\8
\8

\8

\8
\8
\8
\8
\8
\8
\8

\8
\figfin
\eqfig{250pt}{70pt}{
%\ins{155pt}{40pt}{$t$}
\ins{163pt}{42pt}{$\bf\d_v$}
\ins{185pt}{50pt}{$t$}
\ins{185pt}{40pt}{$t$}
\ins{185pt}{8pt}{$t$}
\ins{203pt}{73pt}{$(k_1)$}
\ins{203pt}{53pt}{$(k_2)$}
\ins{203pt}{3pt}{$(k_h)$}
\ins{95pt}{38pt}{$F_0^{(k)}(t)=$}
\ins{165pt}{26pt}{$v$}
%
\ins{-50pt}{38pt}{$\V F^{(k)}(t)=$}
\ins{0pt}{40pt}{$\bf\aa$}
\ins{25pt}{26pt}{$v$}
%\ins{15pt}{40pt}{$t$}
\ins{23pt}{42pt}{$\bf\d_v$}
\ins{45pt}{50pt}{$t$}
\ins{45pt}{40pt}{$t$}
\ins{45pt}{8pt}{$t$}
\ins{63pt}{73pt}{$(k_1)$}
\ins{63pt}{53pt}{$(k_2)$}
\ins{63pt}{3pt}{$(k_h)$}
}{kfig2}{Fig.2}
\*
\0where it is $\sum_j k_j=k-\d_v$, see \equ(3.3); the label $\d_v$ can
be $0$ or $1$: the first drawing represents the term with $\d=\d_v$ in
the expression for $\V F^{(k)}$ in \equ(3.3), and the second drawing
represents the contribution to $F_0^{(k)}$ with $\d=\d_v$.
The node $v$ represents $-\dpr^{h+1}_\f f_{\d_v}$ times $\fra1{h!}$ in
the second graph and $-\dpr_\aa\dpr^h_\f f_1$ times $\fra1{h!}$ in the
first. Because of the $\dpr_\aa$ derivative we can imagine that in the
first graph thelabel $\d_v$ on the node $v$ is {\it constrained} to be
$1$.
We can in the same way represent $\f^{(k)}(t)$ and $\AA^{(k)}(t)$: we
can in fact change the labels $t$ on the lines merging into the node
$v$ into labels $\t$ and interpret the node $v$ as representing an
integration operation over the time $\t$; one get in this way
the following graphs:
\*
\eqfig{250pt}{70pt}{
\ins{155pt}{40pt}{$t$}
\ins{163pt}{42pt}{$\bf\d_v$}
\ins{185pt}{50pt}{$\t$}
\ins{185pt}{40pt}{$\t$}
\ins{185pt}{8pt}{$\t$}
\ins{203pt}{73pt}{$(k_1)$}
\ins{203pt}{53pt}{$(k_2)$}
\ins{203pt}{3pt}{$(k_h)$}
\ins{95pt}{38pt}{$\f^{(k)}(t)=$}
\ins{165pt}{26pt}{$v$}
%
\ins{0pt}{40pt}{$\aa$}
\ins{-50pt}{38pt}{$\AA^{(k)}(t)=$}
\ins{25pt}{26pt}{$v$}
\ins{15pt}{40pt}{$t$}
\ins{23pt}{42pt}{$\bf \d_v$}
\ins{45pt}{50pt}{$\t$}
\ins{45pt}{40pt}{$\t$}
\ins{45pt}{8pt}{$\t$}
\ins{63pt}{73pt}{$(k_1)$}
\ins{63pt}{53pt}{$(k_2)$}
\ins{63pt}{3pt}{$(k_h)$}
}{kfig2}{Fig.3}
\*
The node $v$ with the label $\d_v$, which we noted that it must be $1$
in the first drawing and that can be either $0$ or $1$ in the second,
has to be thought as representing the operations acting on a generic function
$F$:
$$\eqalignno{\II_\s F(t)=&\ig_{\s\io}^t F(\t) \, d\t&\eq(3.6)\cr
\OO_\s F(t)=&\ig_{\s\io}^t\Big(w_{01}(t)w_{00}(\t)-w_{00}(t)w_{01}(\t)\Big)
F(\t)\,d\t\,- \,
w_{00}(t)\ig^{\s\io}_0 w_{01}(\t)F(\t)\,d\t \cr}$$
%
where $\s=+$ if we study the stable manifold and $\s=-$ if we study
the unstable one.
In this way the graphs of Fig.3 represent respectively the {\it values}:
$$
\fra1{h!}\II_\s\Big(-\dpr_\aa\dpr^h_\f f_1[\t] \prod_{j=1}^h
\f^{(k_j)}[\t]\Big)(t),\qquad
\fra1{h!}\OO_\s\Big(-\dpr^{h+1}_\f f_\d[\t]\prod_{j=1}^h
\f^{(k_j)}[\t]\Big)(t))x\Eq(3.7)$$
%
where an argument in square brackets means a dummy integration
variable, inserted just as a reminder of the integration operation
involved; furthermore $f_\d[t]$ abbreviates $f_\d(\aa+\oo t,\f^0(t))$.
Clearly $I^{(k)},\AA^{(k)}$ can be expressed simply by summing over
the labels $k_j$ and $\d$ the values of the graphs in Fig.3: the
summations should run over the same ranges appearing in \equ(3.2), \ie
$h$ between $2-\d$ and $k$, and $k_j\ge1$ such that $\sum_j k_j=k-\d$
and $\d=0,1$). If we study the stable manifold we must take $\s=+$ and
if we study the unstable one me must take $\s=-$ and
$I^{(k)},\AA^{(k)}$ become respectively $I^{s,(k)},\AA^{s,(k)}$ or
$I^{u,(k)},\AA^{u,(k)}$.
It is now immediate to iterate the above representation; one simply
recalls that each symbol:
\eqfig{25pt}{0pt}{\ins{15pt}{8pt}{$t$}\ins{33pt}{3pt}{$(k_j)$}}{kfig1}{Fig.4}
\*
\0represents $\f^{(k_j)}(t)$ and that \equ(3.7) is multi linear in the
$\f^{(k_j)}(t)$. This leads to representing $\AA^{(k)}(t)$ as sum of
values of graphs $\th$ of the form:
\*
\figini{bggmfig0}
\8
\8<%!PS-Giovanni-1.13>
\8
\8<0.83333 0.83333 scale 0 90 punto >
\8<70 90 punto >
\8<120 60 punto >
\8<160 130 punto >
\8<200 110 punto >
\8<240 170 punto >
\8<240 130 punto >
\8<240 90 punto >
\8<240 0 punto >
\8<240 30 punto >
\8<210 70 punto >
\8<240 70 punto >
\8<240 50 punto >
\8<0 90 moveto 70 90 lineto>
\8<70 90 moveto 120 60 lineto>
\8<70 90 moveto 160 130 lineto>
\8<160 130 moveto 200 110 lineto>
\8<160 130 moveto 240 170 lineto>
\8<200 110 moveto 240 130 lineto>
\8<200 110 moveto 240 90 lineto>
\8<120 60 moveto 240 0 lineto>
\8<120 60 moveto 240 30 lineto>
\8<120 60 moveto 210 70 lineto>
\8<210 70 moveto 240 70 lineto>
\8<210 70 moveto 240 50 lineto>
\8
\8
\figfin
\eqfig{199.99919pt}{141.666092pt}{
\ins{-29.16655pt}{74.999695pt}{\it root}
\ins{0.00000pt}{91.666298pt}{$\bf\aa$}
\ins{49.99979pt}{70.833046pt}{$v_0$}
\ins{45.83314pt}{91.666298pt}{$\d_{v_0}$}
\ins{126.66615pt}{99.999596pt}{$v_1$}
\ins{120.83284pt}{124.999496pt}{$\d_{v_1}$}
\ins{91.66629pt}{41.666500pt}{$v_2$}
\ins{158.33270pt}{83.333000pt}{$v_3$}
\ins{191.66589pt}{133.332794pt}{$v_5$}
\ins{191.66589pt}{99.999596pt}{$v_6$}
\ins{191.66589pt}{70.833046pt}{$v_7$}
\ins{191.66589pt}{-8.333300pt}{$v_{11}$}
\ins{191.66589pt}{16.666599pt}{$v_{10}$}
\ins{166.66600pt}{54.166447pt}{$v_4$}
\ins{191.66589pt}{54.166447pt}{$v_8$}
\ins{191.66589pt}{37.499847pt}{$v_9$}
}{bggmfig0}{\hskip.6truecm Fig.5}
\kern0.9cm
\didascalia{A graph $\th$ with $p_{v_0}=2,p_{v_1}=2,p_{v_2}=3,
p_{v_3}=2,p_{v_4}=2$ and $k=12$, and some labels. The lines length is
drawn of arbitrary size. The nodes labels $\d_v$ are indicated only
for two nodes. The lines are imagined oriented towards the root and
each line $\l$ carries also a (not marked) label $\t_v$, if $v$ is the
node to which the line leads; the root line carries the label $t$ but
its ``free'' extreme, that we call the ``root'', is not regarded as a
node.}
The meaning of the graph is recursive: all nodes $v$, see Fig.5,
represent $\OO_\s$ operations except the ``first'' node $v_0$ which
instead represents a $\II$ operation; the extreme of integration is
$+\io$ if we study the stable manifold and $-\io$ if we stufy the
unstable one. Furthermore each node represents a factor
$-\fra1{p_v!}\dpr^{p_v+1}_\f f_{\d_v}[\t_v]$ if $p_v$ is the number of
lines merging into $v$, {\it except} the ``first'' node $v_0$ which
represents $-\dpr_\aa\dpr^{p_v}_\f f_{\d_v}[\t_v]$ instead. The
product $\prod_v \fra1{p_v!}$ is the ``combinatorial factor'' for the
node $v$
The lines merging into a node are regarded as distinct: \ie we imagine
that they are labeled from $1$ to $p_v$, but we identify two graphs
that can be overlapped by permuting suitably and independently
the lines merging into the nodes.
It is more convenient to think that all the lines are numbered from
$1$ to $m$, if the graph has $m$ lines, still identifying graphs that
can be overlapped under the above permutation operation (including the
line numbers). In this way a graph with $m$ lines will have a
combinatorial factor simply equal to $\fra1{m!}$ provided we define
$1$ instead of $\prod_v \fra1{p_v!}$ the combinatorial factor of each
node: {\it we shall take the latter numbering option}. Hence in Fig.5
one has to think that each line carries also a number label although
the line numbers, distinguishing the lines, are not shown.
The endnodes $v_i$ should carry a $(k_i)$ label: but clearly unless
$k_i=1$ they would represent a $\f^{(k_i)}$ which could be further
expanded; hence the graphs in Fig.5 should have the labels $(k_i)$
with $k_i=1$: this however carries no information and the labels are
not drawn. The interpretation of the endnodes is easily seen that has
to be: $\OO_\s(-\dpr_\f f_{\d_{v_i}})(\t_{v'_i})$ if $v'_i v_i$ is the
line linking the endnode $v_i$ to the rest of the graph. An exception
is the trivial case of the graph with only one line and one node: this
represents $\II(-\dpr_\aa f_{\d_{v_0}})(t)$ and it will be called the
{\it Melnikov's graph}.
In this way we have a natural decomposition of $\AA^{(a,(k)}(\aa,t)$
as a sum of {\it values} of graphs. It is now easy to represent the
power series expansion of the trajectories on the manifolds
$W^a(\TT(\AA_0))$: one simply collects all graphs with labels $\d_v$
with $\sum_vd_v=k\ge1$ (they can have at most $2k$ lines, if one
looks at the restrictions on the labels) and adds up their ``values''
obtaining the coefficient $\AA^{a,(k)}(t)$. The $\OO_\s$ and $\II_\s$
operations involve integrals with $\s\io$ as an extreme and one has,
obviously, to choose $\s=+$ if $a=s$ and $\s=-$ if $a=u$.
Since all the integration operations $\OO$ or $\II$ are, in general,
improper we see the convenience of the graphical representation and
its analogy with the Feynman graphs of quantum field theory: in fact
this is {\it more than an analogy} as the above graphs can be regarded
as the Feynman graphs of a suitable field theory: see [GGM0] for the
discussion of a similar case (\ie the KAM theory representation as a
quantum field theory).
An essential feature is missing: namely the graphs have no loops (they
are in fact tree graphs). This major simplification is compensated by
the major difficulty that the number of lines per node is {\it
unbounded} (\ie a field theory that generated the graphs would have to
be ``non polynomial'').
Noting that the value of each graph is a function of $\aa$ we now have
to check that each $\AA^{a,(k)}(0)$ has the form
$\AA^{a,(k)}(0)=\dpr_\aa \F^{a,(k)}$.
For this purpose we consider graphs like Fig.5 but with the root
branch {\it deleted} keeping however a {\it mark} on the first node $v_0$ to
remember that the line has been taken away. We call such a graph a
{\it rootless} graph.
{\it It is convenient to define the value $\Val_\s(\th)$ of such
rootless trees}: its is defined as before but the marked node now
represents the operation $\II_{\s,0}(F)\=\ig_{\s\io}^0 d\t F(\t)$ with
$\s=+$ for the analysis of the stable manifold and $\s=-$ for the
unstable, and the function $\dpr^{p_{v_0}} f_{\d_{v_0}}[\t]$ (keeping
in mind that the marked node {\it must} have $\d_{v_0}=1$, by
construction).
The key remark is now the identity (``Chierchia's root
identity'', see [G3]):
$$\II_{\s,0}( F\, \OO_\s(G))=\II_{\s,0}( G \,\OO_\s(F))\Eq(3.8)$$
%
which is an algebraic identity as our improper integrals only involve
functions $F, G$ linear combinations of ``monomials'' of the form
$\s^\ch\,\fra{(\s t g)^j}{j!} e^{-g\s \t h} e^{i\oo\cdot\nn \, t}$ for
some $\ch=0,1,j,h,\nn$ with $\s=\sign \t$, see above, for which both
sides of \equ(3.7) can be explicitly and easily evaluated.
This identity can be used to relate the values of different graphs: it
means that the values of two rootless tress differing only because the
mark is on different nodes and otherwise superposable are {\it
identical}: this can be seen easily by successive applications of the
identity \equ(3.7), see [G3].
Therefore if we define:
$$\F^{\s,(k)}(\aa)=\fra1k \sum_\th \Val_\s(\th), \qquad
\F^\s(\aa)=\sum_{k-1}^\io \e^k \F^{\s,(k)}(\aa)\Eq(3.9)$$
%
we see that the gradient with respect to $\aa$ of $\F^{\s,(k)}(\aa)$ is
precisely $\AA^{a,(k)}(\aa)$. And the splitting $\V
Q(\aa)$ is the gradient of $\F(\aa)=\F^{+}(\aa)-\F^{-}(\aa)$.
One can get directly a graphical representation of $\F^{(k)}$ as:
$$\F^{(k)}(\aa)=\sum_{\th:\,k}\ig_{+\io}^{-\io} dt\,
{\rm Wal}_{\s(t)}(\th)\Eq(3.10)$$
%
where $\s(t)=\sign(t)$ and ${\rm Wal}_{\s(t)}(\th)$ is just the {\it
integrand} in the $\II_{\s,0}$ integral with respect to the first node
variable $\t=\t_{v_0}$ appearing in the evaluation of
$\Val_{\s}(\th)$. This concludes the construction of Eliasson's
potential.
%\ifnum\mgnf=0\pagina\fi
\*
\0{\bf\S4. Properties of the potential.}
\numsec=4\numfor=1\*
Many properties of the gradient $\V Q(\aa)=\dpr_\aa \F(\aa)$ have been
studied in [G3], [GGM1], [GGM2], [GGM3]: they are immediately
translated into properties of the potential $\F$, either by
integration or by following the proofs of the corresponding statements
for $\V Q(\aa)$. We just summarize them:
(1) If $\oo$ is fixed then, generically,
the first order dominates:
$$\F(\aa)=\e\ig_{+\io}^{-\io} d\t\, f(\aa+\oo t,\f^0(t)) + O(\e^2)\Eq(4.1)$$
%
this is the well known Melnikov's result. We shall say that there is
``dominance of Melnikov's term'' for some quantity every time that the
lowest order perturbative term gives the dominant asymptotic behavior
for it in a given limiting situation. Hence in \equ(4.1) domination
refers to $\e\to0$.
In the ``one dimensional'' case not explicitly treated above, but much
easier, of a periodic forcing in which there is only one angle $\aa$
and one action $\AA$, Melnikov's domination remains true {\it even} if
the parameter $g$ becomes small provided the $\e$ is chosen of the
form $\m g^q$ for some $q>0$ (proportional to the degree $N_0$ of $f$
as a trigonometric polynomial in $\f$) and $|\m|$ small enough.
This is somewhat nontrivial: in [G3] there is a proof based on the
above formalism; other proofs are available as the problem is
classical. The nontriviality is due to the necessity of showing the
existence of suitable cancellations that eliminate values of graphs
contributing to $\F(\aa)$ higher order ``corrections'' (corresponding
to special graphs) which are individually present and, in fact, larger
than the first order contributions.
(2) The next case to study is the same case of $g$ small but with the
Hamiltonian \equ(2.1) (\ie quasiperiodically, rather than
periodically, and rapidly forced): let $g^2=\h$ and $\h<1$ be a
parameter that we want to consider near $0$. In this case, too,
convergence requires that $\e=\m\h^q$ for some $q>0$ (proportional to
the degree $N_0$ of $f$ as a trigonometric polynomial in $\f$) and
$|\m|$ small enough.
The problem is discussed already in [G3] and, following it, we consider
the graphs $\th$ that contribute to $\F^{(k)}$ and at each node we
decompose $f_{\d}$ into Fourier harmonics $f_{\d}(\aa,\f)=\sum_{\nn}
f_\nn(\f) e^{i\nn\cdot\aa}$. This leads to considering {\it new graphs}
$\th$ in which at each node $v$ a label $\nn_v$ is added signifying
that in the evaluation of the graph value the functions
$f_{\d_v}(\aa,\f)$ are replaced by $f_{\d_v,\nn}(\f)
e^{i\aa\cdot\nn}$. Of course at the end we shall have to sum over all
the ``momentum labels'' $\nn_v\in Z^2$. We call $F^{(k)}_{\th}$ the
contribution to $\F^{(k)}$ from one such more decorated graph. Then
from \S8 of [G3] one sees that:
$$\eqalign{|\F^{(k)}_\th|\,\le&\, D\, B^{k-1} \,k!^{\,p}\,
\big(\sum_{\nn'}
e^{-|\nn'\cdot\oo|\fra\p{2g}}\big)\,\prod_{v}|f_{\nn_v}|\cr
|\F^{(k)}_\nn|\,\le& (b \h^{-q})^{k}\cr}\Eq(4.2)$$
%
where $g=\hdp$ the sum over $\nn'$ runs over the {\it nonzero} values
of the sums of subsets of $\nn_1,\ldots,\nn_k$. The constants $B,D$
are bounded by an inverse power of $\h$ and $p>0$ is constant
(depending on the degree of $f$ in $\f$); the constants $b,q$ can be
bounded only in terms the maximum of $|f|$ in a strip $|\Im \a_j|,|\Im
\f|<\x$ on which the maximum is finite. The first property follows from the
analysis in \S8 of [G3]; the second is symply the statement that the
stable and unstable manifolds are analytic function of $\e$ with
radius of convergence proportional to $\h^q$ for some $q$ (essentially
a result of Graff, see \S5 of [CG]).
(3) A consequence of \equ(4.2) is that $\F(\aa)$ can be represented
as, see [GGM3] for details on the corresponding statement for the
gradient of $\F$:
$$\eqalignno{
\F(\aa)=&\e\sum_\nn e^{i\aa\cdot\nn}(M_\nn+\e D_\nn(\aa,\e))\cr
M_\nn=& \sum_n f_{\nn,n}\ig_{+\io}^{-\io} (1-\fra{\dot\f^0(t)}{\cosh
gt})\,\cos n\f^0(t)\,dt&\eq(4.3)\cr
|\dpr_\aa^h D_\nn(\aa)|<& \h^{-p_h}C_h |\nn|^h,\qquad p_h,q_h>0\cr}
$$
%
which allows us to say, very easily, that the Eliasson's potential is
``in some sense'' dominated by Melnikov's value at least in the
special cases in which $f_{\nn,n}$ are positive and ``{\it as large as
possible}'', \ie $ f_{\nn,n}= c\,e^{-\k|\nn|}\d_{n,1}$ for $c>0$, and
$\oo$ has good Diophantine properties, \eg if $\o_1/\o_2$ is the
golden mean (here $|\nn|\defi|\n_1|+|\n_2|$).
In the latter instance one verifies that, for all $\aa$,
$|D_\nn(\aa,\e)|<\h^{-q'} M_\nn$ for all $|\nn|<\h^{-1}$ which togheter
with the analyticity in $\e$ of $\F(\aa)$ allows disregarding the
contributions to $\F(\aa)$ from the $\nn$'s exceeding $\h^{-1}$. The
properties of the golden mean allow us immediately to see that in the
sum only one pair $\pm\nn$ dominates at $\aa=\V0$: it is the pair
$\nn_0=(f_k,-f_{k+1})$ if $f_j$ is the Fibonacci's sequence and $k$ such
that $\k |\nn_0|+\fra\p{2\hdp}|\oo\cdot \nn_0|$ is minimum; {\it apart}
from exceptional intervals of values of $\h$ in correspondence of which
there may be two pairs (or more) (see \S2,6 of [DGJS]). The domination
persists for all the $\aa$'s such that $|\sin\nn_0\cdot\aa|>b$ where
$b>0$ is any prefixed constant (the smaller $b$ the smaller has $\e$ to
be to insure dominance).
Also the gradient of $\F(\aa)$, and in fact any derivative of $\F$ is
dominated by the Melnikov's term, by the same type of argument. But
this is somewhat trivial: the real question is, in view of the remarks
in \S2 about the possible applications to heteroclinic strings and to
Arnold's diffusion, whether the homoclinic splitting is dominated by
Melnikov's integral. This seems to be, in the generality considered
here, {\it still an open problem}. The reason is very simple; from
\equ(4.3) one easily deduces that:
$$\D=-\e^2\sum_{\nn,\nn'}
e^{-(|\oo\cdot\nn|+|\oo\cdot\nn'|)\fra\p{2}\hdm}
\big((\nn\wedge\nn')^2 M_\nn M_{\nn'}+ \e
d_{\nn,\nn'}\big)\Eq(4.4)$$
%
{\it hence one realizes that the term that should be leading,
$\nn=\pm\nn'=\pm\nn_0$}, is missing in the lowest order
part. Therefore the main contribution comes, {\it or may come}, from
the remainder $d_{\nn,\nn'}$ on which we have little information
besides the above bounds (which would be plenty if the Melnikov's main
term did not vanish). Curiously the above exceptional cases, \ie when
the value of $\h$ is taken along a sequence $\h_j\to0$ such that for
each $j$ there are {\it two} minimizing vectors $\nn_0$ and $\nn'_0$,
can be, instead, easily solved because $(\nn_0\wedge\nn'_0)^2\ge1$ as
no two Fibonacci's vectors can be parallel.
In the literature there are various claims about ``proofs'' of
dominance of Melnikov's contribution to the splitting: they however
seem to be always proofs of the ``easy part'' namely of the dominance
of the Melnikov's term in {\it some components of} the splitting vector
$\V Q(\aa)$ (implied by the above analysis).
The only known case of generic dominance of the Melnikov term {\it for
the splitting} is the one discussed in [GGM1], see (5) below. And its
analysis is already far more subtle than the above.
(4) In general the estimates \equ(4.2), called ``quasi flat'' in [G3]
are {\it optimal} (see [GGM4]): hence one {\it cannot hope} to have
bounds on the Fourier transform of $\F$ of the form, for some $r>0$:
$|\F_\nn|< const\, \h^{-r} e^{-\fra\p{2g}|\oo\cdot\nn|}
e^{-\k|\nn|}$. Such estimates are called ``exponentially small'' and,
occasionally, have been claimed to be possible.
(5) The above results are {\it very easy} compared to the ones that
can be obtained by taking $g^2$ {\it fixed} and $\oo=(\hdp \o,\hdm
\o')$, discussed in [GGM1] and called the {\it three time scales
problem}, because the system has three time scales of orders
respectively $\hdm,1,\hdp$. In this case we consider the values of
$\h$ for which $\oo$ verifies a Diophantine property of the form
$|\oo\cdot\nn|>\h^\g|\nn|^{-t}$ with some $\g,\t>0$ and we take $\e$
equal to a suitably large power of $\h$ so that the small divisors
problems can be overcome and the invariant tori do exists.
The quasi flat estimates hold (for small $\h$) but they {\it do not
imply that the matrix $\dpr_{\aa,\aa}\F(\aa)|_{\aa=\V0}$ has three
matrix elements of size exponentially small as $\h\to0$}. In fact all
the four matrix elements are of the order of a power of $\h$: this is
so {\it in spite of the fact that the Melnikov term $M(\aa)$
generates a contribution to the $2\times2$ splitting matrix with three
exponentially small entries}.
In other words neither $\F$ nor the Hessian matrix
$\dpr_{\aa,\aa}\F|_{\aa=\V0}$ are dominated by the the Melnikov's
``first order'' contribution. {\it Nevertheless the Melnikov's
contribution to the Hessian gives the leading term in the limit
$\h\to0$}! (generically in the perturbation).
Of course if the above mentioned exponential estimates could be
correct this would follow immediately from them: but they are not
valid (as they would imply the wrong statement that the splitting
matrix has $3$ exponentially small entries) and the result holds {\it
only because remarkable cancellations take place}. Hence, contrary to
what is sometimes stated, the above case requires a delicate analysis,
compared to the one in [G3] which solves easily the problems
(1)$\div$(4) above at least as far as the domination of the first
order in the derivatives of the Eliasson's function (hence the
splitting vector) is concerned. In particular this means that $\F$
{\it is not a good measurement} of the splitting.
It is in the theory of this ``three time scales problem'' that the
analogy with field theory and renormalization theory turns out to be
particularly useful and the methods characteristic of such theories
apply very well and turn into a rather simple matter the check of the
infinitely many identities that are necessary in order that all terms
in the Hessian that dominate the Melnikov contribution cancel each
other leaving out just the Melnikov contribution as the leading one as
$\h\to0$.
(6) The just described graphical technique seems not only very well
suited for the questions analyzed or mentioned above but it seems
quite promising also with respect to the solution of one of the main
standing problems, namely: what is the asymptotic behavior of the
splitting as $g\to0$ and $\oo$ fixed? The case mentioned in (3) above
requires that all the Fourier components of the perturbation do not
vanish: a finer analysis shows that this can be somewhat weakened but
{\it not} to the extent of allowing polynomial perturbations. Hence
such cases seem to have a rather limited interest: they appear in fact
too special. But even so the only thing we know is the Melnikov's
dominance in Eliasson's potential and in its derivatives.
On the other hand the theory discussed in [G3], [GGM3], and in \S3
suggests the following question. First of all let us {\it define} an
extension of Melnikov's function to {\it higher order}. We simply
consider the function $\F^0(\aa)$ which is obtained from the
diagrammatic representation \equ(3.9) {\it but replacing the operators
$\OO$ associated with the nodes of $\t$ by the operator}:
$$\OO_0(F)(t)=\fra12\sum_{\ch=\pm}\ig_{\ch\io}^t
\big(w_{01}(t)w_{00}(\t)-w_{00}(t) w_{01}(\t)\big)\,
F(\t)\,d\t\Eq(4.5)$$
%
then, supposing $\oo_0$ with golden rotation number (or any number
with very good Diophantine nature):
\*
\0{\it Conjecture: In model \equ(2.1) and assuming that $g=\hdp$,
will the Hessian of $\F^0(\aa)$ give the leading asymptotics as
$\h\to0$ of the splitting determinant at $\aa=\V0$ ``generically'' in
$f$ ?}
\*
Here generic means both genericity in the space of trigonometric
polynomial pertubations of fixed degree (arbitrary) and in the space
of the analytic perturbations, possibly with the constraint that the
perturbation is of positive or negative type. However we require that
the perturbation be polynomial in the $\f$ variable, see (9) below. As
far as I know there is no proof even of the convergence of the series
defining $\F^0$ (which is well defined only as a formal series and
which may have to be regarded as an asymptotic series, see [G3], [GGM1]).
The conjecture can be extended to the case of three time scales considered
in (5): in that case it is affirmatively answered in [GGM], where,
however, one also sees that the Eliasson's potential and its derivatives
is {\it not} dominated by the first order. It is only the splitting
determinant that is dominated by the first order: {\it not surprisingly
as this is the only quantity among the ones discussed which has a direct
physical meaning}.
The conjecture could be strengthened by adding, for instance, that
$\F^0$ can be replaced by the function $\tilde F^0$ obtained from $\F^0$
by developing in powers of $\e$ its Fourier coefficients $\F^0_\nn$ and
retaining only the lowest non vanishing order $\tilde\F^0_\nn$ of each
Fourier coefficents to form the Fourier transform of $\tilde\F^0$. In
this stronger form it becomes, in the assumptions of (2) above (fast
forcing and ``maximal size of the Fourier coefficients of the
perturbation), simply the statement that the splitting determinant can
be computed by the first order Melnikov's integral: an open problem (as
mentioned above). However in this form the conjecture is not really
stronger than above because using $\F^0$ instead of $\F$ amounts to
saying that the $\e d{\nn,\nn'}$ in \equ(4.4) has the form $\e
(\nn\wedge\nn')^2 d'_{\nn,\nn'}$.
Clearly in order that the answer to the question be affirmative one
has to show the existence of suitable cancellations: I have checked
that {\it they are indeed present at the order beyond the lowest}
(since the lowest order for the homoclinic determinant is the second,
this means that the answer is affirmative to third order). The check
requires using the results in [GGM1], which may already imply a
positive answer to all orders.
Denoting $\OO$ the operator $\OO(F)(t)\defi\OO_{\s(t)}(F)(t)$ (with
$\s(t)=\sign(t)$) one remarks that in all the expressions involved in
the graphs evaluations one always really uses $\OO$; then it is useful
to note the (algebraic) relation between the operator $\OO$ and
$\OO_0$:
$$\eqalignno{
\OO F(t)=&\OO_0 F(t)+ |w_{01}(t)|\, G_0(F)+ w_{00}(t)\, G(F)\cr
\OO_0 F(t)=&\fra12\sum_{\ch=\pm}\ig_{\ch\io}^t
\big(w_{01}(t)w_{00}(\t)-w_{00}(t) w_{01}(\t)\big)\, F(\t)\,d\t&\eq(4.6)\cr
G_0(F)=& \fra12 \ig_{+\io}^{-\io} d\t\, w_{00}(\t)\, F(\t)\,d\t\,,
\qquad G(F)=
\fra12 \ig_{+\io}^{-\io} d\t\, |w_{01}(\t)|\, F(\t)\,d\t\cr}$$
%
and, as it is clear from [GGM1], the $G,G_0$ factors play the role of
``counterterms'' in the field theory interpretation of the diagrammatic
expansion of $\F$. Hence the above question suggests that the leading
behavior of the splitting determinant is due to graphs without
counterterm contributions.\annota5{Called in field theory ``most
divergent'' graphs: rather improper an expression because in any
reasonable field theory there should be no divergences at all; as it is
the case in the theories that have been actually shown to exist on a
mathematical basis} Here the ``counterterms'' contain non analytic
functions and they are responsible for the impossibility of
exponentially small estimates. A positive answer to the above question
would state that they only give rise to subleading contributions to the
splitting.
An explicit expression for the value contributing to $\F^0$ can be
found in [GGM1]: see (6.2), for the isochronous case \equ(2.1), and
see the paragraph preceding (7.4) for the anisochronous case.
(7) The above theory can be immediately extended to anisochronous
cases: one just has to consider a few new types of graphs, [G3], that
contribute to the splitting vector $\V Q(\aa)$ and to the splitting
potential $\F(\aa)$.
(8) Most of the considerations above do not really use that the
dimension of the quasi periodic motion is $2$: if it is suppoo=sed
larger it is however difficult to see what will be the leading
behavior of the splitting. One reason is that even the analysis of the
Melnikov term is itself a quite difficult task: Diophantine
approximation theory is in a very rudimentary stage if the dimension
of the quasi periodic motion is $\ge3$.
A glimpse of the difficulties that one should expect to meet is given
by the three time scales problem (5) above. In this problem we can
think that the slow frequency of order $\hdp$ is in fact obtained
because the perturbation by a {\it three dimensional} quasi periodic
motion with three {\it fast } frequences $\o_1,\o_2,\o_3$ of order
$\hdm$ contains an almost resonant harmonic $\nn$ such that
$\n_1\o_1+\n_2\o_2=O(\hdp)$. One would then naively think that in this
case the homoclinic splitting can become ``large'' because we can have
$\oo\cdot\nn$ small of order $\hdp$ with not too large $\n_1,\n_2$. But
{\it this is illusory} precisely because from the results of the case (5)
one sees that, although we could expect a large splitting vector and
matrix, its Hessian at the homoclinic point will be exponentially
small as $\h\to0$. Therefore in the three dimensional case we should
expect that the resonances {\it do not enhance the splitting}: they
can make large the spliting matrix but not its determinant! This
remark also explains why the problem (5) above is so unexpectedly
difficult to analyze (see [GGM1]).
(9) Finally there seems to be no reason whatsoever for having a small
homoclinic splitting when the perturbation is not a polynomial (but
``just'' analytic) in the $\f$ variable, not even when the rotation
vector $\oo$ is very fast.
\*
\0{\it Acknowledgments: I am honored to have been asked by Luis
Michel to contribute to this volume. I am also grateful to the
Directors and Members of IHES who made possible the development of
many of my scientific works through the frequent invitations to visit
IHES in the last 33 years. For the technical part of this paper I am
greatly indebted to G.Gentile, V. Mastropietro, G. Benfatto,
G.Benettin, A. Carati: their suggestions and help have been
essential.}
\*
\vskip1truecm
\0{\bf References.}
\*
\vskip1truecm
\0[BCG] Benettin, G., Carati, A., Gallavotti, G.: {\it A rigorous
implementation of the Jeans--Landau--Teller approximation for
adiabatic invariants}, Nonlinearity {\bf 10}, 479--507, 1997.
\*
\0[CG] Chierchia, L., Gallavotti, G.:
{\it Drift and diffusion in phase space},
Annales de l'In\-sti\-tut Henri Poincar\'e B {\bf 60}, 1--144, 1994.
\*
\0[DGJS] Delshams, S., Gelfreich, V.G., Jorba, A., Seara, T.M.:
{\it Exponentially small splitting of separatrices under fast
quasiperiodic forcing}, Communications in Mathematical Physics
{\bf189}, 35--72, 1997.
\*
\0[E] Eliasson, L.H.: {\it Absolutely convergent series expansions
for quasi-periodic motions}, Ma\-the\-ma\-ti\-cal Physics Electronic
Journal, {\bf 2}, 1996.
\*
\0[Ge] Gelfreich, V.G.: {\it A proof of exponentially small
transversality of the sepratrices for the standard map}, in
mp$\_$arc@math. utexas. edu, \#98-270: this recent paper, besides
clarifyng various aspects of previous papers, provides an accurate
exposition of the main ideas (and appropriate references) of the other
papers by the russian school.
\*
\0[GGM0] G. Gallavotti, G. Gentile, V. Mastropietro: {\it Field theory
and KAM tori}, p. 1--9, Mathematical Physics Electronic Journal,
MPEJ, {\bf 1}, 1995 (http:// mpej.unige.ch), .
\*
\0[GGM1] G. Gallavotti, G. Gentile, V. Mastropietro: {\it Pendulum:
separatrix splitting}, in mp$\_$arc@math.utexas.edu, \# 97-472. To
appear with a different title: {\it Separatrix splitting for systems
with three time scales}.
\*
\0[GGM2] G. Gallavotti, G. Gentile, V. Mastropietro: {\it
Hamilton-Jacobi equation, heteroclinic chains and Arnol'd diffusion
in three time scales systems}, mp$\_$arc@math. utexas. edu \#98-4;
chao-dyn@xyz. lanl. gov \#9801004.
\*
\0[GGM3] G. Gallavotti, G. Gentile, V. Mastropietro: {\it Melnikov's
approximation dominance. Some examples}, mp$\_$arc@math. utexas. edu
\#98-331; chao-dyn@xyz. lanl. gov \#9804043.
\*
\0[GGM4] G. Gallavotti, G. Gentile, V. Mastropietro: {\it Homoclinic
splitting, II. A possible counterexample to a claim by Rudnev and
Wiggins on Physica D}, chao-dyn@xyz. lanl. gov \#9804017.
\*
\0[HM] Holmes, P., Marsden, J.: {\it }. See also:
Holmes, P., Marsden, J., Scheurle,J: {\it Exponentially Small
Splittings of Separatrices with applications to KAM Theory and
Degenerate Bifurcations}, Contemporary Mathematics, {\bf81}, 213--244,
1988.
\*
\0[T] Thirring, W.: {\it Course in Mathematical Physics}, vol. 1,
p. 133, Springer, Wien, 1983.
\*
\*
\0{\it Author's preprints at {\tt http://ipparco.roma1.infn.it}}
\0{\sl Address: Fisica, Universit\'a di Roma ``La Sapienza'', P.le Moro
2, 00185 Roma, Italy}
\0{e-mail:\tt \ giovanni@ipparco.roma1.infn.it}
\ciao