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\title[Computation of invariant tori and their whiskers (I)]
{A parameterization method for the computation of
invariant tori and their
whiskers in quasi periodic maps: \\
rigorous results.}
\author{A. Haro}
\address{
Departament de Matem\`atica Aplicada i An\`alisi, Facultat de Matem\`atiques,
Universitat de Barcelona.
Gran Via de les Corts Catalanes 585, 08007 Barcelona (Spain).}
\email[A. Haro]{\tt haro@cerber.mat.ub.es}
%\email{\tt haro@cerber.mat.ub.es}
\author{ R. de la Llave}
\address{
Dept. of Mathematics,
University of Texas at Austin.
Austin TX 78712 (USA)}
\email[R. de la Llave]{llave@math.utexas.edu}
%\email{llave@math.utexas.edu}
\setcounter{tocdepth}{3}
\begin{document}
\maketitle
\contentsline {section}{\tocsection {}{1}{Introduction}}{2}
\contentsline {section}{\tocsection {}{2}{Overview of the paper}}{5}
\contentsline {subsection}{\tocsubsection {}{2.1}{Existence of invariant tori}}{5}
\contentsline {subsection}{\tocsubsection {}{2.2}{Existence of asymptotic invariant manifolds (whiskers)}}{8}
\contentsline {subsection}{\tocsubsection {}{2.3}{One dimensional asymptotic manifolds.}}{10}
\contentsline {subsection}{\tocsubsection {}{2.4}{Notation}}{12}
\contentsline {subsubsection}{\tocsubsubsection {}{2.4.1}{Differential geometry}}{12}
\contentsline {subsubsection}{\tocsubsubsection {}{2.4.2}{Spaces of differentiable functions with anisotropic differentiability}}{13}
\contentsline {subsubsection}{\tocsubsubsection {}{2.4.3}{Spaces of analytic functions}}{14}
\contentsline {subsubsection}{\tocsubsubsection {}{2.4.4}{Polynomial bundle maps}}{15}
\contentsline {subsubsection}{\tocsubsubsection {}{2.4.5}{Transfer operators, cocycles}}{15}
\contentsline {section}{\tocsection {}{3}{Invariant tori}}{16}
\contentsline {subsection}{\tocsubsection {}{3.1}{Existence and persistence of invariant tori}}{16}
\contentsline {subsection}{\tocsubsection {}{3.2}{Bootstrap on the regularity}}{20}
\contentsline {section}{\tocsection {}{4}{Asymptotic invariant manifolds}}{23}
\contentsline {subsection}{\tocsubsection {}{4.1}{Statement of results}}{24}
\contentsline {subsection}{\tocsubsection {}{4.2}{Examples}}{26}
\contentsline {subsection}{\tocsubsection {}{4.3}{Proof of the theorem on invariant manifolds }}{27}
\contentsline {subsubsection}{\tocsubsubsection {}{4.3.1}{Preliminaries}}{27}
\contentsline {subsubsection}{\tocsubsubsection {}{4.3.2}{Finding the dynamics on the manifold}}{28}
\contentsline {subsubsection}{\tocsubsubsection {}{4.3.3}{Standing hypotheses}}{30}
\contentsline {subsubsection}{\tocsubsubsection {}{4.3.4}{The equation for the higher order terms}}{30}
\contentsline {subsubsection}{\tocsubsubsection {}{4.3.5}{Functional spaces and lemmas on derivatives of highly iterated functions}}{31}
\contentsline {subsubsection}{\tocsubsubsection {}{4.3.6}{Solving the linearized equation}}{34}
\contentsline {subsubsection}{\tocsubsubsection {}{4.3.7}{Solving the equation with one less derivative}}{37}
\contentsline {subsubsection}{\tocsubsubsection {}{4.3.8}{The last derivative}}{39}
\contentsline {subsubsection}{\tocsubsubsection {}{4.3.9}{Proof of the uniqueness of the invariant manifold}}{40}
\contentsline {section}{\tocsection {}{5}{Results for flows}}{42}
\contentsline {subsection}{\tocsubsection {}{5.1}{Reduction of the results for flows to results for maps}}{42}
\contentsline {subsection}{\tocsubsection {}{5.2}{The Poincar\'e trick}}{43}
\contentsline {subsection}{\tocsubsection {}{5.3}{A direct treatment of the differential equations case}}{44}
\contentsline {subsubsection}{\tocsubsubsection {}{5.3.1}{The one dimensional case.}}{45}
\contentsline {section}{\tocsection {}{6}{Acknowledgments}}{46}
\contentsline {section}{\tocsection {}{}{References}}{46}
\begin{abstract}
We prove
rigorous results on persistence of invariant tori
and their invariant manifolds for quasiperiodically perturbed
systems. The proofs are based on the parametrization method
of \cite{CabreFL03a,CabreFL03b}.
The invariant manifolds results proved here include
as particular cases
of the usual (strong) stable and (strong)
unstable manifolds, but also include other non-resonant
manifolds.
The method lends itself to numerical
implementations whose analysis and implementation is
studied in \cite{HLlnum}.
\end{abstract}
\section{Introduction}
The goal of this paper is to present a proof of existence of invariant
tori and their invariant manifolds
in quasi-periodically perturbed systems. The proof is designed to be readily
implementable and the implementation is described in the companion paper
\cite{HLlnum}.
The systems we consider are of the form:
\begin{itemize}
\item (discrete time)
\begin{equation}
\label{discrete}
\begin{array}{l}
\bar x= F(x,\th) \ ,\\
\bar \th= \th + \omega \ ,
\end{array}
\end{equation}
where $x\in\nr^n$ and $\th\in\nt^d$ are variables,
and $\omega\in\nr^d$ is the {\em rotation vector}.
\item (continuous time)
\begin{equation}
\label{continuous}
\begin{array}{l}
\dot x= X(x,\th) \ ,\\
\dot \th= \omega \ ,
\end{array}
\end{equation}
where $x\in\nr^n$ and $\th\in\nt^d$ are variables,
and $\omega\in\nr^d$ is the {\em frequency vector}.
\end{itemize}
Systems of the form \eqref{discrete}, \eqref{continuous} are called
skew-products in the mathematical literature. In applications they appear
when one forces a system with a quasi-periodic external perturbation.
An important particular case is when the external forcing is small. That is:
\begin{itemize}
\item (quasi-autonomous discrete time)
The system \eqref{discrete} has the form
\begin{equation}
\label{perturb_discrete}
F(x,\th) = F_0(x) + F_1(x,\th)\ ,
\end{equation}
where $F_1$ is small. For $F_1\equiv 0$ the
dynamics of $x$ and $\th$ are uncoupled
and for a fixed point $x_0$ of $F_0$ the torus
\[
\K_0=\{x_0\}\times\nt^d
\]
is invariant for the whole system \eqref{discrete} given by
\eqref{perturb_discrete}.
\item (quasi-autonomous continuous time)
The system \eqref{continuous} has the form
\begin{equation}
\label{perturb_continuous}
X(x,\th)= X_0(x) + X_1(x,\th)\ ,
\end{equation}
where $X_1$ is small. For $X_1\equiv 0$ the
dynamics of $x$ and $\th$ are uncoupled
and for a fixed point $x_0$ of $X_0$ the torus
\[
\K_0=\{x_0\}\times\nt^d
\]
is invariant for the whole system \eqref{continuous} given by
\eqref{perturb_continuous} when $X_1 = 0$.
Similarly, if $x_0(t)$ is a $T$ periodic orbit for the
autonomous system $X_0$, then
\[
\K_0= \{x_0(t) \st t\in[0,T] \} \times\nt^d
\]
is invariant for $X$ when $X_1 = 0$.
\end{itemize}
As motivation for the results presented here, we note
that in the perturbative case, it is natural to look
for invariant manifolds close to $\K_0$ in
the perturbed systems.
If $x_0$ had invariant manifolds, we can consider
whether there are corresponding objects for
the quasiperiodically excited problem.
We will consider two types of problems for quasiperiodic
systems \eqref{discrete}, \eqref{continuous}.
\begin{itemize}
\item[a)] Existence of invariant tori (of dimension $d$)
which are normally hyperbolic,
and persistence of such tori under perturbations;
\item[b)] Existence of asymptotic manifolds attached to an invariant
torus (also known as whiskers).
\end{itemize}
In the case that the fixed point $x_0$ of $F_0$ (or $X_0$) is hyperbolic
and that the manifolds we are considering are the stable and
unstable manifolds, persistence results follow
from the general theory of normally hyperbolic
manifolds
\cite{Fenichel71, Fenichel74, Fenichel77,HirschP69, HirschPS77}.
In this paper we present an approach which
differs from the classical approach of
normal hyperbolicity. The approach is well suited for
numerical implementations (which we will discuss in \cite{HLlnum}).
For the cases we consider, it also has some mathematical advantages
over the general theory of normal hyperbolicity.
\begin{itemize}
\item
For part a), taking advantage of
the special structure of \eqref{discrete},\eqref{continuous} we will
show in Section~\ref{tori} that the invariant tori
in a) are as smooth as the system (including
analytic) and that they depend smoothly (including analytic)
on parameters. Such results are false for more general systems
(see e.g. \cite{Llave01} for explicit examples).
\item
For part b) we obtain in Section~\ref{whiskers}
asymptotic manifolds (also called {\em whiskers})
associated to
non-resonant parts of the linearization.
This included as particular cases, the strong
stable and the strong unstable manifolds of
\cite{Fenichel71, Fenichel74, Fenichel77,HirschPS77},
but it also includes other cases
associated to other invariant bundles of the linearization.
In some cases, we can consider invariant manifolds that correspond
to the slowest directions. These manifolds have interest in applications
since they dominate the asymptotic behavior of the systems.
Non-resonant invariant manifolds have been considered in
\cite{Llave97,ElBialy01,CabreFL03a,CabreFL03b,Llave03} for fixed points
of diffeomorphisms in Banach spaces
and, using a device introduced in
\cite{HirschP69}, one can extend them
to construct non-resonant manifolds for
an invariant manifold. The results obtained
here take advantage of the special structure of
the system and conclude more regularity, including
analyticity, which does not follow from the general theory.
\end{itemize}
A very important motivation for the work undertaken
here is that the method of proof
can be modified into a numerical algorithm.
The method unifies the calculation of invariant tori
and of invariant manifolds so that the algorithms for
both are based on very similar functional equations, so that
the implementations for both cases can share significant
parts of the code.
In the companion paper \cite{HLlnum},
we discuss implementation issues and results that have
been obtained applying the method to some test cases that have appeared
in the literature.
The results presented here are designed to serve
as justification of numerical calculations. We
have formulated them as \emph{``a posteriori''} estimates.
The existence of invariant objects is equivalent to
the existence of a function satisfying a functional equation.
We will show that given a function that solves it approximately
and that satisfies some non-degeneracy conditions, then
there is a true solution. Moreover the distance from the true
solution to the approximate one can be bounded by the
error in the solution of the functional equation.
One can, for example, take as an approximate solution
the result of a numerical computation. To verify the
reliability of the computed solutions, one does not need
to study the algorithm to compute them. It suffices to check
that they satisfy the equation approximately and that they
satisfy the non-degeneracy conditions.
The method we use has precedents and has been in common use in the
BCN group \cite{Simo90,Simo98,CastellaJ00,LibI,LibII,LibIII,LibIV}.
Other methods are based on
graph transforms appear in \cite{BroerOV97, Osinga98, Lorenz95}.
The proof we present is based on the parameterization method. In general,
the parameterization method formulates a functional equation for a
parameterization of
the invariant manifold as well as for the dynamics on it.
With respect to the general parameterization method,
the cases considered here have the important advantage
that we know that the motion in the angle variables
variables is a rotation with the frequency of the
external perturbation. This simplifies substantially
the functional equations considered and eliminates
the main source of difficulties in the analysis considered in
\cite{CabreFL03a, CabreFL03b}, namely, the existence of unknown functions
that appear as composition in the right.
As will be seen in more detail later,
the functional equation that we need to deal with
is much better behaved for the analysis than those that appear in the
graph transform method (they give rise to a differentiable operator
in $C^r$ spaces that can be studied with the regular implicit function
theorem in Banach spaces). Since the construction will happen in $C^r$
-- even analytic -- spaces, the error bounds will happen in very
differentiable norms and we will be able to obtain numerical control
over high derivatives of the manifolds.
The good analytic properties
of the functional equations
giving the invariant manifolds
seem to translate in numerical stability of the algorithm.
We postpone this discussion till \cite{HLlnum}. Also in \cite{HLlnum}
we will present variants of the results here which, even if
less precise in the number of derivatives used, etc. are
designed to validate numerical computations. In particular, in
\cite{HLlnum} we will present
\emph{a posteriori} bounds that assert that given an approximate
solution of the functional equation that satisfy non-degeneracy
conditions, there is a true solution close by.
In \cite{HLlnum} we will give particular attention to the situation where
the non-resonant
manifolds have one normal direction. This is the situation that has also
been considered in the literature in the stable case
e.g. in \cite{OsingaF00}.
\section{Overview of the paper}
In this section we discuss heuristically the main ideas of the paper and
explain its organization. For the sake of simplicity, we will discuss the
discrete case. Indeed, we will present detailed proofs of the main
results only in the discrete case and in
Section \ref{sec:flows} we will present a short argument
which shows that
the results for flows can be deduced from the
results for maps.
\subsection{Existence of invariant tori}
\label{heuristic_tori}
The existence and persistence of invariant tori
for \eqref{discrete} is developed in
Section~\ref{tori} (see Theorem~\ref{theorem_tori}).
It is based on the equation
\begin{equation}
\label{parameterization_tori}
F(K(\th-\omega),\th-\omega)- K(\th) = 0\ ,
\end{equation}
where $F:\nr^n\times\nt^d\to\nr^n$ and $\omega\in\nr^d$ are given
and we are supposed to find $K:\nt^d\to\nr^n$.
We note that \eqref{parameterization_tori} is equivalent to
\begin{equation}
\label{invariance_torus}
F(K(\th),\th) = K(\th + \omega) \ ,
\end{equation}
which makes it clear that the set of points
\begin{equation}
\label{graph_torus}
\K= \{(K(\th),\th) \ | \ \th\in\nt^d\}
\end{equation}
is invariant under the dynamical system \eqref{discrete}.
Indeed, $K$ is a parameterization of
a torus in which the dynamics is a rotation.
In the ergodic theory literature, situations such as \eqref{invariance_torus}
are described as saying that the rotation by $\omega$ is a subfactor of the
skew-product \eqref{discrete}, or, somewhat less precisely, that
\eqref{discrete} is semiconjugate to a rotation by $\omega$.
\begin{remark} \label{subharmonic}
An interesting phenomenon, widely discussed in applications,
is that of \emph{subharmonic} tori. These are invariant tori
so that their basic frequencies are a submultiple of those
of the perturbation. These can be incorporated in
\eqref{parameterization_tori} simply by considering the
external perturbation as having an internal frequency
which is a submultiple of the original one.
We just note that it suffices to consider, instead of $\th\in\nt^d$,
$\th\in (N\nt)^d$, where $N\in\nn$.
A more refined version is to change
$\th\in \nt^d= \nt \times\dots\times \nt$ by
$\th\in N_1\nt \times \dots \times N_d\nt$,
where $N_1,\dots,N_d\in \nn$ (notice that $N$ is the minimum common
multiple of $N_1,\dots,N_d$).
These coverings transformations are also useful when considering
whiskers which are non orientable.
\end{remark}
\begin{remark}
One can wonder if it is possible to find
some more invariant tori for \eqref{discrete}
on which the motion
is semiconjugate to a rotation.
(In the ergodic theory literature, this is described
saying that \eqref{discrete} \emph{``admits the rotation as a
factor''}. )
We see that the most general form of an invariant embedding of a torus
$\nt^d$ in $\nr^n\times\nt^d$ is given by
$\tilde K: \nt^d \rightarrow \nr^n \times \nt^d$,
where $\tilde K(\varphi) = (\tilde K_x( \varphi), \tilde K_\th(\varphi))$
satisfies
\begin{equation} \label{invarianceextended}
\begin{split}
& F( \tilde K_x ( \varphi), \tilde K_\th( \varphi) ) =
\tilde K_x( \varphi + \omega) \ , \\
& \tilde K_\th( \varphi) + \omega = \tilde K_\th( \varphi + \omega)\ .
\end{split}
\end{equation}
We observe that, when the rotation $\omega$ is ergodic,
the only measurable solutions of
the second equation in \eqref{invarianceextended} are
$\tilde K_\th(\varphi) = \varphi + a$, where $a\in\nr^d$.
Noticing that $\tilde K( \varphi + a)$ is a solution of
\eqref{invarianceextended} if $\tilde K( \varphi )$ is,
we obtain that, when $\omega$ is ergodic, the solutions of
\eqref{parameterization_tori} are in one-to-one correspondence
with the solutions of \eqref{invarianceextended}, i.e. the
existence of rotations as subfactors of
\eqref{discrete}.
\end{remark}
The study we will undertake is based in the
study of the equation
\eqref{parameterization_tori}.
It is quite important to notice that provided
$F\in C^{r+1}(\nr^n\times\nt^d,\nr^n)$, the operator
$\Tau: C^{r}(\nt^d,\nr^n) \to C^{r}(\nt^d,\nr^n)$ defined
by
\begin{equation}
\label{Tau}
\Tau(K)(\th)\equiv F(K(\th-\omega),\th-\omega) - K(\th)
\end{equation}
is a differentiable operator when
$C^{r}(\nt^d,\nr^n)$ is given the $C^r$ topology. See \cite{LlaveO99}.
Hence, as we will see, we can study
\eqref{parameterization_tori} using standard implicit function theorems in
$C^r$ spaces in the case that the invariant manifold is normally
hyperbolic.
Note that a formal calculation -- which is justified in
\cite{LlaveO99} -- gives
\begin{equation}
\label{DTau}
\Dif\Tau(K)\Delta(\th) =\Dif_x F(K(\th-\omega),\th-\omega)
\Delta(\th-\omega) - \Delta(\th) \ .
\end{equation}
Hence, once we show that $\Dif \Tau$ is invertible, it is clear by the
Implicit Function Theorem that the existence of approximate solutions
implies existence of true solutions (see Section~\ref{tori}).
In particular, if we have a true solution for a certain $F$, for which
$\Dif \Tau$ is invertible, it will be an approximate solution if we modify
$F$ slightly and, hence, we have a true solution for the modified $F$.
For an invariant torus, the invertibility of $\Dif \Tau$ is
closely related to the fact that the manifold is normally hyperbolic.
This is an
extension of the theory of the characterization of Anosov diffeomorphisms
in \cite{Mather68}, that has been studied in many places
\cite{Mane78,HirschPS77,Swanson83, trilogy3}.
An elaboration of this theory
for rotations on tori
can be found in \cite{trilogy2}.
In particular, it is very important for us that the invertibility of
$\Dif \Tau$ comes from properties of the
spectrum of the operator $\M_\omega$ defined by
\begin{equation}
\label{transfer}
\M_\omega \Delta (\th)= \Dif_x F(K(\th-\omega),\th-\omega)
\Delta(\th-\omega)\ ,
\end{equation}
which is a shift multiplied by the monodromy matrix
\[
M(\th)= \Dif_x F(K(\th),\th)\ .
\]
The operator $\M_\omega$ is referred to as
the transfer operator associated to
$M_\omega$, which is a vector bundle map over a rotation $\omega$.
The spectral theory of transfer operators, not necessarily over rotations,
has been studied in
\cite{Mather68,LatushkinS90, LatushkinS91,ChiconeL99,trilogy1}.
The spectral theory of transfer operators over rotations has been studied
in \cite{trilogy2}.
One of the main developments in
\cite{trilogy2} is that the equivalence between normal hyperbolicity
and invertibility of
$\Dif \Tau$ in $C^0$ spaces that holds in general,
for systems
of the form \eqref{discrete} becomes
an equivalence between normal
hyperbolicity and
invertibility of
$\Dif \Tau$ in $C^r$
spaces, $r \in \nn \cup \{\infty, a\}$ (where $a$ means analytic).
This is not true in systems which are not of the form \eqref{discrete}.
\begin{remark}
Equation \eqref{parameterization_tori} can also be used to find
invariant tori in some cases where $\K$ is not normally hyperbolic.
Notably, in the case of Hamiltonian systems, equations
very similar to \eqref{parameterization_tori} has been used to compute
KAM tori or lower dimensional tori, including also their existence under
quasiperiodic perturbations.
A version of KAM theory related to the theory developed here is found
in \cite{Russmann76,CellettiC97,JorbaLZ00,Llave01,GonzalezJLlV00}
(see also \cite{JorbaS92} for perturbative results
in the context of lower dimensional tori).
\end{remark}
For the systems \eqref{continuous}, the analogous of
\eqref{parameterization_tori} is
\begin{equation}
\label{parameterization_tori_continuous}
X(K(\th),\th)= \Dif K(\th) \omega \ .
\end{equation}
Due to the appearance of a derivative,
\eqref{parameterization_tori_continuous}
is apparently much worse behaved that
\eqref{parameterization_tori}. Nevertheless, by passing to integral forms
equation \eqref{parameterization_tori_continuous}
can be dealt with also using implicit function theorems.
Also, it can be reduced to taking time-one map or Poincar\'e
sections with transversals. We discuss this in Section
\ref{sec:flows}.
\subsection{Existence of asymptotic invariant manifolds (whiskers)}
\label{heuristic}
The study of the invariant manifolds attached to an invariant torus
$\K$ as in Section~\ref{heuristic_tori} is developed in Section~\ref{whiskers}.
We note that the perturbations in the dynamic variables propagate by the
variational equations of \eqref{discrete} on the torus $\K$:
\begin{equation}
\label{variational}
\begin{split}
\Delta \bar x & = \Dif_x F(K(\th),\th) \Delta x\ ,\\
\bar \th & = \th + \omega \ ,
\end{split}
\end{equation}
where $\Delta x\in\nr^n$ and $\th\in\nt^d$. To study the variational
equations \eqref{variational} it is natural to consider them as acting
in $\nr^n\times \K \simeq \nr^n\times \nt^d$. The points in $\K$ move
according to the rotation $\omega$.
In the general theory of normal hyperbolic manifolds one studies
the action of the variational equations on
$T_\K (\nr^n\times \nt^d)$, the tangent bundle of the (extended) phase
space $\nr^n\times \nt^d$ restricted to the invariant object $\K$.
In this theory one uses a splitting
\begin{equation}
\label{TN}
T_\K (\nr^n\times \nt^d) = T \K \oplus N \K\ ,
\end{equation}
where $T\K$ is the tangent bundle of $\K$ and $N\K$ is a bundle
transversal to $T\K$.
In the general theory of normal hyperbolic manifolds, the splitting
\eqref{TN} is generally not invariant under the action of the variational
equations.
In fact, $T \K$ is invariant while $N \K$ generally is not
(the variational equations have a block triangular structure).
However, in our case we can take $N\K$ to correspond to the directions
along the $\nr^n$ factor of the (extended) phase space. With this
choice, the splitting \eqref{TN} is invariant and the variational
equations are block diagonal. Since the block corresponding to $T\K$ is
just the identity, to study dynamical properties it suffices to study
the cocycle corresponding to \eqref{variational}.
In the language of
global differential geometry, the variational equations can be considered
as a linear vector bundle map on a bundle $N\K$ whose fibbers are $\nr^n$ and
whose base points are the points in $\K\simeq\nt^d$.
We show in Theorem~\ref{theorem_whiskers}
that given a subbundle $E_1$ of $N\K$ invariant under
the variational equations and such that the
spectrum of the linearization
restricted to it satisfies certain non-resonance
conditions, then, there is an invariant manifold tangent to
this subbundle which is invariant under the map.
This result includes, as a particular case, the
usual strong stable and strong unstable invariant
manifold theorems, but it also includes some
more exotic manifolds. In particular, sometimes
we can find invariant manifolds corresponding to the less
contracting part of the spectrum. These are the ``slow'' manifolds,
which dominate the approach to the manifold.
Our study of whiskers is based on the study of the equation
\begin{equation}
\label{parameterization_whisker}
F(W(\eta,\th),\th)=
W(\Lambda(\eta,\th),\th+\omega) \ ,
\end{equation}
where, as before, $F:\nr^n\times\nt^d\to\nr^n$ and $\omega\in\nr^d$ are given
and we are supposed to find $W:\nr^{n_1}\times\nt^d\to\nr^n$ and
$\Lambda:\nr^{n_1}\times\nt^d\to\nr^{n_1}$, where $n_1\leq n$. Moreover,
$W(0,\th)= K(\th)$ is the parameterization obtained in
\eqref{discrete} of the invariant torus and $\Lambda(0,\th)= 0$.
The equation \eqref{parameterization_whisker} implies that
\begin{equation}
\label{graph_whisker}
\W =
\{(W(\eta, \th),\th) \ | \ \th\in\nt^d \ , \eta\in\nr^{n_1}\} \ ,
\end{equation}
is invariant under $F$, and $\Lambda$
is the induced dynamics on the manifold.
The conditions
\[
\begin{split}
&W(0,\th)= K(\th) \ , \\
&\Lambda(0,\th)= 0 \ ,
\end{split}
\]
express that
$\W$ extends the invariant manifold found in \eqref{discrete}.
Notice also that
\[
\Dif_x F(K(\th),\th) W^1(\th) =
W^1(\th+\omega) \Lambda^1(\th)\ ,
\]
where
$W^1(\th)= \Dif_\eta W(0,\th)$,
$\Lambda^1(\th)= \Dif_\eta \Lambda(0,\th)$.
This says that the bundle spanned \
by the columns of $W^1$ is invariant under the variational equations
\eqref{variational}.
Notice that in this formulation such a bundle is trivial.
This is the case which often appears in practice but,
even if the bundles could be non-trivial, using a device in \cite{HirschP69}
one can augment the bundles so that they become trivial.
Notice that $W$ and $\Lambda$ are not uniquely defined,
Nevertheless, as we will see, it is possible
to chose normalizations that make them unique.
We will try to find simple expressions
for $\Lambda$, in particular, polynomial expressions. Notice that the case
$n_1= n$ amounts to find normal forms for the dynamics around the torus
(these are the nonstationary normal forms).
When
\[
|\Dif_\eta \Lambda(0,\th)| \le \lambda < 1
\]
or, more generally, that for some $n \in \nn$,
\[
| \Dif_\eta \Lambda(0, \th + n \omega) \cdots \Dif_\eta \Lambda(0,\th)| \le \lambda < 1
\]
we obtain that the
points of $\W$ close to $\K$ converge to $\K$ upon iteration of the map.
In other words, when $\Lambda(\cdot,\th)$ is a contraction for
all $\th$, all $\eta$ sufficiently small, the manifolds
that we obtain
are submanifolds of the usual stable manifold.
The solution of the equation~\eqref{parameterization_whisker}
is more complicated
than that of \eqref{parameterization_tori}. In general, it involves
non-resonance
conditions on the spectrum of $\Dif\Tau(K)$. Such non-resonance
conditions are automatic
when one studies strong stable manifolds or the classical stable manifolds.
Hence, we obtain the classical theorems as particular cases, but
we can obtain invariant manifolds associated to other non-resonant subbundles.
We also note that \eqref{parameterization_whisker} gives the invariant manifold
$\W$ in a parametric form. In some cases -- notably the
rotating Henon map, which has been extensively used in the literature --
this parameterization is global (it is an entire function).
{From} the numerical point of view, the fact that
the parameterization is global has the advantage that the algorithm
presented here -- in principle -- does not require the step of globalization.
The parameterization \eqref{parameterization_whisker}
does not have any difficulty following
the twists and turns typical of invariant manifolds in
its domain of definition. Of course, these
turns interfere severely with the possibility of studying
the manifolds as graphs.
In \cite{HLlnum}, we present a more detailed discussion of
numerical issues. Even if the numerical parameterization can follow several
folds of the manifold, it sometimes reduces the error
to evaluate it in a small domain -- where the error is very
small -- and then, propagate it.
\subsection{One dimensional asymptotic manifolds.}
\label{one_dimensional}
In this section, we will discuss the simplest case of the method,
which happens when the invariant subbundle is one dimensional ($n_1= 1$)
and trivial (we can get this using covering transformations),
$\omega$ is Diophantine, and $F$ is smooth enough.
This case has been considered in the literature (e.g \cite{OsingaF00}),
and it is also given special attention in \cite{HLlnum}.
Given the special interest of the case, it is worth presenting it in detail.
The presentation includes all the essential ideas of the problem
but avoids some of the technical complications that will be incorporated
later. The discussion will be
informal and we
will not keep track of what are the differentiability assumptions, etc.
Let us consider the invariance equation
\begin{equation}
\label{parameterization_whisker1}
F(W(\eta,\th),\th)= W(\lambda \eta, \th + \omega) \ ,
\end{equation}
where $\lambda\in\nr$ is a number to be determined. Notice that
we are fixing the dynamics on the manifold as the simplest one: it is
linear in the normal direction, and the expansion rate is constant.
\begin{remark}
The fact that the expansion rate is constant
is not too restrictive when the subbundle is one dimensional
and the rotation $\omega$ is Diophantine. In such a case, one can
make the expansion constant by multiplying by a factor chosen
conveniently. This factor satisfies a cohomological equation
involving small divisors, so we loose some regularity in this
formulation.
We emphasize, however, that the transformation to
constant
rates is only done in this pedagogical
section to simplify the notation so that the ideas come across
better.
Later, we will also present results when $\lambda$ is
a function of $\th$. This leads
to optimal results on regularity of \
the invariant manifolds, and does not requires arithmetical assumptions on $\omega$.
\end{remark}
We write
\[
W(\eta,\th)= W^{\leq} (\eta,\th) + W^{>}(\eta, \th) \ ,
\]
where
\[
W^{\leq} (\eta,\th) = \sum_{i= 0}^L W^i(\th) \eta^i
\]
is a polynomial in the variable $\eta$ whose degree $L$ will be made
explicit later, and the high order part of the
function $W$ satisfies
\[
\displaystyle \frac{\partial^i W^{>}}{\partial \eta^i}(0,\th)= 0
\ \mbox{for $i=0,\dots,L$.}
\]
We seek the coefficients $W^0,\dots,W^L$ of $W^{\leq}$ and the remainder
$W^{>}$ from the equation \eqref{parameterization_whisker1},
which leads to a hierarchy
\begin{equation}
\label{hierarchy1}
\begin{array}{l}
F(W^0(\th),\th)= W^0(\th+\omega) \\
\Dif_x F(W^0(\th),\th) W^1(\th) = \lambda W^1(\th+\omega) \\
\Dif_x F(W^0(\th),\th) W^2(\th) + P^2(\th)=
\lambda^2 W^2(\th+\omega) \\
\vdots \\
\Dif_x F(W^0(\th),\th) W^L(\th) + P^L(\th) =
\lambda^L W^L(\th+\omega) \ ,
\end{array}
\end{equation}
where $P^i$ stands for a polynomial expression in $W^1,\dots,W^{i-1}$
for $i=2,\dots L$ whose coefficients are derivatives of $F$ of order
up to $i$ evaluated at $(W^0(\th),\th)$. (For example,
$P^2 = \frac {1}{2} \Dif_x^2 F(W^0(\th),\th) (W^1(\th))^{\otimes 2}$.)
The high order part $W^{>}$ satisfies
\begin{equation}
\label{remainder1}
\Dif_x F(W^0(\th),\th) W^{>}(\eta,\th) +
P^{>}(\eta,\th) =
W^{>}(\lambda \eta, \th+\omega)\ ,
\end{equation}
where $P^{>}$ contains terms which vanish to
order higher than $L$.
The hierarchy of equations \eqref{hierarchy1} can be solved by
recursion in the degree of the polynomials
matched, provided that some non-resonance conditions,
that we will discuss now are satisfied.
The equation for $W^0$ is an equation of the type we have studied
in the theory of existence of invariant tori. Hence, we take
\[
W^0(\th)= K(\th)
\]
to be the solution of the equation for the invariant torus.
The equation for $W^1$ states that $W^1$ is an eigenfunction
for the transfer operator $\M_\omega$ defined in \eqref{transfer},
whose eigenvalue is $\lambda$.
Note that the equation $\M_\omega W^1 = \lambda W^1$ has a very clear
geometric interpretation, namely that the bundle spanned by $W^1$ is invariant
under $M$, and also that the expansion rate in appropriate coordinates
is constant.
Hence, the geometric interpretation of the second equation in
\eqref{hierarchy1}
is that the bundle spanned by $W^1$ is invariant for the linearization
of $F$. As it is apparent from the theory developed in
\cite{Johnson80,JohnsonS81,trilogy2},
in the case that $\omega$ is Diophantine and the bundle is
one dimensional (the case considered in \eqref{parameterization_whisker1})
{\em the geometric and the analytical characterization are equivalent}.
The other equations of the hierarchy \eqref{hierarchy1} have the form
\[
\M_\omega W^i (\th) - \lambda^i W^i(\th) = - P^i(\th) \ ,
\]
where $i=2,\dots L$. Hence, under the assumption that $\lambda^i$ is not in
the spectrum of $\M_\omega$ for $i=2,\dots L$, we can recursively solve the
equations of the hierarchy.
We also note that the equation for $W^{>}$ can be solved automatically
if $L$ is large enough (depending on the
spectrum of $\M_\omega$, see the condition \eqref{Lcondition1} below).
Indeed, since
$W^{>}(\eta,\th)$ vanishes to high order in
$\eta$, we have that the norm defined by
\[
\norm{W^>}_{C^0}
=
\norm{W^>}_{C^0([-\rho,\rho]\times \nt^d, \nr^n)} \\
= \sup_{|\eta|\leq \rho, \th\in\nt^d} |W^>(\eta,\th)|
\]
satisfies, for small enough $\rho>0$,
\[
\norm{W^{>}(\lambda \eta,\th+\omega)}_{C^0}
\leq \lambda^{L+1} \norm{W^{>}(\eta,\th)}_{C^0} \ .
\]
Therefore, if
\begin{equation}
\label{Lcondition1}
\norm{M}_{C^0} \lambda^{L+1} < 1 \ ,
\end{equation}
we see that the equation of the
remainder \eqref{remainder1} is solvable in $C^0$.
As it turns out, one can develop a theory of solutions in higher regularity
that goes along the same lines.
In summary, provided that $\lambda$
is an eigenvalue of $\M_\omega$, $|\lambda| < 1 $, and
such that $\lambda^i$ is not in the spectrum
of $\M_\omega$, for $i= 2,\dots L$, one can obtain that there is a solution
for the hierarchy \eqref{hierarchy1}.
In the case that the functions are analytic, it can be shown (e.g. using
the majorant method) that the series for $W$ converge and, hence, there
is an analytical solution for \eqref{parameterization_whisker1}.
In the finitely differentiable case, we will show that
the equations \eqref{hierarchy1} can
be solved to a finite order, and there is a solution for
\eqref{parameterization_whisker1} which is obtained applying a fixed point
argument starting with the solutions of \eqref{hierarchy1} to a suitable
finite order.
The existence of a solution to \eqref{parameterization_whisker1} shows
that there is a parameterization of the asymptotic manifold associated
to $\lambda$ in such a way the restriction to the manifold is an
exponential contraction toward the invariant torus.
As we will see, if $\lambda^i$ is in the spectrum of $\M_\omega$ for
some $i= 2,\dots L$, we can modify slightly the procedure indicated
by constructing a dynamics
on the manifold that is polynomial in $\eta$.
The hierarchy of equations \eqref{hierarchy1} can be numerically solved
to a finite order. The fixed point argument alluded above shows that,
given one such numerical solution, there is a true solution nearby. Hence,
the fixed point argument ca be though of as ``a posteriori'' validating
estimates in the sense of numerical analysis.
The numerical analysis issues are discussed in \cite{HLlnum}.
\subsection{Notation}
\subsubsection{Differential geometry}
The appropriate language to express the results of this paper is that of
Differential Geometry, in particular vector bundles, Finsler metrics,
bundle maps,
etc.
Related to these geometric
objects, it is natural to define spaces of
functions adapted to them and study the functional equations
in terms of operators acting on these spaces. See \cite{MeyerS89}
for more details on bundles in a context very similar to the
one considered here.
For instance, the quasiperiodic skew product
\[
\begin{array}{l}
\bar x = F(x,\th) \ , \\
\bar \th= \th + \omega \ ,
\end{array}
\]
is a bundle map in $E= \nr^n\times\nt^d$, a trivial bundle
over $\P= \nt^d$. An invariant torus $x=
K(\th)$
is given by a section $K:\nt^d\to\nr^n$ on such a bundle that satisfies the
invariance equation $F(K(\th),\th)= K(\th+\omega)$. A whisker
$x= W(\eta,\th)$ of the
torus is given by a bundle map over the identity
$W:E_1 \to E= \nr^n\times\nt^d$ where $E_1$ is a linear subbundle
of $E$, and its dynamics is a bundle map $\Lambda:E_1\to E_1$ over the
rotation $\omega$ in $E_1$:
$F(W(\eta,\th),\th) = W(\Lambda(\eta,\th),\th+\omega)$.
The whiskers we will obtain can be
topologically non-trivial. Notice also that
we will obtain polynomial approximations of the whiskers, and the
dynamics on the whisker will be polynomial.
The more general set up would be the following.
Given two $C^r$ vector bundles $E_1$ and $E_2$
over the same base manifold $\P$,
we consider bundle maps $F_f:B_1 \to E_2$, where $B_1$ is
a tubular neighborhood of $E_1$. The subindex $f$ denotes the motion on the
base manifold, that is $f:\P\to\P$, the map such that
$\Pi_2\comp F = f\comp\Pi_1$.
We denote the elements
of $E_1$ as $x_\th= (x,\th)$ and of $E_2$ as $y_\th= (y,\th)$, and
pictorially, we write
\[
\begin{array}{l}
y= F(x,\th) \ , \\
\bar \th= f(\th) \ .
\end{array}
\]
Notice that the geometric objects
are $C^r$ in the horizontal variables and $C^\infty$ (in fact, analytic)
in the vertical variables (using trivialization charts). Hence, the regularity
of $F$ in $\th$ is at most $C^r$.
Along this paper, the motion on the base manifold $\P= \nt^d$ is a rotation
$f(\th)= \th+\omega$.
In particular, the system \eqref{discrete} is a bundle map over
the trivial bundle $E= \nr^n\times \nt^d$. This is not a restriction,
since one could uses the trick in \cite{HirschP69} to trivialize the bundle.
On the other hand, we will not use such a trick in order to trivialize the
subbundles of $E$.
\subsubsection{Spaces of differentiable
functions with anisotropic differentiability}
In order to obtain sharp results on regularity of the invariant manifolds,
it will be very important for us to distinguish the regularities of the
functions with respect to the horizontal variables ($\th$) and the
vertical variables ($x$, $\eta$), because
the angle variables parameterizing the torus
and the real variables used to parameterize the
stable directions enter very differently in the functional
equations that we need to solve.
Hence, when one is interested in optimal regularity it is
natural to introduce spaces in which the regularity along these
two variables is not the same.
The following spaces are an adaptation of
the definitions used in \cite{CabreFL03a,CabreFL03b}.
They are designed to make easy induction arguments
for the functional equations.
\begin{Definition}
\label{CSigma}
We consider subsets $\Sigma \subset \nn^2$ is
such that $(i,j) \in \Sigma $
and $\tilde \imath \le i, \tilde \jmath \le j $ implies
$ (\tilde \imath, \tilde \jmath) \in \Sigma$.
We denote $C^\Sigma= C_{\cdot}^\Sigma(B_1,E_2)$ the set of
maps $F$ for which $D^i_\th D^j_x F$ exists, is
continuous and bounded for every $(i,j) \in \Sigma$.
We consider $C^\Sigma$ endowed with the norm
\[
|| F ||_{C^\Sigma(B_1,E_2)} = \sup_{(i,j) \in \Sigma, (x,\th) \in B_1}
|D^i_\th D^j_x F(x,\th) | \ ,
\]
which makes it a Banach space.
We denote by $C_\cdot^\Sigma= C_\cdot^\Sigma(B_1,E_2)$
the Banach subspace of bundle maps with such regularities.
If we fix the base dynamics $f:\P\to\P$,
we consider the Banach subspace $C_f^\Sigma(B_1,E_2)$ of bundle maps
over $f$.
Special cases are:
\begin{itemize}
\item F is $C^{r,s}$ when
$D^i_\th D^j_x F(x,\th)$
exists, is continuous and bounded for
$0 \le i \le r$, $0 \le j \le s$;
\item F is jointly $C^r$ when it is
a $C^r$ mapping
with bounded derivatives up to order $r$.
This is equivalent to the existence, continuity and
boundedness of
$D^i_\th D^j_x F(x,\th)$
for $0 \le i +j \le r$.
\item F is $C^{\Sigma_{r,s}}$ when
$D^i_\th D^j_x F(x,\th)$
exists, is continuous and bounded for
$(i,j) \in {\Sigma_{r,s}}=
\{ (i, j) \in \nn^2 \mid \; i\le r \ , i+j\le r+s \}$.
\end{itemize}
\end{Definition}
We will see that the classes
$C^{\Sigma_{r,s}}$
are well adapted to the
study of optimal regularity with respect to the horizontal and vertical
directions. The results
of this paper also work if one changes $C^{\Sigma_{r,s}}$ by
$C^{r+s}$, because $C^{r+s} \subset C^{\Sigma_{r,s}} \subset C^{r,s}$.
\begin{Definition}\label{lowerorder}
We say that the derivative $D_\th ^kD_x^l$ is of lower order
than the derivative $D_\th ^iD_x^j$ if $(k,l) \in \Sigma_{i,j} $.
\end{Definition}
\subsubsection{Spaces of analytic functions}
Some of the results obtained in the present paper have simpler proofs
when working in the analytic category. For the sake of simplicity, we will
consider functions that
are analytic in both horizontal and vertical variables.
A real-analytic vector bundle is a smooth bundle that has a vector atlas
for which the transition maps between vector charts are real-analytic
(and linear in the vertical components).
In the following, $E_1,E_2$ are two real-analytic vector bundles.
\begin{Definition}
A real-analytic bundle map $F:B_1 \to E_2$ defined in a tubular
neighborhood $B_1$ of $E_1$ is real-analytic if its local representations
(in real-analytic charts) are real-analytic.
\end{Definition}
In order to introduce topology in the space of real-analytic bundle
maps, we fix finite vector atlases in both $E_1$ and $E_2$. Then, we define
the complex neighborhood of $B_1$ of size $\xi$ as
\[
B_1^{\xi}= \{ (x+\ii y,\th+\ii \varphi) \st (x,\th) \in B_1 \ ,
|y|< \xi, |\varphi|< \xi \}\ ,
\]
where the expressions above are understood once one has taken local
charts. Similarly, we define $E_2^{\eta}$.
Once we have fixed a complex neighborhood $E_2^{\eta}$, we can
define a scale of spaces of real-analytic bundle maps from $B_1$ to
$E_2$.
\begin{Definition}
A real-analytic bundle map $F:B_1 \to E_2$ is $C^{a,\xi}$ if
it has an holomorphic extension $\hat F:B_1^\xi \to E_2^\eta$,
and $\hat F$ is continuous in the closure. We equip $C^{a,\xi}$
with the norm
\[
\norm{F}_{C^{a,\xi}(B_1,E_2)} =
\sup_{(\hat x,\hat \th)\in B_1^\xi} |\hat F(\hat x, \hat \th)| =
\norm{\hat F}_{C^0(B_1^\xi,E_2^\eta)}\ ,
\]
where the norms in $E_1$ and $E_2$ have also been complexified.
It is standard that, with the indicated norm, $C^{a,\xi}$ is a Banach
space.
\end{Definition}
\subsubsection{Polynomial bundle maps}
We introduce now the polynomial bundle maps, which are the objects
that we will use to represent the dynamics on an invariant manifold.
\begin{Definition}
A bundle map $P_f:E_1\to E_2$ is said to be a $C^r$ polynomial bundle map
of degree $k$ if $f$ is $C^r$ and $P$ is of the form
\[
P(x,\th)= P^0(\th) + P^1(\th) x + P^2(\th) x^{\otimes 2} +
\dots + P^k(\th) x^{\otimes k}\ ,
\]
where for all $i= 0,\dots,k$, $P^i$ is a $i$-multilineal bundle map over $f$
from $E_1\times \stackrel{i}{\dots}\times E_1$ to $E_2$, and of class $C^r$.
Each $P^i$ can be chosen symmetric. We will say that
$P^i_\th(x) = P^i(x,\th)= P^i(\th) x^{\otimes i}$ is homogeneous
of degree $i$. Notice that $P\in C^{r,\infty}_f$.
Obviously, for $r= a$ this is a real-analytic polynomial bundle map, and
$P\in C^a$.
\end{Definition}
We observe that a $C^r$ $i$-multilineal
bundle map over the identity is identified with a $C^r$ section on the
$i$-multilineal symmetric bundle $L^i_s(E_1;E_2)$.
In particular, a polynomial map
over the identity is equivalent to a section of the bundle
$\bigoplus_{i=0}^k L^i_s(E_1;E_2)$.
The definition and comments in the analytic case corresponds to $r= a$.
\subsubsection{Transfer operators, cocycles}
The variational equations \eqref{variational}
over an invariant torus $\K$ for \eqref{discrete}
define a linear skew-product $M$ given by
\[
\begin{split}
\bar v & = M(\th) v\ ,\\
\bar \th & = \th + \omega \ ,
\end{split}
\]
where $v\in\nr^n$ and $\th\in\nt^d$, and
$M(\th)= \Dif_x F(K(\th),\th)$.
An useful notation is that of cocycles. We will write
\begin{equation}
\label{cocycle}
\begin{array}{l}
M_\th^0= M(\th,0)= \Id \ , \\
M_\th^m= M(\th,m)=
M(\th+({m-1})\omega) \cdot \dots \cdot
M(\th)\ \mbox{ if } m> 0 \ , \\
M_\th^m= M(\th,m)= M(\th+ m \omega)^{-1} \cdot \dots \cdot
M(\th-\omega)^{-1}
\ \mbox{ if } m< 0 \ .
\end{array}
\end{equation}
We will also write $M(v,\th,m)= M_\th^m v= M(\th,m)v$.
We associate to the linear skew-product $M$ a
{\em transfer operator} over the rotation $\omega$.
This transfer operator is the map $\M_\omega:\Sec{}{} \to\Sec{}{}$
on the space of sections
$\Sec{}{}= \{ v:\nt^d\to \nr^n\}$
defined by
\begin{equation}
\label{transfer2}
(\M_\omega v)(\th)= M(\th-\omega) v(\th-\omega)\ .
\end{equation}
Clearly, the linearization of the map near an
invariant torus, \eqref{transfer} is a particular case of
\eqref{transfer}. Other examples that will play a role
in our discussion are the tensor products of
the linearization, which will play a role in the study of
higher derivatives.
\begin{remark}
Notice that we identify a section in the trivial bundle $E= \nr^n\times\nt^d$
over $\nt^d$ with a function from $\nt^d$ to $\nr^n$.
\end{remark}
\begin{remark}
The previous definitions and notations \eqref{transfer2}
and \eqref{cocycle} can be introduced
for general bundle maps $F_f:E\to E$, and in such a case, we will made
explicit the non linear character of the objects produced, saying that
$\F_F$ is a non-linear transfer operators, or the family
$F(x,\th,m)= F_\th^m(x)$ is a non-linear cocycle.
\end{remark}
The transfer operator of an analytic $M$
can be considered as acting on spaces of
sections with different regularities. For example,
it can be considered as acting on bounded sections ($\Sec{}{b}$),
continuous sections ($\Sec{}{C^0}$), $C^r$ sections ($\Sec{}{C^r}$),
analytic sections ($\Sec{}{C^a}$),
etc. In particular, the spectral theory of these operators (in fact,
the complexification of these operators, acting on complex sections
in $E_{\nc}= E \oplus \mbox{\rm\bf i} E \simeq \nc^d \times \nt^d$)
will be very important for us.
In general, it can happen that the spectrum of the operator
depends on the spaces it is considered as acting on.
Nevertheless, for the case that the motion on the base is
a rotation, the spectrum does not depend on the space.
The following result is established in \cite{trilogy2}. Note that it depends
crucially on the fact that the motion on the base is a rotation. It could be
false if the motion in the base is a general map.
\begin{Theorem}\label{eqs_spectrum_Cr}
Let ${M_\omega}:E\to E$ be a $C^r$ $r \in \nn$
vector bundle automorphism
over a rotation.
Then:
\begin{equation}
\label{crucial2}
\Spec({\M_\omega},\Sec{}{b}(E)) =
\Spec({\M_\omega},\Sec{}{C^r}(E)) \ .
\end{equation}
\end{Theorem}
For the proof of Theorem~\ref{eqs_spectrum_Cr} we refer to
\cite{trilogy2}.
The Theorem~\ref{eqs_spectrum_Cr} does not apply to
the case that the spaces are analytic.
The best result for analytic spaces that can
be found in \cite{trilogy2} is:
\begin{Theorem}\label{eqs_spectrum_anal}
Let ${M_\omega}:E\to E$ be a $C^{a, \xi^*}$,
vector bundle automorphism
over a rotation.
Then for all $ \xi < \xi^*$ we
have
\begin{equation}
\label{crucialanal}
\dist( \Spec({\M_\omega},\Sec{}{b}(E)),
\Spec({\M_\omega},\Sec{}{C^{a,\xi}}(E)) ) \le O(\xi)
\end{equation}
where by $\dist$ we mean the Hausdorff distance among sets.
\end{Theorem}
\section{Invariant tori}
\label{tori}
\subsection{Existence and persistence of invariant tori}
In this section, we formulate the result on existence and persistence
of invariant tori. The main result is the following.
\begin{Theorem}
\label{theorem_tori}
Let $U\subset\nr^n$ be an open set.
Let $F:U\times\nt^d \subset \nr^n\times\nt^d \to \nr^n$ be a map of class
$C^{\Sigma_{r,1}}$, with $r\geq 0$
-- including $C^{\Sigma_{r,1}}= C^a$ in the analytic case $r= a$ --,
such that for all $\th\in\nt^d$ the
map $F(\cdot,\th): U\to\nr^n$ is a local diffeomorphism.
Let $\omega\in\nr^d$ be a rotation.
We consider the skew product
\[
\begin{array}{l}
\bar x = F(x,\th) \ , \\
\bar\th= \th + \omega \ ,
\end{array}
\]
that is a vector bundle map on the bundle $E= \nr^n\times\nt^d$.
Let $K:\nt^d\to U\subset\nr^n$ be a $C^r$ map such that:
\begin{itemize}
\item $\K$ is an approximate invariant torus, that is
\begin{equation} \label{almostinvariant}
\norm{F(K(\th),\th)-K(\th+\omega)}_{C^r} <
\ep\ .
\end{equation}
\item For the $C^r$ matrix valued map $M:\nt^d \to \mbox{\rm GL}_n(\nr)$,
defined by
\[
M(\th)= \Dif_x F(K(\th),\th) \ ,
\]
the corresponding transfer operator $\M_\omega$ satisfies the spectral gap
condition
\begin{equation}\label{gapassumption}
\Spec(\M_\omega,\Sec{}{b}(E)) \cap \{ z\in\nc \st |z|= 1 \} = \emptyset \ .
\end{equation}
\end{itemize}
Then:
\begin{itemize}
\item If $\ep$ is small enough,
there exists a $C^r$ map $K_F:\nt^d\to U\subset\nr^n$ such that
\begin{equation}\label{invariantKF}
F(K_F(\th),\th)= K_F(\th+\omega) \ ,
\end{equation}
and $\norm{K_F-K}_{C^r}= \mbox{\rm O}(\ep)$.
\item
The solution $K_F$ above is the only $C^0$ solution
of \eqref{invariantKF} in
a $C^0$ neighborhood of $K$.
\item The torus $K_F$ is normally hyperbolic.
\end{itemize}
Moreover, the map $F \to K_F$ is $C^1$ when $F$ is given the
$C^{\Sigma_{r,1}}$ topology and $K_F$ the $C^r$ topology.
\end{Theorem}
\begin{remark}
Notice that the spectral gap assumption \eqref{gapassumption}
for an invariant torus is equivalent to normal
hyperbolicity (\cite{Mane78,HirschPS77,Swanson83}).
\end{remark}
\begin{remark}
Notice that the spectral gap assumption \eqref{gapassumption}
is formulated in the space of bounded sections -- not necessarily
continuous --. Using the results in \cite{trilogy2}
about the spectrum of transfer operators over
rotations, we obtain that the spectrum over bounded sections is
the same as that over $C^r$ sections. This is what allows
to obtain $C^r$ regularity in the conclusions.
Of course, these results depend very heavily on the fact that the
motion on the torus is a rotation.
\end{remark}
\begin{remark}
\label{hyperbolic}
The linear operator corresponding to $\hat K= K_F$ is
\[
\hat M(\th) = \Dif_x F(\hat K(\th),\th) \ ,
\]
and by the mean value theorem, if $F$ is $C^{\Sigma_{r,2}}$, then
\[
\norm{M - \hat M}_{C^0} \leq
\norm{F}_{C^{\Sigma_{0,2}}} \norm{K-\hat K}_{C^0}\ .
\]
Hence, for the transfer operators $\M_\omega,{\hat \M}_\omega$
corresponding to $M,\hat M$, respectively,
we have
\[
\norm{\M_\omega - {\hat \M}_\omega}_{C^0}
\leq \norm{F}_{C^{\Sigma_{0,2}}} C \ep\ ,
\]
Under the hypothesis of Theorem~\ref{theorem_tori},
$F$ is $C^{\Sigma_{r,1}}$, one has
\[\norm{M - \hat M}_{C^0}
\leq \eta\left(\norm{K-\hat K}_{C^0}\right)\ ,
\]
where $\eta$ is the modulus of continuity of $\Dif_x F$, and
one obtains the estimate
\[
\norm{\M_\omega - {\hat \M}_\omega}_{C^0} \leq \eta(C \ep)\ .
\]
Therefore, by the usual properties of the stability of the
spectrum \cite{Kato76}, we can ensure that ${\hat \M}_\omega$ is also
hyperbolic. Indeed, if we can know that
\[
\Spec({\M}_\omega,\Sec{}{b}) \cap \{ z\in\nc \st
\lambda_- \leq |z| \leq \lambda_+ \}= \emptyset \ ,
\]
with $\lambda_- < 1 < \lambda_+$,
then we can ensure that
\[
\Spec({\hat \M}_\omega,\Sec{}{b}) \cap \{ z\in\nc \st
\lambda_- + C\ep \leq |z| \leq \lambda_+ - C\ep \} = \emptyset \ ,
\]
and we can obtain bounds for the norms of the spectral projections.
Hence, the hyperbolicity properties of the exact torus are similar
to those of the approximate one and their difference is bounded by the
error in the approximation.
\end{remark}
\begin{remark}
The formulation we have presented of Theorem \ref{theorem_tori},
implies the usual formulation of the Theorem on persistence of
normally hyperbolic invariant tori.
If $K_F$ is a parameterization of
a torus invariant under a map $F$, it will be smooth and it will
satisfy \eqref{almostinvariant} for all the maps $G$ close to $F$.
Furthermore, if the torus is normally hyperbolic for $F$,
then, the operator
$M(\th)= \Dif_x F(K(\th),\th) $ is hyperbolic.
By the stability of the spectrum under perturbations, we
will obtain that \eqref{gapassumption} will be satisfied for $G$ close
to $F$.
Hence, we have verified that, given a normally
hyperbolic invariant torus, if we perturb the map slightly, we
have all the assumptions of Theorem~\ref{theorem_tori} for
the perturbed map and the original invariant torus.
The conclusions of Theorem~\ref{theorem_tori} give the persistence of
the invariant torus.
\end{remark}
\begin{remark}
We call attention to the fact that the proof
works for $\omega$ resonant or non-resonant (ergodic). For $\omega$
irrational, the spectral gap condition is equivalent to
$1\notin \Spec(\M_\omega,\Sec{}{b}(E))$, since in such a case the spectrum
is rotationally invariant. See \cite{Mather68,trilogy1}.
\end{remark}
\begin{remark}
\label{parameters}
The results we have formulated here immediately imply smooth dependence
on parameters. If
$F_\gamma:U\times \nt^d \subset \nr^n\times \nt^d \to \nr^n$
depends on a possibly multidimensional parameter $\gamma$
we can, without loss of
generality assume that the range of the parameter is $\nt^\ell$.
The extended map $\tilde F:U\times \nt^d \times \nt^\ell \to \nr^n$
given by $F(x,\th,\gamma) = F_\gamma(x,\th)$ defines an
extended skew product in $\nr^n \times \nt^d \times \nt^\ell$ by
\[
\begin{split}
\bar x & = F_\gamma(x,\th) \ ,\\
(\bar \th, \bar \gamma) & = (\th, \gamma) + (\omega, 0) \ .
\end{split}
\]
We see that verifying the assumptions of Theorem~\ref{theorem_tori}
for $F_\gamma$
uniformly on the parameters $\gamma$
is the same as verifying the assumptions of
Theorem~\ref{theorem_tori} for the extended
system $\tilde F$. The existence of smooth invariant tori for
the extended system is the same as the smooth dependence on
parameters for the family $F_\gamma$.
In this formulation, we obtain that the regularity
of in $\th$ and the parameters is joint. There are examples that show
that this is optimal.
\end{remark}
\begin{remark}
The formulation of Theorem~\ref{theorem_tori}
is very similar to the a-posteriori estimates of
numerical analysis. A numerical method
can produce an approximate solution
that satisfies \eqref{almostinvariant} up to a few units of
round--off error. It is also possible to verify the
other hypothesis of Theorem~\ref{theorem_tori} on the
computed solution. These issues of numerical
analysis, as well and results of implementations
are discussed in more detail in \cite{HLlnum}.
\end{remark}
\begin{remark}
The good analytical properties of the operators appearing in the invariance
equation of the torus seems to translate in stability of numerical methods.
This analysis is completed in the companion paper \cite{HLlnum}.
\end{remark}
\bproof
We will prove first Theorem~\ref{theorem_tori}
for finite differentiable maps.
We have to solve the equation \eqref{parameterization_tori}
\[
F(K(\th-\omega),\th-\omega) - K(\th)= 0
\]
in $C^r$.
In this case,
the map $\T_F: C^{r}(\nt^d,U) \to C^{r}(\nt^d,\nr^n)$ defined by
\begin{equation}
\label{def_TauF}
\Tau_F(K)(\th) = F(K(\th-\omega),\th-\omega) - K(\th)
\end{equation}
is a $C^1$ operator \cite{LlaveO99}. Moreover,
the derivative of $\Tau_F$ is given by
\[
\begin{split}
\Dif\Tau_F(K)\Delta(\th)
& = \Dif_x F(K(\th-\omega),\th-\omega) \Delta(\th-\omega) - \Delta(\th) \\
& = \M_\omega \Delta (\th) -\Delta (\th) \ ,
\end{split}
\]
i.e., $\Dif \Tau_F(K) = \M_\omega - \Id$.
By Theorem~\ref{eqs_spectrum_Cr}
we have that the spectral gap does not depend on the spaces
considered. Therefore, the spectral gap assumption \eqref{gapassumption} on
the bounded sections,
implies that $\Dif \Tau_F(K)$ is invertible as a linear operator acting on
$C^r$ sections $\Delta$.
The existence and uniqueness of $\hat K= K_F$ in $C^r$ spaces
follows from the Inverse Function Theorem. The torus is normally hyperbolic
since the transfer operator ${\hat \M}_\omega$ associated to
$\hat M(\th) = \Dif_x F(\hat K(\th),\th)$ is hyperbolic
(see Remark~\ref{hyperbolic}).
The uniqueness in the $C^0$ space follows from the fact that $\hat K$ is
obviously a $C^0$ invariant tori, and that the transfer operator
${\hat \M}_\omega$ is hyperbolic in $C^0$.
>From the Inverse Function Theorem in $C^0$ spaces, $\hat K$
is the unique $C^0$ invariant torus in a $C^0$ neighborhood of the
$C^r$ torus $\hat K$.
The persistence of the torus under perturbations and the $C^1$ dependence
on $F$ follows by applying the Implicit Function Theorem on
the $C^1$ operator $\Tau:C_\omega^{\Sigma_{r,1}}(U\times\nt^d,\nr^n) \times
C^r(\nt^d, U) \to C^r(\nt^d,\nr^n)$ defined by
\[
\Tau(F,K) (\th)= F(K(\th-\omega),\th-\omega)- K(\th)\ .
\]
The proof of Theorem~\ref{theorem_tori} in the analytic case follows
the same lines, but it is actually much simpler.
The $C^1$ regularity of the composition operator in
\eqref{def_TauF} for analytic
cases follows from the results for
$C^0$ in the complex extension.
For more details, we refer to \cite{Meyer75}.
In the analytic case,
by Theorem~\ref{eqs_spectrum_anal},
we have that if there is a spectral gap
in
$\Spec(\M_\omega,\Sec{}{b}(E))$ then,
for $\xi$ small enough,
$\Spec(\M_\omega,\Sec{}{C^{a,\xi}}(E))$
also has a spectral gap.
The proof, as before, is an application of the Implicit Function
Theorem.
\eproof
\subsection{Bootstrap on the regularity}
\label{bootstrap}
Theorem~\ref{theorem_tori} produces $C^r$ invariant tori from a
$C^r$ approximate invariant tori, and gives estimates in the $C^r$ norms
of the difference between the approximate solutions and the
true ones in terms of the $C^r$ norm of $\Tau(K)$.
The following theorem says that a $C^0$ invariant torus of a
$C^{\Sigma_{r,1}}$ quasiperiodic skew product is necessarily $C^r$.
Henceforth, from the existence of
a $C^0$ approximate invariant torus, we can deduce the
existence of
a $C^r$ invariant torus. Notice, however, that
Theorem~\ref{theorem_bootstrap} does not produce
$C^r$ estimates of the distance between the approximation
and the true solution from $C^0$ estimates on $\Tau$.
As mentioned before, we will find it convenient to
use spaces of functions with anisotropic regularity
introduced in Definition~\ref{CSigma}.
\begin{Theorem}
\label{theorem_bootstrap}
Let $U\subset\nr^n$ be an open set.
Let $F:U\times\nt^d \subset \nr^n\times\nt^d \to \nr^n$ be a map of class
$C^{\Sigma_{r,1}}$, with $r\geq 0$
-- including $C^{\Sigma_{r,1}}= C^a$ in the analytic case $r= a$ --,
such that for all $\th\in\nt^d$ the
map $F(\cdot,\th): U\to\nr^n$ is a local diffeomorphism.
Let $\omega\in\nr^d$ be a rotation.
Let $K:\nt^d\to U\subset\nr^n$ be a $C^0$ parameterization
of a normally hyperbolic invariant torus $\K$.
Then, the parameterization $K$ is $C^r$.
\end{Theorem}
\bproof
For $r= 0$ we has nothing to do.
We will prove first the result in the analytic case.
By smoothing $K$, we can consider tori $\tilde K_{\eta}$ in
$C^{a,\eta}$ with $\norm{K-\tilde K_\eta}_{C^0}$ small enough and
$\eta$ small enough. Since
\[
\norm{F(K(\th-\omega),\th-\omega) - K(\th)}_{C^0} = 0\ ,
\]
by choosing $\eta$ sufficiently small we have
\[
\norm{F(\tilde K_\eta(\th-\omega),\th-\omega) -
\tilde K_\eta(\th)}_{C^{a,\eta}}
< \ep\ .
\]
Since $K$ is normally hyperbolic, by choosing $\eta$ small enough we can
get that the transfer operator associated to
$M_\eta(\th) = \Dif_x F(K_\eta(\th),\th)$ is hyperbolic in
$\Sec{}{C^{a,\eta}}$ \cite{trilogy2}.
So, by the Inverse Function Theorem, there is an
analytic invariant torus $K_\eta$ near $\tilde K_\eta$. By uniqueness,
$K_\eta= K$.
The technique of the proof in the finite differentiable case ($r\geq 1$)
is similar to the proof of the regularity in \cite{LlaveW95}. First, we
show that the
formal equations for derivatives have unique solutions, which are continuous.
Then, under the assumptions of regularity of $F$, we show that the ``Taylor''
expansions obtained with these derivatives satisfy the equations with a
smallness condition which is a power of the displacement. Then, using the
quantitative estimates for the Theorem~\ref{theorem_tori}, we conclude that
the $C^0$ solution $K$ differs from its Taylor approximation less than
a power and, by the converse of Taylor's theorem \cite{AMR,Nelson69}
we conclude that $K$ is indeed $C^r$.
We will work in detail the case $r= 1$.
We will show that the case for $r > 1$ can be deduced
from this by induction. In Remark~\ref{directproofCr}
we will also sketch the
relatively easy modifications that are needed for
a direct proof.
If $K$ were $C^1$, taking derivatives in \eqref{parameterization_tori}
we would obtain
\[
\Dif_x F(K(\th-\omega),\th-\omega) \Dif_\th K(\th-\omega)
+
\Dif_\th F(K(\th-\omega),\th-\omega) -
\Dif_\th K(\th) = 0\ ,
\]
so $\Dif_\th K(\th)$ would solve the equation
\begin{equation}
\label{derivative}
\Dif_x F(K(\th-\omega),\th-\omega) K'(\th-\omega) - K'(\th) =
- \Dif_\th F(K(\th-\omega),\th-\omega)\ ,
\end{equation}
in $C^0(\nt^d,L(\nr^d,\nr^n))$. We note that the right hand side of
\eqref{derivative} is a continuous function.
By the hyperbolicity assumptions, there exists one and only one $C^0$ solution
$K'$ of this equation \eqref{derivative}. We will see that in fact
$K$ is differentiable and $\Dif_\th K= K'$.
To do so, given $\eta\in\nr^d$ sufficiently small, we consider
\begin{equation}
\label{Keta}
\tilde K_\eta(\th) = K(\th-\eta) + K'(\th-\eta) \eta\ ,
\end{equation}
and we will see that
$\norm{K-\tilde K_\eta}_{C^0} \leq |\eta| \gamma(|\eta|)$,
with $\gamma$ converging to zero as $|\eta|$ converges to zero.
Since $K'$ is continuous, this will prove that $K$ is $C^1$ and
$\Dif_\th K= K'$.
We note that, by the uniform continuity of $K$ and the fact that
$K'$ is bounded,
\[
\norm{\tilde K_\eta - K}_{C^0} \leq \gamma(|\eta|)\ .
\]
We now compute $\Tau(\tilde K_\eta)$ using the first order Taylor expansion
of $F$:
\begin{equation}
\label{computation}
\begin{split}
\Tau_F(\tilde K_\eta)(\th)
& =
F(\tilde K_\eta(\th-\omega),\th-\omega) - \tilde K_\eta(\th) \\
& =
F(K(\th-\eta-\omega)+K'(\th-\eta-\omega)\eta,
(\th-\eta-\omega)+\eta) \\
& \phantom{=} - K(\th-\eta) - K'(\th-\eta)\eta \\
& =
F(K(\th-\eta-\omega),\th-\eta-\omega) \\
& \phantom{=} +
\Dif_x F(K(\th-\eta-\omega),\th-\eta-\omega) K'(\th-\eta-\omega)\eta
\\
& \phantom{=} +
\Dif_\th F(K(\th-\eta-\omega),\th-\eta-\omega) \eta +
R(\eta,\th) \\
& \phantom{=} - K(\th-\eta) - K'(\th-\eta)\eta \ ,
\end{split}
\end{equation}
where $R$ is the remainder of the Taylor expansion, and
$|R(\eta,\th)| \leq |\eta| \gamma(|\eta|)$. We also see that,
using the fact that $K$ is invariant and $K'$ satisfies the equation
\eqref{derivative}, we obtain that all the terms in
\eqref{computation} except $R$ cancel. Hence, we obtain
\[
\norm{\Tau_F(\tilde K_\eta)}_{C^0} \leq |\eta| \gamma(|\eta|)\ .
\]
Since $\tilde K_\eta$ is $C^0$ close to $K$, the hyperbolicity property of
the cocycle remain uniform, and as $\tilde K_\eta$ is a $C^0$ approximate
invariant torus, applying Theorem~\ref{theorem_tori} we conclude that there is
a torus $K_\eta$ solving $\Tau_F(K_\eta)= 0$ and
\[
\norm{K_\eta - \tilde K_\eta}_{C^0} \leq |\eta| \gamma(|\eta|)\ .
\]
On the other hand, by the uniqueness statements of Theorem~\ref{theorem_tori}
we conclude that $K_\eta = K$ for $\eta$ small. Hence, for $\eta$ small
\[
\norm{\tilde K_\eta - K}_{C^0} \leq |\eta| \gamma(|\eta|)\ .
\]
This shows that indeed $K'$ is the derivative of $K$.
The case of higher regularity is obtained
by the \emph{``tangent functor trick''}. Later in
Remark~\ref{directproofCr} we will sketch an alternative proof
which avoids the induction.
Let $r\geq 2$.
Assume we have proved that if $F$ is $C^{\Sigma_{r-1,1}}$ then
a normally hyperbolic invariant torus $K$ is $C^{r-1}$.
We will prove now that if $F$ is $C^{\Sigma_{r,1}}$ then $K$ is $C^r$.
We extend the map $F$ to
\[
\hat F(x,Y,\th) =
\left(\begin{array}{l}
F(x,\th) \\
\Dif_x F(K(\th),\th) Y + \Dif_\th F(x,\th)
\end{array}\right) \ ,
\]
where $x\in\nr^n$, $Y\in L(\nr^d,\nr^n)$, $\th\in\nt^d$. Notice that
$\hat F$ is in $C^{\Sigma_{r-1,1}}$ and that
$(K(\th),\Dif_\th K(\th))$ is a $C^{r-2}$ normally hyperbolic
invariant torus of the skew product associated to $\hat F$. By induction,
it is $C^{r-1}$, and, in particular $\Dif_\th K(\th)$ is $C^{r-1}$.
So $K$ is $C^r$, and we are done with the proof of the bootstrap.
\eproof
\begin{remark}\label{directproofCr}
A direct proof follows the following lines. We proceed again by induction
in the order of derivatives. Assume that $r-1$ derivatives of $K$
exist and are continuous. Taking derivatives of \eqref{parameterization_tori}
formally up to order $r$ we obtain that $K^r= \Dif^r K$ satisfies the equation
\begin{equation}
\label{rderivative}
\Dif_x F(K(\th-\omega),\th-\omega) K^r(\th-\omega) -
K^r(\th) = R_r(\th) \ ,
\end{equation}
where
\[
\begin{split}
R_r(\th) = &
- \sum_{j= 0}^{r-2}
\comb{r-1}{j}
\Dif^{r-1-j}\left(\Dif_x F(K(\th-\omega),\th-\omega)\right)
\Dif^{j+1} K(\th-\omega) \\
& - \Dif^{r-1}\left(\Dif_\th F(K(\th-\omega),\th-\omega)\right) \ .
\end{split}
\]
Notice that, since $F$ is $C^{\Sigma_{r,1}}$ and $K$ is $C^{r-1}$,
the right hand side $R_r$ of \eqref{rderivative} is continuous.
Again, by the hyperbolicity of the cocycle, we can find a continuous
solution $K^r$ of \eqref{rderivative}, and we have to see that in fact
$K$ is $C^r$ and $\Dif^r K= K^r$.
It is obvious that for $\eta\in\nr^d$, the torus $K_\eta$ given
by $K_\eta(\th)= K(\th+\eta)$ is invariant under the skew product
$F_\eta$ given by $F_\eta(x,\th)= F(x,\th+\eta)$.
Let us consider the expansion
\[
K^{\leq r}_\eta (\th)=
\sum_{j= 0}^{r-1} \frac{1}{j!} \Dif^j K(\th) \eta^j +
\frac{1}{r!} K^r(\th) \eta^r\ .
\]
Notice that this is a polynomial in $\eta$, and
$\Dif^i_\eta K^{\leq r}_\eta (\th)_{\mid \eta= 0} = \Dif^i K(\th)$
for $iFrom the hypothesis of the theorem we can find a Finsler metric
on $E= \nr^n\times \nt^d$, adapted to the splitting $E= E_1\oplus E_2$,
so that
\begin{equation}
\label{adapted}
\norm{M^{-1}}_{C^0} \left(\norm{M_1}_{C^0}+\ep\right)^{L+1} < 1\ ,
\end{equation}
for $\ep>0$ small enough.
Rather than considering small tubular neighborhoods where the objects
(bundle maps, invariant manifolds, ...) are defined, we will scale the maps
involved in the equations to work in the unit tubular neighborhood of the
zero section $B_E(1)$ (notice that $B_{E_1}(1)\subset B_E(1)$). More
concretely, if we have a bundle map $H:E\to E$ we define
$H^\delta (x,\th)= \frac{1}{\delta} H(\delta x,\th)$, for a
given $\delta>0$. Notice then that the invariance
equation \eqref{invariance_whisker}
\[
F_\th \comp W_\th = W_{\th+\omega} \comp \Delta_\th
\]
holds in $B_E(\delta)$ if and only if
\[
F_\th^\delta \comp W_\th^\delta =
W_{\th+\omega}^\delta \comp \Delta_\th^\delta
\]
holds in $B_E(1)$. Moreover,
\[
F_\th^\delta = M_\th + N_\th^\delta\ ,
\]
where $N^\delta$ satisfies
$N_\th^\delta(0)= 0,
\Dif_x N_\th^\delta(0)= 0$
for all $\th\in\nt^d$ and that $\norm{N^\delta}_{C^{\Sigma_{r,s}}}$ is
small in $B_E(3)$ by taking a small $\delta$.
\subsubsection{Finding the dynamics on the manifold}
\label{sec:dynamics}
In this section we show that, under the non-resonance hypotheses of the
theorem, we can solve the invariance equation \eqref{invariance_whisker} up to
order $L$, that is there exists a polynomial bundle map
$W^{\leq}:E_1\to E$ over the identity
and a polynomial bundle map
$\Lambda:E_1\to E_1$ over $\omega$, both of them of degree $L$
and $C^r$ in $\th$, such that
\[
F(W^{\leq}(\eta,\th),\th) =
W^{\leq}(\Lambda(\eta,\th),\th+\omega) + o(|\eta|^L)\ .
\]
The discussion follows along the lines of Section~\ref{heuristic}.
We write
$W^{\leq}(\eta,\th)= \sum_{k= 0}^L W^k(\eta,\th)$, and
$\Lambda(\eta,\th)= \sum_{k= 1}^L \Lambda^k(\eta,\th)$.
This broken up the invariance equation \eqref{invariance_whisker}
into a hierarchy
of equations, and we can study them recursively.
The zero-order equation is
\[
F(W^0(\th),\th)= W^0 (\th+\omega) \ ,
\]
that amounts to the torus parameterized by $W^0$ is invariant under
$F$. So, we take $W^0= 0$ (after the election of suitable coordinates made
in Section~\ref{preliminaries}, the torus is the zero-section).
The first-order equation is
\[
M(\th) W^1(\th) \eta=
W^1(\th+\omega) \Lambda^1(\th) \eta\ .
\]
We point out that is just the equation of invariance of the bundle generated
by $W^1$, and $\Lambda^1$ is the linearized dynamics on
such a bundle. Hence, we take $W^1= I_1$ (the immersion of $E_1$ into $E$)
and $\Lambda^1(\th)= M_1(\th)$.
In contrast to the Diophantine one-dimensional case discussed briefly in
Section~\ref{heuristic}, there is no simple
uniqueness for $\Lambda^1(\th)$. We will assume that some choice is made.
The subsequent equations matching terms of order
$k=2,\dots L$ are to be considered equations
for $W^k$,$\Lambda^k$, assuming that $W^1,\dots,W^{k-1}$,
$\Lambda^1,\dots,\Lambda^{k-1}$ are known.
More concretely, the equation for the order $k$ is
\begin{equation}
\label{eqkk}
M(\th) W^k(\eta, \th) = W^1(\th+\omega)
\Lambda^k(\eta, \th) +
W^k(\Lambda^1(\th) \eta , \th+\omega) +
R^k(\eta, \th) \ ,
\end{equation}
where $R^k$ is a homogeneous polynomial of degree $k$ over the rotation
$\omega$, depending
polynomially on $W^1,\dots,W^{k-1}$ and $\Lambda^1,\dots,\Lambda^{k-1}$,
and so $R^k$ is $C^r$ in $\th$. We rewrite equation \eqref{eqkk} as:
\begin{equation}
\label{eqk}
M(\th-\omega) W^k(\Lambda^1(\th-\omega)^{-1} \eta,\th-\omega) -
W^1(\th) \hat \Lambda^k(\eta, \th) -
W^k(\eta, \th) = \hat R^k(\eta,\th) \ ,
\end{equation}
where $\hat R^k\in \Sec{}{C^r}(L^k_s(E_1;E))$ is known, and it is
defined by
\[
\hat R^k(\eta, \th) =
R^k(\Lambda^1(\th-\omega)^{-1}\eta, \th-\omega)\ ,
\]
and the unknown terms are
$W^k \in \Sec{}{C^r}(L^k_s(E_1;E))$ and
$\hat \Lambda^k \in \Sec{}{C^r}(L^k_s(E_1;E_1))$, where
$\hat \Lambda^k(\eta, \th)=
\Lambda^k(\Lambda^1(\th-\omega)^{-1}\eta, \th-\omega)$.
Taking projections over $E_1,E_2$ in equation~\eqref{eqk},
and taking into account that $W^1= I_1$, we obtain:
\note{A}{Habiamos perdido la ``B''}
\begin{eqnarray}
\label{eqk1}
\hat R^k_1(\eta, \th) & = &
M_1(\th-\omega) W^k_1(M_1(\th-\omega)^{-1}\eta, \th-\omega)
- W^k_1(\eta, \th)
\\
\notag
& &
+ B(\th-\omega) W^k_2(M_1(\th-\omega)^{-1}\eta, \th-\omega)
- \hat \Lambda^k(\eta, \th) \ ,
\\
\label{eqk2}
\hat R^k_2(\eta, \th) & = &
M_2(\th-\omega) W^k_2(M_1(\th-\omega)^{-1}\eta, \th-\omega) -
W^k_2(\eta, \th)
\ .
\end{eqnarray}
\eqref{eqk1} is an equation for $W^k_1$ and $\hat\Lambda^k_1$ in
$\Sec{}{C^r}(L^k_s(E_1;E_1))$ and \eqref{eqk2} is an equation for $W^k_2$ in
$\Sec{}{C^r}(L^k_s(E_1;E_2))$. Notice we have to solve first \eqref{eqk2},
and the solve \eqref{eqk1}.
\note{A}{He a\~nadido esta frase}
The operators that appear in both equation are
a generalization of the Sylvester operators in
\cite{BeynK98,LlaveW95,CabreFL03a}, and have been studied in great detail in
\cite{trilogy1}.
In general, given vector bundle maps $E,F$ over the same base manifold $\P$,
and two vector bundle maps
$M_f:E\to E$ and $N_f:F\to F$ over the same homeomorphism
$f:\P\to\P$, we construct a vector bundle map over $f$ on the bundle of
$k$-multilinear maps $S^k_f= S^k_{f,M,N}:L^k(F;E)\to L^k(F;E)$, by
\[
(S^k(\th) w_\th) (v_1,\dots,v_k)=
M(\th) w_\th (N(\th)^{-1} v_1,\dots,
N(\th)^{-1} v_k)\ ,
\]
where $w_\th\in L^k(F_\th;E_\th)$, and
$v_1,\dots,v_k\in F_{f(\th)}$.
We will refer to this action as the Sylvester vector bundle map associated
to $M_f$ and $N_f$. The spectrum of the corresponding transfer operator
is clarified in the following proposition (see \cite{trilogy1} for the proof,
a similar argument happens in \cite{CabreFL03a}).
\begin{Proposition}
\label{Sylvester}
Let $M_f:E\to E$, $N_f:F\to F$ be two vector bundle maps over the same
homeomorphism $f:\P\to\P$, and let $S^k_f= S^k_{f,M,N}$ be the corresponding
Sylvester vector bundle map on $L^k(F;E)$, where $k\geq 1$.
Then:
\[
\begin{array}{lcl}
\Spec(\S^k_f,\Sec{}{b}(L^k_s(F;E))) &\subset&
\Spec(\S^k_f,\Sec{}{b}(L^k(F;E))) \\ &\subset&
\Spec(\M_f,\Sec{}{b}(E)) \cdot (\Spec(\N_f,\Sec{}{b}(F)))^{-k}\ .
\end{array}
\]
\end{Proposition}
The idea of the proof is that one can factor the multilinear operators
into linear operators in each of the factors. The spectrum of
these elementary operators can be readily be related
to the operators of the one variable operator. On the other
hand, the action on each of the coordinates commutes with
the others. Hence, we can apply a well known result in Banach
algebra theory that ensures that the spectrum of the
product of two communting operators is contained in the
product of the spectra. We refer to the references above
for complete details.
Another result of \cite{trilogy2} is that for $C^r$ vector bundle maps
over rotations the spectra of transfer operators acting on bounded sections
and on continuous sections coincide with that on acting on $C^r$ sections.
Introducing the Sylvester vector bundle maps
$S_1^k= S_{\omega,M_1,M_1}^k$,
$S_2^k= S_{\omega,M_2,M_1}^k$ and $S_B^k= S_{\omega,B,M_1}^k$,
\eqref{eqk1} and \eqref{eqk2} can be rewritten as
\begin{eqnarray}
\label{eqk1t}
\S_1^k W_1^k - W_1^k - \hat \Lambda^k &=& \hat R^k_1 - S_B^k W_2^k\ , \\
\label{eqk2t}
\S_2^k W_2^k - W_2^k \phantom{\ - \hat \Lambda^k} &=& \hat R^k_2 \ .
\end{eqnarray}
\note{A}{He a\~nadido el termino en ``B'', antes estaba implicito en la
``R''}
Since
\[
\Spec(\S_2^k,\Sec{}{b}(L^k_s(E_2;E_1)))
\subset \A_2 \cdot \A_1^{-k}
\]
by Proposition~\ref{Sylvester} and by assumption H.3
$\A_1^{-k} \cap \A_2 = \emptyset$ for $k= 0,\dots, L$,
we conclude that
\[
1\notin \Spec(\S_2^k,\Sec{}{C^r}(L^k_s(E_2;E_1))) \ .
\]
Therefore, \eqref{eqk2t} admits a unique solution $W^k_2 \in
\Sec{}{C^r}(L^k_s(E_2;E_1)))$.
We solve \eqref{eqk1t} as follows.
If $\A_1 \cap \A_1^k= \emptyset$ we conclude that
\[
1\notin \Spec(\S_1^k,\Sec{}{C^r}(L^k_s(E_1;E_1))) \ ,
\]
and we take $\Lambda^k= 0$ and $W^k_1$ solving
$\S_1^k W_1^k - W_1^k = \hat R^k_1 - S_B^k W_2^k$.
Otherwise we will choose
$W^k_1= 0$ and $\hat \Lambda^k= -\hat R^k_1 + S_B^k W_2^k$.
This proves claim b) of
Theorem~\ref{theorem_whiskers}.
\begin{remark}
\label{graph}
Notice that solutions of \eqref{eqk1t} are not unique.
We could choose
$W^k_1= 0$ and $\hat \Lambda^k= -\hat R^k_1 + S_B^k W_2^k$
(and then we compute $\Lambda^k$),
for which we do not need non-resonance condition such as
$\A_1 \cap \A_1^k= \emptyset$. Notice that this election corresponds to
find the invariant manifold as a graph (over $E_1$).
\end{remark}
\subsubsection{Standing hypotheses}
\label{standing}
Since the coefficients of $W^{\leq}$ and $\Lambda$
are computed recursively from $N$, the smallness condition on $N$
state at the end of Section~\ref{preliminaries}
implies that
$W^{\leq}$ is close to the immersion $I_1:E_1\to E$ and $\Lambda$ is
close to $M_{1,\omega}:E_1\to E_1$. In summary:
\begin{quote}
We assume that
\[
\norm{N}_{C^{\Sigma_{r,s}}(B_{E}(3),E)}\ , \
\norm{W^{\leq}-I_1}_{C^{r,s}(B_{E_1}(1),E)}\ , \
\norm{\Lambda-M_1}_{C^{r,s}(B_{E_1}(1),E_1)}
\]
are as small as we need.
\end{quote}
Using scaling arguments similar to those in Section~\ref{preliminaries}, by
taking the scaling parameter $\delta$ small enough we may assume that
$\Lambda$ is approximately linear and it is a contraction which maps
$B_{E_1}(1)$ in $B_{E_1}(\lambda)$ with $\lambda<1$, and
\[
%\label{LipLambda}
\norm{\Dif_\eta \Lambda}_{C^0(B_{E_1}(1), E_1)} \leq
\norm{M_1}_{C^0} + \varepsilon < 1\
\]
and then
\[
\norm{M^{-1}}_{C^0} \norm{\Dif_\eta \Lambda}_{C^0(B_{E_1}(1), E_1)}^{L+1}
\leq \norm{M^{-1}}_{C^0} (\norm{M_1}_{C^0} + \varepsilon)^{L+1} < 1\ .
\]
\subsubsection{The equation for the higher order terms}
Once we have obtained the polynomial
vector bundle map $\Lambda$ over the rotation $\omega$
and the $L$-order approximation $W^{\leq}$
of the invariant manifold $W$,
we have to find the higher
order terms of the parameterization of the invariant manifold, $W^>$.
We will write
\[
W= W^{\leq} + W^{>},
\]
where $W^{>}:E_1\to E$ is a $C^{\Sigma_{r,s}}$ bundle map
over the identity such that $\Dif^j_\eta W^{>}(0,\th)= 0$ for every $j\leq L$.
The invariance equation \eqref{invariance_whisker} is reformulated in terms
of $W^{>}$ as
\[
\begin{array}{l}
M(\th) \left[ W^{\leq}(\eta,\th) +
W^{>}(\eta,\th)\right] +
N\left(W^{\leq}(\eta,\th) + W^{>}(\eta,\th),\th\right)
\\ =
\ W^{\leq}(\Lambda(\eta,\th),\th+\omega) +
W^{>}(\Lambda(\eta,\th),\th+\omega)
\end{array}
\]
or, with more compact notation,
\begin{equation}
\label{W>equation}
\begin{split}
W^{>}_{\th} - M_{\th}^{-1} \cdot
W^{>}_{\th+\omega} \comp \Lambda_\th = &
-(
W^{\leq}_{\th} -
M_{\th}^{-1} \cdot W^{\leq}_{\th+\omega}\comp\Lambda_\th) \\
& - M_{\th}^{-1} \cdot N_\th \comp (W^{\leq}_\th + W^{>}_\th) \ .
\end{split}
\end{equation}
Equation~\ref{W>equation} is an equation for $W^>$
to be solved in the space of $C^{\Sigma_{r,s}}$
bundle maps from $E_1$ to $E$, over the identity, whose $L$ first vertical
derivatives vanish on the zero section of $E_1$.
Notice that the way we
have constructed $W^{\leq}$ and $\Lambda$ ensures that, if
$W^>$ satisfies the conditions above, then
the right hand side
of \eqref{W>equation} satisfies also the same conditions.
If we define the operator $\S$ by
\begin{equation}
\label{def_S}
(\S H)_\th =
H_\th - M_\th^{-1} H_{\th+\omega}\comp \Lambda_\th \ ,
\end{equation}
then \eqref{W>equation} reduces to the fixed point equation
\begin{equation}
\label{fixedpoint}
W^>_\th = -\S^{-1} (W^{\leq}_{\th} -
M_{\th}^{-1} \cdot W^{\leq}_{\th+\omega}\comp\Lambda_\th
+ M_{\th}^{-1} \cdot N_\th \comp (W^{\leq}_\th + W^{>}_\th)) \ ,
\end{equation}
provided that $\S^{-1}$ exists and it is continuous in suitable spaces.
The existence of $\S^{-1}$ is equivalent to solve the linearized equation
\begin{equation}
\label{linear_equation}
(\S H)_\th = H_\th -
M_\th^{-1} \cdot H_{\th+\omega} \comp \Lambda_\th =
R_\th\ .
\end{equation}
Formally, the solution of \eqref{linear_equation} is
\begin{equation}
\label{solution_linear_equation}
H_\th = \sum_{k= 0}^\infty
M^{-k}_{\th+k\omega} R_{\th+k\omega}\comp \Lambda^k_\th\ .
\end{equation}
To prove the existence of $\S^{-1}$, we will analyze the convergence
of \eqref{solution_linear_equation} is suitable spaces. Then we will show
that \eqref{fixedpoint} defines a solution $W^>$.
The spaces will be introduced in Section~\ref{spaces}. The analysis
of the linearized equation \eqref{linear_equation} will be undertaken
in Section~\ref{linear_solution}. The fixed point equation \eqref{fixedpoint}
will be partially solved in Section~\ref{fixedpointproblem}, since
we will loose one derivative. This derivative will be recovered in
Section~\ref{final}.
\subsubsection{Functional spaces and lemmas on derivatives of highly
iterated functions}
\label{spaces}
With this motivation (cf. \cite{CabreFL03a}), given two $C^r$ vector bundles
$E,F$ over the same manifold $\P$ we define the Banach space
of $l$-flat $C^{\Sigma_{r,s}}$ bundle maps over the identity
\[
\Gamma_{r,s,l}(E; F) =
\{ H\in C^{\Sigma_{r,s}}_{\id} (B_E(1),F) \st
\Dif^j_x H (0,\th) = 0 \mbox{ for $0\leq j\leq l$},\
\norm{H}_{\Gamma_{r,s,l}} < \infty \}
\]
where the norm $\norm{\cdot}_{\Gamma_{r,s,l}}$ is given by
\begin{equation}
\label{conical}
\begin{split}
\norm{H}_{\Gamma_{r,s,l}} &=
\max \left\{
\norm{H}_{C^{\Sigma_{r,s}}(B_E(1),F)} \ , \
\displaystyle \max_{i\leq r}
\sup_{(x,\th)\in B_E(1)\setminus E_0}
\frac{|\Dif^i_\th\Dif^l_x H(x,\th)|}{|x|} \right\} \ .
\end{split}
\end{equation}
The first part in this definition controls the derivatives
in $\Sigma_{r,s}$, and the second part
controls the derivatives of order $l$ in $x$ and any order of
$\th$. We impose that the derivatives in the highest order are
controlled by $x$.
We do not require that the functions have $l+1$ derivatives with respect to
the vertical directions, but we require that the derivatives of order $l$
are estimated by linear functions on $x$. Of course, the functions
whose derivatives up to order $l+1$ in $x$ exist and such that the
derivatives up to order $l$ in $x$ vanish at the zero section $x= 0$ are in
our space. For such functions, the norm
$\norm{\Dif_\th^i \Dif^{l+1}_x H}_{C^0(B_E(1),F)}$ estimates the last
term in \eqref{conical}. Indeed, the arguments will work with this norm
in place of the ``conical'' norm in \eqref{conical}.
In the real-analytic case $r= a$, and once we have fixed a complexification
given by $\xi$, we define
\[
\Gamma_{a,l}(E; F) =
\{ H\in C^{a,\xi}_{\id} (B_E(1),F) \st
\Dif^j_x H (0,\th) = 0 \mbox{ for $0\leq j\leq l$},\
\norm{H}_{\Gamma_{a,l}} < \infty \}
\]
where the norm $\norm{\cdot}_{\Gamma_{a,l}}$ is given by
\begin{equation}
\label{conical_analytic}
\begin{split}
\norm{H}_{\Gamma_{a,l}} &=
\norm{\Dif^{l+1}_x H}_{C^{a,\xi}}\ .
\end{split}
\end{equation}
The following lemmas will be very useful. Although the notation we use
is for the finite differentiable case, they also work for the analytic case.
In fact, in the analytic case the proofs are easier, because we
do not take derivatives with respect the horizontal directions and
the norm \eqref{conical_analytic}
do not involves ``conical'' terms as in \eqref{conical}.
The following lemma follows immediately from Taylor's theorem. We use
the notation $t_+= \max(t,0)$ for $t\in\nr$.
\begin{Lemma}
\label{Taylor}
Given $E,F$ two $C^r$ vector bundles over the same manifold $\P$,
let $H\in\Gamma_{r,s,l}(E; F)$ and $(i,j)\in\Sigma_{r,s}$. Then,
for every $(x,\th)\in B_E(1)$:
\[
|\Dif^i_\th\Dif^j_x H(x,\th)| \leq
\frac{1}{(l-j)_+!} \norm{H}_{\Gamma_{r,s,l}} |x|^{(l-j+1)_+}
\]
\end{Lemma}
If we multiply $H\in \Gamma_{r,s,l}$ by a matrix, we obtain again
a function in $\Gamma_{r,s,l}$. More concretely,
\begin{Lemma}
\label{Leibnitz}
Given $E,F,G$ three $C^r$ vector bundles over the same manifold
$\P$, let $H\in\Gamma_{r,s,l}(E; F)$ and
$P\in C^{\Sigma_{r,s}}_{\id}(B_E(1),L(F,G))$. Then
$P\cdot H\in \Gamma_{r,s,l}(E; G)$, where we define $(P\cdot H)_\th (x) =
P_\th(x) \cdot H_\th(x)$, and
\[
\norm{P\cdot H}_{\Gamma_{r,s,l}} \leq C
\norm{P}_{C^{\Sigma_{r,s}}} \norm{H}_{\Gamma_{r,s,l}}\ ,
\]
where $C$ is a constant.
\end{Lemma}
\bproof Applying Leibnitz's rule to compute
$\Dif^i_\th\Dif^j_\eta (P\cdot H)$ for $(i,j)\in\Sigma_{r,s}$ we
obtain the estimates
\[
\begin{array}{lcl}
|\Dif^i_\th\Dif^j_x (P_\th(x) \cdot H_\th(x))|
&\leq&
\displaystyle
\sum_{m= 0}^i \sum_{n= 0}^j \comb{i}{m} \comb{j}{n}
|\Dif^{i-m}_\th\Dif^{j-n}_x P_\th(x)|
|\Dif^m_\th\Dif^n_\eta H_\th(x)| \\ \\
&\leq&
\displaystyle
\sum_{m= 0}^i \sum_{n= 0}^j \frac{\comb{i}{m} \comb{j}{n}}{(l-n)_+!}
\norm{P}_{C^{\Sigma_{r,s}}} \norm{H}_{\Gamma_{r,s,l}} |x|^{(l-n+1)_+}
\\ \\
&\leq&
C \norm{P}_{C^{\Sigma_{r,s}}} \norm{H}_{\Gamma_{r,s,l}} |x|^{(l-j+1)_+} \\ \\
&\leq&
C \norm{P}_{C^{\Sigma_{r,s}}} \norm{H}_{\Gamma_{r,s,l}} \ .
\end{array}
\]
Notice also that, for $j= l$,
\[
\begin{array}{lcl}
\frac{1}{|x|} |\Dif^i_\th\Dif^l_x (P_\th(x) \cdot H_\th(x))|
&\leq&
C \norm{P}_{C^{\Sigma_{r,s}}} \norm{H}_{\Gamma_{r,s,l}} \ ,
\end{array}
\]
and we are done with the proof of Lemma~\ref{Leibnitz}.
\eproof
\begin{remark}
In particular, if the matrix $P$ only depends on $\th$,
then we have the estimate
$\norm{P\cdot H}_{\Sigma_{r,s,l}} \leq
C \norm{P}_{C^r} \norm{H}_{\Gamma_{r,s,l}}$.
\end{remark}
In the analysis of the convergence of the expansion
\eqref{solution_linear_equation} in $\Gamma_{r,s,l}$ spaces,
we have to estimate the norms of its terms, that involve derivatives in
the horizontal and vertical directions. This kind of problems will also
appear in further arguments.
Following \cite{CabreFL03a,CabreFL03b}, we use the following sets of indices,
closely related to $\Sigma _{i,j}$,
to describe in some detail the structure of the
expression of the derivatives
of the composition in terms of the derivatives of the bundle maps,
with respect to horizontal and vertical directions:
\begin{equation}
\label{defSigma*ij}
\begin{array}{ll}
\Sigma^* _{i,0} & =
\{ (a,b) \in \nn^2 \mid \; a+b\leq i,\, b\ge 1 \}
\cup \{ (i,0) \} \subset \Sigma_{i,0} \ ,\\
\Sigma ^*_{i,j} & = \{ (a,b) \in \nn^2 \mid
\; a+b\leq i+j,\, a \le i,\, b\ge 1 \} \subset \Sigma_{i,j}
\qquad \mbox{ if } j\ge 1 \ ,
\end{array}
\end{equation}
\begin{equation}\label{defSigmatij}
\begin{array}{ll}
\tilde \Sigma _{i,0} & = \{ (a,b) \in \nn^2
\mid \; a+b\leq i \}\ , \\
\tilde \Sigma_{i,j} & = \Sigma^*_{i,j}
\qquad \qquad \mbox{ if } j\ge 1 \ .
\end{array}
\end{equation}
We will also use the notation
\begin{equation}
\label{sigma}
\sigma(t,m)= \sum_{j= 0}^m t^j \leq (1+t)^m
\end{equation}
for $t\geq 0$.
We have the following estimates of the norms of composition of bundle maps.
\begin{Lemma}
\label{composition1}
For $M\in C^{r,\infty}_{\omega}(E,E)$ (linear),
$R\in C^{\Sigma_{r,s}}_{\id}(E_1,E)$, and
$\Lambda\in C^{\Sigma_{r,s}}_{\omega}(E_1,E_1)$, then
$M_\th^{-1} \cdot R_{\th+\omega} \comp \Lambda_\th\in
C^{\Sigma_{r,s}}_{\id}(E_1,E)$ and
for all $(i,j)\in\Sigma_{r,s}$
\[
\begin{array}{l}
\Dif_\th^i \Dif_\eta^j
(M_\th^{-1} \cdot R_{\th+\omega} \comp \Lambda_\th)
= \\ \\
\
\displaystyle
\sum_{m= 0}^i \sum_{(a,b)\in\Sigma_{m,j}^*} \sum_{I,J}
C_{m,j,a,b,I,J} \
\Dif_\th^{i-m} (M_\th^{-1}) \
\Dif_\th^a \Dif_\eta^b
R_{\th+\omega} \comp \Lambda_\th \
\Dif_\th^{i_1}\Dif_\eta^{j_1} \Lambda_\th
\dots
\Dif_\th^{i_b}\Dif_\eta^{j_b} \Lambda_\th \ .
\end{array}
\]
where $I= (i_1,\dots,i_b)$, $J= (j_1,\dots,j_b)$ are multi-indices with
$|I|_1 = m-a$, $|J|_1= j$, $i_l+j_l\geq 1$ for $l= 1,\dots b$,
and $C_{m,j,a,b,I,J}$ is a combinational coefficient depending on the indices.
Moreover, we have a bound
\[
\norm{M_\th^{-1} \cdot R_{\th+\omega} \comp \Lambda_\th}_
{C^{\Sigma_{r,s}}} \leq
C \norm{M^{-1}}_{C^r} \norm{R}_{C^{\Sigma_{r,s}}}
\sigma(\norm{\Lambda}_{C^{\Sigma_{r,s}}},r+s)\ .
\]
\end{Lemma}
\bproof
First compute
$\Dif^m_\th \Dif^j_\eta (R_{\th+\omega} \comp \Lambda_\th)$
by using induction arguments, and
then compute
$\Dif_\th^i \Dif_\eta^j (M_\th^{-1} \cdot R_{\th+\omega}
\comp \Lambda_\th)$.
A key point for obtaining this formulas is that the derivatives of
$\th\to \th+\omega$ are bounded uniformly independently of the order.
Notice that the resulting expression contains derivatives
$D^a_\th D^b_\eta R_{\th+\omega} $ of orders
$(a,b)\in \Sigma ^*_{0,j} \cup \Sigma ^*_{1,j} \cup \dots \cup
\Sigma ^*_{i,j} = \tilde \Sigma _{i,j}$
if $j > 0$, or $\{ (a,b) \mid \, a+b \leq i \}
= \tilde \Sigma _{i,0}$ if $j=0$.
\eproof
\begin{Lemma}
\label{composition2}
For $M\in C^{r,\infty}_{\omega}(E,E)$ (linear),
$F\in C^{\Sigma_{r,s}}_{\omega}(E,E)$, and
$W\in C^{\Sigma_{r,s}}_{\id}(E_1,E)$
we have for all $(i,j)\in\Sigma_{r,s}$
\[
\begin{array}{l}
\Dif_\th^i \Dif_\eta^j
(M_\th^{-1} \cdot F_{\th} \comp W_\th)
= \\ \\
\
\displaystyle
\sum_{m= 0}^i \sum_{(a,b)\in\Sigma_{m,j}^*} \sum_{I,J}
C_{m,j,a,b,I,J} \
\Dif_\th^{i-m} (M_\th^{-1}) \
\Dif_\th^a \Dif_x^b
F_{\th} \comp W_\th \
\Dif_\th^{i_1}\Dif_\eta^{j_1} W_\th
\dots
\Dif_\th^{i_b}\Dif_\eta^{j_b} W_\th \ .
\end{array}
\]
where $I= (i_1,\dots,i_b)$, $J= (j_1,\dots,j_b)$ are multi-indices with
$|I|_1= m-a$, $|J|_1= j$, $i_l+j_l\geq 1$ for $l= 1,\dots b$,
and $C_{m,j,a,b,I,J}$ is a combinational coefficient depending on the indices.
Moreover, we have a bound
\[
\norm{M_\th^{-1} \cdot F_{\th} \comp W_\th}_
{C^{\Sigma_{r,s}}} \leq
C \norm{M^{-1}}_{C^r} \norm{F}_{C^{\Sigma_{r,s}}}
\sigma(\norm{W}_{C^{\Sigma_{r,s}}},r+s)\ .
\]
\end{Lemma}
\subsubsection{Solving the linearized equation}
\label{linear_solution}
In this section we prove the invertibility of the linear operator $\S$
introduced in \eqref{def_S}, which is equivalent to solve
the linear equation \eqref{linear_equation}. We start with two lemmas.
\begin{Lemma}
\label{bound_M}
Let $M:E\to E$ be a $C^r$ vector bundle map over a rotation
$\omega$. Then, for all $0\leq m\leq r$, and $k\geq 0$,
\[
\norm{\Dif^m_\th M_{\th+k\omega}^{-k}}_{C^0} \leq C_r k^m
\norm{M_\th^{-1}}_{C^0}^k\ ,
\]
where $C_r$ is a constant that does not depend on $k$.
\end{Lemma}
\bproof
Applying Leibnitz rule, we have that
\[
\Dif_\th^m (M(\th)^{-1} M(\th+\omega)^{-1} \dots
M(\th+(k-1)\omega)^{-1}) \\
\]
contains a sum of $C_{m,k} \leq k^m$ terms of the form
\[
M(\th)^{-1} \cdots
\Dif^{r_1} M(\th+k_1\omega)^{-1} \cdots
\Dif^{r_j} M(\th+k_j\omega)^{-1} \cdots
M(\th+(k-1)\omega)^{-1} \ ,
\]
where $j\in\{1,\dots,m\}$, $r_1 + \dots + r_j = m$, $r_1,\dots, r_j >0$ and
$0\leq k_1 < \dots }) = \S^{-1} \left(
-(W^{\leq}_{\th} - M_{\th}^{-1} W^{\leq}_{\th+\omega}\comp\Lambda_\th)
- M_{\th}^{-1} N_\th \comp (W^{\leq}_\th + W^{>}_\th) \right) \ ,
\end{equation}
in $\Gamma_{r,s,L}$. First, we will prove that
$\Tau$ is a contraction, but in a closed ball of
$\Gamma_{r,s-1,L}$ (recall that $L\leq s-1$, by hypothesis). The derivative
that we loose in the following result will be recovered later.
\begin{Lemma}
\label{fixedpointlemma}
Under the hypotheses of Theorem~\ref{theorem_whiskers}, and the
standing hypotheses of Section~\ref{standing},
we have that the operator $\Tau:\Gamma_{r,s-1,L}\to \Gamma_{r,s-1,L}$
sends the closed unit ball $\bar B_{\Gamma_{r,s-1,L}}(1)$ into itself,
and it is a contraction there. Obviously,
if $r= a$ then $\Gamma_{r,s-1,L} = \Gamma_{a,L}$.
\end{Lemma}
\bproof
If $W^{>}\in \bar B_{\Gamma_{r,s-1,L}}(1)$, $W^{>}$ is defined in
$B_{E_1}(1)$ and $\norm{W^{>}}_{C^0} \leq 1$, then
for all $\eta_\th\in B_{E_1}(1)$,
\[
\begin{split}
|W^{\leq}_\th (\eta) + W^{>}_\th (\eta)| & \leq
|(W^{\leq}_\th - I^1_\th)(\eta)| + |\eta| + |W^{>}_\th (\eta)| \\
& < 3\ ,
\end{split}
\]
and then $N_\th \comp (W^{\leq}_\th + W^{>}_\th)$ is well defined
and of class $C^{\Sigma_{r,s-1}}$.
Let $H$ be the bundle map over the identity defined by
\[
H_\th =
-(
W^{\leq}_{\th} -
M_{\th}^{-1} W^{\leq}_{\th+\omega}\comp\Lambda_\th
)
- M_{\th}^{-1} N_\th \comp (W^{\leq}_\th + W^{>}_\th)\ .
\]
Notice that $H\in\Gamma_{r,s-1,L}$. We are going to bound
$\norm{H}_{\Gamma_{r,s-1,L}}$. To do so, we bound
$\Dif^i_\th\Dif^j_\eta H_\th(\eta)$ for $i,j\in\Sigma_{r,s-1}$ and
$\eta_\th \in B_{E_1}(1)$.
We write $H_\th= H_\th^1 + H_\th^2$, where
\begin{equation}
\label{H1}
\begin{array}{lcl}
H_\th^1
&=&
- W^{\leq}_{\th}
+ M_{\th}^{-1} W^{\leq}_{\th+\omega}\comp\Lambda_\th
- M_{\th}^{-1} N_\th\comp W^{\leq} \\ \\
&=&
- W^{\leq}_{\th} + M_\th^{-1} I^1_{\th+\omega}\comp \Lambda_\th +
\\
& & M_{\th}^{-1}(W^{\leq}_{\th+\omega}-I^1_{\th+\omega})
\comp \Lambda_\th
- M_{\th}^{-1} N_\th\comp W^{\leq}
\end{array}
\end{equation}
and
\begin{equation}
\label{H2}
\begin{array}{lcl}
H_\th^2
&=&
M_{\th}^{-1} N_\th\comp W^{\leq}_\th -
M_{\th}^{-1} N_\th\comp (W^{\leq}_\th + W^{>}_\th) \\ \\
&=&
\displaystyle -\int_0^1 M_\th^{-1}
\Dif_x N_\th\comp (W^{\leq}_\th + t W^{>}_\th)\ W^{>}_\th\ dt\ .
\end{array}
\end{equation}
Since $W^{\leq}$ and $\Lambda$ are
polynomial of degree $L$ (with $L+1\leq s$), then $H^1\in C^{\Sigma_{r,s}}$.
If $j\leq L$, we apply Taylor's theorem to obtain the estimate
\begin{equation}
|\Dif^i_\th\Dif^j_\eta H^1_\th(\eta)|
\leq
\norm{\Dif^i_\th\Dif^{L+1}_\eta H^1_\th}_{C^0} \cdot
|\eta|^{L+1-j} \ ,
\end{equation}
so we have to estimate $|\Dif^i_\th\Dif^j_\eta H^1_\th(\eta)|$ for
$L}_\th)\
W^{>}_\th\
\right)\right| dt \\ \\
&\leq
C \norm{M^{-1}}_{C^r} \norm{N}_{C^{\Sigma_{r,s}}}
\sigma(\norm{W^{\leq}}_{C^{r,L}}+1,r+s)
\norm{W^{>}}_{\Gamma_{r,s-1,L}}
|\eta|^{(L-j+1)_+} \ ,
\end{split}
\]
where we use the bound
\[
\sup_{t\in[0,1]}
\norm{ W^{\leq} + t W^{>}}_{C^{\Sigma{r,s}}} \leq
\norm{W^{\leq}}_{C^{r,L}}+1 \ ,
\]
and then
\begin{equation}
\label{bound2}
\norm{H^2}_{\Gamma_{r,s-1,L}} \leq
C \norm{M^{-1}}_{C^r} \norm{N}_{C^{\Sigma_{r,s}}}
\sigma_{r+s}(\norm{W^{\leq}}_{C^{r,L}}+1)
\norm{W^{>}}_{\Gamma_{r,s-1,L}}\ .
\end{equation}
\begin{remark}
This is the point in which we loose one derivative with respect to the
vertical direction and we are forced to work in $\Gamma_{r,s-1,L}$ instead
of $\Gamma_{r,s,L}$. Obviously, this drawback has not to do in the analytic
case.
\end{remark}
The standing hypotheses of Section~\ref{standing} about the smallness of
$N$ and $W^{\leq} - I^1$ give that
\eqref{bound1},\eqref{bound2} are small,
so
\[
\norm{\Tau W^{>}}_{\Gamma_{r,s-1,L}} = \norm{\S^{-1} H}_{\Gamma_{r,s-1,L}}
<1\ .
\]
We have to prove now that $\Tau$ is a contraction in the closed unit ball
of $\Gamma_{r,s-1,L}$. To do so, let $W^{>}$ and $\Delta$ such that
$W^{>}$ and $W^{>}+\Delta \in \bar B_{\Gamma_{r,s-1,L}}(1)$. Notice that
\[
\begin{split}
\Tau(W^{>}+\Delta)_\th - \Tau(W^{>})_\th
&=
\displaystyle
-\int_0^1 \S^{-1} M_\th^{-1}
\Dif_x N_\th\comp (W^{\leq}_\th + W^{>}_\th + t \Delta_\th)
\Delta_\th \ dt \ ,
\end{split}
\]
and using the same arguments are those leading to \eqref{bound2} we obtain that
\begin{equation}
\label{Lipchitz}
\begin{split}
& \norm{ \Tau(W^{>}+\Delta) - \Tau(W^{>})}_{\Gamma_{r,s-1,L}} \\
& \leq \norm{\S}_{\Gamma_{r,s-1,L}} \norm{M^{-1}}_{C^r}
\norm{N}_{C^{\Sigma_{r,s}}} \sigma_{r+s}(1+\norm{W^{\leq}}_{C^{r,L}})
\norm{\Delta}_{\Gamma_{r,s-1,L}} \ .
\end{split}
\end{equation}
Under the smallness conditions on $N$ the operator $\Tau$ is a contraction
in $\bar B_{\Gamma_{r,s-1,L}}(1)$.
\eproof
\begin{remark}
The previous lemma close the proof of statements a) and b) of
Theorem~\ref{theorem_whiskers} in the analytic case.
\end{remark}
\subsubsection{The last derivative}
\label{final}
Lemma~\ref{fixedpointlemma} proves that the operator $\Tau$
defined in \eqref{fixedpointoperator} has a fixed point.
It is the result claimed except for the fact that in the finite
differentiable case we obtain that $W$ is $C^{\Sigma_{r,s-1}}$
instead of $C^{\Sigma_{r,s}}$.
In this section we will see that $\Dif_\eta W^> \in
\Gamma_{r,s-1,L-1}$, and as a result we will obtain
$W^>\in\Gamma_{r,s,L}$, ending the proof of
statements a) and b) of Theorem~\ref{theorem_whiskers}.
Recall that $W^>$ solves the equation \eqref{W>equation}
\begin{equation}
\begin{split}
W^{>}_{\th} - M_{\th}^{-1} \cdot
W^{>}_{\th+\omega} \comp \Lambda_\th = &
-(
W^{\leq}_{\th} -
M_{\th}^{-1} \cdot W^{\leq}_{\th+\omega}\comp\Lambda_\th) \\
& - M_{\th}^{-1} \cdot N_\th \comp (W^{\leq}_\th + W^{>}_\th) \ .
\end{split}
\end{equation}
So, $\Dif_\eta W^>$ solves the equation
\begin{equation}
\label{DW>equation}
\begin{split}
\Dif_\eta W^{>}_{\th} - M_{\th}^{-1} \cdot
\Dif_\eta W^{>}_{\th+\omega} \comp \Lambda_\th \cdot \Dif_\eta \Lambda_\th
= & - \Dif_\eta\left(
W^{\leq}_{\th} -
M_{\th}^{-1} \cdot W^{\leq}_{\th+\omega}\comp\Lambda_\th\right) \\
& - M_{\th}^{-1} \cdot \Dif_x N_\th \comp W_\th
\cdot (\Dif_\eta W^\leq_\th + \Dif_\eta W^>_\th)\ .
\end{split}
\end{equation}
Let $U$ be the bundle map
\[
\begin{split}
U_\th = &
- \Dif_\eta\left(
W^{\leq}_{\th} -
M_{\th}^{-1} \cdot W^{\leq}_{\th+\omega}\comp\Lambda_\th\right) \\
&
-M_{\th}^{-1} \cdot \Dif_x N_\th \comp W_\th
\cdot \Dif_\eta W^\leq_\th\ .
\end{split}
\]
Notice that $U\in \Gamma_{r,s-1,L-1}(E_1,L(E_1,E))$.
We consider now the operators $\tilde \S, \tilde \T$ defined by
\[
\begin{split}
(\tilde \S \tilde H)_\th = &
\tilde H_\th - M_{\th}^{-1} \cdot
\tilde H_{\th+\omega} \comp \Lambda_\th \cdot \Dif_\eta \Lambda_\th \\
(\tilde \T \tilde H)_\th = &
M_{\th}^{-1} \cdot \Dif_x N_\th \comp W_\th
\cdot \tilde H_\th \ .
\end{split}
\]
Both operators act on bundle maps
$\tilde H\in \Gamma_{r,s-1,L-1}(E_1,L(E_1,E))$.
\begin{Lemma}
\label{finallemma}
Under the hypotheses of Theorem~\ref{theorem_whiskers}, and the
standing hypotheses of Section~\ref{standing},
we have that the operators
$\tilde \S,\tilde \T:\Gamma_{r,\bar s,L-1}\to \Gamma_{r,\bar s,L-1}$
are bounded for $L-1\leq \bar s \leq s-1$. Moreover, taking
$\norm{N}_{C^{\Sigma_{r,s}}}$ small enough, $\tilde \S$ is invertible
and $\norm{\tilde \S} \norm{\tilde \T} < 1$.
\end{Lemma}
\bproof
For $\tilde H\in \Gamma_{r,\bar s,L-1}$:
\[
\begin{split}
\norm{M_\th^{-1} \Dif_x N \comp W_\th \cdot \tilde H}_{\Gamma_{r,\bar s,L-1}}
\leq &
C \norm{M_\th^{-1} \Dif_x N \comp W_\th}_{C^{\Sigma_{r,s-1}}}
\norm{\tilde H}_{\Gamma_{r,\bar s,L-1}} \\
\leq &
C \norm{M_\th^{-1}}_{C^r} \norm{N}_{C^{\Sigma_{r,s}}}
\norm{\tilde H}_{\Gamma_{r,\bar s,L-1}}
\sigma(\norm{W}_{C^{\Sigma_{r,\bar s}}},r+\bar s)\ .
\end{split}
\]
where we have applied
Lemma~\ref{Leibnitz} and
Lemma~\ref{composition2}.
This proves that $\tilde \T$ is bounded and as small as necessary.
The operator $\tilde \S$ is obviously bounded in $\Gamma_{r,\bar s,L-1}$
(see Proposition~\ref{Sbounded} for the arguments). Given $\tilde G$
in $\Gamma_{r,\bar s,L-1}$, the series
\begin{equation}
\label{tHsol}
\tilde H_\th =
\sum_{k= 0}^\infty M_{\th+k\omega}^{-k} \tilde
G_{\th+k\omega}\comp \Lambda_\th^k \cdot \Dif_\eta \Lambda_\th^k
\end{equation}
provides a formal solution of $\tilde \S\tilde H = \tilde G$. By repeating
again the arguments of Proposition~\ref{Sbounded}, we can bound each term
in \eqref{tHsol} as follows:
\[
\begin{split}
\norm{M_{\th+k\omega}^{-k} \tilde
G_{\th+k\omega}\comp \Lambda_\th^k \cdot
\Dif_\eta \Lambda_\th^k}_{\Gamma_{r,\bar s,L-1}}
\leq
& C \norm{M_{\th+k\omega}^{-k}}_{C^r}
\norm{\tilde G_{\th+k\omega}\comp \Lambda_\th^k}_{\Gamma_{r,\bar s,L-1}}
\norm{\Lambda_\th^k}_{C^{\Sigma_{r,\bar s}}}
\\
\leq
&
C (k+1)^r \norm{M^{-1}}_{C^0}^k (\norm{M_1}_{C^0} + \varepsilon)^{(L+1)k}
\norm{\tilde G}_{\Gamma_{r,\bar s,L-1}}
\end{split}
\]
Since $\norm{M^{-1}}_{C^0} (\norm{M_1}_{C^0} + \varepsilon)^{L+1} <1$
(see Section~\ref{standing}), we are done with
the proof of Lemma~\ref{finallemma}.
\eproof
The following lines are the final arguments of the proof of statements a) and
b) of Theorem~\ref{theorem_whiskers}.
At this point, $W^> \in \Gamma_{r,s-1,L}$, so
$\Dif_\eta W^>\in \Gamma_{r,s-2,L-1}$. Moreover, $\Dif_\eta W^>$ is
a solution of \eqref{DW>equation}, that reads
\[
\tilde \S \Dif_\eta W^> = U - \tilde \T \Dif_\eta W^> \ .
\]
Applying Lemma~\ref{finallemma} with $\bar s= s-2$, we conclude that
\begin{equation}
\label{solDW>}
\Dif_\eta W^> = (\Id + \tilde\S\tilde \T)^{-1} \tilde \S^{-1} U \ ,
\end{equation}
is the only solution in $\Gamma_{r,s-2,L-1}$. Notice that
$U\in \Gamma_{r,s-1,L-1}$,
and the operators $\tilde \S,\tilde \T$ are also well defined in
$\Gamma_{r,s-1,L-1}$, by Lemma~\ref{finallemma} with $\bar s= s-1$.
Much more, since $\norm{\tilde\S\tilde \T)}<1$ in $\Gamma_{r,s-1,L-1}$,
again by Lemma~\ref{finallemma}, we conclude that \eqref{solDW>} is also
well defined in $\Gamma_{r,s-1,L-1}$.
In summary, $\Dif_\eta W^> \in \Gamma_{r,s-1,L-1}$, so
$W^>\in \Gamma_{r,s,L}$ and $W\in C^{\Sigma_{r,s}}$.
\subsubsection{Proof of the uniqueness of the invariant manifold}
\label{uniqueness}
In this section we will prove the uniqueness of the invariant manifold
mentioned in c) of Theorem~\ref{theorem_whiskers} (see \cite{CabreFL03a}).
Notice that if $W= W(\eta,\th)$
is a $C^{\Sigma_{r,L+1}}$-parameterization of an invariant manifold $\W$
attached to the torus and tangent to $E_1$ then,
after the election of coordinates given in Section~\ref{preliminaries},
\[
W(0,\th)= 0 \ , \ \Dif_\eta W(0,\th)= \left(\begin{array}{c}
\Id_{E_1} \\ O \end{array}\right)\ .
\]
So, we can write locally $\eta= W_{1,\th}^{-1} (x_1)$ and
$G_{\th}(x_1)= W_{2,\th} \comp W_{1,\th}^{-1}$.
So, $G: B_1 \subset E_1 \to E_2$ is $C^{\Sigma_{r,L+1}}$,
where $B_1$ is a tubular neighborhood
of the zero section in $E_1$.
Notice that, locally, the manifold $\W$ is a graph $\{x_2= G_\th (x_1)\}$.
Moreover, $G_\th(0) = 0$ and $\Dif_{x_1} G_\th (0) = 0$.
Notice that this graph representation is independent of the former
parameterization of the manifold.
If we see that there is one and only one invariant
$C^{\Sigma_{r,L+1}}$ graph, tangent to $E_1$, we will be done with the proof
of the uniqueness.
The invariance equation of the graph $x_2= G_\th(x_1)$ is
\begin{equation}
\label{invariance_graph}
\begin{split}
G_\th(x_1)
= & A_{2,\th}^{-1}
\left(G_{\th+\omega} (F_{1,\th}(x_1,G_\th(x_1))) - N_{2,\th}(x_1,G_\th(x_1))\right)
\\
= & A_{2,\th}^{-1}
\left(G_{\th+\omega}
(A_{1,\th} x_1 + B_\th G_\th(x_1) + N_{1,\th}(x_1,G_\th(x_1))) -
N_{2,\th}(x_1,G_\th(x_1))\right)
\\
= & {\U}(G)_\th (x_1)\ .
\end{split}
\end{equation}
In Remark~\ref{graph} we showed how to solve this equation up to order $L$.
We found a polynomial $G_\th^{\leq}$ of degree $L$ and coefficients
of class $C^r$ such that
\[
G_\th^{\leq}(x_1) = {\U}(G^{\leq})_\th (x_1) + o(|x_1|^{L})\ .
\]
This polynomial is unique.
So, we obtain a fixed point equation for the higher order terms of
the graph $x_2 = G_\th^{\leq}(x_1) + G_\th^{>}(x_1)$,
\begin{equation}
\label{higher_graph}
\begin{split}
G_\th^{>}(x_1)
& = -G_\th^{\leq}(x_1) + {\U}(G^{\leq} + G^{>})_\th (x_1) \\
& = {\V}(G^{>})_\th (x_1)\ .
\end{split}
\end{equation}
We will see that this equation has at most one solution
$G^{>}: B_1 \subset E_1 \to E_2$ such that
\[
\begin{split}
[G^{>}]_{L+1}
& = \sup_{(x_1,\th)\in B_1} \frac{|G_\th^>(x_1)|}{|x_1|^{L+1}} \\
& < \infty\ .
\end{split}
\]
In fact, we will fix $B_1= B_{E_1}(1)$ and some smallness conditions
on $B$, $N$, etc. using the scaling arguments in Section~\ref{preliminaries}
and Section~\ref{standing}.
\def\Lip{{\rm Lip}}
Assume that there are two solutions $G^1= G^{\leq} + G^{1,>}$ and
$G^2= G^{\leq} + G^{2,>}$ of \eqref{invariance_graph} (or $G^{1,>}$ and
$G^{2,>}$ of \eqref{higher_graph}). Then,
for $(x_1,\th) \in B_1$,
\[
\begin{split}
|({\U}(G^2))_\th (x_1) - & ({\U}(G^1))_\th (x_1)|
\phantom{\leq
\norm{A_{2,\th}^{-1}}
\left[ |G_{\th+\omega}^{\leq}(F_{1,\th}(G_\th^2(x_1))) -
G_{\th+\omega}^{\leq}(F_{1,\th}(G_\th^1(x_1)))| \right.} \\
\leq
&
\norm{A_{2,\th}^{-1}}
\left[ |G_{\th+\omega}^{\leq}(F_{1,\th}(G_\th^2(x_1))) -
G_{\th+\omega}^{\leq}(F_{1,\th}(G_\th^1(x_1)))| \right.\\
&
\phantom{
\norm{A_{2,\th}^{-1}}}
\ + |G^{2,>}_{\th+\omega}(F_{1,\th}(G_\th^2(x_1))) -
G^{1,>}_{\th+\omega}(F_{1,\th}(G_\th^2(x_1)))| \\
&
\phantom{
\norm{A_{2,\th}^{-1}}}
\ + |G^{1,>}_{\th+\omega}(F_{1,\th}(G_\th^2(x_1))) -
G^{1,>}_{\th+\omega}(F_{1,\th}(G_\th^1(x_1)))| \\
&
\phantom{
\norm{A_{2,\th}^{-1}}}
\ + \left. \! |N_{2,\th}(x_1,G_\th^2(x_1)) -
N_{1,\th}(x_1,G_\th^1(x_1))| \right]\\
\leq
&
\norm{A_{2,\th}^{-1}}
\left[ \Lip G_{\th+\omega}^{\leq} (\norm{B_\th} + \Lip_{x_2} N_{1,\th})
|G^{2,>}_\th(x_1) - G^{1,>}_\th(x_1)| \right. \\
&
\phantom{
\norm{A_{2,\th}^{-1}}}
\ + [G^{2,>} - G^{1,>}]_{L+1} |F_{1,\th}(G_\th^2(x_1))|^{L+1} \\
&
\phantom{
\norm{A_{2,\th}^{-1}}}
\ + \Lip G^{1,>}_{\th+\omega} (\norm{B_\th} + \Lip_{x_2} N_{1,\th})
|G^{2,>}_\th(x_1) - G^{1,>}_\th(x_1)| \\
&
\phantom{
\norm{A_{2,\th}^{-1}}}
\ + \left. \! \Lip_{x_2} N_{2,\th} |G^{2,>}_\th(x_1) - G^{1,>}_\th(x_1)|
\right] \ .
\end{split}
\]
Notice that we can get
\[
\begin{split}
|F_{1,\th}(G_\th^2(x_1))|
=
& |A_{1,\th} x_1 + B_\th G_\th^2(x_1) + N_{1,\th}(x_1,G_\th^2(x_1))| \\
\leq
& (\norm{A_1}_{C^0} + \varepsilon) |x_1|
\end{split}
\]
by using smallness assumptions on $B$ and $N$. The bound depends also on
$G^{2,>}$.
So, again using the smallness assumptions that will depend also on
$G^{1,>}$, we have
\[
\begin{split}
[{\V}(G^{2,>}) - {\V}(G^{1,>})]_{L+1}
= & \sup_{(x_1,\th)\in B_1}
\frac{|{\U}(G^2)_\th (x_1) - {\U}(G^1)_\th (x_1)|}{|x_1|^{L+1}}
\\
\leq & \norm{A_2^{-1}}_{C^0}
(\varepsilon + (\norm{A_1}_{C^0} + \varepsilon)^{L+1})
[G^{2,>} - G^{1,>}]_{L+1}
\\
\leq \nu [G^{2,>} - G^{1,>}]_{L+1}
\end{split}
\]
for some $\nu<1$. So $G^{2,>} = G^{1,>}$, and the proof of
Theorem~\ref{theorem_whiskers} is finished.
\section{Results for flows} \label{sec:flows}
\subsection{Reduction of the results for flows to results
for maps}
The results proved for discrete time maps
imply results for the discrete time problem.
If $F_{t,\th}= F_t( \cdot, \th)$
is the time $t$ flow of the the vector field $X$
given in \eqref{continuous}
that is,
\begin{equation}
\begin{split}
&\frac{d}{dt} F_t(x, \th) = X( F_t(x, \th), \th+t\omega)\\
& F_0(x,\th) = x
\end{split}
\end{equation}
we see that if a torus is invariant for the
vector field, it is invariant for the time-one map,
$F_1$.
If $F_1$ has hyperbolicity properties,
then, it is possible to use Theorem \ref{theorem_tori} to
study tori invariant under $F_1$ and
Theorem \ref{theorem_whiskers} to study their invariant manifolds.
This provides with candidates for invariant tori
and invariant manifolds for the vector field.
We want to argue that, given the uniqueness properties that
we have found for invariant tori, the tori and manifolds
which are invariant under $F_1$ have to be invariant for
the whole flow.
Recall that
\begin{equation}\label{evolution}
\begin{split}
F_{{t + s},\th}(x)
= & F_{t,\th+s\omega}\comp F_{s,\th}(x) \\
= & F_{s,\th+t\omega}\comp F_{t,\th}(x)\ .
\end{split}
\end{equation}
If for a fixed $s \in \nr\setminus\{0\}$, $K_s:\nt^d\to\nr^n$
is a solution of \eqref{parameterization_tori} for $F_s$,
that is
\begin{equation} \label{invariance2}
F_s( K_s(\th), \th) = K_s(\th + s \omega),
\end{equation}
then the torus $\K_s= \{(K_s(\th),\th) \st \th \in \nt^d\}$
is invariant under
the $s$-time map $F_s$: $F_{s,\th}(\K_{s,\th}) = \K_{s,\th+s\omega}$.
Then, for any $t \in \nr$ $K_{s,t}:\nt^d\to\nr^n$ defined
by
\[
K_{s,t} (\th) = F_t(K_s(\th-t\omega),\th-t\omega)
\]
parameterizes a torus $\K_{s,t}= F_{t,\th-t\omega} (\K_{s,\th-t\omega})$
such that
\begin{equation}
\begin{split}
F_{s,\th}(\K_{s,t,\th})
& = F_{s,\th}\comp F_{t,\th-t\omega} (\K_{s,\th-t\omega}) \\
& = F_{t,\th+(s-t)\omega} \comp F_{s,\th-t\omega} (\K_{s,\th-t\omega}) \\
& = F_{t,\th+(s-t)\omega} (\K_{s,\th+(s-t)\omega}) \\
& = \K_{s,t,\th+s\omega}\ .
\end{split}
\end{equation}
We see that $K_{s,t}(\th) = F_t(K_s(\th - t \omega), \th - t \omega) $
satisfies the same equation \eqref{invariance2}. Therefore,
given the uniqueness properties of $K$
obtained in Theorem \ref{theorem_tori} we obtain that
for all $|t|$ sufficiently small,
$
F_t( K_s(\th - t \omega), \th - t \omega) = K_s(\th)
$, equivalently,
\[
F_t( K_s(\th), \th) = K_s(\th + t \omega)
\]
Repeating the application of the above
equation, we obtain that for any
integer $n$
\[
F_{nt}( K_s(\th), \th) = K_s(\th + nt \omega)
\]
Hence, using the uniqueness statements in Theorem \ref{theorem_tori},
we have shown that the solutions for one time of
\eqref{invariance2} are invariant under the flow.
We can treat analogously the solutions for
a fixed $s$ of
\begin{equation}
\label{invariance_s}
F_s(W_s(\eta, \th), \th) = W_s(\Lambda_s(\eta, \th),\th+s\omega ).
\end{equation}
The key point is using uniqueness of the invariant manifold (is the
parameterization which is not unique). So, assume we have an invariant
manifold $\W_s= \{ (W_s(\eta,\th) \st (\eta,\th) \in U_1\}$ of $F_s$, where
$U_1$ is a tubular neighborhood of the zero section of $E_1$,
that is $F_{s,\th}(\W_{s,\th}) \subset \W_{s,\th+s\omega}$,
and the tangent bundle of $\W_{s}$ over $\K$ is
$T_\K \W_{s} = E_{1}$.
We assume also that the
subbundle $E_1$ is invariant under the whole flow, not just $F_s$,
which means that $\Dif_x F_{t,\th} (\K_\th) E_{1,\th} = E_{1,\th+t\omega}$
for all time $t$,
and that $E_1$ satisfies the hypothesis of Theorem~\eqref{theorem_whiskers}
for $F_s$.
Then, for $\W_{s,t}$ defined by
$\W_{s,t,\th} = F_{t,\th-t\omega} (\W_{s,\th-t\omega})$ we have
\begin{equation}
\label{W_s-t}
\begin{split}
F_{s,\th}(\W_{s,t,\th}) \subset & \W_{s,t,\th+s\omega}\ .
\end{split}
\end{equation}
So, $\W_{s,t}$ is invariant under $F_s$. Moreover,
\[
\begin{split}
T_{\K_\th} \W_{s,t,\th}
& = \Dif_x F_t(\K_{\th-t\omega}) T_{\K_{\th-t\omega}} \W_{s,\th-t\omega} \\
& = \Dif_x F_t(\K_{\th-t\omega}) E_{1,\th-t\omega} \\
& = E_{1,\th}\ ,
\end{split}
\]
and the uniqueness of the invariant manifold established in
Theorem~\ref{theorem_whiskers} gives that $\W_{s,t}= \W_s$.
Repeating the previous arguments given for $\K_s$ we see that
$\W_s$ is invariant under the flow.
\subsection{The Poincar\'e trick}
We can also derive the results for flows from the results for maps
using the Poincar\'e trick. To do so, we split the angle variables
as $\th = (\varphi, \th_d) \in \nt^{d-1}\times \nt$. We also write
the frequency vector as $\omega= \omega_d (\alpha,1)$.
Hence, the Poincar\'e map, with respect to the angle variable $\th_d$ is
\begin{equation}
\label{Poincare}
\begin{array}{l}
\bar x= f(x,\varphi) = F_{\frac{1}{\omega_d}}(x,(\varphi,0)) \ ,\\
\bar \varphi= \varphi + \alpha \ .
\end{array}
\end{equation}
This in a skew product in $\nr^n\times \nt^{d-1}$, over the rotation
$\alpha\in \nr^{d-1}$.
If $k(\varphi)$ is an invariant torus for \eqref{Poincare}, that is
$f(k(\varphi),\varphi)= k(\varphi+\alpha)$, then the torus $K(\th)$
defined by
\[
K(\varphi,\th_d)=
F_{\frac{\th_d}{\omega_d}}(k(\varphi-\th_d \alpha),
(\varphi-\th_d \alpha,0))
\]
is invariant under the whole system \eqref{continuous}. Notice that the
torus is well defined (the definition does not depend on the representative
of $\th_d\in \nt = \nr/\nz$).
If $w(\eta,\varphi)$ is a whisker of $k(\varphi)$ for \eqref{Poincare},
and the dynamics is given by $\lambda(\eta,\varphi)$, that is
$f(w(\eta,\varphi))= w(\lambda(\eta,\varphi),\varphi+\alpha)$, then the
parameterization
\[
\hat W(\eta,\varphi,s) = F_{\frac{s}{\omega_d}}(w(\eta,\th),(\varphi,0))
\]
covers a whisker of the torus $K$.
\subsection{A direct treatment of the differential equations
case}
In spite of the fact that we have shown that the rigorous results
for flows can be deduced from the results for maps,
it is instructive to sketch a direct treatment. We can obtain the invariance
equations either in integral form or differential form.
A torus $K(\th)$ is invariant under \eqref{continuous} if
\begin{equation}
\label{invariancet}
F_t(K(\th),\th)= K(\th+t\omega) \ ,
\end{equation}
for all $t\in \nr$. Notice that, for $\omega$ ergodic, it is enough
solving the equation for a given $t\neq 0$.
The differential form of \eqref{invariancet} is
\begin{equation}
\label{invariance4}
X(K(\th),\th) = \Dif K(\th) \omega \ .
\end{equation}
The integral form of \eqref{invariancet} is
\begin{equation}
\label{invariance3}
K(\th+t\omega)= K(\th) +
\int_{0}^t X(K(\th+s\omega),\th+s\omega) ds
\end{equation}
For a whisker of the torus, $W(\eta,\th)$, the invariance is given by
\begin{equation}
\label{invariance_whisker_continuous0}
F_t(W(\eta,\th),\th) = W(\Lambda_t(\eta,\th),\th+t\omega)\ ,
\end{equation}
where $\Lambda_t$ is a flow on the manifold, such that
$\Lambda_t(0,\theta)= 0$.
That is, there exists
a vector field
\[
\begin{array}{l}
\dot \eta= A(\eta,\th) \ ,\\
\dot \th= \omega \ ,
\end{array}
\]
such that
\[
\begin{split}
&\frac{d}{dt} \Lambda_t(\eta, \th) = A( \Lambda_t(\eta, \th), \th+t\omega) , \\
& \Lambda_0(\eta,\th) = \eta \ .
\end{split}
\]
The infinitesimal version of \eqref{invariance_whisker_continuous0}
is the equation
\begin{equation}
\label{invariance_whisker_continuous1}
X(W(\eta,\th),\th) = \Dif_\eta W(\eta,\th) A(\eta,\th)
+ \Dif_\th W(\eta,\th) \omega \ ,
\end{equation}
where $W$ and $A$ are unknown functions, and $A(0,\th)= 0$.
The integral form is given by
\begin{equation}
\begin{split}
W(\Lambda_t(\eta,\th),\th+t\omega) =
& W(\eta,\th) + \int_{0}^t X(W(\Lambda_s(\eta,\th),\th+s\omega),
\th+s\omega) ds \ , \\
\Lambda_t(\eta,\th)=
& \eta + \int_{0}^t A(\Lambda_s(\eta,\th),\th+s\omega) ds \ .
\end{split}
\end{equation}
\subsubsection{The one dimensional case.} For the sake of simplicity,
we will analyze here the simplest case in which the
whisker is one dimensional (in the vertical variable), and trivial
as a bundle over the torus. Moreover, we will assume that
the dynamics on the manifold can be reduced to constant coefficients
$\dot \eta= \lambda \eta$. In this case, \eqref{invariance_whisker_continuous1}
reads
\begin{equation}
\label{continuous1}
X(W(\eta,\th),\th) = \Dif_\eta W(\eta,\th) \lambda \eta
+ \Dif_\th W(\eta,\th) \omega\ ,
\end{equation}
where the unknowns are $W$ and $\lambda$.
Similarly as was done in Section~\ref{one_dimensional}, we
write
\[
W(\eta,\th)= W^{\leq} (\eta,\th) + W^{>}(\eta, \th) \ ,
\]
where
\[
W^{\leq} (\eta,\th) = \sum_{i= 0}^L W^i(\th) \eta^i
\]
and the high order part of the
function $W$ satisfies
\[
\displaystyle \frac{\partial^i W^{>}}{\partial \eta^i}(0,\th)= 0
\ \mbox{for $i=0,\dots,L$.}
\]
We seek the coefficients $W^0,\dots,W^L$ of $W^{\leq}$ and the remainder
$W^{>}$ from the equation \eqref{continuous},
which leads to a hierarchy
\begin{equation}
\label{hierarchy1c}
\begin{array}{l}
X(W^0(\th),\th)= \Dif_\th W^0(\th) \omega \ ,
\mbox{which gives $W^0(\th)= K(\th)$}\\
\Dif_x X(K(\th),\th) W^1(\th) = \lambda W^1(\th) + \Dif_\th W^1(\th)\omega \\
\Dif_x X(K(\th),\th) W^2(\th) + P^2(\th)=
2\lambda W^2(\th) + \Dif_\th W^2(\th)\omega \\
\vdots \\
\Dif_x X(K(\th),\th) W^L(\th) + P^L(\th) =
L\lambda W^L(\th) + \Dif_\th W^L(\th)\omega ,
\end{array}
\end{equation}
where $P^i$ stands for a polynomial expression in $W^1,\dots,W^{i-1}$
for $i=2,\dots L$ whose coefficients are derivatives of $X$ of order
up to $i$ evaluated at $(W^0(\th),\th)$.
The high order part $W^{>}$ satisfies
\begin{equation}
\label{remainder1c}
\Dif_x X(K(\th),\th) W^{>}(\eta,\th) +
P^{>}(\eta,\th) =
\lambda \Dif_\eta W^{>}(\eta, \th) + \Dif_\th W^{>}(\eta,\th) \omega\ ,
\end{equation}
where $P^{>}$ contains terms which vanish to
order higher than $L$.
The hierarchy of equations \eqref{hierarchy1c} can be solved by
recursion in the degree of the polynomials
matched, provided that some non-resonance conditions are satisfied.
The analysis of these equations is harder than the corresponding analysis
in the discrete case, due to the appearance of derivatives that
``change'' the space of the functions in the left and right hand side.
In numerical applications, these equations can be solved
by using Fourier expansions, up to arbitrarily high degree $L$.
\section{Acknowledgments}
A main part of this work was carried
out while A. H. was enjoying a
Fulbright Scholarship at
Univ. of Texas at Austin.
A. H. has been supported by the MCyt/FEDER Grant
BFM2000--805 and the Catalan grant 2000SGR--27 and
the INTAS project 00-221. His work is also supported
by the MCyT/FEDER Grant BFM2003-07521-C02-01.
The work of R. L. has been partially supported by
N.~S.~F. grants. R. L. has enjoyed a Dean's Fellowship at
U.T. Austin Spring 03. Visits of R.L. to Barcelona
were supported by FBBV and ICREA.
\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\def\cprime{$'$}
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\end{document}