Content-Type: multipart/mixed; boundary="-------------9905281510372" This is a multi-part message in MIME format. ---------------9905281510372 Content-Type: text/plain; name="99-200.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-200.comments" AMS-Code: 58F22, 58F27 PACS-Code: 05.45.-a, 45.20.Jj E-mail: jabad@math.utexas.edu, koch@math.utexas.edu For possible updates, see ftp://ftp.ma.utexas.edu/pub/papers/koch/ ---------------9905281510372 Content-Type: text/plain; name="99-200.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-200.keywords" renormalization, invariant manifold, graph transform, hamiltonian flow, invariant torus, quasiperiodic motion, periodic orbit, accumulation ---------------9905281510372 Content-Type: application/x-tex; name="periodic.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="periodic.tex" %%%%%%%%%%%%%%%%%%%% param.1 % \magnification=\magstep1 \pageno=0 % %%%%%%%%%%%%%%%%%%%% fonts.4 % \font\huge=cmbx10 scaled\magstep2 \font\xbold=cmbx10 scaled\magstep1 \font\cbold=cmsy10 scaled\magstep1 \font\mbold=cmmi10 scaled\magstep1 \font\sub=cmbx10 \font\small=cmr7 \font\plg=cmtt10 \font\plgt=cmtt10 at 10truept % %%%%%%%%%%%%%%%%%%%% titles.3 % \count5=0 \count6=1 \count7=1 \count8=1 \def\proof{\medskip\noindent{\bf Proof.\ }} \def\qed{\hfill\smallskip \line{\hfill\vrule height 1.8ex width 2ex depth +.2ex \ \ \ \ \ \ }\bigskip} % \def\references{\bigskip\noindent\hbox{\bf References}\medskip} \def\remark{\medskip\noindent{\bf Remark.\ }} \def\nextremark{\smallskip\noindent$\circ$\hskip1.5em} \def\firstremark{\bigskip\noindent{\bf Remarks.}\nextremark} \def\abstract#1\par{\noindent{\sub Abstract.} #1 \par} \def\equ(#1){\hskip-0.03em\csname e#1\endcsname} \def\clm(#1){\csname c#1\endcsname} \def\equation(#1){\eqno\tag(#1)} %\def\equation(#1){\eqno\tag(#1) {\rm #1}} \def\tag(#1){(\number\count5. \number\count6) \expandafter\xdef\csname 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\let\cl=\centerline \let\wh=\widehat \let\wt=\widetilde \let\eps=\varepsilon \let\sss=\scriptscriptstyle % \input amssym.def \input amssym.tex \def\RG{\Re} \def\bigRG{\hbox{\cbold <}} \def\mean{{\Bbb E}} \def\proj{{\Bbb P}} \def\natural{{\Bbb N}} \def\integer{{\Bbb Z}} \def\rational{{\Bbb Q}} \def\real{{\Bbb R}} \def\complex{{\Bbb C}} \def\torus{{\Bbb T}} \def\iso{{\Bbb J}} \def\Id{{\Bbb I}} \def\id{{\rm I}} % \def\half{{1\over 2}} \def\third{{1\over 3}} \def\quarter{{1\over 4}} % \def\AA{{\cal A}} \def\BB{{\cal B}} \def\CC{{\cal C}} \def\DD{{\cal D}} \def\EE{{\cal E}} \def\FF{{\cal F}} \def\GG{{\cal G}} \def\HH{{\cal H}} \def\II{{\cal I}} \def\JJ{{\cal J}} \def\KK{{\cal K}} \def\LL{{\cal L}} \def\MM{{\cal M}} \def\NN{{\cal N}} \def\OO{{\cal O}} \def\PP{{\cal P}} \def\QQ{{\cal Q}} \def\RR{{\cal R}} \def\SS{{\cal S}} \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}} % %%%%%%%%%%%%%%%%%%%% figtex2.tex % \newdimen\texpscorrection \texpscorrection=0truecm %%%%% must be 0.15truecm in ps_fonts \font\eightrm=cmr8 \font\cmrseven=cmr7 \font\cmbxseven=cmbx7 \def \eightpoint{\rm} \def\ifff(#1,#2,#3){\ifundefined{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3}\else \write16{!!!!!doubly defined #1,#2}\fi} \def\NEWDEF #1,#2,#3 {\ifff({#1},{#2},{#3})} \def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax} %%%%% figure psfile height (in cm) width (in cm) caption \newcount\FIGUREcount \FIGUREcount=0 \newskip\ttglue \newdimen\figcenter \def\figure #1 #2 #3 #4\cr{\null\ifundefined{fig#1}\global \advance\FIGUREcount by 1\NEWDEF fig,#1,{Figure~\number\FIGUREcount}\fi \write16{ FIG \number\FIGUREcount: #1} {\goodbreak\figcenter=\hsize\relax \advance\figcenter by -#3truecm \divide\figcenter by 2 \midinsert\vskip #2truecm\noindent\hskip\figcenter \special{psfile=#1}\vskip 0.8truecm\noindent \vbox{\eightpoint\noindent {\bf\fig(#1)}: #4}\endinsert}} %%%%% figurewithtex psfile texfile height (in cm) width (in cm) caption \def\figurewithtex #1 #2 #3 #4 #5\cr{\null\ifundefined{fig#1}\global \advance\FIGUREcount by 1\NEWDEF fig,#1,{Figure~\number\FIGUREcount}\fi \write16{ FIG \number\FIGUREcount: #1} {\goodbreak\figcenter=\hsize\relax \advance\figcenter by -#4truecm \divide\figcenter by 2 %%%%% \midinsert \bigskip\vbox{ \vskip #3truecm\noindent\hskip\figcenter \special{psfile=#1}{\hskip\texpscorrection\input #2 } \vskip 0.4truecm\noindent \centerline{\noindent{\cmbxseven\fig(#1).~\cmrseven #5}} }\bigbreak %%%%% \endinsert }} \def\fig(#1){\ifundefined{fig#1}\global \advance\FIGUREcount by 1\NEWDEF fig,#1,{Figure~\number\FIGUREcount} \fi \csname fig#1\endcsname\relax} % %%%%%%%%%%%%%%%%%%%% periodic5.tex % \def\ssH{{\scriptscriptstyle H}} \def\ssT{{\scriptscriptstyle T}} % \def\rGre{1} \def\rBKa{2} \def\rFDLl{3} \def\rTomi{4} \def\rDDLl{5} \def\rED{6} \def\rMEsc{7} \def\rMcKi{8} \def\rDCGMi{9} \def\rAKW{10} \def\rKi{11} \def\rKos{12} \def\rMMS{13} \def\rMcKiv{14} \def\rCGJ{15} \def\rCGJK{16} \def\rCJBC{17} \def\rKol{18} \def\rMo{19} \def\rArn{20} \def\rTh{21} \def\rDLl{22} \def\rCE{23} \def\rHPS{24} \def\rPal{25} \def\rPDM{26} % %%%%%%%%%%%%%%% \cl{{\huge Renormalization and Periodic Orbits}} \cl{{\huge for Hamiltonian Flows}} \vskip.6in \cl{Juan J.~Abad, Hans Koch \footnote{$^1$} {{\small Supported in Part by the National Science Foundation under Grant No. DMS--9705095.}}} \cl{Department of Mathematics, University of Texas at Austin} \cl{Austin, TX 78712} \vskip.8in \abstract We consider a renormalization group transformation $\RR$ for analytic Hamiltonians in two or more dimensions, and use this transformation to construct invariant tori, as well as sequences of periodic orbits with rotation vectors approaching that of the invariant torus. The construction of periodic and quasiperiodic orbits is limited to near--integrable Hamiltonians. But as a first step toward a non--perturbative analysis, we extend the domain of $\RR$ to include any Hamiltonian for which a certain non--resonance condition holds. \par\vfill\eject \section Introduction and Results In this paper we complement and extend the results given in [\rKi], by using a renormalization group (RG) transformation to construct sequences of periodic orbits for near--integrable Hamiltonians, and by extending the domain of this transformation to a larger set of Hamiltonians. The construction of periodic orbits that approximate quasiperiodic motion is a canonical application of RG ideas; see for example [\rCE,\rED--\rAKW]. It relates observed universal accumulation rates to eigenvalues of the linearized RG transformation. This part of our analysis is restricted to near--integrable Hamiltonians, since RG fixed points relevant to critical cases have not yet been obtained rigorously. But the first part, which includes the definition of the RG transformation, does not require near--integrability. The work presented here is essentially self--contained, but for a motivation of some of our choices, and other background material, the reader is referred to [\rKi]. We start with some definitions. On $\complex^d$, consider the two norms $|v|=\sum_j|v_j|$ and $\|v\|=\max_j|v_j|$. Let $V$ and $W$ be two fixed but arbitrary $d\times d$ matrices over $\complex$, satisfying $V^\ssT W=\id$, where $V^\ssT$ denotes the transposed of $V$. Define $\DD_{\rho,1}=\bigl\{q\in\complex^d\colon|V {\rm Im} q|<\rho\bigr\}$ and $\DD_{\rho,2}=\bigl\{p\in\complex^d\colon\|W p\|<\rho\bigr\}\,$, for every $\rho>0$. Unless stated otherwise, we will identify $\DD_{\rho,1}$ with a complexified $d$--torus. In particular, a function on $\DD_\rho=\DD_{\rho,1}\times\DD_{\rho,2}$ is assumed to be $2\pi$--periodic in each component of its first argument. If analytic, such a function $H$ may be written as a Fourier--Taylor series $$ H(q,p)=\sum_{(\nu,\alpha)\in I}H_{\nu,\alpha}\, (Wp)^\alpha e^{i q\cdot\nu} \,, \qquad (q,p)\in\DD_\rho \,, \equation(FTdef) $$ where $I=\integer^d\!\times\!\natural^d$. Here, and in what follows, $x\cdot y$ denotes the standard dot product of two vectors in $\complex^d$, and $x^\alpha=x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_d^{\alpha_d}\,$. \claim Definition(AArho) Given any $\rho>0$, define $\AA_\rho$ to be the Banach space of all analytic Hamiltonians $H$ on $\DD_\rho$ of the form \equ(FTdef), for which the norm $$ |H|_\rho=\sum_{(\nu,\alpha)\in I} |H_{\nu,\alpha}|\rho^{|\alpha|}e^{\rho\|W\nu\|} \equation(AArhoNorm) $$ is finite. The identity operator on $\AA_\rho$ will be denoted by $\Id$. On the product space $\AA_\rho^d$ we define the two norms $|f|_\rho=\sum_j|f_j|_\rho$ and $\|f\|_\rho=\max_j|f_j|_\rho\,$. Let $\AA'_\rho$ be the space of all functions in $\AA_\rho\,$, whose first partial derivatives belong to $\AA_\rho\,$. On this space, we consider the following two seminorms: Denoting by $\nabla_jH$ the partial gradient of $H$ with respect to the $j$--th argument, $j=1,2$, we define $|H|_\rho'$ and $\|H\|_\rho'$ to be the sum and maximum, respectively, of the numbers $\|W\nabla_1H\|_\rho$ and $|V\nabla_2H|_\rho\,$. Our first result is concerned with the possibility of finding a canonical change of variables that eliminates the component of a Hamiltonian in the direction of some predefined subspace of $\AA_\rho\,$. The coordinate changes are restricted to those canonical transformations $U\colon (q,p)\mapsto(q+Q,p+P)$ for which the one--form $P\cdot dq+p\cdot dQ$ is not just closed, but the differential of a function $S$. The function $\phi=p\cdot Q-S$, expressed in terms of $q$ and $p+P$ (if possible), satisfies the equation $$ \bigl(Q(q,p),P(q,p)\bigr)=(\iso\nabla\phi)\bigl(q,p+P(q,p)\bigr)\,, \equation(GenFun1) $$ where $\iso(q,p)=(p,-q)$. Conversely, given $\phi\in\AA_\rho$ sufficiently small, we can use \equ(GenFun1) to define a canonical transformation $U=U_\phi\,$. Denote by $\wh H$ the Hamiltonian vector field associated with a Hamiltonian $H$, that is, $\wh Hf=(\iso\nabla H)\cdot\nabla f$. \claim Theorem(UUHgeneral) Let $\rho,c>0$, and let $\HH$ be some non--empty open set of Hamiltonians $H\in\AA_\rho$ satisfying $|H|_\rho0$, such that for all values of $r$ in the interval $[\rho',(\rho'+8\rho)/9]$, \hfil\break $(a)\;$ if $\;f\in\AA_r\,$, then $\;\Id^{-}f\in\AA_r\;$ and $\;|\Id^{-}f|_r\le a|f|_r\,$; \phantom{$\Bigm|$} \hfil\break $(b)\;$ $\bigl|\Id^{-}H\bigr|_\rho\!abc\rho^{-1}||\phi||_r'\,$. \phantom{$\Bigm|$} \hfil\break Then there exists a map $\UU$ that assigns to each $H\in\HH$ a canonical transformation $\UU_\ssH$ from $\DD_{\rho'}$ to $\DD_{(\rho'+\rho)/2}\,$, such that $$ H\circ\UU_\ssH\in\AA_{\rho'}\,,\qquad \Id^{-}\bigl(H\circ\UU_\ssH\bigr)=0\,. \equation(UUHgeneral) $$ The map $\NN: H\mapsto H\circ\UU_\ssH$ is analytic from $\HH$ to $\AA_{\rho'}\,$. Furthermore, if $H\in\HH$ satisfies $\Id^{-}H=0$, then $\UU_\ssH=\id$ and $D\NN(H)=\Id^{+}-\Id^{+}\wh H\bigl(\Id^{-}\wh H\Id^{-}\bigr)^{-1}\Id^{-}\,$, where $\Id^{+}=\Id-\Id^{-}\,$. This theorem was proved in [\rKi], in the special case where $\HH$ is a small neighborhood of a linear Hamiltonian $(q,p)\mapsto\omega\cdot p$, and $\Id^{-}$ is a specific projection adapted to the choice of the vector $\omega$. A similar projection will be considered again here. Its usefulness for renormalization derives from a combination of \clm(UUHgeneral), and Lemma 1.3 below. Given a nonzero vector $\omega\in\real^d$, and two positive constants $\sigma$ and $\kappa$, let $$ I^{-}=\bigl\{(\nu,\alpha)\in I: |\omega\cdot\nu|>\sigma\|W\nu\|+\kappa|\alpha|\bigr\}\,,\qquad I^{+}=I-I^{-}\,, \equation(IpDef1) $$ and define $\Id^{\pm}H$ by restricting the sum in \equ(FTdef) to the corresponding index sets $I^{\pm}$, $$ \bigl(\Id^{\pm}H\bigr)(q,p)=\sum_{(\nu,\alpha)\in I^{\pm}}H_{\nu,\alpha} (Wp)^\alpha e^{i q\cdot\nu} \,. \equation(IIpmDef1) $$ The functions $\Id^{-}H$ and $\Id^{+}H$ will be referred to as the non--resonant and resonant part, respectively, of $H$. Clearly, the projection $\Id^{-}$ satisfies the condition $(a)$ of \clm(UUHgeneral), with $a=1$. Notice that if $H(q,p)$ depends on $p$ only, then $\Id^{-}H=0$. For the purpose of renormalization, we now focus on vectors $\omega=(1,\omega_2,\ldots,\omega_d)$ whose components span a real algebraic number field of degree $d$. In particular, $\omega\cdot\nu\not=0$ for every nonzero vector $\nu$ in $\integer^d$. Another consequence of this assumption [\rKi] is that there exists an integral $d\times d$ matrix $T$ such that \vskip0pt plus.1\vsize\penalty-75\vskip0pt plus -.1\vsize \medskip \item{(T1)} $T$ has a simple real eigenvalue $\vartheta>1$, and $T\omega=\vartheta\omega$. \item{(T2)} All other eigenvalues of $T$ are simple, and of modulus less than $1$. \item{(T3)} $\det(T)=\pm 1$. \medskip Such a matrix $T$ provides a way of approximating $\omega$ by vectors with rational components: If $w\in\rational^d$ is nonzero, then the vector $T^nw$, when rescaled such that its first component is one, approaches $\omega$ as $n\to\infty$. The same approximating sequences can also be found in some Hamiltonian systems [\rGre--\rAKW], in the form of periodic orbits that accumulate at an invariant $\omega$--torus. In order to investigate this Hamiltonian ``representation'' of the arithmetic related to $\omega$, we ``lift'' the inverse of $T$, viewed as a map on frequency vectors, to a transformation acting on a space of Hamiltonians. A transformation that has some of the required properties is $H\mapsto\mu^{-1}H\circ T_\mu\,$, where $$ T_\mu(q,p)=\bigl(Tq,\mu(T^\ast)^{-1}p\bigr)\,, \qquad\quad \mu\not=0\,. \equation(TmuDef) $$ It combines a canonical change of variables (case $\mu=1$) with a scaling in $p$, and is part of most RG schemes for Hamiltonians [\rAKW--\rCJBC]. By itself, this transformation is not a dynamical system on any of the spaces $\AA_\rho\,$, since the domain $\DD_\rho$ is not left invariant by $T_\mu\,$. But we can combine it with \clm(UUHgeneral): The following lemma shows that a canonical change of variables, that eliminates the non--resonant part of a Hamiltonian, can ``transfer analyticity'' from the variable $p$ to the variable $q$. After fixing an integral $d\times d$ matrix $T$ satisfying (T1--T3), we adapt the matrices $W$ and $V$ to our choice of $T$, by assuming that the row vectors $W_1,W_2,\ldots,W_d$ of $W$ are an ordered basis of eigenvectors for $T$, $$ TW_j=\vartheta_jW_j\,,\qquad |\vartheta_j|\le|\vartheta_i|\,,\qquad 1\le i\le j\le d\,, \equation(TWj) $$ with $W_1=\omega$. In addition we fix the parameters $\sigma,\kappa>0$, with $\sigma$ restricted by the condition $|\vartheta_2|+\sigma(\vartheta-|\vartheta_2|)<1$. \claim Lemma(TmuSub) Let $0<\rho'<\rho$ and $\mu\in\complex$ be given such that $$ |\vartheta_2|+\sigma\bigl(\vartheta-|\vartheta_2|\bigr) <{\rho'\over\rho}\,,\qquad 0<\left|{\mu\over\vartheta_d}\right|e^{\rho\kappa(\vartheta-|\vartheta_2|)} <{\rho'\over\rho}\,. \equation(SigmaMuCond) $$ Then every function $H\in\Id^{+}\AA_{\rho'}$ extends analytically to $T_\mu\DD_\rho\,$, and $H\mapsto H\circ T_\mu$ is a compact linear map from $\Id^{+}\AA_{\rho'}$ to $\AA_\rho\,$, whose operator norm is $\le 1$. Let now $\HH$ be some subset of $\AA_\rho$ for which the given projection $\Id^{-}$ satisfies a non--resonance condition, as described in \clm(UUHgeneral). \claim Definition(RRDef) Given a nonzero complex number $\mu$ of modulus less than $|\vartheta_d|$, define $$ \RR_\mu(H)={\eta\over\mu}H\circ\UU_\ssH\circ T_\mu\,, \qquad\qquad H\in\HH\,, \equation(RRDef) $$ where $\eta=\eta(H)$ is determined such that the coefficient $\wt H_{0,(1,0,\ldots,0)}$ of the renormalized Hamiltonian $\wt H=\RR_\mu(H)$ is equal to one, if this is possible. The action of this transformation is particularly simple when restricted to functions of the form $$ h_w(q,p)=w\cdot p\,, \qquad\qquad w\in\complex^d\,. \equation(HnullDef) $$ In particular, it is independent of the choice of $\mu$, since $h_w$ is invariant under the scaling $\SS_z: H\mapsto z^{-1}H(.,z.)$, for any $z\not=0$. More explicitly, if $w=\beta_1\omega+\beta_2 W_2+\ldots+\beta_d W_d\,$, with $\beta_1$ nonzero, then $\RR_\mu(h_w)=h_{w'}\,$, where $w'=\eta T^{-1}w$ and $\eta=\vartheta/\beta_1\,$. This shows e.g. that $h_\omega$ is a (trivial) fixed point of $\RR_\mu\,$. For particular choices of the scaling parameter $\mu$, there are other trivial fixed points as well. Such fixed points can be found easily by restricting $\RR_\mu$ to the space of Hamiltonians $H\in\AA_\rho$ for which $\nabla_1H=0$, and thus $\UU_\ssH=\id$. \clm(UUHgeneral) and \clm(TmuSub) can be combined as follows. (Additional symmetries of $\RR_\mu$ are mentioned at the end of Section 3.) \claim Theorem(RRexists) Let $0<\rho<\sigma/\kappa$. If $\rho'<\rho$ is sufficiently close to $\rho$, and $\mu\in\complex$ satisfies \equ(SigmaMuCond), then there exists an open neighborhood $\HH$ of $\{H\in\Id^{+}\AA'_s: H_{0,0}=0,\ |H-h_\omega|'_s<\kappa\rho\}$ in $\AA_\rho\,$, where $s=(\rho'+8\rho)/9$, such that the transformation $\RR_\mu$ is well defined, analytic, and compact, as a map from $\HH$ to $\AA_\rho$. The same holds for $\RR_{z\mu}=\SS_z\circ\RR_\mu\,$, for all $z$ in some open neighborhood $Z\subset\complex$ of the unit circle, and $\SS_z\circ\RR_\mu=\RR_\mu\circ\SS_z\,$, for all $z\in Z$. This theorem is obtained by verifying conditions $(b)$ and $(c)$ of \clm(UUHgeneral) for the given domain $\HH$. For simplicity, we have chosen $b>0$ close to zero. This may not be sufficient to be used in the construction of a nontrivial RG fixed point, although the approximate fixed point found in [\rAKW], for the golden mean case $T=\left[{0 1\atop 1 1}\right]$, does have a relatively small non--resonant part. For such a task, it is likely that all parameters (including $\sigma$, $\kappa$, $\rho$, $\rho'$) will have to be fine--tuned simultaneously, and a sharper version of \clm(UUHgeneral) may be needed as well. As mentioned earlier, $\RR_\mu$ acts trivially on Hamiltonians that only depend on the action variable $p$. This makes it possible to compute all eigenvalues and eigenvectors of the derivative $D\RR_\mu(h_\omega)$ of $\RR_\mu$ at the fixed point $h_\omega\,$. The eigenvectors are precisely the monomials $(q,p)\mapsto(Wp)^\alpha$, and the point $0$ in the spectrum of $D\RR_\mu(h_\omega)$ corresponds to functions with torus--average zero; see [\rKi] for details. In what follows, $\rho$ is a fixed positive real number less than $\sigma\kappa$, and the scaling parameter $\mu$ is assumed to be real, satisfying $$ 0<\mu<\left|{\vartheta_d^2\over\vartheta_1}\right|\,. \equation(muinterval) $$ In this case, $h_\omega$ is an isolated fixed point of $\RR_\mu\,$, and $D\RR_\mu(h_\omega)$ has precisely $d$ eigenvalues outside the open unit disk. One of them is $\lambda_0=\vartheta/\mu$, associated with constant Hamiltonians, and the other $d-1$ eigenvalue--eigenvector pairs are $$ \lambda_j=\vartheta_1/\vartheta_j\,,\qquad h_{_{W_j}}(q,p)=W_{\!j}\cdot p\,,\qquad j=2,\ldots,d\,. \equation(lambdaj) $$ The corresponding local unstable manifold $\WW^u$ of $h_\omega$ is simply the $d$--dimensional affine subspace of $\AA_\rho$ that is tangent to the expanding eigenspace at $h_\omega\,$. As usual in the theory of renormalization, the transformation $\RR_\mu$ is merely a tool for constructing and analyzing certain objects that are of interest outside this theory. We start with a discussion of the local stable manifold $\WW^s$ of $\RR_\mu$ at the fixed point $h_\omega\,$. By definition, if $H$ lies on $\WW^s$, then the sequence of Hamiltonians $H_n=\RR_\mu^n(H)$ converges to $h_\omega\,$, as $n\to\infty$. This fact can be used to define a sequence of canonical transformations $$ \VV_n(H)=V_0(H)\circ V_1(H)\circ\ldots\circ V_{n-1}(H)\,,\qquad V_k(H)=T_\mu^k\circ\UU_{H_k}\circ T_\mu^{-k}\,. \equation(VnDef) $$ Formally, $H\circ\VV_n(H)$ approaches a constant multiple of $h_\omega\,$, as $n$ tends to infinity. But we cannot expect convergence on an open subset of phase space, unless $H$ is integrable. One of the things that can be extracted from the transformations $\VV_n(H)$ is the limit of $\VV_n(H)\circ\Upsilon$, as $n\to\infty$, where $\Upsilon(q)=(q,0)$. This limit yields the function $\Gamma_{\!\ssH}$ described below. \claim Theorem(WsFlow) Given two positive real numbers $r'$ and $r>r'+\rho$, there exists an open neighborhood $B$ of $h_\omega$ in $\AA_r$, and for every $H\in B$ a complex number $c_\ssH$ and an analytic function $\Gamma_{\!\ssH}\colon\DD_{r',1}\to\DD_r\,$, such that the following holds. If $H\in\WW^s\cap B$ then $$ \bigl(\iso\nabla H\bigr)\circ\Gamma_{\!\ssH} =c_\ssH\,\omega\cdot\nabla\Gamma_{\!\ssH} \qquad {\rm on}\ \DD_{r',1}\,. \equation(WsFlow) $$ For $H=h_\omega\,$, the values of $c_\ssH$ and $\Gamma_{\!\ssH}$ are $1$ and $\Upsilon$, respectively. Furthermore, $H\mapsto c_\ssH$ is an analytic function on $B$, and $H\mapsto\Gamma_{\!\ssH}-\Upsilon$ is an analytic map from $B$ to some Banach space of analytic $2\pi$-periodic functions on $\DD_{r',1}\,$. This theorem is essentially identical to [\rKi, Theorem 1.7]. Thus, we will not repeat a proof here. Notice that \equ(WsFlow) is the equation of an invariant $d$--torus for $H$, with rotation vector proportional to $\omega$; or more precisely (if $c_\ssH$ is not real), the equation of an invariant $d$--torus for $c_\ssH^{-1}H$, with rotation vector $\omega$. By construction, this torus lies on the energy surface $H^{-1}(0)$. Another property of this torus is that it is ``centered at $p=0$'', in the sense that the integral $$ K(\gamma)={1\over 2\pi}\oint\limits_\gamma p\cdot dq \equation(KgammaDef) $$ vanishes, if $\gamma$ is any closed curve on the torus. This follows from the fact that $\Gamma_{\!\ssH}$ is the limit of canonical transformations $\VV_n(H)$, that leave $p\cdot dq$ invariant up to a differential of a one--form. The assumption $H\in\WW^s$, used to prove \equ(WsFlow), replaces the non--degeneracy assumption in traditional KAM theory. To discuss the connection between these two conditions, we consider families $\beta\mapsto H_\beta\,$, $$ H_\beta=H\circ R_\beta\,,\qquad R_\beta(q,p)=(q,p+\beta)\,, \equation(HbetaDef) $$ generated from a fixed Hamiltonian $H$ by a translation in the action variable $p$. To be more specific, let $r>\rho$, and consider a Hamiltonian $h\in\AA_r$ of the form $$ h(q,p)=\omega\cdot p+{\textstyle{1\over 2}}\, p\cdot Mp+f(p)\,, \qquad f(p)=\OO(|p|^3)\,, \equation(hint) $$ where $M$ is a real symmetric $d\times d$ matrix, such that the quadratic form $p\mapsto p\cdot Mp$ is non--degenerate, when restricted to the $d-1$ dimensional contracting subspace of $T^\ast$. It is straightforward to check that if $h$ is sufficiently close to $h_\omega\,$, then the family $\beta\mapsto H_\beta\,$, for $H=h$, intersects the stable manifold $\WW^s$ transversally. Since this property persists under small perturbations, every Hamiltonian $H\in\AA_r$ near $h$ has an invariant $c\omega$--torus on the energy surface $H^{-1}(0)$. The torus is given by $\Gamma'=R_{\beta'}\circ\Gamma_{\! H_{\beta'}}\,$, where $\beta'$ is the value of the parameter $\beta$ for which $H_\beta$ belongs to $\WW^s$. Under suitable assumptions on $H$, the renormalization transformation $\RR_\mu$ can also be used to construct periodic orbits for $H$, whose rotation vectors are ``rational approximants'' for $c \omega$. Recall that a curve $\gamma:\real\to\DD_\rho$ is a (lifted) orbit for $H$ if it satisfies the first order differential equation $\gamma'=(\iso\nabla H)\circ\gamma$. A periodic orbit that closes (modulo $2\pi\integer^d$ in the variable $q$) after a time $2\pi\tau>0$, but not earlier, can be parametrized as follows: $$ \gamma(t)=\bigl(tw+Q_0+Q(t/\tau)\,,\, P_0+P(t/\tau)\bigr)\,, \equation(Periodic1) $$ where $w=(\gamma(2\pi\tau)-\gamma(0))/(2\pi\tau)$, and where $Q,P$ are periodic functions with fundamental period $2\pi$ and average zero. The rotation vector $w$ belongs to $\real\integer^d$, that is, $w$ is a real scalar multiple of some vector in $\integer^d$. In what follows, given a nonzero vector $w\in\real\integer^d$, we denote by $\tau(w)$, or $\tau$ for short, the value of the smallest positive real number $t$ such that $tw$ belongs to $\integer^d$. In order to construct periodic orbits that depend continuously on $H$, near the Hamiltonian \equ(hint) that is invariant under translations $q\mapsto q+u$, we will need to limit the number of ways this symmetry can be broken by a perturbation. We shall do this by restricting to Hamiltonians $H(q,p)$ that are even functions of $q$ --- a property that is preserved under the transformation $\RR_\mu\,$. The corresponding ``even'' subspace of $\AA_r\,$, $r>0$, will be denoted by $\BB_r\,$. We note that the approximate RG fixed point of [\rAKW] lies in such a space $\BB_r\,$. Let $\rho'$ and $\eps$ be fixed positive real numbers (to be chosen below). \claim Definition(SigmaDef) For every nonzero vector $w\in\real\integer^d$, we define $\HH(w)$ to be the set of all Hamiltonians $H\in\BB_\rho\,$, such that a constant multiple of $H$ has a periodic orbit $\gamma$ with rotation vector $w$, on the surface of constant energy zero, with $K(\gamma)=0$. The functions $Q$ and $P$ in \equ(Periodic1) are assumed to be $2\pi$--periodic, and to have average zero. In addition, we require $P$ to be even, $Q$ odd, and $Q_0=0$. \hfil\break A subset $\Sigma(w)$ of $\HH(w)$ is defined by requiring also that $P_0=kV^\ast Vw$, with $|k|<\eps^2/\tau$, and that $Q,P$ extend analytically to the strip $|{\rm Im} z|<\rho'/\tau$, satisfying the bounds $|VQ(z)|<\eps$ and $\|WP(z)\|<\eps$. \claim Theorem(SigmaOK) There exist $\rho',\eps>0$ such that the following holds. If $w\in\real\integer^d$ is sufficiently close to $\omega$, and $h_w$ lies on $\WW^u$, then there exists an open neighborhood $B(w)$ of $h_w$ in $\BB_\rho\,$, such that $\Sigma(w)\cap B(w)$ is an analytic manifold of codimension $d$ that intersects $\WW^u$ transversally at $h_w\,$. Consider now such a rotation vector $w\in\real\integer^d$, and let $\Sigma_n(w)=\RR_\mu^{-n}(\Sigma(w)\cap B(w))$, for all $n\ge 0$. By the $\lambda$--Lemma [\rPal,\rPDM], this defines a sequence of codimension $d$ manifolds that accumulate at $\WW^s$ as follows (see also Section 4). Assume that $\beta\mapsto H_\beta$ is an analytic $d$--parameter family of Hamiltonians in the domain of $\RR_\mu\,$, that intersect $\WW^s$ transversally at $\beta=\beta'$. Then there exists an open neighborhood $B'$ of $\beta'$ in $\complex^d$, such that for sufficiently large $n$, the set $\{\beta\in B': H_\beta\in\Sigma_n(w)\}$ contains a single point, say $\beta_n\,$, and the ratio $|\beta_n-\beta'|/|\beta_{n+1}-\beta'|$ converges to $|\lambda_2|$, as $n\to\infty$. %------------------------ Fig.1--------------------------------------- \vskip 1.0truecm \figurewithtex sigma_n.eps sigma_n.aux 6.5 9.5 Accumulation of the hypersurfaces $\Sigma_n$ at the stable manifold $\WW^s$\cr \medskip %------------------------ Fig.1--------------------------------------- The reason for considering the manifolds $\Sigma_n(w)$ is the fact that $\Sigma_n(w)\subset\HH(T^nw)$. Thus, by considering families of the type \equ(HbetaDef), we can construct infinite sequences of periodic orbits for a single Hamiltonian $H$. As part of the proof of the theorem below, and under its assumptions, we will show that $$ -{1\over n}\ln\bigl|\beta_n-\beta'\bigr|=\ln|\lambda_2| +\OO\bigl({\textstyle{1\over n}}\bigr)\,. \equation(betasymptotics) $$ \claim Theorem(AccuOrbits) Let $r>\rho$ and $w\in\real\integer^d$ be given, $w\not=0$. Let $h\in\BB_r$ be a Hamiltonian of the form \equ(hint), with $M$ as described after \equ(hint). If $h$ is sufficiently close to $h_\omega$ in $\BB_r\,$, then there exists an open neighborhood $B$ of $h$ in $\BB_r\,$, and a positive integer $N$, such that for every Hamiltonian $H\in B$, and every $n\ge N$, some constant multiple of $H$ has a periodic orbit $\gamma_n$ with frequency vector $w_n=(\vartheta^{-1}T)^nw$, lying on the energy surface $H^{-1}(0)$, and satisfying $$ -{1\over n} \ln\bigl|\gamma_n(0)-\Gamma'(0)\bigr|=\ln|\lambda_2| +\OO\bigl({\textstyle{1\over n}}\bigr)\,, \equation(gammasymptotics) $$ where $\Gamma'$ is the invariant torus described after \equ(hint). Our reason for using $K(\gamma)=0$ as one of the conditions in the definition of $\Sigma(w)$, is the resulting identity $$ {1\over 2\pi\tau(w_n)}\oint\limits_{\gamma_n}p\cdot dq =w_n\cdot\beta_n\,, \equation(AccuOrbits) $$ valid for large $n$, if $w\in\real\integer^d$ is chosen sufficiently close to $\omega$. The normalized integral in this equation can be regarded as the coordinate of $\gamma_n$ in the direction of $w_n\,$, since it changes by an amount $w_n\cdot v$ under a translation $p\mapsto p+v$. It appears that in this direction, the orbits $\gamma_n$ accumulate faster than in some other directions: A straightforward calculation shows that if $\nabla_1H=0$, then the difference $w_n\cdot\beta_n-\omega\cdot\beta'$ is of the order $|\lambda_2|^{-2n}$. The same might be true more generally, as $K$ is the functional that appears in the variational equation for orbits on fixed energy surfaces. \section Eliminating Non--Resonant Modes Our goal in this section is to prove \clm(UUHgeneral), concerning the existence of a canonical change of coordinates that eliminates the component of a Hamiltonian in the direction of a given ``non--resonant'' subspace of $\AA_\rho\,$. We start by giving some basic estimates involving the evaluation, multiplication, differentiation, and composition of functions in the spaces $\AA_\rho\,$. \claim Proposition(Trivial) Let $\rho,\delta>0$. Consider $f,g\in\AA_\rho\,$, and $P,Q\in\AA_\rho^d\,$, and $h\in\AA_{\rho+\delta}\,$. Define $U(q,p)=\bigl(q+Q(q,p),P(q,p)\bigr)$, for all $(q,p)$ in $\DD_\rho\,$. Then \item{$(a)$} $|f(q,p)| \le|f|_\rho\ $ for all $(q,p)$ in $\DD_\rho\,$. $\phantom{\Bigm|}$ \item{$(b)$} $fg\in\AA_\rho$ and $|fg|_\rho\le|f|_\rho|g|_\rho \,$. $\phantom{\bigm|}$ \item{$(c)$} $|h|_\rho+\delta|h|'_\rho\le|h|_{\rho+\delta}\,$. $\phantom{\Bigm|}$ \item{$(d)$} $h\circ U\in\AA_\rho\,$ and $|h\circ U|_\rho\le|h|_{\rho+\delta}\,$, if $|V Q|_\rho\le\delta$ and $\|WP\|_\rho\le\rho+\delta\,$. The proof of these estimates is straightforward and will be omitted. Define $\{H,\phi\}=\nabla_1 H\cdot\nabla_2\phi-\nabla_2 H\cdot\nabla_1\phi$. \claim Proposition(UUphiComp) Let $r,\delta>0$ and $0<\eps<\half$. Denote by $B'$ the set of all functions $\phi\in\AA'_r$ that satisfy $\|\phi\|'_{r+2\delta}<\eps\delta$. Then for every function $\phi\in B'$, the equation \equ(GenFun1) has a unique solution $Q,P\in\AA_r^d$ satisfying $\|WP\|_r\le\delta\,$. The corresponding canonical transformation $U_\phi\colon(q,p)\mapsto\bigl(q+Q(q,p),p+P(q,p)\bigr)$ is analytic from $\DD_r$ to $\DD_{r+2\delta}\,$. If $H$ is any function in $\AA_{r+2\delta}\,$, then $H\circ U_\phi$ belongs to $\AA_r\,$, and $$ \eqalign{ \bigl|H\circ U_\phi\bigr|_r &\le|H|_{r+2\delta}\,,\cr \bigl|H\circ U_\phi-H\bigr|_r &\le{2\over 3}\eps|H|_{r+2\delta}\,,\cr \bigl|H\circ U_\phi-H-\{H,\phi\}\bigr|_r &\le{1\over 3}\eps^2|H|_{r+2\delta}\,.\cr} \equation(UUphiComp1) $$ Furthermore, the maps $\phi\mapsto(Q,P)$ and $\phi\mapsto H\circ U_\phi$ are analytic on $B'$. \proof Denote by $B$ the set of all $P\in\AA_r^d$ satisfying $\|WP\|_r\le\delta\,$. Let $\phi\in B'$, and define a map $F: B\to\AA_r^d$ by setting $F(P)=-(\nabla_1\phi)\circ G$, where $G(q,p)=(q,p+P(q,p))$. If $P\in B$ then, by using \clm(Trivial), we obtain $$ \eqalign{ \|W DF(P)h\|_r &=\max_i\bigl|h\cdot \bigl(\nabla_2(W\nabla_1\phi)_i\bigr)\circ G\bigr|_r\cr &\le\max_i\|W h\|_r \bigl|V\bigl(\nabla_2(W\nabla_1\phi)_i\bigr)\circ G\bigr|_r\cr &\le\max_i\|W h\|_r \bigl|V\nabla_2(W\nabla_1\phi)_i\bigr|_{r+\delta}\cr &\le\max_i\|W h\|_r \delta^{-1}\bigl|(W\nabla_1\phi)_i\bigr|_{r+2\delta}\cr &=\|W h\|_r\delta_2^{-1} \|W\nabla_1\phi\|_{r+2\delta} \le\|W h\|_r\,,\cr} \equation(DFPbound) $$ for all $h\in\AA_r^d$. This, together with the bound $\|W F(0)\|_r\le\delta/2$, shows that $F$ is a contraction on $B$, for the norm $\|W .\|_r\,$. Thus, equation \equ(GenFun1) has a unique solution $(Q,P)\in\AA_r^d\times B$. By \clm(Trivial), this solution satisfies $$ \eqalign{ |V Q|_r&=\bigl|(V\nabla_2\phi)\circ G\bigr|_r \le|V\nabla_2\phi|_{r+\delta} <\eps\delta\,,\cr \|W P\|_r &=\|(W\nabla_1\phi)\circ G\|_r \le\|W\nabla_1\phi\|_{r+\delta} <\eps\delta\,.\cr} \equation(QPbound) $$ Consider now $H\in\AA_{r+2\delta}\,$. In order to prove \equ(UUphiComp1), let $$ \bigl(f(z)\bigr)(q,p)=H\Bigl(q+z\nabla_2\phi\bigl(q,p+zP(q,p)\bigr)\,,\, p-z\nabla_1\phi\bigl(q,p+zP(q,p)\bigl)\Bigr)\,. \equation(UUphiComp2) $$ From the bounds \equ(QPbound) and \clm(Trivial), it follows that this equation defines a function $f$, from an open neighborhood of the disk $|z|\le 2/\eps$ to $\AA_r\,$, satisfying $|f(z)|_r\le|H|_{r+2\delta}\,$. By using the representation $$ f(s)=f(0)+sf'(0)+{1\over 2\pi i}\oint\limits_{|z|=2/\eps} {f(z)\over z-s}\left({s\over z}\right)^2 dz\,, \equation(fTaylor) $$ we obtain the bound $$ \eqalign{ \bigl|H\circ U_\phi-H-\{H,\phi\}\bigr|_r &=\bigl|f(1)-f(0)-f'(0)\bigr|_r\cr &\le\Biggl|{1\over 2\pi i}\oint\limits_{|z|=2/\eps} {f(z) dz\over(z-1)z^2}\Biggr|_r \le{1\over 3}\eps^2|H|_{r+2\delta}\,.\cr} \equation(UUphiComp3) $$ The first two inequalities in \equ(UUphiComp1) are proved similarly. The analyticity of the maps that assign to $\phi\in B'$ the functions $P,Q\in\AA_r^d$ and $H\circ U_\ssH\in\AA_r\,$, follows by the implicit function theorem and the chain rule. \qed \vskip0pt plus.1\vsize\penalty-75\vskip0pt plus -.1\vsize \proofof(UUHgeneral) We start with an informal description of the proof. Consider $H_0=H\in\HH$. Our goal is to define functions $\phi_0,\phi_1,\phi_2,\ldots$ such that if we set $$ G_n=U_{\phi_0}\circ U_{\phi_1}\circ\ldots\circ U_{\phi_{n-1}}-\id\,,\qquad H_n=H\circ(\id+G_n)\,,\qquad n=1,2,\ldots\,, \equation(HnGnDef) $$ then $G_n\to G_\infty$ and $H_n\to H_\infty=H\circ(\id+G_\infty)$ as $n\to\infty$, with $\Id^{-}H_\infty=0$. Starting with $n=0$, we define $\phi_n=\Id^{-}\phi_n$ to be the solution of the equation $$ \Id^{-}\{H_n,\phi_n\}=-\Id^{-}H_n\,. \equation(PhiDef) $$ If $\Id^{-}H_n$ is small, say of the order $\eps_n\,$, then the same should be true for $\phi_n\,$. By using that $H_{n+1}=H_n\circ U_{\phi_n}\,$, and thus $$ \Id^{-}H_{n+1} =\Id^{-}\bigl(H_n\circ U_{\phi_n}-H_n-\{H_n,\phi_n\}\bigr)\,, \equation(UUphiComp0) $$ we see from equation \equ(UUphiComp1) that $\Id^{-}H_{n+1}$ is of the order $\eps_{n+1}\approx\eps_n^2\,$. Now the process is repeated for $n=1,2,\ldots$. In order to be more precise, assume now that $\HH$ and $\Id^{-}$ satisfy the assumptions of \clm(UUHgeneral). Let $t_0=(1-\rho'/\rho)/9$ and $\rho_0=\rho$. Define $$ t_n=(2/3)^nt_0\,,\quad \delta_n=t_n\rho\,,\quad \rho_n=\rho'+9\delta_n\,,\qquad n=1,2,\ldots\,, \equation(rhonDef) $$ so that $\rho_{n+1}=\rho_n-3\delta_n\,$, for all $n\ge 0$. Given $H\in\HH$, we intend to verify inductively that for all $m>0$, the function $H_m$ belongs to $\AA_{\rho_m}$ and satisfies the bounds $$ \eqalign{ |H_m|_{\rho_m}&< c\,,\cr |H_m-H_{m-1}|_{\rho_m}&<{c\over 4}t_0b(2/3)^{2(3/2)^m}\,,\cr |\Id^{-}H_m|_{\rho_m} &<{ac\over 3}(t_0b)^2(2/3)^{4(3/2)^m}\,.\cr} \equation(FmHmBound) $$ By assumption, these bounds hold for $m=0$, if we set $H_{-1}=0$. Let now $n\ge0$ be fixed, and assume that \equ(FmHmBound) has been verified for all $m\le n$. We start by showing that equation \equ(PhiDef) can be solved. By using \clm(Trivial).c we obtain $$ \eqalign{ |H_n-H|'_{\rho_n-\delta_n} &\le\sum_{m=1}^n|H_m-H_{m-1}|'_{\rho_m-\delta_m} \le\sum_{m=1}^n \delta_m^{-1}|H_m-H_{m-1}|_{\rho_m}\cr &<\sum_{m=1}^n{bc\over 4\rho}(3/2)^n(2/3)^{2(3/2)^m} <{bc\over 2\rho}\,.\cr} \equation(HnDerBound) $$ Thus, if $\phi$ belongs to $\Id^{-}\AA_{\rho_n-\delta_n}'\,$, then $$ \bigl|\Id^{-}\{H_n-H,\phi\}\bigr|_{\rho_n-\delta_n} \le a|H_n-H|'_{\rho_n-\delta_n}\|\phi\|_{\rho_n-\delta_n}' \le{abc\over 2\rho}\|\phi\|_{\rho_n-\delta_n}'\,. \equation(HnHatBound0) $$ This, together with the assumption $(c)$ of \clm(UUHgeneral), implies that $\Id^{-}\wh{H_n}$ maps $\Id^{-}\AA_{\rho_n-\delta_n}'$ onto $\Id^{-}\AA_{\rho_n-\delta_n}\,$, and that $$ \bigl|\Id^{-}\wh{H_n}\phi\bigr|_{\rho_n-\delta_n} \ge\bigl|\Id^{-}\wh H\phi\bigr|_{\rho_n-\delta_n}\! -\bigl|\Id^{-}\{H_n-H,\phi\}\bigr|_{\rho_n-\delta_n} \ge{abc\over 2\rho}\|\phi\|_{\rho_n-\delta_n}'\,, \equation(HnHatBound) $$ for all $\phi\in\Id^{-}\AA'_{\rho_n-\delta_n}\,$. Consequently, equation \equ(PhiDef) has a unique solution $\phi_n\in\Id^{-}\AA_{\rho_n-\delta_n}'\,$, and this solution satisfies the bound $$ \|\phi_n\|_{\rho_n-\delta_n}' \le{2\rho\over abc}|\Id^{-}H_n|_{\rho_n-\delta_n} <{2\rho\over 3}t_0^2b(2/3)^{4(3/2)^n} \le \delta_n\eps_n\,, \equation(PhinBound) $$ where $$ \eps_n={2\over 3}t_0b(2/3)^{3(3/2)^n}\,. \equation(epsilonDef) $$ In fact, the solution can be obtained by using a convergent Neumann series (for the operator being inverted) that is dominated in norm by a geometric series with ratio $1/2$. We note that $t_0b\le 1$, and thus $\eps_n<1/2$. This follows from the fact that $$ \eqalign{ abc\|\phi\|_{\rho_0-\delta_0}' &\le\rho|\Id^{-}\{H,\phi\}|_{\rho_0-\delta_0}\cr &\le\rho a|H|_{\rho_0-\delta_0}'\|\phi\|_{\rho_0-\delta_0}' \le at_0^{-1}c\|\phi\|_{\rho_0-\delta_0}'\,,\cr} \equation(epsilonLThalf) $$ for all $\phi\in\Id^{-}\AA_{\rho_0-\delta_0}'\,$. Now we can use \clm(UUphiComp) to prove the three bounds \equ(FmHmBound) for $m=n+1$. The first of these bounds is straightforward. For the second one we have $$ \eqalign{ |H_{n+1}-H_n|_{\rho_n-3\delta_n} &\le{2\over 3}\eps_n|H_n|_{\rho_n-\delta_n} <(2/3)^2ct_0b(2/3)^{3(3/2)^n}\cr &\le(2/3)^{7/2}ct_0b(2/3)^{(3/2)^{n+1}} <{c\over 4}t_0b(2/3)^{(3/2)^{n+1}}\,,\cr} \equation(NewFmBound) $$ and the bound on $\Id^{-}H_{n+1}$ is obtained by using identity \equ(UUphiComp0): $$ \eqalign{ \bigl|\Id^{-}H_{n+1}\bigr|_{\rho_n-3\delta_n} &\le{a\over 3}\eps_n^2|H_n|_{\rho_n-\delta_n}\cr &<{4ac\over 27}(t_0b)^2(2/3)^{6(3/2)^n} <{ac\over 4}(t_0b)^2(2/3)^{4(3/2)^{n+1}}\,.} \equation(NewHmBound) $$ This proves that \equ(FmHmBound) holds for all positive integers $m$. In particular, we find that the sequence of Hamiltonians $H_n$ converges in $\AA_{\rho'}$ to a function $H_\infty$ that satisfies $\Id^{-}H_\infty=0$. Next, we estimate the functions $G_n$ defined in equation \equ(HnGnDef). The bounds \equ(PhinBound) and \equ(QPbound) show that $g_n=U_{\phi_n}-\id$ satisfies $\|g_n\|_{\rho_n-3\delta_n}<\eps_n\delta_n\,$, for any $n\ge 0$, where $$ \bigl\|(Q,P)\|_r=\max\bigl\{ |V Q|_r\,, \|W P\|_r\bigr\}\,,\qquad (Q,P)\in\AA_r^d\times\AA_r^d\,. \equation(AAprodNorm2) $$ Thus, by using the identity $$ G_n=\sum_{m=0}^{n-1} g_m\circ U_{\phi_{m+1}}\circ\ldots\circ U_{\phi_{n-1}}\,, \equation(GnSum) $$ and \clm(Trivial).d, we obtain the bound $$ \|G_n\|_{\rho_n}\le\sum_{m=0}^{n-1}\|g_m\|_{\rho_{m+1}} <\half\sum_{m=0}^{n-1}\delta_m <{3\over 2}\delta_0\,. \equation(GnBound1) $$ By \clm(UUphiComp), we also have $$ \eqalign{ \|G_{n+1}-G_n\|_{\rho'} &=\bigl\|g_n+(G_n\circ U_{\phi_n}-G_n)\bigr\|_{\rho'}\cr &\le\|g_n\|_{\rho_n-3\delta_n} +\bigl\|G_n\circ U_{\phi_n}-G_n\bigr\|_{\rho_n-3\delta_n}\cr &<\eps_n\delta_n+{2\over 3}\eps_n\|G_n\|_{\rho_n-\delta_n} <2\eps_n\delta_0\,.\cr} \equation(GnBound2) $$ This shows that the sequence $\{G_n\}$ converges in $\AA^d_{\rho'}\times\AA^d_{\rho'}\,$, and that the limit $G_\infty$ defines an analytic map $\UU_\ssH=\id+G_\infty$ from $\DD_{\rho'}$ to $\DD_{(\rho'+\rho)/2}\,$. The identity $H_\infty=H\circ(\id+G_\infty)$ follows from the fact that $H$ is analytic on $\DD_\rho\,$. Since $\id+G_n$ is a canonical transformation, for every $n\ge 0$, the limit $\UU_\ssH$ is a canonical transformation as well. In order to verify the remaining claims in \clm(UUHgeneral), let us consider a fixed but arbitrary curve $z\mapsto H_z\in\HH$, which is analytic on an open neighborhood $Z$ of zero in $\complex$. For each $z\in Z$, our construction of $\UU_{H_z}$ defines a sequence of functions $\phi_{z,n}\in\AA'_{\rho_n-\delta_n}$ and $H_{z,n}\in\AA_{\rho_n}$ and $G_{z,n}\in\AA_{\rho_n}^{2d}\,$. Since we have only used bounds that are uniform on $\HH$, all of these functions depend analytically on $z$. The same applies to the limit $n\to\infty$, since convergence was proved to be uniform on $\HH$. Thus, the maps $H\mapsto H_\infty$ and $H\mapsto G_\infty$ are analytic on $\HH$. Assume now that $|\Id^{-}H_{z,n}|_{\rho_n}=\OO\bigl(|z|^k\bigr)$, for some $n\ge 0$ and $k\ge 1$ Then the inequality \equ(HnHatBound) shows that $\|\phi_{z,n}\|'_{\rho_n-\delta_n}$ is also of order $|z|^k$. This, together with the identity \equ(UUphiComp0), and the last inequality in \equ(UUphiComp1), implies that $|\Id^{-}H_{z,n+1}|_{\rho_{n+1}}=\OO\bigl(|z|^{2k}\bigr)$. By applying this to a curve of the form $H_z=H+zh$ with $\Id^{-}H=0$, we see e.g. that $\UU_\ssH=\id$, and that $$ \eqalign{ H_z\circ\UU_{H_z} &=H_{z,1}+z^2R_1(z)=\Id^{+}H_{z,1}+z^2R_2(z)\cr &=\Id^{+}\bigl(H_z+\{H_z,\phi_{z,0}\}\bigr)+z^2R_3(z)\cr &=\Id^{+}H+z\Id^{+}h-\Id^{+}\wh H\phi_{z,0}+z^2R_4(z)\,,\cr} \equation(FFHzh) $$ with $R_j(z)\in\AA_{\rho'}$ for all $z\in Z$. Furthermore, we have $$ \phi_{z,0}=z\bigl(\Id^{-}\wh{H_z}\Id^{-}\bigr)^{-1}\Id^{-}h =z\bigl(\Id^{-}\wh H\Id^{-}\bigr)^{-1}\Id^{-}h +z^2R_5(z)\,, \equation(Phiz0) $$ with $R_5(z)\in\AA_{\rho-\delta_0}'\,$, for all $z\in Z$. The last two inequalities yield the given formula for the derivative of $H\mapsto H\circ\UU_\ssH\,$. This concludes the proof of \clm(UUHgeneral). \qed We note that \clm(UUHgeneral) could be applied iteratively, using an increasing sequence of projections $\Id_0^{-}\le\Id_1^{-}\le\Id_2^{-}\le\ldots$. The part of $H$ that is resonant at step $k$ but becomes non--resonant (and gets eliminated) at step $k+1$, could be called a resonance of order $k$. The iteration scheme of classical KAM theory [\rKol--\rDLl] is of this type, except e.g. that the canonical transformation $U_{\phi_0}$ is used in place of $\UU_\ssH\,$. This is sufficient for near--integrable Hamiltonians, since the remaining non--resonant term is very small. \section The Renormalization Group Transformation The transformation $\RR_\mu$ implements the abovementioned iteration scheme as a dynamical system. Roughly speaking, the step $H\mapsto H\circ\UU_\ssH$ eliminates the non--resonant part of $H$, or resonance of order zero. In the next step $H\mapsto H\circ T_1\,$, the order of each remaining resonance is lowered by one. Two additional steps are included in the definition of $\RR_\mu\,$, in order to re-normalize the resulting Hamiltonian: A scaling $\SS_\mu$ of the action variable, and a scaling $H\mapsto\eta H$ of the energy (or time). The main effect of iterating $\RR_\mu\,$, besides scaling, is to eliminate resonances of higher and higher order, by successively decreasing orders and eliminating the lowest one. The remaining part of this section contains our proofs of \clm(TmuSub) and \clm(RRexists), followed by some remarks on symmetries. \proofof(TmuSub) For every index $(\nu,\alpha)$ in $I^{+}$ we have $$ \eqalign{ \|W T^\ast\nu\| &=\max_j|W_j\cdot T^\ast\nu| =\max_j|\vartheta_j| |W_j\cdot\nu|\cr &\le\bigl(\vartheta-|\vartheta_2|\bigr)|\omega\cdot\nu| +|\vartheta_2| \|W\nu\|\cr &\le\bigl(\vartheta-|\vartheta_2|\bigr)\kappa|\alpha| +\bigl[|\vartheta_2|+\sigma\bigl(\vartheta-|\vartheta_2|\bigr)\bigr] \|W\nu\|\,.\cr} \equation(TmuSub1) $$ Choose $r>\rho$ in such a way that \equ(SigmaMuCond) remains true if $\rho$ is replaced by $r$. Then the inequality \equ(TmuSub1) implies that $$ e^{r\|W T^\ast\nu\|} \le\left|{\rho'\vartheta_d\over r\mu}\right|^{|\alpha|} e^{\rho'\|W\nu\|}\,. \equation(TmuSub2) $$ Let now $H\in\Id^{+}\AA_{\rho'}\,$. Then for $H\circ T_\mu$ we have the representation $$ \eqalign{ \bigl(H\circ T_\mu\bigr)(q,p) &=H\bigl(Tq,\mu(T^\ast)^{-1}p\bigr)\cr &=\sum_{(\nu,\alpha)\in I^{+}}H_{\nu,\alpha} (Wp)^\alpha \Biggl[\,\prod_{j=1}^d \left({\mu\over\vartheta_j}\right)^{\alpha_j} \Biggr] e^{iq\cdot(T^\ast\nu)}\,.\cr} \equation(TmuSub3) $$ Thus, by using the bound \equ(TmuSub2), we obtain $$ \bigl|H\circ T_\mu\bigr|_r =\sum_{(\nu,\alpha)\in I^{+}}\bigl|H_{\nu,\alpha}\bigr| \bigl(\rho'\bigr)^{|\alpha|}\Biggl[\,\prod_{j=1}^d \left|{r \mu\over\rho'\vartheta_j}\right|^{\alpha_j}\Biggr] e^{r\|W T^\ast\nu\|} \le|H|_{\rho'}\,. \equation(TmuSub4) $$ This shows that $H\mapsto H\circ T_\mu$ defines a bounded linear operator from $\Id^{+}\AA_{\rho'}$ to $\AA_r\,$. The assertion now follows from the fact that the inclusion map from $\AA_r$ into $\AA_\rho$ is compact, and that $\left|H\circ T_\mu\right|_\rho\le\left|H\circ T_\mu\right|_r\,$. \qed \proofof(RRexists) Assume that $\kappa\rho<\sigma$. Then we can choose $\rho'<\rho$ and $\kappa'>\kappa$ such that $\kappa\rho^2<\kappa'\rho\rho'<\sigma\rho'$, and such that the first inequality in \equ(SigmaMuCond) holds, if $\kappa$ is replaced by $\kappa'$. Let $s=(\rho'+8\rho)/9$, and define $B=\{H\in\Id^{+}\AA'_s: H_{0,0}=0,\ |H-h_\omega|'_s<\kappa\rho\}$. If $\rho'\le r\le s$, then for every function $\phi\in\Id^{-}\AA_r$ we have $$ \eqalign{ \bigl|\{h_\omega,\phi\}\bigr|_r&=|\omega\cdot\nabla_1\phi|_r =\sum_{(\nu,\alpha)\in I^{-}}|\phi_{\nu,\alpha}| |\omega\cdot\nu| r^{|\alpha|}e^{r\|W\nu\|}\cr &\ge\sigma\!\!\sum_{(\nu,\alpha)\in I^{-}}|\phi_{\nu,\alpha}| r^{|\alpha|}\|W\nu\|e^{r\|W\nu\|} +\kappa'\!\!\sum_{(\nu,\alpha)\in I^{-}}|\phi_{\nu,\alpha}| |\alpha| r^{|\alpha|}e^{r\|W\nu\|}\cr &\ge\sigma\|W\nabla_1\phi\|_r +\kappa'r|V\nabla_2\phi|_r \ge\kappa'r\|\phi\|'_r\,,\cr} \equation(RGe1) $$ which yields the bound $$ \eqalign{ \bigl|\{H,\phi\}\bigr|_r &\ge\bigl|\{h_\omega,\phi\}\bigr|_r-\bigl|\{H-h_\omega,\phi\}\bigr|_r\cr &\ge\bigl(\kappa'r-|H-h_\omega|_r'\bigr)\|\phi\|'_r \ge(\kappa'\rho'-\kappa\rho)\|\phi\|'_r\,,\cr} \equation(RGe2) $$ for all $H\in B$. This shows that $\Id^{-}$ satisfies a non--resonance condition (with respect to the set $B$), as defined in \clm(UUHgeneral), for some constants $a,b,c>0$. The same resonance condition remains satisfied if we replace $B$ by the set $\HH$ of all Hamiltonians $H\in\AA_\rho$ that lie within a small distance $\eps>0$ of $B$. If necessary, we decrease $\eps>0$ such that $(H\circ\UU_\ssH)_{0,(1,0,\ldots,0)}$ is bounded away from zero, for all $H\in\HH$. This is possible since $|H_{0,(1,0,\ldots,0)}|>1-\sigma>0$, for all $H\in B$. Now \clm(UUHgeneral) and \clm(TmuSub) imply that \equ(RRDef) defines a compact analytic map $\RR_\mu$ from $\HH$ to $\AA_\rho\,$, provided that $\mu$ satisfies the second inequality in \equ(SigmaMuCond). By using the analyticity improving property of the map described in \clm(TmuSub), we obtain the same result for $\SS_z\circ\RR_\mu$ and $\RR_\mu\circ\SS_z\,$, uniformly in $z$, for all $z$ in some open neighborhood of the unit circle in $\complex$. In order to prove that $\SS_z\circ\RR_\mu=\RR_\mu\circ\SS_z\,$, as claimed, it suffices to consider $|z|=1$. In this case, it is straightforward to check that $\SS_z$ ``commutes through'' each of the steps used in the proof of \clm(UUHgeneral), yielding $$ \UU_{\SS_zH}\circ S_z=S_z\circ\UU_\ssH\,, \equation(UUSSz) $$ where $S_z(q,p)=(q,zp)$. This can be done by using the fact that $\SS_z\Id^{-}=\Id^{-}\SS_z$ and $\{\SS_zH,\SS_z\phi\}=\SS_z\{H,\phi\}$, which implies that $U_{\SS_z\phi}\circ S_z=S_z\circ U_\phi\,$. The details of this computation are left to the reader; see also [\rKi]. \qed \firstremark Due to the symmetry $\SS_z\circ\RR_\mu=\RR_\mu\circ\SS_z\,$, the transformation $\RR_\mu$ maps the scaling orbit $z\mapsto\SS_zH$ of a Hamiltonian $H$ to the corresponding scaling orbit for $\RR_\mu(H)$. Thus, in situations where these orbits are non--degenerate, a normalization condition can be used to pick an arbitrary representative from each of them. This leads naturally to a transformation \equ(RRDef) where the scaling $\mu$ is chosen to depend on $H$, in such a way that the given normalization is preserved. \nextremark A relation analogous to \equ(UUSSz) holds if the scaling transformations $S_z$ and $\SS_z$ are replaced by translations of the angles, given by $J_\gamma(q,p)=(q-\gamma,p)$ and $\JJ_\gamma H=H\circ J_\gamma\,$, respectively. The corresponding symmetry $\RR_\mu\circ\JJ_\gamma=\JJ_{T^{-1}\gamma}\circ\RR_\mu$ can be related to observations for certain periodic orbits [\rAKW]. \nextremark As was mentioned in the introduction, if $H$ is an even function of the angle variable $q$, then the same is true for $\RR_\mu(H)$. This can be seen easily from our construction of $\UU_\ssH$ in the proof of \clm(UUHgeneral). \nextremark Another invariance property of $\RR_\mu$ is the following. Given $v\in\complex^d$, denote by $\AA(v)$ the set of Hamiltonians $H$ (in the appropriate domain) for which $v\cdot\nabla_2H$ is constant. Our proof of \clm(UUHgeneral) shows that $H\circ\UU_\ssH$ belongs to $\AA(v)$ whenever $H$ does. Thus, $\RR_\mu$ maps $\AA(v)$ to $\AA((T^*)^{-1}v)$. The same holds if we define $\AA(v)$ by the condition $v\cdot\nabla_2H=0$. \nextremark The symmetry properties mentioned above, either separately or combined, can be used e.g. to restrict the search for nontrivial RG fixed points (or invariant families) to appropriate invariant subspaces. Such a fixed point (or family) may be relevant only for Hamiltonians that share all of its symmetries. But as the trivial fixed point $h_\omega$ and other examples show, the ``domain of relevance'' (universality class) may actually be larger; see also [\rAKW,\rCJBC]. \section Periodic Orbits We begin by showing that under a certain condition on $w\in\real^d$, every Hamiltonian $H$ near $h_w$ determines a ``counterterm'' $\Phi(H)$, within a $d$--dimensional subspace of $\BB_\rho$ that is roughly parallel to $\WW^u$, such that a constant multiple of $H+\Phi(H)$ has a periodic orbit of the type described in \clm(SigmaDef). The subsequent proof of \clm(SigmaOK) is based on identifying $\Sigma(w)$ locally with $\Phi^{-1}(0)$. A proof of \clm(AccuOrbits) is given at this end of this section, after some results on $d$--parameter families. Given any $r>0$, define $A_r$ to be the Banach space of all analytic functions $g$ on the strip $|{\rm Im} z|0$ is sufficiently small such that $B$ is contained in the domain of $\RR_\mu\,$. Denote by $X$ the subspace of $\BB_\rho\,$, consisting of all Hamiltonians of the form $ch_w+f$, with $c\in\complex$, and with $f$ a function in $\BB_\rho\,$, whose Fourier--Taylor coefficients $f_{\nu,\alpha}$ are zero whenever $\nu=0$ and $|\alpha|<2$. In addition, let $Y$ be the space of all Hamiltonians of the form $h_y+C$, with $C$ a constant function, $y\in\complex^d$, and $y\cdot\ov v=0$. Then we can identify $\BB_\rho$ with $X\oplus Y$. Define a map $\phi$ from $X\cap B$ to $Y$, by setting $\phi(H)=h_u+E/\xi$, with $H\mapsto (\xi,u,E)$ as described in \clm(SigmaProp). The graph $\Sigma'$ of this map $\phi$ is clearly an analytic manifold of codimension $d$ in $\BB_\rho\,$. In addition, $\Sigma'$ intersects $\WW^u$ transversally at $h_w\,$. This follows essentially from the fact that $\phi(ch_w)=0$: Due to the identity $D\phi(h_w)h_w=0$, it suffices to verify the transversality property in the $(d+1)$--dimensional subspace of all Hamiltonians of the form $(q,p)\mapsto x\cdot p+C$, where it is trivial. In order to compare $\Sigma'$ with the set $\Sigma(w)$ defined in \clm(SigmaDef), consider two different choices $(\delta_1,r_1)$ and $(\delta_2,r_2)$ for the parameters $(\delta,r)$ in \clm(SigmaProp), with $\delta_1<\delta_20$ sufficiently small, then $\Sigma(w)\cap B_1$ is contained in $\Sigma_2'\,$. And by choosing $\delta_1>0$ sufficiently small, we also have $\Sigma_1'\subset\Sigma(w)$. Thus, $\Sigma(w)$ agrees with $\Sigma_1'$ in the ball $B_1\,$. We note that the same choice of parameters works for all rotation vectors $w\in\real\integer^d$ that are sufficiently close to $\omega$. (This fact is not used later on.) The reason for this is that our bounds depend on $w$ only through the constant $\tau=\tau(w)$. \qed Our proof of \clm(AccuOrbits) is based on the graph transform method; see e.g. [\rHPS]. We start by introducing some notation. Denote by $\XX$ and $\YY$ the stable and unstable subspaces, respectively, of the linearized RG transformation $D\RR_\mu(h_\omega)$, restricted to $\BB_\rho$. Denote by $\psi$ the (analytic) function, defined on an open neighborhood of $h_\omega$ in $\XX$, with values in $\YY$, whose graph is the local stable manifold $\WW^s$ of $\RR_\mu$ at $h_\omega\,$. As was mentioned in the introduction, $\YY$ is spanned by the $d-1$ functions listed in \equ(lambdaj), together with the constant function. The canonical projections onto $\XX$ and $\YY$ will be denoted by $\proj_s$ and $\proj_u\,$, respectively. Due to our choice of norm on $\BB_\rho\,$, both $\proj_s$ and $\proj_u$ have operator norm $1$. Consider the transformation $\NN_\mu\,$, $$ \NN_\mu=\Psi^{-1}\circ\RR_\mu\circ\Psi\,,\qquad \Psi=\Id+\psi\circ\proj_s\,, \equation(NNDef) $$ defined on a neighborhood of $h_\omega$ in $\BB_\rho\,$. Notice that $\NN_\mu$ and $\RR_\mu$ have the same derivatives at the fixed point $h_\omega\,$, and the same local unstable manifolds. But the local stable manifold of $\NN_\mu$ is trivial; that is, it agrees with $\XX$ near $h_\omega\,$. The largest (in modulus) contracting eigenvalue of $D\NN_\mu(h_\omega)$ is $\lambda=\mu\vartheta_1\vartheta_d^{-2}$. Let $\theta$ be some fixed real number larger than $\vartheta_1|\vartheta_d|^{-2}$. In the remaining part of this section, we restrict the possible choices of $\mu$ by imposing the condition $\theta\mu<|\lambda_2|^{-1}\,$. Since $D\NN_\mu(h_\omega)$ is compact, we can choose a norm $\|.\|$ on $\XX$ that is equivalent to $\|.\|_\rho\,$, but for which the restriction of $D\NN_\mu(h_\omega)$ to $\XX$ has (operator) norm less than $\theta\mu$. Assume now that such a norm has been chosen. The extension to $\BB_\rho=\XX\oplus\YY$ is defined by setting $\|x+y\|=\|x\|+\|y\|_\rho\,$, for every $x\in\XX$ and $y\in\YY$. Given any $\delta>0$, let $D_\delta=\{y\in\YY: \|y\|<\delta\}$, and define $\FF_\delta$ to be the Banach space of all analytic $d$--parameter families $F: D_\delta\to\BB_\rho\,$, that extend continuously to the boundary of $D_\delta$ and satisfy $\proj_u F(0)=0$, equipped with the norm $$ \|F\|_\delta=\sup_{y\in D_\delta}\|F(y)\|\,. \equation(GGnorm) $$ Of particular interest is the family $F^\ast$, defined by $F^\ast(y)=h_\omega+y$, as it parametrizes the local unstable manifold of $\RR_\mu\,$. \claim Proposition(MMDef) If $\delta>0$ is sufficiently small, then the equation $$ \MM_\mu(F)=\NN_\mu\circ F\circ Y_F\,,\qquad Y_F=(\proj_u\circ\NN_\mu\circ F)^{-1}\,, \equation(MMDef) $$ defines an analytic contraction mapping $\MM_\mu\,$, on some open neighborhood $B$ of $F^\ast$ in $\FF_\delta\,$, with fixed point $F^\ast$, and contraction rate less than $\theta\mu$. The proof of this proposition is straightforward: $\proj_u\circ\NN_\mu\circ F^\ast$ agrees with the restriction of $D\RR_\mu$ to $\YY$, so that by the inverse function theorem, $(F,y)\mapsto Y_F(y)$ is well defined and analytic near $(F^\ast,0)$ in $\FF_r\times\YY$; and the asserted contraction property of $\MM_\mu$ follows from the identity $$ \bigl(D\MM_\mu(F^\ast)f\bigr)(y) =\proj_s D\NN_\mu\bigl(F^\ast(Y_{F^\ast}(y))\bigr)f(Y_{F^\ast}(y))\,, \qquad f\in\FF_r\,,\ y\in D_r\,, \equation(DMM) $$ which can be verified by an explicit computation. Notice that $Y_{F^\ast}$ is linear, and that $Y_F(0)=0$ for any $F$. The following proposition will be used to estimate the composition of maps $f_n=Y_{F_n}\,$, associated with an orbit $(F_0,F_1,F_2,\ldots)$ of $\MM_\mu\,$. \claim Proposition(fff) Let $U$, $V$ be normed linear spaces, and let $Z$ be an open ball in $U\oplus V$, centered at zero. Let $L$ be a bounded linear operator on $U\oplus V$ that commutes with the projection $(u,v)\mapsto(u,0)$, and that satisfies $$ \|L(u,0)\|=a\|u\|\,,\quad \|L(0,v)\|\le b\|v\|\,,\qquad u\in U,\ v\in V, \equation(fff1) $$ with $00$ is sufficiently small, and if $w\in\real\integer^d$ is sufficiently close to $\omega$ and normalized, such that $h_w-h_\omega$ belongs to $D_{\delta/2}\,$, then there exists an open neighborhood $B'$ of $F^\ast$ in $\FF_\delta\,$, and constants $k_1,k_2,N>0$, such that the following holds. For every $F\in B'$, and for every non--negative integer $n$, the condition $\Psi(F(y))\in\Sigma_n(w)$ defines a unique parameter value $y=y_n$ in $D_\delta\,$, and this value satisfies the bound $$ k_1|\lambda_2|^{-n}<\|y_n\|0$ have been chosen sufficiently small, such that the restriction of $\MM_\mu$ to $B'$ has the properties described in \clm(MMDef), and such that the derivatives of $F\mapsto Y_F$ and $\MM_\mu$ are uniformly bounded on $B'$. Given any $F_0\in B'$, we set $F_n=\MM_\mu^n(F_0)$, for $n=1,2\ldots$. By the definition of $\MM_\mu\,$, the family $F_n$ intersects $\Psi^{-1}(\Sigma_n(w))$ at a single point $y_n\,$, given by the equation $$ y_n=\bigl(Y_{F_0}\circ Y_{F_1}\circ\ldots\circ Y_{F_{n-1}}\bigr)(z_n)\,, \qquad z_n=Z(F_n)\,. \equation(ynId) $$ By \clm(MMDef), the norm of $F_n-F^\ast$ is bounded by $(\theta\mu)^n\|F_0-F^\ast\|$, for all $n\ge 0$. By analyticity, analogous bounds hold for $\|z_n-z\|$ and $\|DY_{F_n}(y)-L\|$, up to constant factors that we can choose to be independent of $n$, $F\in B$, and $y\in D_{\delta/2}\,$. Notice that $L$ is the inverse of the restriction of $D\RR_\mu(h_\omega)$ to $Y$. All eigenvalues of $L$ are either of modulus $a$, or of modulus less than $b$, with $\theta\mu0$ sufficiently small. In what follows, $c_0,\ldots,c_{18}$ denote positive constants that do not depend on the choice of the Hamiltonian $H$. But unless stated otherwise, $c_i$ may depend on $\mu$. We will also introduce constants $b_1,\ldots,b_6$ that only depend on the choice of $T$. We assume that $\mu>0$ has been chosen sufficiently small such that $b_i\mu<|\lambda_2|^{-1}$. Since $r>\rho$, there exists an open neighborhood $\Lambda$ of zero in $\complex^d$, such that $(\beta,H)\mapsto H_\beta$ is analytic and has bounded derivatives, as a map from $\Lambda\times\BB_r$ to $\BB_\rho\,$. When considering families $\beta\mapsto H_\beta\,$, we will implicitly assume that $\beta\in\Lambda$. Let now $h$ be a Hamiltonian in $\BB_r\,$, of the form \equ(hint), with $M$ as described after \equ(hint), and assume that $h$ is sufficiently close to $h_\omega\,$, such that $\NN_\mu$ is well defined and analytic on an open ball in $\BB_\rho$ containing $h$ and $h_\omega\,$. Define $Y(\beta)=\proj_u\Psi^{-1}(h\circ R_\beta)$. Then $Y(0)=0$, and our assumption on $M$ implies that $Y$ is invertible as a map from an open neighborhood of zero in $\complex^d$, to $\YY$. Define $f_0(y)=\Psi^{-1}(h\circ R_\beta)$, with $\beta=Y^{-1}(y)$. An explicit computation shows that for $n=1,2,\ldots$, the equation \equ(MMDef) defines a family $f_n=\MM_\mu(f_{n-1})$, which belongs to $\FF_\delta$ for large $n$, and that $f_n\to F^\ast$ in $\FF_\delta\,$, as $n$ tends to infinity. Thus, given any open neighborhood $B'$ of $F^\ast$ in $\FF_\delta\,$, there exists a positive integer $\ell$, such that if $H\in\BB_r$ is sufficiently close to $h$, then the equation $$ F(y)=\Psi^{-1}\bigl(\RR_\mu^\ell\bigl( H_{\beta'}\circ R_{Z_\ell(y)}\bigr)\bigr)\,, \qquad Z_\ell=Y^{-1}\circ Y_{f_0}\circ\ldots\circ Y_{f_{\ell-1}}\,, \equation(HFamily) $$ defines a family $F\in B'$. Here, $\beta'$ denotes the parameter value where $\beta\mapsto H_\beta$ intersects $\WW^s$. This value is well defined and depends analytically on $H$, since $\beta\mapsto h_\beta$ intersects $\WW^s$ transversally at $h$. By using \clm(AccuY), and the fact that $Z_\ell$ is invertible near the origin, we can find constants $k_1',k_2',N'>0$, such that for all $n\ge N'$, and for all Hamiltonians $H$ in some open neighborhood $B$ of $h$ in $\BB_r\,$, the condition $H_\beta\in\Sigma_n(w)$ defines a unique parameter value $\beta=\beta_n\,$, and this value satisfies the bound $$ k_1'|\lambda_2|^{-n}<|\beta_n-\beta'|0$ is chosen sufficiently small, we find that the range of $T_\mu^n\circ g_n$ is contained in the domain of $\VV_n(H_{\beta_n})$, for large $n$. A more detailed discussion of the transformations $\VV_n(H)$ can be found in [\rKi]. The estimates obtained there are formulated for $H=H_{\beta'}$ only, but they are easy to adapt to the Hamiltonians considered here. To be more precise, the starting point for these estimates is a bound $$ \|\phi(H_{\beta',m})\|'_{r_0} \le c_2\|\Id^{-}H_{\beta',m}\|_\rho \le c_2c_1(b_1\mu)^m\|H_{\beta'}-h_\omega\|_\rho\,, \equation(old) $$ where $\phi(H_{\beta',m})$ is the generating function of the canonical transformation $\UU_{H_{\beta',m}}\,$. The constants $c_2$ and $b_1\,$, and the parameter $r_0$ defining the domain of $\phi_m(H_{\beta',m})$, are independent of $\mu$. We have omitted an additional factor $(b_1\mu)^m$ that appears in equation (5.11) of [\rKi], since it is not used or needed. Concerning the replacement of $H_{\beta'}$ by $H_{\beta_n}\,$, we note that the first inequality in \equ(old) is a general bound on $\phi$ that applies directly to $H_{\beta_n,m}\,$. The second inequality in \equ(old) carries over as well. In fact, it can be improved if $H$ is close to $h$: $$ \|\Id^{-}H_{\beta,m}\|_\rho \le c_3(\theta\mu)^m\|H-h\|_r\,, \qquad H\in B\,, \equation(AccuH) $$ for all $\beta$ in some open set $\Lambda_m\subset\complex^d$ containing $\beta'$ and $\beta_n\,$. This inequality is trivial for $m\le\ell$, and if $m=\ell+k$ with $k$ positive, it follows from the bound $$ \eqalign{ \bigl\|\Id^{-}\Psi\bigl((\MM_\mu^k(F))(y)\bigr)\bigr\|_\rho &=\bigl\|\Id^{-}\Psi\bigl((\MM_\mu^k(F))(y)\bigr) -\Id^{-}\Psi\bigl((\MM_\mu^k(f_\ell))(y)\bigr)\bigr\|_\rho\cr &\le c_4\bigl\|\MM_\mu^k(F)-\MM_\mu^k(f_\ell)\bigr\| \le c_4(\theta\mu)^k\|F-f_\ell\|\cr &\le c_5(\theta\mu)^k\|H-h\|_r\,.\cr} \equation(AccuH) $$ Here, $F$ is the family defined in \equ(HFamily), and $y$ is an arbitrary point in $D_\delta\,$. We note that for $m=\ell+k$, the abovementioned set $\Lambda_m$ can be taken to be the image of $D_\delta$ under the map $Z_\ell\circ Y_{F_0}\circ\ldots\circ Y_{F_{k-1}}\,$, where $F_n=\MM_\mu^n(F)$. Since $Y_{F_n}\to Y_{F^\ast}$ uniformly on $D_\delta\,$, there exists a universal constant $b>0$, such that $\Lambda_m$ contains a union of balls of radii $c_6b^m$, whose centers trace out a path of length $\le c_7b^{-m}|\beta_n-\beta'|$, connecting $\beta_n$ and $\beta'$. Thus, by using \equ(AccuH), together with Cauchy's formula to estimate the derivative of $\beta\mapsto\Id^{-}H_{\beta,m}$ along the path from $\beta_n$ and $\beta'$, we obtain the first of the following two bounds: $$ \eqalign{ \|\Id^{-}H_{\beta_n,m}-\Id^{-}H_{\beta',m}\|_\rho &\le c_8(b_2\mu)^m|\beta_n-\beta'|\,\|H-h\|_r\,,\cr \|H_{\beta_n,m}-H_{\beta',m} \|_\rho &\le c_9b_3^m|\beta_n-\beta'|\,.\cr} \equation(AccuHdiff) $$ The second bound is obtained similarly. Consider now the curve $\gamma_n=R_{\beta_n}\circ G_n\,$. Clearly, $\gamma_n$ is a periodic orbit for a constant multiple of $H$, with rotation vector $w_n\,$. Our next goal is to show that the $n$--dependence of $\gamma_n(0)$ is mostly due to the translations $R_{\beta_n}\,$. To this end, we split $G_n(0)-\Gamma_{H_{\beta'}}(0)$ into three pieces and use the triangle inequality: $$ \eqalign{ \bigl|G_n(0)-\Gamma_{H_{\beta'}}(0)\bigr| &\le\bigl|\VV_n(H_{\beta_n})\bigl(T_\mu^n(g_n(0))\bigr) -\VV_n(H_{\beta_n})(0)\bigr| \cr &+\bigl|\VV_n(H_{\beta_n})(0)-\VV_n(H_{\beta'})(0)\bigr| +\bigl|\VV_n(H_{\beta'})(0)-\Gamma_{H_{\beta'}}(0)\bigr|\,.\cr} \equation(GnZero) $$ The first term on the right hand side of \equ(GnZero) can be estimated by using the results from [\rKi], with $H_{\beta'}$ replaced by $H_{\beta_n}\,$, as mentioned above. The relevant fact is that $\VV_n(H_{\beta_n})$ is uniformly bounded on a domain whose size decreases exponentially in $n$, independently of $\mu$, while $|g_n(0)|\le c_{10}(\theta\mu)^n$, as mentioned earlier. Thus, $$ \bigl|\VV_n(H_{\beta_n})\bigl(T_\mu^n(g_n(0))\bigr) -\VV_n(H_{\beta_n})(0)\bigr| \le c_{11}(b_4\mu)^n\,. \equation(GnZero1) $$ In order to bound the second term on the right hand side of \equ(GnZero), we define an interpolating family of transformations $s\mapsto\VV_{n,s}(H)$, such that $\VV_{n,0}(H)=\VV_n(H_{\beta'})$ and $\VV_{n,1}(H)=\VV_n(H_{\beta_n})$. To obtain $\VV_{n,s}(H)$, each of the transformations $\UU_{H_{\beta',m}}$ that enter the definition \equ(VnDef) of $\VV_n(H_{\beta'})$, is replaced by the canonical transformation with generating function $$ \phi_{n,m,s}(H)=\phi(H_{\beta',m}) +s\bigl[\phi(H_{\beta_n,m})-\phi(H_{\beta',m})\bigr]\,. \equation(phinmsDef) $$ By using the analyticity of the map $H\mapsto\phi(H)$, and the bounds \equ(betasymptotics2), \equ(AccuH), and \equ(AccuHdiff), we find that for $|s|\le|\lambda_2|^n$, $$ \eqalign{ \|\phi_{n,m,s}(H)\|'_{r_0} &\le c_{12}\|\Id^{-}H_{\beta',m}\|_\rho +c_{13}|s|\,\|\Id^{-}H_{\beta_n,m}-\Id^{-}H_{\beta',m}\|_\rho\cr &+c_{14}|s|\,\|\Id^{-}H_{\beta',m}\|_\rho\|H_{\beta_n,m}-H_{\beta',m}\|_\rho\cr &\le c_{15}(b_5\mu)^m\|H-h\|_r\,.\cr} \equation(phinmsBound) $$ With this bound replacing \equ(old), we now obtain the analogue of Lemma 5.5 in [\rKi], which implies that $v(s)=\VV_{n,s}(H)(0)$ is bounded in modulus by $c_{16}\|H-h\|_r\,$, if $n$ is sufficiently large and $|s|\le|\lambda_2|^n$. Thus, by writing $v(1)-v(0)$ as the integral of $v'$ over $[0,1]$, and estimating $v'$ by using Cauchy's formula with contour $|s|=|\lambda_2|^n$, we obtain the bound $$ \bigl|\VV_n(H_{\beta_n})(0)-\VV_n(H_{\beta'})(0)\bigr| \le c_{17}|\lambda_2|^{-n}\|H-h\|_r\,. \equation(GnZero2) $$ The last term in \equ(GnZero) satisfies again a bound of the form \equ(GnZero1), as was shown in [\rKi]. Putting the pieces together, we now have $$ \bigl|G_n(0)-\Gamma_{H_{\beta'}}(0)\bigr| \le c_{17}|\lambda_2|^{-n}\|H-h\|_r+c_{18}(b_6\mu)^n\,, \equation(GnZero0) $$ provided that $H$ is sufficiently close to $h$ in $\BB_r\,$, and $n$ sufficiently large. Since $$ \pm|\gamma_n(0)-\Gamma'(0)| \le\pm|\beta_n-\beta'|+\bigl|G_n(0)-\Gamma_{H_{\beta'}}(0)\bigr|\,, \equation(dbeta) $$ as a result of the identities $\gamma_n(0)=G_n(0)+(0,\beta_n)$ and $\Gamma'(0)=\Gamma_{H_{\beta'}}(0)+(0,\beta')$, the bound \equ(gammasymptotics) now follows from \equ(betasymptotics2) and \equ(GnZero0), if $H$ is sufficiently close to $h$. \qed We conclude with a proof of \equ(AccuOrbits), using the same notation as above. By the definition of $\Sigma(w)$, we have $H_{\beta_n,n}\circ g_n=0$, and thus $H\circ\gamma_n=H_{\beta_n}\circ G_n =a_n H_{\beta_n,n}\circ g_n\circ\Theta^{-n}=0$, where $a_n$ is some nonzero constant. If $U$ is a canonical transformation with globally defined generating function, or a composition of such transformations, then $K(U\circ\gamma)=K(\gamma)$ for any closed curve $\gamma$. The corresponding identity for $T_\mu$ is $K(T_\mu\circ\gamma)=\mu K(\gamma)$. As a result, we have $K(G_n)=\mu^nK(g_n)=0$, and thus $K(\gamma_n)=\tau(w_n)w_n\cdot\beta_n\,$, which is equivalent to equation \equ(AccuOrbits). \bigskip\noindent\leftline{\bf Acknowledgments} We would like to thank R.~de~la~Llave and P.~Wittwer for helpful discussions. \references %%%%%%%%%%%%%%% \item{[\rGre]} J.~M.~Greene, {\it A Method for Determining a Stochastic Transition.} J.~Math.~Phys.~{\bf 20}, 1183--1201 (1979). \item{[\rBKa]} D.~Bernstein, A.~Katok, {\it Birkhoff Periodic Orbits for Small Perturbations of Completely Integrable Hamiltonian Systems with Convex Hamiltonians.} Invent.~Math.~{\bf 88}, 225--241 (1987). \item{[\rFDLl]} C.~Falcolini, R.~de~la~Llave, {\it A Rigorous Partial Justification of Greene's Criterion.} J.~Stat.~Phys.~{\bf 67}, 609--643 (1992). \item{[\rTomi]} S.~Tompaidis, {\it Approximation of Invariant Surfaces by Periodic Orbits in High-Dimen\-sional Maps: some Rigorous Results.} Experimental Math.~{\bf 5}, 197--209 (1996). \item{[\rDDLl]} A.~Delshams, R.~de~la~Llave, {\it KAM Theory and a Partial Justification of Greene's Criterion for Non--Twist maps.} Pre\-print U. 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Syst. {\bf 19}, 1--47 (1999). %%%%%%%%%%%%%%% \item{[\rKos]} D.~Kosygin, {\it Multidimensional KAM Theory from the Renormalization Group Viewpoint.} In ``Dynamical Systems and Statistical Mechanics'', Ya.G.~Sinai (ed), AMS, Adv.~Sov.~Math.~{\bf 3}, 99--129 (1991). \item{[\rMMS]} R.~S.~MacKay, J.~D.~Meiss, J.~Stark, {\it An Approximate Renormalization for the Break-up of Invariant Tori with Three Frequencies.} Phys.~Lett.~A~~{\bf 190}, 417--424 (1994). \item{[\rMcKiv]} R.~S.~MacKay, {\it Three Topics in Hamiltonian Dynamics.} In ``Dynamical Systems and Chaos'', Vol.2, Y.~Aizawa, S.~Saito, K.~Shiraiwa (eds), World Scientific, London (1995). \item{[\rCGJ]} C.~Chandre, M.~Govin, H.~R.~Jauslin, {\it KAM--Renormalization Group Analysis of Stability in Hamiltonian Flows.} Phys.~Rev.~Lett. {\bf 79}, 3881--3884 (1997). \item{[\rCGJK]} C.~Chandre, M.~Govin, H.~R.~Jauslin, H.~Koch, {\it Universality for the Breakup of Invariant Tori in Hamiltonian Flows.} Phys.~Rev.~E~~{\bf 57}, 6612--6617 (1998). \item{[\rCJBC]} C.~Chandre, H.~R.~Jauslin, G.~Benfatto, A.~Celletti, {\it An Approximate Renormalization-Group Transformation for Hamiltonian Systems with Three Degrees of Freedom.} Pre\-print U. Texas, mp\_arc 99--74 (1999). %%%%%%%%%%%%%%% \item{[\rKol]} A.~N.~Kolmogorov, {\it On Conservation of Conditionally Periodic Motions Under Small Perturbations of the Hamiltonian.} Dokl.~Akad.~Nauka SSSR, {\bf 98}, 527--530 (1954). \item{[\rMo]} J.~Moser, {\it On Invariant Curves of Area--Preserving Mappings of an Annulus.} Nachr. Akad. Wiss. G\"ott., II. Math. Phys. Kl 1962, 1-20 (1962). \item{[\rArn]} V.~I.~Arnold, {\it Proof of A.N.~Kolmogorov's Theorem on the Preservation of Quasi--Periodic Motions under Small Perturbations of the Hamiltonian.} Usp. Mat. Nauk, {\bf 18}, No.~5, 13--40 (1963). Russ. Math. Surv., {\bf 18}, No.~5, 9--36 (1963). \item{[\rTh]} W.~Thirring, {\it A Course in Mathematical Physics I: Classical Dynamical Systems.} Springer-Verlag, Berlin $\cdot$ New York $\cdot$ Wien (1978). \item{[\rDLl]} R.~de~la~Llave, {\it Introduction to KAM Theory.} Preprint U. Texas, mp\_arc 93--8 (1993). %%%%%%%%%%%%%%% \item{[\rCE]} P.~Collet, J.-P.~Eckmann, {\it Iterated Maps on the Interval as Dynamical Systems.} Birk\-h\"auser Verlag, Basel $\cdot$ Boston $\cdot$ Berlin (1980). %%%%%%%%%%%%%%% \item{[\rHPS]} M.~W.~Hirsch, C.~C.~Pugh, M.~Shub, {\it Invariant Manifolds.} Lecture Notes in Math. {\bf 583}, Springer-Verlag, Berlin $\cdot$ New York (1977). \item{[\rPal]} J.~Palis, {\it A Note on the Inclination Lemma ($\lambda $-Lemma) and Feigenbaum's Rate of Approach.} In "Geometric dynamics (Rio de Janeiro, 1981)", J.~Palis (ed), Lecture Notes in Math. {\bf 1007}, 630--635, Springer-Verlag, Berlin $\cdot$ New York, 1983. \item{[\rPDM]} J.~Palis, W.~de~Melo, {\it Geometric Theory of Dynamical Systems. 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7700 4564 l 7728 4567 l 7752 4569 l 7770 4572 l 7784 4573 l 7793 4574 l 7798 4575 l 7800 4575 l gs col0 s gr $F2psEnd rs ---------------9905281510372--