\def\day{7/1/97} \catcode`@=11 % %------------------------- comandi riservati --------------------------- % \def\b@lank{ } \newif\if@simboli \newif\if@riferimenti \newif\if@bozze \newif\if@data \def\bozze{\@bozzetrue \immediate\write16{!!! INSERISCE NOME EQUAZIONI !!!}} \newwrite\file@simboli \def\simboli{ \immediate\write16{ !!! Genera il file \jobname.SMB } \@simbolitrue\immediate\openout\file@simboli=\jobname.smb \immediate\write\file@simboli{Simboli di \jobname}} \newwrite\file@ausiliario \def\riferimentifuturi{ \immediate\write16{ !!! Genera il file \jobname.aux } \@riferimentitrue\openin1 \jobname.aux \ifeof1\relax\else\closein1\relax\input\jobname.aux\fi \immediate\openout\file@ausiliario=\jobname.aux} \newcount\eq@num\global\eq@num=0 \newcount\sect@num\global\sect@num=0 \newcount\para@num\global\para@num=0 \newcount\const@num\global\const@num=0 \newcount\lemm@num\global\lemm@num=0 \newif\if@ndoppia \def\numerazionedoppia{\@ndoppiatrue\gdef\la@sezionecorrente{\the\sect@num}} \def\se@indefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\spo@glia#1>{} % si applica a \meaning\xxxxx; butta via tutto quello % che produce \meaning fino al carattere > % (v. manuale TeX, pag. 382, \strip#1>{}). \newif\if@primasezione \@primasezionetrue \def\s@ection#1\par{\immediate \write16{#1}\if@primasezione\global\@primasezionefalse\else\goodbreak \vskip\spaziosoprasez\fi\noindent {\bf#1}\nobreak\vskip\spaziosottosez\nobreak\noindent} %--------------------------- Indice ------------------------------- \newif\if@indice \newif\if@ceindice \newwrite\file@indice \def\indice{ \immediate\write16{Genera il file \jobname.ind} \@indicetrue \immediate\openin2 \jobname.ind \ifeof2\relax\else \closein2\relax \@ceindicetrue\fi \if@ceindice\relax\else \immediate\openout\file@indice=\jobname.ind \immediate\write \file@indice{\string\vskip5pt \string{ \string\bf \string\centerline\string{ Indice \string}\string}\string\par} \fi } \def\quiindice{\if@ceindice\vfill\eject\input\jobname.ind\else\vfill\eject \immediate\write\file@indice{\string{\string\bf\string~ Indice\string}\string\hfill\folio} \null\vfill\eject\null\vfill\eject\relax\fi} % %------------------------------ a disp. dell'utente: sezioni ------------- \def\sezpreset#1{\global\sect@num=#1 \immediate\write16{ !!! sez-preset = #1 } } \def\spaziosoprasez{50pt plus 60pt} \def\spaziosottosez{15pt} \def\sref#1{\se@indefinito{@s@#1}\immediate\write16{ ??? \string\sref{#1} non definita !!!} \expandafter\xdef\csname @s@#1\endcsname{??}\fi\csname @s@#1\endcsname} \def\autosez#1#2\par{ \global\advance\sect@num by 1\if@ndoppia\global\eq@num=0\fi \global\lemm@num=0 \global\para@num=0 \xdef\la@sezionecorrente{\the\sect@num} \def\usa@getta{1}\se@indefinito{@s@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @s@#1\endcsname\la@sezionecorrente\def \usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 { ??? possibili riferimenti errati a \string\sref{#1} !!!}\fi \expandafter\xdef\csname @s@#1\endcsname{\la@sezionecorrente} \immediate\write16{\la@sezionecorrente. #2} \if@simboli \immediate\write\file@simboli{ }\immediate\write\file@simboli{ } \immediate\write\file@simboli{ Sezione \la@sezionecorrente : sref. #1} \immediate\write\file@simboli{ } \fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @s@#1\string\endcsname{\la@sezionecorrente}}\fi \goodbreak\vskip 48pt plus 60pt \noindent{\bf\the\sect@num.\quad #2} \if@bozze {\tt #1}\fi \par\nobreak\vskip 15pt \nobreak} \def\blankii{\blank\blank} \def\destra#1{{\hfill#1}} \font\titfnt=cmssbx10 scaled \magstep2 \font\capfnt=cmss17 scaled \magstep4 \def\blank{\vskip 12pt} \def\capitolo#1#2\par{ \global\advance\sect@num by 1\if@ndoppia\global\eq@num=0\fi \global\lemm@num=0 \global\para@num=0 \xdef\la@sezionecorrente{\the\sect@num} \def\usa@getta{1}\se@indefinito{@s@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @s@#1\endcsname\la@sezionecorrente\def \usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 { ??? possibili riferimenti errati a \string\sref{#1} !!!}\fi \expandafter\xdef\csname @s@#1\endcsname{\la@sezionecorrente} \immediate\write16{\la@sezionecorrente. #2} \if@simboli \immediate\write\file@simboli{ }\immediate\write\file@simboli{ } \immediate\write\file@simboli{ Sezione \la@sezionecorrente : sref. #1} \immediate\write\file@simboli{ } \fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @s@#1\string\endcsname{\la@sezionecorrente}}\fi \par\vfill\eject \destra{\capfnt {\la@sezionecorrente}\hbox to 10pt{\hfil}} \blankii\noindent{\titfnt\baselineskip=20pt \hfill\uppercase{#2}}\blankii \if@indice \if@ceindice\relax\else\immediate\write \file@indice{\string\vskip5pt\string{\string\bf \la@sezionecorrente.#2\string}\string\hfill\folio\string\par}\fi\fi \if@bozze {\tt #1}\par\fi\nobreak} \def\semiautosez#1#2\par{ \gdef\la@sezionecorrente{#1}\if@ndoppia\global\eq@num=0 \fi \global\lemm@num=0 \global\para@num=0 \if@simboli \immediate\write\file@simboli{ }\immediate\write\file@simboli{ } \immediate\write\file@simboli{ Sezione ** : sref. \expandafter\spo@glia\meaning\la@sezionecorrente} \immediate\write\file@simboli{ }\fi \s@ection#2\par} %------------------paragrafi---------------------------------------- \def\pararef#1{\se@indefinito{@ap@#1} \immediate\write16{??? \string\pararef{#1} non definito !!!} \expandafter\xdef\csname @ap@#1\endcsname {#1} \fi\csname @ap@#1\endcsname} \def\autopara#1#2\par{ \global\advance\para@num by 1 \xdef\il@paragrafo{\la@sezionecorrente.\the\para@num} \vskip10pt \noindent {\bf \il@paragrafo\ #2} \def\usa@getta{1}\se@indefinito{@ap@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @ap@#1\endcsname\il@paragrafo\def\usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 {??? possibili riferimenti errati a \string\pararef{#1} !!!}\fi \expandafter\xdef\csname @ap@#1\endcsname{\il@paragrafo} \def\usa@getta{\expandafter\spo@glia\meaning \la@sezionecorrente.\the\para@num} \if@simboli \immediate\write\file@simboli{ }\immediate\write\file@simboli{ } \immediate\write\file@simboli{ paragrafo \il@paragrafo : pararef. #1} \immediate\write\file@simboli{ } \fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @ap@#1\string\endcsname{\il@paragrafo}}\fi \if@indice \if@ceindice\relax\else\immediate\write \file@indice{\string\noindent\string\item\string{ \il@paragrafo.\string}#2\string\dotfill\folio\string\par}\fi\fi \if@bozze {\tt #1}\fi\par\nobreak\vskip .3 cm \nobreak} %------------------------------ a disp. dell'utente: equazioni ----------- \def\eqpreset#1{\global\eq@num=#1 \immediate\write16{ !!! eq-preset = #1 } } \def\eqlabel#1{\global\advance\eq@num by 1 \if@ndoppia\xdef\il@numero{\la@sezionecorrente.\the\eq@num} \else\xdef\il@numero{\the\eq@num}\fi \def\usa@getta{1}\se@indefinito{@eq@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @eq@#1\endcsname\il@numero\def\usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 { ??? possibili riferimenti errati a \string\eqref{#1} !!!}\fi \expandafter\xdef\csname @eq@#1\endcsname{\il@numero} \if@ndoppia \def\usa@getta{\expandafter\spo@glia\meaning \il@numero} \else\def\usa@getta{\il@numero}\fi \if@simboli \immediate\write\file@simboli{ Equazione \usa@getta : eqref. #1}\fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @eq@#1\string\endcsname{\usa@getta}}\fi} \def\eqsref#1{\se@indefinito{@eq@#1} \immediate\write16{ ??? \string\eqref{#1} non definita !!!} \if@riferimenti\relax \else\eqlabel{#1} ???\fi \fi\csname @eq@#1\endcsname } \def\autoeqno#1{\eqlabel{#1}\eqno(\csname @eq@#1\endcsname)\if@bozze {\tt #1}\else\relax\fi} \def\autoleqno#1{\eqlabel{#1}\leqno(\csname @eq@#1\endcsname)} \def\eqref#1{(\eqsref{#1})} %----------- Lemmi automatici: a disposizione dell'utente ---------------- \def\lemmalabel#1{\global\advance\lemm@num by 1 \xdef\il@lemma{\la@sezionecorrente.\the\lemm@num} \def\usa@getta{1}\se@indefinito{@lm@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @lm@#1\endcsname\il@lemma\def\usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 { ??? possibili riferimenti errati a \string\lemmaref{#1} !!!}\fi \expandafter\xdef\csname @lm@#1\endcsname{\il@lemma} \def\usa@getta{\expandafter\spo@glia\meaning \la@sezionecorrente.\the\lemm@num} \if@simboli \immediate\write\file@simboli{ Lemma \usa@getta : lemmaref #1}\fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @lm@#1\string\endcsname{\usa@getta}}\fi} \def\autolemma#1{\lemmalabel{#1}\csname @lm@#1\endcsname\if@bozze {\tt #1}\else\relax\fi} \def\lemmaref#1{\se@indefinito{@lm@#1} \immediate\write16{ ??? \string\lemmaref{#1} non definita !!!} \if@riferimenti\else \lemmalabel{#1}???\fi \fi\csname @lm@#1\endcsname} %--------------- bibliografia automatica: riservati ---------------------- \newcount\cit@num\global\cit@num=0 \newwrite\file@bibliografia \newif\if@bibliografia \@bibliografiafalse \newif\if@corsivo \@corsivofalse \def\title#1{{\it #1}} \def\rivista#1{#1} \def\lp@cite{[} \def\rp@cite{]} \def\trap@cite#1{\lp@cite #1\rp@cite} \def\lp@bibl{[} \def\rp@bibl{]} \def\trap@bibl#1{\lp@bibl #1\rp@bibl} \def\refe@renza#1{\if@bibliografia\immediate % scrive su .BIB \write\file@bibliografia{ \string\item{\trap@bibl{\cref{#1}}}\string \bibl@ref{#1}\string\bibl@skip}\fi} \def\ref@ridefinita#1{\if@bibliografia\immediate\write\file@bibliografia{ \string\item{?? \trap@bibl{\cref{#1}}} ??? tentativo di ridefinire la citazione #1 !!! \string\bibl@skip}\fi} \def\bibl@ref#1{\se@indefinito{@ref@#1}\immediate \write16{ ??? biblitem #1 indefinito !!!}\expandafter\xdef \csname @ref@#1\endcsname{ ??}\fi\csname @ref@#1\endcsname} \def\c@label#1{\global\advance\cit@num by 1\xdef % assegna il numero \la@citazione{\the\cit@num}\expandafter \xdef\csname @c@#1\endcsname{\la@citazione}} \def\bibl@skip{\vskip 5truept} %------------------------ bibl. automatica: a disp. dell'utente ------------ \def\stileincite#1#2{\global\def\lp@cite{#1}\global \def\rp@cite{#2}} \def\stileinbibl#1#2{\global\def\lp@bibl{#1}\global \def\rp@bibl{#2}} \def\corsivo{\global\@corsivotrue} \def\citpreset#1{\global\cit@num=#1 \immediate\write16{ !!! cit-preset = #1 } } \def\autobibliografia{\global\@bibliografiatrue\immediate \write16{ !!! Genera il file \jobname.BIB}\immediate \openout\file@bibliografia=\jobname.bib} \def\cref#1{\se@indefinito % se indefinito definisce {@c@#1}\c@label{#1}\refe@renza{#1}\fi\csname @c@#1\endcsname} \def\upcref#1{\null$^{\,\cref{#1}}$} \def\cite#1{\trap@cite{\cref{#1}}} % [5] \def\ccite#1#2{\trap@cite{\cref{#1},\cref{#2}}} % [5,6] \def\ncite#1#2{\trap@cite{\cref{#1}--\cref{#2}}} % [5-8] senza definire \def\upcite#1{$^{\,\trap@cite{\cref{#1}}}$} % ^[5] \def\upccite#1#2{$^{\,\trap@cite{\cref{#1},\cref{#2}}}$} % ^[5,6] \def\upncite#1#2{$^{\,\trap@cite{\cref{#1}-\cref{#2}}}$} % ^[5-8] senza def. \def\clabel#1{\se@indefinito{@c@#1}\c@label % sola definizione {#1}\refe@renza{#1}\else\c@label{#1}\ref@ridefinita{#1}\fi} \def\cclabel#1#2{\clabel{#1}\clabel{#2}} % def. doppia \def\ccclabel#1#2#3{\clabel{#1}\clabel{#2}\clabel{#3}} % def. tripla \def\biblskip#1{\def\bibl@skip{\vskip #1}} % spaziatura nella bibl. \def\insertbibliografia{\if@bibliografia % scrive la bibliografia \immediate\write\file@bibliografia{ } \immediate\closeout\file@bibliografia \if@indice \if@ceindice\relax\else\immediate\write \file@indice{\string\vskip5pt\string{\string\bf\string~ Bibliografia\string}\string\hfill\folio\string\par}\fi\fi \catcode`@=11\input\jobname.bib\catcode`@=12\fi } %--------- per comporre il file con la bibliografia -------------- \def\commento#1{\relax} \def\biblitem#1#2\par{\expandafter\xdef\csname @ref@#1\endcsname{#2}} % ricordare: una lista in chiaro della bibliografia si % ottiene eseguendo $ TEX BIBLIST %---------------- titolo in cima alla pagina, data.----------------- \def\data{\number\day.\number\month.\number\year} \def\datasotto{\@datatrue \footline={\hfil{\rm \data}\hfil}} \def\titoli#1{\if@data\relax\else\footline={\hfil}\fi \xdef\prima@riga{#1}\voffset+20pt \headline={\ifnum\pageno=1 {\hfil}\else\hfil{\sl \prima@riga}\hfil\folio\fi}} \def\duetitoli#1#2{\if@data\relax\else\footline={\hfil}\fi \voffset=+20pt \headline={\ifnum\pageno=1 {\hfil}\else{\ifodd\pageno\hfil{\sl #2}\hfil\folio \else\folio\hfil{\sl #1}\hfil\fi} \fi} } \def\la@sezionecorrente{0} % ------------------COSTANTI --------------------------------- \def\const@label#1{\global\advance\const@num by 1\xdef \la@costante{\the\const@num}\expandafter \xdef\csname @const@#1\endcsname{\la@costante}} \def\cconlabel#1{\se@indefinito{@const@#1} \const@label{#1}\fi} \def\constnum#1{\se@indefinito{@const@#1} \const@label{#1}\fi\csname @const@#1\endcsname} \def\ccon#1{C_{\constnum{#1}}} \catcode`@=12 %------------------ FORMATI TEOREMI E GENERALI -------------------- \def\abstract{ \vskip48pt plus 60pt \noindent {\bf Abstract.}\quad} \def\summary{ \centerline{{\bf Summary.}}\par} \def\firma{\noindent \centerline{Dario BAMBUSI}\par\noindent \centerline{Dipartimento di Matematica dell'Universit\`a,}\par\noindent \centerline{Via Saldini 50, 20133 Milano, Italy.}\par} \def\theorem#1#2{\par\vskip4pt \noindent {\bf Theorem \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\semitheorem#1{\par\vskip4pt \noindent {\bf Theorem.}{\sl \ #1} \par\vskip10pt} \def\lemma#1#2{\par\vskip4pt \noindent {\bf Lemma \autolemma{#1}.}{\sl \ #2} \par\vskip4pt} \def\proof{\par\noindent{\bf Proof.}\ } \def\proposition#1#2{\par\vskip4pt \noindent {\bf Proposition \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\corollary#1#2{\par\vskip4pt \noindent {\bf Corollary \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\remark#1#2{\par\vskip4pt \noindent {\bf Remark \autolemma{#1}.}{\sl \ #2} \par\vskip4pt} \def\definition#1{\par\vskip2pt \noindent {\bf Definition.}{\sl \ #1} \par\vskip2pt} %------------------- ROUTINE DI USO GENERALE ----------------------- \def\norma#1{\left\Vert#1\right\Vert} \def\perogni{\forall\hskip1pt} \def\meno{\hskip1pt\backslash} \def\frac#1#2{{#1\over #2}} \def\fraz#1#2{{#1\over #2}} \def\interno{\vbox{\hbox{\vbox to .3 truecm{\vfill\hbox to .2 truecm {\hfill\hfill}\vfill}\vrule}\hrule}\hskip 2pt} \def\quadratino{ \hfill\vbox{\hrule\hbox{\vrule\vbox to 7 pt {\vfill\hbox to 7 pt {\hfill\hfill}\vfill}\vrule}\hrule}\par} \font\strana=cmti10 \def\lie{\hbox{\strana \char'44}} \def\ponesotto#1\su#2{\mathrel{\mathop{\kern0pt #1}\limits_{#2}}} \def\Sup{\mathop{{\rm Sup}}} %\def\Sup#1{\hskip2pt\ponesotto{{\rm Sup}}\su{#1}} \def\tdot#1{\hskip2pt\ddot{\null}\hskip2.5pt \dot{\null}\kern -5pt {#1}} \def\diff#1#2{\frac{\partial #1}{\partial #2}} \def\base#1#2{\frac{\partial}{\partial#1^{#2}}} \def\charslash#1{\setbox2=\hbox{$#1$} \dimen2=\wd2 \setbox1=\hbox{/}\dimen1=\wd1 \ifdim\dimen2>\dimen1 \rlap{\hbox to \dimen2{\hfil /\hfil}} #1 \else \rlap{\hbox to \dimen1{\hfil$#1$\hfil}} / \hfil\fi} \def\Re{{\rm \kern 0.4ex I \kern -0.4 ex R}} \def\Sh{{\rm Sh}\hskip1pt} \def\Ch{{\rm Ch}\hskip1pt} \def\poisson#1#2{\left\{#1 ,#2\right\} } \def\toro{{\bf T}} \def\Na{{\bf N}} \def\Ra{{\bf Z}} \def\id{{\bf 1}} \def\Cm{{\bf C}} \def\reale{{\rm Re}\hskip2pt} \def\imma{{\rm Im}\hskip2pt} \def\rin{{\bf Z}} \def\pmb#1{\setbox0=\hbox{#1}\ignorespaces \hbox{\kern-.02em\copy0\kern-\wd0\ignorespaces \kern.05em\copy0\kern-\wd0\ignorespaces \kern-.02em\raise.02em\box0 }} \def\vett#1{\pmb{$#1$}} \def\A{{\cal A}} \def\B{{\cal B}} \def\C{{\cal C}} \def\D{{\cal D}} \def\E{{\cal E}} \def\F{{\cal F}} \def\G{{\cal G}} \def\H{{\cal H}} \def\I{{\cal I}} \def\L{{\cal L}} \def\M{{\cal M}} \def\N{{\cal N}} \def\O{{\cal O}} \def\P{{\cal P}} \def\Q{{\cal Q}} \def\R{{\cal R}} \def\S{{\cal S}} \def\T{{\cal T}} \def\U{{\cal U}} \def\V{{\cal V}} \def\W{{\cal W}} \def\Z{{\cal Z}} \def\sym{\nabla^\Omega} \def\uno{{\kern+.3em {\rm 1} \kern -.22em {\rm l}}} \def\unpo{\vskip3pt} \def\unp{\vskip6pt} \def\a{\`a\ } \def\o{\`o\ } \def\e{\`e\ } \parindent=0pt \parskip=10pt \def\wnote#1{\footnote{$^*$}{{\tt #1}}} \def\section#1{\bigskip \bigskip {\bf #1} \bigskip} \bozze \riferimentifuturi \def\interi{\Ra} \def\adiabat{\cite{bam?}} \def\frac#1#2{{ #1 \over #2 }} \def\toro{{\bf T}} \def\U{{\cal U}} \def\S{{\cal S}} \def\R{{\cal R}} \def\taglio{\left(\frac{1+e^{-\sigma/2}}{1-e^{-\sigma/2}}\right)^m} \def\dif{d} \def\eps{\varepsilon} \def\Fm{\langle f\rangle} \def\Gm{\langle g\rangle} \def\media{\Fm} \def\normac#1{\norma{#1}_{C^1(\U_{\rho/2}\times\toro)}} \def\phis{\Phi_s(J,\psi)} \def\hs{H_s(J,\psi)} \def\arg{\sigma(\psi-\omega T),\psi-\omega T} \def\dom{\U_\rho} \def\sr{\bar\R} \def\rsuno{\bar\R_1} \def\sruno{\bar\R_1} \def\rs{\bar\R} \centerline{\bf PROOF OF PERSISTENCE OF INVARIANT TORI IN} \centerline{\bf NONHAMILTONIAN PERTURBATIONS OF INTEGRABLE SYSTEMS} \footnote{}{Version of \day} \bigskip\bigskip\bigskip \centerline{ Dario Bambusi} \centerline{\it Dipartimento di Matematica, Universit\`a di Milano,} \centerline{\it via Saldini 50, 20133 Milano (Italy)} \bigskip \centerline{ Giuseppe Gaeta} \centerline{\it Department of Mathematics, Loughborough University,} \centerline{\it Loughborough LE11 3TU (England)} \bigskip\bigskip\bigskip \autosez{0}Introduction We study here the dynamics of the system $$ \cases{ \dot I =\eps \ f(I,\phi,\eps) & \cr \dot\phi =\omega_0 (I)+\eps \ g(I,\phi,\eps) & \cr} \autoeqno{1} $$ where $I\in\G\subseteq\Re^n$ (with $\G$ open) are the slow variables, $\phi\in\toro^m$ are the fast angular variables, and $\eps$ is a (small) real parameter. We assume that all functions appearing here are analytic. We will prove that, provided the average of $f$ over the angles has a hyperbolic attractive zero at a point $I_*\in\G$, and the corresponding frequency $\omega_0 (I_*)$ is sufficiently nonresonant, then there exists a normally hyperbolic attractive invariant torus of the system \eqref{1} which is close to the torus $I_*\times\toro^m$. We introduce the set $\Gamma(\tau,\gamma,K)$ of variables which are ``$\tau,\gamma$ diophanitine up to order $K$'', namely we define $$ \Gamma(\tau,\gamma,K):=\left\{I\in\G\ :\ |\omega_0(I)\cdot k|\geq\frac\gamma{|k|^\tau}\ ,\quad\perogni k\in\Ra\meno\left\{0\right\} \ {\rm such\ that}\ |k|:=\sum_{i=1}^m|k_i|\leq K\right\}\ . $$ We will denote by $\Fm$ the average of the main part of $f$ with respect to the angles, namely $$ \Fm(I):=\frac1{(2\pi)^{m}}\int_{\toro^m}f(I,\phi,0)d^m\phi $$ (similarly we will use the notation $\Gm$). Consider $\Fm$: we assume that there exists an attractive zero $I_*$ of $\Fm$, namely a zero such that all the eigenvalues of $d\Fm(I_*)$ are strictly negative \theorem{m}{Assume that there exist constants $\tau,\gamma,K$ such that $I_*\in\Gamma(\tau,gamma,K)$, thne there exist constants $\epsilon_*, K_*,\ccon0$, such that, if $K>K_*\left|\ln\eps_*\right|^\tau$, and $\exp\frac{K}{K_*}<\eps<\eps_*$, then \eqref{1} has a stable invariant torus close to the torus $\toro_*:=I_*\times\toro^m$, precisely, one has $$ \min_{(I,\phi)\in\toro_*}\norma{I-I_*}\leq\ccon0\eps\left|\ln\eps\right| ^{2\tau+1}\autoeqno{dis} $$ } \autosez{nor}The normal form lemma We first give a quantitative form to our smoothness assumptions. Let $D\subset\G$ (strictly) be open. We introduce the complex open set $$ D_\rho:=\bigcup_{I\in\D}B_{\rho}(I)\ , $$ where $B_{\rho}(I)\subset\Cm^{n}$ is the closed ball of radius $\rho$ centered at $I$. We also introduce the complexification of the $m$--dimensional torus $$ (\toro+i\sigma)^m:=\left\{(\phi_i)\in\left(\toro +i\Re\right)^m\ :\ |\imma\phi_i|\leq\sigma \right\}\ . $$ Then, there exist some $\bar\rho,\bar\sigma,\eta>0$ such that the functions $f$, $g$, and $\omega_0$ can be extended to complex analytic functions on $$ \D_{\bar\rho,2\bar\sigma,\eta}:=D_{\bar\rho}\times\left(\toro^m+i2\bar \sigma \right)\times \B_{\eta}(0)\ , $$ where $\B_{\eta}(0):=\left\{\epsilon\in\Cm\ :\ |\eps|\leq\eta\right\}$, which moreover are bounded on such set; moreover, there exist constants $F,G$, and $\Omega$ such that $$ \eqalign{ \sup_{\D_{\bar\rho,2\bar\sigma,\eta}}\norma{f(I,\phi,\eps)}\leq F \cr \sup_{\D_{\bar\rho,2\bar\sigma,\eta}}\norma{g(I,\phi,\eps)}\leq G \cr \sup_{\D_{\bar\rho,2\bar\sigma,\eta}}\norma{d\omega_0 (I)}\leq \Omega\ . } $$ We will also consider the complex extension $$ \Gamma_{\rho}(\tau,\gamma,K):=\bigcup_{I\in\Gamma(\tau,\gamma,K)} B_{\rho}(I)\ . $$ Here we will prove the following \lemma{nf.1}{Fix $\tau\geq m$ and $\gamma>0$, then, provided $\epsilon$ is small enough there exist constants $\ccon1,\cconlabel2\cconlabel3\cconlabel4\cconlabel5\cconlabel6...,\ccon7$ and an analytic coordinate transformation $$ \eqalign{ I&=J+\eps A(J,\psi,\eps)\ , \cr \phi&=\psi+\eps B(J,\psi,\eps)\ , }\autoeqno{c.c} $$ defined on $\Gamma_{\rho/2}(\tau,\gamma,\ccon1\left|\ln\eps\right|^{\tau})\times (\toro+i\bar \sigma/2)^m$, where $$ \rho:=\frac{\ccon2}{|\ln\eps|^{\tau+1}}\ ;\autoeqno{rho} $$ which transforms \eqref{1} into the system $$ \eqalign{ \dot J&=\eps\Fm(J)+\eps^2Z(J,\eps)+R(J,\psi,\eps) \cr \dot\psi&=\omega_0(J)+\eps\Gm(J)+R_1(J,\psi,\eps)\ . }\autoeqno{sys} $$ Moreover, the vector valued functions $A, B,Z,R,R_1$ are analytic on $\Gamma_{\rho/2}(\tau,\gamma,\ccon1\left|\ln\eps\right|^{\tau})\times (\toro+i\bar \sigma/2)^m$, and, satisfy the estimates $$ \eqalign{ \sup_{\Gamma_{\rho/2}(\tau,\gamma,\ccon1\left|\ln\eps\right|^{\tau})\times (\toro+i\bar \sigma/2)^m}\norma{A}\leq\ccon3\left|\ln\eps\right|^\tau \cr \sup_{\Gamma_{\rho/2}(\tau,\gamma,\ccon1\left|\ln\eps\right|^{\tau})\times (\toro+i\bar \sigma/2)^m}\norma{B}\leq\ccon4\left|\ln\eps\right|^{2\tau} \cr \sup_{\Gamma_{\rho/2}(\tau,\gamma,\ccon1\left|\ln\eps\right|^{\tau})\times (\toro+i\bar \sigma/2)^m}\norma{Z}\leq\ccon5\left|\ln\eps\right|^{2\tau+1} \cr \sup_{\Gamma_{\rho/2}(\tau,\gamma,\ccon1\left|\ln\eps\right|^{\tau})\times (\toro+i\bar \sigma/2)^m}\norma{R}\leq\ccon6\eps^3\left|\ln\eps\right|^{4\tau+2} \cr \sup_{\Gamma_{\rho/2}(\tau,\gamma,\ccon1\left|\ln\eps\right|^{\tau})\times (\toro+i\bar \sigma/2)^m}\norma{R_1}\leq\ccon7\eps^2\left|\ln\eps\right|^{3\tau+1} } $$ } \remark{1.2}{The dependence of $A,B,Z,R,R_1$ on $\eps$ is in general discontinuous (see remark \lemmaref{1.1} below).} \unp \noindent {\tt 0) Preliminaries} We will first decompose $f$ and $g$ in parts of different order in $\eps$. So we write $$ \eqalign{ f(I,\phi,\eps)=\tilde f^0(I,\phi)+\eps \tilde f^1(I,\phi)+\eps^2 \tilde f^2(I,\phi,\eps) \cr g(I,\phi,\eps)=\tilde g^0(I,\phi)+\eps \tilde g^1(I,\phi,\eps)\ , } $$ where $$ \eqalign{ \tilde f^0(I,\phi):=f(I,\phi,0)\ ,\quad \tilde f^1:= \left.\diff{f}{\eps} \right|_{\eps=0}\ ,\quad \tilde f^2:=\frac1{\eps^2}\left[f-\tilde f^0-\eps\tilde f^1\right]\ , \cr \tilde g^0(I,\phi):=g(I,\phi,0)\ ,\quad \tilde g^1:=\frac1\eps\left[g-\tilde g^0\right]\ . } $$ Then we perform the so called ``ultraviolet cutoff'', so we fix a positive $K$ (which will be related to $\eps$ in a while), and define $$ \eqalign{ f^0(I,\phi):=\sum_{|k|\leq K}\tilde f^0_k(I)e^{ik\phi} \cr f^1(I,\phi):=\frac1\eps \sum_{K<|k|\leq 2K}\tilde f^0_k(I)e^{ik\phi} +\sum_{|k|\leq 2K}\tilde f^1_k(I)e^{ik\phi} \cr f^2(I,\phi,\eps):=\tilde f^2(I,\phi,\eps)+\frac1{\eps^2}\sum_{|k|>2K} \tilde f^0_k(I)e^{ik\phi}+\frac1\eps\sum_{|k|>2K} \tilde f^1_k(I)e^{ik\phi} \cr g^0(I,\phi):=\sum_{|k|\leq K}\tilde g^0_k(I)e^{ik\phi} \cr g^1(I,\phi,\eps):=\tilde g^1(I,\phi,\eps)+\frac1{\eps}\sum_{|k|>K} \tilde g^0_k(I)e^{ik\phi}\ , }\autoeqno{Ta} $$ where $\tilde f^i_k$ is the $k-th$ fourier coefficient of $\tilde f^i$, and similarly for $\tilde g^i_k$ (and for the Fourier coefficients of other functions incountered below). \lemma{uc}{Define $$ \eqalign{ F_0:=F\taglio \cr F_1 :=F\taglio \frac{e^{-\sigma K/2}}{\eps}+\frac F{\eta}\taglio \cr F_2:=\frac{8F}{\eta}+F\taglio\left(\frac{e^{-\sigma K/2}}{\eps}\right)^2+ \frac F\eta\frac{e^{-\sigma K/2}}{\eps} \cr G_0:=G\taglio \cr G_1:=\frac{2G}\eta+G\taglio\frac{e^{-\sigma K/2}}{\eps}\ . } $$ Then one has $$ \eqalign{ \sup_{\D_{\bar\rho,\bar\sigma,\eta/2}}\norma{f^i}\leq\sum_{|k|\leq(i+1)K} \sup_{I\in D_\rho}\norma{f^i_k(I)}e^{\bar\sigma |k|}\leq F_i\ ,\quad i=0,1 \cr \sup_{\D_{\bar\rho,\bar\sigma,\eta/2}} \norma{g^0}\leq\sum_{|k|\leq K} \sup_{I\in D_\rho}\norma{g^0_k(I)}e^{\bar\sigma |k|}\leq G_0\ , \cr \sup_{\D_{\bar\rho,\bar\sigma,\eta/2}}\norma{f^2}\leq F_2\ ,\quad \sup_{\D_{\bar\rho,\bar\sigma,\eta/2}}\norma{g^1}\leq G_1 }\autoeqno{A.2} $$ } \proof Just use Cauchy inequality to estimate $\tilde f^1$ and $\tilde g^1$, and (following \cite{gio?}, lemma 8) the exponential decay of Fourier coefficients to get the thesis.\quadratino In order to obtain that the constants $F_i$ and $G_i$ do not depend on $\eps$ we choose $$ K:=\frac2\sigma \left|\ln\eps\right|\ .\autoeqno{(k)} $$ We are thus reduced to the system $$ \eqalign{ \dot I=&\eps f^0(I,\phi)+\eps^2 f^1(I,\phi)+\eps^3 f^2(I,\phi,\eps)\ , \cr \dot\phi=&\omega_0 (I)+\eps g^0(I,\phi)+\eps^2 g^1(I,\phi,\eps)\ , \cr} \autoeqno{v} $$ \remark{1.1}{The functions $f^0, f^1, g^0$ (and therefore also the functions $f^2,g^1$) depend on $\epsilon$ through $K$ (cf. \eqref{Ta} and \eqref{(k)}), and such a dependence is in general discontinuous.} \unp \noindent {\tt 1) Formal theory} We look for a formal coordinate transformation $$ \eqalign{ I= & J+\epsilon A_1(J,\psi)+\epsilon^2 A_2(J,\psi,\eps)=J+\eps A(J,\psi,\eps) \cr \phi=& \psi+\epsilon B(J,\psi)\ , \cr }\autoeqno{t} $$ which reduces the system \eqref{v} to the form \eqref{sys}. We substitute the first of \eqref{t} in the first of \eqref{v} and determine $A_1$ in order to eliminate at first order the dependence of $f$ on the angles. We have $$ \dot J=\eps\left[ f^0(J,\psi)-d_\psi{A_1}(J,\psi)\omega_0 (J) \right]+o(\eps)\ \autoeqno{A1} $$ (here and in the following, the notation $d_\psi A \omega_0 $ means the application of the differential with respect to the angles to $\omega_0$, in coordinates it has the form $\omega_0 ^i {\partial A / \partial \psi^i } $, and similarly for equivalent notations to be used below). Since $$ f^0(J,\psi)=\sum_{|k|\leq K}f^0_k(J)e^{ik\cdot\psi}\ , $$ we define (as usual) $$ A_1(J,\psi):=\sum_{0\not=|k|\leq K}\frac{f^0_k(J)}{i\omega_0 (J)\cdot k} e^{ik\cdot\psi}\ , \autoeqno{A.1} $$ which makes sense provided $\omega_0(J)\cdot k\not=0$ $\perogni k$ with $|k|\leq K$ and $\perogni J$ in the domain we are interested in; with this, the square bracket in \eqref{A1} reduces to $\Fm$. We consider now the equation for $\psi$. Substituting \eqref{t} in it we obtain $$ \dot \psi=\omega_0 (J)+\eps \left[ g^0(J,\psi)+d_J\omega_0 (J)A_1(J,\psi)-d_\psi B(J,\psi)\omega_0 (J) \right]+o(\eps)\ .\autoeqno{psip} $$ We choose $B$ as (using an obvious notation) $$ B(J,\psi)=\sum_{0\not=|k|\leq K}\frac1{i\omega_0 (J)\cdot k}\left( g^0_k(J)-d_J\omega_0 (J)\frac{f^0_k(J)}{i\omega_0 (J)\cdot k } \right) e^{ik\cdot\psi}\ ,\autoeqno{B1} $$ which reduces the square bracket of \eqref{psip} to $\Gm$. We consider again the equation for $J$ in order to eliminate at second order the dependence on the angles. \lemma{77}{Define $\Psi$ by $$ \Psi:=f^1+d_\psi f^0B-d_\psi A_1\Gm+d_Jf^0A_1-d_JA_1\Fm\ .\autoeqno{psi} $$ and $$ A_2(J,\psi):= \ \sum_{0\not=|k|\leq 2K}\frac{\Psi_k(J)}{i\omega_0 (J)\cdot k} \ e^{ik\cdot\psi}\ , $$ then the formal coordinate transformation \eqref{t} reduces \eqref{v} to the form \eqref{sys} with $Z$ the average of $\Psi$ over the angles, $$ \eqalign{ R:=\left( \L^{-1}-\uno+\eps d _J A_1 \right)\eps \media+ \eps \L^{-1}d_\psi A_1\left(\omega_0+\eps\Gm -\dot \psi \right)+\eps^2\left(\L^{-1}-\uno\right)\left(Z+d_JA_1\Fm\right) \cr -\L^{-1}\left[\eps\Delta f^0+\eps^2(d_JA_2\dot J-\Delta f^1) +\eps^2 d_\psi A_2 (\dot\psi-\omega_0)-\eps ^3 f^2(I,\phi)-\eps^3 d_Jf^0 A_2 \right] } $$ and $$ R_1:= \Delta\omega_0 +\eps\Delta g_0+\eps^2g^1 + \eps d_\psi B(\omega_0 -\dot\psi)-\eps d_J B\dot J\ . \autoeqno{r1} $$ } We also introduced the notations $$ \eqalign{ \L&:=\uno+\eps d_JA_1 \cr \Delta f^0 =& f^0(I,\phi)-f^0(J,\psi)-\eps\left[d_Jf^0(J,\psi) A_1(J,\psi) +d_{\psi}f^0(J,\psi) B(J,\psi)\right]\ , \cr \Delta f^1 =& f^1(I,\phi)-f^1(J,\psi) \ , \cr \Delta g_0 =& g^0(I,\phi)- g^0(J,\psi)\ , \cr \Delta\omega_0 =& \omega_0 (I)-\omega_0 (J)-\eps d_J\omega_0 (J)A_1(J,\psi) \ . \cr }\autoeqno{delta} $$ \proof Consider the equation for $J$. We have $$ \eqalign{ \L \dot J+\eps^2 d_J A_2\dot J+\eps d_\psi A_1 \dot \psi +\eps^2 d_\psi A_2 \dot \psi&= \cr \eps \left[ f^0+d_Jf^0\eps A_1+d_\psi f^0\eps B +\Delta f^0 \right] &+\eps^2\left[f^1+\Delta f^1\right]+\eps^3 f^2(I,\phi) } $$ here the functions are evalueted in $J,\psi$ unless otherwise specified. Thus we can rewrite the above formula as $$ \eqalign{ \L\dot J=\eps \left[f^0-d_\psi A_1 \omega_0 (J)\right] +\eps^2 \left\{ f_1+d_\psi f^0 B+d_J f^0 A_1- d_\psi A_2 \dot \psi \right\} \cr -\left(\eps\Delta f^0+\eps^2(\Delta f^1- d_J A_2 \dot J +\eps ^3 f^2(I,\phi)+d_J f^0 A_2) \right) \ ,} \autoeqno{pp} $$ Using the fact that $A_1$ satisfies the homological equation, the square bracket in \eqref{pp} is nothing else than $\Fm$ we rewrite \eqref{pp} as $$ \L\dot J=\eps\Fm+\eps^2\left\{ f_1+d_\psi f^0 B+d_J f^0 A_1- d_\psi A_2\omega_0 \right\}+\F_3\ ,\autoeqno{15} $$ where $$ \eqalign{ \F_3:= -\eps\left[d_\psi A_1(\dot \psi-\omega_0 (J)-\eps\Gm) \right] \cr -\eps^2d_JA_2\dot J+\eps\Delta f^0+\eps^2\Delta f^1+\eps^3f^2(I,\phi)-\eps^2d_\psi A_2(\dot \psi-\omega_0 (J))\ . } $$ Multiplying by $\L^{-1}$, and taking into account the definition of $A_2$ we get $$ \eqalign{ \dot J=\eps\Fm-\eps^{2}d_JA_1\Fm+\eps\left(\L^{-1}-\uno+\eps d_JA_1\right)\Fm \cr +\eps^2 \left\{...\right\}+\L^{-1}\F_3 +\eps^2\left(\L^{-1}-\uno\right)\left\{...\right\}\ .} $$ where the curly bracket coincides with that of \eqref{15}. Taking into account that $$ \left\{...\right\}+d_JA_1\media=\Psi+d_\psi A_2\omega_0 $$ and the definition of $A_2$ we get the expression of $R$. \quadratino \bigskip \noindent {\tt 2) Quantitative estimates} We begin by determininig the domain. To this end the following lemma is usefull \lemma{4.1}{For $K$ large enough (so that $\rho\leq\bar\rho$) define $$ \rho =\frac{\gamma}{\Omega 2^{\tau+2} K^{\tau+1}}\ , \autoeqno{4.lem1} $$ then, one has $$ \Gamma_{\rho}(\tau,\gamma,2K)\subset\Gamma(\tau,\frac\gamma2,2K)\ . $$ } \proof Just use Lagrange theorem; for details see \adiabat\ lemma 4.1. \quadratino In what follows we will use the notation $$ \alpha_K:=\frac{\gamma}{2K^{\tau}}\ .\autoeqno{14.1} $$ >From now on we assume that $\rho$ satifies \eqref{4.lem1}. We fix now $\tau$ and $\gamma$ (while $K$ is determined by \eqref{(k)}). For a function $h$ analytic in $\Gamma_{\alpha\rho}(\tau,\gamma,2K)\times\toro^m+i\beta\sigma$, where $\alpha,\beta \in(0,1]$ we define $$ \norma{h}_{\alpha\rho,\beta\sigma}:=\sup_{B_{\alpha\rho}(I_*)\times\toro^m +i\beta\sigma}\norma{h(I,\phi)}\ . $$ We come to the estimates. Clearly, from \eqref{A.1}, \eqref{A.2}, \eqref{B1}, we have $$ \norma{A_1}_{\rho,\sigma}\leq \frac{F_0}{\alpha_K}\ ,\quad \norma{B}_{\rho,\sigma}\leq\left(\frac{\Omega F_0}{\alpha_K}+G_0\right)\frac1{\alpha_K}\leq \frac{2\Omega F_0}{\alpha_K^2} \ , $$ where we used the fact that $\alpha_K\to0$ as $\eps\to0$ cf. \eqref{(k)}, \eqref{14.1} to identify (and retain) only the main term. Then, using the Cauchy inequalities it is easy to see that $$ \norma{\Psi}_{3\rho/4,3\sigma/4}\leq F_1+ \frac{4F_0^2} {\alpha_K}\left(\frac2\rho+\frac{\Omega}{e\sigma\alpha_K}+\frac{G_0}{e\sigma} \right) \leq\frac{16 F_0^2}{\alpha_K\rho}\ , $$ where we used $\rho=o(\alpha_K)$ (cf. \eqref{4.lem1}, \eqref{14.1}) to identify the leading term. >From this we obtain $$ \eqalign{ \norma{A_2}_{3\rho/4,3\sigma/4}\leq\frac{16 F_0^2}{\alpha_K\alpha_{2K}\rho} \cr \norma{A}_{3\rho/4,3\sigma/4}\leq\frac{2F_0}{\alpha_K}\ } $$ (provided $\eps$ is small enough). Before estimating the remainders $R$, $R_1$ we need some preliminary estimates. \lemma{ln.1}{One has $$ \eqalign{ \norma{\dot J}_{3\rho/4,3\sigma/4}\leq 2\eps F_0\ ,\quad \norma{\dot \psi-\omega_0 }_{3\rho/4,3\sigma/4}\leq2\eps \frac{\Omega F_0}{\alpha_K} \cr \norma{\Delta\omega_0}_{\rho/2,\sigma/2}\leq\eps^2\frac{24\Omega F_0^2} {\rho\alpha_K\alpha_{2K}}\ ,\quad \norma{\Delta g_0}_{\rho/2,\sigma/2}\leq\eps\frac{8G_0F_0}{\rho\alpha_K} \cr \norma{\Delta f_0}_{\rho/2,\sigma/2}\leq\eps^2\frac{2^7 F_0^3}{\rho^2\alpha_K\alpha_{2K}}\ ,\quad \norma{\Delta f_1}_{\rho/2,\sigma/2}\leq\eps\frac{8F_1 F_0}{\rho\alpha_K} \cr \norma{\L}_{\rho/2,\sigma/2}\leq2\ ,\quad \norma{\L-\uno }_{\rho/2,\sigma/2}\leq \frac{4\eps}{\rho\alpha_K}F_0 \ ,\quad \norma{\L-\uno+\eps\diff{A_1}J }_{\rho/2,\sigma/2}\leq \frac{8\eps^2 F_0^2}{\rho^2\alpha_K^2}\ .} $$ } \proof We begin by $\Delta \omega_0$: one has $$ \eqalign{ \norma{\Delta\omega_0}\leq \norma{\omega_0(I)-[\omega_0(J)+\eps\diff{\omega_0}J A]}+\eps^2\norma{\diff{\omega_0}J}\norma{A_2} \cr \leq\norma{\frac{\partial^2\omega_0}{\partial J^2} }\eps^2\norma{A_1+\eps A_2}^2+\eps^2 \norma{\diff{\omega_0}J}\norma{A_2}\ , } $$ from which $$ \norma{\Delta\omega_0}_{\rho/2,\sigma/2}\leq \frac 4\rho\Omega\eps^2\left( \frac{F_0}{\alpha_K}+\eps\frac{16 F^2_0}{\alpha_K\alpha_{2K}\rho}\right)^2+\eps^2\Omega \frac{16 F^2_0}{\alpha_K\alpha_{2K}\rho}\leq \frac{24\Omega F_0^2\eps^2}{\rho\alpha_K\alpha_{2K}}\ . $$ Concerning $\Delta g_0$ we have $$ \norma{\Delta g_0}_{\rho/2,\sigma/2}\leq \norma{\diff{g_0}I}\norma{\eps A_1+\eps^2 A_2 }+\norma{\diff{g_0}\psi}\eps\norma{B}\ , $$ which can be easily estimated giving the claimed result. We come to $\dot J$ and $\dot\psi-\omega_0$. It is easy to obtain the following equations $$ \eqalign{ (\uno+\eps d_JA)\dot J=\eps\media-\eps d_\psi A(\dot \psi-\omega_0 ) -\eps^2d_\psi A_2\omega_0 +\eps(f^0(I,\phi)-f^0(J,\psi))+\eps^2f^1+\eps^3f^2 \cr (\uno+\eps d_\psi B)(\dot \psi-\omega_0 )=[\omega_0(I)-\omega_0(J)]+\eps\Gm(J) +\eps\Delta g^0+\eps^2 g^1(I,\phi) -\eps d_J B\dot J\ .} $$ Solving the second with respect to $\dot \psi-\omega_0 $, substituing in the first, and using the definition of $A_2$ to estimate the last term of the so obtained inequality, one gets the wonted estimates for $\dot J$ and $\dot\psi-\omega_0 $. All the other estimates can be obtained almost in the same way. \quadratino To obtain the proof of lemma \lemmaref{nf.1} we proceed now as follows: first use the above lemma to obtain the estimate $$ \norma{R_1}_{\rho/2,\sigma/2}\leq\eps^2\frac{48\Omega F_0}{\rho\alpha_K\alpha_{2K}}\ . \autoeqno{dpsi} $$ Then, notice that the r.h.s. of \eqref{dpsi} gives also an estimate of $\dot\psi-\omega_0-\eps\Gm$. Using such an estimate it is easy to obtain the following estimate for $R$: $$ \norma{R}_{\rho/2,\sigma/2}\leq\eps^3\frac{2^{11}F_0^3}{\rho^2\alpha_K \alpha_{2K}}\ . $$ This gives Lemma \lemmaref{nf.1} with the claimed estimates. \autosez{hyp}Persistence of stable hyperbolic tori We consider here a system of the form \eqref{sys}. We will assume that $\Fm$ has an attractive zero which is suffitiently nonresonant, and we will prove that the correspnding invariant torus of the averaged system can be continued to an invariant tous of system \eqref{sys} We assume that there exists an attractive zero $J_*$ of $\Fm$, namely a zero such that all the eigenvalues of $d\Fm(J_*)$ are strictly negative We first continue it to a zero of $\Fm+\eps Z$. This requires some care since $Z$ is defined only on nonresonant sets, and moreover such function depends discontinuously on $\eps$. \lemma{h.1}{Assume that there exist $\tau,\gamma, K$ such that the zero $J_*$ of $\Fm$ belongs to $\Gamma(\tau,\gamma,K)$, then there exist $\eps_*,\ccon{h3}$ such that the following holds true: if $$ K\geq\ccon1|\ln\eps_{*}|^\tau\autoeqno{s} $$ then for any $\eps$ satisfying $$ e^{-K/\ccon1}\leq\eps<\eps_*\autoeqno{*} $$ the function $\Fm+\eps Z$ has an attractive zero $J_0(\eps)$ close to $J_*$, namely such that $$ \norma{J_0(\eps)-J_*}\leq \ccon{h3} \eps|\ln\eps|^{2\tau+1}\ ;\autoeqno{dj0} $$ moreover there exists $\lambda>0$ such that all the eigenvalues $\lambda_i=\lambda_i(\eps)$ of $d\Fm(J_0)+\eps d_JZ(J_0)$ satisfy $$ \inf_{e^{-K/\ccon1}\leq\eps<\eps_*}|\lambda_1(\eps)|\geq\lambda\ .\autoeqno{la} $$ } \proof We fix $\eps$ and consider the function $$ F(J,\mu):=\Fm(J)+\frac\mu{\eps|\ln\eps|^{2\tau+1}}Z(J,\eps)\ ,\autoeqno{F} $$ to which we can apply the implicit function theorem. So we obtain that there exists $\mu_*$ such that provided $\mu<\mu_*$ such function has a zero $J_1(\mu)$ smoothly (analitically) depending on $\mu$. Define now $\eps_*$ by $\eps_*|\ln\eps_*|^{2\tau+1}=\mu_*$. Provided condition \eqref{*} is satisfyed it is possible to choose $\eps$ so small that the fraction in \eqref{F} is one, and the function $J_1$ is defined. We then put $J_0(\eps):=J_1(\eps|\ln\eps|^{2\tau+1})$. We come to the estimates \eqref{dj0} and \eqref{la}. Remark that $\diff{J_1}\mu$ (which can be computed explicitly) is bounded uniformly on $\eps$, and therefore \eqref{dj0} is a consequence of the smoothness of $J_1(\mu)$. The eigenvalues of $d_JF(J_1(\mu),\mu)$ depend smoothly on $\mu$, and therefore they are close to the eigenvalues of $d\Fm(J_*)$, so that also condition \eqref{la} holds. \quadratino In all the rest of the paper we will denote by $\rho$ the quantity \eqref{rho}. \remark{h.2}{By \eqref{dj0} one has $$ B_{\rho/4}(J_0(\eps))\subset\Gamma_{\rho/2}(\tau\gamma,K) \ , $$ so that $B_{\rho/4}(J_0(\eps))$ is contained in the domain of definition of \eqref{sys}. } \remark{h.3}{Under the hypotheses of lemma \lemmaref{h.1} the system $$ \eqalign{ \dot J=\eps\Fm+\eps^2 Z \cr \dot \psi=\omega+\eps\Gm }\autoeqno{sys.1} $$ has an attractive torus $J_0(\eps)\times\toro^m$ close to the torus $J_*\times\toro^m$. } We are going to continue such a normally hyprebolic torus to a torus of system \eqref{sys}. We will follow almost literaly the proof of persistence of normally hyperbolic manifolds given by Fenichel, just adding quantitative estimates on the size of the threshold. Such estimates are needed in our case (in which the Lyapunov exponents of the invariant manifold and the size of the perturbation are related). {\it In all the rest of the paper we will assume that $\eps$ is small enough and that $J_*$ is suffitiently nonresonant}. By this we mean that $J_*\in\Gamma(\tau,\gamma,K)$ with $K>\ccon1|\ln\eps|^\tau$. We give now a suitable form to system \eqref{sys}. First we make an $\eps$ dependent translation in $J$ space so that the attractive zero $J_0(\eps)$ coincides with 0, namely so that $\Fm(0)+\eps Z(0,\eps)=0$. Then we write our system in the form $$ \eqalign{ \dot J&=A_{\eps}J+\N(J,\eps)+\eps^{2+p}\R(J,\psi,\eps) \cr \dot\psi&=\omega(J,\eps)+\eps^{1+p}\R_1\ , }\autoeqno{h.s} $$ where $p$ is any real number satisfying $0From this, using \eqref{phi} one has $$ \norma{\psi-\psi'}\leq T[4\omega+2\eps^{1+p}\bar\R_1]\ . $$ Inserting in \eqref{34} we get the thesis. \quadratino So, $G$ is a contraction and therefore has a unique fixed point which is the invariant manifold. Moreover, by construction the fixed point is a section $\sigma:\toro^m\to\U_{\rho_1}$, and therefore the perturbed invariant torus is $\O(\rho_1)$ close to the the torus $J=0$, which in turn is $\O(\eps\left|\ln\eps\right|^{2\tau+1})$ close to the invariant torus $J_*\times\toro^m$ of the averaged system. Finally the change of variables \eqref{c.c} is $\eps\left|\ln\eps\right|^{\tau}$ close to the identity, and therefore the invariant torus satisfyies \eqref{dis}. \autosez{hypg}Extension to the general hyperbolic case \def\J{{\cal J}} In the discussion above, we have assumed that the unperturbed invariant torus is not only hyperbolic, but also attractive. We give now a sketch of the proof of persistence of the normally hyperbolic manifold in the general case, in which the unperturbed invariant manifold is not attractive (or repulsive). Following Fenichel and many other authors we first look at the (local) unstable manifold of the torus $J=0$, which is a conctractive manifold, and so we apply the theory of the previous section to prove its persistence. Then we look at the stable manifold, inverting time this becomes an attractive manifold, so we obtain in the same way its persistence. The normally hyperbolic torus is obtained as the intersection of the two manifolds. To obtain quantitative estimates in this case one needs some care. To explain this we introduce some notations. First we consider again the operator $A$, and we assume that it is diagonal. Let $-\lambda_1,...,-\lambda_k$ be the negative eigenvalues of $A$, and $\mu_1,...,\mu_l$ be the positive eigenvalues. Finally denote by $\J$ the coordinates along the attractive directions, and by $\I$ those in the expanding directions, i.e. $$ A \pmatrix{ \J \cr \I \cr} \ = \ \pmatrix{ A_1 & 0 \cr 0 & A_2 \cr} \pmatrix{ \J \cr \I \cr} $$ with $$ \left( A_1 \J \right)_i = - \lambda_i \J_i \quad , \quad \left( A_2 \I \right)_i = \mu_i \I_i \ . $$ We then write the original system as $$ \cases{ \dot \J=\eps A_1\J+ \eps\H_{\J}(\J,\I)+\eps^{2+p}\R_{\J}(\J,\I,\psi) & \cr \dot \I=\eps A_2\I+ \eps \H_{\I} (\J,\I)+\eps^{2+p}\R_{\I}(\J,\I,\psi) & \cr \dot \psi=\omega(\I,\J) +\eps^{1+p}\R_1(\I,\J,\psi)\ & \cr} $$ where the terms $\H_{\J}$ and $\H_{\I}$ are the different components of $\F - A$. Since $\Z$ and $\Fm$ are $C^2$, it follows that $\F$ is also $C^2$, and therefore $\H_{\I}$ and $\H_{\J}$ are $\O(\rho_1)$, which implies that $\eps \H_{\J}$ is of the same order as the remainder $\eps^{2+p}\R$; thus the theory of the previous section applies with minor changes. In particular, one should again obtain the appropriate estimates, in a way adapted to the new form; we mention that, in particular, the estimates corresponding to lemma \lemmaref{l.1} take a slightly worse form. \bye re The quntity $J$ satisfies the equation $$ \dot J=\eps \Fm(J)+\eps^2 Z+R \ ,\autoeqno{eqj} $$ where $Z$ is the solution to the homological equation here $\Psi$ is given by $$ \Psi:=f^1+d_\psi f^0B+d_Jf^0A_1-d_JA_1\Fm\ .\autoeqno{psi} $$ The remainder $R$ is given by } We write it as follows $$ \dot J=\eps \Fm(J)+\eps^2 \left[ \Psi(J,\psi)-d_\psi A_2(J,\psi) \omega (J)\right]+o(\eps^2)\ , $$ where $$ \Psi:=f^1+d_\psi f^0B+d_Jf^0A_1-d_JA_1\Fm\ .\autoeqno{psi} $$ Denoting by $\Psi_k(J)$ the Fourier components of $\Psi$ (since $N\geq 2K$ their order does not exceed $N$) we write so that denoting $\Psi_0(J)$ as $Z(J)$, equations \eqref{1} are reduced to the form where $$ \eqalign{ R:= & (\uno+\eps d_JA)^{-1} \left[\eps^3 f^2+\eps \Delta f^0+\eps^2\Delta f^1 -\eps d_\psi A_1 \R_1 \right] \cr + & \left[ \left(\uno+\eps d_JA \right)^{-1}-\uno\right]\eps^2Z +\left[ \left(\uno+\eps d_JA \right)^{-1}-\uno+\eps d_JA_1 \right] \eps \Fm\ , }\autoeqno{r} $$ Here we introduced the notations Eventualy we will define the quantities $\R$ and $\R_1$ (cf.\eqref{nf}) by $\eps^{1+p}\R=R$ and $\eps^{1+p}\R_1=R_1$, and it will turn out that such quantities are bounded uniformly in $\eps$. \bigskip \noindent {\tt 2) Quantitative estimates} %{[(#@~`$%^&*|%!)]} We begin by determininig the domain. To this end fix $J_*\in\G$ such that $\omega_*:=\omega (J_*)$ is diophantine. It is clear that, provided $\rho$ is small enough, $\omega(J)\cdot k$ is bounded from below for $|k|\leq N$ and $J\in\U_\rho$. In what follows we will need an estimate of the $C^3(\U_{\rho})$ norm of $(\omega(J)\cdot k)^{-1}$, so the following lemma is useful: \lemma{dio}{Fix $M>1$, and let $\Omega$ be defined as in \eqref{de}; then, provided $$ |k|\leq M\ ,\quad \rho\leq\frac{\gamma}{2\Omega M^{\tau+1}}\ ,\autoeqno{Nb} $$ one has $$ \norma{\frac1{\omega (.)\cdot k}}_{C^3(\U_\rho)}\leq\frac1{\alpha_M}\ , $$ with $$ \alpha_M=\frac{\tilde\gamma}{M^\nu}\ ,\quad \nu=4\tau+3\ $$ and $ \tilde\gamma>0$ independent of $M$. } \proof First notice that for $h\in C^3(\Re^n,\Re)$, one has $$ \norma{\frac1{h}}_{C^3}\leq C\frac{\left(\norma{dh}_{C^2}\right)^3}{\inf|h|^4}\ , $$ for some positive $C$. We fix $k\in\Ra^n\meno \{0\}$ with $|k|\leq M$, and take $h=\omega\cdot k$. Clearly we have $$ \norma{d_J(\omega\cdot k)}_{C^2}\leq \Omega M\ . $$ We now estimate $\omega\cdot k$ from below. By Lagrange's theorem there exists a $J'$ such that $$ k\cdot\omega (J) = k\cdot\omega^* + k\cdot \dif_{J}\omega(J')\bigl(J-J^*\bigr)\ , $$ and one has $$ |k\cdot \dif_{J}\omega(J')\bigl(J-J^*\bigr)| \leq \Omega M\rho $$ for every $J,J'\in B_{\rho}(J^*)$. Thus the statement follows in view of the trivial inequality $$ |k\cdot\omega(J)| \geq |k\cdot\omega^*| - |k\cdot \dif_{J}\omega(J')\bigl(J-J^*\bigr)|\ . $$ \quadratino We fix $\rho$, and introduce a family (independent of $\ell$) of norms for functions $h:\dom\times\toro^m\to\Re^\ell$, given by $$ \norma{h}_{j,r}:=\sum_{k}\norma{h_k}_{C^j(\dom)}|k|^{r}\ . $$ Notice that such a norm dominates the $C^{\min\{j,r\}} (\dom\times\toro)$ norm. It is easy to check that, given two functions $h$, $\tilde h$ one has $$ \norma{h\tilde h}_{j,r}\leq\norma{h}_{j,r}\norma{\tilde h}_{j,r} $$ For the sake of brevity we will write $F^1:=F/K^r$, $F^2:=F/N^r$, $G^1:=G/K^r$, ($F^0:=F$), so that $$ F^i:=\norma{f^i}_{3,3}\ , \ G^i:=\norma{g^i}_{3,3}\ .\autoeqno{norme} $$ One has $$ \norma{A_1}_{3,3}\leq\frac{F^0}{\alpha_K}\ , \ \norma{B}_{2,3}\leq\frac 1{\alpha_{K}}\left(G^0+\frac{\Omega F^0}{\alpha_K}\right)\ . $$ In order to get an estimate of the norms of $A_2$ and of $Z$ we need an estimate of the norm of $\Psi$. Using the definition \eqref{psi}, we immediately get $$ \norma{\Psi}_{2,2}\leq F^1+\frac{F^0}{\alpha_{K}} \left(G^0+\frac{\Omega F^0}{\alpha_K}\right)+\frac{F^0}{\alpha_K}(G^0+2F^0)\ . $$ In order to simplify this expression we remark that in what follows we will link $K$ and $N$ to $\eps$ in such a way that $N,K\to \infty$ as $\eps\to0$. So we have $$ \norma{\Psi}_{2,2}\leq F^1+\frac{(F^0)^2\Omega} {\alpha_K^2}+{\rm h.o.t.} $$ (where ``h.o.t.'' denotes higher order terms), from which, $$ \norma{A_2}_{2,2}\leq\frac1{\alpha_N} \left(F^1+\frac{(F^0)^2\Omega} {\alpha_K^2}\right)+{\rm h.o.t.}\ .\autoeqno{A2S} $$ In what follows we will retain only the leading order terms, and the notation ``+ h.o.t.'' will be omitted; at the end of the procedure we will add a factor 2 to take into account higher order terms. Hence, we will write $$ \norma{A_1+\epsilon A^{2}}_{2,2}\leq \frac{F^0}{\alpha_K}\ . $$ We begin now to estimate the $C^1$ norm of the remainders $R$ and $R_1$. Consider $\Delta\omega $: by Lagrange theorem, we have $$ \eqalign{ & \norma{\omega (I) -\omega (J)-\eps d_J\omega (J)[A_1+\eps A_2 ] }_{C^1 ( \U_{\rho/2}\times\toro^m ) } \leq \cr & \leq \sup_{J\in\dom} \norma{d_J \omega (J) }_{C^2} \norma{\eps A}_{C^1}^2 \leq \eps^2 \Omega \frac{(F^0)^2}{\alpha_K^2} \ , \cr} $$ and it follows that $$ \norma{\Delta\omega }_{C^1(\U_{\rho/2}\times \toro^m)} \leq \frac{\eps^2\Omega}{\alpha_N} \left(F^1 +\frac{\Omega (F^0)^2}{\alpha_K^2}\right)\ . $$ Analogously, we have $$ \eqalign{ \norma{\Delta g_0}_{C^1(\U_{\rho/2}\times \toro^m)} \leq \ & \eps \frac{G^0\Omega F^0}{\alpha_K^2} \cr \norma{\Delta f^1}_{C^1(\U_{\rho/2}\times \toro^m)} \leq \ & \eps\frac{F^1\Omega F^0}{\alpha_K^2} \cr \norma{\Delta f^0}_{C^1(\U_{\rho/2}\times \toro^m)} \leq \ & \eps^2F^0\left[\frac{(F^0)^2\Omega^2}{\alpha_K^4}+\frac{(F^0)^2 \Omega}{\alpha_K^2\alpha_N}+\frac{F^1}{\alpha_N} \right] \ . \cr}\autoeqno{deltas} $$ Using also $$ \normac{\left[(\uno+\eps d_JA(J,\psi))^{-1}-\uno\right] }\leq \eps^2\frac{\Omega F^0G^0}{\alpha_K}\ , $$ we obtain $$ \normac{R_1}\leq 2\eps^2\left[ \frac{F^1\Omega}{\alpha_N}+G^1+\frac{\Omega^2 (F^0)^2}{\alpha_N\alpha_K\alpha_{K_1}}\right] \ .\autoeqno{r1s} $$ To get an estimate of the remainder $R$ we need to estimate the various terms of \eqref{r}. Collecting our results, we get $$ \normac{R}\leq 2\eps^3\left[ F^2+\frac{F^0\Omega}{\alpha_K\alpha_N}\left( \frac{(F^0)^2}{\alpha_K^2} +F^1 \right)+\frac{G^1F^0} {\alpha_K} \right]\ .\autoeqno{R.1} $$ Now, we would like to choose $K$ and $N$ in order to minimize the two remainders. We consider only $R$. One can see that the last term of \eqref{R.1} is of higher order, so we impose that the other ones are of the same order. Namely we impose $$ \frac1{N^r\eps^2}=(K^3N)^\nu\ ,\quad K^{2\nu}=\frac1{K^r\eps}\ , $$ which gives $$ N=\eps\string^\left(-\frac{2r+\nu}{(r+\nu)(r+2\nu)}\right)\ ,\quad K=\eps\string^\left(-\frac1{2\nu+r}\right)\ ,\autoeqno{Na} $$ and $$ \norma{R}_{C^1(\U_{\rho/2}\times\toro^m)} \leq C\eps^{2+p}\ ,\quad \norma{R_1}_{C^1(\U_{\rho/2}\times\toro^m)} \leq C\eps^{1+p_2} $$ with $$ p=\frac{r^2-2r\nu-2\nu^2}{(r+\nu)(r+2\nu)}\ ,\quad p_2=\frac{r^2-r\nu-\nu^2}{(r+\nu)(r+2\nu)}\ . $$ Since $p_2>p$ one has also $\norma {R_1}\leq C\eps^{1+p}$. The estimate on $\rho$ is obtained substituing the expression of $N$ cf.\eqref{Na} into \eqref{Nb}. This concludes the proof. \quadratino \autosez{hyp}Persistence of the normally hyperbolic torus We consider here a system of the form \eqref{nf}. To begin with, we assume that: {\tt (i)} $\Fm(J_*)=0$; {\tt (ii)} the real part of all the eigenvalues $d_J\Fm(0)$ is negative; and that {\tt (iii)} $\omega_*:= \omega (J_*)$ is a diophantine number, so that the domain of definition of system \eqref{nf} is an open neighbourhood of the torus $\Fm(J_*)$, of size $\O(\eps^{p_0})$. In what follows we will follow almost literaly the proof of persistence of normally hyperbolic manifolds given by Fenichel, just adding quantitative estimates on the size of the threshold. Such estimates are needed in our case (in which the Lyapunov exponents of the invariant manifold and the size of the perturbation are related). We fix $\eps$ (small) and notice that, by the implicit function theorem and the estimate \eqref{Z}, $\F$ has a zero which is $\O(\eps^{r/(r+2\nu)})$ close to $J_*$. We will make a translation in the $J$ space so that such a zero coincides with $J=0$. We remark that, by our assumption {\tt (iii)} above, the domain of definition of system \eqref{nf} is of order $\eps^{p_0}$, with $p_0$ which by \eqref{P} is smaller than $r/(r+2\nu)$, and therefore $J=0$ is conatined in such a domain toghether with a neighbourhood of size of order $\O(\eps^{p_0})$. We introduce now some notations. We will write $A:=d\F(0)$, denote by $\lambda_1,...,\lambda_n$ the eigenvalues of $A$, and write $\lambda=\inf_{\eps\in[0,\eta]} \min \left\{ -{\rm Re} (\lambda_1) , ... , -{\rm Re} (\lambda_n ) \right\}$; notice that $\lambda>0$. We denote by $F^t(J,\psi)$ the solution of \eqref{nf} starting at the point $(J,\psi)$. With an obvious notation we write $\left( J(t) , \psi(t) \right) = F^t (J,\psi) $. We will also denote $$ H_t(J,\psi):=J(t)\ ,\quad \Phi_t(J,\psi)=\psi(t)\ ,\autoeqno{ev} $$ and by $\sr, \sruno$ the constants bounding the $C^1$ norm of $\R$ and $\R_1$ respectively. We will often use the formula of variation of constants, which gives $$ \eqalign{ H_t (J,\psi) = e^{\eps At}J+ & \eps \int_0^t e^{\eps A(t-s)} \left( \left[ \F (H_s ( J , \psi )) - A H_s (J , \psi ) \right] \right. \cr & \left. + \eps^{1+p}\R(\hs,\phis) \right) ds \ , \cr}\autoeqno{h} $$ and the formula $$ \Phi_t(J,\psi)=\psi+ \int_0^t\left[\omega(\hs)+\eps^{1+p}\R_1(\hs,\phis)\right]ds \ .\autoeqno{phi} $$ In order to avoid repetitions, we stress that {\it all what follows holds provided $\eps$ is small enough, so from now on such a statement will be understood}, and not repeated at each time. We will also fix from now on $$ T:=\eps^{-7p/8}\ .\autoeqno{T.1} $$ Notice that this establishes a relation between the time $T$ and the smallest Lyapunov exponent $\eps \lambda$ of the unperturbed invariant torus $J=0$, in particular one has $\eps\lambda T\ll 1$. We have the following \lemma{inv}{Assume $p_0From \eqref{xi} we have $$ \xi=\psi+\S^{\sigma}(\psi)=\psi'+\S^{\sigma'}(\psi')\ , $$ from which $$ \psi-\psi'=-\S^{\sigma}(\psi)+\S^{\sigma}(\psi')-\S^{\sigma}(\psi')+ \S^{\sigma'}(\psi')\ , $$ and from this $$ \norma{\psi+\S^{\sigma}(\psi)-\psi'-\S^{\sigma}(\psi')}=\norma{ \S^{\sigma}(\psi')-\S^{\sigma'}(\psi')}\ . $$ The first term is estimated from below using \eqref{33}, while the second term is estimated using the explicit form of $\S$. This gives $$ \norma{\psi-\psi'}\leq(1+\Omega dT)\left[ \Omega T +\epsilon^{2+p}T \left(\rs+\rs T\Omega\right)\right]\norma{\sigma-\sigma'}\ . $$ Inserting this in \eqref{34} we get the thesis. \quadratino So, $G$ is a contraction and therefore has a unique fixed point which is the invariant manifold. Moreover, by construction the fixed point is a section $\sigma:\toro^m\to\U_{\rho_1}$, and therefore the perturbed invariant torus is $\O(\rho_1)$ close to the the torus $J=0$, which in turn is $\O(\eps^{p_1})$ close to the invariant torus of the averaged system. \autosez{hypg}Extension to the general hyperbolic case \def\J{{\cal J}} In the discussion above, we have assumed that the unperturbed invariant torus is not only hyperbolic, but also attractive. We give now a sketch of the proof of persistence of the normally hyperbolic manifold in the general case, in which the unperturbed invariant manifold is not attractive (or repulsive). Following Fenichel and many other authors we first look at the (local) unstable manifold of the torus $J=0$, which is a conctractive manifold, and so we apply the theory of the previous section to prove its persistence. Then we look at the stable manifold, inverting time this becomes an attractive manifold, so we obtain in the same way its persistence. The normally hyperbolic torus is obtained as the intersection of the two manifolds. To obtain quantitative estimates in this case one needs some care. To explain this we introduce some notations. First we consider again the operator $A$, and we assume that it is diagonal. Let $-\lambda_1,...,-\lambda_k$ be the negative eigenvalues of $A$, and $\mu_1,...,\mu_l$ be the positive eigenvalues. Finally denote by $\J$ the coordinates along the attractive directions, and by $\I$ those in the expanding directions, i.e. $$ A \pmatrix{ \J \cr \I \cr} \ = \ \pmatrix{ A_1 & 0 \cr 0 & A_2 \cr} \pmatrix{ \J \cr \I \cr} $$ with $$ \left( A_1 \J \right)_i = - \lambda_i \J_i \quad , \quad \left( A_2 \I \right)_i = \mu_i \I_i \ . $$ We then write the original system as $$ \cases{ \dot \J=\eps A_1\J+ \eps\H_{\J}(\J,\I)+\eps^{2+p}\R_{\J}(\J,\I,\psi) & \cr \dot \I=\eps A_2\I+ \eps \H_{\I} (\J,\I)+\eps^{2+p}\R_{\I}(\J,\I,\psi) & \cr \dot \psi=\omega(\I,\J) +\eps^{1+p}\R_1(\I,\J,\psi)\ & \cr} $$ where the terms $\H_{\J}$ and $\H_{\I}$ are the different components of $\F - A$. Since $\Z$ and $\Fm$ are $C^2$, it follows that $\F$ is also $C^2$, and therefore $\H_{\I}$ and $\H_{\J}$ are $\O(\rho_1)$, which implies that $\eps \H_{\J}$ is of the same order as the remainder $\eps^{2+p}\R$; thus the theory of the previous section applies with minor changes. In particular, one should again obtain the appropriate estimates, in a way adapted to the new form; we mention that, in particular, the estimates corresponding to lemma \lemmaref{l.1} take a slightly worse form. \bye