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Normal form, symmetry, reversible
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\author {G.R.Belitskii${}^{\dagger}$, A.Ya.Kopanskii${}^{\ddagger}$}
\title
{Equivariant Sternberg Theorem}


\maketitle

\begin{center}
 {\footnotesize ${}^{\dagger}$
 Department of Mathematics, Ben-Gurion University of the Negev

 Beer Sheva 84105, Israel ${\tt genrich@cs.bgu.ac.il}$
}
\end{center}

\begin{center}
 {\footnotesize ${}^{\ddagger}$
 Institute of Mathematics, Academy of Sciences of Moldova

 5 Academiei str., Kishinev 2028, Moldova ${\tt alex@revel.moldova.su}$
}
\end{center}


\begin{abstract}
Smooth vector fields with linear symmetries and anti-symmetries
are considered. We prove that provided symmetry group is compact then
a smooth conjugacy in Sternberg Theorem can be chosen symmetric.
\end{abstract}

\section*{Introduction}

The well known Sternberg Theorem (see \cite{St}) asserts that if two local
smooth
vector fields are formally conjugate at a hyperbolic singularity
then they are smoothly conjugate. This result reduces local
classification and normalization problems to the formal ones.

In the last years there is a splash of interest to systems with symmetries
and
anti-symmetries ({\em reversible systems}) (for references see
\cite{LR1}). This activity
faces, in particular, a similar ``equivariant`` problem:
could one provide a conjugacy (smooth,
formal) of two symmetric or anti-symmetric vector fields via a
transformation keeping the property?
The linear and the formal aspects of the problem were considered in
\cite{LR2, Ga}.
But, as far as the authors know, a related version of Smooth Conjugacy
Sternberg
Theorem has not been published yet, though some recent papers make a more
or less explicit use of this result (see \cite{AfM, CHa}). Moreover, in
\cite{Bo} the
following problem is posed: does a smooth vector field anti-commuting with
a linear involution $R$ ({\it $R$-reversible}) admits a smooth
$R$-symmetric normalization.

The aim of the present paper is to prove an equivariant version of
Sternberg Theorem (Theorem 2.4 below). As a corollary this gives the
positive answer to the
mentioned
question in \cite{Bo}, and contains simultaneously the main results of
\cite{Bo} and
\cite{S2}. Actually, we prove a more general result (``modulo
central manifold``). First it was stated in \cite{B1} for local
diffeomorphisms
and then in \cite{S1} for vector fields (see also \cite{BrK1, IL}).

We use the so-called {\em deformation method} or {\em homotopy trick}
going back to
J.Mather (see \cite{M, AVG, BrK2}). This method reduces a nonlinear
problem on equivalence of two vector fields
(or more generally, of two local mappings
near a common singularity) to that on solving
a linear functional equation ({\em cohomology equation}). We give
here a complete proof of its solvability in $\:C^{\infty}$ case.
A similar equation related to conjugacy of local diffeomorphisms
was considered first in \cite{B2}.

\section{Definitions}

By a {\em local diffeomorphism} ({\em local vector field}) we mean in what
follows either a germ of diffeomorphisms (vector fields) at a point or a
representative of this germ defined in a neighborhood of the point.
Sometimes, we omit for briefness the word ``local``.

Let $\:\frG\:$ be a group of local diffeomorphisms
$\:\Phi\::\:(\R^n,0)\:\to\:(\R^n,0)\:$ and
$\:\gs\::\:\frG\:\to\:\R\:$ be a {\em multiplicative character} of
$\:\frG,\:$ {\em i.e.}, a homomorphism into the multiplicative group
$\:\R^*.\:$
\\

{\bf Definition.} {\em A local vector field $\:\xi\:$ is
said to be $\:(\frG,\gs)$-equivariant if}
$$
G_*\xi\:=\:\gs(G)\xi\ \ \ (G\:\in\:\frG).
$$

In this definition, $\:G_*\xi\:$ denotes a usual action
of the local diffeomorphism $G$ on the vector field $\xi$:
$$
(G_*\xi)(x)\:=\:DG(x)\xi(G^{-1}x),
$$
where $\:DG(x)\:$ is the Jacobi matrix.

A local transformation $\:H\:$ keeps the property of $\:\xi\:$ to be
$\:(\frG,\gs)$-equivariant if it commutes with every element
$\:G\:\in\:\frG,\:$ {\em i.e.}, $\:GH\:=\:HG.\:$ Such a local
diffeomorphism is said to be $\:\frG$-{\em equivariant}.

Similar definition may be applied to the {\em formal} category.
Namely, let $\:\frG\:$ be a group of {\em formal} transformations of
$\:\R^n\:$ and $\:\gs:\ \frG\:\rightarrow\:\R\:$ be a multiplicative
character. A formal vector field $\:\hat{\xi}\:$ is said to be
$\:(\frG,\gs)$-{\em equivariant} if
$$
G_*\hat{\xi}\:=\:\gs(G)\hat{\xi}\ \ \ (G\:\in\:\frG).
$$
A formal diffeomorphism $\:\hat{T}\:$ is called $\:\frG$-{\em equivariant}
if it commutes with every $\:G\:\in\:\frG.\:$

In what follows, we assume that $\:\frG\:$ is {\em a compact group of
linear operators in $\:\R^n\:$} and $\:\gs\:$ {\em is continuous}. Hence,
$\:\frG\:$ can be equally regarded as a group of local diffeomorphisms
or as of formal ones. Without loss of generality
$\:\frG\:$ is supposed to be a closed subgroup of the group $\:O(n,\R)\:$ of all
orthogonal transformaions. Thus, every element $\:G\:\in\:\frG\:$ keeps
the standard inner product in $\:\R^n.\:$
And besides, the continuity of $\:\gs\:$ implies
$$
\gs(G)\:=\:\pm 1\ \ \ (G\:\in\:\frG).
$$

Let $\:\xi\:$ be a local (or a formal) $\:(\frG,\gs)$-equivariant vector
field, $\:\xi(0)\:=\:0,\:$ and
$$
\gL\:\equiv\:D\xi(0)
$$
be its {\em linearization} at the origin.
Then $\:\gL\:$ is a $\:(\frG,\gs)$-equivariant transformation of
$\:\R^n:\:$
$$
G\gL G^{-1}\:=\:\gs(G)\gL\ \ \ (G\:\in\:\frG).
$$
Since $\:G\:\in\:\frG\:$ keeps the standard inner product, the conjugate
operator $\:\gL^*\:$ is $\:(\frG,\gs)$-equivariant as well.

\section{Main results}

Recall that two local (formal) vector fields $\:\xi\:$ and
$\:\eta\:$ are said to be $\:C^k$ (formally) conjugate if there exists
a $\:C^k$ (formal) diffeomorphism $\:\Phi\:$ such that
$$
\Phi_*\xi\:=\:\eta.
$$
\\
{\large\bf 2.1. Formal normal form.}
The following assertion can be proved via usual formal power series
technique (cf. \cite{B3, Ga}).
\\

{\bf Theorem 2.1.} {\em Let $\:\hat{\xi}\:$ be a $\:(\frG,\gs)$-equivariant
formal vector field,}
$$
\hat{\xi}(0)\:=\:0,\ \ \ D\hat{\xi}(0)\:=\:\gL.
$$
{\em Then there exists a $\:\frG$-equivariant formal diffeomorphism
$\:\Phi\:$ bringing $\:\hat{\xi}\:$ to the normal form}
$$
\Phi_*\hat{\xi}\:=\:\gL\:+\:\tau,
$$
{\em where the formal series $\:\tau\:$ contains no terms of order
less than $2$ and commutes with} $\:\gL^*.\:$

Let
$$
\R^n\:=\:E_s\:\oplus\:E_u\:\oplus\:E_c
$$
be a $\:\gL$-invariant splitting such that all the real parts of the
eigenvalues of the restriction
$\:\gL_s\:\equiv\:\gL|E_s\ \ (\gL_u\:\equiv\:\gL |E_u;\ \gL_c\:\equiv\:\gL
|E_c)\:$ are negative (positive; equal to zero).
If a formal vector field $\:\hat{\xi}\:=\:\gL\:+\:...\:$ is in the normal
form then the subspaces $\:E_s\:\oplus\:E_c\:$ and $\:E_u\:\oplus\:E_c\:$
are invariant with respect to the formal phase flow of $\:\hat{\xi}.\:$

Now we pass to the $\:C^{\infty}$ case making use of
the ``equivariant`` Borel Lemma:

{\em Let $\:\gG\:$ be a formal $\:\frG$-equivariant diffeomorphism. Then
there is a local $\:\frG$-equivariant diffeomorphism
$\:\Phi\:\in\:C^{\infty}\:$ such that its Taylor series $\:\hat{\Phi}\:$
at the origin is equal to $\:\gG.\:$}

Similarly, {\em given a $\:(\frG,\gs)$-equivariant formal vector field
$\:\tau\:$ there is a local $\:(\frG,\gs)$-equivariant vector field
$\:\xi\:\in\:C^{\infty}\:$ such that its Taylor series $\:\hat{\xi}\:$ at
the origin is equal to} $\:\tau.\:$
Moreover, {\em if the subspaces $\:E_s\:\oplus\:E_c\:$ and
$\:E_u\:\oplus\:E_c\:$
are invariant with respect to the formal phase flow of $\:\tau\:$ then
$\:\xi\:$ may be chosen with the same property}.

Since the group $\:\frG\:$ consits of orthogonal transformations one can
prove all assertions of the equivariant Borel Lemma in a usual way (see,
for example, \cite{H}).

As a result we get
\\

{\bf Corollary 2.2.} {\em Given a local $\:(\frG,\gs)$-equivariant
$\:C^{\infty}$ vector field $\:\eta,\:$}
$$
\eta(0)\:=\:0,\ \ \ D\eta(0)\:=\:\gL,
$$
{\em there exist a local $\:\frG$-equivariant $\:C^{\infty}$
diffeomorphism $\:\Phi\:$ and a local $\:(\frG,\gs)$-equivariant
$\:C^{\infty}$ vector field $\:\xi\:$ such that}

a) {\em the formal Taylor series $\:\hat{\xi}\:$ at the origin is in the normal
form};

b) {\em the subspaces $\:E_u\:\oplus\:E_c\:$ and $\:E_s\:\oplus\:E_c\:$ are
invariant with respect to the phase flow of $\:\xi;\:$}

c) {\em the formal Taylor series $\:\widehat{\Phi_*\xi}\:$ and
$\:\hat{\eta}\:$ at the origin coincide}.

Speaking otherwise, the vector fields $\:\xi\:$ and $\:\eta\:$ ``are
conjugate up to a flat remainder``, {\em i.e.}, all derivatives of the difference
$\:\Phi_*\xi\:-\:\eta\:$ vanish at the origin. Usually, the main functional
problem is ``to kill`` the flat remainder.
\\
\\
{\large\bf 2.2. Smooth conjugacy.}
Fix a local $\:(\frG,\gs)$-equivariant $\:C^{\infty}$ vector
field $\:\xi,\ \xi(0)\:=\:0,\ D\xi(0)\:=\:\gL,\:$ such that the subspaces
$\:E_u\:\oplus\:E_c\:$ and $\:E_s\:\oplus\:E_c\:$ are invariant with
respect to the phase flow of $\:\xi.\:$

Denote by $\:\gd_u\:$ and $\:\gD_u\:$ (respectively, by $\:-\gD_s\:$ and
$\:-\gd_s\:$) the minimum and the maximum of the real parts of the
eigenvalues of the restriction $\:\gL|E_u\:$ (respectively,
$\:\gL|E_s\:$). Given an integer $\:k\:$ put
$$
\rho(k,\gL)\:=\:\max\left\{\:\frac{\gD_s\:+\:k\gD_u}{\gd_s},\:
\frac{\gD_u\:+\:k\gD_s}{\gd_u}\:\right\}\:+\:k\:+\:1.
$$
\\

{\bf Theorem 2.3.} {\em Let $\:\eta\:$ be a local $\:(\frG,\gs)$-equivariant $\:C^K$
vector field, $\:\eta(0)\:=\:0,\:$
$\:K\:\geq\:\rho(k,\gL)\:$ and all derivatives of the difference
$\:\xi\:-\:\eta\:$ up to the order $\:K\:$ vanish on $\:E_c.\:$ Then there
is a local $\:\frG$-equivariant $\:C^k$ diffeomorphism $\:H\:$ which
conjugates $\:\xi\:$ and $\:\eta,\:$ i.e. $\:H_*\xi\:=\:\eta.\:$}

In particular, if $\:K\:=\:\infty\:$ then $\:k\:=\:\infty\:$ as well.

This theorem is the main result of the paper. Its proof is given in
Sections 3 -- 5.

It follows from the conditions of Theorem 2.3 that
$\:D\eta(0)\:=\:D\xi(0)\:=\:\gL\:$ and that the subspace $\:E_c\:$ is
invariant with respect to the phase flow of $\:\eta.\:$ It follows from
the conclusion of Theorem 2.3 that $\:\eta\:$ has local invariant
submanifolds $\:M_1\:$ and $\:M_2\:$ which are $\:C^k$ diffeomorphic
to the subspaces $\:E_u\:\oplus\:E_c\:$ and $\:E_s\:\oplus\:E_c\:$
respectively.
\\
\\
{\large\bf 2.3. Hyperbolic case.}
Recall that $\:\xi\:$ said to be {\em hyperbolic} at the origin if
$\:E_c\:=\:0,\:$ {\em i.e.}, $\:\gL\:$ has no eigenvalues in the imaginary axis.

Two local $\:C^{\infty}$ vector fields $\:\xi\:$ and $\:\eta\:$
are said to be {\em formally} $\:\frG$-{\em conjugate} if there exists a
formal $\:\frG$-equivariant transformation $\:\gG\:$ such that
$$
\gG_*\hat{\xi}\:=\:\hat{\eta}.
$$
\\

{\bf Theorem 2.4.} {\em Let $\:\xi\:$ and $\:\eta\:$ be local
$\:(\frG,\gs)$-equivariant $\:C^{\infty}$ vector fields having a
hyperbolic singularity at the origin. If they are formally $\:\frG$-conjugate
then there is a local $\:\frG$-equivariant $\:C^{\infty}$
diffeomorphism $\:H\:$ wich conjugates $\:\xi\:$ and $\:\eta.\:$}
\\

{\bf Proof.} By Corollary 2.2 one can suppose that the subspaces
$\:E_u\:\oplus\:E_c\:$ and $\:E_s\:\oplus\:E_c\:$ are $\xi$-invariant.
Let $\:\gG\:$ be a formal $\:\frG$-equivariant transformation
conjugating $\:\hat{\xi}\:$ and $\:\hat{\eta},\:$ and let $\:T\:$ be a
local $\:C^{\infty}$ diffeomorphism with the Taylor series
$\:\hat{T}\:=\:\gG.\:$ Then the vector field $\:\zeta\:=\:T_*\eta\:$ is
$\:(\frG,\gs)$-equivariant and the difference $\:\zeta\:-\:\xi\:$ is
flat at the origin. According to Theorem 2.3 $\:(E_c\:=\:0),\:$ the vector
fields $\:\zeta\:$ and $\:\xi\:$ are conjugate via a $\:\frG$-equivariant
local $\:C^{\infty}$ diffeomorphism $\:H.\:$ Hence,
$$
(HT)_*\eta\:=\:\xi
$$
and $\:HT\:$ is $\:\frG$-equivariant as well.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Box $

Theorem 2.4 itself is Equivariant Sternberg Theorem. Together with
Corollary 2.2 it provides a construction of equivariant smooth normal
forms.

Samovol \cite{S2} considers the special case of Theorem 2.4 when $\:\frG\:$
is formed by powers of a given operator $\:J\:$ of a finite order $\:(J^m\:=$
identity for some $\:m),\:$ and $\:\gs(G)\:=\:1\:$ for all
$\:G\:\in\:\frG.\:$ He shows also that the assumption of compactness for
$\:\frG\:$ is essential.

As another special case we derive from Theorem 2.1 the following answer to
the question posed in \cite{Bo}.

Given a linear transformation $\:U\:\in\:GL(n,\R),\:$ a vector field
$\:\xi\:$ is called $\:U$-{\em reversible} if
$$
U_*\xi\:=\:-\xi.
$$
\\

{\bf Corollary 2.5.} {\em Let $\:U\:$ be an orthogonal transformation,
$\:\xi\:$ and $\:\eta\:$ be local hyperbolic $\:U$-reversible
$\:C^{\infty}\:$ vector fields.  Assume that there is a formal
transformation $\:\gG\:$ commuting with $\:U\:$ such that
$\:\gG_*\hat{\xi}\:=\:\hat{\eta}.\:$ Then one can find a local
$\:C^{\infty}\:$ diffeomorphism $\:H\:$ which commutes with $\:U\:$ and
conjugates $\:\xi\:$ and $\:\eta.\:$}
\vspace*{.1cm}

{\bf Proof.}
Put
$$
\frG\:\equiv\:\overline{\bigcup\limits_{m\in\Z}\{ U^m\}},\qquad
\frG_0\:\equiv\:\overline{\bigcup\limits_{k\in\Z}\{ U^{2k}\}},\qquad
\A\:\equiv\:\overline{\bigcup\limits_{k\in\Z}\{ U^{2k+1}\}}.
$$
Since $\:U\:$ is orthogonal then $\:\frG\:$ is a compact subgroup of
$\:GL(n,\R)\:$, $\:\frG_0\:$ is a normal subgroup of $\:\frG\:$ and
$\:\frG\:=\:\frG_0\:\cup\:\A.\:$ One can easily see that given
$\:G\:\in\:\frG_0,\:$ $\:H\:\in\:\A\:$ we get $\:G_*\xi\:=\:\xi,\:$
$\:H_*\xi\:=\:-\xi.\:$ Since $\:\xi\:\not\equiv 0\:$ then $\:\frG_0\:$ and
$\:\A\:$ do not intersect.
Besides,
$$
G\:\in\:\frG_0,\ \ \ H\:\in\:\A\ \Rightarrow\ GH\:\in\:\A,\ \ \
HG\:\in\:\A
$$
and
$$
H_1,\:H_2\:\in\:\A\ \Rightarrow\ H_1H_2\:\in\:\frG_0.
$$
Therefore, by putting
$$
\gs(G)\:=\:1\ \ \ (G\:\in\:\frG_0),\ \ \gs(H)\:=\:-1\ \ (H\:\in\:\A)
$$
we get a continuous character $\:\gs\:$ of the compact subgroup
$\:\frG\:\subset\:O(n).\:$

The vector fields $\:\xi\:$ and $\:\eta\:$ turn out to be
$\:(\frG,\gs)$-equivariant,
and the formal conjugacy $\:\gG\:\:$ --- $\:\:\frG$-equivariant. It
remains to apply Theorem 2.4.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \Box $

\section{Deformation method.}
$\ \ \ \ \ \ \ \ $
\\
As we have mentioned above, the deformation method reduces a conjugacy
problem to
that of solving a multidimensional linear equation.

Given vector fields $\:\xi\:$ and $\:\eta\:$ consider the {\em
deformation} $\:h(x,\gep)\:\equiv\:\gep\eta(x)\:+\:(1\:-\:\gep)\xi(x)\:$
and the vector field
$$
\Xi\:\equiv\:h(x,\gep)\frac{\partial}{\partial
x}\:+\:0\cdot\frac{\partial}{\partial\gep}
$$
on $\:\R^{n+1}.\:$ Take another vector field
$$
\Phi(x,\gep)\:\equiv\:\gvp(x,\gep)\frac{\partial}{\partial
x}\:+\:1\cdot\frac{\partial}{\partial\gep}
$$
Let
$$
Q^t(x,\gep)\:=\:(q^t(x,\gep),\gep),\ \ \
P^t(x,\get)\:=\:(p^t(x,\gep),\gep\:+\:t)
$$
be the pase flows of $\:\Xi\:$ and $\:\Phi\:$ respectively.
Put $\:H(x)\:=\:p^1(x,0).\:$
\\

{\bf Lemma 3.1.} {\em Let the vector fields $\:\Phi\:$ and $\:\Xi\:$
commute, i.e.}
$$
[\:\Phi,\Xi\:]\:=\:0.
\eqno (1)
$$
{\em Then $\:H\:$ conjugates $\:\xi\:$ and $\:\eta,\:$ i.e.,
$\:H_*\xi\:=\:\eta.\:$}
\vspace*{.1cm}

{\bf Proof.} Note that the phase flows of $\:\xi\:$ and $\:\eta\:$ are,
respectively, $\:q_0^t(x)\:\equiv\:q^t(x,0)\:$ and
$\:q_1^t(x)\:\equiv\:q^t(x,1).\:$
Since the phase flows $P^t$ and $Q^t$ commute then
$$
Q^{t_2}(P^{t_1}(x,\ve))=(q^{t_2}(p^{t_1}(x,\ve),\ve+t_1),\ve+t_1)=
(p^{t_1}(q^{t_2}(x,\ve),\ve),\ve+t_1)=P^{t_1}(Q^{t_2}(x,\ve)),
$$
or
$$
q^{t_2}(p^{t_1}(x,\ve),\ve+t_1)=p^{t_1}(q^{t_2}(x,\ve),\ve).
$$
For $\ve=0$, $t_1=1$, $t_2=t$ this gives
$$
q^{t}(p^{1}(x,0),1)=p^{1}(q^{t}(x,0),0).
$$
For $H(x)=p^1(x,0)$ we get
$$
q^t(H(x),1)=H(q^t(x,0)),
$$
or
$$
q^t_1(H(x))=H(q^t_0(x)),
$$
{\it i.e.}, $H$ is a conjugacy between $\xi$ and $\eta$.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \Box $
\\

Condition (1) is equivalent to the equation
$$
D_1\gvp(x,\gep)\cdot
h(x,\gep)\:=\:D_1h(x,\gep)\cdot\gvp(x,\gep)\:+\:\eta(x)\:-\:\xi(x),
\eqno (2)
$$
where
$$
D_1\:\equiv\:\frac{\partial}{\partial x}.
$$
This equation is called the {\em cohomology equation}.

\section{$\frG$-equivariant conjugacy}
$\ \ \ \ \ \ \ \ $
\\
Thus, we have shown that the deformation method
reduces a conjugacy problem to that of finding a smooth solution
$\varphi(x,\varepsilon)$ of the linear non-homogeneous partial
differential
equation (2), and then to
integrating the system ${\dot x}=\varphi(x,\varepsilon)$,
${\dot \varepsilon} =1$. The problem of solvability of the cohomology
equation
is considered in Section 5. Here we assume that a local $C^k$ solution
$\vf$
exists, and prove $\frG$-equivariant conjugacy between $\xi$ and $\eta$.

\vspace*{.1cm}
{\bf Lemma 4.1.} {\it If $\vf$ is a $C^k$ solution of equation} (2)
{\it then for every $U\in \frG$ the map
$$
(U_*^{-1}\vf)(x,\ve)\equiv U^{-1}\vf(Ux,\ve),
$$
is a solution of equation} (2) {\it as well.}

{\bf Proof.} Denote $\tilde\vf =U_*^{-1}\vf$. By virtue of
$(\frG,\sigma)$-equivariance of $\xi$ and $\eta$, the following equalities
are true:
\begin{eqnarray}
Uh(x,\ve)&=&\sigma(U)h(Ux,\ve);\nonumber\\
UD_1h(x,\ve)&=&\sigma(U)D_1h(Ux,\ve)U;\nonumber\\
U(\eta-\xi)(x,\ve)&=&\sigma(U)(\eta-\xi)(Ux,\ve).\nonumber
\end{eqnarray}
Hence,
\begin{eqnarray}
D_1\ti\vf(x,\ve)h(x,\ve)&=&U^{-1}D_1\vf(Ux,\ve)Uh(x,\ve)\nonumber\\
&=&\sigma(U)U^{-1}D_1\vf(Ux,\ve)h(Ux,\ve)\nonumber\\
&=&\sigma(U)U^{-1}[D_1h(Ux,\ve)\vf(Ux,\ve)+(\eta-\xi)(Ux,\ve)]\nonumber\\
&=&U^{-1}[\sigma(U)D_1h(Ux,\ve)]\vf(Ux,\ve)+U^{-1}[\sigma(U)
(\eta-\xi)(Ux,\ve)]
\nonumber\\
&=&U^{-1}UD_1h(x,\ve)U^{-1}\vf(Ux,\ve)+U^{-1}U(\eta-\xi)(x,\ve)\nonumber\\
&=&D_1h(x,\ve)\ti\vf(x,\ve)+(\eta-\xi)(x,\ve).\nonumber
\end{eqnarray}
Therefore, $\ti\vf$ is a solution of the cohomology equation.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \Box $

\vspace*{.1cm}
It is well known that since $\frG$ is compact then one can find a
Haar measure $\mu$ on $\frG$ which is right invariant with respect to
$\frG$. We can suppose without loss of generality that $\mu(\frG)=1$.
Let $\vf$ be a $C^k$ solution of equation (2).
Define a $C^k$ function $\psi:\R^{n+1}\to\R^n$ by
$$
\psi(x,\ve)=\int\limits_{\frG}U^{-1}\vf(Ux,\ve)d\mu(U)\qquad(x\in\R^n,\
\ve\in\R).
\eqno (3)
%%\label{DM3}
%%\ee
$$
\\

{\bf Lemma 4.2.} {\em The function $\:\psi\:$ is a $\:\frG$-equivariant
solution of the cohomology equation} (2).

\vspace*{.1cm}
{\bf Proof.} By linearity of the cohomology equation, it follows from
Lemma 5.1
that $\:\psi\:$ is a solution. Since $\:\mu\:$ is right invariant,
for any $\:V\:\in\:\frG\:$ we have
$$
V^{\-1}\Psi(Vx,\gep)\:=\:V^{-1}\int\limits_{\frG}U^{-1}\gvp(UVx,\gep)d\mu(U)
\:=
$$
$$
=\:\int\limits_{\frG}(UV)^{-1}\gvp(UVx,\gep)d\mu(U)\:=\:
\int\limits_{\frG}W^{-1}\gvp(Wx,\gep)d\mu(W)\:=\:\psi(x,\gep).
$$
Here we put $\:W\:=\:UV.\:$ As a result we get
$$
V^{-1}\psi(Vx,\gep)\:=\:\psi(x,\gep),
$$
{\em i.e.}, $\:\psi\:$ is $\:\frG$ - equivariant.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Box $
\\

{\bf Proof of Theorem 2.3.} Take a local $\:C^k\:$ solution $\:\gvp\:$ of
equation (2). Then the function $\:\psi\:$ given by formula (3) is a local
$\:\frG$-equivariant $\:C^k\:$ solution of (2). Let
$\:F^t(x,\gep)\:\equiv\:(f^t(x,\gep),\gep\:+\:t)\:$ be the phase flow
of the vector field
$$
\Psi(x,\gep)\:=\:\psi(x,\gep)\frac{\partial}{\partial x}\:+\:
1\cdot\frac{\partial}{\partial\gep}.
$$
Put $\:H(x)\:=\:f^1(x,0).\:$ Since $\:[\Psi,\Xi]\:=\:0\:$
then, by Lemma 3.1, the local
$\:C^k$ diffeomorphism $\:H\:$ conjugates $\:\xi\:$ and $\:\eta.\:$
Show that $\:H\:$ is $\:\frG$-equivariant. Choose
$\:U\:\in\:\frG\:$ and denote
$\:\tilde{f}^t(x,\gep)\:\equiv\:U^{-1}f^t(Ux,\gep).\:$ Then
$$
\left.\frac{d\tilde{f}^t(x,\gep)}{dt}\right|_{t=0}\:=
\:\left.U^{-1}\frac{df^t(Ux,\gep)}{dt}\right|_{t=0}\:=\:
U^{-1}\psi(Ux,\gep)\:=\:\psi(x,\gep).
$$
Hence $\:\tilde{F}^t(x,\gep)\:\equiv\:(\tilde{f}^t(x,\gep),\gep\:+\:t)\:$
is also a phase flow of the vector field $\:\Psi.\:$ It follows from
Uniqueness Theorem that $\:\tilde{F}^t\:=\:F^t\:$ and
$\:H(x)\:=\:U^{-1}H(Ux).\:$ We conclude that $\:H\:$ is
$\:\frG$-equivariant
completing the proof of Theorem 2.3.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Box $

\section{Solution of the cohomology equation}
$\ \ \ \ \ \ \ \ $
\\
Thus it remains to show that the family of linear non-homogeneous partial
differential equations
$$
D_1\vf(x,\ve)h(x,\ve)=A(x,\ve)\vf(x,\ve)+\gamma(x)\qquad (x\in\R^n,\
\ve\in I=[0,1])
\eqno (4)
$$
has a local solution of the corresponding smoothness class. We consider
here
in detail the $C^{\infty}$ case. It should be noted that the method of
proof
applied can be easily transfered to a finitely smooth case (with some
natural
modifications in definitions of semi-norms and in estimates).

\vspace*{.1cm}
We suppose in what follows that the functions $h$, $A$ and $\gamma$ are
all
$C^{\infty}$ smooth. Since the problem under consideration is local, we
assume
without loss of generality that the functions $A$ and $\gamma$ have
compact
supports, and the vector field $h$ takes the form
$$
h(x,\ve)=\Lambda x+g(x,\ve),
$$
where the function $g$ has a finite support as well.
Remind that the space $\R^n$ is splitted into the direct sum of
$\Lambda$-invariant linear subspaces
$$
\R^n=E_s\oplus E_u\oplus E_c
$$
in such a way that all the real parts of the eigenvalues of the operator
$\Lambda_s\equiv\Lambda|E_s$ ($\Lambda_u\equiv\Lambda|E_u$;
$\Lambda_c\equiv\Lambda|E_c$) are negative (positive; equal to zero).
Accept the notation
$$
x\equiv(u,v,w)\in E_s\oplus E_u\oplus E_c.
$$

Denote by $\:F^t(x,\gep)\:$ the phase flow of the vector field
$$
h(x,\gep)\:=\:(\gL x\:+\:g(x,\gep))\frac{\partial}{\partial x},
$$
and assume that the subspaces $\:E_u\:\oplus\:E_c\:$ and
$\:E_s\:\oplus\:E_c\:$ are $\:F^t$ - invariant for all $\:\gep\:\in\:I.\:$
\\

{\bf Lemma 5.1.} {\em Let all the derivatives of a $\:C^{\infty}\:$ vector
function $\:\gamma\:$ vanish on $\:E_c,\:$ i.e.,}
$$
D^k\gamma(0,0,w)\:=\:0\qquad (k\:=\:0,1,\:...\:).
$$
{\em Then the equation {\rm (4)} has a $\:C^{\infty}\:$ solution
$\:\gvp(x,\gep)\:$ defined in a neighborhood of the interval
$\:I\:\subset\:\R^{n+1}.\:$}
\\

{\bf Proof.} One can choose a number $\alpha>0$ such that
\begin{eqnarray}
\|\pr_{E_s}F^t(x,\ve)\|&\le&\|u\|e^{-\alpha t}\qquad(t>0),\nonumber\\
\|\pr_{E_u}F^t(x,\ve)\|&\le&\|v\|e^{\alpha t}\qquad(t<0).\nonumber
\end{eqnarray}
Show that for each $k=1,2,\dots$ there is a number $C_k>0$ such that
$$
\|D^i_1F^t(x,\ve)\|\le C_ke^{C_kt}\qquad(1\le i\le k;\ x\in\R^n,\ \ve\in
I,\
t>0).
$$
Consider first the case $k=1$. Observe that
$$
{d(D_1F^t(x,\ve))\over dt}=\left(\Lambda+D_1g(F^t(x,\ve),\ve)\right)
D_1F^t(x,\ve).
$$
Therefore
$$
D_1F^t(x,\ve)=\left.D_1F^t(x,\ve)\right|_{t=0}+\int_0^t
\left(\Lambda+D_1g(F^s(x,\ve),\ve)\right)D_1F^s(x,\ve)ds.
$$
Hence for all $t>0$
$$
\|D_1F^t(x,\ve)\|\le B+C\int_0^t\|D_1F^s(x,\ve)\|ds.
$$
By Gronwall inequality, this gives
$$
\|D_1F^t(x,\ve)\|\le Be^{Ct}\le C_1e^{C_1t}\qquad (t>0).
$$
The same arguments work in a general case. Taking into account that
$$
{d(D_1^kF^t(x,\ve))\over dt}=\left(\Lambda+D_1g(F^t(x,\ve),\ve)\right)
D_1^kF^t(x,\ve)+R(x,\ve,t),
$$
where the function $R$ is polynomial with respect to the functions
$D_1^iF^t(x,\ve)$ $(i=1,\dots,k-1)$, we conclude that
$$
\|D_1^kF^t(x,\ve)\|\le De^{Mt}+C\int_0^t\|D_1^kF^s(x,\ve)\|ds.
$$
Apply again Gronwall inequality and get the desired estimate.

\vspace*{.1cm}
Prove that the function $\gamma$ can be decomposed into the sum of two
$C^{\infty}$ functions which are flat, respectively, on $E_u\oplus E_c$
and
on $E_s\oplus E_c$, {\it i.e.}
$$
\gamma=\gamma_++\gamma_-,
$$
and
$$
D^k\gamma_+(0,v,w)=D^k\gamma_-(u,0,w)=0\qquad (k=1,2,\dots;\
(u,v,w)\in\R^n).
$$
In fact, by Whitney Theorem, one can find a $C^{\infty}$ function
$\gamma_+$
such that
$$
D^k\gamma_+(0,v,w)=0,\quad D^k\gamma_+(u,0,w)=D^k\gamma(u,0,w)\qquad
(k=1,2,\dots;\ (u,v,w)\in\R^n).
$$
Put $\gamma_-=\gamma-\gamma_+$. Then
$$
D^k\gamma_-(u,0,w)=0\qquad (k=1,2,\dots;\ (u,v,w)\in\R^n).
$$
Note that since the cohomology equation has an affine structure, suffice
it
to find solutions $\vf_+$ and $\vf_-$ of the equations
$$
D_1\vf_{\pm}(x,\ve)h(x,\ve)=A(x,\ve)\vf_{\pm}(x,\ve)+\gamma_{\pm}(x).
$$
Then a solution we are looking for is given by
$$
\vf=\vf_++\vf_-.
$$

\vspace*{.1cm}
Show that the equation
$$
D_1\vf_+(x,\ve)h(x,\ve)=A(x,\ve)\vf_+(x,\ve)+\gamma_+(x)
\eqno (5)
$$
has a $C^{\infty}$ solution. Define a linear operator $\Lc$ by
$$
(\Lc\tau)(x,\ve)=-\int_0^{\infty}\tau(F^t(x,\ve),\ve)dt.
$$
Let $\frM_+\subset C^{\infty}(\R^n\times I,\R^n)$ be the set of functions
which are flat on $E_u\oplus E_c$, and let $\frN_+\subset\frM_+$ be  the
subset
of
functions having compact supports. For a positive integer $k$ and a
positive
$\nu$ put
$$
\|\vf\|_{k\nu}=\sup_{\r^n\times
I}\max_{j=0,\dots,k}{\|D_1^j\vf(x,\ve)\|\over
\|u\|^{\nu}},\ \ \ x\:=\:(u,v,w).
$$
By $\frM^k_{+\nu}$ ($\frN^k_{+\nu}$) denote the set of $C^{\infty}$
functions
(having compact supports) such that $\|\vf\|_{k\nu}<\infty$.

\vspace*{.1cm}
Let us prove {\it that the operator $\Lc$ transfers $\frN_+$ into
$\frM_+$.}
In fact, let $\tau\in\frN_+$, $\vf=\Lc\tau$. Put
$$
\tau_t(x,\ve)=\tau(F^t(x,\ve),\ve).
$$
Then
\begin{eqnarray}
\|D_1^k\tau_t(x,\ve)\|&\le&\|D_1^k\tau(F^t(x,\ve),\ve)\|\|D_1F^t(x,\ve)\|^k
\nonumber\\
&&+\sum_{j=1}^{k-1}\sum_pE_{jp}\|D_1^j\tau(F^t(x,\ve),\ve)\|\|D_1^{p_1}F^t(x
,\ve)\|
\cdots\|D_1^{p_j}F^t(x,\ve)\|,\nonumber
\end{eqnarray}
where $\:p\:=\:(p_1,\dots,p_j),\:$
$p_1\:+\:...\:p_s\:=\:k\:$ and $\:E_{jp}\:$ are some constants.
Since $\tau\in\frN_+$ then $\tau\in\frN_{+\nu}^k$ for all $k$ and $\nu$.
The prvevious estimate gives
\begin{eqnarray}
{\|D_1^k\tau_t(x,\ve)\|\over\|u\|^{\nu}}&\le&
{\|D_1^k\tau(F^t(x,\ve),\ve)\|\over\|\pr_{E_s}F^t(x,\ve)\|^{\nu}}\cdot
{\|\pr_{E_s}F^t(x,\ve)\|^{\nu}\over\|u\|^{\nu}}
\|D_1F^t(x,\ve)\|^k
\nonumber\\
&&\nonumber\\
&&+\sum_{j=1}^{k-1}\sum_pE_{jp}
{\|D_1^j\tau(F^t(x,\ve),\ve)\|\over\|\pr_{E_s}F^t(x,\ve)\|^{\nu}}\cdot
{\|\pr_{E_s}F^t(x,\ve)\|^{\nu}\over\|u\|^{\nu}}\nonumber\\
&&\nonumber\\
&&\mathop{\phantom{\sum\sum}}
\times\|D_1^{p_1}F^t(x,\ve)\|\cdots\|D_1^{p_j}F^t(x,\ve)\|.\nonumber
\end{eqnarray}
Hence
\begin{eqnarray}
\|\tau_t\|_{k\nu}&\le&\|\tau\|_{k\nu}e^{-\alpha\nu t}C_k^ke^{C_kkt}+
\sum_p\sum_{j=1}^{k-1}E_{jp}\|\tau\|_{k\nu}e^{-\alpha\nu
t}C_k^{k-1}e^{C_k(k-1)t}
\nonumber\\
&&\nonumber\\
&\le& R_k\|\tau\|_{k\nu}e^{(-\alpha\nu+C_kk)t}.\nonumber
\end{eqnarray}
Since
$$
\|D_1^k\tau_t(x,\ve)\|\le\|\tau_t\|_{k\nu}\|u\|^{\nu}
$$
we have
$$
\|D_1^k\vf(x,\ve)\|\le R_k\|\tau\|_{k\nu}\|u\|^{\nu}\int_0^{\infty}
e^{(-\alpha\nu+C_kk)t}dt.
$$
Choose $\nu_k$ in such a way that if $\nu\ge\nu_k$ then
$-\alpha\nu+C_kk<0$.
Then the integral converges and
$$
\|D_1^k\vf(x,\ve)\|\le
R_k\|\tau\|_{k\nu}\|u\|^{\nu}{1\over|-\alpha\nu+C_kk|}.
$$
We conclude that $\vf\in\frM^k_{+\nu}$ for all $k$ and $\nu\ge\nu_k$.
Hence
$\vf\in C^{\infty}$ and $D^k\vf(0,v,w,\ve)=0$ $(k=0,1,\dots;\
(v,w,\ve)\in E_u\oplus E_c\times I)$. Therefore $\vf\in\frM_+$.

Now consider the parial differential equation
$$
D_1\vf(x,\ve)h(x,\ve)=\tau(x,\ve).
\eqno (6)
$$
Chek that the formula
$$
\vf=\Lc\tau
$$
gives a solution of equation (6).

In fact,
\begin{eqnarray}
\vf(F^s(x,\ve),\ve)&=&-\int_0^{\infty}\tau(F^{t+s}(x,\ve),\ve)dt=
-\int_s^{\infty}\tau(F^{t}(x,\ve),\ve)dt\nonumber\\
&&\nonumber\\
&=&-\int_0^{\infty}\tau(F^{t}(x,\ve),\ve)dt+\int_0^s\tau(F^{t}(x,\ve),\ve)dt
\nonumber\\
&&\nonumber\\
&=&\vf(x,\ve)+\int_0^s\tau(F^{t}(x,\ve),\ve)dt.\nonumber
\end{eqnarray}
Differentiate both parts of the equality with respect to $s$, and put
$s=0$.
We get
$$
D_1\vf(F^s(x,\ve),\ve)\cdot\left.{dF^s(x,\ve)\over ds}\right|_{s=0}=
\left.\tau(F^s(x,\ve),\ve)\right|_{s=0},
$$
or
$$D_1\vf(x,\ve)h(x,\ve)=\tau(x,\ve).
$$

\vspace*{.1cm}
Now define a linear operator $\T$ by the formula
$$
(\T\vf)(x,\ve)=A(x,\ve)\vf(x,\ve)+\gamma_+(x).
$$
Thus we reduce our problem to solving the equation
$$
\vf=\Lc\T\vf.
\eqno (7)
$$
Prove that one can find a $C^{\infty}$ solution which is flat on
$E_u\oplus
E_c$.
Since $\gamma_+\in\frN^k_{+\nu}$ for every $k$ and $\nu$, and the function
$A$
has a compact support, the operator $\T$ transfers $\frM^k_{+\nu}$ into
$\frN^k_{+\nu}$, and besides,
$$
\|\T\vf\|_{k\nu}\le P_k\|\vf\|_{k\nu}+\|\gamma_+\|_{k\nu}
$$
for some constants $\:P_k.\:$
Therefore,
$$
\|\Lc\T\vf\|_{k\nu}\le
{Q_k\over|-\alpha\nu+C_kk|}(\|\vf\|_{k\nu}+\|\gamma_+\|_{k\nu})
$$
for some constants $\:Q_k.\:$
Clearly, $\Lc\T$ transfers $\frM_+$ to $\frM_+$. In the space
$C^{\infty}(\R^n\times I,\R^n)$ equiped with the compact-open topology, we
are going to choose a set which is convex and compact with
respect to this topology, and such that
the operator $\Lc\T$ maps it into itself. To this end, for each
$k=1,2,\dots$
take a number $\nu_k$ in such a way that
$$
{Q_k\over|-\alpha\nu+C_kk|}\le {1\over 2}.
$$
Denote
$$
a_k={Q_k\over|-\alpha\nu+C_kk|}\|\gamma_+\|_{k\nu_k}.
$$
Put
$$
M_k=3a_k.
$$
Set
$$
\M_k=\{\vf\in\frM_+:\ \|\vf\|_{k\nu_k}\le M_k\}\qquad (k=1,2,\dots),
$$
and
$$
\M=\bigcap_k\M_k.
$$

The set $\M$ is a convex compact subset of $\frM_+$.
Therefore $\Lc\T(\M)\subset\M$, and by Tihonov Fixed Point Theorem
\cite{T} the
set
$\M$ contains a fixed point $\vf_0$ of the operator $\Lc\T$. Clearly
$\vf_0$
is a $C^{\infty}$ solution of equation (7).

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