Content-Type: multipart/mixed; boundary="-------------0508231616968" This is a multi-part message in MIME format. ---------------0508231616968 Content-Type: text/plain; name="05-285.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-285.comments" AMS-Code: 37C55, 37E45, 37E20 ---------------0508231616968 Content-Type: text/plain; name="05-285.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-285.keywords" skew flow, torus, cocycle, quasiperiodic, reducibility, renormalization, diophantine, continued fraction, stable manifold, ---------------0508231616968 Content-Type: application/x-tex; name="Skew14.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Skew14.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{$\textfont0=\tenrm\textfont2=\tensy \left#1\vbox to7.25pt{}\right.$}}} %PARAM.2 \magnification=\magstep1 \def\firstpage{1} \pageno=\firstpage %fonts.5 \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 % % 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 %titles.5 % requires fonts.5 or higher \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 \eightrm\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.3\vsize\penalty-75 \vskip0pt plus -.3\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.2\vsize\penalty-75 \vskip0pt plus -.2\vsize\bigskip\bigskip \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.\ }} %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}} \def\Im{{\rm Im}} \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}} %\input smallfonts.tex %\input param.2 %\input fonts.5b %\input symbols.1 %\input titles.5 %\input macros.18 % %\font\teneufm=eufm10 \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}}} % %% hack \font\magteneufm=eufm10 scaled\magstep1 \font\magtenmsb=msbm10 scaled\magstep1 \def\bigbfG{{\hbox{\magteneufm G}}} \def\bigreal{{\magtenmsb R}} \def\smallreal{{\sevenmsb R}} % \def\ldot{\,.} \def\iminus{I^{^-}\!} \def\iplus{I^{^+}\!} \def\iplusminus{I^{^\pm}\!} % \def\Iminus{{\Bbb I}^{^-}\!} \def\Iplus{{\Bbb I}^{^+}\!} \def\Iplusminus{{\Bbb I}^{^\pm}\!} % \def\ssf{{\sss f}} \def\ssg{{\sss g}} \def\ssO{{\sss 0}} \def\ssX{{\!\sss X}} \def\ssY{{\!\sss Y}} % \def\DC{{\rm DC}} \def\GL{{\rm GL}} \def\SL{{\rm SL}} \def\SO{{\rm SO}} \def\SU{{\rm SU}} % \addref{DS} \addref{MP} \addref{Ei} \addref{RKi} \addref{RKii} \addref{RKiii} \addref{RKiv} \addref{MR} \addref{Eii} \addref{AK} \addref{JLDiii} \addref{HKi} \addref{JLDi} \addref{JLDii} \addref{CJ} \addref{HKii} \addref{DG} \addref{SK} \addref{KLM} \addref{Cas} \addref{HP} \addref{Fink} \addref{JM} \addref{Lag} \addref{KM} %% \def\leftheadline{\sevenrm\hfil HANS KOCH and JO\~AO LOPES DIAS\hfil} \def\rightheadline{\sevenrm\hfil Renormalization and Reducibility of Diophantine Skew Flows \hfil} %% %% \cl{{\magtenbf Renormalization of Diophantine Skew Flows,}} \cl{{\magtenbf with Applications to the Reducibility Problem}} \bigskip \cl{Hans Koch \footnote{$^1$} {{\sevenrm Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712}} and Jo\~ao Lopes Dias \footnote{$^2$} {{\sevenrm Departamento de Matem\'atica, ISEG, Universidade T\'ecnica de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal}} } %% %% \bigskip \abstract We introduce a renormalization group framework for the study of quasiperiodic skew flows on Lie groups of real or complex $\scriptstyle n\times n$ matrices, for arbitrary diophantine frequency vectors in \smallreal$^{\sss d}$ and dimensions $\scriptstyle d,n$. In cases where the group component of the vector field is small, it is shown that there exists an analytic manifold of reducible skew systems, for each diophantine frequency vector. More general near-linear flows are mapped to this case by increasing the dimension of the torus. This strategy is applied for the group of unimodular $\scriptstyle 2\times 2$ matrices, where the stable manifold is identified with the set of skew systems having a fixed fibered rotation number. Our results apply to vector fields of class C$^{\sss\gamma}$, with $\gamma$ depending on the number of independent frequencies, and on the diophantine exponent. %% %% \section Introduction and main results %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let $\bfG$ be a Lie subgroup of $\GL(n,\complex)$ or $\GL(n,\real)$, and denote by $\bfA$ the corresponding Lie algebra. We consider vector fields on $\Lambda=\torus^d\times\bfG$ of the form $$ X(q,y)=\bigl(\omega,f(q)y\bigr)\,,\qquad f(q)\in\bfA\,,\qquad (q,y)\in\Lambda\,. \equation(XForm) $$ Here, $\torus^d$ denotes the $d$-torus, with $\torus=\real/(2\pi\integer)$. Such a vector field $X$ determines a linear flow on the torus, $q(t)=q_0+t\omega$, and a linear evolution equation on $\bfG$, $$ \dot y(t)=f(q_0+t\omega)y(t)\,,\qquad y(0)=y_0\,, \equation(ydot) $$ whose coefficients are periodic or quasiperiodic functions of $t$, depending on the frequency vector $\omega$. If $t\mapsto\Phi_\ssX^t(q_0)$ denotes the solution of \equ(ydot), for the case where $y_0\in\bfG$ is the identity, then the flow $\Psi_\ssX$ associated with the vector field \equ(XForm) can be written as $$ \Psi_\ssX^t(q_0,y_0)=\bigl(q_0+t\omega,\Phi_\ssX^t(q_0)y_0\bigr)\,, \qquad (q_0,y_0)\in\Lambda\,,\quad t\in\real\,. \equation(XiDef) $$ Such flows are commonly referred to as {\it skew flows}. Classical Floquet theory shows that if $t\mapsto q(t)$ is periodic, and in particular if $d=1$, then the system is reducible. To be more precise, the vector field \equ(XForm) is said to be {\it reducible} if there exists a function $V:\torus^d\to\bfG$, such that $$ \Phi_\ssX^t(q)=V(q+t\omega)^{-1}e^{t C}V(q)\,,\qquad t\in\real\,,\quad q\in\torus^d\,, \equation(reducible) $$ for some constant matrix $C\in\bfA$. If $\omega\in\real^d$ is fixed, we will also refer to $f$ as being reducible. Another characterization of reducibility can be given by considering the map $\VV:\Lambda\to\Lambda$, defined by $$ \VV(q,y)=\bigl(q,V(q)y\bigr)\,. \equation(VVDef) $$ The pushforward of $X=(\omega,f\ldot)$ under this map is given by the equation $$ \bigl(\VV_\ast X\bigr)(q,y)=\bigl(\omega,(\VV_\star f)(q)y\bigr)\,,\qquad \VV_\star f=(D_\omega V+Vf)V^{-1}\,, \equation(VStar) $$ where $D_\omega=\omega\cdot\nabla$. Modulo smoothness assumptions, \equ(reducible) is equivalent to $\VV_\star f\equiv C$. More recent results concern the reducibility of skew systems with rationally independent frequencies $\omega_1,\ldots,\omega_d\,$, where $t\mapsto q(t)$ is quasiperiodic. For such systems, solving $\VV_\star f\equiv C$ leads to small divisor problems, as in classical KAM theory. Results based on KAM type methods have been obtained in the case where $\bfG=\SL(2,\real)$ [\rDS,\rMP,\rEi], motivated by the study of the one-dimensional Schr\"odinger equation with quasiperiodic potential, and for compact Lie groups [\rRKi,\rRKii]. In particular, Eliasson's result [\rEi] for $\bfG=\SL(2,\real)$ guarantees reducibility for analytic vector fields of the form \equ(XForm), with $\omega$ Diophantine, and with the fibered rotation number (associated with a rotation in $\bfG$) being either rational or Diophantine with respect to $\omega$. The vector field is required to be close to constant, but the smallness condition does not depend on further arithmetic properties of the rotation number. By contrast to these results, there are also generic examples of non-reducible systems [\rMR,\rEi,\rEii]. Another approach to the reducibility problem involves renormalization methods. For discrete time cocycles over rotations by an irrational angle $\alpha$, and for $\bfG=\SU(2)$, Rychlik introduced in [\rMR] a renormalization scheme based on a rescaling of first return maps, using the continued fractions expansion of $\alpha$. Later, Krikorian improved the method in [\rRKiii,\rRKiv], where he was able to prove global (non-perturbative) results for compact $C^\infty$ cocycles. The non-compact case $\bfG=\SL(2,\real)$ was treated in [\rAK]. In the context of flows, renormalization techniques were used in [\rJLDiii] to prove a local normal form theorem for analytic skew systems with a Brjuno base flow. Unlike the KAM methods, the renormalization approach has so far been restricted to skew systems with a one-dimensional base map or two-dimensional base flow. In this paper, we overcome this restriction by making use of the multidimensional continued fractions algorithm introduced in [\rKLM]. In addition, we reduce the smoothness condition on the vector field, by requiring only a finite degree of differentiability. Another novel aspect of our method is that the fibered rotation number is included in the frequency re-normalization procedure. We focus on cases where $\omega\in\real^d$ is {\it diophantine}, in the sense that $$ |\omega\cdot\nu|\ge C\|\nu\|^{1-d-\beta}\,,\qquad \nu\in\integer\setminus\{0\}\,, \equation(DCbeta) $$ for some constants $\beta,C>0$. It is well known that for any fixed $\beta>0$, the measure of the set of vectors $\omega$ that violate \equ(DCbeta) approaches zero as $C$ tends to zero [\rCas]. The constants $\beta,C>0$ are considered fixed in the rest of this paper. Our vector fields are assumed to be of class ${\rm C}^\gamma$, with $\gamma$ larger than some constant $\gamma_\ssO(\beta)$ specified below. Given any $\gamma\ge 0$, define $\FF_\gamma$ to be the Banach space of integrable functions $f:\torus^d\to\GL(n,\complex)$, for which the norm $$ \|f\|_\gamma=\|f_0\|+\!\!\sum_{0\not=\nu\in\integer^d} \|f_\nu\|(2\|\nu\|)^\gamma\,,\qquad f_\nu=(2\pi)^{-d}\!\int\limits_{\torus^d}\!f(q)e^{-i\nu\cdot q}dq\,, \equation(FNorm) $$ is finite. Here, and in what follows, we use the standard $\ell^2$ norm on the spaces $\complex^m$, and the corresponding operator norm for $m\times m$ matrices. Define $\mean f$ to be the torus-average $f_0$ of a function $f\in\FF_\gamma\,$. The set of functions in $\FF_\gamma$ that take values in $\bfG$ or $\bfA$ will be denoted by $\GG_\gamma$ or $\AA_\gamma\,$, respectively. Our first result describes a class of vector fields $X=(\omega,f\ldot)$ that are reducible to the trivial vector field $(\omega,0)$. Define $$ \gamma_\ssO(\beta)=(d+\beta)\left[1+2\beta +2\sqrt{\beta\bigl[1+\beta-1/(d+\beta)\bigr]}\;\right]-1\,. \equation(gammaZero) $$ \claim Theorem(GLmain) Given $\gamma\ge\gamma_2>\gamma_\ssO(\beta)$, there exists an open neighborhood $B$ of the origin in $\FF_\gamma\,$, and for each diophantine unit vector $\omega$ satisfying \equ(DCbeta) a manifold $\MM$ in $B$, such that the following holds. $\MM$ is the graph of an analytic map $M:(\Id-\mean)B\to\mean B$, which vanishes together with its derivative at the origin, and which takes values in $\AA_\gamma$ when restricted to $\AA_\gamma\,$. Every function $f$ on $\MM$ is reducible to zero. The corresponding change of coordinates $V$ belongs to $\FF_\eps$ and depends analytically on $f$, where $\eps=\gamma-\gamma_2\,$. If in addition, $f\in\AA_\gamma\,$, then $V$ belongs to $\GG_\eps\,$, and if $f$ is the restriction to $\torus^d$ of an analytic function, then so is $V$. Here, a function $\psi$ defined on $\MM$ is said to be analytic if $\psi\circ M$ is analytic on the domain of $M$. \smallskip This theorem can also be applied to vector fields $Y=(w,g\ldot)$, whose group component $g$ is close to a constant matrix $A$, but not necessarily small. But $w$ and $A$ have to satisfy a certain diophantine condition. More specifically, assume that $A\in\bfA$ admits a spectral decomposition $A=\kappa\cdot J=\kappa_1J_1+\ldots+\kappa_\ell J_\ell\,$, where $\kappa$ is some vector in $\real^\ell$, and where the $J_j$ are linearly independent mutually commuting matrices in $\bfA$, such that $t\mapsto\exp(tJ_j)$ is $2\pi$-periodic. The vector $\kappa$ will be referred to as the frequency vector of $A$. In order to see how \clm(GLmain) can be applied to $g\approx A$, we start with a skew system $Y=(w,g\ldot)$ on $\torus^m\times\bfG$, and then take $d=m+\ell$. Clearly, if $g\equiv A=\kappa\cdot J$, then the flow for $Y$ is equivalent to the flow for $X=(\omega,0)$, with $\omega=(w,\kappa)$. More generally, if $g-A$ is small but not necessarily zero, we consider the function $$ f(q)=e^{-r\cdot J}g(x)e^{r\cdot J}-\kappa\cdot J\,,\qquad q=(x,r)\,\in\torus^{m}\times\torus^\ell\,. \equation(gtof) $$ If $Y$ is regarded as a vector field on $\Lambda$ by identifying $w$ and $x$ with $(w,0)$ and $(x,0)$, respectively, then the above relation between $g$ and $f$ can be written as $$ g=\Theta_\star f\,,\qquad \Theta(q,y)=\bigl(q,e^{r\cdot J}y\bigr)\,. \equation(ThetaDef) $$ In order to simplify the discussion, assume now that $\omega$ has length one. If $\omega=(w,\kappa)$ is diophantine of type \equ(DCbeta) and $g$ belongs to the manifold $\Theta_\star\MM$, then the flow for $X=(\omega,f\ldot)$ can be trivialized with a change of coordinates $V$, as described in \clm(GLmain). The same now holds for $g$. However, the corresponding change of variables $W(q)=V(q)e^{-r\cdot J}$ is not of the desired form, since it still depends on the coordinates $r_j\,$. But as we will see, $$ \Phi_\ssY^t(x)=W(x+t\omega)^{-1}W(x) =V(x+tw)^{-1}e^{tC}V(x)\,, \equation(PhiRestricted) $$ for some matrix $C\in\bfA$ with frequency vector $\kappa$, provided that $V$ is differentiable. What remains to be shown, in specific cases, is that the space of functions of the type \equ(gtof) has a reasonable intersection with the manifold $\MM$. This procedure can be characterized as transforming some circular motion on $\bfG$ into motion on an extended torus. Our motivation for this approach is to try to treat all frequencies of the system in a unified way. In the case discussed below, it also has the advantage that the analysis of near-constant skew flows $Y=(\omega,g\ldot)$ can be reduced to a purely local analysis near $f\equiv 0$. Consider now $\bfG=\SL(2,\real)$. In this case, there is a natural rotation number that can be associated with a skew flow, due to the fact that the fundamental group of $\bfG$ is $\integer$ (as for higher dimensional symplectic groups). To this end, consider the flow for $Y=(w,g\ldot)$ on the product of $\torus^{d-1}$ with $\real^2\setminus\{0\}$, $$ \dot v(t)=g(x_0+tw)v(t)\,,\qquad v(0)=v_0\,. \equation(vdot) $$ Denote by $\alpha(t)$ the angle between $v(t)$ and some fixed unit vector $u_0\,$, and let $\alpha_0=\alpha(0)$. Then the lift of this angle to $\real$ evolves according to the equation $$ \dot\alpha(t)=-\bigl\langle e^{-\alpha(t)J} Jg(x_0+tw)e^{\alpha(t)J}u_0\,,u_0\bigr\rangle\,,\qquad \alpha(0)=\alpha_0\,, \equation(dotalpha) $$ where $\langle .\,,.\rangle$ denotes the standard inner product on $\real^2$. Here, and in the remaining part of this section, $J=\bigl[{0~-1\atop 1~~~0}\bigr]$. If the components of $w$ are rationally independent, then we can define the so-called {\it fibered rotation number} of $Y$, $$ \varrho(Y)=\lim_{t\to\infty}{\alpha(t)\over t}\,. \equation(rhoY) $$ As was shown in [\rJM], this limit exists for all $x_0\in\torus^{d-1}$ and $\alpha_0\in\real$, and it is independent of these initial conditions. {}From the definition of $\Theta$, we see that $\varrho(Y)=\kappa$ if and only if $\varrho(X)=0$. Thus, we may restrict our analysis to skew flows with fibered rotation number zero. \clm(GLmain) deals with precisely such flows. However, the functions \equ(gtof) are of a particular type, and more can be said in this case. In the following theorem, $\bfG=\SL(2,\real)$, and $\bfA$ is the corresponding Lie algebra of real traceless $2\times 2$ matrices. Denote by $\AA_\gamma^0$ the subspace of functions $g$ in $\AA_\gamma$ with the property that $g(q)=g(x)$, for all $q=(x,r)$ in $\torus^{d-1}\times\torus^1$. \claim Theorem(SLTWOmain) Given $\gamma\ge\gamma_2>\gamma_\ssO(\beta)$ and $a>0$, the following holds for some $R>0$. Consider a constant skew system $(w,A)$ on $\torus^{d-1}\times\bfG$, for a matrix $A\in\bfA$ that has purely imaginary eigenvalues, say $\pm\kappa i$. Assume that $\omega=(w,\kappa)$ satisfies the diophantine condition \equ(DCbeta), and that $\|A\|\le a|\kappa|\|\omega\|$. Then there exists an open neighborhood $B_0$ of the constant function $x\mapsto A$ in $\AA_\gamma^0\,$, containing a ball of radius $R$ centered at this function, such that for any $g\in B_0\,$, the one-parameter family $\lambda\mapsto g+\lambda A$ contains a unique member in $B_0\,$, say $g'$, whose associated skew flow has a fibered rotation number $\kappa$. If $\gamma-\gamma_2=\eps\ge 1$, then $g'$ is reducible to a constant $C\in\bfA$, as described by equation \equ(PhiRestricted), via a change of coordinates $V\in\GG_\eps\,$. Furthermore, the function $g'$, and (if $\eps\ge 1$) the quantities $C$ and $V$, depend real analytically on $g$. This theorem is proved by first performing a simple change of coordinates $g\mapsto L^{-1}gL$ with $L\in\bfG$, such that $L^{-1}AL=\kappa J\,$, followed by a constant scaling $Y\mapsto cY$ of the resulting skew system, which converts $(w,\kappa)$ to a unit vector. This is where the condition $\|A\|\le a|\kappa|\|\omega\|$ comes in. After that, the task is reduced via the map $\Theta$ to the study of vector fields $X=(\omega,f\ldot)$ with $f$ of the type \equ(gtof). Thus, in view of \clm(GLmain), it suffices to prove (besides real analyticity) that the family $\lambda\mapsto f+\lambda J$ intersects the manifold $\MM$ in exactly one point, that $\varrho(X)=0$ implies $f\in\MM$, and that \equ(PhiRestricted) holds if $f$ belongs to $\MM$. Our analysis of skew systems near $(\omega,0)$, including the proof of \clm(GLmain), is based on the use of renormalization group (RG) transformations. These transformations are defined in the next section. As described in more detail in Section 4, each diophantine vector $\omega$ determines, via a multidimensional continued fractions expansion [\rKLM], a sequence of matrices $T_n\in\SL(d,\integer)$. The $n$-th step RG transformation $\NN_n$ involves a change of variables $(q,y)\mapsto(T_nq,y)$, and another change of variables of the form \equ(VVDef), which eliminates certain ``nonresonant modes''. This is similar in spirit to the RG transformations used in [\rJLDiii-\rKLM]. The details of the elimination procedure can be found in Section 3. Each transformation $\NN_n$ has $f\equiv 0$ as a fixed point, and the stable/unstable subspaces of $D\NN_n(0)$ are the same for all $n$. Thus, it is possible to define and construct a ``stable manifold'' (the manifold $\MM$ described in \clm(GLmain)) for the sequence $\{\NN_n\}$. This construction is carried out in Section 5, by extending our RG transformations to parametrized families. The reducibility of functions $f\in\MM$ is proved in Section 6, by combining the partial reductions (elimination of nonresonant modes) from the individual RG steps. The remaining results concerning $\bfG=\SL(2,\real)$ are proved in Section 7. \section Renormalization %%%%%%%%%%%%%%%%%%%%%%%% We start by describing a single RG step. A unit vector $\omega\in\real^d$, and a matrix $T$ in $\SL(d,\integer)$ are assumed to be given, subject to certain conditions that will be described below. The matrix $T$ defines a map $\TT:\Lambda\to\Lambda$, $$ \TT(q,y)=\bigl(T(q),y\bigr)\,, \equation(TTDef) $$ and the pushforward of a vector field \equ(XForm) under this map is given by $$ \bigl(\TT_\ast X\bigr)(q,y)=\bigl(T\omega,(\TT_\star f)(q)y\bigr)\,, \qquad \TT_\star f=f\circ T^{-1}\,. \equation(TStar) $$ For every positive $\tau<1$, define $\KK(\tau)$ to be the set of all vectors in $\real^d$ that are contracted by a factor $\le\tau$ under the action of $S=(T^\ast)^{-1}$. Here, $T^\ast$ denotes the transpose of $T$. Given a fixed value for this contraction factor $\tau$, to be specified later, the ``resonant'' part $\Iplus f$ of a function $f\in\FF_\gamma\,$, and its ``nonresonant'' part $\Iminus f$, are defined by the equation $$ \Iplusminus f(q)=\sum_{\nu\in\iplusminus}f_\nu e^{i\nu\cdot q}\,, \equation(Iplusminus) $$ where $\iplus=\KK(\tau)\cap\integer^d$ and $\iminus=\integer^d\setminus\iplus$. As one would expect (see the lemma below), the resonant part of a function $f\in\FF_\gamma$ is contracted under the action of $\TT_\star$. In order to simplify notation, we will drop the subscript $\gamma$ from now on, unless two different choices of $\gamma$ are being considered at the same time. \claim Lemma(TSub) If $f\in\FF$ satisfies $\Iminus f=\mean f=0$, then $\|\TT_\star f\|\le\tau^\gamma\|f\|$. The proof follows immediately from the definitions: $$ \|\TT_\star f\|=\sum_{0\not=\nu\in\iplus} \|f_\nu\|(2\|S\nu\|)^\gamma \le\sum_{0\not=\nu\in\iplus} \|f_\nu\|(2\tau\|\nu\|)^\gamma=\tau^\gamma\|f\|\,. $$ The complementary property of the nonresonant modes is that they can easily be eliminated via a change of variables of the form \equ(VVDef). To be more precise, we assume that the constant $\tau$ can be (and has been) chosen in such a way that $\KK(\tau/2)$ contains the orthogonal complement of $\omega$. Under this assumption, we will show in Section 3 that if $f\in\FF$ is sufficiently close to zero, then it is possible to find $U_\ssf\in\FF$ close to the identity, such that $$ \Iminus(\UU_\ssf)_\star f=0\,. \equation(Elim) $$ By construction, the map $f\mapsto U_\ssf$ is analytic, and $U_\ssf$ belongs to $\GG$ whenever $f\in\AA$. The renormalized function $\NN(f)$ and the renormalized vector field $\RR(X)$ are now defined by the equation $$ \NN(f)=\eta^{-1}\TT_\star(\UU_\ssf)_\star f\,,\qquad \RR(X)=\eta^{-1}\TT_\ast(\UU_\ssf)_\ast X\,, \equation(RGDef) $$ where $\eta$ is the norm of $T\omega$, so that the torus component of $\RR(X)$ is again a unit vector. The corresponding flow is given by $$ \Phi_{\RR(X)}^t=\bigl[U_\ssf(.+\eta^{-1}t\omega) \Phi_X^{\eta^{-1}t} U_\ssf^{-1}\bigr]\circ T^{-1}\,. \equation(RGPhi) $$ In what follows, the RG transformation $\NN$ is regarded as a map from an open domain in $\FF$ to $\FF$. But it should be kept in mind that its restriction to $\AA$ takes values in $\AA$. An explicit bound on the map $f\mapsto U_\ssf$ leads to the following. \claim Theorem(RGStep) Let $f=C+h$, with $C$ constant and $\mean h=0$. Assume that $\|C\|<\sigma/6$ and $\|h\|<2^{-9}\sigma$, with $\sigma$ satisfying $2\sigma\|S\|<\tau$. Then $$ \NN(f)=\eta^{-1}\bigl[C+\tilde h\bigr]\,,\qquad \bigl\|\tilde h\bigr\| \le{\textstyle{3\over 2}}\tau^\gamma\|h\|\,,\qquad \bigl|\mean\tilde h\bigr|\le 16\sigma^{-1}\tau^\gamma\|h\|^2\,. \equation(RGStep) $$ $\NN$ is analytic on the region determined by the given bounds on $C$ and $h$. Furthermore, if $f$ is real-valued, then so is $\NN(f)$. A proof of this theorem will be given in Section 3. Notice that the zero-average part $h$ of $f$ gets contracted by roughly a factor $\tau^\gamma$ relative to the constant part $C$, which is the same factor that appears in \clm(TSub). The restriction on the size of the domain of $\NN$, which is of the order of $\sigma$, comes from the solution of equation \equ(Elim). The goal now is to compose RG transformations of this type, as long as the constant part of $f$ does not become too large. 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 $\omega_0$ in $\real^d$, we define $$ T_n=P_nP_{n-1}^{-1}\,,\quad S_n=(T_n^\ast)^{-1}\,,\quad \lambda_n=\|P_n\omega_0\|\,,\quad \omega_n=\lambda_n^{-1}P_n\omega_0\,, \equation(TSlw) $$ for $n=1,2,\ldots$. The following theorem will be proved in Section 4, using as input certain estimates from [\rKLM]. \claim Theorem(Ptauexist) Given $\gamma_1>\gamma_\ssO(\beta)$, there exist two sequences $n\mapsto\sigma_n$ and $n\mapsto\tau_n$ of positive real numbers less than one, both converging to zero, such that the following holds. If $\omega_0$ is a unit vector in $\real^n$ satisfying the diophantine condition \equ(DCbeta), then there exists a sequence $n\mapsto P_n$ of unimodular integer matrices, such that with $S_n$ and $\lambda_n$ as defined in \equ(TSlw), $$ 2\sigma_n\|S_n\|<\tau_n\,,\qquad \|S_n\xi\|\le{\tau_n\over 2}\|\xi\|\,,\qquad \lambda_n^{-1}\prod_{j=1}^n \bigl(4\tau_j^{\gamma_1}\bigr)\cdot\sigma_1\le\sigma_{n+1}\,, \equation(tauCond) $$ whenever $\omega_{n-1}\cdot\xi=0$, for every positive integer $n$. In order to simplify the discussion, the quantities described in this theorem are considered fixed from now on. We also assume that $\gamma\ge\gamma_1\,$. The $n$-th step RG transformation $\NN_n$ and the composed RG transformation $\wt\NN_n$ are defined by the equation $$ \NN_n(f)=\eta_n^{-1}(\TT_n)_\star(\UU_\ssf)_\star f\,,\qquad \wt\NN_n=\NN_n\circ\NN_{n-1}\circ\ldots\circ\NN_1\,, \equation(RGnDef) $$ where $\eta_n=\lambda_n/\lambda_{n-1}$ for $n\ge 1$, with $\lambda_0=1$. To be more specific, we choose $\tau=\tau_n$ and $\omega=\omega_{n-1}$ in the construction of the map $\UU_\ssf$ that enters the definition of $\NN=\NN_n\,$. By \clm(RGStep), the transformation $\NN_n$ is well defined on the open ball $B_n\subset\FF$ of radius $2^{-9}\sigma_n\,$, centered at the origin. $B_n$ will be referred to as the domain of $\NN_n\,$. The domain of $\wt\NN_n$ is defined recursively as the set of all functions in the domain of $\wt\NN_{n-1}$ that are mapped into $B_n$ by $\wt\NN_{n-1}$. For such a function $f$, define $f_0=f$ and $$ f_n=\wt\NN_n(f_0)\,,\qquad \bar f_n=\mean f_n\,,\qquad h_n=f_n-\bar f_n\,. \equation(FCh) $$ By \clm(RGStep) and \clm(Ptauexist), we have $$ \|h_n\|\le\lambda_n^{-1} \prod_{j=1}^n\bigl(2\tau_j^\gamma\bigr) \cdot\|f_0\| \le 2^{-10}\sigma_{n+1}\,. \equation(hnBoundOne) $$ This shows e.g. that for $f\in\FF$ close to zero, the question of whether or not $f$ is infinitely renormalizable depends only on the size of the averages $\bar f_n\,$. Consider now a sequence $\rho$ of real numbers satisfying $$ 0<\rho_n\le 2^{-10}\sigma_{n+1}\,,\qquad n=0,1,2,\ldots\,. \equation(rhoCond) $$ Given an open set $B(\gamma)\subset B_1$ containing zero, define $\wt B_0=B(\gamma)$ and $$ \wt B_{n+1}=\bigl\{f\in\wt B_n: \|\bar f_n\|<\rho_n\bigr\}\,, \qquad n=0,1,2,\ldots\,. \equation(wtBnDef) $$ The bound \equ(hnBoundOne) shows that $\wt B_{n+1}$ is contained in the domain of $\wt\NN_{n+1}\,$. \claim Theorem(MM) If $\gamma>\gamma_1$ then there exists a sequence $\rho$ satisfying \equ(rhoCond), and a non-empty open neighborhood $B(\gamma)$ of the origin in $\FF$, such that $\MM_\gamma=\bigcap_{n=0}^\infty\wt B_n$ is the graph of an analytic function $M:(\Id-\mean)B(\gamma)\to\mean B(\gamma)$. Both $M$ and its derivative vanish at the origin. A proof of this theorem is given in Section 5. The reducibility of functions $f$ belonging to $\MM=\MM_\gamma$ will be proved in Section 6, by iterating the identity \equ(RGPhi), and using that $f_n\to 0$, in order to estimate the product of the matrices $U_{f_n}\,$. \section Elimination of nonresonant modes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Here we solve equation \equ(Elim) and prove \clm(RGStep). A unit vector $\omega\in\real^d$ and a matrix $T$ in $\SL(d,\integer)$ are assumed to be given. As mentioned in the last section, we also assume that the cone $\KK(\tau/2)$ contains the orthogonal complement of $\omega$, and that $2\sigma\|S\|<\tau$. \claim Proposition(omegadotnu) If $\nu$ belongs to $\iminus$ then $|\omega\cdot\nu|>\sigma$. \proof Given $\nu\in\iminus$, consider its decomposition $\nu=\nu_\parallel+\nu_\perp$ into a vector $\nu_\parallel$ parallel to $\omega$ and a vector $\nu_\perp$ perpendicular to $\omega$. By using that $\|\nu\|, \|S\|, \|\omega\|\ge 1$, we obtain $$ \eqalign{ \sigma&\le\sigma\|\nu\|<\|S\|^{-1}{\tau\over 2}\|\nu\| \le\|S\|^{-1}\bigl(\|S\nu\|-\|S\nu_\perp\|\bigr)\cr &\le\|S\|^{-1}\|S\nu_\parallel\| \le\|\nu_\parallel\|\le|\omega\cdot\nu|\,,\cr} $$ as claimed. \qed Given any $n\times n$ matrix $C$, define $\hat Cf=fC-Cf$ for every function $f\in\FF$. \claim Proposition(DDinv) Assume that $\|C\|\le\sigma/4$. Then the linear operators $D_\omega=\omega\cdot\nabla$ and $\DD=D_\omega+\hat C$ commute with $\Iminus$, have bounded inverses when restricted to $\Iminus\FF$, and satisfy $$ \bigl\|D_\omega^{-1}\Iminus\bigr\|\le\sigma^{-1}\,,\qquad \bigl\|D_\omega\DD^{-1}\Iminus\bigr\|\le 2\,. \equation(DDinv) $$ \proof Clearly, $D_\omega$, $\hat C$, and $\Iminus$ commute with each other. The first inequality in \equ(DDinv) follows immediately from \clm(omegadotnu). It implies $\|D_\omega^{-1}\hat C\Iminus\|\le 2\sigma^{-1}\|C\|\le 1/2$, and the indicated bound on $D_\omega\DD^{-1}\Iminus=(\Id+D_\omega^{-1}\hat C)^{-1}\Iminus$ is now obtained via Neumann series. \qed In the rest of this paper, we will frequently use analyticity arguments. Thus, let us recall at this point some relevant facts [\rHP] about \hfil\smallskip\noindent {\bf analytic maps.} Let $\XX$ and $\YY$ be Banach spaces over $\complex$, and let $B\subset\XX$ be open. We say that $G:B\to\YY$ is analytic if it is Fr\'echet differentiable. Thus, sums, products, and compositions of analytic maps are analytic. Equivalently, $G$ is analytic if it is locally bounded, and if for all continuous linear maps $f:\complex\to\XX$ and $h:\YY\to\complex$, the function $h\circ G\circ f$ is analytic. This shows e.g. that uniform limits of analytic functions are analytic. Assuming that $B$ is a ball of radius $r$ and that $F$ is bounded on $B$, a third equivalent condition is that $G$ has derivatives of all orders at the center of $B$, and that the corresponding Taylor series has a radius of convergence at least $r$ and agrees with $G$ on $B$. \medskip Another fact that we will use repeatedly is that $\FF$ is a Banach algebra, i.e., we have $\|fg\|\le\|f\|\|g\|$ for all $f,g\in\FF$. In the remaining part of this section, $f\in\FF$ is fixed but arbitrary, $C=\mean f$, and $h=f-C$. We seek a solution of equation \equ(Elim) of the form $U=\exp(\DD^{-1}u)$, with $u$ a function in $\Iminus\FF$. In order to simplify notation, $\mean\FF$ will be identified with $\GL(n,\complex)$. A short computation shows that $$ \Iminus\UU_\star f=u-\psi(u)\,, \equation(MMDef) $$ where $$ \eqalign{ \psi(u)&=-\Iminus\bigl[ (D_\omega\DD^{-1}u)E^{-}_1+(D_\omega E^{+}_2)E^{-}_0 +E^{+}_0hE^{-}_0\cr &\qquad\quad+CE^{-}_2+(\DD^{-1}u)CE^{-}_1+E^{+}_2CE^{-}_0\bigr]\cr} \equation(MMDef) $$ and $$ E^\pm_m=\sum_{k=m}^\infty{1\over k!}\bigl(\pm\DD^{-1}u\bigr)^k\,, \qquad m=0,1,\ldots\,. \equation(Epm) $$ \claim Proposition(MFix) Assume that $\|C\|<\sigma/6$ and $\|h\|<2^{-9}\sigma$. Let $r=2^{-8}\sigma$, and denote by $B_r$ the closed ball of radius $r$ in $\Iminus\FF$, centered at the origin. Then $\psi$ has a unique fixed point $u_\ssf$ in $B_r\,$, and $$ \|u_\ssf\|\le{\textstyle{16\over 15}}\|h\|\,. \equation(PhiBound) $$ The map $(C,h)\mapsto u_\ssf$ is analytic on the domain defined by the given bounds on $C$ and $h$. If $f$ is real-valued, then so is $u_\ssf\,$. Furthermore, if $f$ belongs to $\AA$ then $U_\ssf=\exp(\DD^{-1}u_\ssf)$ belongs to $\bfG$. \proof First, recall that $e^x\le(1-x)^{-1}$ whenever $0\le x<1$. This fact will be used below and in subsequent proofs. A straightforward estimate, using \clm(DDinv) and the Banach algebra property of $\FF$, shows that $\psi$ is an analytic map from the space $\Iminus\FF$ to itself, satisfying the bound $$ \bigl\|\psi(u)\bigr\|\le e^{4\sigma^{-1}\|u\|} \bigl(\|h\|+10\sigma^{-1}\|u\|^2\bigr)\,. \equation(MBound) $$ Notice that $\psi(0)=-\Iminus h$ has norm $\le r/2$. Thus, if we prove that $\|D\psi(u)\|\le 1/2$ for all $u\in B_r\,$, then the existence and uniqueness of a fixed point $u_\ssf\in B_r$ follows from the contraction mapping principle. Let $u\in B_r$ and $g\in\FF$ be fixed but arbitrary, with $\|g\|=1$. Define $\varphi:\complex\to\FF$ by the equation $\varphi(z)=\psi(u+zg)$. If $|z|\le R=2^{-6}\sigma$, then $u+zg$ is bounded in norm by $\sigma/48$, and by using \equ(MBound), we find that $$ \|\varphi(z)\|\le{\textstyle{12\over 11}}\|h\|+11\sigma^{-1}\|u+zg\|^20$, such that the following holds. If $\omega=(w,1)$ is any vector of length less than $d$, satisfying the diophantine condition \equ(DCbeta), and if $n\mapsto t_n$ is any sequence of stopping times, with $t_0=0$ and $\delta t_n=t_n-t_{n-1}>0$, then the bounds $$ \eqalign{ \|P_n^{-1}\|&\le c_1\exp\{(d-1+\theta)t_n\},\cr \|S_n\|&\le c_2\exp\{(d-1)(1-\theta)\delta t_n+d\,\theta\,t_n\},\cr \|S_n\xi\|&\le c_3\exp\{-(1-\theta)\delta t_n+d\,\theta t_{n-1}\}\,,\cr} \equation(MPT) $$ hold for all integers $n>0$, and for all unit vectors $\xi\in\real^d$ that are perpendicular to $P_{n-1}\omega$. \remark The condition $\|\omega\|0$ to be determined. By using that $\delta t_{n+1}=\alpha t_n$ and $s_{n-1}\le\alpha^{-1}t_n\,$, we find that $$ \lambda_n^{-1}\prod_{k=1}^n\bigl(4\tau_k^\gamma\bigr) \sigma_0\sigma_{n+1}^{-1} \le c_1 4^n\tau_0^{\gamma n}\exp\{-\eps c(1+\alpha)^n\}\,, \equation(tauSecond) $$ where $\eps=\mu-\gamma d\,\theta\alpha^{-1}-d\,\alpha\,$. The goal is to choose $\alpha$ in such a way that $\eps>0$. Then by taking $c>0$ sufficiently large, the right hand side of \equ(tauSecond) is less than one, for all positive integers $n$, and the third bound in \equ(tauCond) follows. The condition $\eps>0$ is a quadratic inequality for $\alpha$, which is satisfied by $\alpha=\mu/(2d)$, provided that $\mu^2>4\gamma\theta d^2$. An explicit computation shows that $\mu^2$ is larger than $4\gamma\theta d^2$, whenever $$ \gamma>{d\over(1-\theta)^2} \left[(1-\theta+2d\,\theta) +2\sqrt{\theta} \sqrt{d^2-(1-\theta)(d^2-d+1-\theta)}\;\right]-1\,. \equation(gammaCond) $$ The same condition also guarantees that $\mu$, and thus $\alpha$, is positive. Substituting $\theta=\beta/(d+\beta)$ into the inequality \equ(gammaCond), one gets the equivalent condition $\gamma>\gamma_\ssO(\beta)$, with $\gamma_\ssO(\beta)$ as defined in equation \equ(gammaZero). Finally, substituting the bound \equ(gammaCond) on $\gamma$ into the definition of $\mu$ yields $\alpha=\mu/(2d)>d\theta/(1-\theta)$, which shows that $\tau_{n+1}=\tau_0\exp\{[-(1-\theta)\alpha+d\theta]t_n\}$ tends to zero as $n\to\infty$. Taking $c>0$ sufficiently large ensures that $\tau_n<1$ for all $n$. The analogous property $1>\sigma_n\to 0$ follows now from the first inequality in \equ(tauCond). This completes the proof of \clm(Ptauexist). \section The stable manifold %%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section, we define RG transformations for families of functions in $\AA$, parametrized by $\bfA$. These transformations $\bfN_n$ are then used to prove \clm(MM) and some other estimates that are needed later on. $\bfN_n$ acts on a family $F:\bfA\to\AA$ by composing it from the left with $\NN_n\,$, and from the right with a reparametrization map on $\bfA$ that depends on $F$. Recall that $\NN_n$ is naturally defined on an open domain in $\FF$, but that its restriction to $\AA$ takes values in $\AA$. The situation is analogous for $\bfN_n\,$, since, as will be clear from the construction, the reparametrization map takes real values for real arguments, whenever $F$ is real. Thus, no generality is lost by assuming that $\bfG=\GL(n,\complex)$. We will do this in the remaining part of this paper, unless specified otherwise. We start with a preliminary estimate on inverses of some simple maps. Denote by $b$ the open unit ball in $\bfA$, centered at the origin. Consider the space $\UU$ of analytic functions $U:b\to\bfA$, equipped with the sup-norm. \claim Proposition(UandV) Assume $U\in\UU$ satisfies $\|U\|<{1-\lambda\over 2\lambda}$, with $0<\lambda<1$. Define $\Lambda(A)=\lambda A$ for every $A\in\bfA$. Then $\Lambda^{-1}+U$ has a unique right inverse $\Lambda+V$ on $b$, with $V$ belonging to $\UU$ and satisfying $\|V\|\le 2\lambda\|U\|$. The map $U\mapsto V$ is analytic on the domain in $\UU$ defined by the given condition on $U$. \proof Let $u=\|U\|$ and $r=2\lambda u$. If $A$ and $C$ are matrices in $\bfA$ of norm one, then from Cauchy's formula, we obtain $$ \|DU(\lambda A)C\|\le r^{-1}\sup_{|z|=r}\|U(\lambda A+zC)\| \le r^{-1}\|U\|=(2\lambda)^{-1}\,. \equation(UandVone) $$ Now consider the equation for $V$, which can be written as $\psi(V)=V$, with $\psi$ defined by $\psi(V)=-\lambda U\circ(\Lambda+V)$. Denote by $B$ the closed ball of radius $r$ in $\UU$, centered at the origin. Then $\psi$ is analytic on $\UU$, with derivative given by $$ D\psi(V)H=-\lambda\bigl((DU)\circ(\Lambda+V)\bigr)H\,. \equation(UandVtwo) $$ By equation \equ(UandVone), we see that $\|D\psi(V)\|\le 1/2$, for all $V\in B$. Since $\|\psi(0)\|=r/2$, the map $\psi$ is a contraction on $B$, and thus has a (unique) fixed point in $B$. The analyticity of $U\mapsto V$ follows form the uniform convergence of $\psi^n(0)\to V$ for $\|U\|\gamma_1\,$. Let $K\le 1$ be a fixed positive real number satisfying $$ 8^n\pi_n^{\gamma_2-\gamma_1}K\le 1/16\,, \equation(Cgamma) $$ for all integers $n\ge 0$. Such a number $K$ exists by \clm(Ptauexist). %% 1 instead of 1/16 would be sufficient here %% but 1/16 is used later in \clm(AzeroFam) \claim Lemma(Contraction) If $F_0\in\BB_0$ satisfies $\|F_0-F^0\|_00$, then so is $Y_n\,$, as mentioned earlier, and thus also $Y_n^{-1}$ and $F_n\,$. By induction, we see that all matrices $s_{m,n}$ are real whenever $F$ is, and the same holds for the limits $z_m\,$. \qed Denote by $B'(\gamma)$ the ball in $(\Id-\mean)\AA_\gamma$ of radius $K\rho_0\,$, centered at the origin. Define $B(\gamma)=b_0\oplus B'(\gamma)$, that is, $f\in\AA_\gamma$ belongs to $B(\gamma)$ if and only if $\bar f\in b_0$ and $h=f-\bar f$ belongs to $B'(\gamma)$. Consider now the set $\MM_\gamma$ defined in \clm(MM), with $B(\gamma)$ as described above. \claim Corollary(Mintersect) Let $F$ be a family in the domain of $\wt\bfN$, and let $s\in b_0$. Then $F(s)$ belongs to $\MM_\gamma$ if and only if $s=z_0(F)$. \proof Consider first $f=F(z_0)$. Set $f_n=F_n(z_n)$ for each $n>0$. By the definition of $\bfN_n\,$, and by \clm(znDef), we have $f_n=\NN_n(f_{n-1})$ for $n=1,2,\ldots$, and $\bar f_n=\mean F_n(z_n)=z_n$ belongs to $b_n\,$. This shows that $f\in\MM_\gamma\,$. Consider now a fixed $s=s_0$ in $b_0\,$, and assume that $f_0=F(s_0)$ belongs $\MM_\gamma\,$. Then we can define $f_n=\wt\bfN_n(f)$ for all $n>0$, and $s_n=\bar f_n$ belongs to $b_n\,$. Set $F_0=F$. Proceeding by induction, let $n>0$, and assume $f_{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(YnFBound), we have $s_{n-1}=Y_n^{-1}(s_n)$. As a result, $f_n=F_n(s_n)$. This shows that $s_n=Y_n(s_{n-1})$ holds for all $n>0$, and thus $s_n=z_n$ by \clm(znDef). \qed \proofof(MM) To a function $h\in B'(\gamma)$ we associate the family $F:s\mapsto s+h$. This family belongs to the domain of $\wt\bfN$. Now define $M(h)=z_0(F)$. By \clm(Mintersect), $h+s=F(s)$ belongs to $\MM_\gamma$ if and only if $s=M(h)$. This shows that $\MM_\gamma$ is the graph of $M$ over $B'(\gamma)$. The analyticity of $M$ follows from the analyticity of $z_0\,$. Furthermore, we have $M(0)=z_0(F^0)=0$. The identity $DM(0)=0$ follows from the fact that, by \clm(YnFBound), the derivative of $F\mapsto Y_{n,F}^{-1}$ vanishes at $F^0$, for each $n\ge 0$. \qed The following estimate will be used in the next section. Denote by $\id_{n,m}$ the inclusion map from $\bfA_m$ into $\bfA_n\,$. \claim Proposition(DZnPrime) Let $F$ be in the domain of $\wt\bfN$. Then the map $Z_n'=Y_n\circ\cdots\circ Y_1$ satisfies $$ \|DZ_n'(s)-\lambda_n^{-1}\id_{n,0}\| \le 2^{-10}\|\lambda_n^{-1}\id_{n,0}\|\,, \equation(DZnPrime) $$ for all $s$ in the image of $b_{n-1}$ under $Z_{0,n-1}\,$. \proof Define $s_{k-1}=Y_k^{-1}(s_k)$ for $k=n-1,\ldots 2,1$, starting with a fixed but arbitrary $s_{n-1}\in b_{n-1}\,$. By using \clm(YnFBound), and the fact that the inclusion map from $\bfA_{k-1}$ into $\bfA_k$ has norm $\rho_{k-1}/\rho_k=4\eta_k\tau_k^{-\gamma_1}$, we obtain $$ \|DY_k(s_{k-1})\| \le\bigl(1+2^{-4k-8}c_k\bigr)\|\eta_k^{-1}\id_{k,k-1}\|\,, \equation(DYnBound) $$ with $c_k=\tau_k^\gamma\|F_{k-1}-F^0\|_{k-1}<1$. Taking products, the norm of $DZ_k'(s_0)$ can be bounded by twice the norm of $\lambda_k^{-1}\id_{k,0}\,$. Thus, $$ \eqalign{ \|DZ_n'(s_0)-\lambda_n^{-1}\id_{n,0}\| &=\|DY_n(s_{n-1})\cdots DY_2(s_1)DY_1(s_0)-\lambda_n^{-1}\id_{n,0}\|\cr &\le\sum_{k=1}^n \bigl\|\eta_n^{-1}\cdots\eta_{k+1}^{-1}\id_{n,k} \bigl[DY_k(s_{k-1})-\eta_k^{-1}\id_{k,k-1}\bigr] DZ_{k-1}'(s_0)\bigr\|\cr &\le\sum_{k=1}^n2^{-4k-7}c_k \cdot\|\lambda_n^{-1}\id_{n,0}\|\,,\cr} $$ and the inequality \equ(DZnPrime) follows. \qed \section Reducibility %%%%%%%%%%%%%%%%%%%%% The main goal in this section is to prove \clm(GLmain). Consider first the flow $\Phi_\ssX$ for a general vector field $X=(\omega,f\ldot)$ The identity $$ \Phi_\ssX^t(q)=\id+\int_0^t f(q+s\omega)\Phi_\ssX^s(q)\,ds\,. \equation(PhiIdentity) $$ can be used to construct and estimate $\Phi_\ssX$. By applying first the contraction mapping principle, and then the cocycle identity for $\Phi_\ssX$ to improve the result, we obtain $$ \bigl\|\Phi_\ssX^t-\id\bigr\|_\gamma \le e^{\|tf\|_\gamma}-1\,. \equation(generalPhiBound) $$ This bound holds for any $\gamma\ge 0$, provided that $f\in\AA_\gamma\,$. Consider now $f_0\in\MM_\gamma$ and the corresponding renormalized functions $f_n=\wt\NN_n(f_0)$. In order to simplify notation, the transformation $U_{f_n}$ and the flow $\Phi_{(\omega_n,f_n)}$ will be denoted by $U_n$ and $\Phi_n\,$, respectively. \claim Lemma(CzeroVn) Let $f_0\in\MM_\gamma\,$. For each $n\ge 0$ there exists $V_n\in\GG_0$ such that $$ \Phi_n^t(q)=V_n(q+t\omega_n)^{-1}V_n(q)\,, \qquad t\in\real\,. \equation(CzeroVn) $$ These function $V_n$ satisfy the relations $V_{n+1}=(V_n\circ T_{n+1})U_n$ and the bounds $$ \|V_n-\id\|_0 \le 2^{4-n}\pi_n^{\gamma-\gamma_1}\sigma_1^{-1}\|f_0\|_\gamma\,. \equation(CzeroVnBound) $$ Furthermore, the maps $f_0\mapsto V_n$ are analytic. \proof By equation \equ(RGPhi), we have $$ \Phi_n^t(q) =V_{m,n}(q+t\omega_n)^{-1} \Phi_m^{\eta_m\ldots\eta_{n+1}t}(T_m\ldots T_{n+1}q) V_{m,n}(q)\,, \equation(RGnPhi) $$ for $m>n\ge 0$, where $$ V_{m,n}(q) =U_{m-1}(T_{m-1}\cdots T_{n+1}q)\cdots U_{n+1}(T_{n+1}q)U_n(q)\,. \equation(VmnDef) $$ For convenience later on, we also define $V_{n,n}=\id$. Using the notation of Section 5, we have $f_n\in 2b_n$ and thus $$ \|\eta_m\cdots\eta_{n+1}tf_n\|_\gamma \le 2\lambda_n^{-1}\lambda_m \rho_m|t| \le 2\cdot 4^{-m}\lambda_n^{-1}\rho_0|t|\,. \equation(etatfn) $$ If $m$ is sufficiently large, then \equ(generalPhiBound) leads to the bound $$ \bigl\|\Phi_m^{\eta_m\ldots\eta_{n+1}t}-\id\bigr\|_0 \le 4^{1-m}\lambda_n^{-1}\rho_0|t|\,. \equation(Phietatm) $$ Thus, $\Phi_m^{\eta_m\ldots\eta_{n+1}t}$ converges in $\GG_0$ to the identity, as $m\to\infty$, uniformly in $t$ on compact subsets of $\real$. Consider now the factors $U_j$ in the product \equ(VmnDef). By \clm(RGStep) and \clm(Ptauexist), $$ \sigma_{n+1}^{-1}\|h_n\|_\gamma \le 2^{-n-9}\pi_n^{\gamma-\gamma_1}\varepsilon\,, \qquad \varepsilon=2^9\sigma_1^{-1}\|f_0\|_\gamma<1\,. \equation(hnBoundTwo) $$ Combining this with the estimate \equ(generalUfBound), we obtain $$ \|U_n-\id\|_\gamma \le 2^{-n-7}\pi_n^{\gamma-\gamma_1}\varepsilon\,, \qquad \|U_n\|_\gamma\le e^{2^{-n-7}\varepsilon}\,. \equation(UnBound) $$ Notice that $\|U\circ T\|_0=\|U\|_0\le\|U\|_\gamma$ for any matrix $T$ in $\SL(d,\integer)$, and any $U$ in $\FF_\gamma$ with $\gamma\ge 0$. Thus, the bounds \equ(UnBound) can be used to estimate the product \equ(VmnDef) in $\GG_0\,$. We have $$ \|V_{m,n}\|_0\le\prod_{j=n}^{m-1}\|U_j\|_\gamma \le e^{2^{-n-6}\varepsilon}\,, \equation(VmnBound) $$ and as a result, $$ \|V_{k,n}-V_{m,n}\|_0\le \sum_{j=m}^{k-1}\bigl\|(U_j-\id)V_{j,n}\bigr\|_\gamma \le 2^{-m-5}\pi_n^{\gamma-\gamma_1}\eps\,, \equation(VdiffBound) $$ for $k>m\ge n\ge 0$. This shows that the limits $V_n=\lim_{m\to\infty}V_{m,n}$ exist in $\GG_0\,$, and that they have the properties described in \clm(CzeroVn). The analyticity of $f_0\mapsto V_n$ follows from the uniform convergence of $V_{m,n}\to V_n\,$, combined with the fact that the map $M$ defining the manifold $\MM_\gamma\,$, the RG transformations $\NN_n\,$, and the map $f\mapsto U_f$ described in \clm(MFix), are all analytic. \qed \claim Proposition(RpSymm) The manifold $\MM_\gamma$ is invariant under the torus-translations $R_p\,$, and the map $f_0\mapsto V_0$ defined by \clm(CzeroVn) commutes with these translations. \proof First, we note that the translations $R_p$ are isometries on $\FF$ and commute with $\mean$. This shows in particular that $B(\gamma)$ is invariant under $R_p\,$. The identity \equ(NNRp) shows that $R_pf_0$ belongs to the domain of $\wt\NN_n$ whenever $f_0$ does, and that $\wt\NN_n(f_0)$ and $\wt\NN_n(R_pf_0)$ have the same torus-average. {}From the definition \equ(wtBnDef) of the sets $\wt B_n$ whose intersection is $\MM_\gamma\,$, it is now clear that $\MM_\gamma$ is invariant under torus-translations. The fact that $f_0\mapsto V_0$ commutes with $R_p$ follows from an explicit computation, using the identities \equ(NNRp) and \equ(VmnDef). \qed \claim Lemma(VVV) Let $\gamma\ge\gamma_2>\gamma_1$ and $\eps=\gamma-\gamma_2\,$. If $f_0\in\MM_\gamma$ then the function $V_0$ described in \clm(CzeroVn) belongs to $\GG_\eps$ and has a directional derivative $D_{\omega_0}V_0$ in $\FF_\eps\,$. As elements of $\FF_\eps\,$, both $V_0$ and $D_{\omega_0}V_0$ depend analytically on $f_0\,$. Furthermore, if $f_0$ is the restriction to $\torus^d$ of an analytic function, then so is $V_0\,$. \proof In order to avoid possible ambiguities, assume first that $\gamma=\gamma_2\,$. Denote by $H$ and $\HH$ the maps that associate to each $f\in B'(\gamma_2)$ via $f_0=M(f)$ the corresponding function $V_0$ and the value $V_0(0)$, respectively. \clm(RpSymm) implies that $R_p V_0=H(R_p f)$, and thus $$ V_0(p)=\HH(R_p f)\,,\qquad p\in\torus^d\,. \equation(VVV) $$ By \clm(CzeroVn) the function $\HH$ is bounded and analytic on $B'(\gamma_2)$. Consider its Taylor series at zero, $$ \HH(f)=\sum_{n=0}^\infty\HH_n(f,\ldots,f)\,, \equation(VVTaylor) $$ where $\HH_n=D^n\HH(0)/n!\,$. Let $r$ be a fixed but arbitrary positive real number less than $K\rho_0\,$. Then the series \equ(VVTaylor) converges absolutely in the ball $\|f\|_{\gamma_2}\le r$, and the derivatives of $\HH$ satisfy a bound $\|\HH_n\|\le cr^{-n}$ as $n$-linear functionals on $\AA_{\gamma_2}^n$. Next, we allow $\gamma\ge\gamma_2$ but keep $\HH$ as a function on $B'(\gamma_2)$. Concerning the condition $f_0\in\MM_\gamma$ in \clm(VVV), we note that $\MM_\gamma=\MM_{\gamma_2}\cap B(\gamma)$, which follows from the definition \equ(wtBnDef) of the sets $\wt B_n\,$, and from the fact that $B(\gamma)$ is a subset of $B(\gamma_2)$. Assume now that $f$ belongs to $B'(\gamma)$ and satisfies $\|f\|_\gamma0$ sufficiently small. By using this bound to estimate the sum \equ(VzeroRep), one finds that the sum is absolutely convergent in the region $|\Im(p_j)|<\delta/2$. Thus, $V_0$ extends analytically to this region. \qed The following lemma concerns the situation described in the introduction, where $f=f_0$ is of the form \equ(gtof). These functions $f$ define a closed linear subspace $\AA_\gamma^1$ of $\AA_\gamma\,$, which can also be characterized by the identity $$ f(q+(0,r))=e^{-r\cdot J}f(q)e^{r\cdot J}\,,\qquad q\in\torus^d\,, \quad r\in\real^\ell\,. \equation(AAone) $$ \claim Lemma(ZerotoC) Let $\gamma\ge\gamma_2+1$ with $\gamma_2>\gamma_1\,$, and assume that $f_0$ belongs to $\MM_\gamma\cap\AA_\gamma^1\,$. Set $f=f_0$ and $V=V_0\,$. If $g=\Theta_\star f$, then the flow for $Y=(w,g\ldot)$ is given by equation \equ(PhiRestricted), for some $C\in\bfA$. The corresponding map $f_0\mapsto C$ is analytic. \proof The first equality in \equ(PhiRestricted) follows from \clm(CzeroVn) and the definition \equ(ThetaDef). Define $$ \phi^t(x)=V(x+tw)\Phi_\ssY^t(x)V(x)^{-1}\,, \equation(ZCOne) $$ for $t\in\real$ and $x\in\torus^m$. Notice that $\phi$ is the flow for a skew system $Z=(w,h\ldot)$ on $\torus^m\times\bfG$, and since $V\in\GG_1$ by \clm(VVV), the function $h$ belongs to $\AA_0\,$. {}From the first equality in \equ(PhiRestricted), we have $$ \phi^t(x)=V(x+tw)e^{tA}V(x+t\omega)^{-1}\,. \equation(ZCTwo) $$ Consider now an arbitrary sequence $\{t_j\}$ such that $t_j\kappa\to 0$ on the torus $\torus^\ell$, as $j\to\infty$. Then $\exp(t_j\kappa\cdot J)\to\id$. Furthermore, ${\rm dist}(t_j\omega,t_j w)\to 0$ on the torus $\torus^d$, and since $V$ is of class ${\rm C}^1$, we have $\phi^{t+t_j}(x)\to\phi^t(x)$ uniformly in $x$, if $t=0$. By the cocycle identity for the flow $\phi$, the same holds for any $t\in\real$, and the convergence is uniform in $t$. This implies (see e.g. [\rFink]) that the function $t\mapsto\phi^t(x)$ is periodic or quasiperiodic, with frequencies in $K=\{\kappa_1,\ldots,\kappa_\ell\}$. As a result, $$ h(x+tw)=\dot\phi^t(x)\phi^t(x)^{-1} \equation(ZCThree) $$ is also periodic or quasiperiodic in $t$, with frequencies in $K$. But the frequency module of $t\mapsto h(x+tw)$ is clearly a subset of $W=\{w_1,\ldots,w_m\}$, and since $W\cap K$ is empty, $h$ has to be constant. Setting $C=h$, we obtain $\phi^t(x)=e^{t C}$, and the identity \equ(PhiRestricted) now follows from \equ(ZCOne). A computation of $h(x)$ from the equations \equ(ZCThree) and \equ(ZCTwo) yields $C=VAV^{-1}-(D_\kappa V)V^{-1}$, evaluated at $x$. This identity (between matrices, if $x$ is fixed), together with \clm(VVV), shows that $C$ depends analytically on $f$. \qed In order to complete the proof of \clm(GLmain), consider now the case where $\bfG$ is a proper Lie subgroup of $\GL(n,\complex)$. By \clm(MFix), the restrictions to $\AA$ of the transformations $\NN_n$ take again values in $\AA$, and so the transformations $\bfN_n$ preserves the subspace of families taking values in $\AA$. Thus, the map $M$ described in \clm(MM) takes values in $\AA$ when restricted to $\AA$, as claimed in \clm(GLmain). Similarly, the fact that $U_\ssf\in\GG$ whenever $f=f_0\in\AA$ implies that the matrices \equ(VmnDef) belong to $\bfG$, and so the same is true for the limit $V=V_0(q)$. The same arguments apply to the case where $\bfG$ is a Lie subgroup of $\GL(n,\real)$, if we use that by \clm(znDef), the parameter values $z_0$ defining the map $M$ are all real in this case. The remaining claims of \clm(GLmain) now follow from \clm(MM), \clm(CzeroVn), and \clm(VVV). \section The special case \bigbfG=SL(2,\bigreal) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section, $\bfG$ is the group of unimodular $2\times 2$ matrices over $\real$, and $\bfA$ is the corresponding Lie algebra of real traceless $2\times 2$ matrices. As explained in the introduction, our approach to skew flows with nonzero fibered rotation number is to convert them to skew flows with zero (or near-zero) rotation number, which involves increasing the dimension of the torus. As far as renormalization is concerned, the main difficulty with this approach is that the space $\AA^1$ of functions $f$ of the form \equ(gtof) is not invariant under renormalization. Superficially, the fact that the torus-average of $f\in\AA^1$ is necessarily a constant multiple of $J$ may seem to explain the statement about one-parameter families in \clm(SLTWOmain). However, this property is neither invariant under renormalization, nor does is guarantee that the flow for $X=(\omega,f\ldot)$ remains bounded. Below we will introduce an alternative property, that is more closely linked to hyperbolicity, and invariant. First, we give a simple necessary condition for a skew system to have zero fibered rotation number. \claim Proposition(norot) If $\varrho(X)=0$ then $\tr(Jf(q))=0$ or $\det(f(q))\ge 0$, for some $q\in\torus^d$. \proof If we set $\tau(t)=\tr(Jf(q))$ and $\delta(t)=\det(f(q))$, with $q=q_0+t\omega$, then an explicit calculation shows that \equ(dotalpha) can be written as $$ 2\dot\alpha=\tau+\rho\sin(2\alpha+\beta)\,,\qquad \rho=\sqrt{\tau^2+4\delta}\,, \equation(dotalphatwo) $$ for some angle $\beta$ depending on $f(q)$ and on $u_0\,$. If $\tr(Jf)\not=0$ and $\det(f)<0$ on all of $\torus^d$, then $|\tau|>\rho+\eps$ for some $\eps>0$, and thus $\dot\alpha$ is bounded away from zero, implying that $\varrho(X)\not=0$. \qed One of our goals is to show that a vector field $X=(\omega,f\ldot)$ with $f\in\AA^1$ close to zero cannot generate a hyperbolic flow, by excluding the possibility that the renormalized functions $f_n$ have the following property. \claim Definition(cones) Let $S^1$ be the set of unit vectors in $\real^2$. We say that a vector field $X=(\omega,f\ldot)$ has the {\it expanding cone property} if for every $q\in\torus^d$, there exists an open cone $\CC(q)$ in $\real^2$ not intersecting its negative, with vertex at zero, and a unit vector $u(q)$ in this cone, such that the following holds. The map $q\mapsto S^1\cap\CC(q)$ defines two continuous functions from $\torus^d$ to $S^1$. The function $q\mapsto u(q)$ is continuous as well, and homotopic to a constant. Furthermore, for every $q\in\torus^d$, the cone $\Phi_\ssX^t(q)\CC(q)$ is contained in $\CC(q+t\omega)$ for all $t>0$, and the length of $\Phi_\ssX^t(q)u(q)$ tends to infinity as $t\to\infty$. We note that the expanding cone property is invariant under coordinate changes of the form \equ(TTDef) or \equ(VVDef), with $V$ continuous and homotopic to the identity. A simple condition that implies this property is given in the following proposition. \claim Proposition(symmC) Assume that $f:\torus^d\to\bfA$ is continuous and of the form $f=C+h$, with $C\in\bfA$ symmetric and $\|h(q)\|<\|C\|/4$ for all $q\in\torus^d$. Then $X=(\omega,f\ldot)$ has the expanding cone property. \proof Our assumptions imply that the eigenvalues of $C$ are $\pm\|C\|$. Let $u_0$ be a unit eigenvector of $C$ for the eigenvalue $\|C\|$, and define $\CC_0$ to be the set of all nonzero vectors in $\real^2$ whose angle with $u_0$ is less than $\pi/4$. Consider first the case $f\equiv C$. Then for every nonzero $v$ on the boundary of $\CC_0\,$, the vector $fv$ points to the interior of the cone $\CC_0\,$. Thus, the solutions of equation \equ(vdot), with initial condition $v_0$ in $\CC_0\,$, remains in $\CC_0$ for all times $t>0$. A straightforward computation shows that under the given assumptions of $h$, the same remains true for $f=C+h$. Thus, $X$ has the expanding cone property, with the family of cones being $q\mapsto\CC_0\,$, and with $u(q)=u_0$ for all $q$. Notice that no condition on $\omega$ is needed. \qed \claim Lemma(noexpansion) If $f$ belongs to $\AA^1$ then $X=(\omega,f\ldot)$ cannot have the expanding cone property. \proof Consider first an arbitrary $f\in\AA$ such that $X=(\omega,f\ldot)$ has the expanding cone property. Let $q\in\torus^d$ be fixed. Using the notation of \clm(cones), denote by $A(q)$ the set of all nonzero $v_0\in\real^2$ such that $v(t)=\Phi_\ssX^t(q)v_0$ belongs to $\CC(q+tw)$ for some (and thus each sufficiently large) positive $t$. This set is clearly open. Notice that if $v_0$ is any nonzero vector in $\real^2$, with the property that $v(t)=\Phi_\ssX^t(q)v_0$ tends to infinity as $t\to\infty$, then $v_0$ belongs to either $A(q)$ or $-A(q)$. This follows from the fact that $\Phi_\ssX^t(q)$ is area-preserving (so the angle between $v(t)$ and $\Phi_\ssX^t(q)u(q)$ has to approach zero), and that the opening angles of our cones are bounded away from zero. Thus, given that the two disjoint open sets $\pm A(q)$ cannot cover all of $\real^2\setminus\{0\}$, it is not possible that $|v(t)|\to\infty$ as $t\to\infty$, for every nonzero $v_0\in\real^2$. Assume now for contradiction that $f$ belongs to $\AA^1$. Define $z_r(x)=e^{rJ}u(q)$, with $u$ as described in \clm(cones). Then $\Phi_\ssY^t(x)z_r(x)=e^{(r+t\kappa)J}\Phi_\ssX^t(q)u(q)$ tends to infinity as $t\to\infty$. But as $r$ increases from $0$ to $2\pi$, the vectors $z_r(x)$ cover all of $S^1$, since $u$ is homotopic to a constant function. This implies that $\Phi_\ssY^t(x)v_0$ tends to infinity (in length) for each nonzero $v_0\in\real^2$, which was shown above to be impossible. \qed Now we are ready to renormalize. Denote by $\bfJ$ the one-dimensional subspace of $\bfA$, consisting of real multiples of the matrix $J$. \claim Lemma(AzeroFam) Let $h\in\AA^1\cap B'(\gamma)$, and define $F(s)=h+s$ for $s\in b_0\,$. Then the (unique) value $s=z_0(F)$ where the family $F$ intersects $\MM_\gamma$ belongs to $\bfJ$, and it is the unique matrix in $b_0\cap\bfJ$ for which $F(s)$ has a zero fibered rotation number. \proof Recall that $z_0=z_0(F)$ is real, by \clm(znDef). Assume for contradiction that $z_0$ does not belong to $\bfJ$. Then for sufficiently large $m$, the sets $Z_{0,m}(b_m/3)$ have an empty intersection with $\bfJ$. Denote by $n$ the smallest value of $m$ for which this intersection is empty, and define $\bfJ_k=Z_{0,n-1}(b_{n-1}/k)\cap\bfJ$. Let $r=\|\eta_n^{-1}\id_{n,n-1}\|$. The bound \equ(DYnBound) shows that the image under $Y_n$ of ${1\over 3}b_{n-1}$ is contained in ${r\over 2}b_n\,$, and that the image of $b_{n-1}$ contains ${2r\over 3}b_n\,$. The first property implies that $Z_n'(\bfJ_3)$ intersects ${r\over 2}b_n$ at some point $s$ outside ${1\over 3}b_n$. Now consider the connected component of $Z_n'(\bfJ_1)$ containing $s$. By \clm(DZnPrime), this curve is sufficiently ``parallel'' to $\bfJ$ in order to intersect the subspace $\tr(J^\ast s)=0$ at some point $s_n=Z_n'(s_0)$ that lies inside ${2r\over 3}b_n\,$, but outside ${1\over 4}b_n$. The matrix $s_n$ is symmetric with norm $\ge \rho_n/4$, and by \clm(Contraction), we have $\|F_n(s_n)-s_n\|<\rho_n/16$. Thus, by \clm(symmC), the vector field for $F_n(s_n)$ has the expanding cone property. Given that this property is invariant under coordinate changes of the form \equ(TTDef) or \equ(VVDef), with $V$ continuous and homotopic to the identity, $F(s_0)$ has the same property. But since $s_0\in\bfJ$, the function $F(s_0)$ belongs to $\AA^1$, and we get a contradiction with \clm(noexpansion). This shows that $z_0$ belongs to $\bfJ$. \clm(CzeroVn) shows that $\varrho(F(z_0))=0$. Consider now $s_0\in b_0\cap\bfJ$ different from $z_0\,$. Then there exists $n>0$ such that $s_0$ lies in $Z_{0,m}(b_m)$ for all $m2\|z_n\|$. Denote by $B$ and $C$ the symmetric and antisymmetric parts of $s_n-z_n\,$, respectively. By \clm(DZnPrime), we have $\|C\|>10\|B\|$. In addition, $\|F_n(s_n)-s_n\|<\rho_n/16$ by \clm(Contraction). As a result, $\|C\|>\|F_n(s_n)-C\|$, which by \clm(norot) implies that $F_n(s_n)$ cannot have a vanishing fibered rotation number. Thus, we cannot have $\varrho(F(s_0))=0$, since this property is preserved under renormalization. \qed \proofof(SLTWOmain) We can follow the sketch given after the statement of this theorem. A straightforward computation shows that $\|L\|$ and $\|L^{-1}\|$ can be bounded by $2\|\kappa^{-1}A\|^{1/2}$ Thus, the indicated map $g\mapsto f=(\Theta_\star)^{-1}(cL^{-1}gL)$, with $c=\|\omega\|^{-1}$, admits the bound $$ \|f\|\le 2^{2\gamma}4c\cdot 4\|\kappa^{-1}A\|\|g-A\| \le 2^{2\gamma} 16 a\|g-A\|\,. \equation(ThetaInv) $$ This shows that the image $B_0$ under $f\mapsto g$, of the domain $B=B(\gamma)$ for which \clm(GLmain) holds, contains a ball of radius $R=K r_0/(2^{2\gamma}16a)$, centered at the constant function $A$. The remaining claims of \clm(SLTWOmain) are now an immediate consequence of \clm(GLmain), \clm(AzeroFam), and \clm(ZerotoC). \qed \bigskip \references %%%%%%%%%%% {\ninepoint %% skew systems \item{[\rDS]} E.I.~Dinaburg, Ja.~G.~Sinai, {\it The one-dimensional Schr\"odinger equation with quasiperiodic potential}, Funkcional. 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