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KAM, lower-dimensional, invariant tori, quasi-periodic motion, renormalization.
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 \begin{document}
  \title{A renormalization approach to lower-dimensional tori with Brjuno frequency vectors}
  \author{Hans Koch~\footnote{Department of Mathematics, University of Texas at Austin, 1 University Station C 1200, Austin, TX 78712, USA; Email: koch@math.utexas.edu}  and Sa\v sa Koci\' c~\footnote{Department of Mathematics, University of Toronto, 40 St.~George Street,
Toronto, Ontario, Canada M5S 2E4; Email: s.kocic@utoronto.ca}  \\{\small }
 }

\pagestyle{myheadings}\markboth{Lower-dimensional tori}{H. Koch and S. Koci\' c}

\maketitle
\date{}
\begin{abstract}
We extend the renormalization scheme for vector fields on $\Tt^d\times\Rr^m$, in order to construct lower-dimensional invariant tori with Brjuno frequency vectors for near-integrable Hamiltonian flows. For every Brjuno frequency vector $\omega\in\Rr^d$ and every vector $\Omega\in\Rr^D$ satisfying an arithmetic condition with respect to $\omega$, there exists an analytic manifold $\WW$ of infinitely renormalizable Hamiltonian vector fields; each vector field on $\WW$ is shown to have an analytic invariant torus with frequency vector~$\omega$.
\end{abstract}


\section{Introduction}

It is well known that in near integrable Hamiltonian systems one has the persistence of analytic invariant tori on which the motion is conjugate to a rotation with frequency vectors that satisfy a condition weaker than Diophantine, initially considered by Kolmogorov~\cite{Kolmogorov}.  Conditions of this type, such as the one introduced by Brjuno~\cite{Brjuno1,Brjuno2} that we will refer to as the Brjuno condition, were extensively studied by R\" ussmann~\cite{Russmann1}. In the Siegel problem~\cite{Yoccoz} and the case of circle diffeomorphisms~\cite{Yoccoz2}, the Brjuno condition is known to be the optimal condition under which such a conjugacy can be established. In higher-dimensional problems, the question of the weakest possible condition is a fundamental open problem. Recently, we have developed a renormalization approach to the construction of invariant tori with Brjuno frequency vectors for vector fields on $\Tt^d\times\Rr^m$~\cite{KK2}, using two general theorems, a normal form theorem and a stable manifold theorem, that we proved in~\cite{KK1}. The main objective of this paper is to extend the methods developed in~\cite{KK2} to include the lower-dimensional tori of Hamiltonian systems. The term lower-dimensional torus in Hamiltonian systems refers to a torus of dimension lower than the number of degrees of freedom.

The problem of persistence of lower-dimensional tori, under the conditions stated by Melnikov~\cite{Melnikov}, has already been considered by Moser~\cite{Moser}, in the case of dimension lower by one than the number of degrees of freedom. The proof of existence of elliptic (stable) lower-dimensional tori in the general case was given by Eliasson~\cite{Eliasson}. For an overview of the results on lower-dimensional tori, the reader is referred to~\cite{CQ,Eliasson,GG,Gentile,JLZ,LY,Moser,Poschel,Russmann2,XY,You}. Lower-dimensional tori also appear in and are particularly relevant for PDEs (see e.g.~\cite{BKS,CW,Kuksin}).

We consider Hamiltonians $H=H_0+h$ which are small perturbations of the integrable Hamiltonian
    \bel{H0}
H_0(q,p,\xi,\eta)=\omega\cdot
p+\frac{1}{2}\sum_{j=1}^D\Omega_j\bigl(\xi_j^2+\eta_j^2\bigr)\,.
    \ee
The corresponding dynamics on the phase space $\Tt^d\times\Rr^d\times\Rr^D\times\Rr^D$ is
given by Hamilton's equations
    \bea
\dot q_l&=&\omega_l+\partial_{p_l}h\,,\nonumber\\
\dot p_l&=&-\partial_{q_l}h\,,\nonumber\\ \dot
\xi_j&=&\Omega_j \eta_j+\partial_{\eta_j}h\,,\\
\dot \eta_j&=&-\Omega_j \xi_j+\partial_{\xi_j}h\,. \nonumber
    \eea
Here, ``$\cdot$'' denotes the derivative with respect to time.

For $h=0$, the system is integrable and the dynamics are
constrained to invariant tori. These tori are contained in the level
sets of $H_0$. The maximal dimension of these tori is $d+D$, and in that
case the motion on these tori are characterized by frequencies
$\omega_1,\dots,\omega_d$ and $\Omega_1,\dots,\Omega_D$. The
hyper-surface $p=0$, $\xi=\eta=0$, contained in the level set $H_0=0$, is a lower-dimensional torus of dimension $d$.  The motion on this torus is a linear flow $q\mapsto q+\omega t$ characterized by internal frequencies $\omega_1,\dots,\omega_d$.  The normal space is described by coordinates $\xi,\eta$, whose origin is an elliptic fixed point characterized by the normal frequencies $\Omega_1,\dots,\Omega_D$. The problem of persistence of such elliptic lower-dimensional invariant tori under small perturbations Hamiltonians of the form~\eqref{H0} has been considered for example in~\cite{Poschel,Russmann2}.

After the change of variables
    \be
u_j={1\over\sqrt{2}}\bigl(\xi_j+i \eta_j\bigr)\,,\qquad
 w_j={i\over\sqrt{2}}\bigl(\xi_j-i \eta_j\bigr)\,,
    \ee
the Hamiltonian $H_0$ is transformed into
    \be
H_0(q,p,u,w)=\omega\cdot p-i\sum_{j=1}^D\Omega_ju_jw_j\,.
    \ee
To simplify the notation, we have denoted this function by $H_0$ also.

The symplectic form $dq\wedge dp+d\xi\wedge d\eta$ gives rise to the new symplectic form $dq\wedge dp+du\wedge dw$.
The corresponding operator $\wh H_0$, defined by the Poisson bracket $\wh
H_0\Psi=\{\Psi,H_0\}$, is given by
    \be
\wh H_0=\omega\cdot\nabla_q -i\sum_j\Omega_j\bigl(u_j\partial_{u_j}-
w_j\partial_{w_j}\bigr)\,.
    \ee

If $ \Psi_{\nu,\kappa,\alpha,\beta}(q,p,u,v) =e^{i\nu\cdot
q}p^\kappa u^\alpha w^\beta\,,$ where $(\nu,\kappa,\alpha,\beta)\in\Zz^d\times\Nn_0^d\times\Nn_0^D\times\Nn_0^D$ and $\Nn_0=\Nn\cup\{0\}$, then
    \be
\wh H_0 \Psi_{\nu,\kappa,\alpha,\beta}
=i\bigl[\omega\cdot\nu+\Omega\cdot(\beta-\alpha)\bigr]
\Psi_{\nu,\kappa,\alpha,\beta}\,.
    \ee
The moduli of the  eigenvalues $|\omega\cdot\nu+\Omega\cdot(\beta-\alpha)|$ of $\wh H_0$ are precisely the small denominators that appear in the problem of construction of lower-dimensional tori.  In the renormalization approach that we pursue here, the small divisors are transformed into ``large divisors''  by a scaling of the phase space. The terms containing ``large divisors'' are then eliminated by a coordinate change.

Given $\mu>0$, we will perform the following scaling of the non-toral part of the phase space
    \bel{smu}
S_\mu(q,p,u,w) =(q,\mu p,\mu^{1/2}u,\mu^{1/2}w)\,.
    \ee
Under such a transformation, a Hamiltonian $H$ is transformed as $H\mapsto
\frac{1}{\mu} H\circ S_\mu$. Notice that the Hamiltonian $H_0$ is invariant under this transformation.

Given a matrix $T\in\GL(d,\Rr)$, we will also perform the canonical
scaling
    \bel{tscaling}
\TT(q,p,u,w) =(Tq,\bar T^{-1}p,u,w)\,,
    \ee
where $\bar T=T^T$ is the transpose of $T$. This scaling is an essential part of the method and the choice of matrices $T$ has played an important role in the previous schemes. In~\cite{KK1}, we have constructed a renormalization scheme using a sequence of scaling matrices $T\in\SL(d,\Zz)$ generated by a multidimensional continued fraction algorithm by Khanin, Lopes Dias and Marklof~\cite{KLM}, based on an algorithm by Lagarias~\cite{Lagarias}. The results of~\cite{KLM} provide good bounds in the case of Diophantine vectors $\omega$, and allowed for the extension of the results of~\cite{Kocic} for two degree of freedom Hamiltonians to higher dimensions~\cite{KLM2}. In order to extend the results to Brjuno frequency vectors, we have developed a different method~\cite{KK2,Kocic2}, which uses non-integer matrices in $\GL(d,\Rr)$. In the present paper, we will perform this scaling with non-integer matrices in~$\SL(d,\Rr)$. On a technical level, the problem at hand is more complicated than the one considered in~\cite{KK2}, due to the existence of finitely many additional ``resonances''.

Our matrices $T$ will have the property that $T\omega=\eta^{-1}\omega$, where $0<\eta<1$. With an additional time rescaling the Hamiltonians $H_0$ are transformed as
    \be
H_0\mapsto \eta^{-1} \mu^{-1} H_0\circ\TT\circ S_\mu(q,p,u,w)=\omega\cdot
p-i\sum_j\eta^{-1}\Omega_ju_jw_j\,.
    \ee

We assume that $\omega$ satisfies the following Brjuno condition
\cite{Brjuno1,Brjuno2}:

\begin{definition} $\omega\in\Rr^d$ is Brjuno if
    \bel{BrjunoCond}
\sum_{n=1}^\infty 2^{-n}\ln\left(\frac{1}{\min\limits_{0<|\nu|\le 2^n}|\omega\cdot\nu|}\right)<\infty\,,
    \ee
where the minimum is taken over lattice vectors $\nu$ in $\integer^d$.
\end{definition}
We remark that the set of all Brjuno vectors has full Lebesgue measure. It particular, it contains all Diophantine vectors.

We will further assume that the frequencies $\Omega_1,\dots,\Omega_D$ are real, nonzero and all different and that they satisfy the following condition.

\begin{definition}
We say that $\Omega=(\Omega_1,\dots,\Omega_D)\in\Rr^D$ is Diophantine with respect to $\omega\in\Rr^d$ if
there exist constants $\tau>0$ and $\CC>0$ such that
    \be
|\omega\cdot\nu+\Omega\cdot V|>\CC |\nu|^{-\tau},
    \ee
for all $\nu\in\Zz^d\backslash\{0\}$, and all $V\in\Zz^D$ with
$0<|V|\le 2$.
\end{definition}

The objective of this paper is to consider analytic Hamiltonians, close to $H_0$, that can be expanded in Fourier-Taylor series as
    \bel{Fourier-Taylor-Hamiltonians}
H(q,p,u,w)=\sum_{\nu,\kappa,\alpha,\beta}\,H_{{\nu,\kappa,\alpha,\beta}}e^{iq\cdot\nu}p^\kappa u^\alpha w^\beta.
    \ee

Instead of working with Hamiltonian functions, we will actually work with the corresponding Hamiltonian vector fields that they generate.
We will consider Banach spaces $\AA_\rho$, with $\rho>0$, of analytic vector fields (see Section~\ref{sec2}), close to the vector field $K$, defined by $K(q,p,u,w)=(\omega,0,-i\bar\Omega u,
i\bar\Omega w)$, where $\bar\Omega=\rm{diag}
[\Omega_1,\dots,\Omega_D]$.

We note that in paper we are interested in Hamiltonians that are real analytic in the original variables $(q,p,\xi,\eta)$, that is which in these variables satisfy $H\circ C^*=C^*\circ H$, where $C^*$ is the complex conjugation. In the new variables, this symmetry implies that the Hamiltonians have the following property
    \be
    H\circ C(q,p,u,w)=C^*\circ H(q,p,u,w)\,,
     \ee
where $C(q,p,u,w)=(q^*,p^*,-iw^*,iu^*)$. It also translates to the following property of Hamiltonian vector fields $C^* Q X\circ C=X,$ where $Q$ is the linear transformation $Q(q,p,u,w)=(q,p,-iw,iu)$. We will call the vector fields real if they satisfy this property.

The main result of this paper is the following.
\begin{theorem}\label{main}
For every Brjuno vector $\omega\in\Rr^d$ and $\Omega\in\Rr^D$ Diophantine with respect to $\omega$ with different and nonzero components, there exists an open neighborhood $B$ of real analytic Hamiltonian vector fields $X\in\AA_\rho$ around $K$, and an analytic codimension $d+D$ manifold $\WW\subset B$, such that every Hamiltonian vector field $X\in\WW$ has an analytic invariant $d$-dimensional invariant torus with frequency vector $\omega$.
\end{theorem}

\begin{remark}
The manifold $\WW$ is the stable invariant manifold for the sequence of renormalization operators that we will construct. The unstable directions of the renormalization correspond to changes in frequency vectors $\omega$ and $\Omega$. One can use the above result to prove the existence of invariant $d$-dimensional tori in families of Hamiltonian vector fields intersecting the stable manifold. If one is interested in the invariant tori with frequency vectors parallel to $\omega$, the number of the necessary parameters can be reduced by $1$. As usual (see e.g.~\cite{KK1}), close to a non-degenerate Hamiltonian vector field, for which the $q$-component $X_q|_{p=0}$ is transversal to $\omega$, one can further reduce the number of parameters needed to prove the existence of invariant tori by $d-1$, by considering the translations in the $p$-variables.
\end{remark}

The paper is organized as follows. In Section~\ref{sec2}, we define the spaces of vector fields that we consider. Section~\ref{BSINGLE} contains the formulation and estimates on a single renormalization step. In Section~\ref{BCOMPOSE}, we construct a sequence of renormalization parameters and the stable manifold for the sequence of corresponding renormalization operators. Section~\ref{BTORI} contains the construction of lower-dimensional invariant tori for vector fields on the stable manifold of the renormalization transformations.

\section{Spaces of vector fields}\label{sec2}

Since in this paper we will perform the scaling of the torus $\Tt^d=(\Rr/2\pi\Zz)^d$ using non-integer matrices, it will be necessary to consider functions with periodicity of different lattices in $\Rr^d$. Let $\{e_1,\dots,e_d\}$ be a basis and $\ZZ=\{\sum_{i=1}^dz_ie_i|z_i\in\Zz\}$ be a lattice in $\Rr^d$. Let also $\VV$ be its dual
lattice, i.e.\ the set of points $v\in\Rr^d$ satisfying $\exp(iv\cdot
z)=1$ for all $z\in\ZZ$. If $\ZZ=(2\pi\Zz)^d$, then $\VV=\Zz^d$.

On $\complex^n$ we use norms  $\|c\|=\sup_j|c_j|$ and
$|c|=\sum_j|c_j|$. For linear operators between normed linear
spaces, we will always use the operator norm, unless stated
otherwise. Denote by $D_\rho$, with $\rho>0$, the set of all points
$(x,y,u,w)$ in
$\complex^d\times\complex^{d}\times\complex^{D}\times\complex^{D}$
characterized by $\|\im x\|<\rho$, $\|y\|<\rho$, $\|u\|<\rho$ and
$\|w\|<\rho$. Define $\AA_\rho(\VV)$ to be the Banach space of all analytic
vector field $X$ on $D_\rho\,$, with frequency module in $\VV$, that
can be expanding in Fourier-Taylor series
    \bel{Fourier-Taylor}
X(x,y,u,w)=\sum_{v,\kappa,\alpha,\beta}\,X_{{v,\kappa,\alpha,\beta}}e^{ix\cdot
v}y^\kappa u^\alpha w^\beta\,,
    \ee
where $v\cdot x=\sum_iv_ix_i$ and
$y^\kappa=\prod_iy_i^{\kappa_i}\,,$ and have the finite norm
    \bel{BAArhoNorm}
\|X\|_{\rho}=\sum_{v,\kappa,\alpha,\beta}\|X_{{v,\kappa,\alpha,\beta}}
\|e^{\rho|v|}\rho^{|\kappa|+|\alpha|+|\beta|}\,.
    \ee
Here, $\kappa\in\Nn_0^d$, $\alpha,\beta\in\Nn_0^D$. We will consider only Hamiltonian vector fields, and we will use the same symbol $\AA_\rho(\VV)$ to denote the subspace of Hamiltonian vector fields.

We also define $\AA'_\rho(\VV)$ to be the space of all vector fields
$X\in\AA_\rho(\VV)$, whose derivatives $DX$ are bounded linear
operators on $\AA_\rho(\VV)$. When the lattice $\VV$ is fixed, we will simplify the notation, by not explicitly mentioning the dependence on $\VV$.

We end this section with the following proposition, whose proof follows directly from the above definitions.

\begin{proposition}\label{BTrivial} Let $X\in\AA_\rho$ and
$Z\in\AA_{\rho'}\,$, with $0<\rho'\le\rho$. Then
\begin{enumerate}
\item
  $\|X(x,y,u,w)\|\le\|X\|_\rho$, for all $(x,y,u,w)\in D_\rho\,$.
\item
  $(DX)Z\in\AA_{\rho'}$
  and $\|(DX)Z\|_{\rho'}\le(\rho-\rho')^{-1}\|X\|_{\rho}\|Z\|_{\rho'}\,$,
  if $\rho'<\rho$.
\item
  $X\circ(\id+Z)\in\AA_{\rho'}$
  and $\|X\circ(\id+Z)\|_{\rho'}\le\|X\|_\rho\,$,
  if $\rho'+\|Z\|_{\rho'}\le\rho$.
\end{enumerate}
\end{proposition}


\section{One renormalization step}\label{BSINGLE}

The vector $\omega\in\Rr^d$ satisfying the Brjuno condition~\eqref{BrjunoCond} is supposed to be fixed throughout the paper.
We will rotate and scale the coordinate system such that $\omega$ takes the
form $(1,0,\dots,0)$. In the new coordinates, the lattice vectors
$v$ do not necessarily have integer components. We will denote such
a lattice of vectors $v$ by $\VV$. The lattice $\VV$ is therefore
just the lattice $\Zz^d$ expressed in a transformed system of
coordinates. Notice that the Brjuno condition~\eqref{BrjunoCond}
involves scalar products and is invariant under such a transformation. It can be interpreted just as a condition on the lattice
$\VV$.

On a neighborhood of $K$ in $\AA_\rho(\VV)$, the one-step renormalization operator is defined by
    \be
 \RR(X)=\eta^{-1}\SS_\mu^\ast\TT^\ast\UU_\ssX^\ast X,
    \ee
where $\UU^\ast$ denotes the pullback of a vector under a map $\UU$, i.e.\ $\UU^\ast X=(D\UU)^{-1}X\circ \UU$. Here, the map $\UU_\ssX$ is chosen such that its pullback brings the vector field $X$ into a normal form, that we call resonant. The construction of such a map is carried out in~\clml(BElim). The scaling maps $\SS_\mu$ and $\TT$ are defined as in~\eqref{smu} and~\eqref{tscaling}, respectively (with the change of notation $(q,p)\mapsto (x,y)$).

We choose the matrix $T$ in the definition of the scaling $\TT$
to be a diagonal matrix $T=\mbox{diag}
[\eta^{-1},\zeta,\dots,\zeta]$ in the new coordinate system.
Here $0<\eta\,,\zeta<1$. Thus, the action of $T$ on an arbitrary
vector $x\in\real^d$, that admits a  decomposition $x=x_\spa+x_\spe$
into a component $x_\spa$ parallel to $\omega$, and a component
$x_\spe$ perpendicular to $\omega$, is given by
    \bel{BTDef}
T(x)=\eta^{-1}x_\spa+\zeta x_\spe\,.
    \ee

We assume that the components of $\omega=(1,0,\dots,0)$ are
rationally independent with respect to a given lattice $\VV$, in the
sense that the first component $v_\spa$ of any nonzero vector
$v\in\VV$ is nonzero. Then, given any $L\ge 1$, we can find $\ell>0$
such that
    \bel{emptyRectangle}
|v_\spe|>L \quad {\rm or}\quad |v_\spa|\ge\ell\,,\qquad \forall
v\in\VV\setminus\{0\}\,.
    \ee
In other words, all points in $\VV$, except for the origin, lie
outside the disk characterized by $|v_\spe|\le L$ and
$|v_\spa|<\ell$. This condition is technically very important and
will be used later on to replace the trivial bound $\|\nu\|\ge 1$,
valid for $\nu\in\Zz^d\setminus\{0\}$.

Let $\chi=\max\{\|T\|,\|\bar T^{-1}\|\}$. We will assume that the
parameters $\rho,\eta,\zeta,\sigma,\mu,\ell,L$ are given positive
numbers and that the following conditions are satisfied
    \bel{LdeltaCond}
\sigma<1/2\,,\qquad 2\sigma L\le\ell\,,\qquad
0<\eta\le\zeta<1\,,\qquad e^{-\rho(1-\zeta\eta)L}\le\mu^{3/2}\,.
    \ee


\begin{definition}\label{BResonant}
Consider the Fourier-Taylor expansion~\eqref{Fourier-Taylor} of a
single component of a vector field. A mode of this component of the
vector field is the term in this expansion labeled by
$(v,\kappa,\alpha,\beta)$. A mode of a vector field characterized by
fixed $(v,\kappa,\alpha,\beta)$, with $v\in\VV$, $\kappa\in\Nn_0^d$
and $\alpha,\beta\in\Nn_0^D$, is the
vector field whose $x_i$-component is the $(v,\kappa-\delta_{i}^\ell
e_\ell,\alpha,\beta)$ mode of this component of the vector field,
whenever $\kappa_i>0$ or zero otherwise; $y_i$-component is the
$(v,\kappa,\alpha,\beta)$ mode of this component of the vector
field; $u_j$-component is the
$(v,\kappa,\alpha,\beta-\delta_{j}^\ell e_\ell)$ mode of this
component of the vector field, whenever $\beta_j>0$ or zero
otherwise; $w_j$-component is the $(v,\kappa,\alpha-\delta_{j}^\ell
e_\ell,\beta)$ mode of this component of the vector field, whenever
$\alpha_j>0$ or zero otherwise. Here $i=1,\dots,d$, $j=1,\dots,D$;
$\delta_{i}^\ell$ is the Kronecker delta, i.e.\ $\delta_{i}^\ell=1$
if $i=\ell$ and zero otherwise;
$e_\ell=(\delta_{1}^\ell,\dots,\delta_{\dim}^\ell)$, where $\dim=d$ or
$\dim=D$, is the unit vector in the $\ell-$th coordinate direction of
the standard basis. We have used Einstein's notation, i.e.\ we assume
the summation over repeated up and down indices.
\end{definition}


\begin{remark}\label{BResonant}
Notice that the operators $-i\partial_x$ and $\SS_\mu^\ast$ commute
on the space of vector fields considered, and that modes of a vector
field are the joint eigenvectors of $-i\partial_x$ and $S$, where
$S$ is the generator of the one-parameter group of scalings
$\SS_\mu^\ast$.  A mode of a vector field characterized by mode
indices $(v,\kappa,\alpha,\beta)$ is the joint eigenvector for
$(-i\partial_x,S)$ corresponding to the eigenvalues $(v,k)$, where
$k=\frac{|\alpha|+|\beta|}{2}+|\kappa|-1$. Sometimes, we will also
call $(v,k)$ with $v\in\VV$ and $k\in\{-1,-1/2,0,1/2,1,3/2,\dots\}$
the mode indices, refereing to all the mode indices
$(v,\kappa,\alpha,\beta)$, with
$\frac{|\alpha|+|\beta|}{2}+|\kappa|-1$ equal to $k$.
\end{remark}

\begin{remark}\label{BResonant}
Consider the Fourier-Taylor
expansion~\eqref{Fourier-Taylor-Hamiltonians} of a Hamiltonian. Each
mode of this Hamiltonian, i.e.\ each term in this expansion
characterized by a quadruplet
$(v,\kappa,\alpha,\beta)\in\VV\times\Nn_{0}^d\times \Nn_{0}^D\times
\Nn_{0}^D$, generates a single mode Hamiltonian vector field
characterized by $(v,\kappa,\alpha,\beta)$. The $(0,0,0,0)$ mode of any Hamiltonian vector field is zero.
\end{remark}

\begin{definition}
Let $I=\VV\times\Nn_{0}^d\times \Nn_{0}^D\times \Nn_{0}^D$. In the first renormalization step, let
$\iplus$ be the set of all $(v,\kappa,\alpha,\beta)\in I$ satisfying
$|\omega\cdot v|\le\sigma|v|$ when $v\ne 0$, or $\alpha=\beta$ when $v=0$,  or
$k:=\frac{|\alpha|+|\beta|}{2}+|\kappa|-1>0$, and let $\iminus$ be
the complement of $\iplus$ in $I$. Define $\Iplus$ and $\Iminus$ to
be the projection operators onto the subsets of {\it resonant} and
{\it nonresonant} vector fields spanned by modes with mode indices
in $\iplus$ and $\iminus$, respectively. The {\it resonant} and {\it
nonresonant} parts of a vector field $X\in\AA_{\rho}$ are defined as
$\Iplus X$ and $\Iminus X$, respectively. In addition, we define
$\mean_{k}$, for $k\in\{-1,-1/2,0,1/2,1,3/2,\dots\}$, as the
projection operator onto the space spanned by modes with mode
indices $(0,\kappa,\alpha,\beta)$, with
$\frac{|\alpha|+|\beta|}{2}+|\kappa|-1$ equal to $k$. The torus
averaging operator is then defined by $\mean=\sum_{k}\mean_{k}\,$.
\end{definition}


\begin{lemma}\label{BContractionL} If $0<\rho''\le\chi{\rho'}$, with
$\chi<1$, and $\mu<1$, then $\TT^\ast\SS_\mu^\ast$ defines a bounded
linear operator from $\Iplus\AA_{{\rho'}}(\VV)$ to
$\AA_{\rho''}(T\VV)$, with the property that
    \beal{BContraction}
\bigl\|\TT^\ast\SS_\mu^\ast\mean_{k} X\|_{\rho''} &\le&
\chi^{-1}\mu^{k}\|\mean_{k} X\|_{{\rho'}}\,,\nonumber\\
\bigl\|\TT^\ast\SS_\mu^\ast\Iplus(\Id-\mean)X\|_{\rho''} &\le&
\chi^{-1}\mu^{1/2} \|\Iplus(\Id-\mean)X\|_{{\rho'}}\,.
   \eea
\end{lemma}

\proof It suffices to verify the given bounds for a single mode
$X_{v,k}$ characterized by $(v,k)$. From the definitions of the
scaling maps it follows that
    \bel{BJAbound}
\|\TT^\ast S_\mu^\ast X_{v,k}\|_{\rho''} \le
\chi^{-1}e^A\|X_{v,k}\|_{{\rho'}}\,,
    \ee
where $A\le\rho''|T v_\spa|+\rho''|T
v_\spe|-{\rho'}|v_\spa|-{\rho'}|v_\spe|+k\ln(\mu)+(k+1)\ln(\chi^{-1}\rho''/{\rho'})\,.$
Setting $v=0$, and using that $\rho''\le\chi{\rho'}$ and $k\ge-1$,
yields the first bound in \eqref{BContraction}.

In order to prove the second bound, assume that $(v,k)$ belongs to
$\iplus$. Consider first the case $|v_\spa|\le\sigma|v|$,  with
$v\not=0$, and $k\le 0$. This inequality
 implies $|v_\spa|< 2\sigma|v_\spe|$, by using that $\sigma<1/2$,
and does not allow frequencies $v$ that satisfy $|v_\spe|\le L$ and
$|v_\spa|\ge\ell$, due to the condition \eqref{LdeltaCond}. Thus, we
must have $|v_\spe|>L$ by condition \eqref{emptyRectangle}.
Consequently,
    \bel{BAboundOne}
A\le-{\rho'}\Bigl(1-\frac{\rho''}{{\rho'}}\zeta\Bigr)|v_\spe|+k\ln(\mu)+(k+1)\ln\left(\frac{\chi^{-1}\rho''}{{\rho'}}\right)\le-{\rho'}(1-\zeta\eta)L-\ln(\mu)\,,
    \ee
where we have again used that $\rho''\le\chi{\rho'}$, $\mu<1$ and $
-1\le k\le 0$. The second bound in \eqref{BContraction} now follows
by using \eqref{LdeltaCond}.

Next, consider the case $k>0$, i.e $k\ge 1/2$. Since
$\rho''<\chi{\rho'}$, we have $A\le (1/2)\ln(\mu)$ and thus
    \be
\|\TT^\ast\SS_\mu^\ast X_{v,k}\|_{\rho''} \le \chi^{-1} \mu^{1/2}
\|X_{v,k}\|_{{\rho'}}\,.
    \ee
This also implies the second bound in \eqref{BContraction}. \qed


Let $J=\{(v,\kappa,\alpha,\beta)\in\iminus\,:\, |\Omega\cdot
V|>(1/2)|\omega\cdot v|\mbox{ for all }V=\beta-\alpha\in\Zz^D \mbox{ with
}|V|\le 2\}$, and
$$
\gamma^{(1)}=\max_{(v,\kappa,\alpha,\beta)\in
J}\left\{1,\frac{\sigma}{|\omega\cdot v+\Omega\cdot
V|}\right\}\,,\qquad \gamma^{(2)}=\max_{(v,\kappa,\alpha,\beta)\in
J}\left\{2,\frac{\sigma|v|}{|\omega\cdot v+\Omega\cdot V|}\right\}
$$


\begin{proposition}\label{small_denominators}
For all $0\ne v\in\VV$ and $V\in\Zz^D$ with $|V|\le 2$, if
$|\omega\cdot v|>\sigma|v|$ and $|\omega\cdot v+\Omega\cdot
V|\ne 0$, then  $|\omega\cdot v+\Omega\cdot
V|\ge\sigma/\gamma^{(1)}$ and $|\omega\cdot v+\Omega\cdot
V|\ge\sigma/\gamma^{(2)}|v|$.
\end{proposition}

\proof If $|\omega\cdot v|>\sigma|v|$ and $|\Omega\cdot
V|<(1/2)|\omega\cdot v|$ then we have $|\omega\cdot v+\Omega\cdot
V|\ge|\omega\cdot v|-|\Omega\cdot V|> (1/2)|\omega\cdot v|$ and thus
$|\omega\cdot v+\Omega\cdot V|\ge (\sigma/2)| v|$. Using conditions
\eqref{emptyRectangle} and \eqref{LdeltaCond} and $L\ge 1$, we also
obtain $|\omega\cdot v|>\sigma$ and thus $|\omega\cdot v+\Omega\cdot
V|\ge \sigma/2$, in that case.

The number of modes with $|\Omega\cdot V|>(1/2)|\omega\cdot
v|>(1/2)\sigma| v|$ and $|V|\le 2$ is finite. So, if $|\omega\cdot
v+\Omega\cdot V|\ne 0$ then $|\omega\cdot v+\Omega\cdot
V|\ge\sigma/\gamma^{(1)}$ and $|\omega\cdot v+\Omega\cdot
V|\ge\sigma/\gamma^{(2)}| v|$.\qed


Let $\wh K Z=[K,Z]$.
Let $c_\Omega>0$ be the smaller of the $\min_i \{|\Omega_i|\}$ and $\min_{i\ne j} \{|\Omega_i-\Omega_j|\}$.

\begin{proposition}\label{BKXBound} If $\rho>0$, and $Z\in\Iminus\AA'_\rho$,
then
    \bel{BKXBounds}
\|\Iminus[Z,K]\|_\rho\ge{\sigma\over\gamma}\|Z\|_\rho'\,,
    \ee
where
$\gamma=\frac{2(\rho+1)}{\rho}\max\{\gamma^{(1)},\gamma^{(2)}\}$.
\end{proposition}

\proof Assume that $(v,\kappa,\alpha,\beta)$ belongs to $\iminus$. Then, $k\le 0$ and either $|\omega\cdot v|>\sigma|v|$ with $v\ne 0$ or $\alpha\ne\beta$ with $v=0$.  Due to our choice of the
norm, it suffices to verify these bounds for a single mode
$Z_{v,\kappa,\alpha,\beta}$ of vector field $Z\in\Iminus\AA'_\rho$.
Notice that $\wh K Z_{v,\kappa,\alpha,\beta}=i(\omega\cdot\
v+\Omega\cdot(\beta-\alpha))Z_{v,\kappa,\alpha,\beta}$ and that $\wh
K$ commutes with $\Iminus$. If $\alpha\ne\beta$ with $v=0$, then $|\omega\cdot\
v+\Omega\cdot(\beta-\alpha)|>c_\Omega \ge\sigma/\gamma^{(1)}$.  The previous bounds and
\clmp(small_denominators) show that if $Z\in\Iminus\AA'_\rho$ and
$Y=[Z,K]$, then $\|Z\|_\rho\le \gamma^{(1)}/\sigma \|Y\|_\rho$ and
    \bel{BomegadotnuInv}
\sum_{j=1}^d\left\|{\partial_{x_j}}Z\right\|_\rho
\le{\gamma^{(2)}\over\sigma}\|Y\|_\rho\,,\qquad
\sum_{j=1}^{d}\left\|{\partial_{y_j}}Z\right\|_\rho
+\sum_{j=1}^{D}(\left\|{\partial_{u_j}}Z\right\|_\rho +\left\|{\partial_{w_j}}Z\right\|_\rho)\le{2\gamma^{(1)}\over
\rho\sigma}\|Y\|_\rho\,.
    \ee
As a result we obtain \eqref{BKXBounds}. \qed

This proposition allows us to apply the normal form theorem of~\cite{KK1},
which directly implies the following lemma. The positive number
$\varrho<1$ that appears in the following is fixed throughout the
paper. It has the meaning of the step-dependent domain parameter $\rho$ of the initial
vector fields that we would like to renormalize.

Consider the equations
    \bel{BElimEqu}
\Iminus(X+[Z,X])=0\,,\qquad \Iminus\UU_\ssX^\ast X=0\,.
    \ee

Constants that we call universal, do not depend on any renormalization parameters.

\begin{lemma}\label{BElim} Let $\rho>0$, and
let $\rho'=\rho-(\sigma/\gamma)\varrho$. There exist universal
constants $C_1$ and $C_2$ such that for every vector field
$X\in\AA'_\rho\,$, if
    \bel{BElimCond}
\|X-K\|'_\rho\le C_1(\sigma/\gamma)\,,\qquad \|\Iminus X\|_\rho\le
C_1(\sigma/\gamma)^2\,,\qquad
    \ee
then there exists a vector field $Z\in\Iminus\AA'_\rho$ and a change
of coordinates $\UU_\ssX:D_{\rho'}\to D_\rho\,$, solving equation
\eqref{BElimEqu}, such that the vector field $\UU_\ssX^\ast X$
belongs to $\AA_{\rho'}\,$, and
    \beal{BElimBounds}
\|Z\|'_\rho\,,\|\UU_\ssX-\id\|_{\rho'} &\le&
C_2(\gamma/\sigma)\|\Iminus X\|_\rho\,,\nonumber\\
\|\UU_\ssX^\ast X-X\|_{\rho'} &\le&
C_2(\rho-\rho')^{-1}(\gamma/\sigma)\|\Iminus X\|_\rho\,,\\
\|\UU_\ssX^\ast X-X-[Z,X]\|_{\rho'} &\le&
C_2(\rho-\rho')^{-3}(\gamma/\sigma)^3\|\Iminus
X\|_\rho^2\,.\nonumber
    \eea
The map $X\mapsto\UU_\ssX$ is continuous in the region defined by
\eqref{BElimCond}, and analytic in its interior.
\end{lemma}

By construction, the map $X\mapsto\UU_\ssX^\ast X$ is type preserving, in particular Hamiltonian vector fields remain Hamiltonian under this transformation (see the discussion at the end of Section 5 of~\cite{KK1}).

Define the operator $\Pp$ as the projection operator
$\Pp=\Ee_{-1}+\Ee_{0}^{\alpha=\beta}$ onto the space spanned by resonant
vector fields that expand under scaling. Since in this paper we have restricted our consideration to Hamiltonian vector fields, $\mean_{-1}$ is simply the zero operator. Let the restriction of $\RR$ to this subspace be denoted by $\LL$. Here, $\Ee_0=\Ee_{0}^{\alpha=\beta}+\Ee_{0}^{\alpha\ne\beta}$ is the decomposition of $\Ee_0$ into projection operators onto the spaces characterized by $\alpha=\beta$ and $\alpha\ne\beta$, respectively. We will also define $\meanp=\mean\Iplus=\mean-\mean\Iminus$. Notice that the terms $\mean\Iminus X$ are eliminated in the first renormalization step once and for all, i.e.\ after one renormalization step, we have $\meanp=\mean$. In the following, $\AA_\rho$ denotes the subspace of Hamiltonian vector fields.

\begin{theorem}\label{BRRBoundsT} There exist universal constants $C,R>0$, such
that the following holds, under the given assumptions on
$L,\ell,\eta,\zeta,\gamma$ and $\mu$. Let $B$ be the open ball in
$\AA_\rho(\VV)$, with $(\sigma/\gamma)\varrho<\rho\le\varrho$, of
radius $R(\sigma/\gamma)^2$, centered at $K$. Then $\RR$ is a
bounded analytic map from $B$ to
$\AA_{\eta\rho-\eta(\sigma/\gamma)\varrho} (T\VV)$, satisfying
$\|\LL^{-1}\|\le 1$ and
    \beal{BRRBounds}
\|(\Id-\meanp)\RR(X)\|_{\eta\rho-\eta(\sigma/\gamma)\varrho}
&\le&C\eta^{-2}(\gamma/\sigma)^2\mu^{1/2}
\|(\Id-\meanp)X\|_\rho\,,\nonumber\\
\|(\Id-\proj)\RR(X)\|_{\eta\rho-\eta(\sigma/\gamma)\varrho} &\le&
C\eta^{-2}(\gamma/\sigma)^2\mu^{1/2} \|(\Id-\proj)X\|_\rho\,,\\
\|\proj\RR(X)-\RR(\proj X)\|_{\eta\rho-\eta(\sigma/\gamma)\varrho}
&\le& C\eta^{-2}(\gamma/\sigma)^6
\|(\Id-\meanp)X\|_\rho \|(\Id-\proj)X\|_\rho\,.\nonumber
    \eea
\end{theorem}

\proof Let ${\rho'=\rho-(\sigma/\gamma)\varrho}$. There exists a
universal constant $R>0$, such that the conditions \eqref{BElimCond}
in \clml(BElim) hold, whenever $X$ belongs to the domain $B$,
defined by $\|X-K\|_\varrho<R(\sigma/\gamma)^2$.

By \clml(BContractionL), we have
    \beal{BRRBOne}
\|(\Id-\meanp)\RR(X)\|_{\eta\rho'}
=\eta^{-1}\|\TT^\ast\SS_\mu^\ast(\Id-\meanp)\UU_\ssX^\ast X\|_{\eta\rho'}\nonumber\\
\le \eta^{-2}\mu^{1/2} \bigl[\|(\Id-\meanp)X\|_{\rho'}
+\|(\Id-\meanp)(\UU_\ssX^\ast X-X)\|_{\rho'}\bigr]\,.
    \eea
Using the bound in \eqref{BElimBounds} on the norm of $\UU_\ssX^\ast
X-X$, together with the fact that $\Iminus\meanp=0$, we obtain the
first inequality in \eqref{BRRBounds}. Similarly,
\clml(BContractionL) implies that
    \bel{BRRBTwo}
\|\mean_k\RR(X)\|_{\eta\rho'}\le C_1\eta^{-2}\mu^k\bigl[\|\mean_k
X\|_{\rho'} +\|\mean_k(\UU_\ssX^\ast X-X)\|_{\rho'}\bigr]\,,
    \ee
for some universal constant $C_1>0$ and for all
$k=\frac{|\alpha|+|\beta|}{2}+|\kappa|-1\ge 1/2$. Summing over all
$k\ge 1/2$ to get a bound on
$\|(\meanp-\proj)\RR(X)\|_{\eta\rho'}\,$, and then adding
\eqref{BRRBOne}, yields
    \bel{}
\|(\Id-\proj)\RR(X)\|_{\eta\rho'} \le
C_1\eta^{-2}\mu^{1/2}\bigl[\|(\Id-\proj) X\|_{\rho'}
+\|(\Id-\proj)(\UU_\ssX^\ast X-X)\|_{\rho'}\bigr]\,,
    \ee
if $C_1$ is chosen sufficiently large. Using again the second bound
in \eqref{BElimBounds}, and the fact that $\Iminus\proj=0$ and
$\Iminus\meanp=0$, we obtain the second inequality in
\eqref{BRRBounds}.

By \clml(BContractionL), we also have a bound
    \beal{BRRBThreeA}
\|\proj\RR(X)-\RR(\proj X)\|_{\eta\rho'}
&=&\eta^{-1}\|\TT^\ast\SS_\mu^\ast\proj(\UU_\ssX^\ast
X-X)\|_{\rho'}\nonumber\\ &\le& \eta^{-2}
\|\proj(\UU_\ssX^\ast X-X)\|_{\rho'}\,.
    \eea
Using \clml(BElim), the norm on the right hand side of
\eqref{BRRBThreeA} can be estimated as follows:
    \bel{BRRBThreeB}
\|\proj(\UU_\ssX^\ast X-X)\|_{\rho'} \le
C_2(\gamma/\sigma)^6\|(\Id-\meanp)X\|_\rho^2
+\|\proj[Z,X]\|_{\rho'}\,,
    \ee
where $Z=\Iminus Z$ is the vector field described in
\eqref{BElimEqu}. We can check that for Hamiltonian vector fields $\proj[Z,\proj X]=0$,
and thus
    \beal{BRRBThreeC}
\|\proj[Z,X]\|_{\rho'} &=&\|\proj[Z,(\Id-\proj)X]\|_{\rho'} \le
C_3(\gamma/\sigma)\|Z\|'_\rho\|(\Id-\proj)X\|_\rho \nonumber\\&\le&
C_4(\gamma/\sigma)^2 \|(\Id-\meanp)X\|_\rho\|(\Id-\proj)X\|_\rho\,.
    \eea
Here, we have used the bound on $\|Z\|'_\rho$ from \clml(BElim).
Combining the last three equations yields the third inequality in
\eqref{BRRBounds}.

The bound concerning the restriction $\LL$ of $\RR$ to $\Pp\AA_\rho(\VV)$, is obvious if one notices that this restriction is linear operator  $\LL=\eta^{-1}\TT^\ast$.
 \qed

\section{Composed renormalization transformations}\label{BCOMPOSE}

We  express the Brjuno condition on $\omega$ (and thus on $\VV$) in
terms of the summability of the series of numbers
    \bel{BanDef}
a_n=\sum_{k=n}^\infty 2^{n-k}\Bigl[
2^{-k-\kappa}\ln(1/\Omega'_{k+\kappa})+(k+\kappa')^{-2}\Bigr]\,,\qquad
\Omega_n'=\min_{0<|\nu_\spe|<2^n}|\nu_\spa|\,,
    \ee
for all positive integers $n$. Here $\kappa,\kappa'>2$ are two
integer constants to be determined later.

It follows from the definition that $a_{n+1}/2<a_n<2a_{n+1}$, for
all $n\in\Nn$, and thus $a_{n+1}2^{n+1}/4<a_n2^{n}<a_{n+1}2^{n+1}$.
This makes the sequence $a_n2^{n}$ increasing and well-controllable.

Define the scaling parameters
    \bel{BlambdaEtc}
\eta_{n}=\left(\frac{A_{n+1}}{A_n}\right)^{d-1\over d}\,,\quad
\zeta_n=\left({A_{n+1}\over A_n}\right)^{1\over d}\,,\qquad \mbox{
where } \quad A_n=\sum_{k=n}^\infty a_k\,.
    \ee
for all positive integers $n$. Since $\{a_n\}$ is a summable
sequence of positive numbers, the sequence $\{A_n\}$ is decreasing
and converging to zero. Define recursively
$\lambda_n=\eta_n\lambda_{n-1}$, with $\lambda_0=1$. These
definitions imply that $\eta_n<\zeta_n<1$, for $d\ge 2$.

These parameters are used to define the scaling maps $T_n$  and
$P_n=T_n\cdots T_1$ at each renormalization step
    \be
T_n(x)=\eta_n^{-1}x_\spa+\zeta_n x_\spe \,,\qquad
P_n(x)=\lambda_n^{-1}x_\spa+(\prod_{i=1}^{n}\zeta_i) x_\spe\,.
    \ee
We also define $T_0=P_0$ as the identity maps. Notice that the
determinants $|T_n|=|P_n|=1$, for all $n\in\Nn$, by the choice of
the scaling parameters.

The geometric data $\VV$, $L$ and $\ell$ used in the $n$-th
renormalization step are
    \bel{BVVLdeltaDef}
\VV_{n-1}=P_{n-1}\VV_0\,,\qquad
L_{n-1}=2^{n+\kappa}\prod_{i=1}^{n-1}\zeta_i\,,\qquad
\ell_{n-1}=\lambda_{n-1}^{-1}e^{-a_n2^{n+\kappa}}\,.
    \ee
These definitions immediately imply \eqref{LdeltaCond}.

Let $C_\Omega=\max\{(\min
|\Omega\cdot V|)^{-1},\CC^{-1}\max
|\Omega\cdot V|^{\tau+1}\}$, where the $\min$ and $\max$ are taken over all $V\in\Zz^d$ with $0<|V|\le 2$.

\begin{definition} We also define
    \be
\sigma_n=(2\lambda_{n-1}L_{n-1}C_\Omega\kappa')^{-1}e^{-a_n2^{n+\kappa}} =\frac{A_1}{2C_\Omega\kappa'A_n}2^{-(n+\kappa)}e^{-a_n2^{n+\kappa}}.
    \ee
\end{definition}
This immediately implies $\sigma_n>0$ and $2\sigma_nL_{n-1}\le\ell_{n-1}$, for all $n\in\Nn$, if $\kappa'>b^{-1}$.

\begin{proposition}\label{summability}
For any fixed $\kappa'>0$ and sufficiently large $\kappa$, one has
$\sum_{n=1}^\infty\sigma_n<1/2$.
\end{proposition}

\proof Notice that
$$\sigma_n<\frac{A_1}{2C_\Omega\kappa'a_n}2^{-(n+\kappa)}e^{-a_n2^{n+\kappa}}.$$
Since $\{a_n2^n\}$ is a growing sequence, the sequence
$\{\sigma_n\}$ is decreasing.  Notice also that for a fixed
$\kappa'$, and sufficiently large $\kappa$, we have
$2^{n+\kappa}a_n\ge 2^{n+\kappa}(n+\kappa')^{-2}\ge c'2^\kappa n$,
for some constant $c'>0$ depending only on $\kappa'$. This makes the
sum $\sum_{n=1}^\infty\sigma_n$ finite, and by choosing
$\kappa$ sufficiently large, we can make this sum smaller than $1/2$.\qed


\begin{proposition}\label{Bindeed} If $v\in\VV_{n-1}$ is nonzero, then
either $|v_\spa|\ge\ell_{n-1}$ or $|v_\spe|> L_{n-1}\,$.
\end{proposition}

\proof Assume that $v\in\VV_{n-1}$ satisfies $0<|v_\spe|\le
L_{n-1}$. Then the corresponding lattice point $\nu=P_{n-1}^{-1}v$
in $\VV_0$ satisfies $|\nu_\spe|\le
(\prod_{i=1}^{n-1}\zeta_i)^{-1}L_{n-1}= 2^{n+\kappa}$, and thus
$|\nu_\spa|\ge\Omega'_{n+\kappa}$ by \eqref{BanDef}. Since we have
$\Omega'_{n+\kappa}>e^{-a_n2^{n+\kappa}}$, this yields
    \bel{Bindeed}
|v_\spa|=\lambda_{n-1}^{-1}|\nu_\spa|
\ge\lambda_{n-1}^{-1}\Omega'_{n+\kappa}
>\lambda_{n-1}^{-1}e^{-a_n2^{n+\kappa}}=\ell_{n-1}\,,
    \ee
as claimed. \qed

\begin{definition}
Let $\Omega_{n-1}:=\lambda_{n-1}^{-1} \Omega$, for $n\in\Nn$. Let
also
$$J_{n-1}^{^-}=\{(v,\kappa,\alpha,\beta)\in\iminus(\VV_{n-1})\,:\,
|\omega\cdot v|<2|\Omega_{n-1}\cdot V|\mbox{ for all
$V=\beta-\alpha$ with }|V|\le 2\},$$  and
$$
\gamma_{n}:=\frac{8}{\varrho\lambda_{n-1}}\max_{(v,\kappa,\alpha,\beta)\in
J_{n-1}^{^-}}\left\{2,\frac{\sigma_n\max\{1,|v|\}}{|\omega\cdot
v+\Omega_{n-1}\cdot V|}\right\}\,.
$$
\end{definition}

\begin{definition}\label{rhon}
The domain of analyticity of functions that are going to be
renormalized at the $n$-th step is determined by the parameter
    \be
    \rho_{n-1}=\varrho\lambda_{n-1}\left(1-\sum_{k=1}^{n-1}\frac{\sigma_k}{\lambda_{k-1}\gamma_k}\right).
    \ee
\end{definition}
The numbers $\rho_{n}$ are positive due to \clmp(summability) and
the fact that $\gamma_k\lambda_{k-1}>1$. Moreover
$\rho_{n-1}>\varrho\lambda_{n-1}/2$.


\begin{proposition}\label{Gamman}
If $\Omega$ is Diophantine with respect to $\omega$, then there
exists a universal constant $\xi>0$,  such that for all $n\in\Nn$,
    \be
\gamma_n<\xi\varrho^{-1}\max\{(\min
|\Omega\cdot V|)^{-1},\CC^{-1}\max
|\Omega\cdot V|^{\tau+1}\}\lambda_{n-1}^{-(\tau+1)}(\sigma_n\prod_{i=1}^{n-1}\zeta_i)^{-\tau}\,.
    \ee
Here, the $\min$ and $\max$ are taken over all $V\in\Zz^d$ with $0<|V|\le 2$.
\end{proposition}

\proof
Recall that $\nu=P_{n-1}^{-1}v$. By definition, we have
    \bea
\frac{\varrho\lambda_{n-1}}{8}\gamma_{n}&<&\max_{(v,\kappa,\alpha,\beta)\in
J_{n-1}^{^-}}\left\{2,\frac{\sigma_n\max\{1,|v|\}}{|\omega\cdot
v+\Omega_{n-1}\cdot V|}\right\}\nonumber\\ &\le& \max_{(v,\kappa,\alpha,\beta)\in
J_{n-1}^{^-}}\left\{2,\frac{\max\{\sigma_n,|\omega\cdot v|\}}{|\omega\cdot
v+\Omega_{n-1}\cdot V|}\right\}\nonumber\\
&<&\max_{(v,\kappa,\alpha,\beta)\in
J_{n-1}^{^-}}\left\{2,\frac{\max\{\sigma_n,2\max\,|\Omega_{n-1}\cdot
V|\}}{\lambda_{n-1}^{-1}|\omega\cdot\nu+\Omega\cdot
V|}\right\}\nonumber\\ &=&\max_{(v,\kappa,\alpha,\beta)\in
J_{n-1}^{^-}}\left\{2,\frac{\max\{\sigma_n\lambda_{n-1},2\max |\Omega\cdot
V|\}}{|\omega\cdot\nu+\Omega\cdot V|}\right\}\,,
    \eea
and thus
 \be \frac{\varrho\lambda_{n-1}}{8}\gamma_n<\max\limits_{|\nu|<2\max |\Omega\cdot
V|(\lambda_{n-1}(\prod_{i=1}^{n-1}\zeta_i)\sigma_n)^{-1}}\left\{2,\frac{\max\{\sigma_n\lambda_{n-1},2\max
|\Omega\cdot V|\}}{|\omega\cdot\nu+\Omega\cdot V|}\right\}.
    \ee
Here, we have used that
    \be
    |v|=|v_\spa|+|v_\spe|=\lambda_{n-1}^{-1}|\nu_\spa|+(\prod_{i=1}^{n-1}\zeta_i)|\nu_\spe|\ge (\prod_{i=1}^{n-1}\zeta_i)|\nu|,
    \ee
and that the condition $\sigma_n|v|\le|v_\spa|<2\max
|\Omega_{n-1}\cdot V|$ implies the following inequality $|\nu|<2\max
|\Omega\cdot
V|(\lambda_{n-1}(\prod_{i=1}^{n-1}\zeta_i)\sigma_n)^{-1}$.

Using the Diophantine with respect to $\omega$ condition on
$\Omega$, we obtain that
    \be
\gamma_n<\xi\varrho^{-1}\max\{(\min
|\Omega\cdot V|)^{-1},\CC^{-1}\max
|\Omega\cdot V|^{\tau+1}\}\lambda_{n-1}^{-(\tau+1)}(\sigma_n\prod_{i=1}^{n-1}\zeta_i)^{-\tau},
    \ee
where $\xi$ is a universal constant and the $\min$ and $\max$ are taken over all $V\in\Zz^d$ with $0<|V|\le 2$. \qed

The maximum that appears in this proposition defines the function $b$ mentioned earlier in this section.

\begin{definition}
Let
    \bel{BmuDef} \mu_n:=\exp\Bigl\{- {\varrho\over
3}\lambda_{n-1}(1-\zeta_n\eta_n)L_{n-1}\Bigr\}
=\exp\Bigl\{-{\varrho\over 3A_1}a_n2^{n+\kappa}\}\,,\qquad n\ge 1\,.
    \ee
\end{definition}

\begin{proposition}\label{BmuBoundP} For all $n\ge 1$,
$\mu_{n+1}<\mu_n<\mu_{n+1}^{1/4}$. Furthermore, given $C,N>0$, if
$\kappa'$ and then $\kappa$ are chosen sufficiently large, then for
all $n\ge 1$,
    \bel{BmuBound}
\mu_n\le Ce^{-N2^{n+\kappa}a_n}\,,\qquad \mu_n\le C2^{-Nn}\,,\qquad
\mu_n\le C\left(\frac{A_n}{A_1}\right)^N.
    \ee
\end{proposition}


\proof Let $C>0$ and $N>0$ be arbitrary. Since
$a_{n+1}/2<a_n<2a_{n+1}$, for all $n\in\Nn$, we have
$a_{n+1}2^{n+1}/4<a_n2^{n}<a_{n+1}2^{n+1}$, and thus
 $\mu_{n+1}<\mu_n<\mu_{n+1}^{1/4}$. By
choosing $\kappa$ and $\kappa'$ sufficiently large, we have
$1/(3A_1)\ge N$. Increasing them further, if needed, we obtain the
first bound. Keeping $\kappa'$ fixed, and increasing $\kappa$
further, if necessary, we obtain the second two bounds in
\eqref{BmuBound} by using that $2^{n+\kappa}a_n\ge
2^{n+\kappa}(n+\kappa')^{-2}\ge c'2^\kappa n$, for some positive
constant $c'$ depending only on $\kappa'$. The same inequality,
together with $A_n/A_1>
a_n/A_1>(n+\kappa')^{-2}/A_1>e^{-c'n/A_1}>e^{-2^n/A_1}>e^{-2^{n+\kappa}/(NA_1)}$,
where the last inequality is valid for sufficiently large $\kappa$,
implies the third bound in \eqref{BmuBound}. \qed

\clmp(BmuBoundP) directly implies the following.

\begin{corollary}\label{corollary} Given any $C,N>0$, if
$\kappa'$ and then $\kappa$ are chosen sufficiently large, then for all
$n\ge 1$,
 \bel{BmuBound}
\mu_n\le C\sigma_n^N\,,\qquad \mu_n\le C\eta_n^N\,,\qquad \mu_n\le
C\lambda_n^N\le C\eta_n^N\le C\zeta_n^N \,,\qquad\mu_n\le
C\gamma_n^{-N}\,.
    \ee
\end{corollary}

\proof To obtain the first bound, we have also used the fact that for any $C,N>0$ we have $\mu_n<\kappa'^{-1}$,
for sufficiently large $\kappa$.\qed

At this point we have verified all of the assumptions made in
Section~\ref{BSINGLE}.  We can now apply \clmt(BRRBoundsT) to the
$n$-th step renormalization operator $\RR_n$, defined, by the
parameters introduced above, from a subset in
$\AA_{\rho_{n-1}}(\VV_{n-1})$ to $\AA_{\rho_n}(\VV_n)$.  Denote by
$\LL_n$ the corresponding linear operator from
$\proj\AA_{\rho_{n-1}}(\VV_{n-1})$ to $\proj\AA_{\rho_n}(\VV_n)$.


Define $\AA_{\rho_n,n}=\AA_{\rho_n}(\VV_n)$, for all non-negative
integers $n$.  From \clmt(BRRBoundsT) we immediately obtain the following theorem.
To simplify the notation, we will write down explicitly the
dependence of the norm on the lattice $\VV_n$.

\begin{theorem}\label{BRRnBoundsT} There exist constants $C,r>0$, such that the $n$-th step renormalization operator
$\RR_n$ is a bounded analytic map from an open ball $B_{n-1}$ in
$\AA_{\rho_{n-1},n-1}$ of radius $r(\sigma_n/\gamma_n)^2$, centered
at $K$, into $\AA_{\rho_n,n}\,$, satisfying $\|\LL_n^{-1}\|\le 1$
and
    \beal{BRRnBounds}
\|(\Id-\meanp)\RR_n(X)\|_{\rho_n} &\le&
C\eta_n^{-2}(\gamma_n/\sigma_n)^{2}\mu_n^{1/2}
\|(\Id-\meanp)X\|_{\rho_{n-1}}\,,\nonumber\\
\|(\Id-\proj)\RR_n(X)\|_{\rho_n} &\le&
C\eta_n^{-2}(\gamma_n/\sigma_n)^{2}\mu_n^{1/2} \|(\Id-\proj)X\|_{\rho_{n-1}}\,,\\
\|\proj\RR_n(X)-\RR_n(\proj X)\|_{\rho_n}
&\le&C\eta_n^{-2}(\gamma_n/\sigma_n)^{6}
\|(\Id-\meanp)X\|_{\rho_{n-1}}\|(\Id-\proj)X\|_{\rho_{n-1}}\,.\nonumber
    \eea
\end{theorem}


\smallskip
In what follows, a domain $\DD_{n-1}$ for $\RR_n$ is a subset of the
ball $B_{n-1}$ described in \clmt(BRRnBoundsT), that is open in
$\AA_{\rho_{n-1},n-1}$ and contains the vector field $K$. Given a
domain $\DD_{n-1}$ for each $\RR_n\,$, the domain $\wt\DD_n$ of the
``composed'' renormalization operator
$\wt\RR_{n+1}=\RR_{n+1}\circ\wt\RR_{n}$, for $n\in\Nn$, with
$\wt\RR_1=\RR_1$, is defined recursively as the set of all vector
fields in the domain of $\wt\RR_n$ that are mapped under $\wt\RR_n$
into the domain $\DD_n$ of $\RR_{n+1}\,$. By \clmt(BRRnBoundsT),
these domains are open and non-empty, and the transformations
$\wt\RR_n$ are analytic on $\wt\DD_{n-1}$.

\begin{theorem}\label{BRRnComposeT} Let $0<\epsilon<1/2$. If $\kappa'$ and
then $\kappa$ are chosen sufficiently large, then there exist a
sequence of domains $\{\DD_{n-1}\}$ for the renormalization
transformations $\{\RR_n\}$, such that the set $\WW=\cap_n\wt\DD_n$
is the graph of an analytic function
$W:(\Id-\proj)\DD_0\to\proj\DD_0\,$, satisfying $W(0)=K$. For every $X\in\WW$, if $n\ge 1$ and
$\psi_n=\prod_{i=1}^n\mu_i\,$, then
    \beal{BRRnCompose}
 \bigl\|\wt\RR_n(X)-K_n\bigr\|_{\rho_n}
&\le&\psi_n^{1/2-\epsilon}\|(\Id-\proj)X\|_\varrho\,,\nonumber\\
\bigl\|(\Id-\mean)\wt\RR_n(X)\bigr\|_{\rho_n}
&\le&\psi_n^{1/2-\epsilon}\|(\Id-\mean)X\|_\varrho\,.
%\\ \bigl\|\proj[\wt\RR_n(X)-K_n]\bigr\|_{\rho_n}
%&\le&\psi_n\|(\Id-\proj)X\|_\varrho^2\,.\nonumber
    \eea
\end{theorem}
\proof Our goal is to apply the stable manifold theorem of~\cite{KK1}. To
do so, we first rescale our transformations $\RR_n\,$. Let
$r_n=r_{-1}(\sigma_{n+1}/\gamma_{n+1})^2$ for every non-negative
integer $n$, with $r_{-1}>0$ smaller than half the constant $r$ from
\clmt(BRRnBoundsT).

Define the transformations $R_n\,$  by
    \bel{BRnDef}
R_n(Z)=r_n^{-1}\bigl[\RR_n(K+r_{n-1}Z)-K\bigr]\,,\qquad n\in\Nn\,.
    \ee

Define the projection operators $\mean_n=\meanp$, $\proj_n=\proj$, for $n\in\Nn_0$.
The restriction $R_n\proj_{n-1}$ defines a linear map from
$\proj_{n-1}\AA_{\rho_{n-1},n-1}$ to $\proj_n\AA_{\rho_n,n}\,$, which will
be denoted by $L_n\,$. By \clmt(BRRnBoundsT), $R_n$ is analytic and
bounded on the ball $\|Z\|_\varrho<2$, and satisfies
    \beal{BRBounds}
\|(\Id-\mean_n)R_n(Z)\|_{\rho_{n}}
&\le&\eps_n\|(\Id-\mean_{n-1})Z\|_{\rho_{n-1}}\,,\nonumber\\
\|(\Id-\proj_n)R_n(Z)\|_{\rho_{n}}
&\le&\vartheta_n\|(\Id-\proj_{n-1})Z\|_{\rho_{n-1}}\,,\\ \|\proj_n
R_n(Z)-R_n(\proj_{n-1} Z)\|_{\rho_{n}}
&\le&\varphi_n\|(\Id-\mean_{n-1})Z\|_{\rho_{n-1}}\|(\Id-\proj_{n-1})Z\|_{\rho_{n-1}}\,,\nonumber
    \eea
where
    \bel{BetpOne}
\eps_n=\vartheta_n=C\eta_n^{-2}(\gamma_n/\sigma_n)^{2}(\gamma_{n+1}/\sigma_{n+1})^2\mu_n^{1/2}\mbox{
and }
\varphi_n=C\eta_n^{-2}(\gamma_n/\sigma_n)^{6}(\gamma_{n+1}/\sigma_{n+1})^2\,.\nonumber
    \ee
Here, $C\ge 1$ is a constant that may depend on $\varrho$, but not
on any other renormalization parameters. In addition, we have
$\|L_n^{-1}\|<1/4$. We will restrict $R_n$ to the domain
$D_{n-1}\subset\AA_{\rho_{n-1},n-1}\,$, defined by
    \bel{BRnDomainDef}
\|\proj_{n-1} Z\|_{\rho_{n-1}}<1\,,\qquad
\|(\Id-\proj_{n-1})Z\|_{\rho_{n-1}}<\delta_{n-1}\,,\qquad
\|(\Id-\mean_{n-1})Z\|_{\rho_{n-1}}<\delta_{n-1}\,,
    \ee
where $\delta_{n-1}=(6\varphi_n)^{-1}$. By \clmc(corollary), for any $0<\epsilon<1/2$, if
$\kappa'$ and $\kappa$ are chosen sufficiently large, then
$\eps_n\le\mu_{1/2-\epsilon}\le\eps=3/16$, $\vartheta_n\le\mu_{1/2-\epsilon}\le\vartheta=1/4$, and
$$C\eta_{n-1}^2\eta_n^{-4}(\gamma_{n-1}/\sigma_{n-1})^{-6}(\gamma_{n}/\sigma_{n})^{6}(\gamma_{n+1}/\sigma_{n+1})^{4}\mu_n^{1/2}\le
1,$$ for all positive integers $n$ .  The latter inequality implies
    \bel{BetpTwo}
\eps_n\delta_{n-1}\le\delta_n\,,\qquad
\vartheta_n\delta_{n-1}\le\delta_n\,,
    \ee
for all $n\ge 1$. Since, $\delta_{n-1}\varphi_{n}=1/6<\eps$, the last bound in~\eqref{BRBounds} implies
    \be
\|\proj_n R_n(Z)-R_n(\proj_{n-1} Z)\|_{\rho_{n}}
\le \eps\|(\Id-\mean_{n-1})Z\|_{\rho_{n-1}}.
    \ee

The hypotheses of the stable manifold theorem (Theorem 6.1)
of~\cite{KK1} are now verified, with $\vartheta=1/4$ and
$\eps=(1-\vartheta)/4$, and the conclusions of this theorem
imply the statements in \clmt(BRRnComposeT). \qed


\section{Lower-dimensional invariant tori}\label{BTORI}
%%%%%%%%%%%%%%%%%%%%%%%

We say that a vector field $X\in\AA_\rho$ has an invariant $d$-torus
with frequency vector $\omega$ if there is a continuous embedding
$\Gamma$ from $D_0=\Tt^d\times\{0\}$ into $D_\rho$, such that for
all $t\in\Rr$,
    \be
\Phi_\ssX^t\circ\Gamma=\Gamma\circ\Phi_\omega^t,
    \ee
where $\Phi_\omega$ is the flow of the vector field $(\omega,0)$.
Here $0$ is the zero vector in $\Rr^{d+2D}$.

Consider a one-step renormalization operator $\RR$ and a vector
field $X$ in the domain of $\RR$. If $F$ is any map from $D_0$ into
the domain of $\Lambda_\ssX=\UU_\ssX\circ\tilde\TT\,$, where
$\tilde\TT=\SS_\mu\circ\TT$, define the map
    \bel{BMMXDef}
\MM_\ssX(F)=\Lambda_\ssX\circ F\circ\tilde\TT\,.
    \ee
Formally, if $\Gamma_{\RR(X)}$ is an invariant torus for $\RR(X)$,
then an invariant torus for $X$ is given by
$\Gamma_X=\MM_\ssX(\Gamma_{\RR(X)})$. This can be seen easily from
the identity $\Lambda_\ssX\circ\Phi_{\RR(X)}^{\eta
t}=\Phi_\ssX^t\circ\Lambda_\ssX\,$.

Denote by $\AA_0(\VV)$ the Banach space of continuous functions
$F:D_0\to\complex^{2d+{2D}}\,$, with frequency module in $\VV$, for
which the norm $\|F\|_{0,{\VV}}=\sum_v\|F_v\|$ is finite, where
$\{F_v\}$ are the Fourier coefficients of $F$.

Consider now a fixed but arbitrary vector field $X$ on the stable
manifold $\WW$ described in \clmt(BRRnComposeT). Let $X_0=X$, and
$X_n=\RR_n(X_{n-1})$ for $n\ge 1$. In order to simplify notation, we
will write $\UU_k$ and $\MM_{k+1}$ in place of $\UU_{X_k}$ and
$\MM_{X_k}\,$, respectively. Our goal is to construct an appropriate
sequence of functions $\Gamma_k\in\AA_0(\VV_k)$, satisfying
    \bel{BGammanSeq}
\Gamma_{n-1}=\MM_n(\Gamma_n)
=\Lambda_n\circ\Gamma_n\circ\tilde\TT_{n}^{-1}\,,\qquad
\Lambda_n=\UU_{n-1}\circ\tilde\TT_{n}\,,
    \ee
where $\tilde\TT_{n}=\TT_n\circ\SS_{\mu_n}$, for all $n>0$. Then we
will show that $\Gamma_0$ is an invariant torus for $X_0\,$.

For every $n\ge 0$, define $\BB_n$ to be the vector space
$\AA_0(\VV_n)$, equipped with the norm
    \bel{BnNorm}
\|f\|'_n=r_n^{-1}\|f\|_{0,\VV_n}
=r_n^{-1}\sum_{v\in\VV_n}\|f_v\|\,,\qquad
r_n=\frac{\varrho}{4^n}\lambda_n\,.
    \ee
Denote by $B_n$ the unit ball in $\id+\BB_n\,$, centered at the
identity function $\id$, and by $B_n/2$ the ball of radius $1/2$ in
the same space.

\begin{proposition}\label{BMMnContracts} If $\kappa'$ and
then $\kappa$ are chosen sufficiently large, then there exists an
open neighborhood $B$ of $K$ in $\AA_\varrho\,$, such that for every
$X\in\WW\cap B$, and for every $n\ge 1$, the map $\MM_n$ is well
defined and analytic, as a function from $B_n$ to $\BB_{n-1}\,$.
Furthermore, $\MM_n$ takes values in $B_{n-1}/2$, and
$\|D\MM_n(F)\|\le{1/3}\,$, for all $F\in B_n\,$.
\end{proposition}

\proof Clearly, $\MM_n$ is well-defined in some open neighborhood of
$\id$ in $\BB_n\,$, and
    \bel{BMMnExpr}
\MM_n(F)=\id+g+(\UU_{n-1}-\id)\circ(\id+g)\,,\qquad
g=\tilde\TT_{n}\circ f\circ\tilde\TT_{n}^{-1}\,,
    \ee
where $f=F-\id$. In order to estimate the norm of $\UU_{n-1}-\id$,
we can apply \clml(BElim), with $\rho'_{n-1}$ equal to
$\rho_{n-1}-(\sigma_{n-1}/\gamma_{n-1})\varrho\,$.  By \clml(BElim)
and \clmt(BRRnComposeT), there exist a constant $C>0$, such that
    \beal{BMMnCOne}
     \|\UU_{n-1}-\id\|_{\rho_{n-1}'} &\le& C(\gamma_n/\sigma_n)\|\Iminus
X_{n-1}\|_{\rho_{n-1}} \le C(\gamma_n/\sigma_n)\psi_{n-1}^{1/2-\epsilon}
\|(\Id-\meanp)X\|_\varrho\nonumber\\
&\le&\psi_{n-1}^{1/2-2\epsilon}\|(\Id-\meanp)X\|_\varrho
\le\psi_n^{1/9}\,,
    \eea
for some small $\epsilon>0$, for all $n>1$, and for all $X\in\WW\cap B$.
Here, we have used \clmp(BmuBoundP) and assumed that $\kappa'$ and
then $\kappa$ have been chosen sufficiently large, and that the
neighborhood $B$ of $K$ has been chosen sufficiently small
(depending on $\kappa'$ and $\kappa$). Though all steps in
\eqref{BMMnCOne} cannot be carried through if $n=1$, the final
estimate is also valid in that case.

The composition with $\id+g$ in Equation \eqref{BMMnExpr} is
controlled by \clmp(BTrivial), using that
$\|g\|_{0,\VV_{n-1}}\le\eta_n^{-1}r_n\|f\|'_n$ is less than
$\rho_{n-1}'/2$, since  we assume that $F\in B_n$.  By using that
$r_n/r_{n-1}=\eta_n/4\,$, we obtain
$\|g\|'_{n-1}\le\eta_n^{-1}\eta_n/4\le 1/4\,$. From \eqref{BMMnCOne}
we obtain $\|\UU_{n-1}-\id\|_{n-1}'\le r_{n-1}\psi_n^{1/16}\le 1/2$,
if $\kappa'$ and $\kappa$ have been chosen sufficiently large. These
estimates show that $\MM_{n-1}$ maps $B_n$ into $B_{n-1}/2$.

Now, we obtain a bound on the norm of the derivative map
    \be
D\MM_n(F)\bar f=\bar g+D(\UU_{n-1}-I)\circ(I+g) \bar g,
    \ee
where $\bar g=\tilde\TT_{n}\circ\bar f\circ\tilde\TT_{n}^{-1}$.
Since $\|g\|_{0,\VV_{n-1}}\le\rho_{n-1}/2$, and
    \be
\|D(\UU_{n-1}-I)\|_{\rho_{n-1}'/2}\le
\frac{2}{\rho_{n-1}'}\|\UU_{n-1}-I\|_{\rho_{n-1}'}
    \ee
we obtain a bound on this derivative norm analogous to
\eqref{BMMnCOne}. This, together with the fact that the inclusion
map from $B_n$ into $B_{n-1}$ is bounded in norm by $\eta_n/4$,
shows that $\|D\MM_n(F)\|\le{1/3}\,$, for all $n\ge 1$, and for all
$F\in B_n\,$. \qed

Below, we will make use of the following estimate on the difference
between the flow for $X$ and the flow for the constant vector field
$(\omega,0)$.

\begin{proposition}\label{BBasicFlowBoundP} {\rm\bf ~\cite{KK1}}  Let $\tau$ be a positive real number
and $X$ a vector field in $\AA_\varrho\,$, such that
$\tau\|X-\omega\|_\varrho<r<\varrho$. Then for all times  $t$ in the
interval $[-\tau,\tau]$,
    \bel{BBasicFlowBound}
\|\Phi_X^t-\Phi_\omega^t\|_{\varrho-r} \le\|t(X-\omega)\|_\varrho\,.
    \ee
\end{proposition}

Here and in what follows, we write $\omega$ for the constant vector
field $(\omega,0)$, in order to simplify the notation.

Let $\Phi_n$ be the flow for the vector field $X_n$. In order to
prove that a solution to \eqref{BGammanSeq} yields an invariant
torus $\Gamma_0$ for $X$, we will use the identity
    \bel{BMMPhi}
\Phi_{n-1}^t\circ\MM_n(F)\circ\Phi_\omega^{-t}
=\MM_n\bigl(\Phi_n^{\eta_n t}\circ F\circ\Phi_\omega^{-\eta_n
t}\bigr)\,,
    \ee
which follows from the relation described after \eqref{BMMXDef},
between the flow for a vector field and the flow for the
corresponding renormalized vector field. This requires an estimate
of the following type.

\begin{proposition}\label{BPhiFPsi} Under the same assumptions as in
\clmp(BMMnContracts), there exists an open neighborhood $B$ of $K$
in $\AA_\varrho\,$, such that for every $X\in\WW\cap B$, and for
every $n\ge 1$, the function $\Phi_n^s\circ F\circ\Phi_\omega^{-s}$
belongs to $B_n\,$, whenever $F\in B_n/2$ and
$|s|\le\lambda_n/(4\max\{\|\Omega\|,1\})\,$.
\end{proposition}

\proof We will use the following easily verifiable identity
    \bel{BPhiFPsiOne}
\Phi_n^s\circ F\circ\Phi_\omega^{-s} =\id+f\circ\Phi_\omega^{-s}
+\bigl[\Phi_n^s\circ\Phi_\omega^{-s}-\id\bigr]
\circ\bigl(\id+f\circ\Phi_\omega^{-s}\bigr)\,.
    \ee
Since $\|f\|_{n}'\le 1/2$, and
$\|f\circ\Phi_\omega^{-s}\|_{0,\VV_n}=\|f\|_{0,\VV_n}\,$, we have
$\|f\circ\Phi_\omega^{-s}\|_{0,\VV_n}\le r_n/2\,$. The composition
in~\eqref{BPhiFPsiOne} is well-defined since $r_n/2<\rho_n$ (see
Definition \ref{rhon}).

By \clmp(BBasicFlowBoundP), we have the bound
    \bel{BPhiFPsiTwo}
\bigl\|\Phi_n^s\circ\Phi_\omega^{-s}-\id\bigr\|_{r_n/2}
\le\|s(X_n-\omega)\|_{r_n}\le\|s(X_n-K_n)\|_{r_n}
+\|s(K-\omega)\|_{r_n}   \,,
    \ee
provided that the right hand side of this inequality is less than
$r_n/2$. Since $X\in\WW$, using \clmt(BRRnComposeT) we can bound the
first term as $\|s(X_n-K_n)\|_{r_n}\le
|s|\psi_n^{1/2-\epsilon}\|X-K\|_\varrho$ which can be made smaller than
$r_n/4$, for any $n$, if $\|X-K\|_\varrho$ is chosen sufficiently
small. The second term can be bounded as $\|s(K_n-\omega)\|_{r_n}\le
|s|\|\Omega_n\|r_n=|s|\|\Omega\|\lambda_n^{-1}r_n\le r_n/4$. Thus,
the sum on the right hand side of inequality~\eqref{BPhiFPsiTwo} is
indeed smaller than $r_n/2$. Finally,
    \be
\|\Phi_n^s\circ F\circ\Phi_\omega^{-s}-\id\|_{n-1}'\le
r_{n-1}^{-1}r_n\le \eta_n/4\,,
    \ee
which proves the claim.  \qed

Now we are ready to construct invariant tori. A function $f$ defined
on $\WW$ is said to be analytic if $f\circ W$ is analytic on the
domain of $W$.

\begin{theorem}\label{BGammaLimits} Under the same assumptions as in
\clmp(BMMnContracts), there exists an open neighborhood $B$ of $K$
in $\AA_\varrho\,$, such that the following holds. Given any
$X\in\WW\cap B$, and any sequence of functions $F_k\in B_k\,$,
define
    \bel{BGammanmDef}
\Gamma_{n,k}=\bigl(\MM_{n+1}\circ\ldots
\circ\MM_k\bigr)(F_k)\,,\qquad 0\le n<k\,.
    \ee
Then the limits $\Gamma_n=\lim_{k\to\infty}\Gamma_{n,k}$ exist in
$\BB_n\,$, are independent of the choice of $F_0,F_1,\ldots$, and
satisfy the identities \eqref{BGammanSeq}. Furthermore, $\Gamma_0$
is an invariant torus for $X$, and the map
$X\mapsto\Gamma_0$ is analytic and bounded on $\WW\cap B$.
\end{theorem}

\proof By \clmp(BMMnContracts), the map $\MM_n:
B_n\to B_{n-1}/2$ contracts distances by a factor of at least $1/2$.
Thus, if $1\le n<k<k'$, then the difference
$\Gamma_{n,k'}-\Gamma_{n,k}$ is bounded in norm by $2^{n-k+1}$. This
shows that the sequence $k\mapsto\Gamma_{n,k}$ converges in $\BB_n$
to a limit $\Gamma_n\,$, which is independent of the choice of the
functions $F_k\,$. By choosing $F_k=\Gamma_k$ for all $k$, we obtain
the identities \eqref{BGammanSeq}. The analyticity of
$X\mapsto\Gamma_0$ follows via chain rule from the analyticity of
the maps used in our construction, and from uniform convergence.

In order to prove that $\Gamma_0$ is an invariant torus for $X$, we
will use the identity \eqref{BMMPhi}. To be more precise, given a
real number $t$, with $|t|<(4\max\{\|\Omega\|,1\})^{-1}$, define
$t_n=\lambda_nt$ for all $n\ge 0$. \clmp(BPhiFPsi) allows us to
iterate the identity \eqref{BMMPhi}, and get the identity
    \bel{BPhiGammPsi}
\Phi_0^{t}\circ\Gamma_{0,k}\circ\Phi_\omega^{-t}
=\bigl(\MM_1\circ\ldots\circ\MM_k\bigr)
\bigl(\Phi_k^{t_k}\circ\Phi_\omega^{-t_k}\bigr)\,,
    \ee
for all $k>0$. As proved above, the right (and thus left) hand side
of this equation converges in $\AA_0$ to $\Gamma_0\,$. In addition,
$\Gamma_{0,k}\to\Gamma_0$ in $\AA_0\,$, and the convergence is
pointwise as well, by part $(i)$ of \clmp(BTrivial). Thus, since the
flow $\Phi_0^{t}$ is continuous, we have
$\Phi_0^{t}\circ\Gamma_0\circ\Phi_\omega^{-t}=\Gamma_0\,$. This
identity now extends to arbitrary $t\in\real$, due to the group
property of the flow, and the fact that composition with
$\Phi_\omega^s$ is an isometry on $\AA_0\,$.

Let $\AA_\rho^0$ be the subspace of functions $\AA_\rho$ which do
not depend on the variables $y,u,w$. In what follows, the torus
$\Gamma_0$ associated with a vector field $X\in\WW$ will be denoted
by $\Gamma_\ssX\,$. For convenience, we extend the map
$X\mapsto\Gamma_\ssX$ to an open neighborhood of $K$, by setting
$\Gamma_\ssX=\Gamma_\ssXp\,$, where $X'=(\Id+W)(X-\proj X)\in\WW$.

\begin{theorem}\label{BAnalyticTori} Let $\rho>\varrho+\delta$ with $\delta>0$.
Under the same assumptions as in \clmp(BMMnContracts), there exists
an open neighborhood $B$ of $K$ in $\AA_{\rho}(\VV_0)$, such that
$\Gamma_\ssX$ has an analytic continuation to $\|\im x\|<\delta$,
for each $X\in B$. The map $X\mapsto\Gamma_\ssX$ defines (via the
above extension) a bounded analytic map from $B$ to
$\AA_\delta^0(\VV_0)$.
\end{theorem}

\proof The proof of this theorem is completely analogous to the
proof of Theorem 4.5 of \cite{KK1}. For that reason, we will give
only a sketch here.

Consider the translations $R_q(x,y,u,w)=(x+q,y,u,w)$. By examining the
construction of $\WW$ and $\Gamma_\ssX$, one verifies that for any
$q\in\real^d$, the translated vector field $R_q^\ast X$ belongs to
$\WW$ whenever $X$ does, and that
    \bel{BGammaXu}
\Gamma_X(q,0,0,0)=\bigl(R_q\circ\Gamma_{R_q^\ast X}\bigr)(0,0,0,0)\,.
    \ee
The idea now is to use the analyticity of map
$X\mapsto\Gamma_\ssX\,$, to extend the right hand side of equation
\eqref{BGammaXu} to the complex domain $\|\im q\|<\delta$. This
yields the desired analytic continuation of $\Gamma_\ssX\,$. The
remaining parts of \clmt(BAnalyticTori) are proved by using that the
right hand side of identity \eqref{BGammaXu} is jointly analytic in
$X$ and $q$. \qed

The proof of~\clmt(main) follows from~\clmt(BRRnComposeT), \clmt(BGammaLimits) and~\clmt(BAnalyticTori).

\begin{thebibliography}{99}


\bibitem{BKS} J. Bricmont, A. Kupiainen and A. Schenkel, {\it Renormalization group and the Melnikov prob lem for PDE's}, Comm. Math. Phys. {\bf 221} 1 (2001), 101--140.

\bibitem{Brjuno1} A. D.~Brjuno, {\it Analytic form of
differential equations I}, Trudy Moskov. Mat. Obshch. {\bf 25}
(1971), 119-262; Trans. Moscow Math. Soc. {\bf 25} (1973),
131--288.

\bibitem{Brjuno2} A. D.~Brjuno, {\it Analytic form of differential
equations II},  Trudy Moskov. Mat. Obshch. {\bf 26} (1972),
199-239; Trans. Moscow Math. Soc. {\bf 26} (1974), 199--239.

\bibitem{CQ} L. Chierchia and D. Qian, {\it Moser's theorem for lower dimensional tori}, J. Differ. Eq. {\bf 206} 1 (2004), 55--93.

\bibitem{CW} W. Craig and C. Wayne, {\it Newton's methods andperiodic solutions of nonlinear wave equation}, Comm. Pure Appl. Math. {\bf 46}  (1993), 1409--1501.

\bibitem{Eliasson} L. H. Eliasson, {\it Perturbations of stable invariant tori for Hamiltonian systems},
Ann. Scoula Norm. Sup. Pisa, Cl. Sci. {\bf 15} (1988), 115--147.

\bibitem{GG} G. Gallavotti and G. Gentile, {\it Degenerate elliptic resonances}, Comm. Math. Phys. {\bf 257} 2 (2005), 319--362.

\bibitem{Gentile} G. Gentile, {\it G. Gentile
Degenerate lower-dimensional tori under the Bryuno condition},
Ergodic Theory and Dynamical Systems {\bf 27} 2 (2007), 427-457.

\bibitem{JLZ} A. Jorba, R. de la Llave and M. Zou, {\it Lindstedt series for lower-dimesnional tori}, in {\it Hamiltonian systems with three or more degrees of freedom} (S'Agar\' o, 1995), 151-167, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 553 Ed. C. Sim\' o, Kluwer Acad. Publ., Dordrecht (1999).

\bibitem{KLM} K. Khanin, J. Lopes Dias and J. Marklof, {\it Multidimensional continued fractions, dynamic renormalization and KAM theory}, Commun. Math. Phys. {\bf 270} (2007), 197--231.

\bibitem{KLM2} K. Khanin, J. Lopes Dias and J. Marklof, {\it
Renormalization for multidimensional Hamiltonian flows}, Nonlinearity {\bf 19} (2006), 2727--2753.

\bibitem{KK1} H. Koch and S. Koci\' c, {\it Renormalization of vector fields and Diophantine invariant tori}, {
Ergod. Theor. Dynam. Sys.} {\bf 28} 5 (2008), 1559--1585.

\bibitem{KK2} H. Koch and S. Koci\' c, {\it A renormalization group aproach to quasiperiodic motion with Brjuno frequencies}, preprint: mp\_arc 06-269
(2006), Ergod. Theor. Dynam. Sys. to appear.

\bibitem{Kocic} S. Koci\' c, {\it Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori},  Nonlinearity {\bf 18} (2005), 1--32.

\bibitem{Kocic2} S. Koci\' c, {\it Reducibility of skew-product systems with multidimensional Brjuno base flows}, preprint: mp\_arc 09-137 (2007).

\bibitem{Kolmogorov} A. N. Kolmogorov, {\it On conservation of conditionally periodic motions for a small change in Hamilton's function}, Dokl. Akad. Nauk SSSR {\bf 98} (1954), 527--530.

\bibitem{Kuksin} S. B. Kuksin, {\it Nearly integrable infinite-dimensional Hamiltonian systems}, Lecture Notes in Mathematics {\bf 1556} Springer, New York (1992).

\bibitem{Lagarias} J. C. Lagarias, {\it Geodesic multidimensional continued fractions}, Proc. London Math. Soc., {\bf 3} 69 (1994), 464--488.

\bibitem{LY} Y. Li and Y. Yi, {\it Persistence of lower dimensional tori of general types in Hamiltonian systems}, Trans. Amer. Math. Soc. {\bf 357} (2005), 1565--1600.

\bibitem{Melnikov} V. K. Melnikov, {\it On certain cases of conservation of almost periodic motion for a small change in Hamilton's function}, Dokl. Akad. Nauk SSSR {\bf 165} (1965), 1245--1248.

\bibitem{Moser} J. Moser, {\it Convergent Series Expansions for Quasi-Periodic Motions}, Mathematische Annalen {\bf 169} (1967), 136--176.

\bibitem{Poschel} J. P\" oschel, {\it On elliptic lower-dimensional tori in Hamiltonian Systems}, Mathematische Zeitschrift {\bf 202} 4 (1989), 559--608.

\bibitem{Russmann1} H. R\" ussmann, {\it On  the one-dimensional Schr\" odinger equation with a quasiperiodic potential}, Nonlinear Dynamics (Internat. Conf., New York, 1979)  {\bf 6} (2001), 90--107, Ann. New York Acad. Sci. 357, New York Acad. Sci., New York, 1980.

\bibitem{Russmann2} H. R\" ussmann, {\it Invariant tori in non-degenerate nearly integrable Hamiltonian systems}, Regul. Chaotic Dynam.  {\bf 6} (2001), 119--204.

\bibitem{XY} J. Xu and J. You, {\it Persistence of lower dimensional tori under the first Melnikov's non resonance condition}, J. Math. Pures Appl. {\bf 10} 80 (2001), 1045--1067.

\bibitem{Yoccoz} J.-C. Yoccoz, {\it Petits diviseurs en dimension 1}, Ast\' erisque {\bf 231} (1995).

\bibitem{Yoccoz2} J.-C. Yoccoz, {\it Analytic linearization of circle diffeomorphisms}, in {\em Dynamical systems and small divisors}, Ed. Marmi and Yoccoz, Lecture Notes in MAthematics {\bf 1784} Springer-Verlag (2002).

\bibitem{You} J. You, {\it Perturbations of lower-dimensional tori for Hamiltonian systems}, Journal of Differential Equations {\bf 152} 1 (1999), 1--29.

\end{thebibliography}
\end{document}
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