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\newbox\paperbox \newbox\yrbox \newbox\jourbox \newbox\pagesbox \newbox\volbox \newbox\preprintbox \newbox\toappearbox \newbox\bookbox \newbox\bybookbox \newbox\publisherbox \newbox\inprintbox \def\refclear{ \setbox\nobox=\null \gdef\isno{F} \setbox\bybox=\null \gdef\isby{F} \setbox\paperbox=\null \gdef\ispaper{F} \setbox\yrbox=\null \gdef\isyr{F} \setbox\jourbox=\null \gdef\isjour{F} \setbox\pagesbox=\null \gdef\ispages{F} \setbox\volbox=\null \gdef\isvol{F} \setbox\preprintbox=\null \gdef\ispreprint{F} \setbox\toappearbox=\null \gdef\istoappear{F} \setbox\inprintbox=\null \gdef\isinprint{F} \setbox\bookbox=\null \gdef\isbook{F} \gdef\isinbook{F} \setbox\bybookbox=\null \gdef\isbybook{F} \setbox\publisherbox=\null \gdef\ispublisher{F} } \def\ref{\refclear\bgroup} \def\no {\egroup\gdef\isno{T}\setbox\nobox=\hbox\bgroup} \def\by {\egroup\gdef\isby{T}\setbox\bybox=\hbox\bgroup} \def\paper{\egroup\gdef\ispaper{T}\setbox\paperbox=\hbox\bgroup} \def\yr{\egroup\gdef\isyr{T}\setbox\yrbox=\hbox\bgroup} \def\jour{\egroup\gdef\isjour{T}\setbox\jourbox=\hbox\bgroup} \def\pages{\egroup\gdef\ispages{T}\setbox\pagesbox=\hbox\bgroup} \def\vol{\egroup\gdef\isvol{T}\setbox\volbox=\hbox\bgroup\bf} \def\preprint{\egroup\gdef \ispreprint{T}\setbox\preprintbox=\hbox\bgroup} \def\toappear{\egroup\gdef \istoappear{T}\setbox\toappearbox=\hbox\bgroup} \def\inprint{\egroup\gdef \isinprint{T}\setbox\inprintbox=\hbox\bgroup} \def\book{\egroup\gdef\isbook{T}\setbox\bookbox=\hbox\bgroup\em} \def\publisher{\egroup\gdef \ispublisher{T}\setbox\publisherbox=\hbox\bgroup} \def\inbook{\egroup\gdef\isinbook{T}\setbox\bookbox=\hbox\bgroup\em} \def\bybook{\egroup\gdef\isbybook{T}\setbox\bybookbox=\hbox\bgroup} \def\endref{\egroup \sfcode`.=1000 \if T\isno \item{[\unhbox\nobox\unskip]} \else \noindent \fi \if T\isby \unhbox\bybox\unskip: \fi \if T\ispaper \unhbox\paperbox\unskip\period \fi \if T\isbook {\it\unhbox\bookbox\unskip}\if T\ispublisher, \else. \fi\fi \if T\isinbook In {\it\unhbox\bookbox\unskip}\if T\isbybook, \else\period \fi\fi \if T\isbybook (\unhbox\bybookbox\unskip)\period \fi \if T\ispublisher \unhbox\publisherbox\unskip \if T\isjour, \else\if T\isyr \ \else\period \fi\fi\fi \if T\ispreprint Pre\-print\period \fi \if T\isjour \unhbox\jourbox\unskip\ \fi \if T\istoappear (to appear)\period \fi \if T\isvol \unhbox\volbox\unskip\if T\ispages, \else\ \fi\fi \if T\ispages \unhbox\pagesbox\unskip\ \fi \if T\isyr (\unhbox\yrbox\unskip)\period \fi \if T\isinprint (in print)\period \fi \filbreak } %================================================================= % END BIBLIOGRAPHY %================================================================= %================================================================= % MACROS.15 %================================================================= %%%%%%%%%%%%%%% % Divers % %%%%%%%%%%%%%%% \def\\{\backslash} \def\ssmallskip{\vskip 0pt} %%%%%%%%%%%% % Math % %%%%%%%%%%%% 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\def\TT{{\cal T}} \def\UU{{\cal U}} \def\VV{{\cal V}} \def\WW{{\cal W}} \def\XX{{\cal X}} \def\YY{{\cal Y}} \def\ZZ{{\cal Z}} \def\sK{{\eusm K}} \def\sB{{\eusm B}} \def\sO{{\eusm O}} \def\rp{{\eurm p}} \def\rq{{\eurm q}} \def\restr#1{|_{\textstyle#1}} \def\supp{\mathop{\rm supp}\nolimits} \def\Ker{\mathop{\rm Ker}\nolimits} \def\log{\mathop{\rm log}\nolimits} \def\Im{\mathop{\rm Im}\nolimits} \def\sump{\mathop{\sum{\vphantom{\sum}}^\prime}} \def\imun{{\rm i}} %================================================================= % TITLES.3 \count5=0 %section \count6=1 %equation \count7=1 %claim \count8=1 %subsection %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\title#1\par{\centerline{\huge #1}} \def\abstract#1\par{\noindent{\sub Abstract.} #1 \par} \def\proof{\medskip\noindent{\bf Proof.\ }} \def\preuve{\medskip\noindent{\bf Preuve.\ }} \def\qed{\hfill\vskip 1pt \line{\hfill\vrule height 1.8ex width 2ex depth +.2ex \ \ \ \ \ \ }\bigskip} \def\remark{\medskip\noindent{\bf Remark.\ }} \def\remarque{\medskip\noindent{\bf Remarque.\ }} \def\notation{\medskip\noindent{\bf Notations.\ }} \def\renvoi{\medskip\noindent{\bf Programme Fortran.\ }} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Equations label % \def\equs(#1){(\csname e#1\endcsname)} \def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \def\equation(#1){\ifx\draft\undefined \eqno(\tag(#1)) \else \eqno{{\rm #1}(\tag(#1))}\fi} \def\equ(#1){\ifundefined{e#1}$\spadesuit$#1\else (\csname e#1\endcsname)\fi} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Same as \equation, without \eqno: to be used in \eqalignno and % \displaylines with a \qquad in front. \def\equat(#1){\ifx\draft\undefined \hfil(\tag(#1)) \else \hfil{{\rm #1}(\tag(#1))}\fi} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Same as above with a 'A' added in the first place (Appendix) \def\aequation(#1){\ifx\draft\undefined \eqno(\atag(#1)) \else \eqno{{\rm #1}(\atag(#1))}\fi} \def\aequat(#1){\ifx\draft\undefined \hfil(\atag(#1)) \else \hfil{{\rm #1}(\atag(#1))}\fi} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Add a prime to e#1 (already defined) \def\equationp(#1){\ifx\draft\undefined \eqno(\csname e#1\endcsname') \else \eqno{{\rm #1}(\csname e#1\endcsname')}\fi} \def\equp(#1){\ifundefined{e#1}$\spadesuit$#1\else (\csname e#1\endcsname')\fi} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\newdef #1,#2,#3 {\ifundefined{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3}\else \write16{!!!!!doubly defined #1,#2}\fi} \def\tag(#1){\number\count5.\number\count6 \newdef e,#1,\number\count5.\number\count6 \global\advance\count6 by 1} \def\atag(#1){{\rm A.}\number\count6 \newdef e,#1,A.\number\count6 \global\advance\count6 by 1} \def\claim #1(#2) #3\par{ \vskip.1in\medbreak\noindent {\ifx\draft\undefined\else\llap{(#2)}\fi \bf #1\ \number\count5.\number\count7.\ }{\sl #3}\par \newdef c,#2,#1~{\number\count5.\number\count7} \global\advance\count7 by 1 \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} \def\claimrm #1(#2) #3\par{ \vskip.1in\medbreak\noindent {\ifx\draft\undefined\else\llap{(#2)}\fi \bf #1\ \number\count5.\number\count7.\ }{\rm #3}\par \newdef c,#2,#1~{\number\count5.\number\count7} \global\advance\count7 by 1 \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} \def\clm(#1){\ifundefined{c#1}$\spadesuit$#1\else \csname c#1\endcsname\fi} \def\section#1\par{\vskip0pt plus.3\vsize\penalty-75 \vskip0pt plus -.3\vsize\bigskip\bigskip \global\advance\count5 by 1 \message{#1}\leftline {\xbold \number\count5.\ #1} \count6=1 \count7=1 \count8=1 \nobreak\smallskip\noindent} \def\sectionnonr#1\par{\vskip0pt plus.3\vsize\penalty-75 \vskip0pt plus -.3\vsize\bigskip\bigskip \global\advance\count5 by 1 \message{#1}\leftline {\xbold #1} \count6=1 \count7=1 \count8=1 \nobreak\smallskip\noindent} \def\sec(#1){\ifundefined{s#1}$\spadesuit$#1 \else Section \csname s#1\endcsname\fi} \def\subsection#1\par{\vskip0pt plus.2\vsize\penalty-75 \vskip0pt plus -.2\vsize\bigskip\bigskip \message{#1}\leftline{\sub \number\count5.\number\count8.\ #1} \global\advance\count8 by 1 \nobreak\smallskip\noindent} %=================================================================== \footline{ \ifx\draftfoot\undefined {\hss \tencm\folio\hss} \else \hbox to 0pt{\sevenit Draft version -- \the\day.\the\month.\the\year\hss} {\hss \tencm\folio\hss} \fi} %=================================================================== \newdimen\normalskip \newdimen\normalindent \newdimen\greatbaselineskip \normalskip=5pt \normalindent=20pt \greatbaselineskip=5.25mm \normalbaselineskip=\baselineskip \parskip=\normalskip \pageno=1 \hsize=16.0truecm\vsize=22.5truecm \null\vskip 1cm \title Renormalization Group and the Melnikov \title Problem for PDE's \vskip 1cm \centerline {\bf Jean Bricmont$^{1,}$\footnote {$^\dagger$}{Partially supported by ESF/PRODYN.}, Antti Kupiainen$^{2,}$\footnote {$^\ddagger$}{Partially supported by EC grant FMRX-CT98-0175.}, Alain Schenkel$^{2}$ } \vskip 0.5cm {\noindent\eightrm\baselineskip=10pt \raise 1mm\hbox{$\scriptstyle 1$}$\,$UCL, FYMA, 2 chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium\hfill\break \raise 1mm\hbox{$\scriptstyle 2$}$\,$Department of Mathematics, Helsinki University, P.O.~Box~4, 00014 Helsinki, Finland\hfill\break \par} \vskip 1.2cm \abstract We give a new proof of persistence of quasi-periodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of small-amplitude, quasi-periodic solutions of nonlinear wave equations with periodic boundary conditions. \section Introduction In this paper, we address the persistence problem of quasi-periodic, low dimensional, elliptic tori in infinite dimensional systems. A typical example that we will consider is the nonlinear wave equation (NLW) on a bounded interval, $$ \partial_t u=\partial_x^2 u-Vu+f(u), \equation(NLWintro) $$ with Dirichlet or periodic boundary conditions and $f(u)=\OO(u^3)$. The first results concerning the existence of quasi-periodic solutions of \equ(NLWintro) were obtained independently by Kuksin, P\"oschel and Wayne, [K, P1, W]. They extended to infinite dimensional Hamiltonian systems Eliasson's proof, [E], of the so called Melnikov problem, {i.e.}, the persistence of elliptic invariant tori of dimension lower than the number of degrees of freedom. Based on the Kolmogorov-Arnold-Moser (KAM) approach, these results were restricted, however, to Dirichlet or Neumann boundary conditions and to specific classes of potential $V$ excluding, in particular, the case $V=Const.$ In [P2], P\"oschel covered the case of constant potentials by exploiting the existence of a Birkhoff normal form for the Hamiltonian of \equ(NLWintro). The normal form allowed him to control the torus frequencies via amplitude-frequency modulation, and therefore to dispense with outer parameters provided by an adjustable potential $V(x)$. This approach was applied in [KP] to the persistence of quasi-periodic solutions for the nonlinear Schr\"odinger equation (NLS) subject to Dirichlet (or Neumann) boundary conditions. The case of periodic boundary conditions is more delicate due to the fact that the eigenvalues of the Sturm-Liouville operator $L=-d^2/dx^2+V$ are degenerate. This leads to resonances between pairs of frequencies corresponding to motion in directions normal to the torus (the so called normal frequencies). These additional resonances prevents one from controlling quadratic terms in the Hamiltonian of the system and do not seem to be addressable by KAM techniques. (This difficulty also appears in finite-dimensional Melnikov situations.) Developing new techniques based on the Lyapunov-Schmidt method, Craig and Wayne proved in [CW] persistence of periodic solutions of the NLW with periodic boundary conditions. Later, their approach was significantly improved by Bourgain in [B1-2] who constructed quasi-periodic solutions of the NLW and NLS with periodic boundary conditions. Most notably, it is shown in [B2] that solutions of this type can be constructed, in particular, for the NLS on two-dimensional domains. The usual Melnikov nonresonance condition reads, with $\omega\in\real^d$ and $\mu\in\real^n$ denoting the torus and, respectively, the normal frequencies ($n$ is possibly infinite), $$ \langle k,\omega\rangle+\langle l,\mu\rangle\not=0, \quad k\in\integer^d,\,l\in\integer^n \ {\rm with}\ |k|+|l|\not=0,\,|l|\leq2. \equation(Melcond) $$ In Bourgain's approach and at the price of a considerable technical effort, condition \equ(Melcond) is reduced to $$ \langle k,\omega\rangle+\mu_s\not=0, \quad k\in\integer^d,\ s=1,\dots,n, $$ i.e., all nonresonance conditions on pairs of normal frequencies are absent. More recently, Chierchia and You, see [Y,CY], showed that persistence of quasi-periodic solutions of the NLW with periodic boundary conditions is tractable by KAM techniques. Their nonresonance condition, $$ \langle k,\omega\rangle+\langle l,\mu\rangle\not=0, \quad k\in\integer^d\setminus\{0\},\,l\in\integer^n \ {\rm with}\ |l|\leq2, \equation(WMelcond) $$ is weaker than \equ(Melcond), but stronger than Bourgain's condition. However, for reasons related to the availability of a normal form mentioned above, they are unable to cover the case of constant potential $V$. In the present paper, we give a new proof of Bourgain's result for the NLW with periodic boundary conditions. To this end, we will use a renormalization group procedure recently developed in [BGK] for standard KAM problems. The nonresonance condition that we will impose is the same as Chierchia and You's condition, but our technique could in principle accommodate Bourgain's conditon. In order to describe our result further, we start by specifying the infinite dimensio\-nal Hamiltonians we will consider. For $d_k$, $k\geq1$, a sequence of strictly positive integers uniformly bounded by some $\bar d<\infty$, let $\RR^\infty$ denote the set of infinite sequences $x=(x_1,x_2,\dots)$ with $x_k\in\real^{d_k}$. For an integer $d\geq1$, let $\PP=\torus^d\times\real^d\times\RR^\infty\times\RR^\infty$ where $\torus^d$ is the torus $\real^d/(2\pi\integer^d)$. Deno\-ting the coordinates in $\PP$ by $(\phi,I,x,y)$ and endowing $\PP$ with the symplectic structure $d\phi\wedge dI+dx\wedge dy$, we consider perturbations of integrable Hamiltonians of the form $$ H(\phi,I,x,y)=\omega\cdot I+{\onehalf}I\cdot gI +\onehalf\sum_{k\geq1}\bigl(\mu_k^2|x_{k}|^2+|y_{k}|^2\bigr) +\lambda U(\phi,I,x), \equation(hamiltonian) $$ where $\mu_k\in\real$, $k\geq1$, $\omega\in\real^d$, and $g$ is a real symmetric, invertible $d\times d$ matrix. Above, $|v|^2$ for $v\in\real^m$ denotes $\sum_{i=1}^mv_i^2$. The Hamiltonian flow generated by \equ(hamiltonian) is given by the equations of motion $$ \dot I=-\lambda\partial_\phi U\,,\quad \dot\phi=\omega+gI+\lambda\partial_I U, \equation(eqmot1) $$ and $$ \ddot x_{k}=-\mu_k^2x_{k}-\lambda\partial_{x_{k}}U\, \equation(eqmot2) $$ For $\lambda=0$ and the initial condition $I^0=\phi^0=x^0=y^0=0$, the flow $\phi(t)=\omega t$, $I(t)=0$, and $x(t)=0$, is quasi-periodic and spans a $d$-dimensional torus in $\torus^d\times\real^d\times\RR^\infty\times\RR^\infty$. In order to study the case for which the perturbation is turned on, we consider a quasi-periodic solution of the form $$ (\phi(t),I(t),x(t))=(\omega t+\Phi(\omega t),J(\omega t), Z(\omega t)). $$ Then, \equ(eqmot1) and \equ(eqmot2) require that $\TT\equiv(\Phi,J,Z):\torus^d\rightarrow \real^d\times\real^d\times\RR^\infty$ satisfies the equation $$ \DD\TT(\varphi)=-\lambda\partial U(\varphi+\Phi(\varphi), J(\varphi),Z(\varphi)), \equation(eqmot3) $$ where $\partial=(\partial_\phi,\partial_I,\partial_x)$ and, setting $$ \mu\equiv{\rm diag}(\mu_1\identity_{d_1},\mu_2\identity_{d_2},\dots), \equation(defmu) $$ together with $D\equiv\omega\cdot\partial_\phi$, $$ \DD=\pmatrix{0 & D & 0\cr -D & g & 0\cr 0 & 0 & D^2+\mu^2\cr }.\equation(operator) $$ Note that if $\TT$ is a solution of equation \equ(eqmot3), then so is $\TT_\beta$ for $\beta\in\real^d$, where $$ \TT_\beta(\varphi)=\TT(\varphi-\beta)-(\beta,0,0). \equation(transinv) $$ We now state the two hypothesis under which we shall prove existence of a solution $\TT$ of equation \equ(eqmot3), first introducing the following family of Banach spaces $\RR^\infty_s, s\in\real$, $$ \RR^\infty_s=\{Z\in\RR^\infty\ |\ |Z|_s \equiv\sum_{k\geq1}k^s|Z_k|_{\real^{d_k}}<\infty\}. \equation(defRRs) $$ \noindent{\bf(H1) Asymptotics of eigenvalues.} The sequence $\{\mu_k\}_{k\geq1}$ satisfies $\mu_k>0$ and $\mu_k\not=\mu_l$ for all $k\not=l\geq1$, and there exist $\gamma\geq1$ and $c>0$ such that $$ {\mu_k}\geq ck^\gamma \quad{\rm for\ all}\quad k\geq1. \equation(asymptotics1) $$ Furthermore, if $\gamma>1$ then $$ {\mu_{k'}-\mu_{k}}\geq c(k'^\gamma-k^\gamma) \quad{\rm for\ all}\quad k'>k\geq1. \equation(asymptotics2) $$ If $\gamma=1$, then there exist constants $\xi>0$ and $c_l>0$ such that $$ \mu_{k'}-\mu_{k}=c_l(1+\OO(k^{-\xi})) \quad{\rm for\ all}\quad k'-k=l\geq1. \equation(asymptotics3) $$ \noindent{\bf(H2) Regularity of the perturbation.} The map $(\phi,I,x)\mapsto U(\phi,I,x)$ is assumed to be real analytic in $\phi\in\torus^d$ and real analytic in $I$ and $x$ in a neighborhood of the origin of $\real^d$ and $\RR_0^\infty$. In addition, we assume that there exist an $s>0$ and a $\xi>0$ such that for some $\sO_I\subset\real^d$ and $\sO_x\subset\RR^\infty_s$ neighborhoods of the origin, the gradient $\partial_xU$ is bounded as a map from $\torus^d\times\sO_I\times\sO_x$ to $\RR^\infty_{s+\xi-\gamma}$. In the sequel, we will often use the short notation $s'\equiv s+\xi-\gamma$. \claim Theorem(mainthm) Let $\{\mu_k\}$ satisfy (H1) and $U$ satisfy (H2). Then, there exists a set $\Omega^*=\Omega^*(U,\mu)\subset\real^d$ such that for $\omega\in\Omega^*$, equation \equ(eqmot3) has a unique solution (up to translations \equ(transinv)) which is real analytic in $\lambda$ and $\phi$ provided that $|\lambda|$ is small enough. Furthermore, for all bounded $\Omega\subset\real^d$ the set $\Omega^*$ of admissible frequencies satisfies ${\rm meas}(\Omega\setminus\Omega^*)\rightarrow0$ as $\lambda\rightarrow0$. The proof of \clm(mainthm) is based on an inductive procedure developed in [BGK] for standard KAM problems. This renormalization group iteration can be viewed as an iterative resummation of the Lindsedt series, as is explained in more details in [BGK], and was directly inspired by the quantum field theory analogy with KAM problems forcefully emphasized by Gallavotti {\it et al.} [G, GGM]. Melnikov type problems require to deal with the additional resonances arising from the normal frequencies $\mu_k$, and the goal of the present paper is to explain how the procedure of [BGK] can be applied in such cases. In contrast to standard KAM problems, the set $\Omega^*$ of admissible frequencies depends for Melnikov type problems on the perturbation $U$. In our approach, this dependence expresses itself by the fact that under iteration, the normal frequencies are renormalized in a $U$-dependent way and that the set $\Omega^*$ is defined according to the renormalized normal frequencies. As usual, the set $\Omega^*$ is constructed in such a way that nonresonance conditions are fulfilled in order for the inductive scheme to converge. Our scheme is technically simplified if one imposes nonresonance condition of the form \equ(WMelcond), {i.e.}, conditions involving pairs of normal frequencies. Hypothesis (H1) ensures that $\Omega^*$ has large measure under these conditions, and hypothesis (H2) ensures that the asymptotic properties of the normal frequencies stated in (H1) are preserved under renormalization. The requirement $\xi>0$ is needed both in (H1) when $\gamma=1$, and, for $\gamma>1$, in (H2) in order to cover the case of degenerate normal frequencies (more precisely the case where $d_k>1$ for infinitely many $k$). In Section~2, we show how \clm(mainthm) provides a proof of the existence of quasi-periodic solutions of the 1D NLW with periodic boundary conditions. In particular, $\gamma=1$ in (H1) and we will see that (H2) is satisfied with $\xi=1$. In contrast, one has for the 1D NLS $\gamma=2$ and $\xi=0$. Thus, the scheme presented here only applies to NLS with Dirichlet boundary conditions (namely $d_k=1$ for all $k$) or to the persistence of periodic solutions of NLS (namely $d=1$). In order to cover the other situations, one must be able to dispense with nonresonance conditions involving certain pairs of normal frequencies. The remainder of the paper is organized as follows. Section~2 is devoted to the NLW. In Section~3 we explain the renormalization group scheme that will be used to prove \clm(mainthm). Section~4 is devoted to the definition of the spaces we will consider. In Section~5, we state some crucial inductive bounds, which will be shown to hold in Section~6. Section~7 is concerned with the measure estimate of $\Omega^*$, whereas the proof of \clm(mainthm) is carried out in Section~8. Finally, we have collected in the appendix some technical and intermediary results. \section The 1D Wave Equation In this section, we show how \clm(mainthm) implies the existence of small amplitude quasi-periodic solutions of nonlinear 1D wave equations of the form $$ \partial_t^2u=\partial_x^2u-mu-f(u), \equation(nlw) $$ $t>0$$, x\in[0,2\pi]$, with periodic boundary conditions $u(0,t)=u(2\pi,t)$, $\partial_tu(0,t)=\partial_tu(2\pi,t)$. Here, $m>0$ is a real parameter and $f$ is a real analytic function of the form $f(u)=u^3+\OO(u^4)$. For $f\equiv0$, equation \equ(nlw) becomes $$ \partial_t^2u=\partial_x^2u-mu\equiv-Lu. \equation(linw) $$ The operator $L$ with periodic boundary conditions admits a complete orthonormal basis of eigenfunctions $\psi_n\in L^2([0,2\pi])$, $n\in\integer$, with corresponding eigenvalues $$ \zeta_n=n^2+m, \equation(eigen1) $$ if one sets $\psi_0=1/\sqrt{2\pi}$ and for $n\geq1$, $$ \psi_n(x)={1\over\sqrt{\pi}}\cos(nx),\quad \psi_{-n}(x)={1\over\sqrt{\pi}}\sin(nx). \equation(vbasis) $$ Every solution of the linear wave equation \equ(linw) can be written as a superposition of the basic modes $\psi_n$, namely, for $\II$ any subset of $\integer$ and $\mu_n\equiv\sqrt{\zeta_n}$, $$ u(x,t)=\sum_{n\in\II}a_n\cos(\mu_nt+\theta_n)\psi_n(x), \equation(supersol) $$ with amplitudes $a_n>0$ and initial phases $\theta_n$. Regarding existence of solutions for the nonlinear wave equation \equ(nlw), we will prove the \claim Theorem(thmnlw) Let $1\leq d<\infty$ and $\II=\{n_1,\dots,n_d\}\subset\integer$ satisfying $|n_i|\not=|n_j|$ for $i\not=j$. Then, for $\lambda>0$ small enough there is a set $\AA\subset\{a=(a_1,\dots,a_d)\,|\,00$, the gradient $\partial_\rq G$ is real analytic as a map from some neighborhood of the origin in $l_b^s$ into $l_b^s$, with $$ ||\partial_\rq G(\rq)||_s=\OO(||\rq||_s^3). \equation(boundgrad) $$ \proof We first note that $l_b^s$ is a Banach algebra with respect to convolution of sequences, with $$ ||q*p||_s\leq\sum_{i,j\in\integer}[i]^s|q_{j-i}||p_j| \leq\sup_{i,j\in\integer}\Bigl({[i]\over[j-i][j]}\Bigr)^s ||q||_s||p||_s \leq 2^s||q||_s||p||_s. \equation(Balgebra) $$ Therefore, using the analyticity of $f(u)=u^3+\OO(u^4)$, one computes that in a sufficiently small neighborhood of the origin, $$ ||f(u)||_s\leq C||\rq||_s^3. \equation(int111) $$ On the other hand, since $$ \partial_{\rq_n}G(\rq)=\int_0^{2\pi}f(u)\psi_n(x)dx, $$ the components of $\partial_\rq G(\rq)$ are the Fourier components of $f(u)$ and \equ(boundgrad) follows from the estimate \equ(int111). The regularity of $\partial_\rq G$ follows from the regularity of its components and its local boundedness, cf. [PT] p. 138. \qed We now turn to the normal form analysis of \equ(hamilt2). First, since $g(u)=\onefourth u^4+\OO(u^5)$, we find that $$ G(\rq)={1\over4}\sum_{i,j,k,l}g_{ijkl}\,\rq_i\rq_j\rq_k\rq_l +\OO(|\rq|^5), $$ where $$ g_{ijkl}=\int_{0}^{2\pi}\psi_i\psi_j\psi_k\psi_ldx. \equation(Nijkl) $$ An easy computation shows that $g_{ijkl}=0$ unless $i\pm j\pm k\pm l=0$ for at least one combination of plus and minus signs. This will play an important role later on. Next, given a finite subset of indices $\II_d=\{n_1,\dots,n_d\}\subset\integer$ with $|n_i|\not=|n_j|$ if $i\not=j$, we decompose the Hamiltonian \equ(hamilt2) as $$ H=H_d+H_\infty, $$ where $$\eqalignno{ H_d(\rq,\rp)&={1\over2}\sum_{n\in\II_d}(\mu_n^2\rq_n^2+\rp_n^2)+ {1\over4}\sum_{i,j,k,l\in\II_d}g_{ijkl}\,\rq_i\rq_j\rq_k\rq_l \equiv\Lambda_d(\rq,\rp)+G_d(\rq),\quad\qquad&\equat(Hd)\cr H_\infty(\rq,\rp)&={1\over2}\sum_{n\not\in\II_d}(\mu_n^2\rq_n^2 +\rp_n^2)+ G(\rq)-G_d(\rq)\equiv\Lambda_\infty(\rq,\rp) +G_\infty(\rq).&\equat(Hinfty)\cr }$$ Introducing the complex coordinates $z_j$, $j=1,\dots,d$, by $$ z_j={1\over\sqrt{2\mu_{n_j}}}(\mu_{n_j}\rq_{n_j} +\imun\,\rp_{n_j})\,, $$ one obtains the Hamiltonian $H_d(z,\bar z)=\sum_j\mu_{n_j}|z_j|^2+G_d(z,\bar z)$ on $\complex^d$ with symplectic structure $\imun\sum_jdz_j\wedge d\bar z_j$. For the remaining coordinates, one introduces the notation, for $k\geq1$, $$ x_k=\cases{(\rq_k,\rq_{-k})\in\real^2& if\quad $k,-k\not\in\II_d\,$,\cr \rq_{-\tilde k}\in\real & if\quad $k=|\tilde k|$ for some $\tilde k\in\II_d\,$,\cr} $$ and similarly for $\rp_n$, $n\not\in\II_d$, denoted in terms of $y_k\in\real^{d_k}$, $k\geq1$, with $d_k$ as above, namely, $d_k=2$ if both $k,-k\not\in\II_d$ and $d_k=1$ otherwise. Clearly, for $\rq,\rp\in l_b^s$ one has $x,y\in\RR^\infty_s$, where $\RR^\infty_s$ is defined in \equ(defRRs), and $H_\infty$ reads in these notations $$ H_\infty(z,\bar z,x,y)={1\over2}\sum_{k\geq1}(\mu_k^2|x_k|^2+|y_k|^2) +G_\infty(z,\bar z,x), $$ with $|G_\infty|=\OO\bigl(\sum_{l=0}^3|z|^l||x||_s^{4-l}\bigl)$. The next proposition establishes the existence of a symplectic change of coordinates that transforms the Hamiltonian $H_d$ into a Birkhoff normal form. As it will be clear from the proof, this normal form is {\sl not} available for $H=H_d+H_\infty$, since most frequencies in $H_\infty$ are degenerate. This is the main difference with [P2] in the present discussion. \claim Proposition(normalform) For each $m>0$ and each subset $\II_d\,$, $d<\infty$, satisfying $|n_i|\not=|n_j|$ when $i\not=j$, there exists a near identity, real analytic, symplectic change of coordinates $\Gamma_d$ in some neighborhood of the origin in $\complex^d$ that takes the Hamiltonian \equ(Hd) into $$ H_d\circ\Gamma_d=\Lambda_d+\bar G_d+K_d, $$ where $|K_d|=\OO(|z|^5)$ and $$ \bar G_d(z,\bar z)={1\over2}\sum_{i,j=1}^d\bar g_{ij}|z_i|^2|z_j|^2 \quad{\rm with}\quad \bar g_{ij}={3\over\pi}\,{4-\delta_{ij}\over\mu_{n_i}\mu_{n_j}}\,. \equation(Gbar) $$ Furthermore, setting $\Gamma_\infty=\Gamma_d\oplus \identity_{\RR^\infty_s\times\RR^\infty_s}$, one has $H_\infty\circ\Gamma_\infty=\Lambda_\infty+K_\infty$ with $|K_\infty|=\OO\bigl(\sum_{l=0}^3|z|^l||x||_s^{4-l}\bigr)$. \proof Modulo straightforward modifications, the proof is carried out in [P2] and we restrict ourselves here to a quick overview. The possibility to eliminate all terms in $G_d(z,\bar z)$ that are not of the form $|z_i|^2|z_j|^2$ follows from the fact that for integers $i,j,k,l\in\II_d$ satisfying $i\pm j\pm k\pm l=0$ {\rm and} $\{i,j,k,l\}\not=\{n,n,n',n'\}$ one has, as shown in [P2], $$ |\mu_i\pm\mu_j\pm\mu_k\pm\mu_l| \geq c\,{m\over(N^2+m)^{3/2}}>0\,, \equation(crucial) $$ with $c$ some absolute constant and $N=\min\{|i|,\dots,|l|\}$. To see this, it is convenient to adopt the notation $z_j=w_j$ and $\bar z_j=w_{-j}$ in which $G_d$ reads $$ G_d=\sump_{i,j,k,l}\tilde g_{ijkl}w_iw_jw_kw_l+\OO(|z|^5), \quad\tilde g_{ijkl}={g_{n_{|i|}\dots n_{|l|}}\over \sqrt{\mu_{n_{|i|}}\dots\mu_{n_{|l|}}}}\ , $$ where the prime symbol in the summation sign indicates that the sum runs over all indices $i,j,k,l\in\{1,-1,\dots,d,-d\}$ with $n_{|i|}\pm n_{|j|}\pm n_{|k|}\pm n_{|l|}=0$ for at least one combination of plus and minus signs. Defining the transformation $\Gamma_d$ as the time-$1$ map of the flow of the vector field $X_F$ given by a Hamiltonian $F(z,\bar z)$ of order four, namely, $\Gamma_d=X^t_F|_{t=1}$ and $F=\sump F_{ijkl}w_iw_jw_kw_l$, one obtains using Taylor's formula $H_d\circ\Gamma_d=\Lambda_d+G_d+\{\Lambda_d,F\}+\OO(|z|^6)$ with $$ \{\Lambda_d,F\}=-\imun\sump_{i,j,k,l} (\hat\mu_{i}+\hat\mu_{j}+\hat\mu_{k}+\hat\mu_{l}) F_{ijkl}w_iw_jw_kw_l, $$ where $\hat\mu_i\equiv\sign(i)\mu_{n_{|i|}}$. Therefore, \equ(crucial) allows to choose $F_{ijkl}$ in such a way that $$ G_d+\{\Lambda_d,F\}=\sum_{i,j=1}^d\tilde g_{iijj}|z_i|^2|z_j|^2 +\OO(|z|^5)\equiv \bar G_d+\OO(|z|^5)\,. $$ For the rest of the proof, we refer the reader to [P2]. \qed The Hamiltonian $\Lambda_d+\bar G_d$ is integrable with integrals $|z_i|^2, i=1,\dots,d$. Furthermore, the matrix $\bar g=(\bar g_{ij})_{i,j}$ is non degenerate, as can be checked from the explicit formula \equ(Gbar). Hence, introducing the standard action-angle variables $(I,\phi)\in\real^d\times\torus^d$ and linearizing $H$ around a given value for the action, namely, by setting for some $a=(a_1,\dots,a_d)\in\real^d$, $$ z_i\bar z_i=I_i+a_i^2, $$ one finally obtains $$ H_a=\omega\cdot I+\onehalf I\cdot\bar gI +\sum_{k\geq1}(\mu_k^2x_k^2+y_k^2)+U_a(I,\phi,x), \equation(Hfinal) $$ where $U_a$ is just $K_d+K_\infty$ with the variables $z_i,\bar z_i, i=1,\dots,d$, expressed in terms of $I,\phi$, and where $\omega=(\omega_1,\dots,\omega_d)$ is given by $$ \omega_i=\mu_{n_i}+\sum_{j=1}^d\bar g_{ij}a_j^2\,, $$ and covers a cone at $(\mu_{n_1},\dots,\mu_{n_d})$ as $a$ varies in a neighborhood of the origin of $\real^d$. Furthermore, $U_a$ is real analytic in $\phi\in\torus^d$ and real analytic in $I$ in a sufficiently small neighborhood $\sO_I$ of the origin of $\real^d$. As a function of $x$, $U_a$ is real analytic in a neighborhood $\sO_x\subset\RR_s^\infty$ and by \clm(regGq), its gradient $\partial_x U_a$ is bounded as a map from $\torus^d\times\sO_I\times\sO_x$ to $\RR^\infty_s$. Therefore, since hypothesis (H1) is satisfied with $\gamma=1$, $U_a$ satisfies (H2) with $\xi=1$. Finally, the small parameter $\lambda$ is given in terms of $|a|=\delta$. In the Hamilton's equations for $H_a$, rescaling $a$ by $\delta$, $x$ and $y$ by $\delta^2$, and $I$ by $\delta^4$, one obtains an Hamiltonian system given by the rescaled Hamiltonian $$\eqalign{ \tilde H_a(\phi,I,x,y)&= \delta^{-4}H_{\delta a}(\phi,\delta^4I,\delta^2 x,\delta^2 y)\cr &=\omega\cdot I+{\delta^4\over2}I\cdot\bar gI +\sum_{k\geq1}(\mu_k^2x_k^2+y_k^2)+ \tilde U_a(I,\phi,x), }$$ with $\tilde U_a$ analytic in $\delta$ and, as a function of $I$, $$ \tilde U_a=\OO(\delta)+\OO(\delta^3|I|)+\OO(\delta^5|I|^2). $$ Hence, \clm(mainthm) implies the existence of quasi-periodic solutions $I,x$ and $y$ of period $\omega$, real analytic in $\phi$ and $\lambda$. Tracing the coordinate transformations back to the original variables $\rq_n(t)$ in the expression \equ(decompuv) for $u(x,t)$ completes the proof of \clm(thmnlw) with $u(x,t)$ given by \equ(sasol). \section The Renormalization Group Scheme Equation \equ(eqmot3) consists in a system of equations for the variables $(\Phi,J)$ and $Z$ which are coupled through the perturbation $U$ only. Adopting the notation $$\eqalignno{ V(\Phi,J,Z)(\varphi)&= \lambda\pmatrix{\partial_\phi U\cr \partial_I U\cr} (\varphi+\Phi(\varphi),J(\varphi),Z(\varphi)), &\equat(defV)\cr W(\Phi,J,Z)(\varphi)&= \lambda\partial_xU(\varphi+\Phi(\varphi),J(\varphi),Z(\varphi)), &\equat(defW)\cr }$$ one rewrites equation \equ(eqmot3) as $$\eqalignno{ \pmatrix{0&D\cr-D&g}\pmatrix{\Phi\cr J}&=-V(\Phi,J,Z),&\equat(sys1)\cr (D^2+\mu^2)Z&=-W(\Phi,J,Z).&\equat(sys2)\cr }$$ Our strategy will be to consider \equ(sys1) and \equ(sys2) separately, treating the functions $Z$ and $(\Phi,J)$, respectively, as parameters. As we will see in Section~8, existence of a (unique) solution of the original equation \equ(eqmot3) can then be proved by using the implicit function theorem. Note that \equ(sys1) involves only the torus frequencies $\omega$ and is equivalent to a standard KAM problem. Existence of solution for such equations is well known and has been established by various means. One important feature we will use is the regular dependence of the solution $(\Phi,J)$ on the function $Z$. A precise result about the solution of \equ(sys1) will be stated in Section~4, Theorem~4.1, once the required Banach spaces of functions have been introduced. We now focus our attention on equation \equ(sys2), and will suppress from the notation the dependence of the vector field $W$ on the parameters $\Phi$ and $J$. Most of our analysis will be conducted in Fourier space, and we will denote by lower case letters the Fourier transforms of functions of $\varphi$, the latter being denoted by capital letters, namely, $$ F(\varphi)=\sum_{q\in\integer^d} e^{-iq\cdot\varphi}f(q),\quad {\rm where}\quad f(q)=\int_{\torus^d}e^{iq\cdot\varphi} F(\varphi)d\varphi, $$ where $d\varphi$ stands for the normalized Lebesgue measure on $\torus^d$. For $Z(\varphi)\in\RR^\infty$, note that $z(q)\in\hat\RR^\infty$ with $z_{k_i}(q)=\overline{z_{k_i}(-q)}$, where $\hat\RR^\infty$ stands for $\bigoplus_{k\geq1}\complex^{d_k}$ and $k_i$ refers to the $i^{\rm th}$ component of $\complex^{d_k}$. Similarly, $\hat\RR^\infty_s$ will denote the complexification of the Banach space $\RR^\infty_s$ defined in \equ(defRRs). Finally, we will denote the vector space of functions $z(q)\in\hat\RR^\infty$ by $h$, $$ h=\{z=(z(q))\,|\,z(q)\in\hat\RR^\infty,q\in\integer^d\}. $$ In terms of the Fourier transform of $W$, namely, $$ w_0(z)(q)\equiv\lambda\int_{\torus^d}e^{iq\cdot\varphi}\partial_x U(\varphi+\Phi(\varphi),J(\varphi),Z(\varphi))d\varphi, \equation(wrho) $$ equation \equ(sys2) becomes, $$ \sK_0z=w_0(z), \equation(fp1) $$ where the operator $\sK_0$ is given by the diagonal kernel $$ \sK_0(q,q')=\bigl(|\omega\cdot q|^{2}-\mu^2\bigr) \delta_{qq'}.\equation(drho) $$ Solving equation \equ(fp1) requires to invert the operator $\sK_0$. Although the inverse of $\sK_0$ is unbounded for generic frequencies, restricting $\omega$ to a set of admissible frequencies gives sufficient control on the inverse of $\sK_0$ to prove existence of a solution. As is well known for Melnikov problems, this set depends on the perturbation $U$. In order to prove existence of a solution to equation \equ(fp1), we will follow a strategy developed in [BGK] for standard KAM problems, namely, for equations of the type \equ(sys1). This strategy basically consists in inductively reducing \equ(fp1) to a sequence of effective equations involving denominators of decreasing size. One inductive step, say the $n^{\rm th}$ step, consists in splitting the effective equation obtained at the previous step into two equations involving only large and, respectively, small denominators, where large and small are defined with respect to a scale of order $\eta^n$ for some fixed $\eta<1$. This splitting is done in such a way that the nonlinear operator involved in the large denominators equation is a contraction, and this equation can thus be solved by a simple application of the contraction mapping principle. This, in turn, allows to map the small denominators equation into a new effective equation of the type \equ(fp1), with a new right hand side $w_n$ and (eventually) a new linear operator $\sK_n$. In [BGK], it was shown that for equations of the type \equ(sys1), the above mentioned contraction property follows naturally from symmetries specific to this case. In contrast, equation \equ(sys2) involves in addition the normal frequencies $\mu_k$ and does not possess such symmetry. In order to obtain the required contraction, we must make at every inductive step an additional preparation step. As we shall see below, this amounts to renormalizing the linear operator $\sK_{n-1}$ obtained at the previous step into a new operator $\sK_n$, which, in effect, corresponds to renormalizing the normal frequencies. Furthermore, we will see that the renormalized normal frequencies converge to a $U$-dependent set $\{\mu_\alpha^*\},\ \alpha\geq1,$ as $n\rightarrow\infty$. Therefore, since the Diophantine conditions imposed on $\omega$ will eventually be defined relatively to this set, one obtains in a constructive way the dependence of the set of admissible frequencies on the perturbation $U$. We now describe how the renormalization group approach is implemented in practice for Melnikov type problems. First, we proceed with the above mentioned preparation step by decomposing $w_0$ as $$ w_0(z)=\tilde w_0(z)+A_0z, $$ where the linear operator $A_0$ is the dominant part of $Dw_0(z)$ evaluated at $z=0$. With $\sK_1\equiv \sK_0-A_0$, equation \equ(fp1) now reads $$ \sK_1z=\tilde w_0(z). \equation(step1) $$ As explained in more details below, $A_0$ can be chosen in such a way that $\sK_1$ is of the same form as $\sK_0$, cf. \equ(drho), but now given in terms of a new set of frequencies $\tilde\mu_{k_i}\in\real$ which are perturbation of order $\lambda$ of the original normal frequencies $\mu_k$. The notation $\tilde\mu_{k_i}$ reflects the fact that the perturbation $A_0$ may lift some of the degeneracies. Therefore, when inverting $\sK_1$, denominators smaller than $\OO(\eta)$ occur for $q$ such that $||\omega\cdot q|-\tilde\mu_{k_i}|\leq\OO(\eta)$ for some $k_i$. Furthermore, these small denominators only occur, for such $q$, in a specific subspace $h_{k_i}^q$ of $\complex^{d_k}$ depending on which $\tilde\mu_{k_j}$, if any, has been separated from $\tilde\mu_{k_i}$ by more than $\OO(\eta)$. Introducing $P_1$ as the projection of $h$ onto $h^q_{k_i}$ for $q$ such that $||\omega\cdot q|-\tilde\mu_{k_i}|\leq\OO(\eta)$ and defining $Q_1\equiv\identity-P_1$, one thus expects that the restriction of $\sK_1$ to $Q_1h$ is invertible with an inverse of order $\OO(\eta^{-1})$. Multiplying \equ(step1) by $Q_1$ and $P_1$ leads to the small and large denominators equations for $\tilde z_1\equiv Q_1z$ and $z_1\equiv P_1 z$, $$\eqalignno{ \sK_1\tilde z_1&=Q_1\tilde w_0(\tilde z_1+z_1),&\equat(step11)\cr \sK_1z_1&=P_1\tilde w_0(\tilde z_1+z_1),&\equat(step12)\cr }$$ and by definition of $Q_1$, the first equation can be rewritten as a fixed point equation for the functional $R_1$ defined as $R_1(z_1)\equiv\tilde z_1$, namely, $$ R_1(z)=\sK_1^{-1}Q_1\tilde w_0(z+R_1(z)). \equation(step121) $$ By choice of $A_0$, the nonlinear operator $\sK_1^{-1}Q_1\tilde w_0$ is a contraction and one can solve equation \equ(step121) for $R_1$ using the Banach fixed point theorem. (See point (a) of Theorem~5.1 for this part of the inductive step.) Next, with $w_1$ defined as $$ w_1(z)\equiv\tilde w_0(z+R_1(z)), $$ equation \equ(step12) reads $$ \sK_1 z_1=P_1 w_1(z_1), \equation(step13) $$ and the solution $z=z_1+\tilde z_1$ of the original equation \equ(fp1) is now given by $$ z=z_1+R_1(z_1)\equiv F_1(z_1). $$ Hence, the problem of solving \equ(fp1) is reduced to solving the effective equation \equ(step13). To solve this equation one proceeds similarly, starting with our preparation step. After $n$ steps of this inductive process, the solution of \equ(fp1) is given by $$ z=F_{n-1}(z_n+R_n(z_n))\equiv F_{n}(z_n), \equation(deffn) $$ where $R_n$ solves the functional equation $$ R_n(z)=\Gamma_n\tilde w_{n-1}(z+R_n(z)), \equation(equrn) $$ with $$ \Gamma_n\equiv \sK_{n}^{-1}Q_{n}P_{n-1}, \equation(defGamman) $$ and, for some linear operator $A_{n-1}$, $$\eqalignno{ &\tilde w_{n-1}(z)\equiv w_{n-1}(z)-A_{n-1}z,&\equat(defwtilden)\cr &\sK_{n}\equiv \sK_{n-1}-P_{n-1}A_{n-1},&\equat(defAn)\cr }$$ whereas $z_n$ solves the effective equation $$ \sK_{n}z_n=P_n w_{n}(z_n), \equation(stepn3) $$ with $w_{n}$ defined as $$ w_{n}(z)\equiv\tilde w_{n-1}(z+R_n(z)). \equation(defwn) $$ \claimrm Remark(rem0) The point of this inductive procedure is that $P_nw_n(z)$ becomes effectively linear in $z$ for large $n$. More precisely, we will show, cf. Theorem~5.1 below, that the rescaled maps $w_n^r$ defined by $w_n^r(z)=\eta^{-n}r^{-n}w_n(r^{n}z)$ satisfy for $r<\eta$, $$ P_nw_n^r(z)=P_nDw_n^r(0)z+\OO(\lambda r^{2n}\eta^{-n}) \quad{\rm with}\quad P_nDw_n^r(0)=\OO(\lambda), $$ in some appropriate Banach space. Thus, $z_n=0$ becomes a better and better approximation to the solution of \equ(stepn3), and we shall construct the solution $z$ of the original equation \equ(fp1) as the limit of the approximate solutions $$ z=\lim_{n\rightarrow\infty}F_n(0). \equation(zlimit) $$ We now give a precise description of the operators $P_n$. Note that in order to obtain \equ(equrn) and \equ(stepn3), we have tacitly assumed that $P_{n}P_{n-1}=P_n$. The possibility to define $P_n$ satisfying such a property follows from the convergence of the normal frequencies under renormalization. Recall that renormalization occurs because at every inductive step one turns the nonlinear map $w_n$ of the effective functional equation \equ(stepn3) into a contraction by substracting some linear operator $A_n$. Delaying to subsequent sections the discussion of the appropriate choice for the family $A_m$, $m\geq0$, it suffices to point here to the properties of $A_m$ that will ensure convergence of the renormalized normal frequencies. As will be shown, cf. point (c) of Theorem~5.1 for a precise statement, $A_m$ is a perturbation of order $\lambda\eta^{m}$ and is given by a constant kernel $A_m(q,q')=a_m\delta_{qq'}$ with $a_m:\hat\RR^\infty\rightarrow\hat\RR^\infty$ linear and hermitian. As a consequence, the operator $\sK_n=\sK_0-\sum_{m=0}^{n-1}P_mA_m$ has a kernel of the form \equ(drho) with $\mu^2$ essentially replaced by the positive definite matrix $$ \tilde\mu_n^2\equiv\mu^2+\sum_{m=0}^{n-1}a_m, \equation(defmun) $$ with $\tilde\mu_n$ having a discrete spectrum $\sigma(\tilde\mu_n)\subset\real^+$. One easily checks that the singularities of $\sK_n^{-1}$ are given by the eigenvalues of $\tilde\mu_n$, which therefore correspond to renormalized normal frequencies. Since $a_m$ is of order $\lambda\eta^{m}$, one expects the eigenvalues of $\tilde\mu_n$ to converge as $n\rightarrow\infty$ with $|\nu_{n+1}-\nu_{n}|\leq\OO(\lambda\eta^n)$ for $\nu_{n+1}\in\sigma(\tilde\mu_{n+1})$ and $\nu_{n}\in\sigma(\tilde\mu_{n})$. This, in turn, allows us to define scales of denominators in a consistent way by carefully keeping track of the separation properties of $\sigma(\tilde\mu_n)$ as $n$ increases. To this end, one groups the normal frequencies into a hierarchy of clusters satisfying gap conditions that are preserved by the renormalization procedure. We first introduce some notation. For $x\in\real$ and $\CC$ a finite collection of points in $\real$, let $d(x,\CC)$ denote the distance between $x$ and the smallest interval containing all points in $\CC$, and for two finite collections $\CC_1,\CC_2\subset\real$, let $$ d(\CC_1,\CC_2)\equiv\inf_{x\in\CC_1}d(x,\CC_2). $$ Then, one can uniquely decompose $\sigma(\tilde\mu_n)$ into a maximal number of disjoint clusters $\CC^n_{k,i}$, $k\geq1$, $i=1,\dots,M_k^n$, satisfying $d(\mu_k,\CC^n_{k,i})=\OO(\lambda)$ and the gap condition $$ d(\CC^n_{k,i},\CC^n_{k,j})>\eta^n\quad{\rm if}\ i\not=j. \equation(gapn) $$ Note that $M^n_{k}\leq d_k$, where $d_k$ denotes the multiplicity of the original normal frequency $\mu_k$, and that by requiring $M^n_{k}$ to be maximal, the decomposition $$ \sigma(\tilde\mu_{n})=\bigcup_{k\geq1} \bigcup_{i=1}^{M^n_{k}}\CC^n_{k,i} \equation(defcin) $$ is unique. The above observation about the rate of convergence of $\sigma(\tilde\mu_n)$ as $n\rightarrow\infty$ ensures that eigenvalues belonging to different clusters will remain separated. Generically, one expects all degeneracies to be lifted eventually, so that $M^n_k=d_k$ for $n$ sufficiently large and each cluster $\CC^n_{k,i}$ contains a single eigenvalue. Next, defining $S_{n}\subset\integer^d$ as $$ S_{n}=\bigcup_{k\geq1}\bigcup_{i=1}^{M^n_{k}}S^n_{k,i}, \equation(defsn) $$ where $$ S^n_{k,i}=\{q\in\integer^d\ |\ d(|\omega\cdot q|,\CC^n_{k,i})<\onefourth\eta^{n}\}, \equation(defsin) $$ one is ensured that all $q\in\integer^d\setminus S_n$ satisfy $d(|\omega\cdot q|,\sigma(\tilde\mu_{n'}))\geq\OO(\eta^n)$ for $n'\geq n$. Hence, such $q$ can be safely ``integrated out'' in the large denominators equation. Remark that due to \equ(gapn), the sets $S^n_{k,i}$ are pairwise disjoint. In order to achieve the construction of $P_n$, one must isolate for every $q\in S_n$ the subspace of $\RR^\infty$ in which small denominators will occur. For $q\in S^n_{k,i}$, the latter is given by the eigenspace of $\tilde\mu_n$ associated with the eigenvalues belonging to $\CC^n_{k,i}$. This eigenspace will be denoted by $\JJ^n_{k,i}$, whereas the projector onto $\JJ^n_{k,i}$ will be denoted by $\PP^n_{k,i}$. Thus, one defines $P_n$ to be the diagonal operator acting on $h$ given by the kernel $$ P_n(q)=\sum_{k\geq1} \sum_{i=1}^{M^n_k}\chi^n_{k,i}(\omega\cdot q)\PP^n_{k,i}, \equation(defPn) $$ where $\chi^n_{k,i}$ denotes a function in $\in\CC^1(\real)$ which satisfies $$ \chi^n_{k,i}(\kappa)=\cases{ 1&if\ \ $d(|\kappa|,\CC^n_{k,i})\leq\oneeighth\eta^n$,\cr 0&if\ \ $d(|\kappa|,\CC^n_{k,i})\geq\onefourth\eta^n$,\cr } $$ and interpolates monotonically between $0$ and $1$ otherwise, with $$ \sup_{\kappa\in\real} |{\chi^n_{k,i}}'(\kappa)|\leq C\eta^{-n}, \equation(condder) $$ whereas $Q_n$ is defined as $$ Q_n=\identity-P_n\,. \equation(defQn) $$ Note that $P_n$ and $Q_n$ are not projectors. The smooth functions $\chi^n_{k,i}$ have been introduced in order to ensure the continuity of the diagonal kernels $\Gamma_n(q,q)$, cf. the discussion preceding Lemma~5.3 below. However, we will make use later of the projector $$ \hat P_n(q)=\sum_{k\geq1} \sum_{i=1}^{M^n_k}{\rm I}_{S^n_{k,i}}(q)\PP^n_{k,i}, \equation(defPhatn) $$ where ${\rm I}_\Sigma$ denotes the indicator function of a set $\Sigma$. Note that $P_n\hat P_n=P_n$, whereas $Q_n\hat P_n\not=0$. We conclude this section by a few remarks related to the convergence of the inductive scheme. First, setting $I^n_{k,i}\subset\real$ to be the smallest interval covering $\CC^n_{k,i}$, one easily checks that $|I^n_{k,i}|\leq(d_k-1)\eta^n$. Hence, since the multiplicities of the normal frequencies $\mu_k$ were assumed to be uniformly bounded in $k$, {i.e.}, $d_k\leq\bar d$ for all $k\geq1$, one obtains for all $n\geq1$, $k\geq1$, and $i=1,\dots,M^n_{k}$, $$ |I^n_{k,i}|\leq\bar d\eta^n. \equation(bndabsIin) $$ Next, it follows from the gap condition \equ(gapn) being preserved that for all $m0$, $\alpha>0$ and $b<\infty$ such that the Fourier transforms of $U^{\Phi,J}_{m+1}(\varphi)$ satisfy $$ \sum_{q\in\integer^d}e^{\alpha|q|}\,||u_{m+1}^{\Phi,J}(q)||_ {\LL(\hat\RR_s,\dots,\hat\RR_s;\hat\RR_{s'})}\leq b\,m!\,\rho^{-m}. \equation(coeff1) $$ Inserting the Fourier series for $Z$ into \equ(expa1), one obtains the expansion for $w_0$ as defined in \equ(wrho), $$\eqalignno{ w_0(z)(q)&=\lambda\sum_{m=0}^\infty\sum_{{\bf q}}{1\over m!} u^{\Phi,J}_{m+1}\Bigl(q-\sum_{i=1}^mq_i\Bigr)(z(q_1),\dots,z(q_m))&\cr &\equiv\sum_{m=0}^\infty\sum_{{\bf q}} w^{(m)}_0(q;q_1,\dots,q_m)(z(q_1),\dots,z(q_m)),&\equat(expa2)\cr }$$ where ${\bf q}=(q_1,\dots,q_m)\in\integer^{md}$. This formula suggests to consider $w_0$ as an analytic functions of $z\in h_s$. Let $B(r_0)$ be the open ball of radius $r_0$ in $h_s$ centered at the origin and let $H^\infty(B(r_0),h_{s'})$ denote the Banach space of analytic function $w:B(r_0)\rightarrow H_{s'}$ equipped with the supremum norm, which we shall denote by $|||w|||$. Then, bound \equ(coeff1) implies that $w_0\in H^\infty(B(r_0),h_{s'})$ for $r_0$ small enough. It will be convenient to encode the decay property of the kernels $w^{(m)}_0$ inherited from the estimate \equ(coeff1) as a property of the functional $w_0$. Let $\tau_\beta$ denote the translation by $\beta\in\real^d$, i.e., $(\tau_\beta Z)(\varphi)=Z(\varphi-\beta)$. On $h_s$, $\tau_\beta$ is realized by $(\tau_\beta z)(q)=e^{i\beta\cdot q}z(q)$, and it induces a map $w\mapsto w_\beta$ from $H^\infty(B(r_0),h_{s'})$ to itself if we define $$ w_\beta(z)=\tau_\beta w(\tau_{-\beta}z). \equation(taubeta) $$ On the kernels $w_0^{(m)}$, this is given by $$ w_\beta^{(m)}(q;q_1,\dots,q_m)=e^{i\beta\cdot(q-\sum q_i)} w^{(m)}(q;q_1,\dots,q_m), $$ and makes sense also for $\beta\in\complex^d$. Since $$ |||w_{0\beta}|||\leq\sum_{m=0}^\infty r_0^m\sup_{\bf q} \sum_{q\in\integer^d}e^{-{\rm Im}\beta\cdot((q-\sum q_i)} ||w_0^{(m)}(q;q_1,\dots,q_m)||_{\LL(\hat\RR_s,\dots, \hat\RR_s;\hat\RR_{s'})}, $$ it thus follows from \equ(coeff1) that there exist $r_0>0$, $\alpha>0$, and $D<\infty$, such that $w_{0\beta}$ belongs to $H^\infty(B(r_0),h_{s'})$ and extends to an analytic function of $\beta$ in the strip $|\Im\beta|<\alpha$ with values in $H^\infty(B(r_0),h_{s'})$ satisfying the bound $$ |||w_{0\beta}|||\leq D|\lambda|. \equation(nw0beta) $$ Let us now come back to the existence of a solution for equation \equ(sys1), namely for the standard KAM problem. One has the classical result (see for instance [BGK]): \claim Theorem(KAM) Let $U$ satisfy hypothesis (H2) and let $g$ be an invertible matrix. Then, there is a $\lambda_1>0$ small enough such that for $|\lambda|<\lambda_1$ and $\omega$ satisfying a Diophantine condition of the form $$ |\omega\cdot q|>K|q|^{-\nu}\quad{\rm for}\quad q\in\integer^d, \,q\not=0, $$ \equ(sys1) has a solution $(\Phi,J)\in\sB$ which is real analytic in $\varphi$, analytic in $\lambda$, and vanishes for $\lambda=0$. Furthermore, this solution is unique up to translations $(\Phi,J)(\varphi)\mapsto(\Phi-\beta,J)(\varphi-\beta)$ and depends analytically on $Z$, for $Z$ in a small ball centered at the origin of the Banach space $h_s$. To conclude this section, we list some standard properties of bounded analytic functions defined on open balls in Banach spaces. Let $h,h',h''$ be Banach spaces, $B(r)\subset h$, $B(r')\subset h'$, and $w_i\in H^\infty(B(r),h')$, $w\in H^\infty(B(r'),h'')$. First, one has the composition property: If $|||w_i|||0$ and $\{\CC^n_{\bf k}\}_{{\bf k}\in\II^n}$ the clusters described in the previous section, $$\displaylines{\quad \Omega_n(K)=\bigl\{\omega\in\real^d\ |\ \ d(|\omega\cdot q|,\CC^n_{\bf k}),\ d(|\omega\cdot q|,|\CC^n_{\bf k}\pm\CC^n_{\bf k'}|) >K|q|^{-\nu}\hfill\cr \hfill\forall\,|q|K'$. Furthermore, one introduces for $\omega\in\real^d$ the subsets of $\integer^d$ $$ \QQ^+_\omega=\{q\in\integer^d\,|\,\omega\cdot q>0\}\,,\quad \QQ^-_\omega=\{q\in\integer^d\,|\,\omega\cdot q<0\}. \equation(defqpqm) $$ \claim Proposition(itbounds) There exist positive constants $r$ and $\lambda_0$ small enough such that the following is true for $|\lambda|<\lambda_0$, $n\geq1$, and $|\Im\beta|<\alpha_n$, where $\alpha_1=\alpha$ and, for $n\geq2$, $$ \alpha_n=(1-n^{-2})\alpha_{n-1}. \equation(defalphan) $$ There exists $K_\lambda>0$ satisfying $K_\lambda\rightarrow0$ as $\lambda\rightarrow0$ such that one has for $\omega\in\Omega_n(K_\lambda)$ arbitrary but fixed, \item{\rm(a)} Equation \equ(eqrnb) has a solution $R_{n\beta}$ in $H^\infty(B_n,h^{n-1}_{s})$ analytic in $|\lambda|<\lambda_0$ and $(\Phi,J)\in\sO_\sB$. \item{\rm(b)} Defining $w_{n\beta}$ according to \equ(defwnb), one has $w_{n\beta}\in\AA_n$ and, writing $w_{n\beta}(z)\equiv w_n(z)=w_n(0)+Dw_n(0)z +\delta_2w_n(z)$, $$\eqalignno{ &||\hat P_nw_n(0)||_{s'}\leq\varepsilon r^{2n},&\equat(fstbnd)\cr &|||\hat P_n\delta_2w_n|||_{\AA_n}\leq\varepsilon r^{2n}, &\equat(sndbnd)\cr }$$ \item{}where $\varepsilon\rightarrow0$ as $\lambda\rightarrow0$. \item{\rm(c)} There exists $A_n\in\LL(h_s,h_{s'})$ such that $\tilde w_n\equiv w_n-A_n$ obeys for all $z\in B_n$, $$ ||\hat P_nD\tilde w_n(z)||^{(n)}_{s,s'}\leq\varepsilon\eta^{n}. \equation(thdbnd) $$ Furthermore, $$ ||A_n||_{s,s'}\leq3\varepsilon\eta^{n-1}, \equation(deltan) $$ $A_n(q,q')=0$ if $q\not=q'$ and $$ A_n(q,q)=a_n{\rm I}_{\QQ^+_\omega}(q)+ \overline{a_n}\,\,{\rm I}_{\QQ^-_\omega}(q), \equation(defGn) $$ where $a_n\in\LL(\hat\RR_s^\infty,\hat\RR_{s'}^\infty)$ is hermitian, {i.e.}, $a_n=\overline{a_n}^{\rm T}$, and satisfies for all ${\bf k}\in\II^n$, $$ a_n \JJ^n_{\bf k}=\JJ^n_{\bf k}\,. \equation(subinv)\ $$ \item{\rm(d)} The matrix $\tilde\mu_{n+1}^2\equiv\mu^2+\sum_{m=0}^{n}a_m$ is positive definite and the spectrum of $\tilde\mu_{n+1}$ can be uniquely decomposed into a maximal family of pairwise disjoint clusters $\CC^{n+1}_{k,i}$, $k\geq1$, $i=1,\dots,M^{n+1}_k$, with $M^{n+1}_k\geq M^{n}_k$, satisfying for all $k\geq1$ the gap condition $$ d(\CC^{n+1}_{k,i},\CC^{n+1}_{k,j})>\eta^{n+1} \quad{\rm if}\quad i\not=j, \equation(gapn2) $$ and $$ \nu=\mu_k+\OO(\varepsilon k^{-\xi}) \quad{\rm for\ all}\quad \nu\in\CC^{n+1}_{k,i}\,,\ i=1,\dots,M_k^{n+1}. \equation(asymp) $$ Furthermore, the sets $S^{n+1}_{k,i}$ defined according to \equ(defsin) are pairwise disjoint, and \equ(convclust), \equ(PkiPlj) and \equ(proppn) hold with $n$ replaced by $n+1$. \item{} Let us briefly comment on \clm(itbounds), whose proof will be carried out in Section~6. First, we note that point (d) ensures, in particular, that the new set of clusters $\CC^{n+1}_{k,i}$ enjoy the properties required for proceeding to the next step of the induction, cf. the discussion at the end of Section~3. The asymptotic behavior \equ(asymp) concerns the measure estimate of the set $\Omega^*$ of admissible frequencies in \clm(mainthm). Such an asymptotic behavior is required in order to obtain a set of large measure because one imposes Diophantine conditions with respect to differences of the normal frequencies. We will show in Section~7 that \equ(asymp) implies the \claim Proposition(measureest) For $\nu=\nu(d,\xi)$ sufficiently large, the set $$ \Omega^*(K)\equiv\bigcap_{n\geq1}\Omega_n(K) \equation(omegastar) $$ satisfies for all bounded $\Omega\subset\real^d$, ${\rm meas}(\Omega\setminus\Omega^*(K))\rightarrow0$ as $K\rightarrow0$. Note that $\omega\in\Omega^*$ assume a Diophantine condition with respect to zero. Therefore, one has for such $\omega$, $\integer^d\setminus\{0\}=\QQ^+_\omega\cup\QQ^-_\omega$. Next, we turn to bound \equ(thdbnd), the most delicate estimate to establish. To treat the off-diagonal part $Dw_n(q,q')$, $q\not=q'$, we will rely on the fact that the exponential decay of the kernel $Dw_0(q,q')$ in the size of $|q-q'|$ is preserved due to the introduction of the parameter $\beta$. We note that imposing Diophantine conditions on $\omega$ with respect to the differences $\CC^n_{\bf k}\pm\CC^n_{\bf k'}$ ensures that $|q-q'|$ is of order $\OO(\eta^{-n/\nu})$ for $q\not=q'\in S_n$. To treat the diagonal part, we will use that $Dw_n(q,q)$ depends on $q$ through $\omega\cdot q$ only, and is, in some sense, continuous in this variable. More precisely, defining $t_p:\LL(h_s,h_{s'})\rightarrow\LL(h_s,h_{s'})$, $p\in\integer^d$, by $$ (t_p L)(q,q')=L(q+p,q'+p), \equation(deftp) $$ and setting $$ \Delta_p\equiv t_p-\identity, \equation(defDeltap) $$ we will show that $\Delta_p Dw_n$ is of order $\OO(\varepsilon|\omega\cdot p|)$ on the diagonal. Therefore, since $p=q-q'$ satisfies $|\omega\cdot p|\leq\eta^n$ for $q,q'\in S^n_{\bf k}$ such that $\sign(\omega\cdot q)=\sign(\omega\cdot q')$, one has for $q\in S^n_{\bf k}$, $$ \hat P_nDw_n(q,q)\hat P_n=\hat a_{\bf k}+\OO(\varepsilon\eta^{n}), $$ where $\hat a_{\bf k}:\JJ^n_{\bf k}\rightarrow\JJ^n_{\bf k}$ dependents only on the sign of $\omega\cdot q$. The continuity of $Dw_n(q,q)$ ultimately follows from the fact that $\Gamma_n(q)$ is continuous in $\omega\cdot q$, as stated in the following lemma, whose proof can be found in the Appendix. \claim Lemma(intermediary) Let $\sigma\in\real$ and $p\in\integer^d$. Then the operator $\Gamma_n=\sK_n^{-1}Q_nP_{n-1}$ obeys $$\eqalignno{ ||\Gamma_n||_{\sigma,\sigma+\gamma}&\leq C\eta^{-n}, &\equat(bndinvan)\cr ||\Delta_p\Gamma_n||_{\sigma,\sigma+\gamma}&\leq C\eta^{-2n}|\omega\cdot p|. &\equat(DeltapG)\cr }$$ Finally, the perturbation $a_n$ being hermitian will essentially follow from the reality of the original equation \equ(sys2). More precisely, the derivative $Dw_n$ satisfies $$\eqalignno{ Dw_n^{ij}(q,q')&=\overline{Dw_n^{ij}(-q,-q')},&\equat(symm1)\cr Dw_n^{ij}(q,q')&=Dw_n^{ji}(-q',-q).&\equat(symm2)\cr }$$ Thus, the diagonal element $Dw_n(q,q):\hat\RR^\infty\rightarrow\hat\RR^\infty$ is given by an hermitian matrix for all $q$, and $a_n$ hermitian will follow since, as was mentioned above, $a_n$ will be chosen in such a way that its action on each $\JJ^n_{\bf k}$ is the constant approximation of $Dw_n(q,q)$ for $q\in S^n_{\bf k}$. Note that due to \equ(symm1), one expects $Dw_n(-q,-q)$ to be approximated by $\overline{a_n}$, which explains the decomposition in formula \equ(defGn). Identities \equ(symm1) and \equ(symm2) are easily checked to hold for $n=0$. Indeed, the perturbation $U$ in the Hamiltonian \equ(hamiltonian) being real analytic ensures \equ(symm1), whereas \equ(symm2) follows from the fact that $Dw_0$ is the symmetric second derivative of the functional $Z\mapsto\lambda\int U(\varphi+\Phi(\varphi),J(\varphi), Z(\varphi))d\varphi$, cf. \equ(wrho). Using the recursive relations \equ(defwn) and \equ(defwtilden), one obtains \equ(symm1) and \equ(symm2) for $n\geq1$ by iteration. \claimrm Remark(rem1) The choice of constants is as follows. We first fix $\eta$ small enough according, essentially, to the constants entering the asymptotics of the frequencies $\mu_k$ in (H1), cf. Section~6.4. Given $\eta$, $\varepsilon$ and $r$ are chosen small enough, and $\lambda_0$ is chosen in turn according to $\varepsilon$. The latter choice plays a role only in ensuring that the inductive hypothesis of \clm(itbounds) are satisfied for $n=0$, cf. the introduction in Section~6. Finally, $K_\lambda$ is chosen large enough in order for the estimate $$ Ce^{-Cn^{-2}K_\lambda^{1/\nu}\eta^{-n/\nu}}\leq r^{2n}, \equation(expest) $$ to hold for all $n\geq1$. This will be needed in order to iterate the bound \equ(fstbnd) in Section~6.2. Note that due to the double exponential, the dependence of $K_\lambda$ on $\eta$ and $r$ is given by the behavior at small $n$ of the expressions entering \equ(expest). That $K_\lambda$ can be taken smaller as $\lambda$ goes to zero will follow from the fact that $r$ and $\varepsilon$, and thus ultimately $\eta$, can be taken smaller. Finally, we denote by $C$ a generic constant, independent on $n$, $r$, and $\varepsilon$, which may vary from place to place. \section Proof of \clm(itbounds) We proceed by induction and assume that \clm(itbounds) holds up to $n-1\geq1$. Regarding the inductive hypothesis in the case $n=1$, we simply choose $A_0\equiv0$, so that the bounds for $w_0$ in points (b) and (c) of \clm(itbounds) are a simple consequence of \equ(nw0beta). Furthermore, $\tilde\mu_1=\mu$ and point (d) follows immediately from (H1). We note that in Section~6.1 below, point (a) is established for $n=1$ by taking $\varepsilon$, namely $\lambda$, small enough. At some point in the induction, however, one is forced to consider nontrivial $A_n$ in order for the inductive bounds to hold uniformly in $n$ for a given $\lambda$. In the sequel, we adopt the convention, for $B$ a ball of radius $r$ centered at the origin, to denote by $\gamma B$ the ball of radius $\gamma r$ centered at the origin. \subsection Existence of the Functional $R_{n\beta}$ With the notations $R=R_{n\beta}$, $\Gamma=\Gamma_n$ and $\tilde w=\tilde w_{(n-1)\beta}$, equation \equ(eqrnb) reads $$ R(z)=\Gamma\tilde w(z+R(z)). \equation(equrn2) $$ To prove existence in $H^\infty(B_{n},h^{n-1}_{s})$ of a solution $R$ to equation \equ(equrn2), one starts, using the identities $\tilde w(0)=w(0)$ and $\delta_2\tilde w=\delta_2w$, by decomposing $\tilde w$ as $$ \tilde w(z)=w(0)+D\tilde w(0)z +\delta_2w(z), \equation(decomposition) $$ to obtain from \equ(equrn2), $$ R(z)=\Gamma w(0) +\Gamma D\tilde w(0)(z+R(z)) +\Gamma\delta_2w(z+R(z)). \equation(equR) $$ Defining $$ H=\bigl(1-\Gamma D\tilde w(0)\bigr)^{-1}, \equation(defh) $$ and using the identity $1+H\Gamma D\tilde w(0)=H$, one rewrites \equ(equR) as $$ R(z)=H\Gamma w(0) +H\Gamma D\tilde w(0)z +u(z), \equation(equwtilde2) $$ where $$ u(z)=H\Gamma\delta_2 w(\tilde z) \equiv\GG(u)(z), \equation(equu) $$ and $$ \tilde z\equiv z+R(z)=H\bigl(z+\Gamma w(0)\bigr)+u(z). \equation(ytilde) $$ Since $\Gamma=\Gamma\hat P_{n-1}=\hat P_{n-1}\Gamma$, \equ(bndinvan) (with $\sigma=s+\xi-\gamma$) and the recursive bound \equ(thdbnd) (with $n$ replaced by $n-1$) imply $$ ||\Gamma D\tilde w(0)||^{(n-1)}_{s}\leq ||\Gamma D\tilde w(0)||^{(n-1)}_{s,s+\xi}\leq C\varepsilon\eta^{-1}. \equation(int1) $$ Hence, $$ ||H||^{(n-1)}_s\leq2, \equation(boundonh) $$ for $\varepsilon=\varepsilon(\eta)$ small enough. Since $B_{n}\subset B_{n-1}$, $\tilde w(0)=w(0)$, and since bounds \equ(fstbnd) (with $n$ replaced by $n-1$), \equ(bndinvan) and \equ(int1) hold, the existence of $R$ in $H^\infty(B_{n},h^{n-1}_s)$ follows from the existence of $u$ in $H^\infty(B_{n},h^{n-1}_s)$. For reasons that will become clear in the next section, we actually show that \equ(equu) has a solution $u$ in the ball $$ \BB=\Bigl\{u\in H^\infty(\oneeighth B_{n-1},h^{n-1}_s)\ |\ |||u|||\leq\sqrt{\varepsilon}\eta^{-n}r^{2(n-1)}\Bigr\}. \equation(ballBB) $$ This result is stronger, since $B_{n}\subset\oneeighth B_{n-1}$ for $r$ small enough. Let us first check that $\GG$ maps $\BB$ into itself. From \equ(boundonh) and the recursive bound \equ(fstbnd), it follows that for all $z\in\oneeighth B_{n-1}$ and $u\in\BB$, $\tilde z\in h^{n-1}_s$ with $$ ||\tilde z||_s\leq 2(\oneeighth r^{n}+C\varepsilon\eta^{-n}r^{2(n-1)}) +\sqrt{\varepsilon}\eta^{-n}r^{2(n-1)} \leq\onehalf r^{n}, $$ for $\varepsilon=\varepsilon(r,\eta)$ and $r=r(\eta)$ small enough. Hence, $$ \tilde z\in\onehalf B_{n-1}\subset B_{n-1} \quad{\rm for\ all\ }z\in\oneeighth B_{n-1}, \equation(tildeyinbn1) $$ and one uses the bound \equ(sndbnd) to conclude that for all $u\in\BB$, $$ |||\GG(u)|||\leq2C\eta^{-n}\varepsilon r^{2(n-1)} \leq\sqrt{\varepsilon}\eta^{-n}r^{2(n-1)}, $$ for $\varepsilon$ small enough. To show that $\GG$ is a contraction in $\BB$, we apply the estimate \equ(Cauchy2) to the functions $\tilde z_i$ given by \equ(ytilde) in terms of $u_i\in\BB$, $i=1,2$. Noting that $|||\tilde z_i|||\leq\onehalf r^{n}$, which follows from \equ(tildeyinbn1), and using in addition \equ(sndbnd), one obtains, $$\eqalign{ |||\GG(u_1)-\GG(u_2)|||&\leq2C\eta^{-n} \sup_{z\in\oneeighth B_{n-1}}||\hat P_{n-1}\delta_2w(\tilde z_1)- \hat P_{n-1}\delta_2w(\tilde z_2)||_{s'}\cr &\leq4C\eta^{-n}r^{-n}|||\hat P_{n-1}\delta_2w|||_{\AA_{n-1}} \sup_{z\in\oneeighth B_{n-1}}||\tilde z_1-\tilde z_2||_s\cr &\leq4C\varepsilon\eta^{-n}r^{-n}r^{2(n-1)} \sup_{z\in\oneeighth B_{n-1}}||u_1(z)-u_2(z)||_s\cr &\leq\onehalf|||u_1-u_2|||,\cr }$$ for $r=r(\eta)$ and $\varepsilon=\varepsilon(r,\eta)$ small enough. Before turning to part (b) of \clm(itbounds), we make some remarks that shall be useful later. First note that \equ(tildeyinbn1) means $$ z+R_n(z)\in\onehalf B_{n-1} \quad{\rm for\ all}\quad z\in\oneeighth B_{n-1}. \equation(tildeyinbn) $$ Therefore, with $$ \tilde R_m(z)\equiv z+R_m(z), \equation(deftildeRn) $$ and $$ F_n^m(z)\equiv\tilde R_m\circ\tilde R_{m+1}\circ \dots\circ\tilde R_n(z), \equation(deffnm) $$ it follows recursively that for $m=1,\dots,n$, $$ F_n^m(z)\in\onehalf B_{m-1} \quad{\rm for\ all}\quad z\in B_{n}. \equation(fnmin) $$ Furthermore, since $F_n^1=F_n$, where $F_n$ is defined in \equ(deffn), one has $F_n\in\AA_n$, together with the uniform bound $$ |||F_n|||_{\AA_n}\leq|||\tilde R_1|||_{\AA_1}\leq\varepsilon. \equation(unifFn) $$ \subsection Bounds on the Functional $w_n$ According to \equ(defwnb), one defines $$ w_{n\beta}(z)=\tilde w_{(n-1)\beta}(z+R_{n\beta}(z)). $$ Since $R_{n\beta}\in H^\infty(B_{n},h^{n-1}_s)$, it follows from \equ(tildeyinbn) and the inductive bounds that for all $\beta$ with $|\Im\beta|<\alpha_{n-1}$, $w_{n\beta}$ is well defined as a map from $B_n$ to $h_{s'}$, with $w_{n\beta}\in\AA_n$. In the sequel, we adopt the simplified notation $R=R_{n\beta}$, $w=w_{(n-1)\beta}$ and $w'=w_{n\beta}$. We proceed with proving \equ(fstbnd). Using the decomposition \equ(decomposition) at $z=0$, one may write $$ w'(0)=w(0)+D\tilde w(0)R(0)+\delta_2w(R(0)). $$ Since \equ(tildeyinbn) implies that $R(0)\in\onehalf B_{n-1}$, one obtains using the bounds \equ(fstbnd), \equ(sndbnd) and \equ(thdbnd), $$\eqalignno{ ||\hat P_nw'(0)||_{s'}&\leq\varepsilon r^{2(n-1)} +\onehalf\varepsilon\eta^{n-1}r^{n}+ \varepsilon r^{2(n-1)}&\cr &\leq3\varepsilon.&\equat(pbjunk1)\cr }$$ This leads to $$ |\PP^n_{\bf k}w'(0)(q)|_{s'}\leq3\varepsilon, \equation(pbjunk2) $$ for all ${\bf k}\in\II^n$ and $q\in S^n_{\bf k}$. The latter is valid for all $\beta$ with $|\Im\beta|<\alpha_{n-1}$. Let now $\beta'$ with $|\Im\beta'|<\alpha_{n}$. Then, shifting $\beta'$ to $\beta=\beta'-i(\alpha_{n-1}-\alpha_{n})q/|q|$ and using the recursive relation \equ(defalphan) for $\alpha_n$, one obtains $$ w_{\beta'}'(0)(q)=e^{i(\beta'-\beta)\cdot q}w_{\beta}'(0)(q) =e^{-n^{-2}\alpha_{n-1}|q|}w_{\beta}'(0)(q). \equation(betabetap) $$ Since for such $\beta'$ one has $|\Im\beta|<\alpha_{n-1}$, it follows from \equ(pbjunk2) and \equ(betabetap) that $$\eqalignno{ ||\hat P_nw'(0)||_{s'}&\leq3\varepsilon\sum_{{\bf k}\in\II^n} \sum_{q\in S^n_{\bf k}}e^{-n^{-2}\alpha_{n-1}|q|}\ . &\equat(boundonwp)\cr }$$ From the Diophantine conditions satisfied by $\omega\in\Omega_n(K)$, one infers for $q\in S^n_{\bf k}$ that $|q|>\min(K\eta^{-n/\nu},(4K)^{1/\nu}\eta^{-n/\nu})$, cf. \equ(defsin) and \equ(omegank). Therefore, Bound \equ(fstbnd) finally follows by choosing $K$ appropriately, cf. \equ(expest). We now iterate bound \equ(sndbnd). Using again the decomposition \equ(decomposition), one has $$\eqalign{ \delta_2 w'(z)&=D\tilde w(0)\delta_2R(z) +\delta_2\delta_2w(z+R(z)).\cr }$$ The first term on the right hand side is estimated by using $\delta_2R(z)=\delta_2u(z)$ together with \equ(Cauchy3) applied to $u\in\BB$ with $\gamma=8r$, since $B_{n}\subset\oneeighth B_{n-1}$, to obtain $$\eqalign{ |||\hat P_nD\tilde w(0)\delta_2R|||_{\AA_{n}} &\leq||\hat P_{n-1}D\tilde w(0)||^{n-1}_{s,s'} \sup_{z\in B_{n}}||\delta_2 u(z)||_s\cr &\leq\varepsilon\eta^n{(8r)^2\over1-(8r)^2}|||u|||\cr &\leq\varepsilon r^{2n}{\sqrt{\varepsilon}\,8^2\over1-(8r)^2}\cr &\leq\onehalf\varepsilon r^{2n}, }$$ for $\varepsilon$ small enough. In a similar way, one estimates, using \equ(tildeyinbn), that $$ \sup_{z\in B_n}||\hat P_n\delta_2\delta_2w(z+R(z))||_{s'} \leq\onehalf\varepsilon r^{2n}, $$ which finally leads to \equ(sndbnd). \subsection Bounds on the Derivative In this section, we prove the estimates stated in part (c) of \clm(itbounds). The main difficulty consists in controlling the diagonal part of the kernel of the derivative $Dw_n$ evaluated at zero, namely $Dw_n(0)(q,q)$, $q\in\integer^d$. To address this problem, as mentioned in the end of Section~5, we will use the fact that $Dw_n(0)(q,q)$ depends on $q$ through $\omega\cdot q$ only, and satisfies some continuity property when viewed as a function of $\omega\cdot q$. We start by deriving an a priori bound on the norm of $Dw_n$. From \equ(equrn), one infers that $$ DR_n(z)=H_n(\tilde z)\Gamma_n D\tilde w_{n-1}(\tilde z), \equation(DRn) $$ where $$\eqalignno{ H_n(\tilde z)&=\bigl(1-\Gamma_n D\tilde w_{n-1}(\tilde z)\bigr)^{-1}, &\equat(defH)\cr \tilde z&=z+R_n(z).&\equat(defytilde)\cr }$$ Since by definition, cf. \equ(defwn), one has $$ Dw_n(z)=D\tilde w_{n-1}(\tilde z)\bigl(1+DR_n(z)\bigr), $$ \equ(DRn) and the identity $H_n(\tilde z)=1+H_n(\tilde z)\Gamma_n D\tilde w_{n-1}(\tilde z)$, imply the recursive relation $$ Dw_n(z)=D\tilde w_{n-1}(\tilde z)H_n(\tilde z). \equation(recrelDw) $$ As in the previous section, it follows from \equ(bndinvan), \equ(tildeyinbn), and the inductive bounds, that $||H_n(\tilde z)||^{(n-1)}_s\leq2$ for all $\tilde z\in B_{n-1}$. Therefore, one obtains for all $z\in\oneeighth B_{n-1}$, using again the inductive bound \equ(thdbnd), $$ ||\hat P_nDw_n(z)||^{(n)}_{s,s'}\leq ||\hat P_{n-1}Dw_n(z)||^{(n-1)}_{s,s'}\leq2\varepsilon\eta^{n-1}. \equation(aprioriDw) $$ In order to iterate bounds \equ(thdbnd), we decompose $Dw_n(z)$ as follows $$ Dw_n(z)=\sigma_n+\tau_n+\delta_1Dw_n(z), \equation(decompDwn) $$ where $\sigma_n+\tau_n=Dw_n(0)$ and $\sigma_n(q,q')=Dw_n(0)(q,q')\delta_{qq'}$. Let us consider first the last two terms on the right hand side of \equ(decompDwn). One has the \claim Lemma(delta1wn) Let $r$ and $\varepsilon$ be the positive constants of \clm(itbounds). Then, one has for all $n\geq0$ and all $z\in B_n$, $$\eqalignno{ ||\hat P_n\delta_1Dw_n(z)||^{(n)}_{s,s'}& \leq\onehalf\varepsilon r^{{n\over2}}, &\equat(boundd1Dwn)\cr ||\hat P_n\tau_n||^{(n)}_{s,s'}&\leq\varepsilon r^n. &\equat(bndtaufinal)\cr }$$ \proof Proceeding by induction, we suppose that \clm(itbounds) and \clm(delta1wn) are true up to some $n-1$, $n\geq1$. We start with \equ(boundd1Dwn) and compute from $\delta_1Dw_n(z)=Dw_n(z)-Dw_n(0)$ and the recursive relation \equ(recrelDw) that $$ \delta_1Dw_n(z)= \tilde H_n(\tilde z_0) \bigl(D\tilde w_{n-1}(\tilde z)-D\tilde w_{n-1}(\tilde z_0)\bigr) H_n(\tilde z), $$ where $\tilde z_0=R_n(0)$ and $\tilde H_n(\tilde z_0)=1+Dw_{n-1}(\tilde z_0) H_n(\tilde z_0)\Gamma_n$. As previously, the inductive bound \equ(thdbnd) implies $||\hat P_n\tilde H_n(\tilde z_0)||^{(n-1)}_{s'}\leq2$. Using \equ(tildeyinbn) and $\hat P_n\tilde H_n=\hat P_n\tilde H_n\hat P_{n-1}$, one infers from the identity $D\tilde w_{n-1}(\tilde z)-D\tilde w_{n-1}(\tilde z_0)= \delta_1D\tilde w_{n-1}(\tilde z)-\delta_1D\tilde w_{n-1}(\tilde z_0)$ that for all $z\in\oneeighth B_{n-1}$, $$ ||\hat P_n\delta_1Dw_n(z)||^{(n-1,n)}_{s,s'} \leq C\sup_{z'\in\onehalf B_{n-1}}||\hat P_{n-1}\delta_1 D\tilde w_{n-1}(z')||^{(n-1)}_{s,s'}. $$ Since $\delta_1D\tilde w_{n-1}=\delta_1Dw_{n-1}$, the recursive bound \equ(boundd1Dwn) leads to $$ ||\hat P_n\delta_1Dw_n(z)||^{(n-1,n)}_{s,s'} \leq C\varepsilon r^{{n-1\over2}}, $$ for all $z\in\oneeighth B_{n-1}$. Finally, iterating bound \equ(boundd1Dwn) is completed by restricting $z$ to $B_n\subset\oneeighth B_{n-1}$ and using \equ(Cauchy3) with $\gamma=8r$. Next, we turn to \equ(bndtaufinal), the estimate for the off-diagonal part of $Dw_n(0)$. The norm of $\tau_n$ reads $$ ||\hat P_n\tau_n||^{(n)}_{s,s'}= \sup_{{\bf k'}\in\II^n}\sup_{q'\in S^n_{\bf k'}} \sum_{{\bf k}\in\II^n}\sum_{q\in S^n_{\bf k}} |\PP^n_{\bf k}\tau_n(q,q')\PP^n_{\bf k'}|_{s,s'}\,, $$ and one infers from \equ(boundd1Dwn) and the a priori bound \equ(aprioriDw) that $$ |\PP^n_{\bf k}\tau_n(q,q')\PP^n_{\bf k'}|_{s,s'} \leq2\varepsilon\eta^{n-1}+\onehalf\varepsilon r^{n\over2} \leq3\varepsilon. \equation(appitnpj) $$ The latter is valid for all $\beta$ with $|\Im\beta|<\alpha_{n-1}$. Let now $\beta'$ with $|\Im\beta'|<\alpha_{n}$. Then, shifting $\beta'$ to $\beta=\beta'-i(\alpha_{n-1}-\alpha_{n})(q-q')/|q-q'|$, one obtains $$ \tau_{n\beta'}(q,q')=e^{i(\beta'-\beta)\cdot(q-q')}\tau_{n\beta}(q,q') =e^{-n^{-2}\alpha_{n-1}|q-q'|}\tau_{n\beta}(q,q'). \equation(bbprime) $$ Hence, since $|\Im\beta|<\alpha_{n-1}$ for such $\beta'$, \equ(appitnpj) and \equ(bbprime) lead to $$ ||\hat P_n\tau_{n}||^n_{s,s'}\leq3\varepsilon \sup_{{\bf k'}\in\II^n}\sup_{q'\in S^n_{\bf k'}} \sum_{{\bf k}\in\II^n} \sum_{\scriptstyle q\in S^n_{\bf k}\atop\scriptstyle q\not=q'} e^{-n^{-2}\alpha_{n-1}|q-q'|}\ . \equation(bndtau1) $$ We now show that every term in the previous sum yields a super-exponentially small factor. Let $q\in S^n_{\bf k}$ and $q'\in S^n_{\bf k'}$ for some ${\bf k}\in\II^n$, ${\bf k'}\in\II^n$. Then, one estimates using \equ(defsin) and \equ(bndabsIin) that if $\sign(\omega\cdot q)=\sign(\omega\cdot q')$, $$ d\bigl(|\omega\cdot(q-q')|,\CC^n_{\bf k}+\CC^n_{\bf k'}\bigr)\leq\ \onehalf\eta^n+|I^n_{\bf k}|+|I^n_{\bf k'}| \leq3\bar d\eta^n, $$ and that otherwise $$ d\bigl(|\omega\cdot(q-q')|,|\CC^n_{\bf k}-\CC^n_{\bf k'}|\bigr)\leq\ \onehalf\eta^n+|I^n_{\bf k}|+|I^n_{\bf k'}| \leq3\bar d\eta^n. $$ Therefore, since $q\not=q'$, it follows from \equ(omegank) and $\omega\in\Omega_n(K)$ that $$ |q-q'|\geq\min\Bigl(\Bigl({K\over3\bar d}\Bigr)^{1/\nu},K\Bigr) \eta^{-n/\nu}. $$ Hence, the contribution of each term in \equ(bndtau1) is super-exponentially small, and \equ(bndtaufinal) follows for some $r\ll\eta<1$. \qed Finally, we turn to $\sigma_n$, the diagonal part of $Dw_n(0)$ in the decomposition \equ(decompDwn). We first state a result about the continuity properties of the kernel $\sigma_n(q,q)$, namely that $\Delta_p\sigma_n=t_p\sigma_n-\sigma_n$ is of order $|\omega\cdot p|$. More precisely, one has the \claim Proposition(continuity) Suppose that \clm(itbounds) is valid up to $n-1$ for some $n\geq1$. Then, the diagonal part $\sigma_n(z)$ of $Dw_n(z)$ satisfies for all $z\in B_n$ and all $p$ such that $|\omega\cdot p|<{1\over16}\eta^{n-1}$, $$\eqalignno{ ||\hat P_n\Delta_p\sigma_n(z)||^{n}_{s,s'} &\leq\varepsilon^{{3\over2}}|\omega\cdot p|. &\equat(Dpsigman)\cr }$$ Delaying the proof of the above proposition to the end of this section, we now construct a diagonal operator $A_n\in\LL(h_s,h_{s'})$ such that $\tilde\sigma_n\equiv\sigma_n-A_n$ obeys $$ ||\hat P_n\tilde\sigma_n||^n_{s,s'}= \sup_{{\bf k}\in\II^n}\sup_{q\in S^n_{\bf k}} |\PP^n_{\bf k}\tilde\sigma_n(q,q)\PP^n_{\bf k}|_{s,s'} \leq\onehalf\varepsilon\eta^{n}. \equation(bndtildesigma) $$ The equality above follows from the sets $S^n_{\bf k}$ being pairwise disjoint. This will conclude the proof of iterating \equ(thdbnd), since \equ(boundd1Dwn), \equ(bndtaufinal) and \equ(bndtildesigma) imply that the derivative of $\tilde w_n\equiv w_n-A_n$ satisfies the required bound for $r=r(\eta)$ small enough. In order to obtain bound \equ(bndtildesigma) by using the continuity property \equ(Dpsigman), we would like to construct $A_n$ as an approximation of $\sigma_n(q,q)$ for $\omega\cdot q$ close to the normal frequencies in $\CC^n_{\bf k}$, ${\bf k}\in\II^n$. To this end, we set $\bar\mu_{\bf k}$ to be the center of the interval $I^n_{\bf k}$ and, using that $\{\omega\cdot q\,|\,q\in\integer^d\}$ is dense in $\real$, we choose a sequence $\{q_{l,{\bf k}}\}_{l\geq1}\subset S^n_{\bf k}$ such that $\omega\cdot q_{l,{\bf k}}>0$ for all $l\geq1$ and $$ \lim_{l\rightarrow\infty}\omega\cdot q_{l,{\bf k}}=\bar\mu_{\bf k}. $$ Next, one defines the matrix $\hat a_{n,{\bf k}}\in\LL(\JJ^n_{\bf k})$ by $$ \hat a_{n,{\bf k}}\equiv\lim_{l\rightarrow\infty} \PP^n_{\bf k}\sigma_n(q_{l,{\bf k}},q_{l,{\bf k}})\PP^n_{\bf k}. \equation(deltain) $$ Due to \equ(Dpsigman), the limit in \equ(deltain) exists and does not depend on the particular choice of the sequence $\{q_{l,{\bf k}}\}_{l\geq1}$. Finally, setting $$ a_n\equiv \bigoplus_{{\bf k}\in\II^n} \hat a_{n,{\bf k}}, \equation(defsigman0) $$ we define the operator $A_n:h\rightarrow h$ as given by the diagonal kernel $$ A_n(q,q)=a_n{\rm I}_{\QQ^+_\omega}(q)+ \overline{a_n}\,\,{\rm I}_{\QQ^-_\omega}(q) \equation(defsigman) $$ for all $q\in\integer^d$. We note that by construction, \equ(subinv) is clearly satisfied. Furthermore, it follows from \equ(symm1) and \equ(symm2) that $a_n$ is indeed hermitian. Let us check that definition \equ(defsigman) leads to the required bound \equ(bndtildesigma). By construction, one has for all ${\bf k}\in\II^n$, $$ \lim_{l\rightarrow\infty}\PP^n_{\bf k} \tilde\sigma_n(q_{l,{\bf k}},q_{l,{\bf k}})\PP^n_{\bf k}=0. \equation(limtilde) $$ On the other hand, since $\Delta_pA_n=0$, bound \equ(Dpsigman) is also satisfied by $\tilde\sigma_n$, which by definition of the norm implies that $$ |\PP^n_{\bf k}\Delta_p\tilde\sigma_n(q,q)\PP^n_{\bf k}|_{s,s'}\leq \varepsilon^{{3\over2}}|\omega\cdot p|, \equation(contts) $$ for all $q\in S^n_{\bf k}$, ${\bf k}\in\II^n$, and $p\in\integer^d$ with $|\omega\cdot p|<{1\over16}\eta^{n-1}$. The definition of $S^n_{\bf k}$ together with \equ(bndabsIin) implies that $|\omega\cdot(q-q')|\leq2\bar d\eta^n\leq{1\over16}\eta^{n-1}$ for all $q,q'\in S^n_{\bf k}$ with $\sign{(\omega\cdot q)}=\sign{(\omega\cdot q')}$ and $\eta$ small enough. Therefore, using $$ \tilde\sigma_n(q,q)= \tilde\sigma_n(q',q')+ \Delta_{q-q'}\tilde\sigma_n(q',q'), $$ one infers from \equ(contts) that for all $q_{l,{\bf k}}$ and $q\in S^n_{\bf k}$ with $\omega\cdot q>0$, $$ |\PP^n_{\bf k}\tilde\sigma_n(q,q)\PP^n_{\bf k}|_{s,s'}\leq |\PP^n_{\bf k}\tilde\sigma_n(q_{l,{\bf k}},q_{l,{\bf k}}) \PP^n_{\bf k}|_{s,s'}+ \varepsilon^{{3\over2}}|\omega\cdot(q-q_{l,{\bf k}})|, \equation(tsigtmp) $$ which, with \equ(limtilde), leads to $$ |\PP^n_{\bf k}\tilde\sigma_n(q,q)\PP^n_{\bf k}|_{s,s'}\leq 2\bar d\varepsilon^{{3\over2}}\eta^n. \equation(fbnd) $$ For $q\in S^n_{\bf k}$ with $\omega\cdot q<0$, we note that \equ(tsigtmp) is also valid if one replaces $q_{l,{\bf k}}$ by $-q_{l,{\bf k}}$, and, due to \equ(symm1), that the same is true of \equ(limtilde). Therefore, \equ(fbnd) holds for all $q\in S^n_{\bf k}$, ${\bf k}\in\II^n$, and bound \equ(bndtildesigma) follows by taking $\varepsilon$ small enough. Finally, we check that $A_n$ obeys \equ(deltan). The a priori bound \equ(aprioriDw) together with \equ(bndtildesigma) imply that $||\hat P_nA_n||^{(n)}_{s,s'}\leq3\varepsilon\eta^{n-1}$, which, with \equ(subinv) and definition \equ(defsigman), leads to \equ(deltan). To complete the proof of part (c) of \clm(itbounds), we are left with the \noindent{\bf Proof of \clm(continuity).} Denoting $$ Dw_n(z)=\sigma_n(z)+\tau_n(z), $$ with $\sigma_n(z)(q,q')=Dw_n(z)(q,q')\delta_{qq'}$, one computes from \equ(recrelDw) the recursive relation $$ \sigma_n(z)=\bigl(\tilde\sigma_{n-1}(\tilde z)+T_n(z)\bigr) H_n(\tilde z), \equation(recrelsn1) $$ where $$\eqalign{ H_n(\tilde z)&=\bigl(1-\Gamma_n\tilde\sigma_{n-1}(\tilde z) \bigr)^{-1},\cr T_n(z)(q,q')&=\bigl[\tau_n(z)\Gamma_n\tau_{n-1}(\tilde z)\bigr] (q,q')\delta_{qq'}.\cr }$$ Setting $$ \RR_n(z)\equiv\tilde\sigma_{n-1}\bigl(H_n(\tilde z)-1\bigr),\quad \TT_n(z)\equiv T_n(z)H_n(\tilde z), $$ and using $\Delta_p\tilde\sigma_{n-1}=\Delta_p\sigma_{n-1}$ together with the identity $\Delta_p\sigma_0=0$, one applies \equ(recrelsn1) recursively to obtain $$ \Delta_p\sigma_n(z)=\sum_{m=1}^{n}\Delta_p\bigl( \RR_m(z_m)+\TT_m(z_m)\bigr), \equation(recrelsn2) $$ where $z_m=F_n^{m+1}(z)$, cf \equ(deffnm), with $F_n^{n+1}\equiv\identity$. Note that $\RR_m(z)$ is diagonal and can be rewritten as $$ \RR_m(z)=\tilde\sigma_{m-1}(\tilde z) \Gamma_m \tilde\sigma_{m-1}(\tilde z)H_m(\tilde z). \equation(proddiagonal) $$ As shown below, each term in \equ(recrelsn2) is easily seen to be of order $\varepsilon^2|\omega\cdot p|$. Thus, the main issue in obtaining \equ(Dpsigman) is to ensure that taking the sum will deteriorate the bound only slightly. Let us first consider the terms involving the quantities $\Delta_p\TT_m$. They are higher order terms, since $\TT_m$ is quadratic in the off-diagonal part $\tau_m$ which, as shown in \clm(delta1wn), are bounded by powers of $r$. Indeed, as carried out in the Appendix, one has for all $m=1,\dots,n$ and $z\in B_m$, $$ ||\hat P_m\Delta_p\TT_m(z)||^{(m)}_{s,s'}\leq \varepsilon^2\eta^m|\omega\cdot p|, \equation(bndonDpTTm) $$ so that $$ ||\sum_{m=1}^n\hat P_n\Delta_p\TT_m(z)||^{(n)}_{s,s'}\leq \sum_{m=1}^n||\hat P_m\Delta_p\TT_m(z)||^{(m)}_{s,s'}\leq \varepsilon^2|\omega\cdot p|. \equation(DpTTfinal) $$ On the other hand, the terms involving $\Delta_p\RR_m$ are not higher order terms. Since $$ \Delta_pH_m(\tilde z)=t_pH_m(\tilde z)\Bigl(\Delta_p\Gamma_m t_p\tilde\sigma_{m-1}(\tilde z) +\Gamma_m\Delta_p\tilde\sigma_{m-1}(\tilde z) \Bigr)H_m(\tilde z), $$ \equ(DeltapG) with $\sigma=s+\xi-\gamma$ and $n$ replaced by $m$ yields with the recursive bound \equ(Dpsigman) $$ ||\Delta_pH_m(\tilde z)||^{(m-2)}_{s}\leq\eta^{-m} |\omega\cdot p|. \equation(jj3) $$ Thus, using in addition the recursive bounds \equ(thdbnd) and \equ(Dpsigman), together with $$ ||H_m(\tilde z)-1||^{(m-1)}_{s}= ||\Gamma_m\tilde\sigma_{m-1}(\tilde y)H_m(\tilde z)||^{(m-1)}_{s} \leq C\varepsilon, $$ one obtains for all $m=1,\dots,n$ and $z\in B_m$, $$ ||\hat P_n\Delta_p\RR_m(z)||^{(n)}_{s,s'}\leq ||\hat P_m\Delta_p\RR_m(z)||^{(m)}_{s,s'}\leq C\varepsilon^2|\omega\cdot p|, \equation(app11) $$ to be compared with \equ(bndonDpTTm). However, one can actually show that $$\eqalignno{ \Bigr\|\sum_{m=1}^n\hat P_n\Delta_p\RR_m(z)\Bigl\|^{(n)}_{s,s'}&\leq \sup_{{\bf k}\in\II^n}\sup_{q\in S^n_{\bf k}}\sum_{m=1}^n |\Delta_p\RR_m(z)(q)|_{s,s'}&\equat(bbbinter)\cr &\leq C\varepsilon^2|\omega\cdot p|, &\equat(bndonDpRRm)\cr }$$ with another $n$-independent constant $C$. Although \equ(app11) yields the a priori bound $|\Delta_p\RR_m(z)(q)|_{s,s'}\leq C\varepsilon^2|\omega\cdot p|$ for all $q\in S^n_{\bf k}$ and ${\bf k}\in\II^n$, \equ(bndonDpRRm) will follow from the fact that all but a finite number of terms in \equ(bbbinter) are identically zero. More precisely, there is for all ${\bf k}\in\II^n$ a set $\ZZ^n_{\bf k}\subset\{1,\dots,n\}$ with $\#\ZZ^n_{\bf k}$ uniformly bounded in $n$ and ${\bf k}$ such that for all $q\in S^n_{\bf k}$, $$ |\Delta_p\RR_m(z)(q)|_{s,s'}\equiv0 \quad{\rm if}\quad m\not\in\ZZ^n_{\bf k}. \equation(app2) $$ This leads to \equ(bndonDpRRm) and concludes the proof of bound \equ(Dpsigman), since \equ(recrelsn2), \equ(DpTTfinal) and \equ(bndonDpRRm) lead to \equ(Dpsigman) by taking $\varepsilon$ small enough and by noting that $z_m\in B_m$ for all $z\in B_n$, cf. \equ(fnmin). Identity \equ(app2) for some finite set $\ZZ^n_{\bf k}$ follows from the expression \equ(proddiagonal) for $\RR_m$ since by localization of scales $\Gamma_m(q)=(A^{-1}Q_mP_{m-1})(q)=0$ for most $m\leq n$ if $q\in S^n_{\bf k}$. More precisely, one computes that $$ Q_m(q)P_{m-1}(q)= \sum_{\tilde{\bf k}\in\II^m}\bigl(1-\chi^m_{\tilde{\bf k}}(q)\bigr) \chi^{m-1}_{\tilde{\bf k}_{m-1}}(q)\PP^m_{\tilde{\bf k}}, $$ where the index $\tilde{\bf k}_{m-1}$ serves to denote the (unique) subspace $\JJ^{m-1}_{\tilde{\bf k}_{m-1}}$ containing $\JJ^m_{\tilde{\bf k}}$. Fix now some ${\bf k}\in\II^n$. Then one has for all $q\in S^n_{{\bf k}}$ and all $m\eta^{n+1}$ for $i\not=j$. This partition is unique if $M_k^{n+1}$ is required to be maximal. Furthermore, it follows from \equ(asymptotics2) and \equ(asymptotics3) in (H1) that for $\varepsilon=\varepsilon(c)$ small enough, the components $\sigma(\tilde\mu_{n+1}\PP_k)$ are well separated. Therefore, the sets $S^{n+1}_{\bf k}$, ${\bf k}\in\II^{n+1}$, defined according to \equ(defsin) are pairwise disjoint. Next, \equ(ee2) and the gap condition \equ(gapn2) with $n+1$ replaced by $n$ imply that for $\varepsilon=\varepsilon(c,\eta)$ small enough, every cluster $\CC^{n+1}_{k,i}$ is composed of perturbed eigenvalues belonging to a unique $\CC^n_{k,j^{n+1}_i}$. The distance between these two clusters is at most of order $\OO(k^{-\xi}\varepsilon\eta^{n-1})$, so that \equ(convclust) follows for $n+1$ by induction. In order to iterate \equ(PkiPlj), we note that by definition, $\JJ^{n+1}_{\bf k}$ is the eigenspace of $\tilde\mu_{n+1}$ associated with $\CC^{n+1}_{\bf k}$, ${\bf k}\in\II^{n+1}$, and that every $\JJ^n_{{\bf k}'}$, ${\bf k}'\in\II^{n}$, is also an invariant subspace for $\tilde\mu_{n+1}$ by \equ(subinv). Therefore, each $\JJ^{n+1}_{k,i}$ is contained in a unique $\JJ^n_{k,j^{n+1}_i}$, namely, the eigenspace associated with $\CC^n_{k,j^{n+1}_i}$. Finally, we check that \equ(proppn) iterates. This is a simple consequence of \equ(PkiPlj) and $S^{n+1}_{k,i}\subset S^n_{k,j^{n+1}_i}$, the latter following from \equ(ee2) for $\varepsilon$ small enough. \section Measure Estimate In this section, we prove \clm(measureest), namely, that $\Omega^*(K)=\bigcap_{n\geq1}\Omega_n(K)$ satisfies $$ \lim_{K\rightarrow0}{\rm meas}(\Omega\setminus\Omega^*(K))=0, \equation(measest) $$ for all bounded $\Omega\subset\real^d$. The strategy is standard and consists in studying the complementary sets of $\Omega_n(K)$. For $n\geq1$, $b>0$, and $q\in\integer^d$, let us define $$ \Sigma^n_{q,b}\equiv\Bigl(\bigcup_{\bf k\in\II^n} \Sigma^{n;\bf k}_{q,b}\Bigr) \cup\Bigl(\bigcup_{{\bf k,k'}\in\II^n} \Sigma^{n;{\bf k,k'}}_{q,b}\Bigr), $$ where $$\eqalign{ \Sigma^{n;\bf k}_{q,b}&=\{\omega\in\real^d\ |\ d(|\omega\cdot q|,\CC^n_{\bf k})\leq b\},\cr \Sigma^{n;\bf k,k'}_{q,b}&=\{\omega\in\real^d\ |\ d(|\omega\cdot q|,|\CC^n_{\bf k}\pm\CC^n_{\bf k'}|)\leq b\}.\cr }$$ Next, with $$ \ZZ_n\equiv\{q\in\integer^d\ |\ K^{1\over\nu}\eta^{-{n-1\over\nu}}\leq|q|< K^{1\over\nu}\eta^{-{n\over\nu}}\}, $$ and $$ \Sigma^*(K)\equiv\bigcup_{n\geq1}\bigcup_{q\in\ZZ_n} \Sigma^n_{q,2K|q|^{-\nu}}, $$ one shows first, that for all bounded $\Omega\subset\real^d$, $$ {\rm meas}\bigl(\Omega\cap\Sigma^*(K)\bigr)\leq C_\Omega\,K^{\xi\over\xi+1}, \equation(bndmeas) $$ for some constant $C_\Omega$ depending on $\Omega$ only, and, second, that $$ \bigl[\Sigma^*(K)\bigr]^c\subseteq \Omega^*(K). \equation(inclusion) $$ Obviously, \equ(measest) follows from \equ(bndmeas) and \equ(inclusion). Below, $C_\Omega$ will denote a generic constant that may change from place to place but depends on $\Omega$ only. Let us start with the bound \equ(bndmeas). One has $$ {\rm meas}\bigl(\Omega\cap\Sigma^*(K)\bigr) \leq\sum_{n\geq1}\sum_{q\in\ZZ_n}\bigl(T^{n}_{q,2K|q|^{-\nu}} +\hat T^{n}_{q,2K|q|^{-\nu}}\bigr), \equation(decomp1) $$ where $$ T^{n}_{q,b}=\sum_{{\bf k}\in\II^n}{\rm meas} \bigl(\Omega\cap\Sigma^{n;\bf k}_{q,b}\bigr)\,,\ \hat T^n_{q,b}={\rm meas}\Bigl(\Omega\cap \bigcup_{{\bf k,k'}\in\II^n}\Sigma^{n;\bf k,k'}_{q,b}\Bigr). \equation(T1T2) $$ To treat the terms on the right hand side of \equ(decomp1) involving the quantities $T^{n}_{q,b}$, we first use \equ(bndabsIin) to estimate, $$ {\rm meas}\bigl(\Omega\cap\Sigma^{n;{\bf k}}_{q,b}\bigr)\leq C_\Omega(b+\bar d\eta^n). $$ Next, we note that the asymptotic behavior of the clusters $\CC^n_{\bf k}$, cf. \equ(asymptotics1) and \equ(asymp), implies that $\Omega\cap\Sigma^{n;\bf k}_{q,b}$ is empty if ${\bf k}=(k,\cdot)$ satisfies $k\geq C_\Omega|q|$ for some constant $C_\Omega$. Hence, since the number of indices ${\bf k}$ of the form $(k,\cdot)$ is uniformly bounded in $k$, the number of terms which are non-zero in the sum defining $T^{n}_{q,b}$ is proportional to $|q|$, and one obtains the estimate $T^{n}_{q,b}\leq C_{\Omega}|q|(b+\bar d\eta^n)$. Finally, the fact that $q\in\ZZ_n$ satisfies $\eta^n\leq K|q|^{-\nu}$ leads to $$ \sum_{n\geq1}\sum_{q\in\ZZ_n}T^{n}_{q,2K|q|^{-\nu}} \leq C_\Omega\bigl(2K+\bar dK\bigr) \sum_{q\in\integer^d}|q|^{1-\nu} \leq C_\Omega\, K, \equation(f1) $$ for $\nu=\nu(d)$ large enough. To treat the remaining terms in \equ(decomp1), we first note that, as above, $$ {\rm meas}\bigl(\Omega\cap\Sigma^{n;{\bf k,k'}}_{q,b}\bigr)\leq C_\Omega(b+2\bar d\eta^n). \equation(bbb1) $$ Next, one distinguishes the cases $\gamma=1$ and $\gamma>1$. If $\gamma>1$, then for $k'>k$ the inequality $k'^\gamma-k^\gamma>k'^{\gamma-1}$ and the asymptotic \equ(asymptotics2) imply that $\Omega\cap\Sigma^{n;{\bf k,k'}}_{q,b}$ is empty for ${\bf k}=(k,\cdot)$ and ${\bf k'}=(k',\cdot)$ such that $k'\geq C_\Omega|q|^{1/(\gamma-1)}\equiv k_q$. Furthermore, it follows from \equ(asymp) that for $k_b=b^{-{1\over\xi+1}}$, $$ {\rm meas}\Bigl(\bigcup_{\scriptstyle k>Ck_b\atop\scriptstyle i,j} \Sigma^{n;(k,i),(k,j)}_{q,b}\Bigr)\leq Ck_b^{-\xi}. $$ Therefore, one obtains with \equ(bbb1) $$\eqalign{ \hat T^n_{q,b}&\leq Cb^{{\xi\over\xi+1}}+ \sum_{k=1}^{Ck_b}{\rm meas} (\Omega\cap\Sigma^{n;(k,i),(k,j)}_{q,b}) +\sum_{\scriptstyle k'=2\atop\scriptstyle k0$ and $\nu=\nu(d,\xi)$ large enough. We now consider the case $\gamma=1$. From \equ(asymp) and the asymptotic behavior \equ(asymptotics3), it follows first that $\Omega\cap\Sigma^{n;{\bf k,k'}}_{q,b}$ is empty for ${\bf k}=(k,\cdot)$ and ${\bf k'}=(k',\cdot)$ with $k'-k=l\geq C|q|$, and second that for all $l\geq0$ $$ {\rm meas}\Bigl(\bigcup_{\scriptstyle k>Ck_b\atop\scriptstyle i,j} \Sigma^{n;(k,i),(k+l,j)}_{q,b}\Bigr)\leq Ck_b^{-\xi}, $$ where $k_b=b^{-{1\over\xi+1}}$. Therefore, \equ(bbb1) leads to $$ \hat T^n_{q,b}\leq C|q|b^{{\xi\over\xi+1}}+ C_\Omega b^{-{1\over\xi+1}}|q|(b+2\bar d\eta^n), $$ and one finally obtains for $\nu=\nu(d,\xi)$ large enough, $$ \sum_{n\geq1}\sum_{q\in\ZZ_n}\hat T^{n}_{q,2K|q|^{-\nu}}\leq C_\Omega K^{\xi\over1+\xi}\sum_{q\in\integer^d} |q|^{1-\nu{\xi\over1+\xi}} \leq C_\Omega K^{\xi\over1+\xi}. \equation(f3) $$ Inserting \equ(f1) and \equ(f3) into \equ(decomp1) yields \equ(bndmeas). We now check that \equ(inclusion) holds. If $\omega\not\in\Sigma^*(K)$, then the following is true for all $n\geq1$, $q\in\ZZ_n$ and ${\bf k,k'}\in\II^n$, $$\eqalignno{ &d(|\omega\cdot q|,\CC^n_{\bf k})>2K|q|^{-\nu},&\equat(loc1)\cr &d(|\omega\cdot q|,|\CC^n_{\bf k}\pm\CC^n_{\bf k'}|)>2K|q|^{-\nu}. &\equat(loc2)\cr }$$ Next, we verify that for such $\omega$, this implies that bounds \equ(loc1) and \equ(loc2) hold for all $q\in\bigcup_{m=1}^n\ZZ_m$ provided one replaces the constant $2K$ on the right hand side by $K$. This in turn implies that $\omega\in\Omega_n(K)$ for all $n\geq1$, so that $\omega\in\Omega^*(K)$. Let $mK|q|^{-\nu}.&\equat(bbb3)\cr }$$ Since \equ(bbb3) holds for all $q\in\ZZ_m$, $1\leq m\leq n$, one concludes that $d(|\omega\cdot q|,\CC^n_{\bf k})>K|q|^{-\nu}$ whenever $0<|q|0$ small enough such that $Y_s(\lambda,Z)\in\sO_\sB$ for $|\lambda|<\lambda_2$ and $Z\in\sO_s$. It thus follows from the previous discussion that $\FF$ is analytic in $|\lambda|<\lambda_2$ and $\ZZ\in\sO_s$ with $\FF(\lambda,\ZZ)\in h_s$ and $\FF(\lambda,\ZZ)|_{\lambda=0}=0$ for all $\ZZ\in\sO_s$. One infers, in particular, that the solution of \equ(fpe) at $\lambda=0$ is given by $\ZZ(\lambda)|_{\lambda=0}=0$. Next, one computes $$ D_\ZZ\FF(\ZZ,\lambda)=D_Y Z_s(\lambda,Y_s(\lambda,\ZZ)) D_\ZZ Y_s(\lambda,\ZZ). $$ Since $Y_s(\lambda,Z)|_{\lambda=0}=0$ for all $Z\in\sO_s$ and $Z_s(\lambda,Y)|_{\lambda=0}=0$ for all $Y\in\sO_\sB$, it follows that $D_\ZZ Y_s(\lambda,\ZZ)|_{\lambda=0}= D_Y Z_s(\lambda,Y_s(\lambda,\ZZ))|_{\lambda=0}=0$ for $\ZZ\in\sO_s$, which, in turn, implies $$ D_\ZZ\FF(\lambda,\ZZ)|_{\scriptstyle\lambda=0\atop \scriptstyle\ZZ=0}=0. $$ Therefore, the existence for all $|\lambda|<\lambda_2$ of a unique $\ZZ(\lambda)\in\sO_s$ solving \equ(fpe) follows by the implicit function theorem. \sectionnonr Appendix \noindent {\bf Proof of \clm(intermediary).} \noindent We first consider the estimate \equ(bndinvan). Since $\Gamma_n$ has a diagonal kernel, one has $$ ||\Gamma_n||_{\sigma,\sigma+\gamma}= \sup_{q\in\integer^d}|\sK_n^{-1}(q)Q_n(q) P_{n-1}(q)|_{\sigma,\sigma+\gamma}\ , $$ and using \equ(PkiPlj) one easily computes that $$ \Gamma_n(q)= \sum_{{\bf k}\in\II^n}\sK_n^{-1}(q)\hat\chi^n_{\bf k}(q)\PP^n_{\bf k}, \equation(expGn) $$ whith $$ \hat\chi^n_{\bf k}(q)= \bigl(1-\chi^n_{\bf k}(\omega\cdot q)\bigr) \chi^{n-1}_{{\bf k}_{n-1}}(\omega\cdot q), \equation(defhatchi) $$ where ${\bf k}_{n-1}\in\II^{n-1}$ labels the unique subspace $\JJ^{n-1}_{{\bf k}_{n-1}}$ containing $\JJ^n_{\bf k}$. Note that $\hat\chi^n_{\bf k}(q)\not=0$ only for $q$ in the set $$ \hat S^n_{\bf k}=\{q\in\integer^d\ |\ d(|\omega\cdot q|,\CC^n_{\bf k})\geq\oneeighth\eta^n\ {\rm and}\ d(|\omega\cdot q|,\CC^{n-1}_{{\bf k}_{n-1}})\leq\onefourth\eta^{n-1} \}. \equation(defhatSin) $$ Although the sets $\hat S^n_{\bf k}$ are not pairwise disjoint, $\hat S^n_{\bf k}\cap\hat S^n_{\bf k'}\not=\emptyset$ only if ${\bf k}_{n-1}={\bf k'}_{n-1}$. Since for ${\bf k}=(k,i)$ this happens only if ${\bf k'}=(k,j)$, and since the original frequencies $\mu_k$ are by assumption finitely many times degenerate (uniformly in $k$), there are for all ${\bf k}\in\II^n$ no more than $\bar d$ ${\bf k'}$ such that ${\bf k'}_{n-1}={\bf k}_{n-1}$. Therefore, one obtains, with $0\leq\hat\chi^n_{\bf k}\leq1$, $$ ||\Gamma_n||_{\sigma,\sigma+\gamma}\leq \bar d\sup_{{(k,i)}\in\II^n} \sup_{q\in\hat S^n_{(k,i)}}k^\gamma|\sK_n^{-1}(q)\PP^n_{k,i}|. \equation(bndG1) $$ Let us now fix some ${\bf k}=(k,i)\in\II^n$ and $q\in\hat S^n_{\bf k}$ with $\omega\cdot q>0$. Thus, $P_m(q)=\PP^m_{{\bf k}_m}$ for all $0\leq m\leq n-2$, whereas $P_{n-1}(q)=\chi^{n-1}_{{\bf k}_{n-1}}(q)\PP^{n-1}_{{\bf k}_{n-1}}$. This in turn imply, since $\sK_n=\sK_0-\sum_{m=0}^{n-1}A_mP_m$ and $q\in\QQ^+_\omega$, that $\sK_n(q)$ can be rewritten as $$ \sK_n(q)=\hat \sK_n(q)+\hat\mu_n(q), \equation(hatAhatmu) $$ where $$\eqalignno{ \hat \sK_n(q)&=|\omega\cdot q|^2-\mu^2-\sum_{m=0}^{n-1} a_m\PP^m_{{\bf k}_m}, &\equat(defhatAn)\cr \hat\mu_n(q)&=\bigl(1-\chi^{n-1}_{{\bf k}_{n-1}}(\omega\cdot q)\bigr) a_{n-1}\PP^{n-1}_{{\bf k}_{n-1}}. &\equat(defhatmun)\cr }$$ Note that $\hat \sK_n$ is defined in such a way that $(|\omega\cdot q|^2-\hat \sK_n(q))\PP^n_{\bf k}=\tilde\mu_n^2$, cf. \equ(defmun). One thus infers from the definition of $\JJ^n_{\bf k}$ that $\JJ^n_{\bf k}$ is an invariant subspace of $\hat \sK_n(q)$. Therefore, since the spectrum of $(|\omega\cdot q|-\tilde\mu_n)\PP^n_{\bf k}$ is bounded away from zero by $\eta^n/8$, one concludes from the identity $|\omega\cdot q|^2-\tilde\mu_n^2=(|\omega\cdot q|-\tilde\mu_n) (|\omega\cdot q|+\tilde\mu_n)$ and the asymptotic behavior \equ(asymptotics1), \equ(asymp), that $$ |\hat \sK_n^{-1}(q)\PP^n_{\bf k}|\leq Ck^{-\gamma}\eta^{-n}. \equation(l32b1) $$ On the other hand, denoting $f(q)=1-\chi^{n-1}_{{\bf k}_{n-1}}(\omega\cdot q)$, we compute that $$ \hat \sK_n^{-1}(q)\hat\mu_n(q)=f(q) \bigl(|\omega\cdot q|^2-\tilde\mu_n^2\bigr)^{-1} \PP^{n-1}_{{\bf k}_{n-1}}a_{n-1}. $$ Since $f(q)=0$ whenever $d(|\omega\cdot q|,\CC^n_{\bf k'})\leq\eta^{n}$ for all ${\bf k'}$ such that $\JJ^n_{\bf k'}\subset \JJ^{n-1}_{{\bf k}_{n-1}}$, and since \equ(deltan) (with $n$ replaced by $n-1$) implies $|a_{n-1}|\leq3\varepsilon k^{\gamma-\xi}\eta^{n-2}$, one estimates for $\varepsilon$ small enough that $\bigl|\hat \sK_n^{-1}(q)\hat\mu_n(q)\bigr|\leq k^{-\xi}/4\leq1/4$, which leads to $$ \bigl|\bigl(1+\hat \sK_n^{-1}(q)\hat\mu_n(q)\bigr)^{-1}\bigr|\leq2. \equation(l32b2) $$ Bound \equ(bndinvan) finally follows from \equ(bndG1) by applying \equ(l32b1) and \equ(l32b2) to $$ \sK_n^{-1}(q)\PP^n_{\bf k}= \bigl(1+\hat\sK_n^{-1}(q)\hat\mu_n(q)\bigr)^{-1} \hat \sK_n^{-1}(q)\PP^n_{\bf k}, $$ and by noting that for $q\in\hat S^n_{\bf k}\cap\QQ^-_\omega$, the previous analysis must be carried out with $\overline{a_m}$ instead of $a_m$, $m=0,\dots,n-1$, and leads to identical bounds since $a_m$ being hermitian implies $\sigma\bigr(\,\overline{\tilde\mu_n}\,\bigl)=\sigma(\tilde\mu_n)$. To conclude the proof of \clm(intermediary), it remains to check bound \equ(DeltapG). If $p\in\integer^d$ is such that $|\omega\cdot p|\geq\eta^{n+1}$, one can estimate $$ ||\Delta_p\Gamma_n||_{\sigma,\sigma+\gamma} \leq2||\Gamma_n||_{\sigma,\sigma+\gamma} \leq2\eta^{-n-1}||\Gamma_n||_{\sigma,\sigma+\gamma}\,|\omega\cdot p|, $$ which, with \equ(bndinvan), leads to \equ(DeltapG) for some other constant $C$. Let us assume now that $|\omega\cdot p|<\eta^{n+1}$. One computes from \equ(expGn) that $$\eqalignno{ \Delta_p\Gamma_n(q)&= \sum_{{\bf k}\in\II^n}\Bigl(\sK_n^{-1}(q+p)\hat\chi^n_{\bf k}(q+p)- \sK_n^{-1}(q)\hat\chi^n_{\bf k}(q)\Bigr)\PP^n_{\bf k}&\cr &=\sum_{{\bf k}\in\II^n}\Delta_p\sK_n^{-1}(q) \PP^n_{\bf k}t_p\hat\chi^n_{\bf k}(q) +\sum_{{\bf k}\in\II^n}\sK_n^{-1}(q)\PP^n_{\bf k} \Delta_p\hat\chi^n_{\bf k}(q). &\equat(l32b3)\cr }$$ We now fix some ${\bf k}=(k,i)\in\II^n$ and start by considering the second sum on the right hand side of \equ(l32b3). Since $p$ is such that $|\omega\cdot p|<\eta^{n+1}$, $\Delta_p\hat\chi^n_{\bf k}(q)$ is non zero only for $q$ in a set $\tilde S^n_{\bf k}$ that satisfies, with respect to the cluster $\CC^n_{\bf k}$, similar gap condition as $\hat S^n_{\bf k}$. Therefore, the bounds derived previously imply that $|\sK_n^{-1}(q)\PP^n_{\bf k}|\leq Ck^{-\gamma}\eta^{-n}$ for $q\in\tilde S^n_{\bf k}$, and one concludes by noting that $$ |\Delta_p\hat\chi^n_{\bf k}(q)|\leq C\eta^{-n}|\omega\cdot p|, $$ for all $q\in\integer^d$. We now consider the first sum on the right hand side of \equ(l32b3). Let us fix $q$ satisfying $t_p\hat\chi^n_{\bf k}(q)\not=0$. Using the same notation, we decompose $\sK_n$ as in \equ(hatAhatmu) and express $$ \Delta_p\sK_n^{-1}(q)=\Bigr[ \bigl(1+t_p(\hat \sK_n^{-1}\hat\mu_n)\bigr)^{-1} \Delta_p\hat \sK_n^{-1} \bigl(1+\hat\mu_n\hat \sK_n^{-1}\bigr)^{-1}\Bigl](q). $$ Bound \equ(l32b2) implies that $\bigl|\bigl(1+t_p(\hat\sK_n^{-1}\hat\mu_n)(q)\bigr)^{-1} \bigr|\leq2$. Since $|\omega\cdot p|<\eta^{n+1}$, it follows that $\bigl|\bigl(1+\hat \sK_n^{-1}\hat\mu_n(q)\bigr)^{-1}\bigr|$ satisfies a similar bound for $q$ with $t_p\hat\chi^n_{\bf k}(q)\not=0$. Therefore, using in addition \equ(l32b1), one obtains $$\eqalign{ \bigl|\bigl(\Delta_p\sK_n^{-1}\bigr)(q)\PP^n_{\bf k}\bigr| &\leq2\bigl|\Delta_p\hat \sK_n^{-1}(q)\PP^n_{\bf k}\bigr|\cr &\leq2\bigl|t_p\hat \sK_n^{-1}(q)\PP^n_{\bf k}\bigr|\, \bigl||\omega\cdot q|^2-|\omega\cdot(q+p)|^2\bigr|\, \bigl|\hat \sK_n^{-1}(q)\PP^n_{\bf k}\bigr|\cr &\leq Ck^{-\gamma}\eta^{-2n}|\omega\cdot p|,\cr }$$ where $|\omega\cdot p|<\eta^{n+1}$ has been used again to conclude that $|\hat \sK_n^{-1}(q)\PP^n_{\bf k}|\leq Ck^{-\gamma}\eta^{-n}$ is also verified. This concludes the proof of bound \equ(DeltapG) and \clm(intermediary). \qed \noindent {\bf Proof of Bound \equ(bndonDpTTm).} \noindent Bound \equ(bndonDpTTm) is a simple consequence of \clm(intermediary), \clm(delta1wn), and the a priori bound $$ ||\hat P_m\Delta_pDw_m(z)||^{(m)}_{s,s'}\leq\varepsilon4^m| \omega\cdot p|, \equation(apDppn) $$ valid for all $m=1,\dots,n$, $z\in B_m$ and $p$ satisfying $|\omega\cdot p|<{1\over16}\eta^{m-1}$. Indeed, bounds \equ(bndinvan), \equ(boundd1Dwn) and \equ(bndtaufinal) lead to $$ ||H_m(\tilde z)||^{(m-1)}_{s}\leq2\quad{\rm and}\quad ||\hat P_{m-1}T_m(z)||^{(m-2,m-1)}_{s,s'}\leq \varepsilon^2r^{{m\over2}}. \equation(jj1) $$ Using in addition \equ(DeltapG) and the a priori bound \equ(apDppn), one estimates that for $r=r(\eta)$ small enough, $$ ||\hat P_m\Delta_pT_m(z)||^{(m-1,m)}_{s,s'} \leq\varepsilon^2r^{{m\over4}}|\omega\cdot p|. \equation(jj2) $$ Hence, \equ(bndonDpTTm) follows from \equ(jj1), \equ(jj2) and \equ(jj3) by taking $r=r(\eta)$ small enough and noting that if $p$ satisfies $|\omega\cdot p|<\eta^{m-1}/16$, then the following estimate holds for any operator $B\in\LL(h^{m}_s,h_{s'})$, $$ ||\hat P_mt_p B||^{(m)}_{s,s'}\leq||\hat P_{m-1}B||^{(m-1)}_{s,s'}. \equation(veryspecial) $$ It thus remains to check the a priori estimate \equ(apDppn). In the sequel, we use the shorter notation $\pi_n=Dw_n$. Using $\Delta_p(ab)=\Delta_p ab+t_pa\Delta_pb$, one computes from the recursive relation \equ(recrelDw) that for all $m=1,\dots,n$, $$ \Delta_p\pi_m(z)=t_p\tilde H_m(z)\Delta_p\tilde\pi_{m-1}(\tilde z) H_m(\tilde z)+ t_p\pi_m(z)\Delta_p\Gamma_m\pi_m(z), \equation(recDppi) $$ where $H_m(\tilde z)$ is given by \equ(defH), and $$ \tilde H_m(z)=1+\pi_m(z)\Gamma_m. \equation(deftH) $$ To treat the first term on the right hand side of \equ(recDppi), one first estimates, as previously, that for $\varepsilon$ small enough, $||H_m(\tilde z)||^{(m-1)}_s\leq2$ and, using \equ(bndinvan) and the a priori bound \equ(aprioriDw), $$ ||\hat P_{m-1}\tilde H_m(z)||^{(m-1)}_{s'}\leq2. \equation(bndtildeH) $$ Next, remarking that $\hat P_mt_p\tilde H_m=\hat P_mt_p\tilde H_mt_p\hat P_{m-1}$, one computes $$\eqalignno{ \hat P_mt_p\tilde H_m\Delta_p\tilde\pi_{m-1}(\tilde z)&= \hat P_mt_p\Bigl(\tilde H_m\hat P_{m-1}t_{-p}\Delta_p\tilde\pi_{m-1} (\tilde z)\Bigr),&\cr &=-\hat P_mt_p\Bigl(\tilde H_m\hat P_{m-1}\Delta_{-p}\tilde\pi_{m-1} (\tilde z)\Bigr), &\equat(junk1)\cr }$$ which, with \equ(veryspecial) and \equ(bndtildeH), leads to $$ ||\hat P_mt_p\tilde H_m(z)\Delta_p\tilde\pi_{m-1}(\tilde z) H_m(\tilde z)||^{(m)}_{s,s'}\leq 4||\hat P_{m-1}\Delta_{-p}\tilde \pi_{m-1}(\tilde z)||^{(m-1)}_{s,s'}. \equation(firstt) $$ In order to treat the second term on the right hand side of \equ(recDppi), we first note that $\Delta_p\Gamma_m=\hat P_{m-2}\Delta_p\Gamma_m\hat P_{m-2}$. 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