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%TITLES.3
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\def\proof{\medskip\noindent{\bf Proof.\ }}
\def\qed{\hfill\smallskip
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\ \ \ \ \ \ }\bigskip}
%
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\def\firstremark{\bigskip\noindent{\bf Remarks.}\nextremark}
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\def\equ(#1){\hskip-0.03em\csname e#1\endcsname}
\def\clm(#1){\csname c#1\endcsname}
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%MACROS.12b
\footline{\ifnum\pageno=0\hss\else\hss\tenrm\folio\hss\fi}
\let\ov=\overline
\let\cl=\centerline
\let\wh=\widehat
\let\wt=\widetilde
\let\eps=\varepsilon
%
\input amssym.def
\input amssym.tex
\def\RG{\Re}
\def\bigRG{\hbox{\cbold <}}
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%
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%
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%
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\def\VV{{\cal V}}
\def\WW{{\cal W}}
\def\XX{{\cal X}}
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%ABSTRACT.1
%
\def\rKol{1}
\def\rArn{2}
\def\rMo{3}
\def\rRdL{4}
\def\rElia{5}
\def\rED{6}
\def\rMcK{7}
\def\rMEsc{8}
\def\rTom{9}
\def\rKS{10}
\def\rSK{11}
\def\rKos{12}
\def\rFT{13}
\def\rG{14}
\def\rGM{15}
\def\rGGM{16}
\def\rFKS{17}
\def\rORSS{18}
\def\rR{19}
\def\rHP{20}
\def\rBDGPS{21}
\def\rAbad{22}
%
\vskip.4in
\cl{{\huge A Renormalization Group for Hamiltonians,}}
\cl{{\huge with Applications to KAM Tori}}
\vskip.6in
\cl{Hans Koch
\footnote{$^1$}
{{\small Supported in Part by the
National Science Foundation under Grant No. DMS--9401422,
\hfil\break
and by the Texas Advanced Research Program
under Grant No. ARP--035}}}
\cl{Department of Mathematics, University of Texas at Austin}
\cl{Austin, TX 78712}
\vskip.25in
\vskip.8in
\abstract
We describe a family of renormalization group transformations
which apply to the problem of stability of invariant tori
for analytic Hamiltonians.
Each of these transformations corresponds to a fixed rotation vector,
within a certain class of vectors in $\real^d$.
As an application, we prove a KAM--type theorem
that yields analytic invariant $d$--tori
depending analytically on the Hamiltonian.
\par\vfill\eject
%INTRO.5
\section Introduction and Main Results
In this paper we describe a family of renormalization group transformations
for analytic Hamiltonians in dimension $2d\ge 4$.
Each of these transformations is associated
with a vector $\omega=(1,\omega_2,\ldots,\omega_d)$ in $\real^d$
whose components span an algebraic number field of degree $d$,
and is designed to apply to the stability problem
for invariant $d$--tori with rotation vectors proportional to $\omega$.
Our transformations are similar in spirit to the
block spin transformation in statistical mechanics,
in the sense that they use a process of ``elimination and rescaling''.
The elimination (of irrelevant frequencies)
is done by means of canonical transformations,
as in KAM theory [\rKol--\rRdL],
and the scaling (of the frequency lattice)
involves an integral matrix $T$ with eigenvector $\omega$.
In the case $d=2$, with $\omega_2$ equal to the golden mean,
$T$ is related to the composition operator
used in the commuting map approach to dissipative invariant tori [\rFKS--\rR].
Although the concept is non-perturbative,
our transformations are currently defined only
near certain trivial (integrable) fixed points.
Thus, the applications given in this paper
fall in the domain of KAM theory.
Connections between KAM theory and renormalization
have been described before [\rED--\rGGM].
The most recent work in this area
has concentrated on the problem of resumming the Lindstedt series [\rElia],
using techniques from quantum field theory
to control cancelations and obtain
a convergent ``renormalized expansion'' [\rFT--\rGGM].
As in ``ordinary'' KAM theory,
we apply an iterative procedure that
eliminates irrelevant degrees of freedom from a given
non-integrable Hamiltonian $H_0\,$.
At the $k$-th step, $H_{k-1}$ is transformed
into a new Hamiltonian $H_k$
which is in some sense equivalent to $H_{k-1}\,$,
but closer to integrable.
As usual, this step involves a canonical transformation,
with a certain frequency cutoff.
But there are also notable differences:
\smallskip
\item{$\circ$} The transformation $\RG\colon H_{k-1}\mapsto H_k$
acts on a fixed space of Hamiltonians.
\item{$\circ$} There is no loss of analyticity;
in fact, $\RG$ is analyticity improving.
\item{$\circ$} There are no small denominator problems,
and no major cancelations.
\item{$\circ$} On a conceptual level,
the procedure is non-perturbative.
\smallskip\noindent These are of course typical features
of a (non-perturbative) renormalization group transformation.
The absence of small denominators is e.g. due to the fact
that $\RG$ includes a certain shift, which turns the ``next scale''
into the scale that has been considered before.
The main problem here is to implement such a shift,
and at the same time, improve analyticity.
Before we explain in detail the action of $\RG$,
let us try to sketch the main ideas of non-perturbative renormalization.
After that, we will give a simple example,
which in fact motivated the present work.
The typical setup for a renormalization group analysis
is an open set $\BB$ in some infinite dimensional space,
and a densely defined function $F$ on $\BB$ that takes values
in some $n$--dimensional manifold $\YY$.
The function $F$ plays the role of an observable,
and the level sets of $F$ represent objects that are equivalent
with respect to the phenomenon that is being observed.
Let us assume that some countable set $Y\subset\YY$
of possible values of $F$ has been singled out as particularly interesting.
In many cases of interest, the function $F$ has a ``critical surface''
$\WW\subset\BB$, of the following type:
There exist sequences $(y_0,y_1,\ldots)$ in $Y$,
such that locally, the distance between $F^{-1}(y_k)$ and $\WW$
decreases asymptotically like $|\lambda|^{-k}$, for some
universal number $\lambda$ of modulus $>1$.
In experiments
on $n$-parameter families $V: A\subset\real^n\to\BB$,
the surface $\WW$ shows up as a critical point
of the function $f=F\circ V$,
and the behavior of $f$ near this point appears to be
independent of the family $V$.
Due to this ``universality'',
it is possible to determine all of the relevant quantities
from a single function $f: A\to\YY$,
and the system $(\BB,F)$ need not even be known.
In fact, the general picture described above
is usually part of the (desired) explanation
of universal critical behavior,
observed in experiments on $n$-parameter families.
If the critical surface $\WW$ is (believed to be) smooth,
then the renormalization group method
is a potentially powerful tool for analyzing such behavior.
The idea is to construct a transformation $\RR: B\subset\BB\to\BB$
that has a hyperbolic fixed point,
whose local stable manifold coincides with the critical surface $\WW$.
Then every $n$--parameter family $V: A\to B$
that intersects $\WW$ transversally near the fixed point,
converges (under suitable reparametrization)
to an invariant family describing the unstable manifold.
But in order for this to explain universality,
there has to be a certain relationship
between the functions $F\circ\RR$ and $F$.
In all known examples, this holds by construction:
There exists a (natural) map $r$ on $Y$;
and the transformation $\RR$ is chosen
such that $F(\RR(h))=y$, whenever $F(h)=r(y)$.
In this case, if $(y_0,y_1,\ldots)$ is an orbit for $r$,
then the abovementioned universal number $\lambda$
should be the largest eigenvalue (in modulus)
of the derivative of $\RR$ at the fixed point.
In other words: A renormalization group transformation
contracts differences that are irrelevant
(for the phenomenon being observed),
and expands the relevant ones.
It can also be viewed as a tool for introducing
appropriate coordinates for the study of the observable $F$.
As a trivial (but instructive) example,
consider the space $C^2(G)$ of twice continuously differentiable
functions on a given disk $G\subset\real^2$, centered at the origin.
To every function $h\in C^2(G)$,
we can associate a Hamiltonian vector field
$(q,p)\mapsto(\nabla h(p),0)$, and thus a flow,
on the product of the torus $\torus^2$ with $G$.
The corresponding orbit through the origin
can be characterized by its slope $F(h)=w_2/w_1\,$,
where $w=(\nabla h)(0)$.
We now consider $F:\BB\subset C^2(G)\to\real$ to be our observable.
Here, $\BB$ is a small open neighborhood
of the following function $h^\eps$.
Let $\omega=(1,\vartheta)$ and $\omega'=(-\vartheta,1)$,
with $\vartheta$ equal to the golden mean
$\half+\half\sqrt{5}=1.618\ldots\,$,
and define $h^\eps(p)=\omega\cdot p+\eps(\omega'\cdot p)^2$,
for some fixed $\eps\ge 0$.
We note that $\omega$ and $\omega'$ are the two eigenvectors
of the matrix $T=\left({0~1\atop 1~1}\right)$,
for the eigenvalues $\vartheta$ and $\vartheta'=-1/\vartheta$, respectively.
As distinguished values of $F$,
consider the best rational approximates $y_k=b_k/a_k$
for $\vartheta$. That is, $y_0=1$, and $y_k=r(y_{k-1})$ if $k\ge 1$,
where $r(y)=1+1/y$. The corresponding integer vectors $(a_k,b_k)$
can be obtained by iterating the map $T$,
starting with $(a_0,b_0)=(1,1)$.
This shows that the sequence $(y_0,y_1,\ldots)$ converges to $\vartheta$,
and that $y_k-\vartheta$ is asymptotically proportional
to $(-\vartheta^2)^{-k}$.
Thus, if $v\mapsto h_v$ is a ``typical''
differentiable one-parameter family, passing near $h^\eps$,
we will find a value $v=v_\infty$
for which $F(h_v)=\vartheta$,
and a sequence $(v_m,v_{m+1},\ldots)$,
accumulating at $v_\infty\,$,
such that $F(h_{v_k})=y_k\,$, for all $k\ge m$.
Furthermore, since $F$ is actually differentiable near $h^\eps$,
the differences $v_k-v_\infty$
are asymptotically proportional to $(-\vartheta^2)^{-k}$.
Although there is not much left to explain in this example,
it is still interesting to construct
a useful renormalization group transformation.
As a first guess, consider the map $\RR:h\mapsto h\circ T^{-1}$.
This map provides the desired relationship
between $F\circ\RR$ and $F$, described earlier.
But unfortunately, it does not have a fixed point near $h^\eps$.
The following step is intended to correct this and other shortcomings,
while keeping the essential features of our first guess.
This means that $\RR$ will be modified only in ``irrelevant'' directions.
Here, the symmetries of $F$ play an important role:
$F(h)$ does not change if we
add to $h$ a constant function, or if we
apply a scaling transformation $h\mapsto\mu^{-1}h(\mu .)$, with $\mu\not=0$.
In addition, $h$ may be multiplied by a constant $b\not=0$.
The first two transformations
do not affect the motion on the orbit passing through the origin.
The third transformation (scaling of time)
does not change any orbits, but the velocities get multiplied by $b$.
Thus, let us modify our original guess for $\RR$ by setting
$$
\bigl(\RR(h)\bigr)(p)={b\over\mu}
\Bigr[h\bigl(\mu T^{-1}p\bigr)-h(0)\Bigr]\,,
\equation(almostLmu)
$$
where $b=b(h)$ and $\mu=\mu(h)$ are to be determined.
We note that if we choose $b=\vartheta$ constant,
then $h^0$ is a fixed point of $\RR$,
and if in addition $|\mu|=\vartheta^{-3}$,
then each of the functions $h^\eps$ is a fixed point for $\RR$.
(The same holds for any constant multiples of these functions,
if the values of $b$ and $\mu$ are fixed.)
In each of these cases, the derivative of $\RR$,
at the indicated fixed point, has an eigenvalue $-\vartheta^2$,
and the corresponding eigenfunction is $p\mapsto\omega'\cdot p$.
By choosing $b=b(h)$ and $\mu=\mu(h)$ appropriately,
it is possible to make $h^\eps$ an isolated fixed point
of $\RR$, for any given $\eps$,
and to obtain $\lambda=-\vartheta^2$ as the only eigenvalue
of $D\RR(h^\eps)$ of modulus $\ge 1$.
Then the stable manifold of $\RR$ at $h^\eps$
coincides with the ``critical surface''
defined by the equation equation $F(h)=\vartheta'$.
We will not give any proofs or details here,
as this example is essentially a special case
of what will be discussed later on.
Our next goal is to extend the transformation $\RR$,
and analogous transformations for higher dimensional systems,
to ``general'' (but analytic) Hamiltonians $H=H(q,p)$
near a Hamiltonian of the form $H^0(q,p)=\omega\cdot p$.
Conditions on the rotation vector $\omega\in\real^d$
will be specified below.
In this paper, we are mainly interested in
the properties of the renormalization group transformation,
its local stable and unstable manifold,
and in the existence of invariant tori with rotation vectors $c\omega$.
The accumulation of closed orbits at these tori,
and other aspects related to universality,
will be discussed elsewhere [\rAbad].
Since we will no longer mention any observables,
let us give (without proof) a possible characterization of the
``critical'' surface in this more general setting.
A Hamiltonian near $H^0$ lies on $\WW$
if it has an invariant $d$--torus
whose rotation vector is a constant multiple of $\omega$,
and which is ``centered at $p=0$'', in the sense that the integrals
$$
A_j={1\over 2\pi}\oint\limits_{C_j}\sum_{i=1}^d p_i dq_i\,,
\qquad j=1,2,\ldots,d\,,
\equation(AjDef)
$$
vanish. Here, $C_j$ denotes a closed path on the invariant $d$--torus
with winding number $1$ for the $j$-th $1$-torus,
and winding number $0$ for the other $1$-tori.
We note that the quantities $A_j$ are invariant
under canonical transformations $U_\phi$
associated with a globally defined generating function $\phi$.
(For a definition of $U_\phi\,$, see Section 2.)
In what follows, a Hamiltonian is an analytic function
defined on a complex neighborhood of $\torus^d\times\{0\}$,
where $\torus^d$ denotes the $d$--dimensional torus
and $0$ the origin in $\real^d$.
Such a Hamiltonian may be lifted to a function on
a complex neighborhood $\DD$ of $\real^d\times\{0\}$,
which is $2\pi$--periodic in its first $d$ arguments.
The types of domains considered here are of the form
$\DD(\rho)=\DD_1(\rho)\times\DD_2(\rho)$, with
$$
\DD_1(\rho)=\bigl\{q\in\complex^d\colon \|{\rm Im} q\|<\rho\bigr\}\,,
\qquad
\DD_2(\rho)=\bigl\{p\in\complex^d\colon \|p\|<\rho\bigr\}\,,
\equation(DDrhoDef)
$$
where $\|u\|$ denotes the $\ell^\infty$ norm of vector in $\complex^d$;
that is, $\|u\|=\max\{|u_1|,|u_2|,\ldots,|u_d|\}$.
One way of specifying a Hamiltonian $H$ on $\DD(\rho)$
is by giving its Fourier-Taylor coefficients $H_{\nu,\alpha}\,$,
defined e.g. implicitly by the equation
$$
H(q,p)=\sum_{(\nu,\alpha)\in I}H_{\nu,\alpha}\, p^\alpha e^{i\nu\cdot q} \,,
\qquad (q,p)\in\DD(\rho) \,.
\equation(FTdef)
$$
Here, $I\!=\integer^d\!\times\!\natural^d$,
and we have used the notation
$p^\alpha=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_d^{\alpha_d}$
and $q\cdot\nu=q_1\nu_1+q_2\nu_2+\ldots+q_d\nu_d\,$ .
\claim Definition(AArho)
Given any positive real number $\rho$,
define $\AA(\rho)$ to be the Banach space of all
analytic Hamiltonians $H$ on $\DD(\rho)$ which extend continuously
to the boundary of $\DD(\rho)$,
and for which the norm
$$
\|H\|_\rho=\sum_{(\nu,\alpha)\in I}
|H_{\nu,\alpha}|\rho^{|\alpha|}e^{\rho|\nu|}
\equation(AArhoNorm)
$$
is finite.
Here, and in what follows,
$|v|$ denotes the $\ell^1$--norm of a vector or multi--index $v$;
that is, $|v|=|v_1|+|v_2|+\ldots+|v_d|$.
The identity operator on $\AA(\rho)$ will be denoted by $\Id$.
We would like to define a renormalization group transformation $\RG$
on $\AA(\rho)$, which is essentially of the form \equ(almostLmu),
modulo some canonical change of coordinates
that may depend on the Hamiltonian.
The reason for such a choice is that we want to preserve
the relationship, mentioned earlier,
between certain orbits of the original and renormalized Hamiltonian.
In the example where $T=\left({0~1\atop 1~1}\right)$,
this means that $\RG$ should include a step
where $H(q,p)$ is replaced by $H(Tq,p)$.
(Notice that the resulting change of coordinates
is not homotopic to the identity, as in usual KAM theory.)
But this step represents a highly singular operator on $\AA(\rho)$,
since $T$ is expanding in the direction of $\omega$.
On the other hand, there are terms in the expansion \equ(FTdef)
for which the substitution $q\mapsto Tq$ is harmless;
namely, those that are indexed by pairs $(\nu,\alpha)$
for which $\nu$ is approximately orthogonal to $\omega$.
These terms represent ``slow modes'' in the direction
of the flow for $H^0(q,p)=\omega\cdot p$.
They also happen to be the ones that produce
small denominators in KAM theory.
One of the main ideas in our approach,
motivated by an analogy (described below) with statistical mechanics,
it to {\sl completely} eliminate all non-slow modes,
before applying the substitution $q\mapsto Tq$.
This will be done by means of a canonical change of coordinates.
An attempt at solving such an ``elimination equation''
to first order suggests the following definition of a slow mode.
Here, and in the theorem that follows,
$\omega$ can be an arbitrary vector in $\real^d$.
\claim Definition(Iminus)
Given real numbers $\rho,\sigma,\tau>0$,
and a vector $\omega\in\real^d$,
denote by $I^{-}$ the set of all pairs $(\nu,\alpha)$
in $I=\integer^d\times\natural^d$ which satisfy the condition
$$
|\omega\cdot\nu|>\sigma|\nu|\,,\qquad |\alpha|<\tau|\nu| \,.
\equation(Iminus1)
$$
A projection operator $\Id^{-}$ is defined on $\AA(\rho)$,
by setting
$$
\bigl(\Id^{-}H\bigr)(q,p)=
\sum_{(\nu,\alpha)\in I^{-}}H_{\nu,\alpha}\, p^\alpha e^{i\nu\cdot q} \,,
\qquad (q,p)\in\DD(\rho) \,.
\equation(Iminus2)
$$
In addition, define $I^{+}=I\setminus I^{-}$ and $\Id^{+}=\Id-\Id^{-}\,$.
The terms in the expansion \equ(FTdef),
that are indexed by pairs $(\nu,\alpha)$ in $I^{+}$,
will be referred to as the ``slow modes'' of $H$
in the direction of $\omega$.
(The plus sign in $I^{+}$ refers to the large time
it takes for a slow mode to affect the motion
in the direction of $\omega$.)
Notice the strict inequalities in \equ(Iminus1).
According to this definition,
a Hamiltonian $H=H(q,p)$ that only depends on $p$ has only slow modes.
In an analogy with statistical mechanics,
$I$ plays the role of a lattice, indexing the spin variables,
and $I^{+}$ represents a ``coarser'' sublattice,
used to index local spin averages, also called block spins.
The elimination of fast (non-slow) modes corresponds
to the process of integrating out ``irrelevant fluctuations''
around given local averages.
In order to push the analogy further,
consider again the matrix $T=\left({0~1\atop 1~1}\right)$
and its unstable eigenvector $\omega=(1,\vartheta)$.
Then the shift $(\nu,\alpha)\mapsto(T\nu,\alpha)$,
applied to points in $I^{+}$,
corresponds to a rescaling of the coarse lattice,
designed to turn a system of block spins
back into a regular spin system,
so that the entire process can be repeated.
The following theorem describes the elimination of fast
(non-slow) modes
in the direction of a fixed but arbitrary vector $\omega\in\real^d$,
for Hamiltonians near $H^0: (q,p)\mapsto\omega\cdot p$.
\claim Theorem(UUmain)
Let $0<\rho'<\rho$ and $\sigma,\tau>0$.
Then there exists an open neighborhood $B$ of $H^0$ in $\AA(\rho)$,
and for every $H\in B$ a canonical transformation
$\UU_H$ from $\DD(\rho')$ into $\DD(\rho)$,
such that $\Id^{-}(H\circ\UU_H)=0$,
and such that $\UU_H$ is the identity whenever $\Id^{-}H=0$.
The map $H\mapsto H\circ\UU_H$ is analytic from $B$ to $\Id^{+}\AA(\rho')$,
and its derivative at $H^{0}$ is $\Id^{+}$.
In the remaining part of this introduction,
$\rho$ denotes a fixed (but arbitrary) positive real number.
We will also assume that $\omega=(1,\omega_2,\ldots,\omega_d)$
is a given vector in $\real^d$ whose components span
an algebraic number field of degree $d$.
In particular, $\omega\cdot\nu$ is nonzero
for every vector $\nu\in\integer^d\setminus\{0\}$.
Another consequence of this assumption (see Section 4)
is that there exists an integral $d\times d$ matrix $T$
with the following properties:
\medskip
\item{(T1)} $T$ has a simple eigenvalue $\vartheta$ of modulus
larger than $1$, and $T\omega=\vartheta\omega$.
\item{(T2)} All other eigenvalues of $T$ are simple, nonzero,
and of modulus less than $1$.
\item{(T3)} $\det(T)=\pm 1$.
\medskip\noindent For the vector $\omega=(1,\omega_2)$,
with $\omega_2$ the golden mean $\half+\half\sqrt{5}$,
we have $T=\left({0~1\atop 1~1}\right)$ as an example of such a matrix.
Let now $T$ be a fixed integral matrix satisfying (T1--2),
but not necessarily (T3).
Then we define a linear map $T_\mu$ on $\complex^d\times\complex^d$,
depending on a parameter $\mu\in\complex$, by setting
$$
T_\mu(q,p)=\bigl(Tq,\mu(T^\ast)^{-1}p\bigr)\,,
\qquad q,p\in\complex^d\,.
\equation(TmuDef)
$$
Notice that if the matrix $T$ satisfies (T3) as well
(implying that $T^{-1}$ is integral),
then $T_1$ represents a canonical transformation
of the torus $\torus^d$,
preserving the $1$-form integrated in equation \equ(AjDef).
Throughout the rest of this paper, we assume that
$$
\max_{|y|=1\atop\omega\cdot y=0}\bigl|T^\ast y\bigr|\,<\kappa\,,\qquad
0<\,\max_{|y|=1\atop\phantom{0=0}}\bigl|\mu T^{-1}y\bigr|\,<\kappa\,,
\equation(TmuBound)
$$
for some positive real number $\kappa<1$, which will be regarded as fixed.
The first condition in \equ(TmuBound)
is purely technical and could be avoided,
if necessary, by choosing an appropriate norm on $\real^d$,
or by replacing $T$ with a sufficiently large power of $T$.
The second inequality in \equ(TmuBound)
should be viewed a condition on the parameter $\mu$.
Consider now a Hamiltonian $H$ that contains no slow modes.
If the parameter $\sigma$ has been chosen very large,
then all frequencies $\nu$ for which $H_{\nu,\alpha}$ is nonzero
are contracted by the matrix $T^\ast$.
As a result, we have
\claim Lemma(TmuSub)
Let $\rho'>\kappa\rho$ be given.
If $\sigma>0$ is sufficiently small and $\tau$ sufficiently large,
then every function $H\in\Id^{+}\AA(\rho')$ extends analytically
to $T_\mu\DD(\rho)$, and $H\mapsto H\circ T_\mu$
is a compact linear map from $\Id^{+}\AA(\rho')$ to $\AA(\rho)$.
This lemma will be applied later for a fixed choice of $\rho'$.
Thus, the parameters $\sigma$ and $\tau$
will now be considered fixed as well; for details see Section 4.
Now we are ready to define our renormalization group transformation.
Denote by $\Omega$ the eigenvector of $T^\ast$
corresponding to the eigenvalue $\vartheta$,
normalized such that $\Omega\cdot\omega=1$.
If $H$ belongs to $\AA(\rho)$,
denote by $\mean H$ the average of $H$ over the torus $\torus^d$,
that is,
$$
(\mean H)(p)=(2\pi)^{-d}\int\limits_{\torus^d}H(q,p)dq
=\sum_{\alpha\in\natural^d}H_{0,\alpha}p^\alpha\,.
\equation(meanH)
$$
\claim Definition(RRdef)
Given a Hamiltonian $H$ sufficiently close to $H^0$, we define
a renormalized Hamiltonian $\RG_\mu(H)$ by setting
$$
\RG_\mu(H)={1\over(\Omega\cdot\nabla\mean\tilde H)(0)}
\bigl[\tilde H-(\mean\tilde H)(0)\bigr]\,,
\qquad \tilde H={\vartheta\over\mu}H\circ\UU_H\circ T_\mu\,.
\equation(RRdef1)
$$
That is, $\RG_\mu$ is composed of a $H$--dependent
canonical change of coordinates
(for a global generating function that depends analytically on $H$),
a fixed scaling of phase space, and a $H$--dependent scaling of time.
The scaling parameter $\mu$ need not depend on $H$
in this case, due to a symmetry of the fixed point $H^0$ (see below).
Notice that multiplying a Hamiltonian by a constant factor
does not change the orbits for the corresponding flow.
The role of the factor $\vartheta$ in the above definition
is to ensure that $\RG_\mu(H^0)=H^0$.
To see that $H^0$ is indeed a fixed point,
it suffices to consider $\mu=1$, as will become clear below.
In this case, we have
$\wt{H^0}(q,p)=\vartheta\omega\cdot\bigl((T^\ast)^{-1}p\bigr)
=\vartheta\bigl(T^{-1}\omega\bigr)\cdot p=\omega\cdot p$,
and consequently $\RG_1(H^0)=H^0$.
Concerning the the parameter $\mu$ in the definition \equ(RRdef1),
we would like to argue that,
for a renormalization group analysis near $H^0$,
almost any choice is as good as any other.
Given $u\in\complex^d$, and $z\in\complex$ non-zero, define
$$
\bigl(\JJ_uH\bigr)(q,p)=H(q-u,p)\,,\qquad
\bigl(\SS_z H\bigr)(q,p)=zH(q,z^{-1}p)\,,
\equation(DefJJSS)
$$
whenever possible.
Then $\RG_\mu$ can be written as $\SS_\mu^{-1}\circ\RG_1\,$.
Notice that the transformations $\SS_z\,$, for $0\not=z\in\complex$,
form a group under composition (for a suitable class of $H$'s).
They modify the dynamics in a trivial way, by scaling the momentum $p$.
Thus, the Hamiltonians $\SS_zH$,
for different values of $z$, may be regarded as equivalent.
In particular, $\RG_\mu(H)$ is in the same equivalence class as $\RG_1(H)$.
And since the Hamiltonian $H^0$ is scale invariant ($\SS_z H^0=H^0$),
it is a fixed point of $\RG_\mu$ for any choice of $\mu\not=0$.
\claim Theorem(RRexists)
There exists an open ball $B$ in $\AA(\rho)$, centered at $H^{(0)}$,
such that the transformation $\RG_\mu$ is well defined,
analytic, and compact, as a map from $B$ to $\AA(\rho)$.
The same holds for the map
$\RG_\mu\circ\JJ_u=\JJ_{T^{-1}u}\circ\RG_\mu\,$,
if the imaginary part of $u$ is sufficiently small.
Furthermore, $\RG_\mu$ maps $B\cap\SS_z^{-1}B$
to $\AA(\rho)\cap\SS_z^{-1}\AA(\rho)$,
for any nonzero complex number $z$,
and $\SS_z\circ\RG_\mu=\RG_\mu\circ\SS_z$ on $B\cap\SS_z^{-1}B$.
The first part of this theorem
is a straightforward consequence of \clm(UUmain) and \clm(TmuSub).
In the second part, we list two symmetries of $\RG_\mu$
that will play an important role later on,
especially the ``commutation relation'' involving $\JJ_u\,$.
The fact that $\RG_\mu$ commutes with $\SS_z\,$,
is used mainly to improve bounds that are not manifestly
scaling invariant.
In order to describe the action of $\RG_\mu$ near the fixed point $H^0$,
we need to analyze the derivative of $\RG_\mu$ at $H^0$.
This part is easy, as usual, since we are dealing with a trivial fixed point.
Consider the map $H\mapsto\tilde H$ defined by equation \equ(RRdef1).
>From \clm(UUmain) it follows that the derivative of this map
at $H^{0}$ is given by the operator $L_\mu\,$,
$$
L_\mu h={\vartheta\over\mu}\bigl(\Id^{+}h\bigr)\circ T_\mu\,.
\equation(LmuDef)
$$
The eigenvectors of $L_\mu$ can be determined explicitly.
Namely, let $\vartheta_1,\vartheta_2,\ldots,\vartheta_d$
be the eigenvalues of $T$,
labeled such that $|\vartheta_i|\ge|\vartheta_j|$ for $i\le j$,
and let $\omega^{(1)},\omega^{(2)},\ldots,\omega^{(d)}$
be the corresponding eigenvectors.
In particular, $\vartheta_1=\vartheta$ and $\omega^{(1)}=\omega$.
Then the nonzero eigenvalues of $L_\mu\,$,
and corresponding eigenvectors, are given by
$$
\lambda_\beta={\vartheta\over\mu}\prod_{j=1}^d
\left({\mu\over\vartheta_j}\right)^{\beta_j}\,,
\qquad
\phi_\beta(q,p)=\prod_{j=1}^d\bigl(\omega^{(j)}\cdot p\bigr)^{\beta_j}\,,
\equation(LmuEV)
$$
where $\beta=(\beta_1,\beta_2,\ldots,\beta_d)$ and $\beta_j\ge 0$ for all $j$.
The second part of assumption \equ(TmuBound)
implies that $|\mu/\vartheta_j|<1$ for all $j$.
If we impose the additional condition
$$
|\mu|<{\vartheta_d^2\over|\vartheta_1|}\,,
\equation(muCondition)
$$
then all eigenvalues of $L_\mu$ are of modulus less than $1$,
except those corresponding to $|\beta|\le 1$.
The derivative of $\RG_\mu$ at $H^0$ is given by the equation
$$
D\RG_\mu\bigl(H^0\bigr)h
=\tilde h-(\mean\tilde h)(0)
-\bigl(\Omega\cdot\nabla\mean\tilde h\bigr)(0)H^0
\,,\qquad\tilde h=L_\mu h\,.
\equation(DRR1)
$$
It is easy to check that the eigenvectors of $D\RG_\mu(H^0)$
are exactly the same as those of $L_\mu\,$.
The eigenvalues are the same as well, except for two cases:
The constant function $\phi_{(0,\ldots,0)}=1$
and the function $\phi_{(1,0,\ldots,0)}=H^{0}$
are mapped to zero by $D\RG_\mu(H^0)$.
In what follows,
we assume that $\mu$ satisfies the condition \equ(muCondition).
Then the derivative of $\RG_\mu$ at $H^0$ has precisely $d-1$
eigenvalues of modulus $\ge 1$.
These eigenvalues, and the corresponding eigenvectors, are
$$
\lambda'_j={\vartheta_1\over\vartheta_j}\,,\qquad
h_j(q,p)=\omega^{(j)}\cdot p\,,\qquad j=2,\ldots,d.
\equation(RmuEV)
$$
The $d-1$ dimensional affine space
consisting of all functions of the form
$H^0+v_1h_1+\ldots+v_dh_d\,$ coincides in fact (near $H^0$)
with the local unstable manifold $\WW^u$ of the transformation $\RG_\mu\,$.
This holds independently of the value of $\mu$,
except for the precise definition of ``near''.
In the same sense,
the (codimension $d-1$) local stable manifold $\WW^s$
of $\RG_\mu$ at $H^0$ is independent of $\mu$.
This follows essentially from the last part of \clm(RRexists);
see Section 4 for details.
Now we turn to some applications of the transformation $\RG_\mu\,$.
We consider the flow
defined by the vector field $\iso\nabla H$,
associated with a Hamiltonian $H\in\AA(\rho)$.
Here, $\iso$ is the linear transformation on $\complex^{2d}$,
defined by the equation $\iso(q,p)=(p,-q)$.
The goal is to find invariant tori
with rotation vectors proportional to $\omega$.
Since $H$ is not necessarily real-valued for real arguments,
it is natural to allow time to be a complex variable.
But as a result, the flow for a Hamiltonian $H$ near $H^0$
is never defined for all times.
Consider e.g. a domain of the form $\torus^d(r')\times\DD_2(r')$,
where $\torus^d(r')$ is the complexified torus $\DD_1(r')/(2\pi\integer)^d$.
On this domain, consider the Hamiltonian $H=cH^0$,
for some nonzero complex number $c$.
For this Hamiltonian, the sets $\torus^d(r')\times\{p'\}$,
with $p'\in\DD_2(r')$, are invariant
under the time evolution $(\dot q,\dot p)=(\iso\nabla H)(q,p)=(c\omega,0)$.
We will refer to these set as the invariant tori of $H$,
with rotation vector $c\omega$.
The flow on these tori,
with initial conditions in a given compact subset $K$,
is defined only for times $t\in\complex$
such that $q+tc\omega$ belongs to $\DD(r')$, whenever $q\in K$.
However, this includes all times in $c^{-1}\real$.
Along the corresponding orbits,
the motion is quasiperiodic with rotation vector $c\omega$.
\claim Theorem(WsFlow)
Given two positive real numbers $r'$ and $r>r'+\rho$,
there exists an open neighborhood $B$ of $H^0$ in $\AA(r)$,
and for every $H\in B$ a complex number $\xi'(H)$
and an analytic function
$\Gamma_{\!H}\colon\DD_1(r')\to\DD(r)$,
such that the following holds. If $H\in\WW^s\cap B$ then
$$
\bigl(\iso\nabla H\bigr)\circ\Gamma_{\!H}
=\xi'(H)\,\omega\cdot\nabla\Gamma_{\!H}
\qquad {\rm on}\ \DD_1(r')\,.
\equation(WsFlow)
$$
If in addition, $H(q,p)$ depends on $p$ only, then
$\xi'(H)=(\Omega\cdot\nabla\mean H)(0)$
and $\Gamma_{\!H}(q)=(q,0)$, for all $q\in\DD_1(r')$.
Furthermore, $\xi'$ is an analytic function on $B$,
and $H\mapsto\Gamma_{\!H}-\Gamma_{\!H^0}$
is an analytic map from $B$ to some Banach space of
analytic $2\pi$-periodic functions on $\DD_1(r')$.
In other words,
if $H\in\AA(r)$ is sufficiently close to $H^0$
and lies on the local stable manifold $\WW^s$,
then the corresponding time evolution on $\DD(r)$
leaves invariant a ``torus'' $\Gamma_{\!H}(\torus^d(r'))$,
and the flow on this torus is conjugate to the flow
of the Hamiltonian $\xi'(H)H^0$
on the torus $\torus^d(r')\times\{0\}$.
If in addition, $H$ is real-valued for real arguments,
then $\xi'(H)\in\real$ and $\Gamma_{\!H}(\torus^d)\subset\real^d$.
This fact does not follow from \clm(WsFlow) directly,
but from our definition (construction) of the quantities
$\xi'(H)$ and $\Gamma_{\!H}\,$.
We note that, as usual, $\Gamma_H$
is obtained as the limit of a sequence of canonical transformations
(close to the identity).
These transformation are extracted from the product
$\UU_{H_0}\circ T_\mu\circ\UU_{H_1}\circ T_\mu
\circ\ldots\circ\UU_{H_{n-1}}\circ T_\mu$
that appears in the expression for $\RG_\mu^n(H)$,
by commuting all of the factors $T_\mu$ out to the right.
In this process, analyticity is lost.
Fortunately, it suffices to determine the values
of $\Gamma_H$ on the real torus $\torus^d$.
The result can be continued analytically,
by using the values on $\torus^d$, of the functions $\Gamma_{\!\JJ_u(H)}$
for complex translates of the Hamiltonian $H$.
Before we discuss Hamiltonians that lie off
the stable manifold $\WW^s$,
let us first list some that can be shown to lie on $\WW^s$;
for details see Section 4.
In the following examples, we assume that
$H$ is close to $H^0$ in $\AA(\rho)$.
\medskip
\item{(H1)} $H(q,p)=c\omega\cdot p+f(q)$.
\item{(H2)} $H(q,p)=c\omega\cdot p+f(q,p)$,
with $f(q,p)=\OO\bigl(|p|^2\bigr)$.
\item{(H3)} $H^s=H+v_2h_2+\ldots+v_dh_d\,$,
with $v\in\complex^{d-1}$ chosen appropriately (see below).
\medskip\noindent
The Hamiltonian $H^s$ in example (H3) is simply
the projection, in the direction of $\WW^u$,
of $H$ onto the local stable manifold $\WW^s$.
(Since $\WW^s$ is analytic, $v$ depends analytically on $H$.)
In the language of quantum field theory,
$v_2h_2+\ldots+v_dh_d$ is a ``counterterm''.
It is of course possible to project in a different direction
(use different counterterms),
as long as it is transversal to $\WW^s$ near $H^0$.
One can also try to ``project'' along
the orbit of translations in $p$.
To be more specific, consider
$$
H_v=H\circ R_v^{-1}\,,\qquad
R_v(q,p)=(q,p+v)\,,
\equation(RvDef)
$$
whenever defined.
\claim Observation(HvWs)
If there exists $v\in\complex^d$ such that $H_v$
is well defined and belongs to $\WW^s\cap B$,
where $B$ is the set described in \clm(WsFlow),
then $H$ has an invariant torus,
given by the function $R_v\circ\Gamma_{\!H_v}\,$.
Since $\WW^s$ is of codimension $d-1$,
we can expect that \clm(HvWs) applies
to open sets of Hamiltonians in $\AA(r'')$, if $r''>r$,
and that most of these Hamiltonians
have a one-parameter family of invariant tori,
covering an invariant surface of dimension $d$ in phase space.
To give some examples,
denote by $E\subset\complex^d$ the linear span of the eigenvectors
$\omega^{(2)},\ldots,\omega^{(d)}$ of the matrix $T$.
\claim Theorem(GammaStable)
Given $r''>r>r'+\rho$, there exists an open neighborhood $B$
of $H^0$ in $\AA(r'')$, such that the following holds.
Assume $H\in B$ is of the form $H(q,p)=\omega\cdot p+f(p)$,
with $(\nabla f)(0)=0$.
If the range of $(D\nabla f)(0)$ contains $E$,
then for every function $H'$ in some open neighborhood
$B_H\subset B$ of $H$,
the intersection of the set $\{H'_v\colon v\in\DD_2(r''-r)\}$ with $\WW^s$
is non-empty and contains the range of an analytic curve
$\Upsilon_{\!H'}\colon A\to\AA(r)$,
where $A$ is some open set in $\complex$ containing $0$.
The map $H'\mapsto\Upsilon_{\!H'}$ is analytic from $B_H$ to $C^1(A,\AA(r))$,
and $\Upsilon_{\!H}(0)=H$.
Furthermore, if $(D\nabla f)(0)$ is non-singular,
then the derivative of $\xi'\circ\Upsilon_{\!H'}$
is non-vanishing on $A$, for all $H'\in B_H\,$.
The last statement in this theorem implies that
(under the given assumptions)
every $H'\in B_H$ has invariant tori with rotation vectors $c\omega$,
for an open set of values $c\in\complex$.
Since $\xi'(H)=1$, this set contains an interval of real numbers
including $1$, if $H'$ is sufficiently close to $H$.
\firstremark
The last two theorems can be combined into one
that contains no reference to the transformation $\RG_\mu\,$.
In particular: When considering a fixed space $\AA(r'')$,
there are open sets $\BB(r')$, for any $r'0$, it is always possible to find
a nonzero complex (or real) number $z$ of modulus $\le 1$,
such that $\SS_z^{-1}H$ belongs to $B$.
And if $\SS_z^{-1}H'$ has an invariant torus with
rotation vector proportional to $\omega$, then so does $H'$.
\nextremark
The fixed point $H^0$ is trivial
in the sense that it has no fast modes (or fluctuations,
like a mean field model in statistical mechanics),
and that the relevant eigenvalues of $D\RG_\mu(H^0)$
are algebraic expressions in the
eigenvalues of the scaling matrix $T$.
The same is true for the fixed points $H^\eps$
(of a slightly modified transformation)
corresponding to the functions $h^\eps$
discussed after equation \equ(almostLmu).
The behavior near these fixed points cannot really
be considered ``critical''.
\nextremark
A more general class of renormalization group transformations
is obtained if we (choose $\mu=\mu(H)$ and)
replace the step $H\mapsto\vartheta H$
by a nonlinear scaling of time $H\mapsto\Lambda_H\circ H$.
The obvious conjecture would be that some of these
transformations have nontrivial fixed points.
Of course, the map $H\mapsto\UU_H$ first has to be defined properly,
e.g. by using computer--assisted methods,
in some neighborhood of the expected fixed point $H^\ast$.
\nextremark
An approximate renormalization group transformation,
for $d=2$ and Hamiltonians of the form $H(q,p)=p_1+f(q,p_2)$,
was studied in [\rED,\rMEsc].
The results, and work on area--preserving maps [\rMcK,\rTom],
indicate that a nontrivial fixed point $H^\ast$
may be related to the breakup of smooth invariant tori
for the corresponding rotation vector $\omega$.
\bigskip
The remaining part of this paper is organized as follows:
Sections 2 and 3 deal exclusively with the canonical
transformation $\UU_H$ and its generating function.
In Section 4, after a lemma concerning properties (T1--3),
we describe the renormalization group transformation $\RG_\mu\,$,
the spectrum of its derivative at $H^0$,
and its local stable manifold $\WW^s$.
This includes a discussion of some symmetry properties
of $\RG_\mu$ and $\WW^s$,
that are useful for the construction of invariant tori
with rotation vectors $c\omega$.
These invariant tori are the subject of Section 5.
%CANON.5
\section The Canonical Transformation {\cbold U}$_H$
In this section we prove \clm(UUmain).
The following quantities are assumed to be fixed:
Two positive real numbers $\varrho'<\varrho$,
a vector $\omega\in\real^d$,
and the two parameters $\sigma,\tau>0$ that enter
the definition of the projections $\Id^\pm$; see \clm(Iminus).
(The proof of \clm(UUmain) uses $\varrho=\rho$ and $\varrho'=\rho'$;
but in Section 4, we will choose $\rho'<\varrho'<\varrho<\rho$.)
If $X$ and $Y$ are Banach spaces,
we denote by $\LL(X,Y)$ the Banach space of all
continuous linear operators from $X$ to $Y$.
We will often use the following fact (see e.g. [\rHP]),
without mentioning it explicitly.
\claim Theorem(Analytic1)
Let $X$ and $Y$ be Banach spaces over the complex numbers,
and let $\FF$ be a locally bounded map
from some open set $D\subset X$ to $Y$.
Then the following properties are equivalent:
\item{$(a)$} $\FF$ is G\^ateaux differentiable on $D$.
\item{$(b)$} $\FF$ is of class $C^\infty$ on $D$.
\item{$(c)$} $\FF$ is complex analytic on $D$. That is,
for every $x\in D$, $h\in X$, and $\ell\in\LL(Y,\complex)$, the
\item{$\ $} function $z\mapsto\ell\bigl(\FF(x+zh)\bigr)$
is analytic in a neighborhood of zero.
A map $\FF$ of the type described above,
which has (one of) the three properties $(a,b,c)$
will be referred to as a ``differentiable''
or ``analytic'' map from $D$ to $Y$.
By definition, a transformation $U:(q,p)\mapsto(q+Q,p+P)$
is canonical if and only if the $1$-form
$(p+P)\cdot d(q+Q)-p\cdot dq$ is closed.
In what follows, we will only be interested
in cases where this $1$-form is the differential of some function $S$.
The function $\phi=S-Q\cdot p$, expressed in terms of $q+Q$ and $p$,
will be referred to as the generating function for $U$.
It satisfies the equations
$$
\eqalign{
Q(q,p)&=-\nabla_2\phi\bigl(q+Q(q,p),p\bigr),\cr
P(q,p)&=\nabla_1\phi\bigl(q+Q(q,p),p\bigr),\cr}
\equation(GenFun)
$$
where $\nabla_1\phi$ and $\nabla_2\phi$
denote the gradients of the functions
$q\mapsto\phi(q,.)$ and $p\mapsto\phi(.,p)$, respectively.
Conversely, if $\phi$ is given and sufficiently small (see below),
then \equ(GenFun) defines a transformation $U=U_\phi\,$,
$$
U_\phi(q,p)=\bigl(q+Q(q,p),p+P(q,p)\bigr),
\equation(UQP)
$$
and this transformation is canonical.
The main goal of this section is
to find a solution $\UU_H$ of the equation $\Id^{-}(H\circ\UU_H)=0$,
given $H\in\AA(\rho)$ sufficiently close to $H^0$.
We will solve this equation by an iterative procedure,
which is similar to a Newton algorithm;
but it involves certain choices,
since the solution is not unique
(we could e.g. increase $\sigma$ and/or decrease $\tau$).
In the first step, we determine a canonical transformation $U_\phi\,$,
where $\phi$ depends on $H$,
such that $\Id^{-}(H\circ U_\phi)$ vanishes to first order in $\Id^{-}H$.
(Then $H$ is replaced by $H\circ U_\phi$ and the process is iterated.)
We start by giving an informal description of this step.
In order to simplify notation, the image under the projection $\Id^\pm$
of a function $f$ will be denoted by $f^\pm$.
Assume that $H$ is of the form $H=H^0+f$,
and imagine that $f$ depends on a (small) parameter $\eps$
in such a way that $f^{-}$ is of order $\eps$.
The goal is to obtain $\Id^{-}(H\circ U_\phi)=\OO(\eps^2)$.
Starting with $\phi$ of order $\eps$,
we expect equation \equ(GenFun) to have a solution
$Q=-\nabla_2\phi+\OO(\eps^2)$ and
$P=\nabla_1\phi+\OO(\eps^2)$.
Thus, the following quantities should be of order $\eps^2$.
$$
\eqalign{
R_1(q,p)&=\omega\cdot\bigl(P(q,p)-\nabla_1\phi(q,p)\bigr) \,,\cr
R_2(q,p)&=f^{-}\bigl(q+Q(q,p),p+P(q,p)\bigr)-f^{-}(q,p) \,,\cr
R_3(q,p)&=f^{+}\bigl(q+Q(q,p),p+P(q,p)\bigr)
-f^{+}\bigl(q-\nabla_2\phi(q,p),p+\nabla_1\phi(q,p)\bigr) \,,\cr
R_4(q,p)&=f^{+}\bigl(q-\nabla_2\phi(q,p),p+\nabla_1\phi(q,p)\bigr)
-f^{+}(q,p) +\bigl\{f^{+},\phi\bigr\}(q,p) \,,\cr}
\equation(RiDef)
$$
where $\{f,g\}$ denotes the Poisson bracket of two functions $f$ and $g$,
that is, $\{f,g\}=\nabla_1f\cdot\nabla_2g-\nabla_2f\cdot\nabla_1g$.
The functions $R_i$ defined by equation \equ(RiDef) are used
in the following decomposition of $H\circ U_\phi$,
where $U_\phi$ is defined by equation \equ(UQP).
$$
H\circ U_\phi=H^0+f^{+}-\Id^{+}\bigl\{f^{+},\phi\bigr\}
+\Bigl[\omega\cdot\nabla_1\phi+f^{-}-\Id^{-}\bigl\{f^{+},\phi\bigr\}\Bigr]
+\sum_{i=1}^4 R_i \,.
\equation(HcircU1)
$$
We will now prove that there exists a function $\phi$
for which the expression $[\ldots]$ in \equ(HcircU1) vanishes.
Then $H\circ U_\phi$ has the desired properties, as will be shown later.
First we list some basic properties of the spaces $\AA(r)$.
Given any positive integer $n$,
denote by $\AA^n(r)$ the direct sum of $n$ copies of $\AA(r)$,
equipped with the norm
$\|(V_1,\ldots,V_n)\|_r
=\max\bigl\{\|V_1\|_r,\ldots,\|V_n\|_r\bigr\}$.
Elements $V\in\AA^n(r)$ will also be regarded
as maps $x\mapsto(V_1(x),\ldots V_n(x))$
from $\DD(r)$ to $\complex^n$.
In what follows,
$\partial_jf$ denotes the partial derivative of a function $f$
with respect to its $j$--th argument.
\claim Proposition(Trivial)
Let $0<\delta0$, define $\AA'(r)$ to be the the vector space
of all analytic functions on $\DD(r)$
whose first partial derivatives all lie in $\AA(r)$.
On this space we consider the norm $\|.\|'_r$ defined by
$\|f\|'_r=\max\bigl\{\|f\|_r, \|\nabla_1f\|_r, \|\nabla_2f\|_r\bigr\}$.
The following lemma indicates that the non-slow modes
can indeed be eliminated.
This is one of the two places where
\clm(Iminus) enters in a crucial way.
The other place is \clm(TmuSub),
which shows that the slow modes can be ``saved for later''.
\claim Lemma(varphiExists)
Let $r>0$, define $b=\sigma/(1+\tau/r)$,
and denote by $B$ the open ball of radius $b/2$ in $\AA'(r)$.
Then there exists a differentiable map $\varphi$
from $B$ to the Banach space
of all bounded linear operators from $\AA(r)$ to $\Id^{-}\!\AA'(r)$,
such that for every $h\in B$ and every $f\in\AA(r)$,
the function $\phi=\varphi(h)f$ satisfies the equation
$$
\omega\cdot\nabla_1\phi+f^{-}-\Id^{-}\bigl\{h^{+},\phi\bigr\}=0
\equation(varphiProp)
$$
and the bounds
$$
\|\phi\|_r\,,
\|\nabla_1\phi\|_r\le{2\over\sigma}\|f^{-}\|_r \,,\qquad
\|\nabla_2\phi\|_r\le{2\tau\over\sigma r}\|f^{-}\|_r \,.
\equation(varphiBound)
$$
\proof
>From the condition \equ(Iminus1) defining the set $I^{-}$,
it follows that the operator $\omega\cdot\nabla_1$
has a bounded inverse on $\Id^{-}\!\AA(r)$, given by the equation
$$
\bigl((\omega\cdot\nabla_1)^{-1}f\bigr)(q,p)
=\sum_{(\nu,\alpha)\in I^{-}}{1\over i\omega\cdot\nu}\;
f_{\nu,\alpha}\, p^\alpha e^{i\nu\cdot q} \,.
\equation(OdotNinv)
$$
In this sum, the term indexed by $(\nu,\alpha)$
is an eigenfunction for ${\partial\over\partial q_j}$
and for $p_j{\partial\over\partial p_j}$, with eigenvalue
$i\nu_j$ and $\alpha_j\,$, respectively.
Since $\nu$ and $\alpha$ satisfy the condition \equ(Iminus1),
it follows that $\nabla_1(\omega\cdot\nabla_1)^{-1}$,
and $\nabla_2(\omega\cdot\nabla_1)^{-1}$ define continuous
linear operators from $\Id^{-}\!\AA(r)$ to $\Id^{-}\!\AA^d(r)$,
with operator norms bounded by $1/\sigma$
and $\tau/(\sigma r)$, respectively.
Similarly for the operator $(\omega\cdot\nabla_1)^{-1}$ itself.
By \clm(Trivial).c we also get the following:
Let $h\in\AA'(r)$ and define $\wh{h}=\{h,.\}$.
Then $\Id^{-}\wh{h^{+}}(\omega\cdot\nabla_1)^{-1}\Id^{-}$
is a continuous linear operator on $\AA(r)$,
and its operator norm is bounded by $\|h\|'_r/b$.
Thus, for every $h\in B$, the following defines a bounded linear operator
$\varphi(h)$ from $\AA(r)$ to $\Id^{-}\!\AA'(r)$:
$$
\varphi(h)=-(\omega\cdot\nabla_1)^{-1}
\Bigl[\Id-\Id^{-}\wh{h^{+}}(\omega\cdot\nabla_1)^{-1}\Bigr]^{-1}\Id^{-}.
\equation(varphiDef)
$$
Clearly, $\phi=\varphi(h)f$ satisfies the equation \equ(varphiProp)
and the bounds \equ(varphiBound).
The asserted differentiability of the map $h\mapsto\varphi(h)$
follows from the fact that the inverse of
the operator $[\ldots]$ in \equ(varphiDef) is given by
a convergent Neumann series in $\LL\bigl(\Id^{-}\AA(r),\Id^{-}\AA(r)\bigr)$.
\qed
In other words, if $\phi=\varphi(f^{+})f^{-}$,
then the term $[\ldots]$ in equation \equ(HcircU1) vanishes.
Notice that by construction,
the generating function $\phi$ has no slow modes.
We continue by discussing the solution of \equ(GenFun).
\claim Proposition(GenFun)
Let $r>0$ and $0<\deltaFrom \clm(Trivial).d we see that if $\phi$ is any function in $B'$,
then the equation $\FF_\phi(Q)=-(\nabla_2\phi)(.+Q,.)$
defines a map $\FF_\phi$ from $B$ to some closed subset of $B$.
The corresponding map $(\phi,Q)\mapsto\FF_\phi(Q)$
is clearly of class $C^1$ on $B'\times B$.
By using parts $(b,d)$ of \clm(Trivial),
with $\delta/2$ instead of $\delta$,
we obtain the following bound
on the derivative of $\FF_\phi$ at $Q\in B$, for any given $\phi\in B'$:
$$
\eqalign{
\|D\FF_\phi(Q)h\|_{r-\delta}
&=\|(h\cdot\nabla_1\nabla_2\phi)(.+Q,.)\|_{r-\delta}\cr
&\le{2d\over\delta}\|\nabla_2\phi\|_r\|h\|_{r-\delta}
\le{1\over 2}\|h\|_{r-\delta}\,.\cr}
\equation(DFFbound)
$$
Here, $h$ is an arbitrary vector in $\AA^d(r-\delta)$.
Thus, by the contraction mapping principle,
$\FF_\phi$ has a unique fixed point $Q=\QQ(\phi)$ in $B$,
and by the chain rule,
the map $\phi\mapsto\QQ(\phi)$ is of class $C^1$ on $B'$.
The first bound in \equ(Qbound) follows from \clm(Trivial).d.
The second bound is obtained by applying the mean value theorem
to the difference $\FF_\phi\bigl(\QQ(\phi)\bigr)-\FF_\phi(0)$,
using the bound \equ(DFFbound) on the derivative of $\FF_\phi\,$.
Finally, define $\PP(\phi)=\nabla_1\phi(.+\QQ(\phi),.)$.
By the chain rule, $\PP$ is of class $C^1$ on $B'$.
The proof of \equ(Pbound) is similar to that of \equ(Qbound).
\qed
Unless stated otherwise, we assume from now on
that $Q=\QQ(\phi)$, $P=\PP(\phi)$, and $\phi=\varphi(f^{+})f^{-}$.
The results proved so far show, among other things,
that under the appropriate
regularity and smallness assumptions on $f$,
the equation \equ(HcircU1) reduces to
$$
H\circ U_\phi=H^{0}+f^{+}+M_1(f)+M_2(f)\,,
\equation(HcircU2)
$$
where
$$
M_1(f)=\Id^{+}\bigl\{f^{+},\phi\bigr\},\qquad
M_2(f)=\sum_{i=1}^4 R_i\,,
\equation(MRdef)
$$
and where $R_1,R_2,R_3,$ and $R_4$ are the functions defined in \equ(RiDef).
We continue by estimating the ``higher order'' term $M_2(f)$
in equation \equ(HcircU2), by using the following fact:
Let $X$ and $Y$ be Banach spaces over the complex numbers,
let $h\in X$, and let $R$ be a differentiable map
from $D=\{x\in X\colon\;\|x\|\|h\|$.
If $R$ is bounded on $D$
and if $\|R(zh)\|=\OO(|z|^n)$ as $z\to 0$, then
$$
\|R(h)\|\le \bigl(\|h\|/r\bigr)^n\sup_{x\in D}\|R(x)\|\,.
\equation(Analytic)
$$
This bound is a straightforward consequence of \clm(Analytic1),
using the Hahn--Banach theorem and the maximum principle
for analytic functions; see also the proof of Proposition 2.7.
\claim Proposition(RiBounds)
There is a constant $c>0$ such that the following holds.
Let $0<\delta0 .
\equation(MRbound0)
$$
\claim Proposition(MRbound)
There are constants $a,c_0,c_1,c_2>0$ such that the following holds,
whenever $\delta>0$ is sufficiently small and $\varrho'0$, is denoted by $B_{k,m}(a)$.
\claim Definition(AAkDef)
Given $k\in\{0,1\}$,
define $\AA_k$ to be the vector space
of all sequences $F=(F_0,F_1,F_2,\ldots)$
of functions $F_m\in\AA_{k,m}$
such that $\|F_m\|_{k,m}\to 0$ as $m\to\infty$.
We equip $\AA_k$ with the norm
$$
\|F\|_k=\sup_{m\ge 0}\|F_m\|_{k,m}\,,
\equation(kNorm)
$$
which makes $\AA_k$ a Banach space.
The open ball in $\AA_k$ centered at the origin,
and with radius $a>0$, is denoted by $B_k(a)$.
Given $F\in\AA_k\,$,
we define $F^{-}$ to be the sequence $(F_0^{-},F_1^{-},\ldots)$.
The following four maps $\chi$, $\VV$, $\MM_1\,$, and $\MM_2\,$
will be used to define a contraction on $\AA_1$
whose fixed point is the desired sequence $\wt{F}$.
Define $\chi\colon\,\AA_{1,0}\to\AA_1$ by setting
$$
\bigl(\chi(f)\bigr)_0=f\,,\qquad
\bigl(\chi(f)\bigr)_m=0\,,\qquad m=1,2,\ldots\,,
\equation(JJdef)
$$
for every function $f$ in $\AA_{1,0}$,
and define $\VV\colon\,\AA_1\to\AA_0$ by the equation
$$
\bigl(\VV(F)\bigr)_0=0\,,\qquad
\bigl(\VV(F)\bigr)_m=F_m^{-}+\sum_{k=0}^mF_k^{+}\,,\qquad
m=1,2,\ldots\,,
\equation(SSdef)
$$
where $F$ is an arbitrary sequence in $\AA_1\,$.
Clearly, $\chi$ and $\VV$ are linear and bounded.
$\chi$ is of norm one, and if $\delta_0$ is sufficiently small
(which we are assuming) then $\VV$ is of norm less than $2$.
Finally, using the same constant $a>0$ as in \clm(MRbound),
we can define, for every $m\ge 0$, $j\in\{1,2\}$, and $F\in B_0(a)$,
a function $\bigl(\MM_j(F)\bigr)_m$ by setting
$$
\bigl(\MM_j(F)\bigr)_0=0\,,\qquad
\bigl(\MM_j(F)\bigr)_m=M_j(F_{m-1})\,,\qquad m=1,2,\ldots\,,
\equation(MMdef)
$$
\claim Proposition(MMbound)
There are constants $a,c>0$, with $c$
independent of the choice of $\delta_0\,$,
such that the following holds:
The equation \equ(MMdef)
defines two differentiable maps $\MM_1$ and $\MM_2$
from $B_0(a)$ to $\AA_1\,$,
and these maps satisfy the bounds
$$
\|\MM_j(F)\|_1\le c e^{-{1\over 4}\delta_0^{-1}}\|F\|_0^j\,,
\qquad j\in\{1,2\}\,,
\equation(MMbound)
$$
and
$$
\|D\MM_j(F)G\|_1\le c e^{-{1\over 4}\delta_0^{-1}}
\Upsilon_{\!j,1}
\bigl(\|F^{-}\|_0,\|G^{-}\|_0,\|G\|_0\bigr)\,,
\qquad j\in\{1,2\}\,,
\equation(DMMbound)
$$
for every $F\in B_0(a)$ and $G\in\AA_0\,$.
\proof
Let $j\in\{1,2\}$ be fixed.
By \clm(MRbound), if $a>0$ is sufficiently small
then for every $m>0$ and every $n\in\{0,1,2\}$,
we can define a map $\MM_{j,n,m}$
from $B_0(a)\times B_0(a)$ to $\AA_{1,m}\,$, by setting
$$
\eqalign{
\MM_{j,0,m}(F,G)&=M_j(F_{m-1}) \,,\cr
\MM_{j,1,m}(F,G)&=DM_j(F_{m-1})G_{m-1} \,,\cr
\MM_{j,2,m}(F,G)&=M_j(F_{m-1}+G_{m-1})-M_j(F_{m-1})-DM_j(F_{m-1})G_{m-1}\,.\cr}
\equation(MMjnm)
$$
Here, and in what follows, we assume $F,G\in B_0(a)$.
The estimates given in \clm(MRbound) imply that
$$
\|\MM_{j,n,m}(F,G)\|'_{r_m}\le c_n\delta^{-j-n-1}\Upsilon_{\!j,n}
\bigl(\|F_{m-1}^{-}\|'_{r_{m-1}},
\|G_{m-1}^{-}\|'_{r_{m-1}},
\|G_{m-1}\|'_{r_{m-1}}\bigr)\,,
\equation(MMbound1)
$$
for the values of $j,n$ considered here.
We will now rewrite this bound in terms of the norms
introduced in in \clm(AAkmDef) and \clm(AAkDef),
by using that the polynomials $\Upsilon_{j,n}$
have nonnegative coefficients and satisfy
$\Upsilon_{\!j,n}(su,sv,w)=s^j\Upsilon_{\!j,n}(u,v,w)$.
Since $\Id^{-}M_1=0$, we obtain for $j=1$
$$
\|\MM_{1,n,m}(F,G)\|_{1,m}
\le c_n\delta_{m-1}^{-n-2}
e^{{1\over 3}\delta_m^{-1}-\delta_{m-1}^{-1}}\Upsilon_{\!1,n}
\bigl(\|F^{-}\|_0,\|G^{-}\|_0,\|G\|_0\bigr)\,,
\equation(MMbound2)
$$
and in the case $j=2$, the bound \equ(MMbound1) implies that
$$
\|\MM_{2,n,m}(F,G)\|_{1,m}
\le c_n\delta_{m-1}^{-n-3}
e^{\delta_m^{-1}-2\delta_{m-1}^{-1}}\Upsilon_{\!2,n}
\bigl(\|F^{-}\|_0,\|G^{-}\|_0,\|G\|_0\bigr)\,.
\equation(MMbound3)
$$
In both of these estimates, the difference in the exponential
is equal to $-{1\over 2}\delta_{m-1}^{-1}$.
Thus, we can define six maps $\MM_{j,n}$ from
$B_0(a)\times B_0(a)$ to $\AA_1\,$, by setting
$\bigl(\MM_{j,n}(F,G)\bigr)_m$ equal to $\MM_{j,n,m}(F,G)$
if $m>0$, and equal to zero if $m=0$.
Furthermore, we have the bounds
$$
\|\MM_{j,n}(F,G)\|_1\le
c_n\delta_0^{-n-3}e^{-{1\over 2}\delta_0^{-1}}
\Upsilon_{\!j,n}
\bigl(\|F^{-}\|_0,\|G^{-}\|_0,\|G\|_0\bigr)\,,
\equation(MMbound4)
$$
for $j\in\{1,2\}$ and $n\in\{0,1,2\}$.
This shows that $\MM_j$ is differentiable
as a map from $B_0(a)$ to $\AA_1\,$,
and that $D\MM_j(F)=\MM_{j,1}(F,.)$ for all $F\in B_0(a)$.
The bounds \equ(MMbound) and \equ(DMMbound)
follow from \equ(MMbound4) with $n=1$.
\qed
Define $\MM=\MM_1+\MM_2\,$,
and assume that $\delta_0$ has been chosen sufficiently small
such that $c\exp{-{1\over 4}\delta_0^{-1}}<{1\over 4}$,
where $c$ is the second constant mentioned in \clm(MMbound).
\claim Proposition(KKfix)
If $s>0$ is sufficiently small
then for every $f\in B_{1,0}(s)$, the equation $\KK_f(F)=F$, where
$$
\KK_f(F)=\MM\bigl(\VV(\chi(f)+F)\bigr)\,,\qquad \MM=\MM_1+\MM_2\,,
\equation(KKdef)
$$
has a unique solution $F=\wt{F}$ in $B_1(s)$,
and the map $\FF\colon f\mapsto\wt{F}$
is differentiable from $B_{1,0}(s)$ to $B_1(s)$.
\proof
Let $s<\max(a,1)/4$, where $a$ is the first constant
mentioned in \clm(MMbound).
Then, by \clm(MMbound) and the chain rule,
$(f,F)\mapsto\KK_f(F)$ is a differentiable map
from $B_{1,0}(s)\times B_1(s)$ to $B_1(s/2)$,
and the derivative $D\KK_f(F)$ is bounded in norm by $1/2$,
for all $f\in B_{1,0}(s)$ and $F\in B_1(s)$.
Thus, $\KK_f$ has a unique fixed point $\tilde F=\FF(f)$,
and by the implicit function theorem,
$\FF$ is differentiable.
\qed
Now we are ready to prove \clm(UUmain).
In order to avoid confusion later on
(when $\rho'<\varrho'$ and $\rho>\varrho$), let us restate the claim:
\claim Theorem(UUagain)
There exists an open neighborhood $B$ of $H^0$ in $\AA(\varrho)$,
and for every $H\in B$ a canonical transformation
$\UU_H$ from $\DD(\varrho')$ to $\DD(\varrho)$,
such that $\Id^{-}(H\circ\UU_H)=0$,
and such that $\UU_H$ is the identity whenever $\Id^{-}H=0$.
The map $H\mapsto H\circ\UU_H$ is analytic from $B$ to $\Id^{+}\AA(\varrho')$,
and its derivative at $H^{0}$ is $\Id^{+}$.
\proof
Given $0<\varrho'<\varrho$, choose the sequences $\{r_m\}$
and $\{\delta_m\}$ as specified in \equ(rdelta).
Let $s>0$ be sufficiently small
for the conclusions of \clm(KKfix) to hold.
Since $r_0<\varrho$, the set $B_{1,0}(s)$
contains some open neighborhood of zero in the space $\AA(\varrho)$.
Thus, we can assume now
that $H=H^{0}+f$ with $f\in B_{1,0}(s)$.
Let $\wt{F}=\FF(f)$ be the function described in \clm(KKfix),
and consider the sequence $V(f)$ defined by the equation
$$
\bigl(V(f)\bigr)_m=\bigl(\VV(\chi(f)+\wt{F})\bigr)_m\,,
\qquad m=0,1,2,\ldots \,.
\equation(wtFtoS)
$$
The functions $f_m=(V(f))_m$ satisfy the recursion relation
$$
\eqalign{
f_m&=f_{m-1}^{+}+\wt{F}_m
=f_{m-1}^{+}+\bigl(\MM(\VV(\chi(f)+\wt{F}))\bigr)_m\cr
&= f_{m-1}^{+}+M_1(f_{m-1})+M_2(f_{m-1})\,,\cr}
\equation(fmCheck)
$$
for all $m>0$. This shows that
the Hamiltonian $H_m=H^{0}+f_m$ is the $m$--th iterate,
under the map $H\mapsto H\circ U_\phi$
defined by equation \equ(HcircU1),
of the Hamiltonian $H=H^{0}+f$.
In particular, if we set $\phi_m=\varphi(f_{m-1}^{+})f_{m-1}^{-}$ and
$$
U'_m=U_{\phi_1}\circ U_{\phi_2}\circ\ldots\circ U_{\phi_m}\,,
\qquad m=1,2,\ldots\,,
\equation(UprimDef)
$$
then $H_m=H\circ U'_m$ for all $m>0$.
If $f^{-}=0$ then we see from \equ(fmCheck) and \equ(MRbound1)
that $f_m^{-}=0$, for all $m>0$.
In this case, $\phi_1=\phi_2=...=0$,
so that $U'_m$ is the identity map for all $m$.
Similarly, if $f_1^{-}=0$ then $f_m=f_1$ for all $m>1$.
This fact will be used below.
Before discussing the limit as $m\to\infty$,
we note that $V(f)\in B_0(3s)$, and in particular,
$$
\|f_m^{-}\|_{r_m}<3s e^{-\delta_m^{-1}}\,,
\qquad m=0,1,2,\ldots\,.
\equation(fmminusbound)
$$
This follows from the properties of the maps $\chi$ and $\VV$,
mentioned before \clm(MMbound).
For later reference, we also note that by \clm(KKfix),
the map $f\mapsto V(f)$ is differentiable on $B_{1,0}(s)$.
By \clm(varphiExists) and \clm(GenFun),
the transformation $U_{\phi_{m+1}}-U_0$ is an element of
$\AA^{2d}(r_m-\delta_m)$ whose norm is bounded
by a constant times $\|f_m\|_{r_m}\,$.
Thus, there are constants $c,c'>0$ such that
$$
\eqalign{
\|U'_{m+1}-U'_m\|_{r_\infty}
&=\|U'_m\circ U_{\phi_{m+1}}-U'_m\circ U_0\|_{r_\infty}\cr
&\le c\delta_m^{-1}\|U'_m\|_{r_m}\|f_m^{-}\|_{r_m}
\le c'\delta_m^{-1}e^{-\delta_m^{-1}}\,,\cr}
\equation(UprimeDiff)
$$
for all $m\ge 0$.
Here, we have used the mean value theorem, \clm(Trivial).b,
and the bound \equ(fmminusbound).
This last estimate shows that the sequence of transformations $U'_m$
converges in $\AA(r_\infty)$ to a limit
$$
\UU_H=U'_\infty=\lim_{m\to\infty} U'_m\,,
\equation(UUHDef)
$$
and from \clm(Trivial).a we see that $H_n\to H\circ\UU_H$
pointwise on $\DD(r_\infty)$.
Next, we note that the sequence $(f_0,f_1,\ldots)$
converges in $\AA'(r_\infty)$ to the function
$$
\tilde f=f^{+}+\Sigma^{+}\FF(f)\,,
\equation(tfdef)
$$
where
$$
\Sigma^{+}F=\sum_{m=1}^\infty F_m^{+} \,,\qquad F\in\AA_1\,.
\equation(SigmaPlus)
$$
The operator $\Sigma^{+}$ given by equation \equ(SigmaPlus)
is clearly continuous from $\AA_1$ to $\Id^{+}\AA'(r_\infty)$.
Thus, by \clm(KKfix), the equation \equ(tfdef) defines
a differentiable map $f\mapsto\tilde f$
from $B_{1,0}(s)$ to $\Id^{+}\AA'(r_\infty)$.
The differentiability of the map $H\mapsto H\circ\UU_H$
now follows from the identity
$$
H\circ\UU_H=H^{0}+\tilde f\,,
\equation(HcircUid)
$$
and from the fact that the inclusion maps
$\AA(\varrho)\to\AA'(r_0)$ and $\AA'(r_\infty)\to\AA(\varrho')$
are continuous.
In order to see why
$\|\tilde f-f^{+}\|_{\varrho'}=\OO(\|f\|_{\varrho})^2$,
as claimed in \clm(UUagain),
we can write the map $f\mapsto\tilde f$ as the composition
of the first step $f\mapsto f_1$
with the map $f_1\mapsto\tilde f$, where $f_1\in B_{0,1}(3s)$.
Then we can use that
$\|\tilde f-f_1\|_{\varrho'}=\OO(\|f_1^{-}\|_{0,1})$
and $\|f_1-f^{+}\|_{0,1}=\OO(\|f\|_{\varrho})^2$.
The first bound follows from the fact that
$f_1^{-}=0$ implies $\tilde f=f_1\,$,
and the second bound is obtained from the identity
$f_1-f^{+}=\Id^{+}\{f^{+},\varphi(f^{+})f^{-}\}+M_2(f)$,
together with \clm(RiBounds).
\qed
%UPROP.5
\section Properties of the Transformations {\cbold U}$_H$
One of the goals in this section is to describe
the connection between $\UU_H$ and $\UU_{H'}\,$,
if $H$ and $H'$ are related by a translation
(in the torus variable $q$) or scaling.
But first, we prove the following theorem.
\claim Theorem(aboutPsi)
Let $0<\varrho'<\varrho$.
Then there exists an open neighborhood $B$ of $H^{(0)}$ in $\AA(\varrho)$,
a number $\varrho''\in(\varrho',\varrho)$,
and a differentiable map $\psi$ from $B$ to $\AA(\varrho'')$,
such that $U_{\psi(H)}=\UU_H$ for every $H\in B$,
where $\UU_H\colon\DD(\varrho')\to\DD(\varrho)$ is the
canonical transformation described in \clm(UUagain).
Furthermore, $\psi(H)=0$ whenever $\Id^{-}H=0$.
Here, and in what follows,
$\sigma$, $\tau$, and $\varrho'<\varrho$
are assumed to be the same (arbitrary)
positive real numbers that were used in the last section.
Similarly with $\omega\in\real^d$.
In order to prove \clm(aboutPsi),
it turns out to be useful to determine at once
all of the generating functions $\Psi_m=\psi(H_m)$,
for $m=0,1,2,\ldots$.
This will be done by solving an implicit equation
for the sequence of differences
$\tilde G_m=\Psi_m-\Psi_{m+1}-\phi_{m+1}\,$,
where $\phi_1,\phi_2,\ldots$ are the functions
defined in the proof of \clm(UUagain).
The resulting expression for the generating function of $\UU_H$ is
$$
\psi(H)=\Psi_0=
\tilde G_0+\sum_{m=1}^\infty\bigl(\phi_m+\tilde G_m\bigr)\,.
\equation(GmapstoPsi)
$$
In order to estimate the generating function
of a composite transformation $U_{\phi_0}\circ U_{\phi_1}\,$,
we use the following
\claim Proposition(UcompU)
Let $r>0$ and $0<\delta0$, will be denoted by $B'_{1,m}(a)$.
\claim Definition(AApkDef)
Let $\AA'_1$ be the vector space
of all sequences $\Phi=(\Phi_0,\Phi_1,\Phi_2,\ldots)$
of functions $\Phi_m\in\AA'_{1,m}$
such that $\|\Phi_m\|'_{1,m}\to 0$ as $m\to\infty$.
We equip $\AA'_1$ with the norm
$$
\|\Phi\|'_1=\sup_{m\ge 0}\|\Phi_m\|_{1,m}\,,
\equation(pkNorm)
$$
which makes $\AA'_1$ a Banach space.
The open ball in $\AA'_1$ centered at the origin,
and with radius $a>0$, will be denoted by $B'_1(a)$.
\claim Corollary(NNbound)
There are constants $a,c>0$, such that the equation
$$
\bigl(\NN(\Phi,\Psi)\bigr)_m=N(\Phi_{m+1},\Psi_{m+1})\,,
\qquad m=0,1,2,\ldots
\equation(NNdef)
$$
defines a map $\NN$ from $B'_1(a)\times B'_1(a)$ to $\AA'_1\,$.
Furthermore, this map is differentiable and satisfies
$$
\|\NN(\Phi,\Psi)\|'_1\le c \|\Phi\|'_1\|\Psi\|'_1\,,
\qquad \Phi,\Psi\in B'_1(a)\,.
\equation(NNbound)
$$
\proof
The bound \equ(NNbound) follows from \clm(UcompU),
which implies that
$$
\|N(\Phi_{m+1},\Psi_{m+1})\|'_{r'_m}
\le{6d\over\delta_m}e^{-\delta_m^{-1}}
\|\Phi'\|'_1\|\Phi\|'_1\,,
\equation(NNbound1)
$$
for all $\Phi,\Psi\in B'_1(a)$, if $a>0$ is sufficiently small.
Consider the the maps $\NN_n\,$, for $n=1,2,\ldots$,
where $(\NN_n(\cdot))_m$ is either $(\NN(\cdot))_m$ or zero,
depending on whether $m\le n$ or $m>n$.
These maps are clearly differentiable on $B'_1(a)\times B'_1(a)$.
Furthermore, as \equ(NNbound1) shows,
we have $\NN_n(\Phi,\Psi)\to\NN(\Phi,\Psi)$ as $n\to\infty$,
uniformly on $B'_1(a)\times B'_1(a)$.
Thus, $\NN$ is differentiable (complex analytic).
\qed
Consider now the linear operator $\VV'$ on $\AA'_1\,$,
defined by the equation
$$
\bigl(\VV'G\bigr)_m=\sum_{k=m+1}^\infty G_k\,,
\qquad m=0,1,\ldots\,.
\equation(VVpdef)
$$
Clearly, $\VV'$ is continuous,
and of norm less than one, if $\delta_0$ has been chosen
sufficiently small (which we assume).
\claim Proposition(KKpfix)
If $s>0$ is sufficiently small then for every $\Phi\in B'_1(s)$,
the equation $\KK'_\Phi(G)=G$, where
$$
\KK'_\Phi(G)=\NN\bigl(\Phi,G+\VV'(\Phi+G)\bigr)\,,
\equation(KKpdef)
$$
has a unique solution $G=\tilde G$ in $B'_1(s)$,
and the map $\Phi\mapsto\tilde G$ is differentiable on $B'_1(s)$.
\proof
Let $a,c$ be the constants mentioned in \clm(NNbound).
By using the bound \equ(NNbound) and Cauchy's formula
(as e.g. in the proof of \clm(MRbound)),
the partial derivatives of $\NN$ can be estimated as follows:
$$
\|D_1\NN(\Phi,\Psi)\|\le 2c\|\Psi\|'_1\,, \qquad
\|D_2\NN(\Phi,\Psi)\|\le 2c\|\Phi\|'_1\,,
\equation(DNNbound)
$$
for every $\Phi,\Psi\in B'_1(a/2)$.
Thus, if $s>0$ is sufficiently small, $\KK'_\Phi$ is a contraction
from $B'_1(s)$ to $B'_1(s/2)$, for every $\Phi\in B'_1(s)$.
And in particular,
$\KK'_\Phi$ has a unique fixed point $\tilde G$ in $B'_1(s)$.
By the implicit function theorem,
this fixed point depends differentiably on the parameter $\Phi$.
\qed
The sequence $\Phi=\hat\varphi_n(f)$
described in the proposition below will be used
to construct the generating function
for the canonical transformation $U'_n$
considered in the proof of \clm(UUagain).
The maps $\varphi$ and $V$ that enter the definition
of $\hat\varphi_n(f)$ are defined in \clm(varphiExists)
and equation \equ(wtFtoS), respectively.
\claim Proposition(hatphi)
There exists a constant $c>0$, such that
the following holds if $s>0$ is sufficiently small:
For $1\le n\le\infty$, the equation
$$
\bigl(\hat\varphi_n(f)\bigr)_m
=\cases{\varphi(f_{m-1}^{+})f_{m-1}^{-}\,,
&if $\;0\le m-1< n$;\cr
0,&otherwise;\cr}
\equation(hatphiDef)
$$
where $f_k=(V(f))_k$ for all $k\ge 0$,
defines a differentiable map $\hat\varphi_n$
from $B_{1,0}(s)$ to $B'_1(cs)$.
Furthermore, $\hat\varphi_n(f)\to\hat\varphi_\infty(f)$
as $n\to\infty$, uniformly on $B_{1,0}(s)$.
\proof
As was mentioned in the proof of \clm(UUagain),
the map $V$ defined by \equ(wtFtoS)
is differentiable from $B_{1,0}(s)$ to $B_0(3s)$;
and by \clm(varphiExists), there exists a constant $c>0$
such that $g\mapsto\varphi(g^{+})g^{-}$
defines a differentiable map from
$B_{0,m-1}(3s)$ to $B'_{1,m}(c_ms)$, for every $m>0$,
where $c_m=c\exp -{1\over 3}\delta_m^{-1}$.
Here, and in what follows,
we need to assume that $s>0$ is sufficiently small.
By composing these maps according to equation \equ(hatphiDef),
we obtain a differentiable map $\hat\varphi_n$
from $B_{1,0}(s)$ to $B'_1(cs)$,
for every positive integer $n$.
>From the fact that the sequence $\{c_m\}$ tends to zero,
we see that $\hat\varphi_n(f)\to\hat\varphi_\infty(f)$
uniformly on $B_{1,0}(s)$. This in turn implies
that $\hat\varphi_\infty(f)$ is differentiable (complex analytic).
\qed
\proofof(aboutPsi)
Given $0<\varrho'<\varrho$, choose the sequences $\{r_m\}$,
$\{r'_m\}$, and $\{\delta_m\}$ as specified
in \equ(rdelta) and \equ(rpdelta), and let $\varrho''=r'_0\,$.
We define $\psi$ as the composition of three maps
$H\mapsto f\mapsto\Psi\mapsto\Psi_0\,$,
where $f\in\AA_{1,0}$ is given by the equation $f=H-H^{0}$,
and where $\Psi_0\in\AA'_{1,0}$ is the first
component of the sequence $\Psi$ that will be defined below.
Since $\varrho'0$,
and that $\Psi_0$ is the generating function for $\UU_H\,$.
The first part is an immediate consequence
of \clm(KKpfix) and \clm(hatphi).
Namely, if $s>0$ is sufficiently small, then the equation
$$
\Psi=\tilde G+\VV'(\Phi+\tilde G)\,,\qquad
\Phi=\hat\varphi_n(f)\,,
\equation(tildePsiPhi)
$$
where $\tilde G$ is the sequence described in \clm(KKpfix),
defines a differentiable map $f\mapsto\Psi$
from $B_{1,0}(s)$ to $\BB'_1(3cs)$.
Here, $n$ can be an arbitrary positive integer or the symbol $\infty$,
and $c>0$ is some constant that is independent of $n$.
The abovementioned definition $\psi(H)=\Psi_0$
refers to the case $n=\infty$ .
Let now $f\in B_{1,0}(s)$ be fixed,
with $s>0$ sufficiently small for the above to hold.
We also assume that $3cs<\delta_0/(4d)$,
so that \clm(GenFun) can be used to define
the canonical transformation $U_{\Psi_0}$
on the domain $\DD(\varrho''-\delta_0)$.
This domain contains $\DD(\varrho')$, due to our choice of $\delta_0\,$.
By definition, the sequence $\Psi$
satisfies the equation
$$
\eqalign{\Psi_{m-1}
&=\tilde G_{m-1}
+\sum_{k=m}^\infty\bigl(\Phi_k+\tilde G_k\bigr)
=\tilde G_{m-1}+\Phi_m+\Psi_m\cr
&=N(\Phi_m,\Psi_m)+\Phi_m+\Psi_m\,,\cr}
\equation(PsimCheck)
$$
for all $m>0$. In what follows, let $0n$.
>From the definition \equ(KKpdef) of $\KK'_\Phi(G)$
one readily sees that the set of all sequences $G\in B'_1(s)$,
that satisfy $G_m=0$ whenever $m\ge n-1$,
is invariant under the map $\KK'_\Phi$.
Thus, the fixed point $\tilde G$ of $\KK'_\Phi$
also belongs to this invariant set.
As a consequence, we have $\Psi_m=0$ for all $m\ge n$.
The remaining components of $\Psi$ are determined
by the recursion relation \equ(PsimCheck),
with ``initial'' condition $\Psi_n=0$.
By using \clm(UcompU), we can identify $\Psi_0$
as the generating function for the canonical transformation
$U'_n=U_{\Phi_1}\circ U_{\Phi_2}\circ\ldots\circ U_{\Phi_n}\,$.
As was shown in the proof of \clm(UUagain),
$U'_n\to\UU_H$ in $\AA(\varrho')$, as $n\to\infty$.
On the other hand, \clm(hatphi) and \clm(KKpfix)
imply that the sequence $n\mapsto\Psi_0\,$,
as defined by equation \equ(tildePsiPhi),
converges in $B'_1(3cs)$ to $\psi(H)$.
Thus, since the map $\phi\mapsto U_\phi$
is continuous (by \clm(GenFun)),
we conclude that $U_{\psi(H)}=\UU_H\,$.
Finally, if we assume that $\Id^{-}f=0$,
then $\Phi=0$, and thus $\tilde G=0$ (since $\KK'_0(0)=0$),
implying that $\psi(H)=\Psi_0=0$.
\qed
In the remaining part of this section,
we consider the composition of certain changes of coordinates
with the maps $\phi\mapsto U_\phi$ and $H\mapsto\UU_H\,$.
The results will be used in subsequent sections.
In order to estimate the composition of a function $H\in\AA(r)$
with a linear change of coordinates, it is convenient to split $H$
into homogeneous polynomials $H^{(n)}_\nu$ as follows:
$$
H(q,p)=\sum_{\nu\in\integer^d\atop n\ge 0}
H^{(n)}_\nu(p)e^{i\nu\cdot q}\,,\qquad
H^{(n)}_\nu(p)=\sum_{|\alpha|=n}H_{\nu,\alpha}p^\alpha\,,
\equation(HnnuDef)
$$
where $H_{\nu,\alpha}$ are the Fourier--Taylor coefficients of $H$,
as defined by the equation \equ(FTdef).
In this representation, the norm of $H$ can be written as
$$
\|H\|_r=\sum_{\nu\in\integer^d\atop n\ge 0}
\|H^{(n)}_\nu\|r^n e^{r|\nu|}\,,
\qquad
\|H^{(n)}_\nu\|=\sum_{|\alpha|=n}|H_{\nu,\alpha}|\,.
\equation(HnnuNorm)
$$
If $T$ is an arbitrary $d\times d$ matrix,
and if $x$ is a vector in $\real^d$, define
$$
\|T\|_x=\max_{|y|=1\atop x\cdot y=0}|Ty|\,.
\equation(MatNormDef)
$$
Let now $T$ be an integral $d\times d$ matrix with determinant $\pm 1$,
and let $\mu$ be a nonzero complex number.
Then the equation \equ(TmuDef) defines a linear map
$T_\mu$ on $\complex^d\times\complex^d$.
In the following proposition, we assume that $r,r'>0$ satisfy
$$
\|T^\ast\|_0\le{r\over r'},\quad
\bigl\|\mu T^{-1}\bigr\|_0\le{r\over r'},\quad
\|T\|_0\le{r\over r'},\quad
\bigl\|\mu(T^\ast)^{-1}\bigr\|_0\le{r\over r'}.
\equation(TTTT)
$$
The first two of these inequalities guarantee
that $T_\mu$ maps $\DD(r')$ into $\DD(r)$.
The last two are added for convenience.
\claim Proposition(TUT)
The linear operator $\phi\mapsto\phi\circ T_\mu$
is continuous from $\AA(r)$ to $\AA(r')$, and of norm $1$.
It is also continuous from $\AA'(r)$ to $\AA'(r')$,
and in this case, its operator norm is $\le r/r'$.
Let now $\delta$ be a positive number less than $r/4$,
and define $\delta'=\delta r'/r$.
Assume that $\phi\in\AA'(r)$ is of norm less than
$\min(|\mu|(r'/r)^2,1)\delta/(4d)$.
Then, on $\DD(r'-\delta')$, we have the identity
$$
T_\mu^{-1}\circ U_\phi\circ T_\mu=U_{\phi'}\,,\qquad
\phi'=\mu^{-1}\phi\circ T_\mu\,.
\equation(TUT)
$$
\proof
>From the bound
$$
\eqalign{
\bigl\|\phi^{(n)}_\nu\circ\bigl[\mu(T^\ast)^{-1}\bigr]\bigr\|
(r')^ne^{r'|T^\ast\nu|}
&\le\bigl\|\phi^{(n)}_\nu\bigr\|\,\bigl\|\mu T^{-1}\bigr\|_0^n
(r')^ne^{r'\|T^\ast\|_0|\nu|}\cr
&\le\bigl\|\phi^{(n)}_\nu\bigr\|r^ne^{r|\nu|}\,,\cr}
\equation(TUT1)
$$
we see that $\phi\circ T_\mu\in\AA(r')$ whenever $\phi\in\AA(r)$,
and that $\|\phi\circ T_\mu\|_{r'}\le\|\phi\|_r\,$.
Consider now $\phi\in\AA'(r)$
of norm less than $\min(|\mu|(r'/r)^2,1)\delta/(4d)$,
and define $\phi'=\mu^{-1}\phi\circ T_\mu\,$.
Then the bound \equ(TUT1),
with $\phi$ replaced by $\nabla\phi$,
shows that $\|\phi'\|'_{r'}\le|\mu|^{-1}(r/r')\|\phi\|'_r\,$.
And this in turn implies $\|\phi'\|_{r'}<\delta'/(4d)$.
Now we can apply \clm(GenFun) to obtain two canonical transformations
$U_\phi\colon\,\DD(r-\delta)\to\DD(r)$ and
$U_{\phi'}\colon\,\DD(r'-\delta')\to\DD(r')$,
whose generating functions are $\phi$ and $\phi'$, respectively.
Notice that $T_\mu$ maps the domain $\DD(r'-\delta')$ of $U_{\phi'}$
into the domain $\DD(r-\delta)$ of $U_\phi\,$.
In order to prove \equ(TUT),
let $(Q,P)$ be the increments of the map $U_\phi\,$.
By this we mean that $U_\phi(q,p)=(q+Q(q,p),p+P(q,p))$,
for all $(q,p)$ in the domain of $U_\phi\,$.
Recall that $U_\phi$ is uniquely defined by the equation
$$
Q=-(\nabla_2\phi)\circ V(Q)\,,\qquad
P= (\nabla_1\phi)\circ V(Q)\,,
\equation(UQPrecall)
$$
where $V(Q)$ denotes the map with increments $(Q,0)$.
The following computation shows that the increments $(Q',P')$
of the map $T_\mu^{-1}\circ U_\phi\circ T_\mu$ agree
with those of $U_{\phi'}\,$:
$$
\eqalign{Q'
&=T^{-1}Q\circ T_\mu
=-T^{-1}(\nabla_2\phi)\circ V(Q)\circ T_\mu\cr
&=-T^{-1}(\nabla_2\phi)\circ T_\mu\circ V(T^{-1}Q\circ T_\mu)
=-T^{-1}(\nabla_2\phi)\circ T_\mu\circ V(Q')\cr
&=-\bigl(\nabla_2(\mu^{-1}\phi\circ T_\mu)\bigr)\circ V(Q')
=-(\nabla_2\phi')\circ V(Q')\,,\cr
\cr
P'&=\mu^{-1}T^\ast P\circ T_\mu
=\mu^{-1}T^\ast(\nabla_1\phi)\circ V(Q)\circ T_\mu\cr
&=\mu^{-1}T^\ast(\nabla_1\phi)\circ T_\mu\circ V(Q')\cr
&=\bigl(\nabla_1(\mu^{-1}\phi\circ T_\mu)\bigr)\circ V(Q')
=(\nabla_1\phi')\circ V(Q')\,.\cr}
\equation(TUT2)
$$
Thus, the proposition is proved.
\qed
Given any vector $u\in\complex^d$,
and any complex number $z$, define two linear
transformations $J_u$ and $S_z$ on $\complex^d$,
by setting $J_u(q,p)=(q+u,p)$ and $S_z(q,p)=(q,zp)$, respectively.
Assuming that $z\not=0$,
we can rewrite the definition \equ(DefJJSS) in the form
$\JJ_uH=H\circ J_u^{-1}$ and $\SS_zH=zH\circ S_z^{-1}\,$.
Below it will be shown that, under suitable assumptions,
$$
\UU_{\JJ_uH'}=J_u\circ\UU_{H'}\circ J_u^{-1}\,,\qquad
\UU_{\SS_zH'}=S_z\circ\UU_{H'}\circ S_z^{-1}\,.
\equation(UUJJSS)
$$
But first, we need the following
\claim Proposition(UJS)
Consider $u\in\real^d$, and $z\in\complex$ satisfying $|z|=1$.
Under the same assumptions (and using the same notation)
as in \clm(GenFun), the following holds for all $\phi\in B'$.
If $\phi'=\JJ_u^{-1}\phi$ then
$J_u^{-1}\circ U_\phi\circ J_u=U_{\phi'}\,$,
and if $\phi'=\SS_z^{-1}\phi$ then
$S_z^{-1}\circ U_\phi\circ S_z=U_{\phi'}\,$.
The proof of this proposition is straightforward and will be omitted.
Notice that if $u\in\real^d$ and $|z|=1$
then $\JJ_u$ and $\SS_z$ are isometries on $\AA(r)$,
for any given $r>0$.
For every $r,b>0$, denote by $B(r,b)$ the open ball
in $\AA(r)$ of radius $b$ centered at $H^0$.
\claim Lemma(SHUS)
Let $\delta>0$ and $0<\rho'+\delta\le\varrho'<\varrho\le\rho-\delta$.
There exists $b>0$, such that the following holds,
for all $u\in\DD_1(\delta)$, and for all $z\in\complex$
satisfying $0<|z|\le 1$.
If $H\in B(\rho,b)$ and $H'=\JJ_u^{-1}H$,
then $J_u\circ\UU_{H'}=\UU_H\circ J_u$ on $\DD(\rho')$.
Furthermore, if $H\in B(\varrho,|z|b)$
and $H'=\SS_z^{-1}H$,
then $S_z\circ\UU_{H'}=\UU_H\circ S_z$ on $\DD(\varrho')$.
\proof
Choose $b>0$ such that
$H\mapsto\UU_H-\id$ is well defined and analytic
as a map from $\hat B=B(\varrho,2b)$ to $\AA^{2d}(\varrho')$.
Such a choice is possible by \clm(aboutPsi) and \clm(GenFun).
Consider first the scalings $S_z\,$, in the case $|z|=1$.
Let $H_0\in\hat B$ be fixed, but arbitrary.
Define $V_0$ and $V'_0$ to be the identity maps on $\complex^{2d}$.
We recall that $\UU_{H_0}$ is a composition of canonical transformations
$U_{\FF(H_m)}\,$, where $\FF(H^0+f)=\varphi(f^{+})f^{-}$,
with $\varphi$ defined by equation \equ(varphiDef).
To be more precise, define $V_{m+1}=V_m\circ U_{\FF(H_m)}$
and $H_{m+1}=H_0\circ V_{m+1}\,$, for $m=0,1,2,\ldots$.
Then, as was shown in the proof of \clm(UUagain),
$V_m\to\UU_{H_0}$ pointwise on $\DD(\varrho')$.
Similarly, if we set $H'_0=\SS_z^{-1}H_0$
and define $V'_{m+1}=V'_m\circ U_{\FF(H'_m)}$
and $H'_{m+1}=H'_0\circ V'_{m+1}\,$, for $m\ge 0$,
then $V'_m\to\UU_{H'_0}$ pointwise on $\DD(\varrho')$.
Thus, in order to prove the assertion,
it suffices to show that $S_z\circ V'_k=V_k\circ S_z$
and $H'_k=\SS_z^{-1}H_k\,$, for all $k$.
Assume now that these identities holds for $k=m\ge 0$,
and consider $k=m+1$.
By using that $\SS_z^{-1}$ commutes with $\Id^\pm$ and $\nabla_1\,$,
and that $\nabla_2\circ\SS_z^{-1}=z\SS_z^{-1}\circ\nabla_2$,
one easily verifies that $\FF(\SS_z^{-1}H_k)=\SS_z^{-1}\FF(H_k)$.
Now we can apply \clm(UJS) to obtain
$$
S_z\circ V'_{m+1}
=S_z V'_m\circ U_{\FF(H'_m)}
=V_m\circ U_{\FF(H_m)}\circ S_z=V_{m+1}\circ S_z\,,
\equation(SHUS1)
$$
and
$$
H'_{m+1}=z^{-1}H_0\circ S_z\circ V'_{m+1}
=z^{-1}H_0\circ V_{m+1}\circ S_z=\SS_z^{-1}H_{m+1}\,,
\equation(SHUS2)
$$
which completes the induction step.
Thus, $S_z\circ\UU_{H'_0}=\UU_{H_0}\circ S_z$ on $\DD(\varrho')$,
whenever $|z|=1$ and $H_0\in\hat B$.
Our next goal is to extend this identity to $z$'s
in sets of the form $G_\eps=\{z\in\complex\colon \eps<|z|<1+\eps\}$.
To this end, let $(q,p)$ be a fixed but arbitrary point in $\DD(\varrho')$,
and choose $r<\varrho'$ such that $(q,p)$
belongs to $\DD(r)$.
Next, choose $\eps>0$ sufficiently small such that
for each $z\in G_\eps\,$,
$S_z$ maps $\DD(r)$ into $\DD(\varrho')$,
and $\SS_z^{-1}$ maps $B(\varrho,\eps b)$ into $\hat B$.
Let $H\in B(\varrho,\eps b)$, and define
a function $g_H: G_\eps\to\complex^{2d}$ by setting
$$
g_H(z)=\bigl(S_z\circ\UU_{\SS_z^{-1}H}-\UU_H\circ S_z\bigr)(q,p)\,,
\qquad z\in G_\eps\,.
\equation(SHUS3)
$$
All of the maps used in this definition are analytic
(on the appropriate domains, due to our choice of parameters),
and thus $g_H$ itself is analytic.
But as was shown above, $g_H$ vanishes on the circle $|z|=1$.
Thus, $g_H$ is identically zero on $G_\eps$.
Let now $z\in G_\eps$ be fixed.
Since $\SS_z^{-1}$ maps $B(\varrho,|z|b)\subset\hat B$
into the ball $\hat B$,
where $H\mapsto\UU_H-\id$ is analytic,
the function $H\mapsto g_H(z)$ defined by equation \equ(SHUS3)
is analytic on $B(\varrho,|z|b)$.
But this function vanishes on $B(\varrho,\eps b)$,
and thus $g_H(z)=0$, for all $H$ in $B(\varrho,|z|b)$.
This proves the assertion regarding scalings,
since $\eps$ and $(q,p)$ were arbitrary.
The translations $J_u$ can be handled similarly.
Assume first that $u\in\real^d$.
Then we can follow the same steps as in the case of
$|z|=1$ scalings, replacing
$S_z$ and $\SS_z$ by $J_u$ and $\JJ_u\,$, respectively.
Again, the function $\FF$ has a symmetry:
$\FF(\JJ_u^{-1}H_k)=\JJ_u^{-1}\FF(H_k)$,
which follows from the fact that
$\JJ_u$ commutes with $\Id^\pm$ and differentiation.
The conclusion is that
$J_u\circ\UU_{H'_0}=\UU_{H_0}\circ J_u$ on $\DD(\varrho')$,
for all $u\in\real^d$ and $H_0\in\hat B$.
If $u$ belongs to the set $\DD_1(\delta)$,
then $J_u$ maps $\DD(\rho')$ into $\DD(\varrho')$,
and $\JJ_u$ maps $B(\rho,b)$ into $B(\varrho,b)$.
Thus, for any given point $(q,p)$ in $\DD(\rho')$,
and any given $H\in B(\rho,b)$,
the function $f_H:\DD_1(\delta)\to\complex^{2d}$,
$$
f_H(u)=\bigl(J_u\circ\UU_{\JJ_u^{-1}H}-\UU_H\circ J_u\bigr)(q,p)\,,
\equation(SHUS3)
$$
is analytic on $\DD_1(\delta)$.
But $f_H$ vanishes on $\real^d$,
as was shown above, which implies that $f_H$ is identically zero.
Since $(q,p)$ and $H$ were arbitrary,
this proves the assertion regarding translations.
\qed
%RPROP.5
\section Properties of the Transformations $\bigRG_\mu$
After proving a claim made in the introduction
concerning the existence of matrices satisfying (T1--3),
and a corollary on the diophantine property of $\omega$,
we will give a proof of \clm(TmuSub) and \clm(RRexists).
Then some symmetry properties of the transformation $\RG_\mu$
and its local stable manifold $\WW^s$ will be discussed.
Here, and in the remaining part of this paper,
$\rho$ is considered a fixed (but arbitrary) positive real number.
Let $\omega=(1,\omega_2,\ldots,\omega_d)$ be a vector
whose components are real algebraic numbers.
\claim Lemma(Texists)
The algebraic extension of $\rational$
by the numbers $\omega_2,\ldots,\omega_d$
is of degree $d$ if and only if there exists an integral $d\times d$
matrix $T$ with the properties (T1--2).
Furthermore, some of the (integral) matrices satisfying (T1--2)
have determinant $1$.
\proof
Let $K$ be the smallest field containing the numbers
$1,\omega_2,\ldots,\omega_d\,$.
For any $\alpha\in K$, define $\varphi_1(\alpha)=\alpha$,
and denote by $\varphi_2(\alpha),\ldots,\varphi_d(\alpha)$
the field conjugates of $\alpha$.
In order to prove the ``if'' part,
assume that $T$ is an integral $d\times d$ matrix
satisfying (T1) and (T2).
Then the eigenvalue $\vartheta$ of $T$ is an algebraic integer of degree $d$.
>From the equation $(\vartheta\id -T)\omega=0$
we see that the numbers $\omega_2,\ldots,\omega_d$
belong to the field $\rational[\vartheta]$.
But $T^n\omega=(\vartheta^n,\ldots)$ for all $n\ge 0$,
which shows that $\rational[\vartheta]$ is contained in $K$.
Thus, $K$ is equal to $\rational[\vartheta]$ which is of degree $d$.
Conversely, assume now that $K$ is of degree $d$.
Let $\gamma\in K$ be an algebraic integer of degree $d$.
Then there exists a $d\times d$ matrix $C$ over $\rational$
such that $C\omega=\gamma\omega$.
Let $s\in\integer$ be such that $B=sC$ is a matrix over $\integer$,
and define $\beta=s\gamma$.
Clearly, $\beta$ is an algebraic integer in $K$
of degree $d$, and $B\omega=\beta\omega$.
Denote by $D$ the discriminant of the basis $1,\beta,\ldots,\beta^{d-1}$.
The remaining part of the proof is based on the construction,
given in [\rBDGPS, Theorem 5.2.2], of a Pisot--Vijayaraghavan number
(in an arbitrary real algebraic field) that is a unit.
The construction involves a sequence of real numbers
$\delta_1>\delta_2>\ldots>0$ which will be defined later.
Let now $\delta_j$ be an arbitrary positive real number less than $1$.
By Minkowski's theorem on linear forms,
we can find $c_0,\ldots,c_{d-1}\in\integer$, not all zero, such that
$$
\left|\sum_{n=0}^{d-1}c_n\beta^n\right|\le|D|^{1/2}\delta_j^{1-d}\,,
\qquad
\left|\sum_{n=0}^{d-1}c_n\varphi_k(\beta^n)\right|\le\delta_j\,,
\qquad k=2,\ldots,d\,.
\equation(Texists1)
$$
In other words, if $p_j$ is the polynomial defined by the equation
$p_j(x)=c_0+c_1x+\ldots+c_{d-1}x^{d-1}$,
then $\alpha_j=p_j(\beta)$ is an algebraic integer in $K$
whose conjugates are all of modulus less that $\delta_1\,$.
In addition, we have $|\alpha_j|>1$
since $N(\alpha_j)=\varphi_1(\alpha_j)\cdots\varphi_d(\alpha_j)$
is a nonzero element of $\integer$.
Thus, given that $|\varphi_k(\alpha_j)|>1$ only for $k=1$,
the degree of $\alpha_j$ has to be $d$, and in particular,
the numbers $\varphi_1(\alpha_j),\ldots,\varphi_d(\alpha_j)$
are pairwise distinct and nonzero.
Now define $A_j=p_j(B)$. Then $A_j\omega=\alpha_j\omega$.
Since the minimal polynomial for $\alpha_j$ is unique,
and thus equal to $t\mapsto\det(t\id -A_j)$,
the eigenvalues of $A_j$ are precisely the number $\alpha_j$
and its conjugates $\varphi_k(\alpha_j)$.
This shows that if we set $T=A_j\,$,
then $T$ satisfies (T1) and (T2).
In order to find a matrix $T$ that satisfies (T3) as well,
consider the above construction for $j=1,2,\ldots$,
where $\delta_{j+1}>0$ is chosen to be less than the minimum
of the numbers $|\varphi_2(\alpha_j)|,\ldots,|\varphi_d(\alpha_j)|$.
Since $|N(\alpha_j)|\le|D|^{1/2}$ for all $j$,
we can find $m\in\integer$, and an infinite set $J\subset\natural$,
such that $N(\alpha_j)=m$ for all $j\in J$.
Then there exists a pair of indices $i>j$ in $J$,
such that each coefficient of the polynomial $p_i$
is congruent modulo $m$ to the corresponding coefficient of $p_j\,$.
In other words, $p_i=p_j+mq$ for some polynomial $q$
with coefficients in $\integer$.
Let now $T=\id +q(B)A_j^{\rm adj}$,
where $A_j^{\rm adj}$ is the matrix over $\integer$
such that $A_jA_j^{\rm adj}=\det(A_j)\id$.
Since $\det(A_j)=m$, we have $TA_j=A_j+mq(B)=A_i\,$.
Thus, $T=A_iA_j^{-1}$, and in particular, $\det(T)=1$.
The eigenvalues of $T$ are
$\vartheta_k=\varphi_k(\alpha_i)/\varphi_k(\alpha_j)$,
for $k=1,2,\ldots,d$,
and the corresponding eigenvectors are the same as those of $B$.
All eigenvalues except $\vartheta_1=\alpha_i/\alpha_j$
are of modulus less than one, since
$|\varphi_k(\alpha_i)|\le\delta_i<|\varphi_k(\alpha_j)|$ for all $k>1$.
Thus, as in the case of the matrices $A_j\,$,
we conclude that $T$ satisfies (T1) and (T2).
\qed
In what follows, $T$ is assumed to be an integral $d\times d$ matrix
satisfying (T1) and (T2).
Property (T3) will not be needed in this section.
Let $\vartheta_1,\vartheta_2,\ldots,\vartheta_d$
be the eigenvalues of $T$,
labeled such that $|\vartheta_i|\ge|\vartheta_j|$ for $i\le j$,
and let $\omega^{(1)},\omega^{(2)},\ldots,\omega^{(d)}$
be the corresponding eigenvectors.
In particular, $\vartheta_1=\vartheta$ and $\omega^{(1)}=\omega$.
\claim Corollary(diophant)
There exists a constant $c>0$ such that
every nonzero $\nu\in\integer^d$ satisfies
$|\omega\cdot\nu|\ge c|\nu|^{-\gamma}$,
where $\gamma=-\ln|\vartheta_1|/\ln|\vartheta_2|$.
\proof
Consider the norm on $\real^d$ defined by the equation
$$
\I x\I=\max_{1\le j\le d}\bigm|\omega^{(j)}\cdot x\bigm|\,,
\qquad x\in\real^d\,.
\equation(newNormRd)
$$
Since all norms on $\real^d$ are equivalent,
there exists a constant $b>0$ such that $\I\nu\I\ge b$
for every nonzero $\nu\in\integer^d$.
Let now $\nu\in\integer^d$ be fixed,
and assume that $|\omega^{(1)}\cdot\nu|<\I\nu\I$.
Let $k$ be the smallest non-negative integer such that
$$
\biglI(T^\ast)^{k+1}\nu\bigrI
=\bigl|\omega^{(1)}\cdot\bigl[(T^\ast)^{k+1}\nu\bigr]\bigr|\,.
\equation(diophant1)
$$
Then, for some $j>1$, we have
$$
b\le\biglI(T^\ast)^k\nu\bigrI
=\bigl|\omega^{(j)}\cdot\bigl[(T^\ast)^k\nu\bigr]\bigr|\
=|\vartheta_j|^k\bigl|\omega^{(j)}\cdot\nu\bigr|
\le|\vartheta_2|^k\I\nu\I\,,
\equation(diophant2)
$$
which implies that
$$
|\vartheta_1|^{-k}=|\vartheta_2|^{\gamma k}
\ge b^\gamma\I\nu\I^{-\gamma}\,.
\equation(diophant3)
$$
By combining \equ(diophant1) with this last inequality,
we obtain
$$
\eqalign{
\bigl|\omega^{(1)}\cdot\nu\bigr|
&=|\vartheta_1|^{-k-1}
\bigl|\omega^{(1)}\cdot\bigl[(T^\ast)^{k+1}\nu\bigr]\bigr|
=|\vartheta_1|^{-k-1}\biglI(T^\ast)^{k+1}\nu\bigrI\cr
&\ge b|\vartheta_1|^{-k-1}
\ge b^{\gamma+1}|\vartheta_1|^{-1}\I\nu\I^{-\gamma}\,,\cr}
\equation(diophant4)
$$
and the assertion follows.
\qed
\proofof(TmuSub)
Let $r>\rho$ be such that $\rho'>\kappa r$.
Define $M=\mu(T^\ast)^{-1}$.
Then, by assumption \equ(TmuBound), we have
$\|T^\ast\|_\omega<\kappa$ and $\|M^\ast\|_0<\kappa<\rho'/r$,
where $\|.\|_x$ is the seminorm introduced in \equ(MatNormDef).
Choose $\sigma>0$ such that
$|T^\ast x|\le\kappa$ whenever $|x|=1$ and $|\omega\cdot x|\le\sigma$.
In addition, assume that $|\sigma|<1/|\Omega|$.
This condition will not be used until later on.
Furthermore, choose $\tau>0$ such that
$$
\bigl(\kappa r/\rho'\bigr)^\tau e^{r\|T^\ast\|-\rho'} <1\,.
\equation(tauChoice)
$$
The two constants $\sigma$ and $\tau$
define the projections $\Id^\pm$, as described in \clm(Iminus).
Let now $H$ be a function in $\AA(\rho')$, such that $\Id^{-}H=0$.
Consider the representation \equ(HnnuDef) for $H$.
Then $H^{(n)}_\nu=0$, unless
either $|\omega\cdot\nu|\le\sigma|\nu|$ or $n\ge\tau|\nu|$.
In the first case, when $|\omega\cdot\nu|\le\sigma|\nu|$,
we have the bound
$$
\eqalign{
\|H^{(n)}_\nu\circ M\|r^n e^{r|T^\ast\nu|}
&\le\|M^\ast\|_0^n\|H^{(n)}_\nu\|r^n e^{r|T^\ast\nu|}\cr
&\le\bigl(\kappa r/\rho'\bigr)^n e^{(r\kappa-\rho')|\nu|}
\|H^{(n)}_\nu\|(\rho')^n e^{\rho'|\nu|}\cr
&\le \|H^{(n)}_\nu\|(\rho')^n e^{\rho'|\nu|}\,,\cr}
\equation(Aimprove1)
$$
and in the case $n\ge\tau|\nu|$ we obtain
$$
\eqalign{
\|H^{(n)}_\nu\circ M\|r^n e^{r|T^\ast\nu|}
&\le\|M^\ast\|_0^n\|H^{(n)}_\nu\|r^n e^{r|T^\ast\nu|}\cr
&\le\left[\bigl(\kappa r/\rho'\bigr)^\tau
e^{r\|T^\ast\|-\rho'}\right]^{|\nu|}
\|H^{(n)}_\nu\|(\rho')^n e^{\rho'|\nu|}\cr
&\le \|H^{(n)}_\nu\|(\rho')^n e^{\rho'|\nu|}\,,\cr}
\equation(Aimprove2)
$$
by using the inequality \equ(tauChoice).
These two estimates imply that $H\circ T_\mu$ extends
analytically to $\DD(r)$ and continuously to the boundary of $\DD(r)$,
and that
$$
\|H\circ T_\mu\|_r
=\sum_{\nu\in\integer^d\atop n\ge 0}
\|H^{(n)}_\nu\circ M\|r^n e^{r|T^\ast\nu|}
\le \|H\|_{\rho'}\,.
\equation(Aimprove3)
$$
The assertion now follows from the fact that
the inclusion map from $\AA(r)$ to $\AA(\rho)$ is compact.
\qed
In the remaining part of this paper,
the parameters $\sigma$ and $\tau$
that define the projection operators $\Id^\pm$
are considered fixed. To be more precise,
we pick some arbitrary number $\rho'$ in $(\kappa\rho,\rho)$
and choose $\sigma,\tau$ as described in the proof of \clm(TmuSub).
\proofof(RRexists)
Choose two numbers $\varrho'<\varrho$ in the interval $(\rho',\rho)$.
By \clm(UUagain), the map $\RR\colon H\mapsto H\circ\UU_H$
is analytic, from some open ball $\hat B\subset\AA(\varrho)$
centered at $H^0$, to $\AA(\rho')$.
And by \clm(TmuSub),
the linear operator $L_\mu\colon\AA(\rho')\to\AA(\rho)$,
defined by equation \equ(LmuDef), is compact.
Consider now the two linear functionals
$f_0\colon H\mapsto(\mean H)(0)$ and
$f_1\colon H\mapsto(\Omega\cdot\nabla\mean H)(0)$
that enter the definition \equ(RRdef1) of $\RG_\mu\,$.
These functionals are clearly continuous on $\AA(\varrho)$.
Furthermore, if $f_1$ is restricted to a sufficiently
small neighborhood $B'$ of $H^0$ in $\AA(\varrho)$,
then its range is contained in the disk $|z-1|<1/2$.
Thus, the equation $\FF(H)=[H-f_0(H)]/f_1(H)$
defines a bounded and analytic map $\FF$ from $B'$ to $\AA(\varrho)$.
By combining all this,
we see that if the radius of $\hat B$ is chosen sufficiently small,
then $\RG_\mu=\FF\circ L_\mu\circ\RR$
is well defined, analytic, and compact,
as a map from $\hat B$ to $\AA(\rho)$.
Consider now $u\in\complex^d$ such that $|u|<\rho-\varrho$
and $|u|<\varrho'-\rho'$.
Then $\JJ_uH\in\AA(\varrho)$ whenever $H\in\AA(\rho)$,
and $\JJ_u$ is bounded as a linear operator
from $\AA(\rho)$ to $\AA(\varrho)$. In addition, $\JJ_uH^0=H^0$.
Thus, some open ball $B\subset\AA(\rho)$, centered at $H^0$,
is mapped into $\hat B$ by $\JJ_u\,$;
and the map $\RG_\mu\circ\JJ_u$ is analytic and compact,
when restricted to $B$. Let now $H\in B$ be fixed.
Define $H'=\RR(H)$ and $H''=L_\mu H'$.
If $B$ has been chosen sufficiently small (which we assume),
then \clm(SHUS) implies that $\RR(\JJ_uH)=\JJ_u\RR(H)$.
Next, by using that $\JJ_u$ commutes with $\Id^{+}$,
and that $T_\mu^{-1}\circ J_u\circ T_\mu=J_{T^{-1}u}\,$, we obtain
$$
\eqalign{
L_\mu\JJ_uH'&={\vartheta\over\mu}
\bigl(\Id^{+}H'\bigr)\circ J_u^{-1}\circ T_\mu
={\vartheta\over\mu}\bigl(\Id^{+}H'\bigr)\circ T_\mu
\circ\bigl(T_\mu^{-1}\circ J_u\circ T_\mu\bigr)^{-1}\cr
&=\JJ_{T^{-1}u}L_\mu H'\,.\cr}
\equation(LJJL)
$$
In addition,
$\FF(\JJ_{T^{-1}u}H'')=\JJ_{T^{-1}u}\FF(H'')$,
since $\mean\JJ_vH''=\mean H''$ for any $v$.
By combining these ``commutation relations'',
the claim $\RG_\mu(\JJ_uH)=\JJ_{T^{-1}u}\RG_\mu(H)$ follows.
Next, let $z\in\complex\setminus\{0\}$ and
$H\in B\cap\AA(|z|\rho)$ be given, such that $\SS_z^{-1}H\in B$.
Assume first that $|z|\le 1$.
Define $H'=\RR(H)$ and $H''=L_\mu H'$, as before.
By \clm(SHUS) we have
$\RR(\SS_z^{-1}H)=\SS_z^{-1}\RR(H)$.
Two analogous identities for $L_\mu$ and $\FF$
follow directly from the definitions:
$L_\mu\SS_z^{-1}H'=\SS_z^{-1}L_\mu H'$ and
$\FF(\SS_z^{-1}H'')=\SS_z^{-1}\FF(H'')$.
By combining these three identities,
we obtain $\RG_\mu(\SS_z^{-1}H)=\SS_z^{-1}\RG_\mu(H)$,
as claimed in \clm(RRexists).
In particular, $\RG_\mu(H)\in\SS_z\AA(\rho)$.
In the case $|z|>1$, consider the function $H_z=\SS_z^{-1}H$.
Then the argument above shows that
$\RG_\mu(\SS_zH_z)=\SS_z\RG_\mu(H_z)$.
By substituting back $H_z=\SS_z^{-1}H$,
we obtain the same conclusion as in the case $|z|\le 1$.
\qed
In order to study the spectral properties of $D\RG_\mu(H^0)$,
it is convenient to introduce the following projection operators
$\proj_n\,$, $\proj_{\ssge n}\,$, and $\proj_{1,1}\,$.
If $n\in\natural$ and $H\in\AA(\rho)$, define
$$
\bigl(\proj_n H\bigr)(q,p)=H_0^{(n)}(p),\qquad
\bigl(\proj_{\ssge n}H\bigr)(q,p)=\sum_{m\ge n}H_0^{(m)}(p)\,,
\equation(ProjDef)
$$
and in particular, $(\proj_{\ssge 0}H)(q,p)=(\mean H)(p)$,
for all $(q,p)\in\DD(\rho)$.
Here, $H_0^{(m)}$ denotes the part of $H$
that is homogeneous of order $m$ in the second variable,
averaged over the torus $\torus^d$; see the definition \equ(HnnuDef).
Furthermore, let
$\proj_{1,1}H=\bigl(\Omega\cdot\nabla H_0^{(1)}\bigr)H^0$,
where $\Omega$ is the eigenvector of $T^\ast$
corresponding to the eigenvalue $\vartheta$,
normalized such that $\Omega\cdot\omega=1$.
Notice that all these projection operators commute with
$L_\mu\colon h\mapsto{\vartheta\over\mu}(\Id^{+}h)\circ T_\mu\,$.
The following is a spectral theorem for the operator $L_\mu\,$.
\claim Lemma(LmuEV)
The eigenvectors $\phi_\beta$ of $L_\mu$
that are listed in \equ(LmuEV)
span a dense subspace of $\proj_{\ssge 0}\AA(\rho)$.
The spectral projection corresponding to the eigenvalue
zero of $L_\mu$ is $\Id-\proj_{\ssge 0}\,$.
More precisely, there are constants $a>0$ and $c>1$,
such that for all positive integers $m$,
$$
\bigl\|L_\mu^{m+1}\bigl(\Id-\proj_{\ssge 0}\bigr)\bigr\|
\le\left|{\textstyle{\vartheta\over\mu}}\right|
e^{-ac^m}\bigl\|L_\mu^m\bigl(\Id-\proj_{\ssge 0}\bigr)\bigr\|\,.
\equation(Lmu0)
$$
\proof
The first part, including the fact that
$L_\mu\phi_\beta=\lambda_\beta\phi_\beta$
for all $\beta\in\natural^d$, is obvious.
What remains to be proved is the bound \equ(Lmu0). Let $k=m-1$.
For every integer $n\ge 0$, denote by $V_n^{+}$
the set of all vectors $\nu\in\integer^d$
that satisfy $|\omega\cdot\nu|\le\sigma|\nu|$ or $\tau|\nu|\le n$.
By using the representation \equ(HnnuDef) for $h\in\AA(\rho)$,
we can write
$$
\bigl(L_\mu^{k+1}h\bigr)(q,p)
=\sum_{n\ge 0}\bigl(\vartheta\mu^{n-1}\bigr)^{k+1}
\sum_\nu
h_\nu^{(n)}\bigl((T^\ast)^{-k-1}p\bigr)
\exp\Bigl(i\bigr[(T^\ast)^{k+1}\nu\bigr]\cdot q\Bigr)\,,
\equation(LmuEV1)
$$
for all $(q,p)\in\DD(\rho)$,
where the sum $\sum_\nu$ ranges over all $\nu\in\integer^d$
such that $(T^\ast)^j\nu\in V_n^{+}$, for $j=0,1,\ldots,k$.
In particular, if $\proj_{\ssge 0}h=0$, then $L_\mu^{k+1}h$
only depends on those components $h_\nu^{(n)}$ of $h$
that are indexed by pairs $(\nu,n)$
with the property that $(T^\ast)^k\nu\in V_n^{+}\setminus\{0\}$.
Below we will show that these indices satisfy one of the bounds
$$
n\ge b_1|\vartheta_1|^{k/2}\,,\quad {\rm or}\quad
|\nu|\ge b_2|\vartheta_2|^{-k/2}\,,
\equation(LmuEV2)
$$
where $b_1$ and $b_2$ are positive constants independent of $k$.
Consider now the projection operator $\Id_k^{+}$, which
acts on a function $H\in\AA(\rho)$ by restricting
the sum in the expansion \equ(HnnuDef) to index pairs
$(\nu,n)$ satisfying $(T^\ast)^k\nu\in V_n^{+}\setminus\{0\}$.
In order to prove \equ(Lmu0),
we write $L_\mu^{m+1}(\Id-\proj_{\ssge 0})$
as a product $A\Id_k^{+}L_\mu$,
where $A=L_\mu^{k+1}(\Id-\proj_{\ssge 0})$,
and then we bound the operator norm of this product by
$\|A\|\cdot\|\Id_k^{+}L_\mu\|$.
As was shown in the proof of \clm(TmuSub),
there exists $r>\rho$, such that $L_\mu$ is continuous
as a linear operator from $\AA(\rho)$ to $\AA(r)$,
and bounded in norm by $|\vartheta/\mu|$.
Now regard $\Id_k^{+}$ as a linear operator from $\AA(r)$ to $\AA(\rho)$.
By using the inequalities \equ(LmuEV2),
the norm of this operator can be bounded from above by
$e^{-a'c^k}$, with $a'>0$ and $c>1$ independent of $k$.
This proves the bound \equ(Lmu0).
In order to prove that \equ(LmuEV2) holds whenever
$(T^\ast)^k\nu\in V_n^{+}\setminus\{0\}$,
consider first the case where
$\bigl|\omega\cdot(T^\ast)^k\nu|\le\sigma\bigl|(T^\ast)^k\nu\bigr|$.
Then
$$
|\omega\cdot\nu|
=|\vartheta_1|^{-k}\bigl|\omega\cdot(T^\ast)^k\nu\bigr|
\le\sigma\bigl|(\vartheta_1^{-1}T^\ast)^k\nu\bigr|
\le\sigma\bigl|(\omega\cdot\nu)\Omega\bigr|
+\sigma\bigl|(\vartheta_1^{-1}T^\ast)^kP\nu\bigr|\,,
\equation(LmuEV3)
$$
where $P\nu=\nu-(\omega\cdot\nu)\Omega$ is the component
of $\nu$ in the spectral subspace of $T^\ast$ corresponding
to the eigenvalues $\vartheta_2,\ldots,\vartheta_d$
of modulus less than one.
Thus, since $|\sigma|<1/|\Omega|$,
we can bound $|\omega\cdot\nu|$ from above by a constant times
$|\vartheta_2/\vartheta_1|^k|\nu|$.
Combining this with the lower bound from \clm(diophant),
we find that
$$
b_3|\nu|^{-\gamma}\le
\left|\vartheta_2/\vartheta_1\right|^k|\nu|\,,
\equation(LmuEV4)
$$
for some constant $b_3>0$.
By using the definition of $\gamma$, this inequality can be rewritten
as $|\nu|\ge b_2|\vartheta_2|^{-k}$, which implies \equ(LmuEV2).
Next, consider the case where $\tau\bigl|(T^\ast)^k\nu\bigr|\le n$.
Then there exists $b_4>0$ such that
$$
|\omega\cdot\nu|
=|\vartheta_1|^{-k}\bigl|\omega\cdot(T^\ast)^k\nu\bigr|
\le b_4|\vartheta_1|^{-k}\bigl|(T^\ast)^k\nu\bigr|
\le b_4|\vartheta_1|^{-k}n/\tau\,.
\equation(LmuEV5)
$$
This bound, together with \clm(diophant), implies that
$$
n|\nu|^\gamma\ge b_5|\vartheta_1|^k\,,
\equation(LmuEV6)
$$
for some constant $b_5>0$.
The two alternatives in \equ(LmuEV2) are now obtained
by distinguishing the cases
$n\ge|\nu|^\gamma|$ and $|\nu|^\gamma\ge n$.
\qed
This lemma translates easily into a spectral theorem for $D\RG_\mu(H^0)$,
since $D\RG_\mu(H^0)$ differs from $L_\mu$
only on the two-dimensional invariant subspace spanned by
the constant functions and the function $(q,p)\mapsto\omega\cdot p$.
More precisely, we have
$$
D\RG_\mu\bigl(H^0\bigr)=L_\mu\bigl(\Id-\proj_0-\proj_{1,1}\bigr)\,.
\equation(DRR2)
$$
Notice also that, by \clm(UUmain),
the intersection of $\proj_{\ssge 0}\AA(\rho)$
with the domain of $\RG_\mu$ is mapped
into $\proj_{\ssge 0}\AA(\rho)$ by the transformation $\RG_\mu\,$.
Throughout the rest of this paper,
we assume that $\mu$ satisfies the bound \equ(muCondition).
Then the spectrum of $D\RG_\mu\bigl(H^0\bigr)$
splits into an unstable part,
lying in the complex domain $|z|\ge\theta_u>1$,
and a stable part, lying in $|z|\le\theta_s(\mu)<1$, where
$$
\theta_u=|\lambda_{(0,1,0,\ldots,0)}|
=|\vartheta_1/\vartheta_2|\,,\qquad
\theta_s(\mu)=|\lambda_{(0,\ldots,0,2)}|
=|\mu\vartheta_1/\vartheta_d^2|\,.
\equation(thetaSU)
$$
The corresponding spectral projections are
$\proj^u=\proj_1-\proj_{1,1}$ and $\proj^s=\Id-\proj^u$, respectively.
The space $\AA^u=\proj^u\!\AA(\rho)$ is of dimension $d-1$,
spanned by the functions $h_2,\ldots,h_d\,$ listed in \equ(RmuEV).
More explicitly, we have
$$
\proj^uh=\sum_{j=2}
\Omega^{(j)}\cdot\bigl(\nabla\mean h\bigr)(0)h_j\,,
\qquad h\in\AA(\rho)\,,
\equation(PuExplicit)
$$
where $\Omega^{(j)}$, for $1\le j\le d$, denotes
the eigenvector of $T^\ast$ for the eigenvalue $\vartheta_j\,$,
normalized such that $\Omega^{(j)}\cdot\omega^{(j)}=1$.
Let $\AA^s(\rho)$ be the affine spaces consisting
of all functions of the form $H^0+h$, with $h\in\proj^s\!\AA(\rho)$.
Given an two sets $A,B,$ contained
in the domain (as described in \clm(RRexists)) of $\RG_\mu\,$,
define $\WW^s(A,B)$ to be the set of all functions $H\in A$
such that $\RG_\mu^n(H)\in B$ for all $n\ge 1$.
By the local stable manifold theorem [HSP],
there exists an open neighborhood $B$ of $H^0$ in $\AA(\rho)$,
such that $\WW^s(B,B)$ is the graph
of a differentiable map $W\colon B\cap\AA^s(\rho)\to \AA^u$,
whose derivative vanishes at $H^0$
and is bounded in norm by $1$ at every point in its domain.
Furthermore, if $B'\subset B$ then
$\WW^s(A,B')=\WW^s(A,B)$,
for every sufficiently small open neighborhood
$A$ of $H^0$ that is contained in $B'$.
$\WW^s=\WW^s(B,B)$ will be referred to as
the local stable manifold of $\RG_\mu$ at $H^0$.
A similar construction, using inverse images,
yields $\WW^u=\{H\in B\colon H-H^0\in\AA^u\}$
as the local unstable manifold of $\RG_\mu$ at $H^0$.
Having already introduced the necessary notation,
let us also define (for use later on)
the projection $\wp$, in the direction of $\AA^u$,
from $B$ onto $\WW^s$, by setting
$$
\wp(H)=H^s=\proj^sH+W(\proj^sH)\,,\qquad H\in B\,.
\equation(HsDef)
$$
Trivial examples of Hamiltonians on $\WW^s$
are functions $H=cH^0+f$ in $B$,
with $f(q,p)$ depending on $p$ only,
satisfying $(\nabla_2f)(0,0)=0$.
In this case, $\proj^uH=0$, as can be seen from \equ(PuExplicit),
and $\RG_\mu(H)=H^0+c^{-1}D\RG_\mu(H^0)f$.
The following lemma describes a consequence of the fact
that the renormalization group transformation $\RG_\mu$
``commutes'' with scalings $\SS_z\,$; see \clm(RRexists).
\claim Lemma(WSW)
Let $00$, define $B(\eps)$ to be the pointwise sum
$B_1(\eps)\oplus B_2(\eps)$ of the two sets
$$
B_1(\eps)=\bigl\{H\in\AA^s(\rho)\colon
\|H-H^0\|_\rho<\eps\bigr\}\,,\quad
B_2(\eps)=\bigl\{h\in\AA^u\colon\|h\|_\rho<\eps\bigr\}\,.
\equation(BdDef)
$$
The neighborhood $B$ mentioned in \clm(WSW)
will be $B(\delta)$, with $\delta>0$ specified below.
For the proof of $(1)$, choose $0<\delta<\eps$
such that $B(\eps/a)\subset B$
and $\WW^s(B(\delta),B(\eps))=\WW^s(B(\delta),B)$.
Let $z$ be a complex number satisfying $a\le|z|\le 1$,
and let $h\in B_1(\delta)$.
Then the functions $H=h+W(h)$ and $\RG_\mu^n(H)$
belong to $\WW^s\cap B(\delta)$ and $B(\eps)$,
respectively, for all $n\ge 1$.
Consequently, the functions $\SS_z^{-1}H$ and $\SS_z^{-1}\RG_\mu^n(H)$
belong to $B(\delta/a)$ and $B(\eps/a)$,
respectively, for all $n\ge 1$.
This in turn implies that $\SS_z^{-1}H\in\WW^s$,
since $\SS_z^{-1}\RG_\mu^n(H)=\RG_\mu^n(\SS_z^{-1}H)$ by \clm(RRexists).
But $\SS_z^{-1}H=\SS_z^{-1}h+\SS_z^{-1}W(h)=\SS_z^{-1}h+W(h)$,
and $\SS_z^{-1}h\in B_1(\delta/a)$,
which shows that $W(\SS_z^{-1}h)=W(h)$.
Thus, $W\circ\SS_z^{-1}=W$ on $B_1(\delta)$, as claimed.
In order to prove $(2)$, using analyticity arguments,
we first extend the function $z\mapsto \RG_{z\mu}$
to an open region containing the circle $|z|=1$.
To this end, it suffices to note that
in the proof of \clm(RRexists),
the value of $\mu$ enters only through the condition
$0<\|\mu T^{-1}\|_0<\kappa$.
Consequently, we can find constants $\eps>0$ and $c>1$,
such that $\RG_{z\mu}=\SS_z^{-1}\circ\RG_\mu$ is well defined,
analytic, and compact, as a map from
some open neighborhood of $\ov{B(\eps)}$, to $\AA(\rho)$,
for all $z$ in the region $U=\{z\in\complex\colon 0<|z|\eps>0$. Then there exists an open neighborhood
$B$ of $H^0$ in $\AA(r)$, such that
that the following holds, whenever $u\in\DD_1(\eps)$.
The operator $\JJ_u: \AA(r)\to\AA(\rho)$
maps $\AA^s(r)$ into $\AA^s(\rho)$
and $\WW^s\cap B$ into $\WW^s$.
Furthermore, $W\circ\JJ_u=W$ on $\AA^s(\rho)\cap B$.
\proof
Under the given assumption, $\JJ_u$ is
a bounded linear operator from $\AA(r)$ to $\AA(\rho)$,
and it commutes with $\proj^s$, in the sense
that $\JJ_u\proj^sH=\proj^s\JJ_uH$ for all $H\in\AA(r)$.
Consider an open neighborhood
$B(\delta')=B_1(\delta')\oplus B_2(\delta')$ of $H^0$ in $\AA(\rho)$,
as defined in \equ(BdDef),
where $\WW^s$ is the graph of an analytic function
$W: B_1(\delta')\to B_2(\delta')$.
Choose a positive $\delta<\delta'$ such that
$\RG_\mu^n(H)$ belongs to $B(\delta')$,
for all $H\in\WW^s\cap B(\delta)$ and $n\ge 0$.
Then choose an open ball $B$ in $\AA(r)$, centered at $H^0$,
such that $\JJ_u$ maps $B$ into $B(\delta)$,
whenever $u$ belongs to $\DD_1(\eps)$.
Let $H\in\WW^s\cap B$ be fixed. Then
$$
\JJ_uH=\JJ_u\proj^sH+\JJ_uW\bigl(\proj^sH\bigr)
=\proj^s\JJ_uH+W\bigl(\proj^sH\bigr)\,.
\equation(WJJW1)
$$
This equation shows that
$\JJ_uH\in\WW^s$ if and only if $W(\proj^sH)=W(\JJ_u\proj^sH)$.
Assume now that $u\in\real^d$.
Since $\JJ_{T^{-n}u}$ is an isometry,
it follows from \clm(RRexists), that $\RG_\mu^n(\JJ_uH)$
belongs to $B(\delta')$, for all $n\ge 0$.
Thus $\JJ_uH\in\WW^s\cap B$,
which implies that $W(\proj^sH)=W(\JJ_u\proj^sH)$.
Next, consider the function $f\colon\DD_1(\eps)\mapsto\AA^u$,
defined by the equation $f(u)=W(\proj^sH)-W(\JJ_u\proj^sH)$.
This function is analytic,
since $u\mapsto\JJ_uH$ is analytic as a map
from $\DD_1(\eps)$ to $B(\delta)$.
But $f$ vanishes on $\real^d$,
and thus $W(\proj^sH)-W(\JJ_u\proj^sH)=0$ for all $z\in\DD_1(\eps)$.
This in turn implies that $\JJ_uH\in\WW^s\cap B$
for all $z\in\DD_1(\eps)$.
Since $H\in\WW^s\cap B$ was arbitrary, the assertion is proved.
\qed
Finally, we estimate how fast a point on $\WW^s$
approaches $H^0$ and its torus average, respectively,
under the iteration of $\RG_\mu\,$.
\claim Lemma(Wstable)
Let $\mu\in\complex$ and $\theta\in\real$ be given,
satisfying $0<\theta_s(\mu)<\theta<1$.
Then there exists an open neighborhood $B$ of $H^0$ in $\AA(\rho)$,
three constants $K,a>0$ and $c>1$,
and a positive integer $N$, such that
$$
\bigl\|\RG_\mu^m(H)-H^0\bigr\|_\rho
\le K\theta^m\bigl\|H-H^0\bigr\|_\rho\,, \phantom{xxxxxxxxxxi}
\equation(Wstable)
$$
$$
\phantom{xxxxx.}
\bigl\|\bigl(\Id-\proj_{\ssge 0}\bigr)\RG_\mu^n(H)\bigr\|_\rho
\le e^{-ac^n}\bigl\|H-H^0\bigr\|_\rho
+\theta^{2n}\bigl\|H-H^0\bigr\|_\rho^2\,,
\equation(Wstable0)
$$
whenever $m\ge 0$, $n\ge N$, and $H\in\WW^s\cap B$.
\proof
Fix two numbers $t'0$ such that $\|g'\|\le b_1\|h'\|^2$.
Consider now the function $H''=\RG_\mu(H')$.
Then $h''=H''-H^0$ may be written as $h''=f''+g''$, where
$$
\eqalign{
f''&=D\RG_\mu(H^0)f'\,,\cr
g''&=D\RG_\mu(H^0)g'
+\bigl[\RG_\mu(H^0+f'+g')-D\RG_\mu(H^0)(f'+g')\bigr]\,.\cr}
\equation(Wstable1)
$$
Since $\RG_\mu$ is differentiable (complex analytic),
there are fixed constants $b_2$ and $b_3$ such that
$$
\|f''\|\le t'\|f'\|\,,\qquad
\|g''\|\le b_2\|g'\|+b_3\|f'+g'\|^2\,.
\equation(Wstable2)
$$
By combining this with the inequalities
$\|g'\|\le b_1\|h'\|^2$ and $\|f'\|\le\|h'\|+b_1\|h'\|^2$,
we see that there exists $r>0$, such that
$\|h''\|\le t\|h'\|$, whenever $\|h'\|0$ such that
$$
t^{-2\eps}(b_2+9b_3)^\eps<(\theta/t)^2\,.
\equation(Wstable3)
$$
By using \clm(LmuEV), we can find constants $a>0$ and $c>1$,
and a sequence of positive integers $k_n\le\eps n$,
such that $n-k_n\ge M$ and
$$
\bigl\|(\Id-\proj_{\ssge 0})[D\RG_\mu(H^0)]^{k_n}\bigr\|\le e^{-ac^n}\,,
\equation(Wstable4)
$$
whenever $n$ is sufficiently large, say $n\ge N'$.
Given $n\ge N'$,
and an arbitrary function $H=H^0+h$ in $\WW^s\cap B$,
consider $H'=\RG_\mu^m(H)$ with $m=n-k_n\,$.
As was shown in the first part of this proof,
the norm of $h'=H'-H^0$ is bounded by $\delta=t^m\|h\|\le t^mr$.
Starting with the pair $(f',g')=(\proj^sh',\proj^uh')$,
we now iterate the map
$(f',g')\mapsto(f'',g'')$ defined by equation \equ(Wstable1).
The initial pair satisfies
$\|f'\|\le 2\delta$ and $\|g'\|\le b_1\delta^2$.
In order to estimate the pair $(\hat f,\hat g)$
obtained after $k_n$ steps,
we can successively apply the inequalities \equ(Wstable2):
If $\|g'\|\le b\delta^2$ with $\delta\le b\delta\le 1$, then
$\|g''\|\le b_2b\delta^2+b_3(3\delta)^2\le(b_2+9b_3)b\delta^2$.
Here, we have used that $f'$ remains bounded in norm
by $2\delta$ throughout the iteration procedure.
Thus, since $(b_2+9b_3)^{k_n} b_1\delta\le 1$
by our choice of $\eps$, we find that
$$
\hat f=\bigl[D\RG_\mu(H^0)\bigr]^{k_n}f'\,,\qquad
\|\hat g\|\le b_1\delta^2(b_2+9b_3)^{k_n}\,.
\equation(Wstable5)
$$
By using \equ(Wstable3), \equ(Wstable4), \equ(Wstable5),
and the fact that $H^0+\hat f+\hat g=\RG_\mu^n(H)$,
it is now straightforward to check that the function
$(\Id-\proj_{\ssge 0})\RG_\mu^n(H)
=(\Id-\proj_{\ssge 0})\hat f+(\Id-\proj_{\ssge 0})\hat g$
satisfies the bound \equ(Wstable0),
if $n$ is sufficiently large.
\qed
We note that, although the bound \equ(Wstable0)
is used in Section 5,
it is not really needed to prove our main results
(as it turned out).
\remark
Our definition of $\RG_\mu\colon B_\mu\subset\AA(\rho)\to\AA(\rho)$
involves many choices,
such as the positive real numbers $\rho'<\varrho'<\varrho<\rho$,
and the two parameters $\sigma$ and $\mu$
(restricted by the choice of $\rho'$).
All these numbers are now considered fixed.
This does not, however, restrict the possible choices
for the parameter $\mu$,
as long as the condition \equ(TmuBound) is satisfied.
In particular, if we are only interested in $|\mu|\le s$
for some (allowed) constant $s$,
then $\RG_\mu$ can be defined as $\SS_{\mu/s}^{-1}\circ\RG_s$
on an open ball of radius $|\mu|b$ in $\AA(\rho)$
centered at $H^0$, where $b>0$ is determined by $B_s\,$.
%TORI.2
\section Invariant Tori
Our construction of invariant tori involves
iterates of the transformation
$\RG_\mu\colon B\subset\AA(\rho)\to\AA(\rho)$.
Here, $\rho>0$ is fixed,
and the parameter $\mu$ is assumed to satisfy
the condition \equ(muCondition),
which guarantees that the local unstable manifold $\WW^u$
of $\RG_\mu$ at $H^0$ is of dimension $d-1$.
We start by introducing some notation.
Recall from \clm(RRdef) that
$\RG_\mu(H)=[\tilde H-\Xi(H)]/\xi(H)$,
where
$$
\xi(H)=\bigl(\Omega\cdot\nabla\mean\tilde H\bigr)(0)\,,\qquad
\Xi(H)=\bigl(\mean\tilde H\bigr)(0)\,,
\equation(xiDef)
$$
and $\tilde H=(\vartheta/\mu)H\circ\UU_H\circ T_\mu\,$.
For any $n\ge 0$,
define $H_n=\RG_\mu^n(H)$ whenever $H$ belongs to the domain
of $\RG_\mu^n$.
An explicit expression for $H_n$ is given by the equation
$$
H_n={1\over\xi'_n(H)}
\left({\vartheta\over\mu}\right)^n
\Bigl[H_0\circ\UU_{H_0}\circ T_\mu\circ\UU_{H_1}\circ T_\mu
\circ\ldots\circ\UU_{H_{n-1}}\circ T_\mu -\Xi'_n(H)\Bigr]\,,
\equation(RRiter)
$$
where
$$
\xi'_n(H)=\xi(H_0)\cdots\xi(H_{n-1})\,,\qquad
\Xi'_n(H)=\sum_{k=0}^{n-1}\xi'_k(H)
\left({\vartheta\over\mu}\right)^k\!\Xi(H_k)\,.
\equation(RRiter1)
$$
If we define $V_k(H)=T_\mu^k\circ\UU_{H_k}\circ T_\mu^{-k}\,$,
then the relation \equ(RRiter) between $H$ and $H_n$
can be rewritten as follows:
$$
H\circ V_0(H)\circ V_1(H)\circ\ldots\circ V_{n-1}(H)
=\xi'_n(H)\left({\mu\over\vartheta}\right)^n
\!H_n\circ T_\mu^{-n} +\Xi'_n(H)\,.
\equation(RRiter2)
$$
Here, and in what follows, the matrix $T$ is assumed to
satisfy all three conditions (T1-3) given in the introduction.
As we will show later, the transformations $V_k(H)$
are canonical. Thus, equation \equ(RRiter2) shows
that a suitable restriction of $H$ is canonically equivalent
to another Hamiltonian (the right hand side of \equ(RRiter2))
that is generally much closer to $H^0$.
The assumption here is that $H$ belongs to the domain of $\RG_\mu^n$.
Notice that if $H_n$ happens to be equal to $H^0$,
then the image of $\torus^d\times\{0\}$
under $V_0(H)\circ\ldots\circ V_{n-1}(H)$
is an invariant torus for $H$.
Another important property of $V_k(H)$
is that it is (roughly speaking) independent of the parameter $\mu$.
We will prove this below,
in the case where $H$ lies on the stable manifold $\WW^s$ of $\RG_\mu\,$.
Or equivalently, we replace $H$ by $H^s$,
where $H^s$ denotes the projection (in the direction of $\AA^u$)
of $H$ onto $\WW^s$, as defined by equation \equ(HsDef).
The positive number $\rho'<\rho$ that appears
in the proposition below, and later on as well,
is the one that was used to define
the projections $\Id^\pm$ and the transformations $\RG_\mu\,$;
see also the remark at the end of the last section.
\claim Proposition(Vmu)
Given $z\in\complex$ satisfying $0<|z|\le 1$,
there exists an open neighborhood $B$ of $H^0$ in $\AA(\rho)$,
such that for each $k\ge 1$,
$$
T_{z\mu}^k\circ\UU_{\RG_{z\mu}^k(H^s)}\circ T_{z\mu}^{-k}
=T_\mu^k\circ\UU_{\RG_\mu^k(H^s)}\circ T_\mu^{-k}
\qquad {\rm on~~} T_{z\mu}^k\DD(\rho')\,,
\equation(Vmu)
$$
and $\xi(\RG_{z\mu}^k(H^s))=\xi(\RG_\mu^k(H^s))$,
whenever $H\in B$.
\proof
Let $k\ge 1$ be fixed but arbitrary.
If $H'\in\AA(\rho)$ is sufficiently close
(depending on $k$) to $H^0$ then, by \clm(RRexists), we have
$\RG_{z\mu}^k(H')=\SS_z^{-k}\RG_\mu(H')$,
and by \clm(SHUS), this implies that $F_k(H')=0$ on $S_z^k\DD(\rho')$,
where
$$
F_k(H')=S_z^k\circ\UU_{\RG_{z\mu}^k(H')}\circ S_z^{-k}
-\UU_{\RG_\mu^k(H')}\,.
\equation(Vmu1)
$$
Notice that the equation $F(H^s)=0$ on $S_z^k\DD(\rho')$
is equivalent to \equ(Vmu).
The goal is to extend this identity
to a neighborhood $B$ of $H^0$ that is independent of $k$.
To this end, consider an open set $B'\ni H^0$
with the following properties:
The map $H\mapsto\UU_H$ is analytic on $B'$,
and the the local stable manifolds of $\RG_{z\mu}$
and $\RG_\mu$ coincide in $B'$.
By \clm(aboutPsi) and \clm(WSW),
these conditions are satisfied if $B'$ is sufficiently small.
In addition, the map $\wp\colon H\mapsto H^s$ is analytic on $B'$.
Next, we choose an open ball $B\subset B'$,
centered at $H^0$, such that
$\RG_\mu^n(H^s)$ and $\RG_{z\mu}^n(H^s)$
belong to $B'$, for all $H\in B$, and for all $n\ge 1$.
The existence of such a ball follows e.g.
from the bound \equ(Wstable) in \clm(Wstable).
As a consequence,
$\RG_\mu^n\circ\wp$ and $\RG_{z\mu}^n\circ\wp$
are analytic as maps from $B$ to $B'$.
Let now $(q,p)$ be a fixed but arbitrary point in $S_z^k\DD(\rho')$.
Define $f_k\colon B\to\complex^{2d}$ by setting
$f_k(H)=(F_k(H^s))(q,p)$, for all $H\in B$.
This function $f$, being the (difference and) composition
of analytic maps, is analytic on $B$.
But as was shown above, $f$ is identically zero
on an open neighborhood of $H^0$.
Thus, $f=0$ on $B$, and since $(p,q)$ and $k$ were arbitrary,
the assertion regarding $V_k(H^s)$is proved.
The identity
$\xi(\RG_{z\mu}^k(H^s))=\xi(\RG_\mu^k(H^s))$
can be proved similarly, using the fact that
$\xi\circ\SS_z=\xi$ on $B$; see e.g. the proof of \clm(RRexists).
\qed
We note that both sides of equation \equ(RRiter2)
are defined (naturally) as functions on $T_\mu^n\DD(\rho)$.
But for simplicity, we will now restrict these functions to domains
of the form $S_\mu^n\DD(\rho_n)$, which are necessarily much smaller.
Here, $S_z$ denotes (for any $z\in\complex$)
the linear transformation on $\complex^{2d}$,
defined by the equation $S_z(q,p)=(q,zp)$.
Denote by $m(T)$ the largest of the norms
$\|T\|_0\,,\|T^\ast\|_0\,,\|T^{-1}\|_0\,$, and $\|(T^\ast)^{-1}\|_0\,$.
Let $\eta<1/m(T)$ and $\kappa'<\kappa$ be fixed positive real numbers,
where $\kappa$
is the constant that appears in the condition \equ(TmuBound),
and set
$$
\rho_k=\eta^k\kappa'\rho\,,\qquad k=0,1,2,\ldots\,.
\equation(rhokDef)
$$
\claim Definition(knNorm)
Assume $\mu\in\complex\setminus\{0\}$ is given.
For every integer $n\ge 0$ define $A_n=S_\mu^n\DD(\rho_n)$,
and denote by $\AA_n$ the vector space
of all analytic maps $G\colon\,A_n\to\complex^{2d}$,
such that $G\circ S_\mu^n$ belongs to $\AA^{2d}(\rho_n)$.
On the space $\AA_n$ we consider the following (equivalent) norms:
$$
\|G\|_{k,n}=\bigl\|S_\mu^{-k}G\circ S_\mu^n\bigr\|_{\rho_n}\,,
\qquad G\in\AA_n\,.
\equation(knNorm)
$$
The next two propositions will be used to estimate
the composite transformation $V_0(H^s)\circ\ldots\circ V_{n-1}(H^s)$,
and some of its derivatives, on the domain $\AA_n\,$.
\claim Proposition(VnBound)
If $|\mu|$ is chosen sufficiently small (but positive)
then there exists a real number $a>0$,
and an open neighborhood $B$ of $H^0$ in $\AA(\rho)$,
such that the following holds, for every $k\ge 0$.
If $H\in B$ then $V_k(H^s)$ is a canonical transformation
from $A_{k+1}$ into $A_k\,$.
The map $H\mapsto V_k(H^s)-\id$ is analytic
from $B$ to $\AA_{k+1}$ and satisfies the bound
$$
\|V_k(H^s)-\id\|_{k,k+1}
\le a|b\mu|^{2k}\|H^s-H^0\|_\rho\,,\qquad H\in B\,,
\equation(VnBound)
$$
where $b>0$ is some universal constant depending only
on the matrix $T$.
\proof
Choose $\eta'$ in $(\eta,1/m(T))$,
and define $r_k=\eta'\rho_k\,$, for every $k\ge 0$.
By \clm(Wstable), there are positive constants $b'<|\mu|^{-1}$
and $c_1(\mu)$, such that
$$
\|\Id^{-}H^s_k\|_\rho\le c_1(\mu)|b'\mu|^{2k}\|H^s-H^0\|_\rho\,,
\qquad k=0,1,\ldots\,,
\equation(VnB1)
$$
whenever $H$ (and thus $H^s$) is sufficiently close to $H^0$.
Here, $H^s_k=\RG_\mu^k(H^s)$.
Furthermore, since $H\mapsto H^s$ and $\RG_\mu$ are analytic near $H^0$,
the same is true for the map $H\mapsto H^s_k\,$.
The latter is now composed with the (analytic) map $\psi$
described in \clm(aboutPsi), and we get the bound
$$
\|\psi(H^s_k)\|'_{r_0}\le c_2\|\Id^{-}H^s_k\|_\rho
\le c_1(\mu)c_2|b'\mu|^{2k}\|H^s-H^0\|_\rho\,,
\qquad k\ge 0\,,
\equation(VnB2)
$$
for $H$ sufficiently close to $H^0$,
where $c_2$ is some fixed constant.
The identity
$$
S_\mu^{-k}\bigl[V_k(H^s)-\id\bigr]\circ S_\mu^{k+1}
=T_1^k\bigl[U_{\psi(H^s_k)}-\id\bigr]\circ T_1^{-k}S_\mu
\equation(VnB3)
$$
and \clm(GenFun) now show that if $|\mu|>0$ is sufficiently small,
then $H\mapsto V_k(H^s)-\id$
is well defined and analytic, as a map from some open neighborhood
of $H^0$ in $\AA(\rho)$ to $\AA_{k+1}\,$.
Here, we have also used that $\phi\mapsto\phi\circ T_1^{-k}$
is bounded from $\AA(\rho_1)$ to $\AA(\rho_{k+1})$,
as was shown in \clm(TUT).
By \clm(TUT), $S_\mu^{-k}V_k(H^s)\circ S_\mu^k$ is a canonical transformation
defined on $\DD(\rho_k)$, with generating function
$\psi_k(H^s)=\psi(H^s_k)\circ T_1^{-k}$
belonging to $\AA(r_k)$ and satisfying a bound of the form
$$
\bigl\|\psi_k(H^s)\bigr\|'_{r_k}
\le c_3|\eta|^{-k}\|\psi(H^s_k)\|'_{r_0}\,,\qquad k\ge 0\,,
\equation(VnB3)
$$
for some constant $c_3>0$.
Again, this holds under the assumption
that $|\mu|$ is sufficiently small
and $H$ sufficiently close (depending on $\mu$) to $H^0$.
The bound \clm(VnBound) is now obtained
by combining \equ(VnB3) with \equ(VnB2),
and using \clm(GenFun).
\qed
We note that
by keeping track of the constants in the above proof,
one finds that any $b>\vartheta_1\vartheta_d^{-2}$ will do,
and that the assertion of \clm(VnBound) holds
whenever $|\mu|<\eta/b$.
\claim Proposition(Obvious)
Let $00$,
and an open neighborhood $B$ of $H^0$ in $\AA(\rho)$,
such that the following holds.
If $H\in B$ and $n\ge 1$ then $V'_n(H^s)-\id$ belongs to $\AA_n\,$,
and the map $H\mapsto V'_n(H^s)-\id$ is analytic on $B$.
Similarly, the functions $H\mapsto\xi'_n(H^s)$
defined by \equ(RRiter1) and \equ(HsDef), are analytic on $B$.
Furthermore,
$$
\eqalign{
\bigl\|V'_{n+1}(H^s)-V'_n(H^s)\bigr\|_{0,n+1}
&\le a|b\mu|^{2n}\|H^s-H^0\|_\rho\,,\cr
\bigl\|V'_{n+1}(H^s)-\id\bigr\|_{0,n+1}
&\le a'\|H^s-H^0\|_\rho\,,\cr}
\equation(VprimeBound)
$$
and
$$
\eqalign{
\bigl|\xi'_{n+1}(H^s)-\xi'_n(H^s)\bigr|
&\le a|b\mu|^n\|H^s-H^0\|_\rho\,,\cr
\bigl|\xi'_{n+1}(H^s)-1\bigr|
&\le a'\|H^s-H^0\|_\rho\,,\cr}
\equation(xipBound)
$$
for all $H\in B$ and $n\ge 1$, where $b>0$ is some fixed constant.
\proof
By \clm(VnBound), if $|\mu|>0$ is sufficiently small
then there exists an open neighborhood $B'$ of $H^0$ in $\AA(\rho)$,
and constants $a,b,c>0$ (with $b$ independent of $\mu$),
such that the following holds whenever $0<\eps\le c$.
Let $B(\eps)=\{H\in B'\colon\|H^s-H^0\|_\rho<2\eps\}$,
and for every integer $k\ge 0$,
define $d_k=\rho/(a|b\mu|^{2k}\eps)$
and $D_k=\{z\in\complex\colon |z|0$ is ``sufficiently small'',
and that $0<\eps\le c$.
Let $n\ge 1$ be fixed, and define $G_n(z,H)=z[V_n(H^s)-\id]$.
Then for $k=n-1,\ldots,1,0$, the equation
$$
G_k(z,H)=V_k(H^s)\circ\ldots\circ V_{n-1}(H^s)\circ
\bigl(\id+z[V_n(H^s)-\id]\bigr)-\id
\equation(VprimeB2)
$$
defines an analytic map $G_k$ from $D_n\times B(\eps)$
to $\AA_{n+1}\,$, satisfying
$$
\|G_k(z,H)\|_{k,n+1}
\le\|V_k(H^s)-\id\|_{k,k+1}+\|G_{k+1}(z,H)\|_{k+1,n+1}\,,
\equation(GBound)
$$
for all $z\in D_n$ and $H\in B(\eps)$.
This follows by inductively applying \clm(Obvious), using the identity
$G_k(z,H)=[V_k(H^s)-\id]\circ[\id+G_{k+1}(z,H)]+G_{k+1}(z,H)$.
The inductive estimate on $G_k\,$,
obtained from \equ(GBound) and \equ(VprimeB1),
and valid for all $z\in D_n\,$,
is $\|G_k(z,H)\|_{k,n+1}\le\rho_k(1-\eta)/2$.
Here, and in what follows, we assume that $H\in B(\eps)$.
If $z=1$,
then we can use \equ(VnBound) instead of \equ(VprimeB1),
and the resulting bound on the norm of $G_0(1,H)$
implies the second inequality in \equ(VprimeBound).
For a general $z\in D_n\,$,
an upper bound on $\|G_0(z,H)\|_{0,n+1}$ is given by the number $\rho$.
Thus, by using Cauchy's formula
$$
{\partial\over\partial z}G_0(z,H)
={1\over 2\pi i}\oint\limits_{|\zeta|=d_n+1}\!
{d\zeta\over(\zeta-z)^2}\;G_0(\zeta,H)\,,\qquad |z|\le 1\,,
\equation(VprimeB3)
$$
and the mean value theorem, we obtain the bound
$$
\eqalign{
\bigl\|V'_{n+1}(H^s)-V'_n(H^s)\bigr\|_{0,n+1}
&=\|G_0(1,H)-G_0(0,H)\|_{0,n+1}\cr
&\le\rho/d_n=a|b\mu|^{2n}\eps\,.\cr}
\equation(VprimeB4)
$$
Let now $H$ be a fixed but arbitrary function in $B(c)$.
If $H^s\not=H^0$ then
the first inequality in \equ(VprimeBound) is obtained
from \equ(VprimeB4) by choosing $\eps=\|H^s-H^0\|_\rho\,$.
The case $H^s=H^0$ is trivial,
since $H^s=H^0$ implies that $V_k(H^s)=\id$ for all $k\ge 0$.
The analyticity of the functions $H\mapsto\xi'_n(H^s)$,
in an open neighborhood $B\subset B(c)$ of $H^0$,
follows from the analyticity of $H\mapsto H^s$ and $\RG_\mu\,$.
And \equ(xipBound) is a consequence
of the bound \equ(Wstable) in \clm(Wstable),
and of the fact that $\xi(H^0)=1$.
\qed
The following is an approximate version of equation \equ(WsFlow).
\claim Proposition(approxT)
If $|\mu|>0$ is sufficiently small
then there exists a real number $a>0$,
and an open neighborhood $B$ of $H^0$ in $\AA(\rho)$, such that
$$
\bigl\|\{\id,H\}\circ V'_n(H)
-\xi'_n(H)\bigl\{V'_n(H),H^0\bigr\}\bigr\|_{0,n}
\le a|b\mu|^n\bigl\|H-H^0\bigr\|_\rho\,,
\equation(approxT)
$$
for all $H\in\WW^s\cap B$ and $n\ge 1$,
where $b>0$ is some fixed constant.
\proof
First, we note that for each $n\ge 1$, if $r_n=\rho_n$ then
$$
\bigl\|V'_n(H)\circ S_\mu^n\bigr\|_{r_n}\le 2\,,\qquad
\bigl\|\bigl[H_n-H^0\bigr]\circ T_1^{-n}\bigr\|_{r_n}
\le\bigl\|H_n-H^0\bigr\|_\rho\,,
\equation(approxT0)
$$
for all $H\in\WW^s$ sufficiently close to $H^0$.
These two bounds follow from \clm(VprimeBound) and \clm(TUT).
The same is true if we set $r_n=(1+\eps)\rho_n$,
with $\eps>0$ sufficiently small,
since the positive constants $\eta<1/m(T)$ and $\kappa'<\kappa$
in the definition \equ(rhokDef) can be chosen arbitrarily.
Thus, by \clm(Trivial).b, we have
$$
\bigl\|V'_n(H)\circ S_\mu^n\bigr\|'_{\rho_n}\le{2\over\eps\rho_n}\,,
\qquad\bigl\|\bigl[H_n-H^0\bigr]\circ T_1^{-n}\bigr\|'_{\rho_n}
\le{1\over\eps\rho_n}\bigl\|H_n-H^0\bigr\|_\rho\,,
\equation(approxT1)
$$
where $\|.\|'_{\rho_n}$ is the norm described in \clm(AAprime).
To be more precise, this holds for all $n$, and for all $H$
in some (sufficiently small) open neighborhood of $H^0$ in $\AA(\rho)$.
The same comment applies to the statements made below,
but it will not always be mentioned.
By \clm(VnBound), $V'_n(H)$ is a canonical transformation
from $A_n$ into $A_0=\DD(\rho_0)$,
and the bound \equ(VprimeBound) shows that
this transformation is one-to-one. As a consequence, we have
$\{f,g\}\circ V'_n(H)=\{f\circ V'_n(H),g\circ V'_n(H)\}$,
whenever $f$ and $g$ are analytic on $\DD(\rho_0)$.
This, together with \equ(RRiter2), implies that
$$
\{\id,H\}\circ V'_n(H)
=\bigl\{V'_n(H),H\circ V'_n(H)\bigr\}
=\xi'(H)\left({\mu\over\vartheta}\right)^n\!
\bigl\{V'_n(H),H_n\circ T_\mu^{-n}\bigr\}\,.
\equation(approxT2)
$$
By using first the identities
$(\mu/\vartheta)H^0\circ T_\mu^{-1}=H^0$
and $\{\SS_z^{-1}f,\SS_z^{-1}g\}=\SS_z^{-1}\{f,g\}$,
and then the bound \equ(approxT1),
the left hand side of \equ(approxT)
can now be estimated as follows:
$$
\eqalign{
\bigl\|\{\id,H\}\circ V'_n(H)
&-\xi'_n(H)\bigl\{V'_n(H),H^0\bigr\}\bigr\|_{0,n}\cr
&=\bigl\|\xi'_n(H)(\mu/\vartheta)^n\bigl\{V'_n(H),
\bigl[H_n-H^0\bigr]\circ T_\mu^{-n}\bigr\}\bigr\|_{0,n}\cr
&=\xi'_n(H)|\vartheta|^{-n}
\bigl\|\mu^n\bigl\{V'_n(H),\bigl[H_n-H^0\bigr]\circ T_\mu^{-n}
\bigr\}\circ S_\mu^n\bigr\|_{\rho_n}\cr
&=\xi'_n(H)|\vartheta|^{-n}\bigl\|\bigl\{
V'_n(H)\circ S_\mu^n,\bigl[H_n-H^0\bigr]\circ T_1^{-n}
\bigr\}\bigr\|_{\rho_n}\cr
&\le\xi'_n(H)|\vartheta|^{-n}
4d(\eps\rho_n)^{-2}\bigl\|H_n-H^0\bigr\|_\rho\,.\cr}
\equation(approxT3)
$$
This implies \equ(approxT), if we substitute for $\|H_n-H^0\|_\rho$
the bound given in \clm(Wstable).
\qed
In order to translate between the two different notations
used in \equ(approxT) and \equ(WsFlow),
let us write $F=(F_1,F_2)$,
if $F\in\AA_n$ and $F(q,p)=(F_1(q,p),F_2(q,p))$,
for all $(q,p)$ in the domain of $A_n\,$.
In particular, $\id=(\id_1,\id_2)$,
where $\id_1(q,p)=q$ and $\id_2(q,p)=p$ for all $q$ and $p$.
Then
$$
\eqalign{
\{\id,F\}&=\bigl\{(\id_1,\id_2),F\bigr\}
=\bigl(\{\id_1,F\},\{\id_2,F\}\bigr)
=\bigl(\nabla_2F,-\nabla_1F\bigr)=\iso\nabla F\,,\cr
\{F,H^0\}&=\{F,\omega\cdot\id_2\}
=\omega\cdot\{F,\id_2\}=\omega\cdot\nabla_1 F\,,\cr}
\equation(translate)
$$
where $\iso(q,p)=(p,-q)$.
Now we are ready to take the limit of the sequence $\{V'_n(H)\}$,
considered as $C^\ell$ functions on the torus $\torus^d$.
The limit $V'(H)$ is essentially the function $\Gamma_{\!H}$
described in \clm(WsFlow), restricted to real arguments.
Later on, we will show that this function can be analytically
continued to a complex neighborhood of the torus.
\claim Definition(Cell)
For every integer $\ell\ge 0$,
denote by $C^\ell(\torus^d\times\{0\},\complex^{2d})$,
or $C^\ell(\torus^d)$ for short,
the vector space of all $\ell$--times continuously differentiable
functions from $\real^{2d}$ to $\complex^{2d}$
that are $2\pi$--periodic in the first $d$ variables
(when considered as functions of $2d$ real arguments)
and constant in the last $d$ variables.
When equipped with the norm
$$
\|g\|_{(\ell)}=\max_{|\alpha|\le\ell}\sup_{q\in\real^d}
\bigl|\bigl(\partial^\alpha g\bigr)(q,0)\bigr|\,,
\qquad g\in C^\ell\,,
\equation(CellNorm)
$$
$C^\ell(\torus^d)$ is a Banach space.
Here, we have used the notation
$\partial^\alpha=\partial_1^{\alpha_1}\partial_2^{\alpha_2}
\cdots\partial_{2d}^{\alpha_{2d}}$.
Notice that, by definition,
if $g\in C^\ell(\torus^d)$ then $g(q,p)=g(q,0)$ for all $q$.
The extra variable $p\in\real^d$ has been introduced (kept)
only to simplify notation.
In particular, if $G\in\AA_n\,$,
then the restriction of $G\circ S_0$ to $\real^{2d}$,
which we will also denote by $G\circ S_0$,
belongs to $C^\ell(\torus^d)$, for every $\ell$.
Recall that $S_0(q,p)=(q,0)$. The map $G\mapsto G\circ S_0$
is continuous as a linear operator from $\AA_n$ to $C^\ell$,
and it satisfies the bound
$$
\|G\circ S_0\|_{(\ell)}\le
{\ell !\over\rho_n^\ell}\sup_{x\in A_n}|G(x)|
\le K_\ell\,\eta^{-\ell n}\|G\|_{0,n}\,,
\qquad G\in\AA_n\,,
\equation(BoundAtoC)
$$
for some constant $K_\ell$ that is independent of $n$.
The second inequality in \equ(BoundAtoC) is obtained
by applying \clm(Trivial).a
to the function $G\circ S_\mu^n$ in $\AA(\rho_n)$.
\claim Proposition(WsFlow0)
Given $\ell\ge 1$, there exists a constant $a_\ell>0$
and an open neighborhood $B_\ell$ of $H^0$ in $\AA(\rho)$,
such that the following holds.
For every $H\in B_\ell$, the two limits in the equation
$$
V'(H)=S_0+\lim_{n\to\infty}\bigl[V'_n(H)-\id\bigr]\circ S_0\,,\qquad
\xi'(H)=\lim_{n\to\infty}\xi'_n(H)\,,
\equation(VxiLim)
$$
converge in $C^\ell(\torus^d)$ and $\complex$, respectively,
and the convergence is uniform on $B_\ell\,$.
The functions $H\mapsto V'(H)-S_0$ and $H\mapsto\xi'(H)$
are analytic on $B_\ell$, and they satisfy the bounds
$$
\bigl\|V'(H)-S_0\bigr\|_{(\ell)}\le a_\ell\|H^s-H^0\|_\rho\,,\qquad
\bigl|\xi'(H)-1\bigr|\le a_\ell\|H^s-H^0\|_\rho\,,
\equation(VxiBound)
$$
for all $H\in B_\ell\,$.
Furthermore, if $H\in\WW^s\cap B_\ell$ then
$$
\bigl(\iso\nabla H\bigr)\circ V'(H)=\xi'(H)\omega\cdot\nabla_1 V'(H)\,,
\equation(WsFlow0)
$$
and if in addition, $|\mu|>0$ is sufficiently small, then
$$
V'(H)=V'_k(H)\circ T_\mu^k V'\bigl(\RG_\mu^k(H)\bigr)
\circ T_\mu^{-k}\,,\qquad k=1,2,\ldots\,.
\equation(VTVT)
$$
\proof
First we prove the assertions under the additional assumption
that $|\mu|>0$ be sufficiently small.
In this case,
the uniform convergence (and hence also the analyticity)
on some open neighborhood of $H^0$ of the two limits in \equ(VxiLim),
as well as the bounds \equ(VxiBound),
follow directly from \clm(VprimeBound) and inequality \equ(BoundAtoC).
Thus, $X_n=\xi'_n(H)\omega\cdot\nabla_1 V'_n(H)\circ S_0$
converges in $C^{\ell-1}(\torus^d)$ to $X=\xi'(H)\omega\cdot\nabla_1 V'(H)$,
and by the chain rule, $Y_n=(\iso\nabla H)\circ V'_n(H)\circ S_0$
converges in $C^\ell(\torus^d)$ to $Y=(\iso\nabla H)\circ V'(H)$.
Consider now $H\in\WW^s$ sufficiently close to $H^0$
for \clm(approxT) to apply.
Then $Y_n-X_n\to 0$ by \equ(approxT) and \equ(BoundAtoC).
Consequently, $Y=X$, as claimed in \equ(WsFlow0).
In what follows, we assume that $H\in\WW^s\cap B$
where $B$ is some suitable open neighborhood of $H^0$
(sufficiently small, as needed by the propositions used).
We may also assume that $\WW^s\cap B$
is mapped into itself by $\RG_\mu\,$.
Then $V'_{n-k}(H_k)\to V'(H_k)$ as $n\to\infty$, for all $k\ge 1$.
Furthermore, by \clm(WsFlow0) and \clm(Wstable),
there are constants $a,b>0$, with $b$ independent of $\mu$,
such that
$$
\|V'_{n-k}(H_k)\circ S_0-S_0\|_{(0)}\,,
\ \|V'(H_k)-S_0\|_{(0)}\le a|b\mu|^k\|H-H^0\|_\rho\,,
\equation(VTVT3)
$$
whenever $n>k\ge 1$. Let now $k\ge 1$ be fixed.
>From the definition of $V'_1,V'_2,\ldots\,$, it follows that
$$
V'_n(H)=V'_k(H)\circ T_\mu^k V'_{n-k}(H_k)\circ T_\mu^{-k}\,,
\equation(VTVT1)
$$
for all $n>k$, and in particular,
$$
V'_n(H)\circ S_0=V'_k(H)\circ S_\mu^k F_n\,,\qquad
F_n=T_1^k V'_{n-k}(H_k)\circ S_0\circ T_1^{-k}\,.
\equation(VTVT2)
$$
The goal now is to take $n\to\infty$.
Since $[V'_{n-k}(H_k)-\id]\circ S_0$ converges to $V'(H_k)-S_0$
in $C^0(\torus^d)$,
we have $F_n\to F=T_1^k\circ V'(H_k)\circ T_1^{-k}$
pointwise on $\real^{2d}$.
And from the bound \equ(VTVT3) we see that
if $|\mu|>0$ is chosen sufficiently small (as assumed),
then the ranges of $F_1,F_2,\ldots$ and $F$ are contained
in the domain $\DD(\rho_k)$ where $V'_k(H)\circ S_\mu^k$
is analytic.
This shows that $V'_k(H)\circ S_\mu^k F_n$
converges to $V'_k(H)\circ S_\mu^k F$, pointwise on $\real^{2d}$.
Thus, since $V'_n(H)\circ S_0\to V'(H)$,
we conclude from \equ(VTVT2) that $V'(H)=V'_k(H)\circ S_\mu^k F$,
which is equivalent to \equ(VTVT).
In order to complete the proof,
we need to eliminate the smallness condition on $\mu$.
Let $\mu_0$ be a nonzero complex number
such that $\mu=\mu_0$ satisfies the condition \equ(muCondition).
As was shown above, there exists $z\not=0$ of modulus $\le 1$,
such that for the choice $\mu=z\mu_0\,$,
the limits in \equ(VxiLim) converge in
$C^\ell(\torus^d)$ and $\complex$, respectively,
for every $H\in B'$.
But by \clm(Vmu) (which also uses \clm(WSW)),
the two choices $\mu=\mu_0$ and $\mu=z\mu_0$
lead to the same functions $V'_n(H)\circ S_0\,$,
and thus to the same limit $V'(H)$,
for all $H$ in some open set $B_\ell\subset B$ containing $H^0$.
And by \clm(WSW), the condition $H\in\WW^s\cap B_\ell$
needed for \equ(WsFlow0)
is independent of the choice ($\mu_0$ or $z\mu_0$) of $\mu$,
if $B_\ell$ is sufficiently small.
(The identity \equ(VTVT) could also be extended
from $\mu=z\mu_0$ to $\mu=\mu_0\,$,
but this will not be needed later.)
\qed
Consider $H\in\WW^s\cap B_1\,$.
Then for every positive integer $\ell$,
the function $H_k=\RG_\mu^k(H)$ belongs to $B_\ell\,$,
if $k$ is sufficiently large.
Thus, the identity \equ(VTVT) suggests that $V'(H)$
is in fact of class $C^\ell$.
And since $\ell>0$ was arbitrary, $\ldots$
\claim Lemma(Cinfty)
The map $V'\colon B_1\to C^1(\torus^d)$,
described in \clm(WsFlow0) has the following property:
If $H\in B_1$ then $V'(H)$ is of class $C^\infty$,
and for every $\ell\ge 0$, $V'$ is analytic as a map
from $B_1$ to $C^\ell(\torus^d)$.
\proof
Let $\ell\ge 2$ be fixed but arbitrary.
As in the proof of \clm(WsFlow0), we can (will)
choose $|\mu|>0$ as small as needed, without loosing generality.
Let $m\ge 1$ be such that $\RG_\mu^m(H)$
belongs to $B=B_1\cap B_\ell$ whenever $H\in\WW^s\cap B_1\,$.
The existence of such an integer $m$ follows e.g.
from the bound \equ(Wstable) in \clm(Wstable).
By \clm(WsFlow0) and \clm(Wstable), there are real numbers $a,a'>0$,
and a universal constant $b>0$, such that
$$
\bigl\|V'(H_k^s)-S_0\bigr\|_{(\ell)}
\le a\bigl\|H_k^s-H_0\bigr\|_\rho
\le a'|b\mu|^k\|H-H^0\|_\rho\,,
\equation(Cinfty1)
$$
for all $H\in B_1$ and $k\ge m$, where $H_k^s=\RG_\mu^k(H^s)$.
This bound shows that if $|\mu|>0$
has been chosen sufficiently small,
then there exists $k\ge m$ (sufficiently large) such that
the range of $F(H)=T_1^kV'(H_k^s)\circ T_1^{-k}$
is contained in $\DD(\rho_k)$, for all $H\in B_1\,$.
With this choice of $k$, define $G(H)=[V'_k(H)-\id]\circ S_\mu^k$.
Then
$$
\eqalign{
V'(H)&=V'(H^s)=\bigl[S_\mu^k+G(H)\bigr]\circ F(H)\cr
&=\id+ S_\mu^k\bigl[F(H)-S_0\bigr]+G(H)\circ F(H)\,,\cr}
\equation(Cinfty2)
$$
for all $H\in B_1\,$, as was shown in \clm(WsFlow0).
Since $G(H)\colon\DD(\rho_k)\to\complex^{2d}$ is analytic
and $F(H)\colon\real^{2d}\to\DD(\rho_k)$ is of class $C^\ell$,
equation \equ(Cinfty2) shows that $V'(H)$ is of class $C^\ell$.
By \clm(approxT), $G\colon B_1\to\AA^{2d}(\rho_k)$ is analytic.
Furthermore, $H\mapsto[F(H)-\id]$
is analytic as a map from $B_1$ to $C^\ell(\torus^d)$,
since it is the composition of $H\in B_1\mapsto H_k^s\in B$
with $H\in B\mapsto[V'(H^s)-S_0]\in C^\ell(\torus^d)$,
and both of these maps are analytic (the second one by \clm(WsFlow0)).
By using the last expression for $V'(H)$ in \equ(Cinfty2),
it is now straightforward to show that $H\mapsto[V'(H)-S_0]$ is analytic
as a map from $B_1$ to $C^\ell(\torus^d)$, as claimed.
\qed
If $H\in\WW^s$ has some ``excess analyticity''
then the relationship between $V'(H)$ and $V'(\JJ_q^{-1}H)$,
valid a priori only for real $q$'s,
can be used to analytically continue $V'(H)$ away from $\real^{2d}$.
Given two sets $X$, $Y$, and a point $x\in X$,
define the evaluation functional $\EE_x$ on $Y^X$
by setting $\EE_xf=f(x)$, for all $f\in Y^X$.
\claim Lemma(VVprop)
Let $r-\rho>\eps>0$.
Then there exists an open neighborhood $B$
of $H^0$ in $\AA(r)$ such that the following holds.
For every $H\in B$, the equation
$$
\VV_H(q,p)=\EE_0J_q\circ V'\bigl(\JJ_q^{-1}H\bigr)\,,
\qquad q\in\DD_1(\eps)\,,\ p\in\complex^d\,,
\equation(VVdef)
$$
defines an analytic function $\VV_H$ on $\DD_1(\eps)\times\complex^d$
whose restriction to $\real^{2d}$ coincides with $V'(H)$.
Furthermore, $\VV_H-S_0$ belongs to $\AA^{2d}(\eps)$,
for every $H\in B$, and $H\mapsto\VV_H-S_0$
is analytic as a map from $\AA(r)$ to $\AA^{2d}(\eps)$.
\proof
Choose $\delta$ in $(\eps,r-\rho)$.
Clearly, $(q,H)\mapsto\JJ_q^{-1}H$ is differentiable (analytic)
from $\DD_1(\delta)\times\AA(r)$ to $\AA(\rho)$.
This follows from the fact that
$\JJ_q^{-1}\colon\AA(\rho+\delta)\to\AA(\rho)$
is bounded (of norm $1$) for all $q\in\DD_1(\delta)$,
and that differentiation is bounded from
$\AA(r)$ to $\AA(\rho+\delta)$.
Define $G\colon B_1\to\CC^1(\torus^d)$ by setting
$G(H)=V'(H)-S_0\,$, for all $H\in B_1\,$; see \clm(WsFlow0).
The boundedness of $\JJ_q^{-1}\colon\AA(r)\to\AA(\rho)$
also implies that we can find $B\ni H^0$ open in $\AA(r)$
such that both $H$ and $\JJ_q^{-1}H$ belong to the
domain $B_1$ of the map $V'$.
Thus, since $G$ is analytic on $B_1$,
and since $\EE_0$ is bounded on $C^1(\torus^d)$,
we conclude that $(q,H)\mapsto E_0G(\JJ_q^{-1}H)$
is analytic on $\DD_1(\delta)\times B$.
The analyticity of $\VV_H\,$, for $H\in B$,
follows from the identity $[\VV_H-S_0](q,p)=E_0G(\JJ_q^{-1}H)$.
Let $\eps<\delta'<\delta$.
Since $\JJ_q^{-1}H$ is $2\pi$-periodic in each component of $q$,
the same is true for $\EE_0G(\JJ_q^{-1}H)$.
Thus, $\VV_H-S_0$ belongs to $\AA^{2d}(\delta')$, for all $H\in B$.
Consider now the sup-norm on $\AA^{2d}(\delta')$.
Then the analyticity of $H\mapsto \VV_H-S_0\,$,
as a map from $B$ to $\AA^{2d}(\delta')$,
follows from the fact that all derivatives with respect
to the second argument of $(q,H)\mapsto\EE_0G(\JJ_q^{-1}H)$,
at any $H\in B$,
are bounded uniformly in the first argument, $q\in\DD_1(\delta')$.
Consider now a fixed but arbitrary $q\in\real^d$,
and assume that $H\in\WW^s$.
Recall that $\JJ_q^{-1}$ is an isometry on $\AA(\rho)$.
Thus, given any open neighborhood $B''$ of $H^0$ in $\AA(\rho)$,
if $\|H-H^0\|_\rho$ is sufficiently small,
then the functions $H'=\JJ_q^{-1}H\,$,
$H'_k=\RG_\mu^k(H')$, and $H_k=\RG_\mu^k(H)$ belong to $B''$,
for all $k\ge 0$.
Here, we have also used \clm(WJJW),
and the bound \equ(Wstable) in \clm(Wstable).
By using \clm(RRexists) and \clm(SHUS),
together with the identity $T_\mu^k\circ J_{T^{-k}q}=J_q\circ T_\mu^k$,
we first obtain $H'_k=\JJ_{T^{-k}q}^{-1}H_k$,
and then $V_k(H')=J_q^{-1}\circ V_k(H)\circ J_q\,$, for all $k\ge 0$.
This holds for all $H$ in some open neighborhood $B'$
of $H^0$ in $\AA(\rho)$, that is independent of $q\in\real^d$.
The conclusion is that
$V'\bigl(\JJ_q^{-1}H\bigr)=J_q^{-1}\circ V'(H)\circ J_q\,$,
and thus
$$
\EE_{(q,p)}V'(H)=\EE_0\bigl(V'(H)\circ J_q\bigr)
=\EE_0J_q\circ V'\bigl(\JJ_q^{-1}H\bigr)=\VV_H(q,p)\,,
\equation(VV1)
$$
for all $q\in\real^d$ and $H\in B'$.
This shows that $\VV_H$ and $V'(H)$ agree on $\real^d$, as claimed.
\qed
\proofof(WsFlow)
Define $\Gamma_{\!H}(q)=\VV_H(q,0)$, for all $q\in\DD_1(r')$,
and for all $H$ in the domain of the map
$H\mapsto\VV_H$ described in \clm(VVprop) (with $\eps=r'$).
Consider first a function $H\in\WW^s$ in this domain,
with the property that $H(q,p)$ only depends on $p$,
or equivalently, that $\JJ_q^{-1}H=H$ for all $q$.
In this case, $\xi'(H)=\xi(H)$ and $V'(H)=S_0\,$,
as can be verified form the definitions.
Thus, $\xi'(H)=(\Omega\cdot\nabla\mean H)(0)$ and
$\Gamma_{\!H^0}(u)=E_0J_q\circ S_0=(q,0)$, for all $q\in\DD_1(r')$,
as claimed.
The asserted analyticity properties of the functions $\Gamma_{\!H}\,$,
and of the map $H\mapsto\Gamma_{\!H}-\Gamma_{\!H^0}$
follow directly from \clm(VVprop).
(The Banach space of analytic $2\pi$-periodic functions,
mentioned in \clm(WsFlow),
consists of all functions $g$ on $\DD_1(r')$
such that $G\colon(q,p)\mapsto g(q)$ belongs to $\AA(r')$;
in particular, if $g=\Gamma_{\!H}$ then $G=\VV_H\,$.)
Since $\Gamma_{\!H^0}$ maps $\DD_1(r')$ into $\DD(r')$,
the analyticity of $H\mapsto\Gamma_{\!H}-\Gamma_{\!H^0}$
also implies that $\Gamma_{\!H}$ maps $\DD_1(r')$ into $\DD(r)$,
whenever $\|H-H^0\|_{r'}$ is sufficiently small (which we assume).
As was shown in \clm(VVprop),
the restriction of $\VV_H$ to $\real^d$ is equal to $V'(H)$,
Thus, by \clm(WsFlow0), if $H\in\WW^s$ is sufficiently close to $H^0$,
then the function $\Gamma_{\!H}$ satisfies the equation
$(\iso\nabla H)\circ\Gamma_{\!H}=\xi'(H)\omega\cdot\nabla\Gamma_{\!H}$
on $\real^d$.
But since both sides of this equation
are analytic functions on $\DD_1(r')$,
the same equation holds on $\DD_1(r')$.
This proves equation \equ(WsFlow).
The analyticity of the function $\xi'$ was shown in \clm(WsFlow0).
\qed
\proofof(GammaStable)
Assume that the hypotheses of the theorem are satisfied,
including the conditions on $\nabla f$.
Let $E_H$ be a $d-1$ dimensional subspace of $\complex^d$
that is mapped onto $E$ by $(D\nabla f)(0)$,
and for every $\eps>0$, define $G_\eps=\{v\in E_H\colon\|v\|<\eps\}$.
Choose a positive real number $2\eps