INSTRUCTIONS




 The text between the lines BODY and ENDBODY is made  of


  2384 lines and    85250 bytes (not counting <line feed> or <carriage return>)



 In the following table this count is broken down by ASCII  code;
 immediately following the code is the corresponding  character.

     49192  lowercase letters
      3033  uppercase letters
      3125  digits
        10  ASCII characters   9

      8769  ASCII characters  32
        41  ASCII characters  33 !
       279  ASCII characters  34 "
       110  ASCII characters  35 #
      2190  ASCII characters  36 $
       880  ASCII characters  37 %
        64  ASCII characters  38 &
       170  ASCII characters  39 '
       939  ASCII characters  40 (
       944  ASCII characters  41 )
        17  ASCII characters  42 *
       290  ASCII characters  43 +
       796  ASCII characters  44 ,
       590  ASCII characters  45 -
       800  ASCII characters  46 .
        31  ASCII characters  47 /
       169  ASCII characters  58 :
        61  ASCII characters  59 ;
        75  ASCII characters  60 <
       728  ASCII characters  61 =
        37  ASCII characters  62 >
         9  ASCII characters  63 ?
       278  ASCII characters  64 @
       147  ASCII characters  91 [
      6219  ASCII characters  92 \
       142  ASCII characters  93 ]
       650  ASCII characters  94 ^
      1052  ASCII characters  95 _
       112  ASCII characters  96 `
      1419  ASCII characters 123 {
       462  ASCII characters 124 |
      1420  ASCII characters 125 }

 BODY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%   This is a Plain TeX File. No special care is needed.
%%%   (The program will give a harmless "underfull" to be ignored.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\magnification=\magstep1\hoffset=0.cm
\lineskip=4pt\lineskiplimit=0.1pt
%%%%%%%
\tolerance=1600
\hfuzz=1pt
\vsize=23.truecm
\voffset=0.truecm
\hsize=15.8 truecm
\hoffset=0.4 truecm
\normalbaselineskip=5.25mm
%\baselineskip=5.25mm
\baselineskip=14pt plus0.1pt minus0.1pt \parindent=19pt
\parskip=0.1pt plus1pt
\font\titlefont=cmbx10 scaled\magstep1
\font\sectionfont=cmbx10 scaled\magstep1
\font\subsectionfont=cmbx10
\font\small=cmr7
%%%%%%%%                   FONTS %%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%define sym
\font\tenmsy=msym10
\font\sevenmsy=msym7
\font\fivemsy=msym5
\newfam\msyfam
\font\ninerm=cmr9
\font\ninei=cmmi9
\font\ninesy=cmsy9
\font\ninebf=cmbx9
\font\ninett=cmtt9
\font\ninesl=cmsl9
\font\nineit=cmti9
\font\eightrm=cmr8
\font\eighti=cmmi8
\font\eightsy=cmsy8
\font\eightbf=cmbx8
\font\eighttt=cmtt8
\font\eightsl=cmsl8
\font\eightit=cmti8
\font\sixrm=cmr6
\font\sixbf=cmbx6
\font\sixi=cmmi6
\font\sixsy=cmsy6
\def \eightpoint{\def\rm{\fam0\eightrm}% switch to 8-point type
\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
\setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}%
\normalbaselineskip=9pt
\let\sc=\sixrm  \let\big=\eightbig  \normalbaselines\rm
}
%
%%%%% constant subscript positions %%%%%
%
\fontdimen16\tensy=2.7pt
%\fontdimen13\tensy=2.7pt
\fontdimen13\tensy=4.3pt
\fontdimen17\tensy=2.7pt
\fontdimen14\tensy=4.3pt
\fontdimen18\tensy=4.3pt
\fontdimen16\eightsy=2.7pt
\fontdimen13\eightsy=4.3pt
\fontdimen17\eightsy=2.7pt
\fontdimen14\eightsy=4.3pt
\fontdimen18\eightsy=4.3pt
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\global\newcount\numsec\global\newcount\numfor
\gdef\profonditastruttura{\dp\strutbox}
\def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax}
\def\SIA #1,#2,#3 {\senondefinito{#1#2}
\expandafter\xdef\csname #1#2\endcsname{#3} \else
%\write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi}
\write-1{???? ma #1,#2 e' gia' stato definito !!!!} \fi}
\def\etichetta(#1){(\veroparagrafo.\veraformula)
\SIA e,#1,(\veroparagrafo.\veraformula)
 \global\advance\numfor by 1
% \write15{\string\FU (#1){\equ(#1)}}
% \write16{ EQ \equ(#1) == #1  }}
 \write-1{ EQ \equ(#1) == #1  }}
\def \FU(#1)#2{\SIA fu,#1,#2 }
\def\etichettaa(#1){(A\veroparagrafo.\veraformula)
 \SIA e,#1,(A\veroparagrafo.\veraformula)
 \global\advance\numfor by 1
% \write15{\string\FU (#1){\equ(#1)}}
% \write16{ EQ \equ(#1) == #1  }}
 \write-1{ EQ \equ(#1) == #1  }}
\def\BOZZA{\def\alato(##1){
 {\vtop to \profonditastruttura{\baselineskip
 \profonditastruttura\vss
 \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}}}
\def\alato(#1){}
\def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor}
\def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}}
\def\eq(#1){\etichetta(#1)\alato(#1)}
\def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}}
\def\eqa(#1){\etichettaa(#1)\alato(#1)}
\def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\write-1{No translation for #1}%
\else\csname fu#1\endcsname\fi}
%\def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\else\csname fu#1\endcsname\fi}
\def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi}
%%%%%%%%%%%%%%%%%%
%\openin14=d.aux \ifeof14 \relax \else
%\input d.aux \fi
%\openout15=d.aux
%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%GRAFICA
\def\ZzZ{\hbox{\vrule height0.4pt width0.4pt depth0.pt}}\newdimen\u
\def\pp #1 #2 {\rlap{\kern#1\u\raise#2\u\ZzZ}}
\def\hhh{\rlap{\hbox{{\vrule height1.cm width0.pt depth1.cm}}}}
\def\ins #1 #2 #3 {\rlap{\kern#1\u\raise#2\u\hbox{$#3$}}}
\def\alt#1#2{\rlap{\hbox{{\vrule height#1truecm width0.pt depth#2truecm}}}}
\def\ds{\displaystyle}\def\st{\scriptstyle}\def\sst{\scriptscriptstyle}
\def\pallina{{\kern-0.4mm\raise-0.02cm\hbox{$\scriptscriptstyle\bullet$}}}
\def\palla{{\kern-0.6mm\raise-0.04cm\hbox{$\textstyle\bullet$}}}
\def\pallona{{\kern-0.7mm\raise-0.06cm\hbox{$\displaystyle\bullet$}}}
\newcount\page
\def\figurainizio{\page=\pageno\nopagenumbers\special{xop=b}}
\def\figurafine{\pageno=\page\special{xop=b}
\footline={\hss\tenrm\folio\hss}\pageno=\page}
%%%%%%%%%%%%%%%%%%%%%%
%
\catcode`@=11
%
%
%    Simboli di minore o circa uguale, maggiore o circa uguale.
%
\def\lsim{\mathchoice
  {\mathrel{\lower.8ex\hbox{$\displaystyle\buildrel<\over\sim$}}}
  {\mathrel{\lower.8ex\hbox{$\textstyle\buildrel<\over\sim$}}}
  {\mathrel{\lower.8ex\hbox{$\scriptstyle\buildrel<\over\sim$}}}
  {\mathrel{\lower.8ex\hbox{$\scriptscriptstyle\buildrel<\over\sim$}}} }
\def\gsim{\mathchoice
  {\mathrel{\lower.8ex\hbox{$\displaystyle\buildrel>\over\sim$}}}
  {\mathrel{\lower.8ex\hbox{$\textstyle\buildrel>\over\sim$}}}
  {\mathrel{\lower.8ex\hbox{$\scriptstyle\buildrel>\over\sim$}}}
  {\mathrel{\lower.8ex\hbox{$\scriptscriptstyle\buildrel>\over\sim$}}} }
%
%
%
%
%\def\quad@rato#1#2{{\vcenter{\vbox{
%        \hrule height#2pt
%        \hbox{\vrule width#2pt height#1pt \kern#1pt \vrule width#2pt}
%        \hrule height#2pt} }}}
%\def\quadratello{\mathchoice
%\quad@rato5{.5}\quad@rato5{.5}\quad@rato{3.5}{.35}\quad@rato{2.5}{.25} }
%
%
\font\s@=cmss10\font\s@b=cmss8
\def\reali{{\hbox{\s@ l\kern-.5mm R}}}
\def\naturali{{\hbox{\s@ l\kern-.5mm N}}}
\def\interi{{\mathchoice
 {\hbox{\s@ Z\kern-1.5mm Z}}
 {\hbox{\s@ Z\kern-1.5mm Z}}
 {\hbox{{\s@b Z\kern-1.2mm Z}}}
 {\hbox{{\s@b Z\kern-1.2mm Z}}}  }}
\def\complessi{{\hbox{\s@ C\kern-1.4mm\raise.2mm\hbox{\s@b l}\kern.8mm}}}
\def\toro{{\hbox{\s@ T\kern-1.9mm T}}}
\def\unita{{\hbox{\s@ 1\kern-.8mm l}}}
%
%
\font\bold@mit=cmmib10
\def\setbmit{\textfont1=\bold@mit}
\def\bmit#1{\hbox{\textfont1=\bold@mit$#1$}}
%
\catcode`@=12
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Subj: tex-style/mssymb.tex
%
%               *****     MSSYMB.TeX    *****                  31 Mar 88
%
%       This file contains the definitions for the symbols in the two
%       "extra symbols" fonts created at the American Math. Society.

\catcode`\@=11

\font\tenmsx=msxm10
\font\sevenmsx=msxm7
\font\fivemsx=msxm5
\font\tenmsy=msym10
\font\sevenmsy=msym7
\font\fivemsy=msym5
\newfam\msxfam
\newfam\msyfam
\textfont\msxfam=\tenmsx  \scriptfont\msxfam=\sevenmsx
  \scriptscriptfont\msxfam=\fivemsx
\textfont\msyfam=\tenmsy  \scriptfont\msyfam=\sevenmsy
  \scriptscriptfont\msyfam=\fivemsy

\def\hexnumber@#1{\ifcase#1 0\or1\or2\or3\or4\or5\or6\or7\or8\or9\or
        A\or B\or C\or D\or E\or F\fi }

\def\relaxnext@{\let\next\relax}
\def\noaccents@{\def\accentfam@{0}}

%  The following 13 lines establish the use of the Euler Fraktur font.
%  To use this font, remove % from beginning of these lines.
%\font\teneuf=eufm10
%\font\seveneuf=eufm7
%\font\fiveeuf=eufm5
%\newfam\euffam
%\textfont\euffam=\teneuf
%\scriptfont\euffam=\seveneuf
%\scriptscriptfont\euffam=\fiveeuf
%\def\frak{\relaxnext@\ifmmode\let\next\frak@\else
% \def\next{\Err@{Use \string\frak\space only in math mode}}\fi\next}
%\def\goth{\relaxnext@\ifmmode\let\next\frak@\else
% \def\next{\Err@{Use \string\goth\space only in math mode}}\fi\next}
%\def\frak@#1{{\frak@@{#1}}}
%\def\frak@@#1{\noaccents@\fam\euffam#1}
%  End definition of Euler Fraktur font.

\edef\msx@{\hexnumber@\msxfam}
\edef\msy@{\hexnumber@\msyfam}

\mathchardef\boxdot="2\msx@00
\mathchardef\boxplus="2\msx@01
\mathchardef\boxtimes="2\msx@02
\mathchardef\square="0\msx@03
\mathchardef\blacksquare="0\msx@04
\mathchardef\centerdot="2\msx@05
\mathchardef\lozenge="0\msx@06
\mathchardef\blacklozenge="0\msx@07
\mathchardef\circlearrowright="3\msx@08
\mathchardef\circlearrowleft="3\msx@09
\mathchardef\rightleftharpoons="3\msx@0A
\mathchardef\leftrightharpoons="3\msx@0B
\mathchardef\boxminus="2\msx@0C
\mathchardef\Vdash="3\msx@0D
\mathchardef\Vvdash="3\msx@0E
\mathchardef\vDash="3\msx@0F
\mathchardef\twoheadrightarrow="3\msx@10
\mathchardef\twoheadleftarrow="3\msx@11
\mathchardef\leftleftarrows="3\msx@12
\mathchardef\rightrightarrows="3\msx@13
\mathchardef\upuparrows="3\msx@14
\mathchardef\downdownarrows="3\msx@15
\mathchardef\upharpoonright="3\msx@16
\let\restriction=\upharpoonright
\mathchardef\downharpoonright="3\msx@17
\mathchardef\upharpoonleft="3\msx@18
\mathchardef\downharpoonleft="3\msx@19
\mathchardef\rightarrowtail="3\msx@1A
\mathchardef\leftarrowtail="3\msx@1B
\mathchardef\leftrightarrows="3\msx@1C
\mathchardef\rightleftarrows="3\msx@1D
\mathchardef\Lsh="3\msx@1E
\mathchardef\Rsh="3\msx@1F
\mathchardef\rightsquigarrow="3\msx@20
\mathchardef\leftrightsquigarrow="3\msx@21
\mathchardef\looparrowleft="3\msx@22
\mathchardef\looparrowright="3\msx@23
\mathchardef\circeq="3\msx@24
\mathchardef\succsim="3\msx@25
\mathchardef\gtrsim="3\msx@26
\mathchardef\gtrapprox="3\msx@27
\mathchardef\multimap="3\msx@28
\mathchardef\therefore="3\msx@29
\mathchardef\because="3\msx@2A
\mathchardef\doteqdot="3\msx@2B
\let\Doteq=\doteqdot
\mathchardef\triangleq="3\msx@2C
\mathchardef\precsim="3\msx@2D
\mathchardef\lesssim="3\msx@2E
\mathchardef\lessapprox="3\msx@2F
\mathchardef\eqslantless="3\msx@30
\mathchardef\eqslantgtr="3\msx@31
\mathchardef\curlyeqprec="3\msx@32
\mathchardef\curlyeqsucc="3\msx@33
\mathchardef\preccurlyeq="3\msx@34
\mathchardef\leqq="3\msx@35
\mathchardef\leqslant="3\msx@36
\mathchardef\lessgtr="3\msx@37
\mathchardef\backprime="0\msx@38
\mathchardef\risingdotseq="3\msx@3A
\mathchardef\fallingdotseq="3\msx@3B
\mathchardef\succcurlyeq="3\msx@3C
\mathchardef\geqq="3\msx@3D
\mathchardef\geqslant="3\msx@3E
\mathchardef\gtrless="3\msx@3F
\mathchardef\sqsubset="3\msx@40
\mathchardef\sqsupset="3\msx@41
\mathchardef\vartriangleright="3\msx@42
\mathchardef\vartriangleleft="3\msx@43
\mathchardef\trianglerighteq="3\msx@44
\mathchardef\trianglelefteq="3\msx@45
\mathchardef\bigstar="0\msx@46
\mathchardef\between="3\msx@47
\mathchardef\blacktriangledown="0\msx@48
\mathchardef\blacktriangleright="3\msx@49
\mathchardef\blacktriangleleft="3\msx@4A
\mathchardef\vartriangle="0\msx@4D
\mathchardef\blacktriangle="0\msx@4E
\mathchardef\triangledown="0\msx@4F
\mathchardef\eqcirc="3\msx@50
\mathchardef\lesseqgtr="3\msx@51
\mathchardef\gtreqless="3\msx@52
\mathchardef\lesseqqgtr="3\msx@53
\mathchardef\gtreqqless="3\msx@54
\mathchardef\Rrightarrow="3\msx@56
\mathchardef\Lleftarrow="3\msx@57
\mathchardef\veebar="2\msx@59
\mathchardef\barwedge="2\msx@5A
\mathchardef\doublebarwedge="2\msx@5B
\mathchardef\angle="0\msx@5C
\mathchardef\measuredangle="0\msx@5D
\mathchardef\sphericalangle="0\msx@5E
\mathchardef\varpropto="3\msx@5F
\mathchardef\smallsmile="3\msx@60
\mathchardef\smallfrown="3\msx@61
\mathchardef\Subset="3\msx@62
\mathchardef\Supset="3\msx@63
\mathchardef\Cup="2\msx@64
\let\doublecup=\Cup
\mathchardef\Cap="2\msx@65
\let\doublecap=\Cap
\mathchardef\curlywedge="2\msx@66
\mathchardef\curlyvee="2\msx@67
\mathchardef\leftthreetimes="2\msx@68
\mathchardef\rightthreetimes="2\msx@69
\mathchardef\subseteqq="3\msx@6A
\mathchardef\supseteqq="3\msx@6B
\mathchardef\bumpeq="3\msx@6C
\mathchardef\Bumpeq="3\msx@6D
\mathchardef\lll="3\msx@6E
\let\llless=\lll
\mathchardef\ggg="3\msx@6F
\let\gggtr=\ggg
\mathchardef\circledS="0\msx@73
\mathchardef\pitchfork="3\msx@74
\mathchardef\dotplus="2\msx@75
\mathchardef\backsim="3\msx@76
\mathchardef\backsimeq="3\msx@77
\mathchardef\complement="0\msx@7B
\mathchardef\intercal="2\msx@7C
\mathchardef\circledcirc="2\msx@7D
\mathchardef\circledast="2\msx@7E
\mathchardef\circleddash="2\msx@7F
\def\ulcorner{\delimiter"4\msx@70\msx@70 }
\def\urcorner{\delimiter"5\msx@71\msx@71 }
\def\llcorner{\delimiter"4\msx@78\msx@78 }
\def\lrcorner{\delimiter"5\msx@79\msx@79 }
\def\yen{\mathhexbox\msx@55 }
\def\checkmark{\mathhexbox\msx@58 }
\def\circledR{\mathhexbox\msx@72 }
\def\maltese{\mathhexbox\msx@7A }
\mathchardef\lvertneqq="3\msy@00
\mathchardef\gvertneqq="3\msy@01
\mathchardef\nleq="3\msy@02
\mathchardef\ngeq="3\msy@03
\mathchardef\nless="3\msy@04
\mathchardef\ngtr="3\msy@05
\mathchardef\nprec="3\msy@06
\mathchardef\nsucc="3\msy@07
\mathchardef\lneqq="3\msy@08
\mathchardef\gneqq="3\msy@09
\mathchardef\nleqslant="3\msy@0A
\mathchardef\ngeqslant="3\msy@0B
\mathchardef\lneq="3\msy@0C
\mathchardef\gneq="3\msy@0D
\mathchardef\npreceq="3\msy@0E
\mathchardef\nsucceq="3\msy@0F
\mathchardef\precnsim="3\msy@10
\mathchardef\succnsim="3\msy@11
\mathchardef\lnsim="3\msy@12
\mathchardef\gnsim="3\msy@13
\mathchardef\nleqq="3\msy@14
\mathchardef\ngeqq="3\msy@15
\mathchardef\precneqq="3\msy@16
\mathchardef\succneqq="3\msy@17
\mathchardef\precnapprox="3\msy@18
\mathchardef\succnapprox="3\msy@19
\mathchardef\lnapprox="3\msy@1A
\mathchardef\gnapprox="3\msy@1B
\mathchardef\nsim="3\msy@1C
%\mathchardef\napprox="3\msy@1D
\mathchardef\ncong="3\msy@1D
\def\napprox{\not\approx}
\mathchardef\varsubsetneq="3\msy@20
\mathchardef\varsupsetneq="3\msy@21
\mathchardef\nsubseteqq="3\msy@22
\mathchardef\nsupseteqq="3\msy@23
\mathchardef\subsetneqq="3\msy@24
\mathchardef\supsetneqq="3\msy@25
\mathchardef\varsubsetneqq="3\msy@26
\mathchardef\varsupsetneqq="3\msy@27
\mathchardef\subsetneq="3\msy@28
\mathchardef\supsetneq="3\msy@29
\mathchardef\nsubseteq="3\msy@2A
\mathchardef\nsupseteq="3\msy@2B
\mathchardef\nparallel="3\msy@2C
\mathchardef\nmid="3\msy@2D
\mathchardef\nshortmid="3\msy@2E
\mathchardef\nshortparallel="3\msy@2F
\mathchardef\nvdash="3\msy@30
\mathchardef\nVdash="3\msy@31
\mathchardef\nvDash="3\msy@32
\mathchardef\nVDash="3\msy@33
\mathchardef\ntrianglerighteq="3\msy@34
\mathchardef\ntrianglelefteq="3\msy@35
\mathchardef\ntriangleleft="3\msy@36
\mathchardef\ntriangleright="3\msy@37
\mathchardef\nleftarrow="3\msy@38
\mathchardef\nrightarrow="3\msy@39
\mathchardef\nLeftarrow="3\msy@3A
\mathchardef\nRightarrow="3\msy@3B
\mathchardef\nLeftrightarrow="3\msy@3C
\mathchardef\nleftrightarrow="3\msy@3D
\mathchardef\divideontimes="2\msy@3E
\mathchardef\varnothing="0\msy@3F
\mathchardef\nexists="0\msy@40
\mathchardef\mho="0\msy@66
\mathchardef\eth="0\msy@67
\mathchardef\eqsim="3\msy@68
\mathchardef\beth="0\msy@69
\mathchardef\gimel="0\msy@6A
\mathchardef\daleth="0\msy@6B
\mathchardef\lessdot="3\msy@6C
\mathchardef\gtrdot="3\msy@6D
\mathchardef\ltimes="2\msy@6E
\mathchardef\rtimes="2\msy@6F
\mathchardef\shortmid="3\msy@70
\mathchardef\shortparallel="3\msy@71
\mathchardef\smallsetminus="2\msy@72
\mathchardef\thicksim="3\msy@73
\mathchardef\thickapprox="3\msy@74
\mathchardef\approxeq="3\msy@75
\mathchardef\succapprox="3\msy@76
\mathchardef\precapprox="3\msy@77
\mathchardef\curvearrowleft="3\msy@78
\mathchardef\curvearrowright="3\msy@79
\mathchardef\digamma="0\msy@7A
\mathchardef\varkappa="0\msy@7B
\mathchardef\hslash="0\msy@7D
\mathchardef\hbar="0\msy@7E
\mathchardef\backepsilon="3\msy@7F
% Use the next 4 lines with AMS-TeX:
%\def\Bbb{\relaxnext@\ifmmode\let\next\Bbb@\else
% \def\next{\Err@{Use \string\Bbb\space only in math mode}}\fi\next}
%\def\Bbb@#1{{\Bbb@@{#1}}}
%\def\Bbb@@#1{\noaccents@\fam\msyfam#1}
% Use the next 4 lines if NOT using AMS-TeX:
\def\Bbb{\ifmmode\let\next\Bbb@\else
 \def\next{\errmessage{Use \string\Bbb\space only in math mode}}\fi\next}
\def\Bbb@#1{{\Bbb@@{#1}}}
\def\Bbb@@#1{\fam\msyfam#1}

\catcode`\@=12
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\HB {\hfill\break}
\def\HALF{{\textstyle{1\over 2}}}
\def\half{{1\over 2}}
\def\lis{\overline}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\def\qed{\vrule width 1.7truemm height 3.5truemm depth 0.truemm}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%          Text of the Compuscript follows
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\titlefont
Second Order Hamiltonian Equations on $\T^\io$}

\centerline{\titlefont and}

\centerline{\titlefont  Almost--Periodic Solutions}

\vskip1.5truecm
\centerline{\subsectionfont
Luigi Chierchia$^*$
and Paolo Perfetti$^{**}$}

\vskip.5truecm
\centerline{\ninerm ${}^*$
Dipartimento di Matematica, Universit\`a di Genova, via L. B. Alberti 4,
16132 Genova, and}
\centerline{\ninerm Centro ``Vito Volterra",
II Universit\`a di Roma
``Tor Vergata", 00173 Roma}
\centerline{ \ninerm ${}^{**}$ Dipartimento di Matematica,
II Universit\`a di Roma
``Tor Vergata", 00173 Roma}
\vskip1.5truecm\noindent
{\subsectionfont Abstract}\quad
{\ninerm Motivated by problems arising in
nonlinear PDE's with a Hamiltonian structure and in high dimensional
dynamical systems, we study a suitable generalization to infinite
dimensions of second order Hamiltonian equations of the type
$\ddot x=\dpr_x V$, [$x\in\TN$, $\dpr_x\=(\dpr_{x_1},...,
\dpr_{x_N})$].
Extending methods from quantitative perturbation theory
(Kolmogorov--Arnold--Moser theory, Nash--Moser implicit function
theorem,
\etc) we construct uncountably many almost--periodic solutions  for the
infinite dimensional system $\ddot x_i=f_i(x)$, $i\in \Zd$, $x\in
\T^\Zd$ (endowed with the compact topology); the Hamiltonian structure
is reflected by $f$ being a ``generalized gradient". Such result is
derived under (suitable) analyticity assumptions on $f_i$ but without
requiring any ``smallness conditions".}

\vsk
\vsk
{\small
{\bf Table of Contents}
\vsk
\nin
1. {Introduction\quad\dotfill\quad 2}

\nin
2. {Second Order ODE's on $\T^\io$\quad\dotfill\quad 5}

\nin
3. {Hamiltonian Structure and Averages\quad\dotfill\quad 7}

\nin
4. {Almost--Periodic Solutions (Definitions)\quad\dotfill\quad 9}

\nin
5. {Diophantine Sequences\quad\dotfill\quad 10}

\nin
6. {Main Results\quad\dotfill\quad 12}

\nin
7. {A (Finite--Dimensional) Average Theorem\quad\dotfill\quad 14}

\nin
8. {Proof of Existence of Almost--Periodic Solutions\quad\dotfill\quad 19}

\nin
Appendix 1. {Full Measure of Diophantine vectors in $\RN$\quad\dotfill\quad
23}

\nin
Appendix 2. {A Short KAM Theory\quad\dotfill\quad 23}

\nin
References {\quad\dotfill\quad 29}

\nin
}
\pagina
\vglue2.truecm\penalty-200

\nin{\sectionfont 1. Introduction}

\penalty10000
\vskip0.5truecm
\numsec=1
\numfor=1
\penalty10000

\nin
A natural approach to qualitative theory of  nonlinear partial
differential equations
``with a Hamiltonian structure", is to regard
such PDE's as infinite dimensional conservative dynamical systems and to
try to extend, whenever possible,  results and methods from the well
developed finite dimensional theory.

One of the basic results in finite dimensions is the existence, under
suitable assumptions, of maximal quasi--periodic solutions (see [A]
and references therein). These are solutions, which, up to a change of
coordinates, are described by a linear flow: $t\to\o t$, with
$\o\in\RN$, $N\=$ number of degrees of freedom $\=$ $1/2$ dimension of
the phase space (associated to the Hamiltonian system under
consideration).

Some generalizations to infinite dimensions of
the existence of quasi--periodic solutions have
been studied by several authors; see, \eg, [FSW], [VB], [Wa2],
[P\"o], [Ku], [GW]. All these papers make use of quite stringent
``smallness assumptions" (a drawback already present in finite
dimensions).

\vsk
In this paper we study the following infinite dimensional
``second order Hamiltonian" system of equations on
$\T^\io$, ($\T\=\R/2\p \Z$):
$$
\ddot x_i=f_i(x)\ ,\qquad i\in \Zd\ ,\qquad x\in\T^\Zd\=\TT
\Eq(1.1)$$
where $\TT$, equipped with the weak topology, is regarded as an
infinite dimensional (analytic) manifold and $f$ is a (uniformly)
Lipschitz map from $\TT$ to its tangent space; the Hamiltonian structure
comes from $f$ being a {\it generalized gradient} (see below).

For such systems we will construct uncountably many almost--periodic
solutions under suitably analyticity assumptions on $f$ but {\it
without requiring any smallness condition}. Such solutions will have the
form:
$$
t\to x(t)\=\{x_i(t)\}_{i\in \Zd}\ ,\quad{\rm with}\quad x_i(t)=
[\o_i t + u_i([\o t])]
\Eq(1.2)$$
where $[\cdot]$ denotes the standard projection of $\R$ onto $\T$,
$u$ is a smooth function, and
$\o\in\R^\Zd$ is a ``Diophantine sequence" [ \ie $\exists$ $\g>0$
and a isomorphism, $i_k$, from $\Z_+$ onto $\Zd$ such that,
$\forall$ $N\ge 1$,
$|\sum_{k=1}^N \o_{i_k} n_{i_k}|^{-1}\le \g (\sum_{k=1}^N|n_{i_k}|)^N$,
with $n_{i_k}\in \Z$, $\sum_{k=1}^N|n_{i_k}|>0$ ].
Notice that, as a consequence of a classical
number theoretical theorem  by Liouville, $|\o_i|\to\io$ as
$|i|\to\io$.

For the construction of quasi--periodic solutions for the finite
dimensional situation, namely:
$$
\ddot x=V_x(x)\ ,\quad\qquad x\in\TN
\Eq(1.3)$$
we refer the reader to [A] (and references therein), [M1], [M2],
[SZ], [CC1], [CC2] and especially [CZ].

Our interest in a qualitative analysis of \equ(1.1) has been motivated
mainly by: (i) regular motions for nearly--integrable PDE's with a
Hamiltonian structure;
(ii) discrete approximations of, say, nonlinear
wave equations in $\R^d$ (\eg Sine--Gordon); (iii) many particle systems
interacting via conservative forces.

\vsk
(i) Important examples of nonlinear PDE's, such as the
Korteweg--de--Vries equation and the non--linear Schr\"odinger
equation, fall in the class of (infinite dimensional) {\it integrable
Hamiltonian systems} (see, \eg, [AN] and references therein).
An integrable aspect of these equations (with suitable boundary
conditions) may be described, up to a change of coordinates,
by a linear flow $t\to(\o_1 t,...,\o_i t,...)$ $\in \T^\io$, where the
``frequencies" $\o_i\su\io$ as $i\to\io$ (typically $\o_i\sim
C i^k$ with $k>1$). Recently, it has been established ([Ku])
the existence, for perturbation of the above models,
of special solutions described by a {\it finite number}
of frequencies $t\to (\o'_1 t,...,\o'_N t)$. The problem of the persistence
(under small perturbations) of solutions with {\it infinitely many}
frequencies is open.

The model we study is {\it not} directly related  to
the above models but it might be
regarded as a {\it model problem}  mimicking some of the basic features
coming into play:
Hamiltonian structure, ``compactness" of the configuration
space, regularity, ``frequency growth".
We point out, however, that,
in our model, the frequencies associated to the constructed
almost--periodic solutions, grow {\it very} rapidly as $|i|\to\io$:
Such a fast  growth is related to the Diophantine property and it is
conceivable that relaxing this property one could establish, for
\equ(1.1), the
existence of almost--periodic solutions having frequencies growing
at ``more interesting" rates (such as the polynomial rate mentioned
above).

\vsk
(ii)
It is well known since Lagrange that the $d$--dimensional wave
equation can be derived as (a suitable) limit, as $\e\to 0$, of
harmonic oscillators vibrating orthogonally to a lattice $\Z^d_\e$
$\=$ $\{ n\e$, $n\in \Zd\}$ (see, \eg, [G]).
Such limit is not affected by replacing the harmonic potential
$(x_{i+1}-x_i)^2/2$, $i\in \Z^d_\e$, by $-\cos(x_{i+1}-x_i)$.
For example, the so--called Sine--Gordon equation on
(a domain of) $\R^d$,
$$
w_{tt}=\D w + \sin w\ ,\qquad  w=w(\x,t)\ ,\ \x\in\R^d\ ,\
t\in \R
\Eq(SG)$$
is obtained as limit as $\e\to 0$ of
$$
\ddot x_i= \e^{-2} \sum_{\|j-i\|=1} \sin(x_j-x_i)\ +\ \sin x_i\ ,\quad
i\in \Zd
\Eq(SGa)$$
if one sets, for $\x\in\R^d$, $\x\=i \e$, ($i\in \Zd$), and
$w(\x,t)=x_i(t)$.

The finite approximation \equ(SGa) is, for any fixed $\e>0$, an example
of system \equ(1.1) treatable with our techniques.

We do not discuss here boundary conditions [ and hence a proper
formulation of a problem associated to \equ(SG) ].

\vsk
(iii) Models of many particles interacting via conservative forces provide
natural concrete examples of systems \equ(1.1). More precisely, one can
regard \equ(1.1) as describing a system of {\it infinitely many coupled
rotators} (\ie particles ideally constrained to move on ``circles")
centered on the site $i\in \Zd$ and interacting (``coupled") via the
``forces" $f_i$; ``conservative" meaning that such forces are, in a
suitable sense (see below), gradients of ``potentials".

An example generalizing \equ(SGa) is the following. Fix $L\ge 1$
and consider, for $j\in\Zd$, a collection of functions $g_j$
depending on sites of the lattice within (Euclidean) distance $L$ from
$j$; in formulae:
$$
g_j\=g_j(x^{(L)})\ ,\quad x^{(L)}\=\{x_k\}_{k\in B_j(L)}\ ,
\quad B_j(L)\=\{k:\|k-j\|\le L\}
\Eq(gj)$$
We assume that the ``localized potentials" $g_j$ are {\it
real--analytic}
functions from $\T^{|B_j(L)|}$
$\to\R$
and that, for some positive $M$:
$$
\sup_{j,x^{(L)}\in\T^{|B_j(L)|}} |g_j(x^{(L)})|\le M
\Eq(M)$$
Then we set
$$
f_i\=\sum_{\|j-i\|\le L}\dpr_{x_i} g_j
\Eq(Ex1)$$
The system \equ(1.1) with such $f_i$ is called a finite range system
of infinitely many coupled rotators (see also [Wa1] and [VB]).
Example \equ(SGa) is obtained by letting
$$
g_j\= \e^{-2} \sum_{h=1}^d \cos(x_{j+e_h}-x_j) \ \ -\cos x_j
\Eq(Ex1')$$
where $e_1\=(1,0,...,0)$,...,$e_d\=(0,...,0,1)$.

An example, with $d=1$, of ``long range interaction" is given by
\equ(1.1) with:
$$
f_i\=\cos x_i \sum_{j\in\Z} a_j\prod_{k\ne 0} (1+a_{j+k} \sin x_{i+k})
\ ,\quad\sum_{j\in\Z}|a_j|<\io
\Eq(Ex2)$$
We remark that to the above examples one could associated
the {\it formal Hamiltonians}:
$$
H_{\rm short}(y,x)\=\sum_{j} {y_j^2 \over 2}
-\sum_j g_j
\Eq(H1)$$
for \equ(1.1), \equ(Ex1), and:
$$
H_{\rm long}(y,x)\=\sum_{j} {y_j^2 \over  2}
-\sum_j\prod_k (1+a_k \sin x_{j+k})
\Eq(H2)$$
for \equ(1.1), \equ(Ex2);
in fact, it is immediate to check that the {\it formal Hamilton
equations} (associated to the {\it formal symplectic form}
$\sum_{i\in\Zd}dy_i\wedge dx_i$) yield the respective \equ(1.1).
Rather than trying to give a precise meaning to such formal objects,
we shall use the notion of {\it generalized gradients}, which
shall allow us to treat directly the differential equations \equ(1.1).
Roughly speaking, a generalized gradient is a vector field (\ie
a continuous map from $\TT$ to its tangent space) such that
when {\it averaged} (with respect to the natural probability measure
associated to $\TT$) over $x_j$ with $j\notin I$ with $I$ a finite subset of
$\Zd$, the function (of finite variables) thus obtained, is a
gradient of a periodic function.

\vsk
The rest of the paper is organized as follows: In \S 2 we give a precise
notion of solutions of \equ(1.1) for Lipschitz vector field
(so that existence and uniqueness for all time of the Cauchy problem
trivially follow); in \S 3 we introduce the Hamiltonian structure via
generalized gradients; in \S 4 we define (maximal) almost--periodic
solutions and in \S 5 we define Diophantine sequences and prove their
abundance; \S 6 contains the statements of the main results; \S 7
is devoted to a finite dimensional ``average theorem", which allows
to construct quasi-periodic solutions for a system over $\T^{N+1}$
starting from a quasi--periodic solution of a subsystem over $\TN$
obtained by averaging the potential associated to the original
$(N+1)$--dimensional system: iterating suitably one obtains the proof
in infinite dimensions, which is spelled out in \S 8.
The average theorem of \S 7 is, in turn, based on tools from
perturbation theory (such as Kolmogorov--Arnold--Moser theory,
Nash--Moser implicit function theorems, \etc): a short
summary (with complete proofs) of KAM theory is given in Appendix 2
while Appendix 1 contains a classical results concerning the
full (Lebesgue) measure of Diophantine vectors in $\RN$.
\vglue2.truecm\penalty-200

\nin{\sectionfont 2. Second Order ODE's on $\torus^\io$}

\penalty10000
\vskip0.5truecm
\numsec=2\numfor=1
\penalty10000

\nin
Denote by $\TT$ the Cartesian product of infinitely many copies of the one
dimensional (flat) torus
$$
\TT\=\bigotimes_{i\in \Zd} \=\T^\Zd\ ,\qquad
\T_i\=\T\=\R/2\p\Z
\Eq(2.1)$$
($d$ being a positive integer) and endow $\TT$ with the standard weak
topology (see, \eg,  [ Ke ] ). Such topology is also induced by {\it
metrics}: To any summable positive sequence
$$
w: \ \Zd\to(0,\io) \quad {\rm s.t.} \quad \sum_{i\in\Zd} w_i<\io\ ,
\qquad (w_i>0\ \forall i)
\Eq(2.2)$$
which we shall call a {\it weight}, we can associate a metric $\r_w$
by setting, $\forall$ $x,y$ $\in \TT$ ($x=\{x_i\}_{i\in\Zd}$,
$y=\{y_i\}_{i\in\Zd}$, $x_i,y_i\in \T$):
$$
\r_w(x,y)\=\sum_{i\in \Zd} \r(x_i,y_i) \ w_i
\Eq(2.3)$$
where $\r$ is the standard (flat) metric on $\T\=\T_i$:
$$
\r( [ a ] , [ b ] )\= \inf_{n\in \Z} |a-b+2\p n|
\Eq(2.4)$$
here $a,b\in\R$ and $ [ \cdot ] $ denote equivalence (mod. $2\p$) class.

The pair $(\TT,\r_w)$, denoted also $\TT_w$, is actually a (real--analytic)
{\it infinite dimensional manifold} (with respect to the obvious atlas
obtained by taking the $\bigotimes_{i\in\Zd}$ of the atlas making
$\T$ a real--analytic one--dimensional manifold; for general informations
see [La]). The tangent space of
$\TT_w$ is the Banach space $\BB_w$ formed by the sequences $a\in \R^\Zd$
having finite norm
$$
\|a\|_w\=\sum_\iZd |a_i|\ w_i <\io
\Eq(2.5)$$
We can now give a precise meaning to second order ODE's on $\Tw$:
Given a continuous map $f:\Tw\to\Bw$ consider the system:
$$
\ddot x_i = f_i(x)\ ,\qquad \iZd
\Eq(2.6)$$
where $f_i\=\p_i\circ f$ ($\p_i:\Tw\to\T_i$ being the standard projection).
A {\it solution of \equ(2.6)} is just a continuous map
$t\in\R\to x(t)\in\Tw$, with $x_i\in C^2(\R)$, $\forall i$, and
satisfying the system \equ(2.6).

\vsk
{\bf Remarks 2.1 \quad}
(i) In some sense the notion of solution we have just introduced is a
``weak notion" (as opposed to looking at $\ddot x=f$ as an equation on the
cotangent bundle of $\Tw$).

(ii) Let $f\=0$ and $x(t)\= [ \o t ] \=\{ [ \o_i t ] \}_\iZd$. Then $x(t)$ is a
solution for {\rm any} $\o\in\R^\Zd$ (not necessarily in $\Bw$). Notice in
particular that $t\in\R\to [ \o t ] \in \Tw$ is continuous for any
$\o\in\R^\Zd$. These facts are no longer true if one considers
stronger topologies; for example, if $|\o_i|\to\io$ as $|i|\to\io$,
$ [ \o t ] $ is not continuous with respect to the uniform
topology   [ $\r_{\rm uniform}(x,y)\=\sup_\iZd \r(x_i,y_i)$ ] .
Thus, our interest in solutions $x(t)$ with $x_i$ ``close" to
$\o_i t$ with $|\o_i|$ $\to \io$ as $|i|\to\io$
explains the choice of the compact topology.

\vsk
Global existence and uniqueness for the Cauchy problem associated to
\equ(2.6), with $f$ Lipschitz, are an elementary application
of standard contraction techniques (see [Pe]).
We just stress that the ``initial
velocities" can be taken to be completely arbitrary (and not necessarily
in the tangent space $\Bw$):

\vsk
{\bf Proposition 2.2\quad}
{\sl Let $f\colon{\cal T}_w\to{\cal B}_w$ be a
Lipschitz map (\ie $\exists$ $C>0$ s.t.
$\Vert f(x)-f(y)\Vert_w\le C\rho_w(x,y),\ \forall x,y\in{\cal T}_w$).
Given any $x^0\in{\cal T}_w$ and any $y^0\in{\R}^{{\Z}^d}$,
there exists a unique solution, global in time, of the Cauchy problem}
$$
\left\{\eqalign{
&\ddot x_i(t)=f_i(x(t))\ ,\quad\quad\quad i\in{\Z}^d\cr
&x_i(0)=x_i^0\ ,\quad\quad\dot x_i(0)=y_i^0.\cr}\right.
\Eq(2.7)$$

\vsk
The property of being Lipschitz depends of course from the metric,
as shown by example \equ(Ex2) of \S 1.
In fact, consider two cases: (1) $a_j\=b^{-|j|}$, $b>1$; (2)
$a_j=(1+|j|^p)^{-1}$, $p>1$. Then, in the first case (1),
$f$ is Lipschitz if we take $w_j\=c^{-|j|}$ with any $1<c<b$,
while in case (2) we can take $w_j\=a_j$.
\vglue2.truecm\penalty-200

\nin{\sectionfont 3. Hamiltonian Structure and Averages}

\penalty10000
\vskip0.5truecm
\numsec=3
\numfor=1
\penalty10000

\nin
In finite dimensions, the Hamiltonian (or Lagrangian) structure in
second order ODE's, $\ddot x=f(x)$, is expressed by the vector field
$f$ being a {\it gradient}, $f=\dpr_x V$; the Hamiltonian function is then
$H\=\half \dot x^2- V(x)$ (and the Lagrangian is
$L\=\half \dot x^2+ V(x)$; see [A] for generalities).

In infinite dimensions, one could require as well that $f$ be a
gradient of a $C^1(\Tw;\R)$ function. However this attitude is much too
restrictive as it imposes strong decay properties (as $|i|\to\io$) to the
components $f_i$ of the field and even the simple examples \equ(Ex1),
\equ(Ex2)
of the introduction would not fit in the picture. Indeed what one really
needs is that $f$ is a ``local gradient" (where ``local" refers to localized
portions of $\Zd$).

We shall therefore introduce the notion of {\it generalized gradient} (or
``{\it g--gradient} ") bringing in the local structure with the help of
finite--dimensional projectors, which just {\it average out} the dependence
upon variables $x_i$ with $i$ varying in the complementary of a finite
subset of $\Zd$.

To construct such operators, we introduce a {\it probability measure} on
$\Tw$: Consider the $\s$--algebra, $\RR$, generated by the ``cylinders"
(see, \eg,  [Ha])
$$
\RR_I\=\bigotimes_{i\in I} A_i \ \bigotimes_{j\notin I} \T_i
\Eq(3.1)$$
where $A_i$ is an open subset of $\T_i$ and $I\subset \Zd$
is a finite subset of $\Zd$: $|I|<\io$ ($|\cdot|$ denoting here
cardinality).  Then there exists (see [Ha], \S 38) a unique (probability)
measure $\m$ on $\RR$ such that
$$
\m(\RR_I) = \prod_{i\in I} \m_i(A_i)
\Eq(3.2)$$
where $\m_i$ is the normalized ``Lebesgue measure" on $\T_i$.

Now, given any (not necessarily finite) subset $I$ of $\Zd$,
we can construct, as above,  a product measure $\m_I$
$\=$ $\bigotimes_{i\in I} d\m_i$ on $\bigotimes_{i\in I} \T_i$
(endowed with the topology induced by the metric $\r_I\=\sum
_{i\in I}\r_i$). Then if $J\=I^c\=\Zd\backslash I$ the product measure
$d\m_I\otimes d\m_J$ coincide with $d\m$ and Fubini's theorem holds.
Thus, to any bounded measurable function $g$ on $\TT$ we can naturally
associate a measurable function, $g^{[I]}$, on $\bigotimes_{i\in I}\T_i$,
by setting (for $x_I\in \bigotimes_{i\in I} \T_i$,  $d\m_i$--almost
everywhere):
$$
g^{[I]}\=\ii g \ d\m_J\ ,\qquad J\=\Zd\backslash I
\Eq(3.3)$$
If $|I|<\io$, $g^{[I]}$ is a honest measurable function on $\T^{|I|}$.
We can now define g--gradients.

\vsk
{\bf Definition 3.1\quad}
{\sl A continuous function $f\colon{\cal T}_w\to{\cal B}_w$ is a
{\rm g-gradient }if
 for any finite $I\subset{\Z}^d$ there exists  a ${C}^1
({\T}^{\left\vert I\right\vert};{\R})$ function, $V^{(I)}(x)$, so that }
$$
f_i^{[I]}(x)=\partial_{x_i}V^{(I)}(x),\quad\forall\ i\in I\ ,
\quad\forall\ x\in
{\T}^{\vert I\vert}
\Eq(3.4)$$

\vsk
It is easy to check that the examples \equ(Ex1), \equ(Ex2) in \S 1
are g--gradients (see [Pe]).


\vsk
{\it We shall speak of second order Hamiltonian
equations on $\Tw$ whenever $f$ in \equ(2.6)
is a g--gradient.}

\vsk
Let us conclude this section by introducing the (strong) {\it regularity
class} we shall work with.

\vsk
{\bf Definition 3.2\quad}
{\sl A   g-gradient $f$ is called {\rm uniformly weakly real-analytic}
if there exists a real number  $\xi>0$
such  that for any finite set $I\subset{\Z}^d$, $V^{(I)}(x)$ is
real-analytic on ${\T}^{\vert I\vert}$ and can be analytically continued
to the set }$\{z\in {\C}^{\left\vert I\right\vert}:
\ \left\vert{\rm Im}\,z_i\right\vert\le\xi\}$.

\vsk
{\bf Remarks 3.3\quad} (i) In fact we could deal with more general
classes of vector fields allowing the width of analyticity
of $V^{(I)}$ to tend to zero as $|I|\to\io$ (the allowed rate of decay
would then be dictated by the quantitative analysis carried out below).

(ii) Example \equ(Ex2) in \S 1 is uniformly weakly analytic and as
parameter $\x$ one could take any positive number; example \equ(Ex1)
is uniformly weakly analytic for some (small enough) $\x>0$.
\vglue2.truecm\penalty-200

\nin{\sectionfont 4. Almost--Periodic Solutions (Definitions)}

\penalty10000
\vskip0.5truecm
\numsec=4
\numfor=1
\penalty10000

\nin
We start with the definition of maximal almost--periodic functions
with ``rationally independent" frequency $\o\in\R^\Zd$.

\vsk
{\bf Definition 4.1\quad} {\sl A sequence $\o\in\R^\Zd$ is said to be
{\rm rationally independent} if for any finite subset $I$ of $\Zd$
and for any $n_i\in\Z$
$$
\sum_{i\in I} \o_i n_i\neq 0 \quad {\rm unless} \quad n_i=0
\ \forall\ i\in I
\Eq(4.1)$$
}

\vsk
In other words, $\o$ is rationally independent if any finite vector
$\o^{(I)}\=\{\o_i\}_{i\in I}$ $\in \R^{|I|}$ is rationally independent.

\vsk
{\bf Definition 4.2\quad} {\sl A continuous real function $q(t)$ is called
{\rm almost--periodic over $\Tw$} (with frequency $\o$)
if there exist a rationally independent
sequence $\o\in\R^\Zd$ and a continuous function $Q:\Tw\to\R$ such that
$q(t)=Q([\o t])$. A solution $x(t)$ of \equ(2.6) is called
{\rm maximal almost--periodic} if $x_i(t)-[\o_i t]$ is, for all $i$,
almost--periodic over $\Tw$ with frequency $\o$.}

\vsk
{\bf Remarks 4.3\quad} (i) Recall [see Remark 2.1, (i)] that $t\to[\o t]$
is continuous $\forall$ $\o\in\R^\Zd$.

(ii) A function $q$  almost--periodic over $\Tw$ is almost--periodic
in the sense of H. Bohr with frequency modulus given by
$$
\sigma(q)=\big\{r\in \R: \
r=\sum_{i\in I}\omega_i n_i\quad {\rm for\quad some\quad}
\ I\subset{\Z}^d,
\ \vert I\vert<\infty,\ n_i\in {\Z}\big\}
\Eq(4.2)$$
(see [Ka]).

(iii) The word ``maximal" in the above definition refers to the rationally
independence of the frequency $\o$. Indeed, one can consider {\it
quasi--periodic} solutions of \equ(2.6): these are almost--periodic
solutions with the associated frequency modulus being generated by a fixed
vector $\o^{(N)}\in \R^N$. The existence of such solutions has been
established in a somewhat different context by [Wa2] and [Ku], (see also
[P\"o]).
\vglue2.truecm\penalty-200

\nin{\sectionfont 5. Diophantine Sequences}

\penalty10000
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\numsec=5
\numfor=1
\penalty10000

\nin
Actually the almost--periodic solutions constructed below will have
frequencies $\o$ verifying much stronger numerical properties than just being
rationally independent: {\it They will be Diophantine} in the sense of the
following definition.

\vsk
{\bf Definition 5.1\quad}
{\sl A rationally independent sequence $\o\in\R^\Zd$
is called {\rm Diophantine} if for any finite set $I\subset \Zd$,
there exist constants $\g>0$ and $\t$ ($\ge |I|$)
such that for any choice of $n_i\in \Z$ with $\sum_{i\in I}|n_i|>0$
it is:
$$
|\sum_{i\in I} \o_i n_i|\ge {1 \over \g \big(\sum_{i\in I} |n_i|\big)^\t}
\Eq(5.1)$$
}

\vsk
It is well known that Diophantine vectors in $\R^N$ form a set of full
Lebesgue measure (for completeness we reproduce this elementary result in
Appendix 1), but also in infinite dimensions Diophantine frequencies
are rather abundant. One can in fact construct many Diophantine sequences
with the help of the following Lemma (compare with Lemma 3 of [CZ]).

We recall that a vector $\o\in\R^N$ is called {\it $(\g,\t)$--Diophantine}
if
$$
|\o\cdot n|\ge {1\over \g|n|^\t}\ ; \quad \forall n\in \ZN\backslash\{0\}
\Eq(5.2)$$
where $\o\cdot n$ denotes the standard scalar product in $\R^N$ and $|n|\=
\sum_{i=1}^N|n_i|$.

\vsk
{\bf Lemma 5.2\quad}
{\sl Let $\o\in \R^N$ be  $(\g,N)$--Diophantine; let $\O$
be a positive number satisfying
$$
\O\ge 4 \sqrt{N} |\o|\ ;\qquad\qquad \big(|\o|\=\sum_{i=1}^N|\o_i|\big)
\Eq(5.3)$$
and define the following subset of $[\O,\io)$:
$$
\AA_N\=\AA_N(\o,\O)\=\{\a\ge \O:\
|\o\cdot n+\a h|\ge \O/[ 2(|n|+|h|)^{N+1}]\ ,\
\forall n\in \Z^N, \forall 0\neq h\in \Z\}
\Eq(5.4)$$
There exists a universal number $K>1$ such that:
$$
\ell\big( [\O,\io)\backslash \AA_N \big) < K |\o|
\Eq(5.5)$$
where $\ell$ denotes Lebesgue measure.}

\vsk
{\bf Remarks 5.3\quad} (i) Since by definition  $\O> 2/\g$,  [as
$|\o|\ge |\o_i|\ge
\g^{-1}$ by \equ(5.2)], $(\o,\a)$ is $(\g,N+1)$--Diophantine
whenever $\a\in\AA_N$.

(ii) Whence, in particular, the above Lemma tells us that given a
$(\g,N)$--Diophantine vector $\o$ in $\R^N$ (and almost all vector
in $\R^N$ are $(\g,N)$--Diophantine for some $\g$: see Appendix 1)
we can pick $\a$ in $\AA_N\subset [\O,\io)$ [ whose complementary
measure in $[\O,\io)$ is of $O(|\o|)$ ] so that the vector $(\o,\a)$
is $(\g,N+1)$--Diophantine.

(iii) In the above statement and in its proof we could replace $N$
with any $\t>N-1$, however to avoid introducing too many parameters
we consider only the case $\t=N$.

(iv) It is now easy to construct many Diophantine sequences. Fix
$I_0\in\Zd$, $|I_0|=N<\io$; pick a $(\g_0,N)$--Diophantine vector
$\o^{(N)}\in\R^N$ (with some $\g_0>0$); and fix a one--to--one map, $j_h$,
from $\Z_+$ onto $\Zd\backslash I_0$. Now set $\o_i\= \o_i^{(N)}$
$\forall i\in I_0$ and define $\o_{j_h}$, $h\ge 1$, inductively as follows.
Let $\O_1\=4\sqrt{N}|\o^{(N)}|$ and choose $\o_{j_1}$ in
$\AA_N(\o^{(N)},\O_1)$ so that, by the above Lemma,
$\o^{(N+1)}\=(\o^{(N)},\o_{j_1})$ is $(\g_0,N+1)$--Diophantine.
Analogously, given $\o^{(N+h)}\=(\o^{(N)},...,\o_{j_h})$
$\in \R^{N+h}$, $(\g_0, N+h)$--Diophantine, we let $\O_h\=
4\sqrt{N+h} \big(|\o^{(N)}|+\sum_{k=1}^h|\o_{j_k}|\big)$
and pick $\o_{j_{h+1}}\in \AA_{N+h}(\o^{(N+h)},\O_h)$.
It is clear that in this way one constructs many Diophantine sequences
satisfying \equ(5.1) with
$\g\=\g_0$ and $\t=N+k$ where $k\=0$ if $I\subset I_0$, and otherwise
$k\=\max\{h: {j_h}\in I\}$.

\vsk
{\bf Proof of Lemma 5.2\quad}
For vectors in $\R^m$, we denote by $\|\cdot\|$ the Euclidean norm
and by $|\cdot|$ the $1$--norm (sum of absolute values of components).
Set $a\=\O/(2\|\o\|)$, $e\=\o/\|\o\|$,
and let $\s_N$ be the area of the unit-sphere
$S^{N-1}\=\{x\in \R^N: \|x\|=1\}$. Finally denote by $n$ a generic vector
in $\Z^N$ and by $h$ a generic integer number and for a vector $y\in \R^N$,
let $\bar y$ $\=$  $(y_2,...,y_N)$. Now, let first $N\ge 2$; then:
$$\eqalignno{
\ell \big([\O,\io)\backslash \AA_N\big)
&= \ell \{\a\ge\O : |\o\cdot n+\a h|< {\O \over 2(|n|+|h|)^{N+1}}
\ {\rm for\ \ some\ } h\neq 0\}\cr
&\le \sum_{n\neq 0, h\neq 0} \ell \{\a \ge \O: |{\o\cdot n\over h}+\a|
< {\O\over 2 |h|\  (|n|+|h|)^{N+1}} \} \cr
&\le 2 \O \sum_{h\ge 1} {1\over h} \sum_{|\o\cdot {n\over h}|\ge {\O\over
2}} {1 \over (|n|+|h|)^{N+1}}\cr
&= 2 \O \sum_{h\ge 1} {1\over h} \sum_{|e\cdot n|\ge a h}
{1 \over (|n|+h)^{N+1}}\cr
&\le 2 \O \sum_{h\ge 1} {1\over h} \
\ii_{\{ x\in \R^N:\ |e\cdot x|\ge a h -\sqrt{N}\}} \
{dx \over (|x|+h)^{N+1}}\cr
&\le 2 \O \sum_{h\ge 1} {1\over h} \
\ii_{\{ y\in \R^N:\ |y_1|\ge a h -\sqrt{N}\}} \
{dy \over (\|y\|+h)^{N+1}}\cr
&\le 2 \O \sum_{h\ge 1} {1\over h} \
2^{{N+1\over 2}}
\ii_{\{ y\in \R^N:\ |y_1|\ge a h -\sqrt{N}\}} \
{dy \over (|y_1| + \|\bar y\|+h)^{N+1}}\cr
&= 2 \O \sum_{h\ge 1} {1\over h} \
{2^{{N+3\over 2}} \over N}
\ii_{\R^{N-1}}
{d\bar y \over (\|\bar y\|+(a+1)h - \sqrt{N})^{N}}\cr
&\le  2 \O \sum_{h\ge 1} {1\over h} \
{2^{{N+3\over 2}} \over N} \s_{N-1}
\ii_0^\io
{r^{N-2} \over (r +{a\over 2}h )^{N}} \ dr\cr
&\le  2 \O \sum_{h\ge 1} {1\over h} \
{2^{{N+3\over 2}} \over N} \s_{N-1}
\ii_0^\io
{1 \over (r +{a\over 2}h )^{2}} \ dr\cr
&\le K |\o|\cr}
$$
where: in the sum after the
first inequality we have $n\neq 0$ as $\O >2/\g$ [see
Remark 5.3, (i)];
in the fourth inequality we set $By=x$ with $B$ a unitary matrix
sending the vector $(1,0,...,0)$ to $e$;
in the sixth inequality we used the assumption that
$\O\ge 4 |\o| \sqrt{N}$ and we have taken:
$$
K\= 2^{9/2} \big( \sum_{h\ge 1} h^{-2}\big) \ \sup_N
\{ {2^{N/2} \s_{N-1} \over N} \}
\Eq(5.K)$$
The case $N=1$ is just shorter.
\qed
\vglue2.truecm\penalty-200

\nin{\sectionfont 6. Main Results}

\penalty10000
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\numsec=6
\numfor=1
\penalty10000

\nin
{\bf Theorem 6.1\quad} {\sl
Let $f:\Tw\to\Bw$ be a uniformly weakly analytic g--gradient
(Definitions 3.1, 3.2). Then there exist uncountably many
maximal almost--periodic solutions of \equ(2.6) (Definition 4.2)
with Diophantine frequencies (Definition 5.1).}

\vsk
This result is a  simple corollary of the next more detailed
theorem.

\vsk
Recall that a {\it non--degenerate (maximal) quasi--periodic solution}
of
$$
\ddot x = V_x(x)\ ,\qquad\quad x\in \T^N
\Eq(6.1)$$
with frequency $\o\in \R^N$ is a solution of the form
$$
x(t)=\big[\o t + u([\o t])\big]
\Eq(6.2)$$
for a suitable function $u:\T^N\to\R^N$ satisfying
$$
\det \big( \IN +u_\th(\th) \big) \neq 0\ ,
\qquad\quad \forall \ \th\in\T^N
\Eq(6.3)$$
where $\IN$ is the identity $(N\times N)$--matrix and $(u_\th)_{hk}\=$
$\dpr_{\th_k} u_h$, $h,k=1,...,N$; the word maximal refers to the
maximal dimension of the frequency $\o$.

\vsk
{\bf Remarks 6.2 (On quasi--periodic solutions)\quad} (i) To any
quasi--periodic solution with $\o$ rationally independent
(\ie $\o\cdot n=0$ for some $n\in\Z^N$ $\implies n=0$)
one can associate an $N$--parameter family of solutions obtained by
``phase--translation": for each $\th\in \T^N$,
$x(t;\th)\=$ $[\th+\o t + u( [\th + \o t])]$ is still a solution (as
the flow $t\to [\o t]$ is dense in $\T^N$).

(ii) From now on we shall often omit the projection $[\cdot]$
[see \equ(1.2), \equ(2.4) and (ii) of Remark 2.1 ]
in the notation.

(iii) Indeed, the above family corresponds to an {\it invariant
$N$--torus}, embedded in the phase--space $\R^N\times \T^N$,
given by
$$
\TT^N_\o\= \{ (y,x) \in \R^N\times \T^N :
(y,x)=\big(\o+ D_\o u(\th), \th + u(\th)\big)\ ,\
\th \in \T^N\}
\Eq(6.4)$$
where $D_\o\=\o\cdot\dpr_\th\=\sum_{i=1}^N\o_i\dpr_{\th_i}$; the
$y$--component is just the velocity vector corresponding to the
point $x$ (as $D_\o$ corresponds to ${d\over dt}$ along the linear flow
$t\to \th +\o t$). The non--degeneracy \equ(6.3) of the solution \equ(6.2)
allows to see $\TT^N_\o$ as a regular embedded torus in the ambient
space
$\R^N\times\T^N$.

(iv) In view of the above observations it is clear that to find
non--degenerate quasi--periodic solutions of \equ(6.1) {\it is
equivalent to find solutions of the following non--linear PDE on $\T^N$}:
$$
D_\o^2 u(\th)=V_x(\th+u(\th))\ ,\quad\qquad \min_{\th\in\T^N}
|\det (\IN + u_\th)|>0
\Eq(6.5)$$
(just substitute  \equ(6.2) in \equ(6.1) and use the rationally
independence of $\o$ to replace $\o t$ with the generic point $\th$).
For investigations on \equ(6.5) see [M2], [SZ], [CC1], [CC2], [CZ].

\vsk
{\bf Theorem 6.3\quad} {\sl Let $f$ be as in Theorem 6.1 and assume that
for some finite $I_0\subset \Zd$ the equation
$$
\ddot x^{(0)} = \dpr_\xo V^{(0)} (\xo)\ , \quad\qquad \xo\in \T^{N_0}
\ ,\quad N_0\=|I_0|
\Eq(6.6)$$
admits a (maximal) non--degenerate
quasi--periodic solution with a $(\g,N_0)$--
Diophantine frequency vector $\o^{(0)}\in\R^{N_0}$
[see \equ(6.1) $\div$ \equ(6.3)
and \equ(5.2)]. Then for any $\e>0$ there exist uncountably many maximal
almost--periodic solutions, $x(t)$, of \equ(2.6)
with Diophantine frequencies $\o\in\R^\Zd$
such that $\o_i=\o_i^{(0)}$ for $i\in I_0$ ($|\o_i|\to\io$ as
$|i|\to\io$) and:
$$\eqalign{
& \sup_{{t\in \R}\atop{ p=0,1}}
|{d^p\over dt^p}  [x_i(t)-x_i^{(0)}(t)]|\le
\e\qquad {\rm for}\quad i\in I_0\cr
&\sup_{{t\in \R}\atop{p=0,1}} |{d^p\over dt^p}  [x_i(t)-\o_i t]|\le \e
\quad\qquad\  {\rm for}\quad i\notin I_0\cr}
\Eq(6.7)$$
}

\vsk
The proof of the above theorems will be given in \S 8.
\vglue2.truecm\penalty-200

\nin{\sectionfont 7. A (Finite--Dimensional) Average Theorem}

\penalty10000
\vskip0.5truecm
\numsec=7
\numfor=1
\penalty10000

\nin
In this section we discuss a finite--dimensional problem, which may have
some interest by itself, and whose solution will constitute
the main step of the proof of Theorem 6.3.

Roughly speaking the question is how to construct quasi--periodic
solutions for a (second order) Hamiltonian system
on $\T^{N+1}$ if you know the
existence of quasi--periodic solutions for a ``subsystem" obtained from
the original one by averaging out some variables.

More precisely, let $V:\TNN\to\R$, ($N\ge 1$), be real--analytic and
consider the ``Hamiltonian" equations:
$$
\ddot x'=V_{x'} (x')\ ,\qquad\quad x' \in \TNN
\Eq(7.1)$$
Let also $\Vb(x)$, $x\in \TN$, denote the ``averaged potential"
$$
\Vb(x)\={1\over 2\p} \ii_0^{2\p} V(x')dx_{N+1}\ ,
\quad\qquad x'\=(x,x_{N+1})\in\TN\times\T
\Eq(7.2)$$
and consider the Hamiltonian equations in $\TN$ associated to
$\Vb$:
$$
\ddot x=\Vb_{x} (x)\ ,\qquad\quad x \in \TN
\Eq(7.3)$$
Now {\it assume that \equ(7.3) admits a non--degenerate (maximal)
quasi--periodic solution}
$$
x=\o t + u(\o t)\ ,\quad\quad \o\in\RN\ ,\qquad u:\TN\to\RN
\Eq(7.4)$$
with $\o$ $(\g,N)$--Diophantine and $u$ real--analytic. The question is:
Can one find quasi--periodic solutions for \equ(7.1) ``close"
(in some sense) to \equ(7.4)?

The answer is positive {\it provided} the looked after quasi--periodic
solutions have frequencies $\o'\=(\o,\a)$ with $\a\ggt |\o|$ and
suitable.

To formulate a precise and {\it quantitative} result, we need some
notations.

Given $N,M\ge 1$ and a function $g:\TN\to\R^M$ real--analytic on
$$
\D^N_\x\=\{\th\in\CN:\ |\Im \th_i|\le \x\ ,\quad i=1,...,N\}
\Eq(7.5)$$
[ this means that the components $g_h$, for $h=1,...,M$, admit a
holomorphic extension to some open domain containing $\D^N_\x$ ]
we set:
$$
\|g\|_{\D^N_\x}\=\|g\|_\x\=\sum_{h=1}^M\  \sup_{\D^N_\x} |g_h|
\Eq(7.6)$$
Now let $\LL^p(\CN)$, for $p\in\N$,
be the space of linear maps from $\CN$ into $\LL^{p-1}(\CN)$,
($\LL^0(\CN)\=\CN$,
$\LL^1(\CN)\=\LL(\CN)\=$ $(N\times N)$--matrices,...). If
$T:\TN\to\LL^p(\CN)$ is real--analytic on $\D^N_\x$
[this means that $\forall$ $c_1,...,c_p\in\CN$, the function
$\th\to(...((Tc_1)c_2)...c_N)$ is a real--analytic $\CN$--valued
function on $\D^N_\x$ ] then we set (inductively):
$$
\|T\|_{\D^N_\x}\=\|T\|_\x\=\sup_{ {c\in\CN}\atop{|c|=1}}
\|Tc\|_\x\ ,\quad\qquad \big(|c|\=\sum_{h=1}^N|c_h|\
{\rm for}\ \ c\in\CN\big)
\Eq(7.7)$$
Finally, observe that, without loss of generality,
{\it we can assume that $u$ in \equ(7.4) has vanishing mean value over
$\TN$} (as we can replace $u(\th)$ with $c+u(c+\th)$ with
$c=-\langle u \rangle$, $\langle\cdot\rangle$ denoting average over
$\TN$).

\vsk
{\bf Proposition 7.1\quad} {\sl Let $V:\TNN\to\R$ be a real--analytic
function, let $\Vb$ be as in \equ(7.2) and assume that \equ(7.3)
admits a (maximal) non--degenerate quasi--periodic solution \equ(7.4)
with a $(\g,N)$--Diophantine frequency vector $\o$. Fix $0<r<\r<\x_0$
so that: (i)  $V$ is real--analytic on $\D^{N+1}_{\x_0}$, (ii)
$u$ is real--analytic on $\D^N_r$, (iii) $\{\th +u(\th) :$
$\th\in \D^N_r\}$ $\subset \D^N_\r$ and (iv):
$$
\|\IN+\dpr_\th u\|_r\= U\ ,\qquad
\|(\IN+\dpr_\th u)^{-1}\|_r\=\Ub<\io
\Eq(7.8)$$
Finally fix $0<r'<r$ and let $\O$ be such that
$$\eqalign{
& \O\ge \max \{4\sqrt{N} |\o|\ , \ \g^{-1} \d \}\cr
& \d^2\= C (N+1)!^6 \ 2^{60(N+1)}\ U^{10} \Ub^8\
\b^3\ (r-r')^{-[11(N+1)+1]}
\ (\x_0-\r)^{-1}\cr
& \bar \d\= \d^2(\x_0-\r)\cr
& \b\= \max_{p=1,2,3} \{1\ ,\ \g^2 \|\dpr^p_{x'} V\|_{\x_0}\} \cr}
\Eq(7.9)$$
where $C>1$ is a suitable universal constant.
Then for any $\a\in\AA_N(\o,\O)$ [see \equ(5.4)] there exists
a (maximal) quasi--periodic solution of \equ(7.1), $x'=\o' t+
u'(\o' t)$, with $\o'\=(\o,\a)$ and $u':\TNN\to\RNN$ real--analytic on
$\D^{N+1}_{r'}$. Furthermore $\langle u'\rangle=0$ and:
$$\eqalign{
& \{\th'+u'(\th'):\ \th'\in \D^{N+1}_{r'}\}\subset \D^{N+1}_{\r'}\ ,
\qquad \r'\=\r+{\bar\d\over  (\g\O)^2}<\x_0\cr
& \max_{p=0,1} \|\dpr^p_{\th'} [u'-(u,0)]\|_{r'} \le
\bar \d (\g\O)^{-2}\cr
& \| D_{\o'}  [u'-(u,0)]\|_{r'} \le \d (\g \O)^{-1}\cr
}
\Eq(7.10)$$
where $D_{\o'}\=\o'\cdot \dpr_{\th'}\=\sum_{h=1}^N \o_h\dpr_{\th_h}
+\a\dpr_{\th_{N+1}}$.
}

\vsk
In the following we will not need the explicit (certainly not optimal)
dependence upon $N$ given in \equ(7.9), nevertheless we shall
pay some attention to constants for the sake of concreteness and also
because it may help the reader to keep track of the various estimates.

The proof is based on a result \`a la Nash--Moser from KAM theory
which guarantees the existence of solutions of the equation
\equ(6.5) provided the frequency vector $\o$ is Diophantine
and provided one can find a ``good enough {\it approximate solution}":

\vsk
{\bf Lemma 7.2\quad}
{\sl Let: $\x<\x_0'<\x_0<1$, $V:\TN\to\R$ be a real--analytic
function on $\D^N_{\x_0}$, $v:\TN\to\RN$ be real--analytic on $\D^N_\x$
and such that $\{\th+v(\th): \th\in \D^N_\x\}$ $\subset \D^N_{\x_0'}$,
$\o\in\RN$ be a $(\g,N)$--Diophantine vector; finally let also:
$$\eqalign{
&\|\IN+v_\th\|_\x\=\h\ ,\qquad \|(\IN+v_\th)^{-1}\|_\x\=\hb <\io\cr
&\g^2\|D^2v-V_x(\th+v)\|_\x\=\e\ ,\qquad
\max_{p=1,2,3} \{1,\g^2\|\dpr^p_xV\|_{\x_0}\}\=\b\cr}
\Eq(7.11)$$
($D\=D_\o\=\o\cdot\dpr_\th$). Fix $0<\x'<\x$. There exists a universal
constant $B>1$ such that if
$$
B\ N!^4\ 2^{40 N} \h^{10} \hb^8\ \b\ (\x-\x')^{-8N} (\x_0-\x_0')^{-1}\
\e\le 1
\Eq(7.12)$$
then there exists a function $u:\TN\to\RN$,
real--analytic on $\D^N_{\x'}$, which is solution of
$$
D^2u=V_x(\th+u)\ ,\qquad\quad \langle u \rangle =\langle v \rangle
\Eq(7.13)$$
Furthermore the following estimates hold:
$$\eqalign{
& \|\IN+u_\th\|_{\x'}\le 2\h\ ,\qquad
\|(\IN+u_\th)^{-1}\|_{\x'}\le2\hb \cr
& \max_{{p=0,1}\atop{q=0,1,2}}\{\|\dpr_\th^p(u-v)\|_{\x'}\ ,\
\g^q\| D^q(u-v)\|_{\x'}\}\le A\e\cr}
\Eq(7.14)$$
where $A\=B\ N!^4 2^{40 N} \h^{10}\hb^8\b (\x-\x')^{-8 N}$.}

\vsk
One can actually show that the above solution $u$ is ``locally" unique
(see [CC2]).

This lemma is a refinement of Lemma 6 of [CC1] (see also [M2],
[SZ], [CC2]); the main difference is that we need here to leave the
width of the domain of analyticity of the solution $u$ as a {\it free
parameter} (while in [CC1] was fixed to be half of $\x$). Rather than
indicating the (tiny but dense) adjustments to the proof in [CC1]
we present a complete (and short) proof in Appendix 2.

\vsk
{\bf Proof of Proposition 7.1\quad}
We shall use Lemma 7.2 [ with $N,\o$
replaced by, respectively, $N+1,\o'\=(\o,\a)$]:
We shall construct an approximate solution
$v$ so that the ``error function" $e\=D_{\o'}^2 v-V_{x'}(\th'+v)$
has norm bounded by $O(1/\O^2)$ when $\o'\=(\o,\a)$, $\a\in\AA_N(\o,\O)$
with $\O$ chosen so large [\equ(7.9)] that the condition \equ(7.12)
is satisfied.

We start by observing that the average over $\TNN$ of the vector
--valued function
$$
W(\th')\=V_{x'}(\th+u,\th_{N+1})\ ,\qquad \th'\=(\th,\th_{N+1})\in \TN
\times\T
\Eq(7.15)$$
is zero; in fact
$\langle W_{N+1}\rangle=0$  because
$$
\ii_0^{2\p} \dpr_{x_{_{N+1}}} V(\th+u,\th_{N+1})d\th_{N+1}=0
\Eq(7.16)$$
while the average of the first $N$ component of $W$ are given by
($x'\=(x,x_{N+1})\in\TN\times\T$ and $\th'\=(\th,\th_{N+1})\in\TN\times\T$):
$$\eqalign{
\ii_\TNN \dpr_x V(\th+u,\th_{N+1}) {d\th'\over (2\p)^{N+1}}
&\=\ii_\TN \dpr_x \Vb(\th+u) {d\th\over (2\p)^N}\cr
&= \ii_\TN D^2_\o u(\th) {d\th\over(2\p)^N}=0\cr}
\Eq(7.17)$$
Now, denoting, for a function $G$ with $\langle G\rangle=0$,
$D^{-p}G$ the unique solution with zero average of $D^p g=G$,
we set
$$
v(\th') \= D_{\o'}^{-2} W(\th')\= -\sum_{{n'=(n,h)\in\ZN\times\Z}\atop
{n'\neq 0}} {W_{n'}\over (\o\cdot n+\a h)^2} e^{in'\cdot\th'}
\Eq(7.18)$$
where $W_n$ denote Fourier coefficients;
[ and recall that $\o'\=(\o,\a)$ with a fixed $\a\in\AA_N(\o,\O)$].
Notice that, because $u$ satisfies $D_\o^2u=\Vb(\th+u)$, $v$ can be
written in the form:
$$
v=-\sum_{n\in\ZN,h\neq 0} {W_{n'}\over (\o\cdot n+\a h)^2}
e^{i n'\cdot \th'} +(u,0)\=\tilde v + (u,0)
\Eq(7.19)$$
and therefore
$$
e\=D^2_{\o'} v- V_{x'}(\th'+v)=
V_{x'}(\th+u,\th_{N+1})-V_{x'}\big((\th+u,\th_{N+1})+\tilde v\big)
\Eq(7.20)$$
Next, we estimate $\tilde v$, $v$ and $e$ on $\D^{N+1}_\rb$
for any $r'<\rb<r$. The hypothesis (iii) of Proposition 7.1
and the analyticity assumptions yield:
$$
\|W\|_\rb\le \|V_{x'}\|_{\x_0}\qquad
\implies\qquad |W_{n'}|\le \|V_{x'}\|_{\x_0}\  e^{-|n'|r}
\Eq(7.21)$$
Then, by \equ(7.19), the definition of $\AA_N$ and (A2.3)
we obtain ($p=0,1,...$):
$$\eqalign{
\|\dpr_{\th'}^p
\tilde v\|_\rb &\le 4 {\|V_{x'}\|_{\x_0}\over \O^2}\
\sum_{n'\in\ZNN} |n'|^{2(N+1)+p} e^{-|n'|(r-\rb)}\cr
&\le 4^{3N+4+p}
{\|V_{x'}\|_{\x_0}\over \O^2}\ [2(N+1)+p]!\ (r-\rb)^{-(3N+3+p)}\cr}
\Eq(7.21+)$$
Next we have to evaluate the norms $\h,\hb$ in the text of Lemma 7.2. It
is easy to check that:
$$
\h \le U+\|\dpr_{\th'}\tilde v\|_\rb\ ,\qquad\quad \hb\le\Ub
(1- \Ub\|\dpr_{\th'}\tilde v\|_\rb)^{-1}
\Eq(7.22)$$
Notice also that, since
$$
\{\th'+v:\th'\in\D^{N+1}_\rb\}\subset\D^{N+1}_{\r+\|\tilde v\|_\rb}
\Eq(7.23)$$
we can take (in applying Lemma 7.2)
the parameter $\x_0'$ $=\r+\|\tilde v\|_\rb$. Now,
choosing $\rb=(r+r')/2$ we easily obtain, for $p=0,1$,
the bounds [recall the
definition of $\b$ in \equ(7.9) and see \equ(7.20)]
$$
\|\dpr_{\th'}^p\tilde v\|_\rb\le \hat \d (\g \O)^{-2}\ ,
\qquad\e\=\g^2\|e\|_\rb\le \b\hat \d (\g \O)^{-2}
\Eq(7.24)$$
with
$$
\hat \d\= 2^{9N+14} \ (2N+3)!\ \b (r-r')^{-(3N+4)}
\Eq(7.24+)$$
>From $\d\le(\g\O)$, [see \equ(7.9)], it then follows
[ recall that $U,\Ub\ge 1$ while $\x_0<1$;
and again $p=0,1$]:
$$
\|\dpr_{\th'}^p\tilde v\|_\rb \le {\hat \d \over (\g \O)^2}
<{\x_0-\r\over2}\ ,\qquad \h\le 2U\ ,\qquad
\hb\le 2\Ub
\Eq(7.25)$$
We are now in a position to apply Lemma 7.2 with $\x\= \bar r\=
(r+r')/2$, $\x_0'\=\r+\|\tilde v\|_{\bar r}$ $\le (\r+\x_0)/2$,
$\x'\=r'$, and $N$ replaced by $(N+1)$ and $u$ by $u'$:
using the above estimates [ and bounding $(2N+3)!$ by a constant times
$2^{3N} (N+1)!^2$ ]
we see that the condition \equ(7.12) in
Lemma 7.2 is implied (for a suitable $C>B$)
by $\d^2(\g\O)^{-2}\le 1$, which is verified
because of our choice of $\O$: In fact, by \equ(7.14), \equ(7.25),
\equ(7.24) it is:
$$
A \e \le {\d^2\over 2} (\g \O)^{-2} (\x_0-\r)
\= {\bar \d \over 2} (\g \O)^{-2}
\Eq(7.26)$$
and condition \equ(7.12) reads just $A\e(\x_0-\r)^{-1}\le 1$.
Furthermore we see that
$$
\{\th'+u':\th'\in\D^{N+1}_{r'}\}\subset\D^{N+1}_{\r+\|\tilde v\|_\rb+A\e}
\subset\D^{N+1}_{\r+{\bar \d \over (\g\O)^2}}\=
\D^{N+1}_{\r'}
\Eq(7.27)$$
[ recall \equ(7.14), \equ(7.23), \equ(7.24) and
the definition of $\bar \d$ in \equ(7.10) ], where we used
$\hat \d<\d^2 (\x_0-\r)/2\=\bar \d/2$; the fact
that $\r'<\x_0$ [ see \equ(7.10) ] follows from
$\d^2(\g \O)^{-2}\le 1$ and from the first of \equ(7.25). Now, for
$p=0,1$, using \equ(7.14), \equ(7.26) and $\hat \d< \bar \d/2$, we get
$$\eqalign{
\|\dpr_{\th'}^p \big( u'-(u,0) \big)\|_{r'} & = \|\dpr_{\th'}^p \big
(u'-v+\tilde v \big)\|_{r'}\cr
&\le A\e + {\hat \d \over (\g \O)^2} \le {\bar \d \over (\g\O)^2}\cr}
\Eq(7.28)$$
Finally, mimicking the bounds \equ(7.21)$\div$\equ(7.24), one
obtains
$$
\|D_{\o'}\tilde v\|_{\bar r} \le \g^{-1} {\hat \d \over \g\O}
\Eq(7.29)$$
thus, by \equ(7.14), \equ(7.28) and [ see \equ(7.9) ]
using $\d (\g \O)^{-1}\le 1$, $\hat \d \le \d/2$, also
the bound on $\|D_{\o'}\big(u'-(u,0)\big)\|$ follows easily.
\qquad \qed
\vglue2.truecm\penalty-200

\nin{\sectionfont 8. Proof of Existence of Almost--Periodic Solutions}

\penalty10000
\vskip0.5truecm
\numsec=8
\numfor=1
\penalty10000

\nin
Here we prove Theorem 6.3. Theorem 6.1 follows immediately from Theorem
6.3 and from Corollary A2.4 (see Appendix 2).

\vsk
{\bf Proof of Theorem 6.3}\quad
The idea is to use iteratively the results of \S 7 to construct
quasi--periodic solutions for larger and larger subsystems and
to obtain, in the limit, almost periodic--solutions.

By hypothesis we are given a real--analytic quasi--periodic solution
$y(t)\=\o^\po t+u^\po(\o^\po t)$ of the subsystem:
$$
\ddot y=V_y^{(I_0)}(y)\ ,\qquad y\in \T^{N_0}\ ,\quad N_0\=|I_0|
\Eq(8.1)$$
the frequency vector $\o^\po$ being $(\g,N_0)$--Diophantine.

Thus calling $\x_0$ the analyticity parameter associated to the
field $f$ (see Definition 3.2),
we can assume that there exist $0<r_0<\r_0<\x_0$ so that $u^\po$ is
real--analytic on $\D^{N_0}_{r_0}$ and such that
$$\eqalign{
&\{\th^\po+u^\po(\th^\po):\ \th^\po\in \D^{N_0}_{r_0}\}\subset
\D^{N_0}_{\r_0}\cr
&\|\INo+\dpr_{\th^\po} u^\po\|_{r_0}\=U_0\ ,\qquad
\|(\INo+\dpr_{\th^\po} u^\po)^{-1}\|_{r_0}\=\Ub_0<\io\cr}
\Eq(8.2)$$
For concreteness we shall fix $\r_0\=\x_0/2$
(and take $r_0$ small enough;
notice that $\x_0$ is a fixed parameter which shall not change
in the iteration).

In the following construction there is quite a bit of freedom (whence
the uncountability of the solutions) as the extra frequencies
$\o_i$, $i\notin I_0$ will be arbitrarily chosen in  sets of
positive measure. There is also some (less substantial) freedom in the
{\it order} of ``invading" $\Zd$. More precisely, fix (arbitrarily)
a one--to--one and onto map, $k\in\Z_+\to j_k\in\Zd\backslash I_0$,
and set, recursively, for $k\ge 1$:
$$
I_k\=I_{k-1} \cup \{j_k\}
\Eq(8.3)$$
so that
$$
I_{k-1}\subset I_k\ ,\qquad|I_k|=N_0+k\=N_k\ ,\qquad I_k
\su \Zd
\Eq(8.4)$$
We shall use the following notations. Denote by $(x_1,...,x_{N_0})$
coordinates for $\bigotimes_{i\in I_0} \T_i$ and by $x_{N_k}$ the
coordinate associated to $\T_{j_k}$. Then set, for $k\ge 0$:
$$\eqalign{
&V^{(k)}\= V^{(I_k)}\ , \quad x^{(k)}\=(x_1,...,x_{N_k})\in \T^{N_k}\cr
&\b_k\=\max_{p=1,2,3}
\{1\ ,\ \g^2 \|\dpr^p_{x^{(k)}} V^{(k)}\|_{\x_0}\}\cr}
\Eq(8.5)$$
The $(k+1)^{\rm th}$ step will consist in constructing  non--degenerate
quasi--periodic solutions with $u^{(k+1)}:\T^{N_{k+1}}\to\R^{N_{k+1}}$
and frequency $\o^{({k+1})}\in\R^{N_{k+1}}$ of the form $(\o^{(k)},\a_{k})$
with
$\a_{k}\in\AA_{N_{k}}(\o^{(k)},\O_{k})$ for suitable $\O_{k}\ggt
\O_{k-1}$.

Notice that the functions $V^{(k)}$ are defined up to an additive
constant which we shall choose so that, for all $k\ge 0$:
$$
V^{(k)}={1\over 2\p}\ii_0^{2\p} V^{(k+1)} dx_{j_{k+1}}\=\Vb^{(k+1)}
:\ \T^{N_k}\to\R
\Eq(8.6)$$
Next, we fix the sequence $r_k$ measuring the analyticity domains of the
$u^{(k)}$'s (recall that, in Proposition 7.1, $r'$ is any number between
zero and $r$): Also here the choice is rather arbitrary as the only
requirement is that $r_k\giu r_\io>0$. We shall take:
$$
r_k\= r_0 \big(1-\m \sum_{h=1}^k {1\over h^{\n}}\big)\ ,\qquad
\m\= \big(2 \sum_{h=1}^\io {1\over h^{\n}}\big)^{-1}
\Eq(8.7)$$
where $\n>1$ is a prefixed number (thus $r_\io=r_0/2$).

Now, imagine that $u^{(k)}$, ($k\ge 0$), is given together with
$\o^\pk$ $\in$ $\R^\Nk$, that $u^\pk$ is real--analytic on $\D^\Nk_\rk$,
that $\o^\pk$ is $(\g,N_k)$--Diophantine
and denote $N_k, U_k, \Ub_k, \r_k, \d_k, \bar \d_k, \b_k$
the (obviously) corresponding
objects (see Proposition 7.1; notice that $C,\g,\x_0$ remain fixed in the
construction). Finally, let, for $0<\s\le 3\sqrt{5}/\p^2$ ($\=[2\sum_{k\ge 1}
k^{-4}]^{-{1\over 2}}$)
$$
\Ok\=\max\{4\sqrt{\Nk}\ |\ok|\ ,\  {\d_k (1+k)^2\over
\g \s}\}
\Eq(8.8)$$
and choose
$$
\ak\in\hat \AA_k\=[\Ok, \Ok +K |\ok|]\cap\AA_\Nk(\ok,\Ok)
\Eq(8.9)$$
(recall from \equ(5.5) that $\hat\AA_k$ has positive Lebesgue measure).
>From \equ(8.8) it follows
$$
\sum_{k\ge 0} {\bar \d _k\over (\g\O_k)^2}
\= \sum_{k\ge 0} \Big( {\d_k\over \g \O_k} \Big)^2 (\x_0-\r_k)
< \sum_{k\ge 0} {\d_k\over \g\O_k} \le \s{\p^2\over 6}
\Eq(+)$$
Under the above positions, Proposition 7.1 guarantees the existence of a
non--degenerate quasi--periodic solution with $\ukk:\R^\Nkk\to\T^\Nkk$
(in Proposition 7.1 we set $u\=\uk$, $u'\=\ukk$, $r\=\rk$, $r'\=\rkk$,
\etc) and $\okk\=(\ok,\ak)$; notice in fact that the first of \equ(+)
implies that (recall that $\r_0\=\x_0/2$):
$$
\r_{k+1} \=\r_k + {\bar \d_k \over (\g \O_k)^2} <
{\x_0\over 2} + \x_0 \s^2 \sum_{k\ge 1} {1\over k^4} \le
\x_0\ ,\qquad
\forall\ k
\Eq(*)$$
Now, if we denote by $L$ either the identity operator, or
$\dpr_\thkk$, [$\thkk\=(\th_1,...,\th_\Nkk)$], or $\g D_\okk$,
then the bounds in \equ(7.10) and \equ(+) yield:
$$\eqalign{
\|L\ukk\|_\rkk & \le
\|L u^\po\| +\sum_{h=0}^k\| L\big(u^{(h+1)}-(u^{(h)},0) \big)
\ \|_\rkk\cr
&\le \|Lu^\po\|_{r_0}+ \sum_{k\ge 0} {\d_k\over \g\O_k}\cr
&\le \|L u^\po\|_{r_0} + \s {\p^2\over 6}\cr}
\Eq(8.11)$$
This bound implies, in particular, that, for any $h>0$:
$$
\lim_{k\su \io}\ \sup_{\T^{N_{h+k}}}
\|
L[ u^{(h+k)}-(u^{(k)},\underbrace{0,...,0}_{h \rm \ times})]
\|=0
\Eq(8.12)$$
(recall that $\|a\|\=\sum_{h=1}^N|a_h|$ for $a\in \RN$).
Thus we can {\it define two functions}, $u,Du$, on $\Tw$
as the uniform limits of, respectively,
$\uk$, $D_\ok \uk$: here $Du$ is just a symbol for a function
and $D$ must not be interpreted as a differential operator.
The functions $u,Du$ are
continuous from $\Tw$ into $\Bw$ {\it for any weight $w$}.
In fact, if $\th_h\in\Tw$ converges (in the metric $\r_w$) to $\th$,
and if $\tu$ denote here either $u$ or $Du$,
letting $\bar w$ denote $\sup_i w_i$, we see that:
$$\eqalign{
{1\over\bar w} \sum_{i\in\Zd}|\tu_i(\th_h)-\tu_i(\th)| w_i& \le
\sum_{i\in\Zd}|\tu_i(\th_h)-\tu_i(\th)| \cr
&\le \sum_{i\in\Zd}|\tu_i(\th_h)-\tu_i^{(k)}(\th_h)| +
\sum_{i\in I_k}|\tu_i^{(k)}(\th_h)-\tu_i^{(k)}(\th)| \cr
&\qquad\quad + \sum_{i\in\Zd}|\tu_i^{(k)}(\th)-\tu_i(\th)|\cr}
\Eq(8.13)$$
a quantity that can be made small as we please by taking,
first, $k$ large enough [recall \equ(8.12)] and, then, considering $h$
large enough.
Define also, for each $i\in\Zd$, a function $D^2u_i$ as:
$$
D^2u_i(\th) \= f_i(\th+ u(\th))\ , \qquad \th\in \Tw
\Eq(@)$$
As above $D^2u_i$ is just a symbol for a new function; notice that the
functions $D^2u_i$ are continuous from $\Tw$ into $\R$
(but we dot consider the ``vector--valued"
function $\{D^2u_i\}_{i\in\Zd}$).
Finally define, for any $\th\in\Tw$:
$$
x(t)\=x(t;\th)\=[\th+\o t+ u([\th + \o t])]
\Eq(8.14)$$
where
$$
\o_i\=\left\{
\eqalign{& \o^\po_i\ ,\quad\qquad\qquad i\in I_0\cr
         & \o_{j_k}\=\a_{k}\ ,\qquad i\=j_k\notin I_0\cr}
\right.
\Eq(8.15)$$
>From the above construction follow immediately the following facts:
$$
{\rm (i):}\quad{d\over dt} x_i(t)=\o_i+Du_i([\th + \o t])\ ,\qquad
{d^2\over dt^2} x_i(t)= D^2u_i([\th + \o t])
\Eq(8.15+)$$
(ii): $x(t;\th)$ is solution of \equ(2.6); and (iii):
denoting, for $i\in I_0$, $x_i^\po(t;\th)\=\th_i+\o^\po_i t+
u_i^\po(\th^{(I_0)}+\o^\po t)$
[see \equ(7.10), \equ(8.11), \equ(8.13)]:
$$\eqalign{
& \sup_{{t\in\R}\atop{p=0,1}} |{d^p\over dt^p}
\big(x_i(t;\th)-x_i^\po(t;\th)\big)|\le \s {\p^2\over 6}
\ ,\qquad i\in I_0\cr
& \sup_{{t\in\R}\atop{p=0,1}} |{d^p\over dt^p}
\big(x_i(t;\th)-\o_i t\big)|\le \s {\p^2\over 6}
\ ,\ \qquad\qquad i\notin I_0\cr}
\Eq(8.17)$$
The proof of Theorem 6.3 is completed by taking
$\s\= \min\{ 6\e/\p^2, 3\sqrt{5}/\p^2\}$.
\qed

\vsk
{\bf Remarks 8.1}\quad (i) The above estimates imply that
(choosing $\n$ in \equ(8.7) close enough to $1$) $\Ok\ge (\e\g)^{-1}
k!^{9}$, for all $k$ big enough (the exponent $9$ comes from the
fact that $\g\O_k\ge \d_k (1+k)^2/\s>\d_k \e^{-1}$, from the
$6^{\rm th}$
power of the factorial in \equ(7.9) and from the factor
$(r-r')^{-[11(N+1)+1]}$ $\sim$ $k!^{11 \n}$).
It is also easy to see that, for the
examples \equ(SGa) and \equ(Ex2)
discussed in \S 1, it holds the upper bound $\Ok\le$
$(\e\g)^{-1} b^k$ $k!^{9}$ for a suitable constant $b$ (depending also on
$x^\po(t)$; see [Pe]); thus, in such examples
$$
(\e\g)^{-1} k!^{9}<\o_{j_{k+1}}<(\e\g)^{-1} \bar b^k k!^{9}
\Eq(8.18)$$
for $k$ large enough and for a suitable constant $\bar b$
(recall that $\e>0$ is arbitrary).
As already noticed, this fast growth is intimately related to the
property of $\o$ of being a Diophantine sequence; and (obviously)
$9$ is far from optimal.

(ii) The regularity properties of the
almost--periodic solutions, $x(t)$,  constructed above
are much stronger than just being continuous and having $C^2(\R)$
components $x_i(t)$ (such properties are best reflected by
the approximations via the real--analytic functions $u^{(k)}$).
Notice however that $x_i(t)$ {\it is not $C^3$}.
\vglue2.truecm\penalty-200

\nin{\sectionfont  Appendix 1. Full Measure of Diophantine vectors in
$\RN$}

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\numsec=1
\numfor=1
\penalty10000

\nin
Let $\DD_\t$ denote the set of vectors $\o$ in $\RN$ which
are $(\g,\t)$--Diophantine [see \equ(5.3)] for some $\g>0$
and let $\ell$ be Lebesgue measure.

\vsk
{\bf Proposition A1.1\quad} {\sl $\ell(\RN\backslash
\DD_\t)$=0 provided $\t>N-1$.}

\vsk
{\bf Proof}\quad
It is enough to check that $\lis \DD_{R,\t}\=\{\o:\|\o\|\le R$,
$\o\notin \DD_\t\}$ has Lebesgue measure zero for any $R>0$.
Now, if
$$
\CC_{R,\g}\=\{\o: \|\o\|\le R\  \ {\rm and}\ \ \exist\ 0\neq n\in \ZN:
\ |\o\cdot n|<{1\over \g |n|^\t} \}
\Eqa(a1.1)$$
(recall that $\|\cdot\|$ is the Euclidean norm in $\RN$ while $|\cdot|$
is the sum of absolute values), then $\lis \DD_{R,\t}
=\cap_{\g>0}\CC_{R,\g}$
and the claim follows from the following estimate:
$$\eqalign{
\ell(\CC_{R,\g}) & \le \sum_{n\neq 0} \ell (\{ \o: \|\o\|\le R\ ,\
|\o\cdot n|<{1\over \g |n|^\t}\})\cr
& \le \sum_{n\neq 0} {b_1 R^{N-1}\over \g |n|^\t \|n\|} <
{b_2 R^{N-1}\over \g}\cr}
\Eqa(a1.2)$$
where $b_1,b_2$ are suitable ($N,\t$--dependent) positive constants.
\qed
\vglue2.truecm\penalty-200

\nin{\sectionfont  Appendix 2. A Short KAM Theory}

\penalty10000
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\numsec=2
\numfor=1
\penalty10000

\nin
Here we want to prove in all details Lemma 7.2.

As usual, for vectors in $\CN$ (or $\RN$ or $\ZN$) we denote by
$\|\cdot\|$ the Euclidean norm and by $|\cdot|$ the $1$--norm (sum
of absolute values of components);
recall also the definition of norms of analytic
functions given in \S 7 [\equ(7.5)$\div$\equ(7.7)].

We start with a basic (elementary) tool (a ``Cauchy estimate"):

\vsk
{\bf Lemma A2.1}\quad{\sl Let $g$ be an analytic function from $\DD\subset
\CN\to\C$ ($\dpr \DD$ smooth). Then for any subdomain $\DD'\subset \DD$
with $\d\=$ dist.$(\DD',\dpr \DD)>0$ and any $n\in \NaN$, one has:
$$
\|\dpr_z^n g\|_{\DD'}\=\sup_{\DD'}|{\dpr^{|n|} g\over
\dpr_{z_1}^{n_1}\cdot\cdot\cdot\dpr_{z_N}^{n_N}}|\le |n|!\ \d^{-|n|} \
\|g\|_\DD
\Eqa(a2.1)$$
If $g$ is analytic from $\DD$ into $\LL^p(\CN)$, $p\in\N$, then for any
$q\in \Z_+$, $\dpr^q_zg \in \LL^{p+q}(\CN)$ and
$$\|\dpr_z^q g\|_{\DD'}\le q!\ \d^{-q}\ \|g\|_\DD
\Eqa(a2.2)$$}

\vsk
The proof is a straightforward exercise based upon Cauchy's integral
formula using the contour $|\z_h-z_h|=\d$, $h=1,...,N$ ($z\in \DD'$ fixed
and $\z\in \DD$ variable of integration. The exercise is carried out,
\eg, in [CC2]).

As an application of Lemma A2.1 we prove a useful bound.
If $r\in(0,1)$, $p\in \N$ and $N\in \Z_+$,
one has:
$$
\Si^N_p(r)\=\sum_{n\in\ZN} |n|^pe^{-r|n|}\ \le\  p!\
\big({4\over r}\big)^{p+N}
\Eqa(a2.4)$$

\vsk
{\bf Proof of \equ(a2.4)}\quad For any $r>0$
$$
\Si_p^N=(-1)^p \dpr^p_r \ \sum_{n\in\ZN} e^{-r|n|}= (-1)^p\dpr^p_r
\big({e^r+1\over e^r-1}\big)^N \= (-1)^p \dpr_r^p E(r)^N
\Eqa(a2.5)$$
Now, the function $g(x)\=(-1)^pE(x)^N$ is analytic
and bounded in $B_{r-s}(r)\=\{x\in \C:\ |x-r|<r-s\}$, for any
$0<s<r$. Applying Lemma A2.1with $\DD'\=B_\e(r)$ ($0<\e<r-s$ arbitrary),
$\DD\=B_{r-s}(r)$ and noting that
$\|E\|_\DD=E(s)<1+{2\over s}$,
one gets:
$$
\Si_p^N<\|\dpr^p_x g\|_{B_\e(r)}\le {p!\over (r-s-\e)^p}
\big( 1+{2\over s}\big)^N
\Eqa(a2.6)$$
Inequality \equ(a2.4) follows by taking $s={3\over 4}r$ and using the
arbitrariness of $\e$. \qed

\vsk
An immediate consequence of this bound is:

\vsk
{\bf Lemma A2.2}\quad {\sl Let $p\in \N$; $N,M\in \Z_+$ and consider a
function $g:\TN\to\LL^p(\C^M)$ with zero average and analytic on
$\D^N_\x$ for some $\x<1$. Let $\o\in \RN$ be a $(\g,N)$--Diophantine
vector and denote by $D^{-1}g$ the unique solution with zero average of
the equation $Df=g$ ($D\=\sum_{h=1}^N\o_h\dpr_{\th_h}$). Then, for any
$0<\d<\x$ one has:
$$
\|D^{-1}g\|_{\x-\d}\le \g \  2^{4N} \ N! \d^{-2N} \|g\|_\x
\Eqa(a2.7)$$}

\vsk
{\bf Remark A2.4}\quad R\"u{\ss}mann [R\"u] obtains (with a much more subtle
proof) $N$ as exponent of $\d^{-1}$.

\vsk
{\bf Proof}\quad Using the standard bound on the Fourier coefficients of
analytic functions, $|g_n|\le\|g\|_\x e^{-\x|n|}$, by definition of
$(\g,N)$--Diophantine and by \equ(a2.4) one sees that:
$$\eqalign{
\|D^{-1}g\|_{\x-\d} & = \|\sum_{n\neq 0} {g_n \over i\o\cdot n}
e^{i n\cdot \th}\|_{\x-\d}\le \|g\|_\x \sum_{n\neq 0}
{e^{-\d|n|}\over |\o\cdot n|}\cr
&\le \g \|g\|_\x\sum_{n\neq 0}
|n|^Ne^{-\d|n|}\le \g \|g\|_\x \ 2^{4N}\ N!\ \d^{-2N}\quad
\qed
\cr}
\Eqa(a2.8)$$

\vsk
Lemma 7.2 has an immediate corollary, which has been used in deriving
Theorem 6.1 from Theorem 6.3:

\vsk
{\bf Corollary A2.4}\quad {\sl Let $V$ be as in Lemma 7.2 and let $\o_0$ be a
$(\g,N)$--Diophantine vector. There exists a $0<\m_0<1$ such that, for
any $0<\m<\m_0$, the equation $D_\o^2u=V_x(\th+u)$, with $\o\=\o_0/\m$,
admits a solution.}

\vsk
{\bf Proof}\quad First observe that, since $\m<1$, $\o$ is
$(\g,N)$--Diophantine. Now, take $v\=D^{-2}_\o V_x(\th)$ $\=\m^2
D^{-2}_{\o_0} V_x(\th)$. Then, taking, say, $\x\=\x_0/2$ one
has
$$
\|e\|_\x\=\|D^2v-V_x(\th +v)\|_\x=\|V_x(\th)-V_x(\th+v)\|_\x\le
c \m^2
\Eqa(a2.9)$$
for a suitable $c$ (depending on $\o_0$ and $V$). It is then
obvious that \equ(7.12) can then be achieved by taking $\m$ small
enough.
\qed

\vsk
The proof of Lemma 7.2 is based on a ``Newton scheme" obtained by
iterating the construction of a new ``approximate solution", $v'$ ,
(given a starting approximate solution $v$), producing a
``quadratically smaller" error. Such a Newton scheme is summarized in the
following

\vsk
{\bf Lemma A2.5}\quad {\sl Set $\MM\=(\IN+v_\th)$,
$e\=D^2v-V_x(\th+v)$  and let $z$ be a solution
of
$$
D(\MM^T\MM Dz)=-\MM^Te
\Eqa(a2.10)$$
where $(\cdot)^T$ denotes matrix transposition.
Then, setting:
$$
w\=\MM z\ ,\quad\qquad v'\=v+w
\Eqa(a2.11)$$
the following equation holds:
$$
D^2v'-V_x(\th+v')=e_\th z+q_1+q_2\=e'
\Eqa(a2.12)$$
where
$$\eqalign{
&q_1\=-V_x(\th+v+w)+V_x(\th+v)+V_{xx}(\th+v) w\cr
&q_2\=(\MM^T)^{-1} \AA D z\ ,\qquad\AA\=\MM^T D\MM-(D\MM^T)\MM\cr}
\Eqa(a2.13)$$
Furthermore the matrix--valued function $\AA$ satisfies
$$
D\AA=\MM^T e_\th- e_\th^T \MM\ ,\qquad\quad \langle\AA\rangle=0
\Eqa(a2.14)$$}

\vsk
This is Lemma 1 of [CC1] (with $f$ replaced here by $-V$;
see also [M2], [SZ]), however for completeness we sketch the proof.

\vsk
{\bf Proof of Lemma A2.5}\quad First, using the definition of $e(\th)$, one
checks that \equ(a2.10) makes sense (\ie that the right hand side has
zero mean value over $\TN$). Then, \equ(a2.12) follows easily from the
definitions of $q_1,q_2$ and from
$$
D^2v-V_x(\th+v)=e\ ,\quad\qquad
D^2 \MM-V_{xx}(\th+v)\MM=e_\th
\Eqa(a2.15)$$
the first equation being the definition of $e$ and the second one
being obtained by taking the $\th$--gradient of the first equation.
Finally equation \equ(a2.14) follows easily from the second equation in
\equ(a2.15) and from its transposed; the vanishing of the average of
$\AA$ is checked using the definition of $\AA$ and integration by parts.
\qed

\vsk
{\bf Remark A2.6}\quad The general solution of \equ(a2.10) depends upon an
arbitrary constant $c\in\RN$, which we shall fix by imposing $\langle
w\rangle=0$ (compare (2.6)$\div$(2.8) of [CC1]).

\vsk
{\bf Proof of Lemma 7.2}\quad The idea is to estimate the objects in
Lemma A2.5 and then to iterate the construction infinitely many times
generating a sequence of approximations, $v_j$, ($v_0\=v$,
$v_1\=v+w$,...): Under the ``convergence condition" \equ(7.12),
the ``error function" at the $j^{\rm th}$ step,
$e_j\=D^2v_j-V_x(\th+v_j)$, will be quadratically smaller than the
preceding error function $e_{j-1}$, so that $\|e_j\|\giu 0$
and $v_j$ converges to the solution $u$.

We start by performing estimates on $w,v',e'$ defined in Lemma A2.5
(but keep in mind that the corresponding estimates at the $j^{\rm th}$
step of the procedure are basically identical, having replaced $v$ with
$v_j$, $e$ with $e_j$ and $\x$ with $\hat \x_j\giu \x'$ to be suitably
defined below).

In the following bounds $1\le B_1\le B_2\le...\le B$ will denote suitable
{\it universal constants}.

Fix $0<\d<\x$ and let $z$ be the unique solution of \equ(a2.10) such
that $\langle w\rangle =0$ (see the above Remark A2.6); then,
using Lemma A2.1 and Lemma A2.2, one obtains,
with suitable constants $B_h$:
$$\eqalign{
&\|Dz\|_{\x-{\d\over 2}}\le B_1\  2^{6N}\  N!\ \d^{-2N} \ \h^2\hb^4\ \g^{-1}
\ \e\cr
&\|z\|_{\x-{\d}}\le B_2 \ 2^{12N}\  N!^2\ \d^{-4N} \ \h^4\hb^4\ \e\cr
&\|w\|_{\x-{\d}}\le B_3 \ 2^{12N}\  N!^2\ \d^{-4N} \ \h^5\hb^4\
\ \e\cr}
\Eqa(a2.16)$$
(to check the powers of $\h$ and $\hb$, one needs to take into account the
``constants of integration" coming from solving $D f=g$ with $\langle
g\rangle=0$; compare (2.7), (2.8) of [CC1]).
Now, observe that \equ(7.12) implies that $\|w\|_{\x-\d}<\x_0-\x_0'$ so
that  $e'$ is analytic on $\D^N_{\x-\d}$ and, using again Lemma A2.1 and
Lemma A2.2, one obtains the bounds:
$$\eqalign{
&\|e_\th z\|_{\x-\d}\le B_4\  2^{12N}\
N!^2 \ \d^{-(4N+1)} (\h\hb)^4\  \g^{-2}
\e^2\cr
&\|q_1\|_{\x-\d}\le\|V_{xxx}\|_{\x_0}\  \|w\|^2_{\x-\d}\le
B_5\|V_{xxx}\|_{\x_0} \ 2^{24N} N!^4\  \d^{-8N} \h^{10}\hb^8\e^2\cr
&\|\AA\|_{\x-\d}\le \|D^{-1}(\MM^T e_\th-e_\th^T\MM)\|_{\x-\d}\le
B_6 2^{6N}\  N!\  \d^{-(2N+1)} \h\  \g^{-1} \e\cr
&\|q_2\|_{\x-\d}\le B_7\  2^{10N} N!^2\
\d^{-(4N+1)}\  \h^3\hb^5 \g^{-2}
\e^2\cr}
\Eqa(a2.17)$$
where to bound $\|\AA\|$ we used Lemma A2.2 with $\d$ replaced by $\d/2$
(so as to be able to bound subsequently $\|e_\th\|$).
Thus,  observing that $\d^{-1},\h,\hb$ are $\ge 1$, we get [see
\equ(7.11)]:
$$
\e'\=\g^2\|e'\|_{\x-\d}\le B_8\   2^{24N}\  N!^4\  \d^{-8N} \b\
\h^{10} \hb^8\ \e^2
\Eqa(a2.18)$$
On $\D^N_{\x-2\d}$ we can bound immediately, by Lemma A2.1, the
derivatives of $w$; hence we get:
$$
\max\{\| w\|_{\x-\d}\ ,\ \|w_\th\|_{\x-2\d}\}\le B_3 \
2^{12N} N!^2\  \d^{-(4N+1)} \ \h^5\hb^4\
\e
\Eqa(a2.19)$$
As already mentioned, we iterate now the above construction and relative
estimates, setting $w_0\=v$, $v_j\=\sum_{i=0}^j w_i$, $\d\=\d_0$ and:
$$
\d_j\={\x-\x'\over 2^{j+2}}\ ,\qquad \hat \x_j\=\hat \x_{j-1}-2\d_{j-1}\=
\x-2\sum_{i=0}^{j-1} \d_i=\x'+{\x-\x'\over 2^j}
\Eqa(a2.20)$$
(notice that $\hat \x_0\=\x$ and that $\hat \x_j\giu \x'$; the choice of the
``analyticity--losses" $\d_j$ is rather arbitrary, however the choice
made here turns out to be particularly simple and also good for ``sharp
estimates", see [CC1]). We also set $\h_j\=\|\IN+\dpr_\th v_j\|_{\hat \x_j}$,
$\hb_j\=\|(\IN+\dpr_\th v_j)^{-1}\|_{\hat \x_j}$ and assume
inductively that $\forall$  $0\le j\le j_0$:
$$
\h_j<2\h\ ,\qquad \hb_j<2\hb\ ,\qquad
\sum_{i=1}^j\|w_j\|_{\hat \x_j}<\x_0-\x_0'
\Eqa(a2.21)$$
last inequality being necessary for the definition (and analyticity)
of $e_{j+1}$. By the (analogous of the) above estimates
(where $\e\to \e_j$, $\e'\to \e_{j+1}$, $\d\to\d_j$, $\h\to\h_j$,
$\bar \h\to\bar \h_j$, $\x\to\hat \x_j$, $\x-2\d\to
\hat \x_{j+1}$), we see that,
setting
$$
F\=B_9 2^{40 N} N!^4 (\x-\x')^{-8N} \ \h^{10}\hb^8\ \b\ ,\quad G\=2^{8N}
\Eqa(a2.22)$$
we obtain, for $j\le j_0$:
$$\eqalign{
\e_{j+1} & \=
\g^2\|e_{j+1}\|_{\hat \x_{j+1}}\=\g^2\|D^2v_j-V_x(\th+v_j)\|_{\hat \x_{j+1}}
\le F G^j \e_j^2\cr
&\le \e^{2^{j+1}}\prod_{i=0}^j(F G^{j-i})^{2^j}\cr
&=\big[\e F^{(\sum_{i=1}^{j+1}2^{-i})}
G^{(\sum_{i=1}^{j+1}(i-1)2^{-i})}\big]^{2^{j+1}}<(\e F G)^{2^{j+1}}\cr}
\Eqa(a2.23)$$
(notice that both sums in the last line of the above formula converge to
one as $j\to\io$). Analogously:
$$\eqalign{
\max\{\|w_{j+1}\|_{\hat \x_j-\d_j}\ ,\ \|\dpr_\th w_{j+1}\|_{\hat \x_{j+1}}
\} &\le
B_{10} 2^{12N} N!^2 \big({2^{j+2}\over \x-\x'}\big)^{4N}\
\h^5 \hb^4 \ \e_j\cr
&\= F_0 G_0^j\e_j\le (\e FG)^{2^j}\cr}
\Eqa(a2.24)$$
having used that $F_0\le F$, $G_0\le G$ (and that $\sum_{i\ge j+1}
2^{-i}=2^{-j}$, $\sum_{i\ge j+1}(i-1)2^{-i}=(j+1) 2^{-j}$).
We check now the induction hypotheses \equ(a2.21) for $j=j_0+1$.
We find:
$$
\h_{j_0+1}\le \h+\sum_{j=0}^{j_0} \|\dpr_\th w_j\|_{\hat \x_j}\ ,\qquad
\hb_{j_0+1}\le\hb \big( 1- \hb \sum_{j=1}^{j_0} \|\dpr_\th w_j\|_{\hat \x_j}
\big)^{-1}
\Eqa(a2.25)$$
Thus, (recalling that $\x_0<1$), all we need to complete the check is:
$$
\sum_{j=0}^\io(\e FG)^{2^j}<\x_0-\x_0'
\Eqa(a2.26)$$
which is easily seen to be implied, for a suitable
$B_{11}\ge 2B_{10}$, by
$$
B_{11} N!^4 2^{40N} (\x-\x')^{-8N} \ \h^{10} \hb^8
\b\ (\x_0-\x_0')^{-1}\e<1
\Eqa(a2.27)$$
This shows that the sequences $\dpr^p_\th v_j
\=\sum_{i=0}^j\dpr^p_\th w_i$,
($p=0,1$), converge uniformly on $\D^N_{\x'}$ and, for $p=0$,
$v_j$ converge to a function $u$, which satisfies (by construction)
\equ(7.13).  Let us now estimate $\|D^pu\|_{\x'}$. First, notice that
$w_{j+1}$ and
$e_{j+1}$ are analytic and bounded on $\hat \x_j-\d_j$ so that
$D^2w_{j+1}$,
which is equal to $V_x(\th+v_j+w_{j+1})-V_x(\th+v_j)+e_{j+1}-e_j$,
can be bounded by:
$$
\g^2\|D^2 w_{j+1}\|_{\hat \x_j-\d_j}\le B_{12} \b\ 2^{12N} N!^2 \ \d_j^{-4N}
\ \h^5 \hb^4 \e_j
\Eqa(a2.28)$$
having used that $\h_j< 2\h$, $\hb_j<2\hb$ and that $B_8 2^{12N}
N!^2 \d_j^{-4N} \h^5\ \hb^4\e_j\le 1$. Then, by Lemma A2.2, it follows that
$$
\g\|Dw_j\|_{\hat \x_j-2\d_j}=\g\|D^{-1}D^2w_j\|_{\hat \x_{j+1}}\le
B_{12} \ \b\ 2^{16N} N!^3 \d_j^{-6N} \ \h^5\hb^4 \e_j
\Eqa(a2.29)$$
The bounds \equ(7.14) now follow easily.
\qed
\vglue2.truecm\penalty-200

\nin{\ninebf Acknowledgements} {\ninerm We are deeply indebted
with C. Liverani for many helpful discussions and especially
for suggesting the right form of the ``approximate solution"
in Theorem 7.1. We are also grateful to C. E. Wayne for
useful remarks on a previous version of this paper.}

\vglue2.truecm\penalty-200

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\penalty10000
\vskip0.5truecm
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