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Arnold diffusion, homoclinic splitting, KAM
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% hj2.tex
\def\Di{27 Novembre 1998}
\headline{\hss \ottorm Draft \#20}
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\bibl=0
% Per poter cambiare a piacimento il formato dei riferimenti
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%
% 1: citare nella forma esemplificata da \ref{B}{2}{20}}
% ove XXX e' un simbolo per le iniziali e 2 distingue i lavori con
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\*
\0{\it Internet:
Authors' preprints at: {\tt http://ipparco.roma1.infn.it}
\0\sl e-mail: {\it users:} giovanni, gentile, vieri,
{\it address}: {\tt @ipparco.roma1.infn.it}
}}
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%**end of header
\fiat
\null
\hskip1.truecm
\centerline{\titolone Lindstedt series and Hamilton--Jacobi equation}
\centerline{\titolone for hyperbolic tori in three time scales problems}
\*\*
\centerline{\titolo G. Gallavotti, G. Gentile, V. Mastropietro}
\*
\centerline{Universit\`a di Roma 1,2,3 }
\centerline{\Di}
\vskip.8truecm
\line{\vtop{
\line{\hskip1.5truecm\vbox{\advance \hsize by -3.1 truecm
\\{\cs Abstract.}
{\it Interacting systems consisting of two rotators and a pendulum are
considered, in a case in which the uncoupled systems have three very
different characteristic time scales. The abundance of unstable quasi
periodic motions in phase space is studied via Lindstedt series. The
result is a strong improvement, compared to our previous results, on
the domain of validity of bounds that imply existence of invariant
tori, large homoclinic angles, long heteroclinic chains and
drift--diffusion in phase space.}}\hfill} }}
\vskip1.5truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0{\titolo \S 1. The Hamiltonian system.}
\numsec=1\numfor=1\*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0{\bf 1.1.}
Let $(\f,\a_1,\a_2)=(\f,\aa)\in \TTT^3$ be three angles (\ie
positions on circles); let $(I,A_1,A_2)=(I,\AA)\in \RRR^3$ be their
conjugate momenta (or ``{\it actions}''). We consider the Hamiltonian
function, depending on two parameters $\e,\h>0$, defined by
%
$$ \HH = \hdp \O_{1}A_1 + \h \fra{A_1^2}{2J} + \hdm \O_{2} A_2
+ \fra{I^2}{2J_0} +J_0 g_0^2 (\cos\f-1)+\e f(\f,\a_1,\a_2)
\; , \Eq(1.1) $$
%
with $f$ an {\it even} trigonometric polynomial of degree $N,N_0$ in
$\aa,\f$ respectively; $\O_{1},\O_{2},J,J_0,g_0$ are positive constants.
This system describes two rotators (one anisochronous, labeled $\#1$,
and one isochronous, labeled $\#2$) interacting with a pendulum which
has its free (\ie with $\e=0$) unstable equilibrium position at
$I=0,\f=0$ and the stable one at $I=0,\f=\p$.
The scale of frequency of the pendulum is $O(1)$ in $\h$;
at the same time the two rotators rotate at
constant speed $O(\hdp)$ and $O(\hdm)$ respectively. Hence the system has
three time scales: we assume $\h<1$ so that the {\it slow} rotator is
the \#1 rotator.
The free motion admits invariant tori of dimension $2$ (namely
parameterized by $\AA$, a constant vector, by $\aa$ arbitrary, and
with $I=0$, $\f=0$) which are unstable and possess stable (labeled
$+$) and unstable (labeled $-$) $3$--dimensional manifolds
(parameterized by $\AA$, the same constant vector, by $\aa,\f$ arbitrary,
and with $I=\pm J_0g_0\sqrt{2(1-\cos\f)}$).
We shall study properties that eventually hold when $\h\to0$. It is
well known ([HMS,CG] for instance) that if $\e$ is small most of the
unperturbed tori and their manifolds still exist, just a little
deformed. This means that (under the condition stated below) there
exist functions $\UU^\pm_{\AA'}(\f,\aa)$ and $V^\pm_{\AA'}(\f,\aa)$
which are divisible by $\e$ and analytic in $\aa,\f,\e$, for $\aa\in
\TTT^2$, $|\f|<2\p$, $|\e|<\e_0$, with $\e_0$ small enough, such that
an initial datum starting on the ($3$--dimensional) surfaces
$W_\e^{\s}$, $\s=\pm$, defined as
%
$$ \AA^\s(\f,\aa)=\AA'+\UU^\s_{\AA'}(\f,\aa) \; , \qquad I^\s(\f,\aa)
=\pm J_0g_0\,\sqrt{2(1-\cos\f)}+ V^\s_{\AA'}(\f,\aa) \; , \Eq(1.2)$$
%
evolves, when the time $t\to\pm\io$, tending to be confused with a quasi
periodic motion on a invariant torus $\TT(\AA')$, with rotation vector
%
$$ \oo'=(\o'_1,\o'_2) \; , \qquad %{\buildrel def\over \to} \qquad
\o'_1 \defi \hdp\O_{1}+\h J^{-1} A'_1 \; ,
\qquad \o'_2 \defi \hdm\O_{2} \; , \Eq(1.3)$$
%
and furthermore such asymptotic motion takes place with $\AA$ moving
quasi periodically {\it with average} $\AA'$.
{\it All this holds if $\oo'$ verifies the Diophantine condition}
%
$$|\oo'\cdot\nn|> C |\nn|^{-\t} \; , \qquad
\forall \nn\in\ZZZ^2\setminus\{\V0\} \; , \Eq(1.4)$$
%
for $C,\t>0$ (which may depend also on $\h$).
The values of $\e$ for which we
shall be able to prove the above will be so small that the part of the
stable and unstable manifolds with $|\f|< \fra32\p$ {\it can be
represented as a graph of $\AA,I$ over $\aa,\f$}.\annota1{Note that if
$\e=0$ they are graphs over $\aa,\f$ for $|\f|$ smaller than {\it any}
prefixed quantity $<2\p$.} Hence we look, since the beginning, for
invariant tori which have the latter property.
The approach to the invariant tori, of the points that lie on their
stable manifolds, will be exponential in the sense that their distances
$d(t)$ to the tori will be such that
%
$$ \lim_{t\to+\io} t^{-1}\log d(t)^{-1}=\lis g_0 \; , \qquad
\lis g_0\= \lis g_0(\e)\defi(1+\G(\e,g_0))\,g_0 \; , \Eq(1.5) $$
%
for a suitable analytic function $\G(\e,g_0)$, divisible by $\e$. We
shall call $\lis g_0$ the {\it Lyapunov exponent} of the torus (it will
depend on $\e$ as well as on the considered torus, \ie on $\oo'$ and
on $\h$). The exponent relative to the approach to the same torus
along its unstable manifold (as $t\to-\io$) will be the same, by time
reversal symmetry defined below.
We fix throughout the paper $\t$ ($\t\ge1$) and {\it we shall mainly
study the dependence of $\e_0$, \ie our {\it estimate} for the
analyticity radius, as a function of $\h$: $\e_0=\e_0(C,\h)$.}
\*
\0{\bf 1.2.}
{\cs Remark.} The relation \equ(1.3) between the value of the average
action and the rotation vector is non trivial and it has been named in
[G1,G2] (where it was pointed out) by saying that the tori of
\equ(1.1) are ``torsion free'' or ``twistless''. It is a remarkable
symmetry property of \equ(1.1), see [G1,Ge2,GGM3].
\*
\0{\bf 1.3.}
If $\e=0$ the stable and unstable manifolds coincide (because the
pendulum separatrix is degenerate); it is a degeneracy that is lost when
$\e\ne0$ and generically the manifolds will have only pairwise
isolated trajectories in common, called {\it homoclinic trajectories}.
Nevertheless time reversal symmetry and parity symmetry\annota2{The
latter symmetry is due to the assumption of evenness of $f$.} hold for
\equ(1.1). If $S^t$ denotes the time evolution and the
involution map $i$ (composition of parity and time reversal) is
defined by $i(\f,\aa,I,\AA)=(2\p-\f,-\aa,I,\AA)$, then $iS^t=S^{-t}i$
and there are relations between the stable and unstable
manifolds that are preserved even for $\e\ne0$. Namely
%
$$ \eqalign{
\UU^+_{\AA'}(\f,\aa) = & \UU^-_{\AA'}(2\p-\f,-\aa) \; , \cr
V^+_{\AA'}(\f,\aa) = & V^-_{\AA'}(2\p-\f,-\aa) \; , \cr} \Eq(1.6) $$
%
where care must be exercised because the manifolds contain {\it two}
points over each $\aa,\f$.\annota3{This is in fact already so for
$\e=0$.} Hence if $\f\simeq \p$ the relations in \equ(1.6) concern
points that lie on different connected manifolds; to understand
what happens one should try a drawing taking into account that the
above representations are considered only for $|\f|<\fra32\p$.
Looking at the manifolds at $\f=\p$, {\it assuming their existence and
that they are graphs above $\aa,\f$ for $|\f|<\fra32\p$}, equations
\equ(1.6) imply that, fixed $\AA'$,
%
$$\QQ(\aa)\defi \UU^+_{\AA'}(\p,\aa)
-\UU^-_{\AA'}(\p,\aa) = -\QQ(-\aa) \; , \Eq(1.7)$$
%
so that $\QQ(\V0)=\V0$; but, in general, $\QQ(\aa)\ne \V0$ for $\aa\ne\V0$.
The function $\QQ(\aa)$ is called the {\it homoclinic splitting} (or
simply {\it splitting}) {\it vector} at $\f=\p$, and the
determinant of the matrix with entries $\dpr_{\a_i}Q_j(\V 0)$
(splitting matrix) is called the {\it splitting}.
One can more generally consider
$\Zz\=\Zz(\f,\aa)= (\UU^+_{\AA'} (\f,\aa)-\UU^-_{\AA'}(\f,\aa),
V^+_{\AA'}(\f,\aa)- V^-_{\AA'}(\f,\aa))$ which would be the splitting
vector at $\f$. Here and henceforth the vectors in $\RRR^\ell$ will be
denoted with an underlined letter (while the boldface is used for vectors in
$\RRR^{\ell-1}$); so far $\ell=3$, but shortly we shall consider
$\ell\ge 3$. The function $\Zz$ can be written as the gradient
of a generating function $\F$, \ie $\Zz=(\dpr_\f \F, \dpr_\aa \F)$.
This is a result due to Eliasson who points out that it follows
immediately from the Lagrangian nature of the stable and unstable
manifolds. It is a further symmetry property.\annota4{It can
alternatively be easily seen from the explicit expressions for the
stable and unstable manifolds equations derived in [G1], which also
provide a general algorithm for constructing the function $\F$ as a
convergent series in $\e$ for $\e$ small; see [G3].}
The symmetry of \equ(1.1) (hence the consequent oddness of
$\QQ(\aa)$) implies that there is one trajectory which swings through
$\f=\p$ when $\aa$ is exactly $\V0$: it tends to the same invariant
torus as $t\to\pm\io$, provided the torus exists and its stable and
unstable manifolds are graphs over $\aa,\f$ over an interval of $\f$
greater than $|\f|<\p$.
In this paper we prove the following result.
\*
\0{\bf 1.4.}
{\cs Theorem.} {\it Given the Hamiltonian \equ(1.1), given constants
$s,\O>0$ and given $\h>0$ small enough, the following assertions
hold.\\
\pallino
There are invariant tori with rotation vectors $\oo'$ for all $\oo'$
verifying the Diophantine condition \equ(1.4) with constant
$C=C(\h)=\O e^{-s\h^{-1/2}}$ and
$|\o'_1|\in[\fra12\O_1\hdp,2\O_1\hdp]$.\\
\pallino
Such tori exist for $|\e|<\e_0=O(\h^2)$ and for $\h$ small enough.\\
\pallino
They can be parameterized by their average actions $\AA'$; the angular
velocity is then given by the rotation vector $\oo'\=(\O_1\hdp+\h
J_1^{-1}A'_1,\O_2\hdm)$ (\ie they are ``twistless'') and the Lyapunov
exponents have the form $\lis g_0= (1+\G(\e,g_0))\,g_0$, with
$\G(\e,g_0)$ analytic in $\e$ and divisible by $\e$.\\
\pallino
The parametric equations for such tori and for their stable and
unstable manifolds (``whis\-kers'') can be computed by a convergent
perturbation series in powers of $\e$ around the unperturbed tori with the same
rotation vector and their corresponding stable and unstable manifolds.\\
\pallino
At the homoclinic intersection with $\f=\p$ (existing by symmetry),
between the stable manifold and the unstable manifold of each torus,
the splitting is generically given by the Mel'nikov
integral which is of order $O(\e^2\h^{-\b}e^{-\fra\p2
g_0^{-1}\hdm})$, for $\e$ small enough, with $\b$ depending on the
degree $N_0$ in $\f$ of the perturbation $f$: one can take $\b=2N_0-1$
and the asymptotic formula holds if $|\e|< \h^{\z}$,
$\z=2(N_0+3)$ and $\h$ is small enough.}
\*
\0{\bf 1.5.}
{\cs Remark.} The novelty of the theorem is the ``sharp'' bound
$\e_0=O(\h^2)$. If we ``only'' require $\e_0=O(\h^{\fra92+})$ where
$\fra92+$ is any prefixed positive number $>\fra92$ the result is
proved in [GGM3] (see also [CG] or [Ge2]). The improvement is made
possible by the {\it totally different technique} used (with respect
to [GGM3]): a technique that has interest in its own right and, we
think, beyond the result itself.
In fact the proof of the last assertion of the theorem is
the content of [GGM2], and the values of the constants $\b$ and
$\z$ are taken from Appendix A2 of [GGM2].
\*
\0{\bf 1.6.} Theorem 1.4 will be proved by a further extension of
Eliasson's method, [E,G2,Ge1,Ge2], for the KAM theorem. The following
discussion will show the correctness of the intuition that ``new''
small divisors appear in the perturbation expansion {\it at orders
spaced by} $O(\h^{-1})$. So that the coupling constant is effectively
$O(\e^{\h^{-1}})$ and the analyticity condition is expected to be
$\e^{\h^{-1}} C(\h)^{-q}$ small (for some $q>0$, determined as in the
discussion in Remark 5.16, item (4), below). Hence the analyticity
condition will be $\e C(\h)^{-q\h}$ small rather than the far worse
$\e C(\h)^{-q}$ small, that is implied directly from lemma 1 in [CG]
(where $q=6$ is an estimate).
In the one degree of freedom case the corresponding problem is studied
in [N]: it is a problem that arises naturally in the context of
Nekhoroshev theory. In our case the rotation vector is not
one-dimensional, so that the cancellations between resonances typical
of small divisors problems, [E,G1,Ge1,Ge2], have to be exploited in
order to prove convergence of the perturbative series. The fact that
the two components of the rotation vector \equ(1.3) are so different
in scale has the consequence that small divisors can appear only at
high orders, so that the dependence of the radius convergence on the
Diophantine constant $C(\h)$ is highly improvable with respect the
``na\"{\i}ve'' one, as explained above: the proof of such an assertion
is the subject of the present paper (as, in the weaker form, already of
[GGM3]).
\*
\0{\bf 1.7.} The paper is organized as follows.
In \S 2,3,4 the formalism is concisely illustrated and the graphic
representations of the whiskers in terms of tree graphs is exhibited
(for systems more general than \equ(1.1); see \equ(2.1) below). The
analysis is brief but selfcontained, with references to [G1,Ge2] only
given for further insight and details. The basic formalism is in \S 2,
then we work out in \S 3 two specific examples to explain the origin
of the graphical interpretation, and in \S 4 we set up the general
Feynman rules for evaluating the equations of the whiskers
(and the splitting vector as a particular case) as a sum of
quantities that can be elementarily evaluated. In \S 5 bounds are
derived, assuring the convergence of the perturbative series defining
the whiskers in the more general system in \equ(2.1) below and leading
to Theorem 1.4, when restricted to the Hamiltonian \equ(1.1).
The bounds are derived along the lines of [G1,Ge2]: the main part is
the derivation of the bounds for the part of the expansion
corresponding to what we call the contributions due to ``trees without
leaves'': this is done fully and self consistently in \S 5 and in the
related appendices. Once the bounds on the contributions from trees
without leaves are established, {\it which is the real difficulty},
the same analysis can be applied to bound the other
contributions. Since this is simply reduced, without any further
technical problems, to the case of contributions from the simpler
trees with no leaves we do not repeat this part of the discussion
which is done in [Ge2] following the corresponding analysis done in
[G1,Ge1].
To Appendix A1 we relegate some technical details, while Appendix A3
concerns the cancellation analysis of [Ge2], needed in order to treat
the small divisors problems, with more details with respect to the
quoted paper. An original technical part is also in Appendix A2 and
deals with the improvement of the dependence of the convergence radius
on the Diophantine constant $C(\h)$.
We do not comment here on the obvious relevance of the above results for
the theory of Arnol'd diffusion: see [GGM3,GGM4].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1.truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0{\titolo \S 2. Lindstedt series for whiskered tori.}
\numsec=2\numfor=1\*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0We use the formalism of [Ge2]: it would be pointless to repeat
here the technical work required to motivate the necessity or
usefulness of the notations, and we cannot imagine that any reader may
have interest in the matter that follows unless he has some experience
with Eliasson's method, as exposed in [E] and complemented in
[G1,G2,Ge1,Ge2]. The references to [G1,Ge2] are given only to point at
places where further details on the motivations of the assertions can
be found.
The following analysis innovates [Ge2] in \S 5 because of the
extension of Siegel-Bryuno's bound described in Appendix A2 below:
this section and the next two provide a {\it self contained} description of
the graphical algorithm exploited in \S 5 and Appendix A2.
\*
\0{\bf 2.1.} In the following we shall consider a Hamiltonian
(``Thirring model'') more general than the one in \equ(1.1), \ie a
Hamiltonian which couples a pendulum with $\ell-1$ rotators via a
perturbation $f_1$ which is always an {\it even trigonometric polynomial},
%
$$ \HH = \oo\cdot\AA + \fra{1}{2J}\AA\cdot\AA
+ \fra{I^2}{2J_0} +J_0g_0^2\, f_0(\f) +\e J_0g_0^2\, f_1(\f,\aa)
+ J_0g_0^2\,\g(\e, g_0)\,f_0(\f) \; , \Eq(2.1) $$
%
where $(\aa,\AA)\in\TTT^{\ell-1}\times\RRR^{\ell-1}$,
$(\f,I)\in \TTT^1\times\RRR^1$, $J_0>0$, $J$ is a diagonal matrix,
%$\pmatrix{J'&0\cr0&J''\cr}$, $00$. Going back
to the original Hamiltonian \equ(1.1) we {\it therefore} set $g_0^2=
\lis g_0^2\,(1+\g(\e,\lis g_0))$ and we can invert the latter relation
as $\lis g_0^2= (1+\G(\e,g_0))\, g_0^2$ for $\e$ small enough (this
will mean: for $|\e|0$; see \equ(1.4).
We look for an invariant torus and for its stable
and unstable manifolds with the property that the quasi periodic
rotation on the torus takes place at velocity $\oo'$ and, {\it at the
same time}, the action variables oscillate with an average position $\AA'$.
\*
\pallino Before proceeding we remark that the above {\it two}
requirements may seem contradictory as there may seem to be
no reason for being able to
prescribe simultaneously the ``spectrum'' $\oo'$ and the ``average
action'' $\AA'$ of the invariant tori. In fact this property of ``{\it
twistless}'' motion on the tori or of ``{\it absence of torsion}'' is
very remarkable (see the Remark 1.2 and [G1]): it will appear as due to the
special symmetries of the system \equ(2.1) and to the separation of
the energy into a quadratic part involving actions only and an angular
part involving only the angles.
\*
Note also that we could confine ourselves to study the torus with
average position $\AA'=\V0$, as in [G1,Ge2], because any torus
can be reduced to that one through a trivial canonical
transformation (a translation in the action variables).
This explains why in the quoted papers only the torus
covered with rotation vector $\oo$ is explicitly considered:
however in the following we consider also $\AA'\neq\V0$,
as we are interested in showing the abundance of such tori
in phase space (see the Remark 1.5).
The quantity $X_j^{\s}(t;\aa)$ can be graphically represented as
sum of {\it values} which can be associated with tree graphs, that we
shall call ``Feynman graphs'' or ``trees'' {\it tout court}, see
Fig.\equ(2.4) below. The trees are partially ordered sets of points,
called {\it nodes}, connected by unit lines, called {\it branches},
and they are ``oriented'' towards a point called {\it root}, which is
reached by a single branch of the tree. Given two nodes $v$ and $w$
of a tree, we say that $w$ precedes $v$ ($w\le v$) if there is a path
connecting $w$ to $v$, oriented from $w$ to $v$. With an abuse of
notations we shall sometimes consider a tree as the collection of its
nodes, sometimes as the collection of its branches and sometimes as
the collection of both nodes and branches. The root {\it will not} be
considered a node.
A typical tree considered below can be drawn as in Fig.\equ(2.4):
the labels meaning and the caption of such a drawing
(which has to be interpreted as a mathematical formula) will be
elucidated in the coming sections.
\*
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\eqfig{199.99919pt}{141.666092pt}{
\ins{-29.16655pt}{74.999695pt}{\rm root}
\ins{0.00000pt}{91.666298pt}{$j$}
\ins{49.99979pt}{70.833046pt}{$v_0$}
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\ins{91.66629pt}{41.666500pt}{$v_2$}
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\ins{191.66589pt}{133.332794pt}{$v_5$}
\ins{191.66589pt}{99.999596pt}{$v_6$}
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\ins{191.66589pt}{37.499847pt}{$v_9$}
}{bggmfig0}{\hskip.6truecm\eq(2.4)}
\kern0.9cm
\didascalia{A tree $\th$ with $m=12$,
and some labels. The line numbers,
distinguishing the lines, and their orientation pointing at the root,
are not shown. The lines length should be the same but it is drawn of
arbitrary size. The nodes labels $\d_v$ are indicated only for two
nodes.}
The branch starting at the node $v$ and linking it to the uniquely
determined next node (or to the root), which we call $v'$, will be
denoted by $\l_{v}$: there is a unique correspondence between nodes
and branches starting at them. We shall say that $\l_v$ exits from
$v$ and enters $v'$; given a node $v$ we shall say that a branch $\l$
{\it pertains} to $v$ if either $\l$ enters $v$ or $\l$ exits from
$v$; \eg in Fig.\equ(2.4) the line $v_1v_0\=\l_{v_1}$ ``exits'' $v_1$
and ``enters'' $v_0$, hence it pertains to both.
In [G1] two expansions are considered for the functions
$X^{\s}_j(t;\aa)$ representing the stable and unstable manifolds: one of
them is used to exhibit cancellations taking place at all orders in the
sums that express the coefficients of the power series in $\e$ of the
splitting vector, [G1,BCG,GGM2];
it is somewhat more involved than the other one that
is convenient to just discuss convergence of the perturbation series for
the splitting vector and that we shall use here. This is the reason why (as
in [Ge2]) we shall not have trees whose lowest nodes carry a graphical
decoration called {\it form factor}, or {\it fruit} in [G1,GGM2].
Nevertheless some of the nodes will still have a particular
structure: to characterize them we introduce, below as in [Ge2], the
notion of ``{\it leaf}\/'', which is related to the notion of fruit in
[G1], from which it differs (and it, even, differs slightly from the
similar notion of leaf in [Ge2]), see below for the motivation of the
name.
%\ifnum\mgnf=0\pagina\fi
\*
\0{\bf 2.3.} As mentioned the drawing Fig.\equ(2.4) has to be regarded
as a mathematical formula expressing a function of the labels and of
the topological structure of the trees. We now prepare the notation
for the definition of ``value'' of a tree (following [Ge2]) (see
[G1] for a simpler case): the derivation is not difficult but somewhat
long and unusual for the subject (the breakthrough work [E] still does
not seem to be well known in its technical aspects!). We discuss it in
detail not only for completeness but in the attempt to clarify a
construction that has generated quite a few new results starting from
the work of [E], see [G1,GGM2,BGGM].
Let us consider the unperturbed motion $ X^0(t)\=(\f^0(t),\aa+\oo'
t,I^0(t),\AA')$, where $(\f^0(t),I^0(t))$ is the separatrix motion,
generated by the pendulum in \equ(2.1) starting at $t=0$ in $\f=\p,\,
\AA=\AA',I=-2J_0 g_0$, so that $\f^0(t)=4 \arctan e^{-g_0t}$. Let
$X^\s(t;\a)$, $\s={\rm sign}\,t=\pm$, be the evolution, under the flow
generated by
\equ(1.1), of the point on $W^\s_\e$ which at time $t=0$ is
$(\p,\aa,I^\s(\aa,\p),\AA^\s(\aa,\p))$, see \equ(1.2); let
%
$$X^\s(t)\=X^\s(t;\aa)\equiv \sum_{h\ge 0} X^{h\s}(t;\aa) \e^h=
\sum_{h\ge 0} X^{h\s}(t) \e^h,\qquad \s=\pm \; , \Eq(2.5)$$
%
be the power series in $\e$ of $X^\s$, (which we want to show to be
convergent for $\e$ small); note that $X^{0\s}\=X^0$ is the
unperturbed whisker. We shall often omit writing explicitly the $\aa$
variable among the arguments of various $\aa$-dependent functions, to
simplify the notations, and we shall regard the two functions
$X^{h\s}(t)$, as forming a single function $X^h(t)$, which is
$X^{h+}(t)$ if $\s=+,\, t>0$, and $X^{h-}(t)$ if $\s=-,\,t<0$.
Components of $X$ will be labeled $j$, $j=0,\ldots,2\ell-1$,
consistently with \equ(2.3), with the
convention that $X_0\defi X_-$ describes the coordinate $\f$,
$(X_j)_{j=1,\ldots,\ell-1}\defi\XX_\giu$ describes the $\aa$
coordinates, $X_\ell\defi X_+$ describes the $I$ coordinate and
$(X_j)_{j=\ell+1,\ldots,2\ell-1}\defi \XX_\su$
describes the $\AA$ coordinates,
%
$$ X \defi\, (X_j)_{j=0,\ldots,2\ell-1}\defi\,
(X_-,\XX_\giu,X_+,\XX_\su) \; , \Eq(2.6)$$
%
\ie we write first the angle and then the action components, first
the pendulum and then the rotators. The ${\bf \su}$ (``{\it up}'') and ${\bf
\giu}$ (``{\it down}'') labels recall that the components with labels ${\bf
\giu}$ ($0< j<\ell$) have ``lower'' index than the variables with
labels ${\bf \su}$ ($\ell0$; however their sums
have {\it no singularity} at $t=0$ and can be anaytically continued
for $\s t<0$ (\ie $x\ge1$). More precisely the functions that one has to
integrate are contained in an {\it algebra} $\hat \MM$ on which the
integration operations that we need can be given a meaning.
%To describe such class we introduce the algebra $\hat \MM$ of the
%functions of $t$ defined as follows.
\*
\0{\cs Definition} ([G1]).
{\it Let $\hat\MM$ be the space of the functions
of $t$ which can be represented, for some $k\ge 0$, as
%
$$M(t)=\sum_{j=0}^k{(\s t g_0)^j\over j!} M_j^\s(x,\oo t)\ ,\quad
x\=e^{-\s g_0t}\ ,\quad \s={\rm sign}\, t \; , \Eq(2.21)$$
%
with $M_j^\s(x,\pps)$ a trigonometric polynomial in $\pps$ with
coefficients holomorphic in the $x$-plane in the annulus $0<|x|<1$,
with possible singularities, outside the open unit disk, in a closed
cone centered at the origin, with axis of symmetry on the imaginary
axis and half opening $<\fra\p2$, and possible polar singularities at
$x=0$. The smallest cone containing the singularities will be called
the {\it singularity cone} of $M$.}
\*
The proper interpretation of the improper
integrals $\ig_{\s\io}^{g_0t} M(\t) d g_0 \t$,
which henceforth will be denoted by $\igb_{\s\io}^{g_0t} M(\t)
dg_0\t$, is simply the {\it residuum} at
$R=0$ of the analytic function
%
$$ \II_R M\defi\ig_{\s\io+i\theta}^{g_0t}e^{-Rg_0\s z} M(z)\,d\,g_0z \; ,
\Eq(2.22) $$
%
(where $\theta$ is arbitrarily prefixed) which is
defined and holomorphic for $\Re R>0$ and large enough, \ie
%
$$\II M(t) \= \igb_{\s\io}^{g_0t} dg_0\t \, M(\t)
\defi \oint\fra{d R}{2\p i R} \,\II_R M(t) \; . \Eq(2.23)$$
%
By linear extension this defines the integration of function in $\hat
\MM$ for $|x|<1$. The analyticity in $x$ around $x=\pm1$ and the
remarks that $\fra{d}{d g_0 t}\II M(t)\= M(t)$, \ie $\II M(t)\=
\II M(t')+\ig_{g_0t'}^{g_0t} d\,g_0\t\, M(\t)$, so that $\II M(t)$ is
a special primitive of $M(t)$ (at fixed $\s$), allow us to
analytically continue the result of the integration to a function in
$\hat \MM$. The operator $\II$ maps the algebra $\hat \MM$ into itself
because one checks that on the monomial \equ(2.19) one has
%
$$\II M(t)=\cases{- g_0^{-1} \s^{\chi +1}e^{i\oo'\cdot\nn t-pg_0\s t}
\sum_{h=0}^j (g_0\s t)^{j-h} {1\over(j-h)!}
{1 \over(p- i \s g_0^{-1} \oo'\cdot\nn)^{h+1}}
\; , & if $|p|+|\nn|>0 \; , $\cr
g_0^{-1}\s^{\ch+1}\fra{(\s g_0 t)^{j+1}}{(j+1)!} \; ,
& otherwise $\; , $ \cr}\Eq(2.24)$$
%
showing, in particular, that the radius of convergence in $x$ of $\II
M$, for a general $M$, is the same as that of $M$. But in general the
singularities will not be polar, even when those of the
$M_j^\s$'s were such.
We shall see that the cases $|p|+|\nn|=0$ do not enter in the
discussion (a feature of the method of [Ge2]). The complete
expression of $X^{h\s}(t)$ becomes
%
$$\eqalignno{
&\X^{h\s}_-(t) = w_{0\ell}(t)\II(w_{00}\F^{h\s}_+)(t)-w_{00}(t)\big(
\II(w_{0\ell}\F^{h\s}_+)(t)-\II(w_{0\ell}\F^{h\s}_+)(0^\s)\Big)\defi
\OO(\F^{h\s}_+)(t) \; , \cr
&
\XXX^{h\s}_\giu(t) = J^{-1}J_0 \Big(\II^2(\FFF^{h\s}_\su)(t)-\II^2(
\FFF^{h\s}_\su)(0^\s)\Big) \defi
\lis\II^2(\FFF^{h\s}_\su(t)) \; , & \eq(2.25)
\cr
& \X^{h\s}_+(t) = w_{\ell\ell}(t)\II(w_{00}\F^{h\s}_+)(t)-w_{\ell0}(t)
\Big(\II(w_{0\ell}\F^{h\s}_+)
(t)-\II(w_{0\ell}\F^{h\s}_+)(0^\s)\Big)\defi\OO_+(\F^{h\s}_+)(t)
\; , \cr
&
\XXX^{h\s}_\su(t) = \II(\FFF^{h\s}_\su)(t) \; , \cr} $$
%
where $\OO,\OO_+,\lis\II^2$ are implicitly defined here (and $\II^2$
is $\II$ applied twice); and
$\X^{h\s},\F^{h\s}\=(0,\V0,\F_+^{h\s},$ $\FFF^{h\s}_\su)$ are introduced in
\equ(2.9). While $\X^{h\s}$ has non zero components over both the
{\it angle} ($j=0,\ldots,\ell-1$) and over the {\it action}
($j=\ell,\ldots,2\ell-1$) components, the $\F^{h\s}$ has, as already
noted, only the action directions non zero; the notation $0^\s$ means
the limit as $t\to0$ from the left ($\s=-$) or from the right
($\s=+$), but below we shall drop the superscript on $0$ (always clear
from he context because it is the same as the superscript $\s$ of
the functions $\X^{h\s}$). Furthermore, with
the definitions \equ(2.20) of $\tilde \FFF_\su^{h\s}(\nn,p)$ one
finds also the property (with the notations
in \equ(2.1))
%
$$ \tilde \FFF_{\su}^{h\s}(\V0,0) = \V0\; , \Eq(2.26)$$
%
for all $h\ge1$.
We shall repeatedly use that in order to compute $\X^{h\s}_j$ we only
need $\X^{h'\s}_{j'}$ with $0\le j'<\ell$ (\ie only
$\X^{h'\s}_+, \X^{h'\s}_\su$) and $h'
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\figfin
\eqfig{250pt}{100pt}{
\ins{9pt}{56pt}{$j $}
\ins{0pt}{38pt}{$r $}
\ins{33 pt}{38 pt}{$v_0 $}
\ins{57pt}{64pt}{$p $}
\ins{57pt}{34pt}{$q $}
\ins{233pt}{64pt}{$0 $}
\ins{233 pt}{34pt}{$0 $}
\ins{186pt}{56pt}{$0 $}
\ins{151pt}{56pt}{$j $}
\ins{77pt}{76pt}{$v_1 $}
\ins{77pt}{5pt}{$v_2 $}
\ins{140pt}{38pt}{$r $}
\ins{173pt}{38pt}{$v_0 $}
\ins{209pt}{38pt}{$v_1 $}
\ins{255pt}{5pt}{$v_2 $}
\ins{255pt}{76pt}{$v_3 $}
\ins{19pt}{62pt}{$1, j_{v_0}$}
\ins{65pt}{94pt}{$1, j_{v_1}$}
\ins{65pt}{25pt}{$1, j_{v_2}$}
\ins{164pt}{63pt}{$1, j_{v_0}$}
\ins{196pt}{63pt}{$0, j_{v_1}$}
\ins{244pt}{94pt}{$1, j_{v_3} $}
\ins{244pt}{25pt}{$1, j_{v_2} $}
}{albero1}{\eq(3.6)}
\*
\0where the labels on the nodes $v$ are denoted $\d_v,j_v$ and those
on the lines $\l_v$ are denoted $j_{\l_v}$.
The label $\d_v=0,1$ on the node $v$ indicates selection of
$f_{\d_v}$, \ie of $f_0$ or $f_1$, the label $j_v$ denotes a
derivative with respect to $\f$ if $j_{v}=\ell$ or with respect to
$\a_{j_v}$ if $j_v=\ell+1,\ldots,2\ell-1$. For the label
$j_{\l_v}$ associated with the branch $\l_v$ following $v$, one has
$j_{\l_v}=j_v-\ell$ for all $v$ except for the highest node $v_0$, for
which one has $j_{\l_{v_0}}=j_{v_0}$. In the examples above,
\equ(3.3) and \equ(3.5) correspond, respectively,
to the first figure in \equ(3.6) with $j_{v_0}=j,j_{v_1}=p+\ell,
j_{v_2}=q+\ell$ and to the second with
$j_{v_1}=j_{v_2}=j_{v_3}=\ell,j_{v_0}=j$, (hence
$j_{\l_{v_1}}=j_{\l_{v_2}}=j_{\l_{v_3}}=0$, $j_{v_0}=j$). In the
examples the labels $p,q$ correspond to $\dpr_{\a_p},\dpr_{\a_q}$ in
\equ(3.3).
\*
\0{\bf 3.2.} {\cs Remark.}
The exception for the meaning of $j_{\l_{v_0}}$ is
convenient, in the above cases, as the
integration over $\t_{v_0}$ differs from the others: the inner
ones evaluate $\X^{h\s}_j$ for $j=0,\ldots,\ell-1$, because the
functions $f_0,f_1$ only depend on the angle variables (see the last
paragraph in \S 2.4); the last integral, however, evaluates in the
examples a component of $\XXX^{h\s}_\su$ (which is labeled
$j=\ell+1,\ldots,2\ell-1$), but, in general, $j$ can be any value
$j=0,\ldots,2\ell-1$. Note that this is not so for the inner labels
$j_\l$ which must be angle labels $j_\l=0,\ldots,\ell-1$. So, in
general, we shall have that the value of a tree with $j_{\l_{v_0}}=j$
contributes to $\X^{h\s}_j$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1.truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0{\titolo \S 4. Trees and Feynman graphs approach to whiskers construction:
the general case.}
\numsec=4\numfor=1\*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0We now proceed to describe the general case.
\*
\0{\bf 4.1.} To compute the splitting vector we only need to consider the
variable $t$ equal to $0$. However we shall be also interested in
$\X^{h\s}(\aa,t)$ for $\s t>0$, for instance in order to study how
fast the invariant torus is approached by the motions on its stable
and unstable manifolds (to obtain its Lyapunov exponent). Hence it
will be natural to attribute the label $t$ to the root: this will also
remind that the integral over $\t_{v_0}$ has to be performed between
$\s\io$ and $t$, (the value $\s=-$ corresponds to the unstable
manifold and the value $\s=+$ corresponds to the stable one). Since we
shall {\it never} consider the stable manifold for $t>0$ or the
unstable for $t<0$ the value of $\s$ will be the same as that of the
sign of $t$.
We shall be interested in computing not only $\XX_{\su}^\s(0;\aa)-\AA'$
(or $\XX^\s_{\su}(t;\aa)-\AA'$), as in [GGM2], but, more
generally, $X^\s(t;\aa)-X^0(t;\aa)$, with $\s=\hbox{sign}\,t$, (here
$X^{0}$ denotes the unperturbed motion).
In general the rules to express $X^\s(t;\aa)-X^0(t;\aa)$ as sum of
``values'' associated with trees will be described now, assuming that the
reader follows us by applying and checking them to the special cases
\equ(3.3), \equ(3.5), illustrated in \equ(3.6).
\*
The reader might be helped in following the construction of the
algorithm to express the stable and unstable manifolds below, by
keeping in mind that we simply decompose the (quite involved and
recursively defined by \equ(2.25),\equ(2.12)) expressions for the
whiskers, so far obtained, {\it further}.
The purpose being of reducing their evaluation to {\it very elementary
algebraic operations}: ultimately just products of simple factors
associated with the nodes (and their labels) of a tree, that we shall
call ``coupling constants'', and of factors associated with the
branches (and their labels), that we shall call ``propagators'', each of
which can be trivially evaluated and trivially bounded.
The reader familiar with Quantum Field Theory will realize the
striking analogy between the algorithms discussed below and the {\it
Feynman graphs}: in fact a ``tree'' will turn out as an analog to a
(loopless) Feynman graph and {\it very likely it is} a Feynman graph
of a suitable (non trivial) field theory. Our analysis amounts to a
renormalization group analysis of it and it partially extends, to the
case of the theory of the stable and unstable manifolds of hyperbolic
tori in nearly integrale systems, the field theoretic interpretation
already discussed in detail in previous works, see [GGM1] and appended
references, in the study of KAM tori.
\*
\pallino To each node we attach an {\it order label} $\d_v=0,1$,
see Fig.\equ(3.6), and a corresponding function $f_{\d_v}$: if a node
$v$ bears a label $\d_v=1$ the associated function is $f_1$ and if it
bears a label $\d_v=0$ it is $f_0$.
\*
\pallino To each node $v$ of a tree $\th$, see the Figure \equ(2.4) above,
we associate an integration {\it time variable} $\t_v$ and an {\it
integration operation}, which corresponds to $\lis\II^2$ or $\OO$ if
the node {\it is not the highest node} $v_0$ and to
$\lis\II^2$ or $\OO$ or $\II$ or $\OO_+$
if the node $v$ {\it is the highest}, \ie $v=v_0$.
This is so because in the first case (a ``lower node'')
one must use the first two equations in \equ(2.25) because in
\equ(2.12) only angle components of $X^{h\s}$ appear, while in the
second case (that of the highest node) one can use all of
\equ(2.25) since we can evaluate either an angle coordinate
$\X^{h\s}_j(\aa,t)$, $j<\ell$, or an action coordinate, $j\ge\ell$.
When $v< v_0$ the choice between the two possibilities will be marked
by an {\it action label} $j_v$ associated with each node:
if $j_v=\ell$, $v\ell$, we choose $\lis\II^2$ if $j_{\l_{v_0}}=j_{v_0}-\ell$ and
$\II$ if $j_{\l_{v_0}}=j_{v_0}$, see \equ(2.25).
As said in Remark 3.2, the meaning of the branch label
is that a tree with $j_{\l_{v_0}}=j$ is a graphic
representation of a ``contribution'' to $\X_j^{h\s}$. Therefore if
$j_{\l_{v_0}}\ge\ell$ we call the branch an {\it action branch} and if
$j_{\l_{v_0}}<\ell$ we call it an {\it angle branch}.
In the first of the figures in \equ(3.6) integrals with respect to
the nodes $v_1,v_2$ are of the type $\lis
\II^2$. In the second the integrals over the $\t_{v_n}$, $n=1,2,3$, are
all of the type $\OO$. In both cases the integrals over $\t_{v_0}$ are
of the form $\II$ because we fixed $j_{\l_{v_0}}=j>\ell$ to be an
action label. We can associate a branch label $j_{\l_v}$ also to the
inner branches with $v\t_{v'}$ if $\s=+$ and $\t_{v}<\t_{v'}$ if $\s=-$, while
$\t_v,\t_{v'}$ have the same sign but are otherwise unrelated if
$\r_v=0$; see \equ(2.25) and check this in the examples.
Besides the labels already introduced also the labels $\r_v=0,1$, just
described but not shown in \equ(3.6), should be imagined carried by
each node.
\*
\pallino Given a tree labeled as above we pick up the nodes $v$
with $\r_v=0$ which are closest to the root, and consider the
subtrees having such nodes as highest nodes.
We call each such subtree, \ie each such node {\it together with the
subtrees ending in it} (and its labels), a {\it leaf}.\annota5{This
definition is slightly different from the one given in [Ge2], where
the leaf represents a collection of trees and, as explained below, is
related to a resummation operation (see also comments in \S 4.2, item
(v), below, and \equ(4.27)), that we do not consider here.} The name
is natural if one imagines to enclose the part of the tree including
the node $v$ itself and half of the line $\l_v$ into a circle (or,
more pictorially, into a leaf shaped contour): hence, to whom tries
the drawing, it will look like the {\it venations} of a leaf and the
half line outside it will look like its {\it stalk}.
\*
\pallino
All nodes which do not belong to any leaves will be called
{\it free nodes}; they carry, by construction, a label $\r_v=1$, so that
the corresponding time variables are hierarchically ordered from the
lowest nodes up to the root: \ie if $w\t_v$ if $\s=-$. Given a tree $\th$ let us call $\th_f$ the
set of free nodes in $\th$, and call $\Th_L$ the set of highest nodes of the
leaves.
\*
\pallino Each
$f_{\d_v}$ function, associated with the node $v$ with order label
$\d_v$, can be decomposed into its Fourier harmonics. This can be done
graphically by adding to each node $v$ a {\it mode} label
$\nvec_v=(\n_{0v},\nn_v)\= (n_v,\nn_v)\in\ZZZ^{\ell}$, with
$|\nn_v|\le N$ and $|n_v|\le N_0$, that denotes the particular
harmonic selected for the node $v$. If $(j_v,\d_v,\r_v,\nvec_v)$ are
the labels of $v$ we will associate with $v$ the quantity
$f^{\d_v}_{\nvec_v}\,e^{i(\oo\cdot\nn_v
\t_{v}+n_v\f^0(\t_v))}$ multiplied by appropriate products of factors
$i n_v$ (one per $\f$--derivative) and $i\n_{vj}$ (one per
$\a_{j}$--derivative, $j=j_{\l_v}$).
If the mode labels $\nvec_v$ are specified for each $v$ we shall define
the {\it momentum} $\nn(v)$ ``flowing'' on a branch $\l_v$ as the sum
of all the angle mode components $\nn_w$ of the nodes $w$ {\it preceding}
the branch, with $v$ included,
%
$$ \nn(v) \defi\sum_{w\in\th,\ w\le v}\nn_w \; ; \Eq(4.1) $$
%
the momentum $\nn(v_0)$ flowing through the root branch will
be called the {\it total momentum} (of the tree).
We shall define also the {\it total free momentum} of the tree
as the sum of the mode labels of the free nodes: more generally,
for any free node $v$ we can define the {\it free momentum}
flowing through the branch $\l_v$ as
%
$$ \nn_0(v) = \sum_{w\in\th_f,\ w \le v} \nn_w \; . \Eq(4.2) $$
%
For instance in the above examples the two contributions \equ(3.3),
\equ(3.5) (represented by figure \equ(3.6)) are decomposed into sums of
several distinct contributions once the $\r_v$ and the mode labels
$\nvec_v$ are specified.
Likewise we can look at a leaf as a tree: the momentum $\nn'$
flowing through its stalk will then be called the internal {\it leaf
momentum}. Note that its value gives {\it no contribution} to the
total free momentum of the tree to which the leaf belongs.
\*
\pallino The free
momenta will turn out to describe the harmonics of the time dependent
quasi periodic motion around the invariant tori, while the Fourier
expansion modes of $X^{h\s}(t;\aa)$ as a function of
$\aa$ are related to the sum of the free momenta {\it and} of all the
internal leaf momenta. This is an important difference: it is a
property stressed in [G1] where it is referred as ``quasi flatness'',
source of the main difficulties and interest in the theory of
homoclinic splitting, see [G1,GGM2,GGM3,G3].
\*
\0{\bf 4.2.} The trees contributions of the
examples of \S 3 will be sums over the various labels of ``values'' of
trees decorated by more labels:
%
$$\eqalignno{ &\fra12\igb_{\s\io}^{g_0t} dg_0\t_{v_0}
(-i\n_{v_0j})(i\n_{v_0p})(i\n_{v_0q})\,f^1_{\nvec_{v_0}}\,
e^{i(\nn_{v_0}\cdot\oo'\t_{v_0}+ n_{v_0}\f^0(\t_{v_0}))}\cdot & \eq(4.3)
\cr &\quad\cdot\lis\II^2\big((-i\n_{v_1p})\,f^1_{\nvec_{v_1}}\,
e^{i(\nn_{v_1}\cdot\oo'\t_{v_1}+
n_{v_1}\f^0(\t_{v_1}))}\big)(\r_{v_1}\t_{v_0})\,
\lis\II^2\big((-i\n_{v_2q}) \,f^1_{\nvec_{v_2}}\,
e^{i(\nn_{v_2}\cdot\oo'\t_{v_2}+
n_{v_2}\f^0(\t_{v_2}))}\big)(\r_{v_2}\t_{v_0}) \; , \cr}$$
%
for \equ(3.3) and
%
$$\eqalignno{
&\fra12 \igb_{\s\io}^{g_0t} dg_0\t_{v_0}
(-i\n_{v_0j})(in_{v_0})\,f^1_{\nvec_{v_0}}\,
e^{i(\nn_{v_0}\cdot\oo' \t_{v_0}+ n_{v_0}\f^0(\t_{v_0}))}\,
\OO\Big((-in_{v_1})\,f^0_{\nvec_{v_1}}\, e^{i n_{v_1}\f^0(\t_{v_1})}
& \eq(4.4) \cr
& \qquad \OO\big((-i n_{v_2})\,f^1_{\nvec_{v_2}}\,
e^{i(\nn_{v_2}\cdot\oo' \t_{v_2}+ n_{v_2}\f^0(\t_{v_2}))} \big)
(\r_{v_2}\t_{v_1}) \,
\OO\big((-i n_{v_3})\,f^1_{\nvec_{v_3}}\,
e^{i(\nn_{v_3}\cdot\oo' \t_{v_3}+ n_{v_3} \f^0(\t_{v_3}))}\big)
(\r_{v_3}\t_{v_1}) \Big) \; , \cr} $$
%
for \equ(3.5), with the conventions following \equ(3.3)
about the dummy integration variables.
\*
\pallino The integration operations are still fairly involved, as it
can be seen from \equ(2.25) and from the expressions for $\lis \II^2$ and
$\OO$. With the above conventions for the dummy variables and noting that,
for any function $F$ in $\hat\MM$,
%
$$ \lis \II^2 \big( F(\t)\big)(t)=J^{-1}J_0\, \Big(
\II(g_0(t-\t)F(\t) )(t) - \II(g_0(t-\t)F(\t))(0) \Big) \; ,\Eq(4.5)$$
%
we see that the integration over the $\t_v$ has (by \equ(2.25)) one of
the two forms, when $\r_v=1$ and $v'$ is not the root (so that
$j_{\l_v}=j_{v}-\ell$),
%
$$ \eqalign{
(1)\quad& \II\big(
( w_{0\ell}(\t_{v'}) w_{00} (\t_v) -
w_{00}(\t_{v'}) w_{0\ell}(\t_v) )
e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))}
G_v(\t_v)\big)(\t_{v'}), \qquad j_{\l_v}=0 \; , \cr
%
(2)\quad&\II\big(g_0(\t_{v'}-\t_v)\,
e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\big)(\t_{v'}),
\qquad \kern3.6truecm 0\ell$ and $t=0$;
in such a case the last two of \equ(2.25)
are relevant and setting $v=v_0$ the integration over $\t_{v_0}$ is
the value for $\t_{v'}$ of
%
$$\eqalign{
(1)\quad&\II\left(w_{00}(\t_v) e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))}
G_v(\t_v)\right)(0) \; , \qquad j_{\l_v} = \ell \; , \cr
%
(2)\quad&\II\left(e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))}
G_v(\t_v)\right)(0) \; ,
\qquad \kern1.2truecm j_{\l_v}>\ell \; . \cr} \Eq(4.8)$$
%
because, if $\t_{v'}=0$, one has $w_{\ell\ell}(0)=1$ and
$w_{\ell 0}(0)=0$; see \equ(2.15) and the last two of \equ(2.25).
More generally, if $\t_{v'_0}=t\neq 0$, setting $v=v_0$ and
$r=v'_0$, one defines for $\r_{v_0}=1$
%
$$\eqalign{
(1)\quad& \II\big(
( w_{0\ell}(\t_{v'}) w_{00} (\t_v) -
w_{00}(\t_{v'}) w_{0\ell}(\t_v) )
e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))}
G_v(\t_v)\big)(\t_{v'}), \qquad j_{\l_v}=0 \; , \cr
%
(2)\quad&\II\big(g_0(\t_{v'}-\t_v)\,
e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\big)(\t_{v'}),
\qquad \kern3.6truecm 0< j_{\l_v}<\ell \; , \cr
%
(3)\quad& \II\big(
(w_{\ell\ell}(\t_{v'}) w_{00} (\t_v) -
w_{\ell 0}(\t_{v'}) w_{0\ell} (\t_v) )
e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))}
G_v(\t_v)\big)(\t_{v'}) \; , \qquad j_{\l_v}=\ell \; , \cr
%
(4)\quad&\II\big(
e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))}
G_v(\t_v)\big)(\t_{v'}) \; , \qquad \kern5.3truecm j_{\l_v}>\ell \; ,
\cr} \Eq(4.9)$$
%
(see the last two relations in \equ(2.25)) and for $\r_{v_0}=0$
%
$$\eqalign{
(1)\quad & w_{00}(\t_{v'})\II\big(w_{0\ell}(\t_v)
e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\big)(0)\,,\qquad
j_{\l_v}=0 \; , \cr
%
(2)\quad&\II\big(g_0\t_v\,e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))}
G_v(\t_v)\big)(0),\qquad \kern1.7truecm 00
\; ; \cr}\Eq(4.10)$$
%
note that, for $\t_{v'}=t=0$ and $j_{\l_{v_0}}\ge \ell$,
\equ(4.9) and \equ(4.10), summed together, give \equ(4.8).
\*
\pallino Hence each node still describes a rather complicated set
of operations: it is, therefore, convenient to consider separately the
terms that appear in \equ(4.6)$\div$\equ(4.10). This can be done by
simply adding further labels at each node. To this end, looking at the
integrals in \equ(4.7) and \equ(4.10), at $\r_v=0$, and in
\equ(4.6) and \equ(4.9), at $\r_v=1$, we see that the
following kernels are involved in the integrals
%
$$ \eqalignno{
w^0_{j_{\l_v}}(\t_{v'},\t_v) & = \cases{
w_{00}(\t_{v'}) w_{0\ell}(\t_v) ,
& \kern2.8truecm$v>v_0\, , j_v=\ell \, $ $\to$ $j_{\l_v}=0\,,$ \cr
g_0\t_v , & \kern2.8truecm $v>v_0\, , j_v>\ell \,$
$\to$ $0\ell ,
\quad 0\ell,\quad j_{\l_{v_0}}>\ell \, , $ \cr}
& \eq(4.11) \cr
%
w^1_{j_{\l_v}}(\t_{v'},\t_v) & = \cases{
w_{0\ell}(\t_{v'}) w_{00}(\t_v) - w_{00}(\t_{v'}) w_{0\ell}(\t_v), &
$v>v_0\ , j_v=\ell \,$ $\to$ $j_{\l_v}=0\,,$ \cr g_0(\t_{v'}-\t_v), &
$v>v_0\ , j_v>\ell
\,$ $\to$ $0\ell , \quad 0\ell, \,j_{\l_{v_0}}>\ell , $\cr} \cr}$$
%
respectively appearing in \equ(4.7) and \equ(4.10), at $\r_v=0$,
and in \equ(4.6) and \equ(4.9), at $\r_v=1$.
The function in \equ(4.11) involving the Wronskian matrix elements can
be computed from \equ(2.15) and one finds, for instance, that the function
in the seventh row on the r.h.s. is
%
$$
w_{0\ell}(\t_{v'}) w_{00}(\t_v) - w_{00}(\t_{v'}) w_{0\ell}(\t_v) =
\fra12 \left\{\fra{g_0(\t_{v'}-\t_{v})}{\cosh g_0\t_{v'}\,
\cosh g_0\t_{v}}+\fra{\sinh g_0\t_{v'}}
{\cosh g_0\t_{v}}-\fra{\sinh g_0\t_{v}}{\cosh g_0\t_{v'}}
\right\} \; ; \Eq(4.12)$$
%
hence if we consider \equ(4.6)$\div$\equ(4.10) we note that
the integrals over $\t_v$ involve functions that can be written, for
$\r=\r_v,\t=\t_v,\t'=\t_{v'}$ and for suitable coefficients
$c_j(\r,\a,v)$, ($\r=1$ if we consider \equ(4.6), \equ(4.9) and
$\r=0$ if we consider \equ(4.7), \equ(4.10)),
%
$$ \sum_{\a=-1}^2 T_\r^{(\a)}(\r\t',\t)\,Y^{(\a)}(\t',\t)
\, c_j(\r,\a,v) \; , \Eq(4.13)$$
%
where $Y^{(\a)}(\t',\t)$ are given, if $x=e^{-\s g_0 \t}$
and $x'=e^{-\s g_0 \t'}$, by
%
$$ \eqalignno{
Y^{(-1)}(\t',\t) =
&\fra12 {\sinh g_0\t\over\cosh g_0\t'} \,
\exp[in \f^0(\t)] =
\sum_{k'=1}^{\io}\sum_{k=-1}^{\io}
y_n^{(-1)}(k',k) {x'}^{k'}x^{k} \; ,\qquad k'\ {\rm odd}\,, \cr
%
Y^{(0)}(\t',\t) =
&\fra12 {\exp[in\f^0(\t)]\over\cosh g_0\t'\cosh g_0\t}
=\sum_{k'=1}^{\io}\sum_{k=1}^{\io}
y_n^{(0)}(k',k) x'^{k'}x^{k} \; , \qquad \kern1.2truecm k'\
{\rm odd}\,, &\eq(4.14)\cr
%
Y^{(1)}(\t',\t) =
&\fra12 {\sinh g_0\t'\over\cosh g_0\t} \,
\exp[in\f^0(\t)]= \sum_{k'=-1}^{\io}\sum_{k=1}^{\io}
y_n^{(1)}(k',k) {x'}^{k'}x^{k} \; , \qquad \kern.3truecm
k'\ {\rm odd} \, , \cr
%
Y^{(2)}(\t',\t) =
&\exp[in\f^0(\t)] = \sum_{k=0}^{\io} \tilde y_n^{(2)}(0,k) x^{k}
\; , \qquad \kern3.6truecm k'\=0\cr} $$
%
which define the coefficients $y_n^{(\a)}(k',k)$ for $\a=-1,0,1,2$ (it
is easily checked that $k'$ is {\it odd} in the first three relations)
and we set, for $\a=-1,0,1,2$,
%
$$ T^{(\a)}_\r(\r\t',\t) = \cases{
g_0(\t'-\t) & if $\a$ is either $0$ or $2$ and $\r=1\;$, \cr
g_0\t & if $\a$ is either $0$ or $2$ and $\r=0\;,$ \cr
1 & if $\a$ is either $-1$ or $1\;.$ \cr} \Eq(4.15) $$
%
{\it Likewise} we shall set, defining the coefficients $\tilde
y_n^{(\a)}(k',k)$, for $\a=-1,0,1$, and $\lis y_n^{(-1)}(k',k)$,
%
$$\eqalign{
&\tilde Y^{(\a)}(\t',\t)=-\tanh
g_0\t'\, Y^{(\a)}(\t',\t)\defi \sum_{k'=-\a}^{\io}\sum_{k=\a}^\io
\tilde y^{(\a)}_n(k',k) {x'}^{k'}x^{k}\; , \qquad\a=\pm1,k'={\rm odd}\;, \cr
%
&\tilde Y^{(0)}(\t',\t)=-\tanh
g_0\t'\, Y^{(0)}(\t',\t)\defi \sum_{k'=1}^{\io}\sum_{k=1}^\io
\tilde y^{(0)}_n(k',k) {x'}^{k'}x^{k} \;,\qquad\qquad k'\ {\rm odd}\, \cr
%
&\tilde Y^{(2)}(\t',\t)= Y^{(2}(\t',\t)\defi\sum_{k=1}^\io
\tilde y^{(2)}_n(0,k) x^{k} \;,\cr
%
&\lis Y^{(1)}(\t',\t)= {\cosh g_0\t'\over \cosh g_0\t}
\exp[in\f^0(\t)]\defi\sum_{k'=-1}^{\io}\sum_{k=1}^\io
\lis y^{(1)}_n(k',k)
{x'}^{k'}x^{k},\qquad\quad k'\ {\rm odd}\,,\cr
%
&\tilde T^{(0)}_1(\t',\t)=
g_0(\t'-\t),\qquad \tilde T^{(2)}_1(\t',\t)\=1,
\qquad \tilde T^{(0)}_0(0,\t)=T^{(0)}_0 \, ; \cr}
\Eq(4.16) $$
%
in all other cases the $T,\tilde T, \lis T$--functions will be defined
$1$ (no matter which is the value of the labels that we attribute to
them: this is done to uniformize the notation.
The label $k$ will be called the {\it incoming hyperbolic mode} and
$k'$ the {\it outgoing hyperbolic mode} for reasons that become clear
by contemplating \equ(4.19) below.
In terms of \equ(4.14)$\div$\equ(4.16) the functions
\equ(4.11) multiplied by $\exp[in_v\f^0(\t_v)]$ can be expressed
as in \equ(4.13),
thus defining implicitly the coefficients $c_j(\r,\a,v)$ in \equ(4.13):
%
$$ \eqalignno{
w^0_{j_{\l_v}}(\t_{v'},\t_v) \, \exp[in_v\f^0(\t_v)] & = \cases{
T^{(0)}_0(0,\t_v)\,Y^{(0)} (\t_{v'},\t_v) + Y^{(-1)} (\t_{v'},\t_v),
& $j_{\l_v}=j_v-\ell=0, $ \cr
T^{(2)}_0(0,\t_v)\, Y^{(2)} (\t_{v'},\t_v),
& $0\ell,$ \cr} & \eq(4.17) \cr
%
w^1_{j_{\l_v}}(\t_{v'},\t_v) \, \exp[in_v\f^0(\t_v)] & = \cases{
T^{(0)}_1(\t_{v'},\t_v)\,Y^{(0)} (\t_{v'},\t_v)
+ Y^{(1)} (\t_{v'},\t_v) + & \cr \qquad - Y^{(-1)} (\t_{v'},\t_v),
& $j_{\l_v}=j_v-\ell=0,$ \cr
T^{(2)}_1(\t_{v'},\t_v)\,Y^{(2)} (\t_{v'},\t_v),
& $0\ell . $\cr} \cr} $$
%
One could avoid introducing the $\tilde T$ functions as they are
simply related to the $T$ functions or are just identically $1$:
however it is convenient to introduce them to make the above formulae
more symmetric and therefore easier to keep in mind while working with.
Finally we define the coefficients $\x_j(k',0)$ by the power series
expansion
%
$$ \eqalign{
{1 \over \cosh g_0\t{'}} & = \sum_{k'=1}^{\io} \x_\ell(k',0) x{'}^{k'}
\; , \qquad \ k'\ge1\,, \hbox{ odd } \; , \cr
1 & = \x_j(0,0) \; , \qquad j>\ell \; , \cr}\Eq(4.18) $$
%
where $x'=e^{-\s g_0 \t'}$ and $k'$ is odd, which occurs as
coefficient $w_{00}(\t')$ in \equ(4.7) (when $\r_v=0$, \ie
$v\in\Th_L$).
The above definitions (taken from (42) and (45) in [Ge2]) suffice to
discuss the whiskers (and therefore the splitting in the action variables).
\*
\pallino The \equ(4.13) allow us to introduce a
``relatively simple notation'': we can add to each node a {\it badge}
label $\a_v=(-1,0,1,2)$ that will distinguish which choice we make
between the possibilities in \equ(4.14) and \equ(4.16) and two
{\it hyperbolic mode} labels $k'_v,k_v$ which select which particular
term we choose in the sums in \equ(4.14) and \equ(4.16);
they are integer numbers $\ge-1$. We shall
not have to introduce labels to distinguish terms coming from the
expansions of $Y^{(\a)},\tilde Y^{(\a)}, \lis Y^{(\a)}$ bearing the
same badge $\a$ because one can check that the labels $\a_v$ together
with $j_v$ and $v$ itself uniquely determine which choice has to be made.
In terms of the latter labels we can define a {\it hyperbolic
momentum} of a line $\l_v$ as a label $p(v)\in \ZZZ$ which will be the
sum of all the hyperbolic modes of the nodes that precede $v$ {\it
plus} the incoming hyperbolic mode of the node $v$ itself: this
is the sum of the labels $k_w$ associated with all {\it free} nodes
$w\le v$, with $v$ included, and of the labels $k_w'$ associated with
all the {\it free} nodes $w\ell$ it is natural
to collect together the terms with $p(v_0)=0$: for them, since
$j_{\l_{v_0}}>\ell$, in \equ(4.30)
one must have $k'_{v_0}+p(v_0)=0$ by the last of
\equ(4.14). Note also that by \equ(2.26) the case
$(\nn_0(v),p(v_0))=(\V0,0)$ is excluded. If $j_{\l_{v_0}}\le \ell$ we,
likewise, collect the terms with $k'_{v_0}+p(v_0)=0$ and, for similar
reasons the term with $k'_{v_0}+p(v_0)=-1$ cannot be present
(see again \equ(4.14) and \equ(4.16), and use
$p(v_0)\ge -1$ supplemented by the relations between the labels
$p(v_0)$ and $k_{v_0}'$ which will be exhibited in \S 5.1).
Hence the cases with $p(v_0)+k'_{v_0}=-1$ are excluded by
construction\annota{6}{\rm The initial data $\X^{h\s}(0,\aa)$ were
determined precisely by imposing boundedness at $\s t=+\io$, \ie by
imposing the absence of divergent terms in the expansion in powers of
$x=e^{-g_0\s t}$ which would correspond to the terms with $p(v_0)=-1$.} and
we see that the sum of the values of the trees with
$p(v_0)+k_{v'_0}=0$ give us the equations for the actions and the
angles of the invariant torus to which the whiskers considered are
asymptotic: the terms with $p(v_0)+k_{v'_0}=0$ asymptote to quasi
periodic functions of $\oo t$ so that replacing $\oo t$ by $\pps\in
T^{\ell-1}$ one gets a parameterization of the points on the tori in
terms of a point $\pps\in T^{\ell-1}$ on a ``standard torus''.
And the terms with $k'_{v_0}+p(v_0)=1$ provide the leading
corrections. Since such terms are present already to order $0$ (as one
sees from the expression of the pendulum separatrix) the distance
between a point moving on the stable manifold of the torus and the
torus itself will be proportional to $x=e^{-g_0\s t}$ as $\s t\to\io$
so that $g_0$ has the interpretation of Lyapunov exponent of the
invariant torus; see \equ(2.27) in \S 2.27.
(c) Summarizing: {\it the case $(\nn_0(v),p(v))=(\V0,0)$ has to be
ruled out as a consequence of \equ(2.26) and of \equ(4.32),
respectively for the contributions to $\XXX_{\su}^{h\s}$ and to
$\X_+^{h\s}$ (see the last constraint listed at the beginning of \S
4.3). All cases with $k'_{v_0}+p(v_0)=-1$ are also excluded}.
%\ifnum\mgnf=0\pagina\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1.truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0{\titolo \S 5. Bounds.}
\numsec=5\numfor=1\*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0{\bf 5.1.}
We now discuss how to bound the value of a tree or of a sum of a small
number of trees which we take for simplicity without leaves and
without counterterms.
The more general case will be eventually reduced, see
below, to the one we consider here.
We shall discuss first how to bound values of trees without leaves
and counterterms such that $p(v_0)=0$ if $v_0$ is the highest node;
hence we shall consider trees, always without leaves
and counterterms, with $p(v_0)= 0$. At the end we shall see
how the presence of leaves and counterterms modifies the analysis.
The following discussion is ``locally'' simple, but ``globally''
delicate and repeats that in [Ge2], \S4:
the conclusions are also summarized in the table 0,1,2,3 below.
{}From \equ(4.19) it follows that the hyperbolic momentum $p(v)$ is
$p(v)\ge -1$ and, as remarked after
\equ(4.19), $p(v)=0$ can occur only in special cases:
more precisely if $p(v)=0$, then $k_v$ is either $-1$ or $0$, and\\
(1) if $k_{v}=0$, all free nodes $w$ preceding $v$
(whether immediately or not) have $k_w'+k_w=0$, while\\
(2) if $k_v=-1$, all free nodes $w$ preceding $v$ have $k_w'+k_w=0$, {\it
except} for a single node $\tilde w\ell$, because $k_w'$ must be $0$ in
such a case, so that the second of \equ(4.18) applies;
\0(2) if $k_v=-1$, then all the leaves again must have the highest node
$w$ with $j_w>\ell$, except at most one leaf with highest node
$\tilde w$ with $j_{\tilde w}=\ell$ and $k_{\tilde w}'=1$.
\*
\0{\cs (Vertical) Table 0.} Possible cases when $p(v)=0,-1$.
\*
\halign{\strut\vrule\kern2truemm$#\quad$&
\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#$\quad\hfill\vrule\cr
\noalign{\hrule}
p(v) & k_v & k'_v & \a_v & j_{v} \cr
-1 & -1 & {\rm odd}\ge 1 & -1 & \ell \cr
0 & -1 & {\rm odd}\ge 1 & -1 & \ell \cr
0 & 0 & {\ge1} & -1 & \ell \cr
0 & 0 & {\ge0} & 2 & {>\ell} \cr
\noalign{\hrule}
}
\*
\0{\cs (Horizontal) Table 1.} Cases $p(v)=0$, $w\notin \PP$.
\*
\halign{\strut\vrule\kern2truemm$#\quad$&
\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#$\quad\hfill\vrule\cr
\noalign{\hrule}
\a_w & -1 & 0 & 1 & 2 \cr
(k'_w,k_w) & (1,-1) & \hbox{impossible} & (-1,1) & (0,0) \cr
p(w) & -1 & \hbox{impossible} & 1 & 0 \cr
j_w & \ell & \hbox{impossible} & \ell & {>\ell} \cr
\noalign{\hrule}
}
\*
%\ifnum\mgnf=0\pagina\fi
\0{\cs (Horizontal) Table 2.} Cases $p(v)=0$, $w\in \PP, w> \tilde w$.
\*
\halign{\strut\vrule\kern2truemm$#\quad$&
\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#$\quad\hfill\vrule\cr
\noalign{\hrule}
\a_w & -1 & 0 & 1 & 2 \cr
(k'_w,k_w)& (1,-1) & {\rm impossible} & (-1,1) & (0,0) \cr
p(w) & 0 & {\rm impossible} & 2 & 1 \cr
j_w & \ell & {\rm impossible} & \ell & {> \ell} \cr
\noalign{\hrule}
}
\*
\0{\cs (Horizontal) Table 3.} Cases $p(v)=0,\, w=\tilde w$.
\*
\halign{\strut\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#\quad$&
\vrule\kern2truemm$\quad#$\quad\hfill\vrule\cr
\noalign{\hrule}
\a_w & -1 & 0 & 1 & 2 \cr
(k'_w,k_w)& (1,0) & {\rm impossible} & (-1,2) & (0,1)\cr
p(w) & 0 & {\rm impossible} & 2 & 1 \cr
j_w & \ell & {\rm impossible} & \ell & {> \ell} \cr
\noalign{\hrule}
}
\*
\pallino We extend the definition of path also to the case $p(v)=0,
k_v=0$, by setting $\PP\defi\emptyset$ if $j_v>\ell$ and $\PP\defi v$ if
$j_v=\ell$, only for purposes of notational convenience (see \equ(5.3)
below). This is consistent with the above tables and does not change
them.
\*
\0{\bf 5.2.} {\cs Remark.}
Note that, if a tree (or subtree) $\th_0$ with highest node $v_0$
has total hyperbolic momentum $p(v_0)=0$, then there is one and
only one path $\PP$, and, if $\PP\neq\emptyset$, then $\PP$
connects the node $v_0$ to some node $\tilde w\ell$, \cr
-\s g_0^2\left[ g_0^2+ ( \oo'\cdot\nn_0(v) )^2 \right]^{-1}
& if $v\notin\PP$, $j_v=\ell$, \cr
g_0^{2-\d_{j_v,\ell}}\left[-\s\,( g_0p(v)-i\s \oo'\cdot\nn_0(v))
\right]^{-(2-\d_{j_v,\ell})}
& if $v\in\PP$, $\a_v\neq-1$, $\to p(v)\ne0$ \cr
g_0\left[i\oo'\cdot\nn_0(v)\right]^{-1} &
if $v\in\PP$, $\a_v=-1$, \cr} \Eq(5.3)$$
%
because:
\0(a) The first line is such because if $j_v>\ell$ one has necessarily
$\a_v=2$, see Tables 0,1,2,3 and we have to integrate a function
$g_0(\t_{v'}-\t_v) e^{i n_v\f^0(\t_v)}$ so that $k_v\ge0$: hence
$k_v=p(v)=0$ and we have the second function in \equ(4.29) to integrate.
\0(b) The second line is such because if $v\not\in\PP,\, j_v=\ell$ we have
$w^1_\ell(\t_{v'},\t_v) e^{i n_{v}\f^0(\t_{v})}$ which is a sum of three
terms (see the third of \equ(4.17)): the first has $k_v+k_{v'}\ge2$ so
is excluded (recall that $p(v_0)=0$ and $v\le v_0$); while the second
only sees the contribution to $Y^{(1)}$ with $k_v'=-1,k_v=1$, see
\equ(4.14), and the third only contributes by the term with
$k_v'=1,k_v=-1$ in $Y^{(-1)}$. In the two cases one has $p(v)=1$ or
$p(v)=-1$ respectively; adding up together the latter two contributions
and using the first of
\equ(5.2) to compute the sum of the coefficients we get
%
$$ \fra{-\s g_0 y_{n_v}^{(1)}(-1,1) }{g_0-i\s\oo'\cdot\nn_0(v)}
- \fra{-\s g_0 y_{n_v}^{(-1)}(1,-1) }{-g_0-i\s\oo'\cdot\nn_0(v)}
= \fra\s2 \fra{-2\s g_0^2}{g_0^2
+(\oo'\cdot\nn_0(v))^2} \; , \Eq(5.4) $$
%
as it can be read from the coefficients in the intermediate
column of \equ(4.24) and from $p(v)=\a_v=\pm1$.
\0(c) The third line of \equ(5.3) is obtained by noting that, if
$v\in\PP$, $v> \tilde w$ one has $p(v)=1+k_v$, so that, if $j_v=\ell$
and $\a_v\ne-1$, then $p(v)>0$, see Table 2; if $v=\tilde w$ and
$\a_v\ne-1$, one has $p(v)\ne0$, see Table 3 (note that $\a_v\ne2,0$ so
that we have to consider the first integrand in \equ(4.29)).
If $j_v>\ell$ then $\a_v=2$, and, by the Tables 2,3, one has
$k_{v'}=0,k_v\ge0$ and $k_v+k_{v'}=1$, so that
$k_v=1$ and $p(v)=2$; while, if $v=\tilde w$, then $k_{v'}=0,
k_v\ge0$ and $k_{v}'+k_v=1$ imply $k_v=1$, so that $p(v)=1$.
So in both cases $p(v)\ge 1$.
\0(d) The fourth line is found by looking at the Tables 2,3 as follows: if
$\a_v=-1, v\in\PP, v>\tilde w$, one has $k_v+k_v'=0$, hence
$k_v=-1,k_{v'}=1$ and $p(v)=0$; this happens only if $j_v=\ell$ so that
we have to consider the first integral in
\equ(4.29) and we get the fourth relation.
\*
This shows that the only trees that do not have a value tending to $0$
as $t\to\s\io$, \ie are those with $p(v_0)+k_{v_0}'=0$ (all the others
tend to $0$ as a power of $x=e^{-g_0\s t}$), have propagators that are
even functions of the momenta flowing in them. In fact the observation
on the absence of paths preceding $v_0$ implies that only the first
two propagators in \equ(5.3) appear in such trees. Since, as already
remarked, the trees with $p(v_0)+k_{v_0}'=0$ give the equations of the
tori this is an interesting check that the tori equations so obtained
at $t=+\io$ and $t=-\io$ do {\it coincide}. A similar analysis, and
check, holds for the cases $j_{\l_{v_0}}\le \ell$.
\*
\0{\bf 5.4.} {\cs Remark.}
Collecting together the contributions from $\a_v=-1$ and $\a_v=1$, for
$v\notin\PP$, is a convenient operation and has nothing to do with the
deeper resummations that imply the cancellations necessary for
convergence estimates: the systematic use
of this operation should be described by adding a label to the trees
on the nodes $v\notin \PP$ and replacing on the branches which give
rise to one of the two propagators in \equ(5.4) the $\a_v$ label by
the new label (\eg a $*$ label which indicates that we consider the
sum of the values of a tree with $\a_v=1$ and one with $\a_v=-1$).
We shall do this without explicitly mentioning the new label, to
simplify the notation. Moreover we can no more associate a label
$p(v)$ to a node of this kind, as two factors with different $p(v)$
label ($p(v)=\pm 1$ for $\a_v=\pm 1$) have been considered together;
nevertheless we shall modify slightly the definition of $p(v)$ by
setting $p(v)\defi 1$ in such a case (and letting it unchanged in all
the other cases).
We shall continue to call $G_v[\oo'\cdot\nn_0(v)]$ a {\it propagator}
as, for the purposes of the following analysis, only
such modified version of the original propagators
appearing in \equ(4.30) plays a r\^ole.
\*
\0{\bf 5.5.}
Furthermore we define the {\it degree} $D$ of a propagator to be $D=2$
if either $v\notin\PP$ or $v\in\PP,j_v>\ell$ (hence $\a_v\ne-1$), and
$D=1$ otherwise (the constraint, see \equ(4.3), $1\le r_v\le 2$
implies that the power to which the divisors appear raised is either
$1$ or $2$); by extension we shall say that a branch $\l$ has degree
$D_{\l}=D$ if the corresponding propagator has degree $D$.
The coefficients $\bar F_{\nvec_v}$ and $y'_{v}$
in \equ(5.1) satisfy the bounds
%
$$ |y_{v}'|\le 4N \; , \qquad \prod_{v\le v_0}
|\bar F_{\nvec_v}|\le (\CC N^2)^m \; , \Eq(5.5) $$
%
for some constant $\CC$ depending on the perturbation $f_1$ in
\equ(2.1); see \equ(2.6), \equ(2.13) and \equ(2.18).
For instance one can take
%
$$ \CC=\max\{|J^{-1}|J_0,1\} \max_{|n|\le N_0 ,
\, |\nn|\le N} |f_\nvec| \; ; \Eq(5.6) $$
%
see \equ(4.23), where $|J^{-1}|$ is the maximum of the matrix elements
of the (diagonal) matrix $J^{-1}$.
To bound the product in \equ(5.1), we shall consider simultaneously the
cases $k_{v_0}=0,-1$; if $k_{v_0}=0$ the path $\PP$ is supposed to be
reduced to a single node, $v_0$, or to the empty set, $\emptyset$,
depending on the value of $j_{v_0}$, (respectively $j_{v_0}=\ell$, and
$j_{v_0}>\ell$, see above).
What follows below and in Appendix A2 really goes beyond [Ge2],
although it constitutes a natural extension of it. From now
now let us consider the case $\ell=3$ and the Hamiltonian \equ(1.1).
We shall assume first a condition on the rotation vectors stronger than
the Diophantine one, as done in [G1,GG,Ge2], \ie we suppose that they
satisfy a {\it strong Diophantine condition}
%
$$ \eqalignno{
(1) & \quad C_0 | \oo_0 \cdot \nn| \ge |\nn|^{-\t} \; , \quad\quad
\V0 \neq \nn \in \ZZZ^{2} , \qquad C_0^{-1}=\hdm C(\h) \; ,
&\eq(5.7) \cr
(2) & \quad
\min_{0\ge p\ge n} \Big| C_0 |\oo_0 \cdot \nn| -
2^p \Big| \ge 2^{n+1} \; , \quad\hbox{if} \quad n \le 0,
\; \; 0 < |\nn| \le (2^{n+3})^{-1/\t} , \cr} $$
%(3) & \quad |\oo_0\cdot\nn|\ne 1
%\qquad {\rm for\ all}\ \nn\in Z^{\ell-1}\cr} $$
%
where $n, p \in \ZZZ$, $n\le 0$, and
%
$$ \oo_0\= \h^{-1/2}(\O_1+\hdp J^{-1}A_1)^{-1}\oo'=
\big(1,\h^{-1}(\O_1+\hdp J^{-1}A_1)^{-1}\O_2\big) \; , \Eq(5.8) $$
%
so that $\oo'\cdot\nn=\h^{1/2}(\O_1+\hdp J^{-1}A_1)\oo_0\cdot\nn$.
We suppose also that $A_1\in[-\hdm R,\hdm R]$, with $R\le J\O_1/2$,
so that $\h^{1/2}(\O_1+\hdp J^{-1}A_1)\ge \h^{1/2}\O_1/2$.
If we write $\oo'=(\hdp\O_1+\h J^{-1}A'_1,\hdm\O_2)$
then the measure of the set of
$A'_1$'s such that $\oo_0$ verifies the strong Diophantine condition
\equ(5.7) has measure of size $O(C_0^{-1}\h^{-3/2})$.
%The third condition in \equ(5.7) does not affect the measure
%size because it is verfied outside a set of zero measure.
By reasoning as in [GG], once the case of strong Diophantine vectors
has been understood, it can be extended to cover also the case of the
usual (weaker) Diophantine condition (expressed by (1) in \equ(5.7)
above). Alternatively one could follow the approach in [GM] avoiding
completely considering condition (2) in \equ(5.7) and assuming only
the ``usual'' condition (1) in \equ(5.7). We shall not perform such
an analysis (which can be easily adapted from the quoted papers), and
we shall confine ourselves to the case of strongly Diophantine
vectors.\annota{7}{\rm Basically the argument is the following: the
analysis that we present does not change if $2^p,2^n$ are replaced by
exponentials in another base $q$ (larger than $1$) or even if they are
replaced by $\g(p),\g(n)$, where $\g(p)/q^p\tende{p\to-\io}1$, and if
in the second of \equ(5.7) we substitute $2^p, 2^{n+1}, 2^{n+3}$ by,
respectively, $\g(p), \g(n+1), \g(n+3)$. One then proves a simple
arithmetic lemma (see [GG]), whereby it follows that, if the
first of \equ(5.7) is verified and if $\g(p)$ is suitably chosen,
then the second holds with $\g(p), \g(n+1), \g(n+3)$
replacing $2^p, 2^{n+1}, 2^{n+3}$.}
Keeping in mind that $C_0=\hdm e^{+s \hdm}$ is enormous we shall say that
%
$$\eqalign{
(1)\ & \ G_v[\oo'\cdot\nn_0(v)] \qquad\hbox{is on scale 1,
if $C_0|\oo_0 \cdot \nn_0(v) | > C_0/4 $, or if $p(v)\ne0$;}\cr
(2)\ & \ G_v[\oo'\cdot\nn_0(v)] \qquad\hbox{is on scale 0,
if $1/2\ell$ (see the first of
\equ(5.3)) and $p(v_0)=0$ implies
that $\a_{v_0}=2, k_{v_0}=0, k'_{v_0}=0$ so any path preceding $v_0$
would necessarily imply the contradiction $p(v_0)=1$. Also if
$D_{\l_{v_1}}=1$, $D_{\l_{v_0}}=2$ one must have $j_{v_1}=\ell$ hence
$k_{v_1}=0$ (otherwise $p(v_0)>0$) so that $k'_{v_1}=0$: {\it but}
$\a_{v_1}<2$ and $k'_{v_1}$ must be odd. The cases $D_{\l_{v_0}}=1$,
$D_{\l_{v_1}}=1,2$ are both allowed.
\*
\0{\bf 5.9.}
Given a tree $\th$, let $V$ be a resonance (if there are any) with
entering branch $\l_{v_1}$ of degree $D_{\l_{v_1}}=2$. Then consider
the family of all trees which can be obtained from $\th$ by detaching
the part of the tree having $\l_{v_1}$ as root branch and reattaching
it to all the remaining nodes {\it internal to $V$ but external to the
resonances contained inside the cluster $V$} (if any); to the just
defined set of trees we add all the trees obtained by reversing
simultaneously the signs of the latter modes of the nodes
(this can be done as the sum of the mode vectors $\nn_w$ of such nodes,
$w\in V$, vanishes). The set of all the so obtained trees will be
denoted $\FFFF_V(\th)$.
The definition of resonance and the strong Diophantine condition
insures that all the trees so constructed have a well defined value
(\ie no division by zero occurs in evaluating it with the above
rules); see the Remark 5.10, (1), below.
If the entering branch $\l_{v_1}$ of the resonance
has degree $D_{\l_{v_1}}=1$ then also the exiting branch $\l_{v_0}$
has degree $D_{\l_{v_0}}=1$, and we
collect together with the considered tree also the tree which is
obtained from $\th$ through the following operation. Replace the
resonance $V$ with {\it a single node} $v$ carrying labels $\d_v=0$
and $\k_v=k_V$, if $k_V$ is the order of the resonance. The set of
all the so obtained trees will be denoted by $\FFFF_V(\th)$: the
definition of the class $\FFFF_V(\th)$ will therefore depend on the
degree of the branch entering $V$.
Then repeat the above operations for all resonances in $\th$.
Thus a class $\FFFF(\th)$
has been constructed and the number of elements of $\FFFF(\th)$ is
bounded by the product $\prod_V 2 \NN_V$ of the numbers $\NN_V$ of
branches in each resonance $V$ {\it which are not} contained inside
inner resonances. The latter product is bounded by
$\exp \sum_V 2\NN_V\le \exp 2m $; the $\FFFF(\th)$ can be obtained
starting from any of its elements (which therefore we shall call
{\it representatives} of the class): this is again a {\it consequence of
the strong Diophantine condition}, see [Ge2].
\*
\0{\bf 5.10.} {\cs Remarks.}
(1) The strong Diophantine condition plays a r\^ole here that should be
stressed. In fact one checks that because of it the scale of a branch
inside a resonance {\it cannot} change too much, as one considers the
different members of a given family. Not enough to change the sets of
branches that belong to a given resonance and insures that the different
families of trees {\it do not overlap}: for this reason the strong
Diophantine condition leads to a simplification of the analysis (the
simplification in the simpler case of the KAM theory). A simplification
that is however not major (as explained informally in [G1] and as shown
in [GG], see footnote 7 above).
(2) To see how the above difficulty is bypassed by using the
alternative approach of [GM1,GM2], we refer to the conclusive comments
in [GM1], \S 3.
\*
\0{\bf 5.11.} Consider trees with $p(v_0)=0$, if $v_0$ is the
highest node of the tree; then the expression of each tree value
contains a product like \equ(5.1). As mentioned in the introduction {\it
we consider only trees without leaves.}
Since the leaf values factorize with respect the product \equ(5.1),
they can be dealt with separately, and no overlap arises with the
cancellation mechanisms acting on the product
\equ(5.1): so that leaves can be easily taken into
account; see \S A3.3 in Appendix A3 (see also [G1,Ge1,Ge2]).
The counterterms can also be explicitly expanded in terms of tree
values, according to \equ(3.2), which again we can imagine to have no
leaves, (see however the comments in \S A3.3 below).
The cancellation mechanisms described in [Ge1,Ge2] (and recalled in
Appendix A3) lead to the bound (on a given family $\FFFF(\th)$
described above, in \S 5.9), see \equ(5.1), \equ(4.30), \equ(4.23)
%
$$ \eqalignno{
\left( {1\over \h^{1/2}} \right)^{2m}
& \Big[ (4N^3\CC')^{m} 2^{4m}e^{2m}
\prod_{n\le0} \big( C_0^{2N_n^2} 2^{-2nN^2_n} \big)
\big( C_0^{N_n^1} 2^{-nN^1_n} \big) \Big] \; \cdot & \eq(5.11)\cr
& \cdot \Big[\prod_{n\le 0}\;\prod_{T,\,n_T=n}
\prod_{i=1}^{m^1_T(n)}\,2^{(n-n_{i}+3)}
\prod_{i=1}^{m^2_T(n)}\,2^{2(n-n_{i}+3)}
\Big] \; , \cr} $$
%
where
\0$\bullet$ $\CC'=\max\{(2g_0/\O_1)^2,4^2\}\CC$, with $\CC$ the
dimensionless constant defined in \equ(5.6);
\0$\bullet$ $m$ is the number of nodes $v\ge v_0$;
\0$\bullet$ $N^j_n$ is the number of propagators on scale $n$ and of degree
$j$ in $\th$, which can be written as
%
$$ N^j_n=\bar N^j_n+\sum_{T \atop n_T=n, D_T=j}
(-1) + \sum_{T \atop n_T=n} m^j_T(n) \; , \Eq(5.12) $$
%
where $m^j_T(n)$ is the number of resonances on scale $n$ and degree
$j$ (\ie with entering branch having a propagator of degree $j$)
contained inside the cluster $T$;
\pallino the terms $\bar N^j_n$, $j=1,2$, which count the number of
propagators {\it which do not correspond to resonant branches} plus
the number of clusters on scale $n$ and of degree $j$ in $\th$,
satisfy the bounds
%
$$ \sum_{j=1}^2 \bar N^j_n \le 4 m N 2^{(n+3)/\t},\qquad
\sum_{n=-\io}^{0} \sum_{j=1}^2 \bar N^j_n \le
4 m \g N \h \; , \Eq(5.13) $$
%
(with $\g=4\O_1/\O_2$) which are proven in Appendix A2;
\0$\bullet$ the first square bracket in \equ(5.11) is the bound on
the product of individual elements in the family $\FFFF(\th)$ times the
bound on their number $\prod_V 2\NN_V< e^{2m}$, see above.
\0$\bullet$ the second square bracket term is the part coming from the maximum
principle, (in the form of Schwarz's lemma), applied to bound the
sums of the tree values (``{\it resummations}'') over the classes
$\FFFF(\th)$
introduced above: this is a {\it non trivial product of small factors}
that arise from the
cancellations associated with the resummations, see Appendix A3. In
\equ(5.11) $n_i$ is the scale of the cluster $V_i$ which is the $i$--th
resonance inside $T$, as in [Ge2];
\pallino the $\eta^{-m/2}$
arises as a lower bound on the small divisors of the form $\oo'\cdot\nn$
on scale $n=1$ (for $n=1$ we use the better bound $|\oo_0\cdot\nn|\ge
2^2\hdp$).
\*
\0{\bf 5.12.}
{\cs Remark.} The first bound \equ(5.13) holds for all $n$ and for all
Hamiltonians of the form \equ(2.1). On the contrary the second bound in
\equ(5.13) will follow from the fact that the rotation vector $\oo_0$
has the form \equ(5.8), with $\h$ small, and will be used to control
the (huge) factors $C_0$ in \equ(5.11).
\*
\0{\bf 5.13.}
Hence by substituting \equ(5.12) and the first of \equ(5.13) into
\equ(5.11) we see that, for $j=1,2$, the $m^j_T(n)$ is taken away by
the first factor in $\,2^{jn} 2^{-jn_{i}}$, while the remaining
$\,2^{-jn_i}$ are compensated by the $-1$ before the $+m^j_T(n)$ in
\equ(5.11) taken from the factors with $T=V_i$ (note that there are
always enough $-1$'s), and therefore
\equ(5.11) is bounded by
%
$$ \eqalign{
& \left( {2\over \h^{1/2} } \right)^{2m} (4N^3\CC')^{m}
e^{m} 2^{4m}2^{8m} C_0^{8m\g N\h}
%2^{8mN\g} \Big] \cr & \Big[ \prod_{n=-\io}^{n_0-1} C_0^{8mN2^{(n+3)/\t}}
\prod_{n=-\io}^{0} 2^{-8 m N n 2^{(n+3)/\t} } \; , \cr} \Eq(5.14) $$
%
because the product of the factors $C_0$ in \equ(5.11) can be bounded by
using the second of \equ(5.13), since the product does not contain the
$n=1$ factor). The last product in \equ(5.14) is bounded by
%
%$$ \eqalign{
%\prod_{n=-\io}^{n_0-1} & C_0^{8mN2^{(n+3)/\t}} 2^{-8mNn2^{(n+3)/\t})}
%\cr & \qquad \le \exp \left[ 8mN 2^{3/\t}
%\sum_{p=[\g/\h]+1}^{\io} (\log C_0+p\log2) 2^{-p/\t} \right]
%\; , \cr} \Eq(5.15) $$
$$ \prod_{n=-\io}^{0} 2^{-8mNn 2^{(n+3)/\t}} \le \exp \Big[ 8mN2^{3/\t}
\log2 \sum_{p=1}^{\io} p 2^{-p/\t} \Big] \; , \Eq(5.15) $$
%
hence, by adding the remark that the perturbation degree $k$ and the
number of tree nodes $m$ are related by $m<2k$, a bound on the
sum over all the subtrees of order $k$ with $p(v_0)=0$, $\nn(v_0)=\nn$
(recalling that the number of trees with $m$ nodes is $<4^m m!$) is
%
$$ \D_k\defi \Big| \fra1{|\FFFF(\th)|}
\sum_{\th'\in\FFFF(\th)} \prod_{v\in\th'}
\bar F_{\nvec_v} G_v[\oo'\cdot\nn_0(v)]\,\tilde y_{v}\Big|
\le B_0^{2k} \h^{-2k}\; , \Eq(5.16)$$
%
for some positive constant $B_0$.
The normalization constant $|\FFFF(\th)|$ is
introduced in order to avoid overcountings: in fact if
$\th'\in\FFFF(\th)$ then $\th\in\FFFF(\th')$, so that,
without dividing by $|\FFF(\th)|$ in \equ(5.16), each tree would
be counted $|\FFFF(\th)|$ times.
If $C_0^{-1}=\hdm C(\h)$ is chosen
as in the statement of Theorem 1.4, an explicit calculation gives
the bound on \equ(5.11) of the form $(\hdm)^{4k} B_0^k$, $k\ge1$, and
%
$$ B_0 = 2^{18}(4N^3\CC') \, \exp \Big[ 2+4\g N\h\log\h+ 8s\g N\hdp
+ 8N 2^{3/\t}\log 2 \sum_{p=1}^{\io} p 2^{-p/\t} \Big] \; , \Eq(5.17) $$
%
which is bounded uniformly in $\h$ (for $\h\le1$).
\*
\0{\bf 5.14.} In the previous section trees with $p(v_0)=0$
have been considered; in particular only the contributions
\equ(5.1) arising from the value \equ(4.30), once the
corresponding tree has been deprived of leaves and counterterms,
have been bounded and the bound \equ(5.16) has been obtained
through a suitable resummation operation.
In such a case the sum over the labels
$(k_v',k_v)$ is trivial because the condition $p(v_0)=0$ imposes that
only a few values (up to three per node) can be assumed by the
hyperbolic mode labels; also the sum over the mode labels $\nvec_v$
cannot create any problems. In fact for any node $v$ one has
$|\nvec_v|\le N$ and $|n_v|\le N_0$ (see the eighth item in \S 4.1).
The cases $p(v_0)\ne0$ as well as those involving graphs containing
leaves or counterterms can be treated in the same manner as already
done in [G1,Ge2]. We provide, in Appendix A4, a quick description
of the construction of the analyticity bound $\e_0=D^{-1}$ with
%
$$e_0^{-1}=D=\left[ B 2^6\ell (2N+1)^{2\ell-1}(2N_0+1) \right]^2 \; ,
\qquad B=\max(B_0\h^{-1},B_1)\Eq(5.18)$$
%
and $B_1$ is a suitable numerical constant.
The part of Theorem 1.4 not concerning the connection between the
average action $\AA'$ and the rotation vector $\oo'$ nor the splitting
size follows.
\*
\0{\bf 5.15.}
Determining the exact splitting size (\ie the leading behavior
asymptotically as $\h\to0$ with $\e< B\h^2$) is {\it not} trivial
because of the existence of major cancellations in the evaluation of
the determinant of the splitting matrix; however the analysis in
[GGM2] dealt with this question in detail: in the latter paper
remarkable cancellations are exhibited and an exact formula for the
splitting angles is derived (see (7.19) of [GGM2]).
One gets the results in the last item of Theorem 1.4 simply if
[GGM2] and the first part of Theorem 1.4 (to estimate the
remainders) are used: then the claimed bounds on the splitting follow
immediately (see Remark 1.5). In [GGM3] an improvement of lemma 1 and
lemma 1' of [CG] was used instead to control the density of
tori in phase space (the lemmata in [CG] were,
as such, useless already in the case in [GGM2]
because they would require that $\e$ be far smaller than the $\e_0$ of
Theorem 1.4); see [GGM3], where this is discussed in detail and differs
from our case only because it relied on a theorem weaker than
Theorem 1.4 above (as the radius of convergence estimate there is
proportional to $\h$ to the power $\fra92+$ rather than our $2$).
\*
\0{\bf 5.16.} {\cs Remarks.} (1)
The bound \equ(5.16) and the discussion in \S 5.14
imply the convergence of the perturbative
expansions for the parametric equations of the invariant tori (for the
Hamiltonian \equ(1.1)), if $|\e|<\e_0=O(\h^2)$. This bound on the
convergence radius should be compared with the value given by
[GGM3], which, for $\NN=O(\hdm)$, gives $\e_0=O(\h^{\fra92}/
\log^2\h^{-1})$. {\it As usual the Lindstedt series gives a much
better estimate than the classical method} (\ie an exponent $2$ versus
$\sim 4.5$). We do not see immediately how to improve substantially the
classical estimate without important changes in the architecture of the
proof of [GGM3], although this should be possible; on the other hand,
from the above analysis, $\e_0=O(\h^2)$ might be close to an optimal
result. If so it should be no surprise that our analysis is so delicate.
(2) In the Hamiltonian \equ(1.1),\equ(2.1) the polynomial dependence
of the interaction on the rotators angles has very likely a purely
technical motivation (as it simplifies the analysis) and could
probably be relaxed into a more general analytical dependence, as in
[BCG]. On the contrary the hypothesis that the perturbation is a
trigonometric polynomial of degree $N_0$ in $\f$ is fundamental to
get the correct asymptotic behavior, in order to apply the results in
[GGM2], where the dominance of Mel'nikov integral is proven {\it
provided the perturbation is polynomially small in a power of
$\h^{N_0}$} (so that the results of [GGM2] become meaningless for
$N_0\to\io$).
(3) A bound of the form \equ(5.16)
holds under the weaker condition that $C(\h)\le e^{-s\h^{-a}}$, with
%$a<1$. Also the case $a=1$ can be included provided that, in such a
%case, a condition on $s$ has to be imposed.
$a\le 1$ (see \equ(5.17)).
(4) If $q$ is defined as in \S1.6 so that $|\e| C(\h)^{q\h}< 1$ implies
analyticity in $\e$, the above analysis gives that $q$ can be taken
$q=8\g N$.
In general all the bounds found so far are not uniform in $N$; in order
to deal with the analytical case in the frame of the exploited formalism
one should bound the small divisors by using the results of [GM2] or
Eliasson-Siegel's bound (see for instance [BGGM]), and use explicitly as
in [BCG] the exponential decay in $\nn$ of the Fourier coefficients
$f^1_{\nvec}$.
(5) Note that we have convergence for $|\e| T_c$ the asymptotic
formula that we can prove only for $T> \z$ holds, but for
$T=T_c$ it is modified remaining qualitatively of the same size
$O(e^{-\fra12\hdm})$ and for $TT_c$ is described by a trivial fixed point; a non trivial
fixed point describes the case $T=T_c$ and another ``low temperature''
fixed point describes the cases $T\ell$ or $j_w=\ell$ and internal
momenta $(\nn',p')$. Such terms would give rise to $\aa$--dependent
counterterms which of course are not allowed: however it turns out
that the sum over all contributions to tree values of trees with
$(\nn(v_0),p(v_0))=(\V0,0)$ from such trees cancel {\it exactly}:
this is explained, together with the other cancellations built in
our algorithm, in Appendix A3 (see \S A3.5 in particular).
\*
\0{\bf A1.3.}
Let us consider the first integral in \equ(A1.1). Corresponding to the
node $v_0$ of each tree whose value contributes to $\X_-^{h\s}(t)$ there
is a coefficient $\tilde y_{n_{v_0}}(k_{v_0}',k_{v_0})$, see \equ(4.14),
\equ(4.17). Then from \equ(A1.1) and the just formulated
condition to impose we obtain
%
$$ \sum_{\th\in\TT_{\V0,h} , \a_{v_0}=-1 \atop p(v_0)=0 ,k_{v_0}'=1}
\overline{{\rm Val}}(\th) + \g_h(g_0) \left.
w_{\ell}^1(t,\t)\,
\sin\f_0(\t) \right|_{k'=1,p=0} = 0 \; , \Eqa(A1.4) $$
%
where the sum is over the set $\TT_{\V0,h}$ of
all trees of order $h$ and momentum $\nn(v_0)=\V0$,
with $\nn_0(v_0)=\V0$ (see Remark A1.2),
$p(v_0)=0$, $j_{v_0}=\ell$ and $k_{v_0}'=1$; hence if $p(v_0)=0,
j_{v_0}=\ell$, one must have $k_{v_0}=-1$, hence $\a_{v_0}=-1$ and
$k'_{v_0}=1$ which is a possible case indeed.
A trivial calculation (just take into account that
$y_{n_v}^{(-1)}(1,-1)$ $=$ $\s/2$ and
$\sin\f^0(\t)=4\s x + O(x^3)$) gives
%
$$ \left. w_{\ell}^1(t,\t)\,
\sin\f_0(\t) \right|_{k'=1,p=0} = 2 \; , \Eqa(A1.5) $$
%
so that \equ(4.32) follows; the above follows [Ge2], page 287.
\*
%\0{\bf A1.3.} {\cs Remark.} In defining the counterterms
%in \S A1.1 no condition has been imposed on the label $\nn'$ appearing
%in \equ(A1.2) and \equ(A1.3). So it is not {\it a priori} evident that
%the contributions in \equ(A1.4), for which the {\it total momentum}
%$\nn(v_0)$ {\it vanishes}, \ie $\nn(v_0)=\V0$, are all contributions
%with $(\nn_0(v_0),p(v_0))=(\V0,0)$. On the other hand it is
%immediately understood that this is necessary, in order that the
%counterterms be independent of $\aa$. As a matter of fact all
%contributions with $(\nn_0(v_0),p(v_0))=(\V0,0)$ which have not also
%$\nn(v_0)=
%\V0$ automatically vanish when summed together,
%as we shall see in \S A3.5.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1.truecm
%\ifnum\mgnf=0\pagina\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0{\titolo Appendix A2. (Improved) resonant Siegel-Bryuno's bound}
\numsec=2\numfor=1\*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0{\bf A2.1.} We follow the idea of P\"oschel, [P\"o] (see also [G1,
GG,Ge2]). In the discussion, we focus on the scale labels, so
that it is quite irrelevant which value the $p(v)$'s, $v \in \th$,
assume, and therefore which resonances are strong and which are not.
Calling $N^*_n(\th)$ the number of non resonant branches
carrying a scale label $\le n$, in a tree $\th$ with $m$ nodes,
we shall prove first that
%
$$ N^*_n(\th) \le 2m E_n - 1 \; , \qquad
E_n \defi N2^{(3+n)/\t},\qquad n\le1\; , \Eqa(A2.1) $$
%
provided that $N^*_n(\th)>0$, and
%
$$ N^*_{0}(\th) \le 2m \g N \h - 1 \; , \qquad
\g \defi 4\O_1/\O_2 \; , \Eqa(A2.2) $$
%
if $N_{0}^*(\th)>0$.
Define, as in \S 5.7, $\oo_0=(1,\h^{-1}(\O_1+\hdp J^{-1}A_1)^{-1}\O_2)$.
Then $C_0|\oo_0\cdot\nn|>|\nn|^{-\t}$ for all
$\V0\neq\nn\in\ZZZ^{\ell-1}$; see \equ(5.7).
Assume also $\h$ so small that $C_0\ge 2$,
%%%%, \fra14\fra{\O_2}{\h \O_1},\O_1+\hdp J^{-1}A_1>\O_1/2$
(this is not restrictive as we are interested in $\h\to 0$).
Set $E_n\= N2^{(n+3)/\t}$ as in \equ(A2.1). Note that if $m\le
E_n^{-1}$ one has $N_n^*(\th)=0$. In fact $m\le E_n^{-1}$ implies
that, for all $v\in\th$, $|\nn_0(v)| \le N E_n^{-1}$, \ie
$C_0|\oo_0\cdot\nn_0(v)|\ge (N^{-1}E_n)^{\t}$ $=$ $2^{n+3}$, so that
there are {\rm no} clusters $T$ with $n_T=n$. Note also that if
$m\le (\g N \h)^{-1}$, with $\g=4\O_1/\O_2$, then $N_{0}^*=0$, as
$|\oo_0\cdot\nn_0(v)|\ge 1$ for all $v\in\th$ in such a case.
\*
\0{\bf A2.2.} Let us prove first the inequality \equ(A2.1).
If $\th$ has the root branch either with scale $>n$,
or with scale $\le n$ and resonant,
then calling $\th_1,\th_2,\ldots,\th_k$
the subtrees of $\th$ ending into the highest node $v_0$ of $\th$
and with $m_j>E_n^{-1}$ nodes, $j=1,\ldots,k$,
one has $N_n^*(\th)=N_n^*(\th_1)+\ldots+N_n^*(\th_k)$ and the statement is
inductively implied from its validity for $m' m- (2E_n)^{-1}$: but in the
latter case we shall show that the root branch of $\th_1$ has scale $>n$.
Accepting the last statement (which will be proved below), one will
obtain $N_n^*(\th)=1+N_n^*(\th_1)= 1+N_n^*(\th'_1)+\ldots+N_n^*
(\th'_{k'})$, where $\th'_j$'s are the $k'$ subtrees ending into the
highest node of $\th'_1$ with orders $m'_j>E_n^{-1}$,
$j=1,\ldots,k'$. Going once more through the analysis the only non trivial
case is if $k'=1$ with the root branch of $\th'_1$ non resonant;
and in such case
$N_n^*(\th'_1)=N_n^*(\th^{\prime \prime}_1) + \ldots +
N_n(\th^{\prime \prime}_{k^{\prime \prime}})$, \etc., until we reach a
trivial case or a tree of order $\le m-(2E_n)^{-1}$.
It remains to check that if $m-m_1<(2E_n)^{-1}$ then the root branch
of $\th_1$ has scale $>n$. Let us proceed by {\it reductio ad absurdum}.
Suppose that the root branch of $\th_1$ is on scale $\le n$. Then
$C_0|\oo_0\cdot\nn_0(v_0)|\le\,2^n$ and $C_0|\oo_0\cdot\nn_0(v_1)|\le
\,2^n$, if $v_1$ is the highest node of $\th_1$.
Hence $C_0|\oo_0\cdot(\nn_0(v)-\nn_0(v_1))|< 2^{n+1}$ (equality
would imply violation of the strong Diophantine property, \equ(5.7)), and
the Diophantine condition implies that
%
$$ |\nn_0(v_0)-\nn_0(v_1)|> 2^{-(n+1)/\t} \= \d \; , \Eqa(A2.3) $$
%
because $\nn_0(v_0)\neq\nn_0(v_1)$ (the root branch of $\th$ being
supposed non resonant).
But $m-m_1<(2E_n)^{-1}$, so that $|\nn_0(v_0)-\nn_0(v_1)|<
(2E_n)^{-1}N < 2^{-1}2^{-(n+3)/\t}$ $=$ $2^{-(1+2/\t)}\d<\d$,
which contradicts inequality \equ(A2.3).
\*
\0{\bf A2.3.} Let us prove now \equ(A2.2).
If $\th$ has the root branch either with scale $1$,
or with scale $\le 0$ and resonant,
then calling $\th_1,\th_2,\ldots,\th_k$
the subtrees of $\th$ ending into the highest node $v_0$ of $\th$
and with $m_j>(\g N\h)^{-1}$ nodes, $j=1,\ldots,k$,
one has $N_{0}^*(\th)=N_{0}^*(\th_1)+\ldots+N_{0}^*(\th_k)$
and the statement is inductively implied from its validity for $m'm- (2\g N\h)^{-1}$: but in the
latter case the root branch of $\th_1$ has scale $1$.
Accepting the last statement (which will be proved below),
one will obtain $N_{0}^*(\th)=1+
N_{0}^*(\th_1)= 1+N_{0}^*(\th'_1)+\ldots+N_{0}^*(\th'_{k'})$,
where $\th'_j$'s are the $k'$ subtrees ending into the highest
node of $\th'_1$ with orders $m'_j>(2\g N\h)^{-1}$.
Going once more through the analysis the only non trivial
case is if $k'=1$ and in that case
$N_{0}^*(\th'_1)=N_{0}^*(\th^{\prime \prime}_1) + \ldots +
N_{0}(\th^{\prime \prime}_{k^{\prime \prime}})$, \etc., until
we reach a trivial case or a tree of order $\le m-(2\g N\h)^{-1}$.
It remains to check that, if $m-m_1<(2\g N\h)^{-1}$,
then the root branch of $\th_1$ has scale $1$.
Suppose that the root branch of $\th_1$ is on scale $\le 0$. Then
$p(v_1)\neq0$ and $|\oo_0\cdot\nn_0(v_0)|\le1/4$,
$|\oo_0\cdot\nn_0(v_1)|
\le 1/4$, if $v_1$ is the highest node of $\th_1$, \ie
%
$$ |\oo_0\cdot(\nn_0(v_0)-\nn_0(v_1))| \le 1/2 \; . \Eqa(A2.4) $$
%
As the root branch of $\th$ is supposed non resonant, then $m-m_1<(2\g
N\h)^{-1}$ implies that $0$ $<$ $|\nn_0(v_0)-\nn_0(v_1)|$ $<$ $(2\g
N\h)^{-1}N = (2\g\h)^{-1}$, so that one would have
$|\oo_0\cdot(\nn(v_0)-\nn(v_1))|\ge 1$, which is contradictory with the
inequality \equ(A2.4).
%(the case $|\oo_0\cdot(\nn(v_0)-\nn(v_1))|= 1$
%would imply $|\oo_0\cdot\nn_0(v_0)|$ $=$ $|\oo_0\cdot\nn_0(v_1)|$ $=$
%$1/2$, so that it is not possible because, given the form of $\oo_0$,
%would imply that $\pm \oo_0\cdot(\nn_0(v_0)\pm\nn_0(v_1))=1$ for some
%choice of the signs which is impossible because, if $n=0$,
%$C_0|\oo_0\cdot\nn_0(v_0)|0$ then
the number $p_n(\th)$ of clusters of scale $n$ verifies the bound
%
$$ p_n(\th) \le 2m N 2^{(n+3)/\t}-1 \; . \Eqa(A2.5) $$
%
In fact this is true for $m\le E_n^{-1}$, if
$E_n$ is defined as in \S A2.1. Otherwise,
if the highest tree node $v_0$ is not in a cluster on scale $n$,
one calls $\th_1,\ldots,\th_k$ the subtrees ending into $v_0$, and
one has $p_n(\th)=p_n(\th_1)+\ldots+p_n(\th_k)$,
so that the statement follows by induction.
If $v_0$ is in a cluster $V$ of scale $n$, and $\th_1$, $\ldots$, $\th_k$
are the subtrees entering the cluster containing $v_0$ and with
orders $m_j> E_n^{-1}$, one will find
$p_n(\th)=1+p_n(\th_1)+\ldots+p_n(\th_k)$.
Again we can assume that $k=1$, the other cases being trivial.
But in such case there will be only one branch entering the cluster $V$
and it will have a propagator of
scale $\le n-1$. Therefore the cluster $V$ must contain at least
$E_n^{-1}$ nodes. This means that $m_1\le m-(2E_n)^{-1}$.
Finally, the bound
%
$$ \sum_{n=-\io}^0 p_n(\th) \le 2m\g N\h-1 \Eqa(A2.6) $$
%
is a trivial consequence of \equ(A2.2).
\*
\0{\bf A2.5.} Let $\bar N^*_n\le N^*_n$ be the number of non
resonant branches on scale $n$.
Then if $\bar N_n$ is the number of non resonant branches {\it plus}
the number of clusters on scale $n$, $\bar N^*_n$ verifies the bounds
%
$$ \bar N_n^* = \big( \bar N_n^* + p_n \big) - p_n \= \bar N_n-p_n
\le 4m N 2^{(n+3)/\t}-2- \sum_{T \atop n_T=n} (1)
\le 4m N 2^{(n+3)/\t}+\sum_{T\atop n_T=n} (-1) \; . \Eqa(A2.7) $$
%
This proves that \equ(A2.1) and \equ(A2.5) imply
an inequality analogous to the first of \equ(5.13);
likewise one derives an inequality similar to the second of \equ(5.13)
by combining \equ(A2.2) and \equ(A2.6).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1.truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0{\titolo Appendix A3. Cancellations between resonances}
\numsec=3\numfor=1\*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0In this appendix we recall briefly the cancellation mechanisms of
[Ge2]. We provide this as a guide to the reader and as a tune up of a
fine points of the analysis of [Ge2] (the analysis in A3.2 is given
here in full details while in [Ge2] it was left out).
\*
\0{\bf A3.1.}
Consider a tree $\th$ with a strong resonance $V$ of order $k_V$.
Let $\l_{v_0}$ and $\l_{v_1}$ be, respectively, the exiting
and entering branches of $V$.
There are two possibilities: either the degree of the propagator
corresponding to the branch exiting from $V$ is $D_{\l_{v_0}}=2$
or it is $D_{\l_{v_0}}=1$ (equivalently the degree of the
resonance is either $D_V=2$ or $D_V=1$).
Let us discuss first the case in which the degree $D_V$
of the resonance is $D_V=2$. Then $j_{v_0}>\ell$ (see \equ(5.3)
and the comments after the definition of strong resonance in \S 5.9)
and, by following the notations of \S 5.1, we shall say that
$\PP=\emptyset$, \ie there is no path $\PP$ ending into $v_0$.
It follows, from the properties of $\PP$ discussed at the beginning of
\S 5.1 above, that $p(v_1)=0$ implies $j\=j_{v_1}>\ell$
and $D_{\l_{v_1}}=2$ (see again \equ(5.3)).
Consider all the trees belonging to the class $\FFFF(\th)$ which
are obtained from $\th$ by detaching the subtree
having as branch root the entering branch $\l_{v_1}$ of the resonance
and attaching it to all the remaining nodes of $V$
(see the definition of the class $\FFFF_V(\th)$ in \S 5.9).
As a consequence of such an operation\\
$\bullet$ some of the branches
internal to the resonance have changed the
free momentum
by an amount $\nn_0(v_1)$, and
\\
$\bullet$ if $w$ is the node inside $V$ to which
the branch $\l_{v_1}$ is attached and $j_{v_1}-\ell>0$, then
$\bar F_{\nvec_w}$ (see \equ(4.23)),
has the form of an even function of $\nvec_w$ times
a factor $(i\n_{wj})$.
We shall call {\it resonance value} $\RR_V$
the product of factors appearing in
the definition of tree value and relative only to the nodes and branches
internal to the resonance $V$:
%
$$ \RR_V = \bar F_{\nvec_{v_0}} y_{v_0}'
\prod_{v\in V \atop v 0$.
If we sum also on an overall change of signs of the mode labels of the
nodes internal to the resonance (by following the definition of the
class $\FFFF(\th)$ given in \S 5.9), we obtain a zero contribution also to
first order in $\m$ (here the even parity of the perturbation $f$ is
essential, see [G1,Ge2]).
This can be seen by using the explicit form of the functions in
\equ(4.21), \ie the coefficients listed in \equ(4.24).
Noting that in the present case {\it there cannot be any $\PP$ inside
$V$} the only propagators we can associate with the branches internal to
$V$ have the form of the two first terms of \equ(5.3), so that, for
$\m=0$, {\it they are even functions of the mode labels}. Moreover in
such a case the analysis in \S 5.1 shows that $\a_v=-1$, $\a_v=1$ and
$\a_v=2$ imply, respectively, $k_v'=-k_v=1$, $k_v'=-k_v=-1$ and
$k_v'=-k_v=0$ (the case $\a_v=0$ is not possible here): then no $n_v$
labels appear in the coefficients $y_{n_v}^{(\a_v)}(k_v',k_v)$
corresponding to the nodes $v\le v_0$ (see the list of coefficients in
\equ(4.24)). Therefore all the dependence on the $n_v$ labels is
through the factors $\bar F_{\nvec_v}$ in \equ(4.23). This yields that
there is an even number of the $n_v$ (if there are any) corresponding to
the nodes $v\in V$: two for each branch $\l_v$ with $j_v=\ell$, by
taking into account that $j_{v_0},j_{v_1}>\ell$, so that no change is
produced by the sign reversal (since, by the parity properties of the
Hamiltonians \equ(1.1) and \equ(2.1), one has also
$f^{\d_v}_{\nvec_v}=f^{\d_v}_{-\nvec_v}$). This means that the resonance
value is an even function of $\m$.
\*
\0{\bf A3.2.}
Let us now consider the case in which the strong resonance is of degree
$D_V=1$ and the tree $\th$ has no leaves inside $V$. In such a case
$\a_{v_0}=-1$ and $j_{v_0}=\ell$, hence $D_{\l_{v_0}}=1$ (see
\equ(5.3)): then a first order zero in $\m$ will be enough. Moreover
there is a $\PP$ inside the resonance: we shall distinguish between
the cases $v_1\notin\PP$ and $v_1\in\PP$.
Let us consider first the case $v_1\notin\PP$ (in particular this is
the case when $\PP=v_0$, $k_{v_0}=0$, provided $k_V\ge 2$). In such a
case $j_{v_1}>\ell$ and we can reason as above to obtain a first order
zero. Note that in such a case there would be no cancellations
between tree values of trees obtained by the sign reversal operation.
On the contrary, if $v_1\in\PP$, then $k_{v_0}=-1$,
and one has also $\a_{v_1}=-1$ and $j_{v_1}=\ell$.
%in particular this is the case when $\PP=v_0$, with $k_{v_0}=-1$.
In this case consider together with
the tree $\th$ also the tree $\th'$ obtained from
$\th$ by performing the following operation
(recall the definition of $\FFFF_V(\th)$):
replace the resonance $V$ with a single node $v$
carrying labels $\d_v=0$ and $\k_v=k_V$
(if $k_V$ is the order of the resonance $V$),
then express the counterterm $\g_{\k_v}(g_0)$ associated with
the node $v$ in terms of trees.
If $\th_1$ is the subtree having $\l_{v_1}$ as root branch,
then the values of the two considered trees
$\th$ and $\th'$ can be written, respectively,
as ${\rm Val}(\th)=A(\th)\RR_V{\rm Val}(\th_1)$
and ${\rm Val}(\th')=A(\th)[\g_{k_V}(g_0)\s/2]{\rm Val}(\th_1)$,
where $\s/2=y_{v}^{(-1)}(1,-1)$ and $A(\th)$ takes into account the
factors corresponding to all nodes {\it not} preceding $v_0$,
and has the same value for both $\th$ and $\th'$.
The resonance value $\RR_V$, for $\m=0$, can be written as
%
$$ \RR_V = \overline{{\rm Val}}(\th_0)\, in_{v_1'} \; , \qquad
\hbox{ for some } \th_0\in\TT_{\V0,k} \hbox{ with } p(v_0)=0
\hbox{, } k_{v_0}=-1 \; , \Eqa(A3.2)$$
%
see the definitions \equ(4.30) and \equ(4.31)
of tree value and the definition \equ(A3.1) of resonance value:
remember that we are considering resonances $V$ with degree
$D_V=1$, so that $k_{v_0}=-1$ and, as a consequence, $k_{v_0}'\ge 1$;
see \equ(4.14). The counterterm $\g_\k(g_0)$ can be represented
in terms of trees as in \equ(4.32); note that, if
the tree contributing to $\g_\k(g_0)$ has $k_{v_0}=-1$,
the condition $\a_{v_0}=-1$ implies that such a tree
has a node $w>v_0$ with $k_w+k_w'=1$,
while all the other nodes $v\neq w$ have $k_v+k_v'=0$.
Among the contributions in \equ(4.32) to $\g_{k_V}(g_0)$
there will be a quantity $\overline{{\rm Val}}(\th_2)$, where $\th_2$
will have the same topological form of $\th_0$ in \equ(A3.2)
with the node $w$ such that $k_w+k_w'=1$ corresponding
to the node $v_1'\in V$; then we denote both nodes by $w$.
Then $\overline{{\rm Val}}(\th_0)$ will be related to
$\overline{{\rm Val}}(\th_2)$ by
%
$$ \overline{{\rm Val}}(\th_2) = -\left[
{y_{n_v}^{(\a_v)}(k_w',k_w)|_{k_w'+k_w=1}\over
y_{n_v}^{(\a_v)}(k_w',k_w)|_{k_w'+k_w=0}} \right]
\overline{{\rm Val}}(\th_0) \; , \Eqa(A3.3) $$
%
so that a look at the coefficients listed in and after \equ(4.24) shows
that the factor in square brackets in \equ(A3.3) (when it is not
vanishing) is equal to $4in_w\s$. The quantity $\overline{{\rm
Val}}(\th_2)$, in order to contribute to $\g_{k_V}(g_0)\s/2$, has to be
multiplied by a factor $-4\s$ extra with respect to $\overline{{\rm
Val}}(\th_0)$, which, on the other hand, has to be multiplied by $in_w$
in order to contribute to the resonance value $\RR_V$ (see \equ(4.32).
Then, for $\m=0$, by summing the values
of the two considered contributions one obtain
%
$$ A(\th) \Big[ \overline{{\rm Val}}(\th_0)\,in_w-{1\over4\s}
\overline{{\rm Val}}(\th_2)\Big]{\rm Val}(\th_1) \; , \Eqa(A3.4) $$
%
which is zero by \equ(A3.3), so that a first order zero is obtained.
\*
\0{\bf A3.3.} If there are leaves, nothing changes in the discussion
of \S A3.1, as $k_{v_0}=0$ implies that only leaves $w$ with
$j_w>\ell$ are possible, and $\x_w(k_{w}',0)\=1$ in such a case
(see \equ(4.18)).
%The only real difference is that the presence of
%the leaves modifies the combinatorial factor of the tree,
%but it is a trivial change and in the end nothing changes.
In \S A3.2, when discussing the case $v_1\in\PP$, one has to take care of
the case in which there is a leaf with highest node
$\tilde w$ with $k_{\tilde w}'=1$
(such a leaf will be at the end of the path $\PP$).
In fact the resonances having as entering branch a branch of the
path $\PP$ cannot have any leaves with $k_w'=1$, while
when considering the graphical representation for $\g_\k(g_0)$, there
will be also contributions arising from trees containing a leaf:
such contributions will be either of the form \equ(4.32) with
$\overline{{\rm Val}}(\th_2)=
\overline{{\rm Val}}(\th_2)in_{v_1'}\x_{v_1}(1,0)
L^{h_1\s}_{\ell\nn(v_1)}(\th_2)$, or of the form $\g_\k(g_0)=
\g_{\k-h_1}(g_0)\,
in_{v_1'}\x_{v_1}(1,0) L^{h_1\s}_{\ell\nn(v_1)}(\th_2)$,
where $h_1\ge 1$, and $\th_2'$ is a suitable tree of order $k-h_1$.
Then one realizes that the two contributions cancel exactly,
so that no new case has to be discussed with respect to the analysis
of \S A3.2.
\*
\0{\bf A3.4.} The above discussion completes the proof of
approximate cancellations of resonance values (\ie of cancellations to
first and second order, according to the degree of the resonant
branch). The existence of cancellations, approximate to the first or
second order, is all is needed to obtain the bound \equ(5.16): the
analysis continues exactly as in [Ge2] and is based on simple
analyticity arguments that allow us to exploit, via the maximum
principle, the fact that in a resonance with momentum $\nn$
the functions of $\m=\oo_0\cdot\nn$ that have been considered above have
a zero in $\m$ of order $1$ or $2$.
A complete analysis showing that the higher orders contributions (\ie
the part which does not cancel) can be performed as in [Ge2], Appendix
B, and the final result is given by the bound
\equ(5.16) in \S 5.11.
\*
\0{\bf A3.5.} We shall show now that all contributions
with $(\nn_0(v_0),p(v_0))=(\V0,0)$ involved in the
definition of the counterterms (see Appendix A1)
must have automatically also $\nn(v_0)=\V0$.
The analysis performed in \S 5.1 shows that in order
to have $p(v_0)=0$ (for $j_{v_0}=\ell$), there can be
any number of leaves with highest nodes $w$ such that
$j_w>\ell$ and only one leaf $w$ with $j_w=\ell$
(contributing, respectively, $k_{w}'=0$ and $k_{w}'=1$
to $p(v_0)$).
Each time a leaf with $j_w>\ell$ appears, if we sum together the
values of all trees obtained by detaching the leaf with its stalk,
then reattaching it to all the other nodes of $\th_f$, we obtain
a vanishing contribution: simply by the cancellation mechanism
described in \S A3.1 (assuring there the first order zero),
{\it which, now, is an exact cancellation as the leaf does
not contribute to the free momenta of the branches of $\th_f$,
so that it does not modify the propagators.}
So we can suppose that no leaf with $j_{w}>\ell$ is possible in trees
involved in the determination of the counterterms.
In the same way, if we have a tree $\th$ having a leaf with $j_w=\ell$
and $h_w=h-h_1$ (for some $h_1$),
we can reason as in \S A3.2 and consider, together with
$\th$, also the tree formed by only one free node,
carrying a counterterm label $h_1$ and bearing the same leaf as $\th$.
The same cancellation mechanism described in \S A3.2 apply now:
again the only difference is that now the
cancellation is exact (by the same reason as before).
This shows that no tree with leaves can contribute to
$(\nn_0(v_0),p(v_0))=(\V0,0)$, so that for such trees
one has $\nn(v_0)=\nn_0(v_0)=\V0$. This, together
with the analysis in \S A1.1, proves \equ(A1.4) in \S A1.2.
\*
\ifnum\mgnf=0\pagina\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1.truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0{\titolo Appendix A4. Graphs with non zero total hyperbolic
momentum, with leaves or with counterterms}
\numsec=4\numfor=1\*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0{\bf A4.1}
Consider first the cases $p(v_0)\neq 0$. In this case we consider the
nodes $w1$, the discussion is easier
as no splitting of the integration domains is needed.
So we can conclude that a final bound $(2B)^{2k}$
is obtained for $\Val(\th)$; so far neither leaves
nor counterterms have been considered.
\*
\0{\bf A4.2}
Introducing the leaves and the counterterms, one sees (recall
Remark 4.4) that the value of any tree $\th$ can be always be
written as the product of a factor like \equ(4.28) times
the product of the counterterms and of the leaf values;
each counterterm can be decomposed in turn as sum of values of
amputated trees (see \equ(4.32)). As each leaf and each
amputated tree can contain other leaves and counterterms
we can iterate such a decomposition procedure, until, at the end,
the value of the tree $\th$, with highest node $v_0$,
turns out to be given by the product
of factorizing terms which\\
{(1)} either are of the form \equ(4.28),
with $\r_{w}=0$ for any subtree with highest node $wh
(N+N_0)$ vanish at order $h$ as a consequence of the trigonometric
assumption on the perturbation $f_1$, see \equ(2.2), the bound
$(2B)^{2k}$ is a bound both for the Fourier coefficients of
$X^\s(t;\aa)$ and for the function $X^\s(t;\aa)$ itself.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1.truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\0{\titolo References}
\*\*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\*
\FINE
\ciao
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