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\begin{opening}
\title{A POSSIBLE MECHANISM FOR THE KAM TORI BREAKDOWN}
\date{today}
\author{Guido Gentile}
\institute{IHES, 35 Route de Chartres, 91440 Bures sur Yvette, France}
\author{Vieri Mastropietro}
\institute{Dipartimento di Matematica, $II^a$ Universit\`a di Roma,
00133 Roma, Italia}
\end{opening}
\runningtitle{TORI BREAKDOWN}
\begin{document}
\begin{abstract}
{\it Inspired by the quantum field theory, a resummation is performed
in the Lindstedt series defining the KAM tori, and a possible mechanism
for universality of the tori breakdown is discussed.
This work is mostly based on a joint paper with G. Gallavotti.}
\end{abstract}
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\let\z=\zeta \let\h=\eta \let\th=\vartheta\let\k=\kappa \let\l=\lambda
\let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho
\let\s=\sigma \let\t=\tau \let\iu=\upsilon \let\f=\varphi\let\ch=\chi
\let\ps=\psi \let\o=\omega \let\y=\upsilon
\let\G=\Gamma \let\D=\Delta \let\Th=\Theta \let\L=\Lambda\let\X=\Xi
\let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi \let\O=\Omega
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\def\AA{{\V A}}\def\aa{{\V\a}}\def\nn{{\V\n}}\def\oo{{\V\o}}
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\def\lis#1{{\overline #1}}
\def\NN{{\cal N}}\def\FF{{\cal F}}\def\Im{{\rm\,Im\,}}\def\Re{{\rm\,Re\,}}
\def\={{ \; \equiv \; }}\def\Dpr{{\V \dpr}\,}
\def\sign{{\rm sign\,}}\def\atan{{\,\rm arctg\,}}
\def\hh{\V h}\def\pps{\V \psi}
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\\{\bf 1. Introduction.} The parametric equations of KAM tori
for a $\ell$ degrees of freedom quasi integrable system can
be shown to be one point--Schwinger functions of
a suitable euclidean quantum field theory (QFT), \cite{G}, \cite{GGM}.
This shows that it is very natural to use the powerful techniques
developed in QFT (see for instance \cite{BG}) to study the convergence
of the perturbative series for the KAM tori ({\it Lindstedt series}),
and in fact this has been done in recent proofs of the KAM theorem,
(indeed in the first of such proofs, due to Eliasson, \cite{E},
this is only implicit but it appears more and more clear in following
simplified proofs, \cite{G}, \cite{GM3}, \cite{GM1}, \cite{GM2}).
In QFT it is often necessary to perform a resummation in the
perturbative expansion for the Schwinger functions, so that the original
series in the perturbative parameter $\e$ is replaced by a different
power series in suitable functions of $\e$, called {\it form factors}:
inspired by the analogy with QFT, we perform a resummation of the
Lindstedt series, (see \S 3), and obtain a new expression for the
sum of the series, which we call the {\it resummed series}.
By the non resummed series, one estimates the tori
analyticity domain $D$ around $\e=0$, with boundary $\e=\e_D(\theta)$,
$\theta\in [0,2\pi)$, with a disk of radius $R$ not larger than
the largest disk contained in $D$, \ie $R\le\min_{\theta}
|\e_D(\theta)|$.
Hence, if for some system $\e(0),\e(\pi)\gg \min_{\theta} |\e_D(\theta)|$,
then, although the convergence radius of the non resummed series can be
well estimated by $R$, the invariant tori exist for real $\e$ bigger
than $R$, \ie $R$ is a bad estimate of the value $\e_c$ of $\e$ at which
there is the tori breakdown.
We will see that the resummed series, at least in principle,
could allow us to estimate the domain $D$ with a domain less simple
than a disk and this might allow to obtain better estimates of the
value of $\e_c$.
Finally the resummation leads to a conjecture about the mechanism of
the phenomenon of the tori breakdown, (see \S 4).
\*
\\{\bf 2. Formalism.} We consider for simplicity $\ell$
rotators with inertia momenta $J$, angular momenta
$\vec A=(A_1,\ldots,A_{\ell})\in {\bf R}^{\ell}$, and angular positions
$\vec a=(\a_1,\ldots,\a_{\ell})\in {\bf T}^{\ell}$, described by the
Hamiltonian $ {\cal H} = {1\over 2 J} \vec A \cdot \vec A +
\e f(\vec\a)$, where $\vec A\in {\bf R}^{\ell}$, $\vec\a\in {\bf T}^{\ell}$
and $f(\vec\a)$ is analytic and even in its argument.
Similar considerations holds for much more general hamiltonians.
Let $\oo_0=J^{-1}\vec A_0$ be a rotation vector
verifying for $C_0, \t>0$ the {\sl diophantine property}
$C_0|\oo_0\cdot\vec\nu|>|\vec\nu|^{-\t}$, $\vec 0 \neq
\vec\nu \in {\bf Z}^{\ell} $.
The KAM theorem states the existence of a one parameter family
$\e\to T_\e$ of tori with parametric equations
$ \vec A=\vec A_0+\vec H(\vec\psi)$,
$\vec a=\vec \psi+\vec h(\vec\psi)$, $\vec \psi\in{\bf T}^{\ell}$,
where $\vec H(\vec \psi)$ and $\vec h(\vec \psi)$ are analytic functions
of $\e$, $\psi_j$, $j=1,\ldots,\ell$, divisible by $\e$,
defined for $|\e|$, $|\hbox{Im}\psi_j|$ small enough.
Writing $\vec H(\pps)=\sum_{k=1}^\io\sum_\nn e^{i\nn\cdot\pps}
\e^k\vec H^{(k)}_\nn$, $\vec h(\pps)=\sum_{k=1}^\io \sum_\nn
e^{i\nn\cdot\pps}\e^k\vec h^{(k)}_\nn$, it is easy to define
a graphical representation for $\vec h_{\nn}^{(k)}$
and $\vec H_{\nn}^{(k)}$ in the following way.
Consider $k$ oriented lines, labeled from $1$ to $k$: the final extreme
$v'$ of a line is called the root and the other extreme
$v$ the vertex, and the line is denoted $v'\leftarrow v$.
The lines are arranged on a plane by attaching in all possible ways
the vertices of some lines to the roots of others, to form a
connected tree: each tree, following a therminology mutuated from QFT,
can be called a {\it Feynman graph}.
In this way only one root $r$ remains unmatched and it will be called
the {\it root} of the graph, whose lines will be called {\it branches}
and whose vertices other than the root will be called {\it nodes}.
Between the nodes there is an ordering relation $\le$;
we write $w0$ such that
the coniugacy is described by two functions $\V h,\V H$ and written as
$\V\a=(\psi,0)+(h_1(\psi),h_2(\psi))$ and $\vec A=(H_1(\psi),H_2(\psi))$
with $\vec h$ H\"older continuous with exponent $\d'<\d$ and $\vec H$ of
class $C^{1+\d'}$. Furthermore the above conjugacy has a H\"older
continuous regularity $\d'<\d$ in the $\e-\e_c$ variable.}
In particular $\vec H$ is ``once more differentiable'' than $\vec h$,
for $\e=\e_c$; this is quite surprising and, perhaps, quite unexpected as
$\V H$ is obtained from $\V h$, by applying to it
$\oo\cdot\partial_{\pps}$. The idea of studying the
universality of the tori breakdown from a tree expansion is a refined
version of an important idea in \cite{PV}: except that we have
{\it not} made here the simplifying assumption of
absence of resonances (\ie we {\it allow} for non zero Fourier
components of opposite wave label $\pm\nn$,
and find resummations that in some sense eliminate them).
If one accepts that the above pendulum system has the same critical
exponents for the golden mean torus in the standard map then it follows
that $\d=0.7120834$ by the scaling argument on p. 207 of \cite{M}.\
\begin{thebibliography}{}
\bibitem{BG}
{Benfatto, G., Gallavotti, G. (1995) {\it Renormalization group}, Princeton
University Press, Princeton. }
%
\bibitem{E}
{Eliasson, L.H. (1988)
Absolutely convergent series expansions
for quasi-periodic motions, Report 2--88,
Department of Mathematics, University of Stockholm. }
%
\bibitem{ED}
{Escande, D.F., Doveil, F. (1981)
Renormalization method for computing the threshold of the
large--scale stochastic instability in two degrees of freedom
Hamiltonian systems,
{\it Journal of Statistical Physics}
{\bf 26}, 257--284. }
%
\bibitem{G}
{Gallavotti, G. (1994) Twistless KAM tori,
{\it Communications in Mathematical Physics}
{\bf 164}, 145--156. }
%
\bibitem{GGM}
{Gallavotti, G., Gentile, G., Mastropietro V. (1995) Field theory and
KAM tori,
{\it Mathematical Physics Electronic Journal}, {\bf 1}. No. 4.}
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\bibitem{GM3}
{Gentile, G., Mastropietro V. (1994) KAM theorem revisited,
submitted to {\it Physica D}.}
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\bibitem{GM1}
{Gentile, G., Mastropietro, V. (1995)
Tree expansion and multiscale analysis for KAM tori,
{\it Nonlinearity}, {\bf 8}, 1-20.}
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\bibitem{GM2}
{Gentile, G., Mastropietro, V. (1995)
Methods for the analysis of the Lindstedt series for KAM tori
and renormalizability in classical mechanics. A review with
some applications,
to appear in {\it Reviews in Mathematical Physics}. }
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\bibitem{K}
{Katznelson, Y. (1976) {\it An introduction to harmonic analysis}, Dover.}
%
\bibitem{M}
{MacKay, R.S. (1993) {\it Renormalization in area preserving maps}, World
Scientific, London.}
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\bibitem{PV}
{Percival, I., Vivaldi, F. (1988) Critical dynamics and
diagrams, {\it Physica} {\bf 33D}, 304--313. }
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\end{thebibliography}
\end{document}
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