\vspace{0.5cm}
\noindent
{\bf $BJd0d(B}

$BJdBj(B7.2$B$r>ZL@$7$?$$!%(BSacker and Sell[10]$B$NH/A[$rMxMQ$7$F$+$J$jD>@\E*$J>ZL@$,$G$-$k!%$7$+$7!"(BConley[2]$B$NF3F~$7$?J}K!$r;H$&$3$H$K$9$k!%E@$N(B$\omega$$B6K8B=89g$rFCD'IU$1$k$d$jJ}$@$+$i$G$"$k!%$3$NFCD'IU$1$O0J2<$NDjM}(BA.2$B$G9T$J$&!%$3$l$O$=$l<+?H$H$7$F$bLLGr$$!%(B

$B2f!9$NJ}K!$O!V:?6K>.!WN.$l$N35G0$rF3F~$7$F!"E@$N(B$\omega$$B6K8B=89g$,:?6K>.$G$"$k$3$H$r>ZL@$9$k!%$=$3$G8{G[E*:?6K>.N.$l$OC10lE@$G$"$k$3$H$r<($9!%$3$&$7$FJdBj$,>ZL@$5$l$k!%(B

$B0J2<$N(B2$B$D$NDj5A$O(BConley[2]$B$K$h$jE83+$5$l$?!%(B

\vspace{0.3cm}
\noindent
\underbar{$BDj5A(B.} $\varphi$$B$O5wN%(B$d$$B$r;}$D40Hw5wN%6u4V(B$X$$B>e$NN.$l$H$9$k!%(B$x,y \in X$$B$H$7!"(B$\varepsilon$$B$H(B$T$$B$r@5?t$H$9$k!%=8$^$j(B$(x_1, \ldots,x_{n+1}, t_1, \ldots, t_n)$$B$O!"(B$i=1, \ldots, n$$B$KBP$7$F0J2<$N>r7o$rK~$?$9$H$-(B$x$$B$+$i(B$y$$B$X$N(B$(\varepsilon,T,\varphi)$$B:?$H8F$P$l$k!%(B

(a) $x_i \in X, x=x_1, y=x_{n+1},$

(b) $t_i \geq T,$

(c) $d(\varphi(x_i,t_i), x_{i+1}) \leq \varepsilon.$

\vspace{0.3cm}
\noindent
\underbar{$BDj5A(B.} 
$x,y \in X$$B$H$9$k!%$9$Y$F$N@5$N(B$\varepsilon$$B$H(B$T$$B$KBP$7$F(B$x$$B$+$i(B$y$$B$X$N(B
$(\varepsilon,T,\varphi)$$B:?$,$"$k$H$-!"(B$x>y$$B$H=q$/!%(B

\vspace{0.3cm}
Conley$B$O!V(B$>$$B!W$O(B$X$$B>e$N?d0\4X78$G$"$k$,H?<ME*$G$bH?BP>NE*$G$b$J$$$3$H$r<($7$?!%$3$3$G$O$d$dIT@53N$G$O$"$k$,!V(B$>$$B!W$r=g=x$H8F$V$3$H$K$9$k!%8{G[E*$JN.$l$N>l9g!"$3$N=g=x$O<!$N0UL#$G8{G[4X?t$H0lCW$9$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BJdBj(BA.1.} $\varphi$$B$O%3%s%Q%/%H$J5wN%6u4V(B$X$$B>e$NN.$l$H$9$k!%(B$\varphi$$B$O(B$g$$B$K4X$7$F8{G[E*$G$"$k$H$7!"(B$x,y \in X$$B$H$9$k!%$3$N$H$-(B
$$
   x > y \Rightarrow g(x) \geq g(y).
$$

\vspace{0.3cm}
\noindent
\underbar{$B>ZL@(B.} $x>y$$B$+$D(B$g(x) < g(y)$$B$H2>Dj$9$k!%<B?t(B$a_1$$B$H(B$a_2$$B$rA*$s$G(B
$$
   g(x) < a_1 < a_2 < g(y),
$$
$B$H$7!"$^$?(B$K=g^{-1}([a_1,a_2])$$B$OITF0E@$r4^$^$J$$$h$&$K$9$k!%(B
$$ \begin{array}{cc}
  K_1 = g^{-1}((-\infty,a_1]), &  K_2 = g^{-1}([a_2,-\infty)), 
   \end{array}
$$
$B$H$*$/!%$3$N$H$-(B$K_1$$B$H(B$K_2$$B$O8_$$$KAG$J%3%s%Q%/%H$J=89g$G$"$k!%(B$\delta = d(K_1,K_2)$$B$H$*$/!%(B$K$$B$O%3%s%Q%/%H$GITF0E@$r4^$^$J$$$+$i!"@5$N(B$T$$B$,$"$C$F(B
$$ \begin{array}{rl}
  x \in K, & t \geq T \Rightarrow \varphi(x,t) \in K_1.
   \end{array}
$$
$B$@$+$i(B$\varepsilon < \delta$$B$*$h$S(B$T' > T$$B$KBP$7$F(B$x$$B$+$i(B$y$$B$X$N(B$(\varepsilon,T',\varphi)$$B:?$O$J$$!%$3$l$O(B$x>y$$B$KL7=b$9$k$+$iJdBj$O>ZL@$5$l$?!%(B\hfill $\Box$

\vspace{0.3cm}
$B$3$3$G6K>.N.$l$r<!$N$h$&$K0lHL2=$7$h$&!%(B

\vspace{0.3cm}
\noindent
\underbar{$BDj5A(B.} $\varphi$$B$O40Hw5wN%6u4V(B$X$$B>e$NN.$l$H$9$k!%$9$Y$F$N(B$x,y \in X$$B$KBP$7$F(B$x>y$$B$J$i(B$\varphi$$B$r(B\underline{$B:?6K>.(B}$B$H8F$V!%(B

\vspace{0.3cm}
Conley[2]$B$O$9$Y$F$N(B$x \in X$$B$KBP$7$F(B$x > x$$B$N$H$-!"N.$l$r!V:?2s5"E*!W$H8F$s$@!%%3%s%Q%/%H$JO"7k6u4V>e$G$O:?2s5"@-$H:?6K>.@-$,F1CM$G$"$k$3$H$,<($;$k!%(B

$\omega$$B6K8B=89g$N@-<A$H$7$F<!$rF@$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(BA.2.} $\varphi$$B$O6I=j%3%s%Q%/%H$J5wN%6u4V(B$X$$B>e$NN.$l$H$9$k!%(B$x_0 \in X$$B$O(B$\omega(x_0)$$B$,6u$G$J$/%3%s%Q%/%H$J=89g$G$"$k$h$&$JE@$H$9$k!%$3$N$H$-(B$\omega(x_0)$$B$K@)8B$7$?(B$\varphi$$B$O:?6K>.$G$"$k!%(B

\vspace{0.3cm}
$B$$$^(B$\omega$$B6K8B=89g$K@)8B$7$?N.$l$r9M$($F$$$k$3$H$KCm0U$9$k$3$H$,=EMW$G$"$k!%4JC1$K<($;$k$h$&$K!"(B$x,y \in \omega(x_0)$$B$J$i(B$X$$B>e$N(B$\varphi$$B$K$h$C$FM?$($i$l$k=g=x$KBP$7$F(B$x>y$$B$G$"$k!%$7$+$7@)8B$7$?N.$l$K$h$C$FM?$($i$l$k(B$\omega(x_0)$$B>e$N=g=x$O@)8B$7$?=g=x$H0[$J$k!%$3$N0c$$$OHsM7AvE@$r9M$($l$P0lL\N;A3$G$"$k!%(B$\omega(x_0)$$B$NE@$O$9$Y$FN.$l(B$\varphi$$B$K4X$7$FHsM7AvE*$G$"$k!%$7$+$7$3$l$i$O(B$\omega(x_0)$$B$K@)8B$7$?N.$l$NCf$G$OM7AvE*$GM-$jF@$k!%(B

$X$$B$,%3%s%Q%/%H$J$i!"(B$\omega(x_0)$$B$O<+F0E*$K6u$G$J$/$J$j%3%s%Q%/%H$G$"$k$3$H$K$bCm0U$9$Y$-$G$"$k!%$@$+$i(B$\omega(x_0)$$B$O%3%s%Q%/%H5wN%6u4V>e$NN.$l$K$*$$$F$O$D$M$K:?6K>.$G$"$k!%(B

$BDjM}(BA.2$B$r>ZL@$9$k$N$K<!$NL?Bj$,I,MW$G$"$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BL?Bj(BA.3.} $\varphi$$B$O6I=j%3%s%Q%/%H$J5wN%6u4V(B$X$$B>e$NN.$l$H$9$k!%(B
$x_0 \in X$$B$O(B$\omega(x_0)$$B$,6u$G$J$/%3%s%Q%/%H$J=89g$G$"$k$h$&$JE@$H$9$k!%$3$N$H$-(B$\omega(x_0)$$B$OO"7k$G$"$k!%$5$i$K(B$\omega(x_0)$$B$r4^$`3F3+=89g(B$U$$B$KBP$7$F@5$N(B$t'$$B$,$"$C$F!"(B$t \geq t'$$B$KBP$7$F(B$\varphi(x_0,t) \in U$$B$G$"$k!%(B

\vspace{0.3cm}
$X$$B$,%3%s%Q%/%H$J$iL?Bj(BA.3$B$OI8=`E*$J7k2L$G$"$k(B[7]$B!%(B$\omega(x_0)$$B$,6u$G$J$/$F%3%s%Q%/%H$G$"$k$H$$$&2>Dj$OI8=`E*>ZL@$,(B$X$$B$,6I=j%3%s%Q%/%H$N$H$-$K@5$7$$$3$H$rJ]>Z$7$F$$$k!%(B

\vspace{0.3cm}
\noindent
\underbar{$BDjM}(BA.2$B$N>ZL@(B.} $\Omega=\omega(x_0)$$B$H$*$-!"(B$\psi$$B$O(B$\varphi$$B$r(B$\Omega$$B$K@)8B$7$?N.$l$H$9$k!%(B$y,y' \in \Omega$$B$H$7!"(B$\varepsilon$$B$H(B$T$$B$,M?$($i$l$?$H$9$k!%(B$y$$B$+$i(B$y'$$B$X$N(B$(\varepsilon,T,\psi)$$B:?$r9=@.$;$M$P$J$i$J$$!%(B

$U$$B$O(B$X$$B$N3+ItJ,=89g$G$"$C$F(B$\Omega$$B$r4^$_JDJq$O%3%s%Q%/%H$G$"$k$H$9$k(B(?)$B!%(B$\delta < \varepsilon/2$$B$rA*$s$G(B
$$ \begin{array}{ccc}
  u_1, u_2 \in \bar{U}, & d(u_1,u_2) 
\Rightarrow d(\varphi(u_1,t), \varphi(u_2,t)) < \varepsilon/2 & 
$BG$0U$N(B t \in [0,2T],
   \end{array}                 \eqno(A.1)
$$
$B$H$J$k$h$&$K$9$k!%(B$V$$B$O(B$U$$B$N3+ItJ,=89g$G$"$C$F(B$\Omega \subset U$$B$+$D$9$Y$F$N(B$x \in V$$B$KBP$7$F(B$d(x, \Omega) < \delta$$B$G$"$k$H$9$k!%L?Bj(BA.3$B$h$j!"(B$t'$$B$,$"$C$F(B$\varphi(x_0,[t',\infty)) \subset V$$B$G$"$k!%(B

$x_1 \in \varphi(x_0,[t',\infty))$$B$rA*$s$G(B$d(x_1,y) < \delta$$B$H$9$k!%(B$T'>T$$B$rA*$s$G(B$d(\varphi(x_1,T'),y') < \varepsilon/2$$B$H$9$k!%(B$n$$B$O(B$T'/T$$BFb$N:GBg@0?t$H$7!"(B$i=2, \ldots,n$$B$KBP$7$F(B$x_i=\varphi(x_{i-1},T)$$B$H$*$/!%(B$x_{n+1} = \varphi(x_n, T'-(n-1)T) = \varphi(x_1,T')$$B$H$*$/!%(B$i=2, \ldots,n$$B$KBP$7$F(B$y_i \in \Omega$$B$rA*$s$G(B$d(y_i,x_i) < \delta$$B$H$9$k!%(B$y_1=y$$B$*$h$S(B$y_{n+1}=y'$$B$H$*$/!%(B$i=1, \ldots,n-1$$B$KBP$7$F(B$t_i=T$$B$H$7!"$^$?(B$t_n=T'-(n-1)T$$B$H$9$k!%(B

$(y_1, \ldots,y_{n+1}, t_1, \ldots,t_n)$$B$O(B$y$$B$+$i(B$y'$$B$X$N(B$(\varepsilon,T,\psi)$$B:?$G$"$k$3$H$r<gD%$9$k!%Dj5A$N(B(a)$B$H(B(b)$B$O9=@.K!$K$h$jK~$?$5$l$F$$$k$+$i<($9$Y$-$O(B(c)$B!"$9$J$o$A!"(B
$$ \begin{array}{ccc}
 d(\psi(y_i,t_i), y_{i+1}) \leq \varepsilon & {\rm for} & 
i=1, \ldots, n,
   \end{array}                 \eqno(A.2)
$$
$B$G$"$k!%$H$3$m$,(B$d(y_i,x_i) < \delta$$B$+$D(B$t_i < 2T$$B$G$"$k$+$i!"(B(A.1)$B<0$h$j(B
$$ 
 d(\psi(y_i,t_i), y_{i+1}) = 
 d(\varphi(y_i,t_i), \varphi(x_i,t_i)) < \varepsilon/2.
$$
$d(y_{i+1},x_{i+1}) < \delta< \varepsilon/2$$B$G$"$k$+$i!";03QITEy<0$h$j(B(A.2)$B$,=P$k!%$@$+$i(B$(y_1, \ldots,y_{n+1}, t_1, \ldots,$ $t_n)$$B$O(B$y$$B$+$i(B$y'$$B$X$N(B$(\varepsilon,T,\psi)$$B:?$G$"$j!"DjM}(BA.2$B$O>ZL@$5$l$?!%(B\hfill $\Box$

\vspace{0.3cm}
$BJdBj(B6.2$B$ODjM}(BA.2$B!"L?Bj(BA.3$B!"$*$h$S<!$NL?Bj$N7O$H$7$F=P$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BL?Bj(BA.4.} $\varphi$$B$O%3%s%Q%/%H!"O"7k$+$D6u$G$J$$5wN%6u4V(B$X$$B>e$N:?6K>.$J8{G[E*N.$l$H$9$k!%$3$N$H$-(B$X$$B$O$?$@(B1$BE@$+$i$J$k!%(B

\vspace{0.3cm}
\noindent
\underbar{$B>ZL@(B.} $x,y \in X$$B$H$9$k!%(B$x>y$$B$G$"$k$+$i(B$g(x) \geq g(y)$$B$G$"$k!%(B$y>x$$B$G$"$k$+$i(B$g(y) \geq g(x)$$B$G$"$k!%$@$+$i(B$g$$B$O(B$X$$B>e$G0lDj$G$"$j!"$h$C$F(B$X$$B$OITF0E@$N$_4^$`!%ITF0E@$O8IN)$7$F$$$F(B$X$$B$O%3%s%Q%/%H$G$"$k$+$i(B$X$$B$OM-8B8D$NE@$7$+4^$^$J$$!%(B$X$$B$OO"7k$G6u$G$J$$$+$i!"(B1$BE@$@$1$+$i$J$k!%(B\hfill $\Box$

\vspace{0.5cm}
%\newpage
{\bf References}

\re
1. Churchill, R.: Isolated invariant sets in compact metric spaces. {\it J. Differential Equations}  {\bf 12}, 330-352(1972).

\re
2. Conley, C.: The gradient structure of a flow: I. {\it IBM Report}, Yorktown Heights, N.Y., 1972.

\re
3. Conley, C., Easton, R.: Isolated invariant sets and isolating blocks. {\it Trans. A.M.S.} {\bf 158}, 35-61(1971).

\re
4. Easton, R.: Regularization of vector fields by surgery. {\it J. Differential Equation} {\bf 10}, 92-99(1971).

\re
5. Easton, C.: The topology of the regularized integral surfaces of the 3-body problem. {\it J. Differential Equation} {\bf 12}, 361-384(1972).

\re
6. Moser, J.: regularization of Kepler's problem and averaging method on a manifold. {\it Comm. Pure Appl. Math.} {\bf XXIII}, 609-636(1970).

\re
7. Nemytskii, V., Stepanov, V.: {\it Qualitative theory of Differential equations}. Princeton Univ. Press, 1960.

\re
8. Painlev\'{e}, P.: Lecon sur la th\'{e}orie analytique des \'{e}quations duff\'{e}rentielles(Stockholm 1895). Hermann, Paris, 1897.

\re
9. Saari, D.: Singularities and collisions of newtonian gravitationl systems. {\it Arch. rational Mech. Anal.} {\bf 49},311-320(1973).

\re
10. Sacker, R., Sell, G.: On the existence of nontrivial recurrent motions. {\it Duke math. J.}, to appear.

\re
11. Siegel, C.: Der Dreierstoss. {\it Acta math.} {\bf 42}, 127-168(1941).

\re
12. Siegel, C., Moser, J.: {\it Lectures on Celestial Mechanics}. Berlin, Springer, 1971. 

\re
13. Sperling, H.: On the real singularities of the n-body problem. {\it J. Reine Angew. math.} {\bf 245}, 14-50(1970).

\re
14. Sundman, K.: M\'{e}moire sur le probl\'{e}me des trois corps. {\it Acta Math.} {\bf 36}, 105-179(1912).

\re
15. Sundman, K.: Nouvelles recherches sur le probl\'{e}me des trois corps. {\it Acta Soc. Sci. Fenn.} {\bf 35}, No.9(1909).

\re
16. Zeipel, H. von: Sur les singularit\'{e}s du probl\'{e}me des n corps. {\it Ark. Mat. Astr. Fys.} {\bf 4}, No.32(1908).

\re
17. Wintner, A.: {\it The analytical foundations of celestial mechanics.} Princeton Univ. Press, 1941.
