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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% SOME SMALL TRICKS%%%%%%%%%%
\def \breakline{\vskip 0em}
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\TITLE COHOMOLOGY EQUATIONS NEAR HYPERBOLIC POINTS AND 
GEOMETRIC VERSIONS OF STERNBERG LINEARIZATION THEOREM.
\ENDTITLE
\AUTHOR
A. Banyaga
\FROM
Math Dept.
Penn State Univ.
University Park, PA 16802
\AUTHOR
R. de la Llave
\FROM
Math Dept. 
Univ. of Texas 
Austin, TX 78712
\AUTHOR
C.E. Wayne
\FROM
Math Dept.
Penn State Univ.
University Park, PA 16802
\ENDTITLE
\ABSTRACT 

We prove that if two germs of diffeomorphisms 
preserving a volume, symplectic or contact structure are 
tangent to a high enough order and the linearization is 
hyperbolic, it is possible to find a smooth change of 
variables preserving the same structure that sends one into  
the other.  This result is a geometric version of Sternberg's
linearization theorem which we recover as a particular case. 

An analogous result is also proved for flows. 

\ENDABSTRACT

\def\cite#1{{\rm [#1]}}
\def\bref#1{{\rm [~\enspace~]}}		% blank ref cite
\def\circX{\mathop{\buildrel \circ\over X}\nolimits} 
\def\Lip{\mathop{\rm Lip}\nolimits}
\def\spec{\mathop{\rm spec}\nolimits}
\def\Spec{\mathop{\rm Spec}\nolimits}

\SECTION Introduction and statement of results.

The celebrated Sternberg linearization theorem states that, given a local 
diffeomorphism with a fixed point, if the eigenvalues of its linearization 
satisfy certain non-resonance conditions, it is possible to find a 
differentiable change of variables making it linear. That is, given $f$, 
$f(0) =0$, it is possible to find $h$ in such a way that 
$h^{-1}\circ f\circ h (x) = Df(0)x$. 

In many applications, $f$ preserves a geometric structure --- symplectic, 
volume or  contact --- and it is natural to require that $h$ also does. 

In the case that $f$ preserves a geometric structure, it necessarily 
violates several of the non-resonance conditions and, there are easy 
examples where it is impossible to reduce to the linear part. Nevertheless, 
very frequently, one can hope to reduce to a much simpler form --- usually 
called  a ``normal form.'' 

Typically, it is not difficult to find polynomial germs $h_k$ of 
degree~$k$ in such a way that 
$$h_k^{-1} \circ f\circ h_k (x) = N_k (x) + o(\| x\|^k)$$ 
with $N_k$ a much simpler diffeomorphism. (These eliminations usually 
entail only power matchings.) 

The main goal of this paper is to prove a theorem stating that if
the formal eliminations can be carried to a large enough 
order, and that $Df(0)$ is hyperbolic, there is a diffeomorphism $h$, 
which reduces $f$ to exactly the normal form $N_n$. In  the case that $f$ 
preserves a symplectic or volume form or a contact  structure so does $h$. 

We also discuss an analogue for flows. 

Such theorems were sketched in \cite{St3}. A proof by other methods 
appeared in \cite{Ch}. 

The method of proof that we use is based on the deformation method of 
singularity theory \cite{Ma}.
This method is ideally suited to discussing conjugacy problems 
in which a geometric structure is preserved. In principle, the 
preservation of a geometric structure is a non-linear non-local problem,
but with the use of deformations, the preservation of the geometric 
structure is implemented by considering equations in an appropriate
linear space.


In a first section, we describe the basic formalism of deformations for 
general diffeomorphisms as well as for diffeomorphisms preserving symplectic 
or volume forms or contact structures.
The basic idea of the deformation method is to 
embed the problem into a family of problems, that includes also a 
trivially solvable one. Then,
we study
the derivatives of the quantities involved.
The advantage is that the equations involved are
always linear cohomology equations. (If we think of derivatives as
infinitesimal quantities, it is clear that the only equations we can
form among infinitesimal quantities are linear.) The method is also
well suited for geometric problems since the non-linear and non-local
constraints which are imposed by the preservation of the geometric
structure also become linear constraints, which can be implemented by
considering the cohomology equations among linear spaces of local objects.

In a second section, we provide estimates for the cohomology equations 
of the previous sections and, establish the main theorems of this paper.


We point out that from the point of view of group theory, the problem 
just discussed is the problem of classifying the conjugacy classes of 
the group  of germs of diffeomorphisms 
preserving a geometric structure, or classifying the orbits under the natural action. 

The reduction of the problem to the study of cohomology equations in the 
Lie algebra is quite standard in finite dimensional Lie groups, so in a 
certain sense, the method considered here is the extension to  an infinite
dimensional situation of methods that had been successful in the finite 
dimensional situation.  This point of view is emphasized in [St2]. 

For problems of conjugacy, the deformation method was introduced in the
context of singularity theory. Some early refences are
\cite{Ma} and the notes of the lectures
of R. Thom by H. Levine \cite{Le}.

For the symplectic case, a discussion 
of the formal normal forms using this method can be found in \cite{Mo}. 
 For problems like ours, the main technical tools of the method  are $C^k$ estimates for solutions 
of cohomology equations. In a global context, these estimates were 
introduced in \cite{LMM} and, in a context very similar to ours in 
\cite{BLW}. The paper [Il] uses the deformation method to prove the 
Sternberg theorem for general diffeomorphisms without regard to 
geometric structures.

We  point out that the deformation method
can also be used to study the formal 
eliminations and the classification 
of normal forms. For example,  the papers \cite{Ta} and \cite{Ro}
used the deformation methods to classify germs of 
vector fields, forms, and diffeomorphisms to a finite order and
the paper \cite{Mo} discusses normal forms of symplectic 
diffeomorphisms using the deformation method.
The advantages of the method appear not only in theoretical treatments, but it
also allows effective numerical implementations \cite{DL}. 

In relation with the problem of convergence of
symplectic  normal forms 
of hyperbolic
maps we also point out that
for analytic, two dimensional,
symplectic mappings a proof of the existence of analytic changes of
variables reducing to a normal form has also been presented in \cite{Mo3}.
The method of proof is, nevertheless quite different and many of the 
techniques we use such as cut-off functions etc. clearly do
not apply for analytic regularities.



\REMARK
We will consider the volume preserving problem only on
manifolds of dimension $3$ or bigger.
The case when the dimension is $1$ is completely
trivial and, for the case in which the dimension is 
$2$ 
volume preserving is the same as preserving 
a symplectic structure. The results for 
symplectic structures are sharper than those we obtain for
general volume preserving case particularized for dimension $2$.

More precisely, we will prove: 

\CLAIM {Theorem}(maindiff) 
Let $f,N$ be $C^r$ diffeomorphisms of $\real^n$, $f(0)=N(0)=0$, and let 
$A$ and $B$ be as defined below. 
\vskip1pt
Assume 
\smallskip
\item{i)} $D^i f(0) = D^i N(0) $ \quad $i=0,\ldots,k<r-1$,
\item{ii)} $\Spec Df(0) \subset \{ z\in \complex \mid \lambda_-^{-1} \le 
|z| \le \lambda_+\} \cup \{z\in \complex \mid \mu_-^{-1}\le |z| \le \mu_+\}$
\item{} for some $0<\lambda_-^{-1} < \lambda_+ < 1 <\mu_-^{-1} < \mu_+.$
\vskip1pt
Then, provided that $1 \le \ell < kA -B$, for some
integer $\ell$, we can find a $C^\ell$ 
diffeomorphism $h$ such that $h^{-1}\circ f\circ h=N$ on a neighborhood 
of the origin, $h(0) =0$, $Dh (0)=Id$. 
\vskip1pt
If both $f,N$ preserve symplectic, contact or volume structures 
$h$ can be chosen to preserve the same structure. 

\REMARK
In this paper we will show in detail that, in all the cases we can take 
$$
A= {| \ln \lambda_+| \over \ln \mu_+}\ {|\ln \mu_- |\over ( \ln\lambda_- +
| \ln \mu_-|)}.
$$ 
Morever,
we can take :
$$
B= B_{\rm no-structure} \equiv 
{ (\ln \mu_+)^2 + |\ln \mu_-|(\ln \lambda_- - |\ln \lambda_+|) 
\over
(\ln \mu_+) ( \ln \lambda_- + | \ln \mu_-|)}
$$ 
in the case that there is no  geometric structure to be preserved.

$$
 B = B_{\rm symplectic} \equiv 1 - 2{|\ln\lambda_+| \over (\ln \mu_+)} 
{|\ln \mu_{-} |\over ( \ln \lambda_- + | \ln \mu_-|)}
$$
when we require preservation of a symplectic structure ;
$$
 B = B_{\rm volume} \equiv 1 + (n-2){{\ln \lambda_- - |\ln \mu_- |}\over{
(\ln \mu_+)(\ln \lambda_- + |\ln \mu_-|) }} +(n-2) {{\ln \mu_+}\over{
(\ln \lambda_- + |\ln \mu_-|) }}
$$
for volume preserving case and 
$$
B= B_{\rm contact}\equiv 1 + B_{\rm no-structure}$$
in the contact case.

Analogously, we have for vector fields:

\CLAIM {Theorem}(mainvec) 
Let $X,Y$ be $C^r$ vector fields of $\real^n$, $X(0)= Y(0)=0$. 
\vskip1pt
Assume 
\smallskip
\item{i)} $D^i X(0) = D^iY(0)$\quad $i=0,\ldots,k<r-1$ 
\item{ii)} $\Spec DX(0) \subset \{ z\in\complex\mid -\lambda_- \le 
Re (z) \le \lambda_+\} 
\cup \{z\in\complex \mid -\mu_- \le Re (z) \le \mu_+\}$
\item{} for some $-\lambda_- < \lambda_+ < 0 < -\mu_- < \mu_+$
\vskip1pt
Then, provided that $\ell = kA - B\ge1$ we can find a $C^\ell$ 
diffeomorphism $h$ such that $h_*X=Y$ on a neighborhood of the origin, 
$h(0)=0$, $Dh(0) = Id$.
\vskip 0pt
In case that both $X,Y$ preserve a volume, symplectic or contact structure 
$h$ can be chosen to preserve the same structure. 
\REMARK
In all the above cases, we can take the values of $A$ and $B$
to be equal to their values in the diffeomorphism case, except
that we omit all logarithms.  For example,
$$
A= { \lambda_ + \over  \mu_+}\ { |\mu_-|\over (\lambda_- + |\mu_-|)}~~,
$$ 

$$
B= B_{\rm no-structure} \equiv  {{ (\mu_+)^2 + |\mu_- | (\lambda_- -
|\lambda_+|)}\over{ (\mu_+) (\lambda_- + |\mu_-|) }} ~~,
$$ 
and so on.

\REMARK 
The values of $A$ and, especially, of $B$ that we have obtained 
are not optimal even using the method 
of proof in this paper.
At the end of the proof we indicate in remarks 
how the results could have been
improved by using more complicated
spaces to measure the regularity.
For example, one can obtain better results if 
one keeps track separately of the derivatives with respect to parameters
and with respect to the variables in the manifold. A further improvement 
can be obtained if one keeps track of the derivatives along stable and 
unstable manifolds in the way that it was done in [LMM].
We also point out that in the contact and 
volume cases, it is possible to obtain more precise formulas that 
involve the
eigenvalues  of the derivative at $0$
and not just bounds on their size.
Since these improvements require a lot of technical complication and are
not central  to the main goal of this paper, we have just relegated
them to remarks along the side of the main proof.

\REMARK 
Notice that the expressions for $B$ in \clm(maindiff) and \clm(mainvec) 
change if  we substitute for $f$ and $N$, $f^{-1}$, $N^{-1}$ respectively. 
The $h$ obtained in one case would work in the other. Given a particular 
pair $f,N$ we could choose  whichever of the expressions gives  the more
favorable values of $A$ and $B$.

\REMARK 
The method of proof we present carries over to infinite dimensions. 
The only properties of $\real^n$ which are used is that it is a Banach 
space and that it has the ``approximation property'' --- that is, that 
there exist $C^\infty$ functions with bounded support and taking 
the value 1 in an open ball. 

Even if the generalization of \clm(mainvec) and \clm(maindiff) to Banach 
spaces satisfying the approximation property is quite straightforward, 
and natural when there are no geometric structures present, we point 
out that generalization of geometric structures to infinite dimensions 
is somewhat delicate. There is no natural definition of volume forms and, 
even if it is possible to define symplectic and contact forms by 
extending straightforwardly the finite dimensional definitions, these 
extensions are not natural in many applications (see e.g., [CM], [M] 
for a discussion of natural extensions of symplectic forms to infinite 
dimensions). 

\SECTION The deformation method 

The basic idea of the deformation method is to consider a family of problems 
that interpolates between a trivial one and the problem we want to solve. 

Assuming that the problem is solved for all values
of the deformation parameter, we can derive equations 
between the infinitesimal deformations of the problem and the solution. 

 From a heuristic point of view, the deformation
method can be considered as an ``infinitesimal'' 
version of induction. Assuming that the problem is solved for all values 
of the parameter between $[0, \epsilon]$ we study what it would take  to 
solve it for $[0,\epsilon +{\rm infinitesimal}]$.
(This amounts to first order 
perturbation theory.)

The advantage of this procedure is that the equations between the 
infinitesimal deformations are  linear. 
More importantly from our 
point of view, they are geometrically natural and it is possible to perform 
geometrically natural constructions on them  that force the preservation 
of geometric structures. The preservation of geometric structures is,
in principle, a non-linear and non-local problem. Nevertheless, by
considering the infinitesimal equation, we can formulate it as
a problem about quantities belonging to appropriate linear spaces.

 We will be concerned
with  the so-called {\sl hyperbolic \/} diffeomorphisms
and flows. The following definition is quite standard.

\CLAIM {Definition}(hyperbolic)
We say that a diffeomorphism $f:\real^n\to\real^n$ with $f(0)=0$ 
is hyperbolic at $0$ if the spectrum of $Df(0)$ does not intersect 
the unit circle.
We say that a vector field $L(x)$ in $\real^n$ with $L(0)=0$, is 
hyperbolic if the spectrum of $DL(0)$ does not intersect the imaginary axis. 

As is well known, (see {\it e.g.} [Ni]) $f$ is a hyperbolic diffeomorphism 
if and only if one can write $\real^n = E^s\oplus E^u$, and, denoting 
$A= Df(0)$, we have $AE^s=E^s$, $AE^u = E^u$ and, for some norm 
$\| A|_{E^s}\| <1$; $\| (A|_{E^u})^{-1}\| <1$. 

Similarly, a flow is hyperbolic if, denoting $A= DL(0)$ we have 
$\real^n = E^s\oplus E^u$, $AE^s = E^s$, $AE^u = E^u$ and 
$\|\exp (A|_{E^s})\|<1$, $\| \exp (- A|_{E^u})\|<1$,  for some 
norm.  

This characterization makes it clear that a vector field is hyperbolic 
at zero if and only if its time $t$ map defines a hyperbolic map. 



In  this paper, we will consider mainly  two problems: 

\CLAIM {Problem}(diffeos)
Given $f,N$, germs of $C^r$ diffeomorphisms in $(\real^n,0)$, $f$ 
-- and therefore $N$-
hyperbolic diffeomorphisms at $0$,
 $\| f(x) - N(x)\| = o(\|x\|^s)$, $s \ge 1$, determine $g$,
the germ of a
diffeomorphism in $(\real^n,0)$  with $Dg(0) = Id$, in such a way that 
$g^{-1}\circ f\circ g=N$.

\CLAIM {Problem}(vectors)
Given $\XX,\YY$ ,
germs of 
$C^r$ 
vector
fields at $0\in \real^n$, $\XX(0)= \YY(0)=0$, and  
$\|  \XX(x) - \YY(x)\| \le o(\|x\|^s)$, with  $\XX$ --
and hence $\YY$ -- hyperbolic flows at $0$,
determine $g$ a germ of a diffeomorphism $Dg(0)=Id$ such that 
$$g_* \XX = \YY\ .$$

If we have a differentiable family $f_\epsilon$ with $f_0=N$, $f_1 =f$, 
it is natural to 
try to find a family $g_\epsilon$ with $g_0 = Id$ satisfying 
$$g_\epsilon^{-1} \circ f_\epsilon \circ g_\epsilon = f_0\ .
\EQ(conjugacydiff)$$

Analogously, for a family of vector fields, ${\cal X}_0 = {\cal X}$, 
${\cal X}_1 = {\cal Y}$ (for example, ${\cal X}_\epsilon = \epsilon  
{\cal Y} + (1-\epsilon){\cal X}$) it is natural to try to find $g_\epsilon$ 
with $g_0 = Id$ in such a way that 
$${g_\epsilon}_* {\cal X}_\epsilon = {\cal X}_0\ .  
\EQ(conjugacyvec)$$ 

Since $f_\epsilon$ is a smooth family of diffeomorphisms, we can find a 
family of vector fields ${\cal F}_\epsilon$ defined by 
$${d\over d\epsilon} f_\epsilon = {\cal F}_\epsilon \circ f_\epsilon\ .
\EQ(deformation)$$

By the uniqueness theorem of differential equations, if $f_0,{\cal F}_\epsilon$ 
are $C^1$, there is one and only one family $f_\epsilon$ with the given 
 $f_0$ and ${\cal F}_\epsilon$.  We also note that,
if $D^i f_\epsilon(0) = D^i f_0(0),\ i=1,\cdots,k$, then, 
$D^i {\cal F}_\epsilon(0) = 0~,\ i = 1,\cdots,k$.

If we assume that $g_\epsilon$ can also be written in  the form 
\equ(deformation), taking derivatives with respect to $\epsilon$ in 
\equ(conjugacydiff), we obtain 
$${d\over d\epsilon} (g_\epsilon^{-1}\circ f_\epsilon \circ g_\epsilon) 
= (g_\epsilon^{-1})_* ({\cal F}_\epsilon - {\cal G}_\epsilon + 
{f_\epsilon}_* {\cal G}_\epsilon)
\circ(g_\epsilon^{-1} \circ f_\epsilon^{-1}\circ g_\epsilon)\ .$$

So that, if, given $\calF_\epsilon$, we can find a $C^1$ family of vector fields ${\cal G}_\epsilon$ 
satisfying 
$$
{\cal F}_\epsilon - {\cal G}_\epsilon + (f_\epsilon)_* {\cal G}_\epsilon =0\ , 
\EQ(cohomologydiff)
$$ 
the family of diffeomorphisms $g_\epsilon$ determined by 
$g_0 = Id$; \ ${d\over d\epsilon} g_\epsilon = {\cal G}_\epsilon \circ g_\epsilon$ --- 
whose existence is guaranteed by the fundamental existence theorem of 
O.D.E.'s --- will satisfy: 
$$
{d\over d\epsilon} g_\epsilon^{-1} \circ f_\epsilon \circ g_\epsilon = 0
\quad ;\quad 
g_0^{-1} \circ f_0 \circ g_0 = f_0~~,
$$ 
hence, $g_\epsilon^{-1} \circ f_\epsilon \circ g_\epsilon = f_0$. 

Analogously, denoting 
$\dot{\cal X}_\epsilon = {d\over d\epsilon} {\cal X}_\epsilon$,
we have: 
$${d\over d\epsilon} (g_\epsilon)_* {\cal X}_\epsilon = (g_\epsilon)_* 
(L_{{\cal G}_\epsilon} {\cal X}_\epsilon + \dot{\cal X}_\epsilon).$$ 
Where, we recall, we have 
$L_{{\cal G}_\epsilon} {\cal X}_\epsilon = [{\cal G}_\epsilon , 
{\cal X}_\epsilon] = -  L_{{\cal X}_\epsilon} {\cal G}_\epsilon\ .$ 
Therefore, finding $C^1$ solutions of 
$$-L_{{\cal X}_\epsilon} {\cal G}_\epsilon + \dot{\cal X}_\epsilon =0,
\EQ(cohomologyvec)$$
solves the original problem. 

Notice that both \equ(cohomologydiff) and \equ(cohomologyvec) 
are linear in the unknowns --- the original problems \equ(conjugacydiff) and 
\equ(conjugacyvec) are not --- and are geometrically natural. 
The fact that the equations are linear, is to be expected since the vector fields
can be considered as infinitesimal diffeomorphisms and 
the equations between infinitesimal quantities are 
typically linear.

We emphasize that both \equ(cohomologydiff) and \equ(cohomologyvec)
were shown to be  equivalent to the original problems 
independently of the fact that there were geometric structures present.
Our next task is to 
introduce geometric structures and to show that the 
deformation method provides 
a natural calculus to study them.

Notice that all the calculations we have done  so far
only require that one can take
derivatives and that the solutions of O.D.E's are unique.
If we consider higher regularities we find it convenient to define precisely.

\CLAIM {Definition}(families)
We say that $f_\epsilon$ is a $C^r$ family if $f_\epsilon (x)$ is $C^r$ 
jointly in $\epsilon$ and $x$ and, moreover 
$$\sup_{x,\epsilon} \sup_{\ell + |\alpha| \le r} 
\left| \left( {\partial \over\partial\epsilon}\right)^\ell {\partial^\alpha 
\over \partial x^\alpha} f_\epsilon (x)\right| 
\equiv \| f_\epsilon\|_{C^r} < \infty\ .$$ 

\REMARK 
Notice that, with this definition when the domain is $\real^n$, we can have 
infinitely differentiable functions  such as $x^2$ which are not $C^1$ in the sense
above since we require uniform control of the derivatives 
over all of the space.
Also, it could have been natural to require a different number of
derivatives with respect to the parameters and with respect to the variables.

\REMARK
We  recall that it is a result in harmonic analysis (see [Kr]) that if
$r+1$ derivatives with respect to the parameter exist and are continuous and
$r+1$ derivatives with respect to the variables exist and are continuous, 
then the function is jointly $C^r$. Unfortunately, (see again [Kr]) it is not
true that the function is jointly $C^{r+1}$. These complications in the 
regularity are due to the fact that the $C^r$ spaces $r \in \natural$ are not 
well behaved under many of the operations that appear naturally in harmonic
analysis. Nevertheless, they are quite well behaved under composition and,
for the purposes of this paper, this is quite important.

\REMARK
We also recall that the theorem of existence and regularity on
initial conditions for O.D.E's shows that if
$\calF_\epsilon(x)$ is $C^r$ in $\epsilon$ and $C^\ell$ in $x$,
then the solutions $f_\epsilon$ of
${d \over d\epsilon} f_\epsilon(x) = \calF_\epsilon( f_\epsilon(x)); \quad
f_0(x) = x$
exist and are unique when $\ell \ge 1$, $r \ge 0$. Moreover
they are $C^{r+1}$ in $\epsilon$, $C^\ell$ in $x$.
(See [Ha], Chapter 5.)

\SECTION Volume and symplectic geometry

A $k$-form $\gamma$ on an $n$-dimensional manifold $M$ is said to be 
{\it nondegenerate\/} if the mapping $X\to i(X)\gamma$ sending a 
vector field $X$ to the $(k-1)$-form $i(X)\gamma$ is an isomorphism: 
this means that for all $x\in M$, $X_{(x)} \to i(X_{(x)})[\gamma_{(x)}]$ 
is an isomorphism of the tangent space $T_xM$ with the vector space 
$\Lambda^{k-1} (T_x^* M)$ of alternating $(k-1)$-linear forms on $T_xM$. 
This can happen only if $ n = \dim (T_xM) =\dim (\Lambda^{k-1}(T_x^*M)) = 
{n\choose k-1}$, i.e., if $k=2$ or $n$.

Let $\gamma$ be a nondegenerate $k$-form; then any $(k-1)$-form $\omega$ 
defines a unique vector field $X$ such that $i(X)\gamma=\omega$. 

A $(k-2)$-form $F$ is called a {\it hamiltonian\/} of the unique vector 
field $X_F$ such that 
$$i(X_F)\gamma = dF\ .$$ 
Clearly, $X_F = X_{F'}$ if and only if $F-F'$ is a closed form.
The vector 
field $X_F$ is called a {\it globally hamiltonian vector field with 
hamiltonian\/} $F$.

The vector field $X^\omega$ defined by a closed $(k-1)$-form 
$\omega$  according to $i(X^\omega)\gamma = \omega $ is said to be a
 {\it locally hamiltonian\/} vector field. 
Indeed if $U$ is any contractible open subset of $M$, the Poincar\'e lemma 
says that $\omega|_U = dF$, for some $(k-2)$-form, $F$, on $U$.
 Hence on $U$,
$X^\omega =X_F$.

Recall that on an $n$-dimensional manifold only 2-forms or $n$-forms may 
be nondegenerate. A nondegenerate $n$-form is called a volume form. This 
form is automatically closed. For 2-forms, 
if some critical calculations are to go through, 
we must require this 
as an independent hypothesis and get 
the notion of symplectic form: i.e., a nondegenerate closed 2-form. From 
now on, $\gamma$ stands for a nondegenerate closed $k$-form (which is a 
symplectic form, or a volume form); to emphasize the distinction, we 
reserve the letter $\omega$ for symplectic forms and $\mu$ for volume forms. 

Note that even if a hamiltonian determines a unique vector field, the
hamiltonian of a vector field is only determined up to forms 
with zero exterior derivative. When $\gamma$ is a 2-form,
the hamiltonians are 0-forms, that is functions and the 
the only functions with zero differential are constants. Nevertheless,
if $\gamma$ is a $k$ form with
$k \ge 3$, there are much more complicated $(k-2)$-forms with zero 
exterior derivative.


Let $\gamma$ be a $C^\infty$ nondegenerate $k$-form and consider the 
equation:
$$i(X)\gamma = dF$$ 
If $X$ is $C^r$, so is $i(X)\gamma = dF$, hence, using the 
same construction  of integrating along lines, used in the 
standard proof of
Poincar\'e lemma (see [Sp] among others),
we obtain that  the hamiltonian $F$ is 
$C^r$ and the $C^r$-norm of $F$ can be estimated by a constant times the 
$C^r$-norm of $X$. In the symplectic case, it is possible 
to do better. If $F$ is a function, the fact that 
$dF$ is $C^r$ allows us to conclude that $F$ is
$C^{r+1}$ and that the $C^{r+1}$ norm of $F$
is bounded by the $C^r$ norm of $dF$.
Conversely if the hamiltonian $F$ is $C^r$, the vector 
field is $C^{r-1}$ and the $C^{r-1}$ norm of $X$ is estimated by the 
$C^r$-norm of the hamiltonian times a constant (depending on $\gamma$). 


Using the standard formulas relating the interior product $i(\cdot)$, the 
exterior derivative $d$, and the Lie derivative,
$$\eqalign{
i\bigl( [X,Y]\bigr)\alpha & = L_X \bigl( i(Y)\alpha\bigr) 
- i(Y) (L_X\alpha)\cr
L_X\alpha & = i(X)\,d\alpha + d\, i(X)\alpha\cr}$$ 
for all $k$-forms $\alpha$, we get the well known: 

\CLAIM {Lemma}(commutator) 
Let $\gamma$ be a nondegenerate closed $k$-form. 
\smallskip
\item{(i)} If $X,Y$ are locally hamiltonian vector fields, then $[X,Y]$ is 
a globally hamiltonian vector field with hamiltonian  $i(X)(i(Y)\gamma)$. 
\item{(ii)} If $X$ has $G$ as hamiltonian and $Y$ is locally hamiltonian, 
then $L_YG$ is a hamiltonian for $[Y,X]$. 
\item{(iii)} If $Y$ has $F$ as a hamiltonian, then $-L_XF$ is a 
hamiltonian of $[Y,X]$.
\smallskip

\REMARK 
In the Hamiltonian case,
$i(X)(i(Y)\omega) =  \omega(Y,X)$ 

\PROOF
Observe first that a locally hamiltonian vector field $X$ satisfies: 
$$L_X\gamma =0\ .$$ 
Indeed 
$$L_X\gamma = d\, i(X)\gamma + i(X)\,d\gamma =0$$ 
since $d\gamma =0$ and $i(X)\gamma $ is closed. 

To prove (i), compute: 
$$\eqalign{i\bigl([X,Y]\bigr)\gamma & = L_X \bigl( i(Y)\gamma\bigr) 
- i(Y)(L_X\gamma)\cr 
&= L_X i(Y)\gamma\cr
&= di\bigl( i_X i_Y \gamma \bigr) + i_X d\bigl(i(Y)\gamma\bigr)\cr
&= d\bigl[ i(X)i(Y)\gamma\bigr]\cr}\ .$$ 
This proves (i). 

For (ii) we know that $i(X)\gamma = dG$, so 
$$\eqalign{d(L_Y G) & = L_Y(dG) + L_Y\bigl[ i(X)\gamma\bigr]\cr
&= i\bigl({[Y,X]}\bigr) \gamma + i\bigl(X \bigr)(L_Y\gamma\bigr)\cr
& = i\bigl([Y,X]\bigr)\gamma\cr}\ .$$ 

For (iii), $i(Y)\gamma = dF$, so:
$$d(L_XF) = L_X\,dF = L_Xi\bigl(Y\bigr)\gamma 
= i\bigl([X,Y]\bigr)\gamma + i\bigl(Y\bigr)\bigl(L_X\gamma\bigr)
 = i\bigl([X,Y]\bigr)\gamma\ .$$

A differentiable family  $f_\epsilon$ of diffeomorphisms preserves $\gamma$ 
if and only if the associated family $\calF_\epsilon$ of vector fields is 
locally hamiltonian: indeed, if $f_\epsilon^* \gamma =\gamma$, taking 
derivatives with respect to $\epsilon$, we get: 
$$0 = {d\over d\epsilon} (f_\epsilon^* \gamma) = f_\epsilon^* (L_{\calF_
\epsilon} \gamma)$$ 
and since $d\gamma =0$, we conclude: 
$$L_{\calF_\epsilon} \gamma = d\, i(\calF_\epsilon)\gamma =0\ .$$ 

In this paper we are interested in what happens near a critical  point. 
Hence, in a contractible neighborhood of the point, $\calF_
\epsilon$ is globally hamiltonian and we denote by $F_\epsilon$ its 
hamiltonian: 
$$i(\calF_\epsilon)\gamma =  dF_\epsilon$$ 
The hamiltonian $F_\epsilon$ determines
$\calF_\epsilon$ which in turn determines 
$f_\epsilon$ given $f_0$.

If $f_\epsilon$ is a family of $\gamma$-preserving local diffeomorphisms 
with $f_0 =N$, $f_1=f$ it is natural to try to find $\gamma$-preserving 
$g_\epsilon$ such that $g_\epsilon^{-1}\circ f_\epsilon\circ g_\epsilon 
= f_0$, by first determining its hamiltonian $G_\epsilon$.

We recall the following property of the interior product whose proof is 
a straightforward computation:

\CLAIM Proposition (computation)
If $f$ is a diffeomorphism, $\gamma$ a $p$-form, and $X$ 
a vector field then,
$$
i\bigl( f_* X\bigr) \gamma =
(f^{-1})^*\left[ i\bigl(X\bigr)\bigl( f^*\gamma\bigr)\right]\ . 
$$

If we apply \clm(computation) to simplify the result of taking the
interior product of all the terms in \equ(cohomologydiff) we obtain:
$$F_\epsilon -G_\epsilon + (f_\epsilon^{-1})_* G_\epsilon =0\ .
\EQ(cohomologyhamil)$$
This equation is equivalent to \equ(cohomologydiff) and hence to
the solution of the main problem.
In the symplectic case, $G_\epsilon$ 
is a function and \equ(cohomologyhamil) can be written as: 
$$F_\epsilon - G_\epsilon + G_\epsilon \circ f_\epsilon^{-1}=0\ .
\EQ(cohomologysymp)$$ 

An analogous calculation shows that if
one is given a family  $\calX_\epsilon$ of $\gamma$-preserving 
vector fields 
with hamiltonian $X_\epsilon$, it is 
equivalent to find a family $g_\epsilon$ of 
$\gamma$-preserving maps solving \equ(conjugacyvec) and to
solve:
$$d\left(-L_{\calX_\epsilon} G_\epsilon + \dot X_\epsilon\right) =0
\EQ(cohomologySV1)$$
(Where by $\dot X_\epsilon$
we mean ${\partial \over \partial \epsilon } X_\epsilon$.)
and then determine $g_\epsilon$ by 
$$
{dg_\epsilon \over d\epsilon} = G_\epsilon \circ g_\epsilon \ ;\qquad 
g_0 = \hbox{ identity.}$$

Clearly, a $G_\epsilon$  solving
$$-L_{\calX_\epsilon} G_\epsilon + \dot X_\epsilon =0
\EQ(cohomologySV)$$
is a solution of \equ(cohomologySV1) and, therefore, we will concentrate
our efforts in solving \equ(cohomologySV).
This does not incurr any loss of generality for the original problem
because hamiltonians are really defined up to exact forms.
If $-L_{\calX_\epsilon} G_\epsilon + \dot X_\epsilon = R_\epsilon$ with 
$d R_\epsilon = 0$, then $X_\epsilon + \int_0^\epsilon R_s ds$ generates the 
same vector field that $X_\epsilon$.

The previous discusion shows that we can reduce the 
problem of conjugacy of families, all of whose
elements preserve a geometric structure to the study of 
cohomology equations.
To study  \clm(diffeos) and \clm(vectors),
where we are not given families but just 
two diffeomorphisms or vector fields, 
we need only to show that  given such problems, we can 
find families whose initial and final points 
are the given diffeomorphisms or vector 
fields and  such that all the intermediate vector 
diffeomorphisms and vector fields in the family 
preserve the same geometric structure.

For 
vector fields
we observe that the space of locally or globally hamiltonian vector fields is a 
subspace of the vector space of all vector fields.
Given $\calX$ and $\calY$ locally or globally hamiltonian 
vector fields, then: 
$$\calX_\epsilon = \epsilon \calY + (1-\epsilon) \calX$$ 
is a smooth path of locally or globally hamiltonian vector fields such that 
$\calX_0=\calX$ and $\calX_1 = \calY$. 

For
diffeomorphisms,
we need to interpolate $f,N:\real^n\to \real^n$ with the properties: 
$$f(0) = N(0)=0\qquad (D^if)(0)=(D^iN)(0)\ ,\qquad i=0,\ldots,k<r\ .$$
In addition we may require that the interpolating maps preserve the 
form $\gamma$ when $\gamma$ is either the standard volume form 
$dx_1\wedge \cdots\wedge dx_n$ or the standard symplectic form 
$dx_1\wedge dx_2+\cdots +dx_{2p-1}\wedge dx_{2p}$ $(n=2p)$. 

Simply consider $\hat f = f\circ N^{-1}$ and $\hat f_\epsilon :\real^n \to 
\real^n$:
$$\hat f_\epsilon (x) = \cases{
\displaystyle {\epsilon^{-1}\hat f(\epsilon x)} &if $\epsilon \ne 0$\ .\cr
\noalign{\vskip6pt}
x&if $\epsilon =0$\ .\cr}
\EQ(interpolationeasy)$$
It is easy to see that $\hat f_\epsilon$ is a smooth family of 
diffeomorphisms interpolating between  the identity and $\hat f$
[Mi] 
and preserving the volume or symplectic form $\gamma$. 
The required family $f_\epsilon$ is defined by:
$$f_\epsilon = \hat f_\epsilon \circ N$$ 
Clearly 
$$f_1 = f\quad ;\quad  f_0=N; \qquad f_\epsilon (0) = N(0) = 0$$ 
and 
$$(D^i f_\epsilon) (0) = (D^iN) (0)\ ,\qquad i=0,1,\ldots,k<r\ .$$ 
Moreover if $f$ and $N$ were $\gamma$-preserving, so are $f_\epsilon$ 
for all $\epsilon$. 
If $f$ and $N$ are $C^r$, $r \ge 1$, $f_\epsilon$ is a $C^{r-1}$ family.


\SECTION Contact geometry

A contact form on a $(2n+1)$-dimensional manifold is a 1-form $\theta$ 
such that $\theta \wedge (d\theta)^n$ is a volume form. The 
codimension~1 distribution $D_\theta$ whose sections are vector fields $X$ with 
$i(X) \theta =0$, is the contact structure defined by $\theta$. A 
diffeomorphism $f$ preserves the contact structure $D_\theta$ if and only 
if $f^*\theta = \lambda\theta$, for some nowhere zero function $\lambda$. 

If we have a family of diffeomorphisms $f_\epsilon$ preserving the contact 
structure $D_\theta$, then we have: $f_\epsilon^* \theta=\lambda_\epsilon 
\theta$ and taking derivatives with respect to $\epsilon$: 
$$L_{\calF_\epsilon}\theta =
(f_\epsilon^{-1})^* \left[ {d\lambda_\epsilon \over 
d\epsilon} \cdot \theta\right]\ .$$ 
Since $(f_\epsilon^{-1})^*\theta = 
(\lambda_\epsilon\circ f_\epsilon^{-1})^{-1}\theta$, we have 
$$
L_{\calF_\epsilon}\theta = \mu_\epsilon \theta
\EQ(contactvector)
$$ 
with 
$$\mu_\epsilon = \left( {1\over\lambda_\epsilon} \cdot {d\lambda_\epsilon 
\over d\epsilon}\right) \circ f_\epsilon^{-1}\ .$$ 
Since the factor $\mu_\epsilon$ 
and $\lambda_\epsilon$ obviously depend on the family 
$f_\epsilon$, when there is risk of confusion, we will denote them by 
$\mu_\epsilon^{f_\epsilon},\  \lambda_\epsilon^{f_\epsilon}$ if
there is any danger of confusion.

A vector field $\calF$ satisfying \equ(contactvector) is called a
contact vector field. Again, we will denote the dependence of the factor
$\mu_\epsilon$ on the vector field by $\mu_\epsilon^{\calF_\epsilon}$

We recall the following fundamental result in contact geometry.


\CLAIM {Lemma}(Lieberman)
Let $(M,\theta)$ be a contact manifold.
\vskip1pt
Given a function $F$ there is one and only one contact vector
field $\cal F$ satisfying
$$F= \theta ({\cal F})\ .$$


Such an $F$ is called the generating function of the vector
 field ${ {\cal F}}$.

\PROOF
On the distribution $D_\theta$, $d\theta$ is a symplectic structure. Hence 
it defines an isomorphism $\widetilde{d\theta}$ between sections of 
$D_\theta$ (called horizontal vector fields) and sections  of its dual 
$D_\theta^*$, which can be identified to 1-forms $\alpha$ vanishing on the 
orthogonal complement of $D_\theta$. These forms  are called semi-basic 
forms. The orthogonal complement of $D_\theta$ is spanned by the Reeb 
field $Z$: i.e., the unique vector field $Z$ such that $i(Z)\theta =1$ 
and $i(Z)\,d\theta =0$. 

Given $F$ a smooth function then $\alpha_F = (i(Z)\,dF)\theta -dF$ is a 
semi-basic form. Hence $(\widetilde{d\theta})^{-1} [\alpha_F]$ is a 
well defined horizontal vector field. One verifies that 
$$\calF = FZ + (d\tilde\theta)^{-1} \Bigl[ \bigl( i(Z)\,dF\bigr) \theta 
- dF\Bigr]$$ 
is a contact vector field with $\theta (\calF) =F$.
\QED 

For a 1-form $\alpha$ and a vector field $X$ we write $i(X)\alpha$ or 
$\alpha (X)$ or $\langle \alpha,X\rangle$ interchangeably. 
If $f_\epsilon$ and $\calG_\epsilon$ are 
families of contact diffeomorphisms and vector fields respectively,
we have:
$$\eqalign{
\theta (f_{\epsilon *} \calG_\epsilon) (x) 
& = \langle (f_\epsilon^* \theta)(f_\epsilon^{-1}(x),
	\calG_\epsilon (f_\epsilon^{-1} (x)\rangle \cr
& = \lambda_\epsilon (f_\epsilon^{-1}(x))\ .\ 
	(\theta (\calG_\epsilon))(f_\epsilon^{-1}(x))\cr
& = \lambda_\epsilon (f_\epsilon^{-1}(x))\ \ 
G_\epsilon (f_\epsilon^{-1}(x))\cr 
&= [(\lambda_\epsilon G_\epsilon)\circ f_\epsilon^{-1}](x)\cr}$$ 
Therefore taking generating functions \equ(cohomologydiff) becomes: 
$$F_\epsilon - G_\epsilon + (\lambda^{f_\epsilon} G_\epsilon) \circ 
f_\epsilon^{-1} = 0\ .
\EQ(cohomologyconstant)$$  
Analogously, for vector fields, we observe that the generating function of 
$L_{\calF}\calG = [\calF,\calG]$ is 
$$\eqalign{
i([\calF,\calG]) \theta &= L_{\calF} i(\calG)\theta -
i(\calG)L_{\calF}\theta\cr
&= L_{\calF} G- i(\calG) [\mu_{\calF}\theta]\cr
&=L_{\calF} G- \mu^{\calF} G\cr}\ .$$
Hence \equ(cohomologyvec) becomes, for contact fields: 
$$-L_{\calX_\epsilon} G + \mu^{\calX_\epsilon} + \dot{\calX}_\epsilon=0\ .
\EQ(cohomologyCV)$$ 

To connect a contact diffeomorphism $f:\real^{2n+1} \to \real^{2n+1}$ 
with $f(0) = 0$, and $Df(0) = Id$ to the identity, the simple formula
\equ(interpolationeasy)
that worked for symplectic and volume preserving 
diffeomorphisms,
does not work since $f_\epsilon$ is not a contact diffeomorphism. 

Instead we use a chart on a 
neighborhood $\calU$ 
of the set of contact diffeomorphisms
with the set ${\cal J}^1_M$ of 1-jets of functions on $M$. 
This chart [Ly] is the equivalent of the Weinstein chart for symplectic 
diffeomorphisms and it is built exactly the same way, that we now describe.

Let $(M,\theta)$ be a contact manifold. Consider $\hat M= M\times M\times 
\real$ and denote by $\pi_1$, $\pi_2$, $t$, the projection of $\hat M$ 
onto the first, second, and third factors. 
On $\hat M$ we have the contact form $\hat\theta = t\cdot
 \left(\pi_1^* \theta\right) 
- \pi_2^* \theta$. Let $f$ be a contact diffeomorphism of $(M,\theta)$, 
i.e., $f^*\theta = \lambda\theta$. We define the graph $\Gamma_f$ of $f$ 
as the mapping: 
$$\Gamma_f : M\to \hat M : x\mapsto (x,f(x),\lambda (x))\ .$$ 
Clearly, $\Gamma_f$ is a Legendre embedding, i.e., 
$$\Gamma_f^* \hat\theta = 0$$ 
Let $L= \Gamma_f(M) \subseteq \hat M$ and $L_0 = \Gamma_{id}(M)\subseteq 
\hat M$. Both $L,L_0$ are Legendre submanifolds. We identify $L_0$ with $M$ 
as the zero section of the 1-jet bundle $J^1 M$ over $M$. There exists a 
diffeomorphism $\sigma$ of a neighborhood $M_1$ of $L_0$ in $\hat M$ with 
a neighborhood $M_2$ of $M=L_0$ in $J^1 M$ such that $\sigma|_{M_0} =$ 
identity and $\sigma^* \theta_M = \hat\theta$, where $\theta_M$ is the 
canonical contact form of $J^1 M$. Then if $f$ is $C^2$ close enough to the 
identity, $L\subseteq M_1$ and $\sigma(L)$ is a Legendre submanifold of 
$J^1 M$, which is $C^1$-close to $L_0$, hence $\sigma (L) = j^1 (\mu_f)$ 
for some function on $M$. The desired contact isotopy $f_\epsilon$ will 
have as graph: 
$$L_\epsilon = \sigma^{-1} [j^1 (\epsilon \mu_{id} + (1-\epsilon)\mu_f)]$$ 
where 
$$\sigma (L) = j^1 (\mu_f)\quad\hbox{and}\quad\sigma (L_0) = j(\mu_{id})\ .$$
Observe that for points $x\in M$ such that 
$$f(x_0) = x_0\quad\hbox{and}\quad (Df)(x_0)= \hbox{ Identity,}\quad 
(f^*\theta) (x_0) = \theta (x_0)$$ 
(i.e., $\lambda (x_0)=1$), hence $\Gamma_{id}(x_0) = \Gamma_f(x_0)$. 
Since $\sigma$ is a diffeomorphism the corresponding functions have 
the same jet at the point $x_0$. Therefore the deformation $L_\epsilon$ 
is fixed at $x_0$, i.e., $L_\epsilon (x_0) = \Gamma_{id} (x_0) = \Gamma_f 
(x_0)$ for all $\epsilon$. Therefore $D^\ell f_\epsilon (x_0) = D^\ell f(x_0)$, 
for all $\ell= 0,\cdots,k$. If $f$ and $N$ are $C^r$, $r \ge 2$, then
$f_\epsilon$ will be a $C^{r-1}$ family.

\SECTION Regularity results for cohomology equations 

In this section we study the solution of the cohomology equations. We 
emphasize that the cohomology equations are linear and  relate
functions taking values in linear spaces. The geometric properties 
derived for the objects we are interested in are consequences of the 
choices of spaces. In particular, it is not necessary to produce solutions 
using only geometrically natural methods. From the point of view of 
analysis, it is useful to take coordinates, add elements
or apply smoothing operators. Although those operations 
are not geometrically natural, they can be used in the present
context.


We observe that the cohomology equations we have 
encountered
( \equ(cohomologydiff), \equ(cohomologyhamil), \equ(cohomologysymp),
\equ(cohomologyconstant) )
can be written in the form 
$$\varphi_\epsilon (x) - M_\epsilon(x) \varphi_\epsilon \circ f_\epsilon 
(x) = \eta_\epsilon (x)
\EQ(discretecohomology)$$ 
where $\eta_{\epsilon} : \real^n \to \real^m$, 
$M: \real^n \to {\cal M}^{m \times m}$ 
(We denote by ${\cal M}^{m \times m}$ the space of $m \times m$ real valued
matrices.)
and $f_{\epsilon}: \real^n \to \real^n$ 
are given and we have to determine $\varphi_{\epsilon}$. 


Moreover, for the applications 
we are considering, we have that $f_{\epsilon}(0) =0$ and that 
$\eta_{\epsilon}$ vanishes up to some  
given order.

More generally, in the following theorems, we will give conditions under
which we can solve \equ(discretecohomology) assuming only that for
some Banach space $E$, with the approximation property mentioned  in
the introduction, $\eta_\epsilon: \real^n \mapsto E$, $M: \real^n
\mapsto {\cal L}(E,E)$, $f_\epsilon: \real^n \mapsto \real^n$.
(Here ${\cal L}(E,E)$ is the set of bounded linear operators from
$E$ to itself.) This additional generality will be of use later in
the construction.

The cohomology equations for vector fields can, likewise be written 
$$L_\epsilon\varphi_\epsilon (x) + M_\epsilon(x) 
\varphi_\epsilon (x) = \eta_\epsilon (x)
\EQ(contcohomology)$$ 
where $L = \sum_i A_i(x) {\partial \over \partial {x_i} }$ where 
 $A(x)\in {\cal M}^{m \times m}$.

We will start discussing both equations in the contractive case 
($\| Df\|<1$).  This will have direct applications for the general 
deformation method and    for the contact 
cases. It will also be an intermediate step in the symplectic and 
volume preserving cases. 

\CLAIM {Theorem}(contraction)
Assume that in \equ(discretecohomology) we have: 
\smallskip
\item{i)} $f_\epsilon ,\eta_\epsilon ,M_\epsilon $ are $C^r$ 
and $f_\epsilon(0) = 0$ for all $\epsilon$,
 $D^if_\epsilon (0) = D^if_0 (0)$, $i\le k$,
\item{ii)} $D^i\eta_\epsilon (0) =0$, $i\le k<r$,
\item{iii)} $\| Df_\epsilon(0)\| \le \tilde\alpha <1$,
\item{iv)} $\| M_\epsilon(0)\| \le \tilde\beta$,
\item{v)} $\tilde\alpha^{k+1}\tilde\beta <1$.
\smallskip
Then there exists one and only one continuous solution $\varphi_\epsilon$ of 
\equ(discretecohomology) defined in a small enough neighborhood of zero 
which satisfies 
\smallskip
\item{a)} $\| \varphi_\epsilon (x)\| \le K||x||^{k+1}$.
Moreover
\smallskip
\item{b)} $\varphi_\epsilon $ is a $C^r$ family. 


\REMARK
Since it is always possible to choose norms in $\real^n$, $\real^m$ in 
such a way that the norms of matrices are as close as desired to the 
spectral radius, conditions iii), iv) are really conditions about the 
spectrum of $Df_{\epsilon}(0)$ and $M_{\epsilon}(0)$. 


\PROOF
Valuable intuition about the proof
can be obtained by considering first the particular case
in which $M_\epsilon(x) \equiv M$ does not depend 
on $x$ or on $\epsilon$ and, likewise, 
$f_\epsilon(x) \equiv A x$ is just a linear map.


In that case, we claim that 
$$
\varphi_\epsilon = \sum_{n=0}^\infty M^n \eta_\epsilon( A^n x)
\EQ(solutioneasy)
$$
is a solution of \equ(discretecohomology).
Since $\| A^n x\| \le  \|A\|^n \,\|x\|$,
we have $\|\eta_\epsilon(A^n x)\|\le K\,\|A\|^{n(k+1)}\,\|x\|$
so that the series converges uniformly
whenever $\| M\|\,\|A\|^{k+1} < 1$. The uniform convergence
justifies the rearrangements needed to show it is 
indeed a true solution.

The norm of $\ell^{\rm th}$ derivative with respect to the variables
of the $n$ term in \equ(solutioneasy)
$\| M^n D^\ell \eta_\epsilon( A^n x)
\left( A^{\otimes \ell} \right) ^n\|$
can be bounded by
$ \|M\|^n\,\|A\|^{n(k+1 -\ell)} K\|x\|^{k+1-\ell} \|A\|^{\ell n}$
if $\ell \le k+1$ and by
$ \|M\|^n\,K \|A\|^{\ell n}$ if $\ell \ge k+1$.
Hence, the series obtained by differentiating term by term 
the series \equ(solutioneasy) converges uniformly and,
hence the series defines a $C^r$ function.
The proof in the general case will follow the same pattern,
even if some of the estimates are considerably more involved.




 
We can choose a ball $B_\rho$ of  of radius $\rho \le 1$ centered at  
$0\in\real^n$ in such a way that 
$$\eqalign{& \sup_{|x|\le \rho} \| Df_\epsilon (x)\| \le \alpha <1\ ,\cr
&\sup_{|x| \le \rho} \| M_\epsilon (x)\| \le \beta,  \qquad 
\beta\alpha^{k+1} <1. \cr}
\EQ(approxbound)$$



Since, by applying \equ(discretecohomology) $N$ times we have 
$$\varphi_\epsilon  (x) = \sum_{i=0}^N 
\left[ \prod_{j=0}^{i-1} M_\epsilon \bigl( f_\epsilon ^j (x)\bigr)
\right] \eta_\epsilon \bigl( f_\epsilon^i (x)\bigr) 
+ \left[ \prod_{ j=0}^N M_\epsilon \bigl(f_{\epsilon}^j(x)\bigr)\right] 
\varphi_\epsilon \bigl( f_{\epsilon}^{N+1} (x)\bigr).$$ 

By  \equ(approxbound) we have 
$$\left|\left| \prod_{j=0}^N M_{\epsilon}\bigl(f_{\epsilon}^j(x)\bigr)\right|\right|
\le \beta^{N+1},$$ 
and if $\varphi_{\epsilon}$ is to satisfy a) 
$| \varphi_{\epsilon} (f_{\epsilon}^{N+1}(x)) |\le K|\alpha|^{(N+1)(k+1)}$. 
The only possible solution of 
of \equ(discretecohomology) satisfying a) is 
$$\varphi_\epsilon (x) = \sum_{i=0}^\infty 
\left[ \prod_{j=0}^{i-1} M_\epsilon\bigl( f_\epsilon^j (x) 
\bigr)\right] \eta_\epsilon \bigl( f_\epsilon^i (x)\bigr)\ .
\EQ(solution)$$ 

The proof of \clm(contraction) will be complete once we show that the sum
in \equ(solution) is well defined and that it satisfies a) and b).
In the course of this proof we will denote by $K$ constants that are 
independent of $i$ and $x$. The actual value of the constants may
change from line to line.


We start by observing that the series \equ(solution)
converges uniformly on $B_\rho$. 
Since $\| f^i(x)\| \le \rho\alpha^i$,
hypothesis ii) implies that we have 
$\|\eta_\epsilon (f_\epsilon^i(x))\| \le K(\rho\alpha^i)^{k+1}$.
On the other hand, 
$\| \prod_{j=0}^{i-1} M_\epsilon(f_\epsilon^j(x))\| \le \beta^i$, 
so that the supremum of the 
general term can be estimated by $K\rho^{k+1} (\alpha^{k+1} \beta)^i$, 
which, by \equ(approxbound) is a geometric series of ratio smaller than 1. 
By the Weierstrass $M$-test, the series converges uniformly. 

Once  we have established uniform convergence, it is easy to show that 
it satisfies \equ(discretecohomology) by substituting it in  the equation and 
rearranging the order of the terms.

Moreover, 
$$ \| \varphi_\epsilon (x)\| 
 \le \sum_{i \ge 0} \beta^i K\bigl( \| x\| \alpha^i\bigr)
^{k+1}
\le \| x\|^{k+1} K\ .
$$ 

To establish existence and continuity of higher derivatives, we take 
derivatives term by term in \equ(solution) and show that the
resulting  series converge uniformly. 

We observe that by the product rule for derivatives,
$$\eqalign{
 D ( \left[ \prod_{j=0}^{i-1} M_\epsilon
\bigl( f_\epsilon ^j (x)\bigr)  \right] &
\eta_\epsilon  \bigl( f_\epsilon ^i(x)\bigr) )  =\cr
 & \sum_{j'=0}^{i-1} \left[\prod_{j=0}^{j'-1} 
M_{\epsilon}\bigl( f_\epsilon ^j (x)\bigr) \right]
DM_{\epsilon}(f_{\epsilon}^{j'}(x)) Df_\epsilon (f_\epsilon^{j'-1}(x))
\cdots Df_\epsilon (x) \cr
& \qquad \times  \left[\prod_{j=j'+1}^{i-1} 
M\bigl(f_\epsilon ^j(x)\bigr) \right]
\eta_\epsilon  \bigl(f_\epsilon ^i(x)\bigr)\cr
& +\prod_{j=0}^{i-1} M\bigl( f_\epsilon^j (x)\bigr) 
 D\eta_\epsilon (f_\epsilon^i(x)) 
Df_\epsilon(f_\epsilon^{i-1}(x)) \cdots Df_\epsilon(x),\cr}
\EQ(derivatives)$$ 
and that the supremum over $B_\rho$ of each of the terms in the sum can 
be bounded by $\beta^{i-1}K(\rho\alpha^i)^{k+1}$. The last term can be 
bounded by $\beta^i K(\rho \alpha^i)^k\alpha^i$. Hence, the derivative 
can be bounded by 
$$(\beta\alpha^{k+1})^i [i\rho^{k+1} K\beta^{-1} + \tilde K \rho^k]$$ 
which  converges absolutely if summed over $i$.

We also observe that
$\| D\varphi (x)\| \le K\| x\|^k$. 

For fixed $\epsilon$
we have established that
the solution is $C^1$. 

To prove that  the solution is 
$C^r$ as claimed in the theorem,
we can use the ``tangent functor trick.'' 
We observe that by taking derivatives of \equ(discretecohomology) we obtain: 
$$D\varphi_\epsilon (x) - M_\epsilon (x) D\varphi_\epsilon
\bigl( f_\epsilon(x)\bigr) Df_\epsilon(x) 
= D\eta_\epsilon (x) + DM_\epsilon(x) 
\varphi_\epsilon\bigl( f_\epsilon(x)\bigr)\ .
\EQ(derivative)$$ 

Note that the equation for $D\varphi_{\epsilon}$
is of the form of cohomology equations considered in the theorem.
It satisfies conditions
analogous to those satisfied by $\varphi_{\epsilon}$, except 
that $\beta$
is replaced by $\beta \cdot \alpha$ (since the linear operator is replaced 
by $D \varphi_{\epsilon} \to M_{\epsilon}(x) D \varphi_{\epsilon}(f_{\epsilon}(x) ) D f_{\epsilon}(x) )$ and
condition $(ii)$ is replaced by $D^i R_{\epsilon} (0) = 0$,
$ i \le (k-1)_+ < r$, where $R_{\epsilon}(x)$ refers to the right
hand side of \equ(derivative) and $(k-1)_+ =
\max(k-1,-1)$.  This in turn means that condition $(v)$ is replaced by
$\alpha^{(k-1)_+ +1}(\beta \alpha) < 1$.  Clearly if 
$\alpha^{k+1} \beta < 1$ then $\alpha^{(k-1)_+ +1}(\beta \alpha) < 1$,
so we can repeat the same construction as outlined above to 
construct $D \varphi_{\epsilon}$.  Uniqueness implies that the solution
of \equ(derivative) is indeed the derivative of $\varphi_{\epsilon}(x)$.
Then a calculation like \equ(derivatives) implies $\varphi_{\epsilon}$
is $C^2$.

If $r \ge 2$, we can repeat this procedure, differentiating \equ(derivative)
a second time with respect to $x$.
In this case $\beta$ is replaced by $\beta \alpha^2$, while in
condition $(ii)$, $k$ is replaced by
$(k-2)_+$.  Once again if $\beta \alpha^{k+1} < 1$, then
$(\alpha^2 \beta) \alpha^{(k-2)_+ + 1} < 1$, so condition $(v)$ is
satisfied and we can construct $D^2 \varphi_{\epsilon}$ and using a calculation
like \equ(derivatives) conclude that $\varphi_{\epsilon}$ is $C^3$.

Similarly, we will show that we can bootstrap the 
number of derivatives
with respect to $\epsilon$.

We compute the derivative with respect to $\epsilon$ of one of the terms 
in the sum \equ(solution) 
$$\eqalign{
&{\partial\over\partial\epsilon} \left[ \prod_{j=0}^{i-1} M_\epsilon 
\bigl( f_\epsilon^i (x)\bigr)\right] \eta_\epsilon \bigl( f_\epsilon^i 
(x)\bigr) = \cr
&= \sum_{j_1=0}^{i-1} \prod_{j=0}^{j_1-1} M_\epsilon
 \bigl( f_\epsilon^j (x)\bigr) 
\Biggl[  \left( {\partial\over\partial \epsilon} M_\epsilon\right) 
\bigl( f_\epsilon^{j_1} (x)\bigr) + \cr
&\qquad\qquad DM_\epsilon \bigl( f_\epsilon^{j_1} (x)\bigr) 
\sum_{0\le j_2\le j_1-1} Df_\epsilon^{j_1-j_2-1} (f_\epsilon^{j_2+1}(x)) 
\left( {\partial\over\partial \epsilon} f_\epsilon\right) 
\bigl( f_\epsilon^{j_2} (x)\bigr) \Biggr]\cr
&\qquad \prod_{j=j_1+1}^{i-1} M_\epsilon \bigl( f_\epsilon^j (x)\bigr) 
\eta_\epsilon \bigl( f_\epsilon^i (x)\bigr) + \cr
&+ \left[ \prod_{j=0}^{i-1} M_\epsilon \bigl( f_\epsilon^j (x)\bigr) \right] 
\Biggl[{\partial \eta_\epsilon\over \partial\epsilon} 
\bigl( f_\epsilon^i (x)\bigr) + \cr
&\qquad\qquad D\eta_\epsilon \bigl( f_\epsilon^i (x)\bigr) 
\sum_{j_1=0}^{i-1} Df_\epsilon^{i-j_1-1} (f_\epsilon^{j_1+1}(x)) 
\left( {\partial\over\partial \epsilon} f_\epsilon\right) 
\bigl( f_\epsilon^{j_1} (x)\bigr) \Biggr] \cr}\ .
\EQ(parderivatives)$$
Observe that since $D^i\eta_\epsilon(0)=0\  i= 0,\cdots,k$
for all values of $\epsilon$,
we have ${\partial^\ell \over \partial\epsilon^\ell} D^i\eta_\epsilon (0)=0$, 
$D^i {\partial^\ell\over\partial \epsilon^\ell} \eta_\epsilon (0) =0$. Hence, 
$$\left\| \left( {\partial^\ell \over\partial \epsilon^\ell} \eta_\epsilon 
\right) \bigl( f^j (x)\bigr)\right\| 
\le K\bigl( \alpha^j \| x\|\bigr)^{k+1}\ .$$ 

Similarly,
$$\left\| \left( {\partial^\ell \over \partial\epsilon^\ell} f_\epsilon\right) 
\bigl( f^j (x)\bigr)\right\| 
\le K\bigl( \alpha^j \| x\|\bigr)^{k+1}\ .$$ 

Hence, it is not difficult to establish again that the norm of the
derivative in \equ(parderivatives) is less than 
$$K\| x\|^{k+1} (\alpha^{k+1}\beta)^i i^2\ . $$ 

Now we observe that the derivative of $\varphi$ with respect to parameters 
satisfies 
$$\dot\varphi_\epsilon (x) - M_\epsilon (x)\dot\varphi_\epsilon 
\bigl( f_\epsilon (x)\bigr) = 
\dot M_\epsilon (x)\varphi_\epsilon \bigl( f_\epsilon (x)\bigr) 
+ M_\epsilon (x) D\varphi_\epsilon \bigl( f_\epsilon (x)\bigr) 
\dot f_\epsilon (x) +\dot  \eta_\epsilon(x).
\EQ(derivativepar)$$ 

The first term in the right hand side vanishes at zero
to order $k$ and, using
that $\dot f_\epsilon$ vanishes to order $k$, we see that
the second one vanishes 
up to order $2k-1 \ge k$ and that it is is $C^{r-1}$.  Since we established 
that $\dot\varphi_\epsilon$ vanishes up to order $k$, we conclude that 
$\dot\varphi_\epsilon$ is $C^{r-1}$ in the $x$ variables. 

Applying the result just established to \equ(derivativepar) we will 
conclude that $\dot\varphi_\epsilon$ admits one derivative with respect to 
$\epsilon$.

Hence, we can proceed by induction. If 
$D_\epsilon^a D^b_x \varphi_\epsilon $
exist and 
are continuous and vanish at the origin, we conclude that
$D_\epsilon^{a+1} D^b_x \varphi_\epsilon $
and
$D_\epsilon^a D^{b+1}_x \varphi_\epsilon $
exist. The induction stops only 
when the corresponding 
derivatives of $\eta_\epsilon$ cease to exist.



\REMARK
It is also possible to prove that the function is $C^r$ by
taking formal derivatives term by term and 
showing that the resulting series converges uniformly.
The estimates needed are developed in [BLW].

\QED

\REMARK
Notice that the size of the ball on which the solution is defined is 
independent of $\eta$. It depends only on $f$ and $M$. This is, of course, to
be expected since $\eta_\epsilon$ enters only linearly.


To study the solution of \equ(discretecohomology)
in the general hyperbolic case, we will find it convenient to perform 
several initial reductions that will make it possible to 
simplify the notation in the 
proof, but which do not incurr a loss of generality.

We observe that, if \equ(discretecohomology) holds and $h_\epsilon$  is a 
diffeomorphism, we have 
$$\varphi_\epsilon \circ h (x) -M_\epsilon \circ h_\epsilon(x)
\varphi_\epsilon\circ 
h_\epsilon(h_\epsilon^{-1} \circ f_\epsilon \circ h_\epsilon(x)\bigr) = 
\eta_\epsilon \circ h_\epsilon(x) 
\EQ(change)$$ 
so that $\tilde \varphi_\epsilon = \varphi_\epsilon \circ h_\epsilon$ satisfies 
\equ(discretecohomology) with data $\tilde M_\epsilon = M_\epsilon \circ 
h_\epsilon(x)$, $\tilde f_\epsilon = h_\epsilon^{-1} 
\circ f_\epsilon \circ h_\epsilon$, 
$\tilde \eta_\epsilon = \eta_\epsilon \circ h_\epsilon$. 

In particular, if we take $h_\epsilon(x)=ax$ for some small positive
number, $a$,
we have  $D\tilde f_\epsilon (0) = Df_\epsilon (0)$ and $D^i \tilde f_\epsilon 
(x) = a^{i-1} D^i f_\epsilon (ax)$, 
$$\left( {\partial\over\partial\epsilon}\right)^\ell D^i\tilde f_\epsilon 
(x) = a^{i-1} \left({\partial\over\partial\epsilon}\right)^\ell 
D^i f_\epsilon (ax)\ ,$$ 
so that, for $i>1$, $i+\ell \le r$ 
$$\sup_{\scriptstyle \epsilon \in [0,1]\atop \scriptstyle \| x\| <1} 
\left\| \left( {\partial\over\partial\epsilon}\right)^\ell D^i 
\tilde f_\epsilon (x)\right\|~~,$$ 
can be made as small as we please by taking $a$ sufficiently small. 

Furthermore, 
$$\sup_{\scriptstyle \epsilon\in [0,1]\atop \scriptstyle \| x\| <1} 
\left\| \left( {\partial\over\partial\epsilon}\right)^\ell 
\bigl[ D\tilde f_\epsilon (x) - D\tilde f_\epsilon (0)\bigr]\right\| 
\le \sup_{\scriptstyle \epsilon \in [0,1]\atop\scriptstyle \| x\|\le a} 
\left\| \left( {\partial\over\partial\epsilon}\right)^\ell 
\bigl[ Df_\epsilon (x) - Df_\epsilon (0)\bigr]\right\|~~,$$ 
which also can be made arbitrarily small.


If $\alpha (x)$ is a $C^\infty$ function with $\alpha (x) =1$ when $\| x\|
\le1$, $\alpha (x) =0$, when  $\|x\| \ge 3/2$, we see that defining, 
$$\eqalign{
{\buildrel\approx\over f}_\epsilon (x) & = \alpha (x) \tilde f_\epsilon (x) 
+ \bigl( 1-\alpha (x)\bigr) Df_\epsilon (0) x ~~,\cr 
{\buildrel\approx\over M}_\epsilon (x) & = \alpha (x) \tilde M_\epsilon (x) 
+ \bigl( 1-\alpha (x)\bigr) \tilde M_\epsilon (0) ~~, \cr
{\buildrel\approx\over\eta}_\epsilon(x) & = \alpha (x) \tilde \eta_\epsilon (x) 
+
\bigl( 1-\alpha (x)\bigr) \tilde \eta_\epsilon (0)~~,\cr}
\EQ(cutoff)$$ 
finding a solution of \equ(discretecohomology) defined in a sufficiently 
small neighborhood of the origin can be accomplished by finding global 
solutions of \equ(discretecohomology) under the assumption that 
$\| f_\epsilon (x) - D f_\epsilon (0)x\|_{C^r (\real^n \times [0,1])}$ 
is sufficiently small and that $M_\epsilon,\eta_\epsilon$ 
are in $C^r (\real^n\times [0,1])$. 

We emphasize that the solutions we obtain in this case may depend 
on the cut-off function used.
There are easy examples that show that indeed 
different cut-off functions lead to different solutions.
Hence,  the solutions produced by \clm(contraction) 
are highly non-unique.

There are some further reductions that we will use.

We recall the stable/unstable manifold theorem for diffeomorphisms.

\CLAIM {Theorem}(stableman)
Let $A:\real^s \oplus \real^u$ be a linear map 
and $f_\epsilon$ a $C^r$ family of diffeomorphisms
$f_\epsilon(0) = 0$
in such a way that 
\smallskip
\item{i)} The splitting $\real^s \oplus\real^u$ is invariant under $A$,
\item{ii)} $\| A|_{\real^s}\| <1$, $\| A^{-1}|_{\real^u}\| <1$.
\item{iii)} $\| f_\epsilon -A\|_{C^r (\real^s\oplus\real^u \times [0,1])} 
\le\delta$ 
where $\delta > 0$ is a number that can be computed 
explicitley depending only on $A$.
\vskip1pt
Then, there exist unique $C^r$ families $W_\epsilon^s: \real^s\to \real^u$,  
$W_\epsilon^u :\real^u\to \real^s$ such that 
\smallskip
\item{a)} The graphs of $W_\epsilon^s$, $W_\epsilon^u$ are invariant under 
$f_\epsilon$,
\item{b)} $\| W_\epsilon^s \|_{C^r (\real^s\times [0,1])} \le K\delta$,
\item{} $\| W_\epsilon^u \|_{C^r (\real^u \times [0,1])} \le K\delta$.
\smallskip
where $K$, can be chosen independently of $\delta$.

Proofs of invariant manifold theorems with dependence on parameters can be 
found in \cite{La}  using the graph transform method and in
\cite{LW} using Irwin's method.

We note that a  corollary of the estimates b) in \clm(stableman) is that if
$D^i f_\epsilon(0) = 0$ for $i=1,\cdots,k$ then,
$D^i W^s_\epsilon(0) = 0,
D^i W^u_\epsilon(0) = 0, i = 1,\cdots,k$.
In effect, if we consider $\tilde f_\epsilon(x) = a^{-1} f_\epsilon(a x)$
we see that $|| \tilde f_\epsilon - A||_{C^r} \le K a^{k-1}$.
On the other hand, the function
$\tilde W_\epsilon^s(y) = a^{-1}W_\epsilon^s(a y)$ has a graph invariant under
$\tilde f_\epsilon$ and, by the uniqueness in the conclusions of
\clm(stableman), we obtain that it should satisfy estimates b).
We conclude that  $|| a^{-1} W_\epsilon^s(a y)||_{C^r} \le K a^{k-1}$
which,  can only happen if the derivatives of $W_\epsilon^s$
of order less or equal than $k$ vanish at zero. Of course, an analogous
argument works for $W_\epsilon^u$.
The same result can be obtained by examining the graph transform
equations in [La]. By repeated differentiation we can obtain
equations satisfied by the derivatives of $W_\epsilon^s$ and
we can check by inspection that if the derivatives
at the origin of
$f_\epsilon$ vanish so do the derivatives of $W^s_\epsilon$. 

This theorem allows us to choose the change of variables 
\equ(change) in such a way that the invariant manifolds are the coordinate 
axes. 

That is, $\real^n = \real^s \oplus \real^u$ and $f_\epsilon (x,0) = 
(f_\epsilon^s (x),0)$, $f_\epsilon (0,y) = (0,f_\epsilon^u (y))$ 
$\forall x\in \real^s$, $y\in \real^u$. 

Therefore, we have established: 

\CLAIM {Lemma}(coordinates)
Let $f_\epsilon$ be a $C^r$ family of diffeomorphisms
defined in a neighborhood of the origin  
in $\real^n$ such that 
\smallskip
\item{i)} $f_\epsilon (0)=0$,\quad $Df_\epsilon (0) =A$, 
$D^if_\epsilon (0) = D^i f_0 (0)$,\quad $2\le i\le k$,
\item{ii)} $A$ is hyperbolic.
\vskip1pt
Then, if $s$ and $u$ are the dimensions of the stable and unstable subspaces 
for $A$, $\real^n = \real^s \oplus \real^u$, for every $\delta >0$ we can 
find a $C^r$ family $h_\epsilon$ and a $C^r$ family $\tilde f_\epsilon$ 
defined on the whole of $\real^n$ such that:
\smallskip
\item{a)} $h_\epsilon(0) =0$,\quad $Dh_\epsilon (0)=Id$,\quad 
$D^i h_\epsilon (0)=0$,\quad $2\le i\le k$, 
\item{b)} $h_\epsilon^{-1} \circ f_\epsilon \circ h_\epsilon = 
\tilde f_\epsilon$ on a neighborhood of the origin,
\item{c)} $\| \tilde f_\epsilon -A\|_{C^r} \le \delta ~~,$ 
\item{d)} $\tilde f_\epsilon (\real^s \oplus \{0\}) = \real^s \oplus\{0\}$, 
\item{}  $\tilde f_\epsilon^{-1} (\{0\} \oplus\real^u) = \{0\} \oplus
\real^u~~.$
\smallskip

Since for our purposes it is enough to have solutions of 
\equ(discretecohomology) on a neighborhood of the origin, we can
use cut-off functions and study
\equ(discretecohomology) defined in the whole $\real^n$, but nevertheless 
assume that $f_\epsilon$ satisfies the conclusions of \clm(coordinates) 
and that $\eta_\epsilon$ has compact support around the origin. 

Since the stable and unstable directions will play an important role, we 
will introduce the notation
 $x= (x_s,x_u)$, with $x\in \real^n$, $x_s\in \real^s$, 
$x_u\in \real^u$. We will also write 
$$A(x_s,0) = (A_sx_s,0),\quad  A(0,x_u) = (0,A_ux_u)\ .$$ 

Inspired by the method used in \cite{St} we start by solving the equation 
on the neighborhood of $\real^s$ up to high orders in $x_u$. 

\CLAIM {Lemma}(stable)
Let $f_\epsilon$, $\eta_\epsilon$ be $C^r$ families as above. Assume that:
\smallskip
\item{i)} $\| A_s\| \le \lambda_+ < 1$,
\item{} $\| A_u\| \le \mu_+ > 1$,
\item{} $D^i\eta_\epsilon (0) = 0$; $i=0,\ldots,k$,
\item{} $\| M_\epsilon (0)\| \le\beta_+$.
\vskip1pt 
Let $\ell$ be such that:
\item{ii)} $\lambda_+^{k+1} \beta_+ \mu_+^\ell < 1$~,~~{\it i.e.}
$\ell < -{{\ln \beta_+}\over{\ln\mu_+}} + (k+1) {|\ln\lambda_+|\over 
\ln \mu_+}~~.$ 
\vskip1pt
Then, we can find a $C^\ell$ family $\tilde\varphi_\epsilon$
of compactly supported functions such that 
\smallskip
\item{a)} ${\partial^j \over \partial x_u^j} (\tilde\varphi_\epsilon (x) - 
M_\epsilon (x) \tilde\varphi_\epsilon \circ
 f_{\epsilon}(x)-\eta_\epsilon(x))|_{x_u=0} =0, \quad j = 0, \cdots,\ell~~.$ 
\smallskip


\PROOF
Since we have good control of the 
problem on the stable manifold,
our first goal is to rewrite the problem 
in an expansion in powers of $x_u$.
We will expand the original equation in powers of 
$x_u$ and try to match the coefficients of 
corresponding powers.

Using a partial Taylor expansion 
$$\eta_\epsilon (x) = \sum_{|i|=0}^\ell \eta_\epsilon^{[i]} (x_s) x^{\otimes i}_u 
+ \eta_\epsilon^{[>]} (x).$$
Here
$\eta_\epsilon^{[i]} (x_s) \in {\cal L}( (\real^u)^{\otimes i}, R^m)$
is given by Taylor's formula,
$\eta_\epsilon^{[i]}(x_s) = {1 \over i!}{ \partial^i \over {\partial x_u}^i}
\eta_\epsilon( x_s,0)$.

Note further that the Taylor remainder satisfies:
$${\partial^i \over \partial x_u^i} \eta_\epsilon^{[>]} (x_s,0) = 0
\qquad i=0,\ldots,\ell\ ,$$
and that $\eta_\epsilon^{[i]}$ is an $r-i$ family with
$\|\eta_\epsilon^{[i]}\|_{C^{r-i}} \le \|\eta_\epsilon\|_{C^r}$.
We introduce 
similar notations 
for $M_\epsilon (x)$ and its Taylor expansion.
We  will prove that one can solve recursively for the coefficients in
an analogous expansion of $\tilde \varphi_\epsilon$.
Once we succeed in doing this, we can cut off the Taylor
polynomial by a function tangent to the identity to infinite order. Since
the conclusion a) depends only on the Taylor
expansions up to finite order, it will also be satisfied.


If $\tilde\varphi_\epsilon^i (x) = \tilde\varphi_\epsilon^{[i]}
(x_s) x_u^{\otimes i}$ is the monomial of degree $i$ in the 
Taylor expansion, then 
$$
\eqalign{
\tilde\varphi_\epsilon^i \bigl(f_\epsilon (x)\bigr) 
&= \tilde\varphi_\epsilon^{[i]} \bigl( f_\epsilon^s (x_s,x_u)\bigr) 
f_\epsilon^u (x_s,x_u)^{\otimes i}\cr
&= \tilde\varphi_\epsilon^{[i]} \bigl( f_\epsilon^s (x_s,0)\bigr)
\bigl({\partial \over \partial x_u} f_\epsilon^u(x_s,0)\bigr)^{\otimes i}
x_u^{\otimes i} + R_\epsilon^{[i]}(x)~~,
}
$$

where 
$
\left( {\partial\over\partial x_u}\right)^j R_\epsilon^{[i]} 
(x_s,0) = 0\qquad j < i+1\ .
$

The operator 
$ \tilde\varphi_\epsilon^{[i]} \bigl( f_\epsilon^s (x_s,0)\bigr)
\bigl({\partial \over \partial x_u} f_\epsilon^u(x_s,0)\bigr)^{\otimes i}$
belongs to ${\cal L}( (\real^u)^{\otimes i}, \real^m)$
and maps $x_u^{\otimes i}$ to
$ \tilde\varphi_\epsilon^{[i]} \bigl( f_\epsilon^s (x_s,0)\bigr)
\bigl({\partial \over \partial x_u} f_\epsilon^u(x_s,0) x_u\bigr)^{\otimes i}$
If $\tilde \varphi_\epsilon^i (x)$ is a $C^\ell$ family,
then  $R_\epsilon(x)$ is a $C^{\ell -1}$ family,
and if
$$
\left( {\partial\over\partial x_s}\right)^j \tilde \varphi_\epsilon^{[i]}
(0)=0 \qquad j\le k\ ,$$ 
then 
$$\left( {\partial\over\partial x_s}\right)^j R_\epsilon^{[i]} (0,0) =0
\qquad j\le k\ .$$ 

The derivatives  of order $\ell$ of $R_\epsilon^{[i]}$ with respect to 
$\partial\over\partial x_u$ can also be evaluated in terms of tensor  
products of derivatives of $f$ up to order $\ell$ and $\varphi_\epsilon^{[i]} 
(x_s)$.

Therefore, if
$$\varphi_\epsilon^{[\le]} (x) = \sum^\ell \varphi_\epsilon^{[i]} (x_s) 
x_u^{\otimes i}, 
\EQ(polynomial)$$ 
then, 
$$\varphi_\epsilon^{[\le]} \bigl(f_\epsilon (x)\bigr) = \sum_{i=0}^\ell 
T_\epsilon^{[i]} (x_s) x_u^{\otimes i} + R_\epsilon^{[\ge]}(x)$$
where, $R_\epsilon^{[\ge]}$ is of high order and 
$$T_\epsilon^{[i]} (x_s) = \varphi_\epsilon^{[i]} \bigl(f_\epsilon^s (x_s,0)\bigr) 
\bigl( {\partial\over\partial x_u} f_\epsilon^u(x_s,0)\bigr)^{\otimes i} 
+ C_\epsilon^{[i]} (x_s)~~,$$ 
where $C^{[i]}$ is an expression that involves derivatives of $f$ and 
$\varphi_\epsilon^{[j]}$ for $j<i$. 

If we substitute  the expression \equ(polynomial) for 
$\varphi_\epsilon^{[\le]}$ into 
\equ(discretecohomology) and equate similar powers we obtain for $i \le \ell$
fixed, 
$$
\eqalign{
 \eta_\epsilon^{[i]} (x_s) &=
\varphi_\epsilon^{[i]} (x_s) - M_\epsilon^{[0]} (x_s) \varphi_\epsilon^{[i]} 
\bigl( f_\epsilon^s (x_s,0)\bigr)
\bigl( {\partial \over \partial x_u} f^u_\epsilon(x_s, 0) \bigr)^{\otimes i}
\cr
& -\sum_{j=0}^{i-1} M_\epsilon^{[i-j]} (x_s) 
{C}_\epsilon^{[j]} (x_s)~~. 
}
\EQ(recursion)
$$ 

\REMARK The $\tilde{C}_\epsilon^{[i]}$'s in \equ(recursion) are redefined
with respect to the ${C}_\epsilon^{[i]}$'s in the previous equation
by absorbing additional terms of lower order in the derivatives of
$f$ and $\varphi^{[j]}$ with $j < i$.

We observe that \equ(recursion) is of the form of the cohomology
equations we considered in \clm(contraction). The data,
$C^{[i]}$, $\eta^{[i]}_\epsilon$ and the unknowns,
$\varphi^{[i]}$ are defined on the stable manifold and
$f^s_\epsilon( x_s,0)$ maps the stable manifold into itself
and is a contraction. Therefore, 
we want to verify the quantitative
hypothesis of \clm(contraction).

Note that the linear operator $M_\epsilon$ that appears in
\clm(contraction) is replaced by the operator
$
\varphi^{[i]}(\cdot) \mapsto M^{[0]}_\epsilon(x_s) 
\varphi^{[i]}(\cdot) 
\bigl( {\partial \over \partial x_u} f^u_\epsilon(x_s, 0) \bigr)^{\otimes i}
$
The norm of this operator is bounded in a neighborhood
of the origin by supremum of the norm of $M^{[0]}_\epsilon$
and by the supremum of the norm of
${\partial \over \partial x_u} f^u_\epsilon(x_s, 0)$
raised to the power $i$. Hence, by choosing the neighborhood
sufficiently small, we can bound the operator norm
by $\beta \mu^i$ where $\beta$ and $\mu$ are as close as one likes
to $\beta_+$ and $\mu_+$. Similarly, we can assume that
$|| {\partial \over \partial x_s} f^s_\epsilon(x_s,0)|| \le \lambda$,
where $\lambda$ can be made arbitrily close to $\lambda_+$ 
by considering a sufficiently small neighborhood of the origin.
Hypothesis $(v)$ of \clm(contraction) becomes
$\lambda^{k+1}\beta\mu^i < 1$, which is satisfied because of
$ii)$ of the hypothesis of the Lemma, provided that 
we consider only a sufficiently small neighborhood of the origin.

Once we have candidates for what the derivatives should be, 
the rest of the proof of \clm(stable) is just 
verifying that we can extend them using a cutoff function.

If $\gamma :\real^u \to \real$ is a $C^\infty$ function taking the 
value $1$ in a neighborhood of zero and with compact support, then 
$$\tilde\varphi_\epsilon (x) = \varphi_\epsilon^{[\le]} (x) \gamma (x_s)\ ,$$ 
where $\varphi_\epsilon^{[\le]}$ is as in \equ(polynomial) and 
the coefficients are obtained solving \equ(recursion) is a solution of 
the problem. 


We observe that, if 
$$\varphi_\epsilon (x) - M_\epsilon (x) \varphi_\epsilon 
\bigl( f_\epsilon (x)\bigr) = \eta_\epsilon (x)$$ 
and 
$$\Phi_\epsilon (x) - M_\epsilon (x) \Phi_\epsilon \bigl( f_\epsilon (x)\bigr) 
= \alpha_\epsilon (x)\ ,$$ 
then 
$$(\varphi_\epsilon - \Phi_\epsilon) (x) - M_\epsilon (x) (\varphi_\epsilon 
- \Phi_\epsilon) \bigl( f_\epsilon (x)\bigr) = \eta_\epsilon (x) 
- \alpha_\epsilon (x)\ .
\EQ(diff)
$$ 

Hence, to find solutions of \equ(discretecohomology), by considering 
$\varphi_\epsilon - \tilde\varphi_\epsilon$ where $\varphi_\epsilon$ 
is produced using \clm(stable) we can consider only the case where 
$({\partial\over\partial x_u})^i \eta_\epsilon (x)$ vanishes on the 
stable direction for $|i| \le \ell$. 
\QED

Now, we
try to solve the equation \equ(cohomologydiff)  for functions which
vanish to a high order in the unstable direction.

\CLAIM {Lemma}(unstable)
Let $f_\epsilon$, $M_\epsilon$ be as before. Assume that $\eta_\epsilon$ 
is a $C^r$ family with support in a neighborhood of the origin
and that, in addition to the hypothesis of
\clm(stable) we have:
\smallskip
\item{i)} $\|A_s^{-1}\| \le \lambda_- > 1$
\item{} $\|A_u^{-1}\| \le \mu_- < 1$
\item{ii)} $D^i \eta_\epsilon (x_s,x_u) \le K|x_u|^{k-i+1}$
\item{iii)} $\| M_\epsilon (0)^{-1}\| \le \beta_-$ 
\smallskip
Let $\ell\in \natural$ be such that
\smallskip
\item{iv)} $\lambda_-^\ell \beta_-\mu_-^{k-\ell+1} < 1\ ;\
{\rm that\ is,\ } 
\ell < {(k+1) |\ln\mu_-|- \ln \beta_- \over (\ln \lambda_- + \ln \mu_-)}\ .$
\smallskip
Then, there exists a $C^\ell$ family $\varphi_\epsilon$ solving 
\equ(discretecohomology) on a neighborhood of the origin. 

\REMARK
The conditions imposed in $iv)$ are too conservative.
The natural conditions for the method
of proof presented here seem to be:
\item{$iv')$} $\lambda_-^\ell \beta_-\mu_-^{k+1} < 1\ ;\ 
\ell < {-\ln \beta_-\over\ln \lambda_-} + (k+1) {|\ln\mu_-|\over 
\ln \lambda_-}\ .$

At the end of the proof we will sketch how to improve the argument
to get this result.

\PROOF 
The proof is very similar to \clm(contraction).

Again, considerable intuition can be obtained considering the case where
$M$ is constant and, likewise, $f_\epsilon \equiv A$.

We just check that 
$$
\varphi_\epsilon(x) = \sum_{i=1}^\infty M^{-n} \eta_\epsilon(A^{-n}x)
\EQ(simple2)
$$
is a solution.

In effect, since $\| (A^{-n} x)_u \| \le \mu_-^n \| x_u\|$,
we have that $\eta_\epsilon( A^{-n}x) \le K \mu_-^{n(k+1)}\|x_u\|$.
So that the series \equ(simple2) converges provided that
$\mu_-^{k+1} \beta_{-} < 1$.

As before, if we take $a$ derivatives with respect to
$x$ and $b$ derivatives with respect to $\epsilon$ in the general term,
we obtain a general term:
$$
M^{-n} D^a_x D^b_\epsilon \eta_\epsilon( A^{-n}x)
\bigr( A^{\otimes a} \bigl)^{-n}
$$
whose norm can be bounded by
$ \left( \beta_{-} \mu_{-}^{k+1-a} \lambda_-^a \right)^n K \| x_u\| $
if $a \le k+1$. 

To prove the general case,
 observe that, by cutting off the function $M$ and $f_\epsilon$ 
as we did in \equ(cutoff) we can assume that the bounds assumed for 
the derivative at zero are valid globally, with slightly worse 
constants. We can arrange that these new constants also satisfy  
assumption iv).

We also observe that \equ(discretecohomology) can also be written 
$$\varphi_\epsilon (x) - M_\epsilon^{-1} \bigl( f_\epsilon^{-1}(x)\bigr) 
\varphi_\epsilon( f_\epsilon^{-1} (x)) = - M_\epsilon^{-1} 
\bigl( f_\epsilon^{-1} (x)\bigr) \eta_\epsilon \bigl( f_\epsilon^{-1} 
(x)\bigr) ~~.
\EQ(discretecohomology2)$$



We claim that a solution of \equ(discretecohomology2)
can be obtained by setting 
$$\varphi_\epsilon (x) = \sum_{i=1}^{\infty}
 \left[ \prod_{j=1}^i M_\epsilon^{-1} 
\circ f_\epsilon^{-j} (x)\right] \eta_\epsilon \circ f_\epsilon^{-i} (x) ~~.
\EQ(solution2)$$ 

The estimates to show that 
the sum in \equ(solution2) converges are very similar to those
used in the study of \equ(solution). Hence, it will
be important to find out how fast 
$f_\epsilon^{-i}(x)$ approaches the set where 
$\eta_\epsilon$ vanishes.


\CLAIM {Proposition}(lambda) 
Under the conditions of \clm(unstable) denote $f_\epsilon^{-n} (x_s,x_u) = 
(x_n^s,x_n^u)$. Provided that $f_\epsilon$ is $C^1$-close enough 
to $A$, we can find $\alpha$ as close as desired to $|A^{-1}|_{E^u}|$ 
and $K>1$ so that $|x_n^u| \le K\alpha^n\|x_0^u\|$. 

\PROOF 
We denote: 
$$
f_\epsilon^{-1}(x_s,x_u) = (A_s^{-1} x_s+N^s (x_s,x_u)\ ,\ 
A_u^{-1} x_u + N^u (x_s,x_u) )~~.
\EQ(fm1)$$

We note that $N^u(x_s,0) = 0$ due to the invariance of the 
unstable manifold and that the $C^1$ norm of $N^s$ and 
$N^u$ can be assumed to be as small as desired.

Hence, we can estimate
$$
\eqalign{
| A_u^{-1}x_u + N^u(x_s,x_u)| &\le
| A_u^{-1}x_u + N^u(x_s,x_u) - 
  N^u(x_s, 0)| \cr &
\le ( ||A_u^{-1}|| + || N^u||_{C^1} ) ||x_u|| ~~.
}
$$
\QED

To show that the R.H.S. of \equ(solution2) 
converges, we just observe that 
$$\left\| \prod_{j=1}^i M_\epsilon^{-1} \circ 
f_\epsilon^{-j} (x)\right\| \le (\beta_- + \delta)^i $$
where $\delta$ can be made arbitrarily small by
making the neighborhood under consideration sufficiently small.
Moreover, by \clm(lambda) and the assumptions 
on $\eta$ we have 
$$\| \eta_\epsilon \circ f_\epsilon^{-i} (x)\| 
\le K(\mu_- +\delta)^{i(k+1)}\ .$$
By assumption iv) the general term of \equ(solution2) 
is bounded by a geometric series of ratio less than 1 
if $\delta$ is sufficiently small.

The rest of the argument is very similar to that in the 
proof of \clm(contraction). 
We observe that \equ(solution2) 
has the same form as \equ(solution) 
with $M_\epsilon$ replaced by $M_\epsilon^{-1}$, 
$f_\epsilon$ replaced by $f_\epsilon^{-1}$ so that, 
if we take derivatives term by term we obtain 
analogues of \equ(derivatives), \equ(parderivatives). 

Consider, for example, the analogue of \equ(derivatives).
We have:
$$
\eqalign{
D&\left( \prod_{j=1}^i M^{-1}_\epsilon\circ f_\epsilon(x) \right)
\eta \circ f_\epsilon^{-i}(x) = \cr
=  & \sum_{j' = 0}^{i-1} 
\left[\prod_{j=0}^{j'-1} M_\epsilon^{-1}(f^{-j}_\epsilon(x)) \right]
DM_\epsilon^{-1}(f_\epsilon^{-j'}(x)) \times \cr
& \times Df_\epsilon^{-1}( f_\epsilon^{-(j'-1)}(x)\cdots Df_\epsilon^{-1}(x))
\quad
\left[\prod_{j = j'+1}^{i-1} M_\epsilon^{-1}(f_\epsilon^{-j}(x)) \right]
\eta_\epsilon(f_\epsilon^{-1}(x)) +
\cr
& +\prod_{j=0}^{i-1} M_\epsilon^{-1}(f_\epsilon^{-j}(x))
D\eta_\epsilon(f_\epsilon^{-i}(x))
Df_\epsilon^{-1}(f_\epsilon^{-i+1}(x))
\cdots
Df_\epsilon^{-1}(f_\epsilon^{-1}(x))
\cr}
\EQ(fullder)
$$
Most factors in this expression have been estimated before.
The only additional comment we need to make is that, in 
any neighborhood of the origin, we can bound
$\| Df_\epsilon^{-1}\| \le (\lambda_- + \delta)$ and 
$\delta$ can be chosen arbitrarily small by taking the 
neighborhood of the origin small. Thus, the norm of
\equ(fullder) can be bounded by
$$ \sum_{j' =0}^{i-1} K_1 (\beta_- + \delta)^{i-1}
(\lambda_- + \delta)^{j'}(\mu_- + \delta)^{i(k+1)}
+ K_2(\beta_- + \delta)^i (\lambda_{-} + \delta)^i (\mu_{-} +\delta)^{ik}
\EQ(firstbound)
$$
Note that, to estimate $D\eta_\epsilon(f_\epsilon^{-i})$
we have used hypothesis ii) and \clm(lambda).

We can bound \equ(firstbound) by $K(\beta_- + \delta)^i
(\lambda_{-} + \delta)^i (\mu_{-} +\delta)^{ik}$
and, if $\delta$ is sufficiently small, hypothesis 
iv) will ensure that the sum over $i$ converges.
Higher derivatives and derivatives with respect to $\epsilon$
are handled in a similar fashion.
\QED


As remarked before, the conditions imposed in \clm(unstable)
are too conservative.

In the following paragraphs, we 
will give a proof only in the case where $f$ is linear and just
sketch the modifications needed in the general situation.
Even if these imporvements lead to better values of $A$,
we did not think it was worth to write a whole proof.
Of course, these remarks
are only meant for the very motivated reader.

For the case that $M_\epsilon(x) \equiv M$ and
$f_\epsilon(x) = A x$, we observe that, if we denote by $D_s$ and
$D_u$ derivatives along the stable and unstable directions
respectively,  the results in \clm(stable),
allow us to assume:
\item{$ii')$}
 $\| D_u^\ell \eta_\epsilon(x) \| \le K \| x_u\|^{k+1-\ell}$; 
\item{} $\| D_s^\ell \eta_\epsilon(x) \| \le K \| x_u\|^{k+1}$; 
rather than the more conservative $ii)$ in \clm(unstable).

As before, the solution of \equ(cohomologydiff) is given by
$\sum_{i=1}^\infty M^{-n} \eta_\epsilon(A^{-n} x)$.
If we estimate the terms that we obtain when we 
apply $D_s^\ell$ and $D_u^\ell$ to the general term, we obtain:

$$\eqalign{
&\| M^{-n} D_s^\ell \eta_\epsilon( A^{-n}x) ( A_s^{\otimes
\ell})^{-n} || \le
(\beta_- \mu_-^{k+1} \lambda_-^\ell)^n K \|x_u\|^{k+1} \cr
&\| M^{-n} D_u^\ell \eta_\epsilon( A^{-n}x) ( A_u^{\otimes
\ell})^{-n} || \le
(\beta_- \mu_-^{k+1 - \ell} \mu_-^\ell)^n K \|x_u\|^{k+1-\ell}
}
$$

Hence, under the hypothesis $iv')$, we know that $\ell$ derivatives
along complementary directions exist and are uniformly bounded.
It is well known (see e.g. \cite{Kr})
that, if a function has $\ell$ derivatives along
complementary directions and those are uniformly bounded, the
function is $C^{\ell - \epsilon}$. Of course, if we consider 
$\ell \notin \natural$ with the usual meaning of H\"older regularity
for the derivative of order $[\ell]$, then this lemma allows us to
remove the $\epsilon$ in the conclusions.

In the case that $f_\epsilon$ is not a constant linear map, 
the argument can be
generalized. We remark that, since $f_\epsilon$ is uniformly close to 
$A$ in the whole space, then, one can prove stable and unstable
foliation theorems
completely similar to the ones usually stated for compact manifolds.
These foliations, have smooth leaves but are not 
very smooth in transverse directions. 
Nevertheless, it is shown in \cite{LMM} that one can
consider differential operators along the leaves. Moreover, one has
regularity results analogous to those for coordinate foliations,
namely that regularity along  both the stable and  unstable
foliations implies global regularity -- with a loss of $\epsilon$ in
the integer case and no loss in the non-integer case -- (See 
\cite{LMM}, \cite{Jo}.)
The existence of derivatives along the stable and unstable directions
can be used by estimating the derivatives term by term. The details
of these estimates are in \cite{LMM}. Moreover, one can use
regularity lemmas that show that when a function is differentiable
along the stable and unstable leaves, then it is differentiable.
This completes the sketch of the proof that $(iv)$ in \clm(unstable)
can be replaced by $(iv')$.


As a consequence of \clm(stable) and \clm(unstable)
we have the main result of the section.

\CLAIM {Theorem}(cohomology) 
Let $f_\epsilon$, $M_\epsilon$, $\eta_\epsilon$ be $C^r$ 
families as before. Assume that $Df_\epsilon (0) =A$ is hyperbolic, and that 
\smallskip
\item{i)} $\|A_s\| \le\lambda_+ \quad ,\quad  
\|A_s^{-1}\| \le \lambda_- \ .$
\item{} $\|A_u\| \le \mu_+ \quad ,\quad 
\|A_u^{-1}\| \le \mu_- \ ,$
\item{} $\| M_\epsilon (0)\| \le\beta_+\quad ,\quad 
\| M_\epsilon (0)^{-1} \| \le \beta_-\ ,$
\smallskip
where $\lambda_+, \mu_- < 1$,
$\lambda_-, \mu_+ > 1$.
\item{ii)} $D^i \eta_\epsilon (0)=0\quad ,\quad i\le k<r.$ 
\smallskip
Let  $\ell \in \natural$ be such that:
$$
\ell   <
 (k+1) {|\ln\lambda_+| |\ln \mu_-| \over \ln \mu_+
( \ln \lambda_- + |\ln \mu_-| ) }
-{ \ln \beta_+ | \ln \mu_-| \over \ln \mu_+(\ln \lambda_- + | \ln \mu_-|)}
-{ \ln \beta_- \over (\ln \lambda_- + |\ln \mu_{-}|)}\ .
$$ 
Then,  it is possible to find a $C^\ell$ family $\varphi_\epsilon$ 
satisfying \equ(discretecohomology) on a neighborhood of the origin. 

\def\tell{{\tilde \ell}}

\PROOF
Using \clm(stable) we can construct a  $C^\tell$
family $\tilde \phi_\epsilon$
that solves \equ(discretecohomology)
when restricted to the stable manifold
and that satisfies $a)$ of \clm(stable)
vanishing up to order $\tell$,
where $\tell$ is any integer satisfying:
$$
\tell < {-\ln \beta_+ \over \ln \mu_+}  + (k +1)
 {|\ln \lambda_+ |\over \ln \mu_+}.
\EQ(ineq1)
$$
Hence, we can choose $\tell$ to be: 
$$
\tell \ge {-\ln \beta_+ \over \ln \mu_+}  + (k +1) {|\ln \lambda_+| \over \ln \mu_+} -1.
\EQ(ineq2)
$$

If we now consider the 
family $\phi_\epsilon - \tilde \phi_\epsilon$,
we see that it will satisfy the same equation 
\equ(discretecohomology)
but with a right hand side that satisfies the hypothesis 
of \clm(unstable).

We can now obtain a solution which is 
$C^\ell$ for any $\ell$ such that:
$$
\eqalign{
\ell \le {-\ln \beta_-\over (\ln \lambda_-
+ \ln \mu_{-}|)} + (\tell+1) {\ln\mu_-\over 
(\ln \lambda_- + |\ln \mu_{-}|)}
}\ .
\EQ(ineq3)
$$
If we now take a $\tell$ satisfying 
\equ(ineq2), we see that we can take
$\ell$ as claimed in the 
statement of
\clm(cohomology).


\QED

\REMARK
The assumption that $\ell \in \natural$ does not enter in the study of the
cohomology equation. It is quite possible to develop a theory of 
cohomology equations when the coefficients are in the usual H\"older spaces.
To use them in the deformation method, we would need a theory of 
solutions of ordinary differential equations when the coefficients are 
H\"older.

>From \clm(cohomology) we can deduce the results claimed in the
main theorems if
we observe that in all cases, 
if we have a $C^r$ family of mappings,
the generators of the flow are a $C^{r-1}$ family and,
hence, the hamiltonians are also a $C^{r-1}$ family.
(This is the reason why the condition $k < r - 1$
appears in \clm(maindiff) rather than 
$k < r$ in \clm(cohomology).)
Furthermore, if our family of diffeomorphisms $f_\epsilon$
is tangent to order $k$ at the origin, the 
vector field ${\cal F}$ will vanish also to order $k$.
In the symplectic case, this implies that the hamiltonian 
$F_\epsilon$ vanishes up to order $k+1$. In the volume preserving case,
since we need to integrate, we can only conclude that the hamiltonian 
vanishes up to order $(k-1)$ at the origin. In the contact case, since
the hamiltonian is obtained just taking interior products, it 
vanishes up to order $k$.

Now we turn to compute the values 
for $A$, $B$ for the different 
geometric structures that we claimed in the 
remarks after \clm(maindiff)
To do that, we just need to relate the 
bounds on $M$ to the bounds  of the 
derivatives in the original
problem  -- the operators $M$ are
constructed out of the derivatives of the
original problem -- and to study the relation of
the regularity of the hamiltonian to the regularity 
of the original problem.

If we look at the way that the cohomology equations were derived,
we see that in the case that no structure is preserved,
the equation \equ(cohomologydiff)
can be written in components
as
$\FF_\epsilon(x) - \GG_\epsilon(x) + (Df_\epsilon\circ f_\epsilon^{-1}(x) )
 \GG_\epsilon( f_\epsilon^{-1}(x) ) = 0$. 
In order to compare it more easily with the equation \equ(discretecohomology)
discussed in
\clm(cohomology),
we write it as
$\FF_\epsilon(f_\epsilon(x)) - Df_\epsilon^{-1}(x) 
\GG_\epsilon( f_\epsilon(x)) + \GG_\epsilon( x ) = 0$ 
so that $\lambda_{\pm}, \mu_{\pm}$ have the same meaning 
in \clm(cohomology) and in \clm(maindiff).
In that case, $M = Df^{-1}$ and, hence we can take
$\beta_+ = \lambda_-$ and $\beta_- = \mu_+$.
Substituting this into the claim of
\clm(cohomology) leads to the value 
in the remarks after \clm(maindiff).


Similarly, in the case that the mapping 
is symplectic, we take 
$M_\epsilon=1$ and hence $\beta_+ = \beta_- = 1$.
The order of tangency in the cohomology equation
is one more than that appearing in the 
hypothesis of \clm(maindiff).

In the case that the flow is 
volume preserving, we take $M_\epsilon=
 \left(Df_\epsilon^{-1}\right)^{\wedge (n-2)}$
-- notice that we are acting on antisymmetric forms --
hence, we can take $\beta_+ =  \lambda_-^{(n-2)}$,
$\beta_- = \mu_+^{(n-2)}$.
Notice that the previous bounds are very conservative and 
that we could take as $\beta_+$ the product of the $(n-2)$
-- maybe after changing the norm to an equivalent one --
largest absolute values of points in the spectrum.
An analogous statement holds for $\beta_-$.
This is the improvement alluded to in the remarks 
at the end of the statement of \clm(maindiff).
The order of tangency in  the cohomology equation is one less than
the order of tangency of the diffeomorphisms. So that the $k$ 
appearing in \clm(cohomology) in this case is one less than the 
$k$ appearing in the hypothesis of \clm(maindiff).

In the contact case,
we take $M$ to be the 
factor by which  the push forward multiplies a form.
hence, we can take $\beta_+ = \lambda_-$,
$\beta_- = \mu_+$

Note that in the symplectic, volume-preserving, and contact cases,
to construct the flow from the solution of the cohomology equation
we have to take one derivative.  Thus, if we know that solution
of the cohomology equation is $C^{\ell}$, we can conclude that
the conjugating diffeomorphism is $C^{\ell -1}$.


For the case of flows, it is quite possible to give a very similar 
treatment which we only sketch.

We observe that if $L_\epsilon$, $\varphi_\epsilon$, $M_\epsilon$, 
$\eta_\epsilon$ are as in \equ(contcohomology), if we call 
$f_\epsilon^t$ the time $t$ map of the flow and introduce the matrix 
valued function $\Gamma_\epsilon^t$ by: 
$$\left.\eqalign{ {d\over dt} \Gamma_\epsilon^t (x)
& = \Gamma_\epsilon^t (x) M_\epsilon (f_\epsilon^t (x))\cr
\Gamma_\epsilon^0 (x) &= Id\cr}\right\}\ .$$ 
then \equ(contcohomology) can be written as 
$$\eqalign{ {d\over dt} \Gamma_\epsilon^t (x) \varphi_\epsilon 
(f_\epsilon^t (x)) 
& = \Gamma_\epsilon^t (x) M_\epsilon (f_\epsilon^t (x)) 
\varphi_\epsilon (f_\epsilon^t (x))\cr
&\qquad +  \Gamma_\epsilon^t (x) [L_{\epsilon} \varphi_\epsilon] \circ 
f_\epsilon^t (x) = \cr 
& = \Gamma_\epsilon^t (x)\, \eta_\epsilon (f_\epsilon^t x)\cr}\ .$$
Hence 
$$\varphi_\epsilon (x) = - \int_0^s \Gamma_\epsilon^s (x) 
\eta_\epsilon (f_\epsilon^s (x)) \, ds 
+ \Gamma_\epsilon^t (x) \varphi_\epsilon (f_\epsilon^t(x)) ~~.$$ 
As in the case of the discrete equation this suggests that, 
if $f_\epsilon^t (x) $ is a contradiction we can write a solution as 
$$\varphi_\epsilon (x) = - \int_0^\infty \Gamma_\epsilon^s (x) 
\eta_\epsilon (f_\epsilon^s (x)) \, ds\ .
\EQ(solutiondiff)$$ 

Notice that this equation is quite similar to \equ(solution) 
and an analysis  analogous to the one we performed there can establish 
that the improper integral above 
converges and is differentiable with respect to $x$ and 
to parameters. 

For the case of flows, it is also possible to prove an invariant 
manifold theorem and the same argument we gave can be adapted 
without difficulty. 
We leave the details to the reader. 

\CLAIM {Theorem}(cohomologyvec) 
Let $L_\epsilon$, $\varphi_\epsilon$, $M_\epsilon$, $\eta_\epsilon$ 
be $C^r$ families of vector fields, vector valued  as before. 
Assume that $DL_\epsilon(0) =A$ is hyperbolic
in the sense of flows and that if $\sigma (A)$ denotes
the spectrum of the operator $A$, we have:
\smallskip
\item{i)} $-\lambda_- \le Re ( \sigma (A_s)) \le\lambda_+$,
\item{} $-\mu_- \le Re (\sigma (A_u)) \le \mu_+$,
\item{} $\| M_\epsilon (0) \| \le \beta_+$,
\item{} $\| M_\epsilon (0)^{-1} \| \le\beta$,
\item{ii)} $D^i \eta_\epsilon (0) = 0\quad i\le k<r$.
\smallskip
Let 
$$\ell < (k+1) {{|\lambda_+| |\mu_{-}| }\over{(\mu_{+})
(\lambda_{-} + |\mu_{-}|) }} - {{\beta_+ |\mu_{-}|}\over{
(\mu_{+}) (\lambda_{-} + |\mu_{-}|) }} - {{\beta_-}\over{
(\lambda_{-} + |\mu_{-}|) }}\ .
$$
Then, it is possible to find a $C^\ell$ family $\varphi_\epsilon$ 
satisfying \equ(cohomologyvec) on a neighborhood of the origin. 

>From this, we can derive the values of
$A$ and $B$ in  the 
remarks after \clm(mainvec) in a way analogous 
to the way we derived the values of
in \clm(maindiff)  from those in
\clm(cohomology).


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