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\markboth{E. Fontich, R. de la Llave, P. Mart\'{\i}n}{Prefoliations for
nonresonant systems}
\title[Prefoliations for nonresonant systems]{Invariant
prefoliations for nonresonant
nonuniformly hyperbolic systems}
\author[E. Fontich] {Ernest Fontich}
\address{
Departament de Matem\`atica Aplicada i An\`alisi \\
Universitat de Barcelona \\
Gran Via, 585, 08007 Barcelona, Spain \\
}
\author[R. de la Llave]{Rafael de la Llave}
\address{
Department of Mathematics\\
The University of Texas at Austin\\
Austin, TX 787121082, U.S.A
}
\author[P. Mart\'{\i}n ]{Pau Mart\'{\i}n }
\address{
Departament de Matem\`atica Aplicada IV\\
Universitat Polit\`ecnica de Catalunya \\
EdC3, Jordi Girona, 13, 08034 Barcelona Spain
}
\date{April 14 2003}
\begin{abstract}
Let $\{x_i\}_{i \in \N}$ be a regular orbit of a $C^2$
dynamical system $f$. Let $S$ be a subset of
its Lyapunov exponents.
Assume that all the Lyapunov exponents in $S$
are negative and that
the sums of Lyapunov exponents in $S$
do not agree with any Lyapunov exponent in the complement
of $S$.
Denote by $E^S_{x_i}$ the linear spaces spanned by the
spaces associated to the Lyapunov exponents in $S$.
We show that there are smooth manifolds $W^S_{x_i}$ such that
$f(W^S_{x_i}) \subset W^S_{x_{i+1}}$ and $T_{x_i} W^S_{x_i} = E^S_{x_i}$.
We establish the same results for orbits satisfying dichotomies
and whose rates of growth satisfy similar nonresonance conditions.
These systems of invariant manifolds are, in general, not a foliation.
\end{abstract}
\maketitle
\section{Introduction and statement of results}
When studying the behavior of an orbit of a dynamical system
$f$ it is natural to study also the behavior of its linearization
and wonder whether there are nonlinear analogues for the
features found in the study of the linearization.
Very often we can classify the tangent vectors along an orbit into
subspaces with different rates of exponential growth either in the
future
or in the past
(we will discuss later several precise definitions of
\emph{rates of growth}).
Since these subspaces and combinations of them are invariant,
the question of the existence of invariant objects for the full
system related to this linear spaces naturally arises.
The goal of this paper is to show that, under appropriate nonresonance
conditions for the rates of growth, indeed one can find manifolds
tangent to the spaces that grow at the indicated rates.
As it will become clear, the nonresonance conditions
are necessary for the existence of such invariant
manifolds.
For the case of a fixed point of a local diffeomorphism
of a Banach space near a fixed point,
invariant manifolds associated to nonresonant subsets
of the spectrum have been
considered in \cite{dlL97} (See also \cite{ElB98}, \cite{CFdlL02}).
Particular cases of nonresonant manifolds for uniformly hyperbolic
systems were considered in \cite{Pes73},\cite{LlaveW95},
\cite{JiangPL95}.
As remarked in Section 8.3 of \cite{dlL97} and in
Section 2 of \cite{CFdlL02}, the results for fixed points,
imply results for more general sets using the device of
lifting,
see \cite{HirschP68}. Given a dynamical system,
$f$ we consider the action $\tilde f$ on the Banach space of
$C^0$ vector fields defined by
\begin{equation}\label{lift}
[ \tilde f v] (x) =\exp^{1}_x f ( \exp_{f^{1}(x)} v( f^{1}(x)) )
\end{equation}
The zero vector field is, clearly, a fixed point of $\tilde f$.
Moreover,
the linearization of $\tilde f$ at the zero field
is $f_*$, the push forward of $f$ acting on $C^0$ vector fields.
Hence, under assumptions on the spectrum of $f_*$, we can associate
invariant manifolds for $\tilde f$. As it is well known
from \cite{Mat68}, the spectral properties of $f_*$ are
related to the growth rates of the linearized system.
In this paper, we develop a theory of nonresonant manifolds
for orbits that satisfy some rather weak notion of hyperbolicity.
This notion is based on properties of each individual orbit
and does not require the uniformity assumptions that are required
in the lifting approach.
Indeed, we introduce a very weak notion of rates of growth
 see Definition \ref{def1}  which generalizes at the same time
the notions of exponential dichotomies and the notions of
Lyapunov exponents of nonuniform hyperbolic theory.
We note that the nonresonant conditions are automatically
satisfied by the most contractive part of the stable spectrum.
Hence, the results here generalize the classical results on
strong stable manifolds. Nevertheless, in contrast with
the strong stable manifolds, the nonresonant manifolds
constructed here do not integrate to foliations 
see \cite{JiangPL95} .
For the sake of simplicity, we will formulate the results only
for dynamical systems with discrete time $\Z$.
Analogous results are true for flows of time $\R$.
The results for continuous times follow by taking time
one maps of the flows. Of course, it is possible to
give a direct proof of the results for flows
following the arguments presented here for diffeomorphisms.
As a motivation for the study of the manifolds considered
here, we will mention that they give one possible precise meaning to
the idea of \emph{slow manifolds} which is used in many heuristic
calculations of asymptotic behavior in dynamics 
see \cite{Fraser88, MaasP92} for
a discussion of several possible meanings of \emph{slow manifolds} .
{From} a mathematical point of view, we point that one can prove
by following the arguments in \cite{delaLlaveMM86} that the
solutions of cohomology equations are smooth on the manifolds considered
here.
The fact that the manifolds considered here do not lead to
a foliation provides an obstruction to smooth equivalence of
dynamical systems which is not related to periodic orbits
and which is not captured by nonautonomous linearization
 see \cite{Llave92}, \cite{JiangPL95} .
\subsection{Notions of rates of growth}
There are two widely
used methods to formalize the rates of growth of vectors
along an orbit.
One possibility  considered in \cite{SS74}, \cite{Fenichel74},
\cite{Fenichel77} 
is to require that there is an uniform
expansion or contraction, but that there is an spread on
the rates.
That is, for $\mu_1 < \lambda_1 < \cdots < \mu_p < \lambda_p$,
one characterizes the bundles $E^i_x$ by:
\begin{equation} \label{conddichotomy}
v\in E_x^i \iff Df^n( x ) v \le \left\{
\begin{array}{ll}
C\exp( \lambda _i n ) v & n\ge 0 \\
C\exp( \mu _i n) v & n\le 0 .
\end{array} \right.
\end{equation}
We note that a consequence of
\eqref{conddichotomy} is that the angle between the different
spaces $E^i_x$ is bounded from below.
\begin{remark}
As it is well known (see e.g. \cite{Mat68})
if the spectrum of $f_*$  the pushforward by
$f$  acting on $C^0$ vector fields is contained in
annuli of outer radius $e^{\lambda_i}$ and inner radius $e^{\mu_i}$,
then, every orbit satisfies \eqref{conddichotomy}.
Under conditions \eqref{conddichotomy}, it is possible to develop a theory of
invariant manifolds and foliations based on lifting to actions of
bundles as in \eqref{lift}. See \cite{HirschP68},
\cite{HirschPS77} for the origins of the
theory and \cite{CFdlL02} for results
for nonresonant invariant manifolds.
In this paper, we will, however,
base our study on properties of individual orbits
rather than on lifting them to bundles.
\end{remark}
Another characterization of rates of growth is the existence of Lyapunov
exponents considered in \cite{Oseledec68}.
\begin{equation}\label{conditionexponent}
v \in E^i_x \iff
\lim _{n\to \pm \infty} \frac{1}{n} \ln (Df^n(x) v ) = \lambda_i.
\end{equation}
for some real numbers $ \lambda_1, \dots, \lambda_p$.
Notice that Definition \eqref{conditionexponent}
ignores polynomial terms in the rate and it can be quite nonuniform
along the orbit. As it is well known, Oseledecs' theorem
ensures that, given a measure $\rho$ invariant under the system,
one has \eqref{conditionexponent} $\rho$almost everywhere and,
moreover, one can find sets of measure arbitrarily close
to full measure where
there is some uniformity in the deterioration of the hyperbolicity
properties.
One can consider that condition \eqref{conddichotomy} requires
that there is an exponential rate which is uniform along the orbit
but one has to allow a spread on the rate. On the other hand,
in condition \eqref{conditionexponent}, there is an
exponential rate but one only requires that the exponential rate
happens in an averaged sense and that it does not need to
be uniform along the orbit.
Neither of the characterizations
of rates of growth \eqref{conddichotomy}
and \eqref{conditionexponent}
is more general than the other.
Even if a vector is in one of the subbundles in \eqref{conddichotomy},
it may fail to have a Lyapunov exponent if the rates of
growth keep on oscillating. Such points are easy to
construct in hyperbolic systems (e.g. horseshoes) that have
a symbolic dynamics. On the other hand,
systems admitting Lyapunov exponents may have fluctuations that destroy
the possibility of uniformity.
We introduce a new definition that encompasses both of the previous
definitions \eqref{conddichotomy}
and \eqref{conditionexponent}.
We allow the existence of an spread in the exponential rate
as well as a deterioration of the constants along the orbit.
\begin{defi}\label{def1}
Given $\lambda = \{\lambda_1, \dots, \lambda_p\}$ and
$\mu = \{\mu_1, \dots, \mu_p\}$ such that
$ \mu_1 \le \lambda_1 < \mu_2 \le \lambda_2 < \dots <
\mu_p \le \lambda_p $,
$\ell > 1$, $ \eps > 0$, we say that a point $x$ has a
$\lambda,\mu, \eps, \ell$ regular orbit if we can find
invariant decompositions
$T_{f^m(x)}M = \oplus _{i=1}^p E^i_{f^m(x)} $
such that:
\begin{itemize}
\item[(i)]
$v\in E^i_{f^m(x)}$ implies
for $n\ge 0$,
\begin{equation}\label{positiverate}
\begin{split}
& \ell^{1} \exp(\mu _i n) \exp (\eps m) v \\
& \le Df^n(f^m(x))v \\
& \le \ell \exp(\lambda _i n) \exp (\eps m) v .
\end{split}
\end{equation}
and
for $n < 0$,
\begin{equation}\label{negativerate}
\begin{split}
& \ell^{1} \exp(\lambda _i n) \exp (\eps m) v \\
& \le Df^n(f^m(x))v \\
& \le \ell \exp( \mu_i n) \exp (\eps m) v .
\end{split}
\end{equation}
\item[(ii)]
{\rm angle }$(E^i_{f^m(x)}, E^j_{f^m(x)} )
\ge \ell^{1} \exp(\eps m), \quad i\ne j$.
\end{itemize}
\end{defi}
We denote $I_i=[\mu_i, \lambda_i]$. We
also denote by $\Lambda_{\lambda, \mu, \eps, \ell}$
the set of $\lambda, \mu, \eps, \ell$ regular orbits.
We recall that if $f$ is a $C^1$ system and $\rho$ is an ergodic
invariant probability measure, Oseledecs multiplicative ergodic theorem
implies that, if the Lyapunov exponents are $\gamma_i$, $i=1, \dots, p$,
then, for $\eps >0$, the sets
$\Lambda_{\gamma \eps, \gamma + \eps, \eps, \ell}$
can be made to have measure as close to full as desired by choosing
$\ell$ big enough.
That is, $\rho$almost all orbits are regular but the constant
$\ell$ cannot be chosen uniformly.
The sets $\Lambda_{\gamma +\eps, \gamma  \eps, \eps, \ell}$
are often called Pesin sets.
Condition
\eqref{conddichotomy} clearly implies
Definition \ref{def1} taking $\eps = 0 $ and $\ell$ a suitable
large number.
A fortiori, it is shown in \cite{Mat68}
that the fact that the spectrum of the pushforward
is contained in annuli of inner radii $\exp{\mu_i}$ and
outer radii $\exp \lambda_i$
is equivalent to the fact that all orbits
satisfy Definition \ref{def1}
with $\eps = 0$, $\ell$ chosen uniformly for all points.
\subsection{Nonresonance}
Given two intervals $I_1, I_2 \subset \R$, we denote
\[
I_1 + I_2 = \{ t = t_1 + t_2 :\; t_1 \in I_1, t_2 \in I_2\}.
\]
Of course, when the intervals consist of one number,
the above operation corresponds to
the sum of numbers.
\begin{defi}\label{def:nonresonant}
Let $\{I_i\}_{i=1}^p $ be a collection of
intervals. Given $S\subset \{1,2,\dots, p\} $,
we say that a subset $\{I_i\}_{i\in S} $ is nonresonant if
for $j \in \N$, $j \ge 2$, and any collection $i_1, \ldots, i_j \in S$
of indices (perhaps repeated),
we have:
\begin{equation} \label{eq:nonresonant}
I_{i_1} + \cdots + I_{i_j} \cap \big(\bigcup_{i \in S^c} I_i\big) = \emptyset.
\end{equation}
We say that a subset $\{I_i\}_{i\in S} $ is contractive when
\begin{equation}\label{contractive}
\bigcup_{i \in S} I_i \subset \R^.
\end{equation}
\end{defi}
We denote $I^S = \cup_{i \in S} I_i$
and $I^{S^c} = \bigcup_{i \in S^c} I_i$.
We denote by $E^S_x = \bigoplus_{i\in S} E^i_x$
and $E^{S^c}_x = \bigoplus _{i\in S^{c}} E^i_x$.
We clearly have $T_xM = E^S_x \oplus E^{S^c}_x $.
We denote $\Pi ^S_x$, $\Pi ^{S^c}_x$ the projections associated to this
decomposition.
We call $d_S(x) = \dim E^S_x$ and $d_{S^c}(x) = \dim E^{S^c}_x$.
For collections of intervals $I_i$ and
subsets $S$ satisfying \eqref{eq:nonresonant} and
\eqref{contractive},
we denote by $N_S$ the integer defined by
\begin{equation}\label{NS}
N_S = \left[ \frac{\min \{ t \in I^{S^c} \}}{ \max \{ t
\in I^S\} }\right]
\end{equation}
where $[t]$ denotes integer part of $t$.
It is clear that for collections satisfying
\eqref{contractive}, then \eqref{eq:nonresonant} is
automatically verified if $j > \max\{N_S, 0\}$.
\begin{remark}
Note that in part (i) of Definition \ref{def1} we have
only assumed that in the invariant space $E^i_x$
we have the rates of growth \eqref{positiverate}, \eqref{negativerate}.
In the literature, it is often assumed that the
space $E^i_x$ is precisely characterized by
\eqref{positiverate}, \eqref{negativerate}.
One interesting example of spaces where assumption (i)
of Definition applies
but which is not characterized by
\eqref{positiverate}, \eqref{negativerate}
is Cartesian product systems in which the two factors
overlap.
For example, if we take the Cartesian product of a system
by itself, $F(x,x) = ( f(x), f(y) ) $ and
$E^i_x$ is a spectral decomposition for $f$, we see
that $\tilde E^i_{x,y} = E^i_x \times \{0\}$ is
an space admissible for our definition. On the other
hand, the only spectral space is
$ (E^i_x \times \{0\}) \oplus ( \{0\} \times E^i_y)$.
It is perfectly possible to have situations such as
those mentioned here satisfy the nonresonance conditions
in Definition \ref{def:nonresonant}. Note that, there we are
only assuming that \eqref{eq:nonresonant} happens for
$j \ge 2$.
\end{remark}
\begin{remark}
For collections which satisfy \eqref{contractive}, property
\eqref{eq:nonresonant} amounts to a finite number
of conditions. It is clear that if we enlarge slightly the intervals
both conditions will remain valid.
\end{remark}
\begin{remark}
When the intervals are points, it is instructive to
compare condition
\eqref{eq:nonresonant} to the conditions in
Sternberg linearization theorem. The conditions in Sternberg
linearization theorem require that
no interval contains sums of points in other intervals.
Here we only require that the numbers of
the set $I^{S^c}$ cannot be obtained as sums of numbers in the set $I^{S}$.
Indeed, the proof presented here has some similarities with
the proof of Sternberg theorem. We start by computing a polynomial
approximation to the desired object and, then, we use an analytic
argument to show that the very approximate polynomial solutions
can be modified to become a true solution of the problem.
A Sternberg theorem along orbits under full nonresonance
conditions can be found in \cite{Yomdin88}. Similar
results are crucial for \cite{Yomdin87}. A related theory is the
theory of the nonautonomous normal forms \cite{GuysinskyK98}
develop under uniformity conditions on the bundle. It seems
clear that one could develop a similar theory
under assumptions on the behavior of individual orbits.
\end{remark}
\begin{remark}\label{stronglystable}
An important example of nonresonant set is when $S$ includes the $I_i$
contained in $(\infty,l)$ where $l < 0$, $ l \not\in \bigcup_{i=1}^p
I_i$. In such a case, the bundle $E^S_x$ is the strongly stable
bundle and our results give the usual strongly stable invariant
manifolds.
Note that the strong stable manifold admits a
characterization
\begin{equation}\label{strongstable}
W^{ss}_s = \{ y  d( f^n(x), f^n(y) ) \le C_{y,x} \lambda^n \}
\end{equation}
which makes it clear that $y \in W^{ss}_x$ is an equivalence
relation. This makes it clear that the set of
strong stable manifolds is a lamination.
For more general nonresonant manifolds, there is not a
characterization in terms of rates of growth. In \cite{JiangPL95}
one can find examples of situations where the nonresonant
manifolds of neighboring points have nontrivial intersections.
\end{remark}
\subsection{Statement of main results}
\begin{defi}
We will say that a family of maps $\{w_n\}_{n\in \Z}$ is uniformly $k$
differentiable if the maps are $k$ differentiable and
$\sup _{n\in \Z} \D^k w_n \_{C^0} < \infty$. We will also say
that a family of parameterized
manifolds $\{W_n\}_{n\in \Z}$ is uniformly
$k$ differentiable if there are parameterizations of $W_n$ that are
uniformly $k$ differentiable.
\end{defi}
The main result of this paper is:
\begin{thm} \label{mainthm}
Let $f$ be a $C^r$ diffeomorphism, $r\in \N \cup\{\infty\}$, $r\ge 2$,
of a compact manifold $M$.
Let $x\in \Lambda_{\lambda, \mu, \eps, \ell}$ and
$\{I_i\}_{i\in S}$ be a nonresonant set. Assume that $r\ge N_S+1$
and that $\eps < \delta/2$ and $\delta$ is small enough such that if
we consider the enlarged intervals
$ \tilde I_i = I_i+ [2 \delta, 2 \delta]$, the set $\{\tilde I_i\}_{i\in S}$ is
still a nonresonant and contractive set.
Then, there exist maps
$ w_n : B(0,1) \subset\R^d \longrightarrow M $, $ n \in \Z $,
in such a way that
\begin{itemize}
\item [(a)] $w_n (0) = f^n (x)$.
\item [(b)] $w_n $ are uniformly $C^r$.
\end{itemize}
If we denote by $W_n$ the range of $w_n$ we have:
\begin{itemize}
\item [(c)] $f(W_n) \subset W_{n+1}$.
\item [(d)] $T_{f^n(x)} W_n = E^S_{f^n(x)} $.
\end{itemize}
Moreover, there exists $K$ such that the manifolds $W_n$
contain a disk
of radius
\[
K\ell^{2} \sqrt{\tanh(\delta )}\exp ( (2 \eps+\delta) n)
\]
and
\begin{itemize}
\item [(e)] If $W_n$, $\hat W_n$ are families of manifolds
satisfying $\mbox{\rm{(c)}}$, $\mbox{\rm{(d)}}$,
and both of them are uniformly $C^m$ for
some $m \le r$ then $T^i _{f^n(x)} W_n = T^i _{f^n(x)} \hat W_n$, $i\le m$.
\item [(f)]
If $W_n$, $\hat W_n$ are sequences of manifolds satisfying
$\mbox{\rm{(c)}}$ and $\mbox{\rm{(d)}}$, and
both of them are uniformly $C^k$, $k>N_S$, then,
$W_n \cap B_n = \hat W_n \cap B_n$ for some balls $B_n$ around
$f^n(x)$.
In particular,
if the manifolds are uniformly $C^{N_S +1}$, they have to
agree with the manifolds $\range w_n$ and hence
they are uniformly $C^r$.
\item[(g)]
If $W_n$ is a set of manifolds satisfying
$\mbox{\rm{(c)}}$, $\mbox{\rm{(d)}}$,
$\sup _{n\in \N} \W_{n} \ _{C^{N_s}} < \infty$ and
$W_0$ is a $C^k$ manifold for some $N_S < k \le r$, then
$\sup _{n\in \N} \W_{n} \ _{C^k} < \infty$.
\end{itemize}
\end{thm}
The meaning of the above result is that if the set
$S$ of rates of growth is nonresonant,
we can find some collections of smooth manifolds
that are nonlinear analogues of the linear subspaces that
are invariant for the manifold $S$.
The final conclusions of Theorem \ref{mainthm}
are uniqueness conclusions that say that these systems of leaves
are unique under some regularity properties.
Roughly,
part (f) tells us that the manifolds are unique when
they are regular beyond a critical value.
In particular, when we know that they are more regular
than this critical value, they are as regular
as the map. Hence, part (f) is a bootstrap of
regularity argument which starts working when the
regularity is higher than a critical value
related to rates of contraction. Similar bootstrap arguments
appear in rigidity theory. See, for example, \cite{Llave92}.
Part (g) tells us that the invariant manifolds
are unique provided that we have
some low regularity in the past. Then, they are roughly as
regular as the map.
In Section \ref{sec:example}, we will provide an example that
illustrates the role of the critical regularity in the uniqueness
properties.
It is important to remark that the proof will only use constructions
in a neighborhood of an orbit.
\begin{remark} We have formulated Theorem \ref{mainthm} only for
maps on finite dimensional manifolds.
Nevertheless, there are versions, roughly along the same
lines for maps on infinite dimensional Banach spaces
which have smooth cutoff functions (e.g. Hilbert spaces).
However, we note that some of the arguments we present  notably
the construction of the Lyapunov metric  require some serious
modification (it is somewhat easier for Hilbert spaces).
We will not consider these situations in this paper.
\end{remark}
\begin{remark}\label{deterioration}
Given the deterioration of the hyperbolicity properties
(measured by $\eps$) the sizes of these manifolds cannot
be chosen uniformly and they could become very small.
Nevertheless, we can bound the size by an amount that is
bounded from below by $\eps$. In particular, in the case that the
exponential rate properties do not deteriorate, i.e.
$\eps = 0$  as it is assumed in the results in \cite{Fenichel74},
\cite{Fenichel77} 
then, we obtain that the sizes are uniform.
\end{remark}
\begin{remark} Even in the case that the orbit is a
fixed point, the nonresonance condition is necessary for the existence
of an invariant manifold.
As we will see, the candidates for a jet satisfy functional equations
which may fail to have solutions if the nonresonance conditions
are violated. Hence, in those examples where there is no candidate for
a jet, there cannot be a smooth invariant manifold.
We refer to Example 5.5 of \cite{dlL97} for more details.
\end{remark}
\begin{remark}
Note that, in contrast with many of the results in
uniform normal hyperbolicity
which are valid for $r \ge 1$, our results are only valid
for $r \ge 2$.
It seems to us that the proof will work for
$r = 1 + \delta $ with mainly notational difficulties.
Nevertheless, the proof with $r = 1$ does not work in general
since the paper \cite{Pugh84} contains an example of
a $C^1$ system satisfying Definition \ref{def1} for which there is
no $C^1$ invariant manifold.
The paper \cite{Pugh84} contains the conjecture that in the case
that the stable manifold is onedimensional, indeed one
could get stable manifolds even for $r = 1$.
\end{remark}
\begin{remark}\label{integrability}
If one has hyperbolicity properties for
all the orbits, one could, perhaps hope that the results given
here for individual orbits could be made coherent to
integrate the distribution $E^S_x$ to give a foliation.
For the stable foliation, such procedure is carried out
in \cite{Fenichel74} \cite{Fenichel77}.
Nevertheless, in the generality considered
here, the leaves produced here do not integrate to a foliation.
Indeed, even in the very uniform case when $\eps = 0$,
in \cite{JiangPL95} one can find examples  $C^\omega$ close to
linear automorphisms of the torus  where these invariant manifolds
cross in arbitrarily small neighborhoods.
In some particular cases  e.g. maps of the torus close to
linear and when $S$ corresponds to the intervals contained in
$\R^$  in \cite{LlaveW95},
there is a way of integrating the foliations
based not on local properties but on global behavior.
The leaves produced in \cite{LlaveW95} are not very
smooth and, hence, are very different from the ones considered
here. In particular, in the proof of \cite{LlaveW95}
one has to take into account global properties of the manifold.
\end{remark}
\section{Proof of Theorem \ref{mainthm}}
The proof of Theorem \ref{mainthm} starts very similar to the proof
of the invariant manifold in \cite{Pes76} \cite{Pes77}
and subsequent papers. We start by defining a Lyapunov
metric, which is singular with respect to the Euclidean
metric but makes the hyperbolicity properties
of the problem uniform.
This allows us to choose a convenient system of coordinates
in neighborhoods of each of the points in the orbit.
In a second step we see that, writing $W_n= \graph V_n$,
under the nonresonance
conditions, it is possible to determine uniquely candidates $V^0_n$ for
the jet of the functions $V_n$.
That is, if the functions $V_n$ were differentiable enough,
we could take derivatives of the invariance equation and
obtain functional equations satisfied by the sequence of jets.
We will show that these functional equations admit unique solutions.
We emphasize that at this point we only require
the nonresonance condition
\eqref{eq:nonresonant} and not the contractive hypothesis
\eqref{contractive}.
In a third and final step, we use the computed candidates for
jets to show that we can transform the equation satisfied
by $U_n \equiv V_n  V_n^0$ into an equation that can be dealt with by
the contraction mapping principle in some appropriate
spaces by using the assumption (\ref{contractive}).
The motivation of this scheme is that for functions
that vanish at high order at the origin, by taking norms
based on derivatives of order $N_S$ the
action of a contraction $\lambda$ on the right is a contraction
by $\lambda^{N_S}$.
One of the consequences of this study will be that the $U_n$
vanish to high order at the origin so that the $V_n^0$ are indeed
the jets of the $V_n$.
\subsection{The Lyapunov metric and coordinates around the orbit}
The main goal of this section is
to establish Lemma \ref{coordinates}
which provides us with a system of coordinates around an orbit.
The main idea  rather standard in the study of nonuniformly
hyperbolic systems  is that one can define a Lyapunov metric
around an orbit which makes the hyperbolicity
uniform. Once we express all the properties in this
metric, many of the methods of the theory of uniformly
hyperbolic systems start to apply. In our case, we will
reduce the problem to the study of systems with exponential
dichotomies.
We will use different norms and scalar
products. The subindexes E, L, R will stand for Euclidean, Lyapunov and
Riemannian norms or scalar products respectively.
We also recall that,
given a set $D$, the modulus of continuity of $h_{\mid D}$ is
\[
\omega(h,\eta) = \sup_{y,z\in D, \,yz\le \eta} \h(y) h(z)\.
\]
\begin{lem}\label{coordinates}
Let $M$ be a compact $C^\infty$ ndimensional manifold.
Given a $C^r$ map $f:M\to M$, $r\in \N \cup \{\infty \}$,
$ r \ge 2$, $x\in \Lambda _{\lambda, \mu, \eps, \ell}$,
$\delta > 2\eps$, $\tau >0$
and a fixed (Euclidean) orthogonal decomposition
$\R^d = \bigoplus _{i=1}^p E^i$
such that $\dim E^i = \dim E^i_{f^k(x)}$,
there exists a sequence
of $C^\infty $ maps
\[
\Phi _k :B(0,1)\subset \R^d \rightarrow M, \quad k\in \Z
\] such that
\begin{itemize}
\item [(i)] $\Phi _k (0) = f^k(x)$.
\item [(ii)]
$
D\Phi_k(0) E^i = E^i_{f^k(x)} .
$
\end{itemize}
If we denote by $f_k = \Phi^{1}_{k+1} \circ f \circ \Phi_{k}$ we have:
\begin{itemize}
\item [(iii)]
$\exp (\mu_i2 \delta) \v\_E \le \Df_k(0) v \_E \le
\exp (\lambda_i+2 \delta) \v\_E,
\;\; v \in E^i$.
\item [(iv)] \range $(\Phi_k) \supset \{ y:\; d_R(f^k(x), y) \le
(2/\pi) \Gamma p^{1/2} \ell^{2}
\sqrt{\tanh \delta } \exp ((2\eps+ \delta) k) \}$,
where $\Gamma$ is some constant.
\item [(v)] $ \sup _{k, \; x \le 1, \; 2\le j\le r }
\D^jf_k(x)\ \le \tau $.
\item [(vi)] The modulus of continuity of $D^rf_k$ is bounded
independently of $k$ and can be made as small as we want.
\end{itemize}
\end{lem}
\proof For a regular orbit we define the Lyapunov
inner product in
$E^i_{f^k(x)}$ by
\begin{equation}
\begin{split}
\langle v, w\rangle_L = & \sum_{n\ge 0} e^{2(\lambda_i + \delta)n }
\langle Df^n (f^k(x)) v, Df^n (f^k(x)) w \rangle _R\\
& + \sum_{n< 0} e^{2(\mu_i  \delta)n }
\langle Df^n (f^k(x)) v, Df^n (f^k(x)) w \rangle_R, \\
& \quad v,w \in E^i_{f^k(x)}
\end{split}
\end{equation}
and in $T_{f^k(x)}M $
\[
\langle v, w\rangle_L = \sum_{i=1}^{p}
\langle \Pi^i_{f^k(x)} v, \Pi^i_{f^k(x)} w\rangle_L ,
\quad v,w \in T_{f^k(x)}M,
\]
where $\Pi^i_{f^k(x)}$ are the projectors to $E^i_{f^k(x)}$.
Note that the subspaces $E^i_{f^k(x)}$ are orthogonal with
respect to the Lyapunov metric,
and that, by Definition~\ref{def1},
denoting by $\varphi_k$ the minimum angle between the
subspaces $E^i_{f^k(x)}$,
$ \\Pi^i_{f^k(x)}\ \le \frac{1}{\sin \varphi_k}
\le \frac{\pi}{2 \varphi_k} \le\frac{\pi}{2} \ell \exp (\eps k)$.
By the previous definitions, if $v\in E^i_{f^k(x)}$,
\begin{eqnarray*}
\ v\^2_L & \le & \sum_{n\ge 0} e^{2(\lambda_i + \delta)n }
\Df^n (f^k(x)) v\_R^2
+ \sum_{n < 0} e^{2(\mu_i  \delta)n }
\Df^n (f^k(x)) v\_R^2 \nonumber\\
& \le & \Big[ \sum_{n\ge 0} e^{2(\lambda_i + \delta)n}
\ell^2 e^{2\lambda_i n } e^{2\eps k }
+ \sum_{n < 0} e^{2(\mu_i  \delta)n }
\ell^2 e^{2\mu_i n } e^{2\eps k } \Big] \ v\^2_R \nonumber\\
&=& (\ell e^{\eps k})^2 /\tanh \delta \,
\ v\_R^2
\end{eqnarray*}
and if $ v\in T_{f^k(x)}M$,
\[
\ v\^2_L =\sum_{i=1}^{p} \\Pi^i_{f^k(x)} v\^2_L
\le \Big(\sum_{i=1}^{p} \frac{\pi^2}{4}\ell^4e^{4\eps k}
/\tanh \delta \Big) \ v\^2_R
\]
and hence
\begin{equation}\label{fitavL}
\ v\_L \le \Big(\sqrt{p} (\pi/2)\ell^2 e^{2\eps k} /\sqrt{\tanh \delta}
\Big) \ v\_R.
\end{equation}
Next, we define a family of linear maps $C_k$.
Given the
orthogonal decomposition $\R^d = \bigoplus_{i=1}^{p} E^i$ we define
$C_{k\, \mid E^i}: E^i \rightarrow E^i_{f^k(x)} $ as follows.
We take an orthonormal basis
$\{ e_1^i, \dots , e_{d_i}^i \}$ in each $E^i$ with respect to the
Euclidean scalar product and an orthonormal basis
$\{ u_1^i, \dots , u_{d_i}^i \}$ in each
$E^i_{f^k(x)}$
with respect to $\langle \cdot, \cdot \rangle_L $.
We just define $C _{k \mid E^i} $ by the relations
$C_k e_j^i = u_j^i $, $ 1\le j\le d_i$, $1\leq i \leq p$.
With this definition $C_k$ becomes an isometry from
$(\R^d, \\cdot \_E)$ to $(T_{f^k(x)}M,\\cdot \_L )$. Indeed,
$\ C_k(\sum \lambda _j e_j) \^2_L =\\sum \lambda _j u_j \^2_L
= \sum \lambda_j^2 = \\sum \lambda _j e_j \^2_E$.
Moreover, from \eqref{fitavL} we have
\[
\C_k^{1}v\_E =\v\_L \le \frac{\sqrt{p}(\pi/2) \ell^2
e^{2\eps k}}{\sqrt{\tanh \delta}}\v\_R.
\]
We also have
\begin{eqnarray}
\C_k v\ _R &\le& \sum _{i=1}^{p}\\Pi^i_{f^k(x)}C_k v\_R
\le \sum _{i=1}^{p}\\Pi^i_{f^k(x)}C_k v\_L \nonumber \\
&\le& \sqrt{p} (\sum _{i=1}^{p}\\Pi^i_{f^k(x)}C_k v\^2_L)^{1/2}
= \sqrt{p} \ C_kv\_ L = \sqrt{p} \ v\_ E.\label{cotasupC_R}
\end{eqnarray}
Let $\exp $ be the exponential map of the Riemannian metric in $M$.
We define
\[
\Phi_k(y) = \exp _{f^k(x)}(\gamma_k C_k y)
\]
and we claim that if we choose $\gamma _k$ in a suitable way,
$\Phi_k$ satisfies
all the conclusions of Lemma \ref{coordinates}.
Indeed, conclusions (i) and (ii) are immediately satisfied.
Since $D\Phi_k(0) = \gamma _k C_k $,
$D\Phi_k(0) $ transforms the Euclidean metric into $\gamma_k$
times the Lyapunov metric.
We have
\begin{eqnarray*}
Df_k(0) &=& D\Phi_{k+1} ^{1}(f^{k+1}(x)) Df (f^k(x)) D\Phi_k(0) \\
&=&
\frac{1}{\gamma_{k+1}} C^{1}_{k+1} Df(f^k(x)) \gamma _k C_k
\end{eqnarray*}
and if $v\in E^i$
\[
\Df_k(0)v\_E = \frac{\gamma _k}{\gamma_{k+1}} \Df(f^k(x)) C_k v\_L .
\]
Furthermore , if $v \in E^i$
\begin{eqnarray*}
\lefteqn{
\Df(f^k(x)) C_k v\^2_L }\\
& = & \sum_{n\ge 0} e^{2(\lambda _i + \delta)n}
\ Df^n (f^{k+1}(x)) Df(f^k(x))C_k v \^2_R \\
&& + \sum_{n < 0} e^{2(\mu _i  \delta)n}
\ Df^n (f^{k+1}(x)) Df(f^k(x))C_k v \^2_R \\
& = & \sum_{n\ge 0} e^{2(\lambda _i + \delta)n}
\ Df^{n+1}(f^{k}(x)) C_k v \^2_R \\
&& + \sum_{n < 0} e^{2(\mu _i  \delta)n}
\ Df^{n+1} (f^{k}(x)) C_k v \^2_R \\
& = & e^{2(\lambda _i + \delta)}\Big[
\sum_{n\ge 0} e^{2(\lambda _i + \delta)n}
\ Df^{n}(f^{k}(x)) C_k v \^2_R  \C_k v \_R^2\Big]\\
&& + e^{2(\mu _i  \delta)}\Big[\sum_{n < 0} e^{2(\mu _i  \delta)n}
\ Df^{n} (f^{k}(x)) C_k v \^2_R + \C_k v \_R^2\Big]\\
&\le & e^{2(\lambda _i + \delta)} \C_k v \_L^2.
\end{eqnarray*}
Then
\[
\Df_k(0)v\_E \le \frac{\gamma _k}{\gamma_{k+1}}
e^{(\lambda _i + \delta)} \C_k v \_L .
\]
Recall that $ \C_k v \_L = \v\_E$.
This proves the upper bound claimed in (iii) in
Lemma \ref{coordinates}.
The proof of the lower bound is completely analogous.
Following the previous computations we have
\[
\ Df(f^k(x)) C_k v \ _L \ge e^{\mu_i \delta} \C_k v\_L
\]
and hence
\[
\exp(\mu_i \delta) \frac{\gamma _k}{\gamma_{k+1}}\v\_E
\le \ Df_k(0) v \ _E \le
\exp(\lambda_i +\delta) \frac{\gamma _k}{\gamma_{k+1}}\v\_E .
\]
We take $\gamma _k=\Gamma e^{\delta  k}$ with $\Gamma \le 1$
to be determined later on.
Then, if $k\ge 0$, $\frac{\gamma _k}{\gamma_{k+1}}= e^\delta$ and,
if $k<0$, $\frac{\gamma _k}{\gamma_{k+1}}= e^{\delta}$.
To prove (iv) in Lemma \ref{coordinates},
we observe that $\gamma _k C_k$ sends the unit ball of $\R^d$ to
$\gamma_k $ times the unit ball of $T_{f^k(x)}M $ with respect
to the Lyapunov metric.
{From} \eqref{fitavL} the unit Lyapunov ball contains the
Riemannian ball of radius
$\gamma_k \frac{2}{\sqrt{p}\pi} \ell^{2} e^{2\epsk }
\sqrt{\tanh \delta}$ and finally
$\exp_{f^k(x)}$ sends this ball to $M$.
For (v) we evaluate the derivative
$D^rf_k = D^r(\Phi^{1}_{k+1} \circ f\circ \Phi _k)$.
Recall the Faa di Bruno formulas
\begin{equation}\label{fdb1}
D^rf_k = \sum_{j=1}^{r} \sum_{l_1 + \dots +l_j = r \atop
1\le l_1, \dots ,l_j \le r}
c_{l_1,\dots,l_j}^{r,j}
D^j \Phi^{1}_{k+1} \circ (f\circ \Phi_k) D^{l_1 }
(f\circ \Phi_k)\cdots D^{l_j }(f\circ \Phi_k)
\end{equation}
and
\begin{equation} \label{fdb2}
D^{l }(f\circ \Phi_k) = \sum_{j=1}^{l} \sum_{m_1 + \dots +m_j = l\atop
1\le m_1, \dots ,m_j \le l}
c_{m_1,\dots,m_j}^{l,j}
D^j f \circ \Phi_k D^{m_1 }
\Phi_k\cdots D^{m_j }\Phi_k
\end{equation}
where the $c$'s are combinatorial coefficients which depend on the indices.
In our case, we were interested mainly in
\[
\Phi_k= \exp_{f^{k}(x)} \circ (\gamma_{k} C_{k})
\]
and
\[ \Phi^{1}_{k+1} =
\frac{1}{\gamma_{k+1}} C^{1} _{k+1} \exp^{1}_{f^{k+1}(x)}.
\]
{From} now on $K$ will stand for a constant which only depends on $M$
and the properties of the exponential map.
We observe that
\[
\ D^m \Phi_k\ \le K(\gamma _k \C_k\)^m.
\]
Similarly,
\[
\ D^j \Phi^{1}_{k+1}\ \le K \frac{1}{\gamma _{k+1}} \C^{1}_{k+1}\.
\]
By \eqref{cotasupC_R} the norm of $C_k$ as a linear map from
$(\R^d, \\cdot\_E) $ to $(T_{f^{k+1}(x)} M, \\cdot\_R)$ is smaller than $\sqrt{p}$.
We deduce that
\[
\begin{split}
\D^r f_k \ &\le K \ell^2\frac{\gamma ^r_k}{\gamma _{k+1}}
\frac{e^{2\eps k}}
{\sqrt{\tanh \delta}}\f \_{C^r} \\
&= \frac{K\ell^2 \Gamma ^{r1}}
{\sqrt{\tanh \delta}}e^{(2\eps  (r1)\delta)k} \ f \_{C^r} .
\end{split}
\]
Therefore if we take $\Gamma$ small enough we can adjust the smallness
of the derivatives of $f_k$ of order $r\ge 2$.
Finally we prove (vi) of Lemma \ref{coordinates}.
Since $M$ is compact, all derivatives of $f$ are uniformly continuous.
Let $B_z$ be the ball of center 0 and radius $\sqrt{p}\,\Gamma $ in $T_z M$
(the radius $\sqrt{p}\,\Gamma $ is motivated by \eqref{cotasupC_R}).
Since $M$ is $C^{\infty}$ and compact
\[
\sup _{0\le m\le r+1}\sup _{z\in M}\sup_{x\in B_z} \D^m \exp_z(x)\ < \infty.
\]
For $m\le r$ and $\y\_E,\z\_E \le 1$
we have
\[
\D^m \Phi _k (y) \_R=\D^m \exp _{f^{k}(x)}(\gamma _k C_k y )
(\gamma _k C_k )^{\otimes m} \\le K\Gamma^m e^{\delta k m} ,
\]
and
\begin{eqnarray*}
\D^m \Phi _k(y)  D^m \Phi _k(z) \
&\le &
\D^{m+1} \exp _{f^{k+1}(x)}\\,\\gamma _k C_k\^{m+1}\yz \ \\
& \le & K \Gamma ^{m+1}e^{\delta k (m+1)}\yz \ .
\end{eqnarray*}
Then by \eqref{fdb2}
and the fact that $D^jf$, $0\le j\le r$, are bounded and uniformly continuous,
we get that
\[
\D^l (f\circ \Phi_k) \\le K \Gamma^l e^{\delta kl}\ f \_{C^l}
\]
and
\begin{eqnarray*}
\omega(D^l (f\circ \Phi_k), \eta)
&=& \sum_{j=1}^{l}K\Gamma^l e^{\delta kl}
\omega(D^jf, K \Gamma \eta)
+ K \Gamma ^{l+1}
e^{\delta k(l+1)}\ f \_{C^l} \eta \\
&=& K \Gamma^l e^{\delta k l} o(1) + K\Gamma^{l+1}
e^{\delta k (l+1)} \eta.
\end{eqnarray*}
Next we consider the modulus of continuity of $D^j\Phi^{1}_{k+1}
\circ(f\circ\Phi_k)$.
For simplicity, we just bound the modulus of continuity using
the mean value theorem assuming that $D^{j+1}f$ is uniformly bounded.
We get
\[
\omega(D^j\Phi^{1}_{k+1}
\circ(f\circ\Phi_k), \eta) \le K \ell^2 \frac{e^\delta}{\sqrt{\tanh \delta}}
e^{2\eps k+1}\ f \_{C^1}\eta .
\]
Finally by \eqref{fdb1}
\begin{equation*}
\begin{split}
\omega(D^r f_k, \eta) \le & K \ell^2
\frac{e^\delta}{\sqrt{\tanh \delta}}
e^{2\eps k+1} \Gamma^r e^{\delta kr}\ f \_{C^1} \ f \_{C^r}\eta\\
& + K\sum_{j=1}^{r} \sum _{l_j}\Gamma^{1}e^{\delta k+1} \ell^2
\frac{e^{2\eps k+1}}{\sqrt{\tanh \delta}}\Gamma^{rl_j}
e^{\delta k(rl_j)} \\
& \quad \quad \big[\Gamma^{l_j}e^{\delta kl_j}\omega(D^{l_j} f, K\Gamma \eta)
+ \Gamma ^{l_j+1} e^{\delta k(l_j+1)}\ f \_{C^{l_j}}\eta\big] \\
\le &
K\ell^2 \Gamma ^{r1}\frac{e^{(r+1)\delta}}{\sqrt{\tanh \delta}}
\Big( e^{[2\eps (r1)\delta ]k+1}\omega(D^{r} f, K\Gamma \eta)
+ \Gamma e^{[2\eps  r\delta ]k+1}\ f \_{C^{r}}\eta\Big).
\end{split}
\end{equation*}
Then, if $\delta > 2\eps/(r1)$, $ \omega(D^r f_k, \eta)$
is bounded independently of $k$, and, if $r\ge 2$, can be made small
taking $\Gamma$ small.
\qed
\begin{defi}
Let $B_2$ be the ball around the origin in $\R^d$ of radius 2. Given
a sequence $\mathcal{F} = \{ f_n\}_{n\in \Z}$, $f_n \in C^r(B_2,\R^d)$,
satisfying $f_n(0) = 0$, we denote
\begin{eqnarray*}
A_n & = & Df_n(0), \\
\tilde f_n & = & f_n  A_n, \\
\tilde \F & = & \{ \tilde f_n \}_{n\in \Z},\\
\ \mathcal{F} \_{C^r(B_2)} & = & \sup_{n\in \Z} \ f_n \_{C^r(B_2)}.
\end{eqnarray*}
\end{defi}
\begin{remark}
Given a family of $C^r$ maps, $\F = \{ f_n\}_{n\in \Z}$ with $f_n(0) = 0$,
we will consider the family $\F_{\lambda}
= \{ f_{n,\lambda}\}_{n\in \Z}$, with $\lambda >0$ and
\[
f_{n,\lambda}(x) = \lambda^{1} f_n(\lambda x).
\]
Notice that, if $r\geq 2$ and $\\F\_{C^r(B_2)}< \infty$, then
$\ \tilde \F_{\lambda}\_{C^r(B_2)}$ can be assumed to be
arbitrarily small by taking $\lambda$ small enough.
Indeed, if $\F = \{f_n\}_{n\in \Z}$, it is clear that
\[
\begin{split}
\tilde f_{n,\lambda}(x)  &=
\lambda^{1} f_n(\lambda x)  Df_n(0) \lambda x \\
& \leq
\lambda \D^2 f_n \circ \lambda \_{C^0(B_2)} \leq
\lambda \ \F\_{C^r(B_2)},
\end{split}
\]
and that
\[
\begin{split}
D\tilde f_{n,\lambda} (x)  &=
 Df_n(\lambda x)  Df_n(0)  \\
& \leq
\lambda \D^2 f_n \circ \lambda \_{C^0(B_2)} \leq
\lambda \ \F\_{C^r(B_2)}.
\end{split}
\]
For $2 \leq i \leq r$, one has
\[
D^i\tilde f_{n,\lambda} (x)  =
\lambda^{i1}  D^i f_n(\lambda x)  \leq
\lambda^{i1} \ \F \_{C^r(B_2)}.
\]
Moreover if $\{\graph V_n\}$ is a sequence of manifolds such that
$V_n(0)= 0 $ and $f_n(\graph (V_n)) \subset\graph (V_{n+1}) $, then,
defining $V_{n, \lambda}(x) = \lambda^{1} V_n(\lambda x)$,
we have that $\graph V_{n, \lambda}$ is sequence of manifolds
such that $f_{n, \lambda} (\graph (V_{n, \lambda})) \subset\graph (V_{n+1,
\lambda}) $.
\end{remark}
\begin{remark}
If $r = 1+\alpha$, $0<\alpha<1$, and $Df_n$ are uniformly H\"older,
that is, there exists $K>0$ independent of $n$
such that $Df_n(x) Df_n(y) \leq
K xy^{\alpha}$, then $\\tilde \F\_{C^{1+\alpha}(B_2)}$ can
also be assumed to be small taking $\lambda$ small. This can be easily
checked since
\[
\tilde f_{n,\lambda}(x)  =
\left \int_0^1 (Df_n(t\lambda x)  Df_n(0))x\,dt \right
\leq
K \lambda^{\alpha} /(\alpha+1) ,
\]
\[
D\tilde f_{n,\lambda}(x)  \leq K \lambda^{\alpha} ,
\]
and
\[
D\tilde f_{n,\lambda}(x)  D\tilde f_{n,\lambda}(y)
=
Df_n(\lambda x)  Df_n(\lambda y) \leq K \lambda^{\alpha}
xy^{\alpha}.
\]
However, if $r=1$, it may not be possible to make
$\\tilde \F_{\lambda}\_{C^1(B_2)}$ small. As an example, consider the family
$\{f_n\}_{n\geq 1}$, where
\[
f_n(x) = \frac{\sin (n x)}{n}.
\]
It is clear that $\sup_n \f_n\_{C^1} = 1$.
But, for all $\lambda > 0$,
\[
\tilde f_{n,\lambda}(x) = \lambda^{1}
\left( \frac{\sin(\lambda n x)}{n}  \lambda x \right)
\to x, \qquad \mbox{when $n\to \infty$},
\]
which implies that $\\tilde \F_{\lambda}\_{C^1} \geq 1$, for all
$\lambda >0$.
\end{remark}
As a consequence of Lemma \ref{coordinates}, Theorem \ref{mainthm} will be
immediately implied
by the following theorem.
\begin{thm} \label{manifoldsforfamilies}
Let $\F = \{f_n \}_{n\in \Z}$ be a family of $C^r$
maps in $\R^q$, $r\in \N\cup \{ \infty\}$, $r\ge 2$.
Assume that there is a
decomposition $\R^q = E^1 \oplus \cdots \oplus E^p$ invariant
under $A_n$, that is, $A_n E^i = E^i$ and a set of real numbers
$ \tmu_1 \le \tlambda_1 < \tmu_2 \le \cdots \le
\tlambda_{p 1} < \tmu_p \le \tlambda_p$, such that
\begin{equation}
\label{subspaces}
v \in E^i \iff e^{\tmu_i} v \leq
A_n v  \leq e^{\tlambda_i} v \qquad \mbox{for all }\; n\in \Z.
\end{equation}
Denote by $I_i = [\tmu_i, \tlambda_i] $.
Let $S \subset \{1, \dots,p\}$ be such that $\{I _i: \; i \in S\}$
satisfies \eqref{eq:nonresonant} and \eqref{contractive}.
Denote $E^S = \bigoplus_{i \in S} E^i$, $E^{S^c} =
\bigoplus_{i \in S^c} E^i$.
Assume that $\ \tilde \F \_{C^{N_S+1}(B_2)}$ is sufficiently small,
the modulus of continuity of $ D^r \tilde f_n$ is uniformly bounded
with respect to $n$ and
$r\geq N_S +1$.
Let $B_1$ be the unit ball in $E^S$.
Then, there exist $C^r$ maps $V_n:B_1\subset E^S \to E^{S^c}$
in such a way that
\begin{itemize}
\item[(a)]
\label{aofthm}
$\sup_{n\in \Z} \ V_n \_{C^r(B_1)} < \infty$.
\item[(b)]
\label{bofthm}
$V_n(0) = 0$, $\quad f_n \big( \mbox{\rm graph} (V_n) \big) \subset
\mbox{\rm graph} (V_{n+1})$.
\item[(c)] $DV_n(0) = 0$.
\end{itemize}
Moreover, we have that
\begin{enumerate}
\item[(d)] If $V_n$, $\hat V_n$ are families of maps satisfying
$\mbox{\rm{(b)}}$
and
\[
\sup_{n\in \Z}\ V_n \_{C^m(B_1)} < \infty, \qquad
\sup_{n\in \Z}\ \hat V_n \_{C^m(B_1)} < \infty
\]
for some $m\leq r$, then $D^i V_n(0) = D^i \hat V_n (0)$ for all
$i\leq m$.
\item[(e)]
If $V_n$, $\hat V_n$ satisfy $\mbox{\rm{(b)}}$,
$\sup_{n\in \Z} \ V_n \_{C^{N_S +1}(B_1)} \le 1$,
$\sup_{n\in \Z} \ \hat V_n \_{C^{N_S +1}(B_1)} \le 1$, then
$V_n = \hat V_n$. In particular, if $\hat V_n$ satisfies $\mbox{\rm{(b)}}$
and
$\sup_{n\in \Z} \ \hat V_n \_{C^{N_S +1}(B_1)} \le 1$,
$\hat V_n$ has to agree with the $V_n$ produced by this theorem and hence
$\hat V$ is $C^r$ and
$\sup_{n\in \Z} \ \hat V_n \_{C^{r}(B_1)} < \infty$.
\item[(f)]
If $V_n$ satisfies $\mbox{\rm{(b)}}$,
$\sup_{n\in \Z} \ V_n \_{C^{N_S}(B_1)} < \infty$ and
$V_0 \in C^k(B_1)$ for some $N_S < k \leq r$ then
$\sup_{n\in \N} \ V_{n} \_{C^{k}(B_1)} < \infty$.
\end{enumerate}
\end{thm}
\begin{remark}
If we assume in (e) that $\sup_{n\in \Z}\ V_n \_{C^{N_S+1}(B_1)} $,
$\sup_{n\in \Z}\ \hat V_n \_{C^{N_S+1}(B_1)} $
are just bounded, scaling the maps we get the suprema to be smaller than 1.
This implies to work in a smaller domain and hence we obtain
uniqueness in this smaller domain.
\end{remark}
To apply Theorem \ref{manifoldsforfamilies}
in proving Theorem \ref{mainthm}, we
take $\tlambda_i = \lambda_i + 2 \delta$,
$\tmu_i = \mu_i  2 \delta$, where $\delta$ is the
small number we had to use in the
estimates (iii) in Lemma \ref{coordinates}.
Recall that it is enough to take $\delta > 2 \eps$.
Note that if a set of intervals is nonresonant and
contractive, then if we enlarge them by a sufficiently
small quantity, then the enlarged intervals also satisfy the
same properties. Hence, when applying Lemma \ref{coordinates}
to the original situation, we have to pay attention to getting that
$\delta$ is small enough that the nonresonance conditions
and contractivity are still satisfied.
Then, taking $w_n(y)=\phi(y,V_n(y))$ all the conclusions
follow.
The rest of the paper will be devoted to a proof of
Theorem \ref{manifoldsforfamilies}.
We denote by $A_n^S$ and $A_n^{S^c}$ the restrictions of $A_n$ to
$E^S$ and $E^{S^c}$ respectively, and by
$\Pi ^S$ and $\Pi ^{S^c} $ the projectors onto $E^S$ and $E^{S^c} $
respectively.
Note that in this setting they are independent of $n$.
Conclusion (b) of Theorem~\ref{manifoldsforfamilies}
can be expressed more explicitly as:
\begin{equation}\label{graphinvariance}
V_{n+1}\big( A_n^S x + \Pi^S \tilde f_n (x, V_n(x))\big)
=
A_n^{S^c} V_n(x) + \Pi^{S^c} \tilde f_n (x, V_n(x)).
\end{equation}
\subsection{Obtaining a polynomial approximation}
To show that such a sequence of maps $V_n$ exists and satisfies
the uniqueness statements, we will start by determining their
candidates to derivatives at zero.
That is, if a sequence $\{ V_n\}_{n\in \Z}$ satisfies
\eqref{graphinvariance}, we will show that the derivatives
at the origin have to satisfy a functional equation.
By studying this functional equation, we will show that, under the
nonresonance conditions, there
is one and only one bounded solution. Hence, if there is a sequence of
differentiable maps, there is only one possibility for their
jets at the origin. We will denote this only possibility by
$V^0_n$. Incidentally, the analysis of this equation will show
that if the nonresonance conditions are not met, it is possible that the
equations for the jet do not have any solution. Hence, a fortiori, that
there are no smooth invariant manifolds satisfying the conclusions of
Theorem \ref{mainthm}.
In later sections we will use $V^0_n$
to transform \eqref{graphinvariance} into another equation
which has better contraction properties.
Taking formally $i$ derivatives of
\eqref{graphinvariance} at zero we obtain that, if the derivatives
exist, they must satisfy
\begin{equation}
\label{derivativesat0}
D^i V_{n+1}(0) {(A_n^S)}^{\otimes i} = A_n^{S^c} D^i V_n (0) + R_{n,i},
\end{equation}
where $R_{n,i}$ is an expression that involves only derivatives of
$V_n$ up to order $i1$ and derivatives of $f_n$ up to order $i$.
We claim that the hierarchy of equations \eqref{derivativesat0} can be
solved recursively and that the solution is unique.
In order to solve \eqref{derivativesat0}, we recall that
$D^i V_n(0)$ is a linear
map from $(E^S)^{\otimes i}$ to $E^{S^c}$. We claim
\begin{lem} \label{jetsfound}
Let $\F = \{f_n\}_{n\in \Z}$ be a family of maps,
$\ \F \_{C^{k}(B_2)} < \infty$,
such that $D f_n (0) = A_n$ satisfies the hypotheses of
Theorem \ref{manifoldsforfamilies}. Then, for
all $1 \leq i \leq k$, the family of equations \eqref{derivativesat0}
have a unique solution in the Banach space
\[
\{\mathcal{T}^i = (T_n^i): \; T_n^i \in L( (E^S)^{\otimes i},E^{S^c}), \;
\\mathcal{T}^i \ = \sup_n \T_n^i \<\infty \} .
\]
Moreover the norm of the solution can be made as small as we want by
taking $\\tilde \F\_{C^{k}(B_2)}$ small.
For $i=1$ the only bounded solution is $DV_n(0) = 0$.
\end{lem}
This result proves parts (c) and (d) of Theorem \ref{manifoldsforfamilies}.
\textbf{Proof of the lemma} We will solve equations \eqref{derivativesat0}
in the components corresponding to the decomposition $E^i$.
Notice that the decomposition
\[
(E^S)^{\otimes i} = \bigoplus_{j_1, \dots, j_i\in S }
E^{j_1} \otimes \cdots \otimes E^{j_i}
\]
is preserved by the linear map
\[
(A_n^S)^{\otimes i} = \bigoplus_{{j_1}, \dots, {j_i}\in S }
A_n^{j_1} \otimes \cdots \otimes A_n^{j_i}:
(E^S)^{\otimes i} \to (E^S)^{\otimes i},
\]
where
\[
A_n^{j_1} \otimes \cdots \otimes A_n^{j_i} :
E^{j_1} \otimes \cdots \otimes E^{j_i}
\to E^{j_1} \otimes \cdots \otimes E^{j_i}
\]
and $\ A_n^{j_1} \otimes \cdots \otimes A_n^{j_i} \ =
\ A_n^{j_1} \ \cdots \A_n^{j_i} \$.
If $T^i_n \in L((E^S)^{\otimes i}, E^{S^c})$, we can use the
decomposition of $(E^S)^{\otimes i}$ and
$E^{S^c} = \bigoplus_{i \in S^c} E^i$ to write
\[
T^i_n = \bigoplus_{l\in S^c,\; j_1, \dots, j_i \in S}
\tau_{n, j_1, \dots, j_i}^l
\]
where
\[
\tau_{n, j_1, \dots, j_i}^l : E^{j_1} \otimes \cdots \otimes E^{j_i}
\to E^l
\]
is linear. The same decomposition is made to the terms $R_{n,i}$
of \eqref{derivativesat0},
which also belong to $L((E^S)^{\otimes i}, E^{S^c})$,
to obtain
$R_{n,i} = \bigoplus_{l\in S^c,\; j_1, \dots, j_i \in S}
R_{n, j_1, \dots, j_i}^l $.
Then, equations \eqref{derivativesat0}
are equivalent to the set of equations
\begin{equation}\label{formal1}
\tau_{n+1,j_1,\dots, j_i}^l A_n^{j_1} \otimes \cdots \otimes A_n^{j_i}
= A_n^l \tau_{n,j_1,\dots, j_i}^l
+ R_{n, j_1, \dots, j_i}^l,
\end{equation}
with $n \in \Z, j_1, \dots, j_i \in S,
l \in S^c$.
Equations \eqref{formal1} can be rewritten in either the form
\begin{equation}\label{formal2}
\tau_{n,j_1,\dots, j_i}^l
= (A_n^l)^{1}
\tau_{n+1,j_1,\dots, j_i}^l A_n^{j_1}\otimes \cdots \otimes A_n^{j_i}
 (A_n^l)^{1} R_{n, j_1, \dots, j_i}^l
\end{equation}
or in the form
\begin{equation}\label{formal3}
\tau_{n+1,j_1,\dots, j_i}^l
= A_n^l \tau_{n,j_1,\dots, j_i}^l
(A_n^{j_1} \otimes \cdots \otimes A_n^{j_i})^{1}
+ R_{n, j_1, \dots, j_i}^l
(A_n^{j_1} \otimes \cdots \otimes A_n^{j_i})^{1}.
\end{equation}
Both equations \eqref{formal2}, \eqref{formal3}
can be considered as fixed point equations
and the right hand side of each of them defines continuous affine maps on
the Banach space
\begin{eqnarray*}
\{\tau_{j_1,\dots, j_i}^l = (\tau_{n+1,j_1,\dots, j_i}^l )_{n\in \Z}:
\; \tau_{n, j_1, \dots, j_i}^l \in
L( E^{j_1} \otimes \cdots \otimes E^{j_i},E^l), \\
\\tau_{j_1,\dots, j_i}^l\ = \sup_{n\in \Z}
\ \tau_{n+1,j_1,\dots, j_i}^l \ < \infty
\} .
\end{eqnarray*}
The map associated to \eqref{formal2} has a Lipschitz
constant which is bounded from above by
\begin{equation}\label{number1}
\exp( \tlambda_{j_1}+ \cdots +\tlambda_{j_i}{\tmu_l})
\end{equation}
while the map associated to \eqref{formal3} has a Lipschitz
constant which is bounded from above by
\begin{equation}\label{number2}
\exp( \tlambda_l \tmu_{j_1} \cdots \tmu_{j_i}).
\end{equation}
Since the intervals $\{ I_i \}_{i\in S}$
are nonresonant, one of the two numbers
\eqref{number1}, \eqref{number2} is smaller than $1$.
Then, one of the two equations \eqref{formal2}, \eqref{formal3}
has a right hand side which defines a contraction. Hence,
one of \eqref{formal2}, \eqref{formal3} can be solved by the
contraction mapping principle.
Since both equations \eqref{formal2}, \eqref{formal3}
are equivalent to \eqref{formal1},
we have proved that the equations \eqref{formal1}
have a unique solution.
All terms $R_{n,i}$ have a factor $D^j\tilde f(0,0)$, $2\le j\le i$,
which is small if $\\tilde \F\_{C^{k}(B_2)}$ is small. Therefore
$R_{n, j_1, \dots, j_i}$ are also small and hence the nonhomogeneous
parts of (\ref{formal2}), (\ref{formal3}) are small. {From} that we deduce that the unique solutions of (\ref{formal1}) are small.
Notice that the equations for the first derivative are homogeneous.
It follows, then, that $DV_n(0) = 0$ is the unique bounded solution
of equations \eqref{derivativesat0} for $i=1$.
This establishes Lemma \ref{jetsfound}. \qed
\subsection{The high order part of the manifolds}
We now rewrite the equations of invariance in such a way
that we can take advantage of the fact that we
already know the low order terms
of the expansion of the solution of the invariance equation.
We will write the sequence of maps
$\VV = \{V_n\}_{n\in \Z}$ in \eqref{graphinvariance} in the form
$\VV = \VV^0 + \UU$ with $\VV^0 = \{V_n^0\}_{n\in \Z}$,
$\UU = \{U_n\}_{n\in \Z}$ and
\begin{equation}\label{substitution}
V_n = V_n^0 + U_n,
\end{equation}
where $V_n^0$ is the unique polynomial of degree $N_S$ whose derivatives
at 0 satisfy equations \eqref{derivativesat0} and $D^i U_n(0) = 0$, for
$0 \leq i \leq N_S$. Note
that $\\VV^0\_{C^r}$ $r \in \N$ can be made as small as we need.
Next we reformulate our original problem in
terms of the $U_n$.
As it turns out, this will be very convenient since
we will have that the $U_n$ are maps that vanish to high order.
For such kind of maps, composing with contractions on
the right, leads to a rather strong contraction factor.
If we substitute \eqref{substitution} in
the equation of invariance \eqref{graphinvariance}
we obtain a fixed point equation for
the operator $\sigma$
acting on sequences of
maps $\UU = \{U_n\}_{n\in \Z}$, defined by
\begin{eqnarray}
\sigma(\UU)_n (x) & = &
(A_n^{S^c})^{1} \big[
(V^0_{n+1}+U_{n+1} ) \circ \psi(\UU)_n (x)
\nonumber \\
\label{graphtransform}
& & \mbox{\hspace{3em}}
 \Pi^{S^c} \tilde f_n (x, V^0_n(x)+U_n(x))
\big]  V_n ^0(x),
\end{eqnarray}
where
\begin{equation}
\label{psi}
\psi(\UU)_n (x) = A_n^S x + \Pi^S \tilde f_n (x, V^0_n(x)+U_n(x)).
\end{equation}
A simple algebraic manipulation shows that,
at least formally  ignoring questions of
whether the compositions can be defined , it is equivalent to say that $\UU$ is a fixed point
of \eqref{graphtransform} than to say that $\VV^0 + \UU$ is a solution
of \eqref{graphinvariance}.
The plan will be to produce fixed points of
\eqref{graphtransform} in a space of well behaved functions for
which the algebraic manipulations involved in
transforming \eqref{graphinvariance} into \eqref{graphtransform}
can be justified.
For the moment, we will consider only the case that $r \in \N$.
Later we will see how this result can be extended to
the cases $r =\omega, \infty$.
More precisely, we consider $\sigma$ acting on the
following spaces:
\begin{eqnarray}
\X_k & = & \{ \UU = \{U_n\}_{n\in \Z}: \; U_n \in C^k(B_1), \; D^i U_n(0) =
0,
\; 0\leq i \leq N_S,
\nonumber \\
\label{Xsubk}
& &
\quad \\UU \_{\X_k} \equiv
\sup_{n\in \Z} \max_{N_S < i \leq k} \ D^i U_n\_{C^0} < \infty \},
\qquad \quad \mbox{for $k > N_S$},
\end{eqnarray}
and
\begin{eqnarray}
\X^0 & = & \{ \UU = \{U_n\}_{n\in \Z}: \; U_n \in C^{N_S}(B_1), \;
D^i U_n(0) = 0, \; 0\leq i \leq N_S,
\nonumber \\
\label{Xsuper0}
& & \quad
\\UU \_{\X^0} \equiv
\sup_{n\in \Z} \sup_{x \in B_1\setminus \{0\}} x ^{1}  D^{N_S} U_n(x) < \infty \}.
\end{eqnarray}
Note that, by the mean value theorem, if $k> N_S$,
$\X_k \subset \X^0$. Moreover, if $\UU \in \X_k$,
$\\UU\_{\X^0} \le \ \UU \_{\X_k}$.
Let $\B_{N_S+1} = \{\UU \in \X _{N_S+1}: \ \UU \_{\X _{N_S+1}} \le 1\}$
and $\B(\rho_{N_S+2}, \dots, \rho_k)$ be the set of families
$\UU = \{ U_n\} _{n\in \Z}\in \X_k$ such that
\begin{itemize}
\item [] $\sup_{n\in \Z}\ D^{N_S+1}U_n\_{C^0} \le 1 $.
\item [] $\sup_{n\in \Z}\ D^jU_n\_{C^0} \le \rho_j $, $\qquad N_S+1 0$
such that
\begin{enumerate}
\item [(i)]$\sigma(\B_{N_S+1}) \subset \B_{N_S+1}$.
\item [(ii)] $\sigma$ restricted to
$\B_{N_S+1} $
is a contraction in the
$\X^0$ norm.
\end{enumerate}
Moreover, under the same smallness conditions on
$\\tilde \F \_{C^{N_S+1}(B_2)}$ needed for $\mbox{\rm{(i)}}$,
$\mbox{\rm{(ii)}}$
\begin{enumerate}
\item [(iii)]
$\sigma(\B(\rho_{N_S+2}, \dots, \rho_k)) \subset \B(\rho_{N_S+2}, \dots, \rho_k)$, for $N_S+2 \leq k \leq r$.
\end{enumerate}
\end{lem}
For future reference, it will be important to note that the
smallness conditions assumed in Lemma \ref{lemaux}
are only smallness assumptions
$\\tilde \F \_{C^{N_S+1}(B_2)}$ and do not change as we
increase $r$.
Before proving the lemma we first establish some bounds for $\psi$.
Given $\UU, \hat \UU \in \B_{N_S+1}$ we have
\begin{eqnarray}
\label{devpsi1}
\D \psi(\UU)_n \ & \leq &\A_n^S\ + K \ \tilde \F \_{C^1},\\
\hat \UU\_{C^0},
\label{psi0}
x^{1}\\psi(\UU)_n(x)\
& \leq & \A_n^S\+ K \\tilde \F \_{C^1}, \\
\label{divpsi0}
x^{1}\(\psi(\UU)_n  \psi(\hat \UU)_n)(x)\
& \leq & K \\tilde \F\_{C^1} \\UU \hat \UU\_{\X^0}, \\
\label{difdevpsi0}
x^{1}\D (\psi(\UU)_n  \psi(\hat \UU)_n)(x)\
& \leq & K \\tilde \F\_{C^2} \\UU \hat \UU\_{\X^0},
\end{eqnarray}
where $K$ is a constant independent of $n$. Formula \eqref{devpsi1}
is straightforward from the definition of $\psi$, in \eqref{psi},
while \eqref{psi0}\eqref{difdevpsi0} follow from the
fact that $\tilde f_n(0) = 0$, $D\tilde f_n (0) = 0$,
$V_n^0 (0) = 0$
and the derivatives
of $V_n^0$ and $U_n$ are uniformly bounded.
{\bf Proof of Lemma \ref{lemaux}.\,}
It is clear that, by the choice of the polynomials $V_n^0$,
\[
D^i \sigma(\UU)_n(0) = 0, \qquad 0 \leq i \leq N_S.
\]
We observe that, for $k\ge 2$, by the Faadi Bruno formula,
\begin{equation} \label{formulafaa}
D^k(\sigma \UU)_n = (A_n^{S^c})^{1}D^k U_{n+1}
\circ \psi(\UU)_n D\psi(\UU)_n^{\otimes k}
+ B_{n}(\UU) D^k U_n + R_{n,k}(\UU),
\end{equation}
where
\begin{eqnarray}
B_{n}(\UU) & = & (A_n^{S^c})^{1} D (V^0_{n+1}+U_{n+1})
\circ \psi(\UU)_n
\Pi^S D_2 \tilde f_n \circ \eta(\UU)_n
\nonumber \\
& &  (A_n^{S^c})^{1}\Pi^{S^c} D_2 \tilde f_n \circ \eta(\UU)_n
\end{eqnarray}
is linear,
\begin{equation}
\label{eta}
\eta(\UU)_n = (\Id, V_n^0 + U_n)
\end{equation}
and, since $k\geq N_S+1$,
\begin{equation}
\begin{split}
R_{n,k}& (\UU) = \\
& (A_n^{S^c})^{1} \Big(
\sum_{l=
2}^{N_S}
\sum_{*}
c_{j_1,\dots,j_l}^{k,l}
D^l V_{n+1}^0 \circ \psi(\UU)_n
D^{j_1}\psi(\UU)_n \otimes \cdots \otimes
D^{j_l}\psi(\UU)_n
\\
&
+ \sum_{l=2}^{k1} \sum_{*}
c_{j_1,\dots,j_l}^{k,l}
D^l U_{n+1} \circ \psi(\UU)_n
D^{j_1}\psi(\UU)_n \otimes \cdots \otimes
D^{j_l}\psi(\UU)_n
\\
&
+ D(V_{n+1}^0+U_{n+1})\circ \psi(\UU)_n \times
\\
&\quad \quad\times
\Pi^S
\sum_{l=2}^{k} \sum_{*}
c_{j_1,\dots,j_l}^{k,l}
D^l \tilde f_n \circ \eta(\UU)_n
D^{j_1}\eta(\UU)_n \otimes \cdots \otimes
D^{j_l}\eta(\UU)_n
\\
&  \Pi^{S^c}
\sum_{l=2}^{k} \sum_{*}
c_{j_1,\dots,j_l}^{k,l}
D^l \tilde f_n \circ \eta(\UU)_n
D^{j_1}\eta(\UU)_n \otimes \cdots \otimes
D^{j_l}\eta(\UU)_n \Big),
\end{split}
\end{equation}
where $ \sum_{*}$ stands for the sum over the indices $j_i$
such that $1 \leq j_1, \dots, j_l \leq k $ and $j_1+ \cdots +j_l = k$
and the coefficients
$c_{j_1,\dots,j_l}^{k,l}$ are combinatorial numbers.
Notice that a first derivative of $\tilde f_n$ appears as a factor in
each term in $B_{n}(\UU)$. Also, $R_{n,k}(\UU)$ consists on a finite
sum of terms. Some of them have explicitly a derivative of $\tilde f_n$
as a factor.
The other terms have a factor of the form
\begin{equation}\label{factor}
D^{j_1}\psi(\UU)_n \otimes \cdots \otimes
D^{j_l}\psi(\UU)_n
\end{equation}
where
\begin{equation}\label{index}
j_1 + \cdots + j_l = k \quad \mbox{and} \quad l \leq k1.
\end{equation}
Because of \eqref{index},
there is some $i$ such that $j_i \geq 2$ and therefore,
among the factors in \eqref{factor}, there is a factor
of the form
\[
D^{j_i}\psi(\UU)_n = D^{j_i} (\tilde f_n \circ \eta_n(\UU)).
\]
Now we consider the case $k=N_S+1$.
The previous factors can be made arbitrarily small by assuming
that $\ \tilde \F\_{C^{N_S+1}}$ is sufficiently small.
As a consequence,
for any given $\nu>0$, if $\ \tilde \F\_{C^{N_S+1}}$ is small enough,
we have that
\[
\B_{n}(\UU)\,\, \R_{n,N_S+1}(\UU) \_{C^0} < \nu.
\]
Since $\sup_n
\(A_n^{S^c})^{1}\ \, \A_n^S\^{N_S+1} < 1
$,
there exists $\nu >0$ such that
\[
\gamma \equiv \(A_n^{S^c})^{1}\ \, (\A_n^S\+\nu)^{N_S+1} + 2\nu < 1
\]
and hence
\[
\ D^{N_S+1}(\sigma \UU)_n \_{C^0} \leq
\sup_n \big(
\(A_n^{S^c})^{1}\ \,( \A_n^S\+\nu)^{N_S+1} + \nu \big)
\D^{N_S+1}\UU\_{C^0}
+ \nu \leq \gamma,
\]
for all $\UU $ in $\B_{N_S+1}$.
This proves (i).
If $N_S+1 < k \le r$, from (\ref{formulafaa}) we have that
\[
\ D^{k}(\sigma \UU)_n \ \leq
\gamma \sup_n
\D^kU_n \ + Q(\rho_{N_S+2}, \dots , \rho_{k1}) ,
\]
where $Q$ is a polynomial. Therefore, since $\gamma < 1$, there
exists $\rho_k>0$ such that
$\rho_k = \gamma\rho_k + Q(\rho_{N_S+2}, \dots , \rho_{k1})$
and
$\sigma(\B(\rho_{N_S+2}, \dots, \rho_k)) \subset
\B(\rho_{N_S+2}, \dots, \rho_k)$.
To prove (ii) we
take derivatives, and we obtain
\begin{eqnarray}
\lefteqn{x ^{1} \ D^{N_S} (\sigma(\UU)_n \sigma(\hat \UU)_n)(x)\}
& & \nonumber \\
\label{linearterm1}
& \leq &
x ^{1}
\(A_n^{S^c})^{1}D^{N_S} (U_{n+1} \hat U_{n+1})
\circ \psi(\UU)_n(x) D\psi(\UU)_n(x)^{\otimes {N_S}} \ \\
\label{dif1}
& \mbox{}+ &
x ^{1}
\(A_n^{S^c})^{1}
(D^{N_S}\hat U_{n+1} \circ \psi(\UU)_n(x) \\
\nonumber
& & \mbox{\hspace{5em}} D^{N_S}\hat U_{n+1}\circ \psi(\hat \UU)_n)(x)
D\psi(\UU)_n(x)^{\otimes {N_S}} \
\\
\label{difcontraction1}
& \mbox{}+ &
x ^{1}
\(A_n^{S^c})^{1}
D^{N_S}\hat U_{n+1}\circ \psi(\hat \UU)_n(x) \times \\
&& \mbox{\hspace{5em}} \nonumber \times
(D\psi(\UU)_n(x)^{\otimes {N_S}}D\psi(\hat \UU)_n(x)^{\otimes {N_S}}) \
\\
\label{linearterm2}
& \mbox{}& + x ^{1}\B_{n}(\UU)(x) D^{N_S} (U_n \hat U_n)(x)\ \\
\label{difcontraction2}
& \mbox{}& +
x ^{1}\(B_{n}(\UU)B_{n}(\hat \UU))(x) D^{N_S} \hat U_n (x)\ \\
\label{rest}
& \mbox{}& +
x ^{1}\ (R_{n,{N_S}}(\UU)R_{n,{N_S}}(\hat \UU))(x)\.
\end{eqnarray}
Taking into account inequalities \eqref{devpsi1} and \eqref{psi0},
we bound \eqref{linearterm1} as follows,
\begin{eqnarray*}
\lefteqn{x ^{1}
\(A_n^{S^c})^{1}D^{N_S} (U_{n+1} \hat U_{n+1})
\circ \psi(\UU)_n(x) D\psi(\UU)_n(x)^{\otimes N_S} \ }& & \\
& \leq &
x ^{1}\psi(\UU)_n(x)\,
\(A_n^{S^c})^{1}\ \,\D\psi(\UU)_n \^{N_S}
\U_{n+1} \hat U_{n+1}\_{\X^0} \\
& \leq &
\(A_n^{S^c})^{1}\ \big( \A_n^S \+K\\tilde \F\_{C^1}
\big)^{N_S+1}
\U_{n+1} \hat U_{n+1}\_{\X^0}.
\end{eqnarray*}
To bound \eqref{dif1}, we use inequality \eqref{divpsi0} and
the fact that $\UU, \hat \UU \in \B_{N_S+1}$. In this way,
\begin{eqnarray*}
\lefteqn{x ^{1}
\(A_n^{S^c})^{1}
(D^{N_S}\hat U_{n+1} \circ \psi(\UU)_n(x)
 D^{N_S}\hat U_{n+1}\circ \psi(\hat \UU)_n(x))
D\psi(\UU)_n(x)^{\otimes {N_S}} \} & & \\
& \leq &
\(A_n^{S^c})^{1}\\,\ D\psi(\UU)_n\^{N_S}
\\hat \UU\_{C^{N_S+1}}x ^{1}
\(\psi(\UU)_n
\psi(\hat \UU)_n)(x)\ \\
& \leq &
K \(A_n^{S^c})^{1}\
\big( \A_n^S \+K\\tilde \F\_{C^1} \big)^{N_S}
\\hat \UU\_{C^{N_S+1}}\\tilde \F\_{C^1}
\\UU
\hat \UU\_{\X^0}
\end{eqnarray*}
Term \eqref{difcontraction1} can be bounded in the
following way, using inequalities \eqref{difdevpsi0} and
\eqref{devpsi1},
\begin{eqnarray*}
\lefteqn{x ^{1}
\(A_n^{S^c})^{1}
D^{N_S}\hat U_{n+1}\circ \psi(\hat \UU)_n(x)
(D\psi(\UU)_n(x)^{\otimes {N_S}}D\psi(\hat \UU)_n(x)^{\otimes {N_S}}) \
} && \\
& \leq &
\(A_n^{S^c})^{1}\
\big( \A_n^S \+K\\tilde \F\_{C^1} \big)
\ \hat U_{n+1} \_{C^{N_S}} \\
&& \quad x ^{1}
\D\psi(\UU)_n(x)^{\otimes {N_S}}D\psi(\hat \UU)_n(x)^{\otimes {N_S}} \
\\
& \leq &
N_S K \(A_n^{S^c})^{1}\
\big( \A_n^S \+K\\tilde \F\_{C^1} \big)^{N_S}
\ \hat U_{n+1} \_{C^{N_S}} \ \tilde \F \_{C^2}
\\UU \hat \UU \_{\X^0},
\end{eqnarray*}
where we have bounded
\begin{eqnarray*}
\lefteqn{
x ^{1}
\D\psi(\UU)_n(x)^{\otimes {N_S}}D\psi(\hat \UU)_n(x)^{\otimes {N_S}} \} \\
& \leq &
\sum_{j=1}^{N_S}
x ^{1}\D\psi(\UU)_n(x)D\psi(\hat \UU)_n(x)\ \,
\D\psi(\UU)_n\^{N_Sj} \D\psi(\hat \UU)_n\^{j1}
\\
& \leq & N_S K \big( \A_n^S \+K\\tilde \F\_{C^1} \big)^{N_S1}
\ \tilde \F \_{C^2}
\\UU \hat \UU \_{\X^0}.
\end{eqnarray*}
Term \eqref{linearterm2} can be easily bounded taking into
account that $\ B_n(\UU)\ \leq \tilde K \\tilde \F\_{C^1}$,
for some $\tilde K >0$.
To obtain a
bound
for \eqref{difcontraction2} we proceed in the following way:
\begin{equation}\label{separated}
\begin{split}
\B_{n}(\UU)B_{n}(\hat \UU)\ \leq &
\(A_n^{S^c})^{1}\\, \ D (V^0_{n+1}+U_{n+1})
\circ \psi(\UU)_n
\Pi^S D_2 \tilde f_n \circ \eta(\UU)_n\\
& 
D (V^0_{n+1}+\hat U_{n+1})
\circ \psi(\hat \UU)_n
\Pi^S D_2 \tilde f_n \circ \eta(\hat \UU)_n \ \\
+ &
\ \Pi^{S^c} D_2 \tilde f_n \circ \eta(\UU)_n
 \Pi^{S^c} D_2 \tilde f_n \circ \eta(\hat \UU)_n \.
\end{split}
\end{equation}
Since
$
\\eta(\UU)_n(x)\eta(\hat \UU)_n(x)\ = \ U_n(x) \hat U_n(x)\
$,
the last term of \eqref{separated} can be bounded immediately by
\[
\\tilde \F\_{C^2} \\UU  \hat \UU \_{\X^0}.
\]
The first term in \eqref{separated}
can be split into the three
following terms that can be easily bounded
\begin{eqnarray*}
& \ D (U_{n+1}\hat U_{n+1})
\circ \psi(\UU)_n
\Pi^S D_2 \tilde f_n \circ \eta(\UU)_n \, & \\
& \ \big( D (V^0_{n+1}+\hat U_{n+1})
\circ \psi(\UU)_n

D (V^0_{n+1}+\hat U_{n+1})
\circ \psi(\hat \UU)_n\big )
\Pi^S D_2 \tilde f_n \circ \eta(\UU)_n \, & \\
& \ D (V^0_{n+1}+ \hat U_{n+1})
\circ \psi(\hat\UU)_n \big(
\Pi^S D_2 \tilde f_n \circ \eta(\hat \UU)_n

\Pi^S D_2 \tilde f_n \circ \eta(\UU)_n \big) \. &
\end{eqnarray*}
Finally, to bound \eqref{rest}, we recall that
$R_{n,N_S}(\UU)$ is a finite sum of terms having expressions of
the following forms:
\begin{enumerate}
\item
Terms of the form
\[
D^l V_{n+1}^0 \circ \psi(\UU)_n
D^{j_1}\psi(\UU)_n \otimes \cdots \otimes
D^{j_l}\psi(\UU)_n, \qquad 2 \leq l \leq N_S, \; j_1+ \cdots +j_l = k.
\]
The norm of their difference is easily bounded by
\begin{eqnarray*}
& & x^{1}\
\big( D^l V_{n+1}^0 \circ \psi(\UU)_n

D^l V_{n+1}^0 \circ \psi(\hat \UU)_n \big)
D^{j_1}\psi(\UU)_n \otimes \cdots \otimes
D^{j_l}\psi( \UU)_n \ \\
& & +
x^{1}\
D^l V_{n+1}^0 \circ \psi(\hat \UU)_n \big(
D^{j_1}\psi(\UU)_n \otimes \cdots \otimes
D^{j_l}\psi(\UU)_n
\\
& & \mbox{\hspace{3em}}
D^{j_1}\psi(\hat \UU)_n \otimes \cdots \otimes
D^{j_l}\psi(\hat \UU)_n \big)\.
\end{eqnarray*}
Both differences can be bounded by some constant times
$
\\tilde \F\_{C^2} \\UU  \hat \UU \_{\X^0}.
$
\item
Terms of the form
\[
D^l U_{n+1} \circ \psi(\UU)_n
D^{j_1}\psi(\UU)_n \otimes \cdots \otimes
D^{j_l}\psi(\UU)_n, \quad 2 \leq l \leq N_S1, \; j_1+ \cdots +j_l = k.
\]
Since $\UU \in \B_{N_S+1}$, their difference can be bounded
as in the previous case.
\item
Terms of the form
\[
D^l \tilde f_n \circ \eta(\UU)_n
D^{j_1}\eta(\UU)_n \otimes \cdots \otimes
D^{j_l}\eta(\UU)_n,
\qquad 1 \leq l \leq N_S, \; j_1+ \cdots +j_l = k,
\]
which can be treated as before, since $\tilde f_n$ is $C^{N_S+1}$.
\item
Terms of the form
\[
D (V^0_{n+1}+ U_{n+1})
\circ \psi(\UU)_n
\Pi^S
D^l \tilde f_n \circ \eta(\UU)_n
D^{j_1}\eta(\UU)_n \otimes \cdots \otimes
D^{j_l}\eta(\UU)_n,
\]
where $2 \leq l \leq N_S, \; j_1+ \cdots +j_l = k$,
that can be bounded like the preceding ones.
\end{enumerate}
It follows, then, that
the Lipschitz constant of $\sigma$ is less than
\[
\(A_n^{S^c})^{1}\\big( \A_n^{S}\ + \nu \big)^{N_S+1}+\nu,
\]
where $\nu$ is as small as we need taking
$\\tilde \F\_{C^{N_s+1}}$ small enough. Then,
$\sigma$ is a contraction.
\qed
By (i) and (ii) it is clear that there is a fixed point of
$\sigma$ in $\X^0$  hence, a solution of \eqref{graphinvariance} .
Moreover, because of (i), this fixed point of $\sigma$ also belongs
to the $\X^0$closure of $\B(\rho_{N_S+2}, \dots, \rho_k)$,
$N_S+1 \leq k \leq r$.
By Lemma \ref{lemaux} and Proposition A2 in \cite{Lan},
we get the existence
of a $C^{r1+\mbox{\scriptsize lip}}$ solution of the
invariance equation. Furthermore, given any
$\UU \in \B(\rho_{N_S+2}, \dots, \rho_k)$,
the sequence $\sigma^j(\UU)$ tends in the $C^{r1}$ norm
to the fixed point of $\sigma$.
Now, to check that this solution is in fact $C^r$,
we note first that
\[
\R_{n,r}(\UU)  R_{n,r}(\hat \UU)\_{C^0} \to 0,
\]
when $ \ \UU  \hat \UU\_{C^{r1}} \to 0$ and
$\UU, \hat \UU \in \B(\rho_{N_S+2}, \dots, \rho_k)$.
This fact is trivial for all the terms involving derivatives
up to order $r1$ of $\tilde \F$, $\UU$ and $\hat \UU$, since these
derivatives are in fact Lipschitz. The only terms involving
$r$ derivatives are
\[
D (V^0_{n+1}+ U_{n+1})
\circ \psi(\UU)_n
\Pi^S
D^r \tilde f_n \circ \eta(\UU)_n D\eta(\UU)_n^{\otimes r}
\]
and
\[
\Pi ^{S^c} D^r \tilde f_n \circ \eta(\UU)_n D\eta(\UU)_n^{\otimes r}
\]
which are continuous in $\UU$ since $D^r \tilde f_n$ and $\eta$ are
continuous with modulus of continuity independent of $n$.
Next we consider the sequence of maps
$\UU^l=\{U^l_n\}_{n\in \Z}$ defined by
\[
U_n^0 = 0, \; \; n\in \Z, \qquad \UU^{l+1} = \sigma(\UU^l).
\]
The preceding arguments show that the sequence
$\UU^l \in \B(\rho_{N_S+2}, \dots, \rho_k)$ and
converges in the $C^{r1}$ norm to a function
$\UU^{\infty}=\{U^\infty_n\}_{n\in \Z}$.
The sequence of $r$derivatives satisfies the recurrence relation
\[
D^r U_n^{l+1} = (A_n^{S^c})^{1}D^r U_{n+1}^l
\circ \psi(\UU^l)_n D\psi(\UU^l)_n^{\otimes r}
+ B_{n}(\UU^l) D^r U_n^l + R_{n,r}(\UU^l).
\]
That is, denoting $D^r U_{n}^l$ by $T_n^l$,
$\TT^l = \{ T_n^l\}_{n\in \Z}$, we have that
\begin{equation}
\TT^{l+1} = \AA(\UU^l) \TT^l + \RR(\UU^l),
\end{equation}
with
\begin{eqnarray*}
(\AA(\UU^l) \TT^l )_n & = & (A_n^{S^c})^{1} T_{n+1}^l
\circ \psi(\UU^l)_n D\psi(\UU^l)_n^{\otimes r}
+ B_{n}(\UU^l) T_n^l, \\
\RR(\UU^l)_n & = & R_{n,r}(\UU^l).
\end{eqnarray*}
We have that $\AA(\UU^l)$ is a linear map
from
\begin{eqnarray*}
\Xi & = & \{
\TT = \{T_n\}_{n\in \Z}:\; T_n\in C^0(B(0,1), L^r(\R^d;\R^d), \\
&& \qquad\qquad \omega(T_n ,\eta) \mbox{ uniformly bounded in } n\}
\end{eqnarray*}
to itself. Notice that the terms of the sequence
$\TT^l$ belong to $\Xi$.
Moreover
both $\AA(\UU)$ and $\RR(\UU)$ are continuous in
the $C^0$ norm when $\UU$ is $C^{r1}$ with modulus of continuity
of the $r1$ derivative bounded.
We claim that the sequence $\TT^l$ converges in the $C^0$ norm to
a continuous map. Indeed, this limit will be the only bounded solution,
$\TT^{\infty}$,
of the equation
\begin{equation}
\label{atthelimit}
\TT = \AA(\UU^{\infty}) \TT + \RR(\UU^{\infty}).
\end{equation}
This equation has a unique solution since,
\[
\ \AA(\UU^{\infty}) \ \leq \sup_{n\in \Z} \(A_n^{S^c})^{1}\
\big( \A_n^{S} \ + \nu \big)^r + \nu < \gamma < 1,
\]
and, hence, the right hand side of (\ref{atthelimit}) is a
contraction. We can also assume that
$\ \AA(\UU^{l}) \ \leq \gamma$.
To prove the claim we check that $\TT^l \to \TT^{\infty}$,
when $l \to \infty$, in the $C^0$ norm.
\begin{eqnarray*}
\\TT^l  \TT^{\infty} \ & \leq &
\ \AA(\UU^{l1})\TT^{l1}  \AA(\UU^{\infty})\TT^{\infty}\
+\ \RR(\UU^{l1}) \RR(\UU^{\infty}) \ \\
& \leq & \gamma \\TT^{l1}  \TT^{\infty}\ + d_l,
\end{eqnarray*}
where
$$
d_l = \ (\AA(\UU^{l1})  \AA(\UU^{\infty}))\TT^{\infty}\
+\ \RR(\UU^{l1}) \RR(\UU^{\infty}) \.
$$
Notice that, by the continuity of $\AA$ and $\RR$,
$d_l \to 0$, when $l \to \infty$. Then it is clear that
$$
\\TT^l  \TT^{\infty} \ \leq
\gamma^{l1} \\TT^{1}  \TT^{\infty}\ +
\sum_{j=0}^{l1} \gamma^j d_{lj},
$$
which tends to 0 when $l \to \infty$
since $\gamma <1$ and $d_l \to 0$.
This proves that
$D^r U_{n}^l$ tends in the $C^0$ norm to $T_n^{\infty}$.
To check that this map is the $r$ derivative
of $\UU^{\infty}$, we simply note that for all $n$
\[
D^{r1}U^l_n(y)  D^{r1}U^l_n(x) =
\int_0^1 D^r U_n^l(x+s(yx))(yx)\, ds.
\]
Since the integrand in the right hand side converges uniformly
to the continuous map $T_n^{\infty}$, we have
\begin{eqnarray*}
D^{r1}U^\infty_n(y)  D^{r1}U^\infty_n(x) & = &
T^\infty_n(x)(yx) \\&& +
\int_0^1 [T^\infty_n(x+s(yx))T^\infty_n(x) ](yx)\, ds
\end{eqnarray*}
and hence $D^r U^{\infty}_n (x)= T_n^{\infty}(x)$.
To prove the case when $r = \infty$,
we note that when $f_n\in C^\infty$, we can find a sequence
of positive numbers $\rho_{N_S+2}, \dots, \rho_k $ such that
$\sigma(\B(\rho_{N_S+2}, \dots, \rho_k)) \subset \B(\rho_{N_S+2}, \dots, \rho_k)$,
for all $k \ge N_S+2$.
According to the preceding arguments the fixed point of $\sigma$ is
$C^{r1+\mbox{\scriptsize lip}}$ for all $k$, and hence $C^\infty$.
The case $r = \omega$ is much easier. It just suffices to observe that the
previous arguments work exactly the same in a complex ball. We consider
the Banach space of functions, analytic in the open ball, continuous
on the closed ball, vanishing to order $N_S$ at the origin topologized
with the supremum of the $N_S + 1$ derivative.
We have established the existence
claim in the theorem, (a) and (b). To prove
the uniqueness claim (e) consider $\VV = \{V_n\}_{n\in \Z}$,
$\hat \VV = \{\hat V_n\}_{n\in \Z}$. From (b),
$\VV= \sigma(\VV)$ and $\hat \VV= \sigma(\hat \VV)$.
Then
$$
\\VV \hat \VV\_{\X^0} = \\sigma(\VV ) \sigma(\hat \VV) \_{\X^0} \le
\lip \sigma_{\mid \B_{N_S+1}} \\VV \hat \VV\_{\X^0}
$$
and $\lip \sigma_{\mid \B_{N_S+1}}< 1$ in the $\X^0$ norm.
This shows that $\VV= \hat \VV$.
To prove claim (f), we observe that \eqref{graphinvariance}
shows that if $\hat V_n$ is $C^{k}$, $ k \leq r$, for some $n$,
then it is $C^{k}$ for all $n$. To prove the claim, we just
have to obtain uniform estimates for the derivatives, assuming
they exist. We shall consider the case $k= N_S+1$. The other
cases follow by induction.
If we take $N_S+1$ derivatives of \eqref{graphinvariance}, we obtain
in a similar way as for (\ref{formulafaa})
\begin{eqnarray}
\label{derivativek}
D^{N_S+1} \hat V_n(x) & = &
(A_n^{S^c})^{1} D^{N_S+1} \hat V_{n+1} \big( A_n^S x +
\Pi^S \tilde f_n(x, \hat V_n(x))\big) (A_n^{S})^{\otimes {(N_S+1)}}
\nonumber \\
& & \mbox{}
+ Q_{n,{N_S+1}}(\hat \VV),
\end{eqnarray}
where $Q_{n,{N_S+1}}(\hat \VV)$ contains terms with derivatives up
to order ${N_S+1}$ of $\hat V_n$ and $\hat V_{n+1}$, but all
of them multiplied
by factors which involve derivatives of $\tilde f_n$ up
to order ${N_S+1}$.
{From} that, we conclude that
\begin{eqnarray}
\label{boundderivativek}
\ D^{N_S+1} \hat V_n\_{C^0} & \leq &
\(A_n^{S^c})^{1}\ \,\A_n^S\^{{N_S+1}} \D^{N_S+1} \hat V_{n+1}\_{C^0}
\nonumber \\
& & \mbox{}
+ \nu ( \\hat V_n\_{C^{N_S+1}} + \\hat V_{n+1}\_{C^{N_S+1}}),
\end{eqnarray}
where $\nu$ can be made as small as we need by assuming that
$\\tilde \F\_{C^r}$ is sufficiently small.
We observe that
\begin{eqnarray}
\label{norm}
\\hat V_n \_{C^{N_S+1}}
& \leq & \sup ( \\hat V_n\_{C^{N_S}}, \D^{{N_S+1}}\hat V_n\_{C^0}) \nonumber
\\
& \leq & \\hat \VV \_{C^{N_S}} + \D^{{N_S+1}}\hat V_n\_{C^0}.
\end{eqnarray}
Substituting \eqref{norm} into \eqref{boundderivativek} and
using that $\(A_n^{S^c})^{1}\ \,\A_n^S\^{{N_S+1}}< 1$ we obtain
that
\[
\ D^{N_S+1} \hat V_n\_{C^0} \leq \gamma \ D^{N_S+1} \hat V_{n+1}\_{C^0}
+D,
\]
where $\gamma <1$ and $D$ is some constant
independent of $n$. {From} that, claim (f) follows easily.
\qed
Notice that the lengths of the manifolds and the uniform bounds
on their derivatives depend only on the properties of the orbit
and the $C^r$ properties of the map.
For the case that the orbits come from a splitting of the Mather
spectrum, we can choose these constants uniformly for all the
orbits in the manifold.
By AscoliArzel\'a theorem, the mapping $x \mapsto V_x$ maps the
compact set on $M$ into a $C^{r1+\mbox{\scriptsize lip}}$ compact set of submanifolds.
Moreover, the uniqueness statements we have proved show that
in these topologies the mapping has a closed graph.
Hence, it should be continuous. We point out that the
methods of \cite{Pes73} allow to prove that the mapping
$x \mapsto V_x$ is continuous when the manifolds are given
in the $C^r$ topology. To obtain a similar improvement with
our methods, it would have been necessary to show the existence
of a uniform modulus of continuity for the
$r$ derivatives in Theorem \ref{manifoldsforfamilies}.
\section{An example} \label{sec:example}
The following example
illustrates some of the
subtle phenomena involved in the
slow manifolds showing that uniqueness
may hold or not depending very much on the details of
the conditions. In particular, it shows that some of the
limitations in Theorem \ref{manifoldsforfamilies} do belong.
The example is presented as a family of maps, as in the setting of
Theorem~\ref{manifoldsforfamilies}, but, as it will be shown at the end of
this section, this family of maps can be lifted to a
smooth map from a four dimensional compact manifold
to itself.
The construction of such a lift is explicit, and
follows the one in \cite{Pugh84}.
\begin{example}\label{mainexample}
Consider the sequence of maps $f_n : \R^2 \rightarrow \R^2$,
$n \in \Z$, defined by:
\begin{equation}\label{sequence}
\begin{split}
& f_n (x_1, x_2) = (\frac{1}{3} x_1, \frac{1}{20} x_2), \qquad n \ne 0, \\
& f_0 (x_1, x_2) = (\frac{1}{3} x_1, \frac{1}{20} x_2 + \vp(x_1,x_2) ),
\end{split}
\end{equation}
where $\vp$ is a $C^\infty$ real valued function with compact support  which
we will think of as very small  not including
$(0,0)$ and
\begin{equation} \label{conditiononvp}
\left\sup_{(x_1,x_2) \in \R^2}
\frac{\partial \vp}{\partial x_2}(x_1,x_2) \right
< \frac{1}{20}.
\end{equation}
\end{example}
Clearly, we have
\[
D f_n(0,0) =
\begin{pmatrix}
& \frac{1}{3} & 0 \\
& 0 & \frac{1}{20}
\end{pmatrix}, \qquad n \in \Z.
\]
Moreover, condition (\ref{conditiononvp}) ensures that each $f_n$,
$n\in \Z$, is a bijective map. Indeed, this assertion is trivial for $n\neq
0$. For $n=0$, we remark that, given $(z_1,z_2) \in \R^2$, the
equation
$$
f_0(x_1,x_2) = (z_1,z_2)
$$
is equivalent to
$$
x_1 = 3z_1, \qquad x_2 = 20 z_2  20 \vp (3 z_1, x_2).
$$
The second condition
has a unique solution, since its right hand side defines
a contraction on $\R$. Hence, because $\det Df_n(x_1, x_2) \neq 0$,
for all $(x_1,x_2) \in \R^2$, the functions
$f_n$, $n\in \Z$, are global diffeomorphisms.
The sequence of maps \eqref{sequence} verifies
Definition \ref{def1} if we take as $E^i_n$ the coordinate axes
and we set
$\lambda_1 = \mu_1 = \log(1/3)$,
$\lambda_2 = \mu_2 = \log(1/20)$, $ \ell = 1$, $\eps = 0 $.
We take $E^S_n$ to consist just of the
first coordinate axis.
In such case, $N_S = [ \log 20/ \log 3] = 2$.
The set of manifolds $\graph( V_n )$ satisfies the
condition
\[
f_n( \graph(V_n )) =\graph(V_{n+1}), \qquad \mbox{for all }n \in \Z
\]
if and and only if
the functions $V_n$ satisfy
\begin{equation}\label{solution}
V_{n+1}(x_1) = \frac{1}{20} V_n( 3 x_1) + \delta_{n,0}
\vp(3 x_1, V_n(3 x_1))
\end{equation}
where $\delta$ is the $\delta$ of Kronecker.
Furthermore, it is easy to verify by
induction in $n$ that a sequence of functions $V_n$ satisfying
\eqref{solution}
satisfies also the initial condition
\[
V_0 = \Psi,
\]
where $\Psi: \R \rightarrow \R$ is a $C^\infty$ function with
compact support such that $\Psi (0) = 0$,
if and only if
it is of the form
\begin{equation}\label{fullsolution}
V_n( x_1) =
\begin{cases}
\left( \frac{1}{20}\right)^n
(\Psi + 20 \vp \circ (\mbox{Id},\Psi))( 3^n x_1), &\qquad n \ge 1, \\
\left( \frac{1}{20}\right)^n \Psi( 3^n x_1), &\qquad n \le 0.
\end{cases}
\end{equation}
We see that
\begin{equation} \label{derivative}
\frac{d^j}{d x_1^j}
V_n( x_1) =
\begin{cases}
\left( \frac{3^j}{20} \right)^n
\frac{d^j}{d x_1^j}
(\Psi + 20 \vp \circ (\mbox{Id},\Psi))
( 3^n x_1), &\qquad n \ge 1, \\
\left( \frac{3^j}{20} \right)^n
\frac{d^j}{d x_1^j}
(\Psi)( 3^n x_1), &\qquad n \le 0.
\end{cases}
\end{equation}
We make the following observations.
\begin{itemize}
\item[(i)]
Since $V_n'(0) = \left( \frac{3}{20} \right)^n \Psi'(0) $ we see that
the derivative is unbounded unless
$\Psi'(0) = 0$.
In such a case, $V_n'(0) = 0$, for all $n \in \Z$.
This phenomenon of boundedness of first derivatives implying
tangency is an illustration of part (d) of Theorem \ref{mainthm}.
\item[(ii)]
Suppose that $\Psi$ has support not containing $0$.
If $j \le N_S$  equivalently, $3^j/20 < 1$ , we have that
$ V_n^{(j)}( x_1)  $
is bounded uniformly on $n$
in a ball around the origin.
This follows by observing that for $n > 0$, we have uniform boundedness
in \eqref{derivative} because of the factor
$\left( \frac{3^j}{20} \right)^n$.
For $n < 0$, we have boundedness because for
$n$ sufficiently negative, the support of $V_n$ is
outside of the unit ball.
This illustrates that we cannot expect uniqueness by only assuming
boundedness of derivatives of order less than $N_S$.
\item[(iii)]
Let $j >N_S$. Choose $\vp \equiv 0$ and $\Psi$ such that
$\Psi(0) = 0 $, $\Psi'(0) = 0$,
$\Psi^{(j)}(0) \ne 0 $.
Then, it is easy to see that
$V_n_{C^j}$ is bounded for $n$ negative but not
for $n$ positive.
\item[(iv)]
If $j > N_S$,
the only possibility of having uniform bounds for
$
V_n^{(j)}( x_1)  $
for $n > 0 $ is that
\begin{equation} \label{functionalequation}
\Psi =  20 \varphi \circ (\mbox{Id},\Psi).
\end{equation}
We claim that this equation has a unique continuous solution
which is $C^{\infty}$
and has compact support.
Indeed, for any $x\in \R$, condition (\ref{conditiononvp}) implies that
the right hand side of the equation
$$
y = 20 \vp (x, y)
$$
is a contraction, and, hence, has a unique solution $\Psi(x)$.
Since $\vp$ has compact support, $\Psi(x)$
also has compact support. The standard implicit function theorem
ensures that $\Psi$ is $C^{\infty}$.
This determines uniquely the functions $V_n$ for all values of
$n$.
Note that for these functions
$ V_n^{(j)}( x_1)  $ is uniformly bounded
for $n < 0$.
This illustrates the fact that we have uniqueness under the assumption of
uniformly boundedness of
the derivatives of order bigger than $N_S$.
\end{itemize}
Now we show how to embed this family in a smooth map.
In the following, we will denote by $\SS^2$ the two sphere.
\begin{prop} \label{liftsequencetomap}
Consider the family of maps $\{f_n\}_{n \in \Z}$ of
Example~\ref{mainexample}.
Then,
there exist a two dimensional compact smooth manifold, $M^2$,
a smooth map $F:\SS^2 \times M^2 \to \SS^2 \times M^2$,
a point $z_0 \in \SS^2 \times M^2$ with orbit
$\{z_n = F^n (z_0)\}$, and smooth two dimensional
submanifolds $N_n\subset \SS^2 \times M^2$ such that
\begin{itemize}
\item[i)] $z_n \in N_n$, $F(N_n) \subset N_{n+1}$,
\item[ii)] there exists a diffeomorphism
$\sigma$ such that $F_{\mid N_n} = \sigma^{1} \circ f_n \circ
\sigma$, $n\in \Z$.
\end{itemize}
\end{prop}
\Proof
The construction of the map $F$ is performed in two steps.
The first one consists in lifting the discrete family $\{f_n\}$ to
a smooth family of maps
$\{\tilde g_{\lambda}\}_{\lambda \in [0,1]}$, with
$\tilde g_{\lambda}:\SS^2 \to \SS^2$.
In the second one,
with the aid of an auxiliary map on a compact smooth manifold,
exhibiting a chosen hyperbolic dynamics  a Smale horseshoe, for instance
, the map $F$ and the orbit are explicitly given.
We introduce some notations that will be used later.
We fix $\SS^2$ to be $\{(x,y,z) \in \R^3: \; x^2+y^2 +z^2 = 1\}$,
$\SS^2_+ = \SS^2 \cap \{z >0\}$, $\SS^2_ = \SS^2 \cap \{z <0\}$ and
$E = \SS^2 \cap \{z =0\}$. Let $\pi: \SS^2_ \to \R^2$ be the map
\begin{equation} \label{projection}
\pi (x, y, z ) = \left(  \frac{x}{z},  \frac{y}{z} \right).
\end{equation}
Notice that $\pi$ is the projection of $\SS^2$ from the center
of the sphere onto the plane $\{z = 1\}$. We remark that
$\pi$ is a diffeomorphism. We denote $\pi^{1}$ by $\sigma$.
It is clear that
\begin{equation} \label{chart}
\sigma (x_1,x_2) =
\left( \frac{x_1}{\sqrt{1+x_1^2+x_2^2}},
\frac{x_2}{\sqrt{1+x_1^2+x_2^2}},
\frac{1}{\sqrt{1+x_1^2+x_2^2}}
\right).
\end{equation}
We also consider the antipodal map $\mu: \SS^2 \to \SS^2$, that is,
$\mu(p) = p$. In this way, $\sigma$ is a chart of $\SS^2$ covering
$\SS^2_$ and $\mu \circ \sigma$ is a chart covering $\SS^2_+$.
We define the one parameter family of maps $g_{\lambda}:\R^2 \to \R^2$
by
\begin{equation} \label{glambda}
g_{\lambda}(x_1,x_2) = (a x_1, b x_2 + \lambda
\varphi(x_1,x_2)),
\end{equation}
where $a= 1/3$ and $b=1/20$.
We have that $g_1 = f_0$ and $g_0 = f_n$, $n\in \Z \setminus \{0\}$.
Now we define a lift of $g_{\lambda}$ to $\SS^2_$ by
\begin{equation} \label{gtilde}
\tilde g_{\lambda}^ = \sigma \circ g_{\lambda} \circ \pi,
\end{equation}
to $\SS^2_+$ by
\begin{equation} \label{gtilde+}
\tilde g_{\lambda}^+ = \mu \circ \tilde g_{\lambda}^ \circ \mu^{1},
\end{equation}
and finally
$$
\tilde g_{\lambda}(p) =
\begin{cases}
\tilde g_{\lambda}^+(p), & \qquad p \in \SS^2_+, \\
\tilde g_{\lambda}^(p), & \qquad p \in \SS^2_ .
\end{cases}
$$
{From} (\ref{projection}) and (\ref{chart}) we have that,
for $(x,y,z) \in \SS^2$, $z\neq 0$,
$$
\tilde g_{\lambda}(x,y,z) =
\left(
\frac{ax}{\rho(x,y,z)},
\frac{by\lambda z \varphi(x/z,y/z)}{\rho(x,y,z)},
\frac{z}{\rho(x,y,z)}
\right),
$$
where
$$
\rho(x,y,z) = \sqrt{z^2+a^2 x^2+(by\lambda z \varphi(x/z,y/z))^2}.
$$
We remark that $\rho$
never vanishes.
Since $\varphi$ has compact support, $\tilde g_{\lambda}$ extends
to a $C^{\infty}$ map
defined in the whole sphere. Indeed, if we take
$R = \inf_{r\in \R} \{r:\, \mbox{supp}(\vp) \subset D_r(0)\}$,
we have that $\varphi(u,v) = 0$, for $(u,v)>R$. Then,
for any $(x,y,z)\in \SS^2$ such that
$z<1/\sqrt{1+R^2}$, we have $\rho(x,y,z) = \sqrt{z^2+a^2 x^2+b^2
y^2}$ and
$$
\tilde g_{\lambda}(x,y,z) =
\left(
\frac{ax}{\rho(x,y,z)},
\frac{by}{\rho(x,y,z)},
\frac{z}{\rho(x,y,z)}
\right),
$$
which extends to a unique $C^{\infty}$ function in a neighborhood
of $E$ and, hence, to $\SS^2$.
Notice that, for each $\lambda \in [0,1]$,
$\tilde g_{\lambda}: \SS^2 \to \SS^2$ is a diffeomorphism,
which preserves $\SS^2_$ and $\SS^2_+$, and that, restricted to $\SS^2_$,
is conjugated to $g_{\lambda}$ through $\pi$ and $\sigma$. Moreover,
the dependence on the parameter $\lambda$ is smooth.
Now, we consider any compact two dimensional smooth manifold, $M^2$,
and a diffeomorphism $h:M^2\to M^2$ with an invariant hyperbolic
subset $\Sigma$ such that $h_{\mid \Sigma}$ is conjugated to
the Bernoulli shift with two symbols. We can, for example, take $h$
to be a Smale horseshoe. We can also assume that the Lyapunov exponents of
this hyperbolic set are bigger than $\log 3$, in order to avoid
resonances. Let $q_0 \in \Sigma$ be
the point corresponding to the sequence $(\cdots 11011 \cdots)$.
Clearly, there exists a $C^{\infty}$ function with compact support,
$\eta:M^2 \to [0,1]$, such that $\eta \equiv 1$ in a neighborhood of
$q_0$ and vanishes outside a compact set which does not include
$h^n(q_0)$, $n\neq 0$, that is, $\eta(h^n(q_0)) = \delta_{n,0}$, $n\in \Z$.
We define $F:\SS^2 \times M^2 \to \SS^2 \times M^2$ by
$$
F(p,q) = (\tilde g_{\eta(q)}(p), h(q)),
$$
and we check that $F$ satisfies the properties listed in Proposition
\ref{liftsequencetomap}. We consider the orbit of the
point $z_0 = (\sigma(0,0), q_0)$. It is clear that
$z_n = F^n(z_0) = (\sigma(0,0), q_n)$, where $q_n = h^n(q_0)$.
By definition, the submanifolds $N_n = \SS^2 \times \{q_n\} \equiv \SS^2$
verify that $F(N_n) = N_{n+1}$ and $F_{\mid N_n} = \tilde
g_{\eta(q_n)}$. Finally, $ \tilde g_{\eta(q_n)}$ is conjugated by
$\sigma$ to $f_n$, which establishes the claim. \qed
\centerline{\sc Acknowledgments}
\medskip
E.F and P.M. acknowledge the
partial support of the
the Spanish Grant DGICYT BFM20000805 and
the Catalan grant CIRIT 2001SGR70.
R.L. has been partially supported by
NSF and acknowledges the hospitality of
UPC and UB as well as ICREA.
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