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\centerline{\titlefont
Maximal almost-periodic solutions for Lagrangian equations}
\centerline{\titlefont on infinite dimensional tori}
\vskip0.5truecm

\centerline{
Luigi Chierchia\footnote*{Dip. di Matematica, II
Universit\`a di Roma, ``Tor Vergata", 00133 Roma, Italy}
\footnote\dag{Partially supported by CNR--GNFM grant, n. 2398/91}
and Paolo Perfetti\footnote{**}{Dip. di Fisica,
Universit\`a di Roma, ``La Sapienza", 00185 Roma, Italy}
}
\vskip2.0truecm\noindent
{\sectionfont 1. Introduction }

\vskip1truecm\noindent
We shall briefly discuss the generalization to infinite dimensions of the
existence of {\it maximal quasi-periodic } solutions for the following
Euler-Lagrange equations on ${\bf T}^N\equiv({\bf R}/2\pi{\bf Z})^N:$
$$
\ddot x_i=V_{x_i}(x)\ ,\ \ \ \ i=1,\ldots,N
\eqno(1.1)
$$
associated to the Lagrangian
$$
L(\dot x,x)={1\over2}\sum_{i=1}^N{\dot x_i}^2+V(x)
\eqno(1.2)
$$
$V(x)=V(x_1,\ldots,x_N)$ being a smooth function $2\pi$-periodic in $x_i$;
$(\cdot)_{x_i}$ denotes partial differentiation:
$V_{x_i}\equiv{\partial V\over \partial x_i}.$
If $\omega\in{\bf R}^N$ is a \lq\lq Diophantine vector"
and if $V$ is small enough it
follows from KAM theory (see e.g. [SZ], [CC1], [CC2] for Lagrangian KAM
theory) that (1.1) admits {\it quasi-periodic }solutions
$x(t)=\omega t+u(\omega t)$
where $u\colon{\bf T}^N\to{\bf R}^N$ is a smooth function. We recall that
a vector $\omega\in{\bf R}^N$ is called $(\gamma,\tau)$-Diophantine if
$$
\Bigl\vert\sum_{i=1}^N\omega_in_i\Bigr\vert\ge{1\over\gamma\vert n\vert
^{\tau}},\qquad\forall\ n\in\,{\bf Z}^N\backslash\{0\}
\eqno(1.3)
$$
for some positive numbers $\gamma$ and $\tau$;
$\vert n\vert\equiv\sum_{i=1}^N\vert n_i\vert$.

A simple rescaling argument $(\omega\to\omega/\epsilon,\quad 0<\epsilon<<1)$
shows that
{\it for any} potential (smooth enough) there always exist plenty of
 quasi-periodic
solutions with {\it large }frequencies.

In this note we present some results concerning  the existence
of many {\it maximal almost-periodic }solutions in infinite dimensions. Such
 results are, in a suitable sense, a generalization to infinite dimensions of
[CZ] and complete proofs will appear elsewhere.

Before introducing the precise infinite dimensional set-up that generalizes
the Lagrangian system (1.1)-(1.2), let us consider two examples
to which our techniques will apply. The first is a so-called \lq\lq
finite-range
system
of infinitely many coupled rotators", which can be viewed as an infinite system
of coupled second order differential equations:
$$
{d^2\over dt^2}x_i(t)=a_i\cos(x_i-x_{i-1})-a_{i+1}\cos(x_{i+1}
-x_i)
\eqno(1.4)
$$
where $i\in{\bf Z}$; the $a_i$'s are real constants with $\sup_{i\in{\bf Z}}
\left\vert a_i\right\vert<\infty$.

Systems of this kind (and their finite dimensional approximations) are of
interest in statistical mechanics ( from which the above terminology has been
borrowed) and have been extensively studied by many
authors; see e.g.  [VB], [W], [FSW], [P].

The second example involves a \lq\lq {\it long-range }interaction"
and is given by
the following equations:
$$
{d^2\over dt^2}x_i(t)=\cos x_i\sum_{n\in{\bf Z}}a_n
\prod_{0\ne m\in{\bf Z}}(1+a_{n+m}\sin x_{i+m})
\eqno(1.5)
$$
where $a_n\in{\bf R}$ and  $\sum_{n\in{\bf Z}}
\left\vert a_n\right\vert<\infty.$
Notice that {\it formally } (1.4) and (1.5) are the Euler-Lagrange
equations associated to the {\it formal} Lagrangians
$$
L(x,\dot x)={1\over2}\sum_i{\dot x_i}^2+\sum_ia_i\sin(x_i-x_{i-1})
$$
and
$$
L(x,\dot x)={1\over2}\sum_i{\dot x_i}^2+\sum_i\prod_j
[1+a_j\sin x_{i+j}].
$$
\vskip1.5truecm\noindent
{\sectionfont 2. Set up }

\vskip0.5truecm\noindent
As \lq\lq configuration space'' we shall consider the Cartesian product
$$
{\bf T}^{{\bf Z}^d}\equiv\bigotimes_{i\in{\bf Z}^d}
{\bf T}_i\equiv{\cal T},\qquad
{\bf T}_i\equiv{\bf T}\equiv{\bf R}/2\pi{\bf Z}
$$
 endowed with the standard
 weak (compact) topology. Such a topology is induced by metrics:
given a {\it weight }$w$ (i.e. $w_i>0$ for all $i$ and $\sum_{i\in{\bf Z}^{^d}}
w_i<\infty$) we introduce on ${\cal T}$ the metric
$$
\rho_w(x,y)\equiv\sum_{i\in{\bf Z}^d}w_i\ \rho(x_i,y_i),\qquad (x,y\in{\cal T})
$$
where $\rho(x_i,y_i)$ is the standard flat metric on ${\bf T}$.
The tangent space
of ${\cal T}_w\equiv({\cal T},\rho_w)$ is the Banach space, ${\cal B}_w,$
of vectors
$a\in{\bf R}^{{\bf Z}^{^d}}$ with finite norm:
$$
\left\Vert a\right\Vert_w=\sum_{i\in{\bf Z}^d}w_i\left\vert a_i
\right\vert<\infty.
$$
Given a continuous map $f\colon {\cal T}_w\to{\cal B}_w$
we can now consider the
{\it second order system }
$$
\ddot x_i=f_i(x(t))\ \ ,\qquad\qquad i\in{\bf Z}^d
\eqno(2.1)
$$
and a {\it continuous map }$t\in{\bf R}\to x(t)\in{\cal T}_w$ will be called a
{\it solution }of (2.1) if (for any $i\in{\bf Z}^d$) $x_i(t)$ belongs
to ${\rm C}^2({\bf R})$ and satisfy (2.1).

Equation (2.1) is an equation in ${\cal B}_w$ (having identified the tangent
space of ${\cal B}_w$ with ${\cal B}_w$ itself). However for a solution of
(2.1)
it might
 (and will) happen that $\dot x(t)$ {\it does not }belong to ${\cal B}_w$
(e.g., it may happen that $\dot x_i(t)=\omega_i+\partial_t u_i(t)$ with
$\omega_i$
growing arbitrarily fast as
$\vert i\vert\to\infty$ and $\vert u_i\vert+\vert\partial_t u_i\vert$
bounded).

With standard contraction arguments one can prove the following elementary
\vskip0.5truecm\noindent
{\bf Proposition. }\ \ \ \
{\sl Let $f\colon{\cal T}_w\to{\cal B}_w$ be a
Lipschitz map (i.e. $\exists C$ s.t.
$\Vert f(x)-f(y)\Vert_w\le C\rho_w(x,y),\ \forall x,y\in{\cal T}_w$).
Given any $x^o\in{\cal T}_w$ and any $y^o\in{\bf R}^{{\bf Z}^{^d}}$,
there exists a unique solution, global in time, of the Cauchy problem}
$$
\left\{\eqalign{
&\ddot x_i(t)=f_i(x(t)),\quad\quad\quad i\in{\bf Z}^d\cr
&x_i(0)=x_i^o,\quad\quad\dot x_i(0)=y_i^o.\cr}\right.\quad
\eqno(2.2)
$$
\vskip0.5truecm\noindent
We remark that the initial \lq\lq velocity" $y^o$ is arbitrary and is {\it not }
required to belong to the tangent space ${\cal B}_w$.
\vskip1truecm\noindent
Notice that the examples (1.4) and (1.5) in the introduction have a
Lipschitz right hand side, {\it provided} one chooses suitably the
weight $w$.

In this context the \lq\lq Lagrangian structure'' will be reflected by $f$
being, in a suitable sense, a {\it gradient.}

As it is clear even
from the \lq\lq finite-range'' example above we need a {\it generalized }notion
of gradient. Such a notion will use {\it averages. }

The Haar measure
$d\mu_i\equiv(2\pi)^{-1}dx_i$ on ${\bf T}_i\equiv{\bf T}$ induces,
in a natural way, a
product measure
$d\mu\equiv\bigotimes_{i\in{\bf Z}^d}d\mu_i $
on ${\cal T}_w$. More precisely $d\mu$ is the unique extension on the
$\sigma$-algebra generated by the cylinders,
$$
{\cal R}_I\equiv\bigotimes_{i\in I}A_i\bigotimes_{j\in{\bf Z}^d\backslash I}
{\bf T}_j,\qquad
A_i\equiv\hbox{ open subset of }\ {\bf T}_i,\ \ \ \vert I\vert<\infty
$$
($\vert I\vert\equiv$ cardinality of $I$),
satisfying
$$
\mu({\cal R}_I)=\prod_{i\in I}\mu_i(A_i).
$$
Fubini's theorem holds and one can use integrations as projection operators
on finite dimensional function spaces. If $g\colon{\cal T}_w\to{\bf R}$
is a bounded
measurable
function and $I$ is a finite subset of ${\bf Z}^d,$ we define $g^{[I]}(x^{(I)})$
as the bounded measurable function of $x^{(I)}\in{\bf T}^{\vert I\vert}\equiv
\bigotimes_{i\in I}{\bf T}_i$ obtained by integrating over
$\bigotimes_{i\not\in I}{\bf T}_i$:
$$
g^{[I]}\equiv\int g\bigotimes_{i\not\in I}d\mu_i
\equiv\lim_{j\to\infty}\int g\bigotimes_{k=1}^jd\mu_{n_k}
$$
where $k\to n_k$ is any one-to-one map from ${\bf N}$
onto ${\bf Z}^d\backslash I.$
By Fubini's theorem the function $ g^{[I]}$ does not depend on the choice
of the sequence $\{n_k\}$.

\vskip0.5truecm\noindent
{\bf Definition 2.1 }
{\sl A continuous function $f\colon{\cal T}_w\to{\cal B}_w$ is a
{\it g-gradient }if
 for any finite $I\subset{\bf Z}^d$ there exists  a ${\rm C}^1
({\bf T}^{\left\vert I\right\vert},{\bf R})$ function $V^{(I)}(x)$ so that }
$$
f_i^{[I]}(x)=\partial_{x_i}V^{(I)}(x),\quad\forall\ i\in I\quad\forall\ x\in
{\bf T}^{\vert I\vert}\quad.
$$
We remark that the right hand side of (1.4), (1.5) are
{\it g-gradients }.

\vskip1.5truecm
\noindent
{\sectionfont 3. Results }\vskip0.5truecm\noindent
To state our main results we need some more definitions .

\noindent
{\bf Definition 3.1}
{\sl A vector $\omega\in{\bf R}^{{\bf Z}^{^d}}$ is said to be
{\it rationally independent} if
$$
\sum_{i\in I}\omega_in_i=0\ \hbox{ for some finite }I\subset{\bf Z}^d
\Rightarrow n_i=0\quad\forall\ i\in I.
$$
A rationally independent vector $\omega$ will be also called a
${\cal T}$-frequency
vector.
}

\noindent
{\bf Definition 3.2 }
{\sl Given a ${\cal T}$-frequency vector $\omega,$
a continuous function $q(t)$ is called $\omega$-{\it almost-periodic }
if there exists
a function $Q\in{\rm C}({\cal T}_w,{\bf R})$ such that $q(t)=Q(\omega t)$.
A solution  $x(t)$ of (2.1) is called
{\it maximal almost-periodic} (with frequencies
$\omega\in{\bf R}^{{\bf Z}^{^d}}$)
if $x_i(t)-\omega_it$ is $\omega$-
{\it almost-periodic }
for all $i\in{\bf Z}^d.$}\hfill\break
{\bf Remarks: }\hfill\break
1) The word {\it maximal } refers to the rational independence
property of
$\omega$.\hfill\break
2) An $\omega$-{\it almost-periodic} function $q$ is just an
{\it almost-periodic }function in the sense of H.Bohr with frequency modulus
$$
\sigma(q)=\left\{\sum_{i\in I}\omega_in_i:\ I\subset{\bf Z}^d,
\ \vert I\vert<\infty,\ n_i\in {\bf Z}\right\}.
$$
Finally we introduce the smoothness properties we shall consider.
\hfill\break
{\bf Definition 3.3 }
{\sl A {\it  g-gradient }$f$ is called uniformly weakly real-analytic if there
exists a real number
$\xi>0$
such  that for any finite set $I\subset{\bf Z}^d$, $V^{(I)}(x)$ is
real-analytic on ${\bf T}^{\vert I\vert}$ and can be analytically continued
to the set }$\{z\in {\bf C}^{\left\vert I\right\vert},
\ \left\vert{\rm Im}\,z_i\right\vert\le\xi\}$.

The {\it g-gradients }of the examples in the introduction are uniformly
weakly real-analytic and as parameter $\xi$ one can take any positive
number.
\vskip0.5truecm\noindent
{\bf Theorem 1. }{\sl
Let $f$ be a uniformly weakly real analytic {\it g-gradient.} Then
there exist
uncountably many {\it maximal almost-periodic }solutions of (2.1)}.
\vskip0.5truecm\noindent
This Theorem is an immediate corollary of the following more detailed
statement
\vskip0.5truecm\noindent
{\bf Theorem 2. }{\sl Let $f$ as in {\bf Theorem 1 } (recall Definition 2.1)
and assume that for some
finite $I\subset{\bf Z}^d$ the equation
$$
\ddot x^{(I)}=\partial_{x^{(I)}}V^{(I)}(x^{(I)}),\qquad (x^{(I)}\in
{\bf T}^{\vert I\vert})
\eqno(3.1)
$$
admits a {\it non-degenerate} real-analytic quasi-periodic solution with
a $(\gamma,\tau)$-Diophantine frequency
$\omega^{(I)}\in{\bf R}^{\vert I\vert}$ (i.e. $x^{(I)}(t)=\omega^{(I)}t+
u^{(I)}(\omega^{(I)}t)$ for a suitable real-analytic function $u^{(I)}:
{\bf T}^{\vert I\vert}\to{\bf R}^{\vert I\vert}$ such that
$\det({\rm Id}+\partial u^{(I)})\ne0).$ Then  there exist
 uncountably many $\omega$-{\it almost-periodic} solutions
with $\omega_i=\omega^{(I)}_i,\forall i\in I$. All such frequencies $\omega$
 are \lq\lq $\gamma$-Diophantine" in the sense that for all $J\supset I$ with
$\vert J\backslash I\vert=m$ it is:
$$
\Bigl\vert \sum_{j\in J}\omega_jn_j\Bigr\vert\ge{1\over\gamma(\sum_{j\in J}
\vert n_j
\vert)^{\tau+m}},\quad\qquad \forall\ (\{n_j\}_{j\in J})\in
{\bf Z}^{\vert J\vert}
\backslash\{0\}.
\eqno(3.2)
$$
Moreover for a suitable (small) constant $\epsilon>0$ one has
$$
\Vert x_i-x_i^{(I)}\Vert_{\rm C^2}\le\epsilon \ \ \ (i\in I)\qquad
,\quad\left\Vert x_i-\omega_it\right\Vert_{\rm C^2}\le\epsilon\ \ \
(i\not\in I)
$$
where $\Vert \cdot\Vert_{{\rm C}^2}$ is the standard $ {\rm C}^2$ norm.
}
\vskip0.5truecm\noindent
The frequencies in the above theorems will be such that $\vert\omega_j\vert$
grows rapidly as $\vert j\vert\to\infty$. In fact rough estimates based on
the technique used to prove the above results, indicate the behaviour
$\vert\omega_j\vert\sim(\vert j\vert!)^c.$ Of course better estimates can be
obtained by softening the numerical properties (3.2) of the frequencies.

\noindent
Establishing the existence of maximal almost-periodic solutions with
slowly growing frequencies is an interesting (and difficult) problem
especially in view
of more ``realistic"  models such as perturbed integrable P.D.E.'s (where,
typically, the
integrable frequencies grow as $\vert j\vert^c$ compare [K]).

The reason for the abundance of maximal almost-periodic solutions with
$\vert \omega_j\vert$ growing rapidly may be justified as follows. Imagine
that a subsystem of our model admits a quasi-periodic solution
(see the assumption of
Theorem 2), and imagine to \lq\lq turn on one mode at time". Then,
intuitively speaking, if the ``turned--on mode" rotates fast enough, its effect
will be mostly \lq\lq averaged out" and it will weakly interact with the
pre-existing motion.

In the rest of this note we present the main ideas
beyond the proof of the above theorems.

\vskip1.5truecm
\noindent
{\sectionfont 4. Sketch of proofs}

\vskip0.25truecm\noindent
Theorems 1 is an immediate corollary of Theorem 2: For example one could take
$I=\{i_o\}$ (with any $i_o\in{\bf Z}^d$) in Theorem 2: in this case
(3.1) becomes trivial and it always admits non-degenerate quasi-periodic
(in this case simply periodic) solutions.

\noindent
The proof of Theorem 2 is based on a recursive argument.

Fix once for all a
one-to-one map, $j_n$, from ${\bf N}$ onto ${\bf Z}^d\backslash I$ and let
$I_o\equiv I,$
$I_1\equiv I_o\cup\{j_1\},$  $I_{n+1}\equiv I_n\cup\{j_{n+1}\},$ (so that
$\vert I_n\vert=\vert I\vert+n$ and $I_o\subset I_1\subset\ldots I_n\uparrow
{\bf Z}^d$).

The {\bf main step} consists in showing how, assuming to have a non-degenerate
quasi-periodic solution of
$$
\ddot x=\partial_x V^{(I_n)}(x),\qquad (x\in{\bf T}^{\vert I_n\vert})
\eqno(4.1)_n
$$
with frequencies $\omega\equiv\omega^{(n)}\equiv(\{\omega_j\}_{j\in I_n}),$
one can construct, via a \lq\lq KAM theorem", a quasi-periodic non-degenerate
solution of of (4.1)$_{n+1}$ with frequencies
$\omega^\prime\equiv\omega^{(n+1)}\equiv(\omega,\alpha)\equiv
(\{\omega_j\}_{j\in I_{n+1}})$ provided $\alpha\equiv\omega_{j_{n+1}}$ is
large enough and suitably chosen.

Once this step is carried out and {\it made quantitative} a solution $x(t)$
of (2.1) (with $f$ as in Theorem 2) will be easily obtained by
taking, in a suitable (but natural) sense the limit
$$
x_j(t)\equiv\lim_{n\to\infty}x^{(n)}_j(t)
\eqno(4.2)
$$
where $x^{(n)}_j(t)\equiv\omega^{(n)}_j t+u_j^{(n)}(\omega^{(n)}t)$ is the
quasi-periodic solution of (4.1)$_n$.

The abundance of (rapidly
oscillating) almost-periodic solutions is related to the choice of
$\omega_{j_{n+1}}$ {\it given} $\omega^{(n)}$: $\omega_{j_{n+1}}$ can be
{\it arbitrarily chosen }in a Cantor set $\Omega^{(n+1)}\subset[\overline\alpha,
\infty)$ (for a suitable $\overline\alpha=\overline\alpha(\omega^{(n)})$
large enough) having the property that
$$
\lim_{R\to\infty}\ell(\Omega^{(n+1)}\cap[R,R+1])=1
\eqno(4.3)
$$
where $\ell$ denotes Lebesgue measure. We shall not work out here
the straightforward details necessary to give meaning to (4.2), while we shall
concentrate on the main step.

As already mentioned the main technical tool comes from KAM theory: we shall
use a minor modification (see [Pe]) of the \lq\lq configurational KAM
theorem" given
in [CC1] (see also [SZ] and [CC2]).

Consider (1.1) with $V$ real-analytic
on ${\bf T}^N$ admitting a holomorphic extension to
$$
\Delta^N_{\xi_o}\equiv\{x\in{\bf C}^N\,:\vert{\rm Im}x_j\vert\le\xi_o
\ j=1,\ldots,N\}.
$$
It is easy to see that having a non-degenerate quasi-periodic solution
$x(t)=\omega t+u(\omega t)$ (with $u\colon{\bf T}^N\to{\bf R}^N$ and
$\omega\in{\bf R}^N$ rationally independent) is {\it equivalent} to
require that $u$ satisfies
$$
D^2u(\theta)=V_x(\theta+u(\theta))
\eqno(4.4)
$$
where $Du\equiv\omega\cdot\partial_{\theta}u\equiv\sum_{k=1}^N\omega_k
\partial_{\theta_k}u$;
``non-degenerate" means that $\det ({\rm Id}+u_\theta)\ne0.$

\vskip0.5truecm\noindent
{\bf Theorem K} (cfr. [CC1]).\ \ \ \ {\sl Let $\omega$ verify (1.3) and let
$v\colon{\bf T}^N\to{\bf R}^N$ be a holomorphic function on
$\Delta^N_{\xi_{*}},$ with $\xi_{_*}<\xi_o,$ such that:

i) $\{\theta+v(\theta):\theta\in\Delta^N_{\xi_{_*}}\}\subset
\Delta^N_{\xi_o},$ \qquad
ii) $\sup_{\Delta^N_{\xi_{_*}}}\Vert({\rm Id}+v_{\theta})^{-1}\Vert\equiv
\widetilde M<\infty.$

\noindent
Finally let $0<\xi^\prime<\xi_{_*}.$ Let be
a constant $K=K(\Vert V\Vert_{C^3,\Delta^N_{\xi_o}},\widetilde M,\gamma,
\tau,\xi_{_*}-\xi^\prime)>1$ such that if
$$
K\ \Vert D^2v-V_x(\theta+v)\Vert_{\Delta^N_{\xi_{_*}}}\equiv KE\le1
\eqno(4.5)
$$
then there exists a solution
$u\colon{\bf T}^N\to{\bf R}^N$ of (4.4) with a holomorphic extension to
$\Delta^N_{\xi^\prime};$ such a solution is (locally) unique if we require
that $\int ud\theta=\int vd\theta.$ The constant $K$ can be taken to be
$$
K\equiv\lambda\gamma^2{\overline M^{10}{\widetilde M}^8 (N!)^4\,2^{34N+12}
\over(\xi_{_*}-\xi^\prime)^{8N+1}}
\eqno(4.6)
$$
with
$$
\lambda\equiv{\rm max}\Bigl\{1\ ,\gamma^2\,\Vert V_{xxx}
\Vert_{\Delta^N_{\xi_o}}\Bigr\},\quad
M\equiv\sup_{\Delta^N_{\xi_{_*}}}\Vert{\rm Id}+v_{\theta}\Vert.
\eqno(4.7)
$$
\vskip0.5truecm\noindent
The solution $u$ is close to $v$ in the following sense:
$$
\eqalign{
&\max\Bigl\{\left\Vert u\,-\,v\right\Vert_{\Delta^N_{\xi^{\prime}}},\
\left\Vert u_{\theta}\,-\,v_{\theta}\right\Vert_{\Delta^N_{\xi^{\prime}}}
\Bigr\}\le KE\cr
&\max\Bigl\{\gamma\left\Vert Du\,-\,Dv\right\Vert_{\Delta^N_{\xi^{\prime}}},
\ \gamma^2\left\Vert D^2u\,-\,D^2v\right\Vert_{\Delta^N_{\xi^{\prime}}}
\Bigr\}\le KE
\Lambda\cr}
\eqno(4.8)
$$
$\hbox{where\ \ \ \ }\Lambda={\rm max}\Bigl\{1,\,
\Vert V_{xx}\Vert_{\Delta^N_{\xi_o}}/
\Vert V_{xxx}\Vert_{\Delta^N_{\xi_o}}\Bigr\}.$
}
\vskip1.0truecm\noindent
{\bf Remarks}:

\noindent
1) In other words: near any (non-degenerate)
{\it approximate} solution of (4.4) there is a true (non-degenerate)
solution, provided the approximate solution solves equation (4.4) up to a
small {\it enough} error.

\noindent
2) The (minor) changes with respect to the version in [CC1] are the estimates
in (4.6), (4.8) and the introduction of the parameter $\xi^\prime$ (the domain
of holomorphy of $u$) which, here, is left free.

Assume now to have a non-degenerate solution of (4.1)$_n,$
$x^{(n)}(t)=\theta^\prime+\omega^{(n)}t+u^\prime(\theta^\prime+
\omega^{(n)}t),$
$\theta^\prime\in{\bf T}^{\vert I\vert+n},$
$u^\prime\colon{\bf T}^{\vert I\vert+n}\to{\bf R}^{\vert I\vert+n}.$ Let
$N\equiv\vert I\vert+n+1$ and let $\xi_o$ be the parameter measuring
the analyticity of the {\it g-gradient} $f$ (see Definition 3.3:
$\xi_o\equiv\xi$). The function
$u^\prime\colon{\bf T}^{\vert I\vert+n}\to{\bf R}^{\vert I\vert+n}$ is
assumed to be
such that $\vert {\rm Im}(\theta^\prime_i+u_i^\prime(\theta^\prime))\vert
\le\xi_o$ for
any $i\in I_n,$ for any $\theta^\prime\in\Delta^{N-1}_\xi,$ for a suitable
$\xi;$ the non-degeneracy of $x^{(n)}(t)$ means that
$\Vert({\rm Id}+u^\prime_{\theta^\prime})^{-1}\Vert_{\Delta^{N-1}_\xi}
<\infty.$ To attack the problem with one more degree-of-freedom on
${\bf T}^N$ we shall look directly at the P.D.E. (4.4) for a new
quasi-periodic function
$x^{(n+1)}(t)=\omega^{(n+1)}t+u(\omega^{(n+1)}t),$ $u$ {\it and}
$\alpha\equiv\omega_{j_{n+1}}$ being the
{\it unknowns}: of course $V$ in (4.4) is now $V^{(I_{n+1})}$
(recall that $V^{(I_{n})}=\int_0^{2\pi}V^{(I_{n+1})}d\theta_{j_{n+1}}/2\pi$ up t
o
a constant playing no role).

The new frequency $\alpha=\omega_{j_{n+1}}$ is taken to be in the (Cantor) set
$$
\Omega_{\alpha_o}\equiv\Bigl\{\alpha\in{\bf R}:
\vert\omega^{(n)}\cdot n+\alpha m\vert\ge{\alpha_o\over\vert n
\vert^{N}},\ \alpha\ge\alpha_o,\quad\forall \ n\in{\bf Z}^{N-1},\quad
\forall\ m\in{\bf Z}\backslash\{0\}\Bigr\}
$$
where $\alpha_o$ is (at the moment) an {arbitrary} parameter larger than
$1/\gamma$ ($\gamma,$ $\tau$ are the Diophantine constants associated to
$\omega^{(o)}$: cfr. hypotheses of Theorem 2). It is not difficult to see
that $\ell([\alpha_o,\infty)\backslash\Omega_{\alpha_o})\le c\vert
\omega^{(n)}\vert$ with a suitable constant c depending only on $N,\tau.$
Thus the set $\Omega_{\alpha_o}$ verifies
(4.3).

The idea now is to try to construct out of
$u^\prime(\theta^\prime)$ an approximate solution $v\colon{\bf T}^N\to
{\bf R}^N$ for (4.4) with $V=V^{(I_{n+1})}$ and to use Theorem K
to show the existence of a solution, $u,$ close to $u^\prime.$ The job is
done by setting
$(\theta\equiv(\theta^\prime,\theta_{j_{n+1}})\in{\bf T}^N)$:
$$
\eqalign{
v(\theta)\equiv&D_{\omega^{(n+1)}}^{-2}V_{x}^{(I_{n+1})}(\theta^\prime+
u^\prime(\theta^\prime),
\theta_{j_{n+1}})\cr
=&-\sum_{\scriptstyle n\in{\bf Z}^{N-1}, m\in{\bf Z}\atop
\scriptstyle( n,m)\ne(0,0)}e^{(i n\cdot\theta^\prime+ m
\theta_{j_{n+1}})}{C_{n,m}
\over(\omega^{(n)}\cdot n+\alpha m)^2}\cr}
\eqno(4.9)
$$
where $x\in{\bf T}^N$ ($x=\{x_i\}_{i\in I_{n+1}}$), $C_{n,m}$ denotes the
Fourier
coefficient of the (vector-valued) function
$C(\theta)=V_{x}^{(I_{n+1})}(\theta^\prime+u^\prime(\theta^\prime),
\theta_{j_{n+1}}),$ $D_{\omega^{(n+1)}}\equiv(\omega^{(n)}
,\alpha)\cdot\partial_\theta\equiv\omega^{(n)}\cdot\partial_{\theta^\prime}
+\alpha\partial_{\theta_{j_{n+1}}}$ and
if $g(\theta)$ is a function with zero average we denote by
$D^{-2}g$ the {\it unique} solution of $D^2f=g$ with vanishing mean value.

\noindent
Notice that in (4.9) it is
implicitly stated that
$$
\int_{{\bf T}^N}
V_{x}^{(I_{n+1})}(\theta^\prime+u^\prime(\theta^\prime),\theta_{j_{n+1}})
{d\theta\over(2\pi)^N}=0,
\eqno(4.10)
$$
a fact that is deduced by recalling that $u^\prime$ solves equation (4.4)
with $V=V^{(I_n)}$ and that
$$
\int_0^{2\pi}
V_{x}^{(I_{n+1})}(\theta^\prime+u^\prime(\theta^\prime),\theta_{j_{n+1}})
{d\theta_{j_{n+1}}\over(2\pi)}=V^{(I_n)}_{x^\prime}(\theta^\prime+
u^\prime(\theta^\prime)).
$$
This fact justify the application of the operator $D^{-2}.$

Now, notice that
since $u^\prime$ solves (4.4) with $V=V^{(I_n)}$, it is
$u^\prime(\theta^\prime)=D^{-2}_{\omega^{(n)}}[V_{x^\prime}^{(I_n)}
(\theta^\prime+u^\prime)]$ thus
$$
v=-\sum_{m\ne 0}e^{(\imath n\cdot\theta^\prime+m\theta_{j_{n+1}})}{C_{n,m}\over
(\omega^{(n)}\cdot n+\alpha m)^2}+(u^\prime,0)\equiv\tilde v+(u^\prime,0).
$$
But recalling the definition of $\Omega_{\alpha_o}$ it is easy to see that
the norm of $\tilde v$ is $O(1/(\alpha_o)^2)$ and it can be made
arbitrarily small by choosing the free parameter $\alpha_o$ large enough.
How large
one has to choose $\alpha_o$ is dictated by the conditions
(4.5) in Theorem K. In fact it is not difficult to show that
also $\Vert D^2v-V^{(I_{n+1})}(\theta+v)\Vert$ is of size
$O(1/(\alpha_o^2))$; thus if one takes $\alpha_o=O(K^{1/2})$ one can apply
 Theorem K obtaining in such a way, a quasi-periodic solution
$x^{(n+1)}(t).$ This concludes our discussion of the main step.

\noindent
Complete
detailed proofs will appear elsewhere.
\vskip2.0truecm\noindent
{\sectionfont References }\vskip1.0truecm
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\bye