Plain TeX 8 pages. 1 figure. 106 K. BODY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %TO PRINT THE POSTCRIPT FIGURES THE DRIVER NUMBER MIGHT HAVE TO BE %ADJUSTED. IF the 4 choices 0,1,2,3 do not work set in the following line %the \driver variable to =5. Setting it =0 works with dvilaser setting it %=1 works with dvips, =2 with psprint, =3 with dvitps, (hopefully). %Using =5 prints incomplete figures (but still understandable from the %text). The value MUST be set =5 if the printer is not a postscript one. \newcount\driver \driver=2 %%%this is the value to set!!! %%% the values =0,1 have been tested. The figures are automatically %%% generated. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FORMATO \newcount\mgnf\newcount\tipi\newcount\tipoformule \mgnf=0 %ingrandimento \tipi=2 %uso caratteri: 2=cmcompleti, 1=cmparziali, 0=amparziali \tipoformule=1 %=0 da numeroparagrafo.numeroformula; se no numero %assoluto %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INCIPIT\ \ifnum\mgnf=0 \magnification=\magstep0\hoffset=0.cm \voffset=-0.5truecm\hsize=16.5truecm\vsize=24.truecm \parindent=4.pt\fi \ifnum\mgnf=1 \magnification=\magstep1\hoffset=0.truecm \voffset=-0.5truecm\hsize=16.5truecm\vsize=24.truecm \baselineskip=14pt plus0.1pt minus0.1pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt \parskip=0.1pt plus1pt\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\overfullrule=10pt % %%%%%GRECO%%%%%%%%% % \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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% Numerazione pagine %%%%%%%%%%%%%%%%%%%%% NUMERAZIONE PAGINE {\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 } \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} \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} %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglio\hss} %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglioa\hss} % %%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI %%% %%% per assegnare un nome simbolico ad una equazione basta %%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o, %%% nelle appendici, \Eqa(...) o \eqa(...): %%% dentro le parentesi e al posto dei ... %%% si puo' scrivere qualsiasi commento; %%% per assegnare un nome simbolico ad una figura, basta scrivere %%% \geq(...); per avere i nomi %%% simbolici segnati a sinistra delle formule e delle figure si deve %%% dichiarare il documento come bozza, iniziando il testo con %%% \BOZZA. Sinonimi \Eq,\EQ,\EQS; \eq,\eqs; \Eqa,\Eqas;\eqa,\eqas. %%% All' inizio di ogni paragrafo si devono definire il %%% numero del paragrafo e della prima formula dichiarando %%% \numsec=... \numfor=... (brevetto Eckmannn); all'inizio del lavoro %%% bisogna porre \numfig=1 (il numero delle figure non contiene la sezione. %%% Si possono citare formule o figure seguenti; le corrispondenze fra nomi %%% simbolici e numeri effettivi sono memorizzate nel file \jobname.aux, che %%% viene letto all'inizio, se gia' presente. E' possibile citare anche %%% formule o figure che appaiono in altri file, purche' sia presente il %%% corrispondente file .aux; basta includere all'inizio l'istruzione %%% \include{nomefile} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 % \write15{\string\FU (#1){\equ(#1)}} \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 % \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} \global\advance\numfig by 1 % \write15{\string\FU (#1){\equ(#1)}} \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{$\scriptstyle##1$}}}}} \footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglio\hss} } \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} \let\EQS=\Eq\let\EQ=\Eq \let\eqs=\eq \let\Eqas=\Eqa \let\eqas=\eqa %%%%%%%%% %\newcount\tipoformule %\tipoformule=1 %=0 da numeroparagrafo.numeroformula; se no numero % %assegnato \ifnum\tipoformule=1\let\Eq=\eqno\def\eq{}\let\Eqa=\eqno\def\eqa{} \def\equ{{}}\fi \def\include#1{ \openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13 \fi} \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi %\openout15=\jobname.aux %\write15 % %%%%%%%%%%% GRAFICA %%%%%%%%% % % Inizializza le macro postscript e il tipo di driver di stampa. % Attualmente le istruzioni postscript vengono utilizzate solo se il driver % e' DVILASER ( \driver=0 ), DVIPS ( \driver=1) o PSPRINT ( \driver=2); % o DVITPS (\driver=3) % qualunque altro valore di \driver produce un output in cui le figure % contengono solo i caratteri inseriti con istruzioni TEX (vedi avanti). % %\newcount\driver \driver=1 %\ifnum\driver=0 \special{ps: plotfile ini.pst global} \fi %\ifnum\driver=1 \special{header=ini.pst} \fi \newdimen\xshift \newdimen\xwidth % % inserisce una scatola contenente #3 in modo che l'angolo superiore sinistro % occupi la posizione (#1,#2) % \def\ins#1#2#3{\vbox to0pt{\kern-#2 \hbox{\kern#1 #3}\vss}\nointerlineskip} % % Crea una scatola di dimensioni #1x#2 contenente il disegno descritto in % #4.pst; in questo disegno si possono introdurre delle stringhe usando \ins % e mettendo le istruzioni relative nel file #4.tex (che puo' anche mancare); % al disotto del disegno, al centro, e' inserito il numero della figura % calcolato tramite \geq(#3). % Il file #4.pst contiene le istruzioni postscript, che devono essere scritte % presupponendo che l'origine sia nell'angolo inferiore sinistro della % scatola, mentre per il resto l'ambiente grafico e' quello standard. % Se \driver=2, e' necessario dilatare la figura in accordo al valore di % \magnification, correggendo i parametri P1 e P2 nell'istruzione % \special{#4.ps P1 P2 scale} % \def\insertplot#1#2#3#4{ \par \xwidth=#1 \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2 \vbox{ \line{} \hbox{ \hskip\xshift \vbox to #2{\vfil \ifnum\driver=0 #3 \special{ps::[local,begin] gsave currentpoint translate} \special{ps: plotfile #4.ps} \special{ps::[end]grestore} \fi \ifnum\driver=1 #3 \special{psfile=#4.ps} \fi \ifnum\driver=2 #3 \ifnum\mgnf=0 \special{#4.ps 1. 1. scale}\fi \ifnum\mgnf=1 \special{#4.ps 1.2 1.2 scale}\fi\fi \ifnum\driver=3 \ifnum\mgnf=0 \psfig{figure=#4.ps,height=#2,width=#1,scale=1.} \kern-\baselineskip #3\fi \ifnum\mgnf=1 \psfig{figure=#4.ps,height=#2,width=#1,scale=1.2} \kern-\baselineskip #3\fi \ifnum\driver=5 #3 \fi \fi} \hfil}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newskip\ttglue %%cm semplificato \def\TIPI{ \font\ottorm=cmr8 \font\ottoi=cmmi8 \font\ottosy=cmsy8 \font\ottobf=cmbx8 \font\ottott=cmtt8 %\font\ottosl=cmsl8 \font\ottoit=cmti8 %%%%% cambiamento di formato%%%%%% \def \ottopunti{\def\rm{\fam0\ottorm}% passaggio a tipi da 8-punti \textfont0=\ottorm \textfont1=\ottoi \textfont2=\ottosy \textfont3=\ottoit \textfont4=\ottott \textfont\itfam=\ottoit \def\it{\fam\itfam\ottoit}% \textfont\ttfam=\ottott \def\tt{\fam\ttfam\ottott}% \textfont\bffam=\ottobf \normalbaselineskip=9pt\normalbaselines\rm} \let\nota=\ottopunti} %%%%%%%% %%am \def\TIPIO{ \font\setterm=amr7 %\font\settei=ammi7 \font\settesy=amsy7 \font\settebf=ambx7 %\font\setteit=amit7 %%%%% cambiamenti di formato %%% \def \settepunti{\def\rm{\fam0\setterm}% passaggio a tipi da 7-punti \textfont0=\setterm %\textfont1=\settei \textfont2=\settesy %\textfont3=\setteit %\textfont\itfam=\setteit \def\it{\fam\itfam\setteit} \textfont\bffam=\settebf \def\bf{\fam\bffam\settebf} \normalbaselineskip=9pt\normalbaselines\rm }\let\nota=\settepunti} %%%%%%% %%cm completo \def\TIPITOT{ \font\twelverm=cmr12 \font\twelvei=cmmi12 \font\twelvesy=cmsy10 scaled\magstep1 \font\twelveex=cmex10 scaled\magstep1 \font\twelveit=cmti12 \font\twelvett=cmtt12 \font\twelvebf=cmbx12 \font\twelvesl=cmsl12 \font\ninerm=cmr9 \font\ninesy=cmsy9 \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightsl=cmsl8 \font\eightit=cmti8 \font\sixrm=cmr6 \font\sixbf=cmbx6 \font\sixi=cmmi6 \font\sixsy=cmsy6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \font\twelvetruecmr=cmr10 scaled\magstep1 \font\twelvetruecmsy=cmsy10 scaled\magstep1 \font\tentruecmr=cmr10 \font\tentruecmsy=cmsy10 \font\eighttruecmr=cmr8 \font\eighttruecmsy=cmsy8 \font\seventruecmr=cmr7 \font\seventruecmsy=cmsy7 \font\sixtruecmr=cmr6 \font\sixtruecmsy=cmsy6 \font\fivetruecmr=cmr5 \font\fivetruecmsy=cmsy5 %%%% definizioni per 10pt %%%%%%%% \textfont\truecmr=\tentruecmr \scriptfont\truecmr=\seventruecmr \scriptscriptfont\truecmr=\fivetruecmr \textfont\truecmsy=\tentruecmsy \scriptfont\truecmsy=\seventruecmsy \scriptscriptfont\truecmr=\fivetruecmr \scriptscriptfont\truecmsy=\fivetruecmsy %%%%% cambio grandezza %%%%%% \def \eightpoint{\def\rm{\fam0\eightrm}% switch to 8-point type \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \tt \ttglue=.5em plus.25em minus.15em \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \normalbaselineskip=9pt \let\sc=\sixrm \let\big=\eightbig \normalbaselines\rm \textfont\truecmr=\eighttruecmr \scriptfont\truecmr=\sixtruecmr \scriptscriptfont\truecmr=\fivetruecmr \textfont\truecmsy=\eighttruecmsy \scriptfont\truecmsy=\sixtruecmsy }\let\nota=\eightpoint} \newfam\msbfam %per uso in \TIPITOT \newfam\truecmr %per uso in \TIPITOT \newfam\truecmsy %per uso in \TIPITOT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%Scelta dei caratteri %\newcount\tipi \tipi=0 %e' definito all'inizio \newskip\ttglue \ifnum\tipi=0\TIPIO \else\ifnum\tipi=1 \TIPI\else \TIPITOT\fi\fi \def\didascalia#1{\vbox{\nota\0#1\hfill}\vskip0.3truecm} %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DEFINIZIONI VARIE % \def\V#1{\vec#1} \def\T#1{#1\kern-4pt\lower9pt\hbox{$\widetilde{}$}\kern4pt{}} \let\dpr=\partial\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}} \let\implica=\Rightarrow\def\tto{{\Rightarrow}} \def\pagina{\vfill\eject}\def\acapo{\hfill\break} \let\ciao=\bye %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%LATINORUM \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{{}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%DEFINIZIONI LOCALI \def\AA{{\V A}}\def\aa{{\V\a}}\def\nn{{\V\n}}\def\oo{{\V\o}} \def\mm{{\V m}}\def\nn{{\V\n}}\def\lis#1{{\overline #1}} \def\NN{{\cal N}}\def\FF{{\cal F}} \def\={{ \; \equiv \; }}\def\su{{\uparrow}}\def\giu{{\downarrow}} \def\II{{\cal I}} \def\Dpr{{\V \dpr}\,} \def\Im{{\rm\,Im\,}}\def\Re{{\rm\,Re\,}} \def\sign{{\rm sign\,}} \def\atan{{\,\rm arctg\,}} \def\pps{{\V\ps{\,}}} \let\dt=\displaystyle \def\2{{1\over2}} \def\txt{\textstyle}\def\OO{{\cal O}} \def\tst{\textstyle} \def\st{\scriptscriptstyle} \let\\=\noindent \def\*{\vskip0.3truecm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \catcode`\%=12\catcode`\}=12\catcode`\{=12 \catcode`\<=1\catcode`\>=2 \openout13=gvnn.ps \write13<%%BoundingBox: 0 0 240 170> \write13<% fig.pst> \write13 \write13<0 90 punto > \write13<70 90 punto > \write13<120 60 punto > \write13<160 130 punto > \write13<200 110 punto > \write13<240 170 punto > \write13<240 130 punto > \write13<240 90 punto > \write13<240 0 punto > \write13<240 30 punto > \write13<210 70 punto > \write13<240 70 punto > \write13<240 50 punto > \write13<0 90 moveto 70 90 lineto> \write13<70 90 moveto 120 60 lineto> \write13<70 90 moveto 160 130 lineto> \write13<160 130 moveto 200 110 lineto> \write13<160 130 moveto 240 170 lineto> \write13<200 110 moveto 240 130 lineto> \write13<200 110 moveto 240 90 lineto> \write13<120 60 moveto 240 0 lineto> \write13<120 60 moveto 240 30 lineto> \write13<120 60 moveto 210 70 lineto> \write13<210 70 moveto 240 70 lineto> \write13<210 70 moveto 240 50 lineto> \write13 \closeout13 \catcode`\%=14\catcode`\{=1 \catcode`\}=2\catcode`\<=12\catcode`\>=12 %\input cfiat %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglio\hss} \vglue0.truecm %\BOZZA \0{\bf Non recursive proof of the KAM theorem} \footnote{${}^*$}{\nota Archived in {\tt mp\_arc@math.utexas.edu}; %\#93-172 available also by e-mail request to the authors.} \vskip.8truecm \0{\bf Giovanni Gallavotti, Guido Gentile \footnote{${}^1$}{\nota Dipartimento di Fisica, Universit\`a di Roma, ``La Sa\-pi\-en\-za", P.le Moro 2, 00185 Roma, Italia. E-mail: {\tt gallavotti\%40221.} {\tt hepnet@lbl.gov}, {\tt gentileg\%39943.hepnet@lbl.gov}. } } \vskip0.5truecm \0{\bf Abstract:} {\sl A selfcontained proof of the KAM theorem in the Thirring model is discussed, completely relaxing the ``strong diophantine property'' hypothesis used in previous papers.} \vskip0.5truecm \0{\sl Keywords:\it\ KAM, invariant tori, classical mechanics, perturbation theory, chaos} \vskip1.3truecm \\{\bf 1. Introduction} \vglue.5truecm\numsec=1\numfor=1\pgn=1 \\In [G] a selfcontained proof of the KAM theorem in the Thirring model is discussed, under the hypothesis that the rotation vectors $\oo_0$ verify a {\it strong diophantine property}. At the end of the same paper a heuristic argument is given to show that in fact such a hypothesis can be relaxed. In the present work we develop the heuristic argument into an extension of the KAM theorem proof described in [G]; the extension applies to rotations vectors verifying only the usual diophantine condition. It is a proof again based on Eliasson's method, [E]. In our opinion this shows that a hypothesis like the strong diophantine property of [G], or something similar to it, is very natural as it simplifies the structure of the proof by separating from it the analysis of a simple arithmetic property, whose untimely analysis would obscure the proof. For an introductory discussion of the model and a more organic exposition of the problem, we refer to [G], [G1], and to the references there reported. In the remaining part of this section we confine ourselves to define the model, to introduce the basic notation, and to give the result we have obtained. The Thirring model, [T], is described by the hamiltonian: % $$ \fra12 J^{-1}\AA\cdot\AA\,+\,\e f(\aa) \Eq(1.1) $$ % where $J$ is the (diagonal) matrix of the inertia moments, $\AA=(A_1,\ldots,A_l)\in R^l$ are their angular momenta and $\aa=(\a_1,\ldots,\a_l)\in T^l$ are the angles describing their positions: the matrix $J$ will be supposed non singular; but we only suppose that $\min_{j=1,\ldots,l}J_j=J_0>0$, and no assumption is made on the size of the {\it twist rate} $T=\min J_j^{-1}$: the results will be uniform in $T$ (hence they can be called ``twistless results''). We suppose $f$ to be an even trigonometric polynomial of degree $N$: % $$ f(\aa)=\sum_{0<|\nn|\le N} f_\nn\,\cos\nn\cdot\aa, \qquad \qquad f_\nn=f_{-\nn} \; , \qquad |\nn| = \sum_{j=1}^l |\nn_j| \Eq(1.2) $$ % We shall consider a ``rotation vector'' $\oo_0=(\o_1, \ldots,\o_l)\in R^l$ verifying the {\it diophantine condition}: % $$ \bar C_0|\oo_0\cdot\nn|\ge |\nn|^{-\t}, \kern1.5cm\V0\ne\nn\in Z^l \Eq(1.3) $$ % with diophantine constants $\bar C_0, \t$. The {\it diophantine vectors} have full measure in $R^l$ if $\t$ is fixed $\t>l-1$. We shall set $\AA_0=J\oo_0$. As in [G], we prove the following result. \* \\{\bf Theorem}:{\it The system described by the Hamiltonian \equ(1.1) admits an $\e$--analytic family of motions starting at $\aa=\V0$ and having the form: % $$ \AA = \AA_0 + \V H(\oo_0t;\e),\qquad\aa= \oo_0t+\V h(\oo_0 t;\e) \Eq(1.4) $$ % with $\V H(\pps;\e),\V h(\pps;\e)$ analytic, divisible by $\e$, for $|\Im \psi_j|<\x$, $\pps\in T^l$, and for $|\e|<\e_0$ with: % $$ \e_0^{-1} = b J_0^{-1} (2^\t \bar C_0 )^2 f_0 N^{2+l} e^{c N}e^{\x N} \Eq(1.5) $$ % where $b,c$ are $l$--dependent positive constants, $f_0=\max_\nn |f_\nn|$.} \* This means that the set $\AA=\AA_0+\V H(\pps;\e), \, \aa=\pps+\V h(\pps;\e)$ described as $\pps$ varies in $T^l$ is, for $\e$ small enough, an invariant torus for \equ(1.1), which is run quasi periodically with angular velocity vector $\oo_0$. It is a family of invariant tori coinciding, for $\e=0$, with the torus $\AA=\AA_0,\,\aa=\pps\in T^l$. The presence of the factor $2^\t$ marks the only difference from the analogous result in [G]. Calling $\V H^{(k)}(\pps),\V h^{(k)}(\pps)$ the $k$-th order coefficients of the Taylor expansion of $\V H,\V h$ in powers of $\e$ and writing the equation of motion as $\dot\aa=J^{-1}\AA$ and $\dot\AA=-\e\dpr_\aa f(\aa)$ we get immediately recursion relations for $\V H^{(k)},\V h^{(k)}$, namely, for $k>1$: % $$ \eqalign{ {\V\o}_0 \cdot\V\dpr\,h^{(k)}_j & = J^{-1}_j H^{(k)}_j \cr {\V\o}_0\cdot\V\dpr\,H^{(k)}_j & = - \sum_{m_1,\ldots,m_l\atop|\mm|>0}\fra1{\prod_{s=1}^l m_s!} \dpr_{\a_j}\, \dpr^{m_1+\ldots+m_l}_{\a_1^{m_1}\ldots\a_l^{m_l}} f(\oo_0 t) \cdot {\sum}^* \prod_{s=1}^l\prod_{j=1}^{m_s} h^{(k^s_j)}_s(\oo_0 t) \cr} \Eq(1.6) $$ % where the $\sum^*$ denotes summation over the integers $k^s_j\ge1$ with: $\sum_{s=1}^l\sum_{j=1}^{m_s}k^s_j=k-1$. The trigonometric polynomial $\V h^{(k)}(\pps)$ will be completely determined (if possible at all) by requiring it to have $\V0$ average over $\pps$, (note that $\V H^{(k)}$ has to have zero average over $\pps$). For $k=1$ it is: % $$ \tst\V h^{(1)}(\pps)=-\sum_{\nn\ne\V0} \fra{iJ^{-1}\nn}{(i\oo_0\cdot\nn)^2} f_\nn\,e^{i\nn\cdot\pps} \Eq(1.7) $$ % One easily finds that the equation for $\V h^{(k)}$ can be solved and its solution is a trigonometric polynomial in $\pps$, of degree $\le k N$, odd if $\V h^{(k)}$ is determined by imposing that its average over $\pps$ vanishes. The remaining part of the paper is structured as follows: in section 2 we set a diagrammatic expansion of $\V h^{(k)}$, as in [G]. In section 3 we discuss a proposition which leads to the original result of this paper, and in section 4 we prove the theorem, repeating the discussion in [G], with some minor changes due to the weakening of the strong diophantine property hypothesis. The above theorem fully reproduces, in the model \equ(1.1), the theorem of Eliasson: for another alternative proof of the same theorem with no assumption of parity or of finite degree on the trigonometric polynomial $f$, see [CF]. \vskip1.truecm \\{\bf 2. Diagrammatic expansion} \vglue.5truecm\numsec=2\numfor=1\pgn=1 \\Let $\th$ be a tree diagram: it will consist of a family of ``lines'' (\ie segments) numbered from $1$ to $k$ arranged to form a (rooted) tree diagram as in the figure: % \insertplot{240pt}{170pt}{%gvnn.tex \ins{-35pt}{90pt}{\it root} \ins{25pt}{110pt}{$j$} \ins{60pt}{85pt}{$v_0$} \ins{55pt}{115pt}{$\nn_{v_0}$} \ins{115pt}{132pt}{$j_{1}$} \ins{152pt}{120pt}{$v_1$} \ins{145pt}{155pt}{$\nn_{v_1}$} \ins{110pt}{50pt}{$v_2$} \ins{190pt}{100pt}{$v_3$} \ins{230pt}{160pt}{$v_5$} \ins{230pt}{120pt}{$v_6$} \ins{230pt}{85pt}{$v_7$} \ins{230pt}{-10pt}{$v_{11}$} \ins{230pt}{20pt}{$v_{10}$} \ins{200pt}{65pt}{$v_4$} \ins{230pt}{65pt}{$v_8$} \ins{230pt}{45pt}{$v_9$} }{gvnn} \kern1.3cm \didascalia{fig. 1: A tree diagram $\th$ with $m_{v_0}=2,m_{v_1}=2,m_{v_2}=3,m_{v_3}=2,m_{v_4}=2$ and $m=12$, $\prod m_v!=2^4\cdot6$, and some decorations. The line numbers, distinguishing the lines, are not shown.} To each vertex $v$ we attach a ``mode label'' $\nn_v\in Z^l,\,|\nn_v|\le N$ and to each branch leading to $v$ we attach a ``branch label'' $j_v=1,\ldots,l$. The order of the diagram will be $k=$ number of vertices $=$ number of branches (the tree root will not be regarded as a vertex). We imagine that all the diagram lines have the same length (even though they are drawn with arbitrary length in fig.1). A group acts on the set of diagrams, generated by the permutations of the subdiagrams having the same vertex as root. Two diagrams that can be superposed by the action of a transformation of the group will be regarded as identical (recall however that the diagram lines are numbered, \ie are regarded as distinct, and the superpositon has to be such that all the decorations of the diagram match). Tree diagrams are regarded as partially ordered sets of vertices (or lines) with a minimal element given by the root (or the root line). We shall imagine that each branch carries also an arrow pointing to the root (``gravity'' direction, opposite to the order). We define the ``momentum'' entering $v$ as $\nn(v)=\sum_{w\ge v}\nn_w$: therefore the momentum entering a vertex $v$ is given by the sum of the momenta entering the immediately following vertices plus the ``momentum emitted'' by $v$ (\ie the mode $\nn_v$). If from a vertex $v$ emerge $m_1$ lines carrying a label $j=1$, $m_2$ lines carrying $j=2$, $\ldots$, it follows that \equ(1.6) can be rewritten: % $$ h^{(k)}_{\nn j}=\fra1{k!} {\sum}^*\prod_{v\in\th}\fra{(-i J^{-1}\nn_v)_{j_v}\, f_{\nn_v}\prod_{s=1}^l(i\nn_v)^{m_s}_s}{(i\oo_0\cdot\nn(v))^2} \Eq(2.1) $$ % with the sum running over the diagrams $\th$ of order $k$ and with $\nn(v_0)=\nn$; and the combinatorics can be checked from \equ(1.6), by taking into account that we regard the diagram lines as all different (to fix the factorials). The ${}^*$ recalls that the diagram $\th$ can and will be supposed such that $ \nn(v) \ne \V0 $ for all $ v\in\th $ (by the remarked parity properties of $\V h^{(k)}$). As in [G], according to Eliasson's terminology, we can define the {\it resonant diagrams} as the diagrams with vertices $v',v$, with $v'0$ we define $C_0\=2^\t \bar C_0$: this leaves \equ(1.3) still valid. We define the set $B_n$ of the values of $|\oo\cdot\nn|$ as $\nn$ varies in the set $0\le|\nn|< (2^{n+3})^{-1/\t}$, with $n=0,-1,-2,\ldots$. The sets $B_n$ verify the inclusion relation $B_n\subset B_m$ if $m2^{n+3}$, if $x\ne0$. More abstractly let $B_n$, $n=0,-1,\ldots$, be a sequence of sets such that i) $0\in B_n$, ii) $B_n\subset B_m$ if $mp-3$, hence we can suppose $p\ge n+3$. \def\PP{{\cal P}} Fix $p\le0$ and let $G=[a,b]$ be an interval verifying what we shall call below the {\it property $\PP_n$}: % $$\PP_n\,:\kern1.cm |x-\g|\ge 2^{m+1}\quad {\rm for\ all}\quad n\le m\le-3,\qquad \g\in G, \qquad |G|\ge 2^{n+1}\Eq(3.2)$$ % Let $G_{p-3}\=[2^{p-1},b_{p-3}]$, with $b_{p-3}\in [2^{p-1},2^p]$, be a {\it maximal} interval verifying property $\PP_{p-3}$ (note that $G_{p-3}$ exists because $x\in B_{p-3},\,x\ne0$ implies $x\ge 2^p$ by the spacing property, and it is $b_{p-3} \ge 2^{p-1} + 2^{p-2}$). Assume inductively that the intervals $[a_n,b_n]=G_n$ can be so chosen that $G_{n'}\subseteq G_{n\prime\prime}$ if $n'b_n$, \ie $y=b_n+2^{n+1}$: in such case it cannot be, again by the spacing property, that $x+2^{n+2}>y=b_n+2^{n+1}$, so that $b_n-x\ge 2^{n+1}$ and we can take $G=[b_n-2^n,b_n]$. If $x$ is in the second half the roles of left and right are exchanged. This completes the analysis of the case in which only one point of $B_{n-1}$ falls in $G_n$. The cases in which either no point or at least two points of $B_n$ fall in $G_n$ are analogous but easier. % \footnote{${}^2$}{\nota If two consecutive points $xb_n$: the spacing property implies that the interval $(y-2^{n+2},y)$ is free of points of $B_{n-1}$. Hence if $a=\max\{a_n,y-2^{n+2}+2^n\}$, $b=y-2^n$ then $G=[a,b]$ has the property $\PP_{n-1}$. If, instead, $ya_n$ and $b=\min\{b_n,y+2^{n+2}-2^n\}$ and $G=[a,b]$ enjoys property $\PP_{n-1}$. } % \* \0{\it Remark}: note that the above proof is constructive. \* Consider the special case in which the sets $B_n$ are the ones defined at the beginning of the section. The above lemma can then be translated into the following arithmetic proposition. \* \0{\bf Proposition}: {\it Given a diophantine vector $\oo_0$, \ie a vector verifying \equ(1.3), let $C_0=2^\t \bar C_0$ and $\oo= C_0\oo_0$; it is possible to find a sequence $\g_p\in[2^{p-1},2^p]$ such that $\g_{p-1}\le\g_p$ and: % $$\big||\oo\cdot\nn|-\g_p\big|\ge 2^{n+1},\qquad{\rm if}\quad 0<|\nn|\le (2^{n+3})^{-\t^{-1}}\Eq(3.3)$$ % for all $n\le0$ and for all $p\ge n$. Furthermore $|\oo\cdot\nn|\ne\g_n$, for all $n\le0$.} \* {\it Remark:} 1) The sequence $\g_p$ is constructively defined by the proof above. 2) The \equ(3.3) is very similar to the condition added in [G] to the \equ(1.3) to define the {\it strong diophantine property}. The point of the present paper is that all that is really needed (see the following \S4) to prove the theorem in \S1 are \equ(1.3) and \equ(3.3): the latter is a simple arithmetic property which is in fact a consequence of \equ(1.3). 3) As remarked in [G] almost all $\oo_0$ verify \equ(1.3) for some $\bar C_0$ and some $\t>l-1$ with a sequence $\g_p$ that can be prescribed {\it a priori} as $\g_p=2^p$. This, however, leaves out important cases like $l=2$ and $\oo_0$ with a quadratic irrational as rotation number. And it has the very unfortunate drawback of being non constructive, as the set of full measure of the $\oo$ verifying the strong diophantine property is constructed by abstract nonsense arguments (\eg the Borel Cantelli lemma). Nevertheless considering strongly diophantine vectors is natural as it leads to a simplified proof, [G], of the KAM theorem, with Eliasson's method, eliminating one side difficulty. 4) For the purpose of comparison note that the final comment of ref [G] conjectures the above proposition: however the constant $C_0$ introduced there is $2 C_0$ in the above notations and, therefore, the quantities called there $\g_p$ are $2$ times the ones in \equ(3.3) (in other words the first inequality in \equ(3.3) has to be multiplied side by side by $2$ to become the statement of [G]). \vskip1.truecm \\{\bf 4. Proof of the theorem} \vglue.5truecm\numsec=2\numfor=1\pgn=1 \\Let us consider a diagram $\th$ and its clusters. We wish to estimate the number $N_n$ of lines with scale $n\le0$ in it, assuming $N_n>0$ (we remind that a line is on scale $n$, if the line propagator $(\oo \cdot \nn)^{-2}$ is such that $\g_{n-1} < |\oo \cdot \nn| < \g_n$, see also the remark after \equ(3.1)). Denoting $T$ a cluster of scale $n$ let $m_T$ be the number of resonances of scale $n$ contained in $T$ (\ie with incoming lines of scale $n$); we have the following inequality, valid for any diagram $\th$: % $$ N_n\le\fra{4k}{E\,2^{-\e n}}+\sum_{T, \,n_T=n} (-1+m_T)\Eq(4.1) $$ % with $E=N^{-1}2^{-3\e},\e=\t^{-1}$. This is an extension of Brjuno's lemma, [B], [P], called in [G] ``resonant Siegel-Brjuno bound'': the proof, extracted from [G], can be found in appendix. Consider a diagram $\th^1$; we define the family $\FF(\th^1)$ generated by $\th^1$ as follows. Given a resonance $V$ of $\th^1$ we detach the part of $\th^1$ above $\l_V$ and attach it successively to the points $w\in\tilde V$, where $\tilde V$ is the set of vertices of $V$ (including the endpoint $w_1$ of $\l_V$ contained in $V$) outside the resonances contained in $V$. We say that a line is in $\tilde V$, if it is contained in $V$ and has at least one point in $\tilde V$. Note that all the lines $\l$ in $\tilde V$ have a scale $n_\l\ge n_V$. For each resonance $V$ of $\th^1$ we shall call $M_V$ the number of vertices in $\tilde V$. To the just defined set of diagrams we add the diagrams obtained by reversing simoultaneously the signs of the vertex modes $\nn_w$, for $w\in \tilde V$\footnote{${}^3$}{\nota This can be done without breaking the relationship which has to exist between the lines, as it can be easily checked by observing that $ \sum_{w \in \tilde V} \nn_w = \vec0 $, since, for any resonance $V$, $ \sum_{v \in V} \nn_v = \vec0 $. }: the change of sign is performed independently for the various resonant clusters. This defines a family of $\prod 2M_V$ diagrams that we call $\FF(\th_1)$. The number $\prod 2M_V$ will be bounded by $\exp\sum2M_V\le e^{2k}$. Let $\l$ be a line, in a cluster $T$, contained inside the resonances $V=V_1\subset V_2\subset\ldots$ of scales $n=n_1>n_2>\ldots$; then the shifting of the lines $\l_{V_i}$ can cause a change in the size of the propagator of $\l$ by at most $\g_{n_1}+\g_{n_2}+\ldots < 2^{n_1} + 2^{n_2} + \ldots < 2^{n+1}$. Since the number of lines inside $V$ is smaller than $\lis N_n \equiv E 2^{-n \t^{-1}}$, ($E=2^{-3 \t^{-1}} N^{-1}$), the quantity $\oo\cdot\nn_\l$ of $\l$ has the form $\oo\cdot\nn^0_\l+\s_\l\oo\cdot\nn_{\l_V}$ if $\nn^0_\l$ is the momentum of the line $\l$ ``inside the resonance $V$'', \ie it is the sum of all the vertex modes of the vertices preceding $\l$ in the sense of the line arrows, but contained in $V$; and $\s_\l=0,\pm1$. Therefore not only $|\oo\cdot\nn^0_\l| > 2^{n+3}$ (because $\nn^0_\l$ is a sum of $\le \lis N_n$ vertex modes, so that $|\nn^0_\l|\le N\lis N_n$) but $\oo\cdot\nn^0_\l$ is ``in the middle'' of the interval of scales containing it and, by the proposition in section 3 (in [G] this was a consequence of the strong diophantine property), does not get out of it if we add a quantity bounded by $2^{n+1}$ (like $\s_\l\oo\cdot\nn_{\l_V}$). {\it Hence no line changes scale as $\th$ varies in $\FF(\th^1)$}. Let $\th^2$ be a diagram not in $\FF(\th^1)$ and construct $\FF(\th^2)$, \etc. We define a collection $\{\FF(\th^i)\}_{i=1,2,\ldots}$ of pairwise disjoint families of diagrams. We shall sum all the contributions to $\V h^{(k)}$ coming from the individual members of each family. This is similar to the {\it Eliasson's resummation}. We call $\e_V$ the quantity $\oo\cdot\nn_{\l_V}$ associated with the resonance $V$ with scale $n$. If $\l$ is a line in $\tilde V$, (see paragraphs following \equ(4.1)), we can imagine to write the quantity $\oo\cdot\nn_\l$ as $\oo\cdot\nn^0_\l+\s_\l\e_V$, with $\s_\l=0,\pm1$. We want to show that the product of the propagators is holomorphic in $\e_V$ for $|\e_V|< \g_{n_V-3}$. Let us reason in the following way. If $\l$ is a line on scale $n_V$, $\g_{n_V} > |\oo\cdot\nn_\l| > \g_{n_V-1} $; remarking that it is $|\oo\cdot\nn^0_\l| > 2^{n+3}$, we obtain immediately $ | \oo\cdot\nn_\l| > 2^{n+3} - 2^n > 2^{n+2}$, so that $n_V \ge n+3$. On the other hand, if $n_V > n+3$, \ie $n_V = n+m$, for some $m>3$, we note that $ |\oo\cdot\nn^0_\l| > \g_{n_V-1}-\g_n $, because the resonance scales and the scales of the resonant clusters (and of all the lines) do not change, so that it follows that, for $| \e_V | < \g_{n_V-3}$, $ \, |\oo \cdot \nn_\l^0 + \s_\l \e_V | \ge \g_{n_V-1}-\g_n-\g_{n_V-3} $ $ \ge (2^{n_V-2} - 2^{n_V-m}) -\g_{n_V-3} $ $ \ge (2^{n_V-3} + 2^{n_V-4} + \ldots + 2^{n_V-m+1}) - 2^{n_V-3} $ $ \ge 2^{n_V-4}$; otherwise, if $n_V=n+3$, we note that $|\oo \cdot \nn_\l^0 | > 2^{n+3}$, so that $ |\oo \cdot \nn_\l^0 + \s_\l \e_V | > 2^{n+3} - \g_{n_V-3} $ $\ge 2^{n_V-1}$, for $| \e_V | < \g_{n_V-3}$. Therefore we can conclude that, while $\e_V$ varies in a complex disk of radius $\g_{n_V-3}$ and center in $0$, the quantity $|\oo\cdot\nn^0_\l+\s_\l\e_V|$ does not become smaller than $2^{n_V-4}$. Note the main point here: the quantity $\g_{n_V-3}$ will usually be $\gg \g_{n_{\l_V}}$ which is the value $\e_V$ actually can reach in every diagram in $\FF(\th^1)$; this can be exploited in applying the maximum principle, as done below. It follows that, calling $n_\l$ the scale of the line $\l$ in $\th^1$, each of the $\prod 2 M_V\le e^{2k}$ products of propagators of the members of the family $\FF(\th^1)$ can be bounded above by $ \prod_\l\,2^{-2(n_\l-4)}=2^{8k}\prod_\l\,2^{-2n_\l}$, if regarded as a function of the quantities $\e_V=\oo\cdot\nn_{\l_V}$, for $|\e_V|\le \,\g_{n_V-3}$, associated with the resonant clusters $V$. This even holds if the $\e_V$ are regarded as independent complex parameters. By construction it is clear that the sum of the $\prod 2M_V\le e^{2k}$ terms, giving the contribution to $\V h^{(k)}$ from the trees in $\FF(\th^1)$, vanishes to second order in the $\e_V$ parameters (by the approximate cancellation discussed above). Hence by the maximum principle, and recalling that each of the scalar products in \equ(8) can be bounded by $N^2$, we can bound the contribution from the family $\FF(\th^1)$ by: % $$ \left[\fra1{k!} \Big(\fra{f_0 2^{2\t} C_0^2 N^2}{J_0}\Big)^k 2^{8k} e^{2k} \prod_{n\le0}2^{-2nN_n}\right]\left[\prod_{n\le0}\prod_{T,\,n_T=n} \prod_{i=1}^{m_T}\,2^{2(n-n_{i}+4)}\right] \Eq(4.2) $$ % where: % \acapo 1) $N_n$ is the number of propagators of scale $n$ in $\th^1$ ($n=1$ does not appear as $|\oo\cdot\nn| \ge \g_0 \ge 2^2 $, in such cases, and $ 2^8 \ge 2^4 $); % \acapo 2) the first square bracket is the bound on the product of individual elements in the family $\FF(\th^1)$ times the bound $e^{2k}$ on their number; % \acapo 3) the second term is the part coming from the maximum principle (in the form of Schwarz's lemma), applied to bound the resummations, and is explained as follows: % \acapo i) the dependence on the variables $\e_{V_i}\=\e_i$ relative to resonances $V_i\subset T$ with scale $n_{\l_{V_i}}=n$ is holomorphic for $|\e_i|< \,\g_{ n_i-3}$ if $n_i\=n_{V_i}$, provided $n_i \ge n+3$ (see above); % \acapo ii) the resummation says that the dependence on the $\e_i$'s has a second order zero in each. Hence the maximum principle tells us that we can improve the bound given by the first factor in \equ(4.2) by the product of factors $ (|\e_i| \, \g_{n_i-3}^{-1})^2 \le 2^{2(n - n_i + 4)} $, if $n_i \ge n+3$ (of course the gain factor can be important only when $\ll1$). Hence substituting \equ(4.1) into \equ(4.2) we see that the $m_T$ is taken away by the first factor in $\,2^{2n}2^{-2n_{i}}$, while the remaining $\,2^{-2n_i}$ are compensated by the $-1$ before the $+m_T$ in \equ(4.1), taken from the factors with $T=V_i$ (note that there are always enough $-1$'s). Hence the product \equ(4.2) is bounded by: % $$ \fra1{k!}\,(2^{2\t} C_0^2 J_0^{-1}f_0 N^2)^k e^{2k}2^{8k}2^{8k} \prod_n\,2^{-8 n k E^{-1}\,2^{\e n}}\le \fra1{k!} B_0^k \Eq(4.3) $$ % with $B_0 = 2^{18} e^2 ( 2^{2\t} C_0^2 f_0 J_0^{-1} ) N^2 \exp [ 2^{3 \t^{-1}} ( 8 N \log 2 ) \sum_{n=1}^{\io} n 2^{-n \t^{-1} } ] $. To sum over the trees we note that, fixed $\th$ the collection of clusters is fixed. Therefore we only have to multiply \equ(4.3) by the number of diagram shapes for $\th$, ($\le 2^{2k}k!$), by the number of ways of attaching mode labels, ($\le (3N)^{lk}$), so that we can bound $|h^{(k)}_{\nn j}|$ by \equ(1.5). \* \penalty-200\vskip1.truecm \\{\bf Appendix : Resonant Siegel-Brjuno bound.} \penalty10000\vskip0.5truecm Calling $N^*_n$ the number of non resonant lines carrying a scale label $\le n$. We shall prove first that $N^*_n\le 2k (E 2^{-\e n})^{-1}-1$ if $N_n>0$. We fix $n$ and denote $N^*_n$ as $N^*(\th)$. If $\th$ has the root line with scale $>n$ then calling $\th_1,\th_2,\ldots,\th_m$ the subdiagrams of $\th$ emerging from the first vertex of $\th$ and with $k_j>E\,2^{-\e n}$ lines, it is $N^*(\th)=N^*(\th_1)+\ldots+N^*(\th_m)$ and the statement is inductively implied from its validity for $k'k-\fra12 E\,2^{-n\e}$. Finally, and this is the real problem as the analysis of a few examples shows, we claim that in the latter case the root line of $\th_1$ is either a resonant line or it has scale $>n$. Accepting the last statement it will be: $N^*(\th)=1+N^*(\th_1)= 1+N^*(\th'_1)+\ldots+N^*(\th'_{m'})$, with $\th'_j$ being the $m'$ subdiagrams emerging from the first node of $\th'_1$ with orders $k'_j>E\,2^{-\e n}$: this is so because the root line of $\th_1$ will not contribute its unit to $N^*(\th_1)$. Going once more through the analysis the only non trivial case is if $m'=1$ and in that case $N^*(\th'_1)=N^*(\th^{\prime \prime}_1) + \ldots + N^*( \th^{\prime \prime}_{m^{\prime \prime} })$, \etc., until we reach a trivial case or a diagram of order $\le k-\fra12 E\,2^{-n\e}$. It remains to check that if $k_1>k-\fra12E\,2^{-n\e}$ then the root line of $\th_1$ has scale $>n$, unless it is entering a resonance. Suppose that the root line of $\th_1$ has scale $\le n$ and is not entering a resonance. Note that $|\oo\cdot\nn(v_0)|\le\,\g_n,|\oo\cdot\nn(v_1)|\le \,\g_n$, if $v_0,v_1$ are the first vertices of $\th$ and $\th_1$ respectively. Hence $\d\=|(\oo\cdot(\nn(v_0)-\nn(v_1))|\le2\,2^n$ and the diophantine assumption implies that $|\nn(v_0)-\nn(v_1)|> (2\,2^n)^{-\t^{-1}}$, or $\nn(v_0)=\nn(v_1)$. The latter case being discarded as $k-k_1<\fra12E\,2^{-n\e}$ (and we are not considering the resonances: note also that in such case the lines in $\th/\th_1$ different from the root of $\th$ must be inside a cluster), it follows that $k-k_1<\fra12E\,2^{-n\e}$ is inconsistent: it would in fact imply that $\nn(v_0)-\nn(v_1)$ is a sum of $k-k_1$ vertex modes and therefore $|\nn(v_0)-\nn(v_1)|< \fra12NE\,2^{-n\e}$ hence $\d>2^3\,2^n$ which is contradictory with the above opposite inequality. A similar, far easier, induction can be used to prove that if $N^*_n>0$ then the number $p$ of clusters of scale $n$ verifies the bound $p\le 2k \,(E2^{-\e n})^{-1}-1$. In fact this is true for $k\le E2^{-\e n}$, (see footnote 4). Let, therefore, $p(\th)$ be the number of clusters of scale $n$: if the first tree node $v_0$ is not in a cluster of scale $n$ it is $p(\th)=p(\th_1)+\ldots+p(\th_m)$, with the above notation, and the statement follows by induction. If $v_0$ is in a cluster of scale $n$ we call $\th_1$, $\ldots$, $\th_m$ the subdiagrams emerging from the cluster containing $v_0$ and with orders $k_j> E2^{-\e n}$. It will be $p(\th)=1+p(\th_1)+\ldots+p(\th_m)$. Again we can assume that $m=1$, the other cases being trivial. But in such case there will be only one branch entering the cluster $V$ of scale $n$ containing $v_0$ and it will have a momentum of scale $\le n-1$. Therefore the cluster $V$ must contain at least $E2^{-\e n}$ nodes. This means that $k_1\le k-E2^{-\e n}$: thus \equ(4.1) is proved. \penalty-200 \* %{\it Acknowledgements: I am indebted to J. Bricmont %and M. Vittot for many clarifying discussions; and in particular %to S. Miracle-Sol\'e, G. Benfatto and G. Gentile %for the same reasons and for their %interest and suggestions.} \* \vskip0.5truecm \penalty-200 {\bf References} \vskip0.5truecm \penalty10000 \item{[A] } Arnold, V.: {\it Proof of a A.N. Kolmogorov theorem on conservation of conditionally periodic motions under small perturbations of the hamiltonian function}, Uspeki Matematicheskii Nauk, 18, 13-- 40, 1963. \item{[B] } Brjuno, A.: {\it The analytic form of differential equations}, I: Transactions of the Moscow Mathematical Society, 25, 131-- 288, 1971; and II: 26, 199--239, 1972. \item{[CF] } Chierchia, L., Falcolini, C.: {\it A direct proof of the Theorem by Kolmogorov in Hamiltonian systems}, preprint, archived in {\tt mp\_arc@math.utexas.edu}, \#93-171. \item{[E] } Eliasson, L. H.: {\it Hamiltonian systems with linear normal form near an invariant torus}, ed. G. Turchetti, Bologna Conference, 30/5 to 3/6 1988, World Scientific, 1989. And: {\it Generalization of an estimate of small divisors by Siegel}, ed. E. Zehnder, P. Rabinowitz, book in honor of J. Moser, Academic press, 1990. But mainly: {\it Absolutely convergent series expansions for quasi--periodic motions}, report 2--88, Dept. of Math., University of Stockholm, 1988. \item{[G] } Gallavotti, G.: {\it Twistless KAM tori.}, archived in {\tt mp\_arc@math.utexas.edu}, \#93-172. See also: Gallavotti, G.: {\it Invariant tori: a field theoretic point of view on Eliasson's work}, talk at the meeting in honor of the 60-th birthday of G. Dell'Antonio (Capri, may 1993), Marseille, CNRS-CPT, preprint \item{[G2] } 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.}, deposited in the archive {\tt mp\_arc@math.utexas.edu}, \#93-164. \item{[G3] } Gallavotti, G.: {\it The elements of Mechanics}, Springer, 1983. \item{[G4] } Gallavotti, G.: {\it Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods}, Reviews in Modern Physics, 57, 471- 572, 1985. See also: Gallavotti, G.: {\it Quasi integrable mechanical systems}, Les Houches, XLIII (1984), vol. II, p. 539-- 624, Ed. K. Osterwalder, R. Stora, North Holland, 1986. \item{[K] } Kolmogorov, N.: {\it On the preservation of conditionally periodic motions}, Doklady Aka\-de\-mia Nauk SSSR, 96, 527-- 530, 1954. See also: Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: {\it A proof of Kolmogorov theorem on invariant tori using canonical transormations defined by the Lie method}, Nuovo Cimento, 79B, 201-- 223, 1984. \item{[M] } Moser, J.: {\it On invariant curves of an area preserving mapping of the annulus}, Nach\-rich\-ten Akademie Wiss. G\"ottingen, 11, 1-- 20, 1962. \item{[P] } P\"oschel, J.: {\it Invariant manifolds of complex analytic mappings}, Les Houches, XLIII (1984), vol. II, p. 949-- 964, Ed. K. Osterwalder, R. Stora, North Holland, 1986. \item{[PV] } Percival, I., Vivaldi, F.: {\it Critical dynamics and diagrams}, Physica, D33, 304-- 313, 1988. \item{[S] } Siegel, K.: {\it Iterations of analytic functions}, Annals of Mathematics, {\bf 43}, 607-- 612, 1943. \penalty10000 \item{[T] } Thirring, W.: {\it Course in Mathematical Physics}, vol. 1, p. 133, Springer, Wien, 1983. \ciao ENDBODY