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%%%%%%%%
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\def\TIPIO{
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\newfam\truecmsy %per uso in \TIPITOT
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\\{\bf A proof of existence of whiskered tori with
quasi flat homoclinic
intersections in a class of almost integrable hamiltonian systems}
\footnote{${}^*$}{\nota This
paper is deposited in the archive {\tt mp\_arc@math.utexas.edu},
\#94-??.}
\vskip1.truecm
\0{\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 Rotators interacting with a pendulum via small,
velocity independent, potentials are considered: the invariant tori
with diophantine rotation numbers are unstable and have stable and
unstable manifolds ({\it ``whiskers''}), whose intersections define
a set of homoclinic points. The homoclinic splitting can be introduced
as a measure of the splitting of the stable and unstable manifolds near
to any homoclinic point. In a previous paper, [G1], cancellation
mechanisms in the perturbative series of the homoclinic splitting have
been investigated. This led to the result that, under suitable conditions,
if the frequencies of the quasi periodic motion on the tori are large,
the homoclinic splitting is smaller than any power in the frequency of the
forcing (``quasi flat homoclinic intersections"). In the case $l=2$ the
result was uniform in the twist size: for $l>2$ the discussion
relied on a recursive proof, of KAM type, of the whiskers
existence, (so loosing the uniformity in the twist size). Here we extend
the non recursive proof of existence of whiskered tori to the more than
two dimensional cases, by developing some ideas illustrated in the quoted
reference.}
%
\*
%
\\{\bf Key words:} {\it KAM, homoclinic points, cancellations,
perturbation theory, classical mechanics, renormalization}
\vskip1.truecm
\\{\bf 1. Introduction}
\vskip.5truecm\pgn=1\numfig=1\numsec=1\numfor=1
\\The existence of whiskered tori is known from the works
of Melnikov, [Me], Moser, [Mo], Graff [Gr]; a
general theory can be found in [LW]. In this paper we discuss
the existence of whiskered tori in a special class of almost
integrable hamiltonian systems.
As in [G1], we consider a model consisting of a family of
rotators, say $l-1$ in number, interacting with a pendulum via a
conservative force (the model can be called, as in [G1],
{\it rotator--pendulum model}, or {\it simple
resonance model}, or {\it Arnold model}).\footnote{${}^2$}{\nota
In [LW] the generation of whiskered tori is studied
for systems whose hamiltonian can be expressed, in terms of
action-angle variables $(\AA,\aa)$, as $H(\AA,\aa)=H_0(\AA)+\m
f(\AA,\aa)$, so that there is no hyperbolicity in the
unperturbed problem. Then, under the hypothesis that the non
degeneration condition $\Vert \dpr_{A_i}\dpr_{A_j} H_0 \Vert \ge c >0$
is fulfilled, invariant whiskered tori are constructed near perturbed
periodic orbits. A case in which the above condition does not hold
is studied in [CG], in connection to a celestial mechanics
problem (D'Alembert procession).}
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.
The model can be described by the $l$ degrees of freedom hamiltonian
$H_\m\=H_0+\m f$ given by
%
$$ \oo\cdot\V A+{1\over2}J^{-1}\AA\cdot\AA+{I^2\over2J_0}+g^2J_0
(\cos\f-1)+\m \sum_{{|\n|\le N}\atop{\nn\neq \V 0}}
f_\n\cos(\aa\cdot\nn+n\f) \Eq(1.1)$$
%
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, and $f_\n$ are fixed constants. We suppose $f_{n,\V0}\=0$,
for all $n$: this will be clearly not restrictive.
A natural {\it energy scale} for the model will be $E=J_0g^2$.
We suppose {\it a priori} that:
\*
\penalty-500
\\ {\bf Hypothesis H$_1$: \it the parameters $\oo,\m$ verify, in general:
%
$$\oo=\fra{\oo_0}{\sqrt\h},\qquad |\m|\le b\h^Q, \qquad \h\le1\Eq(1.2)$$
%
with $Q$ and $b^{-1}$ which will be restricted to be large enough in
the course of the analysis. }
\*%
\penalty10000
\0and:
\*
\\{\bf Hypothesis H$_2$: \it $\oo_0$ is a {\it diophantine vector}, \ie:
%
$$\tst C_0 |\oo_0\cdot\nn|\ge |\nn|^{-\t}\ ,\qquad \hbox{for all}
\ \V0\ne\nn\in {\bf Z}^{l-1} \Eq(1.3)$$
%
for some {\it diophantine constant} $C_0$ and some {\it diophantine
exponent} $\t>0$.}
\*
The $l=2$ and $J=+\io$ case will {\it not} be excluded and corresponds
to the ``pendulum in a periodic force field''.
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:
%
$$\tst W_0^{\pm}\=W_0\=\{
{I^2\over2J_0}+g^2J_0(\cos\f-1)=0 \} \times \{\AA\=\AA^0\ ,\aa\in
{\bf T}^{l-1}\} \Eq(1.5)$$
%
Then, it follows ``from KAM theory'', [Me],[E],[CG], that ``many''
unperturbed tori around the torus $\V A^0=\V0$ (including the one $\V
A^0=\V0$ itself) can be {\it continued analytically} (in $\m$), togheter
with their whiskers, into invariant tori with the same $\oo$,
for all $|\m|**0$; hence they can be
written as:
%
$$W^\pm_\m=\{(I,\AA,\f,\aa)=(I^\pm_\m(\aa,\f),\AA^\pm_\m(\aa,\f),\f,\aa):
\aa\in {\bf T}^{l-1}, |\f|<2\p-\d\}\Eq(1.6)$$
%
for suitable real--analytic (in $(\aa,\f)$ {\it and} $\m$) functions
$\AA^\pm,I^\pm$.
In this context, it is natural to {\it measure the splitting between
$W^+_\m$ and $W^-_\m$} at $\f=\p$ and $\aa=\V 0$ by the quantity:
$\d(\aa)\=\det \dpr_\aa(\AA^+_\m-\AA^-_\m)|_{\f=\p}$
at $\aa=0$, and its $\aa$-derivatives at $\aa=\V 0$.
In [CG] an algorithm for the computation of the $\m$-expansion
coefficients of the functions $\V A^\pm$, $\V I^\pm$ is
introduced, and in [G1] it is used in deriving the result
that the homoclinic splitting is smaller than any power in
$\h$, as $\h \to 0$.\footnote{${}^3$}{\nota
In [CF] the persistence of quasiperiodic solutions for nearly
integrable hamiltonian systems, described by hamiltonians
of the form ${1\over2}\AA\cdot\AA-\m f(\aa)$,
is proven with similar tools.}
Such a result is obtained by
checking several cancellation mechanisms, operating to
all orders of perturbation theory. However
the problem is solved selfconstistently only in the case
$l=2$, the solution of the cases $l >2$ relying on results
inherited from the KAM theory approach of [CG]: in such cases
one looses the uniformity in the twist size,
defined as $t_w=\min_{j=0, \ldots, l-1} J^{-1}_j$, see [G2].
It would, therefore, be nice to have a proof completely
freed from KAM-type results. In [G1] the conjecture that this can be
done is advanced (and motivated): in this paper we extend the
selfconsistent treatment to the more general case $l\ge2$, by using
some extra cancellations which can be seen as an extension of those
exposed in [G1], [G2], [GG], so obtaining a theory fully independent on
KAM-type results. To be more precise, we prove the
existence of the whiskered tori in a selfconsistent way, and
assuring the uniformity in the twist size. The steps through
which the proof proceeds are the following: 1) starting from the
unpertubed motion on the separatrix, one perturbatively finds
the equations of the motion on two $l$-dimension manifolds,
one stable and the other unstable, expressed by a formal series
expansion in the perturbation parameter; 2) under the hypothesis that the
series converges, the motions become asymptotic to motions
on $(l-1)$-dimension invariant tori; 3) one checks the series
convergence.
The paper is selfcontained: \S2$\div$\S5 have a definitory nature,
however, and they are almost literally taken from the review article
[G1], with some abstraction effort, while
the original work is in \S6$\div$\S8 (and in the appendices) and, as it
has been said, it develops the ideas of [G1], [G2], [GG]. The above
illustrated steps are approached in \S 4 and \S 8,
where they receive a more mathematical statement.
Propositions 4.1 and 4.2 at the end of \S4 provide formal statements of
the above results. In \S6 besides quoting our key estimate (the first of
(\seidue)) we briefly discuss the connection of this work with the
theory of the homoclinic splitting.
\vskip1.truecm
\\{\bf Acknowledgements}: I am indebted to G. Gallavotti
for having originally proposed this work, and
for encouragement and many clarifying discussions and suggestions
all along during its draft.
\vskip1.truecm
\\{\bf 2. Recursive formulae}
\vskip.5truecm\pgn=1\numfig=1\numsec=2\numfor=1
\\In this section 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).
The unperturbed motion is simply:
%
$$ X^0(t)\=(I^0(t),\V 0,\f^0(t), \aa+\oo t)\Eq(2.1) $$
%
where $(I^0(t),\f^0(t))$ is the separatrix motion, generated by the
pendulum in \equ(1.1) starting at, say, $\f=\p$, and
$\f^0(t)=4 \arctan e^{-gt}$.
Let $X^\s_\m(t;\a)$, $\s=\pm$, be the evolution, under the flow
generated by \equ(1.1), of the point on $W^\s_\m$ given by
$(I^\s_\m(\aa,\p),\AA^\s_\m(\aa,\p),\p,\aa)$ (see \equ(1.6); let:
%
$$X^\s_\m(t)\=X^\s_\m(t;\aa)\equiv \sum_{k\ge 0} X^{k\s}(t;\aa) \m^k=
\sum_{k\ge 0} X^{k\s}(t) \m^k,\qquad \s=\pm\Eq(2.2)$$
%
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 $X^{k\s}(t)$, as forming a single function $X^k(t)$,
which is $X^{k+}(t)$ if $\s={\rm sign}\,t=+$, and
$X^{k-}(t)$ if $\s={\rm sign}\,t =-$.
Inserting \equ(2.2) into the Hamilton equation associated with
\equ(1.1), we see that the coefficients $X^{k\s}(t)$
satisfy the hierarchy of equations:
%
$${d\over dt} X^{k\s}\= \dot {X}^{k\s}=L X^{k\s}+F^{k\s}\Eq(2.3)$$
%
where:
%
$$\tst
L\=L(t)=\pmatrix{
0 &\V 0 & J_0^{-1} &\V 0 \cr
\V 0 &0 &\V 0 &J^{-1} \cr
g^2J_0 \cos\f^0(t) &\V 0 &0 &\V 0 \cr
\V 0 &0 &\V 0 &0 \cr},\quad
F^1(t)=\pmatrix{0\cr0\cr
-\dpr_\f f(\f^0(t),\aa+\oo t)\cr
-\dpr_\aa f(\f^0(t),\aa+\oo t)\cr}\Eq(2.4)$$
%
and where $F^{k\s}$ depends upon $X^0,...,X^{k-1\s}$ but not on
$X^{k\s}$; 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.
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:
%
$$\eqalign{
& {d\over dt} X_+^{k\s}= (g^2 J_0 \cos\f^0) X_-^{k\s}+ F_+^{k\s}
\ ,\quad\quad\quad
{d\over dt} X^{k\s}_\su=\V F^{k\s}_\su\cr
&{d\over dt} X_-^{k\s} = J_0^{-1} X_+^{k\s}\ ,\quad\kern3.cm
{d\over dt} \V X^{k\s}_\giu=J^{-1} \V X^{k\s}_\su\cr}
\Eq(2.6)$$
%
as $F^{k\s}_-$, $\V F^{k\s}_\giu$ vanish identically, for $k\ge 1$.
And, for all $k\ge1$, we can write the following formula for $F^{k\s}$
in terms of the coefficients $X^0,...,X^{k-1\s}$ and of the
derivatives of $H_0$ and $f$:
%
$$ \eqalignno{
& F_-^{k\s}\=0\ ,\quad\quad \V F_\giu^{k\s}\=\V 0\ ,\quad\quad \V
F_\su^{k\s}= -\sum_{|\V m|\ge0} (\dpr_\aa f)_{\V m}(\f^0,\aa+\oo t)
\sum_{(k^i_j)_{\V m,k-1}} \prod_{i=0}^{l-1}\prod_{j=1}^{m_i}
X^{k^i_j \s}_i & \eq(2.7) \cr
& F_+^{k\s} \= \sum_{|\V m|\ge 2} (g^2 J_0 \sin \f)_{\V m} (\f^0)
\sum_{(k_j)_{\V m,k}} \prod_{j=1}^{m}X^{k_j\s}_- -
\sum_{|\V m|\ge0} (\dpr_\f f)_{\V m}(\f^0,\aa+\oo t)
\sum_{(k^i_j)_{\V m,k-1}} \prod_{i=0}^{l-1}\prod_{j=1}^{m_i}
X^{k^i_j\s}_i\cr}$$
%
where $(G)_{\V m}(\cdot)$, with $G= \dpr_\aa f, \dpr_\f f,
g^2 J_0 \sin \f$, and
$(k^i_j)_{\V m,p}$, with $k^i_j\ge 1$, $ m_i\ge0$, $p=k,k-1$,
are defined as:
%
$$\eqalign{
(G)_{\V m}(\cdot)\=&\Bigl(
{\dpr^{m_0}_\f\dpr^{m_1}_{\a_1}
\ldots\dpr^{m_{l-1}}_{\a_{l-1}}\dpr^{m_l}_I\dpr^{m_{l+1}}_{A_1}
\ldots\dpr^{m_{2l-1}}_{A_{l-1}}\,G\over m_0!\,m_1!\,\ldots\,
m_{l-1}!\,m_l!\,m_{l+1}!\,\ldots\,m_{2l-1}!}\Bigr)(\cdot)\cr
(k^i_j)_{\V m,p}\=&(k^0_1,\ldots,k^0_{m_0},k^1_1,\ldots,k^1_{m_1},
\ldots,k^{2l-1}_1,\ldots,k^{2l-1}_{m_{2l-1}})\qquad {\rm
s.t.\ }\sum k^i_j=p\cr}\Eq(2.8)$$
Note that the first sum in the expression for $\V F^h_+$ can only
involve vectors $\V m$ with $m_j=0$ if $j\ge1$, because the function $J_0
g^2\cos\f$ depends only on $\f$ and not on $\aa$, (hence also $k^i_j=0$
if $i>0$). We use here the above notation to uniformize the notations.
The evolution of $X^k$ is determined by integrating
\equ(2.6), if the initial data are known. The $k=1$ case requires a
suitable interpretation of the symbols, in according to
equation \equ(2.4).
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 $\f=\p,I=2g
J_0$:
%
$$ W(t)=\pmatrix{
{1\over\cosh gt}&{{\bar w}\over4J_0g}\cr
-J_0g{\sinh gt\over\cosh^2 gt}&
(1-{{\bar w}\over4}{\sinh gt\over\cosh^2gt})\cosh gt\cr},
\qquad{\bar w}\={2gt+\sinh 2gt\over\cosh gt}\Eq(2.9)$$
%
And the evolution of the $\pm$ (\ie $I,\f$) components can be determined by
using the above wronskian:
%
$$\pmatrix{X^{k\s}_-\cr X^{k\s}_+\cr}= W(t)
\pmatrix{0\cr X^{k\s}_+(0)\cr} +
W(t)\ii_0^t{W\,}^{-1}(\t)\pmatrix{0\cr F^{k\s}_+(\t)\cr}\ d\t
\Eq(2.10)$$
%
Thus, denoting by $w_{ij}$ ($i,j=0,l$) the entries of $W$
we see immediately that:
%
$$\eqalign{ & X^{k\s}_+(t)=w_{ll}(t)X^{k\s}_+(0)+w_{ll}(t)
\ig^t_0 w_{00}(\t) F^{k\s}_+(\t) d\t-w_{l0}(t)\ig^t_0
w_{0l}(\t) F^{k\s}_+(\t)\,d\t\cr & X^{k\s}_-(t)=
w_{0l}(t)X^{k\s}_+(0)+w_{0l}(t) \ig^t_0w_{00}(\t)
F^{k\s}_+(\t) d\t-w_{00}(t)\ig^t_0w_{0l}(\t)
F^{k\s}_+(\t)\,d\t\cr} \Eq(2.11)$$
%
The integration of the equations
\equ(2.6) for the $\su,\giu$ components yields:
%
$$\eqalign{ \V X_\su^{k\s}(t)=&\V X_\su^{k\s}(0)+\ig_0^t\V
F^{k\s}_\su(\t)d\t\cr \V X_\giu^{k\s}(t)=&J^{-1}\Big(t \V
X_\su^{k\s}(0)+ \ig_0^td\t \,(t-\t)\V F^{k\s}_\su(\t)\Big)\cr}\Eq(2.12)$$
%
having used that the $\V X^{k\s}_\giu (0)\=\V 0$ 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 $k\ge1$ of the perturbation expansion.
\vskip1.truecm
\\{\bf 3. The improper integration $\II$.}
\vskip.5truecm\pgn=1\numfig=1\numsec=3\numfor=1
\\We introduce some integrations operations that can be performed
on the functions introduced in \S 2. The operation is simply the
integration over $t$ from $\s\io$ to $t$, $\s=\sign t$. In general
such 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 $X^k(t)$, 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 3.1}: {\it Let $\hat\MM$ be the space of the functions
of $t$ which can be represented, for some $k\ge 0$, as:
%
$$M(t)=\sum_{j=0}^k{(\s t g)^j\over j!} M_j^\s(x,\oo t)\ ,\quad
x\=e^{-\s gt}\ ,\quad \s={\rm sign}\, t\Eq(3.1)$$
%
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) $M_k^\s\ne0$. The number $k$ 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 3.2}: {\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 3.3}: {\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 3.4}: {\it Let $\hat\MM^k,\hat\MM_0^k,\MM^k,\MM_0^k$
denote the subspaces of $\hat\MM,\hat\MM_0,\MM,\MM_0$, respectively,
containing the functions of $t$--degree $\le k$.}
\*
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(3.1), 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:
%
$$\tst \s^\ch\,\fra{(\s t g)^j}{j!} x^h e^{i\oo\cdot\nn \, t}\Eq(3.2)$$
%
where $\ch=0,1$ (\ie the \equ(3.2) 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) 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 3.5}: {\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:
%
$$\II_R M(t)\=\ig_{\s\io+i\th}^te^{-Rg\s z} M(z)\,dz\Eq(3.3)$$
%
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(3.3) 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:
%
$$\II M(t)\=\oint\fra{d R}{2\p i R} \,\II_R M(t)\Eq(3.4)$$
%
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(3.2)
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, $\II:\hat\MM^k\to \hat\MM^{k+1}$;
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:
%
$$\igb_{(\s)}^tM(\t)d\t\=\II M(t)\ ,\qquad M\in\hat\MM,\
\s=\hbox{sign}\,\Re t\Eq(3.5)$$
%
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:
%
$$ \igb_{\s\io}^t M(\t) d\t \= \II M(0^\s) + \ig_0^t M(\t) d\t \quad .
\Eq(3.6) $$
\vskip1.truecm
\\{\bf 4. Analytic expressions of the expansion coefficients for the
whiskers}
\vskip.5truecm\pgn=1\numfig=1\numsec=4\numfor=1
\\We will show that the $X^{k}$'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 \S2 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}$, $h < k$, and
$ F^h \in\MM^{2(h-1)}$, $\V F^{h}_\su\in \MM^{2(h-1)}_0 $, $ h \le k $,
and, furthermore, that the singularity cone consists of just the
imaginary axis (\ie the singularities of the functions
defining $X^k,F^k$ 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:
%
$$\eqalign{
F^h(x,\V\psi,t)=&\sum_{j=0}^{2(h-1)}\fra{(\s t g)^j}{j!}
F^{h\s}_j(x,\V\psi),\qquad h=1,\ldots,k\cr
X^h(x,\V\psi,t)=&\sum_{j=0}^{2h-1}\fra{(\s t g)^j}{j!}
X^{h\s}_j(x,\V\psi),\qquad h=1,\ldots,k-1\cr}\Eq(4.1)$$
%
by setting $\pps=\oo t$, $\s=\sign t$, $x=e^{-g\s t}$, with
$F^{k\s}_j,X^{k\s}_j$ 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 $F^h_j,X^h_j$ 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:
%
$$\V F^{h\s}_{\su\V0}(\s\io)=\V0, \qquad {\rm for\ all}\ h\le
k\Eq(4.2)$$
%
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 $X^k$.
This does not imply the convergence of the series: however
in \S 8 we prove such a result, so justifying the
boundedness hypothesis and completing the research of bounded motions.
We note that, since $F^{k\s}\in\MM^{2(k-1)}$ and $\V
F^{k\s}_{\su\V0} (\s \io)=\V0$ hold, the function $\V X^{k\s}_\su(t)$,
given by the first of \equ(2.12), is in fact in $\MM^{2(k-1)}$
(by integration). But of course we do not know (yet)
the initial data $X^{k\s}(0)$.
To find expressions for $X^k_\su$ 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(3.5), so that:
%
$$\V X^{k\s}_\su(t)=\V X^{k\s}_{\su}(T)+\II \V F^{k\s}_{\su}(t)-\II
\V F^{k\s}_{\su}(T)\Eq(4.3)$$
%
where $\s={\rm\,sign\,}t,$ and $T$ has the same sign of $t$.
The function $\V X_\su^{k\s}(T)$ 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(4.1), at $x=0$: $\V X^{k\s}_{0\su}(0,\oo T)$,
(in the sense that the difference tends to $0$, bounded proportionally
to $(g|T|)^{2k-1}e^{-g|T|}$). The function $\II \V F^{k\s}_\su(T)$ also
becomes asymptotically quasi periodic with exponential speed {\it and
$\V0$ average}, because $\V F^{k\s}_\su\in \MM_0^{2(k-1)}$ and by the
definition of $\II$: therefore the two quasi periodic functions of $T$
must cancel modulo a constant equal to $\media{\V
X^{k\s}_{0\su}(0,\cdot)}\=\V X_{\su\V0}^{k\s}(\s\io)$.
Hence it follows that:
%
$$\V X^{k\s}_\su (t)=\V X_{\su \V 0}^{k\s} (\s\io)+ \II \V F^{k\s}_\su (t)
\Eq(4.4)$$
%
and, by inserting \equ(4.4) into the second of \equ(2.12), (considering
also that $\ig_0^t\t\V F^{k\s}_\su(\t)\,d\t= t\II\V F^{k\s}_\su(t)+$ a
$t$-bounded function), we see that the $\V X_\giu^{k\s}(t)$ can be
bounded only if:
%
$$\V X_{\su\V 0}^{k\s}(\s\io)=\V 0,\kern 1.truecm\hbox{hence:}
\kern1.truecm \V X_\su^{k\s}(t)=\II \V F_\su^{k\s}(t)\Eq(4.5)$$
%
yielding, setting $t=0^\s$, the initial values of $X_\su^k$ {\it and}
a simple form for its time evolution. Analogously, recalling that $\V
X_\giu^{k\s}(0)=\V 0$, essentially by definition, one finds:
%
$$
\V X_\giu^{k\s}(t)= J^{-1}\big( \II^2 \V F_\su^{k\s}(t)-
\II^2\V F_\su^{k\s}(0^\s)\big)\=J^{-1}\bar\II^2 \V F_\su^{k\s}(t)
\Eq(4.6)$$
%
which gives a simple form to the time evolution of the $\aa$ (\ie $\giu$)
component of $X^k$ in terms of the operator $\lis\II^2$ defined by the
r.h.s. of \equ(4.6).
Likewise considering the \equ(2.11) and the behaviour at $\s \io$ of
$W$ in \equ(2.9), if $X^{k\s}(t)$ has to be
bounded at $\s\io$, we see from the second of \equ(2.11)
that:
%
$$X_+^{k\s}(0)=-\igb_0^{\s\io} w_{00}(\t) F_+^{k\s}(\t)\ d\t
\Eq(4.7)$$
%
Thus we get (defining at the same time also $\OO$ and $\OO_+$):
%
$$\eqalign{
& X^{k\s}_+(t)=w_{ll}(t)\igb_{(\s)}^t
w_{00}(\t)F^{k\s}_+(\t)d\t-w_{l0}(t)\ig^t_0w_{0l}(\t)
F^{k\s}_+(\t)d\t\=\OO_+ F^{k\s}_+(t)\cr
&
X^{k\s}_-(t)= w_{0l}(t)\igb_{(\s)}^t w_{00}(\t)
F^{k\s}_+(\t)d\t-w_{00}(t)\ig^t_0w_{0l}(\t)
F^{k\s}_+(\t)d\t\=\OO F_+^{k\s}(t)\cr}
\Eq(4.8)$$
%
The \equ(4.5),\ equ(4.6), \equ(4.8) and the boundedness request
imply \equ(4.1) for $h=k+1$, as we can show by reasoning as in [G1].
As already remarked before \equ(4.3) we note again that, since
$F^{h\s}_{\su\V0}(\s\io)=\V0$ for $h\le k$, the $\V X_\su^k,\V X^k_\giu$
functions are in fact in $\MM^{2(k-1)}$, as the $\II$ operation, on
such $\V F^k_\su$ 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(4.1) is proved for $X^k$, and it remains to check it for $F^{k+1}$.
This follows from the expression of $F^{k+1}$, see \equ(2.7), in terms
of the $X^h$ with $h\le k$: 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 $k$ and verify the
inductive hypothesis); and, furthermore, the coefficients
$(\dpr_\aa f)_{\V m}(\f_0,\oo t)$ or $g^2 J_0\sin\f_0$ or
$g^2 J_0\cos\f_0$ do not contain any terms that can possibly increase
the degree. Hence $ F^{k+1}\in \MM^{2k} $. To see that
$\V F^{(k+1)\s}_\su\in \MM^{2k}_0$, \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:
%
$$\tst
\V F_{\su\V 0}^{k\s}(\s\io)\=\ii_{{\bf T}^{l-1}}
\V F^{k\s}_\su(\V\ps,\s\io){d\V\ps\over (2\p)^{l-1}}\=
\langle \V F_\su^{k\s}(\cdot,\s \io)\rangle=\V 0\Eq(4.9)$$
%
for all $k\ge1$, and, still for all $k\ge1$, as:
%
$$\eqalign{\tst
X^h_-(t)=&w_{0l}(t)\II(w_{00}F^h_+)(t)-w_{00}(t)\big(\II(w_{0l}F^h_+)(t
)-\II(w_{0l}F^h_+)(0^\s)\Big)\=\OO(F^h_+)(t)\cr
\V X^h_\giu(t)=&J^{-1}\,\Big(\II^2(\V F^h_\su)(t)-\II^2(\V
F^h_\su)(0^\s)\Big)\=J^{-1}\lis\II^2(\V F^h_\su(t))\cr
X^h_+(t)=&w_{ll}(t)\II(w_{00}F^h_+)(t)-w_{l0}(t)\Big(\II(w_{0l}F^h_+)(t
)-\II(w_{0l}F^h_+)(0^\s)\Big)\=\OO_+(F^h_+)(t)\cr
\V X^h_\su(t)=&\II(\V F^h_\su)(t)\cr} \Eq(4.10) $$
%
where $\OO,\OO_+,\lis\II^2,\II$ are defined here and in \S3; 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}
($j=0,\ldots,l-1$) components and over the {\it action}
($j=l,\ldots,2l-1$) the $F^h$ has only the action components non zero.
{\it Furthermore the above functions describe a motion on
the whisker $W^\s$ with initial data at some $\aa$ and $\f=\p$.}
We can give the above discussion a more formal statement through
the following propositions:
\*
\\{\bf Proposition 4.1}: {\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
scattering and homoclinic phase shifts, see [CG], [G1].
\*
\\{\bf Proposition 4.2}: {\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 4.1. And in fact the
convergence is exponential in the sense that for $\s t\ge 0$:
%
$$ \left| X^\s (x,\V\psi+\oo t,t)-X^\s(0,\V\psi,\s\io)
\right| \le C e^{- \fra12 g\s t} \Eq(4.11) $$
%
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 8.
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 the analysis in this paper should also allow us
to find the expansion of $g(\m)$ in a convergent power series in $\m$:
however we do not discuss this further.
\vskip1.truecm
\\{\bf 5. Tree formalism: part I}
\vskip.5truecm\pgn=1\numfig=1\numsec=5\numfor=1
\\In this section we review the graphical formalism
developed in [G1], \S 5, in order to represent, via
equations \equ(4.10) and \equ(2.7), the generic $h$-th
order contribution to the homoclinic splitting.
We introduce a label $\n$ to split the functions appearing
in \equ(2.7) as sums of their Fourier
components; let:
%
$$\eqalign{ f^{\d}(\f,\aa)\=& \sum_{\n=(n,\nn)} \fra{f^\d_\n}2 \,
e^{i(n\f+\nn\cdot\aa)} ,\qquad \d=0,1\cr
f^0(\f,\aa)\=&J_0
g^2\cos\f=\sum_{\n,\,\nn=\V0\atop n=\pm1} \fra{f^0_\n}2 \,e^{i n\f}
,\qquad f^1(\f,\aa)\=f(\f,\aa)=\sum_\n\fra{f_\n^1}2\, e^{i
(n\f+\nn\cdot\aa)}\cr}\Eq(5.1)$$
%
(the introduction of 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 will 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}{$j_\l$}
%\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}{$j_{\l_1}$}
\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.5.1: 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 $j_{\l}=j$.}
\*
\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{${}^4$}{\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:
%
$$ { 1\over 2}(-i\n_v)_{j_v-l}f^{\d_v}_{\n_v} \; e^{i(n_v\f^0(\t_v)+
(\aa+\oo\t_v)\cdot\nn_v)}\prod_{s=0}^{l-1}(i\n_{vs})^{m_s}\Eq(5.2)$$
%
(the last product is missing if no nodes follow $v$)
which is univoquely determined by the sets of labels attached to $v$,
and to each branch $\l$ we associate an improper integration operation
with upper limit
$t$, denoted $\OO,$ $J^{-1}\lis\II^2$, $\OO_+$, $\II$ in \equ(4.10),
and the branch label will be $j_{\l}=0$
when representing $\OO$, $j_\l=1,\ldots,l-1$ for $J^{-1}\lis
\II^2$, $j_\l =l$ 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(5.2).
\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 $\l_v$:
%
%$$\tst
%\EE_{v}^T \Big(_v \cdot \Big)_v \= \cases{\OO\Big(_v \cdot\Big)_v
%(\t_{v'}),\quad &if $v>v_0\ ,\quad j_{\l_v}=0$; \cr J^{-1} \lis \II^2
%\Big(_v \cdot \Big)_v (\t_{v'})\ ,\quad &if $v>v_0\ ,\quad 1\le
%j_{\l_v}\le l-1$;\cr}$$
%%
%for $v>v_0$, otherwise:
%%
$$\tst\EE_{v}^T \Big(_v \cdot \ \Big)_v
\= \cases{\OO\Big(_v \cdot\Big)_v (\t_{v'})\ , & if $v \ge v_0\ ,
\quad j_{\l_v}= 0$\ ,\cr
J^{-1} \lis\II^2
\Big(_v \cdot \ \Big)_v (\t_{v'}) &if $v \ge v_0\ ,\quad 1\le j_{\l_v}
\le l-1$\ ,\cr \OO_+\Big(_v \cdot\ \Big)_v (\t_{v'})\ ,\quad &if $v=v_0\
,\quad j_{\l_{v}}=l$; \cr \II \Big(_v \cdot\ \Big)_v (\t_{v'})\ ,\quad
&if $v=v_0\ ,\quad l+1\le j_{\l_v}\le 2l-1$\cr} \Eq(5.4)$$
%
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(5.4), if $v=v_0$, has to be interpreted as the limit
as $t\to0^\s$.
Then it follows that $X_j^h(t)$ can be written as:
%
$$X^h_j(t)=\sum_{\th \in trees}\fra1{m(\th)!}\sum_{labels;\,
\sum_v\d_v=h} \, \tilde V_j(t;\th) \Eq(5.5)$$
%
where $m(\th)=$ number of branches of $\th=$ number of
nodes of $\th$.
\*
\\{\it Remark} : If we do not perform the operation $\EE^T$ 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 $F^h_{j_{v_0}}(t)$.
In particular, from \equ(4.10), we deduce that the whiskers
splitting $\D_j^h(\aa)=X_j^{h+}(0;\aa)-X_j^{h-}(0;\aa)$ is given by:
%
$$ \D_-^h(\aa) \= 0\ , \quad\;
\V \D_\giu^h (\aa) \= \V 0\ , \quad\;
\D_+^h(\aa)= -\igb_{-\io}^{+\io} d\t \; w_{00}(\t) F_+^{h \s}(\t)\ ,
\quad\; \V \D_\su^h(\aa)= -\igb_{-\io}^{+\io} d\t \;
\V F_\su^{h \s}(\t)\ \; \Eq(5.6) $$
%
where $F_j^{h\s}$ is defined as above prescribed. Note that if
$\aa=\V0$ then we are at a homoclinic point, because the hamiltonian
\equ(1.1) is even: so that \equ(5.6) is identically vanishing also
for the components $j=l,\ldots 2l-1$.
\vskip1.truecm
\\{\bf 6. Theory of the homoclinic splitting: results}
\vskip.5truecm\pgn=1\numfig=1\numsec=6\numfor=1
\\As a consequence of the above analysis and the analysis in [G1],
we get that, in general, the angles of homoclinic splitting,
(or $\d(\a)$, introduced in \S 1), are smaller
than any power in $\h$. Let us denote $\D_\nn^h$ 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 splitting $(\m,\aa)\to$
$\D(\aa)\=X^+_\m(0;\aa)-X^-_\m(0;\aa)$; then the property of smallness
is an immediate consequence of the following bounds.
Let $d\in(0,\fra\p2)$, and let:
%
$$\e_h\=\e_h(d)\=\sup_{0<|\nn_0|\le Nh}
e^{-|\oo\cdot\nn_0|g^{-1}(\fra\p2-d)},\qquad\b=4(N_0+1),\qquad
p=4\t\Eq(6.1)$$
%
where $N_0$ is the maximal $\f$--harmonic of the perturbation $f$ in
\equ(1.1). Note that, if $l=2$, it is $\e_h\=\e_1$.
Then, for $j \ge l$ and for all $J\in [J_0,+\io)$ and $h\ge1$:
%
$$|\D^h_{j\nn}|\le g J_0 D B^{h-1} \quad , \qquad\qquad|\D^h_{j\nn}|
\le g J_0 D d^{-\b}(B d^{-\b })^{h-1} (h-1)!^{p+2} \e_h\Eq(\seidue)$$
%
where $D$ and $B$ are suitable dimensionless constants depending
on the various parameters describing \equ(1.1), {\it but not on the
perturbation parameters $\h,\m$}. Note also that since we always
suppose that $f$ is a trigonometric polynomial of degree $N$, it is
actually $\D_{j\nn}^h=0$ if $|\nn|>Nh$. Both bounds in (\seidue) are
uniform in $J\ge J_0$ and one can take $J\to+\io$. The second
equation in (\seidue) has been proven in [G1], \S 8 and Appendix A1,
by using some cancellation mechanisms operating to all orders in the
perturbative series of the homoclinic splitting. To the first one
the following section is devoted, as it represents the original
result with respect to [CG], [G1].
In this section we confine ourselves to show that, by reasoning as
in [G1], the bounds (\seidue) imply that the splitting
is smaller than any power, so justifying the expression ``quasi flat
homoclinic interesections''.
By (\seidue), the splitting can be bounded, for any multiindex $\V a$,
by:
%
$$|\dpr_\aa^{\V a} \V\D_\su(\V 0)|\le g J_0 D \sum_{h=1}^\io
\sum_{0<|\nn|\le Nh} |\m|^h |\nn|^{|\V a|}\min\{B^{h-1},
B_h \e_h(d)\}\Eq(6.3)$$
%
having denoted $B_h=B^{h-1}d^{-\b h}(h-1)!^{p+2}$. Note that, if $N$
is the trigonometric degree of the polynomial $f$ in \equ(1.1), the
sums over $\nn$ can be suppressed by multiplying the $h$-th term by
the mode counting factor $\bar C^h \= (2N+1)^{h(l-1)+h|\V a|}$
(where $\bar C$ is the maximum number of non zero Fourier components
times the maximum of $|\nn|^{\V a}$).
>From this bound it follows that $|\dpr_\aa^{\V a}\V\D_\su|$ is
smaller than any power in $\h$ (see \equ(1.2)).
In fact we can split the sum over $h$ in \equ(6.3) into
a finite sum, $\sum_{1\le h\le h_0}(\cdot)$ and a ``remainder",
$\sum_{h> h_0}(\cdot)$; then, if $\h$ is small enough, and $\h, Q$
in \equ(1.2) are such that $b \h^Q = \m_0$, $\m_0^{-1} > B \bar C$,
and $|\m|<\m_0/2$, we find
%
$$ \sum_{h>h_0}(\cdot) \le { g J_0 D \over B }
\sum_{h> h_0} (|\m| B \bar C)^h
\le { 2 g J_0 D \over B }
\Big( {|\m|\over\m_0} \Big)^{h_0} \Eq(6.4)$$
%
and:
%
$$
\sum_{h=1}^{h_0} (\cdot)\le g J_0 D \, h_0|\m|\,
\bar C^{h_0} d^{-\b h_0}
B^{h_0-1}(h_0-1)!^{p+2} \e_{h_0}(d)\Eq(6.5)$$
%
Thus if $\m=\h^{Q+s}$, $d=\sqrt\h$, and $s\ge1$ we see that fixing
$h_0=r/s$, for any $r>1$, the $|\dpr_\aa^{\V a} \V\D_\su|$ is bounded
by a ($r$-dependent) constant times $\h^r$ (as in such a case
\equ(6.5) is just a remainder, exponentially small in $\h^{-1/2}$).
\vskip1.truecm
\\{\bf 7. Tree formalism: part II}
\vskip.5truecm\pgn=1\numfig=1\numsec=7\numfor=1
\\We introduce
the dimensionless quantities related to the homoclinic splitting by:
%
$$ \D^h_{l\nn} = J_0 g\,\bar \D^h_{l\nn}\ ,\qquad \qquad
\D^h_{j\nn}= J g\,\bar \D^h_{j\nn}\ ,\quad (l+1 \le j \le 2l-1)
\Eq(7.1) $$
%
and denote $\X^{h\s}_j (t)$, $\s=\pm$, $0 \le j < 2l$, the dimensionless
quantities corresponding to the perturbed motions $ X^{h\s}_j (t)$:
obviously $\bar\D^h_{j\nn}=\X^{h+}_{j\nn}(0)-\X^{h-}_{j\nn}(0)$, $j\ge l$.
Given a tree $\th$, with $m(\th)=m$, we can write its contribution
to $\X_{j\nn}^{h\s_{v_0}}(t)$, $j \ge l$, as:
%
$$ \eqalign{
{1\over m!}\;\tilde V_j(t;\th)= & {1\over m!} \prod_{v_0 \le v \in \th}
\oint\fra{dR_v}{2\p iR_v}\sum_{\r_v=0,1}\ig_{\s_{v'}\io}^{\r_v g\t_{v'}}
d\,g\t_v \, e^{-\s_{v}g R_{v}\t_v} \; w^{\r_v}_{j_{v}}(\t_{v'},\t_v) \cr
& \cdot \Big[{(-i\n_v)_{j_v-l}\over2}\;c_{\n_v}\;e^{i(n_v\f^0(\t_v)+\nn_v
\cdot\oo\t_v)}\prod_{s=0}^{l-1}(i\n_{vs})^{m^s_v}\Big]\cr} \Eq(7.2) $$
%
where $\t_{v_0}'=t$, $j_{v_0}=j$,
and we have defined the dimensionless coefficients $c_{\n_v}$ as:
%
$$ c_{\n_v} \= [ (J_0 g^2)^{-1} \d_{j_v,l} +(J g^2)^{-1}
\big( 1 - \d_{j_v,l} \big) \d_v ] f_{\n_v}^{\d_v} \; , $$
%
where $\d_{j_v,l}$ is $1$ if $j_v=l$, and $0$ otherwise
(\ie $j_v=l+1,\ldots,2l-1$), and used \equ(4.10), by setting:
%
$$ \eqalign{
w^0_{j_v}(\t_{v'},\t_v) & = \cases{
w_{00}(\t_{v'}) \bar w_{0l}(\t_v) , & $v>v_0\ , j_v=l$ \cr
g\t_v , & $v>v_0\ , j_v>l$ \cr} \cr
%
w^0_{j_{v_0}}(t,\t_{v_0}) & = \cases{
\bar w_{l0}(t) \bar w_{0l}(\t_{v_0}) , & $j_v=l$ \cr
0 , & $j_v>l$ \cr} \cr
%
w^1_{j_v}(\t_{v'},\t_v) & = \cases{
\bar w_{0l}(\t_{v'}) w_{00}(\t_v) - w_{00}(\t_{v'}) \bar w_{0l}(\t_v),
& $v>v_0\ , j_v=l$\cr g(\t_{v'}-\t_v), & $v>v_0\ , j_v>l$\cr} \cr
%
w^1_{j_{v_0}}(t,\t_{v_0}) & = \cases{
w_{ll}(t)w_{00}(\t_{v_0}) - \bar w_{l0}(t)\bar w_{0l}(\t_{v_0}),
& $j_{v_0}=l$\cr
1 , & $j_{v_0}>l$\cr}
\cr} \Eq(7.5) $$
%
with the dimensionless matrix elements $\bar w_{0l}$,
$\bar w_{l0}$ given, respectively, by
$ \bar w_{0l} =(J_0g)^{-1} w_{0l} = \bar w/4$,
$ \bar w_{l0} =-(Jg)^{-1} w_{l0}$, and $m$ is the total number of branches
(root branch included) and the integers $m_v^s$ decompose $m_v$ and count
the number of branches emerging from $v$ and carrying the labels
$s=0,\ldots,l-1$. If $j v_0$.
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:
%
$$ \eqalign{
w^0_{j_v}(\t_{v'},\t_v) = & \fra12 \left\{ { g \t_v \over
\cosh g\t_{v'} \; \cosh g\t_v } + { \sinh g\t_v \over
\cosh g\t_{v'} } \right\} \cr
%
w^1_{j_v}(\t_{v'},\t_v) = & \fra12 \left\{ { g (\t_{v'}-\t_v) \over
\cosh g\t_{v'} \; \cosh g\t_v } + { \sinh g\t_{v'} \over
\cosh g\t_v } - { \sinh g\t_v \over
\cosh g\t_{v'} } \right\} \cr} \Eq(7.6) $$
%
Then we can write:
%
$$ \eqalign{
w^0_{j_v}(\t_{v'},\t_v) \, e^{i n_v \f^0(\t_v)} & = \cases{
g \t_v \, y_v^{(0)} (\t_{v'},\t_v) + y_v^{(-1)} (\t_{v'},\t_v) \; ,
& if $j_v=l$ \cr
g \t_v \, y_v^{(2)} (\t_v) \; ,
& if $j_v>l$ \cr} \cr
%
w^1_{j_v}(\t_{v'},\t_v) \, e^{i n_v \f^0(\t_v)} & = \cases{
g (\t_{v'}-\t_v) \, y_v^{(0)} (\t_{v'},\t_v) +
y_v^{(1)} (\t_{v'},\t_v) - y_v^{(-1)} (\t_{v'},\t_v) \; ,
& if $j_v=l$ \cr
g (\t_{v'}-\t_v) \, y_v^{(2)} (\t_v) \; ,
& if $j_v>l$ \cr} \cr} \Eq(7.7) $$
%
where the functions $y_v^{(\a)}$, $\a=-1,0,1, 2$, are elements of a
finite set of functions:
%
$$ \eqalign{&
y_v^{(-1)} (\t_{v'},\t_v) = \fra12 {\sinh g\t_v\over\cosh g\t_{v'}}
\;e^{i n_v \f^0(\t_v)} \hskip3.2truecm
%
y_v^{(1)} (\t_{v'},\t_v) = \fra12 { \sinh g\t_{v'} \over
\cosh g \t_v} \; e^{i n_v \f^0(\t_v)} \cr
%
& y_v^{(0)} (\t_{v'},\t_v) = \fra12 { 1\over\cosh g\t_v\cosh g \t_{v'}}
\; e^{i n_v \f^0(\t_v)} \hskip2.truecm
%
y_v^{(2)}(\t_{v'},\t_v) = e^{i n_v \f^0(\t_v)} \cr } \Eq(7.10) $$
%
and admit the following Laurent expansion:
%
$$ \eqalign{
y_v^{(-1)} (\t_{v'},\t_v) & =
\sum_{k_v'=1}^\io \sum_{k_v=-1}^\io
y_v^{(-1)} (k_v',k_v) x_{v'}^{k_v'} x_v^{k_v} \hskip1.2truecm
%
y_v^{(1)} (\t_{v'},\t_v) =
\sum_{k_v'=-1}^\io \sum_{k_v=1}^\io
y_v^{(1)} (k_v',k_v) x_{v'}^{k_v'} x_v^{k_v} \cr
%
y_v^{(0)} (\t_{v'},\t_v) & =
\sum_{k_v'=1}^\io \sum_{k_v=1}^\io
y_v^{(0)} (k_v',k_v) x_{v'}^{k_v'} x_v^{k_v}
\hskip1.5truecm
%
y_v^{(2)} (\t_v) =
\sum_{k_v=0}^\io
y_v^{(2)} (0,k_v) x_v^{k_v} \cr} \Eq(7.9) $$
%
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: $\L_\a=\{ v \in \th \,
: \a_v=\a \}$.
Then, for each tree node, we have four more labels, $k_v,k_v',\r_v,\a_v$,
to add to the previous ones $\t_v, \n_v, \d_v, j_v$,
and, in the end, we have to sum over all the possible consistent
collections of such labels, (note that the just introduced labels are not
quite independent on each other: {\it e.g.} $\a_v=1$ is possible
only if $\r_v=1$, and if an action label is $j_v>l$, then necessarily
it is $\a_v=2$). Therefore the tree value $\tilde V_j(t;\th)$ introduced
in \S 5 can be replaced with a new tree value, $V_j(t;\th)$, taking into
account also the new labels, and \equ(5.5) holds still provided
$\tilde V_j(t;\th)$ is replaced with $V_j(t;\th)$. The generic contribution
$(1/m!)\;V_j(t;\th)$ to \equ(7.2), corresponding to a given tree $\th$,
with $m(\th)=m$, is:
%
$$ {1\over m!} \; V_j(t;\th) = {1\over m!}
\prod_{v_0 \le v \in \th} \oint\fra{d R_v}{2\p i R_v}
\ig_{\s_{v'} \io}^{\r_v g \t_{v'}} d\,g\t_v
\; \VV_v(\th) \Eq(7.11) $$
%
where we have defined the {\it node function} $\VV_v(\th)$, (depending
on the tree which the node $v$ belongs to), as:
%
$$ \VV_v(\th) \=
F_{\n_v} \; T_v ( g\t_{v'}, g\t_v ) \;
e^{-\s_v R_{v} g \t_v}
\; e^{i \o_v \t_v} \; x_v^{k_v}
\prod_{j=1}^{m_v} x_v^{k_{v_j}'} \; , \Eq(7.11a) $$
%
$\o_v = \oo \cdot \nn_v$, $m_v$ 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
is missing if $v$ is a top node), and $T_v ( g\t_{v'}, g\t_v )$
is defined as:
%
$$ T_v ( g\t_{v'}, g\t_v ) = \left( \d_{\a_v,2}+\d_{\a_v,0} \right)
\left[ (1-\r_v) g\t_v + \r_v g(\t_{v'}-\t_v) \right] +
\left( \d_{\a_v,-1}+\d_{\a_v,1} \right) \Eq(7.a) $$
%
(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 $\r_v=1$). We have set:
%
$$ F_{\n_v} = { (-i\n_v)_{j_v-l} \over 2 } \; c_{\n_v} \;
\Big[ \prod_{s=0}^{l-1} (i\n_{vs})^{m^s_v} \Big]
\, (-1)^{\d_{\a_v,-1} \d_{\r_v,1} } \; y_v^{(\a_v)} (k_v', k_v) \=
\Phi_{\n_v} \; (-1)^{\d_{\a_v,-1} \d_{\r_v,1} } \; y_v^{(\a_v)} (k_v', k_v)
\Eq(7.12) $$
%
where the coefficients $\Phi_{\n_v}$ satisfy the following bound:
%
$$ \Big| \prod_{v \ge v_0} \Phi_{\n_v} \Big| \le
\Big( {N \over 2 } F_0 N \Big)^m \= {\CC}^m \Eq(A1.2)$$
%
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:
$$ \left| \prod_{v \ge v_0} y_v^{(\a_v)} (k_v', k_v) \right|
\le M^{2m} \prod_{v \ge v_0} \l^{k_v+k_v'} \Eq(7.12a) $$
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 $y_v^{(v)}$'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 equal to a constant $M_3=M_2 \l^{-1}=1$, so that we can
set $M=\max \{ M_1,M_2 \}$.\footnote{${}^5$}{\nota
The request that {\it all} the $x$ satisfy the property $|x|<\l$ is not
so strong: in the cases 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 Lemma 8.3 below).}
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.
Now we give three definitions about trees which perhaps do not deserve really
a their own name, since they do not correspond to any object admitting a
natural interpretation, (expecially the second and third ones), but they
will appear in the following discussion, and therefore it will be
useful to have a name to label them.
\*
\\{\bf Definition 7.1} : {\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 we sum their values:
the so obtained quantity $\bar L_{j_v}^{h_v\s_v}(\t_{v'})$
will be associated to 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 can associate
to a reduced tree $\bar\th$ a value $V_j(t;\bar\th)$, where, corresponding
to each free node $v$, there is a factor $\VV_v(\bar\th)\=\VV_v(\th)$ as in
\equ(7.11a), and, corresponding to each leaf $v$, there is factor
$\bar L_{j_v}^{h_v\s_v}(\t_{v'})$.}
\*
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 contribution to
$\X_{j_{\l_v}\nn(v)}^{h_v\s_v}(\t_{v'})$, $j_{\l_v}=j_v-l$,
($\nn(v)$ is the momentum of the node $v$, as it is defined in \S 5),
whose dependence on $\t_{v'}$ reveals itself only through the factor,
(see the third line in \equ(7.5)):
%
$$\x_v(\t_{v'})=[w_{00}(\t_{v'})\d_{j_v,l}+(1-\d_{j_v,l})]\Eq(7.13a)\;,$$
%
so that we can write $\bar L_{j_v}^{h_v\s_v}(\t_{v'})=\x_v(\t_{v'}) \;
\bar L_{j_v}^{h_v\s_v}(0)$. We define $\bar L_{j_v}^{h_v\s_v}(0)$ as the
{\it value of the leaf} $v$ of the reduced tree. Also the factor
\equ(7.13a) admits a series expansion like the functions $y_v^{(\a_v)}$'s
in \equ(7.9):
%
$$\x_v(\t_{v'})=\sum_{k_v'=1}^{\io}\x_v(k_v',0)x_{v'}^{k_v'}\Eq(7.13b)$$
%
We can use explicitly the order of the integration variables,
so defining:
%
$$ \o(v) = \sum_{\bar \th_f \ni w \ge v } \o_w \; , \qquad
k(v) = \sum_{ \bar \th_f \ni w \ge v} k_w \; , \qquad
k'(v) = \sum_{\bar \th \ni w > v} k_w' \; , \qquad
p(v) = k(v)+k'(v) $$
%
and writing:
%
$$ \eqalign{
\prod_{\bar \th_f \ni v \ge v_0} e^{-k_v g \s \t_v } & =
e^{-k(v_0) g \s \t_{v_0} } \; \cdot \;
\prod_{\bar \th_f \ni v > v_0} e^{-k(v) g \s ( \t_v - \t_{v'} ) } \cr
\prod_{\bar \th \ni v \ge v_0} e^{-k_v' g \s \t_v } & =
e^{-[ k'(v_0) + k_{v_0}'] g \s \t_{v_0} } \; \cdot \;
\prod_{\bar \th \ni v > v_0} e^{-k'(v) g \s ( \t_v - \t_{v'} ) } \cr
\prod_{\bar \th_f \ni v \ge v_0} e^{-R_v g \s \t_v } & =
e^{- \sum_{w \ge v_0} R_w g \s \t_{v_0} } \; \cdot \;
\prod_{\bar \th_f \ni v>v_0}e^{-\sum_{w \ge v}R_w g\s(\t_v-\t_{v'})}\cr
\prod_{\bar \th_f \ni v \ge v_0} e^{i \o_v \t_v } & =
e^{i \o(v_0) \t_{v_0} } \; \cdot \;
\prod_{\bar \th_f \ni v>v_0} e^{i\o(v)(\t_v-\t_{v'})}\cr} \Eq(7.13) $$
%
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 $\bar L_{j_v}^{h_v\s_v}(0)$, which is independent on
$\t_{v'}$, times a factor \equ(7.13a), which one has to take into account in
the computation of $p(\tilde v)$, for each $\tilde v < v$. Note that
only the free nodes contribute to $k(v)$ and $\o(v)$; we can write
$\o(v)=\oo\cdot\nn_0(v)$, where $\nn_0(v)$ is the ``free
momentum" of the reduced tree. Note also that
the leaves with $j_v=l$ are such that, in \equ(7.13),
$k_v' \ge 1$, see \equ(7.13b), \equ(7.9), 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.
\*
\\{\bf Definition 7.2} : {\it Given a tree $\th$, we set $\LL_{-1}\=
\{ v \in \th : v \in \L_{-1}, \hbox{ and } p(v)=0 \}$. We define the
{\rm generalized reduced tree} $\bar \th^G$ in the following way.
Let us draw a bubble encircling each node $v>v_0$, $v \notin \LL_{-1}$,
with $\r_v=0$, and the entire subtree emerging from it, and let us delete
all the so obtained bubbles, except 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, we consider all the possible trees with the same
labels attached to the node $v$, (in particular with the same $h_v$), and
we sum their values: the so obtained quantity $L_{j_v}^{h_v\s_v}(\t_{v'})$
will be associated to a fat point, replacing the original bubble, which
will be called a {\rm leaf} (of the generalized reduced tree). We still
call {\it leaves} the fat points, and {\rm free nodes} the
generalized 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 generalized 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 to the fat point representing it.
We can associate to a generalized reduced tree $\bar\th^G$ a value
$V_j(t;\bar\th^G)$, where, corresponding to each free node $v$, there is a
factor $\VV_v(\bar\th^G)\=\VV_v(\th)$ as in \equ(7.11a), and, corresponding
to each leaf $v$, there is factor $L_{j_v}^{h_v\s_v}(\t_{v'})$.}
\*
\\{\it Remark 1} : The Definition 7.1 is only a preliminary definition
preluding to Definition 7.2, which is more involved, but a useful one.
The generalized reduced trees are different from the reduced trees as to the
resummation procedure of the leaves, (for instance, a tree contributing to
a generalized reduced tree with $\r_v=0$, for one $v \in \LL_{-1}$, can be
counted also among the trees contributing to the reduced tree in which $v$
is a leaf). So the leaves of the reduced trees are different
from the leaves of the generalized reduced trees, (that's why we
have used different symbols to label their values). The more natural
notion is the first one, since it allows us to order the time variables;
but this is not sufficient to prove our result, and so the
introduction of the generalized reduced trees is necessary to become aware
of some cancellation mechanisms which can be implemented only by considering
together the nodes $v \in \th$ in $\LL_{-1}$, with $\r_v=0,1$. This will
be explicitly exploited in the proof of Lemma 8.2.
\\{\it Remark 2} : The reduced degree is so defined that the degree of a
generalized 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 3} : Note that, unlike what happened in \S 5, now only
to the free nodes an integration time variable is associated. This could
be found a little misleading as to the notion of node, with respect
with the usual terminology, (see [G1], [G2], [GG]); 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 of the proof of Lemma 8.1 below).
\*
We remark also that it is still possible write
%
$$ L_{j_v}^{h_v\s_v}(\t_{v'})=\x_v(\t_{v'})\;L_{j_v}^{h_v\s_v}(0)\;,
\Eq(7.14a) $$
%
being $\x_v(\t_{v'})$ defined in \equ(7.13a). Again we call
$L_{j_v}^{h_v\s_v}(0)$ the {\it value of the leaf} $v$ of the
generalized reduced tree.
Eventually we define the {\it free momentum} of the generalized reduced
tree with first node $v_0$ as $\nn_0(v_0)=\sum_{\bar\th_f^G\ni w \ge v_0}
\nn_w$. Note that, if $(1/m!)V_j(t;\bar\th^G)$ is a contribution
to $\X_{j\nn}^{h\s_{v_0}}(t)$, $\nn\,\=\,\nn(v_0)$,
then it is $\nn_0(v_0)\neq\nn$, since
$\nn_0(v_0)$ takes into account only the free nodes of $\bar\th^G$,
while $\nn$ depends also on the momentum labels affixed to the leaves.
\*
\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.7.1: A generalized reduced tree $\bar\th^G$ 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
\*
As done in the case of the reduced trees, we can define also for a
generalized reduced tree $\bar \th^G$ the sets
$\bar \th_f^G \= \{ v \in \bar \th^G : v \hbox{ is a free node } \}$ and
$\bar \th_L^G \= \{ v \in \bar \th^G : v \hbox{ is a leaf } \}$,
verifying the properties $\bar \th^G = \bar \th_f^G \cup \bar \th_L^G$ and
$\bar \th_f^G \cap \bar \th_L^G = \emptyset$.
We note that the equalities \equ(7.13) cannot be used for generalized
reduced trees, since the time variables are no longer ordered.
Nevertheless it is still possible to exploit them partially.
In fact, let us consider a generalized reduced tree, and let us single
out the nodes $v$'s in $\LL_{-1}$: for each such node $v$ we introduce a
label $D(v)$, the {\it depth label}, counting the maximum number of nodes in
$\LL_{-1}$ we can meet moving forward along any path connecting $v$ to the
top nodes.
Let us start from the nodes $v^{(0)}$'s in $\LL_{-1}$ with $D(v^{(0)})=0$:
all the following free nodes $v$'s have $\r_v=1$, so that their time variables
are ordered, and we can use the relations \equ(7.13) from $v^{(0)}$ to the
top nodes following it. Then we sum the two contributions with $\r_{v^{(0)}}=0$
and $\r_{v^{(0)}}=1$, and we obtain a function of $\t_{{v^{(0)}}'}$.
(Note that the sum over such two contributions corresponds to perform
an integration from $0$ to $\t_{{v^{(0)}}'}$, instead of two
improper integrations, since the functions
which we integrate are equal up to the sign, see the $y_v^{(-1)}$ term
in \equ(7.6) and \equ(7.10)).
As second step, we consider the nodes $v^{(1)}$'s in $\LL_{-1}$ with
$D(v^{(1)})=1$: all the following nodes have $\r_v=1$, since the
nodes with depth zero have disappeared, (\ie we have integrated already
over them), and so the relations \equ(7.13) can be exploited again.
Then we sum over the two contributions $\r_{v^{(1)}}=0$ and $\r_{v^{(1)}}=1$,
and we obtain a function of $\t_{{v^{(1)}}'}$. And so on: we iterate the
procedure until the first node of the generalized reduced tree is reached.
The result of the whole procedure will be found inductively when
explaining the proof of Lemma 8.2.
\*
\\{\bf Definition 7.3} : {\it Given a generalized reduced tree $\bar\th^G$,
we define the {\rm stripped value} of the generalized reduced tree
$V^S_j(t;\bar \th^G)$ as the value we obtain by associating to each free
node a factor $\VV_v(\bar\th^G)\=\VV_v(\th)$
as in \equ(7.11a), but retaining for each leaf
only the factor $\x_v(\t_{v'})$ in \equ(7.14a). Note that the
discarded contribution of the leaf $v$ is nothing else but its value,
$L_{j_v}^{h_v\s_v}(0)$, as it is defined after \equ(7.14a).}
\*
\\{\it Remark} : The just given definition may appear too involved.
Perhaps it is so, but it turns out to be notationally useful,
as will become clear along the proof of Lemma 8.1, see in particular
(\setteb) below. In particular we note that the contribution of a leaf
$v\in\bar\th^G$ to a stripped value $V_j^S(t;\bar\th^G)$ 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.
\vskip1.truecm
\\{\bf 8. Analyticity of the homoclinic splitting}
\vskip.5truecm\pgn=1\numfig=1\numsec=8\numfor=1
\\It can be useful to elucidate the problems arising in
the treatment and to sketch the strategy followed in order to
solve them. If all the nodes $v$ had $p(v) \neq 0$, then all the
integrals would trivially factorize, (there would be no need to
distinguish between reduced trees and generalized reduced trees),
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 checking that each time a power $t^p$ appears, it comes together with
a factor $1/p!$, (and it is $p \le m$, since the case $p(v)=\o(v)=0$
is not possible when $T_v(g\t_{v'},g\t_v)\neq 1$, see below);
the second is treated in part by exploiting some new cancellations
related to the particular structure of the kernels \equ(7.6), which can
be very easily visualized in terms of the generalized reduced trees
introduced in Definition 7.2. Other cancellation mechanisms will
be used in Appendices A1 and A2, and are essentially taken from [G1].
To do explicitly what has been said, it will be necessary to single out
the cases in which such problems can really arise. Therefore, in order to
study the contributions to $\X_{j\nn}^{h\s}(t)$, it will turn out to be
useful to distinguish between several cases, according
to the value of the labels $p(v_0)$ and $k_{v_0}$. For each
considered case we obtain a lemma giving us a convergence result:
as a consequence of such lemmata, Theorem 8.1 below will follow.
The idea is the following. 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(7.5), $p(v)$ 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 -1 or to 1, (according to some rules
which will appear clearer along the below discussion). If $p(v)=0$,
then, as we shall see, $k_v$ can assume only the values either $k_v=0$
or $k_v=-1$. If $k_v=0$, 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, (see Lemma 8.2); while, if $k_v=-1$, the integrals over
the $\t_w$'s, $w\ge v$, can be inductively studied, by exploiting
also the previous result, (see Lemma 8.2).\footnote{${}^6$}{\nota
It is important to stress that a subtree with first node $v$ represents a
contribution to $\X^{h_v\s_v}_{j_{\l_v}\nn(v)}(\t_{v'})$, so that it is
possible to express $\X^{h_{v_0}\s_{v_0}}_{j_{\l_{v_0}}\nn(v_0)}(t)$ in
terms of analogous functions of lower order, with $j_{\l_v} < l$. This
allows us to look for an inductive proof about the structure of
a tree with $p(v_0)=0$, $k_{v_0}=-1$, since the case in which there is
no node $v>v_0$ with $p(v)=0$, $k_v=-1$, is easy, (if the assertion
about the case $p(v)=0$, $k_v=0$, is accepted).}
It remains to study
the cases $p(v) \neq 0$, but they follow quite easily, if we use the
two above results, by explicit calculations, (see Lemma 8.3). 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.
\*
\\{\bf Lemma 8.1}: {\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) If $0 < j \le l-1$, such a sum can be written as:
%
$$ e^{i\oo\cdot\nn_0(v_0)\r_{v_0}t} \prod_{\bar\th_f^G \ni v \ge v_0}
\Phi_{\n_v} G_v[\o(v)] \Eq(7.15) $$
%
where $\F_v$ is defined in \equ(7.12),
$0<|\nn_0(v_0)|\le m_0N$, $m_0$ being the number of free nodes in
$\bar\th^G$, and $G_v[\o(v)]$ is defined to be:
%
$$ G_v[\o(v)] = \cases{ [i g^{-1}\o(v)]^{-2} & if $j_v > l$
\cr [1 + g^{-2} \o^2(v)]^{-1} & if $j_v = l$ \cr} \Eq(7.16) $$
%
with $j_{v_0} >l$.
\acapo
2) If $j=0$, the sum over $\r_{v_0}=0,1$ gives:
%
$$ (-in_{v_0}) {gt \over \cosh gt } \, \Phi_{\n_{v_0}} \,
\Big(\prod_{\bar\th_f^G \ni v>v_0} \Phi_{\n_v} G_v[\o(v)]\Big)\,
\ig_0^1 ds e^{i s\oo\cdot\nn_0(v_0)t}
\Eq(7.17) $$
%
where $|\nn_0(v_0)| \le m_0N$, and the function $G[\o(v)]$ is defined in
\equ(7.16).
\acapo
3) The sum over all the generalized reduced trees with labels
$p(v_0)$ and $k_{v_0}$ fixed to be zero, of the expressions
\equ(7.15) or \equ (7.17), 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 $\bar\th^G$.}
\*
\\{\it Remark 1} : Note that the first two statements are easy consequences
of the definitions, while the third one is rather deep, being essentially
equal to the KAM theorem, as it appears from the proof, (see also [G2]
and [GG]).
\\{\it Remark 2}: We note in advance that, as will be shown along the
proof of the lemma, when contributions with $\a_v=1$ and $\a_v=-1$
are summed together, the corresponding nodes $v$ turn out to have, in the
respective cases, $p(v)=1$ and $p(v)=-1$, so that $p(v)\neq0$, \ie
$v\notin \LL_{-1}$. Therefore, since the cancellation implemented in
Lemma 8.2 below occurs between contributions with a different label
$\r_v$ affixed to a node $v \in \LL_{-1}$, no cancellations overlapping
can arise.
\*
\\{\bf Lemma 8.2}: {\it The contribution to
$\X^{h\s}_{j\nn}(t)$, $\nn\in{\bf Z}^{l-1}$, $\s=\pm1$,
$j=0$, arising from the sum of the stripped values of all the
generalized reduced trees of reduced degree $m_0$, with labels $p(v_0)=0$
and $k_{v_0}=-1$, can be written as:
%
$$ \sum_{r=1}^{m_0} Q^r_{v_0}(x) {(gt)^r\over r!}\;E(m_0-r) \ig \m_r(ds)
e^{i\oo(s)\cdot\nn_0(v_0)t} A_{v_0}(\oo\cdot\nn_0(v_0),r,s) \Eq(7.19) $$
%
where $|\nn_0(v_0)|\le m_0N $, $r$ is the number of nodes in $\LL_{-1}$,
$s=\{s_1,\ldots,s_r\}$, with $s_i\in[0,1]$, $i=1,\ldots,r$,
being ``interpolation parameters", and
$\m_r(ds)$ is a suitable normalized positive measure:
%
$$ \m_r(ds) = ds_1 ds_2 \ldots ds_{r-1} ds_r
\; [r \, s_1^{r-1}] \, [ (r-1) \, s_2^{r-2}] \ldots [s_{r-1}] \; , $$
%
and the nodes in $\LL_{-1}$ are totally ordered so that $w_i < w_j$
for any $i < j$, with $w_1=v_0$, $i=1,\ldots,r$.\footnote{${}^7$}{\nota
{\rm That is the nodes $w_1,\ldots, w_r$ belong to
a connected monotone path.}}
The function $\oo(s)\cdot\nn_0(v_0) \= \oo(v_0,s)$ is defined in the
following way. Let us call $\th(w_i)$ the
(generalized reduced) tree with first node $w_i$, and
$\th(w_i)\setminus\th(w_{i+1})$ the tree obtained from $\th(w_i)$
by deleting the entire subtree emerging from $w_{i+1}$ (recall that
$w_{i+1}>w_i$), the node $w_{i+1}$ included. Then:
%
$$ \o(v_0,s) = \sum_{i=1}^r s_1 \ldots s_i \sum_{w \in
\th(w_i)\setminus\th(w_{i+1}) } \o_w \Eq(7.18) $$
%
Note that $\o(v_0,s)$ satisfies the property that
$0 \le | \o(v_0,s)| \le m_0N$, as $\o(v_0)$ did. The functions
$Q^r_{v_0}(x)$ are defined as:
$Q^r_{v_0}(x)= \sum_{k\ge 1}^{\io} Q^r_{v_0}(k) x^k$, $x=\exp[-\s gt]$,
and the functions $E(m_0-r)$ and $A_{v_0}(\oo \cdot \nn_0(v_0),r,s)$ verify
the bounds: $E(p) \le e^{2p} $, and $\left| A_{v_0}(\oo\cdot\nn_0(v_0),r,s)
\right| \le D_1 C_1^{m_0-1} $ for some constants $C_1, D_1 > 0$.}
\*
Let us consider a generalized reduced tree with given
shape and collection of indices, and let us consider the $p(v)$
labels. Let us single out the nodes $v$'s, with $p(v)=0$: then each such
node will be enclosed, together with all the generalized reduced subtree
emerging from it, inside a bubble $\b_v$
which will be wiggly if $j_v>0$, and smooth if $j_v=l$.
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 generalized 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 withered flowers} the wiggly bubbles, and
{\it fresh flowers} the smooth ones; unlike the leaves,
the flowers will not be considered nodes. A generalized 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
either: i) the sum over all the generalized reduced trees of the
stripped values \equ(7.15), times the product of the leaf values,
(if the flower is withered), or: ii) the sum over
all the generalized reduced trees of the stripped values \equ(7.17),
times the product of the leaf values,
(if the flower is fresh, and $k_v=0$), or: iii) an expression
differing from \equ(7.19) inasmuch it lumps together also the leaf values,
(if the flower is fresh, and $k_v=-1$). We shall see later that, in order
to obtain the latter expression, it will be enough to substitute
the function $A_{v_0}(\oo\cdot\nn_0(v_0),r,s)$ in \equ(7.19) with a
function which admits the same bound, being $m_0$ replaced with $m$,
(see also note 9).
The degree of a generalized reduced tree is given by the number of its free
nodes plus the sum of the degrees of its withered and fresh flowers, and
of its leaves; analogously, the order of a generalized reduced tree
is given by the sum of the order labels of its nodes plus
the sum of the orders of its flowers.
All the withered flowers give a contribution to the stripped value of
the generalized reduced tree of the form \equ(7.15),
(by Lemma 8.1), and the dependence on
the time variable reveals itself only through the exponential
factor $\exp [ i \oo\cdot\nn(v) \t_v ]$. As to the fresh flowers,
they contribute to the stripped value
a factor \equ(7.19), (we can imagine to rewrite \equ(7.17) in
the same form, with the constraints $Q_v^1(x)=-in_v(\cosh gt)^{-1}$ and
$Q_v^r(x)(x)=0$ if $r\ge 2$). Obviously in both cases we have to take into
account the leaf values too.
\*
\\{\bf Lemma 8.3}: {\it The contribution to
$\X^{h\s}_{j\nn}(t)$, $\nn\in{\bf Z}^{l-1}$, $\s=\pm1$,
$2l>j\ge 0$, arising from the sum of the values
of all the generalized reduced trees of degree $m$, with labels
$p(v_0) \neq 0$, can be written as:
%
$$ \sum_{r_0=0}^{m-1} \sum_{r=0}^{m-1} Q^r_{v_0}(x)\, { (gtr_0)^r
\over r! } \, E(m-1-r) \, \ig \m_r(ds) e^{i\oo(s)\cdot\nn_0(v_0)t}
B_{v_0}(\oo \cdot \nn_0(v_0),r,s)\Eq(7.20) $$
%
where $|\nn_0(v_0)|\le mN$,
$r_0$ is the number of fresh flowers, $r$ is the sum of the powers
of the time variables the fresh flowers contribute, $\m_r(ds)$ and
$\oo(s)\cdot\nn_0(v_0)$ are defined as in Lemma 8.2, $r_0^r$
is meant as $1$ when $r=r_0=0$,
and $Q^r_{v_0}(x)= \sum_{k \ge r_0 }^{\io} Q^r_{v_0}(k)x^k$,
$x=\exp[-\s gt]$;
the function $E(m-1-r)$ admits the same bound as the
homonymous one in
Lemma 8.2, and $ \left| B_{v_0}(\oo \cdot \nn_0(v_0),r,s)\right| \le D_2
C_2^{m-1} $ for some constants $ D_2 ,C_2 > 0$.}
\*
\\{\it Proof of Lemma 8.1} : Let us consider a generalized reduced
tree $\bar\th^G$; if $p(v_0)=0$, $k_{v_0}=0$, the root branch
can be $j_{v_0}=l$, or $j_{v_0} >l$. If $v_0$ is the only tree node (\ie
if $\bar\th^G$ is the {\it trivial tree}), the result is obvious, by direct
check. Otherwise, for each $\bar v \ge v_0$, $\bar v \in \bar \th^G$, it is
$p(\bar v)=k_{\bar v} + \sum_{\bar\th^G\ni w>\bar v}(k_w + k_w')$, see
\equ(7.13), where $k_w+k_w' \ge 0$, for each $w$, see \equ(7.9), and
$k_w\=0$ if $w$ is a leaf, see \equ(7.13b).
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 $k_{\tilde w} + k_{\tilde w}'=1$.
Under the hypothesis of the lemma, only the first case
must be considered here. 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 $k_w'=0$. We note that the case $p(\bar v)=0$ and $\a_{\bar v}=-1$
is not possible: {\it this means that, in the case we are
studying, as far as the free nodes are concerned,
the generalized reduced trees behave in the same way as the
reduced trees, and, in particular, the time variables are ordered
and \equ(7.13) can be directly applied, (in particular
we can set $\s_w = \s_{v_0} \,\=\, \s$, $\forall$ $w \ge v_0$,
$w\in\bar\th^G_f$)}. Then we can write:
%
$$ \sum_\th V_j(t;\th)=\sum_{\bar\th^G} V_j^S(t;\bar\th^G)
\prod_{i=1}^{\NN_L}L_{j_i}^{h_{v_i}\s_{v_i}}(0) \Eq(\setteb) $$
%
where $\NN_L$ is the number of leaves of the generalized reduced
tree $\bar\th^G$, and $j_i\=j_{\l_{v_i}}$,
where $v_i$ is the $i$-th leaf. Note that (\setteb) is
the product of factorizing terms, which can be treated separately, being
independent on each other; each $L_{j_i}^{h_{v_i}\s_{v_i}}(0)$, $i>0$,
corresponds to a leaf and has as first node a node $v_i$ with
$\r_{v_i}=0$, while $V_j^S(t;\bar\th^G)$ can have either $\r_{v_0}$ or
$\r_{v_0}=1$. Moreover each
$L_{j_i}^{h_{v_i}\s_{v_i}}(0)$, $i>0$, 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
$V_j^S(t;\bar\th^G)$, 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: see Lemma 8.3). Note that $V_j^S(t;\bar\th^G)$ corresponds to the
stripped value of a generalized reduced tree, so that the hypothesis of
Lemma 8.1 applies to it.
As indicated in the statement of the lemma, if $j_w=l$ we consider together
the cases $w \in \L_{-1}$ and $w \in \L_1$: they give a contribution
to \equ(7.11), containing, as far as the $w$ node is concerned,
a factor $\Phi_{\n_w} \exp [i\o(w)(\t_w-\t_w')]$ 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)= $
$(\s/2) [ e^{-g\s(\t_w-\t_{w'})} - e^{g\s (\t_w-\t_{w'})} ]$.
>From \equ(7.11) and \equ(7.13) we can obtain a sequence of
factorizing integrals; then, for the top nodes different from
the leaves (top free nodes), we have
%
$$\oint\fra{d R_v}{2\p i R_v}\ig_{\s\io}^0 d\,g\t_v\;
\,T_v(-g\t_v)\,e^{-gR_v\sum_{w\le v}\s\t_w}\,
e^{i\t_v\o_v}\,e^{-gk_v\s\t_v}\Eq(7.21)$$
%
where $T_v(-g\t_v)=(-g\t_v)^{1-\d_{j_v,l}}$, see \equ(7.a).
The time integration is trivial and yields:
%
$$ (-\s)^{\d_{j_v,l}} \oint\fra{d R_v}{2\p i R_v} \;
{ e^{- g R_v \sum_{w < v } \s \t_w } \over \big(
R_v + k_v - i \s g^{-1} \o_v \big)^{2-\d_{j_v,l}} } $$
%
where $k_v = k(v) = p(v)$ and $\o_v = \o(v)$.
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(4.9) requires
in such a case $\o(v)\neq 0$. If $j_v=l$, we have to 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
%
$$ \cases{ \left[ i g^{-1}\o(v) \right]^{-2}
& if $j_v > l $ \cr \left[ 1 + g^{-2} \o^2 (v) \right]^{-1}
& if $j_v = l $ \cr} \Eq(7.22)$$
%
(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 generalized reduced tree obtained from $\bar\th^G$
by deleting the original top free nodes, and we have again to consider
an expression like \equ(7.21), so that all the integrations
can be performed in the same way, for each $v \neq v_0$,
if only we take 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(7.22).
In the end, only the node $v_0$ is left. Since $k_{v_0}=0$, if $j_{v_0}>l$,
we have a coefficient $y^{(2)}(0,0)=1$: 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 $\exp[i \o(v_0) \t_{v_0}]$,
and we obtain \equ(7.15), if $G_v[\o(v)]$ is defined as in \equ(7.16).
Otherwise, if $j_{v_0}=l$, then $k_{v_0}=0$ requires $v_0 \in \L_{-1}$,
and we have a coefficient (see \equ(7.10)):
%
$$(-1)^{\r_{v_0}} \sum_{k_{v_0}'=1}^\io y_{v_0}^{(-1)}
(k_{v_0}',0) x^{k_{v_0'}}={(-1)^{\r_{v_0}}\over 2}
{2i n_{v_0}\over\cosh gt}\;\Eq(7.22a)$$
%
and, if we integrate in $\t_{v_0}$ and
sum together the contributions $\r_{v_0}=0,1$, we obtain
\equ(7.17). So Lemma 8.1 is proven if we show that the bound
$D_0 C_0^{m_0-1}$, in the statement 3) of Lemma 8.1,
holds. This will be done in Appendices A1, A2 and A3. \qed
\*
\\{\it Proof of Lemma 8.2} : The expression \equ(7.19) can be
checked by induction. The case $p(v_0)=0$ and $k_{v_0}=-1$ is
the case put aside in the above discussion, (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.8.1: A path $\PP$ connecting the first node $v_0$ of
the generalized reduced tree $\bar\th^G$, (single path tree), 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
Given a generalized reduced tree $\bar\th^G$ with $p(v_0)=0$, and
$k_{v_0}=-1$, then it {\it will} have a path $\PP$: so we call it a {\it
single path tree}. For each $z_i$, 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
case: $z_i \in \L_{-1}$, $z_i \in \L_{2}$, $z_i \in \L_{1}$.
Note that $\LL_{-1} \cap [\bar\th^G \setminus \PP]=\emptyset$, as can be
seen by {\it reductio ad absurdum}: in fact, if $w\in\LL_{-1}$ 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.
In particular this shows that the nodes in $\LL_{-1}$ are totally ordered
as it is said in the statement of the lemma.
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 proof of Lemma 8.1, (\setteb), which are of the same type of
before, up to the first factor, which is given by the stripped
value of a generalized reduced tree with a fixed shape, and
labels $p(v_0)=0$, $k_{v_0}=-1$.
Therefore we have to study only this term.
For each $z_i$ we consider separately the generalized reduced subtree with
root equal to $z_i$ and first node $z_{i+1}$, and the remaining $m_{z_i}-1$
generalized reduced subtrees $\bar\th_{ij}^G$, with root $z_i$, and
first node $v_{ij}$, $j=1, \ldots, m_{z_i}-1$, if $\{v_{ij}\}$ is the
set of nodes immediately following $z_i$, different from $z_{i+1}$.
We treat in a different way the case in which there is no node with
$p(z_i)=0$, and the case in which there is at least one such node. In the
first case, if $\tilde w$ is not a leaf, since the {\it a priori}
possible situations are either $k_{\tilde w}=1$ and $k_{\tilde w}'=0$, or
$k_{\tilde w}=0$ and $k_{\tilde w}'=1$, it must be $k_{\tilde w}=0$
and $k_{\tilde w}'=1$, because $y_v^{(1)}(0,1)=0$; if $\tilde w$ is
a leaf, then again $k_{\tilde w}\=0$ and $k_{\tilde w}'=1$.
Therefore the node $\tilde w$ can be treated as in the proof of Lemma 8.1,
and so we can study the generalized reduced subtrees $\bar\th_{ij}^G$,
$\forall$ $z_i$, so obtaining from each of them a contribution of the form
either $\exp [i\sum_j \o(v_{ij}) \t_{z_i}]$ times
$\prod_{w \in \cup_j \bar\th_{ij}^G } G_w[\o(w)]$, if $v_{ij}$ is a free
node, or $L^{h_{v_{ij}}\s_{v_{ij}}}_{j_{\l_{v_{ij}}}}(0)$, if $v_{ij}$
is a leaf.
Therefore we are left with the integrations along the path $\PP$: but it
is always $p(z_i) \neq 0$, so that we can factorize the
integrations and obtain a product of terms
$( p(z_i) - i \s g^{-1}\o(z_i) )$ to some negative power (1 or 2),
which can be bounded by $1$.
Otherwise, if there are nodes $z_i\in\PP$ with $p(z_i)=0$, \equ(7.19)
can be verified by induction: this is done in Appendix A4, so
that the proof of Lemma 8.2 lacks only the control of the sums
over all the generalized reduced trees. But the number of addends is
trivially bounded, if $m_0$ is the reduced degree of the generalized
reduced tree, 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, ($\le (3N)^{lm_0} \cdot 2^{m_0} \cdot 3^{m_0} \cdot 3^{m_0}$).
\qed
\*
\\{\it Proof of Lemma 8.3} : 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 8.1, 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 fresh flower a contribution
\equ(7.19) arises, and, if $v$ is the end point of the flower stem,
we can decompose the powers of $\t_{v'}$
as in the proof of Lemma 8.2, so constructing several paths along
the generalized reduced tree, (which will be called a {\it multiple paths
tree}), where the paths are uniquely determined by the request that they
connect the first node $v_0$ to the fresh flowers stems.
Then we can explicitly perform the integrations over the
time variables of the nodes belonging to the paths, and
it can be checked that no factorials arise, by reasoning as in the
proof of Lemma 8.2, (the details can be found in Appendix A5).
Nevertheless we must be careful, because we still have to sum over the
labels $p(v)$, (the sum over the other labels can be treated as in the
previous cases). We can resolve this (apparent) problem as follows. If
$\r_{v_0}=1$, $\s t \le g^{-1}$, we split the integral over $\t_{v_0}$:
%
$$\int_{\s\io}^{gt}d\,g\t_{v_0}\;(\ldots)=\int_{\s\io}^{\s}
d\,g\t_{v_0}\;(\ldots)+\int_{\s}^{gt}d\,g\t_{v_0}\;(\ldots)
\= I_m + \int_{\s}^{gt}d\,g\t_{v_0}\;(\ldots) \Eq(7.14)$$
%
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(7.12a)
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(7.9)), and the convergence follows.
We have left the term in \equ(7.14) 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:
%
$$ I_{m_1} \ldots I_{m_p} \prod_{v \in \tilde\th^G_f}
\igb_{\s}^{g \t_{v'}} d g \t_v ( \ldots) $$
%
(all the free nodes $v$'s have $p(v)\neq0$, so that $\r_v=1$)
where $\tilde\th^G$ is a subtree of $\bar\th^G$ 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 generalized 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 generalized 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.\footnote{${}^8$}{\nota
If we recall the proof of the convergence bound
of Lemma 8.1, (as it is carried out in Appendices A1, A2, A3), we can
note that it was obtained by exploiting some cancellations
we could implemented by summing together different generalized
reduced trees, (inside the same family $\FF(\th)$, see Appendix A2);
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 generalized 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 generalized reduced tree $\bar\th^G$, with all its
leaves of fixed orders; then we sum over all the terms of the family
$\FF(\th)$, in which $\bar\th^G$ is contained,
so that the cancellation mechanism is implemented.}
Therefore the bound \equ(7.20), in the statement of Lemma 8.3,
can be inductively checked, exploiting the results of Lemmata 8.1 and 8.2
too, as far as the leaves with label $p(v)=0$ are
concerned.\footnote{${}^9$}{\nota Note that the leaves can have $p(v)=0$,
so that, if this is the case, the bounds of Lemma 8.1 and Lemma 8.2 have to
be implemented. A leaf $v$ with $p(v)=0$ contributing, \eg, to the
generalized reduced tree value (\setteb) through the factor
$L_j^{h_{v_j}\s_{v_j}}(0)$ admits a representation
analogous to the same (\setteb), and
can be expressed as a sum of terms, which are given by the
product of the stripped value of the generalized
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 $p(v)\neq 0$. Then the bound \equ(7.20)
can be assumed to hold, and an inductive proof can be performed.}
This completes the proof of Lemma 8.3. \qed
\*
We can now state the fundamental result giving the convergence
property of the series defining the whiskered tori, and so
completing the proof of Proposition 4.1.
\*
\\{\bf Theorem 8.1}: {\it Let us denote by $\X^{h\s}_{j\nn}(t)$
the dimensionless perturbed motion, $0\le j < 2l$.
We can always write it in the form:
%
$$ \X^{h\s}_{j\nn}(t) = \sum_{r=0}^{2h-1}
\tilde \X^{h\s}_{j\nn}(x,\oo t;r)\;{(gt)^r\over r!}
\Eq(7.30) $$
%
where $|\nn|\le(2h-1)N$, and
$\tilde \X^{h\s}_{j\nn}(x,\oo t;r)$ is an analytic function
in $x$, $\tilde \X^{h\s}_{j\nn}(x,\oo t;r)=$
$\sum_{p=0}^{\io} \tilde \X^{h\s}_{j\nn}(p,\oo t;r) x^p$, with
$|(gt)^r$ $\tilde \X^{h\s}_{j\nn}(p,\oo t;r)|$ $\le$
$\bar D\bar C^{2h-1}\, r!$, for some constant $\bar C, \bar D >0$,
and for all $r\ge 0$, $p \ge 0$, for any $\s t\ge0$.}
\*
The result stated in Theorem 8.1 follows directly from Lemma 8.3,
as far the contribution $| p(v_0)| \ge 1$ is concerned, if we take into
account the inequalities $x^p e^{-px} \le 1$, $x^p e^{-x} \le p!$,
$\forall$ $p \ge 0$, $x \ge 0$,
%so that $ \left|(gt)^r[\min\{p,2h-1
%\}]^{r+1} \exp [-p g\s t] \right| / [(r+1)!] \le 2h-1 $.
and we explicitly bound the sum over $r_0$ in \equ(7.20).
For the contributions $p(v_0)=0$, it follows from Lemma 8.1 and Lemma 8.2,
or better from their proof, as we have to estimate also the contribution
to $\X^{h\pm}_{j\nn}(t)$, with $j\ge l$: it is easily seen that the
discussion can be repeated essentially unchanged and leads to the same
convergence result. The leaves can be treated as in the proof of
Lemma 8.3, so that the writing \equ(7.30) is proven.
\qed
\*
Obviously, if we want to find a bound on the
homoclinic splitting, we can write $\bar \D^h_{j\nn}=\X^{h+}_{j\nn}(0)-
\X^{h-}_{j\nn}(0)$, so obtaining the same bound of Theorem 8.1, up to
a factor $2$. This proves the first of (\seidue), which therefore can
be considered a corollary of Theorem 8.1.
\vskip1.truecm
\\{\bf Appendix A1: Proof of the convergence bound in Lemma 8.1}
\vskip.5truecm\pgn=1\numfig=1\numsec=1\numfor=1
\\As we have seen in \S 8,
from the case $p(v_0)=k_{v_0}=0$ we obtain a contribution
to $\X^{h\s}_{j\nn}(t)$ containing a factor:
%
$$ \prod_{v\ge v_0} \Phi_{\n_v} G_v[\o(v)] \Eqa(A1.1) $$
and we want to find a bound on the sum of \equ(A1.1) over all the
generalized reduced trees
with $p(v_0)$ and $k_{v_0}$ fixed to the above values.
Given a generalized reduced tree $\bar\th^G$, it will be characterized
by its shape and by a collections of labels, as shown in \S 5 and \S 7.
Let us proceed as in [G2], and let us suppose a
condition over the rotation vectors stronger than the hypothesis H$_2$,
\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 [GG]
to eliminate such an unneeded hypothesis.
Therefore we shall make the assumption that the rotation vectors
$\oo$'s satisfy the {\it strong diophantine condition}:
%
$$ \eqalign{
1) & \quad\quad\quad
C_0 | \oo \cdot \nn| \ge |\nn|^{-\t} \quad\quad\quad\quad
\V 0 \neq \nn \in {\bf Z}^{l-1} \cr
2) & \quad\quad\quad
\min_{0 \ge p \ge n} \Big| C_0 |\oo \cdot \nn| -
2^p \Big| \ge 2^{n+1} \quad\quad\quad\quad \hbox{if} \quad n \le 0,
\; \; 0 < |\nn| \le (2^{n+3})^{-\t^{-1}} \cr} \Eqa(A1.4) $$
%
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:
%
$$ G \equiv G_v [\o(v)] = \cases{
- (g C_0 )^2 [ \oo_0\cdot\nn_0(v) ]^{-2} & if $j_v>l$ \cr
- (g C_0 )^2 \left[ (g C_0 )^2 [ 1 + ( \oo_0\cdot\nn_0(v) )^2 ]
\right]^{-1} & if $j_v=l$ \cr} \Eqa(A1.5)$$
%
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 $j_v>l$, 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 $j_v = l$, then $|G|\le1$.
Such a definition, despite its asymmetry, turns out to be useful in
the following estimates, and allows us to use, nearly without changes,
the results of [G1]; we can get rid of the new factor $(2gC_0)^2$,
by defining $C_1=\max\{ 1, (2gC_0)^2 \}$, and introducing a coefficient
$C_1^m$ in the bound \equ(A1.2). This implies a simple redefinition of the
constant ${\CC}$ in \equ(A1.2), and we can say that, if $G$
is on scale $n$, then $|G|<2^{-2n}$, $\forall$ $n \le 0$.
% 2^{-2(n-1)}
{\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 ``generalized reduced tree",
and the symbol $\th$ instead of $\bar\th^G$; however it is
always in the meaning of the latter that the first one has to be
interpretated. In Appendix A4 we will come back to the complete name.
Moreover we call momentum {\rm tout court} the free momentum $\nn_0(v_0)$.}
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 5), 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 incoming 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 entering 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).
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).
\*
The key remark is that the resonant trees 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 total momentum by the amount $\nn_0(v)$: this means that
some of the propagators $\big[ i \o_0(w) \big]^{-2}$ have become
$\big[ i (\o_0(w)+\e) \big]^{-2}$, and some of the
propagators $\big[ - (gC_0)^2 ( 1+ \o_0^2(w)) \big]^{-1}$ have become
$\big[ - (gC_0)^2(1+(\o_0(w)+\e)^2) \big]^{-1}$, if $\e\=\o_0(v)$, and:
\acapo
2) there is one of the node factors which changes by taking successively
the values $\n_{wj}$, $j$ being the branch label of the branch
leading to $v$, and $w\in V$ is the node to which such 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 $n$, let $m_T(n)$ be the number of
resonances of scale $n$ contained in $T$, (\ie with incoming branches of
scale $n$), we have the inequality \equ(A1.7), which is an
adaptation presented in [G1] of the version of {\it Brjuno's lemma}
as it is exposed in [P]: a proof is in Appendix A3.
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 $\l_V$ 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 $\tilde V$. To the just defined set of trees we add the
trees obtained by reversing simoultaneously the signs of the node
modes $\nn_w$, for $w\in \tilde V$: the change of sign is performed
independently for the various resonant clusters. This defines a family
of $\prod 2M_V$ 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(A1.4), \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, as it is explained
in note 8.
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 $\prod 2 M_V\le e^{2m}$ products of propagators
of the members of the family $\FF(\th^1)$ can be bounded above by
$\prod_\l\,2^{-2(n_\l-2)}=2^{4m}\prod_\l\,2^{-2n_\l}$, 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 second order 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
i) 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 for
$|\e_i|<\,2^{ n_i-3}$ if $n_i\=n_{V_i}$, provided $n_i>n+3$.
\acapo
ii) the resummation says that the dependence on the $\e_i$'s has a
second order zero in each. Hence the maximum principle tells us that
we can improve the bound given by the first factor in \equ(A1.6) by the
product of factors $(|\e_i|\,2^{-n_i+3})^2$ 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(A1.6) $>1$, we can leave them in it to simplify
the notation.
{\it The main point here (and the main difference with respect to the
otherwise identical discussion of [G1]) is that, for $n\le0$, not all the
resonances are strong resonances, so that $m_T(n)$ is a bound on the number
of strong resonances, to which all the cancellations exploited in
Appendix A1 apply.}
\vskip1.truecm
\\{\bf Appendix A3: Resonant Siegel-Brjuno bound}
\vskip.5truecm\pgn=1\numfig=1\numsec=3\numfor=1
\\In the following discussion, which is taken from [G1],
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 with scale $>n$ then calling
$\th_1,\th_2,\ldots,\th_k$ the subtrees of $\th$ emerging from the
first vertex of $\th$ and with $m_j>E\,2^{-\e n}$ lines, 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 is either a resonance or it 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 $>n$, unless it is entering a resonance.
Suppose that the root line of $\th_1$ is not entering a resonance. Note
that $|\oo\cdot\nn_0(v_0)|\le\,2^n,|\oo\cdot\nn_0(v)|\le
\,2^n$, if $v_0,v_1$ are the first vertices 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 $m-m_1<\fra12E\,2^{-n\e}$ (and we are not considering the
resonances), 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 $m-m_1$
vertex 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 $m_1\le m-E2^{-\e n}$.
Then \equ(A1.7) is proved.
\vskip1.truecm
\\{\bf Appendix A4: Bound on the single path trees}
\vskip.5truecm\pgn=1\numfig=1\numsec=4\numfor=1
\\Let us consider a single path tree, and let us denote by $z>v_0$ the
first node with $p(z)=0$, (the case in which such a node does not exist has
been considered already in \S 8). Let us suppose inductively that \equ(7.19)
holds (for $m=1$ it can be checked easily to be valid, for some constant
$C_1$). Then the generalized reduced subtree $\bar\th_1^G$ with
root $z$ has $m_1 \le m-q$ nodes, $q\ge1$ being the number of nodes of $\PP$
preceding $z$, and \equ(7.19) is supposed to hold for it by the inductive
hypothesis. We treat as in the case considerd in \S 8 all the generalized
reduced subtrees $\bar\th_{ij}^G$, $z_i 1$, and $\o(z_i,s)$ depends on $s$ through the addend $\o(z,s)$.
The node $v_0\=z_1$ has $p(v_0)=0$, and $\r_{v_0}=0,1$, so that we
sum over the two possible values of the latter label. We decompose:
%
$$ \eqalign{
\sum_{r_1=0}^{m_1} & {(g\t_z)^{r_1} \over {r_1}!} E(m_1-r_1)
\prod_{1 **