Content-Type: multipart/mixed; boundary="-------------0609270800247" This is a multi-part message in MIME format. ---------------0609270800247 Content-Type: text/plain; name="06-269.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-269.comments" AMS-Code: 37C55, 37E20, 37D10 E-mail: koch@math.utexas.edu, kocic@physics.utexas.edu For possible updates see ftp://ftp.ma.utexas.edu/pub/papers/koch/ ---------------0609270800247 Content-Type: text/plain; name="06-269.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-269.keywords" vector fields, flows, analytic, Hamiltonian, reversible, divergence free, symmetric, quasi periodic, invariant torus, elliptic, frequencies, Brjuno vectors, Diophantine, normal form, renormalization group, stable manifold ---------------0609270800247 Content-Type: application/x-tex; name="rg+brjuno8.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="rg+brjuno8.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including smallfonts.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %smallfonts.tex % \newskip\ttglue % \font\fiverm=cmr5 \font\fivei=cmmi5 \font\fivesy=cmsy5 \font\fivebf=cmbx5 \font\sixrm=cmr6 \font\sixi=cmmi6 \font\sixsy=cmsy6 \font\sixbf=cmbx6 \font\sevenrm=cmr7 \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightit=cmti8 \font\eightsl=cmsl8 \font\eighttt=cmtt8 \font\eightbf=cmbx8 \font\ninerm=cmr9 \font\ninei=cmmi9 \font\ninesy=cmsy9 \font\nineit=cmti9 \font\ninesl=cmsl9 \font\ninett=cmtt9 \font\ninebf=cmbx9 % \font\twelverm=cmr12 \font\twelvei=cmmi12 \font\twelvesy=cmsy12 \font\twelveit=cmti12 \font\twelvesl=cmsl12 \font\twelvett=cmtt12 \font\twelvebf=cmbx12 %% EIGHT POINT FONT FAMILY \def\eightpoint{\def\rm{\fam0\eightrm} \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\eightit \def\it{\fam\itfam\eightit} \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl} \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt} \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf} \tt \ttglue=.5em plus.25em minus.15em \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt} \let\sc=\sixrm \let\big=\eightbig \normalbaselines\rm} \def\eightbig#1{{\hbox{$\textfont0=\ninerm\textfont2=\ninesy \left#1\vbox to6.5pt{}\right.$}}} %% NINE POINT FONT FAMILY \def\ninepoint{\def\rm{\fam0\ninerm} \textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\nineit \def\it{\fam\itfam\nineit} \textfont\slfam=\ninesl \def\sl{\fam\slfam\ninesl} \textfont\ttfam=\ninett \def\tt{\fam\ttfam\ninett} \textfont\bffam=\ninebf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\ninebf} \tt \ttglue=.5em plus.25em minus.15em \normalbaselineskip=11pt \setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt} \let\sc=\sevenrm \let\big=\ninebig \normalbaselines\rm} \def\ninebig#1{{\hbox{$\textfont0=\tenrm\textfont2=\tensy \left#1\vbox to7.25pt{}\right.$}}} %% TWELVE POINT FONT FAMILY --- not really small \def\twelvepoint{\def\rm{\fam0\twelverm} \textfont0=\twelverm \scriptfont0=\eightrm \scriptscriptfont0=\sixrm \textfont1=\twelvei \scriptfont1=\eighti \scriptscriptfont1=\sixi \textfont2=\twelvesy \scriptfont2=\eightsy \scriptscriptfont2=\sixsy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\twelveit \def\it{\fam\itfam\twelveit} \textfont\slfam=\twelvesl \def\sl{\fam\slfam\twelvesl} \textfont\ttfam=\twelvett \def\tt{\fam\ttfam\twelvett} \textfont\bffam=\twelvebf \scriptfont\bffam=\eightbf \scriptscriptfont\bffam=\sixbf \def\bf{\fam\bffam\twelvebf} \tt \ttglue=.5em plus.25em minus.15em \normalbaselineskip=11pt \setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt} \let\sc=\sevenrm \let\big=\twelvebig \normalbaselines\rm} \def\twelvebig#1{{\hbox{$\textfont0=\tenrm\textfont2=\tensy \left#1\vbox to7.25pt{}\right.$}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of smallfonts.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including param.2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %param.2 \magnification=\magstep1 \def\firstpage{1} \pageno=\firstpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of param.2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including fonts.5b %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %fonts.5b \font\fiverm=cmr5 \font\sevenrm=cmr7 \font\sevenbf=cmbx7 \font\eightrm=cmr8 \font\eightbf=cmbx8 \font\ninerm=cmr9 \font\ninebf=cmbx9 \font\tenbf=cmbx10 \font\magtenbf=cmbx10 scaled\magstep1 \font\magtensy=cmsy10 scaled\magstep1 \font\magtenib=cmmib10 scaled\magstep1 \font\magmagtenbf=cmbx10 scaled\magstep2 % \font\eightmsb=msbm8 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of fonts.5b %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including symbols.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % symbols.1 % % just concatenated amssym.def version 2.2 and amssym.tex version 2.2b % and commented out stuff: lines now beginning with %# % % use with fonts.5b instead of fonts.5 % %%% ==================================================================== %%% @TeX-file{ %%% filename = "amssym.def", %%% version = "2.2", %%% date = "22-Dec-1994", %%% time = "10:14:01 EST", %%% checksum = "28096 117 438 4924", %%% author = "American Mathematical Society", %%% copyright = "Copyright (C) 1994 American Mathematical Society, %%% all rights reserved. Copying of this file is %%% authorized only if either: %%% (1) you make absolutely no changes to your copy, %%% including name; OR %%% (2) if you do make changes, you first rename it %%% to some other name.", %%% address = "American Mathematical Society, %%% Technical Support, %%% Electronic Products and Services, %%% P. O. Box 6248, %%% Providence, RI 02940, %%% USA", %%% telephone = "401-455-4080 or (in the USA and Canada) %%% 800-321-4AMS (321-4267)", %%% FAX = "401-331-3842", %%% email = "tech-support@math.ams.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "amsfonts, msam, msbm, math symbols", %%% supported = "yes", %%% abstract = "This is part of the AMSFonts distribution, %%% It is the plain TeX source file for the %%% AMSFonts user's guide.", %%% docstring = "The checksum field above contains a CRC-16 %%% checksum as the first value, followed by the %%% equivalent of the standard UNIX wc (word %%% count) utility output of lines, words, and %%% characters. This is produced by Robert %%% Solovay's checksum utility.", %%% } %%% ==================================================================== %#\expandafter\ifx\csname amssym.def\endcsname\relax \else\endinput\fi % % Store the catcode of the @ in the csname so that it can be restored later. %#\expandafter\edef\csname amssym.def\endcsname{% %# \catcode`\noexpand\@=\the\catcode`\@\space} % Set the catcode to 11 for use in private control sequence names. \catcode`\@=11 % % Include all definitions related to the fonts msam, msbm and eufm, so that % when this file is used by itself, the results with respect to those fonts % are equivalent to what they would have been using AMS-TeX. % Most symbols in fonts msam and msbm are defined using \newsymbol; % however, a few symbols that replace composites defined in plain must be % defined with \mathchardef. \def\undefine#1{\let#1\undefined} \def\newsymbol#1#2#3#4#5{\let\next@\relax \ifnum#2=\@ne\let\next@\msafam@\else \ifnum#2=\tw@\let\next@\msbfam@\fi\fi \mathchardef#1="#3\next@#4#5} \def\mathhexbox@#1#2#3{\relax \ifmmode\mathpalette{}{\m@th\mathchar"#1#2#3}% \else\leavevmode\hbox{$\m@th\mathchar"#1#2#3$}\fi} \def\hexnumber@#1{\ifcase#1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8\or 9\or A\or B\or C\or D\or E\or F\fi} \font\tenmsa=msam10 \font\sevenmsa=msam7 \font\fivemsa=msam5 \newfam\msafam \textfont\msafam=\tenmsa \scriptfont\msafam=\sevenmsa \scriptscriptfont\msafam=\fivemsa \edef\msafam@{\hexnumber@\msafam} \mathchardef\dabar@"0\msafam@39 \def\dashrightarrow{\mathrel{\dabar@\dabar@\mathchar"0\msafam@4B}} \def\dashleftarrow{\mathrel{\mathchar"0\msafam@4C\dabar@\dabar@}} \let\dasharrow\dashrightarrow \def\ulcorner{\delimiter"4\msafam@70\msafam@70 } \def\urcorner{\delimiter"5\msafam@71\msafam@71 } \def\llcorner{\delimiter"4\msafam@78\msafam@78 } \def\lrcorner{\delimiter"5\msafam@79\msafam@79 } % Note that there should not be a final space after the digits for a % \mathhexbox@. \def\yen{{\mathhexbox@\msafam@55}} \def\checkmark{{\mathhexbox@\msafam@58}} \def\circledR{{\mathhexbox@\msafam@72}} \def\maltese{{\mathhexbox@\msafam@7A}} \font\tenmsb=msbm10 \font\sevenmsb=msbm7 \font\fivemsb=msbm5 \newfam\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \edef\msbfam@{\hexnumber@\msbfam} \def\Bbb#1{{\fam\msbfam\relax#1}} \def\widehat#1{\setbox\z@\hbox{$\m@th#1$}% \ifdim\wd\z@>\tw@ em\mathaccent"0\msbfam@5B{#1}% \else\mathaccent"0362{#1}\fi} \def\widetilde#1{\setbox\z@\hbox{$\m@th#1$}% \ifdim\wd\z@>\tw@ em\mathaccent"0\msbfam@5D{#1}% \else\mathaccent"0365{#1}\fi} \font\teneufm=eufm10 \font\seveneufm=eufm7 \font\fiveeufm=eufm5 \newfam\eufmfam \textfont\eufmfam=\teneufm \scriptfont\eufmfam=\seveneufm \scriptscriptfont\eufmfam=\fiveeufm \def\frak#1{{\fam\eufmfam\relax#1}} \let\goth\frak % Restore the catcode value for @ that was previously saved. %#\csname amssym.def\endcsname %#\endinput %%% ==================================================================== %%% @TeX-file{ %%% filename = "amssym.tex", %%% version = "2.2b", %%% date = "26 February 1997", %%% time = "13:14:29 EST", %%% checksum = "61515 286 903 9155", %%% author = "American Mathematical Society", %%% copyright = "Copyright (C) 1997 American Mathematical Society, %%% all rights reserved. Copying of this file is %%% authorized only if either: %%% (1) you make absolutely no changes to your copy, %%% including name; OR %%% (2) if you do make changes, you first rename it %%% to some other name.", %%% address = "American Mathematical Society, %%% Technical Support, %%% Electronic Products and Services, %%% P. O. Box 6248, %%% Providence, RI 02940, %%% USA", %%% telephone = "401-455-4080 or (in the USA and Canada) %%% 800-321-4AMS (321-4267)", %%% FAX = "401-331-3842", %%% email = "tech-support@ams.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "amsfonts, msam, msbm, math symbols", %%% supported = "yes", %%% abstract = "This is part of the AMSFonts distribution. %%% It contains the plain TeX source file for loading %%% the AMS extra symbols and Euler fraktur fonts.", %%% docstring = "The checksum field above contains a CRC-16 checksum %%% as the first value, followed by the equivalent of %%% the standard UNIX wc (word count) utility output %%% of lines, words, and characters. This is produced %%% by Robert Solovay's checksum utility.", %%% } %%% ==================================================================== %% Save the current value of the @-sign catcode so that it can %% be restored afterwards. This allows us to call amssym.tex %% either within an AMS-TeX document style file or by itself, in %% addition to providing a means of testing whether the file has %% been previously loaded. We want to avoid inputting this file %% twice because when AMSTeX is being used \newsymbol will give an %% error message if used to define a control sequence name that is %% already defined. %% %% If the csname is not equal to \relax, we assume this file has %% already been loaded and \endinput immediately. %#\expandafter\ifx\csname pre amssym.tex at\endcsname\relax \else\endinput\fi %% Otherwise we store the catcode of the @ in the csname. %#\expandafter\chardef\csname pre amssym.tex at\endcsname=\the\catcode`\@ %% Set the catcode to 11 for use in private control sequence names. \catcode`\@=11 %% Load amssym.def if necessary: If \newsymbol is undefined, do nothing %% and the following \input statement will be executed; otherwise %% change \input to a temporary no-op. %#\ifx\undefined\newsymbol \else \begingroup\def\input#1 {\endgroup}\fi %#\input amssym.def \relax %% Most symbols in fonts msam and msbm are defined using \newsymbol. A few %% that are delimiters or otherwise require special treatment have already %% been defined as soon as the fonts were loaded. Finally, a few symbols %% that replace composites defined in plain must be undefined first. \newsymbol\boxdot 1200 \newsymbol\boxplus 1201 \newsymbol\boxtimes 1202 \newsymbol\square 1003 \newsymbol\blacksquare 1004 \newsymbol\centerdot 1205 \newsymbol\lozenge 1006 \newsymbol\blacklozenge 1007 \newsymbol\circlearrowright 1308 \newsymbol\circlearrowleft 1309 \undefine\rightleftharpoons \newsymbol\rightleftharpoons 130A \newsymbol\leftrightharpoons 130B \newsymbol\boxminus 120C \newsymbol\Vdash 130D \newsymbol\Vvdash 130E \newsymbol\vDash 130F \newsymbol\twoheadrightarrow 1310 \newsymbol\twoheadleftarrow 1311 \newsymbol\leftleftarrows 1312 \newsymbol\rightrightarrows 1313 \newsymbol\upuparrows 1314 \newsymbol\downdownarrows 1315 \newsymbol\upharpoonright 1316 \let\restriction\upharpoonright \newsymbol\downharpoonright 1317 \newsymbol\upharpoonleft 1318 \newsymbol\downharpoonleft 1319 \newsymbol\rightarrowtail 131A \newsymbol\leftarrowtail 131B \newsymbol\leftrightarrows 131C \newsymbol\rightleftarrows 131D \newsymbol\Lsh 131E \newsymbol\Rsh 131F \newsymbol\rightsquigarrow 1320 \newsymbol\leftrightsquigarrow 1321 \newsymbol\looparrowleft 1322 \newsymbol\looparrowright 1323 \newsymbol\circeq 1324 \newsymbol\succsim 1325 \newsymbol\gtrsim 1326 \newsymbol\gtrapprox 1327 \newsymbol\multimap 1328 \newsymbol\therefore 1329 \newsymbol\because 132A \newsymbol\doteqdot 132B \let\Doteq\doteqdot \newsymbol\triangleq 132C \newsymbol\precsim 132D \newsymbol\lesssim 132E \newsymbol\lessapprox 132F \newsymbol\eqslantless 1330 \newsymbol\eqslantgtr 1331 \newsymbol\curlyeqprec 1332 \newsymbol\curlyeqsucc 1333 \newsymbol\preccurlyeq 1334 \newsymbol\leqq 1335 \newsymbol\leqslant 1336 \newsymbol\lessgtr 1337 \newsymbol\backprime 1038 \newsymbol\risingdotseq 133A \newsymbol\fallingdotseq 133B \newsymbol\succcurlyeq 133C \newsymbol\geqq 133D \newsymbol\geqslant 133E \newsymbol\gtrless 133F \newsymbol\sqsubset 1340 \newsymbol\sqsupset 1341 \newsymbol\vartriangleright 1342 \newsymbol\vartriangleleft 1343 \newsymbol\trianglerighteq 1344 \newsymbol\trianglelefteq 1345 \newsymbol\bigstar 1046 \newsymbol\between 1347 \newsymbol\blacktriangledown 1048 \newsymbol\blacktriangleright 1349 \newsymbol\blacktriangleleft 134A \newsymbol\vartriangle 134D \newsymbol\blacktriangle 104E \newsymbol\triangledown 104F \newsymbol\eqcirc 1350 \newsymbol\lesseqgtr 1351 \newsymbol\gtreqless 1352 \newsymbol\lesseqqgtr 1353 \newsymbol\gtreqqless 1354 \newsymbol\Rrightarrow 1356 \newsymbol\Lleftarrow 1357 \newsymbol\veebar 1259 \newsymbol\barwedge 125A \newsymbol\doublebarwedge 125B \undefine\angle \newsymbol\angle 105C \newsymbol\measuredangle 105D \newsymbol\sphericalangle 105E \newsymbol\varpropto 135F \newsymbol\smallsmile 1360 \newsymbol\smallfrown 1361 \newsymbol\Subset 1362 \newsymbol\Supset 1363 \newsymbol\Cup 1264 \let\doublecup\Cup \newsymbol\Cap 1265 \let\doublecap\Cap \newsymbol\curlywedge 1266 \newsymbol\curlyvee 1267 \newsymbol\leftthreetimes 1268 \newsymbol\rightthreetimes 1269 \newsymbol\subseteqq 136A \newsymbol\supseteqq 136B \newsymbol\bumpeq 136C \newsymbol\Bumpeq 136D \newsymbol\lll 136E \let\llless\lll \newsymbol\ggg 136F \let\gggtr\ggg \newsymbol\circledS 1073 \newsymbol\pitchfork 1374 \newsymbol\dotplus 1275 \newsymbol\backsim 1376 \newsymbol\backsimeq 1377 \newsymbol\complement 107B \newsymbol\intercal 127C \newsymbol\circledcirc 127D \newsymbol\circledast 127E \newsymbol\circleddash 127F \newsymbol\lvertneqq 2300 \newsymbol\gvertneqq 2301 \newsymbol\nleq 2302 \newsymbol\ngeq 2303 \newsymbol\nless 2304 \newsymbol\ngtr 2305 \newsymbol\nprec 2306 \newsymbol\nsucc 2307 \newsymbol\lneqq 2308 \newsymbol\gneqq 2309 \newsymbol\nleqslant 230A \newsymbol\ngeqslant 230B \newsymbol\lneq 230C \newsymbol\gneq 230D \newsymbol\npreceq 230E \newsymbol\nsucceq 230F \newsymbol\precnsim 2310 \newsymbol\succnsim 2311 \newsymbol\lnsim 2312 \newsymbol\gnsim 2313 \newsymbol\nleqq 2314 \newsymbol\ngeqq 2315 \newsymbol\precneqq 2316 \newsymbol\succneqq 2317 \newsymbol\precnapprox 2318 \newsymbol\succnapprox 2319 \newsymbol\lnapprox 231A \newsymbol\gnapprox 231B \newsymbol\nsim 231C \newsymbol\ncong 231D \newsymbol\diagup 201E \newsymbol\diagdown 201F \newsymbol\varsubsetneq 2320 \newsymbol\varsupsetneq 2321 \newsymbol\nsubseteqq 2322 \newsymbol\nsupseteqq 2323 \newsymbol\subsetneqq 2324 \newsymbol\supsetneqq 2325 \newsymbol\varsubsetneqq 2326 \newsymbol\varsupsetneqq 2327 \newsymbol\subsetneq 2328 \newsymbol\supsetneq 2329 \newsymbol\nsubseteq 232A \newsymbol\nsupseteq 232B \newsymbol\nparallel 232C \newsymbol\nmid 232D \newsymbol\nshortmid 232E \newsymbol\nshortparallel 232F \newsymbol\nvdash 2330 \newsymbol\nVdash 2331 \newsymbol\nvDash 2332 \newsymbol\nVDash 2333 \newsymbol\ntrianglerighteq 2334 \newsymbol\ntrianglelefteq 2335 \newsymbol\ntriangleleft 2336 \newsymbol\ntriangleright 2337 \newsymbol\nleftarrow 2338 \newsymbol\nrightarrow 2339 \newsymbol\nLeftarrow 233A \newsymbol\nRightarrow 233B \newsymbol\nLeftrightarrow 233C \newsymbol\nleftrightarrow 233D \newsymbol\divideontimes 223E \newsymbol\varnothing 203F \newsymbol\nexists 2040 \newsymbol\Finv 2060 \newsymbol\Game 2061 \newsymbol\mho 2066 \newsymbol\eth 2067 \newsymbol\eqsim 2368 \newsymbol\beth 2069 \newsymbol\gimel 206A \newsymbol\daleth 206B \newsymbol\lessdot 236C \newsymbol\gtrdot 236D \newsymbol\ltimes 226E \newsymbol\rtimes 226F \newsymbol\shortmid 2370 \newsymbol\shortparallel 2371 \newsymbol\smallsetminus 2272 \newsymbol\thicksim 2373 \newsymbol\thickapprox 2374 \newsymbol\approxeq 2375 \newsymbol\succapprox 2376 \newsymbol\precapprox 2377 \newsymbol\curvearrowleft 2378 \newsymbol\curvearrowright 2379 \newsymbol\digamma 207A \newsymbol\varkappa 207B \newsymbol\Bbbk 207C \newsymbol\hslash 207D \undefine\hbar \newsymbol\hbar 207E \newsymbol\backepsilon 237F % Restore the catcode value for @ that was previously saved. %#\catcode`\@=\csname pre amssym.tex at\endcsname %\endinput %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of symbols.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including titles.6c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %titles.6b % requires fonts.5 or higher and smallfonts.tex \count5=0 \count6=1 \count7=1 \count8=1 \count9=1 \def\proof{\medskip\noindent{\bf Proof.\ }} \def\qed{\hfill{\sevenbf QED}\par\medskip} \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{{\baselineskip=10pt \eightpoint\narrower\noindent{\eightbf 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 e#1\endcsname{ (\number\count5.\number\count6)} \global\advance\count6 by 1} \def\claim #1(#2) #3\par{ \vskip.1in\medbreak\noindent {\bf #1\ \number\count5.\number\count7.\ }{\sl #3}\par \expandafter\xdef\csname c#2\endcsname{#1~\number\count5.\number\count7} \global\advance\count7 by 1 \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} \def\section#1\par{\vskip0pt plus.1\vsize\penalty-40 \vskip0pt plus -.1\vsize\bigskip\bigskip \global\advance\count5 by 1 \message{#1}\leftline {\magtenbf \number\count5.\ #1} \count6=1 \count7=1 \count8=1 \nobreak\smallskip\noindent} \def\subsection#1\par{\vskip0pt plus.05\vsize\penalty-20 \vskip0pt plus -.05\vsize\medskip\medskip \message{#1}\leftline{\tenbf \number\count5.\number\count8.\ #1} \global\advance\count8 by 1 \nobreak\smallskip\noindent} \def\addref#1{\expandafter\xdef\csname r#1\endcsname{\number\count9} \global\advance\count9 by 1} \def\proofof(#1){\medskip\noindent{\bf Proof of \csname c#1\endcsname.\ }} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of titles.6c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including macros.18 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %macros.18 % requires fonts.5 or later \def\rightheadline{\hfil} \def\leftheadline{\sevenrm\hfil HANS KOCH\hfil} \headline={\ifnum\pageno=\firstpage\hfil\else \ifodd\pageno{{\fiverm\rightheadline}\number\pageno} \else{\number\pageno\fiverm\leftheadline}\fi\fi} \footline={\ifnum\pageno=\firstpage\hss\tenrm\folio\hss\else\hss\fi} % \let\ov=\overline \let\cl=\centerline \let\wh=\widehat \let\wt=\widetilde \let\eps=\varepsilon \let\sss=\scriptscriptstyle % \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\tr{{\rm tr}} \def\modulo{{\rm mod~}} \def\std{{\rm std}} \def\Re{{\rm Re\hskip 0.15em}} \def\Im{{\rm Im\hskip 0.15em}} \def\defeq{\mathrel{\mathop=^{\rm def}}} % \def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}} % \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}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of macros.18 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %brjuno8.tex %\input smallfonts.tex %\input param.2 %\input fonts.5b %\input symbols.1 %\input titles.6c %\input macros.18 % \def\bfA{{\hbox{\teneufm A}}} \def\bfG{{\hbox{\teneufm G}}} \def\bfJ{{\hbox{\teneufm J}}} \def\bfN{{\hbox{\teneufm N}}} \def\bfR{{\hbox{\teneufm R}}} % \def\fquad{\qquad\qquad} \def\bdot{\hbox{\bf .}} \def\ldot{\,.} \def\iminus{I^{\raise 1.8pt\hbox{$\sss -$}\!}} \def\iplus{I^{\raise 1.8pt\hbox{$\sss +$}\!}} \def\iplusminus{I^{\raise 1.8pt\hbox{$\sss\pm$}\!}} % \def\Iminus{{\Bbb I}^{\raise 1.8pt\hbox{$\sss -$}\!}} \def\Iplus{{\Bbb I}^{\raise 1.8pt\hbox{$\sss +$}\!}} \def\Iplusminus{{\Bbb I}^{\raise 1.8pt\hbox{$\sss\pm$}\!}} % \def\ssf{{\sss f}} \def\ssg{{\sss g}} \def\ssF{{\!\sss F}} \def\ssH{{\!\sss H}} \def\ssHp{{\!\sss H'}} \def\ssK{{\!\sss K}} \def\ssO{{\sss 0}} \def\ssR{{\sss R}} \def\ssX{{\sss X}} \def\ssXp{{\sss X'}} \def\ssY{{\sss Y}} \def\ssZ{{\sss Z}} % \def\DC{{\rm DC}} \def\GL{{\rm GL}} \def\SL{{\rm SL}} \def\SO{{\rm SO}} \def\SU{{\rm SU}} % % \def\smallreal{{\eightmsb R}} \def\smalltorus{{\eightmsb T}} % \def\spa{{\sss\parallel}} \def\spe{{\sss\perp}} %% \addref{EscandeDoveilEION} \addref{ShenkerKadanoffEITW} \addref{MacKayEITW} \addref{StirnemannNISE} % \addref{ChandreGovinJauslinNISE} \addref{AbadKochWittwer} \addref{ChandreJauslinZETW} \addref{KochNINI} \addref{KochZETW} \addref{KochZEFOa} \addref{KochZEFOb} \addref{GaidashevZEFI} \addref{LopesDiasZETW} \addref{KocicTWFI} \addref{KhaninLopesDiasMarklovZEFI} \addref{KochXXXX} \addref{KochKocicXXXX} % \addref{KochLopesDiasZEFI} % \addref{LopesDiasZESI} \addref{LopesDiasXXXX} \addref{GentileMastropietroNISI} % \addref{BrjunoSEON} \addref{BrjunoSETW} \addref{PoeschelEINI} \addref{RuessmannNIFO} \addref{EcalleValetNIEI} \addref{BerrettiGentileZEON} \addref{RuessmannZEON} \addref{GentileXXXX} \addref{GentileXXXXRuessmannZEON} \addref{GentileBartuccelliDeaneXXXX} % \addref{LagariasNIFO} \addref{KleinbockMargulisNIEI} %% \def\leftheadline{\sevenrm\hfil HANS KOCH and SA\v SA KOCI\'C\hfil} \def\rightheadline{\sevenrm\hfil Renormalization with Brjuno frequencies \hfil} %% %% \cl{{\magtenbf A renormalization group approach}} \cl{{\magtenbf to quasiperiodic motion with Brjuno frequencies}} \bigskip \cl{Hans Koch \footnote{$^1$} {{\sevenrm Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712}} and Sa\v sa Koci\'c \footnote{$^2$} {{\sevenrm Department of Physics, University of Texas at Austin, 1 University Station C1600, Austin, TX 78712}} } % % \bigskip \abstract We introduce a renormalization group scheme that applies to vector fields on \smalltorus$^d\times$\smallreal$^{m}$ with frequency vectors that satisfy a Brjuno condition. Earlier approaches were restricted to Diophantine frequencies, due to a limited control of multidimensional continued fractions. We get around this restriction by avoiding the use of a continued fractions expansion. Our results concerning invariant tori generalize those of reference [\rKochKocicXXXX] from Diophantine to Brjuno type frequency vectors. In particular, each Brjuno vector $\omega\in\,$\smallreal$^{\sss d}$ determines an analytic manifold $\WW$ of infinitely renormalizable vector fields, and each vector field on $\WW$ is shown to have an elliptic invariant $d$-torus with frequencies $\omega_1,\omega_2,\ldots,\omega_d\,$. \section Introduction and main results %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The renormalization of Hamiltonian flows and related maps has a relatively long history; see e.g. [\rGentileMastropietroNISI,\rChandreJauslinZETW,\rKochXXXX] and references therein. The approach considered here was motivated originally by the problem of describing the breakup of invariant tori for Hamiltonian systems with two degrees of freedom [\rEscandeDoveilEION--\rKochZEFOb]. Some of these questions have been answered in [\rKochZEFOa,\rKochZEFOb]. The same methods also apply to the study of near-integrable Hamiltonians or near-linear flows. In this regime, the renormalization group approach can be viewed as an alternative to purely perturbative methods, based on KAM theory or Lindstedt series. Over the past few years, its scope has been extended from a small set of ``self-similar'' frequency vectors [\rKochNINI] to arbitrary Diophantine frequencies [\rLopesDiasZETW--\rKochLopesDiasZEFI], and from Hamiltonian flows to a large class of vector fields [\rKochKocicXXXX]. As far as the construction of smooth invariant tori is concerned, this work covers the classical KAM results, but not the later extensions to Brjuno type frequency vectors [\rBrjunoSEON--\rGentileBartuccelliDeaneXXXX]. This is due to the fact that the current approach requires good bounds on a continued fractions expansion, such as the ones obtained in [\rKhaninLopesDiasMarklovZEFI] for Diophantine frequency vectors. Unfortunately, there seem to be significant obstacles to obtaining such bounds for Brjuno vectors in dimensions $d>2$. This has motivated us to develop a renormalization scheme that does not rely on continued fractions. As it turns out, it applies quite naturally to rotation vectors $\omega\in\real^d$ that satisfy the following Brjuno condition [\rBrjunoSEON]: $$ \sum_{n=1}^\infty 2^{-n}\ln(1/\Omega_n)<\infty\,,\qquad \Omega_n=\min_{0<|\nu|\le 2^n}|\omega\cdot\nu|\,, \equation(BrjunoCond) $$ where $\nu$ denotes lattice points in $\integer^d$. Our new renormalization group transformations share some important features with those used in [\rKochNINI--\rLopesDiasXXXX]. Thus, before discussing the differences, we shall first describe the transformations used in [\rKochNINI--\rKochKocicXXXX], starting with some general remarks about renormalization. Renormalization can be viewed as a tool for classifying systems by the value of a given observable that describes asymptotic properties of the system. A renormalization transformation is a map on the space of systems being considered, contracting directions within the same equivalence class, and expanding directions along which the observable changes, preferably in a way that induces a natural action on the set of observed values. In the case at hand, the systems are vector fields on $\MM=\torus^d\times\real^{m}$, and the observed quantities are the ratios among the $d$ frequencies of rotation. Among the natural actions on frequency vectors $\omega\in\real^d$ are the steps in a continued fractions algorithm. They typically involve integer matrices with a distinguished expanding direction, such that rational approximants to $\omega$ approach (up to rescaling) the orbit of $\omega$ under iteration of the algorithm. Consider now a fixed matrix $T\in\SL(d,\integer)$ whose transpose $T^\star$ contracts the orthogonal complement of $\omega$. If we restrict to vector fields $X(x,y)=(w,v(y))$, with $w\not=0$ and $v(0)=0$, then the simplest useful renormalization transformation associated with $T$ is given by $\RR(X)=\eta^{-1}\TT_\mu^\ast X$, where $\TT_\mu^\ast$ is the pullback of the map $$ \TT_\mu(x,y)=(Tx,\mu T'y)\,. \equation(TTmuDef) $$ Here, $T'$ denotes the ${m}\times{m}$ identity matrix (or the inverse of $T^\star$, if desired for the renormalization of Hamiltonian vector fields, where ${m}=d$). There is no natural choice for the scaling parameters $\eta$, but since all members of the family $\eta\mapsto\eta^{-1}\TT_\mu^\ast X$ are equivalent, in the sense that they yield the same frequency ratios, it is useful to choose $\eta=\eta(X)$ in such a way that $\RR(X)$ becomes a specific (normalized) representative of the family. This ensures contraction within this family of equivalent systems. The same considerations apply in principle to the choice of $\mu$, but for the vector fields considered here, it suffices to choose for $\mu$ a small positive constant that makes $\mu T'$ a contraction. When considering more general vector fields, we also have to achieve contraction within families $\UU\mapsto\UU^\ast X$, obtained from changes of coordinates $\UU:\MM\to\MM$ close to the identity. This suggests defining $$ \RR(X)=\eta^{-1}\TT_\mu^\ast\UU_\ssX^\ast X\,, \equation(RGDef) $$ where $\UU_\ssX$ is some change of coordinates designed to bring the renormalized vector field into some appropriate normal form. More details about the choice of normal forms will be given later. In particular, $\UU_\ssX$ is the identity map whenever $X(x,y)=(w,v(y))$. If $\omega\in\real^d$ admits a periodic continued fractions expansion, then it suffices to work with a single RG transformation [\rKochNINI--\rGaidashevZEFI]. More general frequency vectors require a sequence of RG transformations $\RR_n\,$, one for each of the matrices $T_n$ in the continued fractions expansion of $\omega$. In the single frequency case ($d=2$), such an analysis was carried out for Diophantine [\rLopesDiasZETW,\rKocicTWFI] and Brjuno type frequencies [\rLopesDiasZESI,\rLopesDiasXXXX]. What makes this case special is that there is a canonical continued fractions expansion, and the corresponding matrices $T_n$ are known explicitly. More recent results [\rKhaninLopesDiasMarklovZEFI--\rKochLopesDiasZEFI] extend the scope of renormalization to Diophantine vectors $\omega\in\real^d$, for arbitrary $d\ge 2$, using a multidimensional continued fractions algorithm developed in [\rLagariasNIFO,\rKleinbockMargulisNIEI,\rKhaninLopesDiasMarklovZEFI]. Here, the matrices $T_n$ are no longer known explicitly. They are the increments $T_n=P_{n-1}^{-1}P_n$ of integer approximants $P_k\in\SL(d,\integer)$ to matrices $WE(t_k)\in\SL(d,\real)$, where $\{t_k\}$ is some appropriately chosen increasing sequence of positive real numbers, $E(t)={\rm diag}(e^{-t},\ldots,e^{-t},e^{(d-1)t})$, and $W$ is a fixed matrix in $\SL(d,\real)$ that maps the expanding eigenvector of $E(t)$ to $\omega$. The approximation is well controlled for Diophantine frequency vectors [\rKhaninLopesDiasMarklovZEFI], but attempts to extend this to Brjuno vectors have not been successful so far. The idea pursued here is to avoid the integer approximation and renormalize directly with real matrices. Since there is no longer any reason to stay with $\SL(d,\real)$, we choose the matrix $T$ in the definition \equ(TTmuDef) of the scaling $\TT_\mu$ to be of the general form $$ T(x)=\eta^{-1}x_\spa+\beta x_\spe\,,\qquad 0<\eta\,,\beta<1\,, \equation(TDef) $$ where $x=x_\spa+x_\spe$ is the decomposition of $x\in\real^d$ into a component $x_\spa$ parallel to $\omega$, and a component $x_\spe$ perpendicular to $\omega$. Notice that, if $d=2$ and $\omega_1/\omega_2=1/(k+1/(k+\ldots))$, then the choice $\eta=\beta=(\omega_1/\omega_2)^2$ makes $T$ in fact an integer matrix. A matrix of the type \equ(TDef) will be referred to as a {\it scaling matrix}. The corresponding RG transformation $\RR$ is taken to be again of the form \equ(RGDef), with $\eta^{-1}$ the expanding eigenvalue of $T$. Our choice of $\mu$ and $\UU_\ssX$ will be described later. Clearly, $K=(\omega,0)$ is a fixed point for $\RR$. We note that, by choosing the time scaling $\eta^{-1}$ in \equ(RGDef) to be the same as the spatial scaling $\eta^{-1}$ in \equ(TDef), which is independent of the vector field $X$, we allow $\RR$ to have a non-contracting direction. However, this direction is trivial and can be taken care of later. With this choice of scaling $T$, it becomes necessary to consider toral domains of the form $\torus^d=\real^d/\ZZ$, where $\ZZ$ is a simple lattice in $\real^d$. Functions on such a torus can be identified with functions on $\real^d$ that are invariant under $\ZZ$-translations, or equivalently, with quasiperiodic functions on $\real^d$ whose frequency module lies in the dual lattice (the set of points $v\in\real^d$ satisfying $\exp(i v\cdot z)=1$, for all $z\in\ZZ$). For convenience, we will now perform a linear change of coordinates in $\real^d$, such that $\omega=(1,0,\ldots,0)$. The lattice obtained from $2\pi\integer^d$ under this transformation will be denoted by $\ZZ_0\,$, and its dual lattice by $\VV_0\,$. The frequencies $\nu$ in \equ(BrjunoCond) now range over $\VV_0\,$. Our analysis applies to vector fields that are close to $K=(\omega,0)$. We assume analyticity on a complex neighborhood $D_\varrho$ of $D_0=\torus^d\times\{0\}$, characterized by the conditions $|\Im x_i|<\varrho$ and $|y_j|<\varrho$. Denote by $\Phi_\ssX$ the flow for a vector field $X$. An invariant torus for $X$, with frequency vector $\omega$, is a continuous embedding $\Gamma$ of $D_0$ into the domain of $X$, such that $\Gamma\circ\Phi_\ssK^t=\Phi_\ssX^t\circ\Gamma$. Denote by $A^u$ the space of all vector fields $Y(x,y)=(w,My+v)$, with $(w,v)$ a vector in $\complex^d\times\complex^{m}$ and $M$ a complex ${m}\times{m}$ matrix. In Section 2, we will introduce Banach spaces $\AA_\varrho(\VV)$ of analytic vector fields on $D_\varrho\,$, having frequency module in $\VV$, and a projection operator $\proj$ from $\AA_\varrho(\VV)$ onto $A^u$. The subspace of functions in $\AA_\varrho(\VV)$, that do not depend on the coordinate $y\in\complex^{m}$, will be denoted by $\AA_\varrho^0(\VV)$. A function will be called ``real'' if it takes real values for real arguments. Our main result is the following. \claim Theorem(RGSummary) Assume that $\omega$ satisfies the Brjuno condition \equ(BrjunoCond). Then there exists a sequence of scaling matrices $T_n\,$, and a corresponding sequence of RG transformations $\RR_n$ of the form \equ(RGDef), such that the following holds. Define $\VV_n=T_n\VV_{n-1}$ for $n=1,2,\ldots\,$. Then $\RR_n$ is an analytic map, from some open neighborhood $\DD_{n-1}$ of $K$ in $\AA_\varrho(\VV_{n-1})$, to $\AA_\varrho(\VV_n)$. The set $\WW$ of infinitely renormalizable vector fields $X_0$ in $\DD_0\,$, characterized by the property that $X_n=\RR_n(X_{n-1})$ belongs to $\DD_n$ for $n=1,2,\ldots$, is the graph of an analytic function $W:(\Id-\proj)\DD_0\to\proj\DD_0\,$, satisfying $W(0)=K$ and $DW(0)=0$. If $\rho>\varrho+\delta$, with $\delta>0$, then every vector field $X\in\WW\cap\AA_\rho(\VV_0)$ has an elliptic invariant torus $\Gamma_\ssX\in\AA_\delta^0(\VV_0)$ with frequency vector $\omega$. The map $X\mapsto\Gamma_\ssX$ is real analytic on $\WW\cap\AA_\rho(\VV_0)$. The bounds obtained in the proof of this theorem are uniform within classes of Bruno vectors $\BB(\Omega')$ described at the end of Section 3. In addition, we obtain results analogous to those in [\rKochKocicXXXX], concerning the restriction of $W$ to special types of vector fields (Hamiltonian, reversible, divergence free, symmetric) and the reduction of the number of parameters via nondegeneracy conditions. Since the proofs are completely analogous as well, we refer to [\rKochKocicXXXX] for details. The main new aspect in this paper is the choice of the scaling matrices $T_n\,$, and the control of the corresponding sequence of RG transformations $\RR_n\,$. The choice of the coordinate change $\UU_\ssX$ is determined by the same considerations as in earlier work [\rKochNINI--\rLopesDiasXXXX]. Its role is to compensate for the loss of analyticity that results from the scaling $X\mapsto\TT_\mu^\ast X$. In this step, we use a normal form theorem proved in [\rKochKocicXXXX]. Thus, controlling a single RG step is quite simple; see Section 2. In Section 3, we define the matrices $T_n$ and give estimates on the transformations $\RR_n\,$. Then we apply a stable manifold theorem for sequences of maps [\rKochKocicXXXX] to obtain the manifold $\WW$ described in \clm(RGSummary). The construction of invariant tori for vector fields $X\in\WW$ is described in Section 4. \section A single renormalization step %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% As mentioned in the introduction, we work with coordinates where the frequency vector is $\omega=(1,0,\ldots,0)$. The torus considered in this section is $\torus^d=\real^d/\ZZ$, where $\ZZ$ is some simple lattice in $\real^d$. The dual lattice will be denoted by $\VV$. Unless specified otherwise, our norm on $\complex^n$ is $\|v\|=\sup_j|v_j|$. Another norm that will be used is $|v|=\sum_j|v_j|$. For linear operators between normed linear spaces, we will always use the operator norm, unless stated otherwise. Denote by $D_\rho$ the set of all vectors $(x,y)$ in $\complex^d\times\complex^{m}$ characterized by $\|\Im x\|<\rho$ and $\|y\|<\rho$. Define $\AA_\rho(\VV)$ to be the space of all analytic vector field $X$ on $D_\rho\,$, with frequency module in $\VV$, and with a finite norm $$ \|X\|_\rho=\sum_{v,\alpha}\|X_{v,\alpha}\| e^{\rho|v|}\rho^{|\alpha|}\,,\qquad X(x,y)=\sum_{v,\alpha}X_{v,\alpha} e^{iv\cdot x} y^\alpha\,, \equation(AArhoNorm) $$ where $v\cdot x=\sum_jv_jx_j$ and $y^\alpha=\prod_jy_j^{\alpha_j}\,$. The sums in this equation range over all $v\in\VV$ and $\alpha\in\natural^{m}$. In Section 4, we will also use functions with domain $D_0=\torus^d\times\{0\}$. Denote by $\AA_0(\VV)$ the Banach space of continuous functions $F:D_0\to\complex^{d+{m}}\,$, with frequency module in $\VV$, for which the norm $\|F\|_0=\sum_v\|F_v\|$ is finite, where $\{F_v\}$ are the Fourier coefficients of $F$. Since the lattice $\VV$ is fixed in this section, we will simply write $\AA_\rho$ in place of $\AA_\rho(\VV)$. \claim Proposition(Trivial) Let $X\in\AA_\rho$ and $Z\in\AA_{\rho'}\,$, with $0\le\rho'\le\rho$. Then \item{$(a)$} $\|X(x,y)\|\le\|X\|_\rho$ for all $(x,y)\in D_\rho\,$. \item{$(b)$} $(DX)Z\in\AA_{\rho'}$ and $\|(DX)Z\|_{\rho'}\le(\rho-\rho')^{-1}\|X\|_{\rho}\|Z\|_{\rho'}\,$, if $\rho'<\rho$. \item{$(c)$} $X\circ(\id+Z)\in\AA_{\rho'}$ and $\|X\circ(\id+Z)\|_{\rho'}\le\|X\|_\rho\,$, if $\rho'+\|Z\|_{\rho'}\le\rho$. The proof of these estimates is straightforward and will be omitted. In what follows, we always assume that $\rho>0$, unless specified otherwise. We assume that the components of $\omega$ are rationally independent with respect to $\VV$, in the sense that the first component $v_\spa$ of any nonzero vector $v\in\VV$ is nonzero. Then, given any $L\ge 1$, we can find $\ell>0$ such that $$ |v_\spe|>L \quad {\rm or}\quad |v_\spa|\ge\ell\,,\qquad \forall v\in\VV\setminus\{0\}\,. \equation(emptyRectangle) $$ In other words, all points in $\VV$, except for the origin, lie outside the rectangle $|v_\spe|\le L$ and $|v_\spa|<\ell$. Notice that the scaling \equ(TDef) shrinks the length $L$ of the excluded rectangle, and expands its width $\ell$. In what follows, the parameters $L,\ell,\eta,\beta$ are assumed to be given, subject to the conditions \equ(TDef) and \equ(emptyRectangle). \claim Definition(Resonant) Denote by $S$ the generator of the one-parameter group of scalings $\mu\mapsto\SS_\mu^\ast$, defined by $\SS_\mu(x,y)=(x,\mu y)$. Given any subset $J$ of $I=\VV\times\{-1,0,1,2,\ldots\}$, define $P(J)$ to be the joint spectral projection in $\AA_\rho(\VV)$ for the operators $(-i\nabla_x,S)$, associated with the eigenvalues $(v,k)$ in $J$. Let $\tau=(1+\beta)/2$. Given $\gamma\ge 1$ to be chosen later, let $\iplus$ be the set of pairs $(v,k)\in I$ satisfying $|Tv|\le\tau|v|$ or $|Tv|\le\tau(k-\gamma)$, and let $\iminus$ be the complement of $\iplus$ in $I$. Define $\Iplusminus=P(\iplusminus)$. The {\it resonant} and {\it nonresonant} parts of a vector field $X\in\AA_\rho$ are defined as $\Iplus X$ and $\Iminus X$, respectively. In addition, we define $\mean_k=P\bigl(\{(0,k)\}\bigr)$, for each $k\ge -1$. The torus averaging operator is then given by $\mean=\sum_k\mean_k\,$. As we will see later, the scaling $X\mapsto\TT_\mu^\ast X$ is well behaved when restricted to resonant vector fields. Thus, before applying this scaling, we try to perform a change of variables $X\mapsto\UU_\ssX^\ast X$ that eliminates the nonresonant part of $X$. Theorem 5.2 in [\rKochKocicXXXX] shows that this is possible, provided that the problem can be solved to first order in the size of $X-K$. The equation for the map $\UU_\ssX\,$, and for the vector field $Z=\Iminus Z$ generating its first order approximation $\Phi_\ssZ^1\,$, is $$ \Iminus(X+[Z,X])=0\,,\qquad \Iminus\UU_\ssX^\ast X=0\,, \equation(ElimEqu) $$ where $[Z,X]=(DX)Z-(DZ)X$. The following proposition is used to solve the first part of this equation. Given any positive real number $r$, denote by $\AA'_r$ the set of vector fields $X\in\AA_r$ whose first partial derivatives belong to $\AA_r\,$. Assume that $$ 2\sigma L<\ell\,,\qquad \sigma={\textstyle\half}(1-\beta)\eta\,. \equation(LdeltaCond) $$ \claim Proposition(KXBound) If $r>0$ and $Z\in\AA'_r$ is nonresonant, then $$ \|[Z,K]\|_r\ge\sigma\|Z\|_r\,,\qquad \|[Z,K]\|_r\ge{\sigma r\over\sigma r+r+\gamma+2}\|DZ\|_r\,. \equation(KXBounds) $$ \proof Assume that $(v,k)$ belongs to $\iminus$. In particular, we have $|Tv|>\tau|v|$, or equivalently, $\eta^{-1}|v_\spa|+\beta|v_\spe|>\tau|v_\spa|+\tau|v_\spe|$. This immediately implies that $|v_\spa|>\sigma|v_\spe|$. Combining this with the condition $|Tv|>\tau(k-\gamma)$, we also have $$ \sigma^{-1}|v_\spa| =\tau^{-1}\bigl(\eta^{-1}|v_\spa|+\beta\sigma^{-1}|v_\spa|\bigr) >\tau^{-1}\bigl(\eta^{-1}|v_\spa|+\beta|v_\spe|\bigr) =\tau^{-1}|Tv|>k-\gamma\,. $$ The inequality $|v_\spa|>\sigma|v_\spe|$, together with \equ(emptyRectangle) and \equ(LdeltaCond), also implies that $|v_\spa|>\sigma$. These bounds show that if $Z\in\Iminus\AA'_r$ and $Y=[Z,K]$, then $\|Z\|_r\le\sigma^{-1}\|Y\|_r$ and $$ \sum_{j=2}^d\left\|{\partial\over\partial x_j}Z\right\|_r \le{1\over\sigma}\|Y\|_r\,,\qquad \sum_{j=1}^{m}\left\|{\partial\over\partial y_j}Z\right\|_r \le{\gamma+2\over\sigma r}\|Y\|_r\,. \equation(omegadotnuInv) $$ As a result we obtain \equ(KXBounds). \qed This proposition, together with \clm(Trivial), allows us to apply the normal form theorem in [\rKochKocicXXXX, Section 5], which directly implies the following lemma. Let $\varrho>0$ be fixed once and for all. What we will call a ``universal constant'' may depend on the choice of $\varrho$, but not on any other parameter. \claim Lemma(Elim) There exist universal constants $C_1$ and $C_2\,$, such that the following holds. Let $\rho'>0$ and $\rho\ge\rho'+\sigma\varrho$. If $X$ is any vector field in $\AA'_\rho\,$, satisfying $$ \|X-K\|'_\rho\le C_1(\sigma/\gamma)\,,\qquad \|\Iminus X\|_\rho\le C_1(\sigma/\gamma)^2\,,\qquad \equation(ElimCond) $$ then there exists a vector field $Z\in\Iminus\AA_\rho$ and a change of coordinates $\UU_\ssX:D_{\rho'}\to D_\rho\,$, solving equation \equ(ElimEqu). The vector field $\UU_\ssX^\ast X$ belongs to $\AA_{\rho'}\,$, and $$ \eqalign{ \|Z\|_\rho\,,\|\UU_\ssX-\id\|_{\rho'} &\le C_2(\gamma/\sigma)\|\Iminus X\|_\rho\,,\cr \|\UU_\ssX^\ast X-X\|_{\rho'} &\le C_2(\rho-\rho')^{-1}(\gamma/\sigma)\|\Iminus X\|_\rho\,,\cr \|\UU_\ssX^\ast X-X-[Z,X]\|_{\rho'} &\le C_2(\rho-\rho')^{-3}(\gamma/\sigma)^3\|\Iminus X\|_\rho^2\,.\cr} \equation(ElimBounds) $$ The map $X\mapsto\UU_\ssX$ is continuous in the region defined by \equ(ElimCond), and analytic in its interior. Next, we assume that the scaling parameters $\eta$, $\beta$ and $\mu$ satisfy $$ \eta<1/2\,,\qquad e^{-\varrho{(1-\beta)\over 6}L}\le(4\mu)^{\gamma+1}\,,\qquad 4\mu\le e^{-\varrho}\,. \equation(LtaumuCond) $$ \claim Lemma(Contraction) If $\varrho(2+\beta)/3\le\rho'\le\varrho$, then $\TT_\mu^\ast$ defines a bounded linear operator from $\Iplus\AA_{\rho'}(\VV)$ to $\AA_\varrho(T\VV)$, with the property that $$ \eqalign{ \bigl\|\TT_\mu^\ast\mean_k X\|_\varrho &\le 8\eta^{-1}(4\mu)^k\|\mean_k X\|_{\rho'}\,,\cr \bigl\|\TT_\mu^\ast\Iplus(\Id-\mean)X\|_\varrho &\le 2\eta^{-1}\bigl(4e^\varrho\mu\bigr)^\gamma \|\Iplus(\Id-\mean)X\|_{\rho'}\,.\cr} \equation(Contraction) $$ \proof By our choice of norm \equ(AArhoNorm), it suffices to verify the given bounds for vector fields $X=P(J)Y$, with $J\subset\iplus$ containing a single point, say $J=\{(v,k)\}$. Let $b=\varrho/(\rho'\beta)$. Then it follows essentially from the definitions that $$ \|\TT_\mu^\ast P(J)Y\|_\varrho \le 2\eta^{-1}e^A\|P(J)Y\|_{\rho'}\,,\qquad A=\varrho|T v|-\rho'|v|+k\ln(b\mu)\,. \equation(JAbound) $$ Setting $v=0$, and using that $1L$ by condition \equ(emptyRectangle). Consequently, $$ A\le-\varrho\Bigl({\rho'\over\varrho}-\tau\Bigr)|v|+k\ln(b\mu) \le-\varrho{1-\beta\over 6}L+k\ln(b\mu)\,, \equation(AboundOne) $$ and the second bound in \equ(Contraction) follows by using \equ(LtaumuCond). Next, consider the case $|T v|\le\tau(k-\gamma)$. Notice that $k>\gamma$ here, since $v$ is nonzero. By using the bound $A\le\varrho(k-\gamma)+k\ln(b\mu)$, together with \equ(JAbound), we obtain $$ \|\TT_\mu^\ast P(J)Y\|_\varrho \le 2\eta^{-1}\bigl(b e^\varrho\mu\bigr)^k \|P(J)Y\|_{\rho'}\,. $$ This again implies the second bound in \equ(Contraction). \qed Combining the preceding two lemmas, we obtain the following theorem. Notice that, by property \equ(ElimBounds), the restriction of $\RR$ to the subspace $\proj\AA_\varrho(\VV)$ defines a linear operator from $\proj\AA_\varrho(\VV)$ to $\proj\AA_\varrho(T\VV)$. This operator will be denoted by $\LL$. \claim Theorem(RRBounds) There exist universal constants $C,R>0$, such that the following holds, under the given assumptions on $L,\ell,\eta,\beta,\gamma$ and $\mu$. Let $B$ be the open ball in $\AA_\varrho(\VV)$ of radius $R(\sigma/\gamma)^2$, centered at $K$. Then $\RR$ is a bounded analytic map from $B$ to $\AA_\varrho(T\VV)$, satisfying $\|\LL^{-1}\|\le 1$ and $$ \eqalign{ \|(\Id-\mean)\RR(X)\|_\varrho &\le\eta^{-2}(1-\beta)^{-1}(\gamma/\sigma)(C\mu)^\gamma \|(\Id-\mean)X\|_\varrho\,,\cr \|(\Id-\proj)\RR(X)\|_\varrho &\le C\eta^{-2}(1-\beta)^{-1}(\gamma/\sigma)\mu \|(\Id-\proj)X\|_\varrho\,,\cr \|\mean\RR(X)-\RR(\mean X)\|_\varrho &\le C\eta^{-2}(1-\beta)^{-3}(\gamma/\sigma)^3\mu^{-1} \|(\Id-\mean)X\|_\varrho^2\,.\cr} \equation(RRBounds) $$ \proof Let $\rho=\varrho-\varrho(1-\beta)/12$ and $\rho'=\rho-\varrho(1-\beta)/4$. Then there exists a universal constant $R>0$, such that the conditions \equ(ElimCond) in \clm(Elim) hold, whenever $X$ belongs to the domain $B$, defined by $\|X-K\|_\varrho2$ are two integer constants to be determined later. Then the Brjuno condition \equ(BrjunoCond) is equivalent to the condition that the resulting sequence $\{a_n\}$ is summable. We remark that the weighted sum has been included in the definition \equ(anDef) in order to limit the local growth of the sequence $\{a_n\}$. And the term $(k+\kappa')^{-2}$ has been included to avoid sequences $\{a_n\}$ that decrease too rapidly. This allows for a more uniform treatment of all Brjuno vectors. Define $\lambda_0=1$ and $$ \lambda_n=2^{-n-\kappa}e^{-2^{n+\kappa}a_n}\,,\quad \eta_n={\lambda_n\over\lambda_{n-1}}\,,\quad A_n=\sum_{k=n}^\infty a_k\,,\quad \beta_n={A_{n+1}\over A_n}\,, \equation(lambdaEtc) $$ for all positive integers $n$. Consider the corresponding scaling transformations $$ P_n(x)=\lambda_n^{-1}x_\spa+A_1^{-1}A_{n+1}x_\spe \,,\qquad T_n(x)=\eta_n^{-1}x_\spa+\beta_n x_\spe\,. $$ Notice that $P_n=T_1 T_2\cdots T_n$ by equation \equ(lambdaEtc). These quantities, will now be used to define the $n$-th step RG transformation $\RR=\RR_n\,$. To this end, we need to verify the assumptions made in Section 2. Clearly, $\beta=\beta_n$ is positive and less than one, since $n\mapsto A_n$ is a decreasing sequence. And the condition on $\eta=\eta_n$ in equation \equ(LtaumuCond) follows from the fact that $a_n>a_{n-1}/2$ for $n>1$, and that $\lambda_1<1/2$. The geometric data $\VV$, $L$ and $\ell$ used in step $n$ are $$ \VV_{n-1}=P_{n-1}\VV_0\,,\qquad L_{n-1}=A_1^{-1}A_n2^{n+\kappa}\,,\qquad \ell_{n-1}=2^{n+\kappa}\eta_n\,. \equation(VVLdeltaDef) $$ These definitions immediately imply \equ(LdeltaCond). The following proposition shows that the condition \equ(emptyRectangle) holds for all $v\in\VV$. \claim Proposition(indeed) If $v\in\VV_{n-1}$ is nonzero, then either $|v_\spa|\ge\ell_{n-1}$ or $|v_\spe|> L_{n-1}\,$. \proof Assume that $v\in\VV_{n-1}$ satisfies $0<|v_\spe|\le L_{n-1}$. Then the corresponding lattice point $\nu=P_{n-1}^{-1}v$ in $\VV_0$ satisfies $|\nu_\spe|\le A_1A_n^{-1}L_{n-1}= 2^{n+\kappa}$, and thus $|\nu_\spa|\ge\Omega'_{n+\kappa}$ by \equ(anDef). Since we have $\lambda_n<2^{-n-\kappa}\Omega'_{n+\kappa}$, this yields $$ |v_\spa|=\lambda_{n-1}^{-1}|\nu_\spa| \ge\eta_n\lambda_n^{-1}\Omega'_{n+\kappa} >\eta_n 2^{n+\kappa}=\ell_{n-1}\,, \equation(indeed) $$ as claimed. \qed The second condition in \equ(LtaumuCond) is satisfied simply by choosing $\mu=\mu_n\,$, where $$ \mu_k=\exp\Bigl\{-{\varrho\over 6}\cdot {1-\beta_k\over\gamma+1}L_{k-1}\Bigr\} =\exp\Bigl\{-{\varrho\over 6(\gamma+1)A_1}\cdot 2^{k+\kappa}a_k\Bigr\}\,,\qquad k\ge 1\,. \equation(muDef) $$ Finally, the third inequality in \equ(LtaumuCond) is taken care of by choosing $\kappa'$ and $\kappa$ sufficiently large, as the following proposition shows. \claim Proposition(muBound) For all $k\ge 1$, $\mu_{k+1}<\mu_k<\mu_{k+1}^{1/4}$. Furthermore, given $\gamma\ge 1$ and $C,N>0$, if $\kappa'$ and then $\kappa$ are chosen sufficiently large, then for all $k\ge 1$, $$ \mu_k\le Ce^{-N2^{k+\kappa}a_k}\,,\qquad \mu_k\le C\eta_k^N\,,\qquad \mu_k\le C(1-\beta_k)^N\,. \equation(muBound) $$ \proof The inequality $\mu_{k+1}<\mu_k<\mu_{k+1}^{1/4}$ follows from the fact that $2a_{k+1}0$. The same inequality, together with $1-\beta_k=a_k/A_k>(k+\kappa')^{-2}2N/c$, implies the third bound in \equ(muBound). \qed Having verified all of the assumptions made in Section 2, we can now apply \clm(RRBounds) to the $n$-th step RG transformation $\RR_n\,$, defined by the parameters introduced above. Denote by $\LL_n$ the corresponding linear operator from $\proj\AA_\varrho(\VV_{n-1})$ to $\proj\AA_\varrho(\VV_n)$. Define $\AA_{\varrho,k}=\AA_\varrho(\VV_k)$, for all non-negative integers $k$. To simplify notation, the norm in $\AA_{\varrho,k}$ and the projections $\mean$ and $\proj$ on this space will not be given indices. {}From \clm(RRBounds) we immediately obtain \claim Theorem(RRnBounds) Let $\gamma\ge 1$. There exist constants $r,C>0$, such that the following holds, for every positive integer $n$. Let $B_{n-1}$ be the open ball in $\AA_{\varrho,n-1}$ of radius $r\sigma_n^2$, centered at $K$, where $\sigma_n=\half(1-\beta_n)\eta_n\,$. Then $\RR_n$ is a bounded analytic map from $B_{n-1}$ to $\AA_{\varrho,n}\,$, satisfying $\|\LL_n^{-1}\|\le 1$ and $$ \eqalign{ \|(\Id-\mean)\RR_n(X)\|_\varrho &\le C\sigma_n^{-3}\mu_n^\gamma \|(\Id-\mean)X\|_\varrho\,,\cr \|(\Id-\proj)\RR_n(X)\|_\varrho &\le C\sigma_n^{-3}\mu_n \|(\Id-\proj)X\|_\varrho\,,\cr \|\mean\RR_n(X)-\RR_n(\mean X)\|_\varrho &\le C\sigma_n^{-6}\mu_n^{-1} \|(\Id-\mean)X\|_\varrho^2\,.\cr} \equation(RRnBounds) $$ \smallskip In what follows, a domain $\DD_{n-1}$ for $\RR_n$ is a subset of the ball $B_{n-1}$ described in \clm(RRnBounds), that is open in $\AA_{\varrho,n-1}$ and contains the vector field $K$. Given a domain $\DD_{n-1}$ for each $\RR_n\,$, the domain $\wt\DD_n$ of the combined RG transformation $\wt\RR_{n+1}=\RR_{n+1}\circ\RR_n\circ\ldots\circ\RR_1$ is defined recursively as the set of all vector fields in the domain of $\wt\RR_n$ that are mapped under $\wt\RR_n$ into the domain $\DD_n$ of $\RR_{n+1}\,$. By \clm(RRnBounds), these domains are open and non-empty, and the transformations $\wt\RR_n$ are analytic. \claim Theorem(RRnCompose) Let $\gamma\ge 4$. If $\kappa'$ and then $\kappa$ are chosen sufficiently large, then there exist a sequence of domains $\DD_0,\DD_1,\ldots$ for the RG transformations $\RR_1,\RR_2,\ldots\,$, such that the set $\WW=\cap_n\wt\DD_n$ is the graph of an analytic function $W:(\Id-\proj)\DD_0\to\proj\DD_0\,$, satisfying $W(0)=K$ and $DW(0)=0$. For every $X\in\WW$, if $n\ge 1$ and $\psi_n=\mu_1\mu_2\cdots\mu_n\,$, then $$ \eqalign{ \bigl\|\wt\RR_n(X)-K_n\bigr\|_\varrho &\le\psi_n^{1/2}\|(\Id-\proj)X\|_\varrho\,,\cr \bigl\|\proj[\wt\RR_n(X)-K_n]\bigr\|_\varrho &\le\psi_n\|(\Id-\proj)X\|_\varrho^2\,,\cr \bigl\|(\Id-\mean)\wt\RR_n(X)\bigr\|_\varrho &\le\psi_n^{\gamma-1/2}\|(\Id-\mean)X\|_\varrho\,.\cr} \equation(RRnCompose) $$ \proof Our goal is to apply the stable manifold theorem in [\rKochKocicXXXX, Section 6]. To do so, we first rescale our transformations $\RR_n\,$. Let $r_n=r_{n-1}\sigma_{n+1}^2$ for every positive integer $n$, with $r_0>0$ smaller than half the constant $r$ from \clm(RRnBounds). Consider the transformations $R_1,R_2,\ldots\,$, given by the equation $$ R_n(Z)=r_n^{-1}\bigl[\RR_n(K+r_{n-1}Z)-K\bigr]\,,\qquad n=1,2,\ldots\,. \equation(RnDef) $$ The restriction $R_n\proj$ defines a linear map from $\proj\AA_{\varrho,n-1}$ to $\proj\AA_{\varrho,n}\,$, which will be denoted by $L_n\,$. By \clm(RRnBounds), $R_n$ is analytic and bounded on the ball $\|Z\|_\varrho<2$, and satisfies $$ \eqalign{ \|(\Id-\mean)R_n(Z)\|_\varrho &\le\eps_n\|(\Id-\mean)Z\|_\varrho\,,\cr \|(\Id-\proj)R_n(Z)\|_\varrho &\le\vartheta_n\|(\Id-\proj)Z\|_\varrho\,,\cr \|\proj R_n(Z)-R'_n(\proj Z)\|_\varrho &\le\varphi_n\|(\Id-\mean)Z\|_\varrho^2\,,\cr} \equation(RBounds) $$ where $$ \eps_n=C\sigma_n^{-3}\sigma_{n+1}^{-2}\mu_n^\gamma\,,\quad \vartheta_n=C\sigma_n^{-3}\sigma_{n+1}^{-2}\mu_n\,,\quad \varphi_n=C\sigma_n^{-6}\sigma_{n+1}^{-2}\mu_n^{-1}\,. \equation(etpOne) $$ Here, $C\ge 1$ is a constant that may depend on $\gamma$, but not on any other RG parameters. In addition, we have $\|L_n^{-1}\|<1/4$. We will restrict $R_n$ to the domain $D_{n-1}\subset\AA_{\varrho,n-1}\,$, defined by $$ \|\proj Z\|_\varrho<1\,,\qquad \|(\Id-\proj)Z\|_\varrho<1\,,\qquad \|(\Id-\mean)Z\|_\varrho<\delta_{n-1}\,, \equation(RnDomainDef) $$ where $\delta_{n-1}=(6\varphi_n)^{-1}$. By \clm(muBound), if $\kappa'$ and $\kappa$ are chosen sufficiently large, then $C\sigma_n^{-3}\sigma_{n+1}^{-2}\mu_n^{1/2}\le 1/6$ and $C\mu_n^{\gamma-3}\le\sigma_{n+1}^6\sigma_{n+2}^2$, for all positive integers $n$. These inequalities imply $$ \eps_n\le\mu_n^{\gamma-1/2}/6\,,\qquad \vartheta_n\le\mu_n^{1/2}/4\,,\qquad \eps_n\delta_{n-1}\le\delta_n\,, \equation(etpTwo) $$ for all $n\ge 1$. The hypotheses of Theorem 6.1 in [\rKochKocicXXXX] are now verified, with $\eps=1/6$ and $\vartheta=1/4$, and the conclusions of this theorem imply the statements in \clm(RRnCompose). \qed We note that the ``min'' in equation \equ(anDef) could be replaced by ``a lower bound'', as long as $n\mapsto\Omega'_n$ is a non-increasing sequence of positive real numbers, converging to zero, and the corresponding numbers $a_n$ are summable. Our estimates are then uniform in the class $\BB(\Omega')$ of vectors $\omega\in\real^d$ that admit the same sequence $n\mapsto\Omega'_n$ of lower bounds. \section Invariant tori %%%%%%%%%%%%%%%%%%%%%%% Our construction of invariant tori follows closely the ideas used in [\rKochZETW,\rKochZEFOb,\rKocicTWFI,\rKochKocicXXXX]. Consider the RG transformation $\RR$ defined in Section 2, and a vector field $X$ in the domain of $\RR$. If $F$ is any map from $D_0$ into the domain of $\Lambda_\ssX=\UU_\ssX\circ\TT_\mu\,$, define $$ \MM_\ssX(F)=\Lambda_\ssX\circ F\circ\TT_\mu^{-1}\,. \equation(MMXDef) $$ Formally, if $\wt\Gamma$ is an invariant torus for $\RR(X)$, then $\Gamma=\MM_\ssX(\wt\Gamma)$ is an invariant torus for $X$. This can be seen easily from the identity $\Lambda_\ssX\circ\Phi_{\RR(X)}^{\eta t}=\Phi_\ssX^t\circ\Lambda_\ssX\,$. In order to make such identities more precise, we estimate the difference between the flow for $X$ and the flow for the constant vector field $K=(\omega,0)$. \claim Proposition(BasicFlowBound) Let $\tau$ be a positive real number and $X$ a vector field in $\AA_\varrho\,$, such that $\tau\|X-K\|_\varrho0$. Then we will show that $\Gamma_0$ is an invariant torus for $X_0\,$. For every $n\ge 0$, define $\BB_n$ to be the vector space $\AA_0(\VV_n)$, equipped with the norm $$ \|f\|'_n=r_n^{-1}\|f\|_0 =r_n^{-1}\sum_{v\in\VV_n}\|f_v\|\,,\qquad r_n=\psi_n^{1/3}\,, \equation(nNorm) $$ where $\psi_0=1$. Denote by $B_n$ the unit ball in $\BB_n\,$, centered at the identity function $\id$. \claim Proposition(MMnContracts) Let $\gamma\ge 5$. If $\kappa'$ and then $\kappa$ are chosen sufficiently large, then there exists an open neighborhood $B$ of $K$ in $\AA_\varrho\,$, such that for every $X\in\WW\cap B$, and for every $n\ge 1$, the map $\MM_n$ is well defined and analytic, as a function from $B_n$ to $\BB_{n-1}\,$. Furthermore, $\MM_n$ takes values in $B_{n-1}/2$, and $\|D\MM_n(F)\|\le\mu_n^{1/4}\,$, for all $F\in B_n\,$. \proof Clearly, $\MM_n$ is well defined in some open neighborhood of $\id$ in $\BB_n\,$, and $$ \MM_n(F)=\id+g+(\UU_{n-1}-\id)\circ(\id+g)\,,\qquad g=\TT_{\mu_n}\circ f\circ\TT_{\mu_n}^{-1}\,, \equation(MMnExpr) $$ where $f=F-\id$. In order to estimate $\UU_{n-1}-\id$, we can apply \clm(Elim), with $\rho'$ equal to $\varrho-\varrho(1-\beta_n)/3\,$, as in the proof of \clm(RRBounds). We will use \clm(muBound) and assume that $\kappa'$ and then $\kappa$ have been chosen sufficiently large, without always mentioning it. By \clm(Elim) and \clm(RRnCompose), there exist a constant $C>0$, such that $$ \eqalign{ \|\UU_{n-1}-\id\|_{\rho'} &\le C\sigma_n^{-1}\|\Iminus X_{n-1}\|_\varrho \le C\sigma_n^{-1}\psi_{n-1}^{\gamma-1/2} \|(\Id-\mean)X\|_\varrho\cr &\le\psi_{n-1}^{\gamma-1}\|(\Id-\mean)X\|_\varrho \le\psi_n^{3/4}\,,\cr} \equation(MMnCOne) $$ for all $n>1$, and for all $X\in\WW\cap B$, provided that the neighborhood $B$ of $K$ has been chosen sufficiently small (depending on $\kappa'$ and $\kappa$). The first inequality in \equ(MMnCOne) and the final bound also hold for $n=1$. The composition with $\id+g$ in equation \equ(MMnExpr) is controlled by \clm(Trivial), using that $\|g\|_0\le\eta_n^{-1}r_n\|f\|'_n$ is less than $\varrho/2$. Here, and in what follows, we assume that $F\in B_n\,$. By using that $r_n/r_{n-1}=\mu_n^{1/3}\,$, we obtain $\|g\|'_{n-1}\le\eta_n^{-1}\mu_n^{1/3}\le\mu_n^{2/7}\,$. When combined with \equ(MMnCOne), this shows that $\MM_{n-1}$ maps $B_n$ into $B_{n-1}/2$. Using now $\rho'=\varrho/2$, we obtain a bound analogous to \equ(MMnCOne) for the derivative of $\UU_{n-1}\,$. This, together with the fact that the inclusion map from $B_n$ into $B_{n-1}$ is bounded in norm by $\mu_n^{1/3}$, shows that $\|D\MM_n(F)\|\le\mu_n^{1/4}\,$, for all $n\ge 1$, and for all $F\in B_n\,$. \qed Denote by $\Phi_n$ and $\Phi_\infty$ the flows for the vector fields $X_n$ and $K$, respectively. In order to prove that a solution to \equ(GammanSeq) yields an invariant torus $\Gamma_0$ for $X$, we will use the identity $$ \Phi_{n-1}^t\circ\MM_n(F)\circ\Phi_\infty^{-t} =\MM_n\bigl(\Phi_n^{\eta_n t}\circ F\circ\Phi_\infty^{-\eta_n t}\bigr)\,, \equation(MMPhi) $$ which follows from the relation described after \equ(MMXDef), between the flow for a vector field and the flow for the corresponding renormalized vector field. This requires an estimate of the following type. \claim Proposition(PhiFPsi) Under the same assumptions as in \clm(MMnContracts), there exists an open neighborhood $B$ of $K$ in $\AA_\varrho\,$, such that for every $X\in\WW\cap B$, and for every $n\ge 1$, the function $\Phi_n^s\circ F\circ\Phi_\infty^{-s}$ belongs to $B_n\,$, whenever $F\in B_n/2$ and $|s|\le\psi_n^{-1/6}\,$. \proof We will use the identity $$ \Phi_n^s\circ F\circ\Phi_\infty^{-s} =\id+f\circ\Phi_\infty^{-s} +\bigl[\Phi_n^s\circ\Phi_\infty^{-s}-\id\bigr] \circ\bigl(\id+f\circ\Phi_\infty^{-s}\bigr)\,. \equation(PhiFPsiOne) $$ By \clm(BasicFlowBound) and \clm(RRnCompose), we have the bound $$ \bigl\|\Phi_n^s\circ\Phi_\infty^{-s}-\id\bigr\|_{\varrho/2} \le\|s(X_n-K)\|_\rho \le C\psi_n^{1/3}\|(\Id-\proj)X\|_\varrho\,, \equation(PhiFPsiTwo). $$ provided e.g. that the right hand side of this inequality is less than $\varrho/2$. This is certainly the case, for any $n$, if $\|X-K\|_\varrho$ is sufficiently small. The composition by $\id+f\circ\Phi_\infty^{-s}$ in equation \equ(PhiFPsiOne) is controlled in the same way as the composition by $\id+g$ in the proof of \clm(MMnContracts), using also that $\|f\circ\Phi_\infty^{-s}\|_0=\|f\|_0\,$. As a result, the third term on the right hand side of \equ(PhiFPsiOne) belongs to $\BB_n$ and is bounded in norm by $C\|X-K\|_\varrho\,$, which is less than $1/2$ for any $n\ge 1$, if $X$ is sufficiently close to $K$. \qed Now we are ready to construct invariant tori. A function $f$ defined on $\WW$ is said to be analytic if $f\circ W$ is analytic on the domain of $W$. \claim Theorem(GammaLimits) Under the same assumptions as in \clm(MMnContracts), there exists an open neighborhood $B$ of $K$ in $\AA_\varrho\,$, such that the following holds. Given any $X\in\WW\cap B$, and any sequence of functions $F_k\in B_k\,$, define $$ \Gamma_{n,k}=\bigl(\MM_{n+1}\circ\ldots \circ\MM_k\bigr)(F_k)\,,\qquad 0\le n0$. As proved above, the right (and thus left) hand side of this equation converges in $\AA_0$ to $\Gamma_0\,$. In addition, $\Gamma_{0,k}\to\Gamma_0$ in $\AA_0\,$, and the convergence is pointwise as well, by part $(a)$ of \clm(Trivial). Thus, since the flow $\Phi_0^{t}$ is continuous, we have $\Phi_0^{t}\circ\Gamma_0\circ\Phi_\infty^{-t}=\Gamma_0\,$. This identity now extends to arbitrary $t\in\real$, due to the group property of the flow, and the fact that composition with $\Phi_\infty^s$ is an isometry on $\AA_0\,$. Finally, notice that $\lambda_n\|DX_n\|_{\varrho/2}$ is an upper bound on the modulus of the Lyapunov exponent for the flow of $\lambda_nX_n$ on the range of $\Gamma_n\,$. Since $X_0$ is obtained from $\lambda_n X_n$ by a change of variables, and $\Gamma_0$ is the corresponding invariant torus for $X_0\,$, the same upper bound applies to the flow for $X_0$ on the torus $\Gamma_0\,$. But by \clm(RRnCompose), $\lambda_n\|DX_n\|_{\varrho/2}\to 0$ as $n\to\infty$. This shows that the torus $\Gamma_0$ is elliptic. \qed In what follows, the torus $\Gamma_0$ associated with a vector field $X\in\WW$ will be denoted by $\Gamma_\ssX\,$. For convenience, we extend the map $X\mapsto\Gamma_\ssX$ to an open neighborhood of $K$, by setting $\Gamma_\ssX=\Gamma_\ssXp\,$, where $X'=(\Id+W)(X-\proj X)$. \claim Theorem(AnalyticTori) Let $\rho>\varrho+\delta$ with $\delta>0$. Under the same assumptions as in \clm(MMnContracts), there exists an open neighborhood $B$ of $K$ in $\AA_{\rho}(\VV_0)$, such that $\Gamma_\ssX$ has an analytic continuation to $\|\Im x\|<\delta$, for each $X\in B$. With this continuation, $X\mapsto\Gamma_\ssX$ defines a bounded analytic map from $B$ to $\AA_\delta^0(\VV_0)$. A proof of this theorem is completely analogous to the proof of Theorem 4.5 in [\rKochKocicXXXX]. Thus, we will just give a sketch here. Consider the translations $R_u(x,y)=(x+u,y)$. By examining the construction of $\WW$ and $\Gamma_\ssX$, one verifies that for any $u\in\real^d$, the translated vector field $R_u^\ast X$ belongs to $\WW$ whenever $X$ does, and that $$ \Gamma_X(u,0)=\bigl(R_u\circ\Gamma_{R_u^\ast X}\bigr)(0,0)\,. \equation(GammaXu) $$ The idea now is to use the analyticity of map $X\mapsto\Gamma_\ssX\,$, to extend the right hand side of equation \equ(GammaXu) to the complex domain $\|\Im u\|<\delta$. This yields the desired analytic continuation of $\Gamma_\ssX\,$. The remaining parts of \clm(AnalyticTori) are proved by using that the right hand side of \equ(GammaXu) is jointly analytic in $X$ and $u$. \medskip This theorem, together with \clm(RRnCompose), implies \clm(RGSummary). \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \smallskip {\ninepoint \baselineskip=9.7pt %%%%%%%%%% RG and breakup etc \item{[\rEscandeDoveilEION]} D.F.~Escande, F.~Doveil, {\it Renormalisation Method for Computing the Threshold of the Large Scale Stochastic Instability in Two Degree of Freedom Hamiltonian Systems}. J.~Stat.~Phys. {\bf 26}, 257--284 (1981). \item{[\rShenkerKadanoffEITW]} S.J.~Shenker, L.P.~Kadanoff, {\it Critical behavior of a KAM surface. I. Empirical results.} J.~Stat.~Phys.~{\bf 27}, 631--656 (1982). \item{[\rMacKayEITW]} R.S.~MacKay, {\it Renormalisation in Area Preserving Maps}. Thesis, Princeton (1982). World Scientific, London (1993). \item{[\rStirnemannNISE]} A.~Stirnemann, {\it Towards an Existence Proof of MacKay's Fixed Point}. Comm.~Math. Phys. {\bf 188}, 723--735 (1997). %%%%%%%%%% RG for flows \item{[\rChandreGovinJauslinNISE]} 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{[\rAbadKochWittwer]} J.J.~Abad, H.~Koch, and P.~Wittwer, {\it A Renormalization Group for Hamiltonians: Numerical Results}. Nonlinearity {\bf 11}, 1185--1194 (1998). \item{[\rChandreJauslinZETW]} C.~Chandre and H.R.~Jauslin, {\it Renormalization--group analysis for the transition to chaos in Hamiltonian systems}. Physics Reports {\bf 365}, 1--64, (2002). %% rigorous \item{[\rKochNINI]} H.~Koch, {\it A renormalization group for Hamiltonians, with applications to KAM tori}. Erg. Theor. Dyn. Syst. {\bf 19}, 1--47 (1999). \item{[\rKochZETW]} H.~Koch, {\it On the renormalization of Hamiltonian flows, and critical invariant tori}. Discrete Contin. Dynam. Systems A, {\bf 8}, 633--646 (2002). \item{[\rKochZEFOa]} H.~Koch, {\it A Renormalization Group Fixed Point Associated with the Breakup of Golden Invariant Tori}. Discrete Contin. Dynam. Systems {\bf 11}, 881--909 (2004). \item{[\rKochZEFOb]} H.~Koch, {\it Existence of Critical Invariant Tori}. Erg. Theor. Dyn. Syst., to appear. %%%%%%%%%% RG for near-linear flows \item{[\rGaidashevZEFI]} D.G.~Gaidashev, {\it Renormalization of isoenergetically degenerate Hamiltonian flows and associated bifurcations of invariant tori}. Discrete Contin. Dynam. Systems A {\bf 13}, 63--102 (2005). \item{[\rLopesDiasZETW]} J.~Lopes Dias, {\it Renormalisation scheme for vector fields on T$\,^2$ with a Diophantine frequency}. Nonlinearity {\bf 15}, 665--679, (2002). \item{[\rKocicTWFI]} S.~Koci\'c, {\it Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori}. Nonlinearity {\bf 18}, 2513--2544 (2005). \item{[\rKhaninLopesDiasMarklovZEFI]} K.~Khanin, J.~Lopes Dias, J.~Marklof, {\it Multidimensional continued fractions, dynamic renormalization and KAM theory}. Preprint {\tt mp\_arc 05-304}, Commun. Math. Phys., to appear. \item{[\rKochXXXX]} H.~Koch, {\it Renormalization of vector fields}. Preprint {\tt mp\_arc 06-89}, Fields Institute Communications, to appear. \item{[\rKochKocicXXXX]} H.~Koch, S.~Koci\'c, {\it Renormalization of vector fields and Diophantine invariant tori}. Preprint {\tt mp\_arc 06-220}. %% skew flows \item{[\rKochLopesDiasZEFI]} H.~Koch, J.~Lopes Dias, {\it Renormalization of Diophantine skew flows, with applications to the reducibility problem}. Preprint {\tt mp\_arc 05-285} (2005). %%%%%%%%%% flow RG and Brjuno \item{[\rLopesDiasZESI]} J.~Lopes Dias, {\it Brjuno condition and renormalization for Poincar\'e flows}. Discrete Contin. Dynam. Systems, {\bf 15}, 641--656 (2006). \item{[\rLopesDiasXXXX]} J.~Lopes Dias, {\it A normal form theorem for Brjuno skew-systems through renormalization}. J. Differential Equations, to appear. %%%%%%%%%% Lindstedt series, renormalization \item{[\rGentileMastropietroNISI]} G.~Gentile, V.~Mastropietro, {\it Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications}. Rev. Math. Phys. {\bf 8}, 393--444 (1996) %%%%%%%%%% Brjuno & KAM \item{[\rBrjunoSEON]} A.D.~Brjuno, {\it Analytic form of differential equations. I}. Trudy Moskov. Mat. Obshch. {\bf 25}, 119--262 (1971). Trans. Moscow Math. Soc. {\bf 25}, 131--288 (1973). \item{[\rBrjunoSETW]} A.D.~Brjuno, {\it Analytic form of differential equations. II}. Trudy Moskov. Mat. Obshch. {\bf 26}, 199--239 (1972). Trans. Moscow Math. Soc. {\bf 26}, 199--239 (1974). \item{[\rPoeschelEINI]} J.~P\"oschel, {\it Integrability of Hamiltonian systems on Cantor sets}. Comm. Pure Appl. Math. {\bf 35}, 653--696 (1989). \item{[\rRuessmannNIFO]} H.~R\"ussmann, {\it On the frequencies of quasi periodic solutions of analytic nearly integrable Hamiltonian systems}. In ``Seminar on Dynamical Systems'', Euler Int. Math. Inst. St. Petersburg 1991, S. Kuksin, V. Lazutkin, J. P\"oschel (eds). PNLDE {\bf 12}, 160--183, Birkh\"auser Verlag (1994). \item{[\rEcalleValetNIEI]} J.~Ecalle, B.~Valet, {\it Bruno correction and linearization of resonant vector fields and diffeomorphisms}. Math. Z. {\bf 229}, 249--318 (1998). \item{[\rBerrettiGentileZEON]} A.~Berretti, G.~Gentile, {\it Bryuno function and the standard map}. Commun. Math. Phys {\bf 220}, 623--656 (2001). \item{[\rRuessmannZEON]} H.~R\"ussmann, {\it Invariant tori in non-degenerate nearly integrable Hamiltonian systems}. Regul. Chaotic Dynam. {\bf 6}, 119--204 (2001). \item{[\rGentileXXXX]} G.~Gentile, {\it Degenerate lower-dimensional tori under the Bryuno condition}. Preprint {\tt mp\_arc 05-256}. \item{[\rGentileBartuccelliDeaneXXXX]} G.~Gentile, M.V.~Bartuccelli, J.H.B.~Deane, {\it Quasi-periodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems}. Preprint {\tt mp\_arc 06-110}. %%%%%%%%%% Misc \item{[\rLagariasNIFO]} J.C.~Lagarias, {\it Geodesic multidimensional continued fractions}, Proc. London Math. Soc. {\bf 69}, 464--488 (1994). \item{[\rKleinbockMargulisNIEI]} D.Y.~Kleinbock, G.A.~Margulis, {\it Flows on homogeneous spaces and Diophantine approximation on manifolds}. Ann. of Math. (2) {\bf 148}, 339--360 (1998). } \bye ---------------0609270800247--