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%%% "A Direct Proof of a Theorem by Kolmogorov
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\begin{document}
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\title{Compensations in small divisor problems}
\author{ L. Chierchia and C. Falcolini\\
{\footnotesize Dipartimento di Matematica} \\
{\footnotesize Universit\`a di Roma ``Tor Vergata"}\\
{\footnotesize via della Ricerca Scientifica,
00133 Roma (Italy)} \\
{\footnotesize (Internet: chierchia@mat.utovrm.it and
falcolini@mat.utovrm.it)}
\thanks{The authors gratefully acknowledge helpful discussions
with C.~Liverani.}
}
\maketitle
\date{August 1994}
\begin{abstract}
\nin
{\footnotesize
Several small divisor problems arising in the perturbative theory
of Hamiltonian and Lagrangian systems are considered. A general
method that allows to prove compensations among the elementary
contributions of the formal power series expansions associated to
invariant surfaces is presented.
}
\end{abstract}
{\footnotesize \tableofcontents}
%********************************************
\newpage
\setcounter{page}{1}
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\section{Introduction}
\label{sec:par1}
\setcounter{equation}{0}
%
C.L.~Siegel \cite{S} was the first to solve a
small divisor problem (linearization of germs of analytic
functions). His solution was based on a ``direct method" consisting in
computing the $k^{\rm th}$ coefficient of the formal power series
solution and estimating such coefficient by a constant to the
$k^{\rm th}$ power. The solution of small divisor problems arising
in perturbative series
in the theory of Hamiltonian systems has been
one of the major contributions to dynamical
systems of this century:
the techniques employed for such a task, started by A.N.~Kolmogorov in the
early fifties and developed in the early sixties by V.I.~Arnold and
J.K.~Moser, are known as
``Kolmogorov--Arnold--Moser (KAM) theory" (see \eg \cite{A1}
and references therein). Such methods show {\sl
indirectly} the convergence of the formal expansions of the perturbative
series
(longly before considered by
astronomers and especially by H.~Poincar\`e \cite{Po}).
In 1988 H.~Eliasson
proposed a direct proof of the convergence of formal expansions
in the Hamiltonian context (\cite{E1}, \cite{E2}, \cite{E3}).
More recently direct proofs (in the Hamiltonian context)
have been reconsidered in several
papers: \cite{CF1}, \cite{Ga1}, \cite{Ga2}, \cite{Ga3}
\cite{GG}, \cite{Ge1}, \cite{Ge2}; a small divisor problem
arising in elliptic systems
of PDE's has been solved by similar methods in \cite{CF2}.
\giu
In this paper
we consider several small divisor problems and discuss a general
method that allows to prove compensations among
``elementary" contributions to the
$k^{\rm th}$ order of the associated formal expansion.
Such compensations,
modulo technical estimates (which have been thoroughly discussed in
\cite{CF1}), yield new proofs of the
{\sl convergence} of the formal power series.
\giu
The starting point in this subject is a formal power series whose
coefficients are {\sl bona fide} functions determined by linear recursive
equations. ``Recursive" means that the $k^{\rm th}$ coefficient can
be expressed (typically by Taylor's expansions and by inversion of
suitable differential operator $D$)
in terms of the derivatives of a given function
(``the Hamiltonian" or ``the Lagrangian")
and in terms of the $h^{\rm th}$ coefficients with $h0$);
the second sum runs over all possible
functions assigning to each vertex
$v\in V$ of a rooted tree $T$ an integer vector $\a_v\in\zN$
with the constrain
$\sum_{v\in V} \a_v=n$;
the third sum runs over a suitable set of
(possibly) vector--valued indices taking a finite number of
values ($\# B<\io$);
$\L$ is a complex
vector depending on $T$, $\{\a_v\}_{v\in V}$,
$\{\b_v\}_{v\in V}$ and on the Hamiltonian (or Lagrangian);
finally $\g_v\in\real$ are {\sl divisors} that are
described as follows. Rooted trees can be naturally equipped with
a partial order:
we say that $v'\le v$ if the path joining the root $r$ of $T$
with $v'$ contains $v$; (obviously $v'< v$ means $v'\le v$ and $v'\neq v$
and $r\ge v$ $\forall$ $v\in V$ \ie the root $r$ is the first vertex of
the rooted tree $T$).
Given a function $\a$, we define
\beq{1.dl}
\d_v\=\d_v(T,\a)\=\sum_{v'\in V:\ v'\le v} \a_v\ .
\eeq
The divisors $\g_v$, which may assume arbitrarily small values,
are defined in terms of a real--valued function
$\lg \cdot \rg$ which satisfies the {\sl Diophantine condition}
\beq{1.dp}
|\lg n \rg|^{-1} \le \g |n|^\t\ ,\qquad \forall\ n\in \zN\backslash\{0\}
\eeq
with suitable positive constants $\g,\t$.\footnote{
Typically, in dynamical systems,
$\lg n \rg = \o \cdot n$ (where the dot denotes the standard
inner product in $\rN$) but in other situations (\eg \cite{CF2})
the function $\lg \cdot \rg$ might be more complicate
(\eg non linear); in the case $\lg n \rg = \o \cdot n$
it is well known that, if $\t>N-1$ , up to a set of Lebesgue measure zero,
all $\o\in\rN$ satisfy \equ{1.dp} for some $\g$.}
Then
\beq{1.ga}
\g_v\=\g_v(T,\a,\b)\=\cases{ \lg \d_v\rg^{-\s_v} & if $\d_v\neq 0$\cr
1 & if $\d_v=0$.\cr}
\eeq
where $\s_v$ is, say, the first component of the index $\b_v$ and
takes value $0$, $1$ or $2$.
In \equ{1.tr} only the second sum runs over an infinite set of indices:
therefore we assume that there exist positive numbers $\xi,\xi',a>0$
such that, if we set
\beq{1.co}
a^k_{n}\=
\frac{1}{k!}\sum_{T\in \cT^k_*}
\sum_{ {\a:V\to \zN}\atop{\Sigma \a_v=n}}
\sum_{\b:V\to B} |\L(T,\a,\b)| \prod_{v\in V} e^{\xi'|\a_v|} \ ,
\eeq
then, for all $k$ and $n$ one has
\beq{1.co'}
a^k_{n}\le a^k e^{-\xi|n|}\ .
\eeq
In the models considered here, such an assumption is an immediate
consequence of the
well--posedness of the (formal) problem and of the analyticity assumptions
on the Hamiltonian (Lagrangian).
>From \equ{1.co} it follows at once that if one could bound the product of the
divisors as\footnote{From now on we will adhere to the
the common abuse of notation
$v\in T$ in place of the more proper $v\in V(T)$ and also if $vv'=v'v$
denotes an edge of $T$, we shall denote $vv'\in T$ rather than the
more proper $vv'\in E(T)$.}
\beq{1.si}
\prod_{v\in T} |\g_v|\le c^k_3 \ \prod_{T} (1+|\a_v|^b)
\eeq
for some $c_3 > 1$ and $b > 0$,
then from \equ{1.ft}, \equ{1.co}
and \equ{1.si} it would follow
\beq{1.5}
|Z^k_{n}| \le \frac{1}{k!}\sum_{T\in \cT^k_*}
\sum_{ {\a:V\to \zN}\atop{\Sigma \a_v=n}}
\sum_{\b:V\to B} |\L(T,\a,\b)| \ c^k_4 \ \prod_{v\in V} (1+|\a_v|^b)
\le c^k_5 e^{-\xi |n|} ,
\eeq
(for suitable $c_1 > 0$) leading to ``absolute" convergence
(\ie convergence without compensations) of the formal expansion $Z$.
Indeed Siegel's original proof is based on a similar argument,
even though the set up is slightly different (and simpler). Technically
Siegel's problem corresponds to $\th$ varying in a small (complex) ball
so that Fourier series is replaced by Taylor series and
$\a_v$ ranges over $\integer^N_+$: in such a case
$\d_v\neq \d_{v'}$ whenever $v>v'$ and Siegel's method \cite{S}
yields the estimates \equ{1.si}\footnote{For a detailed
discussion, in the present language, of Siegel's methods
see Appendix C of \cite{CF1}; for a different approach
see \cite{Br}.}.
The problem with $\a_v\in \zN$ is that
one can have ``resonances" \ie $\d_v=\d_{v'}$ for $v>v'$ which may lead
to obstinate repetitions of particularly small divisors. It is well
known (see \eg \cite{CF1}, Appendix B) that, in general, one has,
for arbitrarily large $k$,
subfamilies $\cF_{\rm div}\subset \cT^k_*$ and
a choice of $\bar \a$ and $\bar \b$ (depending only on the subfamily)
such that, for suitable $\bar a,\bar b>0$,
\beq{1.di}
\frac{1}{k!}\sum_{T\in \cF_{\rm div}}
\L(T,\bar \a,\bar \b) \ \ \prod_{v\in V} \g_v
\ge \bar a^k k!^{\bar b}\ .
\eeq
Such families are obtained by taking
{\sl chains of resonances} which are defined as follows.
Given $T\in \cT^k_*$ and $\a$ (\ie $\{\a_v\}_{v\in T}$),
a {\sl resonance} is
%
a subtree\footnote{When referred to trees the notation $T'\subset
T$ will always mean ``$T'$ (unrooted) subtree of $T$". Other special
conventions we are adopting are the following: a) for rooted trees the root
(usually denoted $r$) may be identified by adding an extra edge $\eta r$
where $\eta$ is a symbol (not a vertex of the tree) sometimes called
the ``earth"; with these positions one has, for trees, $\# E= \#V-1$
and, for rooted trees, $\# E= \#V$; b) the degree of a vertex $v$ is the
number of edges $vv'$ incident with $v$ and if $v=r$ is the root, the edge
$\eta r$ is included in the count;
c) the degree of a subtree
$T'\subset T$ is the number of edges connecting $T'$ with $T\backslash
T'$; if $T$ is rooted and the root $r$ belongs to $T'$ the edge $\eta r$
must be included in the count.}
%
$R\subset T$ such that: i) $R$ is of degree two
(\ie $R$ is connected to $T\backslash R$
by two edges); ii) if $u$ is the first vertex\footnote{Recall that
$T$ is a rooted tree and hence partially ordered
(the order being such that
the first vertex of $T$ is always its root).}
in $R$ and $z$ is the first vertex following $R$, then
$\d_u=\d_z\neq 0$; iii)
$R$ cannot be disconnected, by removal of one edge, into two subtrees
of degree two satisfying i) and ii).
It will be important to consider different choices of the index $\b$
(\ie $\{\b_v\}_{v\in T}$): in particular given a resonance $R$
and given $\b$ we call {\sl order of the resonance} the number (see
\equ{1.ga})
$\s\=\s_u$ ($u$ being the first vertex of $R$).
A {\sl chain of resonances} is a maximal series of resonances
$R_1,...,R_h$ with $R_i$
adjacent\footnote{\ie connected by one edge.}
to $R_{i+1}$; given a choice of $\b$,
the {\sl order of the chain} is defined
to be $\bar \s\=\s_1+\cdots +\s_h$ where $\s_i\=\s_{u_i}$, $u_i$ being the
first vertex of $R_i$.
>From these positions it follows that
if $C$ is a chain of
order $\bar \s$, if $n=\d_z$ where $z$ is the first
vertex following the chain (\ie following the last resonance, which by
convention will be $R_h$), then
\beq{1.ch}
\prod_{v\in C} \g_v = \lg n \rg^{-\bar \s} \prod_{ {v\in C}\atop {v\neq u_i}}
\g_v
\eeq
(where $v\in C$ means $v\in \bigcup_i V(R_i)$).
The examples for which \equ{1.di} holds are based on chains
with $\bar \s\sim k$ and $|\lg n \rg|\sim k^{-1}$.
This phenomenon may be cured through compensations.
To be more precise we introduce the notion of ``compensable chain".
Consider a chain $C=(R_1,...,R_h)$, ($h\ge 1$) and fix the indices $\b_v$.
Let, as above,
$u_i$ be the first vertex in $R_i$,
let $R_i$ be connected to $R_{i+1}$ by the edge $w_i u_{i+1}$
with $w_i\in R_i$ and $u_1\ge w_1>u_2\ge ...\ge w_h$; let
$z$ be the first vertex following the chain (\ie following $w_h$)
and $n\=\d_z$; and, finally,
let $P_i$ be the path joining $u_i$ with $w_i$.
Consider the following function of $x\in \complex$
\beq{1.pi}
\pi_C(x;T,\a,\b)\=\prod_{i=1}^h \pi_{R_i}(x;T,\a,\b)\ ,
\qquad
\pi_{R_i}(x)
\=\prod_{v\in (R_i\backslash P_i)} \g_v\
\prod_{{v\in P_i}\atop{v\neq u_i}} \bar \g_v(x)\ ,
\eeq
where, if $u_i=w_i$ (\ie $P_i=\{u_i\}$), the product over $P_i$ is absent,
while if $v\in P_i\neq \{u_i\}$ we set
\beq{1.pi'}
\bar \g_v(x)\= \lg \sum_{ {v'\in R_i}\atop {v'\le v}} \a_{v'}
+ x n \rg^{-\s_v} \ .
\eeq
Thus, $\g_v=\bar \g_v (1)$ and
\beq{1.10}
\prod_{v\in C} \g_v= \lg n \rg^{-\bar \s}
\pi_C(1)\ .
\eeq
We say that {\sl a chain $C\subset T\in \cT^k_*$
{\rm (\ie $C=(R_1,...,R_h)$ with $R_i\subset T$)}
is compensable}
if there exists a family of trees $\cF_C\subset \cT^k_*$ whose elements
$T'$ have $C$ as common chain of resonance, and, for each $T'$,
there exists a choice of indices $\b'\=\b'(T')$,
such that the function\footnote{
In other words, the elements $T'$ of $\cF_C$ are obtained from $T$ and $C$
by (possibly) changing the edges connecting $T\backslash C$ with $R_1$,
$R_1$ with $R_2$,...,$R_h$ with $T\backslash C$, and by (possibly)
changing the values of the indices $\b$ on $C$.}
\beq{1.11}
\bar \pi_C(x)\=\sum_{T'\in \cF_C} \L(T',\a,\b')\ \pi_C(x;T',\a,\b')
\eeq
has a zero in $x$ of order at least $\bar \s-1$.
>From the detailed estimates in \cite{CF1}, it follows easily
that {\sl if in a small divisor problem all chains of
resonances are compensable then
the associated formal series \equ{1.1} is in fact convergent}.
In the rest of the paper we shall prove the following statements.
\giu
{\bf P1)} Let $H=H_0(y)+\e H_1(x,y)$ be a real--analytic Hamiltonian
with $y\in B({y_0})\subset \rN$, $x\in \tN$
and $(x,y)$ standard symplectic coordinates\footnote{This simply means
that the Hamiltonian (evolution) equations are given by
$\dot x= \dpr_y H$, $\dot y=- \dpr_x H$.}; here $B({y_0})$
denotes some $N$--sphere centered at $y_0$.
Assume the following standard
``non degeneracy conditions" on the ``integrable part" $H_0$:
i) the Hessian matrix $\dpr^2_y H_0(y_0)$ is invertible;
ii) $\o\=\dpr_y H_0\in\rN$
is a Diophantine vector \ie there exists $\g,\t>0$
such that\footnote{Compare with \equ{1.dp}.}
\beq{1.om}
|\o \cdot n|^{-1} \le \g |n|^\t\ ,
\qquad \forall\ n\in \zN\backslash \{0\}\ .
\eeq
Then, quasi--periodic solutions with frequencies $\o$, \ie solutions
of the form $(x(t),y(t))=Z(\o t)$ with $Z^0(\th)=(\th,y_0)$ and
$Z^k:\th\in \tN \to Z^k(\th)\in \real^{2N}$, satisfy the equations
\beq{1.h1}
D Z = J \dpr H\Big( Z(\th) \Big)\ ,
\eeq
where $D\=\o\cdot \dpr_\th$, $J$ is the standard symplectic matrix
$\pmatrix{0 & I\cr -I & 0\cr}$ and $\dpr$ is the gradient
$\dpr_{(x,y)}$ with respect to the variables $(x,y)$.
Then it is well known\footnote{See \cite{Po} or Appendix B of
\cite{CF1}.} that there exists a unique formal solutions
$Z\sim \sum_{k\ge 0} \e^k Z^k$ of
\equ{1.h1} with the normalization condition
\beq{1.av}
\ig_{\tN} \p_1\circ Z^k d\th= 0\ \quad (k\ge 1) ,
\eeq
where $\p_1$ is the projection onto the first coordinates:
$\p_1(x,y)=x$.
We can then prove
\thm{thm1} There exists a tree expansion \equ{1.tr} for $Z$ such
that all chains of resonances are compensable.
\ethm
%
{\bf P2)} The formulation for the Lagrangian case is very similar. We
let $L\=L_0(y)+\e L_1(x,y)$ be a real--analytic Lagrangian\footnote{The
Lagrangian (evolution equations) are $\frac{d}{dt} \dpr_y L(x,\dot x)=
\dpr_x L(x,\dot x)$.}
with $L_i$
satisfying the same hypotheses assumed above for $H_i$ and $\o$, except
that now $\o \= y_0$.
Quasi--periodic solutions $x(t)=Z(\o t)=\sum_{k\ge 0} \e^k Z^k(\o t)$,
where now $Z^0(\th)=\th$ and
$Z^k:\th\in \tN\to Z^k(\th)\in \rN$, satisfy the equations
\beq{1.h2}
D \ \dpr_y L\big(Z(\th), DZ(\th)\big)=
\dpr_x L\big(Z(\th), DZ(\th)\big)
\ ,\qquad
(D\=\o\cdot\dpr_\th)\ .
\eeq
Mimicking the proof of Appendix B in \cite{CF1} one can show that
there exists a unique formal solution
$Z\sim \sum_{k\ge 0} \e^k Z^k$ of \equ{1.h2} with the normalization
condition as in \equ{1.av} but without the projection $\p_1$.
Then, {\sl Theorem~\ref{thm1} holds also in this case}.
\giu
{\bf P3)}
Maximal invariant tori correspond to analytic continuation (in $\e$)
of unperturbed tori having ``all frequencies excited" \ie tori run by
linear flow $t\to \o t$ with $\o\=\dpr_y H_0(y_0)$ rationally independent
over $\zN$ and $N=\#$ of degrees of freedom. More delicate is what happens
to unperturbed
``resonant" tori\footnote{Here, the word ``resonant" which is related
to the ``resonances" of celestial mechanics, is used in a technically
different meaning from the one used throughout this paper.}
\ie invariant unperturbed tori for which there exist $n\in \zNn$
such that $\o\cdot n=0$. We shall argue (see next item P4))
that {\sl in general,
such tori are not analytically continuable for $\e\neq 0$}. Nevertheless
we will show, under suitable conditions,
how to construct, with the (intrinsically analytic)
methods outlined above, lower dimensional invariant (unstable) tori
when $\e\neq 0$.
For simplicity,
we shall discuss only a special case
of ``second order" nearly integrable Hamiltonian systems
with Hamiltonian function
\beq{1.12}
H\= \frac{1}{2} y^2 + \frac{1}{2} p^2 + \e f(x,q)
\eeq
where $(x,y)$ are symplectic variables as above (\ie with $x\in \tN$)
and so are $(q,p)$ with $q\in \tM$; hence the number of degrees of freedom
is $N+M$ and we shall study $N$--dimensional invariant tori.
The specification ``second order"
is due to the particular form in which can be cast
the Hamilton equations for $H$:
\beq{1.13}
\ddot x= -\e \dpr_x f\ ,\qquad \ddot q = -\e \dpr_q f\ .
\eeq
For $\e=0$, $N$--dimensional invariant tori
(up to a trivial linear and
symplectic change of coordinates\footnote{A transformation of
(standard) symplectic coordinates $(q,p)$ (of a $2d$--dimensional phase
space) is called {\sl symplectic} if it preserves the (standard) two form
$\sum_{i=1}^d dq_i\wedge dp_i$.})
are spanned by solutions of the form
$(y,p)=(\o,0)$, $(x,q)=(x_0+\o t, q_0)$.
%
>From classical transformation theory it follows that (if
$\o$ satisfies \equ{1.om}) the
evolution equations of $H$ in \equ{1.12} are equivalent
to the evolution equations for a Hamiltonian of the form
$\frac{y^2}{2} +\frac{p^2}{2} +\e f_0(q) +o(\e)$ where $f_0$ is the average
(w.r.t. $x$) over $\tN$ of $f$. Motivated by this observation, we consider
the Hamiltonian
\beq{1.12'}
\frac{1}{2} y^2 + \frac{1}{2} p^2 + \e f_0(q) +
\e^{2} \tilde f(x,q)
\= H_0(q,y,p;\e) + H_1(x,q;\e) \ ,\qquad H_1\=\e^2 \tilde f\ .
\eeq
Even though $H_0$ is not, in general, integrable,
{\sl if $q_0$ is a critical point for $f_0$}, $H_0$ still admits
the invariant $N$--torus $\cT_0$ spanned by $(y,p)=(\o,0)$, $q=q_0$ and
$x= x_0+ \o t$ and we want to study the persistence of such torus for the
full Hamiltonian. To attack the problem perturbatively,
we introduce a new analyticity parameter $\m$
with respect to which we shall make formal (and eventually convergent
with radius of convergence greater than one)
power series expansion and consider the Hamiltonian
$H_0+\m H_1$ (so that for $\m=1$ we recover the Hamiltonian \equ{1.12'}).
We also
make the following {\sl hyperbolicity assumption}:
we assume that $q_0$ is such that the matrix
$\e \dpr^2_q f_0(q_0)$ is {\sl positive definite}. Under these hypotheses
it is easy to prove
that there exists a (unique) formal expansion
\beq{1.15}
Z\=(Z_1,...,Z_d)\sim \sum_{k\ge 0} Z^k(\th;\e) \m^k\ ,\qquad \th\in
\torus^N\ ,\qquad d=2(N+M)
\eeq
such that $t\to Z(\o t)$ is a formal
quasi--periodic solution of the Hamiltonian equations
governed by $H_0+\m H_1$ and such that the set
$\{ Z^{0}(\th):\th\in \tN\}$ coincides with the unperturbed
torus $\cT_0$ introduced above. Uniqueness is
achieved by requiring \equ{1.av} where $\p_1$ denotes
the projection onto the $x$ variable.
Then, {\sl Theorem~\ref{thm1} holds also in this case}.
>From this result, as already remarked, it follows that the formal
power series is actually convergent but, what is more interesting in this
case, one can show that for $\e\neq 0$
small enough, {\sl the radius of convergence
(in $\m$) is greater than one} so that the set
$\{ Z(\th):\th\in \tN, \m=1\}$ is an invariant $N$--torus for the
Hamiltonian \equ{1.12'}. We finally mention that from \cite{Gr}
it follows easily that these tori are {\sl whiskered} in the sense of
\cite{A2}.
\giu
{\bf P4)}
Consider again the Hamiltonian \equ{1.12}. Indeed, one can consider
formal power series in $\e$ and one can show
that {\sl if $q_0$ is
a non degenerate critical point of the $x$--average of $f$
then there exists a (unique) formal power series
\equ{1.1} (with $d=2(N+M)$) such that $t\to Z(\o t)$
($\o$ satisfying \equ{1.om}) is a formal quasi--periodic solution
for \equ{1.12} and the set $\{Z^{0}(\th):\th\in\zN\}$ coincide
with the torus spanned by $y=\o$, $p=0$, $q=q_0$, $x= x_0 +\o t$.}
Uniqueness, again, is enforced by the requirement \equ{1.av}.
Also for such a formal series one can write down a tree expansion
completely analogous to those referred to in P1)$\div$P3) above.
However, we shall prove that
there exist chains $C=(R_1,...,R_h)$ such that if
$\cF_0$ denotes the family of {\sl all} trees with chain $C$
then
\beq{1.16}
\sum_{T'\in\cF_0} \L (T')\ \p_C(0)\neq 0\ .
\eeq
In view of this fact it seems natural
to {\sl conjecture that in the present case the formal power series
$Z$ is divergent}.
\section{Weighted trees}
\label{sec:par3}
\setcounter{equation}{0}
%
Here we describe the tree family $\cT^k_*$, which appears in the
basic formula \equ{1.tr}.
\giu
For the models P3) and P4) introduced in \S~\ref{sec:par2},
$\cT^k_*$ is simply the family of all labeled,
rooted trees with $k$ vertices\footnote{The basic terminology can
be found in any introductory book on graphs or in Appendix~A of
\cite{CF1}.}, which we will denote by $\cT^k$. In this case,
as is well known (see \eg \cite{Bo}),
$\# \cT^k = k^{k-1}$ and \equ{1.ft} is clearly satisfied.
\giu
To treat the cases P1) and P2) one has to distinguish, in the Taylor's
expansion, the contributions coming from $H_0$ and $L_0$ from those coming
from $H_1$ and $L_1$. To do this
we introduce the following family of ``weighted trees".
Given a rooted (unlabeled) tree $T$ we call a function of the vertices of
$T$
\beq{3.1}
\chi : v\in V(T) \to \chi_v\in\{0,1\}
\eeq
a {\sl weight function}. A {\sl weighted rooted tree} is a couple
$(T,\chi)$ with $T$ a rooted tree and $\chi$ a weight function. We now
denote $\widetilde \cT_k$ the set of weighted rooted trees satisfying
\beq{3.2}
(i) \quad \deg v\le 2 \ \implies \ \chi_v=1\ ,\qquad
(ii) \quad \sum_{v\in T} \chi_v = k\ .
\eeq
Notice that, in particular, all final vertices (\ie vertices
of degree 1) have weight 1
and that, for any $T\in \widetilde \cT_k$,
$\# V(T) \le 2k-1$ as is easy to verify\footnote{
Let $V_i=\{v\in V:\chi_v=i\}$, and let $k_i=\#V_i$, so that
$k=k_1$. It is well known (see, \eg, \cite{Bo} and recall our
convention on degree of the root) that
$\sum\limits_{V_0} \deg v + \sum\limits_{V_1} \deg v= 2(k_0+k_1)-1$.
If $v\in V_0$ then $\deg v\ge 3$, thus
the sum over $V_0$ can be bounded from below by $3 k_0$
while the sum over $V_1$ can be bounded from below by $k_1$. This leads
to $k_0\le k_1-1$ which is the claim. Such inequality is
optimal.}.
We now define the class $\cT_k$ of
{\sl labeled, weighted rooted trees} obtained from $\widetilde \cT_k$
by labelling with $k$ different labels the $k$ vertices with weight 1.
\giu
In cases P1) and P2) we let $\cT^k_*=\cT_k$; it
is easy to check that \equ{1.ft} holds also in this case\footnote{
Since (\cite{Bo})
$\# \widetilde \cT^h\le 4^h$ and $\# V(T)\le 2k-1$ for any
$T\in \widetilde \cT_k$ one sees that $\# \widetilde \cT_k\le 4^{2k}$.
Since the labels are attached to $k$ vertices, it
is $\# \cT_k\le k! 4^{2k}$.}.
\giu
The relation between Taylor's formula and trees may be based
on the following operation $*$ (that we now briefly discuss for the
case of $\cT_k$; for the case $\cT^k$ see
Appendix~B of \cite{CF1}).
If $T\in\cT_k$ we denote by $\widetilde T$ the tree in $\widetilde \cT_k$
obtained by removing the labels from $T$; we also denote
$T^0$ (or $\widetilde T^0$)
the unrooted tree obtained from $T$ (or $\widetilde T$)
by removing the edge $\eta r$ \ie by not distinguishing any more the root
from the other vertices; finally if $T$ (or $\widetilde T$) is an unrooted
labeled (or unlabeled) tree and $r$ is one of its vertices,
we denote by $T_r$ (or $\widetilde T_r$) the rooted tree obtained by adding
the edge $\eta r$ (\ie by decreeing that $r$ is the root).
Let $\chi\in \{0,1\}$ and $\ell\ge 1$,
let $h_i$ be $\ell$
positive integers such that $h_1+\cdots+h_\ell=k-\chi$ and
pick trees $T_i\in\widetilde \cT_{h_i}$.
We can form a tree $T\in \widetilde \cT_k$ with
root $r$ ($r$ being a vertex different from the vertices of $T_i$,
$\forall$ $i$) of weight $\chi_r\=\chi$ by setting
\beq{3.4}
T\=T_{h_1}*\cdots *T_{h_\ell}\=
\Big(T_{h_1}^0\cup\cdots\cup T^0_{h_\ell} \cup \{r\}+
\sum_{i=1}^\ell rr_i\Big)_r
\eeq
where $r_i$ is the root of $T_i$ (and summing an edge $e$ to a tree $S$
means, obviously, to add $e$ to $E(S)$).
Then, one has the following
\pro{pro:1}
Let $F$ be a complex valued function defined on trees in
$\widetilde \cT_k$
(for any $k$). Then
\beq{3.5}
\frac{1}{k!} \sum_{T\in \cT_k} F(\widetilde T) =
\sum_{\chi=0,1} \sum_{\ell=2-\chi}^{k-\chi} \frac{1}{\ell!}
\ \sum_{h_1+\cdots+h_\ell=k-\chi}\prod_{i=1}^\ell \frac{1}{h_i!}
\ \sum_{ T_i\in \cT_{h_i}} F(\widetilde T_{h_1}*\cdots *
\widetilde T_{h_\ell})\ .
\eeq
\epro
For the proof we refer to \cite{CF1}
(Corollary~B.1 of Appendix~B)\footnote{In \cite{CF1}
it is treated the case of $\cT^k$
(the $*$ operation in $\widetilde \cT^k$
is defined as above, replacing $\chi$ systematically by 1);
adapting the proof to $\cT_k$ is a trivial exercise.} .
%
\begin{figure}[hbt]
\begin{center}
\begin{picture}(350,80)
\thicklines
\put(5,20){\begin{picture}(30,60)
\put(0,40){\circle{8}}
\multiput(0,40)(15,0){3}{\circle*{3}}
\multiput(-2,48)(15,0){3}{{\scriptsize $1$}}
\put(0,40){\line(1,0){30}}
\end{picture}}
\put(60,20){\begin{picture}(30,60)
\put(15,40){\circle{8}}
\multiput(0,40)(15,0){3}{\circle*{3}}
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\put(0,40){\line(1,0){30}}
\end{picture}}
\put(115,20){\begin{picture}(30,60)
\put(0,40){\circle{8}}
\multiput(0,40)(15,0){2}{\circle*{3}}
\put(0,40){\line(1,0){15}}
\put(-2,48){{\scriptsize $1$}}
\put(13,48){{\scriptsize $0$}}
\put(35,50){{\scriptsize $1$}}
\put(35,28){{\scriptsize $1$}}
\put(30,50){\circle*{3}}
\put(15,40){\line(3,2){15}}
\put(30,30){\circle*{3}}
\put(15,40){\line(3,-2){15}}
\end{picture}}
\put(170,20){\begin{picture}(30,60)
\put(15,40){\circle{8}}
\multiput(0,40)(15,0){2}{\circle*{3}}
\put(0,40){\line(1,0){15}}
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\put(13,48){{\scriptsize $0$}}
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\put(35,28){{\scriptsize $1$}}
\put(30,50){\circle*{3}}
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\put(30,30){\circle*{3}}
\put(15,40){\line(3,-2){15}}
\end{picture}}
\put(225,20){\begin{picture}(45,60)
\put(15,40){\circle{8}}
\multiput(0,40)(15,0){4}{\circle*{3}}
\put(-2,48){{\scriptsize $1$}}
\put(13,48){{\scriptsize $0$}}
\put(28,48){{\scriptsize $1$}}
\put(43,48){{\scriptsize $1$}}
\put(0,40){\line(1,0){45}}
\end{picture}}
\put(295,20){\begin{picture}(45,60)
\put(15,40){\circle{8}}
\multiput(0,40)(15,0){3}{\circle*{3}}
\put(0,40){\line(1,0){30}}
\put(-2,48){{\scriptsize $1$}}
\put(13,48){{\scriptsize $0$}}
\put(29,48){{\scriptsize $0$}}
\put(45,50){\circle*{3}}
\put(30,40){\line(3,2){15}}
\put(45,30){\circle*{3}}
\put(30,40){\line(3,-2){15}}
\put(50,50){{\scriptsize $1$}}
\put(50,28){{\scriptsize $1$}}
\end{picture}}
\parbox{12cm}{\caption{\label{fig:wt1}The elements of
$\widetilde \cT_3$. (The
numbers $0$, $1$ are the values of the index $\chi$ for the corresponding
vertices and the encircled vertex is the root of the tree).}}
\end{picture}
\end{center}
\end{figure}
%
%
\begin{figure}[hbt]
\begin{center}
\begin{picture}(350,80)
\thicklines
\put(115,20){\begin{picture}(30,60)
\put(0,40){\circle{8}}
\multiput(0,40)(20,0){2}{\circle*{3}}
\multiput(-2,48)(20,0){2}{{\scriptsize $1$}}
\put(-4,25){$u_2$}
\put(16,25){$u_1$}
\put(0,40){\line(1,0){20}}
\end{picture}}
\put(170,20){\begin{picture}(30,60)
\put(0,40){\circle{8}}
\multiput(0,40)(20,0){2}{\circle*{3}}
\multiput(-2,48)(20,0){2}{{\scriptsize $1$}}
\put(-4,25){$u_1$}
\put(16,25){$u_2$}
\put(0,40){\line(1,0){20}}
\end{picture}}
\put(225,20){\begin{picture}(30,60)
\put(0,40){\circle*{3}}
\put(0,40){\circle{8}}
\put(-2,48){{\scriptsize $0$}}
\put(13,55){{\scriptsize $1$}}
\put(13,35){{\scriptsize $1$}}
\put(20,50){$u_1$}
\put(20,30){$u_2$}
\put(15,50){\circle*{3}}
\put(0,40){\line(3,2){15}}
\put(15,30){\circle*{3}}
\put(0,40){\line(3,-2){15}}
\end{picture}}
\parbox{12cm}{\caption{\label{fig:wt2}The elements of $\cT_2$
($u_1$ and $u_2$ are the labels of $\cT_2$).}}
\end{picture}
\end{center}
\end{figure}
%
\section{Compensations I (Maximal Hamiltonian tori)}
\label{sec:par4}
\setcounter{equation}{0}
\subsection{Tree expansion}
\label{subsec:4.1}
%
Consider the model introduced in P1) of \S~\ref{sec:par2}
and recall that there exists a unique formal solutions
$Z\sim\sum_{k\ge 0} \e^k Z^k$ satisfying
\equ{1.h1} and \equ{1.av}. Denote by $Z^{(1)k}$ the $x$--component
(\ie the first $N$ components) of the vector $Z^k$ and by
$Z^{(2)k}$ the $y$--component;
consistently, let $\dpr^{(1)}\=\dpr_x$
and $\dpr^{(2)}\=\dpr_y$. We also let
\beqno
[\ \cdot\ ]_k\= \frac{1}{k!} \frac{d^k}{d\e^k}\ (\cdot)|_{\e=0}
\eeqno
denote the $k^{\rm th}$ order operator which to a (possibly
formal) power
series $a\sim \sum a_k \e^k$ associates its $k^{\rm th}$ order
coefficient: $[ a ]_k\= a_k$.
Finally let
\beq{4.0}
A \= \dpr^2_y H_0(y_0)\ .
\eeq
With these definitions, we can rewrite
\equ{1.h1} as
\beq{4.1}
D Z^{(1)k} = A Z^{{(2)} k} + [\dpr^{(2)} H_0]_k^{(k-1)} +
[\dpr^{(2)} H_1]_{k-1}^{(k-1)}\ ,
\qquad
D Z^{{(2)} k} = - [\dpr^{(1)} H_1]_{k-1}^{(k-1)}\ ,
\eeq
where the suffix $\phantom{[\cdot]}^{(k-1)}$
means that the argument of the
function within square brackets is, for $k\ge 2$, the polynomial
in $\e$ of degree $(k-1)$ given by
\beq{4.pol}
x=\th+\sum_{h=1}^{k-1} \e^h Z^{{(1)} k}\ ,
\qquad y= y_0+ \sum_{h=1}^{k-1} \e^h Z^{{(2)} k}\ ,
\eeq
and, for $k=1$, is $(x,y)=(\th,y_0)$.
Since the average of $[\dpr^{(1)} H_1]_{k-1}^{(k-1)}$ vanishes
(as is clear from the second of \equ{4.1})
we can apply to it the operator $D^{-1}$ obtained by inverting
the constant coefficient operator\footnote{If $f$ is a (smooth) function
on $\tN$ with vanishing mean value we denote by $D^{-1} f$
the unique solution with vanishing mean value of the equation for $g$:
$D g = f$. Expanding in Fourier series one has $g(\th)=D^{-1}f(\th)=
-i \sum\limits_{n\in\zNn} \frac{f_n}{\o\cdot n} \exp({i n\cdot \th})$,
where
$i=\sqrt{-1}$: the inversion of $D$ introduces the small divisors.}
$D$ and rewrite \equ{4.1} in a more compact way as
\beq{4.2}
D Z^{(\r)k}= (2-\r) A Z^{(2)k}+
\sum_{\chi=0,1} (-1)^{3-\r} [\dpr^{(3-\r)} H_\chi]^{(k-1)}_{k-\chi}\ ,
\qquad (\r=1,2)\ .
\eeq
Notice that while, by \equ{1.av}, the average of $Z^{(1)k}$ vanishes,
the average of $Z^{(2)k}$ can be read (for $\r=1$) from \equ{4.2}
by integrating over $\tN$:
\beq{4.3}
\ig_{\tN} Z^{(2)k}=- A^{-1} \sum_{\chi=0,1} \ig_{\tN} [\dpr^{(2)}
H_\chi]^{(k-1)}_{k-\chi}\ .
\eeq
Taking the $n$--Fourier coefficient of \equ{4.2}, \equ{4.3}
one gets
\beq{4.4}
Z^{(\r)k}_n= \sumsud{\s\in \{0,1,2\}^*}{\chi\in\{0,1\} }
\lg n\rg^{-\s}\
\Big\{ [D^{(\s,\r)} H_\chi]_{k-\chi}^{(k-1)}\Big\}_n\ ,\qquad
\big(\lg n\rg\= \o\cdot n\big)\ ,
\eeq
where $D^{(\s,\r)}$ is the vector--valued
operator\footnote{
For example the $j^{\rm th}$ component of $D^{(2,1)}$ is given by
$D_{j}^{(2,1)}=\sum\limits_{s=1}^N
\frac{\dpr^2 H_0}{\dpr y_j\dpr y_s}(y_0) \frac{\dpr}{\dpr x_s}$.}
\beq{4.5}
D^{(\s,\r)}\=(-1)^{1-\s\r} \ i^{-\s}\ A^{\s-1}\
\dpr^{(4-\s-\r)}\ ,
\eeq
and the $*$ attached to the range of $\s$
means that the
following {\sl constraints} have to be satisfied:
$\s+\r\in \{2,3\}$ and $\s=0$ $\iff$ $n=0$
in which case we adopt the convention that $\lg
n\rg^\s=0^0\=1$.
Notice that $D^{(\s,\r)} H_\chi=0$ if $\chi=0$ and $\s+\r=3$
(as in such a case $\dpr^{(4-\s-\r)}=\dpr_x$ and $H_0$ is independent of
$x$).
\giu
We shall now use Taylor's formula in the following form.
If $f:x\in \real^m \to f(x)\in \real$ is a $C^\io$ function and
if $a(\e)\sim\sum_{s\ge 1} \e^s a^{(s)}$ is a $\real^m$--valued
(possibly formal) power series, then
\beq{4.6}
f\big(a(\e)\big) \sim f(0) + \sum_{h\ge 1} \e^h
\sum_{\ell=1}^h \frac{1}{\ell!}
\sum_{{h_1+\cdots+h_\ell=h}\atop{1\le h_i\le \ell}}
\sum_{{j_1,...,j_\ell}\atop{1\le j_i\le m}}
\frac{\dpr^\ell f}{\dpr x_{j_1}\cdots\dpr x_{j_\ell}}(0)\
a^{(h_1)}_{j_1}\cdots a^{(h_\ell)}_{j_\ell}\ .
\eeq
Thus, if $z\=(x,y)$ and $z^{(1)}\=x$, $z^{(2)}\=y$, by \equ{4.4}
and \equ{4.6} we get for the $j^{th}$ component of the vector
$Z^{(\r)k}_n$, and for $k\ge 2$,
\beqa{4.7}
&& Z^{(\r)k}_{nj}= \\
&& \sum_{ {\s\in \{0,1,2\}^*}\atop{\chi\in \{0,1\}}}
\sum_{\ell=2-\chi}^{k-\chi} \frac{1}{\ell!}
\sumsut{1\le h_i\le k-1}{\Sigma h_i=k-\chi}{(1\le i\le \ell)}
\sumsut{n_i\in\zN}{\Sigma n_i=n}{(0\le i\le \ell)}
\sum_{{\r_1,...,\r_\ell}\atop{\r_i\in \{1,2\}}}
\sum_{ {j_1,...,j_\ell}\atop{1\le j_i\le N}}
\lg n\rg^{-\s}
\Big\{ \frac{ \dpr^\ell\ D_{j}^{(\s,\r)} H_\chi}{\dpr z^{(\r_1)}_{j_1}
\cdots \dpr z^{(\r_\ell)}_{j_\ell}} \Big\}_{n_0}
\prod_{i=1}^\ell Z^{(\r_i)h_i}_{n_i j_i}\nonumber\ ,
\eeqa
where the derivatives of $H_\chi$ are evaluated at
$(\th,y_0)$ and then one takes the $n_0$--Fourier coefficient
(with respect to $\th$); for $k=1$, since $[D^{(\s,\r)} H_0]_1^{(0)}=0$,
one has the simple formula
\beq{4.7'}
Z^{(\r)1}_{nj}=
\sum_{\s\in \{0,1,2\}^*} \lg n\rg^{-\s} \{D^{(\s,\r)}_j H_1\}_n\ .
\eeq
We are ready to prove the following {\sl tree expansion formula}
(recall \equ{1.ga}, \equ{1.dl})
\beq{4.8}
Z^{(\r_0)k}_{nj_0} =
\frac{1}{k!} \sum_{T_r\in\cT_k} \quad
\sum_{\a:V\to \a_v\in \zN \atop \Sigma_v \a_v = n}\quad
\sumsud{\b:V\to\b_v\in B}{\r_r=\r_0}
\quad
\sumsud{j:V\to j_v\in\{1,...,N\}}{j_r=j_0}\
\prod_{v\in T_r} \big\{ \L_v(T_r,\b) H_{\chi_v} \big\}_{\a_v}\
\prod_{v\in T_r} \g_v \ ,
\eeq
where: the index set $B$, which
depends on the function $\d_v$,
is defined as
\beqa{4.B}
& B & \=\Big\{\b=(\s,\r):\ \s\in\{0,1,2\};\ \r\in\{1,2\};\
{\rm s.t.}\nonumber\\
&& \qquad
\s+\r\in\{2,3\}\ ,\s=0\ \iff \ \d_v= 0\Big\}\ ;
\eeqa
the scalar operator $\L_v(T_r,\b)$ is given by
\beq{4.9}
\L_v(T_r,\b)\=
D^{(\s_v,\r_v)}_{j_v}
\prod_{v'\in \calN_v} \dpr_{j_{v'}}^{(\r_{v'})}\ ,
\qquad
\calN_v\=\{v'v>w'$.
%
We shall classify resonances by assigning to them an integer
$s\=s_R\in\{0,1,2\}$, which we shall call {\sl index of the
resonance $R$}. Then, to each resonance $R\subset T$ we
shall associate a family $\cF_R$ of trees $T'$ obtained by
(possibly) changing the edges connecting $R$ with $T\bks R$
and choosing a suitable set of indices .
The family $\cF_R$ and the index $\b'\=\b'(T')$ will be chosen so that
(recall \equ{1.pi})
\beq{4.12}
\sum_{T'\in\cF_R} \L(T',\a,\b')\
\pi_R(x;T',\a,\b')=O(x^s)
\eeq
and the family $\cF_C$ will simply be given by
\beq{4.13}
\cF_C=\bigcup_{i=1}^h \cF_{R_i} \ .
\eeq
Let $R$ be a resonance and let $u$ be its first vertex
and $w'$ the first vertex following $R$. We define the {\sl index
of $R$} as
\beq{4.in}
s_R\=\s_u+\r_u-\r_{w'}\ .
\eeq
Thus if $C=(R_1,...,R_h)$ is a chain of resonances and if $\bar \s$
denotes its order (see \S~\ref{sec:par2}), then
\beq{4.14}
\sum_{i=1}^h s_{R_i}= \bar \s +\r_{u_1} - \r_{z}
\eeq
where $z$ is the first vertex following $R_h$ (which, by convention,
is the last resonance in the chain $C$). Hence, if \equ{4.12} holds,
from \equ{4.13}, \equ{4.14}, \equ{1.pi} and the definition of $\L$
\equ{4.11'} it follows easily that
\beq{4.15}
\sum_{T'\in \cF_C} \L(T',\a,\b')\ \pi_{C}(x;T',\a,\b')=
O(x^{\bar \s +\r_{u_1} - \r_{z}})
\eeq
which implies that the chain $C$ is compensable.
\giu
Let $R\subset T$ be a resonance and let $\hat u u$ and $w {w'}$
be the edges connecting $R$ with $T\bks R$, with $\hat u>u\ge w>{w'}$
(hence $u,w\in R$).
We proceed by constructing the family $\cF_R$.
If $\r_{w'}\neq 1$ we set $\cF_R'=\{T\}$; if
$\r_{w'}=1$ we let $\cF_R'$ be the family of all trees
$T'$ obtained from $T$ by replacing the edge $w {w'}$ with the
edge $\bar w {w'}$, as $\bar w$ varies in $R$. Hence,
$T\in\cF_R'$ and if $\r_{w'}=1$, $\# \cF_R'=\# R$.
If $\s_u+\r_u\neq 3$ we set $\cF_R''=\{T\}$; if $\s_u+\r_u=3$
(\ie $(\s_u,\r_u)=(1,2)$ or $(\s_u,\r_u)=(2,1)$)
we let
$\cF_R''$ be the family of all trees
$T'$ obtained from $T$ by replacing the edge $\hat u u$ with the
edge $\hat u\bar u$, as $\bar u$ varies in $R$.
As above,
$T\in\cF_R''$ and if $\s_u+\r_u=3$, $\# \cF_R'=\# R$.
In the first case (\ie for $T'\in\cF_R'$)
we do not modify the values $\b_v$
(\ie $\b'(T')\=\b$). In the second case (\ie for $T'\in \cF_R''$)
we define $\b'\=\b'(T')$ as follows.
If $v\notin R$ then $\b'_v=\b_v$.
Recall that, by definition,
the first vertex of $R$ considered as subtree of $T'$ is
$\bar u$ while the first vertex of $R$ considered as subtree of $T$ is $u$.
Let $\bar u\neq u$ (otherwise, obviously, we set $\b'\=\b$) and
consider the path $P(u,\bar u)$ connecting $u$ with $\bar u$.
The path $P$ will be formed by $p\ge 2$
ordered vertices that we denote $v_i$: the order is such that
$v_1=u$, $v_p=\bar u$ and the edges of the path are $v_1v_2$,...,$v_{p-1}v_p$.
We then set
\beqa{4.be}
& \b_{v_p}'\= \b_{v_1}\ , &\nonumber\\
& \b_{v_i}'\=\big( \s_{v_{i+1}},
4-\s_{v_{i+1}}-\r_{v_{i+1}}\big)\ , &{\rm for}\ 1\le i\le p-1\nonumber\\
& \b_v'\=\b_v\ , &\forall\ v\notin P(u,v)\ .
\eeqa
It is easy to see that this definition is well posed
(see also (ii) of the following Remark).
Finally, we define
\beq{4.fa}
\cF_R\=\cF_R' \cup \cF_R''\ .
\eeq
\rem{rem:4.1} (i) If $s_R=0$, it is $\cF_R=\{T\}$ and $\b'\=\b$.
\noindent
(ii)
The map $\b\to \b'$ is involutive: more precisely,
if we denote by $\b'(\b;u,\bar u)$ the map defined in \equ{4.be}
(definition which depends on the ordered path $P(u,\bar u)$)
then $\b'(\b'(\b;u,\bar u);\bar u,u)=\b$.
\noindent
(iii) One might say that the families $\cF_R'$ and $\cF_R''$ are
constructed ``going around" the resonance $R$ with a ``discrete
curve" obtained by moving, respectively, the edge connecting
$R$ with $w'$ and the edge connecting $\hat u$ with $R$.
This interpretation might explain the name ``index" given to the quantity
$s_R$: $\cF_R$ is obtained by ``going around" $R$ exactly $s_R$ times.
\noindent
(iv) (On the definition of $\cF_R'$)
In practice, moving around the edge $\bar w z$ produces a factor
proportional to $\a_{\bar w}$ in \equ{4.11'} coming from the
$\dpr^{(\r_z)}=\dpr^{(1)}$ appearing in \equ{4.9} (as
$z\in \calN_{\bar w}$). It is easy to see that, for $x=0$,
summing $\L$ $\pi_R(x)$ over the family $\cF_R'$ produces
a {\sl common} factor $\sum_{\bar w\in R}\a_{\bar w}$
which vanishes by definition of resonance.
\noindent
(v) (On the definition of $\cF_R''$)
The idea is similar: one wants to produce a factor proportional
to $\a_{\bar u}$ when moving around the ``first" edge
$\hat u \bar u$ connecting $R$ with $\hat u$. But now the
situation is more delicate as changing the connection $\hat u\bar u$
{\sl changes the order} in $R$
and, consequently, change the small divisors and also
the structure of the derivatives (since both $\g_v$ and $\L_v$
depend on the order).
Since one wants eventually
to set $x=0$ and collect the factor $\sum_{\bar u\in R}\a_{\bar u}$
one sees the necessity of changing the values of the indices $\b$.
In fact $\b'$ is defined in such a way that, when $x=0$, one can
factor out the {\sl product} of divisors and the {\sl products}
of the operators (derivatives) $\L_v$.
\erem
%
With the above remarks in mind, it is not difficult to verify that
\equ{4.12} holds, proving Theorem~\ref{thm1} in case P1).
\section{Compensations II (Maximal Lagrangian tori)}
\label{sec:par5}
\setcounter{equation}{0}
%
Consider the model introduced in P2) of \S~\ref{sec:par2} for a
real--analytic Lagrangian $L\= L_0(y)+\e L_1(x,y)$.
Formal quasi--periodic solutions of the Lagrangian equations have the form
$x(t)=Z(\o t)$ for an $\o$ satisfying \equ{1.om} and a function $Z(\th)$
which is the (unique) formal
solution $Z\sim\sum_{k\ge 0} \e^k Z^k$, with $Z\in\real^N$, of \equ{1.h2}.
The vector valued function $Z^k$
can be put in a form which is amazingly similar to the Hamiltonian case P1).
\nin
Let us denote by $Z^{(1)k}$ the full vector $Z^k \in \real^N$ and by
$Z^{(2)k}$ the vector $DZ^k=DZ^{(1)k}$ (where $D\= \o\cdot\partial_\th$)
and, as before,
let $\dpr^{(1)}\=\dpr_x$, $\dpr^{(2)}\=\dpr_y$ and
$A_L \= \dpr^2_y L_0(y_0)$.
Equation \equ{1.h2} can then be written as
\beq{5.1}
A_L D Z^{{(2)} k} = - D[\dpr^{(2)} L_0]_k^{(k-1)} -
D[\dpr^{(2)} L_1]_{k-1}^{(k-1)} + [\dpr^{(1)} L_1]_{k-1}^{(k-1)}\ .
\eeq
adopting the same convention used in \equ{4.1}: in particular the arguments
of the functions within square brackets are as in \equ{4.pol}.
As for the previous case (see \equ{4.4}), the $n$--Fourier coefficient of
\equ{5.1} is
\beq{5.2}
Z^{(\r)k}_n= \sumsud{\s\in \{0,1,2\}^*}{\chi\in\{0,1\} }
\lg n\rg^{-\s}\
\Big\{ [D_L^{(\s,\r)} L_\chi]_{k-\chi}^{(k-1)}\Big\}_n
\eeq
where $D_L^{(\s,\r)}$ is now the vector--valued
operator
\beq{5.3}
D_L^{(\s,\r)}\=(-1)^{1-(\s+\r)} \ i^{-\s}\ A_L^{-1}\
\dpr^{(4-\s-\r)}\ ,
\eeq
and $\{0,1,2\}^*$ is the same set of equation \equ{4.4}.
\nin
The Lagrangian problem is now in a form which is identical to
the Hamiltonian case, except for the definition of $D_L^{(\s,\r)}$. Therefore
the tree expansion formula for the component $j_0$ of the $n$--Fourier
coefficient of $Z^k$, with
fixed values of $\r_0$ and $j_0$ at the root $r$, is given again by
\equ{4.8} with the only proviso of replacing $D^{(\s_v,\r_v)}_{j_v}$ in
\equ{4.9} with $D^{(\s_v,\r_v)}_{L,j_v}$.
\noindent
Also, the families $\cF$ of trees for which there are compensations are found
exactly as in the Hamiltonian case and we refer to \S~\ref{subsec:4.2}
for details.
\section{Compensations III (Lower dimensional tori)}
\label{sec:par6}
\setcounter{equation}{0}
%
Recall the notations of \S~\ref{sec:par2}, P3). Because of the particular
form of the Hamiltonian, the formal solution\footnote{Recall that here $\e$
is a fixed real number different from zero, while the (complex)
perturbation parameter appearing in the formal power series is $\m$.}
$Z(\th,\m)$ is of the form
$Z\=(X,Q,DX,DQ)$ where, as usual,
$D\=\o\cdot \dpr_\th$, $\th\in\tN$ and $X\in \rN$, $Q\in \rM$.
Denote
\beq{6.2}
Z^{(1)}\=X\ ,\quad
Z^{(2)}\=Q\ ,\quad
\dpr^{(1)}\=\dpr_x\ ,\quad
\dpr^{(2)}\=\dpr_q\ ,\quad
A\=\e \dpr^2_q f(q_0)\ (>0)\ .
\eeq
One checks immediately that the recursive equations for $Z^{(\r)k}$
($\r=1,2$) are
\beq{6.eq}
\Big( -D^2+(\r-1) A\Big) Z^{(\r)k}= \sum_{\chi=0,1}\big[\dpr^{(\r)} H_\chi
\big]^{(k-1)}_{k-\chi}\ ,\qquad \r=1,2
\eeq
where the argument of the derivatives of
$H_\chi$ is $x=\th$, $q=q_0$. From \equ{6.eq} one can see that
the average (over $\tN$) of $Z^{(1)k}$ vanishes, while the average of
$Z^{(2)k}$ is as in \equ{4.3} but with the minus sign replaced by a plus.
Taking Fourier coefficients of \equ{6.eq} we get the analogous of \equ{4.4},
namely
\beq{6.fo}
Z^{(\r)k}_n= \sumsud{\s\in \{0,2\}^*}{\chi\in\{0,1\} }
\lg n\rg^{-\s}\
\Big\{ [D^{(\r)}_n H_\chi]_{k-\chi}^{(k-1)}\Big\}_n\ ,\qquad
\big(\lg n\rg\= \o\cdot n\big)\ ,
\eeq
which differs from \equ{4.4} for the range of $\s$ and relative constraints:
\beq{6.co}
\s \in \{0,2\}^*\ \iff \ \s+\r\in\{2,3\}\ ,\qquad
n=0 \ \implies \ \s=0\ ,
\eeq
and for the definition of the vector valued operator $D^{(\r)}_n$:
\beq{6.dr}
D^{(\r)}_n\= \Big( \lg n\rg^2+ A\Big)^{1-\r} \ \dpr^{(\r)}\ .
\eeq
Notice that the components $D^{(\r)}_{nj}$ are $N$ if $\r=1$ and $M$
if $\r=2$; we therefore let
\beq{6.N}
N_1\=N\ ,\qquad N_2\= M \ \implies j\in\{1,...,N_\r\}\ .
\eeq
Recall that $A=\e \dpr^2 f_0(q_0)$ is
positive definite, and so is $a+A$ for any $a\ge 0$; thus
\beq{6.aa}
\|(a+A)^{-1}\| \le \frac{b}{|\e|}
\eeq
for a suitable constant $b$ depending only on $\dpr^2 f_0(q_0)$.
We also remark that, by \equ{6.co}, $\r=2$ implies $\s=0$ (hence no
divisors)
while if $\r=1$ then $\s=2$ and $n\neq 0$, in which case the
divisors are $\lg n\rg^2$.
In view of these remarks, we see that \equ{4.8} holds also in the present
case provided we change the following items:
$N$ in the fourth sum is replaced (see \equ{6.N})
by $N_{\r_v}$; the index set $B$ is defined as
\beqa{6.B}
& B & \=\Big\{\b=(\s,\r):\ \s\in\{0,2\};\ \r\in\{1,2\};\
{\rm s.t.}\nonumber\\
&& \qquad
\s+\r\in\{2,3\}\ ,\d_v=0\ \implies \ \s=0 \Big\}\ ;
\eeqa
finally the operator $\L_v$ now depends also on $\a$:
$\L_v(T_r,\b)$ in \equ{4.8} is now replaced by
\beq{6.L}
\L_v(T_r,\a,\b)\=
D^{(\r_v)}_{\d_v j_v}
\prod_{v'\in \calN_v} \dpr_{j_{v'}}^{(\r_{v'})}\ .
\eeq
Also \equ{4.11'} is readily adapted replacing $N$ by $N_{\r_v}$
and $\L_v$ with \equ{6.L}.
We can proceed to define the families of trees $\cF$ and relative
indices $\b'$ which exhibit compensations. Given a resonance
$R$ (and a choice of $\a$ and $\b$), we define the index of $R$ and
the family $\cF_R''$ and relative indices $\b'$
exactly in the same way we did in
\S~\ref{subsec:4.2}. Also the family $\cF_R'$
is defined in the same way but the relative indices $\b'$ are now
defined in a slightly different way
(due to the different definition of $D^{(\r)}_n$): we let
(same notations as in \S~\ref{subsec:4.2})
\beqa{6.be}
& \b_{v_p}'\= \b_{v_1}\ , &\nonumber\\
& \b_{v_i}'\=\b_{v_{i+1}}\ ,
&{\rm for}\ 1\le i\le p-1\nonumber\\
& \b_v'\=\b_v\ , &\forall\ v\notin P(u,v)\ .
\eeqa
With these definitions it is easy to check that
\equ{4.12} holds and hence that
Theorem~\ref{thm1} is valid in case P3) too.
\giu
We close by a remark on the $\m$--radius of convergence
of $\sum_{k\ge 1} \m^k Z^k$. It is an easy exercise to
adapt the estimates in \cite{CF1} to the present case
and to check how the radius of convergence depends on $\e$.
In fact, observing that, by \equ{6.aa}, one has
\beq{6.es}
|\lg n\rg^{-\s}|\ |\big(\lg n\rg^2+A\big)^{1-\r}|
\le \max\{\g^2, \frac{b}{|\e|}\}\ \
\max\{|n|^{2\t}\ ,\ 1\}\ ,
\eeq
leading to an estimate on the radius of convergence $\m_0$
of the type
\beq{6.mu}
\m_0\ge {\rm const}\ \min\{\e\ ,\ \g^{-2}\}\ .
\eeq
Thus $\m=\e^2$ (or $\m=\e^c$ with any $c>1$) is within the
domain of analyticity provided $\e$ is small enough.
\section{An example with no compensations}
\label{sec:par7}
\setcounter{equation}{0}
%
Consider the Hamiltonian \equ{1.12} with
\beq{p.1}
f(x,q) =\sum_{s\geq 1} f_s
e^{i (n^{(s)}\cdot x + m^{(s)}\cdot q)}
\eeq
where $n^{(s)} \in \integer^N$, $m^{(s)} \in \integer^M$
are given integer vectors (with $|n^{(s)}|+|m^{(s)}|>0$) and the Fourier
coefficients $f_s$ decay exponentially fast with $|n^{(s)}|+|m^{(s)}|$.
Let $q_0$ be a non degenerate critical point of the $x$--average of $f$
(i.e. of $f_0=\sum_{s\geq 1} f_s e^{i m^{(s)}\cdot q}$); then,
as for the previous cases, there exists a (unique) formal power series
\beq{p.2}
Z \= (Z_1,\dots , Z_d)\ \sim\ \sum_{k\geq 0} Z^k(\th)\e^k\ ,
\ \th\in\torus^N
\eeq
with $d=2(N+M)$,
such that $t\to Z(\o t)$ is a formal quasi--periodic solution for \equ{1.12}
and the set $\{Z^0(\th):\th\in\torus^N\}$ coincide with the torus spanned by
$y=\o$, $p=0$, $q=q_0$, $x=x_0+\o t$, where $\o\in\real^N$ satisfies condition
\equ{1.om}.
\nin
Expanding $Z^k$ in Fourier series also in the variable $q$, besides the variable
$x$ (see \equ{1.2}), it is still possible to write $Z_n^k$ as in
\equ{1.tr} provided one makes the following changes.
$\cT^k_* = \cT^k$; $B$ is the trivial set $B=\{\b =\s = 2\}$;
$\L(T,\a,\b)$ is replaced by
\beq{p.3}
\L(T,\hat \a,\b) = \sum_{\a':V\to \a_v'\in \integer^M}\ \prod_{v\in V}
f_{\hat\a_v} \prod_{vv'\in E} \hat\a_v \cdot \hat\a_{v'}
\eeq
where $\hat\a_v\= (\a_v,\a_v')\in \integer^{N+M}$; finally
$\g_v = \lg \d_v\rg^{-2}\=\bigl(\o\cdot\sum_{v'\leq v} \a_{v'}\bigr)^{-2}$.
\nin
It is then clear that in order to have compensations it would be
sufficient to have resonances in the variable
$\hat\a$ whenever there are resonances in the variable $\a$.
In other words compensations take place if the Fourier modes in \equ{p.1}
are such that
\beq{p.4}
\sum_{s\in I} n^{(s)} = 0 \quad \implies \sum_{s\in I} m^{(s)} = 0
\eeq
$\forall I\subset \natural$; in fact in such a case one could repeat
word--by--word the arguments in \cite{CF1}.
\nin
In general, if \equ{p.4} does not hold, compensations (of the type
described in this paper) do not occur as it is shown by the following
example.
\giu
Take $N=2$, $M=1$, fix a scalar integer $n\neq 0$ and let
\beq{p.5}
f(x_1,x_2,q) = 2\{\cos x_1 + \cos x_2 +
\cos (x_1 + x_2 - q) + \cos (n x_1 + x_2 + q)\}\ .
\eeq
Hence, the range of $\hat\a$ is the set
$\{\pm (1,0,0),\ \pm (0,1,0),\ \pm (1,1,-1),\ \pm (n,1,1)\}$; for
definiteness we also fix $\o = (\sqrt{2},-1)$.
%
\begin{figure}[hbt]
\begin{center}
\begin{picture}(400,100)
\thicklines
\put(5,60){\circle{8}}
\put(5,60){\circle*{3}}
%\put(10,50){$u_1$}
%\put(50,50){$w_1$}
%\put(100,50){$u_2$}
%\put(140,50){$w_2$}
%\put(190,50){$u_3$}
%\put(230,50){$w_3$}
%\put(280,50){$u_4$}
%\put(320,50){$w_4$}
\multiput(65,60)(90,0){4}{\line(1,0){30}}
\multiput(65,60)(90,0){4}{\circle*{3}}
\multiput(95,60)(90,0){4}{\circle*{3}}
\put(35,60){\oval (75,30)}
\put(125,60){\oval (75,30)}
\put(215,60){\oval (75,30)}
\put(305,60){\oval (75,30)}
\put(350,65){{\scriptsize ($n$,1,1)}}
\put(365,45){$z$}
\put(65,35){$R_1$}
\put(155,35){$R_2$}
\put(245,35){$R_3$}
\put(335,35){$R_4$}
\parbox{14cm}{\caption{\label{fig:d1} Divergent contribution: the chain
$C=( R_1,R_2,R_3,R_4 )$ is made of 4 adjacent resonances with $|R_i|=3$ (in
this symbolic picture only two vertex are drawn).}}
\end{picture}
\end{center}
\end{figure}
\nin
We fix a tree $T\in \widetilde \cT^k$, with $k = 3h+1$, and a function
$\hat \a_v:V(T)\to \integer^3$ so that $T$ contains
a chain $C=( R_1,...,R_h )$ made of $h$ identical resonances $R_i\= \{v_1,v_2,
v_3\}$, with $\hat\a_{v_1}=(-1,-1,1)$, $\hat\a_{v_2}=(0,1,0)$,
$\hat\a_{v_3}=(1,0,0)$,
and the last vertex $z$, following the chain, with
$\hat\a_z = (n,1,1)$ (see Figure~\ref{fig:d1}).
\noindent
Let $\cF_C$ be the family of all trees $T\in\cT^k$
which contains the chain $C$.
A lengthy but straightforward computation shows that
the $i^{th}$ component of the vector valued function $\bar \pi_C(x)$
(see \equ{1.11}) for $h = 1$ is:
\beqa{p.6}
&(\bar\pi_{C}(x))_i \=
(\bar \pi_{R_1}(x))_i & \= \prod_{v\in V} f_{\hat\a_v}
\sum_{u,w\in R_1}
\hat e_i \cdot \hat \a_u \ \Big(
\prod_{vv'\in E(R_1)} \hat\a_v \cdot \hat\a_{v'}\Big) \
\hat \a_w \cdot \hat \a_z \
\pi_{R_1}(x)\nonumber \\
&&\ =\ 2\sum_{j=1}^3 (\d_{i3} \d_{j3} + x^2 B_{ij}(x))\hat \a_{zj}
\eeqa
where $\d_{ij}$ is the Kronecker's symbol, $\hat e_i$ is the vector
of $j^{th}$ component $\d_{ij}$ and
\beq{p.7}
B(x)=\pmatrix{{{-2(6-x^2)}\over {(2-x^2)^2}}&
{{8\sqrt{2}+18 x^2 - 9x^4 + x^6}\over {(2-x^2)^2 (1-x^2)^2}}&
{{6-x^2}\over {(2-x^2)^2}}\cr
&\phantom{a} &\cr
{{8\sqrt{2}+18 x^2 - 9x^4 + x^6}\over {(2-x^2)^2 (1-x^2)^2}}&
{{-2(3-x^2)}\over {(1-x^2)^2}}&
{{3-x^2}\over {(1-x^2)^2}}\cr
&\phantom{a} &\cr
{{6-x^2}\over {(2-x^2)^2}}&
{{3-x^2}\over {(1-x^2)^2}}&0} .
\eeq
Hence $\bar \pi_{R_1}(0)\neq 0$. For $h>1$, one simply has:
\beq{p.8}
\bar \pi_{C}(x) = 2^h (A+x^2 B)^h \hat\a_z\quad,\quad A_{ij}=\d_{i3} \d_{j3}\ .
\eeq
Thus, if $x = \o\cdot \a_z = \sqrt{2}n-1$, taking the third component of
\equ{p.8} (the component for which condition \equ{p.4} is violated)
and assuming that $n\geq h$, one easily checks that
\beq{p.9}
\bigl( (A+x^2 B)^h \hat\a_z\bigr)_3 \geq 1 + O({1\over n})
\eeq
which implies that $C$ is a non compensable chain.
%%%%%%%%%%%%%%%%%%% bibliography:
\begin{thebibliography}{99}
%
{\small
\bibitem{A1}
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\end{document}
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