% This TeX file was generated by /usr/local/bin/texpsinclude
% from the original TeX file la-us.tex
% and the PostScript files 6b.eps example.eps.
% TeX writes out the included PostScript to files.
%
% Here is part of la-us.tex:
\documentstyle[11pt,amssymb,epsf]{amsart}
% Here are the
% TeX macros for dumping included Postscript to files.
% Adapted from Knuth's \answer macro in the TeXbook.
% Jamie Stephens, jamies@math.utexas.edu, 28 Nov 94
\def\endofps{EndOfTheIncludedPostscriptMagicCookie}
\chardef\other=12
\newwrite\psdumphandle
\outer\def\psdump#1{\par\medbreak
\immediate\openout\psdumphandle=#1
\copytoblankline}
\def\copytoblankline{\begingroup\setupcopy\copypsline}
\def\setupcopy{\def\do##1{\catcode`##1=\other}\dospecials
\catcode`\\=\other \obeylines}
{\obeylines \gdef\copypsline#1
{\def\next{#1}%
\ifx\next\endofps\let\next=\endgroup %
\else\immediate\write\psdumphandle{\next} \let\next=\copypsline\fi\next}}
\outer\def\closepsdump{
\immediate\closeout\psdumphandle}
% Here is the PostScript for 6b.eps:
\message{Writing file 6b.eps}
\psdump{6b.eps}%!PS-Adobe-2.0 EPSF-2.0
%%Title: 6b.fig
%%Creator: fig2dev Version 3.1 Patchlevel 1
%%CreationDate: Sat Aug 31 22:42:03 1996
%%For: renato@linux (Renato Iturriaga)
%%Orientation: Landscape
%%BoundingBox: 0 0 268 166
%%Pages: 0
%%BeginSetup
%%IncludeFeature: *PageSize Letter
%%EndSetup
%%EndComments
/$F2psDict 200 dict def
$F2psDict begin
$F2psDict /mtrx matrix put
/col-1 {} def
/col0 {0.000 0.000 0.000 srgb} bind def
/col1 {0.000 0.000 1.000 srgb} bind def
/col2 {0.000 1.000 0.000 srgb} bind def
/col3 {0.000 1.000 1.000 srgb} bind def
/col4 {1.000 0.000 0.000 srgb} bind def
/col5 {1.000 0.000 1.000 srgb} bind def
/col6 {1.000 1.000 0.000 srgb} bind def
/col7 {1.000 1.000 1.000 srgb} bind def
/col8 {0.000 0.000 0.560 srgb} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
/col19 {0.690 0.000 0.000 srgb} bind def
/col20 {0.820 0.000 0.000 srgb} bind def
/col21 {0.560 0.000 0.560 srgb} bind def
/col22 {0.690 0.000 0.690 srgb} bind def
/col23 {0.820 0.000 0.820 srgb} bind def
/col24 {0.500 0.190 0.000 srgb} bind def
/col25 {0.630 0.250 0.000 srgb} bind def
/col26 {0.750 0.380 0.000 srgb} bind def
/col27 {1.000 0.500 0.500 srgb} bind def
/col28 {1.000 0.630 0.630 srgb} bind def
/col29 {1.000 0.750 0.750 srgb} bind def
/col30 {1.000 0.880 0.880 srgb} bind def
/col31 {1.000 0.840 0.000 srgb} bind def
end
save
-77.0 -179.0 translate
90 rotate
1 -1 scale
/clp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/l {lineto} bind def
/m {moveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
4 -2 roll dup 1 exch sub 3 -1 roll mul add
4 -2 roll dup 1 exch sub 3 -1 roll mul add
4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
4 -2 roll mul srgb} bind def
/DrawEllipse {
/endangle exch def
/startangle exch def
/yrad exch def
/xrad exch def
/y exch def
/x exch def
/savematrix mtrx currentmatrix def
x y tr xrad yrad sc 0 0 1 startangle endangle arc
closepath
savematrix setmatrix
} def
/DrawSplineSection {
/y3 exch def
/x3 exch def
/y2 exch def
/x2 exch def
/y1 exch def
/x1 exch def
/xa x1 x2 x1 sub 0.666667 mul add def
/ya y1 y2 y1 sub 0.666667 mul add def
/xb x3 x2 x3 sub 0.666667 mul add def
/yb y3 y2 y3 sub 0.666667 mul add def
x1 y1 lineto
xa ya xb yb x3 y3 curveto
} def
/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def
%%EndProlog
$F2psBegin
10 setmiterlimit
0.06000 0.06000 sc
7.500 slw
% Open spline
n 3000.00 1500.00 m 3300.00 1575.00 l
3300.00 1575.00 3600.00 1650.00 3937.50 1950.00 DrawSplineSection
3937.50 1950.00 4275.00 2250.00 4425.00 2512.50 DrawSplineSection
4575.00 2775.00 l gs col0 s gr
n 4541.51 2655.93 m 4575.00 2775.00 l 4489.42 2685.69 l 4515.96 2671.31 l 4541.51 2655.93 l clp gs 0.00 setgray ef gr gs col0 s gr
% Ellipse
n 3672 2358 30 30 0 360 DrawEllipse gs 0.00 setgray ef gr gs col-1 s gr
% Ellipse
n 3834 1863 30 30 0 360 DrawEllipse gs 0.00 setgray ef gr gs col-1 s gr
30.000 slw
% Interp Spline
[116.7] 0 setdash
n 3450 2700 m
3726.99 2145.10 4656.30 1253.84 5175 1500 curveto
5657.76 1729.10 5573.74 2920.07 5325 3450 curveto
5044.46 4047.68 4042.45 4980.55 3525 4725 curveto
3049.64 4490.24 3182.89 3235.11 3450 2700 curveto
clp gs col-1 s gr
[] 0 setdash
7.500 slw
% Open spline
n 3017.00 1350.00 m 3189.50 1387.50 l
3189.50 1387.50 3362.00 1425.00 3579.50 1533.50 DrawSplineSection
3579.50 1533.50 3797.00 1642.00 3879.50 1694.50 DrawSplineSection
3879.50 1694.50 3962.00 1747.00 4123.50 1642.00 DrawSplineSection
4123.50 1642.00 4285.00 1537.00 4491.00 1436.00 DrawSplineSection
4491.00 1436.00 4697.00 1335.00 4914.50 1308.50 DrawSplineSection
4914.50 1308.50 5132.00 1282.00 5327.00 1398.50 DrawSplineSection
5327.00 1398.50 5522.00 1515.00 5608.50 1736.00 DrawSplineSection
5608.50 1736.00 5695.00 1957.00 5717.50 2208.50 DrawSplineSection
5717.50 2208.50 5740.00 2460.00 5728.50 2632.50 DrawSplineSection
5728.50 2632.50 5717.00 2805.00 5683.50 2988.50 DrawSplineSection
5683.50 2988.50 5650.00 3172.00 5575.00 3382.00 DrawSplineSection
5575.00 3382.00 5500.00 3592.00 5353.50 3806.00 DrawSplineSection
5353.50 3806.00 5207.00 4020.00 5083.50 4170.00 DrawSplineSection
5083.50 4170.00 4960.00 4320.00 4742.50 4458.50 DrawSplineSection
4742.50 4458.50 4525.00 4597.00 4281.00 4646.00 DrawSplineSection
4281.00 4646.00 4037.00 4695.00 3804.50 4590.00 DrawSplineSection
3804.50 4590.00 3572.00 4485.00 3512.00 4331.00 DrawSplineSection
3512.00 4331.00 3452.00 4177.00 3444.50 3903.50 DrawSplineSection
3444.50 3903.50 3437.00 3630.00 3478.50 3412.50 DrawSplineSection
3478.50 3412.50 3520.00 3195.00 3602.50 2962.50 DrawSplineSection
3602.50 2962.50 3685.00 2730.00 3838.50 2527.50 DrawSplineSection
3838.50 2527.50 3992.00 2325.00 4108.50 2467.50 DrawSplineSection
4108.50 2467.50 4225.00 2610.00 4307.50 2741.00 DrawSplineSection
4390.00 2872.00 l gs col-1 s gr
n 4351.44 2754.47 m 4390.00 2872.00 l 4300.67 2786.45 l 4326.55 2770.96 l 4351.44 2754.47 l clp gs 0.00 setgray ef gr gs col-1 s gr
/Symbol findfont 270.00 scalefont setfont
3345 4950 m
gs 1 -1 sc 270.0 rot (S) col-1 show gr
/Times-Roman findfont 270.00 scalefont setfont
3495 4980 m
gs 1 -1 sc 270.0 rot (^) col-1 show gr
/Times-Italic findfont 270.00 scalefont setfont
3642 1914 m
gs 1 -1 sc 270.0 rot (q) col-1 show gr
/Times-Italic findfont 270.00 scalefont setfont
3450 2190 m
gs 1 -1 sc 270.0 rot (p) col-1 show gr
$F2psEnd
restore
EndOfTheIncludedPostscriptMagicCookie
\closepsdump
% Here is the PostScript for example.eps:
\message{Writing file example.eps}
\psdump{example.eps}%!PS-Adobe-2.0 EPSF-2.0
%%Title: example.fig
%%Creator: fig2dev Version 3.1 Patchlevel 1
%%CreationDate: Sat Aug 31 22:51:34 1996
%%For: renato@linux (Renato Iturriaga)
%%Orientation: Portrait
%%BoundingBox: 0 0 382 120
%%Pages: 0
%%BeginSetup
%%IncludeFeature: *PageSize Letter
%%EndSetup
%%EndComments
/$F2psDict 200 dict def
$F2psDict begin
$F2psDict /mtrx matrix put
/col-1 {} def
/col0 {0.000 0.000 0.000 srgb} bind def
/col1 {0.000 0.000 1.000 srgb} bind def
/col2 {0.000 1.000 0.000 srgb} bind def
/col3 {0.000 1.000 1.000 srgb} bind def
/col4 {1.000 0.000 0.000 srgb} bind def
/col5 {1.000 0.000 1.000 srgb} bind def
/col6 {1.000 1.000 0.000 srgb} bind def
/col7 {1.000 1.000 1.000 srgb} bind def
/col8 {0.000 0.000 0.560 srgb} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
/col19 {0.690 0.000 0.000 srgb} bind def
/col20 {0.820 0.000 0.000 srgb} bind def
/col21 {0.560 0.000 0.560 srgb} bind def
/col22 {0.690 0.000 0.690 srgb} bind def
/col23 {0.820 0.000 0.820 srgb} bind def
/col24 {0.500 0.190 0.000 srgb} bind def
/col25 {0.630 0.250 0.000 srgb} bind def
/col26 {0.750 0.380 0.000 srgb} bind def
/col27 {1.000 0.500 0.500 srgb} bind def
/col28 {1.000 0.630 0.630 srgb} bind def
/col29 {1.000 0.750 0.750 srgb} bind def
/col30 {1.000 0.880 0.880 srgb} bind def
/col31 {1.000 0.840 0.000 srgb} bind def
end
save
-64.0 206.0 translate
1 -1 scale
/clp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/l {lineto} bind def
/m {moveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
4 -2 roll dup 1 exch sub 3 -1 roll mul add
4 -2 roll dup 1 exch sub 3 -1 roll mul add
4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
4 -2 roll mul srgb} bind def
/DrawEllipse {
/endangle exch def
/startangle exch def
/yrad exch def
/xrad exch def
/y exch def
/x exch def
/savematrix mtrx currentmatrix def
x y tr xrad yrad sc 0 0 1 startangle endangle arc
closepath
savematrix setmatrix
} def
/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def
%%EndProlog
$F2psBegin
10 setmiterlimit
0.06000 0.06000 sc
7.500 slw
% Ellipse
n 3900 2400 600 300 0 360 DrawEllipse gs col-1 s gr
% Ellipse
n 5700 2400 600 300 0 360 DrawEllipse gs col-1 s gr
% Ellipse
n 2115 2415 570 330 0 360 DrawEllipse gs col-1 s gr
% Ellipse
n 2100 2400 30 30 0 360 DrawEllipse gs 0.00 setgray ef gr gs col-1 s gr
% Ellipse
n 3885 2415 30 30 0 360 DrawEllipse gs 0.00 setgray ef gr gs col-1 s gr
% Ellipse
n 5700 2400 30 30 0 360 DrawEllipse gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 2168 1503 m 2013 1458 l 2063 1508 l 2013 1548 l clp gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 3960 1504 m 3805 1459 l 3855 1509 l 3805 1549 l clp gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 5770 1505 m 5615 1460 l 5665 1510 l 5615 1550 l clp gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 5658 3290 m 5813 3335 l 5754 3288 l 5813 3245 l clp gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 3810 3370 m 3965 3415 l 3906 3368 l 3965 3325 l clp gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 2030 3295 m 2185 3340 l 2126 3293 l 2185 3250 l clp gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 1327 2540 m 1384 2630 l 1421 2592 l clp gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 1416 2174 m 1324 2225 l 1370 2260 l 1390 2220 l clp gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 3138 2507 m 3056 2457 l 3093 2537 l gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 2895 2472 m 2857 2551 l 2932 2510 l gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 2889 2246 m 2929 2322 l 2847 2276 l 2859 2285 l gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 3071 2289 m 3135 2251 l 3109 2326 l gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 4942 2507 m 4860 2457 l 4897 2537 l gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 4711 2460 m 4673 2539 l 4748 2498 l gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 4868 2294 m 4932 2256 l 4906 2331 l gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 4694 2252 m 4734 2328 l 4652 2282 l 4664 2291 l gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 6437 2531 m 6399 2610 l 6474 2569 l gs 0.00 setgray ef gr gs col-1 s gr
% Polyline
n 6446 2198 m 6486 2274 l 6404 2228 l 6416 2237 l gs 0.00 setgray ef gr gs col-1 s gr
% Interp Spline
n 1200 2400 m
1615.25 1950.00 1840.25 1800.00 2100 1800 curveto
2792.66 1799.99 3207.34 3000.01 3900 3000 curveto
4592.66 2999.99 5007.34 1800.01 5700 1800 curveto
5959.75 1800.00 6184.75 1950.00 6600 2400 curveto
gs col-1 s gr
% Interp Spline
n 1200 2400 m
1615.25 2850.00 1840.25 3000.00 2100 3000 curveto
2792.66 3000.00 3207.34 1800.01 3900 1800 curveto
4592.66 1799.99 5007.34 3000.01 5700 3000 curveto
5959.75 3000.00 6184.75 2850.00 6600 2400 curveto
gs col-1 s gr
% Interp Spline
n 1200 2850 m
1373.74 2936.87 1448.74 2974.37 1500 3000 curveto
1636.69 3068.34 1908.23 3300.00 2100 3300 curveto
2387.65 3300.00 2704.27 2840.47 3000 2850 curveto
3306.22 2859.87 3575.85 3384.72 3900 3375 curveto
4236.51 3364.91 4463.49 2785.09 4800 2775 curveto
5124.15 2765.28 5396.48 3287.30 5700 3300 curveto
5903.27 3308.51 6188.52 3072.89 6330 3000 curveto
6376.97 2975.80 6444.47 2938.30 6600 2850 curveto
gs col-1 s gr
% Interp Spline
n 1200 1950 m
1373.74 1863.13 1448.74 1825.63 1500 1800 curveto
1636.69 1731.66 1908.23 1500.00 2100 1500 curveto
2387.65 1500.00 2712.35 1950.00 3000 1950 curveto
3287.65 1950.00 3604.27 1490.47 3900 1500 curveto
4206.22 1509.87 4484.58 2025.00 4800 2025 curveto
5115.42 2025.00 5383.89 1499.25 5700 1500 curveto
5906.05 1500.49 6151.19 1767.32 6285 1845 curveto
6338.61 1876.12 6417.36 1921.12 6600 2025 curveto
gs col-1 s gr
/Times-Roman findfont 225.00 scalefont setfont
6600 2685 m
gs 1 -1 sc (A) col-1 show gr
/Times-Roman findfont 225.00 scalefont setfont
4704 2706 m
gs 1 -1 sc (C) col-1 show gr
/Times-Roman findfont 225.00 scalefont setfont
2925 2712 m
gs 1 -1 sc (B) col-1 show gr
/Times-Roman findfont 225.00 scalefont setfont
1080 2616 m
gs 1 -1 sc (A) col-1 show gr
$F2psEnd
restore
EndOfTheIncludedPostscriptMagicCookie
\closepsdump
% Finally, here is the rest of la-us.tex:
% \documentstyle[11pt,amssymb,epsf,verbatim]{amsart}
%\usepackage{showkeys}
\swapnumbers
\openup 3pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\topmargin -0.2cm %* Page Layout
\headheight 0.7cm
\headsep 1.0cm
\topskip 0.0cm
%\textheight 22.0cm
%\evensidemargin 0.6cm
%\oddsidemargin 0.6cm
%\textwidth 15.0cm
\textheight 7.6in
\textwidth 5.7in
\evensidemargin 1.0cm
\oddsidemargin 1.0cm
%\parindent 0.0cm %* No indentation
\parskip 5pt plus2pt minus1pt %* But space between paragraphs
\renewcommand {\thefigure}{\thesubsection.\arabic{figure}}
\renewcommand {\textfraction}{0} %* No text needed on a page
\sloppy
%\frenchspacing %* Uncomment for german and french
\renewcommand {\baselinestretch}{1.0} %* Distance between lines
%\pagestyle{empty} %* Uncomment for no page-numbering
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def \a{\alpha}
\def \A{{\rm A}}
\def \be{\beta}
\def \Be{\rm B}
\def \ga{\gamma}
\def \Ga{\Gamma}
\def \de{\delta}
\def \e{\varepsilon}
\def \la{\lambda}
\def \La{\Lambda}
\def \vr{\varphi}
\def \om{\omega}
\def \Om{\Omega}
\def \lleft?{>}
\def \bpi{\boldsymbol{\pi}}
\def \fs{{\frak S}}
\def \SH{{\widehat{\Sigma}(L)}}
\def \re{{\Bbb R}}
\def \na{{\Bbb N}}
\def \nulo{\o}
\def \lv{\left\vert}
\def \rv{\right\vert}
\def \lV{\left\Vert}
\def \rV{\right\Vert}
\def \ov{\overline}
\def \ul{\underline}
\def \then{\Longrightarrow}
\def \lqqd{\quad\blacksquare}
\def \dsize{\displaystyle}
\def \tsize{\textstyle}
\def \ssize{\scriptstyle}
\def \sssize{\scriptscriptstyle}
\def \HSL{\widehat{\Sigma}(L)}
\def \SLC{S_{L+c}}
\def \M{{\Bbb M}}
\def \tM{{\widetilde{M}}}
\def \tx{{\widetilde{x}}}
\theoremstyle{plain}
\newtheorem{Thm}{\bf Theorem}[section]
\newtheorem{Prop}[Thm]{\bf Proposition}
\newtheorem{Lemma}[Thm]{\bf Lemma}
\newtheorem{Cor}[Thm]{\bf Corollary}
\newtheorem{Corollary}[Thm]{\bf Corollary}
\newtheorem{Theorem}[Thm]{\bf Theorem}
\theoremstyle{definition}
\newtheorem{remark}[Thm]{\bf Remark}
\begin{document}
\title[Lagrangian Flows]
{\large \bf Lagrangian Flows: The Dynamics of \\
Globally Minimizing Orbits - II}
\author[G. Contreras]{Gonzalo Contreras}
\address{Depto. de Matem\'atica. PUC-Rio\\
R. Marqu\^es de S\~ao Vicente, 225 \\
22453-900 Rio de Janeiro \\
Brasil.}
\email{gonzalo@@mat.puc-rio.br}
\author[J. Delgado]{Jorge Delgado}
\address{Depto. de Matem\'atica. PUC-Rio\\
R. Marqu\^es de S\~ao Vicente, 225 \\
22453-900 Rio de Janeiro \\
Brasil.}
\email{jdelgado@@mat.puc-rio.br}
\author[R. Iturriaga]{Renato Iturriaga}
\address{CIMAT \\
A.P. 402, 3600 \\
Guanajuato. Gto. \\
M\'exico.}
\email{renato@@fractal.cimat.mx}
\thanks{G. Contreras was partially supported by CNPq-Brazil.}
\thanks{R. Iturriaga was partially supported by Conacyt-Mexico,
grant 3398E9307.}
\maketitle
\noindent{\it To the memory of Ricardo Ma\~n\'e.}
\bigskip
\noindent
{\large \bf Introduction.}
\bigskip
In this work we prove most of the theorems of Ma\~n\'e's
unfinished work ``Lagrangian Flows the dynamics of
Globally Minimizing Orbits'',~\cite{Ma3}.
Exceptions are theorem III,
whose proof is divided in~\cite{Ma2} and~\cite{CI}
and theorem IV which was proved in \cite{Ma3}. Also, we provide proofs
for slightly different statements of theorems VII, XI and XIV.
We would like to emphasize that all the theorems in this paper are due
to Ma\~n\'e and all the
responsibility of the proofs is ours.
We encourage the reader to use Ma\~n\'e's original paper \cite{Ma3}
as the introduction of this work.
In section 1 we prove theorems I and II, in section 2 we prove theorem V,
in section 3 we prove theorems VI, VII, VIII and IX, in section 4 we prove
theorems X and XI, and in section 5 we prove theorems XII, XIII and XIV.
The first and second authors want to thank the hospitality of CIMAT.
We want to use
this space to say how much we admired Ricardo's clearness and
brightness and how grateful we are to his enormous generosity. This paper is
to his memory.
\stepcounter{section}
%{\large \bf Section \thesection. }
\bigskip
\bigskip
\centerline{\large \bf \thesection. Basic properties of the critical value.}
\bigskip
Let $M$ be a smooth closed manifold. We say that a smooth function
$L:TM\to \re$ is a {\it Lagrangian} if it satisfies the following conditions:
\begin{itemize}
\item[(a)]{\it Convexity:\/} For all $x\in M$, $v\in T_xM$, the Hessian matrix
$\frac{\partial^2 L}{\partial v_i \partial v_j}(x,v)$
(calculated with respect to linear coordinates on $T_xM$)
is positive definite.
\item[(b)] {\it Superlinearity:\/}
$\lim\limits_{\lV v\rV\to +\infty}
\frac{L(x,v)}{\lV v\rV}=+\infty$.
\end{itemize}
Given an
absolutely continuous curve $x:[0,T]\to M$ define its $L$-action by
$$
S_L(x) := \int_a^b L\left(x(t), \dot{x}(t)\right) \, dt \, .
$$
Fixing $p,q\in M$ and $T>0$, the critical points of the action functional
on the set
$$
AC(p,q,T):=\{ \, x:[0,T]\to M\,\vert\, x(0)=p,\, x(T)=q,\, x \text{ absolutely
continuous }\}.
$$
are solutions of the Euler-Lagrange equation, which in local coordinates
is given by
\begin{equation*}
\frac{d}{dt}L_v=L_x \,. \tag{E-L}
\end{equation*}
Because of the convexity of the Lagrangian this equation can be thought
as a first order differential equation on $TM$. The {\it Lagrangian flow}
$f_t$ on $TM$ is defined by $f_t(x,v)=(\ga(t),\dot{\ga}(t))$, where $\ga$ is
the solution of (E-L) with $\ga(0)=x$ and $\dot{\ga}(0)=v$.
Define the {\it energy \/} function $E:TM\to\re$ as
$$
E(x,v) := \frac{\partial L}{\partial v}(x,v) \cdot v - L \,.
$$
%where $v = \frac{d\,}{dt}(tv)\vert_{t=1}$.
It can be seen that the value of $E(x,v)$ is constant along the orbits
of $f_t$. The superlinearity condition implies that the level sets of the
energy function have bounded velocities and hence that are compact.
This, in turn, implies that the solutions of (E-L) are defined for all values
$t\in\re$, i.e., that the flow $f_t$ is {\it complete}.
Finally, for $p,q\in M$, let
$$
AC(p,q) := \bigcup_{T>0} AC(p,q,T)\, ,
$$
and define the {\it action potential \/} as
$$
\Phi_k(p_1,p_2) := \inf \,\left\{ \, S_{L+k}(x) \, \vert \,
x\in AC(p_1,p_2)\, \right\}\, , \, k\in \re\, .
$$
\bigskip
\noindent{\bf Theorem I.}
{\it
There exists $c(L)\in \re$ such that
\begin{itemize}
\item[(a)] $k< c(L) \then \Phi_k(p_1,p_2) = -\infty\, ,
\quad \forall p_1\, ,\, p_2\in M$.
\item[(b)] $k\ge c(L) \then \Phi_k(p_1,p_2) > -\infty \, ,
\quad \forall p_1\, ,\, p_2\in M\,$ and $\Phi_k$ is Lipschitz.
\item[(c)] $k \ge c(L) \then$
\begin{align*}
&\Phi_k(p_1,p_3) \le \Phi_k(p_1,p_2) + \Phi_k(p_2,p_3)
\quad \forall p_1\, ,\, p_2\, ,\, p_3 \in M \\
&\Phi_k(p_1,p_2) + \Phi_k(p_2,p_1) \ge 0
\quad \forall p_1\, ,\, p_2 \in M
\end{align*}
\item[(d)] $k>c(L) \then \Phi_k(p_1,p_2)+\Phi_k(p_2,p_1) > 0
\quad \forall p_1\, \neq \, p_2\,$.
\end{itemize}
}
\medskip
\begin{remark} This theorem, with the same proof, holds for coverings
$\pi: \widehat{M} \to M$ of a compact manifold $M$, with the lifted
Lagrangian $\widehat{L}=L\circ \pi$.
\end{remark}
\noindent{\bf Proof:}
We first prove that if for some $p_1$, $p_2\in M$,
$\Phi_k(p_1,p_2) = - \infty$, then
$\Phi_k(q_1,q_2) = -\infty$ for all $q_1, q_2\in M$. Let $\ga$, $\eta$:
$[0,1]\to M$, $\ga \in AC( q_1,p_1)$, $\eta\in AC(p_2,q_2)$. Let
$x_n\in AC(p_1,p_2)$ be such that $\lim\limits_{n\to\infty}
S_{L+k}(x_n) = -\infty$. Then
$$
\lim_{n\to\infty}S_{L+k}(\ga * x_n * \eta) = S_{L+k}(\ga) + S_{L+k}(\eta)
+\lim_{n\to\infty} S_{L+k}(x_n) = -\infty\, .
$$
Thus the number
$$
c(L) :=\inf\,\left\{ \, k\in\re \, \vert\, \Phi_k(p,q) > -\infty\, \right\}
$$
does not depend on $(p,q)$. We have to see that $-\infty < c(L) < +\infty$.
Since the function $k\mapsto \Phi_k(p,q)$ is nondecreasing, it is enough
to see that there exist $k_1$, $k_2\in\re$ such that $\Phi_{k_1}(p,q)=-\infty$
and $\Phi_{k_2}(p,q)>-\infty$. We first prove the existence of $k_1$.
Since $L$ is bounded on $\left\{ (x,v)\in TM \, \vert \, v\in T_x M\, ,\,
\lv v\rv \le 2\, \right\}\,$, there exists $B>0$ such that
\begin{equation}\label{E:I.1}
\lv L(x,v)\rv < B \quad \text{ if } \quad \lv v \rv <2\, .
\end{equation}
Let
$x_n:[0,n]\to M$, $x_n\in AC(p,q)$ be such that $|\dot x_n|\le 2$.
Then $S_L(x_n)\le B\, n$ and hence, for $k_1 = - B -1$, we have that
\begin{align*}
\Phi_{k_1}(p,q) &\le \lim\inf_n S_{L-B-1}(x_n) \le \lim\inf_n
\left( \int_0^n L(x_n,\dot{x}_n)\, dt - (B+1)n\right)\\
&\le\lim\inf_n \bigl( Bn - (B+1)n\bigr) = -\infty.
\end{align*}
Now we prove the existence of $k_2$. The superlinearity hypothesis
implies that $L$ is bounded below. Let $A$ be a lower bound for $L$ on $TM$.
We claim that it is enough to take $k_2> -A+1$. Indeed
$$
S_{L+k_2}(x) \ge \int_0^T (A + k_2) \, dt \ge 0
\quad\text{ for all } x\in AC(p,q).
$$
hence $\Phi_{k_2}(p,q) \ge 0$.
It remains to prove that $\Phi_c(p,q)>-\infty$ for all $p$, $q\in
M$, where $c=c(L)$.
Suppose not. Take $p\in M$, then $\Phi_c(p,p)=-\infty$. Let
$\ga\in AC(p,p)$ be such that $S_{L+c}(\ga)<-a<0$.
Then there exists $\e>0$ such that $S_{L+c+\e}(\ga)<-\frac 12 a <0$.
Let $\de_N:=\ga *\overset{N}\ldots *\ga$, then
$$
\Phi_{c+\e}(p,p)\le \lim_N S_{L+c+\e}(\de_N)\le\lim_N
-{\textstyle{\frac 12}} \, a \, N = -\infty\, .
$$
This contradicts the definition of $c(L)$. In particular, we have also proven
that $\Phi_c(p,p)\ge 0$ for all $p\in M$. By taking $\ga\in AC(p,p)$ with
bounded velocities and arbitrarily small parameter intervals, we have that
\begin{equation} \label{E:I.phi_c(p,p)=0}
\Phi_c(p,p)=0\quad\text{ for all } p\in M\, .
\end{equation}
Similarly
\begin{equation} \label{E:I.phi_k(p,p)=0}
\Phi_k(p,p)=0\quad\text{ for all } p\in M\, \text{ for all } k\ge c(L).
\end{equation}
We now prove (c). Let $k\ge c(L)$ and $p_1$, $p_2$, $p_3\in M$. Let
$x^{ij}_n :[0,T_n]\to M$, $x^{ij}_n\in AC(p_i,p_j)$, be such that
$$
\lim_n S_{L+k}(x^{ij}_n) = \Phi_k (p_i, p_j) \quad i,j\in\{1,2,3\}.
$$
Then $x^{12}_n*x^{23}_n\in AC(p_1,p_3)$ and
$$
\Phi_k(p_1,p_3)\le S_{L+k}(x^{12}_n * x^{23}_n) =
S_{L+k}(x^{12}_n) + S_{L+k}(x^{23}_n).
$$
Taking the limit when $n\to\infty$ we get that
\begin{equation}\label{E:I.3}
\Phi_k(p_1,p_3)\le \Phi_k(p_1,p_2) + \Phi_k(p_2,p_3)\, .
\end{equation}
Finally using \eqref{E:I.phi_k(p,p)=0} and \eqref{E:I.3} we obtain that
\begin{equation}
\label{4}
0=\Phi_k(p_1,p_1)\le \Phi_k(p_1,p_2)+\Phi_k(p_2,p_1)
\end{equation}
when $k\ge c(L)$.
We prove that $\Phi_k$ is Lipschitz when $k\ge c(L)$.
Let $\ga:[0,d(p,q)]\to M$ be a geodesic joining $p$ and $q$, using
\eqref{E:I.1} we obtain
\begin{align}
\Phi_k(p,q) &\le \int_0^{d(p,q)}
\left[ L\left( \ga(t),\dot{\ga}(t)\right)+k\right]\, dt \notag\\
\Phi_k(p,q) &\le \left( B+k\right) \,\, d(p,q)
\quad \text{ for }\quad k\ge c(L)\, .\label{E:I.5}
\end{align}
Therefore if $k\ge c(L)$,
\begin{align*}
\Phi_k(p_1,p_2) - \Phi_k(q_1,q_2)
&\le \Phi_k(p_1,q_1) + \Phi_k(q_1,p_2) - \Phi_k(q_1,q_2) \\
&\le \Phi_k(p_1,q_1) + \Phi_k(q_1,q_2) + \Phi_k(q_2,p_2) - \Phi_k(q_1,q_2) \\
&\le \Phi_k(p_1,q_1) +\Phi_k(q_2,p_2) \\
&\le (B +k)\bigl( d(p_1,q_1) + d(p_2,q_2) \bigr).
\end{align*}
Changing the roles of $p_i$ and $q_i$, $i=1,2$ we get that
$$
\lv \Phi_k(p_1,p_2) - \Phi_k(q_1,q_2)\rv \le (B+k) \, \bigl( d(p_1,q_1) +
d(p_2,q_2) \bigr)\, .
$$
We now prove that if $k\ge c(L)$ and $p\ne q$, then the function
$k\mapsto \Phi_k(p,q)$ is strictly increasing. By \eqref{4} this
implies (d). Let $p\ne q$ and $\ell>k\ge c(L)$. Let $x_n:[0,T_n]\to
M$, $x_n\in AC(p,q)$ be such that $\lim_n S_{L+\ell}(x_n)
= \Phi_\ell (p,q)$. We have that
\begin{align*}
S_{L+\ell}(x_n) &= S_{L+k}(x_n) + (\ell - k) T_n \\
\Phi_\ell (p,q) &\ge \Phi_k(p,q) + (\ell - k) \, \lim\inf_n T_n
\end{align*}
It is enough to prove that $\lim\inf_n T_n > 0$, because then
$\Phi_\ell(p,q) > \Phi_k (p,q)$. Suppose that $\lim\inf_n T_n = 0$.
By the superlinearity of $L$, for all $B>0$ there exists $A>0$ such that
$$
\lv L(x,v)\rv \ge B\, \lv v\rv - A\, .
$$
Then
\begin{align*}
\Phi_\ell(p,q) &=\lim_n S_{L+\ell}(x_n) \ge \lim\inf_n\left[
\int_0^{T_n}B\,\lv \dot{x}_n\rv \, dt + (k-A)\, T_n\,\right] \\
& \ge B\, d(p,q) + 0\, ,
\end{align*}
for all $B>0$. Therefore $\Phi_{L+\ell}(p,q) = +\infty$, which contradicts
\eqref{E:I.5}.
\qed
Through the rest of the paper we shall neeed the following results:
\begin{Theorem}{\bf (Mather \cite{Mather})}\label{L:MATHER1}
For all $C>0$ there exists $A_1=A_1(C)$ such that if $T>0$, $p,\, q\in M$ and
$x\in AC(p,q,T)$ satisfy
\begin{itemize}\begin{itemize}\begin{itemize}
\item[(a)] $S_L(x) = \min\{\, S_L(y)\,\vert\, y\in AC(p,q,T)\,\}$.
\item[(b)] $S_L(x)\le C\, T$.
\end{itemize}\end{itemize}\end{itemize}
Then
\begin{itemize}\begin{itemize}\begin{itemize}
\item[(c)] $\lV \dot{x}(t)\rV < A_1$ for all $t\in [0,T]$.
\item[(d)] $x\vert_{[0,T]}$ is a solution of (E-L).
\end{itemize}\end{itemize}\end{itemize}
\end{Theorem}
\medskip
\begin{Corollary}\label{lemmavel}
There exists $A>0$ such that if $T>1$, $p,q\in M$ and $x\in AC(p,q,T)$
satisfy
$$
S_L(x) = \min\{\, S_L(y)\, \vert\, y\in AC(p,q,T)\,\}\,,
$$
then $\lV \dot{x}(t)\rV0$ such that if $p,q\in M$ and $x\in AC(p,q,T)$
satisfy
\begin{itemize}\begin{itemize}\begin{itemize}
\item[(a)] $S_L(x) = \min\{\, S_L(y)\,\vert\, y\in AC(p,q,T)\,\}$.
\item[(b)] $S_L(x)<\Phi_c(p,q) + d_M(p,q)$.
\end{itemize}\end{itemize}\end{itemize}
Then
\begin{itemize}\begin{itemize}\begin{itemize}
\item[(c)] $T>\frac 1A\, d_M(p,q)\,.$
\item[(d)] $\lV\dot{x}(t)\rV 0$ there exists $E>0$ such that
$$
\lv L(x,v)\rv > D\, \lv v\rv -E \quad \text{for all } (x,v)\in TM.
$$
We have that
\begin{align*}
B\, d_M(p,q) &> S_L(\ga) \ge \Phi_c(p,q) \ge S_L(x)-d_M(p,q) \\
&> \int_0^T (D\,\lv\dot{x}(t)\rv - E)\, dt - d_M(p,q) \\
&> D\, d_M(p,q) - E\, T - d_M(p,q)\,.
\end{align*}
Hence
\begin{equation}\label{E:C2.1}
T> \frac{(D-B-1)}{E} \, d_M(p,q) = \left(\frac BE\right)\, d_M(p,q)\,.
\end{equation}
Now let
$$
C:= \max\{\,\lv L(x,v)\rv\,\vert\, \lv v\rv\le \tfrac EB\, \}\,.
$$
Let $\eta\in AC(p,q,T)$ be a minimal geodesic. Then, by \eqref{E:C2.1},
$$
\lv \dot{\eta}\rv \equiv \frac{d_M(p,q)}{T} < \frac EB
$$
and hence
$S_L(\eta)\le C\, T$. Let $A_1=A_1(C)$ be from theorem \ref{L:MATHER1}, then
it is enough to use
$$
A= \max\left\{\, A_1(C),\, \tfrac EB\, \right\}.
$$
\qed
\smallskip
Let ${\cal M}(L)$ be the set of invariant borel probability
measures for the Lagrangian flow.
\medskip
\noindent{\bf Theorem II.}
$$
c(L) = - \min \left\{\left. \int L \, d\mu \, \right\vert\,
\mu\in{\cal M}(L)\, \right\}.
$$
\noindent{\bf Proof:}
Let $\mu\in{\cal M}(L)$ be ergodic. Let $(p,v)\in TM$ be such that
$$
\lim_{T\to\infty} \frac 1T \int_0^T L\bigl( f_t(p,v)\bigr) \, dt = \int L \, d\mu\, .
$$
Let $B>0$ be such that
$$
\lv L(x,v) \rv < B \quad \text{ if } \lv v \rv \le 2 \, .
$$
For $N>0$ let $q_N := \pi\bigl(f_N(p,v)\bigr)$ and let
$\ga_N:[0,d(p,q_N)]\to M$ be a geodesic joining $q_N$ to $p$. Let
$x_N:[0,N]\to M$ be defined by $x_N(t) = \pi \bigl( f_t (p,v) \bigr)$.
Then $f_t(p,v) = \bigl( x_N(t), \dot{x}_N(t)\bigr)$. For $k\in \re$,
we have that
\begin{align*}
S_{L+k}(\ga_N) &= \int_0^{d(p,q_N)} L
\bigl( \ga_N(t), \dot{\ga}_N(t) \bigr)\, dt
\le (B+k)\, \text{diam}(M)\, .\\
\lim_N \frac 1N S_{L+k}(x_N * \ga_N) &= \lim_N \frac 1N S_{L+k}(x_N) +0 \\
&= S_{L+k}(\mu) = S_L(\mu) + k \, .
\end{align*}
If $k< - S_L(\mu)$, then
$$
\Phi_k(p,p) \le \lim_N S_{L+k}(x_N * \ga_N) = -\infty\, .
$$
Hence $k\le c(L)$. Therefore
\begin{align*}
c(L) &\ge \sup\, \left\{ - S_L(\mu) \, \vert \,
\mu\in {\cal M}_{\text{erg}}(L)\, \right\} \\
&\ge - \min \left\{ S_L(\mu) \, \vert \, \mu \in{\cal M}(L)\, \right\}\, .
\end{align*}
Now let $k0$ for all $n$,
then $S_L(\mu)+k\le 0$.
For any $kc(L)$ this remark follows from the following estimates:
\begin{align*}
+\infty > \max_{p,q\in M} \Phi_k(p,q)
&\ge S_{L+k}(x\vert_{[0,T]}) = \Phi_k(x(0),x(T)) \\
&\ge \Phi_c(x(0),x(T)) + (k-c)\, T \\
&\ge \max_{p,q\in M}\Phi_c(p,q) + (k-c)\, T.
\end{align*}
The following theorem is proven in Ma\~n\'e \cite{Ma3}.
\medskip
\noindent{\bf Theorem IV. (Characterization of minimizing measures)}
{\it A measure $\mu\in{\cal M}(L)$ is minimizing if and only if
$\text{supp}(\mu)\subseteq \widehat{\Sigma}(L)$.
}
\medskip
We include a proof below, using theorem V(c).
\bigskip
Given an $f_t$ invariant subset $\La\subseteq TM$, and
$\e>0$, $T>0$, an {\it $(\e,T)$-chain joining
$\xi_a\in\La$ and $\xi_b\in\La$} is a finite sequence
$\{(\zeta_i,t_i)\}_{i=1}^N\subset \La\times\re$ such that
$\zeta_1=\xi_a$, $\zeta_{N+1}=\xi_b$, $t_i>T$ and
$d(f_{t_i}(\zeta_i),\zeta_{i+1})<\e$ for $i=1,\ldots,N$.
We say that a set $\La\subset TM$ is {\it chain transitive}
if for all $\xi_a,\,\xi_b\in\La$, and all $\e>0$, $T>0$ there exists
and $(\e,T)$-chain in $\La$ joining $\xi_a$ and $\xi_b$.
When this condition holds only for all $\xi_a=\xi_b\in\La$ we say
that $\La$ is {\it chain recurrent}.
\bigskip
\noindent{\bf Theorem V. (Recurrence Properties)}
{\it
\begin{enumerate}
\item[a)] $\Sigma(L)$ is chain transitive.
\item[b)] $\widehat{\Sigma}(L)$ is chain recurrent.
\item[c)] The $\alpha$ and $\om$-limit set of a semistatic orbit
is contained in $\widehat{\Sigma}(L)$.
\end{enumerate}
}
\noindent {\bf Proof:}
Lets first prove (c).
Let $w\in\Sigma$ and let $u\in \om(w)$.
We prove that $\om(w)\subseteq \widehat{\Sigma}(L)$, the proof
that $\a(w)\subseteq \widehat{\Sigma}(L)$ is similar.
It is enough to prove that for all
$T>0$, $x_u\vert_{[0,T]}$ is static. Let $p:= x_u(0)$, $q:=x_u(T)$. Let
$p_n=x_w(s_n)$, $q_n=x_w(t_n)$ be sequences of points in $M$ with
$s_n0$, $S>0$, $u,\, v\in\Sigma(L)$, we have to find an $(\e,S)$-chain in
$\Sigma(L)$ joining $u$ to $v$. It is easy to see that such
$(\e,S)$-chain exists if $\om(u)\cap\a(v)\ne\o$. Let
\begin{equation}\label{E:A&O}
{\A} := \a(v) \quad , \quad \Om := \om(u)
\end{equation}
and suppose that $\A\cap\Om =\o$. Let
$p\in\pi(\Om)$, $q\in\pi(\A)$.
Let $\eta_n:=[0,T_n]\to M$ be such that
$$
p := \eta_n(0)\in\pi(\Om)
\quad ,\quad
q := \eta_n(T_n)\in\pi(\A)
$$
\begin{equation}\label{estrella}
\SLC (\eta_n) \le \Phi_c(p,q) + \frac 1n
\end{equation}
By corollary \ref{corvel2}, we can assume that
$\eta_n$ is a solution of (E-L) and satisfies
\begin{equation}\label{E:EtaIsBounded-1}
\lv \dot{\eta}_n(t) \rv < \A
\quad \text{ for all }
0\le t \le T_n
\end{equation}
Given $\frac \e 2 > \de_0>0$ there exists $0<\de<\de_0$ such that
if $\lv v\rv, \; \lv w\rv <\A$ and $d_{TM}(v,w)<\de$ then
\begin{equation}\label{sol}
d_{TM}(f_t(v),f_t(w))<\de_0 \text{ for all } \lv t \rv\le S.
\end{equation}
Let $\M$ be the union of $\A$, $\Om$ and the set of accumulation points
of the tangent vectors of the
$\eta_n$'s:
\begin{equation*}
\M := \A \cup \Om \cup \left\{\text{$v$} \in TM \, \left\vert \,
\begin{array}{lc}
\exists\, \langle n_k \rangle\subseteq\na &0\le t_{n_k}\le T_{n_k} \\
\exists\, \langle t_{n_k}\rangle\subseteq\re &v=\lim_k\dot{\eta}_{n_k}(t_{n_k})
\end{array}
\right.\right\}\,.
\end{equation*}
Then
\begin{equation}\label{E:N5.6}
{\Bbb M} \subseteq \{\, v\in TM\,\vert\, \lV v\rV \le A\, \}\,.
\end{equation}
We shall need the following lemmas
\begin{Lemma}\label{L:MinSigma-1}
\end{Lemma}
{\it
\begin{itemize}
\item[(a)] $\M\subseteq \Sigma(L)$.
\item[(b)] $\M$ is invariant.
\end{itemize}
}
Let ${\Bbb K}$ be the set of vectors which are on the $\om$-limit of vectors
of ${\Bbb M}$.
$$
{\Bbb K} := \bigcup_{v\in{\Bbb M}}\bigl[\om(v)\bigr]\, .
$$
Since $\Bbb M$ is closed and foward invariant, then the closure $\ov{\Bbb
K}\subseteq {\Bbb M} \subseteq \Sigma(L)$. Moreover, the vectors in ${\Bbb K}$
are chain recurrent. By \eqref{E:N5.6} ${\ov{\Bbb K}}$ is
compact. Given $\de>0$ let
$$
{\Bbb K}_\de :=
\bigl\{\, v\in TM\, \big\vert\, d_{TM}(v,\ov{\Bbb K})<\de\,\bigr\}\,.
$$
Since $\ov{\Bbb K}$ is compact, the number of connected
components of ${\Bbb K}_\de$ is finite.
Let $\La_i=\La_i(\de)$, $i=0,1,\ldots,N$, $\Om\subseteq \La_0$, be the
connected components of ${\Bbb K}_\de$.
\begin{Lemma}\label{L:N5.1}
Each component $\La_i$ is $(2\de,T)$-chain transitive for any $T>0$.
\end{Lemma}
If $\A\subset \La_0$ the proposition is proved. Suppose that $\A \cap \La_0
=\o$. Consider the oriented graph $\boldsymbol{\Gamma}$ with vertices
$\La_i$, $i=0,1,\ldots,N$
and an edge $\La_i\to\La_j$ if there exists $v\in \M$ and $t_i0$. Since $\La_i = \La_i(\de)$ is open and connected, it is pathwise
connected. Let $\xi$, $\zeta\in\La_i$ and let $\Ga:[0,S]$ be a continuous path
such that $\Ga(0)=\xi$, $\Ga(s)=\zeta$. Let $0=s_0T$ such that
$d(f_{\tau_i}(v_i),v_i)<\frac{\de}2$. We have that
$$
d(f_{\tau_i}(v_i),v_{i+1})
0$ there exists $N=N(\de)>0$ and $S=S(\de)>0$
such that if $n>N$, then for all $c\in[0,T-n]$ we have that
$$
{\Bbb K}_\de \cap \bigl\{(\eta_n(t),\dot{\eta}_n(t))\, \vert \,
t\in[c-S,c+S]\cap [0,T_n]\,\bigr\} \ne \o \,.
$$
\end{Lemma}
\medskip
\noindent{\bf Proof:}
Suppose it is not true. Then there exists $\delta >0$ and sequences
$n_k\to\infty$ and $c_k\in[0,T_{n_k}]$, such that $0N(\de)$ we have that $00$ there exist $T=T(\ga)>0$ and
$\ga = \ga_{\e,s,t}:[0,T]\to M$ such that $\ga(0)=x_w(t)$, $\ga(T)=x_w(s)$ and
$$
\SLC (\ga) \le -\Phi_c\bigl( x_w(t),x_w(s) \bigr) + \e
$$
Let
\begin{equation}\label{E:Etas}
\eta_n := \ga_{\frac 1n , -n , n}
\qquad\qquad , \qquad\qquad
T_n := T\bigl(\ga_{\frac 1n , -n , n}\bigr)
\end{equation}
We can assume that $\lv \dot{\eta}(t)\rv < A$
for all $0\le t\le T_n$.
The rest of the proof is similar to item (a), but now the corresponding
${\Bbb M}\subseteq \widehat{\Sigma}(L)$.
\qed
\stepcounter{section}
\bigskip
\bigskip
\newpage
\centerline{\large \bf \thesection. Graph, covering and coboundary properties. }
\bigskip
\noindent{\bf Theorem VI. (Graph Properties)}
{\it
\begin{itemize}
\item[(a)] If $\ga(t)$, $t\ge 0$ is an orbit in $\Sigma^+(L)$, then, denoting
$\pi:TM\to M$ the canonical projection, the map $\pi\vert_{\{\dot{\ga}\vert
t>0\}}$ is {\it injective with Lipschitz inverse}.
\item[(b)] Denoting $\Sigma_0(L)\subset M$, the projection of $\SH $, for every
$p\in\Sigma_0(L)$ there exists a {\it unique} $\xi(p)\in T_pM$ such that
$$
( p,\xi(p))\in \Sigma^+(L)\, .
$$
Moreover
$$
(p,\xi(p))\in \SH \, ,
$$
and the vector field $\xi$ {\it is Lipschitz}. Obviously
$$
\SH = \text{\rm graph}(\xi)\, .
$$
\end{itemize}
}
\medskip
For a proof of the following lemma see Mather~\cite{Mather}
or Ma\~n\'e \cite{Ma0}.
\begin{Lemma}{\bf (Mather)}\label{L:MATHER2}
Given $A>0$ there exists
$K>0$ $\e_1>0$ and $\de>0$ with the following property: if
$\lv v_i\rv S_L(\ga_1)+S_L(\ga_2)
\end{align*}
\end{Lemma}
\noindent{\bf Proof of theorem VI:}
{\bf (a)}
Since the curve $\ga$ is semistatic, corollary \ref{corvel2} implies that
there exists $A>0$ such that $\lv\dot{\ga}(t)\rv0$, $\e_1=1$, $\de>0$ be from lemma \ref{L:MATHER2}.
We prove that if $(p,v)$, $(q,w)\in \{\,\dot{\ga}(t)\,\vert\, t>0\,\}$ and
$d_M(p,q)<\de$, then
$$
d_{TM}\bigl((p,v),(q,w)\bigr) K\,
d_M\bigl(\ga(t_1),\ga(t_2)\bigr)
\end{align*}
By lemma \ref{L:MATHER2} there exist $0<\e0}\pi\, f_t\,\Ga^+\,\, .
$$
\noindent{\bf Theorem VII. (Covering Property)}
{\it
\begin{itemize}
\item[(a)] $\pi\Sigma^+(L) = M$.
\item[(b)] For all $p\in\Ga^+_0$, there exists a {\it unique}
$\xi_\Ga(p)\in T_pM$ such that
$$
(p,\xi_\Ga(p))\in\Ga^+\, .
$$
Moreover, $\xi_\Ga$ is
Lipschitz.
\end{itemize}
}
Observe that $\pi:\Sigma^+(L)\to M$ is not necessarily injective.
We recall that Ma\~n\'e stated item (a) in a stronger form:
$\pi \Gamma =M$ for every
equivalence class $\Gamma $. This may not be true as
the following example shows. Let
$M=S^1=\re/{\Bbb Z}$ be the unit circle and $L=\frac{1}{2} v^2 -\cos 6\pi x$.
Then the three maximums of the potential $A=0=2\pi$, $B=\frac{2\pi}3$ and
$C=\frac{4\pi}3$ are singular (hyperbolic saddle) points of the Lagrangian
flow.
For {\it mechanical Lagrangians} $L=\frac 12 v^2 - \phi(x)$, we have that
$$
c(L) = e_0 = - \min_{x\in M} L(x,0)\,,
$$
and the static points are the critical points $(x,0)$ of the Lagrangian flow
such that $L(x,0)=-e_0$. In this example $c(L)= 1$ and
$\widehat{\Sigma}(L)=\{(A,0),(B,0),(C,0)\}$.
We shall prove below that $d_1(A,B)=\Phi_1(A,B)+\Phi_1(B,A)>0$. Hence
$(A,0)$ and $(B,0)$ are not in the same equivalence class. By \eqref{$*$},
$\Sigma^+(L)\subseteq E^{-1}\{c(L)\}$, hence if $\Ga$ is the equivalence class
of $(A,0)$, then $\pi\, \Ga^+ \subseteq [0,\frac{2\pi}{3}[\cup
]\frac{2\pi}{3},2\pi]\neq S^1$. This example can easily generalized
to higher dimensions.
\centerline{\epsfxsize=8cm\epsfbox{example.eps}}
%\centerline{Theorem 6b.}
We show now that in the example above $d_1(A,B)>0$.
By theorem V, the set $\Sigma(L)$ is chain transitive and contains
$\widehat{\Sigma}(L)=\{(A,0),(B,0),(C,0)\}$.
Moreover, by \eqref{$*$}, $\Sigma^+(L)\subseteq E^{-1}\{c(L)\}$.
In our example this is only possible if $\Sigma(L)$ contains
complete components
of the saddle connections (so that $\pi\Sigma^+(L)=M$).
\, From the symmetry of this Lagrangian we get that
$\Sigma(L)$ contains all the saddle connections.
Hence $\Sigma(L)=E^{-1}\{ c(L)\}$.
Let $(x,\dot{x})$ be the orbit on $E^{-1}\{c(L)\}$ with $\a$-limit
$(A,0)$ and $\om$-limit $(B,0)$. Then it is semistatic and hence
\begin{gather*}
L(x,\dot{x})+c(L) = \dot{x}\, L_v(x,\dot{x}) = \lv \dot{x}\rv^2,\\
\Phi_1(A,B)=\lim_{T\to +\infty} S_{L+c}(x\vert_{[-T,+T]})
=\int_{-\infty}^{+\infty} \lv\dot{x}(t)\rv^2 dt > 0 \,.
\end{gather*}
The same argument gives that $\Phi_1(B,A)>0$ and thus $d_1(A,B)>0$.
\bigskip
\noindent{\bf Proof of theorem VII:}
We prove (a).
We may assume that $\pi \SH\ne M$ otherwise the proof is trivial.
Let $p\in M\setminus \pi \SH$ and $q\in \pi \SH \ne \o$.
For each $n\in{\Bbb N}$ let $\gamma_n :[0,T_n] \to M$ be such that
\begin{enumerate}
\item $\gamma_n(0)=p$, $\gamma_n (T_n) =q$.
\item $S_{L+c}(\ga_n) \le \Phi_c(p,q) +\frac{1}{n}$.
\item $\gamma_n$ minimizes $L+c$ in $AC(p,q,T_n)$.
\end{enumerate}
Then
\begin{itemize}
\item[(i)] $\ga_n$ is a solution of (E-L).
\item[(ii)] $\lv\dot{\ga}_n\rv < A$ for all $t\in [0,T_n]$.
\end{itemize}
Since $\SH$ is invariant under the Lagrangian flow, item (b) of theorem VI
implies that
\begin{itemize}
\item[(iii)] $T_n\to +\infty$.
\end{itemize}
Let $\eta(t):=\pi f_t(p,v)$. By (i) and (iii), for any fixed $T>0$
we have that $\ga_n\vert_{[0,T]}\to \eta\vert_{[0,T]}$ in the $C^1$
topology, and hence
$$
S_{L+c}(\eta\vert_{[0,T]}) = \lim
S_{L+c}(\ga_n\vert_{[0,T]})\,.
$$
Clearly $\eta(0)=p$. It is enough to show that $\eta\vert_{[0,+\infty[}$
is semistatic. For, we have that
\begin{align*}
S_{L+c}(\ga_n) &=
S_{L+c}(\ga_n\vert_{[0,T]}) +
S_{L+c}(\ga_n\vert_{[T,T_n]}) \\
&\le \Phi_c(p,q) + \tfrac 1n \\
&\le \Phi_c(p,\ga_n(T)) + \Phi_c(\ga_n(T),q) + \tfrac 1n \\
&\le \Phi_c(p,\ga_n(T))
+ S_{L+c}(\ga_n\vert_{[T,T_n]})+\tfrac 1n\,.
\end{align*}
Hence
$$
S_{L+c}(\ga_n\vert_{[0,T]}) \le \Phi_c(p,\ga_n(T)) + \tfrac 1n\,.
$$
Taking the limit when $n\to +\infty$ we obtain that $\eta$ is semistatic.
We now prove item (b). Let $A>0$ be from
corollary \ref{corvel2} and $K=K(A)>0$, $\e_1=\e_1(A)>0$,
$\de=\de(A)>0$ be from
Mather's lemma \ref{L:MATHER2}.
It is enough to prove that if $p_1,\,p_2\in\Ga^+_0$, $v_1,\, v_2\in\Ga^+$,
$\pi(v_i)=p_i$, $i=1,2$ and $d_M(p_1,p_2)<\de$, then
$d_{TM}(v_1,v_2)\le K\,d_M(p_1,p_2)$.
Suppose it is false. Then there exists $p_i\in\Ga^+_0$, $v_i\in\Ga^+$,
$\pi(v_i)=p_i$ such that $d_M(p_1,p_2)<\de$ and
$d_{TM}(v_1,v_2)> K\, d_M(p_1,p_2)$.
Since $p_i\in\Ga^+_0$, there exists $0<\e<\e_1$ such that
$f_t(v_i)\in\Ga^+$ for $t>-\e$. Let $q_i:=\pi f_{-\e}(v_i)$,
$r_i:=f_{+\e}(v_i)$. By Mather's lemma \ref{L:MATHER2}, there exist
$\eta_i:[-\e,\e]\to M$ such that $\eta_i(-\e)=q_i$, $\eta_i(+\e)=r_i$
and
\begin{alignat}{5}
\SLC(\eta_1)\, &+& \SLC(\eta_2)\, &<&
\,\SLC(x_{v_1}\vert_{[-\e,+\e]})\, &+&\, \SLC(x_{v_2}\vert_{[-\e,+\e]}).
\notag \\
\intertext{Thus}
\Phi_c(q_1,r_2)\, &+&\, \Phi_c(q_2,r_1) &<&\, \Phi_c(q_1,r_1)\quad &+&
\Phi_c(q_2,r_2).\quad
\label{E:S3.A}
\end{alignat}
Let $u_i\in\om(v_i)$ and $z_i:=\pi(u_i)$. If
$T^i_n\overset{n}{\longrightarrow}+\infty$
is such that $f_{T^i_n}(v_i)\overset{n}{\longrightarrow}u_i$,
we have that
\begin{align*}
\Phi_c(q_i,z_i) &= \lim_n \Phi_c(q_i,x_{v_i}(T^i_n))
= \lim_n \SLC(x_{v_i}\vert_{[-\e,T^i_n]}) \\
&= \SLC(x_{v_i}\vert_{[-\e,+\e]})
+ \lim_n \SLC(x_{v_i}\vert_{[+\e,T^i_n]}) \\
&\ge \Phi_c(q_i,r_i) + \Phi_c(r_i,z_i)\,.
\end{align*}
By the triangle inequality for $\Phi_c$ we get that
\begin{equation}\label{E:S3.B}
\Phi_c(q_i,z_i) = \Phi_c(q_i,r_i) + \Phi_c(r_i,z_i)
\end{equation}
\, From \eqref{E:S3.B} and \eqref{E:S3.A} we have that
\begin{align}
\Phi_c(q_1,z_1) + \Phi_c(q_2,z_2)
&= \Phi_c(q_1,r_1) + \Phi_c(r_1,z_1)
+ \Phi_c(q_2,r_2) + \Phi_c(r_2,z_2)
\notag \\
&> \Phi_c(q_1,r_2) + \Phi_c(q_2,r_1)
+ \Phi_c(r_1,z_1) + \Phi_c(r_2,z_2)
\notag \\
&\ge \Phi_c(q_1,z_2) + \Phi_c(q_2,z_1)\,.
\label{E:S3.C}
\end{align}
Since $u_i\in \Ga$, then $d_c(z_1,z_2)=\Phi_c(z_1,z_2)+\Phi_c(z_2,z_1)=0$.
Adding $d_c(z_1,z_2)=0$ to the right of inequality \eqref{E:S3.C}, we obtain
that
\begin{align*}
\Phi_c(q_1,z_1) + \Phi_c(q_2,z_2)
&> \Phi_c(q_1,z_2) + \Phi_c(z_2,z_1)
+ \Phi_c(z_1,z_2) + \Phi_c(q_2,z_1) \\
&\ge \Phi_c(q_1,z_1) + \Phi_c(q_2,z_2)\, .
\end{align*}
This is a contradiction.
\qed
\bigskip
\noindent
{\bf Theorem VIII. (Generic Structure of $\SH $)}
{\it
For a generic Lagrangian $L$, $\SH $ is a uniquely ergodic set.
If it is a periodic orbit then it is a hyperbolic periodic orbit.
}
\medskip
\noindent {\bf Proof:}
This should be thought as a corollary of theorems III and IV.
Take the Generic set given by theorem III of Lagrangians $L$
that satisfy $ \# \widehat{\cal M}(L)=1$ and
call this unique minimizing measure $\mu (L)$.
Then if $\mu$ is an invariant measure of $L$ and it is
supported in $ \SH $ then by theorem IV
it is minimizing. Thus $\mu =\mu (L)$. This proves the theorem.
\qed
\bigskip
\newpage
\noindent
{\bf Theorem IX. (Coboundary Property)}
{\it
If $c=c(L)$, then $(L+c)\big\vert_\SH$ is a Lipschitz coboundary. More
precisely, taking any $p\in M$ and defining $G:\SH\to\re$ by
$$
G(w)=\Phi_c\bigl(p,\pi(w)\bigr)
$$
Then
$$
(L+c)\big\vert_\SH = \frac{dG}{df} \, ,
$$
where
$$
\frac{dG}{df}(w):= \lim_{h\to 0}\frac 1h
\bigl[\, G\bigl(f_h(w)\bigr)-G(w)\, \bigr]\, .
$$
}
\medskip
%\newpage
\noindent{\bf Proof:}
Let $w\in\SH$ and define $F_w(v):=\Phi_c\bigl(\pi(w),\pi(v)\bigr)$. We have
that
\begin{align*}
\left.\frac{dF_w}{df}\right\vert_w
&= \lim_{h\to 0}\frac 1h\, \bigl[\, F_w(f_h\, w) - F_w(w)\,\bigr] \\
&= \lim_{h\to 0}\frac 1h\, \bigl[\, \Phi_c(\pi w, \pi f_h\, w)
- \Phi_c(\pi w, \pi w)\, \bigr] \\
&= \lim_{h\to 0}\frac 1h\,
\left[\, S_{L+c}\left(x_w\vert_{[0,h]}\right)\,\right] \\
&= \lim_{h\to 0} \frac 1h\,
\int_0^h \bigl[L\bigl( x_w(s), \dot{x}_w(s)\bigr)\,+c\bigr]\, ds \\
&= L(w)+c
\end{align*}
We claim that for any $p\in M$ and any $w\in\SH$, $h\in\re$,
\begin{align}
G(f_h\, w) &= \Phi_c\bigl(p,\pi(f_h\, w) \bigr)
= \Phi_c\bigl( p,\pi(w)\bigr)
+ \Phi_c\bigl(\pi(w),\pi(f_h \,w)\bigr) \notag \\
G(f_h\, w) &= \Phi_c\bigl( p,\pi(w)\bigr) + F_w\bigl( f_h(w)\bigr) \, .
\label{E:IX.1}
\end{align}
This is enough to prove the theorem because then
$$
\left.\frac{dG}{df}\right\vert_w
= \left.\frac{d\,}{dh} F_h(f_h\, w)\right\vert_{h=0}
= \left.\frac{F_w}{df}\right\vert_w
= L(w)\, +c,
$$
and $G$ is Lipschitz by theorem I.
We now prove \eqref{E:IX.1}. Let $q:=\pi(w)$, $x:=\pi(f_h\, w)$. We have to
prove that
\begin{equation}\label{E:IX.2}
\Phi_c(p,x)=\Phi_c(p,q)+\Phi_c(q,x)\, .
\end{equation}
Since the points $q$ and $x$ can be joined by the static curve
$x_w\vert_{[0,h]}$, then
$$
\Phi_c(x,q) = - \Phi_c(q,x) \, .
$$
Using twice the triangle inequality for $\Phi_c$ we get that
$$
\Phi_c(p,q)\le \Phi_c(p,x)+\Phi_c(x,q)=
\Phi_c(p,x)-\Phi_c(q,x)\le r\Phi_c(p,q)\,
.
$$
This implies \eqref{E:IX.2}.
\qed
\bigskip
\bigskip
%\newpage
\stepcounter{section}
\centerline{\large \bf \thesection. Connecting orbits
inside fixed energy levels.}
\bigskip
We quote a paragraph from Ma\~n\'e \cite{Ma3}:
``Exploiting that the energy, $E:TM\to\re$, defined as usual by
$E(x,v)=\frac{\partial L}{\partial v}\, v - L$, is a first integral
of the flow generated by $L$, leads to {\it information on the position of
$\Sigma^+(L)$}. First observe that it is easy to check that a semistatic
curve $x:[a,b]\to M$ satisfies:
\begin{equation}\label{$*$}
E(x(t),\dot{x}(t)) = c(L)\,.
\end{equation}
This follows from calculating the derivative at $\la=1$ of
the function $F:\re\to\re$ given by:
$$
F(\la)=\int_a^{\frac{b}{\la}} (L+c)(x_\la(t),\dot{x}_\la (t))\, dt\,,
$$
where $x_\la:[a,\frac b{\la}]\to M$ is given by $x_\la(t)=x(\la t)$.
\, From \eqref{$*$} follows that:
$$
\Sigma^+(L)\subset E^{-1}(c)\,,
$$
that together with $\pi\,\Sigma^+(L)=M$ implies:
$$
\pi\, E^{-1}(c)=M\,.
$$
Hence,
$$
c\ge \max_q E(q,0)\,.
$$
Moreover $\widehat{\Sigma}(L)\subset E^{-1}(c)$ implies:''
\begin{Corollary}
$\mu\in {\cal M}(L)$ is minimizing if and only if
\begin{gather*}
\int\left(\frac{\partial L}{\partial v}\right)v\, d\mu = 0\,, \\
\text{supp}(\mu)\subset E^{-1}(c(L))\,.
\end{gather*}
\end{Corollary}
\bigskip
\noindent{\bf Theorem X.}
{\it
If $k>c(L)$ then for all $p,\,q\in M$, $p\ne q$ there exists a solution $x(t)$
of (E-L) such that $x\in AC(p,q)$ and
$$
\Phi_k(p,q)=S_{L+k}(x)
$$
Moreover the solution $(x(t)\dot{x}(t))$ is contained in the energy level
$E^{-1}\{k\}$.
}
\begin{remark}
If $p=q$ then the infimum
$$
\Phi_k(p,p) = \inf\{S_{L+k}(\ga)\,\vert\, \ga\in AC(p,q)\,\} =0
$$ can not be realized by a path defined on an interval with nonzero length.
For otherwise if $\ga:[0,T]\to M$, $T>0$ is a minimum, then
$$
\Phi_c(p,p)\le S_{L+c}(\ga) = \Phi_k(p,p) - (k-c(L))\, T
< \Phi_k(p,p) =0 \,.
$$
Contradicting theorem I.
\end{remark}
\medskip
\begin{remark} This theorem, with the same proof, holds for coverings
$\pi: \widehat{M} \to M$ of a compact manifold $M$, with the lifted
Lagrangian $\widehat{L}=L\circ \pi$.
\end{remark}
\medskip
%\newpage
\noindent {\bf Proof:}
Suppose that $p,\, q\in M$ and $p\ne q$.
For each $T>0$ there exists a minimizer $x_T$ of $S_L$ on
$AC(p,q,T)$. Then
$$
\Phi_k(p,q) = \inf_{T>0} S_L(x_T)+k\, T\, .
$$
Observe that
\begin{align*}
\lim_{T\to +\infty}\bigl(S_L(x_T)+k\,T\bigr)
&= \lim_{T\to +\infty} \bigl( S_{L+c}(x_T)+(k-c)\,T \bigr) \\
&\ge \lim_{T\to +\infty}\bigl(\Phi_c(p,q) + (k-c)\, T \bigr)\\
&= +\infty\, .
\end{align*}
Choose a sequence $\{T_i\}$ such that $\lim_i S_L(x_{T_i}+k\,
T_i)=\Phi_k(p,q)$. Then $\{T_i\}$ must be bounded and we may assume that it has
a limit $T_0$. Since $\lV\dot{x}_{T_i}\rV0$.
We shall prove that we actually have a minimal solution
with time interval $[0,T_0]$. We proceed as in the proof of Tonelli's theorem
(see Mather \cite{Mather} or Ma\~n\'e \cite{Ma0}, lemma 6.1). Observe that
by corollary \ref{lemmavel} the sequence $\{x_{T_i}\}$ is absolutely
equicontinuous because $\{ S_L(x_{T_i}) \}$ is bounded. Since
$T_i\overset{i}{\longrightarrow} T_0$ there is a subsequence of $\{x_{T_i}\}$
which converges to a curve $x_{T_0}\in AC(p,q,T_0)$. It remains to prove that
\begin{equation}\label{E:10.1}
S_{L+k}(x_{T_0})=\Phi_k(p,q)\,.
\end{equation}
The following lemma uses the superlinearity of the Lagrangian. A proof
can be found in Ma\`n\'e \cite{Ma0}. We may assume that $M=\re^n$.
\begin{Lemma} Given $C>0$ and $\e >0$ and a
compact subset $S\subset M=\re^n$ there exist $\delta>0$ such that
if $(x,v),\,(y,w)\in TM = \re^n\times\re^n$ satisfy
$x\in S$, $d(x,y)<\delta $ and $|v|0$ define
$$
S(C):= \{\, t\in [0,T_0]\,\vert\, \lv \dot{x}(t)\rv \le C\, \} \,.
$$
Since $L$ is bounded below, we can assume that $L$ is positive.
We will prove that for all $h>0$ and $C>0$,
\begin{equation}\label{E:10.2}
\int_{S(C)}L(x(t),\dot x(t))dt\le \Delta + h\,.
\end{equation}
Taking the limits when $h\to 0$ and $C\to \infty$ we get that
$S_L(x)\le \Delta$, completing the proof of \eqref{E:10.1}.
If $i$ is big enough
\begin{align*}
\Delta + h &> \int_{S(C)} L(x_{T_i}(t), \dot{x}_{T_i}(t))\, dt \\
&\ge \int_{S(C)} L(x(t),\dot{x}(t))\, dt - \e\, T_0
+ \int_{S(C)}\frac{\partial L}{\partial v}(\dot{x}_{T_i}(t) -\dot{x}(t))\,
dt\,.
\end{align*}
Then
\begin{equation}\label{E:10.3}
\int_{S(C)}L(x(t),\dot x(t))\, dt\le \Delta + h
+ \e\, T-\lim\inf_i \int_{S(C)}
\frac{\partial L}{\partial v}(\dot x_{T_i}(t)-\dot x(t))\,dt\,.
\end{equation}
We claim that
\begin{equation}\label{E:10.4}
\lim \inf \int_{S(C)}\frac{\partial L}{\partial v}(\dot x_{T_i}(t)-\dot
x(t))\,dt=0.
\end{equation}
This follows from the fact that $x$ is absolutely continuous,
$\frac{\partial L}{\partial v}$ is bounded in $S(C)$ and that for
any interval $(a,b)\subset[0,T_0]$
$$
\lim _{i\to\infty}\int_a^b(\dot x_{T_i}(t)-\dot x(t))\,dt
=\lim_{i\to\infty}(x_{T_i}(b)-x(b))-(x_{T_i}(a)-x(a))=0.
$$
Since $\e>0$ is arbitrary \eqref{E:10.4} together with \eqref{E:10.3} prove
\eqref{E:10.2}.
We now prove that minimizers of $S_{L+k}$ are in the energy
level $E=k$. Suppose that $x\in AC(p,q,T)$ is such that
$$
S_{L+k}(x) = \min_{y\in AC(p,q)} S_{L+k}(y)\,.
$$
Define
$$
F(\lambda )=\int_0^{\lambda T}(L+k)(x_\lambda ,\dot x_\lambda )
$$
where $x_\lambda (t):[0,\lambda T]\rightarrow M$ is defined as $x_\lambda
(t)=x(\frac t\lambda ).$ By the minimizing condition $F^{\prime }(1)=0$. On
the other hand
\begin{align*}
F^{\prime }(\lambda )
&=T\,L\bigl(x_\lambda (\lambda T),\dot x_\lambda (\lambda T)\bigr)
+\int_0^{\lambda T}\frac{\partial L}{\partial \lambda }\, dt+ kT \\
\intertext{now}
\frac{\partial L}{\partial \lambda }
&=\frac{\partial L}{\partial x}\frac{
\partial x_\lambda }{\partial \lambda }+
\frac{\partial L}{\partial v}\frac{
\partial \dot x_\lambda }{\partial \lambda } \\
&=-\frac{\partial L}{\partial x}\dot x_\lambda \left(\frac t\lambda \right)
\frac t{\lambda ^2}-\frac{\partial L}{\partial v}
\left(\frac 1{\lambda ^3}\ddot x_\lambda \left(\frac t\lambda \right)t
+\frac{\dot x_\lambda (\frac t\lambda )}{\lambda^2}\right)
\end{align*}
So
\begin{align*}
0&=T\,L\bigl(x(T),\dot x(T)\bigr)+T\, k-\int_0^T
\left(\frac{\partial L}{\partial x}
\dot x+\frac{\partial L}{\partial v}\ddot x\right)t\, dt -\int_0^T
\frac{\partial L}{\partial v}\dot x\,dt \\
&=T\,L\bigl(x(T),\dot x(T)\bigr)+T\,k
-\int_0^T\frac{\partial L}{\partial v}\dot x\, dt
-\int_0^T\frac d{dt}(L)t\, dt \\
&=T\,L\bigl(x(T),\dot x(T)\bigr)+T\,k-\int_0^T
\frac{\partial L}{\partial v}\dot x\,dt+\int_0^TL\,dt -L\,t\big\vert_0^T\\
&=Tk-\int_0^T E\,dt \\
&=T(k-E).
\end{align*}
This proves that the energy level of the solution $x$ is $k.$
\qed
\newpage
Observe that
\begin{equation}\label{E:10.5.0}
L + k = (\partial L /\partial v) v \quad \text{ on } E^{-1}\{k\}\,.
\end{equation}
\begin{Corollary}\label{C:10.5}
\end{Corollary}
{\it \begin{itemize}
\item[(a)] If $k>c(L)$ and $a,b\in M$, there exists a solution $x(t)$ of (E-L)
such that $x(0)=a$, $x(T)=b$ for some $T>0$, $E(x(t),\dot{x}(t))=k$ for all
$t\in\re$, and:
\begin{equation}\label{E:10.5.1}
\int_0^T\frac{\partial L}{\partial v}(x,\dot{x})\, \dot{x}\, dt
=\min \int_0^{T_1} \frac{\partial L}{\partial v}(y,\dot{y})\, \dot{y}\, dt\,,
\end{equation}
where the minimum is taken over all the absolutely continuous $y:[0,T_1]\to M$,
$T_1\ge 0$, with $y(0)=a$, $y(T_1)=b$ and $E(y(t),\dot{y}(t))=k$ for a.e. $t\in
[0,T_1]$.
\item[(b)] Conversely, if given $k>c(L)$ and $a,\, b\in M$, there exists an
absolutely continuous $x:[0,T]\to M$ with $x(0)=a$, $x(T)=b$,
$E(x(t),\dot{x}(t))=k$ for a.e. $t\in [0,T]$ and satisfying the minimization
property \eqref{E:10.5.1}, then $x(t)$ is a solution of (E-L).
\end{itemize}
}
If $p,q\in \pi E^{-1}\{k\}$, define
$$
ACE(p,q;k):=\{ x\in AC(p,q) \, \vert\, E(x,\dot{x}) = k \text{ a.e.}\,\}\,.
$$
Item (a) of corollary \ref{C:10.5} follows from \eqref{E:10.5.0} and item (b)
follows from the fact that since minimizers of $S_{L+k}$ have energy $k$, then
minimizing $S_{L+k}$ on $AC(p,q)$ is equivalent to minimize it on $ACE(p,q;k)$.
\bigskip
Given $\mu\in{\cal M}(L)$ define its {\it homology} or {\it asymptotic cycle}
(cf. Schwartzman \cite{S}), by $\rho(\mu)\in H_1(M,\re)=H^1(M,\re)^*$ such that
$$
\rho(\mu)\cdot [\theta] = \int_{TM}\theta\, d\mu
\quad , \quad
\forall\, [\theta]\in H^1(M,\re)\,,
$$
where $\theta$ is a closed 1-form and $[\theta]$ its homology class.
Define the {\it Mather's betha function} $\beta:H_1(M,\re)\to\re$, by
$$
\beta(h):= \min\{\, S_L(\mu)\,\vert\, \mu\in{\cal M}(L),\; \rho(\mu)=h\,\}\,.
$$
Since, for any $h\in H_1(M,\re)$ the set $K(h):=\{\,\mu\in{\cal
M}(L)\,\vert\,\rho(\mu)=h\,\}$ is convex, it follows that $\beta$ is a convex
function. Let $\beta^*$ be its Legendre transform: $\beta^*:H^1(M,\re)\to\re$,
$$
\beta^*(w):=\max_{h\in H_1(M,\re)} \{\,w(h)-\beta(h)\,\}\,.
$$
The reader can check that
$$
\beta^*([\theta])= -c(L-\theta),
\quad \forall\, [\theta]\in H^1(M,\re)\,.
$$
Define the {\it strict critical value} $c_0(L)$ by
$$
c_0(L) = \min_\theta c(L-\theta)\,.
$$
Then
$$
c_0(L) = -\beta^*(0) = - \min \{\, S_L(\mu)\,\vert\, \mu\in{\cal
M}(L),\;\rho(\mu)=0\,\}\,.
$$
Observe that
\begin{equation}\label{E:S4.D}
\frac{d\,}{dt} \, E(x,tv)\big\vert_{t=1} = v\cdot L_{vv}(x,v)\cdot v > 0\; ,
\end{equation}
Therefore if $(x,v)$ is a critical point of the energy function $E$ then $v=0$
and $\frac{\partial L}{\partial x}(x,0)=0$.
Let
$$
e_0 := \max_{p\in M} E(p,0).
$$
By \eqref{E:S4.D}, we have that
\begin{equation}\label{E:S4.E}
E(p,0)= \min_{v\in T_pM} E(p,v)\, .
\end{equation}
In particular
$$
e_0 = \min\{\,k\in\re\,\vert\, \pi(E^{-1}\{k\}) = M\, \}\,.
$$
Let $\theta_0$ be a closed 1-form such that $c_0(L)=c(L-\theta_0)$.
Then the energy function and the Euler-Lagrange equations for $L-\theta_0$ and
$L$ are the same. Theorem X implies that $\pi(E^{-1}\{k\})=M$ for all
$k>c(L-\theta_0)=c_0(L)$. Hence
$$
e_0 \le c_0(L)\,.
$$
As observed in Ma\~n\'e \cite{Ma3}, for mechanical Lagrangians $L(x,v) =\frac
12 \langle v ,v\rangle_x - \phi(x)$, with $\langle\;,\,\rangle_x$ a riemannian
metric, we have that
$d_0=e_0=c_0(L)=c(L)$. There is an example in \cite{Ma3} of a Lagrangian $L$
with $e_0c_0(L)$, for every free homotopy class $H\ne 0$ of $M$, there exists a
periodic orbit in $E^{-1}\{k\}$ such that its projection on $M$ belongs to that
free homotopy class.
\end{Corollary}
\noindent{\bf Proof:}
Fix $k>c_0(L)$. By adding a closed 1-form we can assume that $c(L)=c_0(L)$.
Let $AC(H)$ be the set of absolutely continuous closed curves in $M$ with free
homotopy class $H$. Let $x_n\in AC(H)$ with $x_n:[0,T_n]\to M$ and
$$
\lim_{n\to \infty}S_{L+k}(x_n) = \inf_{x\in AC(H)}S_{L+k}(x).
$$
Let $\widetilde{x}_n$ be a lift of $x_n$ to the universal cover $\widetilde{M}$
of $M$. We can assume that $\tx_n$ is a minimizer of $_{\widetilde{L}+k}$ on
$AC(\tx_n(0),\tx_n(T_n);T_n)$ and in particular, that it is a solution of
(E-L). Then the same arguments as in the proof theorem X yield that $\{ T_n\}$
must be bounded and $\lV\dot{x}_n\rV\frac 1A \,\min\{\,\text{length}(\ga)\,\vert\, \ga\in AC(H)\,\}>0.
$$
The same arguments as in theorem X give a closed curve $\ga:[0,T_0]\to M$ wich
is a uniform limit of a subsequence of $x_n$ and hence $\ga\in AC(H)$. Moreover
\begin{equation}\label{E:C.X.1}
S_{L+k}(\ga)=\min_{x\in AC(H)} S_{L+k}(x)\,,
\end{equation}
and $(\ga,\dot{\ga})$ is in the energy level $E^{-1}\{k\}$.
It remains to prove that $\dot{\ga}(0)=\dot{\ga}(T_n)$. Suppose that
$\dot{\ga}(0)\ne \dot{\ga}(T_n)$. Let $\widetilde{M}$ be the universal cover of
$M$ and $\widetilde{L}$ the lift of $L$. Let $\widetilde{\ga}$ be a lift of
$\ga$. Consider the
path $\eta\vert_{[-\e,\e]}=\widetilde{\ga}\vert_{[T_0-\e,T_0]} *
\widetilde{\ga}\vert_{[0,\e]}$. We have that $\eta$ is not $C^1$ and hence it
is not a solution of (E-L). Since $k>c_0(L)\ge e_0$, then $k$ is a regular
value of the energy $\widetilde{E}$ of $\widetilde{L}$. By the Maupertius
principle (see theorem 3.8.5 of Abraham \& Marsden \cite{A-M}),
$\eta\vert_{[-\e,\e]}$ is not a minimizer of the $(\widetilde{L}+k)$-action on
$\widetilde{E}^{-1}\{k\}$. Then there exists $\xi\in(\eta(-\e),\eta(\e))$, with
energy $\widetilde{E}(\xi,\dot{\xi})\equiv k$ and $S_{\widetilde{L}+k}(\xi)<
S_{L+k}(\eta)$. Moreover, since $\widetilde{M}$ is simply connected, the paths
$\xi$ and $\eta$ are homotopic by a homotopy which fixes their endpoints. Hence
$\pi(\xi * \widetilde{\ga}\vert_{[-\e, T_0-\e]})$ and
$\dot{\ga}\vert_{[0,T_0]}$ are in the same free homotopy class of $M$. We have
that
\begin{align*}
S_{L+k}\left(\pi(\xi *\widetilde{\ga}\vert_{[\e,T_0-\e]})\right)
&= S_{\widetilde{L}+k}(\xi * \widetilde{\ga}\vert_{[\e,T_0-\e]}) \\
&= S_{\widetilde{L}+k}(\xi) +
S_{L+k}(\widetilde{\ga}\vert_{[\e,T_0-\e]}) \\
&< S_{L+k}(\ga)\,.
\end{align*}
This contradicts the minimizing property \eqref{E:C.X.1} of $\ga$.
\qed
\medskip
\newpage
``An interesting characterization of the critical value $c(L)$, in terms
of an analogous to Tonelli's theorem (Mather \cite{Mather}) in a prescribed
energy level is given by the following result:''
\hfill {\it --- Ma\~n\'e \cite{Ma3}}
\medskip
\noindent{\bf Theorem XI. }
{\it
Assume that $k$ is a regular value of $E$ and $\dim M >1$.
Suppose that $k$ has the following property: for all $a,b$ in
$\pi E^{-1}(k)$ there exists an absolutely continuous
curve $x:[0,T]\to M$ such that:
\begin{itemize}
\item[(i)] $x(0)=a$ and $x(T)=b$.
\item[(ii)] $E(x(t),\dot x(t))=k$ a.e. in $[0,T]$.
\item[(iii)]{$\displaystyle{
\int_0^T \frac{\partial L}{\partial v} (x(t),\dot x(t))\,
\dot x(t)\, dt
=\min
\int_0^{T_1} \frac{\partial L}{\partial v} (y(t),\dot y(t))\,
\dot y(t)\, dt}$,}\newline
where the minimum is taken over all absolutely continuous
$y:[0,T_1]\to M$, with $y(0)=a,y(T_1)=b$ and
$E(x(t),\dot x(t))=k$ a.e. in $[0,T_1]$.
\end{itemize}
Then $k>c(L)$ and $x(t)$ is a solution of the Euler Lagrange equation.
}
\medskip
The hypothesis $\dim M>1$ is necessary as the example of a simple pendulum
shows. Indeed, for $L(x,v) = \frac 12 \lv v\rv^2 - \cos x$ and any regular
value $k < e_0 = -\min_{p\in S^1}L(p,0)=c(L)=1$ there are such
minimizers. This is because a non-empty energy level $E^{-1}(k)$ with
$kc(L)=e_0$. The energy functions and the Lagrangian
flows for ${\Bbb L}$ and $L$ are the same. For $e_00$. Then the same arguments as for $L$ show
that there exists minimizers for $k0$. Hence there are points in $\theta(k)=\partial \pi
E^{-1}(k)$ which can not be joined to $p$ by $(L+k)$-minimizers with
$E(x,\dot{x})\equiv k$.
Suppose now that $k=c(L)=:c$. The arguments of equation \eqref{$*$} show that
minimizers of $S_{L+c}$ in $AC(p,q)$ are in the energy level $E=c(L)$. Thus
minimizers of $S_{L+c}$ on $ACE(p,q;c(L))$ are also minimizers on $AC(p,q)$
and hence semistatic curves. Let $(q,\xi(q))\in\widehat{\Sigma}\subseteq
E^{-1}\{k\}$. Let $p\notin \pi\{f_t(q,\xi(q))\,\vert\, t\in\re\,\}$.
Suppose that there exists a semistatic curve $x\in AC(p,q,T)$.
Then by theorem VI (b), we have that $\dot{x}(T)=\xi(q)$. This
contradicts the choice of $p$.
Now suppose that $e_0e_0$ and $(p,v)\in TM$,
there exists a unique
$\la>0$ such that $E(p,\la\, v)=k$. Moreover,
$\la = \la(p,v,k):TM\times ]e_0,+\infty[\to]0,+\infty[$ is a smooth function.
Let $\mu$ be an invariant measure $\mu\in {\cal M}(L)$ such that
\begin{equation}\label{E:11.2}
\int \bigl( L + c(L) \bigr) \; d\mu =0\; .
\end{equation}
By theorem 4 and the fact that static curves have energy level
$c(L)$ we have that $\text{supp}(\mu)\subseteq E^{-1}(c(L))$, and by the
Poincar\'e recurrence theorem we have that $\mu\in \ov{{\cal C}(c(L))}$.
For $k>e_0$ define the
measure $\nu_k$ on $E^{-1}(k)$ by
\begin{equation}\label{E:11.3}
\int_{E^{-1}(k)} f\, d\nu_k :=
A(k) \int_{E^{-1}(k)}\frac{f(p,\la(p,v,k)v)}{\la(p,v,k)}\,
d\mu (p,v)
\end{equation}
for any continuous function $f:E^{-1}(k)\to\re$, where
$$
A(k) := \left( \int \frac 1{\la(p,v,k)}\, d\mu (p,v)\right)^{-1} .
$$
Then $\nu_k\in\ov{{\cal C}(k)}$ and $\nu_{c(L)}=\mu$.
This measure $\nu_k$ is just the (probability) measure obtained by
reparametrizing the solutions of (E-L) on $\pi(\text{supp}(\mu))$ so
as to have energy $k$. This process is reversible, i.e. we can recover
$\mu$ by reparametrizing $\nu_k$.
Let
$$
g(k):= \int v\cdot L_v\; d\nu_k = k + \int L \; d\nu_k\; .
$$
Then $g$ is a differentiable function with derivative
$$
g'(k) = 1 + A(k) \int \frac{\partial\,}{\partial k}
\left(\frac{L(p,\la v)}{\la}\right)\, d\mu
+ A'(k) \int \frac{L(p,\la v)}{\la}\, d\mu \, .
$$
If we change the reference energy level $c(L)$ to $k_1>e_0$,
we can use $\nu_{k_1}$ instead of $\mu$ on formula \eqref{E:11.3}.
The function $g(k)$ does not change but now $\la(k_1)\equiv 1$,
$A(k_1)=1$ and
$$
g'(k_1) = 1
+ \int\frac{\partial \,}{\partial k}
\left(\frac{\Bbb L}{\la}\right)\; d\nu_{k_1}
+ A'(k_1) \int L\; d\nu_{k_1} \; ,
$$
where ${\Bbb L}(p,v):= L(p,\la\, v)$. We compute this derivative:
\begin{align*}
\frac{\partial\,}{\partial k}\left(\frac{{\Bbb L}}
{\la}\right)\Big\vert_{k=k_1}
&= \frac{{\Bbb L}_k \, \la - {\Bbb L}\, \la_k}{\la^2}
= {\Bbb L}_k - \la_k\, {\Bbb L}\vert_{k=k_1}\; , \\
{\Bbb L}_k &= v \cdot L_{v} \; \la_k\; .
\end{align*}
Since $E(p,\la\, v)=k$, we have that
\begin{gather*}
\frac{\partial E}{\partial k}
= E_v \cdot (\la_k \, v)
= (v\cdot L_{vv}\cdot v)\ \; \la_k
= 1 \; , \\
\la_k = \frac 1{v\cdot L_{vv} \cdot v}
\quad \text{and} \quad
{\Bbb L}_k = \frac{v\cdot L_v}{v\cdot L_{vv}\cdot v}\; .
\end{gather*}
Moreover
\begin{align*}
\frac{\partial \,}{\partial k} \left( \frac 1{\la}\right)\Big\vert_{k=k_1}
&= -\frac{\la_k}{\la^2}\Big\vert_{k=k_1}
= - \frac 1{v\cdot L_{vv}\cdot v} \\
A'(k_1)
&= - \frac 1{A(k_1)} \int \frac{\partial \,}{\partial k}
\left( \frac 1{\la}\right)\; d\mu
= \int \frac 1{v\cdot L_{vv}\cdot v}\; d\nu_{k_1} \; .
\end{align*}
Therefore
\begin{gather}
\begin{aligned}
g'(k) &= 1 + \int \frac{v\cdot L_v}{v\cdot L_{vv}\cdot v}\; d\nu_k
- \int \frac L{v\cdot L_{vv}\cdot v}\; d\nu_k
+ \left( \int \frac 1{v\cdot L_{vv}\cdot v}\; d\nu_k\right)
\left(\int L \; d\nu_k\right)
\notag\\
g'(k) &= 1 + \int \frac k{v\cdot L_{vv}\cdot v}\; d\nu_k
+ \left(\int \frac 1{v\cdot L_{vv}\cdot v}\; d\nu_k\right)
\left(\int L\; d\nu_k\right)
\notag\\
g'(k) &= \left( \int (L+k)\; d\nu_k\right)
\left( \int \frac 1{v\cdot L_{vv}\cdot v}\; d\nu_k\right)
+ 1
\notag
\end{aligned}
\\
g'(k) = b(k)\; g(k) + 1 \; ,
\label{E:11.4}
\end{gather}
where $b(k) := \int (v\cdot L_{vv}\cdot v)^{-1}\, d\nu_k >0$. Let
$$
B(k) := \int_{e_0}^k b(s)\; ds
$$
and $h(k) := e^{-B(k)}\, g(k)$. From \eqref{E:11.4} we get that
$$
h'(k) = e^{-B(k)} > 0\; .
$$
By \eqref{E:11.2}, $h(c(L))=0$, therefore $h(k)<0$
for all $e_00$. We have that
\begin{equation}\label{E:11.2.8}
D_p E(p_0,0) = - L_p(p_0,0) = 0 \; .
\end{equation}
Since the left hand side of \eqref{E:11.2.7} is positive, using
\eqref{E:11.2.8} we have that $\Delta p \, L_{pp}(p_0,0)\, \Delta p > 0$.
Then there exists a function $G(p,v)>A>0$ such that for $v$ small and $p$ near
$p_0$ such that $E(p,v)=k$, we have
\begin{align}
\lv v \rv^2 &= G(p,v)^2\, \lv \Delta p \rv^2 \; ,
\notag \\
\lv v \rv &\ge A \, \lv \Delta p \rv \; .
\label{E:11.2.9}
\end{align}
Suppose that there exists a differentiable curve $p:[0,\de]\to M$ such that
$p(0)=p_0$ and $E(p(t),\dot{p}(t))=k$. For simplicity suppose that $\de =2$.
Let $x(t):=\lv p(t)-p_0\rv$. Writing $v(t)=\dot{p}(t)$, we have that
\begin{align*}
\frac{d\,}{dt} x(t)^2
&= 2\, x\, \dot{x} = 2\, \langle p(t)-p_0, v(t) \rangle \, ,
\\
\dot{x}(t)
&= \left\langle \frac{p(t)-p_0}{\lv p(t)-p_0\rv} , v(t)\right\rangle
\ge -B \, \lv v(t)\rv \; .
\end{align*}
for some $B>0$, because $p(t)$ is differentiable at $t=0$. From
\eqref{E:11.2.9}, we have that
\begin{align*}
\dot{x}(t) &\ge -B \, \lv v(t) \rv \ge -A B\, x(t) \; , \\
x(t) &\ge x(1) \, \exp\bigl( - A B\, (t-1)\bigr) \; .
\end{align*}
In particular $x(0)\ne 0$. This contradicts the choice of $p(t)$.
Now we prove (b).
Since $D_pE(q_0,0)=L_p(q_0,0)\ne 0$, then $\theta(k)$ is locally a codimension
1 submanifold near $q_0$. Let $q(t)\in\theta(k)$ be given by the condition
$$
L_p(q(t),0)\, \bigl( p(t)-q(t)\bigr) = 0 \; .
$$
Using formula \eqref{E:11.2.7} with $\Delta p = p(t) - q(t)$, we have that
$$
F(p,v)\, \lv v \rv^2 = L_p(q(t),0)\, \Delta p
+ {\cal O}\left( \lv \Delta p \rv^2\right)
$$
and then
$$
\lv v \rv \ge A \; \sqrt{\lv \Delta p \rv} \; ,
$$
for some $A>0$. Let $y(t):= \lv p(t) -q(t)\rv$. Computing
$\frac{d\,}{dt}\, y(t)^2$, we obtain
$$
\dy (t)
= \left\langle \frac{p(t)-q(t)}{\lv p(t)-q(t)\rv} \; ,\; v(t)\right\rangle
\le - B \; \lv v(t)\rv
\le - A\, B \,\sqrt{y(t)} \; ,
$$
for some $B>0$. Therefore
$$
\frac{\dy}{2\, y^{\frac 12}} \le - \frac{A \, B} 2
\quad \text{ and } \quad
y(t) \le \frac{A^2\, B^2}{4} \; (t-t_0)^2\; .
$$
For future reference, we also note that
\begin{align}
\lv \dy(t) \rv &\ge b\; \lv v(t)\rv \ge b_1\; \sqrt{y(t)} \; ,
\label{E:11.2.I} \\
y(t) &\ge b_2 \; (t-t_0)^2 \; ,
\label{E:11.2.II}
\end{align}
for some $b,\; b_1,\; b_2 >0$.
\qed
\stepcounter{section}
\bigskip
\bigskip
\centerline{\large \bf \thesection. Properties of weaker global minimizers.}
\bigskip
\noindent{\bf Definition.}
We say that a solution $x(t)$ of (E-L) is a {\it minimizer} (resp.
{\it forward minimizer}) if
$$
S_L(x\vert_{[t_0,t_1]})\le S_L(y)
$$
for every $t_0\le t_1$ (resp. $00$ such that setting $c=c(L)$
$$
|S_{L+c}(x_w\vert_{[s,t]})|\le C
$$
for every $w\in\La^+(L)$ and all $0\le s \le t$.
\item[(b)] If $w\in\La^+(L)$ and $p\in M$ is such that
$p=\lim_{n\to \infty} x_w(t_n)$ for some sequence $t_n \to \infty$
then the limit
$$
\lim_{n\to \infty}S_{L+c}(x\vert_{[0,t_n]})
$$
exists and does not depend in the sequence $\{ t_n\}$.
\end{itemize}
}
\medskip
\noindent{\bf Proof:}
Item (a) is straightforward consequence of the claim and the definition of
$\delta_w$. From equation (\ref{destrianginv}) it follows that the function
$t\to\delta_w(0,t)$ is increasing. Then the claim implies that
$D:=\lim_{t\to +\infty} \de_w(0,t)$ exists. Thus
\begin{align*}
\lim_{n\to \infty}S_{L+c}(x_w\vert_{[0,t_n]})
&= \lim_{n\to \infty}\Phi_c(x_w(0),x_w(t_n))
+ \lim_{n\to \infty}\delta_w(0,t_n) \\
&= \Phi_c(x_w(0),p)+ D\,.
\end{align*}
\qed
\bigskip
\noindent{\bf Theorem XII.}
{\it
\begin{itemize}
\item[(a)] The $\om$-limit set of an orbit in $\La^+(L)$ is contained in $\SH$.
\item[(b)] The $\a$ and $\om$-limit sets of an orbit in $\La(L)$ are contained
in $\SH$.
\end{itemize}
}
\medskip
\noindent{\bf Proof:}
We only prove (a). Let $w\in\La^+(L)$. Let $(p,v)\in\om(w)$,
$T>0$ and let $\langle n_k\rangle$ be a sequence in $\re^+$ such that
$$
n_{k+1}>n_k + T\,,
$$
$$
(p,v) = \lim_k \bigl( x_w(n_k),\dot{x}_w(n_k)\bigr)\,.
$$
Let
$$
p_k := x_w(n_k) \qquad ,\qquad q_k :=x_w(n_k +T)\,,
$$
$$
(q,u) :=
\bigl(x_v(T),\dot{x}_v(T)\bigr)=\lim_k\,\bigl(q_k\,,\,\dot{x}_w(n_k+T)\bigr)\,.
$$
we have that
\begin{align*}
S_{L+c}(x_w\vert_{[n_k,n_k +T]}) &=\Phi_c(p_k,q_k) + \de_w(n_k,n_k+T) \,,\\
S_{L+c}(x_w\vert_{[n_k+T,n_{k+1}]}) &= \Phi_c(q_k,p_k)+\de_w(n_k+T,n_{k+1})\,,\\
S_{L+c}(x_w\vert_{[n_k,n_{k+1}]})&=\Phi_c(p_k,p_{k+1})+\de_w(n_k,n_{k+1})\,.
\end{align*}
By the claim and equation \eqref{destrianginv}
there is a constant $Q$ such that
\begin{align*}
\sum_{k=1}^\infty\bigl( \de_w(n_k,n_k+T)+\de_w(n_k+T,n_{k+1})\bigr)
& \le\sum_{k=1}^{\infty} \de_w(n_k,n_{k+1}) \\
& \le \lim_{t \to \infty} \delta_w(0,t)
\le Q\,.
\end{align*}
Therefore
\begin{equation}\label{E:S5.zero}
\lim_{k\to\infty}\de_w(n_k,n_{k+1})
=\lim_{k\to\infty}\de_w(n_k,n_k+T)
=\lim_{k\to\infty}\de_w(n_k+T,n_{k+1})
=0\,.
\end{equation}
Hence
\begin{align*}
S_{L+c}(x_w\vert_{[0,T]})
&= \lim_k S_{L+c}(x_w\vert_{[n_k,n_k+T]})\\
&=\lim_k \Phi_c(p_k,q_k) = \Phi_c(p,q)\,.
\end{align*}
and from \eqref{E:S5.zero},
\begin{align*}
\Phi_c(p,q)+ \Phi_c(q,p)
& = \lim_k \Phi_c(p_k,q_k) + \lim_k\Phi_c(q_k,p_k) \\
&= \lim_k\,\bigl[\, S_{L+c}(x_w\vert_{[n_k,n_k+T]})
+S_{L+c}(x_w\vert_{[n_k+T,n_{k+1}]})\,\bigr] \\
&= \lim_k \,\bigl[\, \Phi_c(p_k,p_k) + \de_w(n_k,n_{k+1})\,\bigr] \\
&= \Phi_c(p,p) \\
&= 0 \, .
\end{align*}
This implies that $\om(w)\subseteq\SH$.
\qed
\bigskip
\noindent{\bf Theorem XIII.}
\quad
{\it $f_t\vert_{\La(L)}$ is chain transitive.}
\medskip
\noindent{\bf Proof:}
Let $v$, $w\in\La(L)$ and $\e>0$. It is enough to prove that there exists an
$\e$-chain in $\La(L)$ joining $\om(w)$ to $\a(v)$. By theorem XIII,
$\a(v)\cup\om(w)\subseteq\SH\subseteq\Sigma(L)$.
By theorem V there exists such $\e$-chain contained
in $\Sigma(L)\subseteq\La(L)$. This completes the proof.
\qed
\medskip
\noindent {\bf Proof of the claim:}
Let
\begin{align*}
A &:= 2\, \max\,\bigl\{\, L(p,v)+ c(L)\,\big\vert\,
\lV v\rV\le\text{ diam}M + 1 \,\bigr\}\\
B &:= \max\,\bigl\{\,\lv\Phi_c(p,q)\rv\,\big\vert\, p,q\in M\,\bigr\} \\
Q &:= 3\, \max\,\{ \, A , B\,\} + 2 \, .
\end{align*}
Suppose that there exist $\om\in\La^+(L)$, $0\le a\le b$ such that
$\de_w(a,b)>Q$.
Let $\ga:[a,T_b]\to M$ be in $AC(x_w(a),x_w(b))$ and such that
$$
S_{L+c}(\ga) < \Phi_c(x_w(a),x_w(b))+1\,.
$$
We have that
\begin{align}
S_{L+c}(\ga) &< S_{L+c}(x_w\vert_{[a,b]}) -\de_w(a,b) +1\,, \notag\\
S_{L+c}(\ga) &< S_{L+c}(x_w\vert_{[a,b]}) - Q + 1\,, \label{E:XII.2} \\
S_{L+c}(\ga) &< S_{L+c}(x_w\vert_{[a,b]}) - (A+B)\,. \label{E:XII.3}
\end{align}
Suppose that $T_b\ge b-a$. Let $\eta:[0,1]\to M$ be a geodesic on $M$ such that
$$
\eta(0)=x_w(b) \quad ,\quad
\eta(1)=x_w(T_b+1) \quad ,\quad
\lV\dot{\eta}\rV\le \text{ diam}(M)\, .
$$
Then
$$
S_{L+c}(\eta) < A\, .
$$
Since
$$
-B\le \Phi_c(x_w(b),x_w(T_b+1))
\le S_{L+c}(x_w\vert_{[b,T_b+1]})\, ,
$$
using \eqref{E:XII.3} (or \eqref{E:XII.4}), we have that
\begin{align*}
S_{L+c}(\ga *\eta)
&< S_{L+c}(x_w\vert_{[a,b]}) - (A+B) +A \\
&< S_{L+c}(x_w\vert_{[a,b]}) + S_{L+c}(x_w\vert_{[b,T_b +1]}) \\
S_{L+c}(\ga *\eta)
&< S_{L+c}(x_w\vert_{[a,T_b+1]}) \, .
\end{align*}
This contradicts the hypothesis $w\in\La^+(L)$.
Now suppose that $T_b(b-a)-T_b$ and let $\sigma:[0,T_\sigma]\to M$ be a curve in
$AC(x_u(\tau),\pi\, u)$ such that
$$
S_{L+c}(\sigma) < -\Phi_c\bigl(\pi\, u, x_u(\tau)\bigr)+1\, .
$$
Let $\ov{\ga}:= \ga * \la * x_u\vert_{[0,\tau]}*\sigma *\ov{\la}$. This curve
is in $AC(x_w(a),x_w(b))$, it is defined on a time interval of length
$$
T_b + 1 +\tau + T_\sigma + 1 > T_b +\tau > b-a\, ,
$$
and has $(L+c)$-action
\begin{align}
S_{L+c}(\ov{\ga})
&\le S_{L+c}(\ga) + \bigl( S_{L+c}(\la) + S_{L+c}(\ov{\la})\bigr)
+ \bigl( S_{L+c}(x_u\vert_{[0,\tau]}) + S_{L+c}(\sigma)\bigr) \notag \\
&\le S_{L+c}(\ga) + A\, +
\bigl( \Phi_c(\pi\, u, x_u(\tau))
- \Phi_c(\pi\, u, x_u(\tau)) + 1 \bigr) \notag \\
&< \bigl( S_{L+c}(x_w\vert_{[a,b]}) - Q + 1 \bigr) + A + 1\,, \notag \\
S_{L+c}(\ov{\ga})
&< S_{L+c}(x_w\vert_{[a,b]}) - (A+B)\,. \label{E:XII.4}
\end{align}
Now the same argument as in the case $T_b\ge b$, using $\ov{\ga}$ instead of
$\ga$, gives a contradiction.
\qed
\begin{thebibliography}{99}
\bibitem{A-M} R. Abraham \& J. E. Marsden.
Foundations of Mechanics. Benjamin: London, 1978.
\bibitem{Ca} M.J. Dias Carneiro. {\it On Minimizing Measures of the Actions
of Autonomous Lagrangians}. Nonlinearity, {\bf 8} (1995)
no. 6, 1077-1085.
\bibitem{CI} G. Contreras, R. Iturriaga.
{\it Convex Hamiltonians without conjugate points.}
Preprint, available at
http://www.mat.puc-rio.br/pub.html.
\bibitem{Ma0} R. Ma\~n\'e. Global Variational Methods in Conservative
Dynamics. $18^{\underline{\text{o}}}$ Coloquio Bras. de Mat. IMPA.
Rio de Janeiro, 1991.
\bibitem{Ma1} R. Ma\~n\'e.
{\it On the minimizing measures of Lagrangian dynamical
systems}. Nonlinearity, {\bf 5}, (1992), no. 3,
623-638.
\bibitem{Ma2} R. Ma\~n\'e.
{\it Generic properties and problems of minimizing
measures of lagrangian systems.}
Nonlinearity {\bf 9}, (1996), no. 2, 273-310.
\bibitem{Ma3}
R. Ma\~n\'e.
{\it Lagrangian Flows: The Dynamics of Globally
Minimizing Orbits.} To appear. Proc. Int. Congress in Dyn.
Sys., Montevideo 1995, Pitman Research Notes in Math.
\bibitem{Mather} J. Mather.
{\it Action minimizing invariant measures for positive
definite Lagrangian systems}.
Math. Z. {\bf 207}, (1991), no. 2, 169-207.
\bibitem{Morse} M. Morse.
Calculus of variations in the large.
Amer. Math. Soc. Colloquium Publications vol. XVIII, 1934.
\bibitem{S} S. Schwartzman. {\it Asymptotic cycles.} Ann. of Math. (2)
{\bf 66} (1957) 270, 284.
\end{thebibliography}
\end{document}
\date{September, 1996}