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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 $i=0,\ldots,k1$, $i+\ell \le r$
$$\sup_{\scriptstyle \epsilon \in [0,1]\atop \scriptstyle \| x\| <1}
\left\| \left( {\partial\over\partial\epsilon}\right)^\ell D^i
\tilde f_\epsilon (x)\right\|~~,$$
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 $j* 1$
\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 kFrom \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 kFrom 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).
\SECTION References
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\endref
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diffeomorphisms}
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syst\`emes dynamiques}
\jour{Ast\'erisque} \pages{138--139} \yr{1986}\endref
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\by{Chen, K. T.}
\paper{Equivalence and decomposition of vector fields
about an elementary critical point}
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\endref
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\endref
\ref \no{DL} \by{A. Delshams, R. de la Llave}
\paper{Some algorithms for computations of normal forms
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\jour{Manuscript}\endref
\ref \no{Gr} \by{J. Grey}
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*