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\\{\bigtit Whiskered tori with prefixed frequencies and Lyapunov spectrum}
\footnote{${}^*$}{\nota This
paper is deposited in the archive {\tt mp\_arc@math.utexas.edu},
\#94-??.}
\vskip1.truecm
\\{\bf Guido Gentile}\footnote{${}^1$}{\nota
E-mail: {\tt gentileg\%39943.hepnet@lbl.gov}: Dipartimento di Fisica,
Universit\`a di Roma, ``La Sa\-pi\-en\-za", P. Moro 5, 00185 Roma, Italia.}
\vskip.2truecm
\0{\bf Abstract:} {\sl A classical mechanics problem, as the existence
of whiskered tori for an almost integrable hamiltonian system, is analyzed
with techniques reminiscent of the quantum field theory, following the
strategy developed in recent works about the matter. The system consists
in a collection of rotators interacting with a pendulum via a small
potential depending only on the angle variables. The proof of the
existence of the stable and unstable manifolds (``whiskers") of the
rotators invariant tori corresponding to diophantine rotation numbers
is simplified by setting the Lyapunov spectrum to prefixed values via
the introduction, in the hamiltonian function, of ``counterterms"
depending on the strength of the interaction;
this is a feature usual in quantum field theory,
and emphasizes the analogy between the the field theory and the KAM
framework pointed out already in the mentioned works.}
\*
\\{\bf Key words:} {\it KAM, perturbation theory, classical mechanics,
quantum field theory, renormalization group}
\vskip.8truecm
\\{\bf 1. Introduction}
\vskip.4truecm\pgn=1\numfig=1\numsec=1\numfor=1
\\Until the Eliasson's work, [E], proofs of the KAM theorem, ([K], [A1],
[Mo1]), assuring the existence of infinetely many invariant tori
for perturbed systems, were all based on a rapidly convergent iteration
technique leading to solve recursively a sequence of approximate equations.
Recently, direct (\ie non recursive) proofs of the KAM theorem have been
proposed, (see [E], [G1], [G2], [CF]), by exhibiting explicitly
cancellations occuring between contributions to the formal expansion series
of the quasiperiodic solutions. By extending the important ideas of Melnikov,
[Me], the same techniques have been applied to the study of low dimensional
tori and their stable and unstable manifolds, [G1]. In particular in [G1],
[G2], [GGe], [G5], the analogy between the methods implemented in the
direct proofs and those used in quantum field theory, expecially in the
renormalization group approach, have been pointed
out.\footnote{${}^2$}{\nota
In [G5] it has been shown that the function describing the invariant tori
of a system of rotators interacting via a potential which is a
trigonometrical polynomial in the angle variables, (\ie the hamiltonian
system (1.1) without the pendulum), can be represented as the one
point Schwinger function of a quantum field model.}
In this work such ideas will be developed further, by giving a proof of
the existence of the whiskered tori in a class of almost integrable
hamiltonian systems alternative to that proposed in [Ge], and based
on the introduction of ``counterterms" (a technique usual in
quantum field theory) in the hamiltonian function; the advantage of
such a procedure is that some symmetry properties of the
solutions, discussed in [CG] within the KAM theory framework,
can be directly visualized for
the modified hamiltonian, so simplifying in a remarkable way the
topological structure of the diagrams (trees, see below)
in terms of which the solutions can be graphically represented.
Once the problem is solved, the solutions of the
original hamiltonian equations can be
recovered in a trivial way, and, by construction, they exhibit
explicitly the symmetry properties required by the results found
in [CG]. Actually, the original feature with respect to [Ge] resides on
the simplification in the proof and on the connection with
the quantum field theory methods; therefore the present paper can be
considered an introductory work in view of an extension to other similar
dynamical systems problems, whose treatment requires such methods.
\*
We consider a model consisting of a family of rotators, say $l-1$ in number,
interacting with a pendulum via a conservative force, (see [CG], [G1], [GGe]).
The inertia moments $J_j$, $j=1,\ldots,l-1$, of the rotators form a matrix
$J$ which is diagonal, and are supposed to be $ J_j\ge J_0>0$, if $J_0$ is
the inertia of the pendulum, so setting a scale for the size of the inertia
moments; for simplicity's sake we assume $J_j=J$, for all $j=1,\ldots,l-1$.
The model can be described by the $l$ degrees of freedom hamiltonian
$H_\m\=H_0+\m f$ given by
%
$$ \oo\cdot\V A+{1\over2}J^{-1}\AA\cdot\AA+{I^2\over2J_0}+g^2J_0
(\cos\f-1)+\m \sum_{{|\n|\le N}}
f_\n\cos(\aa\cdot\nn+n\f) \Eq(1.1)$$
%
where $(I,\f)\in {\bf R}^2,(\AA,\aa)\in {\bf R}^{2(l-1)}$
are canonically conjugated variables,
$\oo\in {\bf R}^{l-1}$, $\n\=(n,\nn)\in {\bf Z}^l$, $|\n|=|n|+|\nn|=|n|+
\sum_{i=1}^{l-1} |\n_i|$, $g>0$ ($g^2$ is the ``gravity''), $\oo,\m$ are
parameters, ($\oo$ is the rotation vector, and $\m$ is the perturbation
strenght), and $f_\n$ are fixed constants.
A natural {\it energy scale} for the model will be $E=g^2J_0$.
We suppose {\it a priori} that:
\*
\\{\bf Hypothesis H}: {\it The rotation vector $\oo$ is a
{\it diophantine vector}, \ie $C_0 |\oo\cdot\nn|\ge |\nn|^{-\t}$,
for all $\V 0\ne\nn\in{\bf Z}^{l-1}$, for some {\rm diophantine constant}
$C_0$ and some {\rm diophantine exponent} $\t>0$.}
\*
For $\m=0$, the hamiltonian equations generated by \equ(1.1), (\ie
$\dot I=-\dpr_\f H_\m$, $\dot \f=\dpr_I H_\m$, $\dot \AA=-\dpr_\aa
H_\m$, $\dot \aa=\dpr_\AA H_\m$), admit $(l-1)$--dimensional invariant
tori:
%
$$\TT_0\=\{I=0=\f\}\times \{\AA\=\AA^0\ ,\aa\in {\bf T}^{l-1}\} \Eq(1.4) $$
%
possessing homoclinic stable and unstable manifolds, called
{\it whiskers}. The manifolds equations are:
%
$$ \tst W_0^{\pm}\=W_0\=\{
{I^2\over2J_0}+g^2J_0(\cos\f-1)=0 \} \times \{\AA\=\AA^0\ ,\aa\in
{\bf T}^{l-1}\} \Eq(1.5) $$
%
so that the non trivial Lyapunov exponents of $\TT_0$ are $\pm g$.
Under perturbation we expect the tori to be deformed and to remain
unstable with only two non trivial Lyapunov exponents $\pm g'(\m)$,
where $g'(\m)$ is a suitable analytic function of $\m$; this is indeed
proved in [CG] and [Ge]. The change of the Lyapunov exponents has the
consequence that the solutions of the perturbed motion equations
depend naturally on $t$, (besides via $\exp[\pm i\oo\cdot\nn t]$),
via $e^{\pm g'(\m)t}=e^{\pm gt}[1\pm\m g_1 t + \m^2(\pm g_2^2t +
g_1^2t^2/2)+O(\m^3)]$, where $g_i$, $i>0$, are the coefficients of
the series expansion of the function $g'(\m)$.
Therefore, in a perturbation theory, this will generate contributions
to the solutions of the equations of motion which depend on time via
$e^{\pm gt}\,t^k$ for various values of $k\in{\bf N}$.
This indeed does happen, [Ge], and it is the source of rather deep
technical intricacies: then one has to
exhibit some cancellations which ultimately are probably related to the
fact that the terms $e^{\pm gt}\,t^k$ appearing in the solutions of the
equations of motion can be resummed into ``pure exponentials" like
$e^{\pm g'(\m)t}$.
It is therefore natural to ask if one can fix the Lyapunov exponents
{\it a priori}, as one does with the frequencies. This should allow
us to get
rid of the terms $e^{\pm gt}\,t^k$ and leave us with ``pure exponentials",
eliminating at the same time also the necessity of the analysis of
cancellations between terms with dependence on $t$ through powers of $t$.
Then, instead of studying the model \equ(1.1), we consider the modified model:
%
%
$$ \oo\cdot\V A+{1\over2}J^{-1}\AA\cdot\AA+{I^2\over2J_0}+g^2(\m)J_0
(\cos\f-1)+\m \sum_{{|\n|\le N}}
f_\n\cos(\aa\cdot\nn+n\f) \Eq(1.2) $$
%
where $g^2(\m)$ is a function of $\m$ admitting the series expansion:
%
$$ g^2(\m) = g^2 \sum_{h=0}^{\io} \g_h\m^h \Eq(1.3) $$
%
with $\g_0=1$ and the coefficients $\g_h$, $h\ge1$, to be determined,
as it will be explained later, (see \S 5), in such a way to guarantee
that the frequencies and the non trivial Lyapunov exponents
corresponding to the perturbed system whiskered tori are,
respectively, $\oo$ and $\pm g$, (and we shall see that,
if such a request is fulfilled, then the series convergence
automatically follows).
We shall prove that the perturbed system still admits $(l-1)$-dimensional
invariant tori $\TT_\m$, obtained by analytic continuation from the
unpertured ones, with their stable and unstable whiskers
%
$$ W^\pm_\m = \{(I,\AA,\f,\aa)=(I^\pm_\m(\aa,\f),\AA^\pm_\m(\aa,\f),\f,\aa):
\aa\in {\bf T}^{l-1}, |\f|<2\p\}\Eq(1.6)$$
%
characterized by the sets of initial data $X^\pm_\m(0)$ such that the
distance$(S^t_\m X^\pm_\m(0),\TT_\m) \to 0$ fast as $e^{-g'(\m)|t|}$
$t \to\pm \io$; here $S^t_\m$ is the hamiltonian flow generated by
\equ(1.2).
\*
The paper is selfcontained: \S 2 has a definitory nature and partially
overlaps with \S 2, 3, 4 of [Ge], (which in turn were often literally
taken from the review article [G1], with some abstraction effort), while
\S 3 introduces the diagrammatic formalism, which is similar but not
quite identical to the corresponding one of [Ge], and in fact is
less involved, (less definitions have to be given). The original
work is in \S4$\div$\S5 (and in the appendices). The theory of
the homoclinic splitting, discussed in [G1] and briefly mentioned
in [Ge], will be not even barely touched: here we confine ourselves
to the problem of existence of the whiskered tori.
\vskip.8truecm
\\{\bf Acknowledgements} : The author would thank G. Gallavotti for
having proposed the argument and indicated to him the way how to
approach it.
\vskip.8truecm
\\{\bf 2. Analytic formalism}
\vskip.4truecm\pgn=1\numfig=1\numsec=2\numfor=1
\\In this section we review the formalism which will be used in the proof
of existence of the whiskered tori. With respect to [G1] and [Ge], there
are a few differences here and there, but only about the notations.
\vskip.4truecm
\\{\bf 2.1 Recursive formulae}
\vskip.3truecm
\\We derive simple recursive formulae for the
functions $I^\pm_\m$, $\AA^\pm_\m$ in \equ(1.6) and their time
evolution (see also [G1], \S 2, and [CG], Appendix A10).
Let us consider the unperturbed motion:
%
$$ X^0(t)\=(\f^0(t),\aa+\oo t,I^0(t),\V 0)\Eq(2.1) $$
%
where $(\f^0(t),I^0(t))$ is the separatrix motion, generated by the
pendulum in \equ(1.1) starting, say, at $t=0$ in $\f=\p$, and
$\f^0(t)=4 \arctan e^{-gt}$.
Let $X^\s_\m(t;\a)$, $\s={\rm sign}\,t=\pm$, be the evolution, under the flow
generated by \equ(1.1), of the point on $W^\s_\m$ given by
$(\p,\aa,I^\s_\m(\aa,\p),\AA^\s_\m(\aa,\p))$, see \equ(1.6); let:
%
$$X^\s_\m(t)\=X^\s_\m(t;\aa)\equiv \sum_{h\ge 0} X^{h\s}(t;\aa) \m^h=
\sum_{h\ge 0} X^{h\s}(t) \m^h,\qquad \s=\pm\Eq(2.2)$$
%
be the power series in $\m$ of $X^\s_\m$, (which we will show to
be convergent for $\m$ small); note that $X^{0\s}\=X^0$ is the
unperturbed whisker. We shall often not write 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 $\s=+$, and
$X^{h-}(t)$ if $\s=-$.
Inserting \equ(2.2) into the Hamilton equation associated with
\equ(1.2), we see that the coefficients $X^{h\s}(t)$
satisfy the hierarchy of equations:
%
$${d\over dt} X^{h\s}\= \dot {X}^{h\s}=L X^{h\s}+F^{h\s}\Eq(2.3)$$
%
where:
%
$$\tst
L(t)=\pmatrix{
0 &\V 0 & J_0^{-1} &\V 0 \cr
\V 0 &0 &\V 0 &J^{-1} \cr
g^2J_0 \cos\f^0(t) &\V 0 &0 &\V 0 \cr
\V 0 &0 &\V 0 &0 \cr},\quad
F^1(t)=\pmatrix{0\cr0\cr
-\dpr_\f f(\f^0(t),\aa+\oo t) + g^2\g_1J_0\sin(\f^0(t))\cr
-\dpr_\aa f(\f^0(t),\aa+\oo t)\cr}\Eq(2.4)$$
%
and where $F^{h\s}$ depends upon $X^0,...,X^{h-1\s}$ but not on
$X^{h\s}$; here (as everywhere else) the arrows denote $(l-1)$--vectors.
The entries of the $(2l\times 2l)$ matrix $L$ have different
meaning according to their position: the $\V 0$'s in the first and third
row are $(l-1)$ (row) vectors, the $\V 0$'s in the first and third
column are $(l-1)$ (column) vectors, and the $0$'s
and $J^{-1}$ in the second and fourth column are $(l-1)\times (l-1)$
matrices, while the $0$'s in the first and third columns are scalars.
Then, if we number the components of $X$ with a label $j$,
$j=0,\ldots,2l-1$, with the convention that:
%
$$X_0=X_-,\quad (X_j)_{j=1,\ldots,l-1}=\V X_\giu,\quad
X_l=X_+,\quad (X_j)_{j=l+1,\ldots,2l-1}=\V X_\su\Eq(2.5)$$
%
(\ie we write first the angle and then the action components; first
the pendulum and then the rotators), we see that
\equ(2.3) takes the form:
%
$$ \eqalign{
& {d\over dt} X^{h\s}_+= (g^2J_0\cos\f^0)X^{h\s}_- + F^{h\s}_+
\ ,\quad\quad\quad
{d\over dt} \V X^{h\s}_\su=\V F^{h\s}_\su\cr
& {d\over dt} X^{h\s}_- = J_0^{-1} X^{h\s}_+ \ ,\quad\kern3.cm
{d\over dt} \V X^{h\s}_\giu = J^{-1} \V X^{h\s}_\su \cr}
\Eq(2.6) $$
%
as $F^{h\s}_-$, $\V F^{h\s}_\giu$ vanish identically, for $h\ge 1$.
And, for all $h\ge1$, we can write the following formula for $F^{h\s}$
in terms of the coefficients $X^0,\ldots,X^{h-1\s}$ and of the
derivatives of $H_0$ and $f$:
%
$$ \eqalign{
F_-^{h\s} \= & 0\; ,\quad\quad \V F_\giu^{h\s} \= 0\; ,\cr
\V F_\su^{h\s} = & -\sum_{|\V m|\ge0} (\dpr_\aa f)_{\V m}(\f^0,\aa+\oo t)
\sum_{(h^i_j)_{\V m,h-1}} \prod_{i=0}^{l-1}\prod_{j=1}^{m_i}
X^{h^i_j \s}_i \; ,\cr
F_+^{h\s} \= & \sum_{|\V m|\ge 2} (g^2J_0\sin\f)_{\V m} (\f^0)
\sum_{(h_j^0)_{\V m,h}} \prod_{j=1}^{m} X^{h_j^0\s}_-
+ \sum_{|\V m|\ge 0} \sum_{p=1}^h (g^2\g_pJ_0\sin\f)_{\V m} (\f^0)
\sum_{(h_j^0)_{\V m,h-p}} \prod_{j=1}^{m} X^{h_j^0\s}_- \cr
& - \sum_{|\V m|\ge0} (\dpr_\f f)_{\V m}(\f^0,\aa+\oo t)
\sum_{(h^i_j)_{\V m,h-1}} \prod_{i=0}^{l-1}\prod_{j=1}^{m_i}
X^{h^i_j\s}_i \; , \cr} \Eq(2.7) $$
%
where $(G)_{\V m}(\cdot)$, with $G= \dpr_\aa f, \dpr_\f f, g^2J_0\sin \f$,
and $(h^i_j)_{\V m,q}$, with $h^i_j\ge 1$, $m_i\ge0$, $\V m=(m_0,\ldots,
m_{2l-1})$, are defined as:
%
$$\eqalign{
(G)_{\V m}(\cdot)\=&\Bigl(
{\dpr^{m_0}_\f\dpr^{m_1}_{\a_1}
\ldots\dpr^{m_{l-1}}_{\a_{l-1}}\dpr^{m_l}_I\dpr^{m_{l+1}}_{A_1}
\ldots\dpr^{m_{2l-1}}_{A_{l-1}}\,G\over m_0!\,m_1!\,\ldots\,
m_{l-1}!\,m_l!\,m_{l+1}!\,\ldots\,m_{2l-1}!}\Bigr)(\cdot)\cr
(h^i_j)_{\V m,q}\=&(h^0_1,\ldots,h^0_{m_0},h^1_1,\ldots,h^1_{m_1},
\ldots,h^{2l-1}_1,\ldots,h^{2l-1}_{m_{2l-1}})\qquad {\rm
s.t.\ }\sum h^i_j=q\cr} \Eq(2.8) $$
%
Note that the first two sums in the expression for $F^{h\s}_+$ can only
involve vectors $\V m$ with $m_j=0$ if $j\ge1$, because the function $J_0
g^2\sin\f$ depends only on $\f$ and not on $\aa$). The evolution of $X^h$
is determined by integrating \equ(2.6), if the initial data are known.
The $h=1$ case requires a suitable interpretation of the symbols, in
according to equation \equ(2.4).
We introduce the dimensionless quantities related to the perturbed
motions by:
%
$$ X^{h\s}_j = \X^{h\s}_j, \quad (0\le j\le l-1) \; ,
\qquad X^{h\s}_l = gJ_0\,\X^{h\s}_l, \qquad X^{h\s}_j= gJ\,\X^{h\s}_j,
\quad (l+1\le j\le2l-1), $$
%
and to the functions $F^{h\s}$ through the transformations $\V F^{h\s}_\su
=(g^2J)^{-1}\V\F^{h\s}_\su$, $F^{h\s}_+=(g^2J_0)^{-1}\F^{h\s}_+$,
(obviously: $\F^{h\s}_j=F^{h\s}_j\,\=\,0$, for $j=0,\ldots,l-1$).
We recall that the {\it wronskian matrix} $W(t)$ of a solution $t\to
x(t)$ of a differential equation $\dot x= f(x)$ in ${\bf R}^n$ is a $n\times
n$ matrix whose columns are formed by $n$ linearly independent
solutions of the linear differential equation obtained by linearizing
$f$ around the solution $x$ and assuming $W(0)=$ identity.
The solubility by elementary quadrature of the free pendulum equations
on the separatrix leads after a
well known classical calculation to the following expression for the
wronskian $ W(t)$ of the separatrix motion of the pendulum
appearing in \equ(1.1), with initial data at $t=0$ given by $\f=\p,I=2g
J_0$, (\ie $\X^0_+=2$):
%
$$ W(t)=\pmatrix{
{1\over\cosh gt}&{\bar w(t)\over4}\cr
-{\sinh gt\over\cosh^2 gt}&
(1-{\bar w(t)\over4}{\sinh gt\over\cosh^2gt})\cosh gt\cr},
\qquad \bar w(t)\={2gt+\sinh 2gt\over\cosh gt}\Eq(2.9)$$
%
And the evolution of the $\pm$ (\ie $I,\f$) components can be determined by
using the above wronskian:
%
$$\pmatrix{\X^{h\s}_-\cr \X^{h\s}_+\cr}= W(t)
\pmatrix{0\cr \X^{h\s}_+(0)\cr} +
W(t)\ig_0^{gt}{W\,}^{-1}(\t)\pmatrix{0\cr \F^{h\s}_+(\t)\cr}\ d\,g\t
\Eq(2.10)$$
%
Thus, denoting by $w_{ij}$ ($i,j=0,l$) the entries of $W$
we see immediately that:
%
$$\eqalign{ & \X^{h\s}_+(t)=w_{ll}(t) \X^{h\s}_+(0)+w_{ll}(t)
\ig_0^{gt} w_{00}(\t) \F^{h\s}_+(\t)\,d\,g\t-w_{l0}(t)\ig_0^{gt}
w_{0l}(\t) \F^{h\s}_+(\t)\,d\,g\t\cr & \X^{h\s}_-(t)=
w_{0l}(t) \X^{h\s}_+(0)+w_{0l}(t) \ig_0^{gt}w_{00}(\t)
\F^{h\s}_+(\t)\,d\,g\t-w_{00}(t)\ig_0^{gt}w_{0l}(\t)
\F^{h\s}_+(\t)\,d\,g\t\cr} \Eq(2.11)$$
%
The integration of the equations
\equ(2.6) for the $\su,\giu$ components yields:
%
$$ \eqalign{ \V \X_\su^{h\s}(t)=&\V \X_\su^{h\s}(0)+\ig_0^{gt}\V
\F^{h\s}_\su(\t)\,d\,g\t \cr
\V\X_\giu^{h\s}(t)=&\Big(gt\V\X_\su^{h\s}(0)+
\ig_0^{gt}g(t-\t)\,\V\F^{h\s}_\su(\t)\,d\,g\t\Big)\cr} \Eq(2.12) $$
%
having used that the $\V \X^{h\s}_\giu (0)\=\V 0$ because the initial datum
is fixed and $\m$ independent; and \equ(2.11), \equ(2.12) can be used
to find a reasonably simple algorithm to represent the whiskers
equations to all orders $h\ge1$ of the perturbation expansion.
\vskip.5truecm
\\{\bf 2.2 The improper integration $\II$.}
\vskip.3truecm
\\We introduce some integrations operations which can be performed
on the functions introduced in \S 2.1. The operation is simply the
integration over $t$ from $\s\io$ to $t$, $\s=\sign t$. In general
such an operation cannot be defined as an ordinary integral of a
summable function, because the functions on which it has to operate
(typically the integrands in \equ(2.11) and \equ(2.12)) do not, in
general, tend to $0$ as $t\to\io$. But the simplicity of the initial
hamiltonian has the consequence that the functions $\X^h(t)$, and the
matrix elements $w_{ij}$ in \equ(2.9), belong to a very special
class of analytic functions on which the integration operations that
we need can be given a meaning.
To describe such class we introduce various spaces of functions; all of
them are subspaces of the space $\hat \MM$ of the functions of $t$
defined as follows.
\*%
\0{\bf Definition 2.1}: {\it Let $\hat\MM$ be the space of the functions
of $t$ which can be represented, for some $h\ge 0$, as:
%
$$M(t)=\sum_{j=0}^h{(\s t g)^j\over j!} M_j^\s(x,\oo t)\ ,\quad
x\=e^{-\s gt}\ ,\quad \s={\rm sign}\, t\Eq(2.13)$$
%
with $M_j^\s(x,\V\psi)$ a trigonometric polynomial in $\V\ps$ with
coefficients holomorphic in the $x$-plane in the annulus $0<|x|<1$,
with: 1) 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 $d <\fra\p2$; 2) possible polar singularities at $x=0$;
3) $M_h^\s\ne0$. The number $h$ will be called the {\it $t$--degree} of
$M$. The smallest cone containing the singularities will be called the
{\rm singularity cone} of $M$.}
\*
\0{\bf Definition 2.2}: {\it Let $\hat\MM_0$ be the subspace
of the functions $M\in\hat\MM$ such that the residuum at $x=0$ of
$x^{-1}\media{M_j^\s(x,\cdot)}$ is zero (here the average is
over $\pps$, \ie it is an ``angle average").}
\*
\0{\bf Definition 2.3}: {\it Let $\MM$ and $\MM_0$ be the subspaces
of the functions $M\in\hat\MM$ and, respectively,
$M\in\hat\MM_0$ bounded near $x=0$.}
\*
\0{\bf Definition 2.4}: {\it Let $\hat\MM^h,\hat\MM_0^h,\MM^h,\MM_0^h$
denote the subspaces of $\hat\MM,\hat\MM_0,\MM,\MM_0$, respectively,
containing the functions of $t$--degree $\le h$.}
\*
In the following part of this section we describe briefly
the properties of the functions contained in the above defined spaces,
referring to [G1] for details:
%
\acapo
1) If a function admits a representation like \equ(2.13), with the above
properties, then such a representation is unique (see [CG], \S 10).
\acapo
2) If $M \in \MM$, or $M \in \MM_0$, then $M_j^\s$ have no pole
at $x=0$ and, furthermore, $M_j^\s(0,\V\psi)=0$ if $j>0$.
\acapo
3) $M\in \hat \MM$ can be written as $M=P+M'$ with $P$ being a polynomial
in $\s t$ (with $\s$ dependent coefficients) and with $M'\in\hat\MM_0$:
this can be done in only one way and we call $P$ the ``polynomial
component'' of $M$, and $M'$ the ``non singular'' component of $M$.
\acapo
4) $M\in \MM$ can be written as $M=p+M'$ with $p$ being a constant
function (with constant value depending on $\s$) and $M'\in\MM_0$: $p$
will be called the ``constant component'' of $M$, and $M'$ will be
the ``non singular'' component of $M$.
\acapo
5) The functions in $\hat\MM$ can be expanded as sums of
the following monomials:
%
$$\tst \s^\ch\,\fra{(\s t g)^j}{j!} x^h e^{i\oo\cdot\nn \, t}\Eq(2.14)$$
%
where $\ch=0,1$ (\ie the \equ(2.14) span the space $\hat\MM$).
\acapo
6) The coefficients of the above mentioned expansions and polynomials
depend on $\s=\pm$, \ie each $M\in\hat\MM$ is, in general, a pair of
functions $M^\s$ defined and holomorphic for $t>0$ and $t<0$,
respectively (and, more specifically, in a domain
$\{\s \Re t>0$, $|\Im gt|<\p/2 - d \equiv \x \}$).
The functions $M^\s(t)$ might sometimes (as in our cases below) be
continued analytically in $t$ but in general $M^+(-t)\ne M^-(-t)$ even
when it makes sense (by analytic continuation) to ask whether equality
holds.
\acapo
7) If $M\in \MM$ the points with $\Re t=0$ and $|\Im
g t|<\x$ ($gt=\pm i\p/2$ corresponds to $x=\mp i$) are, (by our
hypothesis on the location of the singularities of the $M_j$ functions),
regularity points so that the values at $t^\pm$, ``to the right" and
``to the left" of $t$, will be regarded as well defined and given by
$M(t^\pm)\=\lim_{t'\to t,\,\Re t'\to \Re t^\pm} M(t')$; in particular
$M^\pm(0^\pm)\= M_0^\pm(1^-,\V 0)$.
\acapo
8) Since $f$ in \equ(1.1), or \equ(1.2), is a trigonometric polynomial, the
function $F^1$, see \equ(2.4), belongs to $\MM$ and, in fact, the component
$\V F_\su^1$ belongs to $\MM_0$ (as accidentally does $F^1_+$ as well).
\*
On the class $\hat\MM$ we can define the following operation.
\*
\\{\bf Definition 2.5}: {\it If $M\in\hat\MM$, and $t=\t+i\th$, with
$\t,\th$ real, and $\t=\Re t\ne0$, $\s=\sign \Re t$, the function:
%
$$ \II_R M(t)\=\ig_{\s\io+i\th}^{gt}e^{-Rg\s z} M(z)\,d\,gz \Eq(2.15) $$
%
is defined for $\Re R>0$ and large enough, the integral being on an
axis parallel to the real axis.
If $M\in\hat\MM$ then the function of $R$ in \equ(2.15) admits an
analytic continuation to $\Re R<0$ with possible poles at the integer
values of $R$ and at the values $i g^{-1} \oo\cdot\nn$ with $|\nn|<$
(trigonometric degree of $M$ in the angles $\V\psi$); and we can then
set:
%
$$\II M(t)\=\oint\fra{d R}{2\p i R} \,\II_R M(t)\Eq(2.16)$$
%
where the integral is over a small circle of radius $r<1$ and
$r<\min |g^{-1}\oo\cdot\nn|$, the minimum being taken over the
$\nn\ne\V0$ which appear in the Fourier expansion of $M$}.
\*
>From the above definition one can immediately derive
an expression for the action of $\II$ on the monomial \equ(2.14)
and check, in particular, that the radius of convergence in $x$ of $\II
M$, for a general $M$, is the same of that of $M$ (but in general the
singularities at $\pm i$ will no longer be polar, even if those of the
$M_j$'s were such). In general, $\II:\hat\MM^h\to \hat\MM^{h+1}$;
but we note that the $\II$ operation does not increase the degree
in $t$ when $|h|+|\nn| > 0$, (see [G1]).
One readily checks that $\II M$ is a primitive of $M$ (\ie the increment
of $\II M$ between $t_0$ and $t$ is the integral of $M$ between the same
extremes). The similarities of the $\II$ operation with a definite
integral justify the use of the notation:
%
$$\igb_{(\s)}^{gt}M(\t)\,d\,g\t\=\II M(t)\ ,\qquad M\in\hat\MM,\
\s=\hbox{sign}\,\Re t\Eq(2.17)$$
%
In fact many standard properties of integration are, in such a way,
extended to the space $\hat \MM$, see [G1]. In particular we can define:
%
$$ \igb_{\s\io}^{gt}M(\t)\,d\,g\t\=\II M(0^\s)+\ig_0^{gt}M(\t)\,d\,g\t\quad .
\Eq(2.18) $$
\vskip.5truecm
\\{\bf 2.3 Analytic expressions of the expansion coefficients for the whiskers}
\vskip.3truecm
\\We will show that the $\X^h$'s defined through \equ(2.2) admit
rather simple expressions in terms of the operation $\II$ (and other
related operations introduced below). Recall that in \S 2.1 we have fixed
$\aa\in {\bf T}^{l-1}$ and $\f=\p$, and we are looking for the motions, on
the stable ($\s=+$) or unstable ($\s=-$) whisker, which start with the
given $\aa$ and $\f=\p$ at $t=0$; in the following $\aa$ is kept
constant and usually notationally omitted.
We suppose inductively that $\X^{h'} \in\MM^{2h'-1}$, $h' < h$, and
$\F^{h'}\in\MM^{2(h'-1)}$, $\V \F^{h'}_\su\in \MM^{2(h'-1)}_0 $, $ h' \le h $,
and, furthermore, that the singularity cone consists of just the
imaginary axis, \ie the singularities of the functions
defining $\X^h,\F^h$ are on the segments on the imaginary
axis $(-i\io,-i]$ and $[+i,+i\io)$.
This means, in particular, that $\F^{h'},\X^{h'}$ can be represented as:
%
$$\eqalign{
\F^{h'}(x,\V\psi,t)=&\sum_{j=0}^{2(h'-1)}\fra{(\s t g)^j}{j!}
\F^{h'\s}_j(x,\V\psi),\qquad h'=1,\ldots,h\cr
\X^{h'}(x,\V\psi,t)=&\sum_{j=0}^{2h'-1}\fra{(\s t g)^j}{j!}
\X^{h'\s}_j(x,\V\psi),\qquad h'=1,\ldots,h-1\cr}\Eq(2.19)$$
%
by setting $\pps=\oo t$, $\s=\sign t$, $x=e^{-g\s t}$, with
$\F^{h'\s}_j,\X^{h'\s}_j$ holomorphic at $x=0$ and vanishing at $x=0$ if
$j>0$. Hence if $x=e^{- g\s t}$ and $\pps$ is kept fixed,
the $\F^{h'}_j$'s, $\X^{h'}_j$'s tend exponentially to
zero as $t\to\s\io$, if $j>0$; while if $j=0$ they tend exponentially
fast to a limit as $t\to\s\io$ (\ie as $x\to0$), which we denote
$\F^{h'}(\pps,\s\io)$ dropping the subscript $0$ as there is no ambiguity.
Furthermore the inductive hypothesis is enriched by:
%
$$\V \F^{h'\s}_{\su\V0}(\s\io)=\V0, \qquad {\rm for\ all}\ h'\le
h\Eq(2.20)$$
%
recalling that, in general, a subscript $\nn$ affixed to a function
denotes the Fourier component of order $\nn\in {\bf Z}^{l-1}$ of the
considered function: $\X^{h'\s}_{j\nn}(t)$ and $\F^{h'\s}_{j\nn}(t)$
are the Fourier transforms in $\V\psi$ of $\X^{h'\s}_j(t,\V\psi)$ and
$\F^{h'\s}_j(t,\V\psi)$, respectively.
Let us suppose, just as an assumption for the time being, that
$\X^{h'\s}(t)$ and, from \equ(2.7), hence also
$\F^{h'\s}(t)$ are bounded as $t\to\s\io$ for all $h'$, so that
$\X^{h'\s}_j(0,\V\psi)=0$ if $j\ge1$: we show then that the latter
information is very strong and permits us to determine $\X^h$.
This does not imply the convergence of the series: however
in \S 5.2 such a result is proven, so justifying the
boundedness hypothesis and completing the research of bounded motions.
We note that, since $\F^{h\s}\in\MM^{2(h-1)}$ and $\V
\F^{h\s}_{\su\V0} (\s \io)=\V0$ hold, the function $\V \X^{h\s}_\su(t)$,
given by the first of \equ(2.12), is in fact in $\MM^{2(h-1)}$
(by integration). But of course we do not know (yet)
the initial data $\X^{h\s}(0)$.
To find expressions for $\X^h_\su$ we start from the
equations \equ(2.6) with initial time at some instant $T$.
And we use that $\II \F(t)$ is a primitive of the function $\F(t)$, see
comment preceding \equ(2.17), so that:
%
$$\V \X^{h\s}_\su(t)=\V \X^{h\s}_{\su}(T)+\II \V \F^{h\s}_{\su}(t)-\II
\V \F^{h\s}_{\su}(T)\Eq(2.21)$$
%
where $\s={\rm\,sign\,}t,$ and $T$ has the same sign of $t$.
The function $\V \X_\su^{h\s}(T)$ tends to become quasi periodic with
exponential speed as $T\to\s\io$: in fact it becomes asymptotic to the
$j=0$ component, see \equ(2.19), at $x=0$: $\V \X^{h\s}_{0\su}(0,\oo T)$,
(in the sense that the difference tends to $0$, bounded proportionally
to $(g|T|)^{2h-1}e^{-g|T|}$). The function $\II \V \F^{h\s}_\su(T)$ also
becomes asymptotically quasi periodic with exponential speed {\it and
$\V0$ average}, because $\V \F^{h\s}_\su\in \MM_0^{2(h-1)}$ and by the
definition of $\II$: therefore the two quasi periodic functions of $T$
must cancel modulo a constant equal to $\media{\V
\X^{h\s}_{0\su}(0,\cdot)}\=\V \X_{\su\V0}^{h\s}(\s\io)$.
Hence it follows that:
%
$$\V \X^{h\s}_\su (t)=\V\X_{\su\V 0}^{h\s}(\s\io)+\II\V\F^{h\s}_\su (t)
\Eq(2.22)$$
%
and, by inserting \equ(2.22) into the second of \equ(2.12), (considering
also that $\ig_0^{gt}g\t\V\F^{h\s}_\su(\t)\,d\,g\t= gt\II\V\F^{h\s}_\su(t)+$ a
$t$-bounded function), we see that the $\V\X_\giu^{h\s}(t)$ can be
bounded only if:
%
$$\V\X_{\su\V 0}^{h\s}(\s\io)=\V 0,\kern 1.truecm\hbox{hence:}
\kern1.truecm \V\X_\su^{h\s}(t)=\II\V\F_\su^{h\s}(t)\Eq(2.23)$$
%
yielding, setting $t=0^\s$, the initial values of $\X_\su^h$ {\it and}
a simple form for its time evolution. Analogously, recalling that $\V
\X_\giu^{h\s}(0)=\V 0$, essentially by definition, one finds:
%
$$
\V \X_\giu^{h\s}(t)= \big( \II^2 \V \F_\su^{h\s}(t)-
\II^2\V\F_\su^{h\s}(0^\s)\big)\=\bar\II^2 \V\F_\su^{h\s}(t)
\Eq(2.24)$$
%
which gives a simple form to the time evolution of the $\aa$ (\ie $\giu$)
component of $\X^h$ in terms of the operator $\lis\II^2$ defined by the
r.h.s. of \equ(2.24).
Likewise considering the \equ(2.11) and the behaviour at $\s \io$ of
$W$ in \equ(2.9), if $\X^{h\s}(t)$ has to be bounded at $\s\io$, we see
from the second of \equ(2.11) that:
%
$$ \X_+^{h\s}(0)=-\igb_0^{\s\io} w_{00}(\t) \F_+^{h\s}(\t)\ d\,g\t
\Eq(2.25) $$
%
Thus we get (defining at the same time also $\OO$ and $\OO_+$):
%
$$\eqalign{
& \X^{h\s}_+(t)=w_{ll}(t)\igb_{(\s)}^{gt}
w_{00}(\t) \F^{h\s}_+(\t)\,d\,g\t-w_{l0}(t)\ig^{gt}_0w_{0l}(\t)
\F^{h\s}_+(\t)\,d\,g\t\=\OO_+(\F^{h\s}_+)(t)\cr
&
\X^{h\s}_-(t)= w_{0l}(t)\igb_{(\s)}^{gt} w_{00}(\t)
\F^{h\s}_+(\t)\,d\,g\t-w_{00}(t)\ig^{gt}_0w_{0l}(\t)
\F^{h\s}_+(\t)\,d\,g\t\=\OO(\F_+^{h\s})(t)\cr}
\Eq(2.26)$$
%
The \equ(2.23), \equ(2.24) and \equ(2.26), and the boundedness request
imply \equ(2.19) for $h'=h+1$, as we can show by reasoning as in [G1].
As already remarked before \equ(2.21) we note again that, since
$\F^{h'\s}_{\su\V0}(\s\io)=\V0$ for $h'\le h$, the $\V\X_\su^h,\V\X^h_\giu$
functions are in fact in $\MM^{2(h-1)}$, as the $\II$ operation, on
such $\V\F^h_\su$ functions, does not increase the degree. Also, if one
looks carefully at the $\X^{h'\s}_\pm$--evaluation in terms of $\F^{h'\s}_+$,
one realizes that the $\OO,\OO_+$ operations may increase the degree but
by at most $1$. Thus the inductive hypothesis made in connection with
\equ(2.19) is proved for $\X^h$, and it remains to check it for $\F^{h+1}$.
This follows from the expression of $\F^{h+1}$, see \equ(2.7), in terms
of the $\X^{h'}$ with $h'\le h$: see \equ(2.7). One treats separately the
sums in \equ(2.7) with $|\V m|\ge2$ and $|\V m|\ge0$: one just has to
consider that in the first case, which might look dangerous for the
inductive hypothesis, the products of $\X$'s contains at least two factors
(which therefore have order labels smaller than $h$ and verify the
inductive hypothesis); and, furthermore, the coefficients
$(g^2J_0)^{-1}(\dpr_\f f)_{\V m}(\f_0,\oo t)$ or
$(g^2J)^{-1}(\dpr_\aa f)_{\V m}(\f_0,\oo t)$ or $\sin\f_0$ or
$\cos\f_0$ do not contain any terms that can possibly increase
the degree. Hence $\F^{h+1}\in \MM^{2h} $. To see that
$\V\F^{(h+1)\s}_\su\in \MM^{2h}_0$, \ie $\V\F^{(k+1)\s}_{\su\V0}
(\s \io)=\V0$, we simply remark that otherwise the second of \equ(2.12)
could not be bounded in $t$ as $t\to\io$.
We can summarize the above considerations as:
%
$$\tst
\V\F_{\su\V 0}^{h\s}(\s\io)\=\ii_{{\bf T}^{l-1}}
\V\F^{h\s}_\su(\V\ps,\s\io){d\V\ps\over (2\p)^{l-1}}\=
\langle \V\F_\su^{h\s}(\cdot,\s \io)\rangle=\V 0\Eq(2.27)$$
%
for all $h\ge1$, and, still for all $h\ge1$, as:
%
$$\eqalign{\tst
\X^h_-(t)=&w_{0l}(t)\II(w_{00}\F^h_+)(t)-w_{00}(t)\big(\II(w_{0l}\F^h_+)
(t)-\II(w_{0l}\F^h_+)(0^\s)\Big)\=\OO(\F^h_+)(t)\cr
\V \X^h_\giu(t)=&\Big(\II^2(\V \F^h_\su)(t)-\II^2(\V
\F^h_\su)(0^\s)\Big)\=\lis\II^2(\V \F^h_\su(t))\cr
\X^h_+(t)=&w_{ll}(t)\II(w_{00}\F^h_+)(t)-w_{l0}(t)\Big(\II(w_{0l}\F^h_+)
(t)-\II(w_{0l}\F^h_+)(0^\s)\Big)\=\OO_+(\F^h_+)(t)\cr
\V \X^h_\su(t)=&\II(\V \F^h_\su)(t)\cr} \Eq(2.28) $$
%
where $\OO,\OO_+,\lis\II^2,\II$ are defined here and in \S 2.2; and
$\X^h\=(\X_-,\V \X_\giu,\X_+,\V \X_\su)=(\X^h_j)$, $j=0,\ldots 2l-1$,
$\F^h=(0,\V0,\F_+^h,\V \F^h_\su)$. Note
that, while $\X^h$ has non zero components over both the {\it angle}
($j=0,\ldots,l-1$) and over the {\it action} ($j=l,\ldots,2l-1$)
componenents, the $\F^h$ has only the action components non zero.
We can give the above discussion a more formal statement through
the following propositions:
\*
\\{\bf Proposition 2.1}: {\it The series defining the functions
$\V \psi\to X^\s(x,\pps,t)=\sum_{h=0}^\io\m^h\, X^{h\s}(x,\V\psi,t)$ are
convergent for $\m$ small enough and $|x|\le1, \s t\ge0$. And if $x=e^{-g\s
t}$ the surfaces $(\pps,t)\to X^\s(x,\pps,t)$ are stable and unstable
whiskers $W^\pm_\m$, (respectively, if $\s=\pm$). The functions $\pps\to
X^\s(0,\pps,\s\io)$ describe invariant tori $\TT$, on which the motion is
$\pps\to\pps+\oo t$. The two tori coincide as sets, although they may be
parameterized differently (\ie points with the same $\pps$ may be
different in the two parametrizations).}
\*
\\{\it Remark} : The map on such torus defined by the correspondence
established by having the same $\pps$ leads to the notion of homoclinic
splitting, homoclinic
scattering and homoclinic phase shifts, see [CG], [G1].
\*
\\{\bf Proposition 2.2}: {\it If $(I,\AA,\f,\aa) \in W_{\mu}^{\pm}$,
\ie if $(I,\AA,\f,\aa)=X^\s_\m$,
then the evolution $S_t(I,\AA,\f,\aa)$ converges to a quasiperiodic
motion on the torus $\TT$ of Proposition 2.1. And in fact the
convergence is exponential in the sense that, for $\s t\ge 0$,
$ \big| X^\s (x,\V\psi+\oo t,t)-X^\s(0,\V\psi,\s\io)\big|\le Ce^{-\fra12
g\s t}$, for some constant $C>0$, and for $\m$ small enough.}
\*
The above propositions are immediate consequences of the previous
discussion: the only result we have not yet is the convergence
of the series \equ(2.2), but this will be obtained in \S 5.2.
\*
\\{\it Remark} : The reason for the above bound of the
exponential damping constant by
$\fra12g$ is that the true decay is $g(\m)=g+O(\m)$, see [CG], \S5,
Lemma 1. In fact, in next sections, we exploit the $\m$-dependence
of the function $g^2(\m)$, whose coefficients $\g_p$, $p\ge1$,
will be set up precisely to make the powers of the time do not arise:
in other words the terms to be integrated will have the form \equ(2.14),
and the coefficients with $|h|+|\nn|=0$ will be automatically vanishing,
so that the degree of the functions will never increase, and, since,
it was originally zero, it will remain that. This means that a suitable
choise of the ``counterterms" $\g_p$, $p\ge1$, yields that
the functions $\X^{h\s}$, $\F^{h\s}$ are in fact in $\MM^0$, for any
$h\ge1$.
\*
\vskip.8truecm
\\{\bf 3. Tree formalism}
\vskip.4truecm \pgn=1\numfig=1\numsec=3\numfor=1
\\In this section we introduce the graphical formalism, partially
developed in [G1], \S 5, [Ge], \S 5, \S 7, in order to represent, via
equations \equ(2.28) and \equ(2.7), the generic $h$-th
order contribution to the solutions of the perturbed motion equations.
{\it For the time being we ignore the presence of the second sum in
\equ(2.7), (\ie we reason as it was $\g_p\,\=\,0$, $\forall$ $p\ge 1$);
we shall see in \S 5.1 how the discussion has to be modified when also
such terms are taken into account.}
We introduce a label $\n$ to split the functions appearing
in \equ(2.7) as sums of their Fourier
components; let:
%
$$\eqalign{ f^{\d}(\f,\aa)\=& \sum_{\n=(n,\nn)} \fra{f^\d_\n}2 \,
e^{i(n\f+\nn\cdot\aa)} ,\qquad \d=0,1\cr
f^0(\f,\aa)\=&J_0
g^2\cos\f=\sum_{\n,\,\nn=\V0\atop n=\pm1} \fra{f^0_\n}2 \,e^{i n\f}
,\qquad f^1(\f,\aa)\=f(\f,\aa)=\sum_\n\fra{f_\n^1}2\, e^{i
(n\f+\nn\cdot\aa)}\cr}\Eq(3.1)$$
%
(the above Fourier representation is convenient
as it eliminates the derivatives with respect to $\f,\aa$ in the
coefficients of \equ(2.7)).
A {\it tree diagram} (or simply {\it tree}) $\th$ will consist of a
family of lines ({\it branches}) arranged to connect a partially
ordered set of points ({\it nodes}), with the higher nodes to the right.
The branches are naturally ordered as well; all of them have two nodes
at their extremes (possibly one of them is a top node) except the
lowest or {\it first} branch which has only one node, the first node
$v_0$ of the tree. The other extreme $r$ of the first branch will be
called the {\it root} of the tree and it will not be regarded as a node;
moreover we shall call {\it root branch} the branch connecting $r$ to
$v_0$. If $v_1$ and $v_2$ are two nodes we say that $v_1v_0$ can be considered
the first node of the tree constisting of the nodes following $v$: such a
tree will be called a subtree of $\th$.
To each node $v$ we attach a finite set of labels $\t_v$, $\n_v
\= (n_v,\nn_v)$, $\d_v$ and $j_v$, that we call, respectively,
the {\it time label}, the {\it mode label}, the {\it order label}
and the {\it action label}, and to each branch $\l_v$ leading to $v$
we attach a {\it branch label} $j_{\l_v}$. The labels are so defined
that $\n_v \in {\bf Z}^l$, $|\n_v| \le N$, $j_v=l, \ldots, 2l-1$,
$\d_v=0,1$. Each branch different from the root branch, and leading
to $v$, carries an {\it angle label}, $j_{\l_v}\=j_v-l =0,\ldots,l-1$;
the root branch label can be either an angle label, or else an
{\it action label} $j_{\l_v}\ge l$, and in this case $j_{\l_v}=j_v$.
\midinsert
\*
\insertplot{240pt}{170pt}{%fig.tex
\def\nn{{\V \n}}
\ins{-35pt}{90pt}{\it root}
\ins{-10pt}{100pt}{$t^\s$}
\ins{25pt}{110pt}{$j_{\l_{v_0}}$}
%\ins{15pt}{80pt}{$h_{\l_0},\nn_{\l_0}$}
\ins{60pt}{85pt}{$v_0$}
\ins{50pt}{125pt}{$\matrix{\t_{v_0}\,\n_{v_0}\cr\d_{v_0}\,j_{v_0}\cr}$}
%\ins{115pt}{106pt}{$h_{\l_1},\nn_{\l_1}$}
\ins{115pt}{132pt}{$j_{\l_{v_1}}$}
\ins{152pt}{120pt}{$v_1$}
\ins{140pt}{165pt}{$\matrix{\t_{v_1}\,\n_{v_1}\cr\d_{v_1}\,j_{v_1}\cr}$}
\ins{110pt}{50pt}{$v_2$}
\ins{190pt}{100pt}{$v_3$}
\ins{230pt}{160pt}{$v_5$}
\ins{230pt}{120pt}{$v_6$}
\ins{230pt}{85pt}{$v_7$}
\ins{230pt}{-10pt}{$v_{11}$}
\ins{230pt}{20pt}{$v_{10}$}
\ins{200pt}{65pt}{$v_4$}
\ins{230pt}{65pt}{$v_8$}
\ins{230pt}{45pt}{$v_9$}
}{f1}
%
\kern1.truecm
\didascalia{Fig.3.1: A tree $\th$ with
$m_{v_0}=2,m_{v_1}=2,m_{v_2}=3,m_{v_3}=2,m_{v_4}=2$ and $m=12$;
the root branch label is defined to be $j_{\l_{v_0}}=j$.}
\*
\endinsert
The {\it order} $h\=h_{v_0}$ of the tree $\th$ with first node $v_0$ is $h=
\sum_{v\ge v_0} \d_v$, \ie the sum of the order labels of the nodes. The
number of branches emerging from the node $v$ is $1+m_v$, if $m_v$ is the
number nodes immediately following the considered node $v$ (we have to
count also the branch leading to $v$): then $ m = 1 + \sum_{v\ge v_0} m_v$,
if $m$ is the number of nodes in $\th$. Of course, as the order label
$\d_v=0,1$ and as each node $v$ with $\d_v=0$ {\it must} have $m_v\ge2$, it
is $h\le m<2h$. In order to dispose of a label counting the number of
nodes of a subtree, we introduce an extra label, (uniquely determined
by the above ones), by defining the {\it degree} of a node $v$, $d_v$,
as the number of nodes of the subtree having $v$ as first node: then
$d_v=1 + \sum_{\bar v \ge v} m_{\bar v}$, $d_{v_0}=m$.
We imagine that all the branches have the same lenght (even though
they are drawn with arbitrary lenght). A group acts on the
sets of trees, generated by the permutations of the subtrees
having the same root. Two trees that can be superposed by the
action of a transformation of the group will be regarded
as identical (recall however that the branches are
numbered, \ie are regarded as distinct, and the superposition has
to be such that all the decorations of the tree
match.\footnote{${}^3$}{\nota
If we use the terminology of [G1], we can say that we
are considering only {\it labeled trees}, (and not {\it topological}
or {\it semitopological trees}).}
We shall imagine that each branch
carries also an arrow pointing to the root (``gravity
direction'', opposite to the ordering).
We define the {\it momentum} of a node $v$ or of the branch $\l_v$
leading to $v$ as $\nn(v)=\sum_{w\ge v} \nn_w$, if $\n_v=(n_v,\nn_v)$
is the {\it mode label} of $v$. The {\it total momentum} is
$\nn(v_0)=\sum_{v\ge v_0}\nn_v$; we say also that $\nn_v$ is
the momentum ``emitted" by the node $v$.
Then to each node $v$ there corresponds a factor:
%
$$ { 1\over 2}(-i\n_v)_{j_v-l}\,c_{\n_v} \; e^{i(n_v\f^0(\t_v)+
(\aa+\oo\t_v)\cdot\nn_v)}\prod_{s=0}^{l-1}(i\n_{vs})^{m^s_v} \Eq(3.2) $$
%
(the integers $m_v^s$ decompose $m_v$ and count the number of branches
emerging from $v$ and carrying the labels $s=0,\ldots,l-1$, and
the last product in \equ(3.2) is missing if no nodes follow $v$), where:
%
$$ c_{\n_v} \= [ (J_0 g^2)^{-1} \d_{j_v,l} +(J g^2)^{-1}
\big( 1 - \d_{j_v,l} \big) \d_v ] f_{\n_v}^{\d_v} \; , $$
%
(where $\d_{j_v,l}$ is $1$ if $j_v=l$, and $0$ otherwise,
\ie $j_v=l+1,\ldots,2l-1$),
and to each branch $\l$ we associate an improper integration operation
with upper limit $t$, denoted $\OO,$ $\lis\II^2$, $\OO_+$,
$\II$, like in \equ(2.28), and the branch label will be $j_{\l}=0$
when representing $\OO$, $j_\l=1,\ldots,l-1$ for $\lis\II^2$, $j_\l =l$
for $\OO_+$, and $j_\l=l+1,\ldots,2l-1$ for $\II$.
Given all the above decorations on a labeled tree $\th$
we define its value $\tilde V_j(t;\th)$ via the following operations:
%
\acapo
(1) We first lay down a set of parentheses $()$ ordered hierarchically
and reproducing the tree structure (in fact any ordered (topological)
tree can be represented as a set of matching parentheses representing
the tree nodes). Matching parentheses corresponding to a node $v$ will
be made easy to see by appending to them a label $v$. The root will not
be represented by a (unnecessary) parenthesis.
%
\acapo
(2) Inside the parenthesis $(_v$ and next to it we write
the factor \equ(3.2).
\acapo
(3) Furthermore out of $(_v$ and next to it we write a symbol
$\EE^T_{v}$ which we interpret differently, depending on the label
$j_{\l_v}$ on $\l_v$:
%
$$\tst\EE_{v}^T \Big(_v \cdot \ \Big)_v
\= \cases{\OO\Big(_v \cdot\Big)_v (\t_{v'})\ , & if $v \ge v_0\ ,
\quad j_{\l_v}= 0$\ ,\cr
\lis\II^2
\Big(_v \cdot \ \Big)_v (\t_{v'}) &if $v \ge v_0\ ,\quad 1\le j_{\l_v}
\le l-1$\ ,\cr\OO_+\Big(_v\cdot\ \Big)_v(\t_{v'})\ ,\quad &if $v=v_0\
,\quad j_{\l_{v}}=l$; \cr \II \Big(_v \cdot\ \Big)_v (\t_{v'})\ ,\quad
&if $v=v_0\ ,\quad l+1\le j_{\l_v}\le 2l-1$\cr} \Eq(3.3)$$
%
being $\t_{v_0'}$ the root time label $t^\s$ of the tree and the
superscript $\s$ attached to $t$ is important only if $t=0$: in such
case \equ(3.3), if $v=v_0$, has to be interpreted as the limit
as $t\to0^\s$.
Let us denote $\X^{h\s}_{j\nn}(t)$ the coefficient of order $h$ in the
Taylor expansion in powers of $\m$ and of order $\nn$ in the Fourier
expansion in $\aa$ of the function $(\m,\aa)\to\X^\s_\m(t;\aa)$. Then it
follows that $\X^{h\s}_{j\nn}(t)$ can be written as:\footnote{${}^4$}{\nota
The only non trivial property of the representation (3.4) is the
combinatorics; however it is not difficult to check it, say
inductively.}
%
$$ \X^{h\s}_{j\nn}(t)=\sum_{\th \in trees}\fra1{m(\th)!}\sum_{labels:\,
\sum_v\d_v=h} \, \tilde V_{j\nn}(t;\th) \Eq(3.4)$$
%
where $m(\th)=$ number of branches of $\th=$ number of
nodes of $\th$, and, if $j\ge l$:
%
$$ \eqalign{
\tilde V_{j\nn}(t;\th)= & \prod_{v_0 \le v \in \th}
\oint\fra{dR_v}{2\p iR_v}\sum_{\r_v=0,1}\ig_{\s_{v'}\io}^{\r_v g\t_{v'}}
d\,g\t_v \, e^{-\s_{v}g R_{v}\t_v} \; w^{\r_v}_{j_{v}}(\t_{v'},\t_v) \cr
& \cdot \Big[{(-i\n_v)_{j_v-l}\over2}\;c_{\n_v}\;e^{i(n_v\f^0(\t_v)+\nn_v
\cdot\oo\t_v)}\prod_{s=0}^{l-1}(i\n_{vs})^{m^s_v}\Big]\cr} \Eq(3.5) $$
%
where $\t_{v_0'}=t$, ($t=0$ is meant as $t\to 0^\s$, $\s=\pm$, so that
$\s=\s_{v_0'}$ is well defined also for $t=0$, see \equ(3.3)),
$j_{v_0}=j$, and we have used \equ(2.28), by setting:
%
$$ \eqalign{
w^0_{j_v}(\t_{v'},\t_v) & = \cases{ w_{00}(\t_{v'}) w_{0l}(\t_v) ,
& $v>v_0\ , j_v=l$ \cr g\t_v , & $v>v_0\ , j_v>l$ \cr} \cr
%
w^0_{j_{v_0}}(t,\t_{v_0}) & = \cases{ w_{l0}(t) w_{0l}(\t_{v_0}) ,
& $j_v=l$ \cr 0 , & $j_v>l$ \cr} \cr
%
w^1_{j_v}(\t_{v'},\t_v) & = \cases{
w_{0l}(\t_{v'}) w_{00}(\t_v) - w_{00}(\t_{v'}) w_{0l}(\t_v),
& $v>v_0\ , j_v=l$\cr g(\t_{v'}-\t_v), & $v>v_0\ , j_v>l$\cr} \cr
%
w^1_{j_{v_0}}(t,\t_{v_0}) & = \cases{
w_{ll}(t)w_{00}(\t_{v_0}) - w_{l0}(t) w_{0l}(\t_{v_0}),
& $j_{v_0}=l$\cr
1 , & $j_{v_0}>l$\cr}
\cr} \Eq(3.6) $$
%
If $j v_0$.
\*
\\{\it Remark 1} : If we do not perform the operation $\EE^T_{v_0}$
relative to the time $\t_{v_0}$ of the first node $v_0$ and set it to be
equal to $t$, setting also $j\=j_{v_0}$, we see that the result is a
representation of $\F^h_j(t)$.
\*
\\{\it Remark 2}: Note that the tree value $\tilde V_j(t;\th)$, defined
through \equ(3.5), is the same one introduced in [G1]. The analysis
of [G1] applies, and allows us, fixed the perturbative order $h$, to give
a bound on $\X_{j\nn}^{h\s}(t)$, via \equ(3.4). Nevertheless, in order to
obtain our stronger bound $C^h$, for some constant $C>0$, some improvement
is needed. The special form of the kernels in \equ(3.6) has to be
exploited, and the terms from which convergence problems arise
have to be singled out: then we perform essentially an exact
calculation on such terms, so that the involved cancellation mechanisms
can be implemented, while the other ones, harmless with
respect to the estimates, can be easily bounded. In order to
achieve such a goal and distinguish between ``dangerous" and ``harmless"
contributions, some new labels will be introduced: only when
such labels will assume some particular values, a more careful
analysis is required.
\*
We can split $w^{\r_v}_{j_v}(\t_{v'},\t_v)$, $v > v_0$, as follows:
if $j_v > l$ we do nothing, otherwise we decompose it as sum of
two (if $\r_v=0$) or three (if $\r_v=1$) terms:
%
$$ \eqalign{
w^0_{j_v}(\t_{v'},\t_v) = & \fra12 \left\{ { g \t_v \over
\cosh g\t_{v'} \; \cosh g\t_v } + { \sinh g\t_v \over
\cosh g\t_{v'} } \right\} \cr
%
w^1_{j_v}(\t_{v'},\t_v) = & \fra12 \left\{ { g (\t_{v'}-\t_v) \over
\cosh g\t_{v'} \; \cosh g\t_v } + { \sinh g\t_{v'} \over
\cosh g\t_v } - { \sinh g\t_v \over
\cosh g\t_{v'} } \right\} \cr} \Eq(3.7) $$
%
Then we can write:
%
$$ \eqalign{
w^0_{j_v}(\t_{v'},\t_v) \, e^{i n_v \f^0(\t_v)} & = \cases{
g \t_v \, y_v^{(0)} (\t_{v'},\t_v) + y_v^{(-1)} (\t_{v'},\t_v) \; ,
& if $j_v=l$ \cr
g \t_v \, y_v^{(2)} (\t_v) \; ,
& if $j_v>l$ \cr} \cr
%
w^1_{j_v}(\t_{v'},\t_v) \, e^{i n_v \f^0(\t_v)} & = \cases{
g (\t_{v'}-\t_v) \, y_v^{(0)} (\t_{v'},\t_v) +
y_v^{(1)} (\t_{v'},\t_v) - y_v^{(-1)} (\t_{v'},\t_v) \; ,
& if $j_v=l$ \cr
g (\t_{v'}-\t_v) \, y_v^{(2)} (\t_v) \; ,
& if $j_v>l$ \cr} \cr} \Eq(3.8) $$
%
where the functions $y_v^{(\a)}$, $\a=-1,0,1, 2$, are elements of a
finite set of functions:
%
$$ \eqalign{&
y_v^{(-1)} (\t_{v'},\t_v) = \fra12 {\sinh g\t_v\over\cosh g\t_{v'}}
\;e^{i n_v \f^0(\t_v)} \hskip3.2truecm
%
y_v^{(1)} (\t_{v'},\t_v) = \fra12 { \sinh g\t_{v'} \over
\cosh g \t_v} \; e^{i n_v \f^0(\t_v)} \cr
%
& y_v^{(0)} (\t_{v'},\t_v) = \fra12 { 1\over\cosh g\t_v\cosh g \t_{v'}}
\; e^{i n_v \f^0(\t_v)} \hskip2.truecm
%
y_v^{(2)}(\t_{v'},\t_v) = e^{i n_v \f^0(\t_v)} \cr } \Eq(3.9) $$
%
and admit the following Laurent series expansion:
%
$$ \eqalign{
y_v^{(-1)} (\t_{v'},\t_v) & =
\sum_{k_v'=1}^\io \sum_{k_v=-1}^\io
y_v^{(-1)} (k_v',k_v) x_{v'}^{k_v'} x_v^{k_v} \hskip1.2truecm
%
y_v^{(1)} (\t_{v'},\t_v) =
\sum_{k_v'=-1}^\io \sum_{k_v=1}^\io
y_v^{(1)} (k_v',k_v) x_{v'}^{k_v'} x_v^{k_v} \cr
%
y_v^{(0)} (\t_{v'},\t_v) & =
\sum_{k_v'=1}^\io \sum_{k_v=1}^\io
y_v^{(0)} (k_v',k_v) x_{v'}^{k_v'} x_v^{k_v}
\hskip1.5truecm
%
y_v^{(2)} (\t_v) =
\sum_{k_v=0}^\io
y_v^{(2)} (0,k_v) x_v^{k_v} \cr} \Eq(3.10) $$
%
with $x_v=\exp[-\s_v g\t_v]$, $\s_v=\sign \t_v$, and $x_{v'}=
\exp[-\s_{v'} g\t_{v'}]$, $\s_{v'}=\sign \t_{v'}$.
We use the fact that $[ \cosh g\t ]^{-1} = 2x/(1+x^2)$,
$ \sinh g\t = \s (1-x^2)/(2x)$, $\cos \f^0(\t)=1 - 8x^2/(1+x^2)^2$,
and $\sin \f^0(\t)=4 \s x (1-x^2)/(1+x^2)^2$, if $x=\exp[-\s g\t]$.
We can compute some coefficients of the above expansions, which
will turn out to be useful in the following:
$y_v^{(-1)}(1,-1) =\s_v/2$, $y_v^{(-1)}(1,0) = 2i n_v$,
$y_v^{(-1)}(1,1) =-\s_v/2$, $y_v^{(0)}(1,1) = 2$,
$y_v^{(0)}(1,2) = 8i n_v \s_v$, $y_v^{(1)}(-1,1) =\s_{v'}/2$,
$y_v^{(1)}(0,1) =0$, $y_v^{(1)}(1,1) = -\s_{v'}/2$,
$y_v^{(2)}(0,0) = 1$, $y_v^{(2)}(0,1) = 4i n_v \s_v$.
We define the sets $\L_\a$, $\a=-1,0,1,2$, as: $\L_\a=\{ v \in \th \,
: \a_v=\a \}$.
If $jl$, then necessarily
it is $\a_v=2$). Therefore the tree value $\tilde V_{j\nn}(t;\th)$
introduced in \equ(3.3) can be replaced with a new tree value,
$V_{j\nn}(t;\th)$, taking into account also the new labels, and \equ(3.4)
holds still provided $\tilde V_{j\nn}(t;\th)$ is replaced with
$V_{j\nn}(t;\th)$. The generic contribution $(1/m!)\;V_{j\nn}(t;\th)$ to
$\X^{h\s}_{j\nn}$, corresponding to a given tree $\th$, with $m(\th)=m$, is:
%
$$ {1\over m!} \; V_{j\nn}(t;\th) = {1\over m!}
\prod_{v_0 \le v \in \th} \oint\fra{d R_v}{2\p i R_v}
\ig_{\s_{v'} \io}^{\r_v g \t_{v'}} d\,g\t_v
\; \VV_v(\th) \Eq(3.12) $$
%
where we have defined the {\it node function} $\VV_v(\th)$, (depending
on the tree to which the node $v$ belongs), as:
%
$$ \VV_v(\th) \=
F_{\n_v} \; T_v ( g\t_{v'}, g\t_v ) \;
e^{-\s_v R_{v} g \t_v}
\; e^{i \o_v \t_v} \; x_v^{k_v}
\Big[ \prod_{j=1}^{m_v} x_v^{k_{v_j}'} \Big] \; , \Eq(3.13) $$
%
where:
\acapo
\\1) $\o_v = \oo \cdot \nn_v$;
\acapo
\\2) $m_v$ is the number of branches emerging from $v$, and $v_1, \ldots,
v_{m_v}$ are the nodes immediately following $v$ moving along the tree
(so that the product in square brackets in \equ(3.13)
is missing if $v$ is a top node);
\acapo
\\3) $T_v ( g\t_{v'}, g\t_v )$ is defined as:
%
$$ T_v ( g\t_{v'}, g\t_v ) = \left( \d_{\a_v,2}+\d_{\a_v,0} \right)
\left[ (1-\r_v) g\t_v + \r_v g(\t_{v'}-\t_v) \right] +
\left( \d_{\a_v,-1}+\d_{\a_v,1} \right) \Eq(3.14) $$
%
(note that $T_v(g\t_{v'},g\t_v) \= T_v(g\t_v)$, if $\r_v=0$, and
$T_v(g\t_{v'},g\t_v)\=T_v(g\t_{v'}-g\t_v)$, if $\r_v=1$);
\acapo
\\4) $F_{\n_v}$ is given by:
%
$$ F_{\n_v} = { (-i\n_v)_{j_v-l} \over 2 } \; c_{\n_v} \;
\Big[ \prod_{s=0}^{l-1} (i\n_{vs})^{m^s_v} \Big]
\, (-1)^{\d_{\a_v,-1} \d_{\r_v,1} } \; y_v^{(\a_v)} (k_v', k_v) \=
\bar F_{\n_v} \; (-1)^{\d_{\a_v,-1} \d_{\r_v,1} } \; y_v^{(\a_v)} (k_v', k_v)
\Eq(3.15) $$
%
where the coefficients $\bar F_{\n_v}$ satisfy the following bound:
%
$$ \Big| \prod_{v \ge v_0} \bar F_{\n_v} \Big| \le
\Big( {N \over 2 } F_0 N \Big)^m \= {\CC}^m \Eq(3.16)$$
%
with $F_0=(J_0g^2)^{-1} \max_\n \{ f_\n \}$,
and the coefficients $ y_v^{(\a_v)} (k_v', k_v) $ satisfy the
bound:
%
$$ \left| \prod_{v \ge v_0} y_v^{(\a_v)} (k_v', k_v) \right|
\le M^{2m} \prod_{v \ge v_0} \l^{k_v+k_v'} \Eq(3.17) $$
%
if the arguments of the $y_v^{(a)}$'s are all inside an annulus
$0<|x| \le \l < 1$, so that the Laurent series defining the functions
appearing in the $y_v^{(v)}$'s converge: therefore, to order $k \ge 0$,
the coefficients can be bounded by a common value $M_1$ on the maxima of
such functions (there are a finite number of them) in a disk of
radius $\l<1$ times $\l^{-k}$, and, for $k=-1$, their absolute values
are known to be $1$, so that we can
set $M=\max \{ M_1,\l \}$.\footnote{${}^6$}{\nota
The request that {\it all} the $x$ satisfy the property $|x|<\l$
will turn out to be not very strong: in the
cases in which it will be used, the time variables will be
ordered so that, if $|x_{v_0}|\le \l$, then $|x_v|\le \l$ for all
$v>v_0$ (see \S 5.2 below), and this will suffice.}
For each $v$, once we have integrated over the $\t_v$ variable,
we have still to evaluate the residue of the resulting expression
at $R_v=0$, so that, if we consider together the two operations
of integration over the time and of evaluation of the residue, we can
imagine to handle a sequence of hierarchically ordered integrals.
This means that we first integrate with respect either to the
$(\t_v-\t_{v'})$'s, (if $\r_v=1$), or to the $\t_v$'s, (if $\r_v=0$),
the $v$'s being the top nodes, in an arbitrary order, then we evaluate
the corresponding residues, an so on until we reach the tree root.
\*
\\{\bf Definition 3.1} : {\it Given a tree $\th$, let us define the {\rm
reduced tree} $\bar \th$ in the following way. Let us draw a bubble $B_v$
encircling each node $v>v_0$ with $\r_v=0$ and the entire subtree emerging
from it, and let us delete all the so obtained bubbles, but the outer ones;
each remaining bubble encloses a subtree with first node $v$ and $\r_v$ label
fixed to be zero. Then, inside each bubble $B_v$, we consider all the possible
trees with the same labels attached to the node $v$, (in particular with the
same $h_v$ and $\r_v=0$), and we sum their values
\equ(3.12):\footnote{${}^7$}{\nota
A leaf value $L_{j_v\nn(v)}^{h_v\s_{v'}}(\t_{v'})$ contributes to
$\X_{j_{\l_v}\nn(v)}^{h_v\s_{v'}}(\t_{v'})$, $j_{\l_v}=j_v-l$,
where $\nn(v)$ the momentum of the node $v$, and $\s_{v'}$ is
the sign of the time variable corresponding to the node $v'$.}
the so obtained quantity $L_{j_v\nn(v)}^{h_v\s_{v'}}(\t_{v'})$
will be associated with a fat point, replacing the original bubble, which
will be called a {\rm leaf} (of the reduced tree). We call {\rm free nodes}
the reduced tree nodes different from the leaves; the
leaves will be considered a particular type of top
nodes, but they will be distinguished from the free nodes.
We define the {\rm reduced degree} and the {\rm reduced order} of a
reduced tree, respectively, as the number of free nodes and as the sum
of their order labels, and the {\rm order of a leaf} as the label $h_v$
associated with the fat point representing it. We can associate to a reduced
tree $\bar\th$ a value $V_{j\nn}(t;\bar\th)$, where, corresponding
to each free node $v$, there is a factor $\VV_v(\bar\th)\=\VV_v(\th)$ as in
\equ(3.12), and, corresponding to each leaf $v$, there is factor
$L_{j_v\nn(v)}^{h_v\s_{v'}}(\t_{v'})$.}
\*
\\{\it Remark 1} : The reduced degree is so defined that the degree of a
reduced tree turns out to be equal to the reduced degree
increased by the sum of the degrees of its leaves, as it is natural
to set. The analogous property holds for the reduced order.
\\{\it Remark 2} : Note that an integration time variable is associated
only to the free nodes. This could be found a little misleading with
respect to the notion of node in the usual terminology,
(see [G1], [G2], [GGe]); nevertheless we use
the name node also for the leaves for convenience, since we want to affix
to the leaves too the labels $k_v=0$ and $k_v'$, (see, in particular,
the first paragraph in \S 4 below).
\\{\it Remark 3} : With respect to [Ge], we do not introduce the notion
of generalized reduced tree; this allows us to lighten the notations,
(and to avoid using trees which are not easy to visualize), but requires
a refinement of the proof. However we shall see in the following
that not too much work must be done in order to achieve such a goal.
\*
\midinsert
\insertplot{240pt}{170pt}{%fig.tex
\ins{-30pt}{90pt}{\it root}
%\ins{-10pt}{100pt}{$t^\s$}
%\ins{25pt}{110pt}{$j_\l$}
%\ins{15pt}{80pt}{$h_{\l_0},\nn_{\l_0}$}
\ins{60pt}{85pt}{$v_0$}
\ins{50pt}{120pt}{$\matrix{\t_{v_0}\,\n_{v_0}\cr\d_{v_0}\,j_{v_0}\cr}$}
%\ins{50pt}{110pt}{$k_{v_0}\,n_{v_0}$}
%\ins{115pt}{106pt}{$h_{\l_1},\nn_{\l_1}$}
%\ins{115pt}{132pt}{$j_{\l_1}$}
\ins{154pt}{122pt}{$v_1$}
\ins{135pt}{160pt}{$\matrix{\t_{v_1}\,\n_{v_1}\cr\d_{v_1}\,j_{v_1}\cr}$}
%\ins{140pt}{145pt}{$k_{v_1}\,n_{v_1}$}
\ins{110pt}{50pt}{$v_2$}
\ins{190pt}{105pt}{$v_3$}
\ins{210pt}{170pt}{$\matrix{\n_{v_4}\,d_{v_4}\cr h_{v_4}\,j_{v_4}\cr}$}
\ins{200pt}{142pt}{$v_4$}
\ins{235pt}{123pt}{$v_5$}
\ins{235pt}{76pt}{$v_6$}
\ins{200pt}{2pt}{$v_9$}
\ins{180pt}{50pt}{$v_8$}
\ins{200pt}{82pt}{$v_7$}
%\ins{230pt}{66pt}{$v_8$}
%\ins{230pt}{45pt}{$v_9$}
}{f2}
%
\kern1.truecm
\didascalia{Fig.3.2: A reduced tree $\bar\th$ with $\NN_L=3$
leaves, $m_{v_0}=2,m_{v_1}=2,m_{v_2}=3,m_{v_3}=2$, and
reduced degree $d_{v_0}=7$; the branch label is defined to
be $j_{\l}=j$. Each fat point represents a leaf.}
\endinsert
By construction all the free nodes have $\r_v=1$, except the first node
$v_0$ which can have $\r_{v_0}=0, 1 $, while the leaves
have, by definition, $\r_v=0$. Given a reduced tree $\bar \th$, we define
$\bar \th_f \= \{ v \in \bar \th : v \hbox{ is a free node } \}$
and $\bar \th_L \= \{ v \in \bar \th : v \hbox{ is a leaf} \}$;
then $\bar \th = \bar \th_f \cup \bar \th_L$ and
$\bar \th_f \cap \bar \th_L = \emptyset$. Note that, since $\r_v=1$,
$\forall$ free node $v>v_0$, the time variables of a
reduced tree are ordered: if $\s_{v_0}=\s$, then $\s_v=\s$, $\forall$
$v>v_0$, $v\in\bar\th_f$, and $\s_v \t_v > \s_{v'} \t_{v'}$ for any
pair of nodes $v, v'$, with $v'$ immediately preceding $v$.
A leaf $v$ represents a factor contributing to
$\X_{j_{\l_v}\nn(v)}^{h_v\s_{v'}}(\t_{v'})$, $j_{\l_v}=j_v-l$, (see note 7),
whose dependence on $\t_{v'}$ reveals itself only through the factor,
(see the third line in \equ(3.7)):
%
$$\x_v(\t_{v'})=[w_{00}(\t_{v'})\d_{j_v,l}+(1-\d_{j_v,l})]\Eq(3.18)\;,$$
%
so that we can write $L_{j_v\nn(v)}^{h_v\s_{v'}}(\t_{v'})=\x_v(\t_{v'}) \;
L_{j_v\nn(v)}^{h_v\s_{v'}}(0)$, being
$L_{j_v\nn(v)}^{h_v\s_{v'}}(0)$ interpretated as the limit as $\t_{v'}\to
0^{\s_{v'}}$. We define
$L_{j_v\nn(v)}^{h_v\s_{v'}}(0)$ as the
{\it value of the leaf} $v$ of the reduced tree. Also the factor
\equ(3.18) admits a series expansion like the functions $y_v^{(\a_v)}$'s
in \equ(3.10):
%
$$\x_v(\t_{v'})=\sum_{k_v'=1}^{\io}\x_v(k_v',0)x_{v'}^{k_v'}\Eq(3.19)$$
%
We can use explicitly the order of the integration variables,
and define:
%
$$ \o(v) = \sum_{\bar \th_f \ni w \ge v } \o_w \; , \qquad
k(v) = \sum_{ \bar \th_f \ni w \ge v} k_w \; , \qquad
k'(v) = \sum_{\bar \th \ni w > v} k_w' \; , \qquad
p(v) = k(v)+k'(v) $$
%
so that we can write:
%
$$ \eqalign{
\prod_{\bar \th_f \ni v \ge v_0} e^{-k_v g \s \t_v } & =
e^{-k(v_0) g \s \t_{v_0} } \; \cdot \;
\prod_{\bar \th_f \ni v > v_0} e^{-k(v) g \s ( \t_v - \t_{v'} ) } \cr
\prod_{\bar \th \ni v \ge v_0} e^{-k_v' g \s \t_v } & =
e^{-[ k'(v_0) + k_{v_0}'] g \s \t_{v_0} } \; \cdot \;
\prod_{\bar \th \ni v > v_0} e^{-k'(v) g \s ( \t_v - \t_{v'} ) } \cr
\prod_{\bar \th_f \ni v \ge v_0} e^{-R_v g \s \t_v } & =
e^{- \sum_{w \ge v_0} R_w g \s \t_{v_0} } \; \cdot \;
\prod_{\bar \th_f \ni v>v_0}e^{-\sum_{w \ge v}R_w g\s(\t_v-\t_{v'})}\cr
\prod_{\bar \th_f \ni v \ge v_0} e^{i \o_v \t_v } & =
e^{i \o(v_0) \t_{v_0} } \; \cdot \;
\prod_{\bar \th_f \ni v>v_0} e^{i\o(v)(\t_v-\t_{v'})}\cr} \Eq(3.20) $$
%
since $\s_v = \s_{v_0'} \equiv \s$, $\forall$ $v \ge v_0$, $v\in\bar\th_f$.
We have used the fact that each leaf $v$ contributes to the reduced
tree a value $L_{j_v\nn(v)}^{h_v\s_{v'}}(0)$, which is independent on
$\t_{v'}$, times a factor \equ(3.18), which one has to take into account in
the computation of $p(\tilde v)$, for each $\tilde v < v$.
The leaves with $j_v=l$ are such that, in \equ(3.20), $k_v' \ge 1$,
see \equ(3.19), \equ(3.10), while, if $j_v>l$, it is $k_v'\,\=\,0$;
in both cases we can define $k_v$ to be identically vanishing, so
attaching such a label, for convenience, also to the leaves. Note that only
the free nodes contribute to $k(v)$ and $\o(v)$; if we define the {\it free
momentum} of the reduced tree with first node $v_0$ as $\nn(v_0)=\sum_{\bar
\th_f\ni w \ge v_0}\nn_w$, then we can write $\o(v)=\oo\cdot\nn_0(v)$.
Note also that, if $(1/m!)V_j(t;\bar\th)$ is a contribution to
$\X_{j\nn}^{h\s_{v_0'}}(t)$, $\nn\,\=\,\nn(v_0)$, then in general the free
momentum $\nn_0(v_0)$ is different from the ``total
momentum" $\nn$, since $\nn_0(v_0)$ takes into account only the free
nodes of $\bar\th$, while $\nn$ depends also on the momentum labels
affixed to the leaves.
\*
\\{\bf Definition 3.2} : {\it Given a reduced tree $\bar\th$, we define the
{\rm stripped value} of the reduced tree $V^S_{j\nn}(t;\bar \th)$ as the
value we obtain by associating to each free node a factor $\VV_v(\bar\th)
\=\VV_v(\th)$ as in \equ(3.12), but retaining for each leaf
only the factor $\x_v(\t_{v'})$ in \equ(3.18). Note that the
discarded contribution of the leaf $v$ is nothing else but its value,
$L_{j_v\nn(v)}^{h_v\s_{v'}}(0)$, as it is defined after \equ(3.18).}
\*
\\{\it Remark} : Note that the contribution of a leaf $v\in\bar\th$ to a
stripped value $V_{j\nn}^S(t;\bar\th)$ does not depend on its order $h_v$,
but only on the label $j_{\l_v}=j_v-l$ of the branch leading to it,
(see \equ(3.18)).
\vskip.8truecm
\\{\bf 4. Some topological properties of the reduced trees}
\vskip.4truecm\pgn=1\numfig=1\numsec=4\numfor=1
\\As we shall see in \S 5, if it was $p(v)\neq0$ $\forall$ $v\ge v_0$,
no convergence problem would arise. However, obviously,
the case $p(v)=0$ is possible and cannot be ruled out:
to deal with it, we need a very accurate analysis of the integrals
appearing in \equ(3.12).
Basically the reason why the bounds can
be improved is the following. It is true that the case $p(v)=0$
is critical, but, when such value of $p(v)$ occurs for some $v\in\th$,
then the values of the $k_w$ and $k_w'$ labels corresponding
to the nodes $w\ge v$ cannot be arbitrary: on the contrary
they have to be arranged in a very special way. And the fact that
the cancellation mechanisms described in Appendix A1 work
is strictly connected to the possible special configurations:
thereby in this section we study the arrangement of the $k_w$
and $k_w'$ labels, $w\ge v$, when $p(v)=0$.
Let us consider a reduced tree $\bar\th$, with first node $v_0$ and
$j\=j_{\l_{v_0}}v}
(k_w + k_w')$, see \equ(3.20), where $k_w+k_w' \ge 0$, for each $w$, see
\equ(3.10), and $k_w\=0$ if $w$ is a leaf, see \equ(3.19).
Therefore $p(v_0)$ can vanish only if either $k_{v_0}=0$ and
$k_w=-k_w'$ for each $w > v_0$, or $k_{v_0}=-1$ and
$k_w=-k_w'$ for each $w > v'$, except one single node
$\tilde w$ such that $k_{\tilde w} + k_{\tilde w}'=1$.
Let us discuss first the case $k_{v_0}=0$. Then, if $w\in\L_{-1}$, the above
property requires $k_w'=-k_w=1$, because $k_w \ge -1$ and
$k_w' \ge 1$; if $w \in \L_{1}$, then $k_w'=-k_w=-1$,
because $k_w \ge 1$ and $k_w' \ge -1$;
otherwise, if $w \in \L_2$, it must be $k_w=k_w'=0$; the possibility
$w \in \L_0$ has to be excluded as it would imply $k_w+k_w'>0$,
and, for the same reason, if $w$ is a leaf, it must be $j_w>l$,
so that $k_w'=0$. Then we can write:
%
$$ \sum_\th V_{j\nn}(t;\th)=\sum_{\bar\th} V_{j\nn}^S(t;\bar\th)
\prod_{i=1}^{\NN_L(\bar\th)} L_{j_i\nn(v_i)}^{h_{v_i}\s_{v_i'}}(0)
\Eq(4.1) $$
%
where $\NN_L(\bar\th)$ is the number of leaves of the reduced tree $\bar\th$,
and $j_i\=j_{\l_{v_i}}$, where $v_i$ is the $i$-th leaf. Note that \equ(4.1)
is the product of factorizing terms, which can be treated separately, being
independent on each other; each $L_{j_i\nn(v_i)}^{h_{v_i}\s_{v_i'}}(0)$,
$i=1,\ldots,
\NN_L(\bar\th)$, corresponds to a leaf and has as first node a node $v_i$
with $\r_{v_i}=0$, while $V_{j\nn}^S(t;\bar\th)$ can have either $\r_{v_0}$
or $\r_{v_0}=1$. Moreover each $L_{j_i\nn(v_i)}^{h_{v_i}\s_{v_i'}}(0)$ can have
$p(v_i)=0$ only if $k_{v_i}=0$ too; otherwise it is $k_{v_i}=\pm 1$, and,
correspondingly, $p(v_i)=\pm 1$. Then we confine ourselves to the study of
$V_{j\nn}^S(t;\bar\th)$, being the other terms either of the
same form, (and so admitting the same bound), or of a different type,
since $p(v_i)\neq 0$, (and so requiring a different discussion, which we
delay). If $j_w=l$, $w\ge v_0$, we consider together
the cases $w \in \L_{-1}$ and $w \in \L_1$: they give a contribution
to \equ(3.12), containing, as far as the $w$ node is concerned,
a factor $\bar F_{\n_w}$ $\exp [i\o(w)(\t_w-\t_w')]$ times
$ e^{-g\s(\t_w-\t_{w'})} y_w^{(1)}(-1,1)$
$ - e^{g\s(\t_w-\t_{w'})} y_w^{(-1)}(1,-1)= $
$(\s/2) [ e^{-g\s(\t_w-\t_{w'})} - e^{g\s (\t_w-\t_{w'})} ]$.
Let us consider now the case $p(v_0)=0$, $k_{v_0}=-1$, (we note that such a
case arise only if $j_{v_0}=l$). Let us call $\tilde w$ the node such that
$k_{\tilde w} + k_{\tilde w}'=1$, (it is $k_w=-k_w'$ for each $w > v_0$,
$w \neq \tilde w$), and let us denote $\PP$ the path leading from $v_0$ to
$\tilde w$, and $z_i, i=1, \ldots, m_{\PP}$ (with $z_1=v_0$, and $z_{m_{\PP}}
=\tilde w$) the nodes crossed by $\PP$.
\midinsert
\*
\insertplot{240pt}{60pt}{%fig.tex
\ins{5pt}{40pt}{$v_0$}
\ins{60pt}{40pt}{$z_2$}
\ins{110pt}{10pt}{$z_3$}
\ins{190pt}{20pt}{$z_4$}
\ins{230pt}{0pt}{$\tilde w$}
}{f3}
%
\kern.4truecm
\didascalia{Fig.4.1: A path $\PP$ connecting the first node $v_0$ of
the reduced tree $\bar\th$, with the node
$\tilde w$, (defined as the node verifying the condition $k_{\tilde w}+
k_{\tilde w}'=1$), with $m_{\PP}=5$, $z_1=v_0$ and $z_5=\tilde w$.}
\endinsert
For each $z_i\in\PP$, it is $p(z_i)=k_{z_i}+1$, so that the
possible values are $p(z_i)=0,1,2$, corresponding, respectively, to the
cases: $z_i \in \L_{-1}$, $z_i \in \L_{2}$, $z_i \in \L_{1}$.
Note that nodes $w$ with $p(w)=0$ and $\a_w=-1$ can occur only along the path
$\PP$, as can be seen by {\it reductio ad absurdum}: in fact, if such a $w$
is not in $\PP$, it contributes $k_w'\ge1$ to each $p(\tilde v)$,
$\tilde v < w$, so that, in particular, it produces a value $p(v_0)\ge1$,
which is not possible under our assumption.
As a consequence of what has been said, we see that, in order to obtain
the contribution to $\X^{h\s}_{j\nn}(t)$, with $p(v_0)=0$, $k_{v_0}=-1$,
we have to consider the sum of products of several factorizing terms,
as in \equ(4.1), which are of the same type of
before, up to the first factor, which is given by the stripped
value of a reduced tree with a fixed shape, and
labels $p(v_0)=0$, $k_{v_0}=-1$. Therefore, with respect to the previous
situation, only this term is new.
\*
Let us consider a reduced tree, with first node $v_0$ having $p(v_0)\neq0$,
with given shape and collection of indices, and let us consider the $p(v)$
labels, $v>v_0$. Let us single out the nodes $v$'s,
with $p(v)=0$: then each such node will be enclosed, together with all the
reduced subtree emerging from it, inside a bubble $\b_v$.
Each branch leading to a so characterized
node $v$ will be called the {\it stem} of the corresponding bubble.
Let us delete all the bubbles, but the outer ones, after summing the values
of all the possible reduced subtrees of fixed order $h_v$ and fixed $p(v),
k_v$ labels attached to the first node $v$ represented by the end point of
the bubble stem.
We can call {\it flowers} the bubbles; unlike the leaves,
the flowers will not be considered nodes. A reduced tree
with first node $v_0$ having $p(v_0)\neq0$ is decorated with
flowers and leaves, and, by construction, all its free nodes, (\ie
the nodes which are not leaves), have $p(v)\neq0$.
Each flower $\b_v$ will be characterized by the labels $j_v, h_v$,
($h_v$ will be the {\it order of the flower}), and by a {\it flower
function}, which is given by \equ(4.1), where the sum is over the reduced
trees having the first node $v$ with $p(v)=0$, and $k_v=0,-1$.
The degree of a reduced tree is given by the number of its free
nodes plus the sum of the degrees of its flowers, and
of its leaves; analogously, the order of a reduced tree
is given by the sum of the order labels of its nodes, (\ie
free nodes and leaves), plus the sum of the orders of its flowers.
\*
Let us consider now the case $j\ge l$. The only change we have with respect
to the previous situation is that the function
$w_{j_{v_0}}^{\r_{v_0}}(t,\t_{v_0})$ which has to be associated with the first
node is not equal to those of the other nodes. Nevertheless, as we have said
after \equ(3.11), a decomposition like the one
in \equ(3.8) can be obtained, with the only difference that the functions
$y_v^{(\a_v)}$ are repalced with some new functions, \ie $\tilde
y_v^{(\a_v)}$, which admit Laurent series expansions as \equ(3.10)
and, therefore, can be treated in the same way. In other words, no further
difficulty is introduced.
\vskip.8truecm
\\{\bf 5. Analyticity of the whiskers}
\vskip.4truecm\pgn=1\numfig=1\numsec=5\numfor=1
\\If all the nodes $v$ had $p(v) \neq 0$, then all the
integrals would trivially factorize, and give an explicitly computable
result bounded by $C^m$, for some constant $C$.
Yet it can happen that $p(v)=0$, for some $v$: then,
if $\o(v)=0$, the integration would increase by one the power of the time
variable, and, moving backwards until the first node is reached, in the end
we could meet dangerously high powers of the time, say $\t_{v_0}^p$, $p
\le 2m$, so that the last integration would give a $p!$-contribution.
Also the case $\o(v) \neq 0$ would give problems, since the
result of the integration on the corresponding time variable
would be of the form $1/[i\o(v)]^{-n_v}$, for some integer $n_v \ge 1$,
if $n_v$ is the power of $\t_v$ arising as a consequence of the mechanism
previously described. In fact both cases can be handled: the first one
by setting the value of the ``counterterms" $\g_p$, $p\ge1$, so that
the case $p(v)=\o(v)=0$ can never occur, (and, therefore, $n_v\le2$
because of the form of the function \equ(3.14)),
the latter by exploiting some cancellation mechanisms
related to the particular structure of the kernels \equ(3.7), which
are partially taken from [G1] and [Ge], and partially introduced
in this work, (always by following the same strategy of the quoted
references).
The idea is the following. Let us consider only
trees without leaves, for the time being.
We have seen that the only terms we have to
handle carefully are those with label $p(v)=0$; because of the structure
of the kernels \equ(3.7), $p(v)$ can never be ``too negative", and, in
fact, it is always $p(v) \ge -1$; moreover $p(v)$ can be
vanishing only if all the $p(w)$ labels of the following $w$ nodes are
equal either to 0 or to $\pm1$, (according to the rules stated in
\S 4). If $p(v)=0$, as we have seen, $k_v$ can only assume the values
either $k_v=0$ or $k_v=-1$: then the integrals over the $\t_w$'s, $w\ge v$,
can be bounded by using the theory of the twistless KAM tori and the
Eliasson's cancellations, once the values of the ``counterterms"
$\g_p$, $p\ge1$ have been suitably fixed. It remains to study
the cases $p(v) \neq 0$, but they are quite easily dealt with,
by explicit calculations, if we use the results for $p(v)=0$. As far as
the leaf values are concerned, it is enough to note that a leaf
$v$ can be viewed as a contribution to $\X^{h_v\s_{v'}}_{j_{\l_v}\nn(v)}(0)$,
so that it can be studied in the same way as the other terms, and,
therefore, admits the same bound.
\vskip.4truecm
\\{\bf 5.1 Renormalization}
\vskip.3truecm
\\In this section we confine ourselves to the first problem, \ie the
``elimination" of the powers of the time variables. This will lead
to a slightly modified definition of the trees, (and therefore
of their values): to be more precise, a restriction on the
the compatibility of the node labels will be introduced: no node $v$
with label $j_v=l$, $\a_v=-1$, $\r_v=1$, $p(v)=0$, $\nn_0(v)=\V0$ will
be allowed.\footnote{${}^8$}{\nota
Note that when $\r_v=0$, even if $p(v)=0$ and $\nn(v)=\V0$, no problem
arises, as, in such a case, the integral is automatically vanishing.}
In \S 5.2, we shall show that the values of the so modified trees,
to degree $m$, admit the bound $C^m$, so that the convergence
of the formal series \equ(2.2) follows.
\*
Note also that, as it appears clear from the discussion in \S 4,
if $j_v=l$, then the value $p(v)=0$ is possible only if $\a_v=-1$,
\ie if the contribution arising from $y_v^{(-1)}(\t_{v'},\t_v)$,
(or $\tilde y_v^{(-1)}(\t_{v'},\t_v)$, see the end comments in \S 4),
is studied. The dependence of both such functions on the time variable
$\t_v$ is through the factor $\sinh g\t_v\,\exp[in_v\f^0(\t_v)]$
(see note 5 after \equ(3.11)); this
will imply that both cases can be handled at once. Therefore it
will be not restrictive to impose the renormalization condition only
to the first function: the latter will turn out to
satisfy it automatically.
Recall Remark 1 after \equ(3.6): the functions $\F^{h\s}_{j\nn}(t)$ admit
a tree representation like the functions $\X^{h\s}_{j\nn}(t)$, the only
difference being that the operation $\EE^T_{v_0}$ relative to the time
$\t_{v_0}$ of the first node $v_0$ is not performed: on the contrary
$\t_{v_0}$ is set equal to $t$, and $j_{v_0}$ equal to $j$.
Consider the $l$-th component of $\F^{h\s}_{\nn}(t)$, \ie
$\F^{h\s}_{l\nn}(t)\,\=\,\F^{h\s}_{+\nn}(t)$, and let us define
$\F^{h\s}_{+\nn}(t;\g_1,\ldots,\g_h)$ the contribution to
$\F^{h\s}_{+\nn}(t)$ corresponding to fixed values of the
parameters $\g_1, \ldots, \g_h$; from the first equation in \equ(3.9),
we deduce that $y_v^{(-1)}(\t_{v'},\t_v)$ can be written as
$y_v^{(-1)}(\t_{v'},\t_v)=[\cosh g\t_{v'}]^{-1}
\,Y_v^{(-1)}(\t_v)$, with $Y_v^{(-1)}(\t_v)=2^{-1}\sinh g\t_v \;
\exp [i n_v\f^0(\t_v)]$ admitting a Laurent series in powers of
$x=\exp[-\s_v g\t_v]$: then the following result holds.
\*
\\{\bf Proposition 5.1} : {\it By \equ(2.17), \equ(3.8) and \equ(3.10), we
can write $Y^{(-1)}_v(\t_v)\,\F^{h\s}_{+\nn}(\t_v;\g_1,\ldots,\g_h)$
as a function of $x=\exp[-g\s \t_v]$, $\V\psi=\oo\t_v$, and consider
its expansion in $x$ and $\V\psi$. In general, given a function
$f(t)$, with $f(t)=\sum_{p,\qq}\hat f(p,\qq)\,x^p\,e^{i\V\psi\cdot\qq}$,
let us denote by $f(t) \Big|_{p=\oo\cdot\qq=0}$ the coefficient
$\hat f(0,\V 0)$ corresponding
to the contribution $(p,\qq)=(0,\V0)$ in the above
powers expansion, (recall the hypothesis H of \S 1 on $\oo$); note
that $f(t) \Big|_{p=\oo\cdot\qq=0}$ is a constant in $t$.
Then the set of equations:
%
$$ \left\{ \left. Y^{(-1)}_v(\t_v)\, \F^{h\s}_{+\nn}(\t_v;\g_1,\ldots,\g_h)
\right|_{p(v_0)=0 \atop
\o(v_0) =0} \right\} = 0 \Eq(5.1) $$
%
with $h\ge1$, defines recursively the ``counterterms" $\g_p$, $p\ge1$,
satisfying the inequalities $\g_p\le C^p$, for some constant $C>0$.}
\*
\\{\it Remark 1} : The expression in left hand side of \equ(5.1)
means that the function:
$$ f(\t_v;\g_1,\ldots,\g_h) \,\=\,
Y^{(-1)}_v(\t_v)\, \F^{h\s}_{+\nn}(\t_v;\g_1,\ldots,\g_h) \; ,$$
considered as a function of $x_v=\exp[-\s_v g\t_v]$ and $\psi_v=\oo\t_v$,
has to be developed into a power series:
%
$$ f(\t_v;\g_1,\ldots,\g_h) =
\sum_{p(v_0)=-1}^{\io} \sum_{\nn_0(v_0) \atop |\nn(v_0)|\ge 0}
\hat f(p(v_0),\nn(v_0);\g_1,\ldots,\g_h) $$
%
and the contribution $p(v_0)=\oo\cdot\nn(v_0)=0$, (which is
$\t_v$--independent), has to be taken.
\*
\\{\it Remark 2} : Note that \equ(5.1), with $h=1$, fixes $\g_1$ to a
well defined value, then \equ(5.1), with $h=2$, can be resolved giving
$\g_2$ as a function of $\g_1$ which is now known, and so on; therefore
we dispose of a costructive algorithm to compute the ``counterterms".
Note that, although \equ(5.1) has the appearance of an implicit equation
for the $\g_h$, this is not really the case. In fact, if we
recall \equ(2.7) and
explicitly write down the terms in which $\g_h$ appears, \equ(5.1)
takes the form:
%
$$ \sum_{n=\pm1} n^2 f_n^0 \g_h + \tilde R^{h\s}_{\nn}(\g_1,\ldots,
\g_{h-1}) = 0 \Eq(5.2) $$
%
where $\sum_{n=\pm1}n^2f_n^0\,\=\,1$, (see \equ(3.1)), for a suitable
function constant $\tilde R^{h\s}_{\nn}(\g_1,\ldots,\g_{h-1})$,
taking into account all the other terms. In fact the only
contribution to \equ(5.1) depending on $\g_h$ arises from the term of the
second sum in the r.h.s. of \equ(2.7) corresponding to $p=h$, and, if we
take into account also the explicit expression of the coefficients
$Y_v^{(-1)}(-1)$ and $Y_v^{(-1)}(0)$, (which can be deduced,
respectively, from $y_v^{(-1)}(k_{v'},-1)$ and $y_v^{(-1)}(k_{v'},0)$),
see \equ(3.9) and the list of coefficients below
\equ(3.10)), necessary to produce an exponent
$p(v_0)=0$, one immediately sees that such a contribution is given by
the first term in \equ(5.2).
\*
\\{\it Proof of Proposition 5.1} : The statement of the proposition is
a rewriting in terms of the tree expansion of the property:
%
$$\left( Y_v^{(-1)} \; F^{h\s}_{+,\V0}\right)(\s\io)\=0\Eq(5.3)$$
%
analogous to \equ(2.27). The only difference is that \equ(2.27) is
imposed by the motions boundedness request, while \equ(5.2) is
imposed {\it a priori} in order to cancel all the terms which could
generate some time powers. In fact, when $\X^{h_v\s_{v'}}_{j\nn(v)}(t)$,
$j=0,l$, is evaluated from \equ(2.28), according to the equations
\equ(3.12) and \equ(3.20), then the condition
that the case $p(v)=\o(v)=0$ can never occur
implies that the degree of the time power
is never increased by one, and, since $F^{1\s}_{+}(t) \in \MM^0$,
then $F^{h\s}_+(t), X^{h\s}_\pm(t)\in \MM^0$ for all $h>1$. It is easy
to check that \equ(5.1) is of the form \equ(5.2), say inductively,
(it is an obvious consequence of
\equ(2.7); see also the above remark), so that the only property that
remains to be checked is that $\g_p$, $p\ge1$, admits the bound $C^p$.
But this will follow from the discussion of \S 5.2, and from Remark 1
after \equ(3.6), (or from the explicit formula \equ(2.7) expressing
the functions $F^{h\s}$'s in terms of the functions $X^{h\s}$'s).
So that Proposition 5.1 is proven.
\qed
\*
The formal series expansion of $g^2(\m)$ will turn out to be convergent
in $\m$; then the only dependence on $t$ of the functions \equ(2.19),
taking $\V\psi$ to be a parameter, is through
the factor $\exp[-g\s t]$. If we recall that the original (true) model
had $g^2$ instead of $g^2(\m)$, \ie the parameter appearing in the
hamiltonian is not $g(\m)=g(1+\G(\m))$, with $\G(\m)=\sum_{p=1}^{\io}
\G_p\m^p$, but $g\,\=\,g(\m)-g\G(\m)$, we can conclude that
the $t$-dependence of the corresponding functions of the model \equ(1.1)
reveals itself through the factor
$\exp[-g(1-\G(\m))\s t]$.\footnote{${}^9$}{\nota
Note that $g(1-\G(\m))=g'(\m)$, where $g'(\m)$ is the function
introduced in \S 1.}
Obviously we can expand such a factor in powers of $\m$, so reobtaining
the powers of $t$, like in [Ge].
\vskip.5truecm
\\{\bf 5.2 Bound on the tree values}
\vskip.3truecm %\pgn=1\numfig=1\numsec=5\numfor=1
\\In this section we prove the fundamental result of this paper, which
assures the convergence of the series defining the whiskered tori, and so
completes the proof of Proposition 2.1 and Proposition 2.2:
\*
\\{\bf Theorem 5.1}: {\it Let us denote by $\X^{h\s}_{j\nn}(t)$
the dimensionless perturbed motion, $0\le j < 2l$, $\s={\rm sign}t$.
We can always write it in the form
$\X^{h\s}_{j\nn}(t)=\tilde \X^{h\s}_{j\nn}(x,\oo t)$,
where $|\nn|\le(2h-1)N$, $x=e^{-\s gt}$,
and $\tilde \X^{h\s}_{j\nn}(x,\oo t)$ is an analytic function in $x$,
$\tilde \X^{h\s}_{j\nn}(x,\oo t)=$
$\sum_{p=0}^{\io} \tilde \X^{h\s}_{j\nn}(p,\oo t) x^p$,
satisfying the bound
$|\tilde \X^{h\s}_{j\nn}(x,\oo t)|$ $\le$ $\bar D\bar C^{2h-1}$,
for some constants $\bar C, \bar D >0$, and for any $\s t\ge0$.}
\*
\\{\it Proof of Theorem 5.1} : Let us start by studying the term
$V_{j\nn}^S(t;\bar\th)$ in \equ(5.1), where the first node $v_0$ of $\th$
has $p(v_0)=0$, and $k_{v_0}=0,-1$; we can associate to such a tree
a path $\PP$, with the convention that $\PP\,\=\,\emptyset$ if $k_{v_0}=0$,
$j_{v_0}>l$, and $\PP\,\=\,v_0$ if $k_{v_0}=0$, $j_{v_0}>l$.
>From \equ(3.12) and \equ(3.20) we
can obtain a sequence of factorizing integrals; then, for the top nodes
$v\notin\PP$ different from the leaves (top free nodes), we have
%
$$\oint\fra{d R_v}{2\p i R_v}\ig_{\s\io}^0 d\,g\t_v\;
\,T_v(-g\t_v)\,e^{-gR_v\sum_{w\le v}\s\t_w}\,
e^{i\t_v\o(v)}\,e^{-gp(v)\s\t_v}\Eq(5.4)$$
%
where $p(v)=k(v)=k_v$ and $\o(v)=\o_v$, and $T_v(-g\t_v)=
(-g\t_v)^{1-\d_{j_v,l}}$, see \equ(3.14).
The time integration is trivial and yields:
%
$$ (-\s)^{\d_{j_v,l}} \oint\fra{d R_v}{2\p i R_v} \;
{ e^{- g R_v \sum_{w < v } \s \t_w } \over \big(
R_v + p(v) - i \s g^{-1} \o(v) \big)^{2-\d_{j_v,l}} } $$
%
The case $\o(v)=p(v)=0$ can be excluded, since if
$j_v=l$ then $p(v)= \pm 1$, and if $j_v>l$ then $p(v)=0$, but
the property remarked in connection with \equ(2.27) requires
in such a case $\o(v)\neq 0$. If $j_v=l$, as we have said before,
we sum together the two contributions $k_v=\pm1$; if $j_v>l$, we have
a factor $y_v^{(2)}(0,0)=1$. Therefore the residue at $R_v=0$ is
%
$$ \cases{ \left[ i g^{-1}\o(v) \right]^{-2}
& if $j_v > l $ \cr \left[ 1 + g^{-2} \o^2 (v) \right]^{-1}
& if $j_v = l $ \cr} \Eq(5.5)$$
%
(a factor $1/2$ could be introduced in the second expression, in order
to remind us not to overcount the labels
$p(v)=\pm1$, when the sum over the trees is performed).
Next we pass to the nodes immediately preceding the top ones, which can be
seen as top ends of a new reduced tree obtained from $\bar\th$ by deleting
the original top free nodes. For each $v\in\bar\th/\PP$, (\ie $v\in\bar\th$,
$v\notin\PP$), we have again to consider an expression like \equ(5.4),
so that all the integrations can be performed in the same way, if only we
bear in mind that the cases $p(v)=0$, $\o(v)=0$ can be excluded, for the
same reasons as before:
this simply means that the residues are always of the form \equ(5.5).
If $v\in\PP$, (\ie $v=z_i$, $i=1,\ldots,m_{\PP}$), then
the integration \equ(5.4) yields:
%
$$(-\s)^{\d_{j_v,l}}[p(z_i)-i\s g^{-1}\o(z_i)]^{-(2-\d_{j_v,l})}\Eq(5.6)$$
%
where $p(z_i)=0$ is possible only if $z_i\in\L_{-1}$; in such a case
it must be $\o(z_i)\neq0$, because of the renormalization procedure
introduced in \S 5.1, so that \equ(5.6) corresponds to a factor which is
bounded by 1, except for the case $j_{z_i}=l$, $\a_{z_i}=-1$,
(\ie $p(z_i)=0$), which gives:
%
$$ [i g^{-1}\o(z_i)]^{-1} \Eq(5.7) $$
%
with $\o(z_i)\neq0$.
In the end, only the node $v_0$ is left. If $k_{v_0}=0$, $j_{v_0}>l$,
we have a coefficient $y_{v_0}^{(2)}(0,0)=1$, so we have to
integrate the function $g(t-\t_{v_0})$, if $\r_{v_0}=1$, or
$g\t_{v_0}$, if $\r_{v_0}=0$, times $\exp[i \o(v_0) \t_{v_0}]$;
if $j_{v_0}=l$, then $k_{v_0}=0$ requires $v_0 \in \L_{-1}$,
and \equ(5.1) imposes $\o(v_0)\neq0$. If $k_{v_0}=-1$, again \equ(5.1)
requires $\o(v_0)\neq0$.\footnote{${}^{10}$}{\nota
In fact the term with $k_{v_0}=0$, and $p(v_0)=\o(v_0)=0$ vanishes when
summed to the term having $k_{v_0}=-1$, and $p(v_0)=\o(v_0)=0$, for a suitable
choise of the ``counterterms" $\g_p$, $p\ge1$, as it has been shown in \S 4.}
We can summarize the results found so far and state the fundamental
convergence bound in the following lemma.
\*
\\{\bf Lemma 5.1} : {\it Let us consider the contribution to
$\X^{h\s}_{j\nn}(t)$, $\nn\in{\bf Z}^{l-1}$, $\s=\pm$, $jv_0$, the following results hold for
the sum.
\acapo
1) Such a sum can be written as:
%
$$ A_{k_{v_0}}(t)\,
e^{i\oo\cdot\nn_0(v_0)\r_{v_0}t} \prod_{\bar\th_f \ni v \ge v_0}
\bar F_{\n_v} G_v[\o(v)] \Eq(5.8) $$
%
where $\bar F_{\n_v}$ is defined in \equ(3.15), $0<|\nn_0(v_0)|\le m_0N$,
$m_0$ being the number of free nodes in $\bar\th$, $A_{k_{v_0}}(t)$ is
the function:
%
$$ A_{k_{v_0}}(t) = (-1)^{\r_{v_0}}\,in_{v_0}\,[\cosh gt]^{-1}\,
\d_{k_{v_0},-1} + \d_{k_{v_0},0} \; ,$$
%
and $G_v[\o(v)]$ is defined to be:
%
$$ G_v[\o(v)] = \cases{ [i g^{-1}\o(v)]^{-2} & if $v\notin\PP$ and $j_v>l$ \cr
[1 + g^{-2} \o^2(v)]^{-1} & if $v\notin\PP$ and $j_v = l$ \cr
(\s)^{\d_{j_v,l}}
[p(v)-i\s g^{-1}\o(v)]^{-(2-\d_{j_v,l})} \hskip1.truecm &
if $v\in\PP$ and $\a_v\neq-1$ \cr
[i g^{-1}\o(v)]^{-1} & if $v\in\PP$ and $\a_v=-1$ \cr} \Eq(5.9) $$
%
with the third term always bounded by 1, since $|p(v)|\ge1$ in such a case.
\acapo
2) The sum over all the reduced trees with label
$p(v_0)$ fixed to be zero, of the expression
\equ(5.6), admits the bound $D_0C_0^{m_0-1}$ for some
constants $C_0,D_0>0$, if $m_0$ is the number of free nodes, $m_0<2h_0$,
with $h_0\le h$ being the reduced order of $\bar\th$.}
\*
\\{\it Proof of Lemma 5.1} : Note that the first statement is an easy
consequence of the definitions, as it has been shown,
while the second one is rather deep, being essentially
equal to the KAM theorem, as it appears from the proof, (see also [G1],
[G2], [GGe], [Ge]). So Lemma 5.1 is proven if we show that the bound
$D_0 C_0^{m_0-1}$, in the statement 2), holds. The sums of the
stripped values of all the reduced tree can be easily controlled.
If $m_0$ is the reduced degree of the reduced tree, the number of addends
is bounded by the number of tree shapes, ($\le 2^{2m_0}m_0!$), see [HP],
times the number of ways of attaching the $\n_v$, $\r_v$, $\a_v$ and $p(v)$
labels, ($\le (3N)^{lm_0} \cdot 2^{m_0} \cdot 3^{m_0} \cdot 3^{m_0}$).
It remains to check that the ``small divisors" in \equ(5.7) give
no problems. This is the more subtle, and will be done in
Appendices A1, A2 and A3.
\qed
\*
Now we pass to the reduced trees whose first node has $p(v_0)\neq0$.
For the time being, let us neglect the
leaf values. If $p(v_0)=-1$, then it is $k_{v_0}=-1$, and $k_w+k_w'=0$,
$\forall$ $w>v_0$, so that the case can be treated as the case
$p(v_0)=k_{v_0}=0$ of Lemma 5.1, with respect to which only the first
node $v_0$ behaves in a different way; the analysis can be carried out
quite unchanged, and so we do not repeat it here. Therefore in the
following we can suppose $p(v_0)\neq-1$.
>From each flower a contribution \equ(5.8) arises, and we can explicitly
perform the integrations over the time variables of the free nodes: each
integration is a proper one, and gives a factor bounded by 1, so that
no new ``small divisor" can arise.
Nevertheless we must be careful, because we still have to sum over the
labels $p(v)$, $v\ge v_0$, (the sum over the other labels can be treated as
in the previous case). We can resolve this (apparent) problem as follows. If
$\r_{v_0}=1$, $\s t \le g^{-1}$, we split the integral over $\t_{v_0}$:
%
$$\int_{\s\io}^{gt}d\,g\t_{v_0}\;(\ldots)=\int_{\s\io}^{\s}
d\,g\t_{v_0}\;(\ldots)+\int_{\s}^{gt}d\,g\t_{v_0}\;(\ldots)
\= I_m + \int_{\s}^{gt}d\,g\t_{v_0}\;(\ldots) \Eq(5.10)$$
%
and we consider the first term. Once all the integrations
are performed, we are left with a contribution which is the product
of a factor admitting a ``good $m$-bound'' times a factor of the
form $\exp[ - p(v_0) ]$. Then we can choose $\l=1/2$ in \equ(3.17)
in order to get a convergent bound: at worst for every node $v$ we have a
factor $2^{k_v +k_{v'}}$ and a factor $e^{-k_v-k_{v'}}$
so that we can perform the summation over the indices
$k_v,\; k_{v'} \ge -1$, (see \equ(3.10)), and the convergence follows.
We have left the term in \equ(5.10) in which the first time variable
$\t_{v_0}$ has to be integrated between $\s g^{-1}$ and t, but one finds
that, in the more general case, the integrals can be written as:
%
$$ I_{m_1} \ldots I_{m_p} \prod_{v \in \tilde\th_f}
\igb_{\s}^{g \t_{v'}} d g \t_v ( \ldots) $$
%
(all the free nodes $v$'s have $p(v)\neq0$, so that $\r_v=1$)
where $\tilde\th$ is a subtree of $\bar\th$ with first node $v_0$
and $\tilde m$ nodes, with $\tilde m + m_1 + \ldots + m_p = m $,
and the last integral is manifestly bounded (see also
[G1]), so that we see that the only very problem is to
show that $I_m \le C^m$, for some constant $C$.
If $\s t > g^{-1}$, we obtain from the last integration,
(the one corresponding top the first node $v_0$),
the factor $\exp[ - p(v_0) g \s t ]$, so that, since
$\exp[ - p(v_0) g \s t ] \le \exp[ - p(v_0) ]$ we can repeat
the above argument to deduce the convergence. Eventually,
if $\r_{v_0}=0$, the same discussion applies, and, in particular,
only the first case has to be treated.
Obviously we have to take into account also the values of the leaves.
However, if we are interested, say, in the contribution to order $h$, the
reduced order $h_0$ of the reduced tree
and the orders $h_i$, $i=1,\ldots,\NN_L$, of the $\NN_L$ leaves
have to be such that $h=h_0+\sum_{i=1}^{\NN_L}h_i$. So we can arrange
the sums as follows: fixed $h$, we sum over $h_0=1,\ldots,h$, and,
fixed $h_0$, we sum over the orders of the leaves
with the constraint $\sum_{i=1}^{\NN_L}h_i=h-h_0$; then we sum over all
the reduced trees of fixed order $h_0$ with $\NN_L$ leaves
of fixed orders, respectively, $h_i$, $i=1,\ldots,\NN_L$. Since the value of
a leaf of order $h_v$ represents a contribution to
$\X^{h_v\s_v}_{j_{\l_v}\nn(v)}(0)$, it can be treated in the same way,
and therefore admits the same bound.
Therefore we can inductively check, by exploiting the results of Lemma 5.1
too, (as far as the leaves with label $p(v)=0$ are concerned), that the
contribution to $\X^{h\s}_{j\nn}(t)$, $\nn\in{\bf Z}^{l-1}$, $\s=\pm1$,
$2l>j\ge 0$, $|\nn_0(v_0)|\le mN$, arising from the sum of the values of all
the reduced trees of degree $m$, with labels $p(v_0)\neq0$, can be bounded
by $D_2 C_2^{m-1}$ for some constants $D_2,C_2>0$. In fact, a leaf $v$ with
$p(v)=0$ contributing, \eg, to the reduced tree value through the factor
$L_{j_i\nn(v_i)}^{h_{v_i}\s_{v_i}}(0)$ admits a representation analogous to the same
\equ(5.1) and can be expressed as a sum of terms, which are given by the
product of the stripped value of the reduced tree with first node $v$ times
the values of its leaves. The procedure can be iterated for all the leaves
with $p(v)$ labels equal to zero, and in this way we can get rid of them and
are left only with leaves having $p(v)\neq0$. Then the bound $D_2 C_2^{m-1}$
can be assumed to hold, and an inductive proof can be performed.
It remains to study the case $p(v_0)=0$, $j\ge l$, but one immediately see
that this can be discussed as the case $p(v_0)=0$, $jl$; see also the first paragraph in the proof of Theorem 5.1).
Given a reduced tree $\bar\th$, it will be characterized by its shape and by
a collections of labels. Let us proceed as in [G1], [G2], and let
us suppose a condition over the rotation vectors stronger than the
hypothesis H in \S 1, \ie let us suppose that they satisfy a {\it strong
diophantine condition}. This is not really necessary, but it simplifies the
proof, and, once the result is obtained, we can reason as in [GGe]
to eliminate such an unneeded hypothesis; as the discussion
can be repeated quite unchanged with respect to [GGe], we simply
refer to it.
Therefore we shall make the assumption that the rotation vectors
$\oo$'s satisfy the {\it strong diophantine condition}:
%
$$ \eqalign{
1) & \quad\quad\quad
C_0 | \oo \cdot \nn| \ge |\nn|^{-\t} \quad\quad\quad\quad
\V 0 \neq \nn \in {\bf Z}^{l-1} \cr
2) & \quad\quad\quad
\min_{0 \ge p \ge n} \Big| C_0 |\oo \cdot \nn| -
2^p \Big| \ge 2^{n+1} \quad\quad\quad\quad \hbox{if} \quad n \le 0,
\; \; 0 < |\nn| \le (2^{n+3})^{-\t^{-1}} \cr} \Eqa(A1.2) $$
%
where $n, p \in {\bf Z}$, $n\le 0$.
We fix a scaling parameter $\g$, which we take $\g=2$,
and define (in analogy to quantum field theory: see, {\it e.g.},
[BfG], [G4]) a propagator:
%
$$ G \equiv G_v [\o(v)] = \cases{
(g C_0 )^2 [ i\oo_0\cdot\nn_0(v) ]^{-2} & if $v\notin\PP$ and $j_v>l$ \cr
(g C_0 )^2 \left[ (g C_0 )^2 + ( \oo_0\cdot\nn_0(v) )^2 \right]^{-1}
& if $v\notin\PP$ and $j_v=l$ \cr
(\s)^{\d_{j_v,l}} (g C_0 )^{2-\d_{j_v,l}}
\left[ (gC_0)p(v)-i\s \oo_0\cdot\nn_0(v) \right]^{-(2-\d_{j_v,l})}
\hskip.2truecm & if $v\in\PP$ and $\a_v\neq-1$ \cr
(gC_0)\left[ i\oo_0\cdot\nn_0(v)\right]^{-1} &
if $v\in\PP$ and $\a_v=-1$\cr} \Eqa(A1.3)$$
%
where $\oo_0 = C_0 \oo$ is a dimensionless frequency,
and we say that:
\acapo
1) $G$ is on scale 1, if $|\oo_0 \cdot \nn_0(v) | > 1 $;
\acapo
2) $G$ is on scale $n \le 0$, if $2^{n-1}<|\oo_0\cdot\nn_0(v)|\le 2^n$.
Note that, if $v\notin\PP$, $j_v>l$, then, if $G$ is on scale $n \le 0$,
it is $|G|<(g C_0)^2 2^{-2(n-1)}$, and, if it is on scale 1, it is
$|G|<(gC_0)^2$, while, if $v\notin\PP$, $j_v = l$, then $|G|\le1$;
if $v\in\PP$, if $\a_v\neq-1$, then $|G|\le1$, otherwise, if
$\a_v=-1$, then $|G|<(g C_0) 2^{-(n-1)}$.
We can get rid of the new factor $(gC_0)^2$,
by defining $C_1=\max\{ 1, (gC_0)^2 \}$, and introducing a coefficient
$C_1^m$ in the bound \equ(3.16). This implies a simple redefinition of the
constant $\CC$ in \equ(3.16), and we can say that, if $G$
is on scale $n$, then, $\forall$ $n\le1$, $|G|<2^{-2(n-1)}$, if
$v\notin\PP$, and $|G|<2^{-(n-1)}$, if $v\in\PP$.
{\it Henceforth (and in the following two appendices), with an abuse of
notation aiming to not overwhelm the discussion, let us use the term ``tree"
instead of the more cumbersome ``reduced tree", (and the symbol $\th$ instead
of $\bar\th$); however it is always in the meaning of the latter that the
first one has to be interpretated. Moreover we call momentum ({\rm tout
court}) of the node $v$ the free momentum $\nn_0(v)$.}
Given a tree $\th$ we can attach a {\it scale label} to each
branch $v'v$ ($v'$ being the node preceding $v$): it is
equal to $n$ if $n$ is the scale of the branch propagator.
Note that the labels thus attached to a tree are uniquely
determined by the tree: they will have only the function of
helping to visualize the orders of magnitude of the various tree
branches.
Looking at such labels we identify the connected clusters $T$ of
nodes that are linked by a continuous path of branches with the same
scale label $n_T$ or a higher one. We shall say that {\it the cluster
$T$ has scale $n_T$}. Since the tree branches carry an arrow
pointing to the root, (see \S 3), we can associate to each cluster a
collection of incoming branches ({\it branches entering $T$})
and a collection of outgoing branches ({\it branches exiting
from $T$}).
\*
\\{\bf Definition A1.1}: {\it Among the clusters we consider the ones with
the property that there is only one tree branch entering them and only
one exiting and both carry the same momentum. If $V$ is one such cluster,
we denote $\l_V$ the outgoing branch, and $n=n_{\l_V}$ its scale label.
We say that such a $V$ is a {\rm resonance} if the number of branches
contained in $V$ is $\le E\,2^{-n\e}$, where $E,\e$ are defined by:
$E\=2^{-3\e}N^{-1},\,\e=\t^{-1}$. We shall say that $n_{\l_V}$ is the
{\rm resonance scale}, and $\l_V$ a {\rm resonant line}.}
\*
Note that if $\l_V$ is the branch exiting from the resonance $V$, the
branch scale $n_{\l_V}$ is smaller than the smallest
scale $n'=n_V$ of the branches inside $V$.
\*
\\{\bf Definition A1.2}: {\it Given a resonance $V$, let $\l_{v}$ and
$\l_{v'}$ be, respectively, the incoming and outgoing branches, (so that
$\l_V\=\l_{v'}$), and $v$, $v'$ the nodes which $\l_v$, $\l_{v'}$,
respectively, lead to ($v'$ is inside the resonance, and $v$
outside).\footnote{${}^{11}$}{\nota
\rm Recall that the ordering is opposite to the gravity.}
We say that $V$ is a {\rm strong resonance}
if it is $\nn_0(v)=\nn_0(v')$, (as in all resonances),
and $p(v)=p(v')\=0$. A tree with strong resonances will be called a
{\rm resonant tree}.}
\*
\\{\it Remark} : We shall see in the following discussion that only the
strong resonances can give problems, so that in fact they are
the only ``true resonances" (in the usual meaning of the word).
The reason why we have introduced a new name for them
is simply to maintain the definition of resonance
given in [G1], as it will turn out that some properties
which we need follow from the very definition of resonance,
and it will be not important if the considered resonances are
strong or not (see, in particular, Appendix A3).
\*
\\{\bf Definition A1.3}: {\it Given a propagator $G$ as in \equ(A1.3),
we associate to it a label which we call the degree $D_G$ of the propagator,
and which we set equal to 2, if $G$ is given by the first or second
term of the r.h.s. of \equ(A1.3), and equal to 1, if $G$ is given by the
third or fourth term. We associate to each cluster $T$ a label $D_T$,
which we call the {\rm degree} of the cluster $T$: we set $D_T=j$, $j=1,2$,
if the propagator of the corresponding outgoing branch has degree $j$.
Given a strong resonance $V$ on scale $n$, let us consider the outgoing
branch (resonant line); the corresponding propagator is given by either the
first term of the r.h.s. of \equ(A1.3), or the fourth one; then the degree
of a strong resonance is equal to the degree of the corresponding resonant
line, and the propagator $G$ of the resonant line on scale n is such that
$|G|< 2^{-D_V (n-1)}$, if $D_V$ is the degree of the strong resonance.}
\*
The key remark is that the resonant trees (see Definition A1.1) cancel
almost exactly. We have already all is needed to see why this happens.
We can reason in the following way.
Given a tree $\th$ with a strong resonance $V$, we call, as before, $v$
the node which the entering branch leads to, and $v'$ the node
which the exiting branch leads to; moreover
let us call $\th_2$ the subtree with first node $v$.
Imagine to detach from the tree $\th$ the subtree $\th_2$,
then attach it to all the remaining nodes
$ w\in V$. We obtain a family of trees whose contributions to
$\X^{h\s}_{j\nn}(t)$ differ because:
\acapo
1) some of the branches above $v'$
have changed momentum by the amount $\nn_0(v)$: this means that,
if $\e\=\o_0(v)$ and $w\in V$,
some of the propagators $\big[ i \o_0(w) \big]^{-2}$ have become
$\big[ i (\o_0(w)+\e) \big]^{-2}$, some of the
propagators $\big[ (gC_0)^2+ (\o_0^2(w))^2 \big]^{-1}$ have become
$\big[ (gC_0)^2+(\o_0(w)+\e)^2 \big]^{-1}$, some of the propagators
$(\s)^{\d_{j_w,l}}\big[(gC_0)p(w)$
$-i\s \o_0(w)\big]^{-(2-\d_{j_w,l})}$ have become
$(\s)^{\d_{j_w,l}}\big[(gC_0)p(w)$
$-i\s (\o_0(w)+\e)\big]^{-(2-\d_{j_w,l})}$ and some of the propagators
$\big[i\o_0(w)\big]^{-1}$ have become $\big[i\o_0(w)+\e\big]^{-1}$, and:
\acapo
2) there is one of the node factors which changes by taking
successively the values $\n_{wj}$, $j\,\=\,j_{\l_v}$ being the branch
label of the branch leading to $v$, and $w\in V$ is the node to which
such a branch is reattached.
Hence if $\e=0$ we would build in this resummation a quantity
proportional to: $\sum_{w\in V} \n_{wj}= \n_{0j}(v)-\n_{0j}(v')$,
which is zero, because $\nn_0(v')=\nn_0(v)$ means that the sum
of the $\nn_w$'s vanishes, and $00$.
Denoting $T$ a cluster of scale $n$, and $m^j_T(n)$ the number of
resonances of scale $n$ and degree $j$ contained in $T$, (\ie with outgoing
branches of scale $n$ and degree $j$), we have the relation \equ(A1.5)
supplemented by the inequality \equ(A1.6), which is an adaptation of the
version of {\it Brjuno's lemma} as it is exposed in [P]: a proof is in
Appendix A3, and is taken from [G1], with some minor changes.
Recall that, given a tree $\th^1$, we define the family $\FF(\th^1)$
generated by $\th^1$ as follows. If $V$ is a resonance
of $\th^1$ we detach the part of $\th^1$ above $\l_v$, (recall
Definition A1.2), and attach it successively to the points
$w\in\tilde V$, where $\tilde V$ is the set of nodes of $V$
(including the endpoint $w_1$ of $\l_v$ contained in $V$) outside the
resonances contained in $V$. We say that a branch $\l$ is in
$\tilde V$, if $\l$ is contained in $V$ and has at least
one point in $\tilde V$; we denote by $n_\l$ its scale.
For each resonance $V$ of $\th^1$ we shall call $M_V$ the number of
nodes in $\tilde V$. If the resonance degree is $D_V=2$, then
to the just defined set of trees we add the
trees obtained by reversing simoultaneously the signs of the node
modes $\n_w$, for $w\in \tilde V$: the change of sign is performed
independently for the various resonant clusters. This defines a family
of $\le\prod 2M_V$ trees that we call $\FF(\th_1)$. The number
$\prod 2M_V$ will be bounded by $\exp\sum2M_V\le e^{2m}$.
It is important to note that the definition of resonance
given in Definition A1.1
is such that the above operation
(of shift of the node to which the branch entering
$V$ is attached) does not change too much the scales of the tree
branches inside the resonances: the reason is simply that inside a
resonance of scale $n$ the number of branches is not very large being
$\le\lis N_n\=E\,2^{-n\e}$.
Let $\l$ be a branch, in a cluster $T$, contained inside the resonances
$V=V_1\subset V_2\subset\ldots$ of scales $n=n_1>n_2>\ldots$: then the
shifting of the branches $\l_{V_i}$ can cause at most a change in the size
of the propagator of $\l$ by at most $2^{n_1}+2^{n_2}+\ldots< 2^{n+1}$.
Since the number of branches inside $V$ is smaller than $\lis N_n$ the
quantity $\oo_0\cdot\nn_\l$ of $\l$ has the form $\oo_0\cdot\nn^0_\l+
\s_\l\oo_0\cdot\nn_{\l_V}$ if $\nn^0_\l$ is the momentum
of the branch $\l$ ``inside the resonance $V$", \ie it is the sum of all
the $\nn_v$ of the nodes $v$ preceding $\l$ in the sense of the
branch arrows, but contained in $V$; and $\s_\l=0,\pm1$.
Therefore not only $|\oo_0\cdot\nn^0_\l|\ge 2^{n+3}$ (because $\nn^0_\l$
is a sum of $\le \lis N_n$ node modes, so that $|\nn^0_\l|\le N\lis
N_n$), but $\oo_0\cdot\nn^0_\l$ is ``in the middle'' of the diadic interval
containing it and does not get out of it if we add a quantity
bounded by $2^{n+1}$ (like $\s_\l\oo_0\cdot\nn_{\l_V}$): this follows
from the second inequality in \equ(A1.2), \ie from the
strong diophantine condition hypothesis. {\it Hence no branch
changes scale as $\th$ varies in $\FF(\th^1)$, if $\oo$ verifies
a strong diophantine condition.}
Let $\th^2$ be a tree not in $\FF(\th^1)$ and construct
$\FF(\th^2)$, \etc. We define a collection
$\{\FF(\th^i)\}_{i=1,2,\ldots}$ of pairwise disjoint families of
trees. We shall sum all the contributions to $\X_{j\nn}^{h\s}(t)$
coming from the individual members of each family. This is a basic
feature of the summation procedure.\footnote{${}^{14}$}{\nota
The proof of the convergence bound of Lemma 5.1 presented here
is obtained by exploiting some cancellations we can implement by summing
together different reduced trees, (inside the same family $\FF(\th)$);
one could think that the leaf values
give problems, since they introduce an extra difference between
the terms we sum, so making us loose the cancellation mechanism.
This is not the case, because the reduced trees appearing
in $\FF(\th)$ are obtained by shifting a part of $\th$,
{\it with all its leaves}, so that no further
difference is introduced. To be more precise, we rearrange the sums as
follows: fix a reduced tree $\th$, with all its
leaves of fixed orders; then we sum over all the terms of the family
$\FF(\th)$, in which $\th$ is contained,
so that the cancellation mechanism is implemented.}
We call $\e_V$ the quantity $\oo_0\cdot\nn_{\l_V}$ associated with the
resonance $V$. If $\l$ is a line with both extremes in $\tilde V$ we can
imagine to write the quantity $\oo_0\cdot\nn_\l$ as
$\oo_0\cdot\nn^0_\l+\s_\l\e_V$, with $\s_\l=0,\pm1$. Since
$|\oo_0\cdot\nn_\l|>2^{n_V-1}$ we see that the product of the
propagators is holomorphic in $\e_V$ for $|\e_V|<2^{n_V-3}$.
In fact $|\oo_0\cdot\nn^0_\l|\ge 2^{n+3}$ because $V$ is a resonance;
therefore $|\oo_0\cdot\nn_\l|\ge 2^{n+3}-2^n\ge 2^{n+2}$ so that $n_V\ge
n+3$. On the other hand note that $|\oo_0\cdot\nn^0_\l|> 2^{n_V-1}-2^n$
so that $|\oo_0\cdot\nn_\l^0+\s_\l\e_V|\ge 2^{n_V-1}-2^n-2^{n_V-3}\ge
2^{n_V-1}-2\,2^{n_V-3}\ge 2^{n_V-2}$, for $|\e_V|< 2^{n_V-3}$.
While $\e_V$ varies in such complex disk
the quantity $|\oo_0\cdot\nn_\l|$ does not become smaller than
$2^{n_V-1}- 2\,2^{n_V-3}\ge2^{n_V-2}$. Note that the
quantity $2^{n_V-3}$ will usually be $\gg 2^{n_{\l_V}-1}$ which is the
value $\e_V$ actually can reach in every tree in $\FF(\th^1)$; this
can be exploited in applying the maximum priciple, as done below.
It follows that, if $V$ is a strong resonance, calling $n_\l$ the
scale of the branch $\l$ in $\th^1$, each of the $\le\prod 2 M_V\le e^{2m}$
products of propagators of the members of the family $\FF(\th^1)$ can
be bounded above by $\prod_\l\,2^{-D_V(n_\l-2)}\le
2^{4m}\prod_\l\,2^{-D_Vn_\l}$, if regarded as a
function of the quantities $\e_V=\oo_0\cdot\nn_{\l_V}$, for $|\e_V|\le
\,2^{n_V-3}$, associated with the resonant clusters $V$. This even
holds if the $\e_V$ are regarded as independent complex parameters.
By construction it is clear that the sum of the $\prod 2M_V\le e^{2m}$
terms, giving the contribution from the trees in $\FF(\th^1)$, vanishes
to $j$-th order, $j=D_V$, in the $\e_V$ parameters, (by the
approximate cancellation discussed in Appendix A1). Hence we can apply
the maximum principle to bound the contribution from the family
$\FF(\th^1)$, so obtaining the second term in square brackets of
\equ(A1.6); the result is explained as follows:
\acapo
1) the dependence on the variables $\e_{V_i}\=\e_i$ relative to
resonances $V_i\subset T$ with scale $n_{\l_V}=n$ is holomorphic for
$|\e_i|<\,2^{ n_i-3}$ if $n_i\=n_{V_i}$, provided $n_i>n+3$.
\acapo
2) the resummation says that the dependence on the $\e_i$'s has a first
order zero in each, if the strong resonance degree is 1, and a second order
zero in each, if the strong resonance degree is 2. Hence the maximum principle
tells us that we can improve the bound given by the first factor in
\equ(A1.4) by the product of factors $(|\e_i|\,2^{-n_i+3})^j$,
$j=D_V$, if $n_i>n+3$. If $ n_i=n+3$ we cannot gain anything: but since
the contribution to the bound from such terms in \equ(A1.4) is $>1$,
we can leave them in it to simplify the notation. The details can be
found in Appendix A4.
\vskip.8truecm
\\{\bf Appendix A3: Resonant Siegel-Brjuno bound}
\vskip.4truecm\pgn=1\numfig=1\numsec=3\numfor=1
\\In the following discussion, we consider 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 $N^*_n$ the number of non resonant lines carrying a
scale label $\le n$.
We shall prove first that $N^*_n\le 2m (E 2^{-\e n})^{-1}-1$ if
$N^*_n>0$.
If $\th$ has the root line either with scale $>n$, or with scale $n$ and
resonant, then calling $\th_1,\th_2,\ldots,\th_k$
the subtrees of $\th$ emerging from
the first node of $\th$ and with $m_j>E\,2^{-\e n}$ lines, $j=1,\ldots,k$,
it is $N_n^*(\th)=N_n^*(\th_1)+\ldots+N_n^*(\th_k)$ and the statement is
inductively implied from its validity for $m'm-\fra12 E\,2^{-n\e}$. Finally, and this is the real
problem as the analysis of a few examples shows, we claim that in the
latter case the root line of $\th_1$ has scale $>n$.
Accepting the last statement it will be: $N_n^*(\th)=1+N_n^*(\th_1)=
1+N_n^*(\th'_1)+\ldots+N_n^*(\th'_{k'})$, with $\th'_j$ being the $k'$
subtrees emerging from the first node of $\th'_1$ with orders
$m'_j>E\,2^{-\e n}$: this is so because the root line of $\th_1$ will
not contribute its unit to $N^*(\th_1)$. Going once more through the
analysis the only non trivial case is if $k'=1$ and in that 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 $\le m-\fra12 E\,2^{-n\e}$.
It remains to check that if $m_1>m-\fra12E\,2^{-n\e}$ then the root line
of $\th_1$ has scale $>n$. Since the root line of $\th$ is not a
resonant line, the root line of $\th_1$ cannot carry the same momentum.
Suppose that the root line of $\th_1$ is on scale $n$. Then
$|\oo\cdot\nn_0(v_0)|\le\,2^n,|\oo\cdot\nn_0(v_1)|\le
\,2^n$, if $v_0,v_1$ are the first nodes of $\th$ and $\th_1$
respectively. Hence $\d\=|(\oo\cdot(\nn_0(v_0)-\nn_0(v_1))|\le2\,2^n$ and
the diophantine assumption implies that $|\nn_0(v_0)-\nn_0(v_1)|>
(2\,2^n)^{-\t^{-1}}$, or $\nn_0(v_0)=\nn_0(v_1)$. The latter case being
discarded as the root line of $\th$ is non resonant,
it follows that $m-m_1<\fra12E\,2^{-n\e}$ is inconsistent:
it would in fact imply that $\nn_0(v_0)-\nn_0(v_1)$ is a sum of $m-m_1$
node modes and therefore $|\nn_0(v_0)-\nn_0(v_1)|< \fra12NE\,2^{-n\e}$
hence $\d>2^3\,2^n$ which is contradictory with the above opposite
inequality.
A similar induction can be used to prove that if $N^*_n>0$ then the
number $p_n^*$ of clusters of scale $n$ verifies the bound
$p_n^* \le 2 m \,(E2^{-\e n})^{-1}-1$. In fact this is true for
$m\le E2^{-\e n}$. Let, therefore, $p(\th)$ be the number of clusters
of scale $n$: if the first tree node $v_0$ is not in a cluster
of scale $n$ it is $p(\th)=p(\th_1)+\ldots+p(\th_k)$, with the
above notation, and the statement follows by induction. If $v_0$ is in
a cluster of scale $n$ we call $\th_1$, $\ldots$, $\th_k$
the subdiagrams emerging from the cluster containing $v_0$ and with
orders $m_j> E2^{-\e n}$. It will be $p(\th)=1+p(\th_1)+\ldots+p(\th_k)$.
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 $V$
of scale $n$ containing $v_0$ and it will have a propagator of
scale $\le n-1$. Therefore the cluster $V$ must contain at least
$E2^{-\e n}$ nodes. This means that $m_1\le m-E2^{-\e n}$.
Therefore we add and subtract from $N^j_n$ the quantity
$\sum_T m^j_T(n)$, where the sum is over the clusters satisfying the
constraint $n_T=n ,D_T=j$, and exploit the inequality:
%
$$ \sum_{j=1}^2 \Big[ N^j_n + \sum_{T \atop n_T=n ,D_T=j} m^j_T(n)
\; \Big] \le N_n^* + p^*_n $$
%
so that \equ(A1.6) is proven.
\vskip.8truecm
\\{\bf
\font\tenmib=cmmib10
\textfont1=\tenmib
Appendix A4: Dimensional estimate of the order of zero in $\e_i$.}
\vskip.4truecm\pgn=1\numfig=1\numsec=4\numfor=1
\\Consider a family $\FF(\th^1)\,\=\,\FF$.
Let $B$ be the first factor in \equ(A1.4) without the $e^{2k}$, \ie
a ``naive" bound on the sum of the values of each of the trees in the
family.
Between the resonances there exists an inclusion relation; let us define
``first generation resonances" the innermost resonances, \ie the
resonances $V^1_{j_1}$, $j_1\ge1$, containing no other resonances, ``second
generation resonances" the next to innermost resonances $V^2_{j_2}$,
$j_2\ge1$, \ie the resonances which become innermost if all the
original innermost ones are regarded as single nodes, and so on.
Let $\e^i_{j_i}=\oo\cdot\nn_{\l_{V^i_{j_i}}}$: each $\e^i_{j_i}$
is a function of the values $\e^k_{j_k}$, corresponding
to resonances following $V^i_{j_i}$ along the tree.
Consider a first generation resonance $V_1^1$ of scale $n_{V_1^1}$:
it is $|\e_1^1|<\g_{n_{\l_{V_1^1}}}$, and the values of the trees in $\FF$
are analytic in $\e_1^1$ for $|\e_1^1|<\g_{n_{V_1^1}-3}$.
Note that if the other $\e_{j_i}^i$'s vary in their analyticity
domains, $\e_1^1$, considered as a function of them, can assume
a value outside its own analyticity domain when
$n_{V_1^1}=n_{\l_{V_1^1}}+3$,\footnote{${}^{16}$}{\nota
When the $\e^i_{j_i}$'s
are considered as variables defined in a larger analyticity
domain, the scale labels can change but no more than one unity;
this follows from the analysis in Appendix A2, as can be easily checked.}
although the contribution of the factors
corresponding to the tree branches to the first square bracket in
\equ(A1.4) remains as in \equ(A1.4).
Then if we sum the values of the considered trees
collecting them into
families (of $\le 2M_{V_1^1}$ terms) corresponding to the Eliasson's
resummation related to the resonance $V_1^1$, {\it only}, we obtain
a sum of functions each of which has a zero of second order in $\e^1_1$,
independently on the other values $\e_{j_i}^i$'s.
Therefore the considered sums are bounded by:
%
$$ B \left( 2M_{V_1^1}\,2^{D_{V^1_1}
(n_{\l_{V_1^1}}-n_{V_1^1}+3)} \right) \; ,$$
%
when the ``gain factor" can be left also when it is not obtained,
(\ie if $n_{V_1^1}=n_{\l_{V_1^1}}+3$), since, in such a case, the
dimensional bound, which cannot be improved, is given
by $B$, and $n_{\l_{V_1^1}}-n_{V_1^1}+3=0$.
We then consider another innermost resonance $V_2^1$ (if existent). We
perform the same resummation, obtaining a bound:
%
$$ B \prod_{i=1}^2 \left( 2M_{V_i^1}2^{D_{V^1_i}
(n_{\l_{V_i^1}}-n_{V_i^1}+3)} \right) $$
%
on each of the subfamilies of $\FF$ that we consider in this way (each
consisting of $\le 2M_{V_1^1}\,2M_{V_2^1}$ trees).
And we continue until all the $N_1$ innermost resonances have been
considered.
Then we consider a second generation resonance $V_1^2$.
As we perform the resummation related to the $2M_{V_1^2}$ terms associated
with the new resonance, we regard the sum of the values of each of the
groups of trees as a function of $\e_1^2=\oo\cdot\nn_{\l_{V_1^2}}$
(also the values $\e^1_{j_1}$'s corresponding to the
innermost resonances contained in $V^2_1$ are regarded as dependent on
$\e_1^2$): for all
the values of $\e_1^2$, with $|\e_1^2|<\g_{n_{V_1^2}-3}$, such a sum
is analytic if $n_{V_1^1}> n_{\l_{V_1^1}}+3$ for each first
generation resonance, and is bounded, {\it in every case} by:
%
$$ B_1 \= B \prod_{i=1}^{N_1} \left( 2M_{V_i^1}\,2^{D_{V_i^1}
(n_{\l_{V_i^1}}-n_{V_i^1}+3)} \right) $$
%
as it can be argued analogously to the previous discussion.
The further sum over the values of the $\le 2M_{V_1^2}$ elements
involved in the new resummation creates a function of $\e_1^2$ with
a second order zero so that we can improve the bound of such a larger
collection of trees by:
%
$$ B_1 \, 2M_{V_1^2}\,2^{D_{V_i^2}(n_{\l_{V_i^2}}-n_{V_i^2}+3)} $$
%
and we can continue in this way until the second generation of resonances is
exhausted, and so on until no resonances are left, and there is only
a big group of terms collected in the successive resummations (containing
all the values of the trees in $\FF$) and the bound \equ(A1.4) is
consequently obtained.
\vskip1.truecm
\\{\bf References}
\vskip.5truecm%\numsec=0\numfor=1
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\ciao
ENDBODY