%%%%% The paper has been composed in plain TeX. It includes all the necessary 
%%%%% macros (and even more than necessary, but hopelessly harmless).
%%%%% CAUTIONS: At the all the font definitions appear at the neginning of the 
%%%%% paper. If your installation has different fonts, replace them with 
%%%%% (possibly) equivalent ones.
%%%%% 
%%%%% A .ref file will be produced by TeX. You can delete it.
%%%%% 
\catcode`@=11
\magnification \magstep1
\tolerance=1500\frenchspacing
%
% Character fonts
%
\font\titfnt=cmbx10 scaled \magstep1 % title
\font\pagfnt=cmssi10                 % running titles
\font\parfnt=cmbx10 scaled \magstep1 % titles of sections
\font\sprfnt=cmssi10 scaled \magstep1% titles of subsections
\font\teofnt=cmssbx10                % beginning of claims (e.g.: theorem
\font\tenenufnt=cmsl10               % statement of theorems, etc. at 10 pt
\font\nineenufnt=cmsl9               % same at 9 pt
\font\eightenufnt=cmsl8              % same at 8 pt
 
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%
% Page layout
%
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\belowdisplayskip=6pt plus 3pt minus 1pt
\abovedisplayshortskip=0pt plus 3pt
\belowdisplayshortskip=4pt plus 3pt
\def\blank{\vskip 12pt}
\def\blankii{\blank\blank}
\def\blankm{\vskip 6pt}
\def\blankq{\vskip 3pt}
%
% Definizione del corpo dei caratteri e dei formati correnti
%
\def\tenpoint{\def\rm{\fam0\tenrm}%
  \def\enufnt{\tenenufnt}%
  \textfont0=\tenrm \scriptfont0=\sevenrm
\scriptscriptfont0=\fiverm
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  \def\bf{\fam\bffam\tenbf}%
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  \tt \ttglue=.5em plus.25em minus.15em
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  \let\big=\tenbig
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\scriptscriptfont0=\fiverm
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\scriptscriptfont3=\tenex
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  \textfont\itfam=\nineit
  \def\sl{\fam\slfam\ninesl}%
  \textfont\slfam=\ninesl
  \def\bf{\fam\bffam\ninebf}%
  \textfont\bffam=\ninebf \scriptfont\bffam=\sixbf
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  \tt \ttglue=.5em plus.25em minus.15em
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  \let\sc=\sevenrm
  \let\big=\ninebig
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\def\eightpoint{\def\rm{\fam0\eightrm}%
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  \def\bf{\fam\bffam\eightbf}%
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  \let\big=\eightbig
  \setbox\strutbox=\hbox{\vrule height7pt depth2pt width\z@}%
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\def\tenmath{\tenpoint\fam-1 } % use after $ in ninepoint sections
\def\tenbig#1{{\hbox{$\left#1\vbox to8.5pt{}\right.\n@space$}}}
\def\ninebig#1{{\hbox{$\textfont0=\tenrm\textfont2=\tensy
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\def\eightbig#1{{\hbox{$\textfont0=\ninerm\textfont2=\ninesy
  \left#1\vbox to6.5pt{}\right.\n@space$}}}
%
% Definizione macro di allineamento testi
%
\newcount\cont@note
\def\footnote{\advance\cont@note by 1
      \edef\@sf{\spacefactor\the\spacefactor}$^{[\the\cont@note]}$\@sf
      \insert\footins\bgroup\ninepoint
      \interlinepenalty100 \let\par=\endgraf
        \leftskip=\z@skip \rightskip=\z@skip
        \splittopskip=10pt plus 1pt minus 1pt \floatingpenalty=20000
        \smallskip\item{$^{[\the\cont@note]}$}\bgroup\strut\aftergroup\@foot\let\next}
\skip\footins=12pt plus 2pt minus 4pt % space added when footnote is present
%\count\footins=1000 % footnote magnification factor (1 to 1)
\dimen\footins=30pc % maximum footnotes per page
%
%  Registrazione referenze su file esterno e macro di conteggio
%
\newcount\numbibliogr@fi@
\global\numbibliogr@fi@=1
\newwrite\fileref
\immediate\openout\fileref=\jobname.ref
\immediate\write\fileref{\parindent 30pt}
\def\cita#1#2{\def\us@gett@{\the\numbibliogr@fi@}
\expandafter\xdef\csname:bib_#1\endcsname{\us@gett@}
           \immediate\write\fileref
           {\par\noexpand\item{{[\the\numbibliogr@fi@]\enspace}}}\ignorespaces
           \immediate\write\fileref{{#2}}\ignorespaces
           \global\advance\numbibliogr@fi@ by 1\ignorespaces}
\def\bibref#1{\seindefinito{:bib_#1}
          \immediate\write16{ !!! \string\bibref{#1} non definita !!!}
          \expandafter\xdef\csname:bib_#1\endcsname{??}\fi
      {$^{[\csname:bib_#1\endcsname]}$}}
\def\dbiref#1{\seindefinito{:bib_#1}
          \immediate\write16{ !!! \string\bibref{#1} non definita !!!}
          \expandafter\xdef\csname:bib_#1\endcsname{??}\fi
      {[\csname:bib_#1\endcsname]}}
\def\citaref#1#2{\cita{#1}{#2}\bibref{#1}}
\def\references{\immediate\closeout\fileref
                \par\goodbreak
                \blankii
                \centerline{\parfnt References}
                \nobreak\blank\nobreak
                \input \jobname.ref}
\def\element#1{\par\blank\noindent\hangindent=30pt
               \llap{#1\enspace}\ignorespaces}
\def\capo#1{\immediate\write\fileref
           {\par\hangindent\parindent }\ignorespaces
           \immediate\write\fileref{{#1}}\ignorespaces}
\def\acknowledgements{
           \blankii\noindent
           {\parfnt Acknowledgements.\quad\ignorespaces}
           }
%
% Definizione macro di intestazione
%
\def\title#1{\null\blankii\noindent{\titfnt\uppercase{#1}}\blank}
\def\riga{\par\vskip 6pt\noindent}
\def\author#1{\leftskip 1.8cm\smallskip\noindent{#1}\smallskip\leftskip 0pt}
\def\abstract#1{\par\blankii\noindent
          {\ninepoint
            {\bf Abstract. }{\rm #1}}
          \par}
\def\keywords#1{\par\blank\noindent
          {\ninepoint
            {\bf Key words. }{\rm #1}}
          \par}
\def\sunto#1{\par\blankii\noindent
          {\ninepoint
            {\bf Sunto. }{\rm #1}}
          \par}
%
% Definizione macro per la numerazione delle formule, capitoli,
% paragrafi, ecc...
%
%
%  Definizioni generali
\def\seindefinito#1{\expandafter\ifx\csname#1\endcsname\relax}
%
%  Paragrafo
\newdimen\@mpiezz@
\@mpiezz@=\hsize
\newbox\boxp@r@gr@fo
\def\section#1#2{
           \goodbreak\vskip 18pt plus 6pt\noindent\ignorespaces
              {\setbox\boxp@r@gr@fo=\hbox{\parfnt\noindent\ignorespaces
              {\csname:sec_#1\endcsname.\quad}}\ignorespaces
             \advance\@mpiezz@ by -\wd\boxp@r@gr@fo
             \box\boxp@r@gr@fo\vtop{\hsize=\@mpiezz@\noindent\parfnt #2}}
           \par\nobreak\vskip 9pt plus 3pt\nobreak
           \noindent\ignorespaces}
\def\secref#1{\seindefinito{:sec_#1}
          \immediate\write16{ !!! \string\secref{#1} non definita !!!}
 \expandafter\xdef\csname:sec_#1\endcsname{??}\fi
      \csname:sec_#1\endcsname
      }
%
%  Sottoparagrafo
\def\subsection#1#2{
           \goodbreak\vskip 9pt plus 2pt\noindent\ignorespaces
           {\setbox\boxp@r@gr@fo=\hbox{\sprfnt\noindent\ignorespaces
              {\csname:sbs_#1\endcsname\quad}}\ignorespaces
             \advance\@mpiezz@ by -\wd\boxp@r@gr@fo
 \box\boxp@r@gr@fo\vtop{\hsize=\@mpiezz@\noindent\sprfnt#2}}
           \par\nobreak\vskip 3pt plus 1pt\nobreak
           \noindent\ignorespaces}
\def\sbsref#1{\seindefinito{:sbs_#1}
          \immediate\write16{ !!! \string\sbsref{#1} non definita !!!}
 \expandafter\xdef\csname:sbs_#1\endcsname{??}\fi
      \csname:sbs_#1\endcsname
      }
%
%  Formula
\def\eqalignno#1{\leqalignno{#1}}
\def\formula#1{\leqno{(\csname:frm_#1\endcsname)}}
\def\frmref#1{\seindefinito{:frm_#1}
          \immediate\write16{ !!! \string\frmref{#1} non definita !!!}
 \expandafter\xdef\csname:frm_#1\endcsname{??}\fi
      (\csname:frm_#1\endcsname)}
%
%  Teorema
\def\endclaim{\endgroup
            \par\if F\sp@zi@tur@{\blankq}\gdef\sp@zi@tur@{T}\fi}
\def\theorem#1{
   \par\if T\sp@zi@tur@{\gdef\sp@zi@tur@{F}}\else{\blankq}\fi
   \noindent{\teofnt Theorem \csname:thr_#1\endcsname:\quad}\begingroup\enufnt
   \ignorespaces}
\def\theoremnn{
            \par\if T\sp@zi@tur@{\gdef\sp@zi@tur@{F}}\else{\blankq}\fi
            \noindent{\teofnt Theorem:\quad}\begingroup\enufnt
            \ignorespaces}
\def\theoremtx#1#2{
            \noindent{\teofnt Theorem \csname:thr_#1\endcsname:
                {\enufnt #2}.\quad}\begingroup\enufnt
            \ignorespaces}
\def\thrref#1{\seindefinito{:thr_#1}
          \immediate\write16{ !!! \string\thrref{#1} non definita !!!}
 \expandafter\xdef\csname:thr_#1\endcsname{??}\fi
      \csname:thr_#1\endcsname}
%
%  Proposizione
\def\proposition#1{
 \par\if T\sp@zi@tur@{\gdef\sp@zi@tur@{F}}\else{\blankq}\fi
 \noindent{\teofnt Proposition \csname:pro_#1\endcsname:\quad}\begingroup\enufnt
 \ignorespaces}
\def\propositiontx#1#2{
  \par\if T\sp@zi@tur@{\gdef\sp@zi@tur@{F}}\else{\blankq}\fi
  \noindent{\teofnt Proposition \csname:pro_#1\endcsname:
   {\enufnt #2}.\quad}\begingroup\enufnt
  \ignorespaces}
\def\proref#1{\seindefinito{:pro_#1}
          \immediate\write16{ !!! \string\proref{#1} non definita !!!}
 \expandafter\xdef\csname:pro_#1\endcsname{??}\fi
      \csname:pro_#1\endcsname}
%
%  Corollario
\def\corollary#1{
  \par\if T\sp@zi@tur@{\gdef\sp@zi@tur@{F}}\else{\blankq}\fi
  \noindent{\teofnt Corollary \csname:cor_#1\endcsname:\quad}\begingroup\enufnt
  \ignorespaces}
\def\corref#1{\seindefinito{:cor_#1}
          \immediate\write16{ !!! \string\corref{#1} non definita !!!}
 \expandafter\xdef\csname:cor_#1\endcsname{??}\fi
      \csname:cor_#1\endcsname}
%
%  Lemma
\def\lemma#1{
  \par\if T\sp@zi@tur@{\gdef\sp@zi@tur@{F}}\else{\blankq}\fi
  \noindent{\teofnt Lemma \csname:lem_#1\endcsname:\quad}\begingroup\enufnt
  \ignorespaces}
\def\lemref#1{\seindefinito{:lem_#1}
          \immediate\write16{ !!! \string\lemref{#1} non definita !!!}
 \expandafter\xdef\csname:lem_#1\endcsname{??}\fi
      \csname:lem_#1\endcsname}
%
%  Definizione
\def\definition#1{
  \par\if T\sp@zi@tur@{\gdef\sp@zi@tur@{F}}\else{\blankq}\fi
  \noindent{\teofnt Definition \csname:def_#1\endcsname:\quad}\begingroup\enufnt
  \ignorespaces}
\def\definitiontx#1#2{
  \par\if T\sp@zi@tur@{\gdef\sp@zi@tur@{F}}\else{\blankq}\fi
  \noindent{\teofnt Definition \csname:def_#1\endcsname:
  {\enufnt #2}.\quad}\begingroup\enufnt
  \ignorespaces}
\def\defref#1{\seindefinito{:def_#1}
          \immediate\write16{ !!! \string\defref{#1} non definita !!!}
 \expandafter\xdef\csname:def_#1\endcsname{??}\fi
      \csname:def_#1\endcsname}
%
%  Dimostrazione
\def\proof{\par\if T\sp@zi@tur@{\gdef\sp@zi@tur@{F}}\else{\blankq}\fi
    \noindent{\teofnt Proof.\quad}\begingroup\ignorespaces}
\def\prooftx#1{\par\if T\sp@zi@tur@{\gdef\sp@zi@tur@{F}}\else{\blankq}\fi
    \noindent{\teofnt Proof #1.\quad}\begingroup\ignorespaces}
\def\endproof{\nobreak\quad\nobreak\hfill\nobreak{\enufnt Q.E.D.}
    \endclaim}
%
% Figura
\newbox\boxfigur@
\newbox\comfigur@
\newdimen\@mpfigur@
\newdimen\m@rfigur@
%\m@rfigur@=.4 cm
\m@rfigur@=2 pc
\@mpfigur@=\hsize
\advance\@mpfigur@ by -2\m@rfigur@
\def\figure#1#2#3{
      \setbox\boxfigur@\vbox{\centerline{
           \vbox to #2{\vfil
           }
      }}
      \topinsert
         {\vbox{
               \vskip 1pt
               \box\boxfigur@
               \vskip 1pt}}
 \setbox\comfigur@\vtop{\hsize=\@mpfigur@\parindent 0pt
         {\ninepoint
         {\teofnt Figure \csname:fig_#1\endcsname.}\enspace{#3}}
      }
      \centerline{\box\comfigur@}
      \endinsert
      \write16{Figura {\csname:fig_#1\endcsname}.}
      }
\def\figref#1{\seindefinito{:fig_#1}
          \immediate\write16{ !!! \string\figref{#1} non definita !!!}
 \expandafter\xdef\csname:fig_#1\endcsname{??}\fi
      \csname:fig_#1\endcsname}
%
%  Osservazione/i
\def\remarks{\par\if
T\sp@zi@tur@{\gdef\sp@zi@tur@{F}}\else{\blankq}\fi
    \noindent{\teofnt Remarks.\quad}\ignorespaces}
\def\remark{\par\if
T\sp@zi@tur@{\gdef\sp@zi@tur@{F}}\else{\blankq}\fi
    \noindent{\teofnt Renark.\quad}\ignorespaces}
%
%       TAVOLE
%
\newcount \tablecount    %\tablecount is the counter for the tables
\tablecount=1
\def \tempbox {}
%   TABLE
%  *******
% Tables with caption at 9 pt; first argument (#1) is the caption; the
% second (#2) is the table. 
\long\def \table#1#2{\midinsert \vskip 3mm {\par
\noindent
     \centerline{\baselineskip=3.5mm \smallfont\noindent
     TABLE\ {\uppercase \expandafter{\romannumeral\tablecount}}}
     \par \vskip -1mm \noindent
     \setbox \tempbox 0 \hbox{\smallfont #1}
     \ifdim \wd \tempbox 0 > \hsize \unhbox \tempbox 0\par
                  \else {\hfill \box \tempbox 0\hfill}\fi}
     \par \vskip -1mm
     #2
     \endinsert
     \advance \tablecount by 1}
%
% Inibizione spaziatura prima degli enunciati
%
\def\acapo{\par\noindent}
\def\noblank{\gdef\sp@zi@tur@{T}}
%
% Definizioni per l'output
%
\nopagenumbers
\def\testos{\null}
\def\testod{\null}
\headline={\if T\tpage{\gdef\tpage{F}{\hfil}}
 \else{\ifodd\pageno\rightheadline\else\leftheadline\fi}
           \fi}
 
\gdef\tpage{T}
\def\rightheadline{\hfil{\pagfnt\testod}\hfil{\pagfnt\folio}}
\def\leftheadline{{\pagfnt\folio}\hfil{\pagfnt\testos}\hfil}
\voffset=2\baselineskip
%
\everypar={\gdef\sp@zi@tur@{F}}
\catcode`@=12
\tenpoint\rm
%
%  Definizioni simboli e operatori matematici
%
\def\pmb#1{\setbox0=\hbox{#1}\ignorespaces
    \hbox{\kern-.02em\copy0\kern-\wd0\ignorespaces
    \kern.05em\copy0\kern-\wd0\ignorespaces
    \kern-.02em\raise.02em\box0 }}
\def\gt{>}
\def\lt{<}
\def\lequal{\leq}
\def\nequal{\not=}
\def\rho{\varrho}
\def\theta{\vartheta}
\def\phi{\varphi}
\def\epsilon{\varepsilon}
\def\reali{\mathinner{\bf R}}
\def\razionali{\mathinner{\bf Q}}
\def\complessi{\mathinner{\bf C}}
\def\toro{\mathinner{\bf T}}
\def\interi{\mathinner{\bf Z}}
\def\naturali{\mathinner{\bf N}}
\def\vett#1{\pmb{$#1$}}
\def\fraz#1#2{{{#1}\over{#2}}}
\def\frac#1#2{{{#1}\over{#2}}}
\def\area{\mathop{\rm area\,}}
\def\grad{\mathop{\rm grad\,}}
\def\mod{\mathop{\rm mod}}
\def\dist{\mathop{\rm dist}}
\def\dim{\mathop{\rm dim}}
\def\codim{\mathop{\rm codim}}
\def\det{\mathop{\rm det}}
\def\Vol{\mathop{\rm Vol}}
\def\vettor{\wedge}
\def\intersezione{\cap}
\def\unione{\cup}
\def\notin{\not\in}
\def\contenuto{\subset}
\def\contiene{\supset}
\def\modulo#1{\left| #1 \right|}
\def\norma#1{{\left\| #1 \right\|}}
\def\moduloin#1#2{\left| #1 \right|_{#2}}
\def\normain#1#2{{\left\| #1 \right\|_{#2}}}
\def\poisson#1#2{\lbrace#1,#2\rbrace}
\def\diff{\mathop{\rm d}}
\def\opt{{\mathinner{\rm opt}}}
\def\eqdif{\buildrel\hbox{\sevenrm def}\over =}
%
%   Ridefinizione caratteri calligrafici
%
\def\Ascr{{\cal A}}
\def\Bscr{{\cal B}}
\def\Cscr{{\cal C}}
\def\Dscr{{\cal D}}
\def\Escr{{\cal E}}
\def\Fscr{{\cal F}}
\def\Gscr{{\cal G}}
\def\Hscr{{\cal H}}
\def\Iscr{{\cal I}}
\def\Jscr{{\cal J}}
\def\Kscr{{\cal K}}
\def\Lscr{{\cal L}}
\def\Mscr{{\cal M}}
\def\Nscr{{\cal N}}
\def\Oscr{{\cal O}}
\def\Pscr{{\cal P}}
\def\Qscr{{\cal Q}}
\def\Rscr{{\cal R}}
\def\Sscr{{\cal S}}
\def\Tscr{{\cal T}}
\def\Uscr{{\cal U}}
\def\Vscr{{\cal V}}
\def\Wscr{{\cal W}}
\def\Xscr{{\cal X}}
\def\Yscr{{\cal Y}}
\def\Zscr{{\cal Z}}
%%%%%%%%%%%%%%%   Local symbol definitions
\expandafter\edef\csname:sec_1\endcsname{1}
\expandafter\edef\csname:frm_1.1\endcsname{1}
\expandafter\edef\csname:frm_1.30\endcsname{2}
\expandafter\edef\csname:frm_1.31\endcsname{3}
\expandafter\edef\csname:sec_2\endcsname{2}
\expandafter\edef\csname:sbs_2.1\endcsname{2.1}
\expandafter\edef\csname:frm_2.1\endcsname{4}
\expandafter\edef\csname:sbs_2.2\endcsname{2.2}
\expandafter\edef\csname:frm_2.2\endcsname{5}
\expandafter\edef\csname:frm_2.3\endcsname{6}
\expandafter\edef\csname:frm_2.4\endcsname{7}
\expandafter\edef\csname:sbs_2.3\endcsname{2.3}
\expandafter\edef\csname:frm_2.5\endcsname{8}
\expandafter\edef\csname:sbs_2.4\endcsname{2.4}
\expandafter\edef\csname:frm_2.6\endcsname{9}
\expandafter\edef\csname:frm_2.7\endcsname{10}
\expandafter\edef\csname:sec_3\endcsname{3}
\def\csi{\xi}
\def\span{\mathop{\rm span}}
\def\ltsim{\hbox{\lower .5ex \hbox{$\buildrel<\over\sim$}\,}}
%FINE DEFINIZIONI LOCALI
\def\testos{A. Morbidelli and A. Giorgilli}
\def\testod{Superexponential stability of KAM tori}
%
%BIBLIOGRAFIA

\cita{arnold1}{V. I. Arnold: {\it Proof of a theorem of A. N. kolmogorov on the 
invariance of quasi--periodic motions under small perturbations of the 
Hamiltonian.} Russ. Math. Surv., {\bf 18}, 9 (1963).
}

\cita{arnold2}{V. I. Arnold: {\it M\'ethodes math\'ematiques de la m\'echanique 
classique.} Editions MIR (1976).
}

\cita{BGGS}{G. Benettin, L. Galgani, A. Giorgilli, J. M.  Strelcyn: {\it 
A proof of Kolmogorov's theorem on invariant tori using canonical 
transformations defined by the Lie method.} Il Nuovo 
Cimento, {\bf 79}, 201 (1984).
}

\cita{BGG}{G. Benettin, L. Galgani and A. Giorgilli: {\it A proof of 
Nekhoroshev's theorem for the stability times in nearly integrable 
Hamiltonian systems.} Cel. Mech., {\bf 37}, 1 (1985).
}

\cita{BG}{G. Benettin and G. Gallavotti: {\it Stability of motions near 
resonances in quasi--integrable Hamiltonian systems.} Journ. 
Stat. Phys., {\bf 44}, 293 (1986).
}

\cita{CG}{L. Chierchia and G. Gallavotti: {\it Drift and diffusion in phase 
space.} Ann. Inst. Henri Poincar\'e,(Phys. Theor.), {\bf 60}, 
1 (1994).
}

\cita{GG}{A. Giorgilli and L. Galgani: {\it Rigorous estimates for the 
series expansions of Hamiltonian perturbation theory.} 
Cel. Mech., {\bf 37}, 95 (1985).
}

\cita{GDFGS}{A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and 
C. Sim\'o: {\it Effective stability for a Hamiltonian system near an 
elliptic equilibrium point, with an application to the restricted three 
body problem.} J. Diff. Eqs., {\bf 20}, (1989).
}

\cita{kolmogorov}{A. N. Kolmogorov: {\it On the preservation of conditionally
periodic motions.} Dokl. Akad. Nauk SSSR, {\bf
98}, 527 (1954).
}

\cita{licht}{A. J. Lichtenberg and M. A. Lieberman: {\it Regular and 
stochastic motion}, Springer--Verlag, New York (1983).}

\cita{lochak}{P. Lochak: {\it Canonical perturbation theory via simultaneous 
approximations.} Uspekhi Math. Nauk, (1992).
}

\cita{MG}{A. Morbidelli and A. Giorgilli: {\it Quantitative 
perturbation theory by successive eliminations of harmonics,} Cel. 
Mech. {\bf 55}, 131 (1993).}

\cita{neishtadt}{A. I. Neishtadt: {\it Estimates in the Kolmogorov
theorem on conservation of conditionally periodic motions}, PMM U.S.S.R.
{\bf 45}, 766--772 (1982). 
}

\cita{nekh1}{N. N. Nekhoroshev: {\it Exponential estimates of the stability
time of near--integrable Hamiltonian systems.} Russ. Math.
Surveys, {\bf 32}, 1 (1977).
}
 
\cita{nekh2}{N. N. Nekhoroshev: {\it Exponential estimates of the stability
time of near--integrable Hamiltonian systems, 2.} Trudy
Sem. Petrovs., {\bf 5}, 5 (1979).
}

\cita{wiggins}{A. D. Perry and S. Wiggins: {\it KAM tori are very 
sticky: rigorous lower bounds on the time to move away from an invariant 
lagrangian torus with linear flow}, Phys. D {\bf 71}, 102--121 (1994).}

\cita{poeschel}{J. P\"oschel: {\it Nekhoroshev's estimates for
quasi--convex Hamiltonian systems}, Math. Z. {\bf 213}, 187
(1993).
} 

%FINE BIBLIOGRAFIA
%%%%%%%%%%%%%%%%%%%%%%%% Start of text %%%%%%%%%%%%%%%%%%%%%
\title{Superexponential stability of KAM tori}
 
\author{\it ALESSANDRO MORBIDELLI\hfill\break CNRS, Observatoire de la C\^ote 
d'Azur
\hfill\break BP 229, 06304 -- NICE Cedex 4, France.}

\author{\it ANTONIO GIORGILLI\hfill\break Dipartimento di Matematica dell'
Universit\`a di Milano 
\hfill\break and Gruppo Nazionale di Fisica Matematica del 
CNR,
\hfill\break Via Saldini 50, 20133 Milano, Italy.}

 
\abstract{We study the dynamics in the neighbourhood of an invariant
torus of a nearly integrable system. We provide an upper bound to the
diffusion speed, which turns out to be of superexponentially small size
$\exp(\exp(-1/\rho))$, $\rho$ being the distance from
the invariant torus. We also discuss the connection of this result with
the existence of many invariant tori close to the considered one.} 
 
\section{1}{Introduction and results}
We consider the problem of Arnold diffusion in nearly integrable
Hamiltonian systems, with the aim of producing bounds on the diffusion
speed in the spirit of Nekhoroshev's theory. We concentrate our
attention in the neighbourhood of the invariant tori, the existence of
which is guaranteed by KAM theory. We show that a clever use of the
known results of both KAM theory and Nekhoroshev's theory leads to
strong consequences, although of local character. 

The key remark is that in the neighbourhood of an invariant KAM torus it
is natural to introduce the distance from the torus as a new
perturbation parameter. Thanks to this change of perspective we can
apply a Birkhoff procedure, thus reducing the perturbation to an
exponentially small size in $1/\rho$, $\rho$ being the distance from the
invariant torus. The new action variables introduced by the Birkhoff
normalization are good adiabatic invariants, since they remove most of 
the quasiperiodic oscillation of the old action variables due to 
perturbation, namely, the so called deformation. At this point we change
again our perspective, and investigate the dynamics with the global
approach of Arnold on the one hand, and of Nekhoroshev on the other
hand. All the results follow from a direct application of known
theorems. In particular, a relevant consequence is that the speed of the
Arnold diffusion (if any) turns out to be of superexponentially small
size $\exp(\exp(-1/\rho))$. This is our main contribution. 

In view of this result, we can consider an invariant KAM torus as the
head of a structure which dominates the dynamics in its neighbourhood.
It appears that such a structure has a typical radius which depends on
the size of the perturbation, and decreases to zero as the perturbation
increases towards the critical size corresponding to the destruction of
the torus. The neighbourhood of the torus turns out to contain many
other invariant tori, constituting a set the relative volume of which
tends to 1 when approaching the head torus. More precisely, the relative
volume of the complement of the set of invariant tori turns out to be as
small as $\exp(-1/\rho)$. This represents a local improvement with
respect to the estimates more or less explicitly contained in many
previous statements: see for instance ref.~\dbiref{neishtadt}. The new
aspect that we point out is that the dynamics is strongly affected by
the existence of such a structure, inasmuch as the chaotic diffusion of
orbits starting in the gaps between tori is thus forced to require a
very long time. According to the known Nekhoroshev's estimates, such a
time is exponentially large with the inverse of the perturbation. In
our case, as we already remarked, the perturbation is exponentially
small in $1/\rho$. This determines the superexponential estimate for
the diffusion time.

The picture resulting from the discussion above contrasts with the quite
widespread opinion, especially among physicists, that the existence of
invariant tori in systems with more than two degrees of freedom is not
so relevant (see for instance~\dbiref{licht}). Such an opinion is
supported by the known fact that the tori do not isolate separated
regions in phase space, thus allowing for the so called Arnold's
diffusion. More recently, some authors have pointed out that KAM tori
are very sticky, by applying locally the Nekhoroshev theory to the
neighbourhood of an invariant torus: see for instance~\dbiref{wiggins}.
Their approach is actually equivalent to the application of the Birkhoff
normalization procedure using the distance from the invariant torus as a
perturbation parameter. However, we go beyond this level. We show in
fact that although the KAM tori are not isolating, they form
nevertheless a kind of impenetrable structure that the orbits cannot
escape (nor enter, of course) for an exceedingly long time, very large
even with respect to the known Nekhoroshev's estimates. Thus, the
behaviour of the system in the region containing invariant KAM tori can
be said to be effectively integrable. 

\vskip 3pt
We come now to a formal statement of our result. We consider a
canonical system of differential equations with an Hamiltonian of the
form 
$$
H(p,q,\epsilon) = H_0(p) +\epsilon H_1(p,q,\epsilon)\ , 
\formula{1.1}
$$
where $p\in\Gscr\subset\reali^n$, $\Gscr$ being an open set, and
$q\in\toro^n$ are the action--angle variables, and $\epsilon$ a small
parameter. The Hamiltonian is assumed to be a real analytic function of all
its variables. The frequencies of the unperturbed system will be denoted
by $\omega(p)=\frac{\partial H_0}{\partial p}$, and the Hessian matrix of
$H_0$ with respect to $p$ will be denoted by $\Cscr(p)$. Also, we shall
denote by $|\cdot|$ a norm on functions over their domain of
analyticity, for instance the usual supremum norm, and by $\|\cdot\|$ a
norm for vectors in $\reali^n$ or $\complessi^n$, for instance the
Euclidean norm. 

We pick up a point $p^*$ such that the corresponding frequency 
$\omega^*:=\omega(p^*)$ satisfies the diophantine condition
$$
|k\cdot\omega^*|\ge \gamma |k|^{-\tau}\quad {\rm for\ all}\ k\in\interi^n
\ ,\quad k\ne0\ ,
\formula{1.30}
$$
for some $\gamma\gt 0$ and $\tau\gt n-1$; here, we define
$|k|=|k_1|+\ldots+|k_n|$. We also assume that the Hamiltonian admits an
analytic bounded extension to a domain 
$$
D_{\delta}(p^*)=B_{\delta}(p^*) \times \toro^n_\delta\ ,
\formula{1.31}
$$
for some positive $\delta$. Here, $B_{\delta}(p^*)$ is the open ball of
radius $\delta$ and center $p^*$ in $\complessi^n$, and
$\toro^n_\delta=\left\{q\in\complessi^n\>:\>|{\rm
Im\,}(q)|\lt\delta\right\}$. By the analyticity of the Hamiltonian, such a
$\delta$ exists. With this setting we can state our

\theoremnn{Consider the Hamiltonian~\frmref{1.1} in the domain
$D_{\delta}(p^*)$ defined by~\frmref{1.31}, with $\omega^*\equiv\omega(p^*)$
satisfying~\frmref{1.30}, and assume that the following conditions hold
with positive constants $\epsilon$, $d\lt 1$ and $m$:
\item{(a)}$|H_1|\lt \epsilon$ in $D_{\delta}(p^*)$;
\item{(b)}$\Cscr(p)$ satisfies the nondegeneracy 
condition $d \|v\| \lt \|\Cscr\,v\| \lt d^{-1}\|v\|$;
\item{(c)}the matrix $C=\Cscr(p^*)$ satisfies $|Cv\cdot v|\gt m |v\cdot
v|$ for all $v\in\reali^n$ with $v\perp\omega^*$. 
\par\noindent
Then there exists a positive $\epsilon^*$ such that for every
$\epsilon\lt\epsilon^*$ the following statement holds true: there is a
positive $\rho^*$ depending on $\epsilon$ such that for every
$\rho\lt\rho^*$ there is an analytic canonical transformation mapping
$(J, \psi)\in B_{\rho}(0)\times\toro^n$ to $(p,q)\in
B_{\delta}(p^*)\times\toro^n$ with the properties:
\item{(i)}$J=0$ is an invariant torus carrying a quasiperiodic flow
with frequencies $\omega^*$.
\item{(ii)}The domain $D_{\rho}(0)$ contains an infinity of invariant
tori the relative volume of which tends to 1 as $\rho\to 0$, being 
estimated by $1-\exp(-1/\rho)$; the
structure of these invariant tori is close to that of the tori $J={\rm
const}$, in the sense that every invariant torus lies in a neighbourhood
of some torus $J={\rm const}$, the radius of the neighbourhood being 
estimated by $\exp[-(1/\rho)]$.
\item{(iii)}For every initial datum 
$J_0\in B_{\rho}(0)$ one has that $|J(t)-J_0|$ is estimated by 
$\exp\bigl(-(1/\rho)^{1/(\tau+1)}\bigr)$ for $|t|\le T$, where
$$
T \simeq \exp\bigl[\exp(1/\rho)^{1/(\tau+1)}\bigr]\ . 
$$ 
}\endclaim

Let us add a comment concerning the local quasi--convexity
hypothesis~(c). We use this hypothesis in connection with Nekhoroshev's
theorem. It should be remarked that the original formulation of
Nekhoroshev requires the less stringent condition of steepness, which
however is more difficult to handle. Later formulations of the
theorem make use of the easier condition of convexity, i.e.,
$|\Cscr(p)v\cdot v|\gt mv\cdot v$ for all $p\in\Gscr$ and for all
$v\in\reali^n$. However, as stressed by Nekhoroshev himself, it is
enough to require that the latter inequality holds for all
$v\perp\omega(p)$. This is the so called condition of quasi--convexity.
The fact that we restrict our attention to the neighbourhood of the
invariant torus $p^*$ allows us to further relax the quasi--convexity
condition to the form~(c). 


\section{2}{Scheme of the proof}
The proof of our theorem relies on a composition of known results,
namely of KAM theorem\bibref{kolmogorov} and Nekhoroshev's
theorem\bibref{nekh1}\bibref{nekh2}, both in a local and a global
formulation. More precisely, we need two preliminary steps in order to
prove the statement (i), and two independent steps in order to prove the
statements (ii) and (iii) respectively. The first step reduces the
Hamiltonian to Kolmogorov's normal form, thus ensuring the existence of
an invariant torus (the head torus). The second step is the construction
of a Birkhoff normal form up to a finite order in the neighbourhood of
the head torus, with exponential estimates of Nekhoroshev's type. The
mapping $(J, \psi)\to(p, q)$ referred to in the statement of the theorem
is actually the composition of the mapping leading to Kolmogorov's
normal form and of that leading to Birkhoff's normal form. The third
step is the application of KAM theorem in the global version due to
Arnold\bibref{arnold1}. The fourth and last step is the application of
Nekhoroshev's theorem. We perform the four steps above in separate
subsections. We omit all the unnecessary technical details, making a
direct use of the known theorems in some available form, without
attempting to find optimal estimates. 


\subsection{2.1}{Use of Kolmogorov's normal form}
We follow the formulation of Kolmogorov's theorem provided in
\dbiref{BGGS}. The hypotheses $(a)$ and $(b)$ allow to apply the 
theorem. Accordingly, there exists a positive $\epsilon^*$ such that for
every $\epsilon\lt\epsilon^*$ there exists a positive $\delta'\lt\delta$
and an analytic canonical transformation $(p', q')\to (p, q)$, mapping
$D_{\delta'}(0)$ into $D_{\delta}(p^*)$, which gives the Hamiltonian the
Kolmogorov's normal form 
$$ 
H'(p',q')=\sum_i\omega_i p'_i + \frac{1}{2}\sum_{i,j}C'_{i,j}p'_ip'_j 
+f(p',q')
\formula{2.1} 
$$ 
with $f$ at least quadratic in $p'$. 
In particular, the real matrix $C'_{ij}$ satisfies the conditions $(b)$ 
and $(c)$ with new constants $m'<m$ and $d'<d$, while the 
quadratic part of $f$ has zero average over the angles $q'$.


\subsection{2.2}{Use of the local formulation of Nekhoroshev's 
theorem}
We expand now the perturbation $f$ in~\frmref{2.1} in power series 
about the origin (i.e., the invariant torus). We write 
the Hamiltonian as
$$
H(p',q')= \sum_i\omega_i p'_i + H_2(p',q') + H_3(p',q') + \ldots
\formula{2.2}
$$
where $H_s(p',q')$ is for $s\ge 2$ a homogeneous polynomial of degree
$s$ in $p'$. As $f$ is analytic in $D_{\delta'}(0)$, the expansion is
convergent, and the norm of $H_s$ decreases at least geometrically with
$s$, being of order $\delta'^{s}$. Remark that the
Hamiltonian~\frmref{2.2} resembles that of a system of perturbed
harmonic oscillators, as considered for instance in~\dbiref{GDFGS}.
Thus, we can apply the local formulation of Nekhoroshev's theorem. To
this end, as usual, we perform a Birkhoff normalization up to some finite
order $r$, thus giving the Hamiltonian the form 
$$
H(J,\psi)= \sum_i \omega_i J_i + Z^{(r)}(J) + \Rscr^{(r)}(J,\psi)\ ,
\formula{2.3}
$$
which is analytic in $D_{\rho}(0)$ for some positive $\rho<\delta'$, 
depending on $r$. Here, $Z^{(r)}(J)$ is the normalized part of the 
new Hamiltonian, while $\Rscr^{(r)}(J,\psi)$ is the still non 
normalized remainder.
In particular the quadratic part of $Z^{(r)}$ is nothing but 
$(1/2)\sum_{i,j}C'_{ij}J_iJ_j$, i.e., it coincides with the average 
quadratic part of the Kolmogorov's normal form~\frmref{2.1}. Thus, the 
validity of the conditions $(b)$ and $(c)$ is preserved by the Birkhoff's 
normalization procedure. 

Standard estimates (see for instance 
\dbiref{BG}, \dbiref{GG}, 
\dbiref{GDFGS}, \dbiref{poeschel})
allow to prove that in $D_{\rho}(0)$ the size of the remainder is of order 
$(r!)^{\tau+1}\rho^{r}$. An optimal choice of $r$ as a function of 
$\rho$, i.e., $r\simeq (1/\rho)^{1/(\tau+1)}$, allows one to prove that 
there exists a positive $\overline\rho$ 
such that for $\rho\lt\overline\rho$ one has 
$$
\left|\Rscr^{(r+1)}\right| \lt A \left|f\right| 
\exp\left[-\left(\frac{\overline\rho}{\rho}\right)^{1/(\tau+1)}\right]\ ,
\formula{2.4}
$$
with a constant  $A$ depending on the number $n$ of degrees of freedom. 

\subsection{2.3}{Use of Arnold's formulation of KAM theorem}
We rewrite the Hamiltonian~\frmref{2.3} in the form 
$H(J,\psi)=H_0(J)+H_1(J,\psi)$, where 
$$
H_0(J)=\sum_i \omega_i J_i + Z^{(r)}(J)\ ,\quad 
H_1(J,\psi)=\Rscr^{(r)}(J,\psi)\ .
\formula{2.5}
$$
To this Hamiltonian we apply the statement of the main theorem in
\dbiref{arnold1}. The analyticity hypothesis is clearly satisfied, as
well 
as the condition $|H_1|\lt M$ for some $M$. Indeed, in view of the result 
of the previous section, the remainder $\Rscr^{(r)}(p,q)$ is an analytic
function in a domain $D_{\rho}$ for some positive $\rho$, and is 
bounded there by $A \left|f\right| 
\exp\left[-\left({\overline\rho/\rho}\right)^{1/(\tau+1)}\right]$. We stress 
that, choosing $\rho$ small enough, the size of $H_1$ can be made 
arbitrarily small, exponentially with $1/\rho$. Concerning the 
nondegeneracy condition
$$
\det\left|\Cscr'(J)\right| \ne 0\ ,\quad {\rm with}\ 
\Cscr'_{i,j}=\frac{\partial^2 H_0}{\partial J_i\partial J_j} 
\ ,
$$
which must hold for all $J$ in the domain where the theorem is applied,
we remark that it holds in some neighbourhood of the torus $J=0$. Indeed,
it holds for $J=0$ since one clearly has $\Cscr'(0)=C'_{ij}$, where 
$C'_{ij}$ is just 
the matrix appearing in \frmref{2.1}, and so, by continuity,
it holds in a neighbourhood of $J=0$. Thus, a straightforward
application of Arnold's theorem shows that there exists a positive 
$\rho_1<\overline\rho$ such that for $\rho<\rho_1$
the relative volume of the
KAM tori in the neighbourhood of $J=0$ is large, and tends to 1 as 
$\rho\to 0$. According to Neishtadt\bibref{neishtadt}, the volume of the 
complement of the set of invariant tori is estimated by 
$\sqrt{\epsilon}$, where $\epsilon$ is the size of the perturbation. In 
our case we replace $\epsilon$ by $\exp(-1/\rho)$. 
This proves the statement $(ii)$.

\subsection{2.4}{Use of Nekhoroshev's theorem}
As in the last section, we rewrite the Hamiltonian as
$H(J,\psi)=H_0(J)+H_1(J,\psi)$, with $H_0$ and $H_1$ given by~\frmref{2.5}.
We use the formulation of Nekhoroshev's theorem given in \dbiref{BGG}. Again,
the analyticity and nondegeneracy conditions are satisfied, as was
already remarked. The condition on the smallness of the perturbation is
satisfied too, due to the exponential decrease of the remainder.
Instead of the convexity condition required in \dbiref{BGG},
$$
\|\Cscr'(p) v\cdot v\| \ge mv\cdot v\ {\rm for\ all}\ v\in\reali^n
\formula{2.6}
$$
we use the quasi--convexity, i.e., we add the condition 
$v\perp\omega(p)$ as remarked in the introduction. In view of the 
hypothesis $(c)$, such a condition is clearly satisfied in some 
neighbourhood of $p=0$. Thus, we apply Nekhoroshev's theorem, according 
to which, denoting by $\epsilon$ the size of the perturbation, 
for $\epsilon$ smaller than a
positive $\overline\epsilon$ one has
$$
|J(t)-J(0)|\lt \Pscr \epsilon^{1/c}
\quad {\rm for}\quad |t|\le \Tscr \left(\frac{1}{\epsilon}\right)^{1/2} 
\exp\left[\left(\frac{1}{\epsilon}\right)^{1/c}\right]\ ,
\formula{2.7}
$$
with constants $\Pscr$, $\Tscr$ and $c\simeq n^2$.
(We stress that the 
estimates are not optimal: for better estimates see for instance 
\dbiref{lochak}, \dbiref{poeschel}.)
In our case 
the size of the perturbation is $\epsilon=A \left|f\right|
\exp\left[-\left({\overline\rho/\rho}\right)^{1/(\tau+1)}\right]$. 
We conclude that there exists a positive $\rho_2$ such that for every 
$\rho<\rho_2$ the upper bound to the diffusion time is of 
order $\exp\left[\exp(1/\rho)\right]$, as claimed. This proves the 
statement $(ii)$. The constant $\rho^*$ in the statement of the theorem 
must be identified with the minimum between $\rho_1$ and $\rho_2$.

\section{3}{Discussion}
We discuss here some points that we consider relevant for
interpretation and use of our result. We discuss in particular three
points. (i)~The possible optimality of our estimate,
and the relation with some recent results of Chierchia and
Gallavotti\bibref{CG} on the existence of diffusion. (ii)~The possible
interpretation of the known phenomenon of the existence of a quite sharp
threshold separating order from chaos in nearly integrable systems.
(iii)~The applicability of the same approach to different situations, for
instance, the case of an elliptic equilibrium. 

\vskip 2pt
Concerning the first point, namely the optimality of the results, a
remark is in order. There is a widespread opinion that Nekhoroshev's
theorem is in some sense optimal, although evidence of this fact has
never been produced. Thus, the following question naturally comes to
mind: the Nekhoroshev's procedure has been already used in
sect.~\sbsref{2.2}; how can we go further in optimization using again
the Nekhoroshev's approach (as is done in sect.~\sbsref{2.4})? 

To clarify this point, let us consider again the 
Hamiltonian~\frmref{1.1}. Attempting to transform that system into an 
integrable one up to terms of order $\epsilon^2$ is clearly impossible, 
as remarked long ago by Poincar\'e, unless some restriction is imposed 
either on the domain, or on the Fourier terms to be removed, or both. 
Indeed, the existence of resonances actually causes a change of the 
actions (in resonant zones) with speed of order $\epsilon$. 
Nekhoroshev's theory, in its most general formulation, states essentially
that the main effect of the perturbations reduces to an oscillation 
along a direction of fast drift, just due to the resonance. Such an  
oscillation is necessarily bounded as far as the resonant zones do not 
overlap: actually, the strongest condition in the so called geometric 
part of the Nekhoroshev's theorem is precisely the non overlapping of 
resonances. Superimposed to the oscillation there could be a very slow 
diffusion, but the diffusion speed is exponentially slow with the 
inverse of the perturbation. In brief, a perturbation of order $\epsilon$
causes an oscillation of order, e.g., $\epsilon^{1/2}$, but a diffusion 
of order $\exp(-1/\epsilon)$ only. 

Let us now come back to our problem. The procedure illustrated in
sect.~\sbsref{2.2} is an attempt to prove that in the neighbourhood of a
torus the system is still integrable, in Birkhoff's spirit. Such a
procedure is successful up to a certain finite order, because the
diophantine condition on the frequencies ensures that in the
neighbourhood of the torus there are no resonances of low order. This
can be understood on the basis of the following heuristic argument. An
elementary estimate shows that inside a ball of radius $\rho$ there are no
resonances of order lower than $\rho^{-1/(\tau+1)}$. On the other hand,
due to the analyticity of the Hamiltonian, the coefficient of a
resonance of order $s$ is of order $\exp(-s)$. Thus, in a ball of radius
$\rho$, Birkhoff's procedure stops when one encounters a resonance of order
$\rho^{-1/(\tau+1)}$, causing a perturbation of order
$\exp(-\rho^{-1/(\tau+1)})$. This is indeed the exponential remainder
given by the local Nekhoroshev's estimate in the neighbourhood of radius
$\rho$ of the torus. 

At this point comes the new idea that we introduce in the present work:
we add the geometric part of the Nekhoroshev's theorem.
This is justified by the following standard argument. It is known that
the number of resonances of order $s$ increases as $s^{n}$, where $n$
is the number of degrees of freedom; on the other hand, the size of a
resonant region is of order $\exp(-s)$, being essentially controlled by
the coefficient of the resonant term. Thus, approaching the torus,
resonances do accumulate geometrically with their order, but each of
them controls a region of an exponentially small volume. One thus concludes
that the total relative volume of the resonant regions is small, which
precludes overlapping. This fact, on the one hand, leaves enough space
free from resonances to allow the existence of a set of invariant tori
of large relative measure. On the other hand, the general conclusion of
Nekhoroshev's theorem that a perturbation of order $\epsilon$
causes a diffusion at most of order $\exp(-1/\epsilon)$ applies here
too. The superexponential estimate in this case follows from
$\epsilon\simeq\exp(-1/\rho)$.

We stress that the superexponential estimate cannot be directly
obtained from Kolmogorov's normal form, without the Birkhoff
normalization of the second step. Heuristically, this can be justified
as follows. Nekhoroshev's theory, including the geometric part, is
developed making always reference to the unperturbed action variables.
The main contribution to the change in time of such quantities is due
to the so called deformation, which in the case of Kolmogorov's normal
form is estimated by some power of $\rho$. On the other hand, when the
distance $\rho$ from the invariant torus decreases to zero a
deformation bounded only by a power of $\rho$ would forbid a consistent
construction of the geography of resonances, as required by the
geometric part of Nekhoroshev's theorem. The good action variables 
introduced by Birkhoff's normalization procedure allow us to overcome 
this difficulty.

The natural question now is: is our new result optimal? We believe that
the answer is yes. Indeed, the recent paper by Chierchia and
Gallavotti\bibref{CG}, although not applicable to our case in a
straightforward manner, suggests that a resonance of multiplicity one
and size $\epsilon$ gives macroscopic diffusion along the resonance line
in a time proportional to $\exp(1/\epsilon)$. Since in the vicinity of a
KAM torus the size of the strongest resonances is $\exp(-1/\rho)$, one
has to expect that diffusion, if any, actually requires a time of order
$\exp(\exp(1/\rho))$. 

\vskip 2pt
We come now to the point (ii), namely the problem of the existence of
thresholds for transition from order to chaos. Here, our argument is
completely heuristic. The classical exponential bound, as given  by
local Nekhoroshev--like results, suggests that diffusion takes place in
a rather smooth way, although quite rapidly, when the perturbation
parameter is increased. Conversely, the numerical simulations shows
that diffusion takes place in a very sharp fashion, hardly compatible
with the Nekhoroshev's exponential. Here we understand that such a sharp
change from order to chaos corresponds to a transition from resonance
non--overlapping to resonance overlapping, which completely changes  the
structure of orbits in phase space and activates different mechanisms of
chaotic diffusion, the diffusion speed passing from exponentially slow
with respect to resonance strength to directly proportional to resonance
strength.

\vskip 2pt
Concerning the point (iii), we discuss on the one hand 
the application to the case of an 
elliptic equilibrium, and, on the other hand, a possible extension to 
our previous work~\dbiref{MG}.

We first consider the case of an elliptic equilibrium point with the
frequencies of the harmonic part of the Hamiltonian satisfying a
diophantine condition~\frmref{1.30}. The main remark is that such an
equilibrium is nothing but a degenerate invariant torus. Thus, one can
follow the same steps as above, just skipping the construction of 
Kolmogorov's normal form. The application of Arnold's theorem then leads
to the existence of a set of invariant tori, the relative volume of which
tends to 1 as $1-\exp(-1/\rho)$ in the limit $\rho\to 0$, ($\rho$ being 
in this case the 
distance from the equilibrium point); this improves a little, from a 
quantitative viewpoint, the statement in~\dbiref{arnold2}, appendix 8.
The application of Nekhoroshev's theorem is instead new: if the
convexity condition (c) of the theorem 
is satisfied, then one obtains a superexponential estimate of the
diffusion time, instead of the usual exponential one.

We come now to our previous result in~\dbiref{MG}. We considered there
an Hamiltonian of the form~\frmref{1.1}, and proved that the diffusion
speed can be made arbitrarily small provided one starts close enough to
an invariant torus. However, we could not recover a superexponential
estimate because we did not actually use in a complete fashion the
powerfulness of the geometric part of the Nekhoroshev's theorem. A
straightforward application of this idea leads to a quantitative
reformulation of that result, stating that for $\epsilon$ sufficiently
small there exists a domain, $D_\epsilon$ say, which contains open balls
of radius $\epsilon$, characterized by a diffusion speed of order
$\exp\bigl(-\exp(1/\epsilon)\bigr)$. The domain $D_\epsilon$ can be
identified with the nonresonant region of the geometric construction by
Nekhoroshev. 

\acknowledgements
We are grateful to D. Bambusi, G. Benettin, F. Fass\`o and L. Galgani 
(in alphabetical order) for very useful discussions and suggestions 
during the preparation of this manuscript.

\references
\bye