Perturbed Basins of Attraction
Han peters
Let $F$ be an automorphism of $\CC^k$ which has an attracting fixed point. It is well known that the basin of attraction is biholomorphically equivalent to $\CC^k$. We will show that the basin of attraction of a sequence of automorphisms $f_1, f_2, \ldots$ is also biholomorphic to $\CC^k$ if every $f_n$ is a small perturbation of the original map $F$.
math0411339.tex
\documentclass{amsart}

\title[Perturbed Basins]{Perturbed Basins of Attraction}
\author{Han peters}
\date{November 15, 2004}
\subjclass{32H50, 32H02}


\usepackage{amscd}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{definition}
\newtheorem{defin}[theorem]{Definition}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\NN}{\mathbb{N}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\aut}{\mathrm{Aut}_0(\CC^k)}

\def\C{{\bf C}}
\def\B{{\bf B}}
\def\a{{\alpha}}
\def\e{{\epsilon}}
\def\d{{\delta}}
\def\l{{\lambda}}
\def\cO{{\mathcal O}}
\def\r{{\rho}}
\def\s{{\sigma}}
\def\v{{\varrho}}
\def\z{{\zeta}}
\def\N{{\mathcal{N}}}
\begin{document}
\maketitle

\begin{abstract}
Let $F$ be an automorphism of $\CC^k$ which has an attracting
fixed point. It is well known that the basin of attraction is
biholomorphically equivalent to $\CC^k$. We will show that the
basin of attraction of a sequence of automorphisms $f_1, f_2,
\ldots$ is also biholomorphic to $\CC^k$ if every $f_n$ is a small
perturbation of the original map $F$.
\end{abstract}

\section{introduction}

 We are interested in the following conjecture, posed by Eric
 Bedford:

\begin{conjecture}\label{bedford}
 Let $F$ be a holomorphic automorphism of a complex manifold $X$,
 which is hyperbolic on an invariant compact subset $K$. Then for every $p
 \in K$, the stable manifold through $p$ is biholomorphically equivalent to
  complex Euclidean space.
\end{conjecture}

It was proved by Jonsson and Varolin \cite{jv} that Conjecture
\ref{bedford} holds almost everywhere with respect to any
invariant probability measure supported on $K$.

One could show that Conjecture \ref{bedford} holds by solving a
related problem in non-autonomous dynamics. For $z \in K$, let
$\phi_z$ be a biholomorphic mapping that maps the local stable
manifold through $z$ onto the local stable tangent space at $z$
(which after scaling we can identify with the unit ball in
$\CC^k$), where $k$ is the dimension of the stable manifold. We
may assume that $\phi_z^\prime(z)$ equals the identity. Let $p_0 =
p, p_1, p_2, \ldots$ be the orbit of $p \in K$. Then the maps
$$
f_n (z) = \phi_{p_{n+1}} F \phi_{p_n}^{-1},
$$
are biholomorphic mappings from the unit ball into the unit ball
(we may assume that these maps are really contracting, since
otherwise we consider some high iterate of $F$ instead). The maps
satisfy the conditions of the main theorem in \cite{fst}, and thus
there is a basin of attraction, defined in terms of the
{\emph{tail space}}. It is clear from the definition that this
basin of attraction is biholomorphic to the stable manifolds, and
the main theorem of \cite{fst} says that the basin of attraction
is biholomorphic to a basin of attraction of a sequence of global
automorphisms, which are uniformly contracting on some
neighborhood of the origin. Thus, Conjecture \ref{bedford} can be
answered positively by proving that the following conjecture
holds:

\begin{conjecture}\label{compact}
Let $f_1, f_2, \ldots$ be a sequence of automorphisms of $\CC^k$
that fix the origin. Suppose that there exist $a,b \in \RR$ that
satisfy $0 < a< b < 1$, and such that the following holds for
every $ n\in \NN$ and every $z \in \B$:
$$
a\|z\| \le \|f_n(z)\| \le b \|z\|.
$$
Then the basin of attraction of the sequence $f_1, f_2, \ldots$ is
biholomorphic to $\CC^k$.
\end{conjecture}

The {\emph{basin of attraction}} of a sequence $f_1, f_2 \ldots$
of automorphisms that all fix the origin is defined as the set of
all points $z \in \CC^k$ such that $f(n)(z) = f_n \circ \cdots
\circ f_1 (z) \rightarrow 0$.

If the automorphisms $f_1, f_2 \ldots$ do not all fix the origin,
but there does exist a small neighborhood $\N$ of $0$ such that
$f_n(\N) \subset \subset \N$ for every $n \in \NN$, then we define
the basin of attraction of the sequence $f_1, f_2, \ldots$ as
$\bigcup f(n)^{-1} \N$.

If all the maps $f_n$ are equal then the conjecture follows
immediately from the following theorem, which follows from the
work of Sternberg \cite{sg}, and was proved independently by Rosay
and Rudin \cite{rr}:

\begin{theorem}\label{RR} Let $F$ be an automorphism of
$\CC^k$, which has an attracting fixed point at $0$, then the
basin of attraction of $F$ is biholomorphically equivalent to
$\CC^k$.
\end{theorem}

Here, $F$ is attracting at the origin means that all eigenvalues
of $F^\prime(0)$ are smaller than $1$ in absolute value. We will
focus on Conjecture \ref{compact}. Our main result is the
following generalization of Theorem \ref{RR}:\\
\\
{\bf{Main Theorem.}} {\emph{ Let $F$ be an automorphism of $\CC^k$
which has an attracting fixed point at $0$. Then there exists an
$\epsilon>0$ such that for any sequence $f_1, f_2, \ldots$ of
automorphisms which satisfy $\|F(z), f_n(z)\| < \epsilon$ for any
$n \in \NN$ and all $z$ in the unit ball, one has that the basin
of attraction of the sequence $f_1, f_2, \ldots$ is
biholomorphic to $\CC^k$.}}\\

The maps $f_n$ don't necessarily have $0$ as an attracting fixed
point. However, if we choose $\epsilon$ small enough then there
exists a small neighborhood of $\N$ of $0$, such that $f_n(\N)
\subset \subset \N$ for all $n \in \NN$. Notice that the
definition of the basin of attraction is independent of the choice
of $\N$. Also notice that if the maps $f_n$ do fix the origin,
then the two definitions of the basin of attraction are exactly
equal.

Hence the Main Theorem gives sufficient conditions for when the
basin of attraction of a sequence $f_1, f_2, \ldots$ of
automorphisms which have an attracting fixed point at the origin
is biholomorphic to $\CC^k$. The following theorem can be found in
\cite{wd}:
\begin{theorem}\label{square}
Let $f_1, f_2, \ldots$ be a sequence of automorphisms of $\CC^k$
which all fix the origin, and let $a,b \in \RR$ that satisfy $0 <
a < b < 1$ and the additional condition $b^2 < a$. Suppose that
the following holds for every $ n\in \NN$ and every $z \in \B$:
$$
a\|z\| \le \|f_n(z)\| \le b \|z\|.
$$
Then the basin of attraction of the sequence $f_1, f_2, \ldots$ is
biholomorphic to $\CC^k$.
\end{theorem}

The proof of Theorem \ref{square} can be adapted to get the
following result which is slightly stronger:

\begin{theorem}\label{blocks}
Let $f_1, f_2, \ldots$ be a sequence of automorphisms which all
fix the origin, and let $t \in (1,2)$. Suppose that for every $
n\in \NN$ there exists $c_1, c_2, \ldots \in (0,1)$ such that
$\prod c_n = 0$ and the following holds for every $z \in \B$:
$$
c_n^t\|z\| \le \|f_n(z)\| \le c_n \|z\|.
$$
Then the basin of attraction of the sequence $f_1, f_2, \ldots$ is
biholomorphic to $\CC^k$.
\end{theorem}

The idea in the proof of both Theorem \ref{square} and
\ref{blocks} is the following:  one defines the maps $H_n :=
A(n)^{-1}f(n)$, where $A(n) = A_n \circ \cdots \circ A_1$ and $A_n
= f_n^\prime(0)$. As $n \rightarrow \infty$, one gets that $H_n$
converges uniformly on compact subsets of the basin of attraction
to a biholomorphic mapping from the basin onto $\CC^k$.

The following theorem was proved in \cite{pw} and is also a
generalization of Theorem \cite{rr}:

\begin{theorem}\label{repeat}
Let $f_1, f_2, \ldots$ be a sequence of automorphisms of $\CC^k$,
which are all attracting at the origin. Then there exist large
enough integers $n_1, n_2, \ldots$ such that the basin of
attraction of the maps $f_1^{n_1}, f_2^{n_2} \ldots$ is
biholomorphic to $\CC^k$.
\end{theorem}

Here the definition of the basin of attraction must be slightly
changed for this theorem to hold. The exact definition is given in
\cite{pw}, as well as an example that shows that Theorem
\ref{repeat} does not hold if one uses a more straightforward
definition.

In the next section we will set our notation, and introduce the
main ingredient in the proof of our Main Theorem, namely Theorem
\ref{stable}. In the Third section we will prove our Main Theorem
using Theorem \ref{stable}. In section 4 we give the main idea of
the proof of Theorem \ref{stable}, and we finish up by proving
Theorem \ref{stable} in the fifth section.

\section{Preliminaries}

Throughout the paper we write $\aut$ for the set of automorphisms
of $\CC^k$ that fix the origin. We denote by $B(r)$ the ball or
radius $r$ centered at the origin and we write $\B$ for the unit
ball. For a sequence $F_1, F_2, \ldots$ we will use the notation
$F(n)$ for the composition of the first $n$ maps, $F_n \cdots F_1$
and we write $F(m,n) = F_n \cdots F_m = F(n)F(m)^{-1}$.

It follows from a well known fact from linear algebra (called $QR$
factorization) that for any automorphism $F \in \aut$ we can find
a unitary matrix $U \in U_k(\CC)$ such that $(U \circ F)^\prime
(0)$ is lower triangular. The map $U$ is in fact unique up to
multiplying rows with complex numbers of unit length. We say that
$U F$ is in {\emph{lower triangular form}}.

\begin{remark}\label{QR}
Given some sequence $F_1, F_2, \ldots \in \aut$ we can find
unitary matrices $U_1, U_2 \ldots$ such that $U_1 F_1, U_2 F_2
U_1^{-1}, U_3 F_3 U_2^{-1}, \ldots$ are all in lower triangular
form.

If all the maps $F_n$ are attracting at the origin, then we define
$\tilde{F}_n = U_n F_n U_{n-1}^{-1}$. We have that $\tilde{F}(n) =
U_n F(n)$, therefore the basin of attraction of the sequence
$\{\tilde{F}_n\}$ is exactly equal to the basin of the original
maps, so we may assume that our original maps were already in
lower triangular form.
\end{remark}

We will say that $F \in \aut$ which is in lower triangular form is
\emph{correctly ordered} if $F^\prime(0)$ has diagonal entries
(from upper left to lower right) $\lambda_1, \ldots , \lambda_k$
that satisfy the following condition:
$$
|\lambda_j| |\lambda_i| < |\lambda_l|,
$$
for $l \le j$ and any $i$. Note in particular that $|\lambda_j| <
1$ for every $j$, and that $|\lambda_j|^2 < |\lambda_l|$ for $ j
\ge l$. Note also that if the eigenvalues satisfy the ordering
$|\lambda_1| \ge \cdots \ge |\lambda_k|$ then $F$ is correctly
ordered, but the sequence $|\lambda_1|, \ldots ,|\lambda_k|$ may
fail to be decreasing by a relatively small amount.

\begin{defin}\label{attracting}
Let $\mathcal{F} \subset \aut$ be a family of correctly ordered
automorphisms. We say that $\mathcal{F}$ is {\emph{uniformly
attracting}} if there exist $a, b \in \RR$ with $0 < a < b < 1$
and we have that for every $F \in \mathcal{F}$ and every $z \in
\B$,
$$
a \|z\| \le \|F(z)\| \le b \|z\|.
$$
Additionally we require that there exists a uniform $\xi <1$ such
that
\begin{eqnarray} \label{eigenvalues}
|\lambda_i||\lambda_j| \le \xi |\lambda_l|,
\end{eqnarray}
for $l \le j$ and any $i$.
\end{defin}

We will prove the following theorem:
\begin{theorem} \label{stable}
Let $F_1, F_2, \ldots \in \aut$ be a uniformly attracting sequence
of correctly ordered automorphisms. Then the basin of attraction
of $F_1, F_2, \ldots$ is biholomorphic to $\CC^k$.
\end{theorem}

\section{Proof of the Main Theorem}

We will now prove our main Theorem, using Theorem \ref{stable}.
The proof consists of several (but finitely many ) steps, and in
each step we may decrease the value of $\epsilon$.

We may assume that $F$ is such that $F^\prime(0)$ is a lower
triangular matrix with diagonal entries (from upper left to lower
right) $\lambda_1, \ldots ,\lambda_k$ that satisfy $|\lambda_1|
\ge \ldots \ge |\lambda_k|$. We may also assume that the
off-diagonal terms of $F^\prime(0)$ are arbitrarily small.

Now we show that we may assume that all the maps $f_n$ have an
attracting fixed point at $0$. Since $F^\prime(0)$ is arbitrarily
close to a diagonal matrix, we choose the neighborhood of the
origin $\N$ so small that $F$ is contracting on $\N$, i.e. that
there exist some $\theta < 1$ such that for all $x,y \in \N$ we
have that $\|F(x) - F(y)\| < \theta \|x - y\|$. Then we can make
$\epsilon$ small enough such that every $f_n$ is also contracting
on $\N$ (with uniform constant $\theta < 1$ that is slightly
increased if necessary). Let $x_0 = 0, x_1, x_2, \ldots$ be the
orbit of $0$, i.e. $x_n = f_n(x_{n-1})$. We have that
$f_n(\mathcal{N}) \subset \subset \mathcal{N}$ for every $n \in
\NN$, so that $x_n \in \mathcal{N}$ for every $n \in \NN$. Let
$T_n$ be the translation of $\CC^k$ that maps $x_n$ to $0$. Then
define $\tilde{f_n} = T_n \circ f_n \circ T_{n-1}^{-1}$. We have
that $\tilde{f_n}(0) = 0$ for all $n$, $0$ is an attracting fixed
point for every map $f_n$, and the maps $\tilde{f_n}$ are still
arbitrarily close to the original map $F$. Since $\tilde{f}(n) =
T_n f(n)$, we have that the approximate basin of attraction of the
sequence $f_1, f_2, \ldots$ is exactly equal to the basin of
attraction of $\tilde{f}_1, \tilde{f}_2, \ldots$. Therefore, we
may as well assume that all the maps $f_n$ have $0$ has an
attracting fixed point, and we only consider the basin of
attraction of the sequence $f_1, f_2, \ldots$.

We will now prove the Main Theorem in the case that $k = 2$. We
may assume that $|\lambda_2|$ is strictly smaller than
$|\lambda_1|$, otherwise the result follows easily from Theorem
\ref{square}. The fact that $F^\prime(0)$ is lower diagonal means
exactly that $(0,1)$ is an eigenvector of $F^\prime(0)$. Let
$\Phi$ be the action on $\PP^1$ induced by the mapping
$F^\prime(0)$. It follows that $\Phi([0:1]) = [0:1]$, and the
multiplier of $\Phi$ at $[0:1]$ is exactly
$\frac{\lambda_1}{\lambda_2}$. Hence $[0:1]$ is a repelling fixed
point, and there exists an arbitrarily small neighborhood of
$[0:1]$, say $\mathcal{N}$ such that $\N \subset\subset \Phi(\N)$.

Let $\phi_n$ be the action on $\PP^1$ induced by $f_n^\prime(0)$.
We can make sure that $\phi_n$ is arbitrarily close to $\Phi$ such
that $\N \subset\subset \phi_n(\N)$ for all $n \in \NN$. Then we
have that $\N \supset \supset \phi_1^{-1}(\N) \supset\supset
\phi(2)^{-1}(\N) \cdots$. It follows that
$$
\bigcap_{n \in \NN} \phi(n)^{-1}(\N) \neq \emptyset.
$$

Let $v \in \bigcap \phi(n)^{-1}(\N)$, and let $v_0 = v, v_1, v_2,
\ldots$ be the orbit of $v$, i.e. $\phi_n(v_{n-1})= v_n$. We have
that $v_n \in \N$ for every $n \in \NN$.

Let $U_n$ be a unitary $2 \times 2$ matrix that maps some length
$1$-representative of $v_n$ in $\CC^2$ onto $(0,1)$, and define
$\tilde{f}_n := U_n f_n U_{n-1}^{-1}$. Then it follows that
$(0,1)$ is an eigenvector of every map $\tilde{f}_n^\prime(0)$,
and thus we have that $\tilde{f}_n^\prime(0)$ is lower triangular.
The basin of the sequence $\tilde{f}_1, \tilde{f}_2, \ldots$ is
exactly equal to $U_0(\Omega)$, so in particular biholomorphic to
$\Omega$. Since the unitary matrices $U_n$ are arbitrarily close
to the identity matrix, we have that $\tilde{f}_n$ is arbitrarily
close to $F$, and it follows that the sequence $\tilde{f}_1,
\tilde{f}_2, \ldots$ satisfy the properties in Theorem
\ref{stable} and we are done.\\

We will use a similar argument to that for $2$ dimensions to prove
the Main Theorem in the general case.

\begin{lemma}\label{grass}
Let $A$ be a lower triangular $k \times k$ matrix whose diagonal
entries $\lambda_1, \lambda_2, \ldots \lambda_k$ satisfy
$|\lambda_1| \ge \cdots \ge |\lambda_k|$. Suppose that $l \in \{1,
\ldots k-1\} $ is such that $|\lambda_l| > |\lambda_{l+1}|$, and
let $L$ be the linear subspace of $\PP^{k-1}$ defined by
$$
L = \{ [z_1:\cdots : z_k] \in \PP^{k-1} \mid z_1 = \cdots = z_{l}
= 0\}.
$$
Then there exists an arbitrarily small neighborhood $\N \subset
G(k-l-1, k-1)$ of $L$ such that $A(\N) \supset \supset \N$.
\end{lemma}
\begin{proof}
First assume that $A$ is diagonal. For $X \in G(k-l-1, k-1)$ close
enough to $L$ we can write
$$
X = \{ [z_1:\cdots : z_k] \in \PP^{k-1} \mid z_1 = \sum_{j \ge l}
\epsilon_{1,j} z_j, \ldots ,z_l = \sum \epsilon_{l,j} z_j\}.
$$
We define $\mathrm{d}(X, L) = \max |\epsilon_{i,j}|$.

For $\delta >0$ small let $\N_{\delta} = \{X \in G(k-l-1, k-1)
\mid \mathrm{d}(X,L) < \delta$\}. Since $A$ is diagonal and
$|\lambda_l| > |\lambda_{l+1}|$ for all $i \le l$ and $j \ge l+1$,
we have that
$$
A(\N_{\delta}) \supset
\N_{\frac{|\lambda_l|}{|\lambda_{l+1}|}\delta} \supset \supset
\N_{\delta}
$$
Fix some $\delta$. Then $\N_{\delta}$ will also work for
arbitrarily small perturbations of the diagonal matrix $A$.

We can conjugate any $A$ with an invertible linear mapping $T$
such that $T^{-1} A T$ is lower triangular and arbitrarily close
to a diagonal mapping (whose diagonal entries are exactly those of
$A$). Hence the set $T(\N_\delta)$ will suffice for $A$.
\end{proof}

Let $l \in \{1, \ldots ,k-1\}$ be such that $|\lambda_{l}| >
|\lambda_{l+1}| = |\lambda_k|$, and let $\Phi$ be the action of
$F^\prime(0)$ on $\PP^{k-1}$ (as in the 2-dimensional case, we may
assume that such an $l$ exists, otherwise we are done by Theorem
\ref{square}). Then it follows from Lemma \ref{grass} that there
exists a small neighborhood $\N \in G(k-l-1, k-1)$ of $L$ (where
$L$ is as in Lemma \ref{grass}, such that $\Phi(\N) \supset\supset
\N$. As in the 2-dimensional case, we denote by $\phi_n$ the
action of $f_n$ on $\PP^{k-1}$. Then we can decrease the value of
$\epsilon$ if necessary to get that $\phi_n(\N) \supset\supset \N$
for every $n \in \NN$. Therefore we can find an orbit of linear
subspaces $L_0, L_1, \ldots$ in $\N$ with $\phi_n(L_{n-1}) = L_n$
for every $n \in \NN$.

We denote by $U_n$ a unitary matrix arbitrarily close to the
identity that maps a representative of $L_n$ in the unit sphere in
$\CC^k$ onto the set
$$
T = \{ z \in \CC^k \mid \|z\| = 1, z_1 = \cdots = z_l = 0 \}.
$$
We define the maps $\tilde{f}_n$ by
$$
\tilde{f}_n = U_n f_n U_{n-1}^{-1}.
$$
Then we have that for every $v \in T$ there exists a $c \in (0,1)$
such that $\tilde{f}_n^\prime(0)(v) \in c T$, and therefore we
have that for any $n \in \NN$ the entries in the $p$-th rows and
$q$-th columns of the matrix $f_n^\prime(0)$ are equal to $0$ for
$p \le l$ and $q \ge l+1$. Since the matrices $U_n$ are
arbitrarily close to the identity, we have that
$\tilde{f}_n^\prime(0)$ is arbitrarily close to $F^\prime(0)$.

 We may also assume that the $(k-l) \times (k-l)$ blocks
  in the lower right corner of the matrices
$\tilde{f}_n^\prime(0)$ are lower
 triangular, since we can apply QR-factorizations as in Remark \ref{QR}
  to these $(k-l) \times (k-l)$
blocks. The diagonal entries in these blocks must in absolute
value be arbitrarily close to $|\lambda_k|$, since in this last
step have only composed with unitary matrices, and in Step 2 we
made sure that $F^\prime(0)$ is arbitrarily close to a diagonal
matrix. Also, it follows that the off-diagonal terms in the $(k-l)
\times (k-l)$ block in the lower right corner of the matrices
$\tilde{f}_n^\prime(0)$ are arbitrarily small.

 Since the last $k-l$
columns of the matrices $\tilde{f}_n^\prime(0)$ are already lower
triangular, and the corresponding eigenvalues are the smallest in
absolute value, we can restrict ourselves to the first $l$
dimensions and apply the same arguments to the next eigenvalues of
$F^\prime$ which are equal in absolute value.

In finitely many steps we get that all the maps
$\tilde{f}_n^\prime(0)$ are lower triangular matrices. To change
from $f$ to $\tilde{f}$, we only compose with unitary matrices
that are arbitrarily close to the identity. As a result, we get
that the diagonal entries of the maps $\tilde{f}_n^\prime(0)$ are
arbitrarily close to the diagonal entrees of $F^\prime(0)$.
Therefore, all the maps in the sequence $f_1, f_2, \ldots$ are
correctly ordered.

We have constructed unitary matrices $U_0, U_1, \ldots$ such that
for every $n \in \NN$ we have that $\tilde{f}(n) = U_n f(n)
U_0^{-1}$. It follows that the basin of attraction of the sequence
$\tilde{f}_1, \tilde{f}_2, \ldots$ is equal to the image of the
basin of attraction of the original sequence $f_1, f_2 \ldots$
under the map $U_0^{-1}$, which is a biholomorphism. The sequence
$\tilde{f}_1, \tilde{f}_2, \ldots$ satisfies the conditions in
Theorem \ref{stable}, and the Main Theorem.

\section{Main Ideas of Theorem \ref{stable}}

In the proof of Theorem \ref{stable} we will construct a sequence
of maps $\Psi_n := G(n)^{-1} \circ X_n \circ F(n)$ which converges
to the Fatou Bieberbach mapping $\Psi : \Omega \rightarrow \CC^k$.
Here the $G_n$'s are \emph{lower triangular polynomial mappings}
(as in \cite{rr}), and the $X_n$'s are polynomial mappings whose
linear parts are the identity. To be more specific, we will start
with some choice for $X_1$ and then define $X_n = [G_n \circ
X_{n-1} \circ F_n^{-1}]_{p}$. Here $d$ is some large integer and
we mean by $[\cdot]_d$ that we discard all terms of degree $d+1$
and higher. It follows immediately that

\begin{eqnarray} \label{difference}
\|G_n^{-1}X_nF_n(z) - X_{n-1}(z)\| = O(\|z\|^{d+1}).
\end{eqnarray}

The challenge is to choose a bounded sequence of lower triangular
polynomial mappings $\{G_n\}$ such that there exists a bounded
orbit $\{X_n\}$, and to get the integer $d$ as large as necessary.
We first show that we can do this in some simpler cases before we
complete the proof of theorem \ref{stable}.

Recall that a polynomial selfmap $G= (g_1, \ldots g_k)$ of $\CC^k$
with $G(0) = 0$ is called lower triangular if
$$
g_j(z) = c_j z_j + h_j (z_1, \ldots z_{j-1}),
$$
for all $j \in \{1, \ldots , k\}$.

\begin{lemma} \label{upper}
Let $G_1, G_2, \ldots$ be a sequence of lower triangular
polynomial mappings of some fixed degree, whose coefficients are
uniformly bounded. Then we have the following:

(a) The degrees of the maps $G(n)$ are bounded, and there is a
constant $\beta < \infty$ so that
$$
G(n)(\B) \subset B(\beta^n)
$$

(b) If also $|c_i|< \theta < 1$ for all $G_j$, then $G(n)(z)
\rightarrow 0$, uniformly on compact subsets of $\CC^k$, and for
every $R > 0$ and $\epsilon
>0$ there exist an $N \in \NN$ such that for $n \ge N$ we have
that
$$
G(n)(B(R)) \subset B(\epsilon).
$$
\end{lemma}
This lemma is similar to Lemma 1 in the appendix of \cite{rr}, and
so is its proof.

We now prove the following simple lemma.
\begin{lemma} \label{affine}
Let $F_1, F_2, \ldots$ be a uniformly bounded sequence of affine
maps on $\CC$ that are all expanding. Then there exists a bounded
orbit $z_0, z_1, z_2, \ldots$
\end{lemma}

Let $F_n (z) = a z + b$. With uniformly bounded we mean that the
constants $a$ and $b$ are bounded from above, and that the
constants $a$ are also bounded from below by some $c > 1$.

\begin{proof}
Since the sequence is compact, there exists a constant $R$ such
that
\begin{eqnarray*}
F_i(B(R)) \supset \supset B(R)
\end{eqnarray*}
for all $i \in \NN$. Now we have that
\begin{eqnarray*}
B(R) \supset\supset F(1)^{-1}B(R) \supset\supset F(2)^{-1}B(R)
\supset\supset \ldots,
\end{eqnarray*}
So the intersection $\bigcap F(n) ^{-1} \Delta (R)$ contains a
point $z_0$. Then we have that the sequence $z_0, z_1, z_2,
\ldots$, where $z_{n} = F_n(z_{n-1})$, is a bounded orbit. Indeed,
$|z_n| < R$ for all $n \in \NN$.
\end{proof}

\subsection{Bounded Orbits in $\CC$}

The argument that shows that there exist bounded sequences $G_n$
and $X_n$ is somewhat complicated. To make it more clear we will
first prove the existence in the one dimensional case, in which
case it is much easier.

Let $f_1, f_2, \ldots$ be a sequence of polynomials with uniformly
bounded coefficients of the form $f_n (z) = \lambda_n z + h.o.t.$,
where $|\lambda_n|
> \theta
> 1$. Let $g_n(z) = \lambda_n^{-1} z$. Let $\mathcal{P}_d$ be
the space of polynomials of the form $z+ h.o.t.$ of degree at most
$d$. Define the map $\phi_n:\mathcal{P}_d \rightarrow
\mathcal{P}_d$ by $\phi_n(X) = [g_n\circ X \circ f_n]_d$, where we
mean by $[\cdot ]_d$ that we discard all terms of degree strictly
higher than $d$.

\begin{proposition}
There exists a bounded sequence $X_0, X_1, \ldots \in
\mathcal{P}_d$ with $X_n = \phi_n(X_{n-1})$.
\end{proposition}
\begin{proof}

We will use induction on the degree. For degree $1$ the statement
is clear since we take $X_n (z) = z$ for all $n$.

Suppose we have constructed a bounded orbit $Y_n$ that satisfies
condition \eqref{difference} for some $p = d-1 \ge 1$. We will add
terms of degree $p$ so that the sequence satisfies condition
\eqref{difference} for $p = d$. Say $X_0 = Y_0 + c_0 z^p$ for some
$c_0 \in \CC$. Then it is easy to see that $X_1 = \phi_1 (X_0) =
Y_1 + c_1 z^d$, where $c_1$ is equal to $\lambda_1^{d-1} c_0$ plus
some constant depending linearly on the coefficients of $Y_0$ and
the higher order terms of $f_1$ (which are uniformly bounded by
assumption). We see that we get a uniformly bounded sequence of
expanding affine maps $\tilde\psi_n : \CC \rightarrow \CC$ that
take $c_{n-1}$ to $c_n$. It now follows from Lemma \ref{affine}
that we can choose $c_0$ such that the sequence $c_0, c_1, \ldots$
is bounded.

It follows by induction that we can find a bounded sequence $X_n$
for any $d \in \NN$.
\end{proof}

\subsection{Degree $2$ polynomials in $\CC^2$}

Next we show that we can get bounded sequences $\{G_n\}$ and
$\{X_n\}$ in the two dimensional case for degree $2$, where we can
explicitly calculate the maps. We write $F_n (x, y) = (\lambda_n x
, \mu_n y + a_n x) + h.o.t.$ and $(F_n^{-1}) (x, y) =
(\lambda_n^{-1} x , \mu_n^{-1} y + b_n x) + h.o.t.$ and let
$G_n(x,y) = (\lambda_n x , \mu_n y + a_n x + d_n x^2)$ for some
constants $d_n \in \CC$ to be chosen later. We will also write
$X_n(x,y) = (x + \alpha_n y^2 + \beta_n xy + \gamma_n x^2, y +
\delta_n y^2 + \epsilon_n xy + \zeta_n x^2)$. We will identify the
map $X_n$ with $(\alpha_n, \beta_n, \gamma_n, \delta_n,
\epsilon_n, \zeta_n) \in \CC^6$. Consider the map $X_{n-1} \mapsto
X_n = [G_n \circ X_{n-1} \circ F_n^{-1}]_2$. We have that
\begin{align*}
\alpha_n &= \lambda_n \mu_n^{-2} \alpha_{n-1} + l_{1,n}, \\
\beta_n &= \mu_n^{-1} \beta_{n-1} + l_{2,n}(\alpha_{n-1}),\\
\gamma_n &= \lambda_n^{-1} \gamma_{n-1} + l_{3,n}(\alpha_{n-1},
\beta_{n-1}),\\
\delta_n &= \mu_n^{-1} \delta_{n-1} + l_{4,n}(\alpha_{n-1}),\\
\epsilon_n &= \lambda_n^{-1} \epsilon_{n-1} + l_{5,n}(\alpha_{n-1}, \beta_{n-1}, \delta_{n-1}),\\
\zeta_n &= \mu_n \lambda_n^{-2} \zeta_{n-1} +
l_{6,n}(\alpha_{n-1}, \ldots ,\epsilon_{n-1}) + d_n
\lambda_n^{-2}.
\end{align*}
Here the $l_{i,n}$ are linear maps that depend only on the
coefficients of $F_n^{-1}$ (which are uniformly bounded) and on
the given variables.

It follows from equation \eqref{eigenvalues} that $|\lambda_n
\mu_n^{-2}|
>1$ for any $n$, and therefore we get a uniformly bounded sequence of
expanding affine maps $\alpha_{n-1} \mapsto \alpha_n$. Hence it
follows from Lemma \ref{affine} that we can find $\alpha_0$ such
that the sequence $\alpha_0, \alpha_1, \ldots$ is bounded. Having
fixed the $\alpha_n$'s, we can use the same argument for the
$\beta_n$'s, since we also have that $|\mu_n^{-1}|>1$ for all $n$.
After we fix the $\beta_n$'s, we can find a bound on the
$\gamma_n$'s, then the $\delta_n$'s, and finally the
$\epsilon_n$'s.

We can't use the same argument for the $\zeta_n$'s, since we may
not have that $|\mu_n \lambda_n^{-2}| >1$. However, we can choose
$\zeta_0 = 0$, and then choose the constants $d_n$ such that
$\zeta_n = 0$ for every $n$. Hence we get the bounded sequences
$\{G_n\}$ and $\{X_n\}$.

\section{Proof of Theorem \ref{stable}}

The argument that we use to construct bounded sequences $\{X_n\}$
and $\{G_n\}$ for higher dimensions and higher degrees is
essentially the same as the argument for degree $2$ polynomial
mappings in $\CC^2$.

We will write $\lambda_{n,1}, \ldots , \lambda_{n,k}$ for the
diagonal terms of $F_n^\prime(0)$.

\begin{proposition}\label{bounded}
For any $d \ge 2$ we can find a bounded sequence of polynomial
mappings $X_0, X_1, \ldots$, where $X_n = I_k + h.o.t.$, and a
bounded sequence of lower triangular polynomial mappings $G_1,
G_2, \ldots$, where $G_n^\prime(0) = F_n^\prime(0)$, such that
\eqref{difference} holds for any $n \ge 1$.
\end{proposition}
\begin{proof}
We will construct bounded sequences $X_1, X_2, \ldots$ and $G_1,
G_2, \ldots$ such that the following equation holds:
\begin{eqnarray}\label{next}
X_n = [G_n X_{n-1} F_n^{-1}]_d,
\end{eqnarray}
for every $n \in \NN$. Notice that this will imply that Equation
\ref{difference} holds for every $n \in \NN$.

We write
$$
X_n(z) = (x_{n,1}(z), x_{n,2}(z), \ldots, x_{n,k}(z)),
$$
and
$$
x_{n,j} = \sum_{\alpha} c_{n,j,\alpha}z^\alpha,
$$
where $\alpha$ is a $k$-tuple, and $z^\alpha = z_1^{\alpha_1}
\cdots z_k^{\alpha_k}$. We refer to $c_{n,j,\alpha}z^\alpha$ as a
term of degree $|\alpha| = \alpha_1 + \cdots + \alpha_k$, index
$j$, and power $\alpha$. We similarly write $g_{n,j,
\alpha}z^{\alpha}$ for the term of the mapping $G_n$ of degree
$|\alpha|$, index $j$ and power $\alpha$.

For two $k$-tuples $\alpha$ and $\beta$ with $|\alpha| = |\beta|$
we will write $\alpha > \beta$ if $\alpha$ has a higher
lexicographical ordering than $\beta$. That is, $\alpha > \beta$
if and only if there is some $j \in \{1, \ldots , k\}$ such that
$\alpha_j < \beta_j$ and $\alpha_i = \beta_i$ for $i \in \{1,
\ldots j-1\}$ (so that $z_1^d$ comes first in the alphabet and
$z_k^d$ comes last).

We will use induction on the degree and index and reverse
induction on the power to fix all the terms of the sequences
$\{X_n\}$ and $\{G_n\}$.

Let $\alpha$ be some $k$-tuple, and let $j \in \{1, \ldots ,k\}$.
Suppose that we have fixed all the terms of degree up to
$|\alpha|-1$ in the sequences $\{X_n\}$ and $\{G_n\}$, such that
\eqref{next} holds for $d = |\alpha|-1$, and assume that the
corresponding coefficients are uniformly bounded. Also assume that
we have fixed all terms of degree $|\alpha|$ and index up to $j-1$
in the sequences $\{X_n\}$ and $\{G_n\}$ such that \eqref{next}
holds for index $1, \ldots , j-1$ and $d=|\alpha|$. Finally,
assume that all terms of degree $|\alpha|$, index $j$ and powers
$\beta > \alpha$ in the sequences $\{X_n\}$ and $\{G_n\}$ are
fixed such that \eqref{next} holds for the terms of degree
$|\alpha|$, index $j$ and power $\beta$.

We will continue to choose the terms of index $j$ and power
$\alpha$ in the sequences $\{X_n\}$ and $\{G_n\}$. It is clear
that after fixing the $c_{n, j, \alpha}$'s and $g_{n,j,\alpha}$'s,
Equation \eqref{next} will still hold for $d = |\alpha|-1$. Since
the linear parts of the $G_n$'s are lower triangular, it also
follows that \eqref{next} will still hold for degree $d =
|\alpha|$ and index $1, \ldots, j-1$. Furthermore, it follows from
the fact that $F_n^\prime(0)$ is lower triangular that
\eqref{next} will still hold for the terms of degree $d =
|\alpha|$, index $j$ and all powers $\beta$ with $\beta > \alpha$.

Once some $c_{n-1,j, \alpha}$ is chosen, it follows from
\eqref{next} that we must have
\begin{eqnarray}\label{constant}
c_{n, j, \alpha} = \lambda_{n,j} \lambda_n^{-\alpha} c_{n-1,j,
\alpha} + g_{n, j, \alpha} \lambda_n^{-\alpha}+ C_{n,j,\alpha}
\end{eqnarray}
where the constant $C_{n,j,\alpha}$ depends on the terms of
$f_{n}^{-1}$ (which are uniformly bounded for any fixed degree)
and the terms of $G_{n}$ and $X_{n-1}$ that we have already fixed
(which are also uniformly bounded per hypothesis).

If the term of index $j$ and power $\alpha$ is lower triangular
(i.e. if $\alpha_i = 0$ for $i \in \{j, \ldots k\}$), then we can
choose $c_{n,j, \alpha} = 0$ for all $n \in \NN$ and we choose
$g_{n, j, \alpha} = -C_{n, j, \alpha}\lambda_n^\alpha$ such that
\eqref{constant} holds for every $n \in \NN$. It follows that the
constants $g_{n,j, \alpha}$'s are bounded.

If the term of index $j$ and power $\alpha$ is not lower
triangular , then it follows from the hypothesis that the sequence
$\{F_n\}$ is uniformly attracting that $|\lambda_{n,j}
\lambda_n^{-\alpha}| > \xi > 1$ (where $\xi$ is as in Definition
\ref{attracting}). We also have that $g_{n,j, \alpha} = 0$ for all
$n \in \NN$, and it follows that the sequence of maps given by
$c_{n-1,j, \alpha} \mapsto c_{n,j, \alpha}$ is a uniformly bounded
sequence of affine maps on $\CC$ that are all expanding. It
follows from Lemma \ref{affine} that there exists a bounded
sequence $c_{n,j,\alpha}$.

So whether the term is lower triangular or not, we can always
choose bounded sequences $\{x_{n,j,\alpha}\}$ and
$\{g_{n,j,\alpha}\}$ such that \eqref{next} is satisfied.

The Proposition follows by induction.
\end{proof}

Now let $p$ be so large that for every $n \in \NN$ the eigenvalues
$\lambda_{n,1}, \ldots \lambda_{n,k}$ of $F_n^\prime(0)$ satisfy
$\xi |\lambda_{n,i}|^p < |\lambda_{n,j}|$ for $i,j \in \{1, \ldots
, k\}$, where $\xi > 1$. We can do this because the sequence
$\{F_n\}$ is uniformly attracting. We construct sequences
$\{G_n\}$ and $\{X_n\}$ as in Proposition \ref{bounded} for $d =
p$. It follows from part (a) of Lemma \ref{upper} that there
exists a $\gamma>1$ such that
$$
\|G(n)^{-1}(w) - G(n)^{-1}(w^\prime)\| \le \gamma^n \|w -
w^\prime\|,
$$
for any $w, w^\prime \in \B$. Recall that we assumed that there is
a constant $b < 1$ such that $\|F_n(z)\| < b \|z\|$ for any $z \in
\B$. Now fix an integer $q$ such that $\gamma b^q<\alpha<1$. We
change the mappings $X_n$ by adding higher order terms such that
\eqref{difference} holds for $d=q+1$. Since we chose $p \in \NN$
such that $|\lambda_i|^p < \xi |\lambda_j|$ holds for all $i,j$,
we have that $|\lambda_{n,j} \lambda_n^{-\alpha}| > \xi$ in
\eqref{constant} even for the terms that are lower triangular.
Hence we can make sure that the altered sequence $\{X_n\}$ is
bounded without changing the sequence $\{G_n\}$. Therefore we have
that
\begin{align}\label{done}
\|G_n^{-1}X_nF_n(z) - X_{n-1}(z)\| \le C\|z\|^{q+1},
\end{align}
for some $C > 0$ independent of $n \in \NN$ and every $z \in \B$.

The proof of Theorem \ref{stable} now follows quickly from an
argument similar to that of the proof of Theorem \ref{RR} which
can be found in the appendix of \cite{rr}.

Define the maps $\Psi_n : \Omega \rightarrow \CC^k$ by
\begin{eqnarray*}
\Psi_n := G(n)^{-1} \circ X_n \circ F(n).
\end{eqnarray*}
We will show that the maps $\Psi_n$ converge uniformly on compact
subsets of $\Omega$ to a biholomorphic map from $\Omega$ onto
$\CC^k$.

Since the sequence $\{X_n\}$ is bounded there is a radius $r$ such
that all the maps $X_n$ are invertible on $B(r)$.

Let $K$ be a compact set in $\Omega$. Then there is an $m \in \NN$
such that $F(m)(K) \subset B(r)$. Let $n \ge m$. Then we have that
$\|F(m,n)(z)\| \le b^{n-m} \|z\|$, for all $z \in B(r)$.

We notice that
\begin{align}\label{small}
\|\Psi_{n+1}(z) - \Psi_n(z)\| &= \| \left(G(n)^{-1}
G_{n+1}^{-1} X_{n+1} F_{n+1} - G(n)^{-1} X_n \right) F(n)(z)\|\\
&\le  \label{smaller} C \gamma^n (b^{n-m}\|F(m)(z)\| )^{q+1} <
\tilde{C} \alpha^n\|F(m)(z)\|^{q+1}.
\end{align}

Since the sequence $\{\alpha^n\}$ is summable, the maps $\Psi_n$
converge uniformly on compact subsets of $\Omega$ to a holomorphic
map $\Psi$. Also, for any compact subset $K$ of $\Omega$, there is
a large $N \in \NN$ such that for $n \ge N$, $\Psi_n$ is
biholomorphic on K. It is a well known fact that the limit of a
convergent sequence of biholomorphic mappings is either injective
or degenerate everywhere. We have that $\Psi_n^\prime (0) = I$ for
all $n$, therefore we have that  $\Psi^\prime(0) = I$ and the
limit map $\Psi$ is injective.

To prove surjectivity of the map $\Psi$, we may assume that we
have chosen $r$ so small that
$$
r^{q} \sum_{n=1}^\infty (C \alpha^n) < \frac{1}{2}.
$$
It follows from estimates \eqref{small} and \eqref{smaller} that
for $z \in B(r)$ and for any $n > m \ge 1$ we have that
$$
\|G(m,n)^{-1}X_n F(m,n)(z) - z \| \le \frac{\|z\|}{2}
$$
Now let $K_m$ be the compact subset of the basin of attraction
such that $F(m)(K_m)= B(r)$. Then we have that $G(m,n)^{-1}X_n
F(n)(K_m) \supset B(r/2)$ for any $n
>m$. It follows from part (b) of Lemma \ref{upper} that for every
$R>0$ there exist an $N \in \NN$ such that for $m \ge N$ we have
that
$$
B(R) \subset G(m)^{-1}\left(B(r/2)\right).
$$
Therefore $B(R) \subset \Psi(K_m)$ for large enough $m$, and thus
we have that $\Psi(\Omega)$ contains balls around the origin with
arbitrarily large radii. This completes the proof.
\begin{thebibliography}{[FB]}
\bibitem[Be]{be} Bedford, E. Open problem session of the {\emph{Biholomorphic Mappings}}
meeting at the American Institute of Mathematics, Palo Alto, CA,
July 2000.
\bibitem[FSt]{fst} Forn\ae ss, J-E. Stens\o nes, B. {\emph{Stable Manifolds of Holomorphic Hyperbolic Maps.}}
 To appear in I.J. math.
\bibitem[JV]{jv} Jonsson, M, Varolin, D. {\emph{Stable Manifolds of
Holomorphic Diffeomorphisms.}} Invent. Math. 149 (2002), 409-430.
\bibitem[PW]{pw} Peters, H, Wold, E. F. {\emph{Non-Autonomous
Basins of Attraction with $4$-dimensional boundaries.}} preprint.
\bibitem[RR]{rr} Rosay, J-P, Rudin, W. {\emph{Holomorphic maps
from $\CC^n$ to $\CC^n$.}} Trans. Amer. Mat. Soc. 310 (1988),
47-86.
\bibitem[Sg]{sg} Sternberg, S., {\emph{Local contractions and a theorem
of Poincar\'{e}.}} Amer. J. Math. 79, (1957), 809-824.
\bibitem[Wd]{wd} Wold, E. F. {\emph{Fatou-Bieberbach domains.}} preprint.
\end{thebibliography}

\end{document}








We identify the space of polynomials $X = I + h.o.t.$ of degree
$d$ as $\CC^N$ for some large $N$, and we will assign a specific
ordering to the terms of $X$ of degree $2$ and higher.

We write
$$
X_n(z) = I(z) + X_{n,2}(z) + X_{n,3}(z) + \ldots + X_{n,d}(z),
$$
where $X_{n,j}$ is the homogeneous part of $X_n$ of degree $j$.
Similarly we write
$$
G_n(z) = G_{n,1}(z) + G_{n,2}(z) + \ldots + G_{n,d}(z),
$$
where $G_{n,1}(z) = F_n^\prime(0)(z)$.

Suppose that we have fixed the maps for some degree $p$ such that
\eqref{difference} holds for $p$. Then this equation will still
hold when we go on to fix the degree $p+1$ part, because the
equation $X_n = [G_n \circ X_{n-1} \circ f_n^{-1}]_{p}$ will still
hold. Therefore we can fix the maps degree by degree, starting
with all the $X_{n,2}$'s and $G_{n,2}$'s, then the $X_{n,3}$'s and
$G_{n,3}$'s and so forth.

For some degree $p$ we write
$$
X_{n,p}(z) = (x_{n,p,1}(z), x_{n,p,2}(z), \ldots , x_{n,p,k}(z)).
$$
We will say that $x_{n,p,j}$ has index $j$. Similarly we write
$$
G_{n,p}(z) = (g_{n,p,1}(z), g_{n,p,2}(z), \ldots , g_{n,p,k}(z)).
$$
Suppose that we have fixed the first $j$ indices of $X_{n,p}$ and
$G_{n,p}$ such that \eqref{difference} holds for these indices.
Notice that when we change the indices $j+1$ through $k$, the
equation $X_n = [G_n \circ X_{n-1} F_n^{-1}]_{p}$ will still hold
for the first $j$ indices, because the maps $G(n)$ will be lower
triangular. Therefore we can fix the maps index by index for each
degree, again starting with all the $x_{n,p,1}$'s and
$g_{n,p,1}$'s, then the $x_{n,p,2}$'s and $g_{n,p,2}$'s and so
forth.

We still need to order the coefficients of the monomials of the
$X_n$'s and $G_n$'s for some fixed degree and index. We write
$$
\left(F_n^{-1}\right)^\prime(0)(z) = (\lambda_{n,1}^{-1}z_1,
a_{n,2,1}z_1 + \lambda_{n,2}^{-1}z_2, \cdots).
$$
Consider some monomial $P(z) = \beta z^\alpha = \beta
z_1^{\alpha_1}\cdots z_k^{\alpha_k}$. When we compose $P$ with
$F^{-1}$ we get
\begin{eqnarray} \label{composition}
P\circ F^{-1}(z) = \beta (\lambda_{n,1}^{-1} z_1)^{\alpha_1}
\cdots (a_{n,k,1} z_1 + \cdots + \lambda_{n,k}^{-1}
z_k)^{\alpha_k} + h.o.t.
\end{eqnarray}

We get a large sum of monomials of degree $|\alpha|$, but notice
that each monomial comes earlier in the alphabet than $P$ or is a
constant times $P$ (with alphabet we mean that we use the
lexicographical ordering on the monomials of degree $|\alpha|$,
starting with $z_1^{|\alpha|}$ and ending with $z_k^{|\alpha|}$).
Therefore we can use the reverse of the lexicographical ordering
for the terms of some fixed degree and index, and we can fix the
coefficients one by one, similarly as we did for the degrees and
the index.

With this ordering on all the terms of the maps $X_n$ up to some
fixed degree, we have that $X \mapsto [G_n X F_n^{-1}]p$ is a
lower triangular linear map on $\CC^N$, i.e. $(z_1, \ldots z_N)
\mapsto (z_1^\prime, \ldots z_N^\prime)$, where

\begin{align}\label{ordering}
z_j^\prime = c_{n,j} z_j + l_{n,j}(z_1, \ldots z_{j-1}).
\end{align}

Now notice from \eqref{composition} that $c_{n,j} = \lambda_{n,k}
\lambda_n^{-\alpha}$, where $k$ and $\alpha$ are resp. the index
and the power of the corresponding term in the $X_n$'s. We assumed
that the sequence $F_1, F_2, \ldots$ is uniformly attracting,
therefore it follows from \eqref{eigenvalues} that $|c_{n,j}| >
> \eta > 1$ unless the corresponding term in the $X_n$'s is a lower triangular term (in
which case $|c_{n,j}|$ may or may not be larger than $1$). So now
we fix all the coefficients one by one. In the case that for every
$n$ we have that $|c_{n,j}| > \eta > 1$, it follows from Lemma
\ref{bounded} that we can find a bounded sequence of coefficients
in the $X_n$'s, and we choose the corresponding coefficients in
the $G_n$'s equal to $0$. If the corresponding term is lower
triangular then we can choose the coefficients in the $X_n$'s
equal to $0$ and we can choose bounded coefficients in the $G_n$'s
such that the maps still satisfy equation \eqref{difference}.

We have proved the following result:

\begin{proposition}\label{bounded}
For any $p \in \NN$, we can construct bounded sequences $\{G_n\}$
and $\{X_n\}$ that satisfy equation \eqref{difference} for every
$n \NN$.
\end{proposition}

math0411339.extracted_bib
\begin{thebibliography}{[FB]}
\bibitem[Be]{be} Bedford, E. Open problem session of the {\emph{Biholomorphic Mappings}}
meeting at the American Institute of Mathematics, Palo Alto, CA,
July 2000.
\bibitem[FSt]{fst} Forn\ae ss, J-E. Stens\o nes, B. {\emph{Stable Manifolds of Holomorphic Hyperbolic Maps.}}
 To appear in I.J. math.
\bibitem[JV]{jv} Jonsson, M, Varolin, D. {\emph{Stable Manifolds of
Holomorphic Diffeomorphisms.}} Invent. Math. 149 (2002), 409-430.
\bibitem[PW]{pw} Peters, H, Wold, E. F. {\emph{Non-Autonomous
Basins of Attraction with $4$-dimensional boundaries.}} preprint.
\bibitem[RR]{rr} Rosay, J-P, Rudin, W. {\emph{Holomorphic maps
from $\CC^n$ to $\CC^n$.}} Trans. Amer. Mat. Soc. 310 (1988),
47-86.
\bibitem[Sg]{sg} Sternberg, S., {\emph{Local contractions and a theorem
of Poincar\'{e}.}} Amer. J. Math. 79, (1957), 809-824.
\bibitem[Wd]{wd} Wold, E. F. {\emph{Fatou-Bieberbach domains.}} preprint.
\end{thebibliography}