=t_0\}\) whose forward orbit accumulates at a point \((0, z_0)\), where \(z_0\) is a repelling fixed point in the Julia set of p.
The proof of this theorem relies on a parametrization theorem for skew product maps where the map on the parameter fiber is attracting (similar to [3]).
A natural question is whether the same construction can be extended for skewproduct maps that are parabolic, or if as in Lilov’s case, all Fatou components in \(A\times \mathbb C\) are nonwandering. Fatou components on the attracting skewproduct case have been explored also by Peters and Smit [6]. We prove in this article that, in the parabolic case, a similar construction can be done as in the case of geometrically attracting.
The first step is to prove an analogous parametrization theorem for the parabolic case. Under an additional condition, we accomplish this and believe that this result can be useful, independently of the application given here. Let us state the additional condition and the parametrization theorem here.
Definition 1.1
Theorem
After we prove this parametrization theorem, we use a similar strategy to the one in [7] to prove that in the parabolic case, forward orbits of onedimensional disks D that lie above A do not necessarily intersect fattened Fatou components. Therefore, Lilov’s theorem is false in general for the parabolic case. Consequently, we see that the dynamics inside the set \(A\times \mathbb C\) is more complicated in the parabolic case than in the superattracting one. More explicitly, we prove that there exist wandering Fatou disks for skew parabolic maps. We deduce this by proving the following theorem:
Theorem
(Theorem 3.5) There exist skewproduct maps of the form \(F(t, z) = (\frac{t}{1+t}, f_t(z))\) where \(f_t(z) = f(t,z)\) polynomial in two variables, and a vertical holomorphic disk \(D \subset \{t=t_1\}\) for well chosen \(t_1\), whose \(\omega\)limit set contains the parabolic fixed point (0, 0) which is completely contained in the Julia set of \(f_0\).
The organization of the paper is as follows: in the next section we prove the parametrization theorem for parabolic skew product maps (Theorem 2.4). We first prove a result for maps in one dimension and use it to prove the result for skewproducts. In Sect. 3, we construct parabolic skew product maps that have wandering Fatou disks (Theorem 3.5). We also prove that our Fatou disks cannot be enlarged to Fatou components.
2 Parametrization of unstable manifolds of parabolic maps in one dimension
Given a parabolic map \(F(z) = z + a_kz^k+O(z^{k+1}), a_k \ne 0, k \ge 2\), the Leau–Fatou theorem say that there exist \(k1\) regions in which each point is attracted to the origin under iterates by F, and also \(k1\) regions in which the orbits are going towards the origin under the iterates of \(F^{1}\) (see [5] for more details). We can think of these regions as stable and unstable manifolds of F. In each one of these regions is possible to find a Fatou coordinate, that is, a change of coordinates map \(\phi\) such that F is conjugate to a translation. To find the change of coordinates \(\phi\), there is not in general an iterative process as it is the case for hyperbolic maps.
However, for a certain class of maps, the change of coordinates can be recovered using iterations of our parabolic map. Let us introduce a needed condition before we state and prove our theorem.
Definition 2.1
The condition of f being special is equivalent to f being formally conjugated to a translation \(h(z) = z1\) in a whole neighborhood of the origin. It is well known that this formal conjugacy is indeed holomorphic when we restrict to wedges for which the origin is a boundary point (see for example the appendix at [2]).
Our starting point is that this holomorphic conjugacy map restricted to the unstable manifold of f can also be recovered using appropriate iterates of f. Even more, we prove precise estimates on the rate of convergence of these iterates.
Theorem 2.2
Proof
This concludes the proof of the theorem. \(\square\)
Remark 2.3
Now we are ready to prove an analogous theorem for a skew product type of map \((t,z) \rightarrow (g(t),f_t(z))\) where \(g(0)=0, g'(0) =1\) and each \(f_t\) is special.
We see that F has an invariant fiber \(t=0\), and at this invariant fiber the action is given by the onedimensional map \(f_0(z)\). Under the given conditions, there exists an invariant manifold associated to \(f_0\) inside the fixed fiber. Our following theorem gives us a parametrization of this invariant manifold using iterates of F and projecting.
Theorem 2.4
Proof
We first choose R large enough so that for \(w>R\), then \(\theta _u(w) < 1/10\). Clearly, we have that for any u: \(\text {Re}(g_u(w)) > \text {Re}(w)\), and the domain \(W = \{\text {Re}(w) > R\}\) is invariant by \(g_u\). In this domain, we also have the easy estimates, for \(w\in W\), \(w+9/10< g_u(w) <w+11/10,\) and therefore, \(w+9k/10< g^k_u(w) <w+11k/10.\)
Lemma 2.5
Proof
Recall the estimates after Eq. (5): \(\theta _u(w) < A/w^2\text { for }w> R, u>R\).
Lemma 2.6
Proof
Lemma 2.7
Proof
It is well known that the range of \(\phi\) is the whole complex plane \(\mathbb C\). See [5] for a proof.
3 Skew parabolic maps with Fatou disks
We use the parametrization theorem of the last section for a dynamical application. As explained in the introduction, we prove a theorem similar to the one in [7]. That is, we prove that there exists some skew product parabolic maps that have wandering Fatou disks. Our construction, however, does not allow us to fatten the disks. We prove that statement at the end of this section. Let us recall the definition of Fatou disks following Ueda [9].
Definition 3.1
Let \(f: X\rightarrow X\) be a holomorphic endomorphism of a complex manifold X. A holomorphic disks \(D \subset X\) is a Fatou disk for f if the restriction of \(\{f^n\}\) to the disk D is a normal family.
Onedimensional disks contained in the Fatou set of any map are clearly Fatou disks. However, we will prove that we cannot enlarge our Fatou disks into a Fatou component.
We need to add another couple of hypothesis to the ones in Definition 1.1. Since we want to construct Fatou disks, we will make then centered at \(z_0\), a critical point of \(f_0\) and not centered at 0 as the iterates used on the Theorem 2.4. We also add the condition \(\alpha =1\) to simplify the computations. Let us put together those additional conditions on the following definition:
Definition 3.2

There exists \(z_0\ne 0\) a critical point of \(f_0\), such that \(f_t(z_0) = t\) for all t.

\(z_0\) a critical point of \(f_0\) of order at least 4.
The following corollary is an immediate consequence of Theorem 2.4.
Corollary 3.3
Proof
Example
Let \(t_0 \in \mathbb C\) be such that \(\phi (t_0) = z_0\). We refer to the complex lines \(\displaystyle {\{t_n = \frac{t_0}{1+nt_0}\}}\), \(n\ge 1\), as critical fibers.
Definition 3.4
Note that \(t_0\) might not be contained in \(V_\epsilon\). However, there exists \(N'=N(t_0)\) such that \(\frac{t_0}{1+Nt_0}\) is in \(V_\epsilon\), for \(N \ge N'\). From now on, we restrict our estimates to the disks \(D_n\), \(n \ge N'\).
Now we are ready to prove Theorem 3.5. Let us restate here:
Theorem 3.5
There exist skewproduct maps of the form \(F(t, z) = (\frac{t}{1+t}, f_t(z))\) where \(f_t(z) = f(t,z)\) is a polynomial in two variables, and a vertical holomorphic disk \(D_m \subset \{t=t_{m}\}\) whose \(\omega\) limit set contains the parabolic fixed point (0, 0) that is in the Julia set of \(f_0\) .
Proof
Consider a special skew product map that satisfies condition (\(\ddag\)). Then we will prove that the forward orbits of \(D_m\) accumulate at the point \((0,z_0)\), and therefore, we prove that for n sufficiently large, the forward orbits of the disks \(D_n\) all avoid the bulged Fatou components of F.
Note that \(t_0\) might not be in our domain \(V_\epsilon\) above. However, for a fixed \(N'=N(t_0)\), we do have that \(\frac{t_0}{1+Nt_0}\) is in \(V_\epsilon\), for \(N>N'\). Therefore, we can obtain all the estimates for iterates of F after we iterate F, \(N'\) times.
We need the following lemma:
Lemma 3.6
Proof
All we need to complete the proof of Theorem 3.5 is the following lemma.
Lemma 3.7
Proof
An immediate consequence of Lemma 3.7 is that for sufficiently large \(n \in \mathbb N\) there exists a sequence of \(l_n \rightarrow \infty\) so that \(F^{l_n}(D_n) \rightarrow (0,z_0)\) as \(\ell \rightarrow \infty\), and the proof of Theorem 3.5 is complete. \(\square\)
Lemma 3.8
The disks \(D_n\) are Fatou disks for F.
Proof
We need to prove that the sequence \(F^k\) restricted to each of the disks \(D_n\) is a normal family. We see that the t coordinate of each disk \(D_n\) is going to 0 and therefore stays bounded. For the z coordinate we have that the sequence \(\{\pi _2 F^{2^k(n+1)}, k\ge 0\}\). It is easy to see also that if (t, z) is such that \(z > R\) and \(t < 1\) then \(F^n(t,z) \rightarrow \infty\). Then it follows that the entire second coordinate of the iterates of F must stay bounded. By Montel’s Theorem this implies that \(F^k\) is a normal family when restricted to each \(D_n.\) \(\square\)
Remark 3.9
However, we see that we cannot enlarge our onedimensional disks into domains of a polidisk shape. Assume that for each n, there exists a domain \(B_n = U_n\times D_n\) where \(U_n\) is an open ball around each \(t_n\), that is \(U_n = \{tt_n<\delta _n\}\). We will let \(\delta _n\) vary for each n. Clearly, \(D_n \subset B_n\). We will argue by contradiction; assume that there exists a sequence of \(\delta _n >0\) so that \(F^{n+1}(B_n) \subset B_{2n+1}\) for n large enough.
Notes
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