\documentclass[reqno]{amsart}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small {\em Electronic Journal of
Differential Equations}, Vol. 2003(2003), No. 38, pp.1--9.
\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/38\hfil Multidimensional singular $\lambda$-lemma ]
{Multidimensional singular $\lambda$-lemma}

\author[Victoria Rayskin \hfil EJDE--2003/38\hfilneg]
{Victoria Rayskin}

\address{Victoria Rayskin \hfill\break
       Department of Mathematics\\
       MS Bldg, 6363\\
       University of California at Los Angeles, 155505\\
       Los Angeles, CA 90095, USA}
\email{vrayskin@math.ucla.edu}

\date{}
\thanks{Submitted November 4, 2002. Published April 11, 2003.}
\subjclass[2000]{37B10, 37C05, 37C15, 37D10}
\keywords{Homoclinic tangency, invariant manifolds, $\lambda$-Lemma,
\hfill\break\indent
order of contact, resonance}

\begin{abstract}
 The well known $\lambda$-Lemma \cite{Pa} states the  following: 
 Let $f$ be a $C^1$-diffeomorphism of $\mathbb{R}^n$ with a
 hyperbolic fixed point at $0$ and $m$- and $p$-dimensional stable
 and unstable manifolds $W^S$ and $W^U$, respectively ($m+p=n$).
 Let $D$ be a $p$-disk in $W^U$ and $w$ be another $p$-disk in
 $W^U$ meeting $W^S$ at some point $A$ transversely. Then
 $\bigcup_{n\geq 0} f^n(w)$ contains $p$-disks arbitrarily
 $C^1$-close to $D$. In this paper we will show that the same
 assertion still holds outside of an arbitrarily small neighborhood
 of $0$, even in the case of non-transverse homoclinic
 intersections with finite order of contact, if we assume that $0$
 is a low order non-resonant point.
\end{abstract}

\maketitle

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{claim}[theorem]{Claim}

\section{Introduction}

Let $M$ be a smooth manifold without boundary and  $f:M\to M$ be a  $C^1$
map that has a hyperbolic fixed point at the origin.
The well known ${\lambda}$-Lemma \cite{Pa} gives an important description
of chaotic dynamics. The basic assumption of this theorem is the presence
of a transverse homoclinic point.

\begin{theorem}[Palis]\label{Palis}
Let $f$ be a $C^1$ diffeomorphism of $\bf R^n$ with a hyperbolic fixed point at
$0$ and $m$- and $p$-dimensional stable and unstable manifolds $W^S$ and $W^U$
($m+p=n$). Let $D$ be a $p$-disk in $W^U$, and $w$ be another $p$-disk in $W^U$
meeting $W^S$ at some point $A$ {\em transversely}.
Then $\bigcup_{n\geq 0} f^n(w )$ contains $p$-disks arbitrarily
$C^1$-close to $D$.
\end{theorem}

The assumption of transversality is not easy to verify for a concrete
dynamical system. Obviously,  the conclusion of the Theorem of Palis
is not true for an arbitrary degenerate (non-transverse) crossing.
Example by Newhouse illustrates this situation (See picture~\ref{newhousePic}).


\begin{figure}[th]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1.eps}
\end{center}
\caption{Newhouse example. Branches of $W_U$ are not $C^1$-close near $0$}
\label{newhousePic}
\end{figure}

In this paper we prove
an analog of the ${\lambda}$-Lemma for the non-transverse case in arbitrary
dimension. Suppose $W^S$ and $W^U$ are sufficiently smooth and cross
non-transversally at an isolated homoclinic point, i.e. they have a
{\it singular homoclinic crossing}.
In Section~\ref{2} we define the order of contact for this crossing
(Definition~\ref{def-ooc-manifold}) and show that it is preserved under
 a diffeomorphic transformation (Lemma~\ref{multidim-preserv-ooc-lemma}).
We prove Singular ${\lambda}$-Lemma for the case of singular finite
order homoclinic crossing of manifolds which have a graph portion
(see Definition~\ref{def-p-p}), under non-resonance restriction. See
Lemma~\ref{lemma} in Section~\ref{3}.

\section{Definitions and Lemmas}\label{2}

In this section we are considering two immersed $C^r$ manifolds in
$\mathbb{R}^n$, $r>1$. Suppose they meet at an isolated point $A$.
We will discuss the structure of these manifolds in the neighborhood of
the point $A$. First, assume that each manifold is a curve.

 Hirsch in his work \cite{Hi} describes the order of contact for two
 curves and formulates the following definition:

\begin{definition}\label{def-ooc-Hirsch} \rm
Let ${\Lambda}_i$ ($i=1,2$) denote two immersed $C^r$ curves in $\mathbb{R}^2$ , $r>1$. Suppose the two curves meet at point $A$. Let $t\mapsto u_i(t)$ be a $C^r$ parameterization of ${\Lambda}_i$, both defined for $t$ in some interval $I$, with non-vanishing tangent vectors $u_i'(t)$. Suppose $0\in I$ and $A = u_i(0)$. The order of contact of the two curves at $A$ is the unique real number $l$ in the range $1\leq l \leq r$, if it exists, such that $u_1 - u_2$ has a root of order $l$ at $0$.
\end{definition}
For our higher-dimensional proof we can reformulate this definition for
two curves in $\mathbb{R}^n$:

\begin{definition}\label{def-ooc-curve} \rm
Let ${\Lambda}_i$ ($i=1,2$) denote two immersed $C^r$ curves in $\mathbb{R}^n$,
$r>1$. Suppose the two curves meet at point $A$. Let $t\mapsto u_i(t)$
be a $C^r$ parameterization of ${\Lambda}_i$, both defined for $t$ in
some interval $I$, with non-vanishing tangent vectors $u_i'(t)$.
Suppose $0\in I$ and $A = u_i(0)$. The order of contact of the two curves
at $A$ is the unique real number $l$ in the range $1\leq l \leq r$, if it
exists, such that $| u_1 - u_2|$ has a root of order $l$ at $0$.
\end{definition}

Now we can define the order of contact for two manifolds of arbitrary
dimensions.

\begin{definition}\label{def-ooc-manifold} \rm
Let $W^S$ and $W^U$ denote two immersed $C^r$ manifolds in $\mathbb{R}^n$,
$r>1$. Suppose the two manifolds meet at an isolated point $A$. The order
of contact $\alpha$ at $A$ is the unique real number $\alpha$ in the
range $1\leq \alpha \leq r$, if it exists, such that
\begin{align*}
\alpha = \sup \big\{& l| C^r\mbox{-curve } {\gamma}_1 \in W^S
\mbox{ has order of contact } l \mbox{ with another }\\
&C^r\mbox{-curve } {\gamma}_2 \in W^U\ \mbox{and } A\in {\gamma}_1
\cap {\gamma}_2\big\}
\end{align*}
\end{definition}

The order of contact is preserved under a diffeomorphism. This result
is first proven for curves (Lemma~\ref{preserv-ooc-lemma}).

\begin{lemma}\label{preserv-ooc-lemma}
Consider a $C^\infty $ surface without boundary and a  $C^r$
diffeomorphism $\phi$ that maps a neighborhood $N'$ of this surface onto
some neighborhood $N \subset \bf R^2$.
Assume that $u(t), v(t)$ are  $C^r$ curves, such that $u(0)=v(0)$.
Then, $\phi$ preserves the order of contact of these curves.
\end{lemma}

\begin{proof}
Without lost of generality, we assume that $u(0)=v(0)=0$.
We have curves
 \[
 \phi \circ u(t),\quad  \phi \circ v(t),
 \]
transformed by the diffeomorphism $\phi$.
There are positive constants $m$ and $M$ such that
 \[
m\leq \frac {| u(t)-v(t)| }{| t|^{l}}
 \leq M, \quad\mbox{as }t\to 0.
 \]
By the $ C^1 $   Mean  Value  Theorem,
 \[
 \phi (x)- \phi (y) =\Big[ \int_{0}^{1}(D \phi )_{\sigma (s)} ds \Big] (x-y),
 \]
where $\sigma (s) =(1-s)x +sy$.
Then
 \[
 (\phi\circ u )(t)-(\phi\circ v )(t)=\Big[ \int_{0}^{1}(D \phi )_{\sigma (s)} d
s \Big] (u(t)-v(t)),
 \]
where $\sigma (s) =(1-s)u(t)+sv(t)$.
Therefore,
  \[
 \frac{(\phi\circ u )(t)-(\phi\circ v )(t)}{t^l}=\Big[ \int_{0}^{1}(D \phi )_{
\sigma (s)} ds \Big] ( \frac{u(t)-v(t)}{t^l} )
 \]
As $t \to 0$, $\sigma (s) \to u(0)$ and the matrix  $ \int_{0}^{1}(D \phi )_{\sigma (s)} ds $
tends to the invertible matrix $ (D \phi )_{u(0)} $.
The ratio $ \frac{u(t)-v(t)}{t^l} $ is a vector whose norm is bounded by $M$ and $m$, $0<m \leq M <\infty $. Hence
  \[m \leq
  \Big[ \int_{0}^{1}(D \phi )_{\sigma (s)} ds \Big] \big( \frac{u(t)-v(t)}{t
^l} \big)
  \leq M.
  \]
\end{proof}
This lemma can easily be generalized for higher dimensions.

\begin{lemma}\label{multidim-preserv-ooc-lemma}
Consider a $C^\infty $ surface without boundary and a  $C^r$ diffeomorphism
$\phi$ that maps a neighborhood $N'$ of this surface onto some neighborhood
$N \subset \bf R^n$.
Assume that $u(t), v(t)$ are  $C^r$ manifolds, such that $u(0)=v(0)$.
Then, $\phi$ preserves the order of contact of these manifolds.
\end{lemma}

This Lemma follows from Lemma~\ref{preserv-ooc-lemma} and
Definition~\ref{def-ooc-manifold}.

For the estimates in the proof of the Singular $\lambda$-Lemma we need the
following definition of a graph portion.

\begin{definition}\label{def-p-p} \rm
Let $f$ be a $C^r$ diffeomorphism of $\bf R^n$ with a hyperbolic fixed point at
the origin. Denote by $W^S$ (resp., $W^U$) the associated stable (resp.,
unstable) manifold, and by $m$ (resp., $p$) its dimension ($m+p=n$, $p<m$). Let $A$ be
a homoclinic point of $W^S$ and $W^U$. Suppose that there exists a small
$p$-disk in $W^U$ around point $A$ (call it $\mathcal{U}$), and there exists
another small $p$-disk in $W^U$ around the origin (call it $\mathcal{V}$).
Define a local coordinate system $E_1$ at $0$, which spans $\mathcal{V}$. Similarly, define a local coordinate system $E_2$ in some neighborhood of $0$ (we can assume that $A$ belongs to this neighborhood), centered at $0$, which spans $W^S$ in this neighborhood. Let $E = E_1 + E_2$.
If $\mathcal{U}$ is a graph of a bijective (in $E$) function defined on $\mathcal{V}$, then $\mathcal{U}$ will be called a graph portion.
\end{definition}

\begin{figure}[th]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2.eps}
\end{center}
\caption{In this picture the iterated part of the $W^U$ manifold is not 
a graph portion of the manifold $W^U$. It will not become $C^1$-close 
to the bottom part with the iterations.}
\label{fig_proj_part}
\end{figure}

There is another assumption that we have to make for the proof of our
$\lambda$-Lemma. The assumption is stronger than the regular first order
non-resonance condition, but weaker than the second order non-resonance.
We will call our restriction one-and-a-half order resonance.

\begin{definition} \label{def5} \rm
Let $f$ be a $C^2$-diffeomorphism of $\mathbb{R}^n$ with a hyperbolic fixed
point at $0$ and $m$- and $p$-dimensional stable and unstable manifolds,
and $f(x,y): \bf R^n \to \bf R^n $ has the linear part
$( ( \mathcal{A}x )_1,\dots,( \mathcal{A}x )_p,( \mathcal{B}y )_1,
\dots,( \mathcal{B}y )_m)$.
Then, the following condition will be called one-and-a-half order
non-resonance condition:
\\
If $a \in \mathop{\rm spec}\mathcal{A}$ and $b \in \mathop{\rm spec}
\mathcal{B}$, then
$a b \notin ( spec\mathcal{A}\cup  \mathop{\rm spec}\mathcal{B}) $.
\end{definition}

\section{Singular $\lambda$-Lemma}\label{3}

Using the above definitions we formulate the following Singular
${\lambda}$-Lemma.

\begin{lemma}\label{lemma}
Let $f$ be a $C^r$-diffeomorphism of $\mathbb{R}^n$ with a hyperbolic fixed point
at $0$ and $m$- and $p$-dimensional stable and unstable manifolds $W^S$ and
$W^U$ ($p\leq m$, $m+p=n$). Let $\mathcal{V}$ be a $p$-disk in $W^U$ and
${\Lambda}$
be a graph portion in $W^U$ having a homoclinic crossing with $W^S$ at some
point $A$. Assume that ${\Lambda}$ and $W^S$ have order of contact $r$
($1 < r <\infty$) at $A$.
Also assume that $f$ is one-and-a-half order non-resonant.
Then for any $\rho >0$, for an arbitrarily small
$\epsilon$-neighborhood $\mathcal{U} \subset \mathbb{R}^n$ of the origin
and for the
graph portion ${\Lambda}$, $(\bigcup_{n\geq 0} f^n({\Lambda}
))\setminus \mathcal{U}$ contains disks $\rho$-$C^1$ close to
$\mathcal{V} \setminus \mathcal{U}$.
\end{lemma}

\begin{remark} \label{rmk3} \rm
There is no loss of generality to assume that $p\leq m$, because we can always replace $f$ with $f^{-1}$.
\end{remark}

\begin{figure}[th]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(70,54)(-8,-6)
\put(-5,0){\line(1,0){65}}
\put(59,-.85){$\to$}
\put(0,-5){\line(0,1){50}}
\put(-.85,45){$\downarrow$}
\put(-.85,-5){$\uparrow$}
\qbezier(0,10)(0,15)(60,15)
\qbezier(0,20)(2,28)(40,28)
\qbezier(0,30)(4,38)(20,40)
\put(2,45){Y}
\put(-4,29){$A$}
\put(-8,19){$f(A)$}
\put(-13,9){$f(f(A))$}
\put(-8,1.5){$W_U$}
\put(1,-6){$W_S$}
\put(17,42){$\Lambda(x)$}
\put(30,30){$f(x,\Lambda(x))$}
\put(45,17){$f(f(x,\Lambda(x)))$}
\put(60,-4){$X$}
\put(-7,-.85){$\leftarrow$}
\end{picture}
\end{center}
\caption{Iterations of the graph portion $\Lambda$ with the diffeomorphism $f$}
\end{figure}

\begin{proof}[Proof of Lemma \ref{lemma}]
Let $\alpha = 1/l$ ($0 < \alpha <1$).
Since ${\Lambda}$ is a graph portion that has finite order of contact with $W^S$, we can assume that locally ${\Lambda}$ is represented by the graph of the following form:
\[
{\Lambda}(x) = A+r(x):\mathbb{R}^p \to \mathbb{R}^m, \quad r(0)=0,
\]
and for any sufficiently small $\sigma >0$
\[
 | r(x)| \leq \mathop{\rm const} \cdot | x|^\alpha\quad \mbox{and}\quad
 | \frac{\partial}{\partial x_i}r(x)| \leq \mathop{\rm const} \cdot | x|^{\alpha -1}\
\]
for all $| x| < \sigma$, $ i=1,\dots,p$.
Let $x=(x_1,\dots,x_p)\in \bf R^p$, $y=(y_1,\dots,y_m)\in \bf R^m$
($p+m=n$) and $f(x,y): \bf R^n \to \bf R^n $ has the linear part
$$
((\mathcal{A}x )_1,\dots,(\mathcal{A}x )_p,(\mathcal{B}y )_1,
\dots,( \mathcal{B}y )_m).
$$
 Assume that $\| \mathcal{A}^{-1}\|$,
$\| \mathcal{B} \| < {\lambda} <1$. Choose an arbitrarily small $\Delta$.
If there is a cross terms $\mathop{\rm const}\cdot x_i y_j$ in the power
expansion of this map around $0$, then we assume one-and-a-half-order
non-resonance condition. Then, by Flattening Theorem (See \cite{ABZ})
there exists smooth change of coordinates, such that locally $f$ can be
written in the form $f(x,y)=(S_1(x,y), S_2(x,y))$, where
\begin{align*}
S_1(x,y)=
&\Big( \big( (\mathcal{A}x)_1 + {\phi_1}(x) +
\sum_{i=1,\dots,p;j=1,\dots,m}x_i y_j
U_{ij}^1 (x,y)\big),\dots ,\\
&\big( (\mathcal{A}x)_p + {\phi_p}(x) +
\sum_{i=1,\dots,p;j=1,\dots,m}x_i y_jU_{ij}^p (x,y)\big) \Big)
\end{align*}
and
\begin{align*}
S_2(x,y)=
&\Big( \big( (\mathcal{B}y)_1 + {\psi_1}(y) +
\sum_{i=1,\dots,p;j=1,\dots,m}x_i y_j V_{ij}^1 (x,y)\big),\dots ,\\
&\big( (\mathcal{B}y)_m + {\psi_m}(y) +
 \sum_{i=1,\dots,p;j=1,\dots,m}x_i y_jV_{ij}^m (x,y)\big)\Big) .
\end{align*}
Here $U(0) = V(0) = 0$, $\|{\phi} \|_{C^1},\;
\| {\psi} \|_{C^1},\; \| U\|_{C^0},\;  \| V \|_{C^0}\leq \Delta$, and
$\| U\|_{C^1},\  \| V \|_{C^1}$ are bounded.


Consider $f(x, {\Lambda}(x))=(T_1^{\Lambda}(x),T_2^{\Lambda}(x))$.
We will work with
$(x,T_2^{\Lambda} \circ (T_1^{\Lambda})^{-1}(x))$ and deduce that
$f^n(x, {\Lambda}(x))$ is $C^1$-small for $n$ big enough and $\sigma >0$
sufficiently small.
First we will show that in $C^1$-topology $(T_1^{\Lambda})^{-1}$ is
$\Delta$-close to $\mathcal{A}^{-1}$.
For simplicity we will denote $T_1^{\Lambda}$ by $T_1$ and $T_2^{\Lambda}$ by
$T_2$.
\begin{align*}
T_1(x)=
&\Big(  (\mathcal{A}x)_1 + {\phi_1}(x)
+ \sum_{i=1,\dots,p;j=1,\dots,m}x_i {\Lambda}_j(x)
U_{ij}^1 (x,{\Lambda}(x)),\dots, \\
&(\mathcal{A}x)_p + {\phi_p}(x) +
\sum_{i=1,\dots,p;j=1,\dots,m}x_i {\Lambda}_j(x)U_{ij}^p (x,{\Lambda}(x))
\Big).
\end{align*}

\begin{claim}
\[
\| \sum_{i=1,\dots,p;j=1,\dots,m}x_i
{\Lambda}_j(x) U_{ij}^t (x,{\Lambda}(x))\|_{C^1} < K\cdot\Delta
\] for $| x | < \sigma$  ($\sigma >0 $ sufficiently small, $K >0$).
\end{claim}
\begin{proof}
Fix some $l\in \{1,\dots,p\}$. Recall that $\Lambda (x) = A + r(x)$.
\begin{align*}
\big|\frac{\partial}{\partial x_l}x_i {\Lambda}_j(x)\big|
&\leq \delta_{il}| {\Lambda}(x)| + | x_i| \cdot| \frac{\partial}{\partial x_l}{\Lambda}_j(x)|\\
&\leq \delta_{il}(| A|+ | x|^\alpha )+| x| \cdot O(1) | x|^{\alpha-1}\\
&\leq | A|\delta_{il}+(\delta_{il}+O(1))| x|^\alpha
= O(1)
\end{align*}
Here
\[
\delta_{il}= \begin{cases}
1 & \mbox{if }  i=l,\\
0 & \mbox{if }  i\neq l. \end{cases}
\]
Through the proof of this Theorem, $O(1)$ will be the set
\begin{align*}
O(1) =& \big\{ \gamma (\zeta ) : \mathbb{R} \mapsto \mathbb{R}
 \mbox{ such that there exists a positive constant $c$ with }\\
&| \gamma (\zeta )| \leq c \mbox{ for all sufficiently small }\zeta \big\}
\end{align*}
Also,
\[
\big| \frac{\partial}{\partial x_l}U_{ij}^t(x,{\Lambda}(x))\big|
= \big| \frac{\partial}{\partial x_l}U_{ij}^t(x,y) + \sum_{k=1}^{m}
\frac{\partial}{\partial y_k}U_{ij}^t(x,y) \cdot \frac{\partial}{\partial x_l}{\Lambda}_k(x)\big| = O(1)\,.
\]
Therefore,
\begin{align*}
&\big\| \sum_{i=1,\dots,p;j=1,\dots,m}x_i
{\Lambda}_j(x) U_{ij}^t (x,{\Lambda}(x))\big\|_{C^1} \\
&\leq
\sum_{i=1,\dots,p;j=1,\dots,m}\big|
\sum_{l=1}^{p}\frac{\partial}{\partial x_l}(x_i {\Lambda}_j(x)
U_{ij}^t (x,{\Lambda}(x)))\big| \\
&\leq
\sum_{i=1,\dots,p;j=1,\dots,m}\sum_{l=1}^{p}\big|\frac{\partial}{\partial x_l}(x_i {\Lambda}_j(x))\cdot
U_{ij}^t (x,{\Lambda}(x)) + x_i {\Lambda}_j(x)\cdot
 \frac{\partial}{\partial x_l}U_{ij}^t (x,{\Lambda}(x))\big|\\
&\leq
\Delta \cdot O(1),
\end{align*}
if $\sigma$ is sufficiently small and $| x| < \sigma$
(Arbitrarily small $\Delta$ was chosen above). The estimate proves
the claim.
\end{proof}

Now, we continue the proof of Lemma~\ref{lemma}.
As it was noted earlier in the proof, $\|{\phi} \|_{C^1}\leq \Delta$,
by Flattening Theorem.
This estimate and the assertion of the Claim imply that
$\| \mathcal{A} - T_1 \|_{C^1} = O(1) \cdot \Delta$.
This obviously implies $\| \mathcal{A}^{-1} - T_1^{-1} \|_{C^1} = O(1)
 \cdot \Delta$.

Now we can do the main estimate, -- the estimate for $\| T_2 \circ T_1^{-1} \|_{C^k} $ $(k=0,1)$.
\begin{align*}
T_2 \circ T_1^{-1} =
&\Big(( \mathcal{B} {\Lambda}( T_1^{-1}))_1 + {\psi_1}({\Lambda}( T_1^{-1}))
\\
&+\sum_{i=1,\dots,p;j=1,\dots,m}( T_1^{-1})_i ( {\Lambda}
( T_1^{-1}))_j
V_{ij}^1 ( T_1^{-1},{\Lambda}( T_1^{-1})),\dots,\\
&\quad ( \mathcal{B} {\Lambda}( T_1^{-1}))_m + {\psi_m}({\Lambda}( T_1^{-1}))\\
&+
\sum_{i=1,\dots,p;j=1,\dots,m}( T_1^{-1})_i ( {\Lambda}
( T_1^{-1}))_j V_{ij}^m ( T_1^{-1}, {\Lambda}( T_1^{-1}))\Big)
\end{align*}
We will begin by estimating each term of this vector.
\[
 \mathcal{B} {\Lambda}( T_1^{-1}) =  \mathcal{B} \cdot A +
 \mathcal{B} \cdot r( T_1^{-1}(x)).
\]
\[
| \mathcal{B} \cdot r( T_1^{-1}(x)) | =
O(1) \cdot \| \mathcal{B}\|  | T_1^{-1}(x)|^\alpha =
O(1) \cdot \| \mathcal{B}\| ( \| \mathcal{A}^{-1} \| +\Delta )^
\alpha | x|^\alpha.
\]
By the chain rule,
\begin{align*}
&\big| \frac{\partial}{\partial x_l}\mathcal{B} \cdot r( T_1^{-1}(x)) \big|\\
&=O(1) \cdot\| \mathcal{B}\|  \| T_1^{-1}\|_{C^1}
 | T_1^{-1}(x)|^{\alpha -1}\\
&=O(1) \cdot \| \mathcal{B}\|  ( \| \mathcal{A}^{-1}\| + \Delta )
( \| \mathcal{A}^{-1}\| + \Delta )^{\alpha -1}  | x |^{\alpha -1}\\
&=O(1) \cdot \| \mathcal{B}\| ( \| \mathcal{A}^{-1}\|
+ \Delta )^\alpha | x |^{\alpha -1}\\
&= O(1) \cdot{\lambda} | x |^{\alpha -1}
\end{align*}
with $\lambda <1$.
Moreover,
\[
| \frac{\partial}{\partial x_l}\mathcal{B}^n \cdot r( T_1^{-n}(x)) | =O(1)  \cdot\| \mathcal{B}\|^n
( \| \mathcal{A}^{-1}\|^n + \Delta )^\alpha | x |^{\alpha -1}
= O(1) \cdot {\lambda}^n | x |^{\alpha -1}
\]
This term can be made small if we perform enough iterations by the map $f$.
I.e., $( \mathcal{B}^n {\Lambda}T_1^{-n})_m$ is $C^1$-small outside of
a fixed neighborhood of $0$, if $n$ is big enough.
For the estimates of the next term one can use the following expansion:
\[
{\psi_1}({\Lambda}( T_1^{-1}(x))) = {\psi_1} ( A + r ( T_1^{-1}(x)) ) =
{\psi_1} ( A ) + D{\psi_1} ( A ) \cdot r( T_1^{-1}(x)) + R ( T_1^{-1}(x) ),
\]
where $R ( T_1^{-1}(x) ) = o(|( T_1^{-1}(x))^\alpha)|$.
Here the set $o(1)$ is the following set of functions:
\begin{align*}
o(1) =& \big\{ \gamma (\zeta ) : \mathbb{R} \mapsto \mathbb{R}
 \mbox{ such that for any positive constant $c$}\\
&\mbox{and  for all sufficiently small }\zeta <\sigma , | \gamma (\zeta )|
< c \big\}
\end{align*}
Similar to the previous calculations ${\psi_1}({\Lambda}( T_1^{-1}(x)))$
can be made small in $C^1$-norm if we perform enough iterations with the
map $f$.
Finally, we will note that the last term
\[
\sum_{i=1,\dots,p;j=1,\dots,m}( T_1^{-1})_i ( {\Lambda}
( T_1^{-1}))_j
V_{ij}^t ( T_1^{-1},{\Lambda}( T_1^{-1}))
\]
can be written as a composition
$ \Sigma^t \circ T_1^{-1}(x)$, where
\[
\Sigma^t (x) = \sum_{i=1,\dots,p;j=1,\dots,m}x_i {\Lambda}_j(x) V_{ij}^t (x,
{\Lambda}(x)) .
\]
Consider $\frac{\partial}{\partial x_l}\Sigma^t \circ T_1^{-1}(x)$.
\[
\frac{\partial}{\partial x_l}\Sigma^t \circ T_1^{-1}(x) = \sum_{i=1}^{p} \frac{\partial}{\partial x_i}\Sigma^t \circ T_1^{-1}(x)
\cdot \frac{\partial}{\partial x_l}(T_1^{-1}(x))_i.
\]
We have already shown that
\[
\big\| \sum_{i=1,\dots,p;j=1,\dots,m}x_i {\Lambda}_j(x) U_{ij}^t
(x,{\Lambda}(x))\big\|_{C^1}= O(1) \cdot \Delta .
\]
Similar, one can show that
\[
\| \Sigma^t \|_{C^1}=
\big\| \sum_{i=1,\dots,p;j=1,\dots,m}x_i {\Lambda}_j(x) V_{ij}^t
(x,{\Lambda}(x))\big\|_{C^1}= O(1) \cdot \Delta .
\]
Also,
\[
\| T_1^{-1} \|_{C^1} \leq \| \mathcal{A}^{-1} \|_{C^1} + \| T_1^{-1}-\mathcal{A}^{-1} \|_{C^1} \leq \| \mathcal{A}^{-1} \|_{C^1} + O(1) \cdot \Delta .
\]
The estimates on $\| \Sigma^t \|_{C^1}$ and $\| T_1^{-1} \|_{C^1}$, together with the fact that $T(0)= 0$, imply that
\[
\| \Sigma^t \circ T_1^{-1} \|_{C^1} = O(1) \cdot \Delta .
\]
Thus, for any small positive number $\rho$ and for any small (but bigger than a
fixed $\epsilon$) $| x |$ one can find $n$ such that
$(x,(T_2^{\Lambda})^{n} \circ (T_1^{\Lambda})^{-n}(x))$ is
$\rho$-$C^1$-close to $\mathcal{V}$. This implies that for any $\rho >0$
and for an arbitrarily small $\epsilon$-neighborhood
$\mathcal{U} \subset \mathbb{R}^n$ of the origin,
$(\bigcup_{n\geq 0} f^n({\Lambda} ))\setminus \mathcal{U}$
contains $p$-disks $\rho$-$C^1$-close to $\mathcal{V} \setminus \mathcal{U}$.
\end{proof}


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\end{document}