Content-Type: multipart/mixed; boundary="-------------0608111724801" This is a multi-part message in MIME format. ---------------0608111724801 Content-Type: text/plain; name="06-220.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-220.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/ ---------------0608111724801 Content-Type: text/plain; name="06-220.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-220.keywords" vector fields, flows, analytic, Hamiltonian, reversible, divergence free, symmetric, quasi periodic, invariant torus, elliptic, Diophantine, continued fractions, normal form, renormalization group, stable manifold ---------------0608111724801 Content-Type: application/x-tex; name="flows13.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="flows13.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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %notes13.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\smallreal{{\eightmsb R}} \def\smalltorus{{\eightmsb T}} % \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}} %% \addref{DelaLlaveZEON} \addref{BroerZEFO} \addref{MoserSISE} % \addref{SevryukEISI} \addref{BroerHuitemaTakensNIZE} \addref{BroerHuitemaSevryukNISI} %% \addref{EliassonEIEI} \addref{GallavottiGentileMastropietroNIFI} \addref{GentileMastropietroNISI} % \addref{GallavottiGentileZETW} \addref{GentileBartuccelliDeaneZEFI} %% \addref{EscandeDoveilEION} \addref{KhaninSinaiEISI} \addref{McKayNIFI} \addref{ChandreGovinJauslinNISE} \addref{KochNINI} \addref{AbadKochZEZE} \addref{ChandreJauslinZETW} \addref{LopesDiasZETWb} \addref{KochZEFOa} \addref{GaidashevZEFI} \addref{KocicTWFI} \addref{LopesDiasZESI} \addref{KochZEFOb} \addref{KhaninLopesDiasMarklovZEFI} \addref{KochLopesDiasZEFI} % \addref{CasselsFISE} \addref{LagariasNIFO} \addref{KleinbockMargulisNIEI} % % forward references \expandafter\xdef\csname eNonres\endcsname{(5.3)} \expandafter\xdef\csname eElimFullCond\endcsname{(5.4)} \expandafter\xdef\csname eElimFullBound\endcsname{(5.5)} \expandafter\xdef\csname eEpsCond\endcsname{(5.20)} \expandafter\xdef\csname eRnPlus\endcsname{(6.4)} \expandafter\xdef\csname cCompAss\endcsname{Assumption~5.1} \expandafter\xdef\csname cElimFull\endcsname{Theorem~5.2} \expandafter\xdef\csname cWWexists\endcsname{Theorem~6.1} % \def\leftheadline{\sevenrm\hfil HANS KOCH and SA\v SA KOCI\'C\hfil} \def\rightheadline{\sevenrm\hfil Renormalization and Diophantine Invariant Tori \hfil} %% %% \cl{{\magtenbf Renormalization of Vector Fields}} \cl{{\magtenbf and Diophantine Invariant Tori}} \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 extend the renormalization group techniques that were developed originally for Hamiltonian flows to more general vector fields on \smalltorus$^d\times$\smallreal$^\ell$. Each Diophantine 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 $\omega_1,\omega_2,\ldots,\omega_d\,$. Analogous manifolds for particular classes of vector fields (Hamiltonian, divergence free, symmetric, reversible) are obtained simply by restricting $\WW$ to the corresponding subspace. We also discuss nondegeneracy conditions, and the resulting reduction in the number of parameters needed in parametrized families to guarantee the existence of invariant tori. \section Introduction %%%%%%%%%%%%%%%%%%%%% Classical KAM theory [\rDelaLlaveZEON--\rMoserSISE] shows that for every Diophantine vector $\omega\in\real^d$, there exist open sets of $d$-parameter families of Hamiltonian vector fields on $\torus^d\times\real^d$, such that one of the family members has an invariant torus with frequency vector $\omega$. (This applies to an individual Hamiltonian satisfying a non-degeneracy condition, if one considers the family of its translates.) Similar results have been obtained for different classes of near-linear vector fields, mostly by KAM type methods [\rMoserSISE--\rBroerHuitemaSevryukNISI] or resummation of Lindstedt series [\rEliassonEIEI--\rGentileBartuccelliDeaneZEFI]. Our goal is to obtain such results within the framework of renormalization transformations [\rEscandeDoveilEION--\rKochLopesDiasZEFI], and more importantly, to develop appropriate techniques for analyzing quasiperiodic motion in a large class of flows. The renormalization approach combines in a natural way the geometric and arithmetic aspects of the problem. It also applies to non-perturbative problems that are outside the reach of other known methods [\rKochZEFOa,\rKochZEFOb]. Our analysis is centered around three results that should be of independent interest: a normal form theorem for vector fields (Section 5), estimates on a multidimensional continued fractions expansion [\rKhaninLopesDiasMarklovZEFI], and a stable manifold theorem for sequences of maps (Section 6). A renormalization group analysis of Diophantine torus flows and/or Hamiltonian vector fields was carried out in [\rLopesDiasZETWb,\rKocicTWFI] for $d=2$, and more recently in [\rKhaninLopesDiasMarklovZEFI] for $d\ge 2$. Earlier results covered a much smaller set of frequencies [\rKochNINI]. One of our goals is to extend the methods developed in these papers to a large class of vector field on $\MM=\torus^d\times\real^\ell$, and to do this in a way that allows for a unified treatment of Hamiltonian, divergence free, symmetric, reversible, and other types of vector fields. Some of our results are sufficiently general to be used e.g. in other problems involving renormalization. Despite the increase in scope, the analysis has in fact become simpler compared to previous work. We note that the tori considered in this paper are elliptic, in the sense that they have zero Lyapunov exponents. It should be possible to adapt our method to hyperbolic situations, but we will not pursue this question here. Denote by $\Phi_\ssX$ the flow for a vector field $X$. In this paper, an invariant $d$-torus for $X$, with frequency vector $\omega\in\real^d$, is a continuous embedding $\Gamma$ of $D_0=\torus^d\times\{0\}$ into the domain of $X$, with the property that $\Gamma\circ\Phi_K^t=\Phi_X^t\circ\Gamma$ for real times $t$, where $K=(\omega,0)$. Here, $0$ denotes the zero vector in $\real^\ell$. We assume that $\omega$ satisfies a Diophantine condition $$ |\omega\cdot\nu|\ge\zeta\|\nu\|^{1-d-\beta}\,,\qquad \nu\in\integer\setminus\{0\}\,, \equation(DCbeta) $$ for some constants $\beta,\zeta>0$. Our renormalization analysis (Sections 2,3,4) applies to vector fields that are close to $K$, after a change of variables, if necessary. We assume analyticity on a complex neighborhood $D_\rho$ of $D_0\,$, characterized by the conditions $|\Im x_i|<\rho$ and $|y_j|<\rho$. We will also consider certain subclasses of vector fields, including Hamiltonian, divergence free, symmetric, and reversible vector fields. If $G$ is a linear map on $\MM$ that leaves $D_\rho$ invariant, we call a vector field $X$ on $D_\rho$ {\it symmetric} with respect to $G$ if $G^\ast X=X$, where $G^\ast X=G^{-1}X\circ G$ is the pullback of $X$ under $G$. If $G\circ G$ is the identity, a vector field is called {\it reversible} with respect to $G$ if $G^\ast X=-X$. Notice that $G^\ast X=\pm X$ implies that $G\circ\Phi_\ssX^t=\Phi_\ssX^{\pm t}\circ G$. In what follows, we will call a vector field symmetric if it is symmetric with respect to $G(x,y)=(x,-y)$, or reversible if it is reversible with respect to $G(x,y)=(-x,y)$. Denote by $A^u$ the space of all vector fields $Y(x,y)=(u,My+v)$, with $(u,v)$ a vector in $\complex^d\times\complex^\ell$ and $M$ a complex $\ell\times\ell$ matrix. In Section 2, we will introduce Banach spaces $\AA_\rho$ of vector fields that are analytic on $D_\rho\,$, and a projection operator $\proj$ from $\AA_\rho$ onto the subspace $A^u$. The subspace of functions in $\AA_\rho$ that do not depend on the coordinate $y\in\complex^\ell$ will be denoted by $\AA_\rho^0\,$. A function will be called ``real'' if it takes real values for real arguments. \claim Theorem(WWandTori) Let $K=(\omega,0)$ with $\omega\in\real^d$ Diophantine. Given $\rho>\delta>0$, there exists an open neighborhood $B$ of $K$ in $\AA_\rho\,$, and a real analytic map $W:(\id-\proj)B\to\proj B$, satisfying $W(0)=K$ and $DW(0)=0$, such that the following holds. Let $\WW$ be the graph of $W$. Then every vector field $X\in\WW$ has an elliptic invariant torus $\Gamma_\ssX\in\AA_\delta^0$ with frequency vector $\omega$. The map $X\mapsto\Gamma_\ssX$ is real analytic on $\WW$. The restriction of $W$ to symmetric vector fields takes values in the subspace of symmetric vector fields, and similar statement holds for reversible, Hamiltonian, and divergence free vector fields. This theorem, as well as the lemma below on parametrized families, will be proved in Section 4. The size of the neighborhood $B$ is independent of $\omega$, given the Diophantine constants and a lower bound on the norm of $\omega$. We note that, for any fixed $\beta>0$, the measure of the set of vectors $\omega$ that violate \equ(DCbeta) approaches zero as $\zeta$ tends to zero [\rCasselsFISE]. In what follows, $\HH_\rho$ denotes either $\AA_\rho\,$, or the subspace of $\AA_\rho$ consisting of all vector fields in a given class (divergence free, Hamiltonian, symmetric, or reversible). The intersection of $A^u$ with $\HH_\rho$ will be denoted by $H^u$. \clm(WWandTori) has an obvious corollary concerning the existence of vector fields with invariant tori in $N$--parameter families, where $N$ is the dimension of $H^u$. In particular, any analytic family $f: B\cap H^u\to\HH_\rho$ sufficiently close to the family $f_0(s)=K+s$ intersects the manifold $\WW\cap\HH_\rho$ transversally, and \clm(WWandTori) yields an invariant torus $\Gamma_\ssX$ for the vector field $X=f(s)$ in the intersection. If we are just looking for families containing a vector field with frequency vector parallel (but not necessarily equal) to $\omega$, then the number of necessary parameters is reduced by one. A further reduction is possible for vector fields that satisfy a nondegeneracy condition, so that some directions in $A^u$ can be generated via translations. To be more precise, let $V$ be some proper linear subspace of $\complex^\ell$. Let $r>\rho>0$, and let $Z=Z(x,y)$ be a real vector field in $\HH_r$ that does not depend on the coordinate $x$, and that satisfies $\proj Z=0$. Given $\eps>0$, define $$ g_\eps(z,v)=zK+\eps\proj J_v^\ast Z\,,\qquad z\in\complex,\quad v\in V\,, \equation(gzvDef) $$ where $J_v(x,y)=(x,y+v)$. We assume that $g_\eps$ is non-degenerate, in the sense that the derivative $Dg_\eps(0)$ is one-to-one. Let $H^u_0$ be a linear subspace of $H^u$ that is transversal to the range of $Dg_\eps(0)$, and define $$ f_\eps(s)=K+\eps Z+s\,,\qquad s\in H^u_0\,. \equation(FzeroDef) $$ We will see later that $f_\eps(0)$ belongs to $\WW$, for small $\eps>0$. Given an open neighborhood $b$ in some complex Banach space, denote by $\FF(b)$ the space of all bounded analytic functions $f:b\to\HH_r\,$, equipped with the sup-norm. \claim Lemma(GeneralFamilies) If $\eps>0$ is chosen sufficiently small, and if $g_\eps$ is non-degenerate, then, given an open neighborhood $b_2$ of the origin in $H^u_0\,$, there exists an open neighborhood $B_2$ of $f_\eps$ in $\FF(b_2)$, such that the following holds. For every family $f\in B_2$ we can find a parameter value $s_\ssf\in b_2\,$, and a nonzero complex number $c_\ssf\,$, such that $X=c_\ssf f(s_\ssf)$ belongs to $\WW$ and thus has an invariant torus $\Gamma_\ssX\in\AA_\delta^0$ with rotation vector $\omega$. The maps $f\mapsto(c_\ssf,s_\ssf)$ and $f\mapsto\Gamma_\ssX$ are real analytic on $B_2\,$. This lemma includes cases where $Dg_\eps(0)$ is onto and thus $H^u_0$ trivial. In such a case, every vector field near $K+\eps Z$ has an invariant torus with frequency vector $\omega$. Consider e.g. the case of Hamiltonian vector fields, or symmetric vector fields with $\ell=d$. Then $H^u$ is of dimension $\ell=d$. Taking for $V$ some $(\ell-1)$--dimensional subspace of $\complex^\ell$ not containing $\omega$, it is easy to write down examples (see e.g. below) of vector fields $Z\in\proj\HH_\rho$ for which $g_\eps$ is non-degenerate and thus $H^u_0=\{0\}$. Hamiltonian vector fields of this type are also called isoenergetically nondegenerate. The following example covers several classes of vector fields. \medskip\noindent {\bf Example.} Consider a basis $\{w_1,w_2,\ldots,w_d\}$ for $\real^d$, with $w_d=\omega$. Let $k$ be the minimum of $d-1$ and $\ell$. Define $X_j(x,y)=(y_j w_j,0)$ for $1\le j\le k$, and if $k<\ell$, define $$ X_j(x,y)=\bigl(0,(y_j-y_\ell)^2(e_j+e_\ell)\bigr)\,,\qquad X_\ell(x,y)=\bigl(0,(y_1^2+\ldots+y_{\ell-1}^2)e_\ell\bigr)\,, $$ for $kk$. Taking $V$ to be the span of $\{e_1,\ldots,e_k\}$, we get again a non-degenerate function $g_\eps\,$, and the parameter space $H^u_0$ is of dimension $d-1-k$. In particular, if $k=d-1$, then we are in the situation described above, where every vector field near $K+\eps Z$ has an invariant torus with frequency vector $\omega$. \medskip Our proof of \clm(WWandTori) and \clm(GeneralFamilies) is based on renormalization group techniques. The general idea in this approach is to take a continued fractions algorithm, acting on frequency vectors, and to ``lift'' it to a space of vector fields in some appropriate way. We choose a multidimensional continued fractions expansion [\rKhaninLopesDiasMarklovZEFI] which, starting from a Diophantine vector $\omega_0\in\real^d$, produces a sequence of vectors $\omega_n=\eta_n^{-1}T_n^{-1}\omega_{n-1}\,$, where $T_n$ is a matrix in $\SL(d,\integer)$ and $\eta_n$ an appropriate normalization constant. The matrices $T_n$ can be used e.g. to construct successive rational approximants to $\omega_0\,$. Our renormalization group (RG) transformation $\RR_n$ that corresponds to the matrix $T_n^{-1}$ has the property that it maps $K_{n-1}=(\omega_{n-1},0)$ to $K_n=(\omega_n,0)$. Other properties will be given below. We start by describing a single RG step. It involves a ``scaling'' of the torus variable $x$ by a matrix in $\SL(d,\integer)$, whose transpose is strongly contracting on the orthogonal complement of some unit vector $\omega\in\real^d$. Given such a matrix $T$, and a nonzero real number $\mu$, define $$ \SS_\mu(x,y)=(x,\mu y)\,,\qquad \TT(x,y)=\bigl(Tx,\wh T y\bigr)\,,\qquad x\in\torus^d\,,\quad y\in\real^\ell\,. \equation(TTdef) $$ Here, $\wh T$ is either the $\ell\times\ell$ identity matrix, or if desired for the renormalization of Hamiltonian vector fields (where $\ell=d$), the inverse of the transpose of $T$. The scaling of a vector field $X$ on $\MM$ is then given by $(\SS_\mu\TT)^\ast X$, the pullback of $X$ under $\SS_\mu\TT$. Recall that the pullback of a vector field $X$, defined on the range of a differentiable map $U$, is given by $U^\ast X=(DU)^{-1}(X\circ U)$. Notice that scaling by $\TT^\ast$ is a singular operation on spaces of analytic vector fields, since it shrinks the domain of analyticity in the expanding direction of $T$. Although the domain loss is of order one (not small), it is possible to associate with $X\in\AA_\varrho$ a change of variables $\UU_\ssX\,$, which is close to the identity for $X$ close to $K=(\omega,0)$, such that the renormalized vector field $$ \RR(X)=\eta^{-1}\TT^\ast\SS_\mu^\ast\UU_\ssX^\ast X \equation(RGDef) $$ belongs again to $\AA_\varrho\,$. To be more specific, we will identify (in Section 2) a subspace of ``resonant'' vector fields, containing $K$, such that the restriction of $\TT^\ast\SS_\mu^\ast$ to this subspace is compact, and in fact analyticity improving, for small $\mu>0$. Then, using a general result from Section 5, we show that there exists an analytic map $X\mapsto\UU_\ssX\,$, defined near $K$, which makes $\UU_\ssX^\ast X$ resonant. In other words, the resonant vector fields, which behave well under scaling, can be regarded as a local normal form for vector fields. We note that $\UU_\ssK$ is the identity, so the transformation $\RR$ maps $K$ to $\wt K=(\wt\omega,0)$, where $\wt\omega=\eta^{-1}T^{-1}\omega$. \claim Theorem(RGSummary) Let $\varrho>0$. Given a Diophantine unit vector $\omega_0\in\real^d$, there exists a sequence of matrices $T_n\in\SL(d,\integer)$, and a corresponding sequence of transformations $\RR_n$ of the form \equ(RGDef), such that the following holds. For $n=1,2,\ldots$ define $\omega_n=\eta_n^{-1}T_n\omega_{n-1}$, with $\eta_n>0$ chosen in such a way that $\omega_n$ is a unit vector. Then $\RR_n$ is well defined and analytic in some open neighborhood $\DD_{n-1}$ of $K_{n-1}=(\omega_{n-1},0)$ in $\AA_\varrho\,$. 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 $W$ with the properties described in \clm(WWandTori) (if $\varrho+\delta<\rho$), where $\omega=\omega_0$ and $B=\DD_0\,$. The set $\WW$ can be regarded as the (local) stable manifold for the transformations $\RR_1,\RR_2,\ldots\,$. A stable manifold theorem that applies to such sequences of maps will be proved in Section 6. Section 2 deals with a single renormalization group transformation $\RR$, using a normal form theorem proved in Section 5. The composition of such transformations $\RR_n\,$, according to a multidimensional continued fractions expansion [\rKhaninLopesDiasMarklovZEFI], will be described in Section 3. Section 4 is devoted to the construction of invariant tori. \section A single renormalization step %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we give a precise definition RG transformation $\RR$ and describe some of its properties. A matrix $T\in\SL(d,\integer)$ is assumed to be given, subject to certain conditions that will be specified below. \subsection Spaces and basic estimates %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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, including matrices, we will always use the operator norm, unless stated otherwise. Let $C$ be some finite dimensional complex Banach space. Denote by $D_\rho$ the set of all vectors $(x,y)$ in $\complex^d\times\complex^\ell$, characterized by $\|\Im x\|<\rho$ and $\|y\|<\rho$. We consider functions on $\torus^d\times\complex^\ell$ with values in $C$, that extend analytically to $D_\rho$ and continuously to the boundary of $D_\rho\,$. Our norm on the space $\AA_\rho(C)$ of such functions $f$ is given in terms of the Fourier-Taylor series of $f$ as follows: $$ \|f\|_\rho=\sum_{\nu,\alpha}\|f_{\nu,\alpha}\| e^{\rho|\nu|}\rho^{|\alpha|}\,,\qquad f(x,y)=\sum_{\nu,\alpha}f_{\nu,\alpha}e^{i\nu\cdot x} y^\alpha\,, \equation(AArhoVnorm) $$ where $\nu\cdot x=\sum_j\nu_jx_j$ and $y^\alpha=\prod_jy_j^{\alpha_j}\,$. The sum in this equation ranges over all $\nu\in\integer^d$ and $\alpha\in\natural^\ell$. If it is clear what space $C$ is being considered, or irrelevant, we will simply write $\AA_\rho$ in place of $\AA_\rho(C)$. The operator norm of a continuous linear map $L$ on $\AA_\rho$ will be denoted by $\|L\|_\rho\,$. Later on, for the construction of invariant tori, we will also use non-analytic functions, with real domain $D_0=\torus^d\times\{0\}$. Denote by $\AA_0$ the Banach space of continuous functions $f:D_0\to\complex^d$, for which the norm $\|f\|_0=\sum_\nu\|f_\nu\|$ is finite, where $\{f_\nu\}$ are the Fourier coefficients of $f$. This space can be viewed as a $\rho\to 0$ limit of the spaces $\AA_\rho$ defined above. \claim Proposition(Trivial) Let $X\in\AA_\rho(C)$ and $Z\in\AA_{\rho'}(\complex^{d+\ell})$, with $0\le\rho'\le\rho$. Then \item{$(a)$} $\|X(x,y)\|\le\|X\|_\rho$ for all $(x,y)\in D_\rho\,$. \item{$(b)$} $(DX)Z\in\AA_{\rho'}(C)$ and $\|(DX)Z\|_{\rho'}\le(\rho-\rho')^{-1}\|X\|_{\rho}\|Z\|_{\rho'}\,$, if $\rho'<\rho$. \item{$(c)$} $X\circ(\id+Z)\in\AA_{\rho'}(C)$ 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. \subsection Resonant and nonresonant modes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let $0<\rho'<\varrho$ be given. These domain parameters are now considered fixed for the entire RG analysis. Choose $\gamma\ge 1$ and $\chi\ge\|\wh T\|$. Let $\mu$ and $\tau$ be positive real numbers, satisfying $$ e^{\rho'/2}\mu<\hat\mu\equiv{\rho'\over\chi\varrho}\,,\qquad \tau\le{\rho'\over 2\varrho}\,,\qquad \tau\ln\bigl(\hat\mu/\mu\bigr) \le{\rho'\over 2(\gamma+1)}\,. \equation(taumuCond) $$ Consider the matrix norm $|M|=\sup_{|v|=1}|Mv|$. \claim Definition(Projections) Denote by $S$ the generator of the one-parameter group of scalings $\mu\mapsto\SS_\mu^\ast$. Given any subset $J$ of $I=\integer^d\times\{-1,0,1,2,\ldots\}$, define $P(J)$ to be the joint spectral projection in $\AA_\rho$ for the operators $(-i\nabla_x,S)$, associated with the eigenvalues $(\nu,k)$ in $J$. Let now $\iplus$ be the set of all pairs $(\nu,k)\in I$ satisfying $|T^\ast\nu|\le\tau|\nu|$ or $|T^\ast\nu|\le\gamma^{-1}\tau k$, and let $\iminus$ be its complement in $I$. The projection onto the ``resonant'' and ``nonresonant'' subspace of $\AA_\rho$ is defined as $\Iplus=P\bigl(\iplus\bigr)$ and $\Iplus=P\bigl(\iminus\bigr)$, respectively. In addition, we define $\mean_k=P\bigl(\{(0,k)\}\bigr)$, for every integer $k\ge -1$, and $\mean=\sum_k\mean_k\,$. Notice that $\mean X$ is the torus average of $X$. The following proposition shows that, unlike in KAM theory [\rDelaLlaveZEON--\rMoserSISE], resonant modes are easy to deal with in the RG approach. \claim Lemma(Contraction) Consider the two linear transformations $\SS_\mu$ and $\TT$ defined in \equ(TTdef). If the condition \equ(taumuCond) holds, then $\TT^\ast\SS_\mu^\ast$ defines a bounded linear operator from $\Iplus\AA_{\rho'}$ to $\AA_\varrho\,$, satisfying $$ \eqalign{ \bigl\|\TT^\ast\SS_\mu^\ast\mean_k X\|_\varrho &\le N(T)\bigl(\mu/\hat\mu\bigr)^k \|\mean_k X\|_{\rho'}\cr \bigl\|\TT^\ast\SS_\mu^\ast\Iplus(\Id-\mean)X\|_\varrho &\le N(T)\bigl(b\mu/\hat\mu\bigr)^\gamma \|\Iplus(\Id-\mean)X\|_{\rho'}\,,\cr} \equation(ContTwo) $$ where $N(T)=\bigl\|T^{-1}\bigr\|+\bigl|\wh T^{-1}\bigr|\bigl|\wh T\bigr|$. \proof By our choice of norm \equ(AArhoVnorm), it suffices to verify the given bounds for vector fields $X=P(J)Y$, with $J$ containing a single point. Let $$ J=\{(\nu,k)\}\,,\qquad A=\varrho|T^\ast\nu|-\rho'|\nu|+k\ln\bigl(\mu/\hat\mu\bigr)\,. \equation(JAdef) $$ Then it follows essentially from the definitions that $$ \|\TT^\ast\SS_\mu^\ast P(J)Y\|_\varrho \le N(T)e^A\|P(J)Y\|_{\rho'}\,. \equation(JAbound) $$ Setting $\nu=0$ yields the first bound in \equ(ContTwo). In order to prove the second bound, assume that $(\nu,k)$ belongs to $\iplus$, and that $\nu\not=0$. Consider first the case $|T^\ast\nu|\le\tau|\nu|$. Then $|\nu|\ge\tau^{-1}$, and we obtain $$ \eqalign{ A&\le(\varrho\tau-\rho')|\nu|+k\ln\bigl(\mu/\hat\mu\bigr) \le(\varrho-\rho'/\tau)-\ln\bigl(\mu/\hat\mu\bigr)\cr &\le-\rho'/(2\tau)-\ln\bigl(\mu/\hat\mu\bigr) \le\gamma\ln\bigl(\mu/\hat\mu\bigr)\,.\cr} \equation(ContThree) $$ In the last inequality we have used condition \equ(taumuCond). Now consider the case $\tau|\nu|<|T^\ast\nu|\le {\tau\over \gamma}k$. By using that $\varrho{\tau\over \gamma}\le {\rho'\over 2\gamma}=\ln(c)$, and $k>\gamma$, we find that $$ A\le \varrho\tau\gamma^{-1}k+k\ln\bigl(\mu/\hat\mu\bigr) \le k\ln\bigl(b\mu/\hat\mu\bigr) \le\gamma\ln\bigl(b\mu/\hat\mu\bigr)\,. \equation(ContFour) $$ The second bound in \equ(ContTwo) now follows from \equ(ContThree) and \equ(ContFour). \qed This lemma shows that the scaling $\TT^\ast\SS_\mu^\ast$ is a contraction (for small $\mu$) on the resonant subspace of $\AA_\varrho\,$, except for a small number of non-contracting directions. In order to exploit this property, given a vector field $X$ that is not necessarily resonant, we first perform a change of variables $\UU_\ssX\,$, such that $$ \Iminus\UU_\ssX^\ast X=0\,. \equation(ElimCond) $$ The corresponding linearized equation, which needs to be solved in the process, is of the form $\Iminus(X-[X,Z])=0$, where $[X,Z]=(DZ)X-(DX)Z$. In order to guarantee the existence of a solution, we need to make the following assumptions. Assume that there exists a $d-1$ dimensional subspace of $\real^d$ where the transpose $T^\ast$ of $T$ contracts distances by a factor of at least $\tau/2\sqrt{d}$. Let $\omega$ be a unit vector in $\real^d$ that is perpendicular to this subspace, and set $K=(\omega,0)$. \claim Proposition(KZ) Choose $\sigma>0$ such that $2\sqrt{d}\,\sigma\|T\|\le\tau$. If $Z$ belongs to $\Iminus\AA'_r$ then $$ \|[K,Z]\|_r\ge\sigma\|Z\|_r\,,\qquad \|[K,Z]\|_r \ge{\sigma r\over r+\gamma+1}\|DZ\|_r\,. \equation(KZbounds) $$ \proof Assume that $(\nu,k)$ belongs to $\iminus$. In other words, $|T^\ast\nu|>\tau|\nu|$ and $|T^\ast\nu|>{\tau\over\gamma}k$. Consider the decomposition $\nu=\nu_{\sss\parallel}+\nu_{\sss\perp}$ into a vector $\nu_{\sss\parallel}$ parallel to $\omega$ and a vector $\nu_{\sss\perp}$ perpendicular to $\omega$. By using that $|\nu_{\sss\perp}|\le\sqrt{d}|\nu|$, we obtain $$ \sqrt{d}\,\sigma|\nu|<\|T\|^{-1}{\tau\over 2}|\nu| \le\|T\|^{-1}\bigl(|T^\ast\nu| -|T^\ast\nu_{\sss\perp}|\bigr) \le\|T\|^{-1}|T^\ast\nu_{\sss\parallel}| \le|\nu_{\sss\parallel}|\le\sqrt{d}\,|\omega\cdot\nu|\,, $$ and in particular, $\sigma<|\omega\cdot\nu|$. Similarly, we have $$ \sqrt{d}\,{\sigma\over\gamma}k \le\|T\|^{-1}{\tau\over 2\gamma}k \le{1\over 2}\|T\|^{-1}|T^\ast\nu| \le\|T\|^{-1}\bigl(|T^\ast\nu|-|T^\ast\nu_{\sss\perp}|\bigr) \le\sqrt{d}\,|\omega\cdot\nu|\,. $$ This shows that if $Z\in\Iminus\AA'_r$ and $Y=[K,Z]=(\omega\cdot\nabla_x)Z$, then $\|Z\|_r\le\sigma^{-1}\|Y\|_r$ and $$ \sum_{j=1}^d\left\|{\partial\over\partial x_j}Z\right\|_r \le{1\over\sigma}\|Y\|_r\,,\qquad \sum_{j=1}^\ell\left\|{\partial\over\partial y_j}Z\right\|_r \le{\gamma+1\over\sigma r}\|Y\|_r\,. \equation(omegadotnuInv) $$ These bounds imply \equ(KZbounds). \qed This proposition allows us now to apply the results from Section 5, which describe a solution of equation \equ(ElimCond). For convenience later on, let us first restate the assumptions of \clm(KZ) in a slightly stronger form: $$ \|T\|,\|T^{-1}\|\le{\chi\over 2\sqrt{d}}\,,\quad \chi={\tau\over\sigma}\,,\qquad |T^\ast\xi|\le{\tau\over 2\sqrt{d}}|\xi|\,,\quad \xi\in\omega^\perp\,. \equation(omegaperp) $$ \claim Lemma(Elim) There exist positive constants $C$ and $C'$, such that the following holds, whenever \equ(taumuCond) and \equ(omegaperp) are satisfied. Denote by $\DD$ the open ball in $\AA_\varrho$ of radius $\eps=C(\sigma/\gamma)^2$, centered at $K$. Then for every $X\in\DD$, there exists an analytic change of coordinates $\UU_\ssX: D_{\rho'}\to D_\rho\,$, such that $\UU_\ssX^\ast X$ belongs to $\AA_{\rho'}$ and satisfies equation \equ(ElimCond). The map $X\mapsto\UU_\ssX$ is analytic from $\DD$ to the affine space $\id+\AA_{\rho'}$ and satisfies the bounds in \clm(ElimFull), with $\kappa=C'\sigma/\gamma$. \proof The image of $\Iminus\AA'_r$ under $Z\mapsto P[Z,K]=P(\omega\cdot\nabla_x)Z$ contains all nonresonant Fourier-Taylor polynomials, and thus it is dense in $\Iminus\AA_r\,$, for an $r>0$. \clm(CompAss) regarding the spaces $\AA_r$ is satisfied by \clm(Trivial), and the condition \equ(Nonres) on $K$ holds by \clm(KZ), with $\kappa=\rho'(1+\rho')\sigma/(2\gamma)$. The hypotheses \equ(ElimFullCond) and \equ(EpsCond) of \clm(ElimFull) are clearly satisfied on $\DD$, with $\eps=C''\kappa^2$ and $C''$ some constant depending only on $\rho$ and $\rho'$. The claims now follow from \clm(ElimFull). \qed \subsection The transformation $\RR$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Given $T\in\SL(d,\real)$, a unit vector $\omega\in\real^d$, and a real number $\gamma\ge 1$, assume that there exists positive constants $\mu,\sigma,\tau<1$ satisfying \equ(taumuCond) and \equ(omegaperp). In what follows, a quantity will be called {\it universal} if it is independent of the choice of $T$, $\omega$, $\gamma$, $\mu$, $\sigma$, and $\tau$. On the domain $\DD$ described in \clm(Elim), we can now define our RG transformation $\RR$ according to equation \equ(RGDef). The normalization constant $\eta$ is defined as $\eta=\|T^{-1}\omega\|$, so that $\wt\omega=\eta^{-1}T^{-1}\omega$ is again a unit vector. Notice that, by construction, $\UU_\ssX=\id$ whenever $X$ is resonant. Thus, $\RR\circ\Iplus$ is linear, and so is $\RR\circ\mean$. Let $\proj=\mean_{-1}+\mean_0\,$. The subspace $\proj\AA_\varrho$ is spanned by vector fields of the form $Y(x,y)=(u,My+v)$ and is invariant under $\RR$. The restriction of $\RR$ to this subspace, which is linear, will be denoted by $\LL$. In the following theorem, $\HH_\rho$ denotes either $\AA_\rho\,$, or the subspace of Hamiltonian vector fields in $\AA_\rho\,$, provided that $\ell=d$ and we choose for $\wh T$ the inverse of the transpose of $T$. \claim Theorem(RRBounds) There exist universal constants $R,C_0>0$, such that the following holds. Let $\DD$ be the open ball in $\HH_\varrho$ of radius $2R(\sigma/\gamma)^2$, centered at $K$. Then $\RR$ is bounded and analytic on $\DD$, satisfying $$ \eqalign{ \|(\Id-\mean)\RR(X)\|_\varrho &\le C_0\eta^{-1} (\gamma/\sigma)(C_0\tau/\sigma)^{\gamma+2}\mu^\gamma \|(\Id-\mean)X\|_\varrho\,,\cr \|(\Id-\proj)\RR(X)\|_\varrho &\le C_0\eta^{-1}(\gamma/\sigma)(\tau/\sigma)^3\mu \|(\Id-\proj)X\|_\varrho\,,\cr \|\mean\RR(X)-\RR(\mean X)\|_\varrho &\le C_0\eta^{-1}(\gamma/\sigma)^3(\tau/\sigma)\mu^{-1} \|(\Id-\mean)X\|_\varrho^2\,,\cr \|\LL^{-1}\| &\le C_0\eta(\tau/\sigma)\,.\cr} \equation(RRBounds) $$ \proof Let $R$ be half the constant $C$ from \clm(Elim), so that we can apply the estimates from \clm(ElimFull). Let $X$ be some vector field in $\DD$. By \clm(Contraction) we have $$ \eqalign{ \|(\Id-\mean)\RR(X)\|_\varrho &=\eta^{-1}\|\TT^\ast\SS_\mu^\ast(\Id-\mean)\UU_\ssX^\ast X\|_\varrho\cr &\le C_1\eta^{-1}(c\tau/\sigma)^{\gamma+2}\mu^\gamma \bigl[\|(\Id-\mean)X\|_{\rho'} +\|\UU_\ssX^\ast X-X\|_{\rho'}\bigr]\,,\cr} \equation(RRBOne) $$ for $c=\exp(\rho'/2)\varrho/\rho'$ and some constant $C_1>0$. Here, and in what follows, $C_1,C_2,\ldots$ denote positive universal constants. Using the bound \equ(ElimFullBound) on the norm of $\UU_\ssX^\ast X-X$, together with the fact that $P=P(\Id-\mean)$, we obtain the first inequality in \equ(RRBounds). Similarly, \clm(Contraction) implies that $$ \|\mean_k\RR(X)\|_\varrho \le C_2\eta^{-1}(\tau/\sigma)^3\mu \bigl[\|\mean_k X\|_{\rho'} +\|\mean_k(\UU_\ssX^\ast X-X)\|_{\rho'}\bigr]\,, \equation(RRBTwo) $$ for all $k\ge 1$. Summing over $k\ge 1$ to get a bound on $\|(\mean-\proj)\RR(X)\|_\varrho\,$, and then adding \equ(RRBOne), yields a bound analogous to \equ(RRBTwo), but with $\mean_k$ replaced by $\Id-\proj$. Applying again the estimate \equ(ElimFullBound) on the norm of $\UU_\ssX^\ast X-X$, we obtain the second inequality in \equ(RRBounds). By \clm(Contraction), we also have $$ \eqalign{ \|\mean\RR(X)-\RR(\mean X)\|_\varrho &=\eta^{-1}\|\TT^\ast\SS_\mu^\ast\mean(\UU_\ssX^\ast X-X)\|_\varrho\cr &\le C_3\eta^{-1}(\tau/\sigma)\mu^{-1} \|\mean(\UU_\ssX^\ast X-X)\|_{\rho'}\,.\cr} \equation(RRBThreeA) $$ Using the bounds \equ(ElimFullBound), the norm on the right hand side of this inequality can be estimated as follows: $$ \|\mean(\UU_\ssX^\ast X-X)\|_{\rho'} \le C_4(\gamma/\sigma)^3\|(\Id-\mean)X\|_\rho^2 +\|\mean[Z,X]\|_{\rho'}\,, \equation(RRBThreeB) $$ where $Z=\Iminus Z$ is the vector field described in \clm(ElimFull). The fact that $\mean Z=0$ implies $\mean[Z,\mean X]=0$. Thus, $$ \eqalign{ \|\mean[Z,X]\|_{\rho'} &=\|\mean[Z,(\Id-\mean)X]\|_{\rho'}\cr &\le C_5\|Z\|_\rho\|(\Id-\mean)X\|_\rho \le C_6(\gamma/\sigma)\|(\Id-\mean)X\|_\rho^2\,.\cr} \equation(RRBThreeC) $$ In the last step, we have used the bound on $\|Z\|_\rho$ from \clm(ElimFull). Combining the last three equations yields the third inequality in \equ(RRBounds). The analyticity and boundedness of $\RR$ on $\DD$ follows from \clm(Elim). In order to bound the inverse of $\LL$, let $Y$ be a vector field in $\proj\HH_\rho\,$. Then $Y$ can be written as $Y(x,y)=(u,My+v)$, and the last inequality in \equ(RRBounds) now follows from the fact that $(\LL^{-1}Y)(x,y)=\eta(Tu,My+\mu v)$. Here, we have used that $\wh T=\id$, except (optionally) in the Hamiltonian case where $M$ and $v$ are zero. \qed Due to the potentially large factor $\mu^{-1}$ in the third inequality of \equ(RRBounds), we will choose the domain of $\RR$ to be of the form $$ \|\proj(X-K)\|_\varrho0$ small (to be determined later). \claim Definition(proper) Given $\gamma\ge 1$, we will call $(\mu,\sigma,\tau,r,\delta)$ {\it proper RG parameters} if $r\le R(\sigma/\gamma)^2$, and if \equ(taumuCond) holds with $\chi=\tau/\sigma$. The parameters are also assumed to be positive, and $\mu,\sigma,\tau<1$. We say that the pair $(T,\omega)$ is {\it compatible} with these parameters if the condition \equ(omegaperp) is satisfied as well. The open subset $\DD$ of $\AA_\varrho$ defined by equation \equ(RRDomain) will be referred to as the domain of $\RR$. \section Infinitely renormalizable vector fields %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Our goal now is to compose RG transformations of the type described above. Let $\lambda_0=1$. Given a sequence of matrices $P_0,P_1,P_2,\ldots$ in $\SL(d,\integer)$, with $P_0$ the identity, and a unit vector $\omega$ in $\real^d$, we define $\omega_0=\omega$ and $$ T_n=P_{n-1}P_n^{-1}\,,\qquad \lambda_n=\|P_n\omega_0\|\,,\qquad \omega_n=\lambda_n^{-1}P_n\omega_0\,, \equation(TSlw) $$ for all $n\ge 1$. We also define $\lambda_0=1$. Assuming that each of the pairs $(T_n,\omega_{n-1})$ is compatible with some proper set of RG parameters, we can define the corresponding RG transformation $\RR_n:\DD_{n-1}\to\AA_\varrho\,$. Notice that the normalization constant $\eta_n$ for $\RR_n$ is given by $\eta_n=\lambda_n/\lambda_{n-1}\,$. Let now $\wt\RR_n=\RR_n\circ\RR_{n-1}\circ\ldots\circ\RR_1\,$. The domain $\wt\DD_n$ of the combined RG transformation $\wt\RR_{n+1}$ is defined inductively as the set of all vector fields in the domain of $\wt\RR_n$ that are mapped under $\wt\RR_n$ into the domain $\DD_n$ of $\RR_{n+1}\,$. By \clm(RRBounds), these domains are open and non-empty, and the transformations $\wt\RR_n$ are analytic. \claim Theorem(RRnCompose) Let $\alpha>\beta$, $m>2\alpha+7$, and $\gamma\ge 2\alpha+2$ be given. Then there exist real numbers $b,C>0$, a decreasing sequence of proper RG parameters $(\mu_n,\sigma_n,\tau_n,r_{n-1},\delta_{n-1})$ satisfying $$ \sigma_{n+2}=\sigma_{n+1}^{1+\alpha}\,,\qquad \mu_n=\sigma_n^m\,,\qquad r_n=\textstyle{1\over 5}\sigma_{n+1}^2r_{n-1}\,,\qquad n=1,2,\ldots\,, \equation(RGparam) $$ and for every every Diophantine vector $\omega\in\Omega$ a sequence of matrices $P_n\in\SL(d,\integer)$ yielding pairs $(T_n,\omega_{n-1})$ that are compatible with the RG parameters, and an open neighborhood $B$ of $K=(\omega,0)$ in $\AA_\varrho\,$, such that the following holds. $B$ contains a ball of radius $b$, centered at $K$. The set $\WW=B\cap_n\wt\DD_n$ is the graph of an analytic function $W:(\Id-\proj)B\to\proj B$, satisfying $W(0)=K$ and $DW(0)=0$. For each $X\in\WW$ and $n\ge 1$, $$ \eqalign{ \bigl\|\wt\RR_n(X)-K_n\bigr\|_\varrho &\le C\sigma_n^{m-2\alpha-7}r_n\|(\Id-\proj)X\|_\varrho\,,\cr \bigl\|\proj[\wt\RR_n(X)-K_n]\bigr\|_\varrho &\le C\sigma_n^{2(m-2\alpha-7)}r_n^2\|(\Id-\proj)X\|_\varrho^2\,,\cr \bigl\|(\Id-\mean)\wt\RR_n(X)\bigr\|_\varrho &\le C\sigma_n^{(m-1)\gamma-2\alpha-6}r_n\|(\Id-\mean)X\|_\rho\,.\cr} \equation(RRnCompose) $$ A proof of this theorem will be given below. It uses a continued fractions expansion developed in [\rLagariasNIFO,\rKleinbockMargulisNIEI,\rKhaninLopesDiasMarklovZEFI], which we will now describe very briefly, and a stable manifold theorem given in Section 6. We note that the second bound in \equ(RRnCompose) is not strictly needed for our subsequent construction of invariant tori. The first bound, with a larger value of $m$, could be used instead. Let $F$ be a fundamental domain for the left action of $\Gamma=\SL(d,\integer)$ on $G=\SL(d,\real)$. Consider the one-parameter subgroup of $G$, generated by the matrices $$ E^t={\rm diag}\bigl(e^{-t},\ldots,e^{-t},e^{(d-1)t}\bigr)\,,\qquad t\in\real\,. \equation(EtDef) $$ Given a Diophantine vector $\omega\in\real^d$, define $W\in G$ to be the matrix obtained from the $d\times d$ identity matrix by replacing its last column vector by a constant multiple of $\omega$ whose last component is $1$. Then, for every $t\in\real$, there exists a unique matrix $P(t)\in\Gamma$ such that $P(t)WE^t$ belongs to $F$. To a given sequence of ``stopping times'' $00$, depending only on the Diophantine constants $\beta$ and $\zeta$, such that for all $n>0$, and for all vectors $\xi\in\real^d$ that are perpendicular to $\omega_{n-1}\,$, $$ \eqalign{ \|T_n\|&\le c_0\exp\{(d-1)(1-\theta)t'_n+d\,\theta\,t_n\},\cr \|T_n^{-1}\|&\le c_0\exp\{(1-\theta)t'_n+d\,\theta\,t_n\},\cr |T_n^\ast\xi| &\le c_0\exp\{-(1-\theta)t'_n+d\,\theta t_{n-1}\}|\xi|\,.\cr} \equation(MPT) $$ \proofof(RRnCompose) Let $\alpha>\beta$ be fixed. We choose $t_n=c(1+\alpha)^n$ for each positive integer $n$, with $c>0$ to be determined. Define $c_1=2c_0\sqrt{d}$ and $$ \sigma_n=\exp\{-dt'_n\}\,,\qquad \tau_n=c_1\exp\{-(1-\theta)t'_n+d\,\theta t_{n-1}\}\,. \equation(sigmatauchoice) $$ Then \clm(MPT) guarantees that the conditions \equ(omegaperp) are satisfied. By using that $t'_1=t_1$ and $t'_n={\alpha\over 1+\alpha}t_n$ for $n>1$, we obtain the bounds $$ \sigma_n\le\exp\{-d\textstyle{\alpha\over 1+\alpha}t_n\}\,,\qquad \tau_n\le c_1\exp\{-\epsilon t_n\}\,, \equation(taunBound) $$ with $\epsilon={1-\theta\over 1+\alpha}(\alpha-\beta)>0$. Let now $\mu_n=\sigma_n^m$ with $m>1$ fixed. Then it is clear that the conditions \equ(taumuCond) are satisfied as well, for any $\gamma>0$, provided that $c$ is chosen sufficiently large. Here, and in what follows, any condition that is said to hold for large values of $c$ is implicitly being satisfied by choosing $c$ as large as necessary. Next, let $r_0=R(\sigma_1/\gamma)^2$, and define $r_1,r_2,\ldots$ as in \equ(RGparam). Then $r_{n-1}\le R(\sigma_n/\gamma)^2$, for all $n\ge 1$. Thus, we have shown that $(\mu_n,\sigma_n,\tau_n,r_{n-1},\delta_{n-1})$ are proper RG parameters, in the sense of \clm(proper), and that $(T_n,\omega_{n-1})$ is compatible with these parameters. This is independent of the choice of $\delta_{n-1}>0$, which we will describe below. Consider now the rescaled RG transformation $R_n\,$, defined by the equation $$ R_n(Z)=r_n^{-1}\bigl[\RR_n(K_{n-1}+r_{n-1}Z)-K_n\bigr]\,. \equation(ScaledRn) $$ The domain of $R_n$ is given by \equ(RRDomain), with $r=1$ and $\delta=\delta_{n-1}\,$, and with $K$ replaced by the zero vector field. The restriction of $R_n$ to $\proj\AA_\rho\,$, which is linear, will be denoted by $L_n\,$. By using the bound on $\LL_n^{-1}$ from \clm(RRBounds), we obtain $\|L_n^{-1}\|\le 1/5$, for large $c>0$. Here, we have used also that $\sigma_n<\|T_n\|^{-1}\le\eta_n\le\|T_n^{-1}\|<\sigma_n^{-1}$. The same inequalities, and \clm(RRBounds), also imply that $$ \eqalign{ \|(\Id-\mean)R_n(Z)\|_\varrho &\le\eps_n\|(\Id-\mean)Z\|_\varrho\,,\quad \eps_n=5C_0\gamma\sigma_n^{(m-1)\gamma-2\alpha-6}\,,\cr \|(\Id-\proj)R_n(Z)\|_\varrho &\le\vartheta_n\|(\Id-\proj)Z\|_\varrho\,,\quad \vartheta_n=5C_0\gamma\sigma_n^{m-2\alpha-7}\,,\cr \|\proj R_n(Z)-R_n(\proj Z)\|_\varrho &\le\varphi_n\delta_{n-1}\|(\Id-\mean)Z\|_\varrho\,,\quad \varphi_n=C_0\gamma^3\sigma_n^{-m-2\alpha-5}\,,\cr} \equation(RBounds) $$ for all $Z$ in the domain of $R_n\,$. Assume now that $m>2\alpha+7$ and $\gamma\ge 2\alpha+2$. Then $\eps_n\le\vartheta_n\le 1/5$, if $c$ is sufficiently large. Furthermore, by setting $\delta_{n-1}=(5\varphi_n)^{-1}$, it is easy to check that $\eps_n\delta_{n-1}\le\delta_n\,$, provided again that $c$ has been chosen sufficiently large. At this point we have verified the hypotheses of \clm(WWexists) with $\eps=\vartheta=1/5$. This includes the condition \equ(RnPlus), since the third inequality in \equ(RBounds) remains true if $\delta_{n-1}$ is replaced by $\|(\Id-\mean)Z\|_\varrho\,$. The assertions of \clm(RRnCompose) now follow from \clm(WWexists). \qed The above also proves \clm(RGSummary), except for the reference to the statements in \clm(WWandTori) that concern the symmetry properties of $W$, and invariant tori. The latter will be proved in the next section. The fact that $W$ preserves the type of a vector field (Hamiltonian, divergence free, symmetric, or reversible) can be seen as follows. First, recall that $\proj$ projects onto vector fields $Y(x,y)=(u,My+v)$ with with $(u,v)$ a vector in $\complex^d\times\complex^\ell$ and $M$ a complex $\ell\times\ell$ matrix. These vector fields are divergence free and symmetric, so $W$ trivially preserves these two types. What remains to be shown is that $M$ and $v$ are zero if $Y=W(X)$ and $X$ is either Hamiltonian or reversible. The elimination step $X\mapsto\UU_\ssX$ is type-preserving by design; see the discussion at the end of Section 5. The scaling $\TT^\ast\SS_\mu^\ast$ is type-preserving as well, except in the case of Hamiltonian vector fields, if we renormalize with $\wh T_n=\id$. In that case, $X_n=\wt\RR_n(X_0)$ is not Hamiltonian with respect to the standard symplectic form $\sum_j dx_j\wedge dy_j\,$, but with respect to $\sum_j dx_j\wedge(C dy)_j\,$, where $C$ is the inverse of the transpose of $P_n\,$. Still, the second ($\complex^\ell$) component of $X_n$ has a zero torus average. The same is trivially true for reversible vector fields. Thus, our RG analysis could be restricted to vector fields with this property, replacing e.g. $\RR_n$ by $\RR_n'=E\circ\RR_n\,$, where $E$ is the canonical projection onto vector fields whose second component has a zero torus average. Since $E$ does not affect Hamiltonian or reversible vector fields, the resulting stable manifold $\WW'$ coincides with $\WW$ in the corresponding subspaces. This shows that the restriction of $W$ to vector fields of a specific type yields a vector field of the same type. \section Construction of invariant tori %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Following [\rKochZEFOb,\rKocicTWFI], our construction of invariant tori is based on the relation between an invariant torus of a vector field $X$ and the corresponding torus of the renormalized vector field $\RR(X)$. We start with an informal discussion of this relation. Then we prove \clm(WWandTori) and \clm(GeneralFamilies). \subsection Preliminaries %%%%%%%%%%%%%%%%%%%%%%%%% Let $X\in\AA_\varrho\,$. Notice that $\RR(X)$ is obtained from $X$ by a change of coordinates (that depends on $X$), combined with a rescaling of time. Thus, the flow for $\RR(X)$ is related to the flow for $X$ by the equation $$ \Lambda_\ssX\circ\Phi_{\RR(X)}^t =\Phi_X^{\eta^{-1}t}\circ\Lambda_\ssX\,,\qquad \Lambda_\ssX=\UU_\ssX\circ\SS_\mu\circ\TT\,. \equation(RGflowRel) $$ In particular, $\TT\circ\Phi_{\RR(K)}^t=\Phi_K^{\eta^{-1}t}\circ\TT$ on $D_0\,$. The identity \equ(RGflowRel) can also be used to relate an invariant torus for $X$ to an invariant torus for $\RR(X)$. To this end, if $F$ is any map from $D_0$ into the domain of $\Lambda_\ssX\,$, define $$ \MM_\ssX(F)=\Lambda_\ssX\circ F\circ\TT^{-1}\,. \equation(MMXDef) $$ Assume that $\RR(X)$ has an invariant torus $\wt\Gamma$ with frequency vector $\wt\omega=\eta^{-1}T^{-1}\omega$, taking values in the domain of $\Lambda_\ssX\,$, and define $\Gamma=\MM_\ssX(\wt\Gamma)$. Then, by using \equ(RGflowRel), together with the fact that $\RR(K)=(\wt\omega,0)$, we obtain $$ \eqalign{ \Gamma\circ\Phi_K^t &=\Lambda_\ssX\circ\wt\Gamma\circ\TT^{-1}\circ\Phi_K^t =\Lambda_\ssX\circ\wt\Gamma\circ\Phi_{\RR(K)}^{\eta t}\circ\TT^{-1}\cr &=\Lambda_\ssX\circ\Phi_{\RR(X)}^{\eta t}\circ\wt\Gamma\circ\TT^{-1} =\Phi_X^t\circ\Lambda_\ssX\circ\wt\Gamma\circ\TT^{-1} =\Phi_X^t\circ\Gamma\,.\cr} $$ This shows that $\Gamma$ is an invariant torus for $X$ with frequency vector $\omega$. In order to make these identities more precise, we need to estimate the difference $Y(t)=\Phi_X^t-\Phi_K^t$ between the flow for a vector field $X$ and the flow for $K=(\omega,0)$. This can be done by solving the integral equation $$ Y(t)=\int_0^t[(X-K)\circ\Phi_K^s]\circ[\id+Y(s)]\,ds\,. \equation(PsiPlusMinusOne) $$ Notice that $\Phi_K^t=\id+tK$ and $\id+Y(t)=\Phi_K^{-t}\circ\Phi_X^t\,$. \claim Proposition(BasicFlowBound) Let $\tau$ be a positive real number and $X$ a vector field in $\AA_\rho\,$, such that $\tau\|X-K\|_\rho3\alpha+8$. If $a$ has been chosen sufficiently large, then there exists an open neighborhood $B$ of $K$ in $\AA_\varrho\,$, and a universal constant $C_1>0$, such that for every $X\in B\cap\WW$, and for every $n\ge 1$, the map $\MM_n$ is well defined and analytic, as a function from $B_n$ to $\BB_{n-1}\,$, and it takes values in $B_{n-1}/2$. Furthermore, $\|D\MM_n(F)\|\le C_1\sigma_n\,$, for all $F\in B_n\,$. \proof Clearly, $\MM_n$ is well defined in some open neighborhood of $\id$ in $\BB_n\,$, and $$ \MM_n(F)=\id+g+(\UU_{n-1}-\id)\circ(\id+g)\,,\qquad g=\SS_{\mu_n}\circ\TT_n\circ f\circ\TT_n^{-1}\,, \equation(MMnExpr) $$ where $f=F-\id$. By \clm(ElimFull) and \clm(RRnCompose), we have for $n>1$ the bound $$ \eqalign{ \|\UU_{n-1}-\id\|_\rho &\le C_2\sigma_n^{-1}\|\Iminus X_{n-1}\|_\varrho \le C_3\sigma_n^{-1}\sigma_{n-1}^{(m-1)\gamma-2\alpha-6}r_{n-1} \|(\Id-\mean)X\|_\varrho\cr &\le C_4\sigma_{n-1}r_{n-1}\|(\Id-\mean)X\|_\varrho \le C_5\sigma_n r_{n-1}\,,\cr} \equation(MMnCOne) $$ with $C_2,\ldots,C_5$ universal constants. The first inequality and the final bound in \equ(MMnCOne) also hold for $n=1$, if the neighborhood $B$ of $K$ has been chosen sufficiently small. Recall that $\rho'<\rho<\varrho$ have been fixed. The composition with $\id+g$ in equation \equ(MMnExpr) is controlled by \clm(Trivial), using that $\|g\|_0\le\sigma_n^{-1}a^{-1}r_n\|f\|'_n<\rho'$ independently of $n$, if $a$ has been chosen sufficiently large. Here, and in what follows, we assume that $F\in B_n\,$. By using that $r_n/r_{n-1}=\sigma_{n+1}^2/5$, we obtain $\|g\|'_{n-1}\le\sigma_n/5$. When combined with \equ(MMnCOne), this yields the bound $\|\MM_n(F)-\id\|'_{n-1}\le\sigma_n/2$, if the neighborhood $B$ of $K$ has been chosen sufficiently small. When restricting $\UU_{n-1}$ to the domain $D_{\rho'}\,$, we obtain a bound analogous to \equ(MMnCOne) for the derivative of $\UU_{n-1}\,$. This, together with the fact that the inclusion map from $B_n$ into $B_{n-1}$ is bounded in norm by $\sigma_{n+1}^2/5$, shows that $\|D\MM_n(F)\|\le C_1\sigma_n$ for all $n\ge 1$, and for all $F\in B_n\,$, where $C_1$ is again a universal constant. This completes the proof of \clm(MMnContracts). \qed Denote by $\Phi_n$ and $\Psi_n$ the flows for the vector fields $X_n$ and $K_n\,$, respectively. \claim Proposition(PhiFPsi) Assume that $m>2\alpha+7+p$ with $p>0$. If $a$ has been chosen sufficiently large, then there exists an open neighborhood $B$ of $K$ in $\AA_\varrho\,$, such that the following holds, for every $X\in B\cap\WW$, and for every $n\ge 1$. If $F\in B_n/2$ and $|s|\le\sigma_n^{-p}$, then $\Phi_n^s\circ F\circ\Psi_n^{-s}$ belongs to $B_n\,$. \proof We will use the identity $$ \Phi_n^s\circ F\circ\Psi_n^{-s} =\id+f\circ\Psi_n^{-s}+\bigl[\Phi_n^s\circ\Psi_n^{-s}-\id\bigr] \circ\bigl(\id+f\circ\Psi_n^{-s}\bigr)\,. \equation(PhiFPsiOne) $$ Let $\eps=m-2\alpha-7-p$. By \clm(BasicFlowBound) and \clm(RRnCompose), we have the bound $$ \bigl\|\Phi_n^s\circ\Psi_n^{-s}-\id\bigr\|_{\rho'} \le\|s(X_n-K_n)\|_\rho \le C\sigma_n^\eps r_n\|(\Id-\proj)X\|_\varrho\,, \equation(PhiFPsiTwo). $$ provided e.g. that the right hand side of this inequality is bounded by $\rho-\rho'$. This is certainly the case if $\eps$ is positive and $\|X-K\|_\varrho$ sufficiently small, independently of $n$. The composition by $\id+f\circ\Psi_n^{-s}$ in equation \equ(PhiFPsiOne) is controlled the same way as the composition by $\id+g$ in the proof of \clm(MMnContracts), using also that $\|f\circ\Psi_n^{-s}\|_0=\|f\|_0\,$. As a result, the third term on the right hand side of \equ(PhiFPsiOne) belongs to $\BB_n$ and is bounded in norm by $Ca\sigma_n^\eps\|X-K\|_\varrho\,$, which is less than $1/2$ for any $n\ge 1$, if $X$ is sufficiently close to $K$. \qed Assume that $\alpha$, $m$, $\gamma$, and $a$ have been chosen in such a way that the hypotheses of \clm(RRnCompose), \clm(MMnContracts), and \clm(PhiFPsi) are satisfied, with $p=1+1/\alpha$. Let $F_0,F_1,\ldots$ be a fixed but arbitrary sequence of functions in $\AA_0\,$, such that $F_n\in B_n$ for all $n\ge 0$. Then we can define $$ \Gamma_{n,m}=\bigl(\MM_{n+1}\circ\ldots \circ\MM_m\bigr)(F_m)\,,\qquad 0\le n0$ such that $\MM_n: B_n\to B_{n-1}/2$ contracts distances by a factor of at least $1/2$, if $n\ge N$. Thus, if $N\le n0$. Thus, if $|t|\le C_2\,$, then $|t_n|\le\sigma_n^{-p}$ for all $n\ge 1$. \clm(PhiFPsi) now allows us to iterate \equ(MMPhi), to get the identity $$ \Phi_0^{t}\circ\Gamma_{0,m}\circ\Psi_0^{-t} =\bigl(\MM_1\circ\ldots\circ\MM_m\bigr) \bigl(\Phi_m^{t_m}\circ\Psi_m^{-t_m}\bigr)\,, \equation(PhiGammPsi) $$ for all $m>0$. As was shown above, the right hand side of this equation converges in $\AA_0$ to $\Gamma_0\,$, and thus the left hand side converges to $\Gamma_0$ as well. In addition, $\Gamma_{0,m}\to\Gamma_m$ in $\AA_0\,$, and since convergence in $\AA_0$ implies pointwise convergence (see part $(a)$ of \clm(Trivial)), and the flow $\Phi_0^{t}$ is continuous, we have $\Phi_0^{t}\circ\Gamma_0\circ\Psi_0^{-t}=\Gamma_0\,$. This identity now extends to arbitrary $t\in\real$ by using the group property of the flow, together with the fact that composition with $\Psi_0^s$ is an isometry on $\AA_0\,$. Finally, notice that by \clm(RRnCompose), $$ \lambda_n\|DX_n\|_\rho\le C_3\sigma_n^\eps r_n\|(\Id-\proj)X_0\|_\varrho\,, \equation(DXnBound) $$ where $C_3$ is some universal constant and $\eps=m-2\alpha-7-p$. The left (and thus right) hand side of this equation is an upper bound on the modulus of the Lyapunov exponents for the flow of $\lambda_n X_n$ on the the range of $\Gamma_n\,$. Since $X_0$ is obtained from $\lambda_n X_n$ by a change of coordinates, 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 $\Gamma_0\,$. Taking $n\to\infty$ shows that this torus is elliptic. \qed \subsection Analyticity and families %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In what follows, the torus $\Gamma_0$ associated with $X\in B\cap\WW$ will be denoted by $\Gamma_\ssX\,$. The domain parameter $\rho$ used in the introduction is renamed to $\varrho'$, to avoid notational conflicts. The following theorem together with \clm(RRnCompose), and the discussion at the end of Section 3 (concerning the restriction of $W$ to specific types), implies \clm(WWandTori). \claim Theorem(AnalyticTori) Let $\varrho'>\varrho+\delta$ with $\delta>0$. Under the same assumptions as in \clm(GammaLimits), the map $X\mapsto\Gamma_\ssX$ defines (via extension) a bounded analytic map from $B'$ to $\AA_\delta^0\,$, where $B'$ is some open neighborhood of $K$ in $\AA_{\varrho'}\,$. \proof For every $u\in\real^d$, define a translation $R_u$ on $\complex^d\times\complex^\ell$ by setting $R_u(x,y)=(x+u,y)$. If $X$ is a vector field on one of the domains $D_r\,$, then $R_u^\ast X$ denotes the pullback of $X$ under $R_u\,$. And for functions $F:D_0\to D_r$ we define $R_u^\ast F=R_u^{-1}\circ F\circ R_u\,$. An explicit computation shows that the RG transformation $\RR$, and the maps $\MM_\ssX$ defined in \equ(MMXDef) satisfy $$ \RR\circ R_u^\ast=R_{T^{-1}u}^\ast\circ\RR\,,\qquad \MM_{R_u^\ast X}=R_u^\ast\circ\MM_X\circ(R_{T^{-1}u}^\ast)^{-1}\,. \equation(RuastRRMM) $$ Here, we have used that the translations $R_u^\ast$ are isometries on the spaces $\AA_r\,$, and that the domain of $\RR$ is translation invariant; see \clm(proper). This also implies that the manifold $\WW$ is invariant under translations $R_u^\ast\,$, which is used in the second identity in \equ(RuastRRMM). It is convenient to extend the function $X\mapsto\Gamma_\ssX$ to an open neighborhood of $K$ in $\AA_\varrho$ by projecting $X$ onto a point $X'\in\WW$ and defining $\Gamma_\ssX=\Gamma_\ssXp\,$. More specifically, we take $X'=(\Id+W)((\Id-\proj)X)$, where $W$ is the map defining $\WW$, as described in \clm(RRnCompose). If restricted to a sufficiently small open ball $B\subset\AA_\varrho$ centered at $K$, the map $X\mapsto\Gamma_\ssX$ is now analytic and bounded on all of $B$. The construction of $\Gamma_0$ in the proof of \clm(GammaLimits), together with the identities \equ(RuastRRMM), and the invariance property $W=W\circ R_u^\ast\,$, shows that $\Gamma_{R_u^\ast X}=R_u^\ast\Gamma_X\,$, for all $X\in B$. Thus, if $u\in\real^d$ then $$ \Gamma_X(u,0)=\bigl(R_u\circ\Gamma_{R_u^\ast X}\bigr)(0,0)\,, \qquad X\in B. \equation(GammaXu) $$ The idea now is to extend the right hand side of \equ(GammaXu) analytically to complex $u$, by using the analyticity of $X\mapsto\Gamma_\ssX\,$. To this end, choose an open neighborhood $B'$ of $K$ in $\AA_{\varrho'}\,$, such that $R_u^\ast B'\subset B$, for all $u\in\complex^d$ of norm $r=\varrho'-\varrho$ or less. Then the right hand side of \equ(GammaXu), regarded as a function of $(X,u)$, is analytic and bounded on the product of $B'$ with the strip $\|\Im u\|0$ and $b$ are chosen sufficiently small, then $G_\eps'=(\Id-\proj)(G_\eps\circ\phi^{-1})$ is a family in $\FF(b)$ of norm $0$ satisfying the condition \equ(EpsCond) given below, then there exists an analytic change of coordinates $\UU_\ssX: D_{\rho'}\to D_\rho\,$, such that $\UU_\ssX^\ast X$ belongs to $\AA_{\rho'}$ and satisfies equation \equ(ElimCondP). The map $X\mapsto\UU_\ssX-\id$ takes values in $\AA_{\rho'}\,$, is continuous in the region defined by equation \equ(ElimFullCond), analytic in the interior of this region, and satisfies the bounds $$ \eqalign{ \|\UU_\ssX-\id\|_{\rho'} &\le{3\over\kappa}\|PX\|_\rho\,,\cr \|\UU_\ssX^\ast X-X\|_{\rho'} &\le 32R{e^r\over\kappa r}\|PX\|_\rho\,,\cr \bigl\|\UU_\ssX^\ast X-X-[Z,X]\bigr\|_{\rho'} &\le\Bigl(2^{11}{e^r\over\kappa r}+1\Bigr)28R{e^r\over(\kappa r)^2} \|PX\|_\rho^2\,.\cr} \equation(ElimFullBound) $$ Here, $Z\in P\AA'_\rho$ is defined by \equ(elimFirst) and satisfies the bound $\|Z\|_\rho\le{2\over\kappa}\|PX\|_\rho\,$. \medskip We start with some basic estimates on flows. The flow $t\mapsto \Phi_\ssX^t$ associated with a vector field $X\in\AA_\rho$ is obtained by solving ${d\over dt}\Phi_\ssX^t=X\circ\Phi_\ssX^t$ with initial condition $\Phi_\ssX^0=\id$. Writing $\Phi_\ssX^t=\id+Y(t)$, this amounts to solving the integral equation $$ Y(t)=\int_0^t X\circ[\id+Y(s)]\,ds\,. \equation(YtEqu) $$ In what follows, any reference to a space $\AA_\varrho$ implicitly assumes that $\rho'\le\varrho\le\rho$. \claim Proposition(YtBound) Let $\varrho',\varrho$ and $\tau$ be positive real numbers, such that $\varrho'+\tau\|X\|_\varrho<\varrho$. Then the equation \equ(YtEqu) has a unique continuous solution $t\mapsto Y(t)\in\AA_{\varrho'}$ on the interval $|t|\le\tau$, and $$ \|\Phi_X^t-\id\|_{\varrho'} \le\|tX\|_\varrho\,. \equation(YtBound) $$ We will omit the proof of this proposition since it is standard: First, equation \equ(YtEqu) is solved locally, using the contraction mapping principle. Due to the uniqueness of these local solutions, they combine into a continuous solution on all of $[-\tau,\tau]$. What makes things straightforward is that, by \clm(CompAss), the derivative of $X\mapsto X\circ(\id+Z)$ admits a uniform bound on the entire domain needed. As a result, the intervals for the local proof can be taken of uniform size. \claim Proposition(FieldFlowBound) Let $00$ such that $$ \eps\le 2^{-6}\kappa r\,,\qquad \eps\le 2^{-9}\kappa^2e^{-r}(1+r)^{-1}R^{-1}\,. \equation(EpsCond) $$ Let $\rho_0=\rho$, and for $m=0,1,\ldots$ define $\rho_{m+1}=\rho_m-2r_m\,$, where $r_m=2^{-m-2}r$. Our first goal is to prove that \equ(Xnplusone) defines a sequence of vector fields $X_m\in\AA'_{\rho_m}\,$, satisfying $$ \|X_m-X_{m-1}\|'_{\rho_m}\le2^{-m-3}\kappa\,,\qquad \|PX_m\|_{\rho_m}\le 8^{-m}\eps\,. \equation(EFindhyp) $$ If we define $X_{-1}=K$ and $X_0=X$, then these bounds hold for $m=0$ by \equ(ElimFullCond). Assume now that \equ(EFindhyp) holds for $m\le n$. Then, by summing up the bounds on $X_m-X_{m-1}$ for $m\le n$, we obtain the first inequality in $$ \|X_n-K\|'_{\rho_n}\le{1\over 4}\kappa\,,\qquad \|PX_n\|_{\rho_n}\le 4^{-n-2}\kappa r_n\,. \equation(ElimCondn) $$ The second inequality follows from \equ(EFindhyp), by substituting the first bound in \equ(EpsCond) on $\eps$. Thus, \clm(ElimFirst) guarantees a unique solution to \equ(Xnplusone), and it yields the bounds $$ \|X_{n+1}\!-\!X_n\|_{\rho_n-r_n} \le 6{e^r\over\kappa r}4^{-n+1}R\eps\,,\qquad \|PX_{n+1}\|_{\rho_n-r_n} \le 7{e^r\over(\kappa r)^2}4^{-2n+3}R\eps^2\,. \equation(EFindnext) $$ Here, we have used also that $\|X_n\|_{\rho_n}\le R$, which follows from the first inequality in \equ(ElimCondn). By using the second condition in \equ(EpsCond), together with the fact that $\|F\|'_{\rho_n-2r_n}\le r_n^{-1}\|F\|_{\rho_n-r_n}\,$, we now obtain \equ(EFindhyp) for $m=n+1$ from the bounds \equ(EFindnext). Next, consider the functions $\phi_k=\Phi_{Z_k}^1-\id$. By \clm(BasicFlowBound) and \clm(ElimFirst), $$ \|\phi_k\|_{\rho_{k+1}}\le\|Z_k\|_{\rho_k} \le{2\over\kappa}\|PX_k\|_{\rho_k}< r_k\,. \equation(psikboundetc) $$ This shows that $U_{m,n}=\Phi_{Z_m}^1\circ\Phi_{Z_{m+1}}^1\circ\ldots\circ\Phi_{Z_{n-1}}^1$ defines a function in $\id+\AA_{\rho_n}$ that takes values in $D_{\rho_m}\,$. Here, and in what follows, it is assumed that $0\le m0$, let $R_n$ be a bounded analytic map, from an open neighborhood $D_{n-1}$ of the origin in $\XX_{n-1}\,$, to $\XX_n\,$, with the following properties: $R_n\proj_{n-1}$ is linear, and the restriction $L_n$ of this linear operator to $\proj_{n-1}\XX_{n-1}$ is invertible. Furthermore, there exist real numbers $\vartheta_n\le\vartheta<1$ and $\eps_n\le\eps=(1-\vartheta)/4$, such that for all $x\in D_{n-1}\,$, $$ \eqalign{ \|(\Id-\mean_n)R_n(x)\| &\le\eps_n\|(\Id-\mean_{n-1})x\|\,,\cr \|(\Id-\proj_n)R_n(x)\| &\le\vartheta_n\|(\Id-\proj_{n-1})x\|\,,\cr \|\proj_nR_n(x)-L_n\proj_{n-1}x\| &\le\eps\|(\Id-\mean_{n-1})x\|\,,\cr \|L_n^{-1}\| &\le\vartheta\,.\cr} \equation(RnCond) $$ Consider now the composed maps $\wt R_n=R_n\circ R_{n-1}\circ\ldots\circ R_1\,$. The domain of $\wt R_1$ is taken to be $\wt D_0=D_0\,$, and for $n=1,2,\ldots\,$, the domain $\wt D_n$ of $\wt R_{n+1}$ is defined inductively as the subset of $\wt D_{n-1}$ that is mapped into $D_n$ by $\wt R_n\,$. We will assume that the domain $D_{n-1}$ of $R_n$ is given by conditions $$ \|\proj_{n-1}x\|<1\,,\qquad \|(\Id-\proj_{n-1})x\|<1\,,\qquad \|(\Id-\mean_{n-1})x\|<\delta_{n-1}\,, \equation(RnDomainDef) $$ where $\{\delta_k\}$ is a sequence of positive real numbers, such that $\delta_k\ge\eps_k\delta_{k-1}$ for all $k>0$. \claim Theorem(WWexists) {\rm (local stable manifold)} Let $R_1,R_2,\ldots$ be a sequence of maps with the properties described above. Then $\WW_0=\bigcap_{n=0}^\infty\wt D_n$ is the graph of an analytic function $W_0: (\Id-\proj_0)D_0\to\proj_0 D_0\,$, satisfying $W_0(0)=0$. For every $x\in\WW_0$, $$ \eqalign{ \|\wt R_m(x)\| &\le\bigl[\vartheta^{(m)}+\eps^{(m)}\bigr]\|(\Id-\proj_0)x\|\,,\cr \|(\Id-\mean_m)\wt R_m(x)\| &\le\eps^{(m)}\|(\Id-\mean_0)x\|\,,\cr} \equation(WWconv) $$ where $\vartheta^{(m)}=\vartheta_1\vartheta_2\cdots\vartheta_m$ and $\eps^{(m)}=\eps_1\eps_2\cdots\eps_m\,$. Furthermore, if the third condition in \equ(RnCond) is strengthened to $$ \|\proj_nR_n(x)-L_n\proj_{n-1}x\| \le\varphi_n\|(\Id-\mean_{n-1})x\|^2\,, \equation(RnPlus) $$ with $\varphi_n\delta_{n-1}\le\eps$, then $DW_0(0)=0$, and $$ \|\proj_m\wt R_m(x)\| \le\bigl[\vartheta^{(m)}\bigr]^2\|(\Id-\proj_0)x\|^2\,. \equation(WWconvPlus) $$ Notice that, by our assumptions \equ(RnCond), if $x$ belongs to the domain of $R_n\,$, and if $\proj_nR_n(x)$ has norm less than one, then $R_n(x)$ belongs to the domain of $R_{n+1}\,$. This shows that $$ \wt D_n=\{x\in \wt D_{n-1}: \|\proj_n\wt R_n(x)\|<1\}\,,\qquad n=1,2,\ldots\,. \equation(WWBnDef) $$ Let $S_n=\proj_n\XX_n\,$, and denote by $b_n$ the open unit ball in $S_n\,$, centered at the origin. Define $\FF_n$ to be the space of analytic functions $f: b_n\to\XX_n\,$, equipped with the sup-norm $\|f\|=\sup_{s\in b_n}\|f(s)\|\,$. Denote by $I_n$ the inclusion map of $b_n$ into $\XX_n\,$. Notice that, if $f\in\FF_{n-1}$ satisfies $$ \proj_{n-1}f=I_{n-1}\,,\qquad \|f-I_{n-1}\|<1\,,\qquad \|(\Id-\mean_{n-1})\circ f\|<\delta_{n-1}\,, \equation(bfRdomain) $$ then $f(s)$ belongs to the domain of $R_n\,$, for all $s\in b_{n-1}\,$. For such functions $f$, define $$ Y_{n,f}=\proj_n(R_n\circ f)\,. \equation(YnDef) $$ \claim Proposition(YnBounds) Assume that $f\in\FF_{n-1}$ satisfies \equ(bfRdomain). Then $Y_{n,f}:b_{n-1}\to S_n$ has a unique right inverse $Y_{n,f}^{-1}:b_n\to b_{n-1}\,$. Both $Y_{n,f}$ and its inverse depend analytically on $f$, on the domain defined by \equ(bfRdomain). Furthermore, $$ \eqalign{ \|Y_{n,f}-\vartheta_n\| &\le\eps\|(\Id-\mean_{n-1})\circ f\|\,,\cr \|Y_{n,f}^{-1}-L_n^{-1}\| &\le\vartheta\eps\|(\Id-\mean_{n-1})\circ f\|\,.\cr} \equation(YnBounds) $$ \proof Let $U=Y_{n,f}-L_n\,$. By the third condition in \equ(RnCond) we have $$ \|U(s)\| =\|\proj_nR_n(f(s))-L_n\proj_{n-1}f(s)\| \le\eps\|(\Id-\mean_{n-1})f(s)\|\,, \equation(UsBound) $$ for all $s\in b_{n-1}\,$ This implies the first bound in \equ(YnBounds). By our assumption on $f$ and $\eps$, we have $\|U\|\le\eps\le r/2$, where $r=(1-\vartheta)/2$. If $s\in S_{n-1}$ is of norm $\le\vartheta$ and $h\in S$ of norm one, then by Cauchy's formula $$ \|DU(s)h\|\le r^{-1}\sup_{|z|=r}\|U(s+zh)\| \le r^{-1}\|U\|\le 1/2\,. \equation(UandVone) $$ The equation for a right inverse $L_n^{-1}+V$ of $L_n+U$ can be written as $\psi(V)=V$, with $\psi$ defined by $\psi(V)=-L_n^{-1}U\circ(L_n^{-1}+V)$. Consider the space of analytic functions $V:b_n\to S_{n-1}\,$, equipped with the sup-norm. Denote by $B$ the closed ball of radius $r$ in this space, centered at the origin. Then $\psi$ is analytic on $B$, with derivative given by $$ D\psi(V)h=-L_n^{-1}\bigl((DU)\circ(L_n^{-1}+V)\bigr)h\,. \equation(UandVtwo) $$ By equation \equ(UandVone), we see that $\|D\psi(V)\|<1/2$, for all $V\in B$. Since $\|\psi(0)\|\le r/2$, the map $\psi$ is a contraction on $B$, and thus has a (unique) fixed point in $B$. This fixed point $V$ satisfies $\|V\|=\|\psi(V)\|\le\|L_n^{-1}U\|$, which implies the second inequality in \equ(YnBounds). The analyticity of $U\mapsto V$ follows form the uniform convergence of $\psi^n(0)\to V$ for $\|U\|\le r/2$. \qed This proposition allows us to define the maps $$ \bfR_n(f)=R_n\circ f\circ Y_{n,f}^{-1}\,,\qquad \wt\bfR_n=\bfR_n\circ\bfR_{n-1}\circ\ldots\circ\bfR_1\,. \equation(bfRnDef) $$ Notice that $\proj_n\bfR_n(f)=I_n$. In particular, since $R_n\circ\proj_{n-1}=\proj_n\circ R_n\circ\proj_{n-1}$ by the second condition in \equ(RnCond), we have $\bfR_n(I_{n-1})=I_n\,$. The domain of $\bfR_n$ is the set of all $f\in\FF_{n-1}$ satisfying \equ(bfRdomain). \claim Lemma(FamContract) If $f_0$ belongs to the domain of $\bfR_1\,$, then $\wt\bfR_n(f_0)$ is well defined for all $n\ge 1$, and $$ \|\wt\bfR_n(f_0)-I_n\|\le\vartheta^{(n)}\|f_0-I_0\|\,. \equation(FamContract) $$ \proof Let $n\ge 1$. Let $f$ be an arbitrary function in the domain of $\bfR_n\,$, and define $f'=\bfR_n(f)$. Consider a fixed but arbitrary $s\in b_n$ and define $s'=Y_{n,f}^{-1}(s)$. By \clm(YnBounds), $s'$ belongs to $b_{n-1}\,$. Thus, the second condition in \equ(RnCond) implies that $$ \eqalign{ \|f'(s)-s\| &=\|(\Id-\proj_n)R_n(f(s'))\|\cr &\le\vartheta_n\|(\Id-\proj_{n-1})f(s')\| =\vartheta_n\|f(s')-s'\|\,.\cr} $$ This shows that $\|f'-I_n\|\le\vartheta_n\|f-I_{n-1}\|$. In addition, we have $\proj_nf'=I_n$ by the definition of $\bfR_n\,$, and $\|(\Id-\mean_n)\circ f'\|\le\eps_n\delta_{n-1}$ by the first inequality in \equ(RnCond). Thus, since $\vartheta_n<1$ and $\eps_n\delta_{n-1}\le\delta_n\,$, the function $f'$ belongs to the domain of $\bfR_{n+1}\,$. This proves \clm(FamContract). \qed This lemma shows that the domain of $\wt\bfR_n$ can be taken to be the domain of $\bfR_1\,$. If $f_0$ is any function in this domain, define $$ f_n=\wt\bfR_n(f_0)\,,\qquad Y_n=Y_{n,f_{n-1}}\,,\qquad Z_{m,n}=Y_{m+1}^{-1}\circ\ldots\circ Y_{n-1}^{-1}\circ Y_n^{-1}\,, \equation(fnYnZmnDef) $$ whenever $0\le m0$, and two positive real numbers $r,r'<1$, both independent of $f_0\,$, such that $Y_n^{-1}$ maps $b_n$ into to $r'b_{n-1}$ and contracts distances by a factor $\le r$, whenever $n\ge N$. In what follows, we assume that $N\le mn$. This shows that $n\mapsto s_{m,n}$ converges as $n\to\infty$, and that the limit $\hat s_m$ is independent of the sequence $\{s_n\}$. In particular, we see that $\hat s_m=z_m$ by choosing $s_n=0$ for all $n$. The identities \equ(zmProp) are obtained by choosing $s_n=z_n$ for all $n$. By \clm(YnBounds), the maps $f\mapsto s_{m,n}=Z_{m,n}(0)$ are analytic on the domain of $\bfR_1\,$. The analyticity of $f\mapsto z_m$ now follows from the uniform convergence of $s_{m,n}\to z_m\,$. \qed \claim Corollary(Wintersect) Let $f$ be a family in the domain of $\bfR_1\,$, and let $s\in b_0$. Then $f(s)$ belongs to $\WW_0$ if and only if $s=z_0(f)$. \proof Consider first $x=f(z_0)$. Then $x\in D_0\,$, since $f$ belongs to the domain of $\bfR_1\,$, and the following holds for $n=1,2,\ldots$. Set $x_n=f_n(z_n)$. By the definition of $\bfR_n\,$, and by \clm(zm), we have $x_n=R_n(x_{n-1})=\wt R_n(x)$. Furthermore, $\proj_nx_n=\proj_nf_n(z_n)=z_n\in b_n\,$, and thus $x$ belongs to the set $\wt D_n$ described in \equ(WWBnDef). This shows that $x\in\WW_0$. Consider now a fixed $s=s_0$ in $b_0\,$, and assume that $x_0=f(s_0)$ belongs $\WW_0$. Then we can define $x_n=\wt\bfR_n(x)$ for all $n>0$, and $s_n=\proj_nx_n$ belongs to $b_n\,$. Set $f_0=f$. Proceeding by induction, let $n>0$, and assume that $x_{n-1}=f_{n-1}(s_{n-1})$. Since $s_n=Y_n(s_{n-1})$, and since $Y_n$ has a unique right inverse on $b_n$ by \clm(YnBounds), we have $s_{n-1}=Y_n^{-1}(s_n)$. As a result, $x_n=f_n(s_n)$. This shows that $s_{n-1}=Y_n^{-1}(s_n)$ holds for all $n>0$, and thus $s_n=z_n$ by \clm(zm). \qed \proofof(WWexists) Denote by $B'_0$ the unit ball in $(\Id-\proj_0)\XX_0\,$, centered at the origin. To a point $x\in B'_0$ we associate the family $f:s\mapsto s+x$. This family belongs to the domain of $\bfR_1\,$. Now define $W_0(x)=z_0(f)$. By \clm(Wintersect), $x+s=f(s)$ belongs to $\WW_0$ if and only if $s=W_0(x)$. This shows that $\WW_0$ is the graph of $W_0$ over $B'_0\,$. The analyticity of $W_0$ follows from the analyticity of $z_0\,$. Furthermore, we have $W_0(0)=z_0(I_0)=0$. The second bound in \equ(WWconv) follows from the first condition in \equ(RnCond). In order to prove the first bound, consider the family $f_0(s)=s+(\Id-\proj_0)x$, the associated functions $f_n$ and $Y_n$ defined in \equ(fnYnZmnDef), and the parameters $z_n$ described in \clm(zm). Then $\wt R_n(x)=f_n(z_n)$ for all $n\ge 0$. By \clm(FamContract) we have $$ \|f_m(z_m)-z_m\|\le\vartheta^{(m)}\|(\Id-\proj_0)x\|\,, $$ and by \clm(YnBounds) and the second inequality in \equ(WWconv), $$ \eqalign{ \|z_m-L_{m+1}^{-1}\cdots L_n^{-1}z_n\| &=\left\|\sum_{k=m}^{n-1}L_{m+1}^{-1}\cdots L_k^{-1} \bigl[Y_{k+1}^{-1}-L_{k+1}^{-1}\bigr](z_{k+1})\right\|\cr &\le\sum_{k=m}^{n-1}\vartheta^{k-m} \vartheta\eps\,\eps^{(k)}\|(\Id-\mean_0)x\|\cr &\le{\vartheta\eps\over 1-\vartheta\eps}\eps^{(m)}\|(\Id-\mean_0)x\|\,,\cr} $$ whenever $0\le m0$, the map $f\mapsto Y_{n,f}$ has a vanishing derivative at $f=\id_{n-1}\,$. By the definition of $W_0$, this implies that $DW_0(0)=0$. Let now $x_0\in\WW_0$. Then $x_0=u+W_0(u)$ with $u=(\Id-\proj_0)x_0\,$. Assume that $u\not=0$. Let $\ell$ be a continuous linear functional on $\XX_0$ of norm one, such that $\ell(W_0(u))=\|W_0(u)\|$. Define $g(z)=\ell(W_0(zu/\|u\|))$ for all $z$ in the complex unit disk $|z|<1$. Since $W_0$ and $DW_0$ vanish at the origin, $z\mapsto z^{-2}g(z)$ defines an analytic function on the unit disk, and by Schwarz's lemma, this function is bounded in modulus by $1$. Here, we have used that $W_0$ has norm less than one on its domain. This shows that $\|W_0(u)\|=g(\|u\|)\le\|u\|^2$, or in other words, that $\|\proj_0 x_0\|\le\|(\Id-\proj_0)x_0\|^2$. Finally, let $m>0$ and consider the stable manifold $\WW_m$ for the shifted sequence of maps $R_m,R_{m+1},\ldots$. Clearly, $x_m=\wt R_m(x_0)$ belongs to $\WW_m\,$. The same arguments as above show that $\|\proj_m x_m\|\le\|(\Id-\proj_m)x_m\|^2$. 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