Content-Type: multipart/mixed; boundary="-------------9811271558178" This is a multi-part message in MIME format. ---------------9811271558178 Content-Type: text/plain; name="98-734.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-734.comments" Three figures are automatically produced from the tex: they are printed if the dvi file is transformed into a ps file via dvips ---------------9811271558178 Content-Type: text/plain; name="98-734.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-734.keywords" Arnold diffusion, homoclinic splitting, KAM ---------------9811271558178 Content-Type: application/x-tex; name="hj2.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="hj2.tex" %**start of header % hj2.tex \def\Di{27 Novembre 1998} \headline{\hss \ottorm Draft \#20} \newcount\mgnf\newcount\tipi\newcount\tipoformule \newcount\aux\newcount\piepagina\newcount\xdata % \mgnf=0 \aux=1 %1 produce aux \tipoformule=1 %0 usa aux; 1 no (usa i simboli dati) \piepagina=1 %0 =data e #par.#pag; 1=data e #pag; 2=#pag \xdata=1 %0 data del giorno, 1 data fissa da \Di: \ifnum\mgnf=1 \aux=0 \tipoformule =1 \piepagina=1 \xdata=1\fi \newcount\bibl %\bibl= ? % 0= rif [XXX], 1= rif. numerici \ifnum\mgnf=0\bibl=0\else\bibl=1\fi \bibl=0 % Per poter cambiare a piacimento il formato dei riferimenti % bibliografici in .tex: % % 1: citare nella forma esemplificata da \ref{B}{2}{20}} % ove XXX e' un simbolo per le iniziali e 2 distingue i lavori con % le stesse iniziali, 7 e' il numero SIMBOLICO del riferimento per XXX2. % Il numero 7 puo' essere ARBITRARIO e viene automaticamente % riaggiustato al momento della compilazione (vedi punto 4) % % 2: Se si sceglie \bibl=0 si cita nella forma [XXX2]; se si sceglie % \bibl=1 si cita nella forma [numero di ordine di prima citazione]. % % 3: La bibliografia va scritta nella forma \def{\qqq}{} % in ordine alfabetico per autore attribuendo un simbolo % qualsiasi al testo che (usando ref.b) produce fin.tex e .tex % con i riferimenti giusti % in \bibl=1 e la si ricompila e stampa. La scheda iniziale .tex % diventa .old. \ifnum\bibl=0 %\openout8=ref.b \def\ref#1#2#3{[#1#2]\write8{#1@#2}} \def\rif#1#2#3#4{\item{[#1#2]} #3} \fi %\def\rif#1#2#3#4{\write9{\noexpand\raf{#1}{#2}{\noexpand#3}{#4}}} \ifnum\bibl=1 \openout8=ref.b \def\ref#1#2#3{[#3]\write8{#1@#2}} \def\rif#1#2#3#4{} \def\raf#1#2#3#4{\item{[#4]}#3} \fi \def\9#1{\ifnum\aux=1#1\else\relax\fi} \ifnum\piepagina=0 \footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglio\hss}\fi \ifnum\piepagina=1 \footline={\rlap{\hbox{\copy200}} \hss\tenrm \folio\hss}\fi \ifnum\piepagina=2\footline{\hss\tenrm\folio\hss}\fi \ifnum\mgnf=0 \magnification=\magstep0 \hsize=16.4truecm\vsize=21truecm \parindent=4.pt\fi \ifnum\mgnf=1 \magnification=\magstep1 \hsize=16.0truecm\vsize=22.5truecm\baselineskip14pt\vglue5.0truecm \overfullrule=0pt \parindent=4.pt\fi \let\a=\alpha\let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon \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 \let\U=\Upsilon {\count255=\time\divide\count255 by 60 \xdef\oramin{\number\count255} \multiply\count255 by-60\advance\count255 by\time \xdef\oramin{\oramin:\ifnum\count255<10 0\fi\the\count255}} \def\ora{\oramin } %\Di e' definito all' inizio \ifnum\xdata=0 \def\data{\number\day/\ifcase\month\or gennaio \or febbraio \or marzo \or aprile \or maggio \or giugno \or luglio \or agosto \or settembre \or ottobre \or novembre \or dicembre \fi/\number\year;\ \ora} \else \def\data{\Di} \fi \setbox200\hbox{$\scriptscriptstyle \data $} \newcount\pgn \pgn=1 \def\foglio{\number\numsec:\number\pgn \global\advance\pgn by 1} \def\foglioa{A\number\numsec:\number\pgn \global\advance\pgn by 1} \global\newcount\numsec\global\newcount\numfor \global\newcount\numfig \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3} \else \write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi} \def\etichetta(#1){(\veroparagrafo.\veraformula) \SIA e,#1,(\veroparagrafo.\veraformula) \global\advance\numfor by 1 \9{\write15{\string\FU (#1){\equ(#1)}}} \9{ \write16{ EQ \equ(#1) == #1 }}} \def \FU(#1)#2{\SIA fu,#1,#2 } \def\etichettaa(#1){(A\veroparagrafo.\veraformula) \SIA e,#1,(A\veroparagrafo.\veraformula) \global\advance\numfor by 1 \9{\write15{\string\FU (#1){\equ(#1)}}} \9{ \write16{ EQ \equ(#1) == #1 }}} \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} \global\advance\numfig by 1 \9{\write15{\string\FU (#1){\equ(#1)}}} \9{ \write16{ Fig. \equ(#1) ha simbolo #1 }}} \newdimen\gwidth \def\BOZZA{ \def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2 {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}} } \def\alato(#1){} \def\galato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\verafigura{\number\numfig} \def\geq(#1){\getichetta(#1)\galato(#1)} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\write16{No translation for #1} \else\csname fu#1\endcsname\fi} \def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi} \openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13\fi \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi \9{\openout15=\jobname.aux} \newskip\ttglue \font\titolone=cmbx12 scaled \magstep1 \font\titolo=cmbx10 scaled \magstep1 \font\ottorm=cmr8\font\ottoi=cmmi7\font\ottosy=cmsy7 \font\ottobf=cmbx7\font\ottott=cmtt8\font\ottosl=cmsl8\font\ottoit=cmti7 \font\sixrm=cmr6\font\sixbf=cmbx7\font\sixi=cmmi7\font\sixsy=cmsy7 \font\tenmib=cmmib10\font\sevenmib=cmmib10 scaled 700 \font\fivemib=cmmib10 scaled 500 \textfont5=\tenmib \scriptfont5=\sevenmib \scriptscriptfont5=\fivemib %\fivei \font\fiverm=cmr5\font\fivesy=cmsy5\font\fivei=cmmi5\font\fivebf=cmbx5 \def\ottopunti{\def\rm{\fam0\ottorm}\textfont0=\ottorm% \scriptfont0=\sixrm\scriptscriptfont0=\fiverm\textfont1=\ottoi% \scriptfont1=\sixi\scriptscriptfont1=\fivei\textfont2=\ottosy% \scriptfont2=\sixsy\scriptscriptfont2=\fivesy\textfont3=\tenex% \scriptfont3=\tenex\scriptscriptfont3=\tenex\textfont\itfam=\ottoit% \def\it{\fam\itfam\ottoit}\textfont\slfam=\ottosl% \def\sl{\fam\slfam\ottosl}\textfont\ttfam=\ottott% \def\tt{\fam\ttfam\ottott}\textfont\bffam=\ottobf% \scriptfont\bffam=\sixbf\scriptscriptfont\bffam=\fivebf% \textfont5\sevenmib\scriptfont5=\fivei \def\bf{\fam\bffam\ottobf} \tt\ttglue=.5em plus.25em minus.15em% \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \normalbaselineskip=9pt\let\sc=\sixrm\normalbaselines\rm} \catcode`@=11 \def\footnote#1{\edef\@sf{\spacefactor\the\spacefactor}#1\@sf \insert\footins\bgroup\ottopunti\interlinepenalty100\let\par=\endgraf \leftskip=0pt \rightskip=0pt \splittopskip=10pt plus 1pt minus 1pt \floatingpenalty=20000 \smallskip\item{#1}\bgroup\strut\aftergroup\@foot\let\next} \skip\footins=12pt plus 2pt minus 4pt\dimen\footins=30pc\catcode`@=12 \let\nota=\ottopunti\newdimen\xshift \newdimen\xwidth \newdimen\yshift \def\ins#1#2#3{\vbox to0pt{\kern-#2 \hbox{\kern#1 #3}\vss}\nointerlineskip} \def\eqfig#1#2#3#4#5{ \par\xwidth=#1 \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2 \yshift=#2 \divide\yshift by 2 \line{\hglue\xshift \vbox to #2{\vfil #3 \special{psfile=#4.ps} }\hfill\raise\yshift\hbox{#5}}} \def\8{\write13} \def\figini#1{\catcode`\%=12\catcode`\{=12\catcode`\}=12 \catcode`\<=1\catcode`\>=2\openout13=#1.ps} \def\figfin{\closeout13\catcode`\%=14\catcode`\{=1 \catcode`\}=2\catcode`\<=12\catcode`\>=12} \def\didascalia#1{\vbox{\nota\0#1\hfill}\vskip0.3truecm} \def\T#1{#1\kern-4pt\lower9pt\hbox{$\widetilde{}$}\kern4pt{}} \let\dpr=\partial\def\Dpr{{\V\dpr}} \let\io=\infty\let\ig=\int \def\fra#1#2{{#1\over#2}}\def\media#1{\langle{#1}\rangle}\let\0=\noindent \def\guida{\leaders\hbox to 1em{\hss.\hss}\hfill} \def\tende#1{\vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hglue3.pt${\scriptstyle #1}$\hglue3.pt\crcr}}} \def\otto{\ {\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}\ } \def\tto{{\Rightarrow}} \def\pagina{\vfill\eject}\let\ciao=\bye \def\dt{\displaystyle}\def\txt{\textstyle} \def\tst{\textstyle}\def\st{\scriptscriptstyle} \def\*{\vskip0.3truecm} \def\lis#1{{\overline #1}}\def\etc{\hbox{\it etc}}\def\eg{\hbox{\it e.g.\ }} \def\ap{\hbox{\it a priori\ }}\def\aps{\hbox{\it a posteriori\ }} \def\ie{\hbox{\it i.e.\ }} \def\fiat{{}} \def\\{\hfill\break} \def\={{ \; \equiv \; }} \def\Im{{\rm\,Im\,}}\def\Re{{\rm\,Re\,}} \def\sign{{\rm sign\,}}\def\atan{{\,\rm arctg\,}} \def\annota#1{\footnote{${}^#1$}} \ifnum\aux=1\BOZZA\else\relax\fi \ifnum\tipoformule=1\let\Eq=\eqno\def\eq{}\let\Eqa=\eqno\def\eqa{} \def\equ{{}}\fi \def\defi{\,{\buildrel def \over =}\,} \def\pallino{{\0$\bullet\;$}}\def\1{\ifnum\mgnf=0\pagina\else\relax\fi} \def\W#1{#1_{\kern-3pt\lower6.6truept\hbox to 1.1truemm {$\widetilde{}$\hfill}}\kern2pt\,} \def\Re{{\rm Re}\,}\def\Im{{\rm Im}\,}\def\DD{{\cal D}} \def\igb{ \mathop{\raise4.pt\hbox{\vrule height0.2pt depth0.2pt width6.pt} \kern0.3pt\kern-9pt\int}} \def\FINE{ \* \0{\it Internet: Authors' preprints at: {\tt http://ipparco.roma1.infn.it} \0\sl e-mail: {\it users:} giovanni, gentile, vieri, {\it address}: {\tt @ipparco.roma1.infn.it} }} \def\V#1{{\,\underline#1\,}} \def\GG{{\cal G}}\def\JJ{{\cal J}}\def\RR{{\cal R}} \def\NN{{\cal N}}\def\CC{{\cal C}}\def\EE{{\cal E}} \def\KK{{\cal K}}\def\FFFF{{\cal F}}\def\UU{{\cal U}} \def\MM{{\cal M}}\def\HH{{\cal H}}\def\PP{{\cal P}} \def\II{{\cal I}}\def\TT{{\cal T}}\def\TJ{{\tilde J}} \def\OO{{\cal O}}\def\LL{{\cal L}}\def\VV{{\cal V}} \def\KJ{{\bf K}} \def\WW{{\cal W}} \def\dn{{\,{\rm dn}\,}}\def\sn{{\,{\rm sn}\,}} \def\cn{{\,{\rm cn}\,}}\def\am{{\,{\rm am}\,}} \def\atan{{\,{\rm arctg}\,}} \def\giu{{\downarrow}}\def\su{{\uparrow}} \def\Val{{\,{\rm Val}\,}} \def\AA{{\bf A}}\def\XX{{\bf X}}\def\FF{{\bf F}} \def\UU{{\bf U}}\def\QQ{{\bf Q}} \def\V0{{\bf 0}} \def\nvec{{\underline{\nu}}}\def\mmm{{\underline{m}}}\def\Zz{{\underline{Z}}} \mathchardef\aa = "050B \mathchardef\bb = "050C \mathchardef\xxx= "0518 \mathchardef\zz = "0510 \mathchardef\oo = "0521 \mathchardef\ll = "0515 \mathchardef\mm = "0516 \mathchardef\Dp = "0540 \mathchardef\H = "0548 %\mathchardef\FFF= "0546 \mathchardef\ppp= "0570 \mathchardef\nn = "0517 \mathchardef\pps= "0520 \mathchardef\XXX= "0504 \mathchardef\FFF= "0508 \def\ndpr{{\kern1pt\raise 1pt\hbox{$\not$}\kern1pt\dpr\kern1pt}} \def\Ndpr{{\kern1pt\raise 1pt\hbox{$\not$}\kern0.3pt\dpr\kern1pt}} \def\hdp{{\h^{\fra12}}}\def\hdm{{\h^{-\fra12}}} \font\cs=cmcsc10 \font\ss=cmss10 \font\sss=cmss8 \font\crs=cmbx8 \font\msytw=msbm10 scaled\magstep1 \font\msytww=msbm8 scaled\magstep1 \font\msytwww=msbm7 scaled\magstep1 \def\RRR{\hbox{\msytw R}} \def\rrrr{\hbox{\msytww R}} \def\rrr{\hbox{\msytwww R}} \def\CCC{\hbox{\msytw C}} \def\cccc{\hbox{\msytww C}} \def\ccc{\hbox{\msytwww C}} \def\NNN{\hbox{\msytw N}} \def\nnnn{\hbox{\msytww N}} \def\nnn{\hbox{\msytwww N}} \def\ZZZ{\hbox{\msytw Z}} \def\zzzz{\hbox{\msytww Z}} \def\zzz{\hbox{\msytwww Z}} \def\TTT{\hbox{\msytw T}} \def\tttt{\hbox{\msytww T}} \def\ttt{\hbox{\msytwww T}} %**end of header \fiat \null \hskip1.truecm \centerline{\titolone Lindstedt series and Hamilton--Jacobi equation} \centerline{\titolone for hyperbolic tori in three time scales problems} \*\* \centerline{\titolo G. Gallavotti, G. Gentile, V. Mastropietro} \* \centerline{Universit\`a di Roma 1,2,3 } \centerline{\Di} \vskip.8truecm \line{\vtop{ \line{\hskip1.5truecm\vbox{\advance \hsize by -3.1 truecm \\{\cs Abstract.} {\it Interacting systems consisting of two rotators and a pendulum are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of unstable quasi periodic motions in phase space is studied via Lindstedt series. The result is a strong improvement, compared to our previous results, on the domain of validity of bounds that imply existence of invariant tori, large homoclinic angles, long heteroclinic chains and drift--diffusion in phase space.}}\hfill} }} \vskip1.5truecm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0{\titolo \S 1. The Hamiltonian system.} \numsec=1\numfor=1\* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0{\bf 1.1.} Let $(\f,\a_1,\a_2)=(\f,\aa)\in \TTT^3$ be three angles (\ie positions on circles); let $(I,A_1,A_2)=(I,\AA)\in \RRR^3$ be their conjugate momenta (or ``{\it actions}''). We consider the Hamiltonian function, depending on two parameters $\e,\h>0$, defined by % $$ \HH = \hdp \O_{1}A_1 + \h \fra{A_1^2}{2J} + \hdm \O_{2} A_2 + \fra{I^2}{2J_0} +J_0 g_0^2 (\cos\f-1)+\e f(\f,\a_1,\a_2) \; , \Eq(1.1) $$ % with $f$ an {\it even} trigonometric polynomial of degree $N,N_0$ in $\aa,\f$ respectively; $\O_{1},\O_{2},J,J_0,g_0$ are positive constants. This system describes two rotators (one anisochronous, labeled $\#1$, and one isochronous, labeled $\#2$) interacting with a pendulum which has its free (\ie with $\e=0$) unstable equilibrium position at $I=0,\f=0$ and the stable one at $I=0,\f=\p$. The scale of frequency of the pendulum is $O(1)$ in $\h$; at the same time the two rotators rotate at constant speed $O(\hdp)$ and $O(\hdm)$ respectively. Hence the system has three time scales: we assume $\h<1$ so that the {\it slow} rotator is the \#1 rotator. The free motion admits invariant tori of dimension $2$ (namely parameterized by $\AA$, a constant vector, by $\aa$ arbitrary, and with $I=0$, $\f=0$) which are unstable and possess stable (labeled $+$) and unstable (labeled $-$) $3$--dimensional manifolds (parameterized by $\AA$, the same constant vector, by $\aa,\f$ arbitrary, and with $I=\pm J_0g_0\sqrt{2(1-\cos\f)}$). We shall study properties that eventually hold when $\h\to0$. It is well known ([HMS,CG] for instance) that if $\e$ is small most of the unperturbed tori and their manifolds still exist, just a little deformed. This means that (under the condition stated below) there exist functions $\UU^\pm_{\AA'}(\f,\aa)$ and $V^\pm_{\AA'}(\f,\aa)$ which are divisible by $\e$ and analytic in $\aa,\f,\e$, for $\aa\in \TTT^2$, $|\f|<2\p$, $|\e|<\e_0$, with $\e_0$ small enough, such that an initial datum starting on the ($3$--dimensional) surfaces $W_\e^{\s}$, $\s=\pm$, defined as % $$ \AA^\s(\f,\aa)=\AA'+\UU^\s_{\AA'}(\f,\aa) \; , \qquad I^\s(\f,\aa) =\pm J_0g_0\,\sqrt{2(1-\cos\f)}+ V^\s_{\AA'}(\f,\aa) \; , \Eq(1.2)$$ % evolves, when the time $t\to\pm\io$, tending to be confused with a quasi periodic motion on a invariant torus $\TT(\AA')$, with rotation vector % $$ \oo'=(\o'_1,\o'_2) \; , \qquad %{\buildrel def\over \to} \qquad \o'_1 \defi \hdp\O_{1}+\h J^{-1} A'_1 \; , \qquad \o'_2 \defi \hdm\O_{2} \; , \Eq(1.3)$$ % and furthermore such asymptotic motion takes place with $\AA$ moving quasi periodically {\it with average} $\AA'$. {\it All this holds if $\oo'$ verifies the Diophantine condition} % $$|\oo'\cdot\nn|> C |\nn|^{-\t} \; , \qquad \forall \nn\in\ZZZ^2\setminus\{\V0\} \; , \Eq(1.4)$$ % for $C,\t>0$ (which may depend also on $\h$). The values of $\e$ for which we shall be able to prove the above will be so small that the part of the stable and unstable manifolds with $|\f|< \fra32\p$ {\it can be represented as a graph of $\AA,I$ over $\aa,\f$}.\annota1{Note that if $\e=0$ they are graphs over $\aa,\f$ for $|\f|$ smaller than {\it any} prefixed quantity $<2\p$.} Hence we look, since the beginning, for invariant tori which have the latter property. The approach to the invariant tori, of the points that lie on their stable manifolds, will be exponential in the sense that their distances $d(t)$ to the tori will be such that % $$ \lim_{t\to+\io} t^{-1}\log d(t)^{-1}=\lis g_0 \; , \qquad \lis g_0\= \lis g_0(\e)\defi(1+\G(\e,g_0))\,g_0 \; , \Eq(1.5) $$ % for a suitable analytic function $\G(\e,g_0)$, divisible by $\e$. We shall call $\lis g_0$ the {\it Lyapunov exponent} of the torus (it will depend on $\e$ as well as on the considered torus, \ie on $\oo'$ and on $\h$). The exponent relative to the approach to the same torus along its unstable manifold (as $t\to-\io$) will be the same, by time reversal symmetry defined below. We fix throughout the paper $\t$ ($\t\ge1$) and {\it we shall mainly study the dependence of $\e_0$, \ie our {\it estimate} for the analyticity radius, as a function of $\h$: $\e_0=\e_0(C,\h)$.} \* \0{\bf 1.2.} {\cs Remark.} The relation \equ(1.3) between the value of the average action and the rotation vector is non trivial and it has been named in [G1,G2] (where it was pointed out) by saying that the tori of \equ(1.1) are ``torsion free'' or ``twistless''. It is a remarkable symmetry property of \equ(1.1), see [G1,Ge2,GGM3]. \* \0{\bf 1.3.} If $\e=0$ the stable and unstable manifolds coincide (because the pendulum separatrix is degenerate); it is a degeneracy that is lost when $\e\ne0$ and generically the manifolds will have only pairwise isolated trajectories in common, called {\it homoclinic trajectories}. Nevertheless time reversal symmetry and parity symmetry\annota2{The latter symmetry is due to the assumption of evenness of $f$.} hold for \equ(1.1). If $S^t$ denotes the time evolution and the involution map $i$ (composition of parity and time reversal) is defined by $i(\f,\aa,I,\AA)=(2\p-\f,-\aa,I,\AA)$, then $iS^t=S^{-t}i$ and there are relations between the stable and unstable manifolds that are preserved even for $\e\ne0$. Namely % $$ \eqalign{ \UU^+_{\AA'}(\f,\aa) = & \UU^-_{\AA'}(2\p-\f,-\aa) \; , \cr V^+_{\AA'}(\f,\aa) = & V^-_{\AA'}(2\p-\f,-\aa) \; , \cr} \Eq(1.6) $$ % where care must be exercised because the manifolds contain {\it two} points over each $\aa,\f$.\annota3{This is in fact already so for $\e=0$.} Hence if $\f\simeq \p$ the relations in \equ(1.6) concern points that lie on different connected manifolds; to understand what happens one should try a drawing taking into account that the above representations are considered only for $|\f|<\fra32\p$. Looking at the manifolds at $\f=\p$, {\it assuming their existence and that they are graphs above $\aa,\f$ for $|\f|<\fra32\p$}, equations \equ(1.6) imply that, fixed $\AA'$, % $$\QQ(\aa)\defi \UU^+_{\AA'}(\p,\aa) -\UU^-_{\AA'}(\p,\aa) = -\QQ(-\aa) \; , \Eq(1.7)$$ % so that $\QQ(\V0)=\V0$; but, in general, $\QQ(\aa)\ne \V0$ for $\aa\ne\V0$. The function $\QQ(\aa)$ is called the {\it homoclinic splitting} (or simply {\it splitting}) {\it vector} at $\f=\p$, and the determinant of the matrix with entries $\dpr_{\a_i}Q_j(\V 0)$ (splitting matrix) is called the {\it splitting}. One can more generally consider $\Zz\=\Zz(\f,\aa)= (\UU^+_{\AA'} (\f,\aa)-\UU^-_{\AA'}(\f,\aa), V^+_{\AA'}(\f,\aa)- V^-_{\AA'}(\f,\aa))$ which would be the splitting vector at $\f$. Here and henceforth the vectors in $\RRR^\ell$ will be denoted with an underlined letter (while the boldface is used for vectors in $\RRR^{\ell-1}$); so far $\ell=3$, but shortly we shall consider $\ell\ge 3$. The function $\Zz$ can be written as the gradient of a generating function $\F$, \ie $\Zz=(\dpr_\f \F, \dpr_\aa \F)$. This is a result due to Eliasson who points out that it follows immediately from the Lagrangian nature of the stable and unstable manifolds. It is a further symmetry property.\annota4{It can alternatively be easily seen from the explicit expressions for the stable and unstable manifolds equations derived in [G1], which also provide a general algorithm for constructing the function $\F$ as a convergent series in $\e$ for $\e$ small; see [G3].} The symmetry of \equ(1.1) (hence the consequent oddness of $\QQ(\aa)$) implies that there is one trajectory which swings through $\f=\p$ when $\aa$ is exactly $\V0$: it tends to the same invariant torus as $t\to\pm\io$, provided the torus exists and its stable and unstable manifolds are graphs over $\aa,\f$ over an interval of $\f$ greater than $|\f|<\p$. In this paper we prove the following result. \* \0{\bf 1.4.} {\cs Theorem.} {\it Given the Hamiltonian \equ(1.1), given constants $s,\O>0$ and given $\h>0$ small enough, the following assertions hold.\\ \pallino There are invariant tori with rotation vectors $\oo'$ for all $\oo'$ verifying the Diophantine condition \equ(1.4) with constant $C=C(\h)=\O e^{-s\h^{-1/2}}$ and $|\o'_1|\in[\fra12\O_1\hdp,2\O_1\hdp]$.\\ \pallino Such tori exist for $|\e|<\e_0=O(\h^2)$ and for $\h$ small enough.\\ \pallino They can be parameterized by their average actions $\AA'$; the angular velocity is then given by the rotation vector $\oo'\=(\O_1\hdp+\h J_1^{-1}A'_1,\O_2\hdm)$ (\ie they are ``twistless'') and the Lyapunov exponents have the form $\lis g_0= (1+\G(\e,g_0))\,g_0$, with $\G(\e,g_0)$ analytic in $\e$ and divisible by $\e$.\\ \pallino The parametric equations for such tori and for their stable and unstable manifolds (``whis\-kers'') can be computed by a convergent perturbation series in powers of $\e$ around the unperturbed tori with the same rotation vector and their corresponding stable and unstable manifolds.\\ \pallino At the homoclinic intersection with $\f=\p$ (existing by symmetry), between the stable manifold and the unstable manifold of each torus, the splitting is generically given by the Mel'nikov integral which is of order $O(\e^2\h^{-\b}e^{-\fra\p2 g_0^{-1}\hdm})$, for $\e$ small enough, with $\b$ depending on the degree $N_0$ in $\f$ of the perturbation $f$: one can take $\b=2N_0-1$ and the asymptotic formula holds if $|\e|< \h^{\z}$, $\z=2(N_0+3)$ and $\h$ is small enough.} \* \0{\bf 1.5.} {\cs Remark.} The novelty of the theorem is the ``sharp'' bound $\e_0=O(\h^2)$. If we ``only'' require $\e_0=O(\h^{\fra92+})$ where $\fra92+$ is any prefixed positive number $>\fra92$ the result is proved in [GGM3] (see also [CG] or [Ge2]). The improvement is made possible by the {\it totally different technique} used (with respect to [GGM3]): a technique that has interest in its own right and, we think, beyond the result itself. In fact the proof of the last assertion of the theorem is the content of [GGM2], and the values of the constants $\b$ and $\z$ are taken from Appendix A2 of [GGM2]. \* \0{\bf 1.6.} Theorem 1.4 will be proved by a further extension of Eliasson's method, [E,G2,Ge1,Ge2], for the KAM theorem. The following discussion will show the correctness of the intuition that ``new'' small divisors appear in the perturbation expansion {\it at orders spaced by} $O(\h^{-1})$. So that the coupling constant is effectively $O(\e^{\h^{-1}})$ and the analyticity condition is expected to be $\e^{\h^{-1}} C(\h)^{-q}$ small (for some $q>0$, determined as in the discussion in Remark 5.16, item (4), below). Hence the analyticity condition will be $\e C(\h)^{-q\h}$ small rather than the far worse $\e C(\h)^{-q}$ small, that is implied directly from lemma 1 in [CG] (where $q=6$ is an estimate). In the one degree of freedom case the corresponding problem is studied in [N]: it is a problem that arises naturally in the context of Nekhoroshev theory. In our case the rotation vector is not one-dimensional, so that the cancellations between resonances typical of small divisors problems, [E,G1,Ge1,Ge2], have to be exploited in order to prove convergence of the perturbative series. The fact that the two components of the rotation vector \equ(1.3) are so different in scale has the consequence that small divisors can appear only at high orders, so that the dependence of the radius convergence on the Diophantine constant $C(\h)$ is highly improvable with respect the ``na\"{\i}ve'' one, as explained above: the proof of such an assertion is the subject of the present paper (as, in the weaker form, already of [GGM3]). \* \0{\bf 1.7.} The paper is organized as follows. In \S 2,3,4 the formalism is concisely illustrated and the graphic representations of the whiskers in terms of tree graphs is exhibited (for systems more general than \equ(1.1); see \equ(2.1) below). The analysis is brief but selfcontained, with references to [G1,Ge2] only given for further insight and details. The basic formalism is in \S 2, then we work out in \S 3 two specific examples to explain the origin of the graphical interpretation, and in \S 4 we set up the general Feynman rules for evaluating the equations of the whiskers (and the splitting vector as a particular case) as a sum of quantities that can be elementarily evaluated. In \S 5 bounds are derived, assuring the convergence of the perturbative series defining the whiskers in the more general system in \equ(2.1) below and leading to Theorem 1.4, when restricted to the Hamiltonian \equ(1.1). The bounds are derived along the lines of [G1,Ge2]: the main part is the derivation of the bounds for the part of the expansion corresponding to what we call the contributions due to ``trees without leaves'': this is done fully and self consistently in \S 5 and in the related appendices. Once the bounds on the contributions from trees without leaves are established, {\it which is the real difficulty}, the same analysis can be applied to bound the other contributions. Since this is simply reduced, without any further technical problems, to the case of contributions from the simpler trees with no leaves we do not repeat this part of the discussion which is done in [Ge2] following the corresponding analysis done in [G1,Ge1]. To Appendix A1 we relegate some technical details, while Appendix A3 concerns the cancellation analysis of [Ge2], needed in order to treat the small divisors problems, with more details with respect to the quoted paper. An original technical part is also in Appendix A2 and deals with the improvement of the dependence of the convergence radius on the Diophantine constant $C(\h)$. We do not comment here on the obvious relevance of the above results for the theory of Arnol'd diffusion: see [GGM3,GGM4]. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip1.truecm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0{\titolo \S 2. Lindstedt series for whiskered tori.} \numsec=2\numfor=1\* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0We use the formalism of [Ge2]: it would be pointless to repeat here the technical work required to motivate the necessity or usefulness of the notations, and we cannot imagine that any reader may have interest in the matter that follows unless he has some experience with Eliasson's method, as exposed in [E] and complemented in [G1,G2,Ge1,Ge2]. The references to [G1,Ge2] are given only to point at places where further details on the motivations of the assertions can be found. The following analysis innovates [Ge2] in \S 5 because of the extension of Siegel-Bryuno's bound described in Appendix A2 below: this section and the next two provide a {\it self contained} description of the graphical algorithm exploited in \S 5 and Appendix A2. \* \0{\bf 2.1.} In the following we shall consider a Hamiltonian (``Thirring model'') more general than the one in \equ(1.1), \ie a Hamiltonian which couples a pendulum with $\ell-1$ rotators via a perturbation $f_1$ which is always an {\it even trigonometric polynomial}, % $$ \HH = \oo\cdot\AA + \fra{1}{2J}\AA\cdot\AA + \fra{I^2}{2J_0} +J_0g_0^2\, f_0(\f) +\e J_0g_0^2\, f_1(\f,\aa) + J_0g_0^2\,\g(\e, g_0)\,f_0(\f) \; , \Eq(2.1) $$ % where $(\aa,\AA)\in\TTT^{\ell-1}\times\RRR^{\ell-1}$, $(\f,I)\in \TTT^1\times\RRR^1$, $J_0>0$, $J$ is a diagonal matrix, %$\pmatrix{J'&0\cr0&J''\cr}$, $00$. Going back to the original Hamiltonian \equ(1.1) we {\it therefore} set $g_0^2= \lis g_0^2\,(1+\g(\e,\lis g_0))$ and we can invert the latter relation as $\lis g_0^2= (1+\G(\e,g_0))\, g_0^2$ for $\e$ small enough (this will mean: for $|\e|0$; see \equ(1.4). We look for an invariant torus and for its stable and unstable manifolds with the property that the quasi periodic rotation on the torus takes place at velocity $\oo'$ and, {\it at the same time}, the action variables oscillate with an average position $\AA'$. \* \pallino Before proceeding we remark that the above {\it two} requirements may seem contradictory as there may seem to be no reason for being able to prescribe simultaneously the ``spectrum'' $\oo'$ and the ``average action'' $\AA'$ of the invariant tori. In fact this property of ``{\it twistless}'' motion on the tori or of ``{\it absence of torsion}'' is very remarkable (see the Remark 1.2 and [G1]): it will appear as due to the special symmetries of the system \equ(2.1) and to the separation of the energy into a quadratic part involving actions only and an angular part involving only the angles. \* Note also that we could confine ourselves to study the torus with average position $\AA'=\V0$, as in [G1,Ge2], because any torus can be reduced to that one through a trivial canonical transformation (a translation in the action variables). This explains why in the quoted papers only the torus covered with rotation vector $\oo$ is explicitly considered: however in the following we consider also $\AA'\neq\V0$, as we are interested in showing the abundance of such tori in phase space (see the Remark 1.5). The quantity $X_j^{\s}(t;\aa)$ can be graphically represented as sum of {\it values} which can be associated with tree graphs, that we shall call ``Feynman graphs'' or ``trees'' {\it tout court}, see Fig.\equ(2.4) below. The trees are partially ordered sets of points, called {\it nodes}, connected by unit lines, called {\it branches}, and they are ``oriented'' towards a point called {\it root}, which is reached by a single branch of the tree. Given two nodes $v$ and $w$ of a tree, we say that $w$ precedes $v$ ($w\le v$) if there is a path connecting $w$ to $v$, oriented from $w$ to $v$. With an abuse of notations we shall sometimes consider a tree as the collection of its nodes, sometimes as the collection of its branches and sometimes as the collection of both nodes and branches. The root {\it will not} be considered a node. A typical tree considered below can be drawn as in Fig.\equ(2.4): the labels meaning and the caption of such a drawing (which has to be interpreted as a mathematical formula) will be elucidated in the coming sections. \* \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}{\rm root} \ins{0.00000pt}{91.666298pt}{$j$} \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\eq(2.4)} \kern0.9cm \didascalia{A tree $\th$ with $m=12$, and some labels. The line numbers, distinguishing the lines, and their orientation pointing at the root, are not shown. The lines length should be the same but it is drawn of arbitrary size. The nodes labels $\d_v$ are indicated only for two nodes.} The branch starting at the node $v$ and linking it to the uniquely determined next node (or to the root), which we call $v'$, will be denoted by $\l_{v}$: there is a unique correspondence between nodes and branches starting at them. We shall say that $\l_v$ exits from $v$ and enters $v'$; given a node $v$ we shall say that a branch $\l$ {\it pertains} to $v$ if either $\l$ enters $v$ or $\l$ exits from $v$; \eg in Fig.\equ(2.4) the line $v_1v_0\=\l_{v_1}$ ``exits'' $v_1$ and ``enters'' $v_0$, hence it pertains to both. In [G1] two expansions are considered for the functions $X^{\s}_j(t;\aa)$ representing the stable and unstable manifolds: one of them is used to exhibit cancellations taking place at all orders in the sums that express the coefficients of the power series in $\e$ of the splitting vector, [G1,BCG,GGM2]; it is somewhat more involved than the other one that is convenient to just discuss convergence of the perturbation series for the splitting vector and that we shall use here. This is the reason why (as in [Ge2]) we shall not have trees whose lowest nodes carry a graphical decoration called {\it form factor}, or {\it fruit} in [G1,GGM2]. Nevertheless some of the nodes will still have a particular structure: to characterize them we introduce, below as in [Ge2], the notion of ``{\it leaf}\/'', which is related to the notion of fruit in [G1], from which it differs (and it, even, differs slightly from the similar notion of leaf in [Ge2]), see below for the motivation of the name. %\ifnum\mgnf=0\pagina\fi \* \0{\bf 2.3.} As mentioned the drawing Fig.\equ(2.4) has to be regarded as a mathematical formula expressing a function of the labels and of the topological structure of the trees. We now prepare the notation for the definition of ``value'' of a tree (following [Ge2]) (see [G1] for a simpler case): the derivation is not difficult but somewhat long and unusual for the subject (the breakthrough work [E] still does not seem to be well known in its technical aspects!). We discuss it in detail not only for completeness but in the attempt to clarify a construction that has generated quite a few new results starting from the work of [E], see [G1,GGM2,BGGM]. Let us consider the unperturbed motion $ X^0(t)\=(\f^0(t),\aa+\oo' t,I^0(t),\AA')$, where $(\f^0(t),I^0(t))$ is the separatrix motion, generated by the pendulum in \equ(2.1) starting at $t=0$ in $\f=\p,\, \AA=\AA',I=-2J_0 g_0$, so that $\f^0(t)=4 \arctan e^{-g_0t}$. Let $X^\s(t;\a)$, $\s={\rm sign}\,t=\pm$, be the evolution, under the flow generated by \equ(1.1), of the point on $W^\s_\e$ which at time $t=0$ is $(\p,\aa,I^\s(\aa,\p),\AA^\s(\aa,\p))$, see \equ(1.2); let % $$X^\s(t)\=X^\s(t;\aa)\equiv \sum_{h\ge 0} X^{h\s}(t;\aa) \e^h= \sum_{h\ge 0} X^{h\s}(t) \e^h,\qquad \s=\pm \; , \Eq(2.5)$$ % be the power series in $\e$ of $X^\s$, (which we want to show to be convergent for $\e$ small); note that $X^{0\s}\=X^0$ is the unperturbed whisker. We shall often omit writing explicitly the $\aa$ variable among the arguments of various $\aa$-dependent functions, to simplify the notations, and we shall regard the two functions $X^{h\s}(t)$, as forming a single function $X^h(t)$, which is $X^{h+}(t)$ if $\s=+,\, t>0$, and $X^{h-}(t)$ if $\s=-,\,t<0$. Components of $X$ will be labeled $j$, $j=0,\ldots,2\ell-1$, consistently with \equ(2.3), with the convention that $X_0\defi X_-$ describes the coordinate $\f$, $(X_j)_{j=1,\ldots,\ell-1}\defi\XX_\giu$ describes the $\aa$ coordinates, $X_\ell\defi X_+$ describes the $I$ coordinate and $(X_j)_{j=\ell+1,\ldots,2\ell-1}\defi \XX_\su$ describes the $\AA$ coordinates, % $$ X \defi\, (X_j)_{j=0,\ldots,2\ell-1}\defi\, (X_-,\XX_\giu,X_+,\XX_\su) \; , \Eq(2.6)$$ % \ie we write first the angle and then the action components, first the pendulum and then the rotators. The ${\bf \su}$ (``{\it up}'') and ${\bf \giu}$ (``{\it down}'') labels recall that the components with labels ${\bf \giu}$ ($0< j<\ell$) have ``lower'' index than the variables with labels ${\bf \su}$ ($\ell0$; however their sums have {\it no singularity} at $t=0$ and can be anaytically continued for $\s t<0$ (\ie $x\ge1$). More precisely the functions that one has to integrate are contained in an {\it algebra} $\hat \MM$ on which the integration operations that we need can be given a meaning. %To describe such class we introduce the algebra $\hat \MM$ of the %functions of $t$ defined as follows. \* \0{\cs Definition} ([G1]). {\it Let $\hat\MM$ be the space of the functions of $t$ which can be represented, for some $k\ge 0$, as % $$M(t)=\sum_{j=0}^k{(\s t g_0)^j\over j!} M_j^\s(x,\oo t)\ ,\quad x\=e^{-\s g_0t}\ ,\quad \s={\rm sign}\, t \; , \Eq(2.21)$$ % with $M_j^\s(x,\pps)$ a trigonometric polynomial in $\pps$ with coefficients holomorphic in the $x$-plane in the annulus $0<|x|<1$, with 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 $<\fra\p2$, and possible polar singularities at $x=0$. The smallest cone containing the singularities will be called the {\it singularity cone} of $M$.} \* The proper interpretation of the improper integrals $\ig_{\s\io}^{g_0t} M(\t) d g_0 \t$, which henceforth will be denoted by $\igb_{\s\io}^{g_0t} M(\t) dg_0\t$, is simply the {\it residuum} at $R=0$ of the analytic function % $$ \II_R M\defi\ig_{\s\io+i\theta}^{g_0t}e^{-Rg_0\s z} M(z)\,d\,g_0z \; , \Eq(2.22) $$ % (where $\theta$ is arbitrarily prefixed) which is defined and holomorphic for $\Re R>0$ and large enough, \ie % $$\II M(t) \= \igb_{\s\io}^{g_0t} dg_0\t \, M(\t) \defi \oint\fra{d R}{2\p i R} \,\II_R M(t) \; . \Eq(2.23)$$ % By linear extension this defines the integration of function in $\hat \MM$ for $|x|<1$. The analyticity in $x$ around $x=\pm1$ and the remarks that $\fra{d}{d g_0 t}\II M(t)\= M(t)$, \ie $\II M(t)\= \II M(t')+\ig_{g_0t'}^{g_0t} d\,g_0\t\, M(\t)$, so that $\II M(t)$ is a special primitive of $M(t)$ (at fixed $\s$), allow us to analytically continue the result of the integration to a function in $\hat \MM$. The operator $\II$ maps the algebra $\hat \MM$ into itself because one checks that on the monomial \equ(2.19) one has % $$\II M(t)=\cases{- g_0^{-1} \s^{\chi +1}e^{i\oo'\cdot\nn t-pg_0\s t} \sum_{h=0}^j (g_0\s t)^{j-h} {1\over(j-h)!} {1 \over(p- i \s g_0^{-1} \oo'\cdot\nn)^{h+1}} \; , & if $|p|+|\nn|>0 \; , $\cr g_0^{-1}\s^{\ch+1}\fra{(\s g_0 t)^{j+1}}{(j+1)!} \; , & otherwise $\; , $ \cr}\Eq(2.24)$$ % showing, in particular, that the radius of convergence in $x$ of $\II M$, for a general $M$, is the same as that of $M$. But in general the singularities will not be polar, even when those of the $M_j^\s$'s were such. We shall see that the cases $|p|+|\nn|=0$ do not enter in the discussion (a feature of the method of [Ge2]). The complete expression of $X^{h\s}(t)$ becomes % $$\eqalignno{ &\X^{h\s}_-(t) = w_{0\ell}(t)\II(w_{00}\F^{h\s}_+)(t)-w_{00}(t)\big( \II(w_{0\ell}\F^{h\s}_+)(t)-\II(w_{0\ell}\F^{h\s}_+)(0^\s)\Big)\defi \OO(\F^{h\s}_+)(t) \; , \cr & \XXX^{h\s}_\giu(t) = J^{-1}J_0 \Big(\II^2(\FFF^{h\s}_\su)(t)-\II^2( \FFF^{h\s}_\su)(0^\s)\Big) \defi \lis\II^2(\FFF^{h\s}_\su(t)) \; , & \eq(2.25) \cr & \X^{h\s}_+(t) = w_{\ell\ell}(t)\II(w_{00}\F^{h\s}_+)(t)-w_{\ell0}(t) \Big(\II(w_{0\ell}\F^{h\s}_+) (t)-\II(w_{0\ell}\F^{h\s}_+)(0^\s)\Big)\defi\OO_+(\F^{h\s}_+)(t) \; , \cr & \XXX^{h\s}_\su(t) = \II(\FFF^{h\s}_\su)(t) \; , \cr} $$ % where $\OO,\OO_+,\lis\II^2$ are implicitly defined here (and $\II^2$ is $\II$ applied twice); and $\X^{h\s},\F^{h\s}\=(0,\V0,\F_+^{h\s},$ $\FFF^{h\s}_\su)$ are introduced in \equ(2.9). While $\X^{h\s}$ has non zero components over both the {\it angle} ($j=0,\ldots,\ell-1$) and over the {\it action} ($j=\ell,\ldots,2\ell-1$) components, the $\F^{h\s}$ has, as already noted, only the action directions non zero; the notation $0^\s$ means the limit as $t\to0$ from the left ($\s=-$) or from the right ($\s=+$), but below we shall drop the superscript on $0$ (always clear from he context because it is the same as the superscript $\s$ of the functions $\X^{h\s}$). Furthermore, with the definitions \equ(2.20) of $\tilde \FFF_\su^{h\s}(\nn,p)$ one finds also the property (with the notations in \equ(2.1)) % $$ \tilde \FFF_{\su}^{h\s}(\V0,0) = \V0\; , \Eq(2.26)$$ % for all $h\ge1$. We shall repeatedly use that in order to compute $\X^{h\s}_j$ we only need $\X^{h'\s}_{j'}$ with $0\le j'<\ell$ (\ie only $\X^{h'\s}_+, \X^{h'\s}_\su$) and $h' \8< /h {10} def /H {35} def> \8< /p0 {0 h H add} def > \8< /p1 {H h H add} def > \8< /p2 {2 H mul h } def > \8< /p3 {2 H mul h 2 H mul add} def > \8< /p4 {4 H mul h H add} def > \8< /p5 {5 H mul h H add} def > \8< /p6 {6 H mul h H add} def > \8< /p7 {7 H mul h} def > \8< /p8 {7 H mul h 2 H mul add} def> \8< > \8< p0 moveto p1 lineto p2 lineto p1 moveto p3 lineto stroke> \8< p4 moveto p5 lineto p6 lineto p7 lineto p6 moveto p8 lineto stroke> \8< > \8< p0 2 0 360 arc fill stroke> \8< p4 2 0 360 arc fill stroke> \8 \8< p1 r 0 360 arc fill stroke> \8< p2 r 0 360 arc fill stroke> \8< p3 r 0 360 arc fill stroke> \8< p5 r 0 360 arc fill stroke> \8< p6 r 0 360 arc fill stroke> \8< p7 r 0 360 arc fill stroke> \8< p8 r 0 360 arc fill stroke> \8< grestore> \figfin \eqfig{250pt}{100pt}{ \ins{9pt}{56pt}{$j $} \ins{0pt}{38pt}{$r $} \ins{33 pt}{38 pt}{$v_0 $} \ins{57pt}{64pt}{$p $} \ins{57pt}{34pt}{$q $} \ins{233pt}{64pt}{$0 $} \ins{233 pt}{34pt}{$0 $} \ins{186pt}{56pt}{$0 $} \ins{151pt}{56pt}{$j $} \ins{77pt}{76pt}{$v_1 $} \ins{77pt}{5pt}{$v_2 $} \ins{140pt}{38pt}{$r $} \ins{173pt}{38pt}{$v_0 $} \ins{209pt}{38pt}{$v_1 $} \ins{255pt}{5pt}{$v_2 $} \ins{255pt}{76pt}{$v_3 $} \ins{19pt}{62pt}{$1, j_{v_0}$} \ins{65pt}{94pt}{$1, j_{v_1}$} \ins{65pt}{25pt}{$1, j_{v_2}$} \ins{164pt}{63pt}{$1, j_{v_0}$} \ins{196pt}{63pt}{$0, j_{v_1}$} \ins{244pt}{94pt}{$1, j_{v_3} $} \ins{244pt}{25pt}{$1, j_{v_2} $} }{albero1}{\eq(3.6)} \* \0where the labels on the nodes $v$ are denoted $\d_v,j_v$ and those on the lines $\l_v$ are denoted $j_{\l_v}$. The label $\d_v=0,1$ on the node $v$ indicates selection of $f_{\d_v}$, \ie of $f_0$ or $f_1$, the label $j_v$ denotes a derivative with respect to $\f$ if $j_{v}=\ell$ or with respect to $\a_{j_v}$ if $j_v=\ell+1,\ldots,2\ell-1$. For the label $j_{\l_v}$ associated with the branch $\l_v$ following $v$, one has $j_{\l_v}=j_v-\ell$ for all $v$ except for the highest node $v_0$, for which one has $j_{\l_{v_0}}=j_{v_0}$. In the examples above, \equ(3.3) and \equ(3.5) correspond, respectively, to the first figure in \equ(3.6) with $j_{v_0}=j,j_{v_1}=p+\ell, j_{v_2}=q+\ell$ and to the second with $j_{v_1}=j_{v_2}=j_{v_3}=\ell,j_{v_0}=j$, (hence $j_{\l_{v_1}}=j_{\l_{v_2}}=j_{\l_{v_3}}=0$, $j_{v_0}=j$). In the examples the labels $p,q$ correspond to $\dpr_{\a_p},\dpr_{\a_q}$ in \equ(3.3). \* \0{\bf 3.2.} {\cs Remark.} The exception for the meaning of $j_{\l_{v_0}}$ is convenient, in the above cases, as the integration over $\t_{v_0}$ differs from the others: the inner ones evaluate $\X^{h\s}_j$ for $j=0,\ldots,\ell-1$, because the functions $f_0,f_1$ only depend on the angle variables (see the last paragraph in \S 2.4); the last integral, however, evaluates in the examples a component of $\XXX^{h\s}_\su$ (which is labeled $j=\ell+1,\ldots,2\ell-1$), but, in general, $j$ can be any value $j=0,\ldots,2\ell-1$. Note that this is not so for the inner labels $j_\l$ which must be angle labels $j_\l=0,\ldots,\ell-1$. So, in general, we shall have that the value of a tree with $j_{\l_{v_0}}=j$ contributes to $\X^{h\s}_j$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip1.truecm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0{\titolo \S 4. Trees and Feynman graphs approach to whiskers construction: the general case.} \numsec=4\numfor=1\* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0We now proceed to describe the general case. \* \0{\bf 4.1.} To compute the splitting vector we only need to consider the variable $t$ equal to $0$. However we shall be also interested in $\X^{h\s}(\aa,t)$ for $\s t>0$, for instance in order to study how fast the invariant torus is approached by the motions on its stable and unstable manifolds (to obtain its Lyapunov exponent). Hence it will be natural to attribute the label $t$ to the root: this will also remind that the integral over $\t_{v_0}$ has to be performed between $\s\io$ and $t$, (the value $\s=-$ corresponds to the unstable manifold and the value $\s=+$ corresponds to the stable one). Since we shall {\it never} consider the stable manifold for $t>0$ or the unstable for $t<0$ the value of $\s$ will be the same as that of the sign of $t$. We shall be interested in computing not only $\XX_{\su}^\s(0;\aa)-\AA'$ (or $\XX^\s_{\su}(t;\aa)-\AA'$), as in [GGM2], but, more generally, $X^\s(t;\aa)-X^0(t;\aa)$, with $\s=\hbox{sign}\,t$, (here $X^{0}$ denotes the unperturbed motion). In general the rules to express $X^\s(t;\aa)-X^0(t;\aa)$ as sum of ``values'' associated with trees will be described now, assuming that the reader follows us by applying and checking them to the special cases \equ(3.3), \equ(3.5), illustrated in \equ(3.6). \* The reader might be helped in following the construction of the algorithm to express the stable and unstable manifolds below, by keeping in mind that we simply decompose the (quite involved and recursively defined by \equ(2.25),\equ(2.12)) expressions for the whiskers, so far obtained, {\it further}. The purpose being of reducing their evaluation to {\it very elementary algebraic operations}: ultimately just products of simple factors associated with the nodes (and their labels) of a tree, that we shall call ``coupling constants'', and of factors associated with the branches (and their labels), that we shall call ``propagators'', each of which can be trivially evaluated and trivially bounded. The reader familiar with Quantum Field Theory will realize the striking analogy between the algorithms discussed below and the {\it Feynman graphs}: in fact a ``tree'' will turn out as an analog to a (loopless) Feynman graph and {\it very likely it is} a Feynman graph of a suitable (non trivial) field theory. Our analysis amounts to a renormalization group analysis of it and it partially extends, to the case of the theory of the stable and unstable manifolds of hyperbolic tori in nearly integrale systems, the field theoretic interpretation already discussed in detail in previous works, see [GGM1] and appended references, in the study of KAM tori. \* \pallino To each node we attach an {\it order label} $\d_v=0,1$, see Fig.\equ(3.6), and a corresponding function $f_{\d_v}$: if a node $v$ bears a label $\d_v=1$ the associated function is $f_1$ and if it bears a label $\d_v=0$ it is $f_0$. \* \pallino To each node $v$ of a tree $\th$, see the Figure \equ(2.4) above, we associate an integration {\it time variable} $\t_v$ and an {\it integration operation}, which corresponds to $\lis\II^2$ or $\OO$ if the node {\it is not the highest node} $v_0$ and to $\lis\II^2$ or $\OO$ or $\II$ or $\OO_+$ if the node $v$ {\it is the highest}, \ie $v=v_0$. This is so because in the first case (a ``lower node'') one must use the first two equations in \equ(2.25) because in \equ(2.12) only angle components of $X^{h\s}$ appear, while in the second case (that of the highest node) one can use all of \equ(2.25) since we can evaluate either an angle coordinate $\X^{h\s}_j(\aa,t)$, $j<\ell$, or an action coordinate, $j\ge\ell$. When $v< v_0$ the choice between the two possibilities will be marked by an {\it action label} $j_v$ associated with each node: if $j_v=\ell$, $v\ell$, we choose $\lis\II^2$ if $j_{\l_{v_0}}=j_{v_0}-\ell$ and $\II$ if $j_{\l_{v_0}}=j_{v_0}$, see \equ(2.25). As said in Remark 3.2, the meaning of the branch label is that a tree with $j_{\l_{v_0}}=j$ is a graphic representation of a ``contribution'' to $\X_j^{h\s}$. Therefore if $j_{\l_{v_0}}\ge\ell$ we call the branch an {\it action branch} and if $j_{\l_{v_0}}<\ell$ we call it an {\it angle branch}. In the first of the figures in \equ(3.6) integrals with respect to the nodes $v_1,v_2$ are of the type $\lis \II^2$. In the second the integrals over the $\t_{v_n}$, $n=1,2,3$, are all of the type $\OO$. In both cases the integrals over $\t_{v_0}$ are of the form $\II$ because we fixed $j_{\l_{v_0}}=j>\ell$ to be an action label. We can associate a branch label $j_{\l_v}$ also to the inner branches with $v\t_{v'}$ if $\s=+$ and $\t_{v}<\t_{v'}$ if $\s=-$, while $\t_v,\t_{v'}$ have the same sign but are otherwise unrelated if $\r_v=0$; see \equ(2.25) and check this in the examples. Besides the labels already introduced also the labels $\r_v=0,1$, just described but not shown in \equ(3.6), should be imagined carried by each node. \* \pallino Given a tree labeled as above we pick up the nodes $v$ with $\r_v=0$ which are closest to the root, and consider the subtrees having such nodes as highest nodes. We call each such subtree, \ie each such node {\it together with the subtrees ending in it} (and its labels), a {\it leaf}.\annota5{This definition is slightly different from the one given in [Ge2], where the leaf represents a collection of trees and, as explained below, is related to a resummation operation (see also comments in \S 4.2, item (v), below, and \equ(4.27)), that we do not consider here.} The name is natural if one imagines to enclose the part of the tree including the node $v$ itself and half of the line $\l_v$ into a circle (or, more pictorially, into a leaf shaped contour): hence, to whom tries the drawing, it will look like the {\it venations} of a leaf and the half line outside it will look like its {\it stalk}. \* \pallino All nodes which do not belong to any leaves will be called {\it free nodes}; they carry, by construction, a label $\r_v=1$, so that the corresponding time variables are hierarchically ordered from the lowest nodes up to the root: \ie if $w\t_v$ if $\s=-$. Given a tree $\th$ let us call $\th_f$ the set of free nodes in $\th$, and call $\Th_L$ the set of highest nodes of the leaves. \* \pallino Each $f_{\d_v}$ function, associated with the node $v$ with order label $\d_v$, can be decomposed into its Fourier harmonics. This can be done graphically by adding to each node $v$ a {\it mode} label $\nvec_v=(\n_{0v},\nn_v)\= (n_v,\nn_v)\in\ZZZ^{\ell}$, with $|\nn_v|\le N$ and $|n_v|\le N_0$, that denotes the particular harmonic selected for the node $v$. If $(j_v,\d_v,\r_v,\nvec_v)$ are the labels of $v$ we will associate with $v$ the quantity $f^{\d_v}_{\nvec_v}\,e^{i(\oo\cdot\nn_v \t_{v}+n_v\f^0(\t_v))}$ multiplied by appropriate products of factors $i n_v$ (one per $\f$--derivative) and $i\n_{vj}$ (one per $\a_{j}$--derivative, $j=j_{\l_v}$). If the mode labels $\nvec_v$ are specified for each $v$ we shall define the {\it momentum} $\nn(v)$ ``flowing'' on a branch $\l_v$ as the sum of all the angle mode components $\nn_w$ of the nodes $w$ {\it preceding} the branch, with $v$ included, % $$ \nn(v) \defi\sum_{w\in\th,\ w\le v}\nn_w \; ; \Eq(4.1) $$ % the momentum $\nn(v_0)$ flowing through the root branch will be called the {\it total momentum} (of the tree). We shall define also the {\it total free momentum} of the tree as the sum of the mode labels of the free nodes: more generally, for any free node $v$ we can define the {\it free momentum} flowing through the branch $\l_v$ as % $$ \nn_0(v) = \sum_{w\in\th_f,\ w \le v} \nn_w \; . \Eq(4.2) $$ % For instance in the above examples the two contributions \equ(3.3), \equ(3.5) (represented by figure \equ(3.6)) are decomposed into sums of several distinct contributions once the $\r_v$ and the mode labels $\nvec_v$ are specified. Likewise we can look at a leaf as a tree: the momentum $\nn'$ flowing through its stalk will then be called the internal {\it leaf momentum}. Note that its value gives {\it no contribution} to the total free momentum of the tree to which the leaf belongs. \* \pallino The free momenta will turn out to describe the harmonics of the time dependent quasi periodic motion around the invariant tori, while the Fourier expansion modes of $X^{h\s}(t;\aa)$ as a function of $\aa$ are related to the sum of the free momenta {\it and} of all the internal leaf momenta. This is an important difference: it is a property stressed in [G1] where it is referred as ``quasi flatness'', source of the main difficulties and interest in the theory of homoclinic splitting, see [G1,GGM2,GGM3,G3]. \* \0{\bf 4.2.} The trees contributions of the examples of \S 3 will be sums over the various labels of ``values'' of trees decorated by more labels: % $$\eqalignno{ &\fra12\igb_{\s\io}^{g_0t} dg_0\t_{v_0} (-i\n_{v_0j})(i\n_{v_0p})(i\n_{v_0q})\,f^1_{\nvec_{v_0}}\, e^{i(\nn_{v_0}\cdot\oo'\t_{v_0}+ n_{v_0}\f^0(\t_{v_0}))}\cdot & \eq(4.3) \cr &\quad\cdot\lis\II^2\big((-i\n_{v_1p})\,f^1_{\nvec_{v_1}}\, e^{i(\nn_{v_1}\cdot\oo'\t_{v_1}+ n_{v_1}\f^0(\t_{v_1}))}\big)(\r_{v_1}\t_{v_0})\, \lis\II^2\big((-i\n_{v_2q}) \,f^1_{\nvec_{v_2}}\, e^{i(\nn_{v_2}\cdot\oo'\t_{v_2}+ n_{v_2}\f^0(\t_{v_2}))}\big)(\r_{v_2}\t_{v_0}) \; , \cr}$$ % for \equ(3.3) and % $$\eqalignno{ &\fra12 \igb_{\s\io}^{g_0t} dg_0\t_{v_0} (-i\n_{v_0j})(in_{v_0})\,f^1_{\nvec_{v_0}}\, e^{i(\nn_{v_0}\cdot\oo' \t_{v_0}+ n_{v_0}\f^0(\t_{v_0}))}\, \OO\Big((-in_{v_1})\,f^0_{\nvec_{v_1}}\, e^{i n_{v_1}\f^0(\t_{v_1})} & \eq(4.4) \cr & \qquad \OO\big((-i n_{v_2})\,f^1_{\nvec_{v_2}}\, e^{i(\nn_{v_2}\cdot\oo' \t_{v_2}+ n_{v_2}\f^0(\t_{v_2}))} \big) (\r_{v_2}\t_{v_1}) \, \OO\big((-i n_{v_3})\,f^1_{\nvec_{v_3}}\, e^{i(\nn_{v_3}\cdot\oo' \t_{v_3}+ n_{v_3} \f^0(\t_{v_3}))}\big) (\r_{v_3}\t_{v_1}) \Big) \; , \cr} $$ % for \equ(3.5), with the conventions following \equ(3.3) about the dummy integration variables. \* \pallino The integration operations are still fairly involved, as it can be seen from \equ(2.25) and from the expressions for $\lis \II^2$ and $\OO$. With the above conventions for the dummy variables and noting that, for any function $F$ in $\hat\MM$, % $$ \lis \II^2 \big( F(\t)\big)(t)=J^{-1}J_0\, \Big( \II(g_0(t-\t)F(\t) )(t) - \II(g_0(t-\t)F(\t))(0) \Big) \; ,\Eq(4.5)$$ % we see that the integration over the $\t_v$ has (by \equ(2.25)) one of the two forms, when $\r_v=1$ and $v'$ is not the root (so that $j_{\l_v}=j_{v}-\ell$), % $$ \eqalign{ (1)\quad& \II\big( ( w_{0\ell}(\t_{v'}) w_{00} (\t_v) - w_{00}(\t_{v'}) w_{0\ell}(\t_v) ) e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\big)(\t_{v'}), \qquad j_{\l_v}=0 \; , \cr % (2)\quad&\II\big(g_0(\t_{v'}-\t_v)\, e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\big)(\t_{v'}), \qquad \kern3.6truecm 0\ell$ and $t=0$; in such a case the last two of \equ(2.25) are relevant and setting $v=v_0$ the integration over $\t_{v_0}$ is the value for $\t_{v'}$ of % $$\eqalign{ (1)\quad&\II\left(w_{00}(\t_v) e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\right)(0) \; , \qquad j_{\l_v} = \ell \; , \cr % (2)\quad&\II\left(e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\right)(0) \; , \qquad \kern1.2truecm j_{\l_v}>\ell \; . \cr} \Eq(4.8)$$ % because, if $\t_{v'}=0$, one has $w_{\ell\ell}(0)=1$ and $w_{\ell 0}(0)=0$; see \equ(2.15) and the last two of \equ(2.25). More generally, if $\t_{v'_0}=t\neq 0$, setting $v=v_0$ and $r=v'_0$, one defines for $\r_{v_0}=1$ % $$\eqalign{ (1)\quad& \II\big( ( w_{0\ell}(\t_{v'}) w_{00} (\t_v) - w_{00}(\t_{v'}) w_{0\ell}(\t_v) ) e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\big)(\t_{v'}), \qquad j_{\l_v}=0 \; , \cr % (2)\quad&\II\big(g_0(\t_{v'}-\t_v)\, e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\big)(\t_{v'}), \qquad \kern3.6truecm 0< j_{\l_v}<\ell \; , \cr % (3)\quad& \II\big( (w_{\ell\ell}(\t_{v'}) w_{00} (\t_v) - w_{\ell 0}(\t_{v'}) w_{0\ell} (\t_v) ) e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\big)(\t_{v'}) \; , \qquad j_{\l_v}=\ell \; , \cr % (4)\quad&\II\big( e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\big)(\t_{v'}) \; , \qquad \kern5.3truecm j_{\l_v}>\ell \; , \cr} \Eq(4.9)$$ % (see the last two relations in \equ(2.25)) and for $\r_{v_0}=0$ % $$\eqalign{ (1)\quad & w_{00}(\t_{v'})\II\big(w_{0\ell}(\t_v) e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\big)(0)\,,\qquad j_{\l_v}=0 \; , \cr % (2)\quad&\II\big(g_0\t_v\,e^{i(\oo'\cdot\nn_v\t_v+n_v\f^0(\t_v))} G_v(\t_v)\big)(0),\qquad \kern1.7truecm 00 \; ; \cr}\Eq(4.10)$$ % note that, for $\t_{v'}=t=0$ and $j_{\l_{v_0}}\ge \ell$, \equ(4.9) and \equ(4.10), summed together, give \equ(4.8). \* \pallino Hence each node still describes a rather complicated set of operations: it is, therefore, convenient to consider separately the terms that appear in \equ(4.6)$\div$\equ(4.10). This can be done by simply adding further labels at each node. To this end, looking at the integrals in \equ(4.7) and \equ(4.10), at $\r_v=0$, and in \equ(4.6) and \equ(4.9), at $\r_v=1$, we see that the following kernels are involved in the integrals % $$ \eqalignno{ w^0_{j_{\l_v}}(\t_{v'},\t_v) & = \cases{ w_{00}(\t_{v'}) w_{0\ell}(\t_v) , & \kern2.8truecm$v>v_0\, , j_v=\ell \, $ $\to$ $j_{\l_v}=0\,,$ \cr g_0\t_v , & \kern2.8truecm $v>v_0\, , j_v>\ell \,$ $\to$ $0\ell , \quad 0\ell,\quad j_{\l_{v_0}}>\ell \, , $ \cr} & \eq(4.11) \cr % w^1_{j_{\l_v}}(\t_{v'},\t_v) & = \cases{ w_{0\ell}(\t_{v'}) w_{00}(\t_v) - w_{00}(\t_{v'}) w_{0\ell}(\t_v), & $v>v_0\ , j_v=\ell \,$ $\to$ $j_{\l_v}=0\,,$ \cr g_0(\t_{v'}-\t_v), & $v>v_0\ , j_v>\ell \,$ $\to$ $0\ell , \quad 0\ell, \,j_{\l_{v_0}}>\ell , $\cr} \cr}$$ % respectively appearing in \equ(4.7) and \equ(4.10), at $\r_v=0$, and in \equ(4.6) and \equ(4.9), at $\r_v=1$. The function in \equ(4.11) involving the Wronskian matrix elements can be computed from \equ(2.15) and one finds, for instance, that the function in the seventh row on the r.h.s. is % $$ w_{0\ell}(\t_{v'}) w_{00}(\t_v) - w_{00}(\t_{v'}) w_{0\ell}(\t_v) = \fra12 \left\{\fra{g_0(\t_{v'}-\t_{v})}{\cosh g_0\t_{v'}\, \cosh g_0\t_{v}}+\fra{\sinh g_0\t_{v'}} {\cosh g_0\t_{v}}-\fra{\sinh g_0\t_{v}}{\cosh g_0\t_{v'}} \right\} \; ; \Eq(4.12)$$ % hence if we consider \equ(4.6)$\div$\equ(4.10) we note that the integrals over $\t_v$ involve functions that can be written, for $\r=\r_v,\t=\t_v,\t'=\t_{v'}$ and for suitable coefficients $c_j(\r,\a,v)$, ($\r=1$ if we consider \equ(4.6), \equ(4.9) and $\r=0$ if we consider \equ(4.7), \equ(4.10)), % $$ \sum_{\a=-1}^2 T_\r^{(\a)}(\r\t',\t)\,Y^{(\a)}(\t',\t) \, c_j(\r,\a,v) \; , \Eq(4.13)$$ % where $Y^{(\a)}(\t',\t)$ are given, if $x=e^{-\s g_0 \t}$ and $x'=e^{-\s g_0 \t'}$, by % $$ \eqalignno{ Y^{(-1)}(\t',\t) = &\fra12 {\sinh g_0\t\over\cosh g_0\t'} \, \exp[in \f^0(\t)] = \sum_{k'=1}^{\io}\sum_{k=-1}^{\io} y_n^{(-1)}(k',k) {x'}^{k'}x^{k} \; ,\qquad k'\ {\rm odd}\,, \cr % Y^{(0)}(\t',\t) = &\fra12 {\exp[in\f^0(\t)]\over\cosh g_0\t'\cosh g_0\t} =\sum_{k'=1}^{\io}\sum_{k=1}^{\io} y_n^{(0)}(k',k) x'^{k'}x^{k} \; , \qquad \kern1.2truecm k'\ {\rm odd}\,, &\eq(4.14)\cr % Y^{(1)}(\t',\t) = &\fra12 {\sinh g_0\t'\over\cosh g_0\t} \, \exp[in\f^0(\t)]= \sum_{k'=-1}^{\io}\sum_{k=1}^{\io} y_n^{(1)}(k',k) {x'}^{k'}x^{k} \; , \qquad \kern.3truecm k'\ {\rm odd} \, , \cr % Y^{(2)}(\t',\t) = &\exp[in\f^0(\t)] = \sum_{k=0}^{\io} \tilde y_n^{(2)}(0,k) x^{k} \; , \qquad \kern3.6truecm k'\=0\cr} $$ % which define the coefficients $y_n^{(\a)}(k',k)$ for $\a=-1,0,1,2$ (it is easily checked that $k'$ is {\it odd} in the first three relations) and we set, for $\a=-1,0,1,2$, % $$ T^{(\a)}_\r(\r\t',\t) = \cases{ g_0(\t'-\t) & if $\a$ is either $0$ or $2$ and $\r=1\;$, \cr g_0\t & if $\a$ is either $0$ or $2$ and $\r=0\;,$ \cr 1 & if $\a$ is either $-1$ or $1\;.$ \cr} \Eq(4.15) $$ % {\it Likewise} we shall set, defining the coefficients $\tilde y_n^{(\a)}(k',k)$, for $\a=-1,0,1$, and $\lis y_n^{(-1)}(k',k)$, % $$\eqalign{ &\tilde Y^{(\a)}(\t',\t)=-\tanh g_0\t'\, Y^{(\a)}(\t',\t)\defi \sum_{k'=-\a}^{\io}\sum_{k=\a}^\io \tilde y^{(\a)}_n(k',k) {x'}^{k'}x^{k}\; , \qquad\a=\pm1,k'={\rm odd}\;, \cr % &\tilde Y^{(0)}(\t',\t)=-\tanh g_0\t'\, Y^{(0)}(\t',\t)\defi \sum_{k'=1}^{\io}\sum_{k=1}^\io \tilde y^{(0)}_n(k',k) {x'}^{k'}x^{k} \;,\qquad\qquad k'\ {\rm odd}\, \cr % &\tilde Y^{(2)}(\t',\t)= Y^{(2}(\t',\t)\defi\sum_{k=1}^\io \tilde y^{(2)}_n(0,k) x^{k} \;,\cr % &\lis Y^{(1)}(\t',\t)= {\cosh g_0\t'\over \cosh g_0\t} \exp[in\f^0(\t)]\defi\sum_{k'=-1}^{\io}\sum_{k=1}^\io \lis y^{(1)}_n(k',k) {x'}^{k'}x^{k},\qquad\quad k'\ {\rm odd}\,,\cr % &\tilde T^{(0)}_1(\t',\t)= g_0(\t'-\t),\qquad \tilde T^{(2)}_1(\t',\t)\=1, \qquad \tilde T^{(0)}_0(0,\t)=T^{(0)}_0 \, ; \cr} \Eq(4.16) $$ % in all other cases the $T,\tilde T, \lis T$--functions will be defined $1$ (no matter which is the value of the labels that we attribute to them: this is done to uniformize the notation. The label $k$ will be called the {\it incoming hyperbolic mode} and $k'$ the {\it outgoing hyperbolic mode} for reasons that become clear by contemplating \equ(4.19) below. In terms of \equ(4.14)$\div$\equ(4.16) the functions \equ(4.11) multiplied by $\exp[in_v\f^0(\t_v)]$ can be expressed as in \equ(4.13), thus defining implicitly the coefficients $c_j(\r,\a,v)$ in \equ(4.13): % $$ \eqalignno{ w^0_{j_{\l_v}}(\t_{v'},\t_v) \, \exp[in_v\f^0(\t_v)] & = \cases{ T^{(0)}_0(0,\t_v)\,Y^{(0)} (\t_{v'},\t_v) + Y^{(-1)} (\t_{v'},\t_v), & $j_{\l_v}=j_v-\ell=0, $ \cr T^{(2)}_0(0,\t_v)\, Y^{(2)} (\t_{v'},\t_v), & $0\ell,$ \cr} & \eq(4.17) \cr % w^1_{j_{\l_v}}(\t_{v'},\t_v) \, \exp[in_v\f^0(\t_v)] & = \cases{ T^{(0)}_1(\t_{v'},\t_v)\,Y^{(0)} (\t_{v'},\t_v) + Y^{(1)} (\t_{v'},\t_v) + & \cr \qquad - Y^{(-1)} (\t_{v'},\t_v), & $j_{\l_v}=j_v-\ell=0,$ \cr T^{(2)}_1(\t_{v'},\t_v)\,Y^{(2)} (\t_{v'},\t_v), & $0\ell . $\cr} \cr} $$ % One could avoid introducing the $\tilde T$ functions as they are simply related to the $T$ functions or are just identically $1$: however it is convenient to introduce them to make the above formulae more symmetric and therefore easier to keep in mind while working with. Finally we define the coefficients $\x_j(k',0)$ by the power series expansion % $$ \eqalign{ {1 \over \cosh g_0\t{'}} & = \sum_{k'=1}^{\io} \x_\ell(k',0) x{'}^{k'} \; , \qquad \ k'\ge1\,, \hbox{ odd } \; , \cr 1 & = \x_j(0,0) \; , \qquad j>\ell \; , \cr}\Eq(4.18) $$ % where $x'=e^{-\s g_0 \t'}$ and $k'$ is odd, which occurs as coefficient $w_{00}(\t')$ in \equ(4.7) (when $\r_v=0$, \ie $v\in\Th_L$). The above definitions (taken from (42) and (45) in [Ge2]) suffice to discuss the whiskers (and therefore the splitting in the action variables). \* \pallino The \equ(4.13) allow us to introduce a ``relatively simple notation'': we can add to each node a {\it badge} label $\a_v=(-1,0,1,2)$ that will distinguish which choice we make between the possibilities in \equ(4.14) and \equ(4.16) and two {\it hyperbolic mode} labels $k'_v,k_v$ which select which particular term we choose in the sums in \equ(4.14) and \equ(4.16); they are integer numbers $\ge-1$. We shall not have to introduce labels to distinguish terms coming from the expansions of $Y^{(\a)},\tilde Y^{(\a)}, \lis Y^{(\a)}$ bearing the same badge $\a$ because one can check that the labels $\a_v$ together with $j_v$ and $v$ itself uniquely determine which choice has to be made. In terms of the latter labels we can define a {\it hyperbolic momentum} of a line $\l_v$ as a label $p(v)\in \ZZZ$ which will be the sum of all the hyperbolic modes of the nodes that precede $v$ {\it plus} the incoming hyperbolic mode of the node $v$ itself: this is the sum of the labels $k_w$ associated with all {\it free} nodes $w\le v$, with $v$ included, and of the labels $k_w'$ associated with all the {\it free} nodes $w\ell$ it is natural to collect together the terms with $p(v_0)=0$: for them, since $j_{\l_{v_0}}>\ell$, in \equ(4.30) one must have $k'_{v_0}+p(v_0)=0$ by the last of \equ(4.14). Note also that by \equ(2.26) the case $(\nn_0(v),p(v_0))=(\V0,0)$ is excluded. If $j_{\l_{v_0}}\le \ell$ we, likewise, collect the terms with $k'_{v_0}+p(v_0)=0$ and, for similar reasons the term with $k'_{v_0}+p(v_0)=-1$ cannot be present (see again \equ(4.14) and \equ(4.16), and use $p(v_0)\ge -1$ supplemented by the relations between the labels $p(v_0)$ and $k_{v_0}'$ which will be exhibited in \S 5.1). Hence the cases with $p(v_0)+k'_{v_0}=-1$ are excluded by construction\annota{6}{\rm The initial data $\X^{h\s}(0,\aa)$ were determined precisely by imposing boundedness at $\s t=+\io$, \ie by imposing the absence of divergent terms in the expansion in powers of $x=e^{-g_0\s t}$ which would correspond to the terms with $p(v_0)=-1$.} and we see that the sum of the values of the trees with $p(v_0)+k_{v'_0}=0$ give us the equations for the actions and the angles of the invariant torus to which the whiskers considered are asymptotic: the terms with $p(v_0)+k_{v'_0}=0$ asymptote to quasi periodic functions of $\oo t$ so that replacing $\oo t$ by $\pps\in T^{\ell-1}$ one gets a parameterization of the points on the tori in terms of a point $\pps\in T^{\ell-1}$ on a ``standard torus''. And the terms with $k'_{v_0}+p(v_0)=1$ provide the leading corrections. Since such terms are present already to order $0$ (as one sees from the expression of the pendulum separatrix) the distance between a point moving on the stable manifold of the torus and the torus itself will be proportional to $x=e^{-g_0\s t}$ as $\s t\to\io$ so that $g_0$ has the interpretation of Lyapunov exponent of the invariant torus; see \equ(2.27) in \S 2.27. (c) Summarizing: {\it the case $(\nn_0(v),p(v))=(\V0,0)$ has to be ruled out as a consequence of \equ(2.26) and of \equ(4.32), respectively for the contributions to $\XXX_{\su}^{h\s}$ and to $\X_+^{h\s}$ (see the last constraint listed at the beginning of \S 4.3). All cases with $k'_{v_0}+p(v_0)=-1$ are also excluded}. %\ifnum\mgnf=0\pagina\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip1.truecm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0{\titolo \S 5. Bounds.} \numsec=5\numfor=1\* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0{\bf 5.1.} We now discuss how to bound the value of a tree or of a sum of a small number of trees which we take for simplicity without leaves and without counterterms. The more general case will be eventually reduced, see below, to the one we consider here. We shall discuss first how to bound values of trees without leaves and counterterms such that $p(v_0)=0$ if $v_0$ is the highest node; hence we shall consider trees, always without leaves and counterterms, with $p(v_0)= 0$. At the end we shall see how the presence of leaves and counterterms modifies the analysis. The following discussion is ``locally'' simple, but ``globally'' delicate and repeats that in [Ge2], \S4: the conclusions are also summarized in the table 0,1,2,3 below. {}From \equ(4.19) it follows that the hyperbolic momentum $p(v)$ is $p(v)\ge -1$ and, as remarked after \equ(4.19), $p(v)=0$ can occur only in special cases: more precisely if $p(v)=0$, then $k_v$ is either $-1$ or $0$, and\\ (1) if $k_{v}=0$, all free nodes $w$ preceding $v$ (whether immediately or not) have $k_w'+k_w=0$, while\\ (2) if $k_v=-1$, all free nodes $w$ preceding $v$ have $k_w'+k_w=0$, {\it except} for a single node $\tilde w\ell$, because $k_w'$ must be $0$ in such a case, so that the second of \equ(4.18) applies; \0(2) if $k_v=-1$, then all the leaves again must have the highest node $w$ with $j_w>\ell$, except at most one leaf with highest node $\tilde w$ with $j_{\tilde w}=\ell$ and $k_{\tilde w}'=1$. \* \0{\cs (Vertical) Table 0.} Possible cases when $p(v)=0,-1$. \* \halign{\strut\vrule\kern2truemm$#\quad$& \vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#$\quad\hfill\vrule\cr \noalign{\hrule} p(v) & k_v & k'_v & \a_v & j_{v} \cr -1 & -1 & {\rm odd}\ge 1 & -1 & \ell \cr 0 & -1 & {\rm odd}\ge 1 & -1 & \ell \cr 0 & 0 & {\ge1} & -1 & \ell \cr 0 & 0 & {\ge0} & 2 & {>\ell} \cr \noalign{\hrule} } \* \0{\cs (Horizontal) Table 1.} Cases $p(v)=0$, $w\notin \PP$. \* \halign{\strut\vrule\kern2truemm$#\quad$& \vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#$\quad\hfill\vrule\cr \noalign{\hrule} \a_w & -1 & 0 & 1 & 2 \cr (k'_w,k_w) & (1,-1) & \hbox{impossible} & (-1,1) & (0,0) \cr p(w) & -1 & \hbox{impossible} & 1 & 0 \cr j_w & \ell & \hbox{impossible} & \ell & {>\ell} \cr \noalign{\hrule} } \* %\ifnum\mgnf=0\pagina\fi \0{\cs (Horizontal) Table 2.} Cases $p(v)=0$, $w\in \PP, w> \tilde w$. \* \halign{\strut\vrule\kern2truemm$#\quad$& \vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#$\quad\hfill\vrule\cr \noalign{\hrule} \a_w & -1 & 0 & 1 & 2 \cr (k'_w,k_w)& (1,-1) & {\rm impossible} & (-1,1) & (0,0) \cr p(w) & 0 & {\rm impossible} & 2 & 1 \cr j_w & \ell & {\rm impossible} & \ell & {> \ell} \cr \noalign{\hrule} } \* \0{\cs (Horizontal) Table 3.} Cases $p(v)=0,\, w=\tilde w$. \* \halign{\strut\vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#\quad$& \vrule\kern2truemm$\quad#$\quad\hfill\vrule\cr \noalign{\hrule} \a_w & -1 & 0 & 1 & 2 \cr (k'_w,k_w)& (1,0) & {\rm impossible} & (-1,2) & (0,1)\cr p(w) & 0 & {\rm impossible} & 2 & 1 \cr j_w & \ell & {\rm impossible} & \ell & {> \ell} \cr \noalign{\hrule} } \* \pallino We extend the definition of path also to the case $p(v)=0, k_v=0$, by setting $\PP\defi\emptyset$ if $j_v>\ell$ and $\PP\defi v$ if $j_v=\ell$, only for purposes of notational convenience (see \equ(5.3) below). This is consistent with the above tables and does not change them. \* \0{\bf 5.2.} {\cs Remark.} Note that, if a tree (or subtree) $\th_0$ with highest node $v_0$ has total hyperbolic momentum $p(v_0)=0$, then there is one and only one path $\PP$, and, if $\PP\neq\emptyset$, then $\PP$ connects the node $v_0$ to some node $\tilde w\ell$, \cr -\s g_0^2\left[ g_0^2+ ( \oo'\cdot\nn_0(v) )^2 \right]^{-1} & if $v\notin\PP$, $j_v=\ell$, \cr g_0^{2-\d_{j_v,\ell}}\left[-\s\,( g_0p(v)-i\s \oo'\cdot\nn_0(v)) \right]^{-(2-\d_{j_v,\ell})} & if $v\in\PP$, $\a_v\neq-1$, $\to p(v)\ne0$ \cr g_0\left[i\oo'\cdot\nn_0(v)\right]^{-1} & if $v\in\PP$, $\a_v=-1$, \cr} \Eq(5.3)$$ % because: \0(a) The first line is such because if $j_v>\ell$ one has necessarily $\a_v=2$, see Tables 0,1,2,3 and we have to integrate a function $g_0(\t_{v'}-\t_v) e^{i n_v\f^0(\t_v)}$ so that $k_v\ge0$: hence $k_v=p(v)=0$ and we have the second function in \equ(4.29) to integrate. \0(b) The second line is such because if $v\not\in\PP,\, j_v=\ell$ we have $w^1_\ell(\t_{v'},\t_v) e^{i n_{v}\f^0(\t_{v})}$ which is a sum of three terms (see the third of \equ(4.17)): the first has $k_v+k_{v'}\ge2$ so is excluded (recall that $p(v_0)=0$ and $v\le v_0$); while the second only sees the contribution to $Y^{(1)}$ with $k_v'=-1,k_v=1$, see \equ(4.14), and the third only contributes by the term with $k_v'=1,k_v=-1$ in $Y^{(-1)}$. In the two cases one has $p(v)=1$ or $p(v)=-1$ respectively; adding up together the latter two contributions and using the first of \equ(5.2) to compute the sum of the coefficients we get % $$ \fra{-\s g_0 y_{n_v}^{(1)}(-1,1) }{g_0-i\s\oo'\cdot\nn_0(v)} - \fra{-\s g_0 y_{n_v}^{(-1)}(1,-1) }{-g_0-i\s\oo'\cdot\nn_0(v)} = \fra\s2 \fra{-2\s g_0^2}{g_0^2 +(\oo'\cdot\nn_0(v))^2} \; , \Eq(5.4) $$ % as it can be read from the coefficients in the intermediate column of \equ(4.24) and from $p(v)=\a_v=\pm1$. \0(c) The third line of \equ(5.3) is obtained by noting that, if $v\in\PP$, $v> \tilde w$ one has $p(v)=1+k_v$, so that, if $j_v=\ell$ and $\a_v\ne-1$, then $p(v)>0$, see Table 2; if $v=\tilde w$ and $\a_v\ne-1$, one has $p(v)\ne0$, see Table 3 (note that $\a_v\ne2,0$ so that we have to consider the first integrand in \equ(4.29)). If $j_v>\ell$ then $\a_v=2$, and, by the Tables 2,3, one has $k_{v'}=0,k_v\ge0$ and $k_v+k_{v'}=1$, so that $k_v=1$ and $p(v)=2$; while, if $v=\tilde w$, then $k_{v'}=0, k_v\ge0$ and $k_{v}'+k_v=1$ imply $k_v=1$, so that $p(v)=1$. So in both cases $p(v)\ge 1$. \0(d) The fourth line is found by looking at the Tables 2,3 as follows: if $\a_v=-1, v\in\PP, v>\tilde w$, one has $k_v+k_v'=0$, hence $k_v=-1,k_{v'}=1$ and $p(v)=0$; this happens only if $j_v=\ell$ so that we have to consider the first integral in \equ(4.29) and we get the fourth relation. \* This shows that the only trees that do not have a value tending to $0$ as $t\to\s\io$, \ie are those with $p(v_0)+k_{v_0}'=0$ (all the others tend to $0$ as a power of $x=e^{-g_0\s t}$), have propagators that are even functions of the momenta flowing in them. In fact the observation on the absence of paths preceding $v_0$ implies that only the first two propagators in \equ(5.3) appear in such trees. Since, as already remarked, the trees with $p(v_0)+k_{v_0}'=0$ give the equations of the tori this is an interesting check that the tori equations so obtained at $t=+\io$ and $t=-\io$ do {\it coincide}. A similar analysis, and check, holds for the cases $j_{\l_{v_0}}\le \ell$. \* \0{\bf 5.4.} {\cs Remark.} Collecting together the contributions from $\a_v=-1$ and $\a_v=1$, for $v\notin\PP$, is a convenient operation and has nothing to do with the deeper resummations that imply the cancellations necessary for convergence estimates: the systematic use of this operation should be described by adding a label to the trees on the nodes $v\notin \PP$ and replacing on the branches which give rise to one of the two propagators in \equ(5.4) the $\a_v$ label by the new label (\eg a $*$ label which indicates that we consider the sum of the values of a tree with $\a_v=1$ and one with $\a_v=-1$). We shall do this without explicitly mentioning the new label, to simplify the notation. Moreover we can no more associate a label $p(v)$ to a node of this kind, as two factors with different $p(v)$ label ($p(v)=\pm 1$ for $\a_v=\pm 1$) have been considered together; nevertheless we shall modify slightly the definition of $p(v)$ by setting $p(v)\defi 1$ in such a case (and letting it unchanged in all the other cases). We shall continue to call $G_v[\oo'\cdot\nn_0(v)]$ a {\it propagator} as, for the purposes of the following analysis, only such modified version of the original propagators appearing in \equ(4.30) plays a r\^ole. \* \0{\bf 5.5.} Furthermore we define the {\it degree} $D$ of a propagator to be $D=2$ if either $v\notin\PP$ or $v\in\PP,j_v>\ell$ (hence $\a_v\ne-1$), and $D=1$ otherwise (the constraint, see \equ(4.3), $1\le r_v\le 2$ implies that the power to which the divisors appear raised is either $1$ or $2$); by extension we shall say that a branch $\l$ has degree $D_{\l}=D$ if the corresponding propagator has degree $D$. The coefficients $\bar F_{\nvec_v}$ and $y'_{v}$ in \equ(5.1) satisfy the bounds % $$ |y_{v}'|\le 4N \; , \qquad \prod_{v\le v_0} |\bar F_{\nvec_v}|\le (\CC N^2)^m \; , \Eq(5.5) $$ % for some constant $\CC$ depending on the perturbation $f_1$ in \equ(2.1); see \equ(2.6), \equ(2.13) and \equ(2.18). For instance one can take % $$ \CC=\max\{|J^{-1}|J_0,1\} \max_{|n|\le N_0 , \, |\nn|\le N} |f_\nvec| \; ; \Eq(5.6) $$ % see \equ(4.23), where $|J^{-1}|$ is the maximum of the matrix elements of the (diagonal) matrix $J^{-1}$. To bound the product in \equ(5.1), we shall consider simultaneously the cases $k_{v_0}=0,-1$; if $k_{v_0}=0$ the path $\PP$ is supposed to be reduced to a single node, $v_0$, or to the empty set, $\emptyset$, depending on the value of $j_{v_0}$, (respectively $j_{v_0}=\ell$, and $j_{v_0}>\ell$, see above). What follows below and in Appendix A2 really goes beyond [Ge2], although it constitutes a natural extension of it. From now now let us consider the case $\ell=3$ and the Hamiltonian \equ(1.1). We shall assume first a condition on the rotation vectors stronger than the Diophantine one, as done in [G1,GG,Ge2], \ie we suppose that they satisfy a {\it strong Diophantine condition} % $$ \eqalignno{ (1) & \quad C_0 | \oo_0 \cdot \nn| \ge |\nn|^{-\t} \; , \quad\quad \V0 \neq \nn \in \ZZZ^{2} , \qquad C_0^{-1}=\hdm C(\h) \; , &\eq(5.7) \cr (2) & \quad \min_{0\ge p\ge n} \Big| C_0 |\oo_0 \cdot \nn| - 2^p \Big| \ge 2^{n+1} \; , \quad\hbox{if} \quad n \le 0, \; \; 0 < |\nn| \le (2^{n+3})^{-1/\t} , \cr} $$ %(3) & \quad |\oo_0\cdot\nn|\ne 1 %\qquad {\rm for\ all}\ \nn\in Z^{\ell-1}\cr} $$ % where $n, p \in \ZZZ$, $n\le 0$, and % $$ \oo_0\= \h^{-1/2}(\O_1+\hdp J^{-1}A_1)^{-1}\oo'= \big(1,\h^{-1}(\O_1+\hdp J^{-1}A_1)^{-1}\O_2\big) \; , \Eq(5.8) $$ % so that $\oo'\cdot\nn=\h^{1/2}(\O_1+\hdp J^{-1}A_1)\oo_0\cdot\nn$. We suppose also that $A_1\in[-\hdm R,\hdm R]$, with $R\le J\O_1/2$, so that $\h^{1/2}(\O_1+\hdp J^{-1}A_1)\ge \h^{1/2}\O_1/2$. If we write $\oo'=(\hdp\O_1+\h J^{-1}A'_1,\hdm\O_2)$ then the measure of the set of $A'_1$'s such that $\oo_0$ verifies the strong Diophantine condition \equ(5.7) has measure of size $O(C_0^{-1}\h^{-3/2})$. %The third condition in \equ(5.7) does not affect the measure %size because it is verfied outside a set of zero measure. By reasoning as in [GG], once the case of strong Diophantine vectors has been understood, it can be extended to cover also the case of the usual (weaker) Diophantine condition (expressed by (1) in \equ(5.7) above). Alternatively one could follow the approach in [GM] avoiding completely considering condition (2) in \equ(5.7) and assuming only the ``usual'' condition (1) in \equ(5.7). We shall not perform such an analysis (which can be easily adapted from the quoted papers), and we shall confine ourselves to the case of strongly Diophantine vectors.\annota{7}{\rm Basically the argument is the following: the analysis that we present does not change if $2^p,2^n$ are replaced by exponentials in another base $q$ (larger than $1$) or even if they are replaced by $\g(p),\g(n)$, where $\g(p)/q^p\tende{p\to-\io}1$, and if in the second of \equ(5.7) we substitute $2^p, 2^{n+1}, 2^{n+3}$ by, respectively, $\g(p), \g(n+1), \g(n+3)$. One then proves a simple arithmetic lemma (see [GG]), whereby it follows that, if the first of \equ(5.7) is verified and if $\g(p)$ is suitably chosen, then the second holds with $\g(p), \g(n+1), \g(n+3)$ replacing $2^p, 2^{n+1}, 2^{n+3}$.} Keeping in mind that $C_0=\hdm e^{+s \hdm}$ is enormous we shall say that % $$\eqalign{ (1)\ & \ G_v[\oo'\cdot\nn_0(v)] \qquad\hbox{is on scale 1, if $C_0|\oo_0 \cdot \nn_0(v) | > C_0/4 $, or if $p(v)\ne0$;}\cr (2)\ & \ G_v[\oo'\cdot\nn_0(v)] \qquad\hbox{is on scale 0, if $1/2\ell$ (see the first of \equ(5.3)) and $p(v_0)=0$ implies that $\a_{v_0}=2, k_{v_0}=0, k'_{v_0}=0$ so any path preceding $v_0$ would necessarily imply the contradiction $p(v_0)=1$. Also if $D_{\l_{v_1}}=1$, $D_{\l_{v_0}}=2$ one must have $j_{v_1}=\ell$ hence $k_{v_1}=0$ (otherwise $p(v_0)>0$) so that $k'_{v_1}=0$: {\it but} $\a_{v_1}<2$ and $k'_{v_1}$ must be odd. The cases $D_{\l_{v_0}}=1$, $D_{\l_{v_1}}=1,2$ are both allowed. \* \0{\bf 5.9.} Given a tree $\th$, let $V$ be a resonance (if there are any) with entering branch $\l_{v_1}$ of degree $D_{\l_{v_1}}=2$. Then consider the family of all trees which can be obtained from $\th$ by detaching the part of the tree having $\l_{v_1}$ as root branch and reattaching it to all the remaining nodes {\it internal to $V$ but external to the resonances contained inside the cluster $V$} (if any); to the just defined set of trees we add all the trees obtained by reversing simultaneously the signs of the latter modes of the nodes (this can be done as the sum of the mode vectors $\nn_w$ of such nodes, $w\in V$, vanishes). The set of all the so obtained trees will be denoted $\FFFF_V(\th)$. The definition of resonance and the strong Diophantine condition insures that all the trees so constructed have a well defined value (\ie no division by zero occurs in evaluating it with the above rules); see the Remark 5.10, (1), below. If the entering branch $\l_{v_1}$ of the resonance has degree $D_{\l_{v_1}}=1$ then also the exiting branch $\l_{v_0}$ has degree $D_{\l_{v_0}}=1$, and we collect together with the considered tree also the tree which is obtained from $\th$ through the following operation. Replace the resonance $V$ with {\it a single node} $v$ carrying labels $\d_v=0$ and $\k_v=k_V$, if $k_V$ is the order of the resonance. The set of all the so obtained trees will be denoted by $\FFFF_V(\th)$: the definition of the class $\FFFF_V(\th)$ will therefore depend on the degree of the branch entering $V$. Then repeat the above operations for all resonances in $\th$. Thus a class $\FFFF(\th)$ has been constructed and the number of elements of $\FFFF(\th)$ is bounded by the product $\prod_V 2 \NN_V$ of the numbers $\NN_V$ of branches in each resonance $V$ {\it which are not} contained inside inner resonances. The latter product is bounded by $\exp \sum_V 2\NN_V\le \exp 2m $; the $\FFFF(\th)$ can be obtained starting from any of its elements (which therefore we shall call {\it representatives} of the class): this is again a {\it consequence of the strong Diophantine condition}, see [Ge2]. \* \0{\bf 5.10.} {\cs Remarks.} (1) The strong Diophantine condition plays a r\^ole here that should be stressed. In fact one checks that because of it the scale of a branch inside a resonance {\it cannot} change too much, as one considers the different members of a given family. Not enough to change the sets of branches that belong to a given resonance and insures that the different families of trees {\it do not overlap}: for this reason the strong Diophantine condition leads to a simplification of the analysis (the simplification in the simpler case of the KAM theory). A simplification that is however not major (as explained informally in [G1] and as shown in [GG], see footnote 7 above). (2) To see how the above difficulty is bypassed by using the alternative approach of [GM1,GM2], we refer to the conclusive comments in [GM1], \S 3. \* \0{\bf 5.11.} Consider trees with $p(v_0)=0$, if $v_0$ is the highest node of the tree; then the expression of each tree value contains a product like \equ(5.1). As mentioned in the introduction {\it we consider only trees without leaves.} Since the leaf values factorize with respect the product \equ(5.1), they can be dealt with separately, and no overlap arises with the cancellation mechanisms acting on the product \equ(5.1): so that leaves can be easily taken into account; see \S A3.3 in Appendix A3 (see also [G1,Ge1,Ge2]). The counterterms can also be explicitly expanded in terms of tree values, according to \equ(3.2), which again we can imagine to have no leaves, (see however the comments in \S A3.3 below). The cancellation mechanisms described in [Ge1,Ge2] (and recalled in Appendix A3) lead to the bound (on a given family $\FFFF(\th)$ described above, in \S 5.9), see \equ(5.1), \equ(4.30), \equ(4.23) % $$ \eqalignno{ \left( {1\over \h^{1/2}} \right)^{2m} & \Big[ (4N^3\CC')^{m} 2^{4m}e^{2m} \prod_{n\le0} \big( C_0^{2N_n^2} 2^{-2nN^2_n} \big) \big( C_0^{N_n^1} 2^{-nN^1_n} \big) \Big] \; \cdot & \eq(5.11)\cr & \cdot \Big[\prod_{n\le 0}\;\prod_{T,\,n_T=n} \prod_{i=1}^{m^1_T(n)}\,2^{(n-n_{i}+3)} \prod_{i=1}^{m^2_T(n)}\,2^{2(n-n_{i}+3)} \Big] \; , \cr} $$ % where \0$\bullet$ $\CC'=\max\{(2g_0/\O_1)^2,4^2\}\CC$, with $\CC$ the dimensionless constant defined in \equ(5.6); \0$\bullet$ $m$ is the number of nodes $v\ge v_0$; \0$\bullet$ $N^j_n$ is the number of propagators on scale $n$ and of degree $j$ in $\th$, which can be written as % $$ N^j_n=\bar N^j_n+\sum_{T \atop n_T=n, D_T=j} (-1) + \sum_{T \atop n_T=n} m^j_T(n) \; , \Eq(5.12) $$ % where $m^j_T(n)$ is the number of resonances on scale $n$ and degree $j$ (\ie with entering branch having a propagator of degree $j$) contained inside the cluster $T$; \pallino the terms $\bar N^j_n$, $j=1,2$, which count the number of propagators {\it which do not correspond to resonant branches} plus the number of clusters on scale $n$ and of degree $j$ in $\th$, satisfy the bounds % $$ \sum_{j=1}^2 \bar N^j_n \le 4 m N 2^{(n+3)/\t},\qquad \sum_{n=-\io}^{0} \sum_{j=1}^2 \bar N^j_n \le 4 m \g N \h \; , \Eq(5.13) $$ % (with $\g=4\O_1/\O_2$) which are proven in Appendix A2; \0$\bullet$ the first square bracket in \equ(5.11) is the bound on the product of individual elements in the family $\FFFF(\th)$ times the bound on their number $\prod_V 2\NN_V< e^{2m}$, see above. \0$\bullet$ the second square bracket term is the part coming from the maximum principle, (in the form of Schwarz's lemma), applied to bound the sums of the tree values (``{\it resummations}'') over the classes $\FFFF(\th)$ introduced above: this is a {\it non trivial product of small factors} that arise from the cancellations associated with the resummations, see Appendix A3. In \equ(5.11) $n_i$ is the scale of the cluster $V_i$ which is the $i$--th resonance inside $T$, as in [Ge2]; \pallino the $\eta^{-m/2}$ arises as a lower bound on the small divisors of the form $\oo'\cdot\nn$ on scale $n=1$ (for $n=1$ we use the better bound $|\oo_0\cdot\nn|\ge 2^2\hdp$). \* \0{\bf 5.12.} {\cs Remark.} The first bound \equ(5.13) holds for all $n$ and for all Hamiltonians of the form \equ(2.1). On the contrary the second bound in \equ(5.13) will follow from the fact that the rotation vector $\oo_0$ has the form \equ(5.8), with $\h$ small, and will be used to control the (huge) factors $C_0$ in \equ(5.11). \* \0{\bf 5.13.} Hence by substituting \equ(5.12) and the first of \equ(5.13) into \equ(5.11) we see that, for $j=1,2$, the $m^j_T(n)$ is taken away by the first factor in $\,2^{jn} 2^{-jn_{i}}$, while the remaining $\,2^{-jn_i}$ are compensated by the $-1$ before the $+m^j_T(n)$ in \equ(5.11) taken from the factors with $T=V_i$ (note that there are always enough $-1$'s), and therefore \equ(5.11) is bounded by % $$ \eqalign{ & \left( {2\over \h^{1/2} } \right)^{2m} (4N^3\CC')^{m} e^{m} 2^{4m}2^{8m} C_0^{8m\g N\h} %2^{8mN\g} \Big] \cr & \Big[ \prod_{n=-\io}^{n_0-1} C_0^{8mN2^{(n+3)/\t}} \prod_{n=-\io}^{0} 2^{-8 m N n 2^{(n+3)/\t} } \; , \cr} \Eq(5.14) $$ % because the product of the factors $C_0$ in \equ(5.11) can be bounded by using the second of \equ(5.13), since the product does not contain the $n=1$ factor). The last product in \equ(5.14) is bounded by % %$$ \eqalign{ %\prod_{n=-\io}^{n_0-1} & C_0^{8mN2^{(n+3)/\t}} 2^{-8mNn2^{(n+3)/\t})} %\cr & \qquad \le \exp \left[ 8mN 2^{3/\t} %\sum_{p=[\g/\h]+1}^{\io} (\log C_0+p\log2) 2^{-p/\t} \right] %\; , \cr} \Eq(5.15) $$ $$ \prod_{n=-\io}^{0} 2^{-8mNn 2^{(n+3)/\t}} \le \exp \Big[ 8mN2^{3/\t} \log2 \sum_{p=1}^{\io} p 2^{-p/\t} \Big] \; , \Eq(5.15) $$ % hence, by adding the remark that the perturbation degree $k$ and the number of tree nodes $m$ are related by $m<2k$, a bound on the sum over all the subtrees of order $k$ with $p(v_0)=0$, $\nn(v_0)=\nn$ (recalling that the number of trees with $m$ nodes is $<4^m m!$) is % $$ \D_k\defi \Big| \fra1{|\FFFF(\th)|} \sum_{\th'\in\FFFF(\th)} \prod_{v\in\th'} \bar F_{\nvec_v} G_v[\oo'\cdot\nn_0(v)]\,\tilde y_{v}\Big| \le B_0^{2k} \h^{-2k}\; , \Eq(5.16)$$ % for some positive constant $B_0$. The normalization constant $|\FFFF(\th)|$ is introduced in order to avoid overcountings: in fact if $\th'\in\FFFF(\th)$ then $\th\in\FFFF(\th')$, so that, without dividing by $|\FFF(\th)|$ in \equ(5.16), each tree would be counted $|\FFFF(\th)|$ times. If $C_0^{-1}=\hdm C(\h)$ is chosen as in the statement of Theorem 1.4, an explicit calculation gives the bound on \equ(5.11) of the form $(\hdm)^{4k} B_0^k$, $k\ge1$, and % $$ B_0 = 2^{18}(4N^3\CC') \, \exp \Big[ 2+4\g N\h\log\h+ 8s\g N\hdp + 8N 2^{3/\t}\log 2 \sum_{p=1}^{\io} p 2^{-p/\t} \Big] \; , \Eq(5.17) $$ % which is bounded uniformly in $\h$ (for $\h\le1$). \* \0{\bf 5.14.} In the previous section trees with $p(v_0)=0$ have been considered; in particular only the contributions \equ(5.1) arising from the value \equ(4.30), once the corresponding tree has been deprived of leaves and counterterms, have been bounded and the bound \equ(5.16) has been obtained through a suitable resummation operation. In such a case the sum over the labels $(k_v',k_v)$ is trivial because the condition $p(v_0)=0$ imposes that only a few values (up to three per node) can be assumed by the hyperbolic mode labels; also the sum over the mode labels $\nvec_v$ cannot create any problems. In fact for any node $v$ one has $|\nvec_v|\le N$ and $|n_v|\le N_0$ (see the eighth item in \S 4.1). The cases $p(v_0)\ne0$ as well as those involving graphs containing leaves or counterterms can be treated in the same manner as already done in [G1,Ge2]. We provide, in Appendix A4, a quick description of the construction of the analyticity bound $\e_0=D^{-1}$ with % $$e_0^{-1}=D=\left[ B 2^6\ell (2N+1)^{2\ell-1}(2N_0+1) \right]^2 \; , \qquad B=\max(B_0\h^{-1},B_1)\Eq(5.18)$$ % and $B_1$ is a suitable numerical constant. The part of Theorem 1.4 not concerning the connection between the average action $\AA'$ and the rotation vector $\oo'$ nor the splitting size follows. \* \0{\bf 5.15.} Determining the exact splitting size (\ie the leading behavior asymptotically as $\h\to0$ with $\e< B\h^2$) is {\it not} trivial because of the existence of major cancellations in the evaluation of the determinant of the splitting matrix; however the analysis in [GGM2] dealt with this question in detail: in the latter paper remarkable cancellations are exhibited and an exact formula for the splitting angles is derived (see (7.19) of [GGM2]). One gets the results in the last item of Theorem 1.4 simply if [GGM2] and the first part of Theorem 1.4 (to estimate the remainders) are used: then the claimed bounds on the splitting follow immediately (see Remark 1.5). In [GGM3] an improvement of lemma 1 and lemma 1' of [CG] was used instead to control the density of tori in phase space (the lemmata in [CG] were, as such, useless already in the case in [GGM2] because they would require that $\e$ be far smaller than the $\e_0$ of Theorem 1.4); see [GGM3], where this is discussed in detail and differs from our case only because it relied on a theorem weaker than Theorem 1.4 above (as the radius of convergence estimate there is proportional to $\h$ to the power $\fra92+$ rather than our $2$). \* \0{\bf 5.16.} {\cs Remarks.} (1) The bound \equ(5.16) and the discussion in \S 5.14 imply the convergence of the perturbative expansions for the parametric equations of the invariant tori (for the Hamiltonian \equ(1.1)), if $|\e|<\e_0=O(\h^2)$. This bound on the convergence radius should be compared with the value given by [GGM3], which, for $\NN=O(\hdm)$, gives $\e_0=O(\h^{\fra92}/ \log^2\h^{-1})$. {\it As usual the Lindstedt series gives a much better estimate than the classical method} (\ie an exponent $2$ versus $\sim 4.5$). We do not see immediately how to improve substantially the classical estimate without important changes in the architecture of the proof of [GGM3], although this should be possible; on the other hand, from the above analysis, $\e_0=O(\h^2)$ might be close to an optimal result. If so it should be no surprise that our analysis is so delicate. (2) In the Hamiltonian \equ(1.1),\equ(2.1) the polynomial dependence of the interaction on the rotators angles has very likely a purely technical motivation (as it simplifies the analysis) and could probably be relaxed into a more general analytical dependence, as in [BCG]. On the contrary the hypothesis that the perturbation is a trigonometric polynomial of degree $N_0$ in $\f$ is fundamental to get the correct asymptotic behavior, in order to apply the results in [GGM2], where the dominance of Mel'nikov integral is proven {\it provided the perturbation is polynomially small in a power of $\h^{N_0}$} (so that the results of [GGM2] become meaningless for $N_0\to\io$). (3) A bound of the form \equ(5.16) holds under the weaker condition that $C(\h)\le e^{-s\h^{-a}}$, with %$a<1$. Also the case $a=1$ can be included provided that, in such a %case, a condition on $s$ has to be imposed. $a\le 1$ (see \equ(5.17)). (4) If $q$ is defined as in \S1.6 so that $|\e| C(\h)^{q\h}< 1$ implies analyticity in $\e$, the above analysis gives that $q$ can be taken $q=8\g N$. In general all the bounds found so far are not uniform in $N$; in order to deal with the analytical case in the frame of the exploited formalism one should bound the small divisors by using the results of [GM2] or Eliasson-Siegel's bound (see for instance [BGGM]), and use explicitly as in [BCG] the exponential decay in $\nn$ of the Fourier coefficients $f^1_{\nvec}$. (5) Note that we have convergence for $|\e| T_c$ the asymptotic formula that we can prove only for $T> \z$ holds, but for $T=T_c$ it is modified remaining qualitatively of the same size $O(e^{-\fra12\hdm})$ and for $TT_c$ is described by a trivial fixed point; a non trivial fixed point describes the case $T=T_c$ and another ``low temperature'' fixed point describes the cases $T\ell$ or $j_w=\ell$ and internal momenta $(\nn',p')$. Such terms would give rise to $\aa$--dependent counterterms which of course are not allowed: however it turns out that the sum over all contributions to tree values of trees with $(\nn(v_0),p(v_0))=(\V0,0)$ from such trees cancel {\it exactly}: this is explained, together with the other cancellations built in our algorithm, in Appendix A3 (see \S A3.5 in particular). \* \0{\bf A1.3.} Let us consider the first integral in \equ(A1.1). Corresponding to the node $v_0$ of each tree whose value contributes to $\X_-^{h\s}(t)$ there is a coefficient $\tilde y_{n_{v_0}}(k_{v_0}',k_{v_0})$, see \equ(4.14), \equ(4.17). Then from \equ(A1.1) and the just formulated condition to impose we obtain % $$ \sum_{\th\in\TT_{\V0,h} , \a_{v_0}=-1 \atop p(v_0)=0 ,k_{v_0}'=1} \overline{{\rm Val}}(\th) + \g_h(g_0) \left. w_{\ell}^1(t,\t)\, \sin\f_0(\t) \right|_{k'=1,p=0} = 0 \; , \Eqa(A1.4) $$ % where the sum is over the set $\TT_{\V0,h}$ of all trees of order $h$ and momentum $\nn(v_0)=\V0$, with $\nn_0(v_0)=\V0$ (see Remark A1.2), $p(v_0)=0$, $j_{v_0}=\ell$ and $k_{v_0}'=1$; hence if $p(v_0)=0, j_{v_0}=\ell$, one must have $k_{v_0}=-1$, hence $\a_{v_0}=-1$ and $k'_{v_0}=1$ which is a possible case indeed. A trivial calculation (just take into account that $y_{n_v}^{(-1)}(1,-1)$ $=$ $\s/2$ and $\sin\f^0(\t)=4\s x + O(x^3)$) gives % $$ \left. w_{\ell}^1(t,\t)\, \sin\f_0(\t) \right|_{k'=1,p=0} = 2 \; , \Eqa(A1.5) $$ % so that \equ(4.32) follows; the above follows [Ge2], page 287. \* %\0{\bf A1.3.} {\cs Remark.} In defining the counterterms %in \S A1.1 no condition has been imposed on the label $\nn'$ appearing %in \equ(A1.2) and \equ(A1.3). So it is not {\it a priori} evident that %the contributions in \equ(A1.4), for which the {\it total momentum} %$\nn(v_0)$ {\it vanishes}, \ie $\nn(v_0)=\V0$, are all contributions %with $(\nn_0(v_0),p(v_0))=(\V0,0)$. On the other hand it is %immediately understood that this is necessary, in order that the %counterterms be independent of $\aa$. As a matter of fact all %contributions with $(\nn_0(v_0),p(v_0))=(\V0,0)$ which have not also %$\nn(v_0)= %\V0$ automatically vanish when summed together, %as we shall see in \S A3.5. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip1.truecm %\ifnum\mgnf=0\pagina\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0{\titolo Appendix A2. (Improved) resonant Siegel-Bryuno's bound} \numsec=2\numfor=1\* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0{\bf A2.1.} We follow the idea of P\"oschel, [P\"o] (see also [G1, GG,Ge2]). In the discussion, we focus on 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 $N^*_n(\th)$ the number of non resonant branches carrying a scale label $\le n$, in a tree $\th$ with $m$ nodes, we shall prove first that % $$ N^*_n(\th) \le 2m E_n - 1 \; , \qquad E_n \defi N2^{(3+n)/\t},\qquad n\le1\; , \Eqa(A2.1) $$ % provided that $N^*_n(\th)>0$, and % $$ N^*_{0}(\th) \le 2m \g N \h - 1 \; , \qquad \g \defi 4\O_1/\O_2 \; , \Eqa(A2.2) $$ % if $N_{0}^*(\th)>0$. Define, as in \S 5.7, $\oo_0=(1,\h^{-1}(\O_1+\hdp J^{-1}A_1)^{-1}\O_2)$. Then $C_0|\oo_0\cdot\nn|>|\nn|^{-\t}$ for all $\V0\neq\nn\in\ZZZ^{\ell-1}$; see \equ(5.7). Assume also $\h$ so small that $C_0\ge 2$, %%%%, \fra14\fra{\O_2}{\h \O_1},\O_1+\hdp J^{-1}A_1>\O_1/2$ (this is not restrictive as we are interested in $\h\to 0$). Set $E_n\= N2^{(n+3)/\t}$ as in \equ(A2.1). Note that if $m\le E_n^{-1}$ one has $N_n^*(\th)=0$. In fact $m\le E_n^{-1}$ implies that, for all $v\in\th$, $|\nn_0(v)| \le N E_n^{-1}$, \ie $C_0|\oo_0\cdot\nn_0(v)|\ge (N^{-1}E_n)^{\t}$ $=$ $2^{n+3}$, so that there are {\rm no} clusters $T$ with $n_T=n$. Note also that if $m\le (\g N \h)^{-1}$, with $\g=4\O_1/\O_2$, then $N_{0}^*=0$, as $|\oo_0\cdot\nn_0(v)|\ge 1$ for all $v\in\th$ in such a case. \* \0{\bf A2.2.} Let us prove first the inequality \equ(A2.1). If $\th$ has the root branch either with scale $>n$, or with scale $\le n$ and resonant, then calling $\th_1,\th_2,\ldots,\th_k$ the subtrees of $\th$ ending into the highest node $v_0$ of $\th$ and with $m_j>E_n^{-1}$ nodes, $j=1,\ldots,k$, one has $N_n^*(\th)=N_n^*(\th_1)+\ldots+N_n^*(\th_k)$ and the statement is inductively implied from its validity for $m' m- (2E_n)^{-1}$: but in the latter case we shall show that the root branch of $\th_1$ has scale $>n$. Accepting the last statement (which will be proved below), one will obtain $N_n^*(\th)=1+N_n^*(\th_1)= 1+N_n^*(\th'_1)+\ldots+N_n^* (\th'_{k'})$, where $\th'_j$'s are the $k'$ subtrees ending into the highest node of $\th'_1$ with orders $m'_j>E_n^{-1}$, $j=1,\ldots,k'$. Going once more through the analysis the only non trivial case is if $k'=1$ with the root branch of $\th'_1$ non resonant; and in such 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 $\le m-(2E_n)^{-1}$. It remains to check that if $m-m_1<(2E_n)^{-1}$ then the root branch of $\th_1$ has scale $>n$. Let us proceed by {\it reductio ad absurdum}. Suppose that the root branch of $\th_1$ is on scale $\le n$. Then $C_0|\oo_0\cdot\nn_0(v_0)|\le\,2^n$ and $C_0|\oo_0\cdot\nn_0(v_1)|\le \,2^n$, if $v_1$ is the highest node of $\th_1$. Hence $C_0|\oo_0\cdot(\nn_0(v)-\nn_0(v_1))|< 2^{n+1}$ (equality would imply violation of the strong Diophantine property, \equ(5.7)), and the Diophantine condition implies that % $$ |\nn_0(v_0)-\nn_0(v_1)|> 2^{-(n+1)/\t} \= \d \; , \Eqa(A2.3) $$ % because $\nn_0(v_0)\neq\nn_0(v_1)$ (the root branch of $\th$ being supposed non resonant). But $m-m_1<(2E_n)^{-1}$, so that $|\nn_0(v_0)-\nn_0(v_1)|< (2E_n)^{-1}N < 2^{-1}2^{-(n+3)/\t}$ $=$ $2^{-(1+2/\t)}\d<\d$, which contradicts inequality \equ(A2.3). \* \0{\bf A2.3.} Let us prove now \equ(A2.2). If $\th$ has the root branch either with scale $1$, or with scale $\le 0$ and resonant, then calling $\th_1,\th_2,\ldots,\th_k$ the subtrees of $\th$ ending into the highest node $v_0$ of $\th$ and with $m_j>(\g N\h)^{-1}$ nodes, $j=1,\ldots,k$, one has $N_{0}^*(\th)=N_{0}^*(\th_1)+\ldots+N_{0}^*(\th_k)$ and the statement is inductively implied from its validity for $m'm- (2\g N\h)^{-1}$: but in the latter case the root branch of $\th_1$ has scale $1$. Accepting the last statement (which will be proved below), one will obtain $N_{0}^*(\th)=1+ N_{0}^*(\th_1)= 1+N_{0}^*(\th'_1)+\ldots+N_{0}^*(\th'_{k'})$, where $\th'_j$'s are the $k'$ subtrees ending into the highest node of $\th'_1$ with orders $m'_j>(2\g N\h)^{-1}$. Going once more through the analysis the only non trivial case is if $k'=1$ and in that case $N_{0}^*(\th'_1)=N_{0}^*(\th^{\prime \prime}_1) + \ldots + N_{0}(\th^{\prime \prime}_{k^{\prime \prime}})$, \etc., until we reach a trivial case or a tree of order $\le m-(2\g N\h)^{-1}$. It remains to check that, if $m-m_1<(2\g N\h)^{-1}$, then the root branch of $\th_1$ has scale $1$. Suppose that the root branch of $\th_1$ is on scale $\le 0$. Then $p(v_1)\neq0$ and $|\oo_0\cdot\nn_0(v_0)|\le1/4$, $|\oo_0\cdot\nn_0(v_1)| \le 1/4$, if $v_1$ is the highest node of $\th_1$, \ie % $$ |\oo_0\cdot(\nn_0(v_0)-\nn_0(v_1))| \le 1/2 \; . \Eqa(A2.4) $$ % As the root branch of $\th$ is supposed non resonant, then $m-m_1<(2\g N\h)^{-1}$ implies that $0$ $<$ $|\nn_0(v_0)-\nn_0(v_1)|$ $<$ $(2\g N\h)^{-1}N = (2\g\h)^{-1}$, so that one would have $|\oo_0\cdot(\nn(v_0)-\nn(v_1))|\ge 1$, which is contradictory with the inequality \equ(A2.4). %(the case $|\oo_0\cdot(\nn(v_0)-\nn(v_1))|= 1$ %would imply $|\oo_0\cdot\nn_0(v_0)|$ $=$ $|\oo_0\cdot\nn_0(v_1)|$ $=$ %$1/2$, so that it is not possible because, given the form of $\oo_0$, %would imply that $\pm \oo_0\cdot(\nn_0(v_0)\pm\nn_0(v_1))=1$ for some %choice of the signs which is impossible because, if $n=0$, %$C_0|\oo_0\cdot\nn_0(v_0)|0$ then the number $p_n(\th)$ of clusters of scale $n$ verifies the bound % $$ p_n(\th) \le 2m N 2^{(n+3)/\t}-1 \; . \Eqa(A2.5) $$ % In fact this is true for $m\le E_n^{-1}$, if $E_n$ is defined as in \S A2.1. Otherwise, if the highest tree node $v_0$ is not in a cluster on scale $n$, one calls $\th_1,\ldots,\th_k$ the subtrees ending into $v_0$, and one has $p_n(\th)=p_n(\th_1)+\ldots+p_n(\th_k)$, so that the statement follows by induction. If $v_0$ is in a cluster $V$ of scale $n$, and $\th_1$, $\ldots$, $\th_k$ are the subtrees entering the cluster containing $v_0$ and with orders $m_j> E_n^{-1}$, one will find $p_n(\th)=1+p_n(\th_1)+\ldots+p_n(\th_k)$. 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 $V$ and it will have a propagator of scale $\le n-1$. Therefore the cluster $V$ must contain at least $E_n^{-1}$ nodes. This means that $m_1\le m-(2E_n)^{-1}$. Finally, the bound % $$ \sum_{n=-\io}^0 p_n(\th) \le 2m\g N\h-1 \Eqa(A2.6) $$ % is a trivial consequence of \equ(A2.2). \* \0{\bf A2.5.} Let $\bar N^*_n\le N^*_n$ be the number of non resonant branches on scale $n$. Then if $\bar N_n$ is the number of non resonant branches {\it plus} the number of clusters on scale $n$, $\bar N^*_n$ verifies the bounds % $$ \bar N_n^* = \big( \bar N_n^* + p_n \big) - p_n \= \bar N_n-p_n \le 4m N 2^{(n+3)/\t}-2- \sum_{T \atop n_T=n} (1) \le 4m N 2^{(n+3)/\t}+\sum_{T\atop n_T=n} (-1) \; . \Eqa(A2.7) $$ % This proves that \equ(A2.1) and \equ(A2.5) imply an inequality analogous to the first of \equ(5.13); likewise one derives an inequality similar to the second of \equ(5.13) by combining \equ(A2.2) and \equ(A2.6). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip1.truecm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0{\titolo Appendix A3. Cancellations between resonances} \numsec=3\numfor=1\* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0In this appendix we recall briefly the cancellation mechanisms of [Ge2]. We provide this as a guide to the reader and as a tune up of a fine points of the analysis of [Ge2] (the analysis in A3.2 is given here in full details while in [Ge2] it was left out). \* \0{\bf A3.1.} Consider a tree $\th$ with a strong resonance $V$ of order $k_V$. Let $\l_{v_0}$ and $\l_{v_1}$ be, respectively, the exiting and entering branches of $V$. There are two possibilities: either the degree of the propagator corresponding to the branch exiting from $V$ is $D_{\l_{v_0}}=2$ or it is $D_{\l_{v_0}}=1$ (equivalently the degree of the resonance is either $D_V=2$ or $D_V=1$). Let us discuss first the case in which the degree $D_V$ of the resonance is $D_V=2$. Then $j_{v_0}>\ell$ (see \equ(5.3) and the comments after the definition of strong resonance in \S 5.9) and, by following the notations of \S 5.1, we shall say that $\PP=\emptyset$, \ie there is no path $\PP$ ending into $v_0$. It follows, from the properties of $\PP$ discussed at the beginning of \S 5.1 above, that $p(v_1)=0$ implies $j\=j_{v_1}>\ell$ and $D_{\l_{v_1}}=2$ (see again \equ(5.3)). Consider all the trees belonging to the class $\FFFF(\th)$ which are obtained from $\th$ by detaching the subtree having as branch root the entering branch $\l_{v_1}$ of the resonance and attaching it to all the remaining nodes of $V$ (see the definition of the class $\FFFF_V(\th)$ in \S 5.9). As a consequence of such an operation\\ $\bullet$ some of the branches internal to the resonance have changed the free momentum by an amount $\nn_0(v_1)$, and \\ $\bullet$ if $w$ is the node inside $V$ to which the branch $\l_{v_1}$ is attached and $j_{v_1}-\ell>0$, then $\bar F_{\nvec_w}$ (see \equ(4.23)), has the form of an even function of $\nvec_w$ times a factor $(i\n_{wj})$. We shall call {\it resonance value} $\RR_V$ the product of factors appearing in the definition of tree value and relative only to the nodes and branches internal to the resonance $V$: % $$ \RR_V = \bar F_{\nvec_{v_0}} y_{v_0}' \prod_{v\in V \atop v 0$. If we sum also on an overall change of signs of the mode labels of the nodes internal to the resonance (by following the definition of the class $\FFFF(\th)$ given in \S 5.9), we obtain a zero contribution also to first order in $\m$ (here the even parity of the perturbation $f$ is essential, see [G1,Ge2]). This can be seen by using the explicit form of the functions in \equ(4.21), \ie the coefficients listed in \equ(4.24). Noting that in the present case {\it there cannot be any $\PP$ inside $V$} the only propagators we can associate with the branches internal to $V$ have the form of the two first terms of \equ(5.3), so that, for $\m=0$, {\it they are even functions of the mode labels}. Moreover in such a case the analysis in \S 5.1 shows that $\a_v=-1$, $\a_v=1$ and $\a_v=2$ imply, respectively, $k_v'=-k_v=1$, $k_v'=-k_v=-1$ and $k_v'=-k_v=0$ (the case $\a_v=0$ is not possible here): then no $n_v$ labels appear in the coefficients $y_{n_v}^{(\a_v)}(k_v',k_v)$ corresponding to the nodes $v\le v_0$ (see the list of coefficients in \equ(4.24)). Therefore all the dependence on the $n_v$ labels is through the factors $\bar F_{\nvec_v}$ in \equ(4.23). This yields that there is an even number of the $n_v$ (if there are any) corresponding to the nodes $v\in V$: two for each branch $\l_v$ with $j_v=\ell$, by taking into account that $j_{v_0},j_{v_1}>\ell$, so that no change is produced by the sign reversal (since, by the parity properties of the Hamiltonians \equ(1.1) and \equ(2.1), one has also $f^{\d_v}_{\nvec_v}=f^{\d_v}_{-\nvec_v}$). This means that the resonance value is an even function of $\m$. \* \0{\bf A3.2.} Let us now consider the case in which the strong resonance is of degree $D_V=1$ and the tree $\th$ has no leaves inside $V$. In such a case $\a_{v_0}=-1$ and $j_{v_0}=\ell$, hence $D_{\l_{v_0}}=1$ (see \equ(5.3)): then a first order zero in $\m$ will be enough. Moreover there is a $\PP$ inside the resonance: we shall distinguish between the cases $v_1\notin\PP$ and $v_1\in\PP$. Let us consider first the case $v_1\notin\PP$ (in particular this is the case when $\PP=v_0$, $k_{v_0}=0$, provided $k_V\ge 2$). In such a case $j_{v_1}>\ell$ and we can reason as above to obtain a first order zero. Note that in such a case there would be no cancellations between tree values of trees obtained by the sign reversal operation. On the contrary, if $v_1\in\PP$, then $k_{v_0}=-1$, and one has also $\a_{v_1}=-1$ and $j_{v_1}=\ell$. %in particular this is the case when $\PP=v_0$, with $k_{v_0}=-1$. In this case consider together with the tree $\th$ also the tree $\th'$ obtained from $\th$ by performing the following operation (recall the definition of $\FFFF_V(\th)$): replace the resonance $V$ with a single node $v$ carrying labels $\d_v=0$ and $\k_v=k_V$ (if $k_V$ is the order of the resonance $V$), then express the counterterm $\g_{\k_v}(g_0)$ associated with the node $v$ in terms of trees. If $\th_1$ is the subtree having $\l_{v_1}$ as root branch, then the values of the two considered trees $\th$ and $\th'$ can be written, respectively, as ${\rm Val}(\th)=A(\th)\RR_V{\rm Val}(\th_1)$ and ${\rm Val}(\th')=A(\th)[\g_{k_V}(g_0)\s/2]{\rm Val}(\th_1)$, where $\s/2=y_{v}^{(-1)}(1,-1)$ and $A(\th)$ takes into account the factors corresponding to all nodes {\it not} preceding $v_0$, and has the same value for both $\th$ and $\th'$. The resonance value $\RR_V$, for $\m=0$, can be written as % $$ \RR_V = \overline{{\rm Val}}(\th_0)\, in_{v_1'} \; , \qquad \hbox{ for some } \th_0\in\TT_{\V0,k} \hbox{ with } p(v_0)=0 \hbox{, } k_{v_0}=-1 \; , \Eqa(A3.2)$$ % see the definitions \equ(4.30) and \equ(4.31) of tree value and the definition \equ(A3.1) of resonance value: remember that we are considering resonances $V$ with degree $D_V=1$, so that $k_{v_0}=-1$ and, as a consequence, $k_{v_0}'\ge 1$; see \equ(4.14). The counterterm $\g_\k(g_0)$ can be represented in terms of trees as in \equ(4.32); note that, if the tree contributing to $\g_\k(g_0)$ has $k_{v_0}=-1$, the condition $\a_{v_0}=-1$ implies that such a tree has a node $w>v_0$ with $k_w+k_w'=1$, while all the other nodes $v\neq w$ have $k_v+k_v'=0$. Among the contributions in \equ(4.32) to $\g_{k_V}(g_0)$ there will be a quantity $\overline{{\rm Val}}(\th_2)$, where $\th_2$ will have the same topological form of $\th_0$ in \equ(A3.2) with the node $w$ such that $k_w+k_w'=1$ corresponding to the node $v_1'\in V$; then we denote both nodes by $w$. Then $\overline{{\rm Val}}(\th_0)$ will be related to $\overline{{\rm Val}}(\th_2)$ by % $$ \overline{{\rm Val}}(\th_2) = -\left[ {y_{n_v}^{(\a_v)}(k_w',k_w)|_{k_w'+k_w=1}\over y_{n_v}^{(\a_v)}(k_w',k_w)|_{k_w'+k_w=0}} \right] \overline{{\rm Val}}(\th_0) \; , \Eqa(A3.3) $$ % so that a look at the coefficients listed in and after \equ(4.24) shows that the factor in square brackets in \equ(A3.3) (when it is not vanishing) is equal to $4in_w\s$. The quantity $\overline{{\rm Val}}(\th_2)$, in order to contribute to $\g_{k_V}(g_0)\s/2$, has to be multiplied by a factor $-4\s$ extra with respect to $\overline{{\rm Val}}(\th_0)$, which, on the other hand, has to be multiplied by $in_w$ in order to contribute to the resonance value $\RR_V$ (see \equ(4.32). Then, for $\m=0$, by summing the values of the two considered contributions one obtain % $$ A(\th) \Big[ \overline{{\rm Val}}(\th_0)\,in_w-{1\over4\s} \overline{{\rm Val}}(\th_2)\Big]{\rm Val}(\th_1) \; , \Eqa(A3.4) $$ % which is zero by \equ(A3.3), so that a first order zero is obtained. \* \0{\bf A3.3.} If there are leaves, nothing changes in the discussion of \S A3.1, as $k_{v_0}=0$ implies that only leaves $w$ with $j_w>\ell$ are possible, and $\x_w(k_{w}',0)\=1$ in such a case (see \equ(4.18)). %The only real difference is that the presence of %the leaves modifies the combinatorial factor of the tree, %but it is a trivial change and in the end nothing changes. In \S A3.2, when discussing the case $v_1\in\PP$, one has to take care of the case in which there is a leaf with highest node $\tilde w$ with $k_{\tilde w}'=1$ (such a leaf will be at the end of the path $\PP$). In fact the resonances having as entering branch a branch of the path $\PP$ cannot have any leaves with $k_w'=1$, while when considering the graphical representation for $\g_\k(g_0)$, there will be also contributions arising from trees containing a leaf: such contributions will be either of the form \equ(4.32) with $\overline{{\rm Val}}(\th_2)= \overline{{\rm Val}}(\th_2)in_{v_1'}\x_{v_1}(1,0) L^{h_1\s}_{\ell\nn(v_1)}(\th_2)$, or of the form $\g_\k(g_0)= \g_{\k-h_1}(g_0)\, in_{v_1'}\x_{v_1}(1,0) L^{h_1\s}_{\ell\nn(v_1)}(\th_2)$, where $h_1\ge 1$, and $\th_2'$ is a suitable tree of order $k-h_1$. Then one realizes that the two contributions cancel exactly, so that no new case has to be discussed with respect to the analysis of \S A3.2. \* \0{\bf A3.4.} The above discussion completes the proof of approximate cancellations of resonance values (\ie of cancellations to first and second order, according to the degree of the resonant branch). The existence of cancellations, approximate to the first or second order, is all is needed to obtain the bound \equ(5.16): the analysis continues exactly as in [Ge2] and is based on simple analyticity arguments that allow us to exploit, via the maximum principle, the fact that in a resonance with momentum $\nn$ the functions of $\m=\oo_0\cdot\nn$ that have been considered above have a zero in $\m$ of order $1$ or $2$. A complete analysis showing that the higher orders contributions (\ie the part which does not cancel) can be performed as in [Ge2], Appendix B, and the final result is given by the bound \equ(5.16) in \S 5.11. \* \0{\bf A3.5.} We shall show now that all contributions with $(\nn_0(v_0),p(v_0))=(\V0,0)$ involved in the definition of the counterterms (see Appendix A1) must have automatically also $\nn(v_0)=\V0$. The analysis performed in \S 5.1 shows that in order to have $p(v_0)=0$ (for $j_{v_0}=\ell$), there can be any number of leaves with highest nodes $w$ such that $j_w>\ell$ and only one leaf $w$ with $j_w=\ell$ (contributing, respectively, $k_{w}'=0$ and $k_{w}'=1$ to $p(v_0)$). Each time a leaf with $j_w>\ell$ appears, if we sum together the values of all trees obtained by detaching the leaf with its stalk, then reattaching it to all the other nodes of $\th_f$, we obtain a vanishing contribution: simply by the cancellation mechanism described in \S A3.1 (assuring there the first order zero), {\it which, now, is an exact cancellation as the leaf does not contribute to the free momenta of the branches of $\th_f$, so that it does not modify the propagators.} So we can suppose that no leaf with $j_{w}>\ell$ is possible in trees involved in the determination of the counterterms. In the same way, if we have a tree $\th$ having a leaf with $j_w=\ell$ and $h_w=h-h_1$ (for some $h_1$), we can reason as in \S A3.2 and consider, together with $\th$, also the tree formed by only one free node, carrying a counterterm label $h_1$ and bearing the same leaf as $\th$. The same cancellation mechanism described in \S A3.2 apply now: again the only difference is that now the cancellation is exact (by the same reason as before). This shows that no tree with leaves can contribute to $(\nn_0(v_0),p(v_0))=(\V0,0)$, so that for such trees one has $\nn(v_0)=\nn_0(v_0)=\V0$. This, together with the analysis in \S A1.1, proves \equ(A1.4) in \S A1.2. \* \ifnum\mgnf=0\pagina\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip1.truecm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0{\titolo Appendix A4. Graphs with non zero total hyperbolic momentum, with leaves or with counterterms} \numsec=4\numfor=1\* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0{\bf A4.1} Consider first the cases $p(v_0)\neq 0$. In this case we consider the nodes $w1$, the discussion is easier as no splitting of the integration domains is needed. So we can conclude that a final bound $(2B)^{2k}$ is obtained for $\Val(\th)$; so far neither leaves nor counterterms have been considered. \* \0{\bf A4.2} Introducing the leaves and the counterterms, one sees (recall Remark 4.4) that the value of any tree $\th$ can be always be written as the product of a factor like \equ(4.28) times the product of the counterterms and of the leaf values; each counterterm can be decomposed in turn as sum of values of amputated trees (see \equ(4.32)). As each leaf and each amputated tree can contain other leaves and counterterms we can iterate such a decomposition procedure, until, at the end, the value of the tree $\th$, with highest node $v_0$, turns out to be given by the product of factorizing terms which\\ {(1)} either are of the form \equ(4.28), with $\r_{w}=0$ for any subtree with highest node $wh (N+N_0)$ vanish at order $h$ as a consequence of the trigonometric assumption on the perturbation $f_1$, see \equ(2.2), the bound $(2B)^{2k}$ is a bound both for the Fourier coefficients of $X^\s(t;\aa)$ and for the function $X^\s(t;\aa)$ itself. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip1.truecm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \0{\titolo References} \*\* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \item{[BCG]} Benettin, G., Carati, A., Gallavotti, G.: {\it A rigorous implementation of the Jeans--Landau--Teller approximation for adiabatic invariants}, Nonlinearity {\bf 10} (1997), 479--507. \item{[BGGM]} Bonetto, F., Gallavotti, G., Gentile, G., Mastropietro, V.: {\it Lindstedt series, ultraviolet divergences and Moser's Theorem}, Annali della Scuola Normale Superiore di Pisa {\bf 26}, 545--593 (1998). \item{[CG] } Chierchia, L., Gallavotti, G.: {\it Drift and diffusion in phase space}, Annales de l'Institut Henri Poincar\'e, B {\bf 60} (1994), 1--144; see also the {\it Erratum}, B {\bf 68} (1998), 135. \item{[CZ] } Chierchia, L., Zehnder, E.,: {\it Asymptotic expansions of quasi-periodic motions}, Annali della Scuola Normale Superiore di Pisa Cl. Sci. (4) {\bf 16} (1989), 245--258. \item{[E] } Eliasson, L.H.: {\it Absolutely convergent series expansions for quasi-periodic motions}, Mathematical Physics Electronic Journal {\bf 2} (1996), http:// mpej.unige.ch. \item{[G1] } Gallavotti, G.: {\it Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review}, Reviews on Mathematical Physics {\bf 6} (1994), 343--411. \item{[G2] } Gallavotti, G.: {\it Twistless KAM tori}, Communications in Mathematical Physics {\bf 164} (1994), 145--156. \item{[G3] } Gallavotti, G.: {\it Reminiscences on science at I.H.E.S. A problem on homoclinic theory and a brief review}, Publications Scientifiques de l'IHES, Special Volume for the 40-th anniversary (1998), ???--???, [preprint in chao-dyn 9804043]. \item{[GG] } Gallavotti, G. , Gentile, G.: {\it Majorant series convergence for twistless KAM tori}, Ergodic Theory and Dynamical Systems {\bf 15} (1995), 1--69. \item{[GGM1] } Gallavotti, G. , Gentile, G., Mastropietro, V.: {\it Field theory and KAM tori}, p. 1--9, Mathematical Physics Electronic Journal {\bf 1} (1995), http:// mpej.unige.ch. \item{[GGM2] } Gallavotti, G. , Gentile, G., Mastropietro, V.: {\it Separatrix splitting for systems with three degrees of freedom}, to appear in Communications in Mathematical Physics, 1999, [preprint with title {\it Pendullum: separatrix splitting} in chao-dyn 9709004]. \item{[GGM3] } Gallavotti, G., Gentile, G., Mastropietro, V.: {\it Hamilton-Jacobi equation, heteroclinic chains and Arnol'd diffusion in three time scales systems}, submitted for publication [Preprint in chao-dyn \#9801004]. \item{[GGM4] } Gallavotti, G., Gentile, G., Mastropietro, V.: {\it Melnikov's approximation dominance. Some examples}, to appear in Reviews in Mathematical Physics, 1999, [preprint in chao-dyn \#9804043]. \item{[Ge1] } Gentile, G: {\it A proof of existence of whiskered tori with quasi flat homoclinic intersections in a class of almost integrable hamiltonian systems}, Forum Mathematicum {\bf 7} (1995), 709--753. \item{[Ge2] } Gentile, G.: {\it Whiskered tori with prefixed frequencies and Lyapunov spectrum}, Dynamics and Stability of Systems {\bf 10} (1995), 269--308. \item{[Gel1] } Gelfreich, V.: {\it Melnikov method and exponentially small splitting of separatrices}, Physica D {\bf101} (1997), 227--248. \item{[Gel2] } Gelfreich, V.: {\it Reference systems for splitting of separatrices}, Nonlinearity {\bf10} (1997), 175--193. \item{[GM1] } Gentile, G., Mastropietro, V.: {\it KAM theorem revisited}, Physica D {\bf 90} (1996), 225--234. \item{[GM2] } Gentile, G., Mastropietro, V.: {\it Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in Classical mechanics. A review with some applications}, Reviews in Mathematical Physics {\bf 8} (1996), 393--444. \item{[N] } Neishtadt, A.I.: {\it The separation of motions in systems with rapidly rotating phase}, Journal of Applied Mathematics and Mechanics {\bf 48} (1984), 133--139. \item{[HMS] } Holmes, P., Marsden, J., Scheurle, J: {\it Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations}, Contemporary Mathematics {\bf 81} (1989), 213--244. \item{[La] } Lazutkin, V.F.,, {\it Splitting of separatrices for the Chirikov's standard map}, mp$\_$arc@ math.utexas.edu, \#98--421, (annotated english translation of a 1984 paper). \item{[P] } Poincar\'e, H.: {\it Les m\'ethode nouvelles de la M\'ecanique C\'eleste}, Vol. III, Gauthier--Villard, Paris, 1899. \item{[P\"o] } P\"oschel, J.: {\it Invariant manifolds of complex analytic mappings}, Les Houches, XLIII (1984), Vol. II, p. 949-- 964, Eds. K. Osterwalder \& R. Stora, North Holland, 1986. \* \FINE \ciao ---------------9811271558178--