
$B>e$N(B2$B$D$N@QJ,$,$H$b$KM-3&$G$"$k$3$H$r<($7$?$$!%JdBj(B4.2$B$r>ZL@$9$k$N$KMQ$$$?$N$H;w$?5DO@$K$h$j!"(B
$$
  |\tau_{2n+1}^* - \tau_{2n+1}^0| \leq M_6 ,  \leqno(6.16)
$$
$$
\displaystyle 
  \left| \exp \left( \int_{\tau_{2n+1}^0}^{\tau} - 
\frac{1}{2} v(\tau) d \tau \right) \right| 
\leq M_7   \leqno(6.17)
$$
$B$,$9$Y$F$N(B$\tau \in [\tau_{2n+1}^0,\tau_{2n+1}^*]$$B$KBP$7$F@.$jN)$D$3$H$,$o$+$k!%$3$3$G(B$M_6$$B$*$h$S(B$M_7$$B$O@5Dj?t$G$"$k!%(B

(6.15)$B$NBhFs$N@QJ,$K4X$7$F$O!"ITEy<0(B(6.13)$B$rF@$k$N$KMQ$$$?$N$HF1$85DO@$r;H$&!%(B
$$ \begin{array}{rl}
\displaystyle 
  \exp \left( \int_{\tau_{2n+1}^0}^{\tau} - \frac{1}{2} v(\tau) d \tau 
\right) = & \displaystyle \exp \left( 
\int_{\tau_{2n+1}^0}^{\tau_{2n+1}^*} - \frac{1}{2} v(\tau) d \tau 
\right) \\ 
& \displaystyle \cdot \exp \left( 
\int_{\tau_{2n+1}^*}^{\tau} - \frac{1}{2} v(\tau) d \tau \right) 
   \end{array}$B!!!!!!!!!!!!(B  \leqno(6.18)
$$
$B$G$"$k$3$H$KCm0U$9$k!%Bh0l9`$OM-3&$G$"$k!%BhFs9`$K4X$7$F$O!"(B$\tau \geq \tau_{2n+1}^*$$B$J$i(B$v(\tau) \geq v^+$$B$G$"$k$3$H$h$j(B
$$
\displaystyle 
 \exp \left( \int_{\tau_{2n+1}^*}^{\tau} - \frac{1}{2} v(\tau) d \tau 
\right) \leq \displaystyle \exp \left( - \frac{1}{2} 
v^+ (\tau - \tau_{2n+1}^*) \right) 
$$
$B$rF@$k!%ITEy<0(B(6.15)$B!"(B(6.16)$B$*$h$S(B(6.17)$B$r9MN8$9$k$H!"$3$l$+$i(B
$$
\displaystyle 
 \int_{\tau_{2n+1}^0}^{\tau_{2n+1}} 
\frac{m_1m_3 |u \cos \phi|}{4} d \tau 
\leq \displaystyle (|u_0| + 2^{-2n} M_5) M_8        \leqno(6.19)
$$
$B$,!"$"$k(B$M_8>0$$B$*$h$S$9$Y$F$N(B$n$$B$KBP$7$FF@$i$l$k!%$7$?$,$C$F!"ITEy<0(B(6.10)$B!"(B(6.11)$B!"(B(6.12)$B!"$*$h$S(B(6.13)$B$h$j(B
$$ \begin{array}{rl}
  |a_n| & \leq \displaystyle 
(|u_0| + 2^{-2n} M_5) M_8 + 2^{-2n} M_2 + M_3 |u_0| \\
& = \displaystyle (M_3 + M_8)|u_0| + 2^{-2n} (M_2 + M_5M_8) 
   \end{array}  \leqno(6.20)
$$
$B$,@.$jN)$D!%(B$\sum |a_n|$$B$,<}B+Ns$G$"$k$3$H$r<($9$N$K!"(B
$$
  |u_0| \leq 2^{-2n} M_9   \leqno(6.21)
$$
$B$,$"$k(B$M_9>0$$B$KBP$7$F@.$jN)$D$3$H$r<($;$P$h$$!%$=$N$?$a$KITEy<0(B(6.3): 
$|w_{12}(\bar{t}_n)| \leq 2^{-n}M_1$$B$r;H$&!%(B
$BDj5A$h$j(B$|w_{12}| = |h_{12}c_{12}^2|$$B$G$"$C$?$3$H$r;W$$$@$=$&!%$7$?$,$C$F(B
$$
  |c_{12}^2(\overline{t}_n) h_{12}(\overline{t}_n)| 
\leq 2^{-2n} M_1   \leqno(6.22)
$$
$B$G$"$k!%<!$K(B$c_{12}(\bar{t}_n)$$B$NI>2A$r(B$u_0$$B$r;H$C$F9T$J$&!%$U$?$?$S(B$t_{2n+1}^*, t_{2n+1}^* \in [\bar{t}_{2n},t_{2n+1}]$$B$r50F;$,(B$T$$B$KE~Ce$9$k;~9o$G$"$k$H$7!"(B$r(t_{2n+1}^*) = r_*$$B$H$*$/!%(B(6.14)$B$h$j(B
$$ \begin{array}{rl}
\displaystyle 
  |c_{12}(\bar{t}_{2n})| & = \displaystyle 
|r^{1/2}(\bar{t}_{2n}) u(\bar{t}_{2n})| \\
& = \displaystyle 
r^{1/2}(\tau_{2n+1}^0) |u_0 + s(\bar{t}_{2n})| \\ 
& \geq \displaystyle 
M_{10} r_*^{1/2} |u_0 + s(\bar{t}_{2n})| , 
   \end{array}  \leqno(6.23)
$$
$B$G$"$k!%$3$3$G(B
%\frac{\displaystyle }{\displaystyle }
$$
  s(t) = \displaystyle 
\int_{t_{2n+1}^0}^t \exp \left( \int_{\tau_{2n+1}^0}^{\tau} 
\frac{1}{2} v(\tau) d \tau \right) f(t) r^{-1/2} dt . 
$$
(6.23)$BFb$NITEy<0$O!"(B$x(t_{2n+1}^*) \in T$$B$J$k$9$Y$F$N(B$x$$B$*$h$S$"$k(B$M_{10}>0$$B$KBP$7$F(B
$$
\displaystyle 
 \frac{r_*}{r(\tau_{2n+1}^0)} = \displaystyle \exp
\int_{\tau_{2n+1}^0}^{\tau_{2n+1}^*} 
\frac{1}{2} v(\tau) d \tau \leq M_{10}^{-2}
$$
$B$G$"$k$3$H$h$j$7$?$,$&!%$3$3$GITEy<0(B(6.16)$B$r;H$C$?!%(B(6.14)$B$h$j(B
$$
  |s(t)| \leq 2^{-2n} M_5 \ \ {\rm for \ all} \ \ t \in 
[t_{2n+1}^0,t_{2n+1}^*] 
$$
$B$G$"$k$3$H$KCm0U$7$h$&!%(B

$B0lJ}!"JdBj(B4.2$B$*$h$S(B4.3$B$N>ZL@$K$b$"$C$?DL$j!"4JC1$K$o$+$k$h$&$K!"(B$d_7>0$$B$*$h$S(B$d_8>0$$B$,$"$C$F!"==J,>.$5$J(B$r_*$$B$KBP$7$F$O(B
$$
 d_7 \leq \displaystyle 
\frac{|h_{12}(\bar{t}_{2n})|}{|h_{12}(t_{2n+1}^*)|} \leq d_8  
                     \leqno(6.24)
$$
$B$,(B$x(t_{2n+1}^*) \in T$$B$J$k$9$Y$F$N(B$x$$B$KBP$7$F@.$jN)$D!%8x<0(B(6.22)$B!"(B(6.23)$B$*$h$S(B(6.24)$B$rJ;$;$F(B
$$
  2^{-(2n+1)} M_1 \geq |w_{12}(\bar{t}_{2n})| \geq 
d_7 |h_{12}(t_{2n+1}^*)| M_{10}^2 r_* |u_0 + s(\bar{t}_{2n})| 
$$
$B$rF@$k!%$7$?$,$C$F(B
%\frac{\displaystyle }{\displaystyle }
$$
  |u_0 + s(\bar{t}_{2n})|^2 \leq 
\frac{\displaystyle 2^{-(2n+1)}M_1 d_7^{-1} M_0^{-2}}
{\displaystyle |r_*h_{12}(t_{2n+1}^*)|} .
$$
$BITEy<0(B(6.21)$B$r>ZL@$9$k$N$K!"(B$|s(\bar{t}_{2n})| \leq 2^{-2n} M_5$$B$h$j!"(B
$$
  |r_* h_{12}(t_{2n+1}^*)| > M_{11}
$$
$B$,!"$"$k(B$M_{11}>0$$B$*$h$S(B$x(t_{2n+1}^*) \in T$$B$J$k$9$Y$F$N(B$x$$B$KBP$7$F@.$jN)$D$3$H$r<($;$P$h$$!%(B$T_0$$B$O%3%s%Q%/%H=89g$G$"$k$+$i!"G$0U$N(B$x \in T$$B$KBP$7$F(B
$$
  |r h_{12}(x)| \neq 0  \leqno(6.25)
$$
$B$G$"$k$3$H$r<($;$P==J,$G$"$k!%(B$rh_{12}(x)$$B$O(B$r$$B$NCM$K0M$i$J$$$3$H!"$*$h$S(BMcGehee$B$N:BI8$G(B
$$
  |r h_{12}(x)| = \displaystyle \left| g - \frac{1}{2} v^2 \sin^2 \phi - w^2 + 2vw \sin \phi - m_1^{3/2} m_3 (1 + 2 \alpha \sin \phi)^{-1/2} 
\right| 
$$
$B$H=q$1$k$3$H$KCm0U$9$k!%(B

$M$$B$r;0BN>WFMB?MMBN$H$7$F!"(B$x \in M$$B$KBP$7$F(B$g(x)=0$$B$G$"$k!%JdBj(B2.1$B$r>ZL@$9$k$N$K;H$C$?$N$HF1$85DO@$K$h$j!"$9$Y$F$N(B$x \in M$$B$KBP$7$F(B$rh_{12}(x) \neq 0$$B$,$o$+$k!%(B$rh_{12}(x)$$B$NO"B3@-$H(B$T_0 \cap M$$B$N%3%s%Q%/%H@-$h$j!"<gD%(B(6.25)$B$O(B$g$$B$,==J,$K>.$5$1$l$P@5$7$$!%$7$?$,$C$F(B$T_0$$B$r==J,>.$5$/$H$l$P(B$|rh_{12}(x)| \neq 0$$B$G$"$k!%(B$T_0$$B$O9%$-$J$@$1>.$5$/$H$l$k$+$i!"$3$l$,@.$jN)$D$H2>Dj$G$-$k!%$7$?$,$C$FITEy<0(B(6.21)$B$,@.$jN)$D!%$9$J$o$A!"$"$k(B$M_9>0$$B$KBP$7$F(B$|u_0| \leq 2^{-2n}M_9$$B$G$"$k!%$@$+$i(B$\sum a_n$$B$O<}B+Ns$G$"$j!"$7$?$,$C$F(B$t \rightarrow t_{\infty}$$B$N$H$-(B${\bf p}_{12}({\bf q}^*(t))$$B$N6K8B$OB8:_$9$k!%(B

$1/2w_{12} = (1 - |{\bf p}_{12}|^2)^{1/2}$$B$*$h$S(B$n \rightarrow \infty$$B$N$H$-(B$w_{12}(t_n) \rightarrow 0$$B$G$"$k$3$H$h$j!"(B$n \rightarrow \infty$$B$N(B
$B$H$-(B$|{\bf p}_{12}({\bf q}^*(t_n))| \rightarrow 1$$B$G$"$k!%(B
$t \rightarrow t_{\infty}$$B$N$H$-(B${\bf p}_{12}({\bf q}^*(t))$$B$N6K8B$,B8:_$9$k$+$i!"(B$t \rightarrow t_{\infty}$$B$N$H$-(B$|{\bf p}_{12}({\bf q}^*(t))| \rightarrow 1$$B$G$"$k!%$3$l$GJdBj(B6.2$B$,>ZL@$5$l$?!%(B\hfill $\Box$

\vspace{0.3cm}
$B<!$,@.$jN)$D$3$H$r3NG'$7$FM_$7$$!%(B
$$
  w_{12} = 2(1 - |{\bf p}_{12}|^2)^{1/2} \rightarrow 0 .
$$
$B$@$+$i(B$t \rightarrow t_{\infty}$$B$N$H$-(B$w_{12}({\bf q}^*(t)) \rightarrow 0$$B$G$"$j!"LsB+$I$*$j!"JdBj(B6.1$B$O>ZL@$5$l$?!%(B\hfill $\Box$

\vspace{0.3cm}
$B$$$^$d=EMW$JJdBj$r=R$Y$k$3$H$,$G$-$k!%$3$l$K$h$jHs>WFMFC0[@-$NB8:_$N>ZL@$,407k$9$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BJdBj(B 6.3.} ${\bf q}^*(t)$$B$rA0@a$G9=@.$7$?M-8B;~4V$KHsM-3&$J2r$G$"$k$H$9$k!%$^$?(B${\bf p}_{12}^*, {\bf p}_{45}^*$$B$OJdBj(B6.2$B$N$b$N$H$9$k!%(B
${\bf p}_{12}^* \cdot {\bf p}_{45}^* \neq 0$$B$+$D(B${\bf p}_{12}^* \neq \pm {\bf p}_{45}^* \neq 0$$B$J$i!"$9$J$o$A!"(B${\bf p}_{12}^*$$B$H(B${\bf p}_{45}^*$$B$O8_$$$KJ?9T$G$b?bD>$G$b$J$$$J$i!"(B$t^a < t_{\infty}$$B$,$"$C$F!"$9$Y$F$N(B$t \in (t^a, t_{\infty})$$B$KBP$7$F(B$c_{12} \neq 0$$B$G$"$k!%8@$$49$($k$H!";~4V6h4V(B$(t^a,t_{\infty})$$BFb$K(B$m_1,m_2$$B$*$h$S(B$m_4,m_5$$B$NFsBN>WFM$O$J$$!%(B

\vspace{0.3cm}
\noindent
\underline{$B>ZL@(B}. $B$3$NJdBj$N>ZL@$O8+$?L\$h$j$d$5$7$$!%$^$:(B
$$
  \dot{c}_{12} = 2m_1m_4 (x_1y_4 - x_4y_1) (r_{14}^{-3} - r_{15}^{-3}) 
$$
$B$r7W;;$9$k!%(B$\theta$$B$O(B2$B$D$N%Y%/%H%k(B${\bf r}_{12}$$B$H(B${\bf r}_{45}$$B$N$"$$$@$N3QEY$H$9$k!%$9$J$o$A!"(B
$$
  \cos \theta = \displaystyle 
\frac{x_1x_4 + y_1y_4}{(x_1+y_1)^{1/2}(x_4+y_4)^{1/2}}
$$
$B$3$N$H$-(B
$$ \begin{array}{rl}
  \dot{c}_{12} = & \displaystyle 
\frac{1}{2} m_1 m_4 r_{12} r_{45} \sin \theta 
\frac{3}{2} r_{12} r_{45} \cos \theta 
\left( (z_1 - z_4)^{-6} + O((z_1 - z_4)^{-9}) \right) \\
         = & \displaystyle \frac{3}{8} 
m_1 m_4 r_{12}^2 r_{45}^2 \sin 2 \theta 
\left( (z_1 - z_4)^{-6} + O((z_1 - z_4)^{-9}) \right) . 
   \end{array}   \leqno(6.26)
$$

${\bf q}^*(t)$$B$rA0@a$G9=@.$7$?HsM-3&2r$H$9$k!%JdBj(B6.2$B$h$j!"(B
${\bf p}_{12}({\bf q}^*(t))$$B$*$h$S(B${\bf p}_{45}({\bf q}^*(t))$$B$O(B$t \rightarrow t_{\infty}$$B$N$H$-(B${\bf p}_{12}^*$$B$*$h$S(B${\bf p}_{45}^*$$B$r6K8B$H$9$k!%$3$3$G(B$t_{\infty}$$B$O2r(B${\bf q}^*(t)$$B$,=*$o$k;~9o$G$"$k!%(B$\theta^*$$B$r(B${\bf p}_{12}^*$$B$*$h$S(B${\bf p}_{45}^*$$B$N4V$N3QEY$H$9$k!%JdBj$N2>Dj$K$h$j!"(B
$$
     \sin 2 \theta \neq 0  
$$
$B$G$"$k!%(B$\sin 2 \theta >0$$B$H2>Dj$9$k!%(B$\sin 2 \theta <0$$B$N>l9g$bF1MM$K<h$j07$($k$3$H$,H=$k!%(B$\omega$$B$r(B${\bf p}_{12}({\bf q}^*(t))$$B$H(B${\bf p}_{45}({\bf q}^*(t))$$B$N4V$N3QEY$H$9$k!%$3$N$H$-(B
$$ \begin{array}{ccc}
   \omega(t) \rightarrow \theta^* & {\rm as} & t \rightarrow t_{\infty} 
   \end{array}
$$
$B$G$"$k!%(B

$t^b < t^{\infty}$$B$O!"$9$Y$F$N(B$t \in (t^b, t_{\infty})$$B$KBP$7$F(B
$$
     \sin 2 \omega (t) > 0  
$$
$B$*$h$S(B
$$
 (z_1 - z_4)^{-6} + O((z_1 - z_4)^{-9}) > 0  
$$
$B$rK~$?$9;~9o$G$"$k$H$9$k!%$3$3$G9b<!$N9`(B$O((z_1-z_4)^{-9})$$B$OJ}Dx<0(B(6.26)$B$+$iMh$F$$$k!%(B$t \rightarrow t_{\infty}$$B$N$H$-(B$(z_1 - z_4) \rightarrow \infty$$B$@$+$i$3$l$O2DG=$G$"$k!%(B

2$B$D$N>l9g$,$"$k!%(B

(a) $B$9$Y$F$N(B$t \in (t^b, t_{\infty})$$B$KBP$7$F(B$c_{12} \neq 0$$B$J$i!"(B$t^b = t^a$$B$H<h$k!%$9$k$HJdBj$O>ZL@$5$l$?$3$H$K$J$k!%(B

(b) $B$"$k(B$t = t^a, t^a \in (t^b, t_{\infty})$$B$,$"$C$F(B$c_{12}(t^a)=0$$B$J$i!"$9$Y$F$N(B$t \in (t^a, t_{\infty})$$B$KBP$7$F(B$c_{12}(t) \neq 0$$B$G$"$k$3$H$r<($=$&!%$9$k$H!"$3$l$G$bJdBj$,>ZL@$5$l$?$3$H$K$J$k!%(B

$B$7$+$7!"Cm0U$9$Y$-$O!"Dj5A$h$j!"(B
$$
  {\bf p}_{12} = \displaystyle 
\frac{{\bf r}_{12}(t^a)}{r_{12}(t^a)} + 
\frac{4({\bf c}_{12} \times (\dot{x}_1,\dot{y}_1,0))}{m_1} , 
$$
$B$J$k$3$H$G$"$k!%(B${\bf c}_{12}(t^a)=0$$B$G$"$k$+$i(B${\bf p}_{12}(t^a) = {\bf r}_{12}(t^a) r_{12}^{-1}(t^a)$$B$G$"$j!"$7$?$,$C$F(B
$$
   \theta(t^a) = \omega(t^a) 
$$
$B$G$"$k!%$f$($K!"(B$r_{12}(t^a) \neq 0$$B$*$h$S(B$r_{45}(t^a) \neq 0$$B$H$7$F(B
$$ 
  \dot{c}_{12} (t^a) = \frac{3}{8} m_1 m_4 r_{12}^2 r_{45}^2 
\sin (2 \omega(t^a)) ((z_1 - z_4)^{-6} + h.o.t) > 0, 
$$
$B$G$"$k!%$7$?$,$C$F!"(B$r_{12}(t^a) \neq 0$$B$*$h$S(B$r_{45}(t^a) \neq 0$$B$N$H$-0J2<$rF@$k!%(B

(1) $t < t^a$$B$G(B$(t^a-t)$$B$,==J,>.$5$1$l$P(B$c_{12}(t) < 0$; $B$^$?(B

(2) $t > t^a$$B$G(B$(t-t^a)$$B$,==J,>.$5$1$l$P(B$c_{12}(t) > 0$$B!%(B

$B>e$N4X78<0$O(B$r_{12}(t^a) = 0$$B$H(B$r_{45}(t^a) = 0$$B$NJRJ}$^$?$ON>J}$,@.$jN)$D$H$-$b@5$7$$!%$3$s$I$O$9$Y$F$N(B$t \in (t^a,t_{\infty})$$B$KBP$7$F(B$c_{12}(t)>0$$B$r>ZL@$7$h$&!%(B

$BGXM}K!$G9T$J$&!%$3$l$,@5$7$/$J$$$H2>Dj$9$k!%$9$k$H(B$t^c \in (t^a,t_{\infty})$$B$,$"$C$F!"$9$Y$F$N(B$t \in (t^a,t^c)$$B$KBP$7$F(B$c_{12}(t)>0$$B$G$"$j!"(B$c_{12}(t^c)=0$$B$G$"$k!%(B$c_{12}(t^c)=0$$B$@$+$i!">e$N5DO@$h$j(B($t^a$$B$r(B$t^c$$B$GCV$-49$($F(B)$B!"(B$t<t^c$$B$G(B$t^c-t$$B$,==J,>.$5$1$l$P(B$c_{12}(t)<0$$B$H$J$j!"2>Dj$KL7=b$9$k!%(B

$B$3$l$GJdBj(B6.3$B$,>ZL@$5$l$?!%(B\hfill $\Box$

\vspace{0.3cm}
$B:G8e$K<g7k2L$G$"$kHs>WFMFC0[@-$NB8:_$r>ZL@$9$k$H$3$m$K$-$?!%(B

\vspace{0.3cm}
\noindent
\underline{$BDjM}(B 1.2$B$N>ZL@(B}. $BDjM}(B1.1$B$*$h$SJdBj(B6.3$B$h$j!"(B$\Lambda$$B$NHs2D;;ItJ,=89g(B$\Lambda_0$$B$,$"$C$F!"$9$Y$F$N(B$x \in \Lambda_0$$B$KBP$7$F(B${\bf p}_{12}^*(x)$$B$*$h$S(B${\bf p}_{45}^*(x)$$B$,8_$$$KJ?9T$G$b$J$$$7!"D>8r$b$7$J$$$3$H$r<($9$@$1$G$h$$$3$H$,H=$C$F$$$k!%(B

$x^*$$B$rDjM}(B1.1$B$NE@$G(B
$$
  \sin (2 \theta (x^*, t^*)) \neq 0$B!!(B
$$
$B$rK~$?$9$b$N$H$9$k!%$3$3$G(B$\theta$$B$O(B${\bf p}_{12}(x^*,t^*)$$B$H(B${\bf p}_{45}(x^*,t^*)$$B$N4V$N3QEY$G$"$k!%$3$N$h$&$JE@$,B8:_$9$k$3$H$OL@$i$+$G$"$k!%JdBj(B6.2$B$N>ZL@$+$i<!$NH/A[$rF@$k!%G$0U$N(B$\epsilon>0$$B$KBP$7$F==J,>.$5$J$/$5$S(B$W,W_i,W_{ij},W_{ijk}, \ldots$$B$rA*$S!"$9$Y$F$N(B$x \in \Lambda_0 = W \cap W_i \cap W_{ij} \cap W_{ijk} \cap \ldots$$B$KBP$7$F(B
$$
     | {\bf p}_{12}(x^*,t^*) - {\bf p}_{12}(x,t) | < \epsilon
$$
$B$*$h$S(B
$$
     | {\bf p}_{45}(x^*,t^*) - {\bf p}_{45}(x,t) | < \epsilon
$$
$B$,(B$\tilde{t}_2 \leq t \leq t_{\infty}$$B$J$k$9$Y$F$N(B$t$$B$KBP$7$F@.$jN)$D$h$&$K$G$-$k!%$7$?$,$C$F(B$\epsilon>0$$B$r==J,>.$5$/A*$s$G(B
$$
  \sin 2 \theta (x, t_{\infty}) \neq 0 
$$
$B$,$9$Y$F$N(B$x \in \Lambda_0$$B$KBP$7$F@.$jN)$D$h$&$K$G$-$k!%(B

$BDjM}(B1.2$B$O$$$^$dJdBj(B6.3$B$+$i=P$k!%(B \hfill $\Box$

\vspace{0.5cm}
\noindent
$B;29MJ88%(B

\re
[1] D. Anosov, Smooth Dynamical Systems. Introductory Article, {\it A.M.S. Transl.} {\bf 125} (1985), 1-20.

\re
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