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%
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%
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\def\rMEsc{2}
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\cl{{\huge On the Renormalization of Hamiltonian Flows,}}
\cl{{\huge and Critical Invariant Tori}}
\vskip.6in
\cl{Hans Koch
\footnote{$^1$}
{{\small Supported in Part by the
National Science Foundation under Grants No. DMS-9705095 and DMS-0088935.}}}
\cl{Department of Mathematics, The University of Texas at Austin}
\cl{Austin, TX 78712}
\vskip.8in
\abstract
We analyze a renormalization group transformation $\RR$
for partially analytic Hamiltonians,
with emphasis on what seems to be needed
for the construction of non-integrable fixed points.
Under certain assumptions, which are supported by numerical data
in the golden mean case,
we prove that such a fixed point has a critical invariant torus.
The proof is constructive and can be used for numerical computations.
We also relate $\RR$ to a renormalization group transformation
for commuting maps.
\par\vfill\eject
\section Introduction
Renormalization group methods for Hamiltonian flows
were introduced first in [\rED,\rMEsc], as a tool for explaining
the breakup of certain smooth invariant tori.
Our aim here is to make these methods precise,
in a domain where the breakup is expected to occur,
and to obtain information about invariant tori
in the borderline (critical) case.
We are particularly interested in invariant tori
for rotation vectors $\omega\in\real^d$
whose component ratios span an algebraic number field of degree $d$.
In this case, there exists an integer $d\times d$ matrix $T$,
with determinant $\pm 1$ and $d-1$ simple eigenvalues
of modulus less than $1$, for which $\omega$ is an eigenvector
with a real eigenvalue $\vartheta_1>1$.
Given such a matrix $T$, its transpose $T^\ast$,
and a nonzero real number $\mu$,
we define $\TT_\mu(q,p)=\bigl(Tq,\mu(T^\ast)^{-1}p\bigr)$.
Here, and in what follows, $(q,p)$ denotes a point
in the cotangent space of the torus $\torus^d$
or its complexification.
A renormalization group (RG) transformation associated with $T$
will be a transformation $\RR$,
acting on a space of Hamiltonians $H=H(q,p)$, such that
$$
\RR(H)=H\circ\TT_1\quad (\modulo\GG)\,.
\equation(RRmodGG)
$$
Here, $\GG$ is some group (ideally) of transformations
that preserves rotation vectors, up to multiplication by a scalar.
This includes coordinate changes, $H\mapsto\!H\circ\UU\,$,
with $\UU$ canonical (symplectic) and homotopic to the identity.
In addition, we will allow a scaling of the action variables,
$H\mapsto\mu^{-1}H(\,.\,,\mu\,.\,)\,$,
and a re-normalization of the energy, $H\mapsto\tau H-\epsilon\,$.
This leads to the following general form of $\RR$:
$$
\RR(H)=H''\circ\UU_\ssHpp\,,
\quad H''=H'\circ U'\,,
\quad H'={\tau\over\mu}H\circ\TT_\mu-\epsilon\,,
\equation(RRform)
$$
where $\tau$, $\mu$ and $\epsilon$ are allowed to depend on $H$.
In what follows, $\epsilon$ is taken to be zero.
$U'$ is some fixed canonical transformation,
introduced for convenience later on.
The canonical transformation $\UU_\ssHpp$ will be described below.
It is similar to the one introduced in [\rKi],
but differs significantly from those
used in earlier renormalization schemes [\rED--\rMMS].
The problem that needs to be dealt with is that the map $H\mapsto H'$
involves a loss of regularity in the direction $(\omega,0)$,
due to the fact that $\TT_\mu$ expands this direction.
(All other directions are contracted, if $\vartheta_1|\mu|<1$.)
% 4 lines of MacKay's "Three Topics in Hamiltonians Dynamics"
% mention that growing wiggles need to be canceled, and end with:
% One way to do this is [ED]'s "elimination of resonances".
%
% Correct, this is how Escande-Doveil dealt with the wiggles.
% No other thoughts expressed on this issue!
% Instead, the story moves on ...
%
% Now exactly those 4 lines are supposed to have
% pointed out the main open problem in the field, and
% given the main idea on how to solve it ?!
Denote by $\partial_3H$ the directional derivative of $H$
in the direction $(0,\omega')$,
where $\omega'$ is the eigenvector of $T^\ast$
for the eigenvalue $\vartheta_1\,$, normalized such that
$\omega\cdot\omega'=1$.
In this paper, we will only consider Hamiltonians $H$
that satisfy $\partial_3H=1$.
In other words, $\omega'\cdot q$ can be regarded as ``time''.
Consequently, we need to choose $\tau=\vartheta_1$ in equation \equ(RRform),
which ensures that $\partial_3H'=1$.
Furthermore, in order for the renormalized Hamiltonian $\RR(H)$
to satisfy the same condition,
we will restrict the choice of $U'$ and $\UU_\ssHpp$
to canonical transformations that leave $\omega'\cdot q$ invariant.
Before defining $\RR$ more precisely,
let us mention a connection between the transformation \equ(RRform),
and the RG transformations for commuting maps,
considered e.g. in [\rMcK,\rSti] for the golden mean case,
where $d=2$, and $T=\left({0~1\atop 1~1}\right)$ as a possible choice.
Let $H$ be a fixed but arbitrary Hamiltonian,
and denote by $t\mapsto\Phi_t$ the flow associated with $H$.
If $x$ is any vector in $\real^d$,
define $V(x)$ to be the translation $(q,p)\mapsto(q-x,p)$.
Denote by $\delta_k$ the unit vector in $\real^d$
in the direction of the $k$--th coordinate axis,
and consider the maps
$$
\Lambda=\TT_\mu\circ U'\circ\UU_\ssHpp\,,\quad
F_k=\Phi_{2\pi\omega'_k}\circ V(2\pi\delta_k)\,,
\qquad k=1,\ldots,d\,.
\equation(FkDef)
$$
\claim Proposition(mapsRG)
Assume that $H$ is differentiable on $\torus^d\times\real^d$,
and that $\Lambda$ is a diffeomorphism of this space.
Then the maps $F_1,F_2,\ldots,F_d$ commute with each other,
leave the (symplectic) manifold
$\MM_\ssH=\bigl\{(q,p)\in H^{-1}(0)\colon\,\omega'\cdot q=0\bigr\}$ invariant,
and their restriction to $\MM_\ssH$ is symplectic.
If we denote by $\wt F_1,\wt F_2,\ldots,\wt F_d$
the corresponding maps for $\wt H=\RR(H)$, then
$$
\wt F_k=\Lambda^{-1}\circ F_1^{T_{1,k}}\circ F_2^{T_{2,k}}\circ
\ldots\circ F_d^{T_{d,k}}\circ\Lambda\,,
\quad k=1,\ldots d.
\equation(mapsRG)
$$
Assume now that $T$ has a real eigenvalue $\vartheta_2$
of modulus less than $1$.
Let $\Omega$ and $\Omega'$ be eigenvectors of $T$ and $T^\ast$,
respectively, for the eigenvalue $\vartheta_2\,$,
normalized such that $\Omega\cdot\Omega'=1$.
In what follows, we will restrict our attention to Hamiltonians
that are invariant under the reflection $(q,p)\mapsto(-q,p)$,
and to canonical transformations $U'$ and $\UU_\ssHpp$
that commute with this reflection.
In addition, we impose the following
\medskip\noindent{\bf Restriction.\ }
In the remaining part of this paper,
every function on $\torus^d\times\real^d$ being considered
is assumed to depend only on the
variables $q$, $\omega\cdot p$ and $\Omega\cdot p$.
Thus, our phase space is effectively $\torus^d\times\real^2$.
\medskip
Let $H_0(q,p)=\omega\cdot p$. Then all of our Hamiltonians are of the form
$$
H=H_0+h\,,\quad
h(q,p)=\!\sum_{(\nu,k)\in I}h_{\nu,k}\cos(\nu\cdot q)(\Omega\cdot p)^k\,,
\equation(Hqp)
$$
where $I=\integer^d\times\natural$.
We define the {\it resonant} part of such a Hamiltonian $H$ (or $h$)
to be the function $\Iplus H$ (or $\Iplus h$),
which is obtained by restricting the sum in \equ(Hqp)
to the index set
$$
\iplus=\bigl\{(\nu,k)\in I: |\omega\cdot\nu|\le\sigma|\Omega\cdot\nu|
{\rm ~or~ } |\omega\cdot\nu|\le\kappa k\bigr\}\,,
\equation(iplusdef)
$$
where $\sigma$ and $\kappa$ are fixed but arbitrary positive real numbers.
Notice that a Hamiltonian that depends on $p$ only
is resonant, in the sense that it agrees with its resonant part.
For every pair $\rho=(\rho_1,\rho_2)$ of positive real numbers,
denote by $\BB_\rho$ the Banach space of all functions
$h$ of the form \equ(Hqp), which have finite norm
$$
\|h\|_\rho=\!\sum_{(\nu,k)\in I}\left|h_{\nu,k}\right|
\cosh(\rho_1\Omega\cdot\nu)\rho_2^k\,,
\equation(normDef)
$$
and define $\AA_\rho$ to be the affine space $H_0+\BB_\rho$
with the metric given by the norm on $\BB_\rho\,$.
The functions $H\in\AA_\rho$ are continuous on the subset of $\real^{d+2}$
characterized by $|\Omega\cdot p|<\rho_2$,
and $H(q+s\Omega,p)$
can be analytically continued to $|{\rm Im~} s|<\rho_1$
and $|\Omega\cdot p|<\rho_2$.
This domain will be denoted by $D_\rho\,$.
\firstremark
Our choice of $\Iplus$ and norms is different from
the one used previously [\rAKW--\rAK].
It was motivated by a study [\rKW] of Hamiltonians in $d=2$,
near an approximate fixed point of the golden mean RG.
The current choice leads to significantly improved estimates
(close to optimal for near-resonant Hamiltonians)
and at the same time,
allows for Hamiltonians that are non-analytic in $d-1$ directions.
\nextremark
One of the important properties of the projection $\Iplus$
is that it yields analyticity in the time direction $(\omega,0)$:
If the sum in \equ(normDef) is restricted to $\iplus$,
then for sufficiently small $\eps>0$,
an additional factor $\exp|\eps\omega\cdot\nu|$
can be included in the sum,
and compensated for (up to a constant factor)
by decreasing the values of $\rho_1$ and $\rho_2$
slightly (by some minimal amount of order $\eps$).
In the case $d=2$, this implies that resonant Hamiltonians
in $\AA_\rho$ are analytic.
Another important property of $\Iplus$ is described in Proposition 2.2.
\claim Proposition(Trivial)
Assume $0<\rho_1'<\rho_1/|\vartheta_2|$
and $0<\rho_2'<\rho_2|\vartheta_2/\mu|$.
Then the map $H\mapsto H'$ defined in \equ(RRform)
is continuous from $\AA_\rho$ to $\AA_{\rho'}\,$.
In the case $d=2$, the restriction of this map to
$\AA_\rho^{+}=\Iplus\AA_\rho$ is compact.
The proof of this proposition is straightforward.
In particular, the compactness for $d=2$ is due to the abovementioned
regularizing property of $\Iplus$.
This property also suggests that we try to choose
the canonical transformation $\UU_\ssHpp\,$,
which appears in the definition \equ(RRform)
of $\RR$, in such a way that
$$
\Iminus\bigl(H''\circ \UU_\ssHpp\bigr)=0\,,
\equation(UUHcond)
$$
where $\Iminus$ is the projection onto nonresonant Hamiltonians:
$\Iminus H=\Iminus h=h-\Iplus h$.
The invariance properties mentioned earlier
also require that $\UU_\ssHpp$ be admissible, in the following sense.
We say that a map $U$ is {\it admissible} if
$$
U=\id+u\,,\quad
u(q,p)=\bigl(u_2(q,p)\Omega,u_3(q,p)\omega'+u_4(q,p)\Omega'\bigr)\,,
\equation(Uform)
$$
with $u_2$ odd and $u_3,u_4$ even functions of $q$.
Unless defined for $p=0$ only,
an admissible map $\id+u$ is also required to satisfy $\partial_3u=0$.
Denote by $\BB'_\rho$ the space of all functions $h$ in $\BB_\rho$
whose directional derivatives in the directions
$(\Omega,0)$ and $(0,\Omega')$ belong to $\BB_\rho\,$.
These derivatives will be denoted by $\partial_2h$ and $\partial_4h$,
respectively.
As the norm of a function $h$ in $\BB_\rho'$
we take the larger of the two numbers
$$
\|h\|_\rho'=\sup_{\kappa\rho_2\|f\|_\rho\le 1\atop\sigma\|g\|_\rho\le 1}
\|f\partial_2h-g\partial_4h\|_\rho
\equation(normprimeDef)
$$
and $\|h\|_\rho/a$, for some fixed but arbitrary $a>0$.
The corresponding affine space $H_0+\BB_\rho'$
will be denoted by $\AA_\rho'\,$.
\claim Theorem(Elimination)
Let $\rho_1'',\rho_2''>0$ and $00$,
there exists $\varrho=\rho''-\OO(\epsilon)$,
and for every $H''\in B(\epsilon)$ an admissible canonical transformation
$\UU_\ssHpp: D_\varrho\to D_{\rho''}$
that is analytic and solves equation \equ(UUHcond).
If $H''$ takes real values for real arguments, then so does $\UU_\ssHpp\,$.
The map $H''\mapsto H''\circ \UU_\ssHpp$ is analytic
from $B(\epsilon)$ to $\AA_\varrho^{+}$, and
$$
\|H''\circ \UU_\ssHpp-H''\|_\varrho
\le(1-A)^{-1}\|\Iminus H''\|_{\rho''}
+\OO\bigl(\|\Iminus H''\|_{\rho''}^2\bigr)\,.
\equation(Elimination)
$$
A proof of this theorem is given in Section 2,
for a more general class of quasiperiodic Hamiltonians.
The solution of \equ(UUHcond) is based on a Nash-Moser type iteration,
and the convergence of this iteration imposes an
upper bound on the size of $\eps$,
as a function of the parameters $\sigma$, $\kappa$, $a$ and $\rho''$.
Aside from the condition $A<1$, which appears naturally,
\clm(Elimination) imposes no other constraints on these parameters.
By combining \clm(Elimination) with \clm(Trivial),
we immediately get the following.
\claim Corollary(RGdomain)
Assume $0<\rho_1<\rho_1''<\rho_1'<\rho_1/|\vartheta_2|$
and $0<\rho_2<\rho_2''<\rho_2'<\rho_2|\vartheta_2/\mu|$.
Let $U': D_{\rho''}\to D_{\rho'}$ be an admissible canonical transformation
which is analytic in a complex open neighborhood of $D_{\rho''}\,$,
maps real points to real points,
and defines a bounded linear operator $H'\mapsto H'\circ U'$
from $\AA_{\rho'}$ to $\AA_{\rho''}\,$.
Let $H_1$ be Hamiltonian in $\AA_\rho$
such that $H_1''=\mu^{-1}\vartheta_1 H_1\circ\TT_\mu\circ U'$
lies in within some fixed distance $A<1$ of $H_0$ in $\AA_{\rho''}'\,$.
If $\|\Iminus H_1\|_\rho$ is sufficiently small,
then there exists an open neighborhood $B$ of $H_1$ in $\AA_\rho$
such that the transformation $\RR$, given by equation \equ(RRform),
with $H''\mapsto\UU_\ssHpp$ as described in \clm(Elimination),
is well defined and analytic as a map from $B$ to $\AA_\rho^{+}\,$.
If $d=2$ and $\Iminus H_1=0$,
then the restriction of $\RR$ to $\Iplus B$ is compact.
As with RG transformations in general, one of the main goals is to
find fixed points, and to study the action of $\RR$ near these fixed points.
The Hamiltonian $H_0$ is such a fixed point,
with $\UU_{\scriptscriptstyle H_0''}=U'=\id\,$.
The corresponding RG analysis (for Hamiltonians that are
not necessarily even in $q$ or satisfy $\partial_3H=1$, but are analytic)
yields results about invariant $\omega$-tori [\rKi]
and sequences of closed orbits accumulating at these tori [\rAK].
There are other integrable fixed points,
such as $H^\circ(q,p)=\omega\cdot p+(\Omega\cdot p)^2$,
with scaling $\mu=\vartheta_2^2/\vartheta_1\,$.
But more interestingly,
numerical results [\rED--\rAKW] suggest that there exist non-integrable
fixed points as well, for $d=2$, and maybe even for $d>2$.
The most extensively studied example is that of the golden mean.
Numerically, the non-integrable fixed point in this case determines
a scaling $\mu^\ast=0.2304601966125\ldots\,$,
which is one of the universal constants that can be observed
during the breakup of golden invariant tori.
We refer to [\rAKW] for more detail, including a discussion
of the critical $\omega$-torus.
In [\rKW] we have started to investigate the golden mean version of $\RR$,
near an approximation for the expected non-integrable fixed point.
A suitable domain for $\RR$ was determined
by first solving the fixed point problem
for a numerical implementation of $\RR$, using $U'=\id$.
This yields an approximate fixed point $H_1\in\AA_\rho^{+}\,$,
and an approximation $\phi$ for the generating function of the
corresponding canonical transformation $\UU_{\scriptscriptstyle H_1''}\,$.
Then, for an RG analysis in a neighborhood of $H_1\,$,
we choose $U'$ to be the canonical transformation generated by $\phi$.
One of our observations so far is that
\clm(Elimination) and \clm(RGdomain) apply perfectly well to this situation.
In order to make the connection with observable phenomena,
it is necessary to show that a (nontrivial) fixed point of $\RR$
has a (critical) invariant $\omega$-torus.
We will address this question below and in Section 3.
We say that $H$ has a (unique) {\it invariant $\omega$-torus} in $D_\rho$
if there exists a (unique) continuous admissible map
$\Gamma$ from $\torus^d\times\{0\}$ to $D_\rho\,$,
such that
$$
\Phi_t\circ\Gamma=\Gamma\circ\Phi^\circ_t\,,\qquad
\Phi^\circ_t(q,0)=(q+t\omega,0)\,,
\equation(omegaTorus)
$$
for all $t\in\real$, where $\Phi$ denotes the flow for $H$.
Assume now that the transformation $\RR$, as described in \clm(RGdomain),
has a (nontrivial) fixed point $H^\ast\in\AA_\rho^{+}\,$,
and denote by $\Lambda^\ast$ the corresponding scaling map
on $\DD_\rho\,$, as defined by equation \equ(FkDef).
Both $H^\ast$ and $\Lambda^\ast$ are assumed to take real values
when restricted to real arguments.
\claim Theorem(Gamma)
If the fixed point $H^\ast\in\AA_\rho^{+}$ has a
real-valued unique invariant $\omega$-torus in $\DD_\rho\,$,
then this torus $\Gamma^\ast$ satisfies the equation
$$
\Gamma^\ast=\Lambda^\ast\circ\Gamma^\ast\circ\TT_1^{-1}\,.
\equation(GammaEqu)
$$
Conversely, let $\Gamma^\ast$ be a continuous admissible map
from $\torus^d\times\{0\}$ to $\DD_\rho\,$,
that satisfies the equation \equ(GammaEqu).
Clearly, $\Gamma^\ast(0,0)$ is a fixed point of $\Lambda^\ast$.
Assume that the derivative of $\Lambda^\ast$ at this fixed point
has exactly one non-contracting direction,
and that $t\mapsto\Gamma^\ast(t\omega,0)$ is continuously differentiable.
Then $\Gamma^\ast$ is an $\omega$-torus for $H^\ast$.
A proof of this theorem, and of \clm(mapsRG), is given in Section 4.
In Section 3 we show that, under some additional assumptions,
the equation \equ(GammaEqu) has a solution defining
an invariant $\omega$-torus for $H^\ast$.
This solution is of class $C^r$, for some $r>0$.
If we assume also that $H^\ast$ is nontrivial,
in the sense that $\vartheta_2$ is not an eigenvalue
of $D\Lambda^\ast(\Gamma^\ast(0,0))$,
then it is clear from equation \equ(GammaEqu)
that the torus $\Gamma^\ast$ cannot be of class $C^1$,
i.e., that it is ``critical''.
All these assumptions are supported by numerical results
(and seem to be verifiable) in the golden mean case.
\section Near-resonant quasiperiodic Hamiltonians
In this section we prove \clm(Elimination),
for a class of Hamiltonians that are not necessarily periodic
in the variables $q_1,\ldots,q_d\,$.
It is convenient to perform a canonical change of coordinates
$(q,p)=\LL(x,y)$
with $x_1=\omega'\!\cdot q$, $x_2=\Omega'\!\cdot q$, $\ldots$,
and $y_1=\omega\cdot p$, $y_2=\Omega\cdot p$, $\ldots$.
In these coordinates, admissible maps like $\UU_\ssHpp$
only depend on $x$ and $y_2$, and only change $x_2$ and $y$.
Since all but the coordinates $(x_1,x_2)$ and $(y_1,y_2)$
enter in a trivial way, we will only present the case $d=2$ here.
An extension to $d>2$ is purely a matter of notation,
and does not affect any estimate or result.
We will simplify (abuse) notation further by writing
$h(x,y)$ instead of $(h\circ\LL)(x,y)$, and $h\in\BB_\rho$
instead of $h\circ\LL\in\BB_\rho\,$, etc.
In these new coordinates, the Fourier-Taylor series and norm
of $h\in\BB_\rho$ are given by
$$
h(x,y)=\!\!\sum_{(v,k)\in I}\!h_{v,k}\cos(v\cdot x)y_2^k\,,\quad
\|h\|_\rho=\!\!\sum_{(v,k)\in I}\!\left|h_{v,k}\right|
\cosh(\rho_1 v_2)\rho_2^k\,,
\equation(newDef)
$$
where $I=\VV\times\natural$, and $\VV\subset\real^2$ is an index set
that can be determined from the vectors $\omega$ and $\Omega$.
But in what follows, $\VV\subset\real^2$ is allowed to be any
vector space over $\integer$.
The set $\iplus$ defining the resonant projection $\Iplus$
consists of all pairs $(v,k)\in I$
for which $|v_1|\le\sigma|v_2|$ or $|v_1|\le\kappa k$.
If $I$ is uncountable, then a function $h\in\BB_\rho$
has (by definition) at most a countable number of nonzero
coefficients $h_{v,k}\,$.
In addition to the spaces $\BB_\rho$ of functions
that are even in $x$, we will also need the corresponding spaces
$\CC_\rho$ of functions that are odd in $x$.
The analogue of \equ(newDef) for these spaces is obtained
by replacing the cosine by the sine,
and excluding terms with $v=0$ from the sums.
Denote by $\FF_\rho$ the direct sum of $\BB_\rho$ and $\CC_\rho\,$.
The functions in $\FF_\rho$ are defined in the complex domain $D_\rho$
characterized by ${\rm Im}(x_1)=0$, $|{\rm Im}(x_2)|<\rho_1\,$,
and $|y_2|<\rho_2\,$, where they are continuous,
and analytic in $x_2$ and $y$.
We start by stating some basic properties of these spaces.
In accordance with our previous definition,
we denote by $\partial_kh$ the partial derivative
of $u\mapsto h((u_1,u_2),(u_3,u_4))$ with respect to $u_k\,$,
for $k=1,2,3,4$.
\claim Proposition(Basic)
Consider functions $f,g\in\FF_\rho$ and $h\in\FF_{\rho'}\,$,
and the change of variables
$V(x,y)=((x_1,x_2+f(x,y)),(y_1,g(x,y)))$.
Let $\alpha=\displaystyle{\max_{t\ge 0}}\;t[1-\tanh(t)]\,$.
\ \ \ $( =0.2784\ldots)$ \vbox{\vskip-.4em\noindent Then}
\item{$(i)$} $|f(x,y)|\le\|f\|_\rho\,$,
\ for all $(x,y)\in D_\rho\,$.
\item{$(ii)$} $\|fg\|_\rho\le\|f\|_\rho\|g\|_\rho\,$.
\item{$(iii)$} $\|\partial_4 h\|_\rho\le(\rho'_2-\rho_2)^{-1}\|h\|_{\rho'}\,$,
\ if $\rho'_1\ge\rho_1$ and $\rho'_2>\rho_2$.
\item{$(iv)$} $\|h\circ V\|_\rho\le(\rho_1'/\rho_1)^\alpha\|h\|_{\rho'}\,$,
\ if $\rho_1'\ge\rho_1+\|f\|_\rho$ and $\rho_2'\ge\|g\|_\rho\,$.
\item{$(v)$} $\|h\|'_\rho\le s_0^{-1}(\rho_1'/\rho_1)^\alpha\|h\|_{\rho'}\,$,
\ if $\rho_1'\ge\rho_1+s_0/(\kappa\rho_2)$ and $\rho_2'\ge\rho_2+s_0/\sigma$
and $s_0>0$.
\proof
The first three statements are straightforward to check.
In order to prove $(iv)$, consider first the case
$h(x,y)=\cos(v\cdot x)y_2^k\,$, and let $X(x,y)=x$.
Then
$$
\eqalign{
\|h\circ V\|_\rho&=\bigl\|\cos(v\cdot X+v_2f)g^k\bigr\|_\rho\cr
&\le\bigl\|\cos(v\cdot X)\cos(v_2f)
-\sin(v\cdot X)\sin(v_2f)\bigr\|_\rho\|g\|_\rho^k\cr
&\le\bigl[\cosh(\rho_1 v_2)\cosh(\|v_2f\|_\rho)
+\cosh(\rho_1 v_2)\sinh(\|v_2f\|_\rho)\bigr]\bigl(\rho_2'\bigr)^k\cr
&={\cosh(\rho_1 v_2)\over\cosh(\rho_1' v_2)}e^{|v_2|\|f\|_\rho}
\cosh(\rho'_1v_2)\bigl(\rho_2'\bigr)^k
\le(\rho'_1/\rho_1)^\alpha\|h\|_{\rho'}\,,\cr}
\equation(BasicA)
$$
In the last inequality, we used that for
$\lambda=\rho_1/\rho'_1$ and $u\ge 0$,
$$
{\cosh(\lambda u)\over\cosh(u)}\,e^{(1-\lambda)u}
=\exp\left(\int_\lambda^1\!tu\bigl[1-\tanh(tu)\bigr]{dt\over t}\right)
\le\lambda^{-\alpha}\,.
\equation(BasicB)
$$
An analogous bound holds if $h(x,y)=\sin(v\cdot x)y_2^k\,$.
Since we are dealing with weighted $\ell^1$ spaces,
the claim $(iv)$ now follows by taking linear combinations and limits.
The bound $(v)$ is obtained from $(iv)$ by estimating
the derivative in
$$
\|h\|'_\rho=\sup_{\kappa\rho_2\|f\|_\rho\le 1\atop\sigma\|g\|_\rho\le 1}
\biggl\|{d\over ds}h\circ V_{s,f,g}\Bigm|_{s=0}\biggr\|_\rho\,,
\equation(BasicC)
$$
where $V_{s,f,g}(x,y)=((x_1,x_2+sf(x,y)),(y_1,y_2-sg(x,y)))$,
using Cauchy's formula with contour $|s|=s_0$.
\qed
The following is one of the main motivations behind
our definition of $\Iplusminus$.
Let $h\in\BB'_\rho\,$.
If $\psi=\Iminus\psi$ is a Fourier-Taylor polynomial,
then we can define
$$
L(h)\psi=(\partial_2h)\DD_4\psi-(\partial_4h)\DD_2\psi\,,\qquad
\DD_k\psi=\partial_k\partial_1^{-1}\psi\,.
\equation(LDef)
$$
\claim Proposition(DDbounds)
$\DD_2\,$, $\DD_4\,$, and $L(h)$ extend to bounded linear operators
from $\Iminus\BB_\rho$ to $\Iminus\BB_\rho\,$,
$\Iminus\CC_\rho\,$,
and $\BB_\rho\,$, respectively, with operator norms
$$
\|\DD_2\|\le\sigma^{-1}\,,\quad
\|\DD_4\|\le(\kappa\rho_2)^{-1}\,,\quad
\|L(h)\|\le\|h\|'_\rho\,.
\equation(DDbounds)
$$
\proof
Consider first $\psi(x,y)=\cos(v\!\cdot\!x)y_2^k$ with $(v,k)$ belonging
to the complement of $\iplus$ in $I$.
By using that $v_2/v_1\le\sigma^{-1}$ and $k/v_1\le\kappa^{-1}$,
we immediately get the bounds
$\|\DD_2\psi\|_\rho\le\sigma^{-1}\|\psi\|_\rho$
and $\|\DD_2\psi\|_\rho\le(\kappa\rho_2)^{-1}\|\psi\|_\rho\,$.
These bounds extend by linearity to all of of $\Iminus\BB_\rho\,$,
which proves the first two inequalities in \equ(DDbounds).
The third inequality now follows
from the definition \equ(normprimeDef) of $\|h\|_\rho'\,$.
\qed
This allows us to solve
the ``linearized version'' of equation \equ(UUHcond).
Denote by $\{H,\phi\}$ the Poisson bracket of of $H$ with $\phi$.
\claim Corollary(PsiSolve)
Assume $H=H_0+h$ belongs to $\AA_\rho'\,$.
If $\|h\|_\rho'\le A<1$ then the equation
$$
\Iminus\bigl(H+\{H,\phi\}\bigr)=0\,,\qquad \Iplus\phi=0\,,
\equation(PsiEqu)
$$
has a unique solution $\phi$ such that $\psi=\partial_1\phi$
belongs to $\Iminus\BB_\rho\,$, and
$$
\|\psi\|_\rho\le(1-A)^{-1}\|\Iminus H\|_\rho\,,\quad
\|\{H,\phi\}\|_\rho\le(1-A)^{-1}\|\Iminus H\|_\rho\,.
\equation(PsiBound)
$$
\proof
Let $\Lplusminus=\Iplusminus L(h)\Iminus$.
By \clm(DDbounds), the operator norm of $\Lplusminus$ is $\le A<1$.
We can rewrite \equ(PsiEqu) in terms of $\psi=\partial_1\phi$
as $(\Id-\Lminus)\psi=\Iminus h$, and this equation can be solved
by inverting $\Id-\Lminus$ by means of a Neumann series.
This yields the first bound in \equ(PsiBound).
The second bound follows from the first, using the fact that
$\{H,\phi\}=L(h)\psi-\psi=\Lplus\psi-\Iminus h$.
\qed
Our next goal is to define a canonical transformation $U_\psi$
such that $H\circ U_\psi=H+\{H,\phi\}+$
``terms that are quadratic or higher order in $\phi$''.
Our generating functions $\phi$ are nonresonant,
and thus do not depend on the variable $y_1\,$.
The corresponding transformation $U_\psi:(x,y)\mapsto(x',y')$
is defined implicitly by the equations
$$
\matrix{
x_1'=x_1\,,\phantom{=\psi(x,y')} & x_2'=x_2+[\DD_4\psi](x,y')\,,\cr
y_1=y_1'+\psi(x,y')\,,& y_2=y_2'+[\DD_2\psi](x,y')\,.\cr}
\equation(UpsiDef)
$$
In fact, the only implicit equation that needs to be solved
is the last one.
We can write it as a fixed point equation for $g=y_2-y_2'$ as follows:
$$
g=\KK(g)\,,\quad
\bigl(\KK(g)\bigr)(x,y)=\bigl[\DD_2\psi\bigr]\bigl(x,(y_1,y_2-g(x,y))\bigr)\,.
\equation(Implicit)
$$
\claim Proposition(Implicit)
Let $0<\sigma^{-1}\eps^3<\eps r_2<\rho_2\,$.
Denote by $B$ the closed ball in $\BB_{\rho-\eps r}$
of radius $\sigma^{-1}\eps^3$, centered at the zero, with $r=(0,r_2)$.
If $\psi\in\Iminus\BB_\rho$ satisfies $\|\psi\|_\rho\le\eps^3$,
then the equation \equ(Implicit) has a unique solution $g\in B$, and
$$
\|g\|_{\rho-\eps r}\le\sigma^{-1}\|\psi\|_\rho\,.
\equation(gBound)
$$
\proof
By using \clm(DDbounds), \clm(Basic), and the mean value theorem,
we find that for any $g,g'\in B$,
$$
\eqalign{
\|\KK(g)\|_{\rho-\eps r}
&\le\|\DD_2\psi\|_\rho
\le\sigma^{-1}\|\psi\|_\rho\le\sigma^{-1}\eps^3\,,\cr
\|\KK(g')-\KK(g)\|_{\rho-\eps r}
&\le\|\partial_4\DD_2\psi\|_{\rho-\eps r}
\|g'-g\|_{\rho-\eps r}
\le b\|g'-g\|_{\rho-\eps r}\,,\cr}
\equation(gBoundA)
$$
with $b=(\sigma\eps r_2)^{-1}\|\psi\|_\rho<1$.
This shows that $\KK$ is a contraction on $B$,
and thus has a unique fixed point in $B$.
The bound \equ(gBound) follows from the first part of \equ(gBoundA).
\qed
Now we are ready to compose a Hamiltonian $H$, satisfying
$$
\|H-H_0\|_\rho\le b**\sigma^{-1}\eps^2$.
If $H\in\AA_\rho'$ satisfies the bounds \equ(FirstBounds),
then the abovementioned canonical transformation $U_\psi$
maps $D_{\rho-2\eps r}$ into $D_\rho\,$,
and $\wt H=H\circ U_\psi$ belongs to $\AA_{\rho-2\eps r}$ and satisfies
$$
\eqalign{
\bigl\|\wt H-H\bigr\|_{\rho-2\eps r}&\le\Delta b(\eps)
=\bigl[1+\eps C_2(\eps)\bigr]\eps^3\,,\cr
\bigl\|\wt H-H\bigr\|_{\rho-2\eps r}'&\le\Delta A(\eps)
=(\sigma r_2)^{-1}(1+\eps r_1/\rho_1')^\alpha
\bigl[1+\eps C_1(\eps)\bigr]\eps^2\,,\cr
\bigl\|\Iminus\wt H\bigr\|_{\rho-2\eps r}&\le C_2(\eps)\eps^4\,,\cr}
\equation(NextBounds)
$$
where $C_1$ and $C_2$ are two functions defined below.
\proof
Our assumptions guarantee that the function $\psi\in\BB_\rho$
described in \clm(PsiSolve) exists and is bounded in norm by $\eps^3$,
and that \clm(Implicit) applies for this $\psi$,
yielding $g\in\BB_{\rho-\eps r}$ of norm $\le\sigma^{-1}\eps^3$.
We will prove the bounds \equ(NextBounds) by using the following
one-parameter family that interpolates between
$F(0)=H$ and $F(1)=\wt H$ in such a way that $F'(0)=\{H,\phi\}$:
$$
\eqalign{
F(s)(x,y)=&-s\bigl(\psi\circ V_s\bigr)(x,y)\cr
&+h\bigl(\bigl(x_1,x_2+\bigl(s[\DD_4\psi]\circ V_s\bigr)(x,y)\bigr),
\bigl(y_1,y_2-\bigl(s[\DD_2\psi]\circ V_s\bigr)(x,y)\bigr)\bigr)\,,\cr}
\equation(FsDef)
$$
where $V_s(x,y)=\bigl(x,(y_1,y_2-sg(x,y))\bigr)$.
Consider $s\in\complex$ with $|s|\le s_0=n\sigma r_2\eps^{-2}$,
where $n$ can be either $1$ or $2$.
By using \clm(Basic) and \clm(DDbounds), we have the bounds
$$
\eqalign{
\|sg\|_{\rho-n\eps r}&\le s_0\|g\|_{\rho-\eps r}\le n\eps r_2\,,\cr
\|s[\DD_2\psi]\circ V_s\|_{\rho-n\eps r}
&\le\sigma^{-1}s_0\|\psi\|_\rho\le n\eps r_2\,,\cr
\|s[\DD_4\psi]\circ V_s\|_{\rho-n\eps r}
&\le(\kappa\rho'_2)^{-1}s_0\|\psi\|_\rho\le n\eps r_1\,.\cr}
\equation(StepOneA)
$$
\clm(Basic) thus shows that $F(s)$ belongs to $\AA_{\rho-n\eps r}$
whenever $|s|\le s_0\,$.
Furthermore, $s\mapsto F(s)$ is analytic,
not only in the interior of the disk $|s|\le s_0\,$,
but in an open neighborhood of it, since the norm of $\psi$
is strictly less than $\eps^3$, due to the fact that $A0$, $b'>b$, $A'>A$ less than $1$, and $\rho'<\rho$.
Our goal is to iterate the map $H\mapsto\wt H$,
described in \clm(StepOne). To this end, it is useful to compare
the third inequality in \equ(NextBounds) with
the third inequality in \equ(FirstBounds), by writing the former as
$$
\|\Iminus\wt H\|_{\rho-2\eps r}\le(1-A')[f(\eps)]^3\,,\qquad
f(\eps)=\bigl(\eps C_2(\eps)/(1-A')\bigr)^{1/3}\eps\,.
\equation(epsMap)
$$
Clearly, if $\eps>0$ is sufficiently small, then
the sequence of numbers $\eps_n=f^n(\eps)$, for $n=0,1,2\ldots$,
converges to zero, and the sum $\sum 2\eps_n r$ converges to a limit
$\delta=\OO(\eps)$.
The sums of the values $\Delta b(\eps_n)$
and $\Delta A(\eps_n)$, described in \equ(NextBounds),
converge as well, to some positive real numbers
$S_b=\OO(\eps^3)$ and $S_A=\OO(\eps^2)$, respectively.
Thus, if $\eps>0$ is sufficiently small,
and $H$ satisfies the bounds \equ(FirstBounds),
then the map $(\eps,\rho,H)\mapsto(f(\eps),\rho-2\eps r,\wt H)$
can be iterated indefinitely, and the iterates converge
in $\real^3\times\BB_{\rho-\delta}$ to a limit
$(0,\rho-\delta,\wh H)$.
By construction, we have $\Iminus\wh H=0$.
We note that $\Delta b\,$, $\Delta A\,$, and $f$ are all decreasing functions
of $\rho'$, and increasing functions of $b'$ and $A'$.
In practice, this means that the results
$\delta$, $S_b\,$, and $S_A$ from one choice of parameters,
can be used to obtain a new choice
$(\rho',b',A')\to(\rho-\delta,b+S_b,A+S_A)$
that improves the bounds.
Denote by $U_{\psi_n}$ the canonical transformation used at step $n$
of our iteration, and by $\phi_n$ its generating function.
Then the equation \equ(UpsiDef),
together with the inequalities \equ(StepOneA)
and part $(i)$ of \clm(Basic), shows that $U_{\psi_n}\!-\id$
is uniformly bounded on $D_{\rho-\delta}$ by a constant times $\eps_n\,$.
Given that the $\eps_n$'s are summable, this implies that
$$
\UU_\ssH=U_{\psi_1}\circ U_{\psi_2}\circ U_{\psi_3}\circ\ldots
\equation(UUHdef)
$$
converges uniformly on $D_{\rho-\delta}$,
and that $\wh H=H\circ\UU_\ssH\,$.
The analyticity of the map $H\mapsto\wh H$ follows from the
uniform convergence of our iteration schemes, including the ones
used to solve the equations \equ(PsiEqu) and \equ(Implicit).
The estimate \equ(Elimination) can be obtained e.g. by differentiating
the map $F: H\mapsto\wh H-H$ and checking that
$DF(\Iplus H)\Iminus H=\{\Iplus H,\phi_1\}$.
The desired bound then follows from the
second inequality in \equ(PsiBound).
\qed
\remark
The results obtained in this section do not change
if we include an additional factor $\exp|\delta v_1|$
in the definition of our norms \equ(newDef).
If $\delta>0$, then the domain $\DD_\rho$
includes complex values of $x_1$ with $|{\rm Im}(x_1)|<\delta\,$,
and the functions in $\FF_\rho$ are analytic in this domain.
Concerning the transformation $\RR$,
the main change is a loss of analyticity in the step $H\mapsto H'$,
where the domain shrinks to $|{\rm Im}(x_1)|<\vartheta_1^{-1}\delta\,$.
But since the step $H''\mapsto H''\circ\UU_\ssHpp$
generates analyticity in the variable $x_1\,$,
as noted in the introduction,
\clm(RGdomain) remains true for small $\delta>0$.
\section Construction of an invariant \fatomega-torus
The discussion in this section is based on the assumption
that the transformation $\RR$, described in \clm(RGdomain),
has a (nontrivial) fixed point $H^\ast\in\AA_\rho^{+}\,$.
The function $H^\ast$ and the corresponding canonical transformation
$U^\ast=U'\circ\UU_{\scriptscriptstyle(H^\ast)''}$
are assumed to take real values when restricted to real arguments.
We will use the same notation as in the previous section.
In particular, we set $d=2$ to simplify notation;
this does not affect any estimate or result.
Solving equation \equ(GammaEqu) for $\Gamma^\ast$
is equivalent (via the relation $\Gamma^\ast=\id+\gamma^\ast$)
to finding a fixed point $\gamma^\ast$
for the transformation $\NN$, defined by the equation
$$
\NN(\gamma)=\Lambda^\ast\circ\Gamma\circ\TT_1^{-1}-\id
=\TT_\mu\circ\bigl[u^\ast\circ(\id+\gamma)+\gamma\bigr]\circ\TT_1^{-1}\,,
\equation(NNdef)
$$
where $\Lambda^\ast=\TT_\mu\circ U^\ast$ and $u^\ast=U^\ast-\id$.
This transformation is restricted to functions
$\gamma=(0,\gamma_2,\gamma_3,\gamma_4)$ that are admissible,
meaning that $\gamma_2$ is odd, and $\gamma_2,\gamma_3$ are even.
We note that, due to the admissibility property $\partial_3 u^\ast=0$,
the fixed point equation $\gamma=\NN(\gamma)$
can be solved for $\gamma_2$ and $\gamma_4$ first,
independently of $\gamma_3\,$,
and then $\gamma_3$ can be determined by iteration:
$$
\gamma_3=\sum_{n=1}^\infty
(\mu/\vartheta_1)^nu^\ast_3\circ(\id+\gamma)\circ\TT_1^{-n}\,.
\equation(gammaThree)
$$
The right hand side of this (formal) equation is independent of $\gamma_3\,$.
Given that $u^\ast$ and $\gamma$ have zero component
in the expanding direction of of $\TT_\mu\,$,
we expect $\NN$ to be a contraction,
if the derivative of $u^\ast$ is not too large.
To be more precise, we also have to take into account the
composition with $\TT_1^{-1}$.
Since $\TT_1^{-1}$ is expanding in the $x_2$ direction,
we should restrict $\NN$ to a space of functions $\gamma(x,0)$
that are of class $C^r$ in the variable $x_2\,$, with $r<1$.
Given $r>0$, denote by $\BB_0$ the space of all functions $f$
on $D_0=\real^d\times\{0\}$, of the form
$$
f(x,0)=\!\sum_{v\in\VV}f_v\cos(v\cdot x),\quad
\|f\|_0=\!\sum_{v\in\VV}|f_v|\bigl(1+|v_2|\bigr)^r\,<\infty\,.
\equation(yaDef)
$$
The corresponding space of odd functions will be denoted by $\CC_0\,$,
and $\FF_0$ is defined to be the direct sum of $\BB_0$ and $\CC_0\,$.
Let
$$
c_\rho(s)=\sup_{t\ge 0}{(1+t)^r\over\cosh(\rho_1t)}\,e^{st}\,,\qquad
|s|<\rho_1\,.
\equation(Krho)
$$
\claim Proposition(Basics)
Consider functions $f,g\in\FF_0$ and $h\in\FF_\rho\,$,
and the change of variables
$V(x,0)=((x_1,x_2+f(x,0)),(0,g(x,0)))$. Then
\item{$(i)$} $|f(x,0)|\le\|f\|_0\,$,
\ for all $x\in\real^2\,$.
\item{$(ii)$} $\|f\circ\TT_1^{-1}\|_0\le|\vartheta_2|^{-r}\|f\|_0\,$.
\item{$(iii)$} $\|fg\|_0\le\|f\|_0\|g\|_0\,$.
\item{$(iv)$} $\|h\circ V\|_0\le c_\rho(\|f\|_0)\|h\|_\rho\,$,
\ if $\rho_1\ge\|f\|_0$ and $\rho_2\ge\|g\|_0\,$.
The proof of these inequalities is straightforward;
see also the proof of \clm(Basic).
If $u=(u_1,u_2,u_3,u_4)$ is a vector-valued function on $D_{\rho'}$
whose second and fourth components belong to
$\FF_{\rho'}\,$, and if $\rho<\rho'$ component-wise,
then we define
$$
\eqalign{
\|u\|_{\rho'}&=\max\bigl\{\|u_2\|_{\rho'}\,, b\|u_4\|_{\rho'}\bigr\}\,,\cr
\|u\|'_\rho&=\max\bigl\{\|\partial_2u_2\|_\rho+b^{-1}\|\partial_4u_2\|_\rho\,,
b\|\partial_2u_4\|_\rho+\|\partial_4u_4\|_\rho\bigr\}\,,\cr}
\equation(VecNorm)
$$
where $b$ is some fixed positive constant.
A seminorm $\|\gamma\|_0$ is defined analogously
for vector-valued functions $\gamma$ on $D_0\,$,
whose second and fourth components belong to $\FF_0\,$.
\claim Lemma(omegaTorus)
Let $\rho_0=\min\{\rho_1,b\rho_2\}$ and $|\mu|<\vartheta_1|\vartheta_2|^r$.
Assume that $u^\ast_3$ belongs to $\BB_\rho\,$,
that $\|u^\ast\|'_\rho$ is finite, and
that $\NN$ has an admissible approximate fixed point $\gamma'$
with seminorm $\|\gamma'\|_0$ less than $|\vartheta_2|^r\rho_0\,$.
Choose a positive real number $R<|\vartheta_2|^r\rho_0-\|\gamma'\|_0\,$,
and define
$$
\eps=\|\NN(\gamma')-\gamma'\|_0\,,\qquad
K=c_\rho(|\vartheta_2|^{-r}R)|\vartheta_2|^{-r}\|\Lambda^\ast\|'_\rho\,.
\equation(epsK)
$$
If $\eps<(1-K)R$, then $\NN$ has a unique
admissible fixed point $\gamma^\ast$ in $\FF_0^4$
which satisfies $\|\gamma^\ast-\gamma'\|_0\le R$.
\proof
As explained earlier, we consider first the
restricted fixed point problem for $\pi\circ\NN\circ i$
on $\CC_0\times\BB_0\,$, where $\pi\gamma=(\gamma_2,\gamma_4)$.
and $i(f,g)=(0,f,0,g)$.
In order to simplify notation, we will omit $\pi$ and $i$.
The restricted fixed point problem is solved
by using the contraction mapping principle,
on the closed ball $B$ of radius $R$ in $\CC_0\times\BB_0$,
centered at $\gamma'$.
The constant $K$ is an upper bound on the operator norm
of the derivative $D\NN(\gamma)$, for $\gamma$ in $B$:
{}From the right-composition with $\TT_1^{-1}$
we get a factor $|\vartheta_2|^{-r}$, and the factor
$c_\rho(|\vartheta_2|^{-r}R)\|\Lambda^\ast\|'_\rho$
is a bound on the tangent map of $\Lambda^\ast$;
both are obtained directly from \clm(Basics).
Finally, the given inequalities involving $R$ ensure
that $\NN$ maps the ball $B$ into itself, and that $K<1$.
Thus, the restricted fixed point problem has a unique
solution $(\gamma^\ast_2,\gamma^\ast_4)$ in $B$.
The missing component $\gamma^\ast_3$ is now
given by the right hand side of equation \equ(gammaThree),
with $\gamma=i(\gamma^\ast_2,\gamma^\ast_4)$.
The sum in this equation converges in $\BB_0\,$,
due to our assumption on $\mu$;
this follows from part $(ii)$ of \clm(Basics).
\qed
We note that the results obtained in this section
do not change if we include an additional factor $\exp|\delta v_1|$
in the definition of our norms \equ(newDef) and \equ(yaDef).
This can be used to prove the differentiability
assumption in \clm(Gamma) on the curve $t\mapsto\Gamma^\ast(t\omega,0)$.
Using methods of the type described in [\rAKW],
we have computed a numerical approximation $H_1$
(over $10000$ nonzero Fourier-Taylor coefficients)
for the expected non-integrable fixed point $H^\ast$ for
the golden mean version of $\RR$, with $U'=\id$.
The same computation also yields an approximation
for the canonical transformation $U^\ast$.
Then we iterated the transformation $\NN$ a large number of times,
starting with $\gamma=0$, to obtain a numerical approximation $\gamma'$
(over $3000$ nonzero Fourier coefficients)
for the invariant $\omega$-torus for $H^\ast$.
The parameters used in this computation were
$\rho=(0.9,0.15)$, $\sigma=0.79$, $\kappa=2\sigma$, $b=6.5$,
and $\mu$ the approximate value of $\mu^\ast$ given in the introduction.
Additional data can be found in [\rKii].
Numerically, the assumptions of \clm(omegaTorus)
are satisfied with a comfortable margin
(e.g. $\|\gamma'\|_0<0.4$ and $K<0.82$).
We have used $r=\delta=0$, but since the norm of a
Fourier-Taylor polynomial is continuous in these parameters,
the same holds for small $r,\delta>0$.
\section Remaining proofs
\proofof(mapsRG)
It should be noted that the functions considered here are not subject to
the restrictions imposed after \clm(mapsRG).
Clearly, the maps $\Phi_{2\pi\omega'_k}$
commute with each other and leave $H^{-1}(0)$ invariant.
Thus, by the periodicity of $H$,
the same holds for the maps $F_k\,$.
In addition, since $q\cdot\omega'$ plays the role of time,
its value changes by an amount $2\pi\omega'_k$
under the map $\Phi_{2\pi \omega'_k}$.
The change under $V(2\pi\delta_k)$ is exactly the opposite:
$-2\pi\delta_k\cdot\omega'=-2\pi\omega'_k\,$.
This shows that $F_k$ leaves $\WW_\ssH$ invariant.
Since there is a linear symplectic change of coordinates
on $\real^{2d}$ which makes $t=q\cdot\omega'$ one of the coordinates,
the Hamiltonian $H$ admits a global ``flow box'' chart
with the canonical symplectic form
$dt\wedge dH+dq_2'\wedge dp_2'+\ldots dq_d'\wedge dp_d'\,$.
Thus, given that the maps $F_k$ are symplectic on $\real^{2d}$,
and leave $t=H=0$ invariant,
their restriction to $\WW_\ssH$ are symplectic as well.
If $f$ and $g$ are differentiable functions on $\torus^d\times\real^d$,
denote by $\{f,g\}$ their Poisson bracket.
By using the identity
$\{\mu^{-1}f\circ\TT_\mu,\mu^{-1}g\circ\TT_\mu\}
=\mu^{-1}\{f,g\}\circ\TT_\mu\,$,
we find that
$$
\eqalign{
{d\over dt}
f\circ\Phi_{\vartheta_1 t}\circ\Lambda\Bigm|_{t=0}
&=\vartheta_1\bigl\{f,H\bigr\}\circ\Lambda\cr
&=\mu^{-1}\vartheta_1\bigl\{f\circ\Lambda,H\circ\Lambda\}
=\bigl\{f\circ\Lambda,\RR(H)\}\,.\cr}
\equation(flowRG)
$$
This shows that the flow for $\wt H=\RR(H)$ is given by
$\wt\Phi_t=\Lambda^{-1}\circ\Phi_{\vartheta_1 t}\circ\Lambda$.
Thus, we have
$$
\eqalign{
\wt F_k&=\wt\Phi_{2\pi\omega'_k}\circ V(2\pi\delta_k)
=\Lambda^{-1}\circ\Phi_{2\pi\vartheta_1\omega'_k}
\circ\Lambda\circ V(2\pi\delta_k)\cr
&=\Lambda^{-1}\circ\Phi_{2\pi(T^\ast\omega')_k}\circ\Lambda
\circ\Lambda^{-1}\circ V(2\pi T\delta_k)\circ\Lambda\cr
&=\Lambda^{-1}\circ\Phi_{2\pi T_{1,k}\omega'_1}\circ\ldots
\circ\Phi_{2\pi T_{d,k}\omega'_d}
\circ V(2\pi T_{1,k}\delta_1)\circ\ldots
\circ V(2\pi T_{d,k}\delta_d)\circ\Lambda\cr
&=\Lambda^{-1}\circ F_1^{T_{1,k}}\circ\ldots\circ F_d^{T_{d,k}}\circ\Lambda\,\cr}
\equation(mapsRGcheck)
$$
as claimed.
\qed
\proofof(Gamma)
We note that, since $H^\ast$ belongs to the domain of $\RR$,
the scaling transformation $\Lambda^\ast$ maps $D_\rho$ into itself.
Recall that $D_\rho$ is $d+2$ dimensional:
we only consider (functions of) the variables $q$,
$\omega\cdot p$ and $\Omega\cdot p$.
Since $H^\ast\in\AA_\rho^{+}$ is real analytic
in the variable $\omega\cdot q$,
so is the map $\Lambda^\ast$; see the remark at the end of Section 2.
Thus, we can use equation \equ(flowRG) to obtain
$$
\Lambda^\ast\circ\Phi^\ast_t=\Phi^\ast_{\vartheta_1 t}\circ\Lambda^\ast
\equation(LambdaPhi)
$$
on an open neighborhood of any fixed compact set in $\DD_\rho\,$,
for times $t$ in some open interval containing $0$.
Assume now that $H^\ast$ has a
real-valued unique invariant $\omega$-torus $\Gamma^\ast$ in its domain.
By restricting the identity \equ(LambdaPhi) to this torus,
and using that the left hand side is defined for all times $t$,
we can extend it to all $t\in\real$.
When combined with equation \equ(omegaTorus), this yields
$$
\eqalign{
\Lambda^\ast\circ\Gamma^\ast\circ\TT_1^{-1}\circ\Phi^\circ_t
&=\Lambda^\ast\circ\Gamma^\ast\circ\Phi^\circ_{\vartheta_1^{-1}t}\circ\TT_1^{-1}\cr
&=\Lambda^\ast\circ\Phi^\ast_{\vartheta_1^{-1}t}\circ\Gamma^\ast\circ\TT_1^{-1}
=\Phi^\ast_t\circ\Lambda^\ast\circ\Gamma^\ast\circ\TT_1^{-1}\,.\cr}
\equation(GammaEquA)
$$
In other words, $\Gamma=\Lambda^\ast\circ\Gamma^\ast\circ\TT_1^{-1}$
satisfies the equation \equ(omegaTorus) defining an invariant
$\omega$-torus. It is straightforward to check that $\Gamma$ is admissible.
Thus, if $H^\ast$ has a unique invariant $\omega$-torus,
then we must have $\Gamma=\Gamma^\ast$.
This proves the first part of the theorem.
Concerning the second part, notice that
the unstable manifold of $\TT_1$ (acting on $\real^d\times\{0\}$)
at the fixed point $(0,0)$
is the orbit of this fixed point under the flow $\Phi^\circ$.
Similarly, as equation \equ(LambdaPhi) shows,
the unstable manifold of $\Lambda^\ast$
(acting on $\real^d\times\real^2$)
at the fixed point $\Gamma^\ast(0,0)$
is the orbit of this fixed point under the flow $\Phi^\ast$.
Here, and in what follows, $\Gamma^\ast$ is assumed to satisfy
the conditions given after \equ(GammaEqu).
The identity \equ(GammaEqu) implies that
$\Gamma^\ast$ maps the unstable manifold of $\TT_1$
to the unstable manifold of $\Lambda^\ast$.
Thus, for every real number $t$ there exists a real number $\tau$,
such that for $q=0$,
$$
\Gamma^\ast\bigl(\Phi^\circ_t(q,0)\bigr)
=\Phi^\ast_\tau\bigl(\Gamma^\ast(q,0)\bigr)\,.
\equation(TorusExistsA)
$$
By taking the dot product of each side of this equation
with the vector $(\omega,0)$, we find that $t=\tau$.
The identity \equ(TorusExistsA) with $t=\tau$ extends immediately
from $q=0$ to $q=s\omega$, for arbitrary real numbers $s$.
Thus, we have $\Gamma^\ast\circ\Phi^\circ_t=\Phi^\ast_t\circ\Gamma^\ast$
when restricted to the orbit of $(0,0)$ under the flow $\Phi^\circ$.
But since both sides of this equation
are well defined as maps from the torus $\torus^d\times\{0\}$
to $\torus^d\times\real^2$, and are continuous,
and since the orbit of $(0,0)$ under the flow $\Phi^\circ$
is dense in this torus,
the identity $\Gamma^\ast\circ\Phi^\circ_t=\Phi^\ast_t\circ\Gamma^\ast$
holds on all of $\torus^d\times\{0\}$.
This shows that $\Gamma^\ast$ defines an invariant $\omega$-torus
for $H^\ast$, as claimed.
\qed
\bigskip\noindent{\bf Acknowledgments}\hfill\break
The author would like to thank P.~Wittwer and J.J.~Abad
for helpful discussions.
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\bye
**