
%\vspace{0.5cm}
\newpage
\noindent
{\bf 8. $BDjM}72$N>ZL@(B}

$B$3$N@a$G$ODjM}(B6.1$B!"(B7.1$B$*$h$S(B7.2$B$N>ZL@$r9T$J$&!%$^$:DjM}(B7.1$B$r>ZL@$7$h$&!%(B

\vspace{0.3cm}
\noindent
\underline{$BDjM}(B7.1$B$N>ZL@(B.} 

\vspace{0.3cm}
$C_{J^*}$$B$r(B$D_0$$BFb$NITJQ6J@~$H$9$k!%(B$m_3=e=0$$B$J$i(B$J$$B$O1?F0$NDj?t$G$"$j!"$3$N6J@~$O1_(B$J=J^*$$B$G$"$k!%$3$N1_$N0BDj$*$h$SIT0BDjB?MMBN$O1_Fb$G8r$o$j!"$b$A$m$s$I$N8r:9$b2#CGE*$G$J$$!%$7$+$7!"8r:9$O40A4$K$O=LB`$7$F$*$i$:!"4X?t(B$H$$B$NK!@~J}8~$G$O8r:9$O2#CGE*$G$"$k(B(Xia[21])$B!%(BLyapunov-Schmit$B$N(Bruduction$BK!$K$h$j!"(B$e$$B$H(B$m_3$$B$,>.$5$$$H$-$K(B$W^s(C_{J^*})$$B$H(B$W^u(C_{J^*})$$B$N8r:9$,=LB`$7$F$$$J$$$3$H$r<($9$K$O!"(B$J$$B$NK!@~J}8~$X$N8r:9$N<M1F$,!"(B$e$$B$H(B$m_3$$B$,>.$5$$$H$-$K$d$O$jHs=LB`$G$"$k$3$H$r<($;$P$h$$!%$3$NL\E*$N$?$a$K(B$J$$B$NJQJ,(B$\Delta J$$B$r7W;;$9$kI,MW$,$"$k!%$O$8$a$K1?F0J}Dx<0$+$i<!$rF@$k!%(B
$$
 dJ/d\phi = \displaystyle 
\frac{-2m_2ex^4 \sin \phi \cos \psi}{1-x^4 \rho} 
\left( -1 + \frac{3}{(1-2x^2 \cos \phi + x^4)^{5/2}} \right) 
 + O(m_2^2e,m_3,e^2). 
$$

$m_2e$$B$N0l<!$^$G$N(B$J$$B$NJQJ,(B$\Delta J$$B$O!">e$NJ}Dx<0$r@QJ,$9$l$PF@$i$l$k!%$^$:(B$\gamma$$B$K1h$C$F(B$D_0$$B$NE@$+$i%[%b%/%j%K%C%/E@(B$p$$B$^$G$N(B$J$$B$NJQJ,$rF@$?$$!%(B$\phi=0$$B$N$H$-$N(B$\psi$$B$NCM$r(B$\psi_0$$B$H$9$k!%$3$N$H$-<!$rF@$k!%(B
$$ \begin{array}{rl}
 \Delta J = & \displaystyle \int_{-\infty}^0 
\frac{-2m_2ex^4 \sin \phi \cos \psi}{1-x^4 \rho} 
\left( -1 + \frac{3}{(1-2x^2 \cos \phi + x^4)^{5/2}} \right) d \phi \\ 
  = & \displaystyle \cos \psi_0 \int_{-\infty}^0 
\frac{-2m_2ex^4 \sin \phi \cos (\psi - \psi_0)}{1-x^4 \rho} \\
   & \displaystyle \times \left( -1 + 
\frac{3}{(1-2x^2 \cos \phi + x^4)^{5/2}} \right) d \phi \\
   & \displaystyle + \sin \psi_0 \int_{-\infty}^0
\frac{2m_2ex^4 \sin \phi \sin (\psi - \psi_0)}{1-x^4 \rho} \\
   & \displaystyle \times \left( -1 + 
\frac{3}{(1-2x^2 \cos \phi + x^4)^{5/2}} \right) d \phi \\
 = &  A \cos \psi_0 + B \sin \psi_0. 
   \end{array}                       \eqno(21)
$$
$B$3$3$G@QJ,$O$9$Y$F%[%b%/%j%K%C%/50F;(B$\gamma$$B$K1h$C$F9T$J$&!%>e$N<0$K$*$$$F:G8e$NEy<0$O(B$A$$B$H(B$B$$B$NDj5A$K$J$C$F$$$k!%(B$\gamma$$B$N50F;$rFsBNLdBj$NJ|J*50F;(B(15)$B$G6a;w$9$l$P<!$rF@$k!%(B
$$ \begin{array}{rl}
 A = & \displaystyle \int_{-\infty}^0 
\frac{-2m_2e\xi^4 \sin \phi \cos (\psi - \psi_0)}{1+Jx^4} 
\left( -1 + 
\frac{3}{(1-2\xi^2 \cos \phi + \xi^4)^{5/2}} \right) d \phi \\ 
  & + O(m_2^2e), \\
 B = & \displaystyle \int_{-\infty}^0 
\frac{2m_2e\xi^4 \sin \phi \sin (\psi - \psi_0)}{1+Jx^4} 
\left( -1 + 
\frac{3}{(1-2\xi^2 \cos \phi + \xi^4)^{5/2}} \right) d \phi \\ 
  & + O(m_2^2e). 
   \end{array}                       \eqno(22)
$$
$J=\sqrt{2}$$B$G$"$k$+$i!">e$N@QJ,$O(B2$B$D$H$bHsM-3&$G$"$k!%$7$?$,$C$F(B$\delta_2>\delta_1>0$$B$O==J,>.$5$$$H$7$F$9$Y$F$N(B$J \in T = (\sqrt{2}+\delta_1,\sqrt{2}+\delta_2)$$B$KBP$7$F!"(B$A$$B$H(B$B$$B$O$H$b$KHs%<%m$G$"$k!%8e$N;29M$N$?$a!"(B$J \rightarrow \sqrt{2}$$B$N$H$-!"$"$k(B$c_2 \neq 0$$B$KBP$7$F(B$B \sim c_2(J-\sqrt{2})^{-2}$$B$G$"$k$3$H$r;XE&$7$F$*$/!%(B

$D_0 \subset S$$B$rJdBj(B7.1$B$N$b$N$H$7!"(B$C_{J^*}$$B$r(B$D_0$$BFb$NITJQ6J@~$H$9$k!%(B$J=h(\psi)$$B$r1_(B$C_{J^*}$$B$NJ}Dx<0$H$9$k!%(B$h(\psi)$$B$O3j$i$+$G!"(B$\psi$$B$K4X$7<~4|(B$2\pi$$B$G$"$k!%(B$J = h_1(\psi)$$B$O<LA|(B$d_1^{-1}$$B$N$b$H$G$N(B$J=h(\psi)$$B$NA|$G$"$k$H$9$k!%$3$N<LA|$OA0@a$NJdBj(B7.1$B$N>ZL@Cf$GDj5A$5$l$F$$$k!%<LA|(B$d_1$$B$O(B$S$$B$NE@(B$q$$B$r!"<LA|(B$d$$B$NDj5AFb$N(B$W^u(q)$$B$H(B$W^s(S)$$B$H$N2#CGE*8rE@(B$q^*$$B$K<L$9$3$H$r;W$$$@$=$&!%>e$G$N7W;;$+$i!"(B$m_2e$$B$N(B1$B<!$^$G$N6a;w$G!"(B$d_1^{-1}$$B$O<!$N7A$r$H$k!%(B
$$
  d_1^{-1}: (J,\psi) \mapsto ( J + A \cos (\psi + b) + 
        B \sin (\psi + b), \psi + b ). 
$$
$B$3$3$G(B$b$$B$O%[%b%/%j%K%C%/50F;(B$p$$B$K1h$C$F(B$-\infty$$B$+$i(B$0$$B$^$G$N(B$\psi$$B$NJQJ,$G$"$k!%(B
$$
 b = \int_{-\infty}^0 
\frac{\displaystyle 1}{\displaystyle 1-x^4 \rho} d \phi 
= \frac{1}{2} \Delta \alpha (\gamma) + o(m_2^2e) \neq 0. 
$$
$h(\psi)$$B$H(B$h_1(\psi)$$B$N4V$K$O<!$N$h$&$J4X78$,$"$k!%(B
$$
  h_1(\psi) = h(\psi-b) + A \cos \psi + B \sin \psi + o(m_2e).
$$
$BF1MM$K!"(B$J=h_2(\psi)$$B$O<LA|(B$d_2^{-1}$$B$N$b$H$G$N(B$J=h(\psi)$$B$NA|$G$"$k$H$9$k!%$3$N<LA|$OA0@a$NJdBj(B7.1$B$N>ZL@$NCf$GDj5A$5$l$F$$$k!%<!$N4X78$,$"$k!%(B
$$
  h_2(\psi) = h(\psi+b) + A \cos \psi - B \sin \psi + o(m_2e).
$$
$B<LA|(B$d$$B$NDj5A$+$i(B$h_2(\psi)=h_1(\psi)$$B$G$"$k!%$7$?$,$C$F!"<!$,@.$jN)$D!%(B
$$
  h(\psi-b) - h(\psi+b) + 2B \sin \psi = 0.     \eqno(23)
$$
$h(\psi)$$B$O6v$G(B$\psi$$B$K4X$7<~4|(B$2\pi$$B$G$"$k$+$i!"(B$h(\psi)$$B$r<!$N$h$&$K(Bcosine$B5i?t$K%U!<%j%(E83+$G$-$k!%(B
$$
  h(\psi) = \sum_{n=0}^{\infty} a_n \cos (n\psi).
$$
$B$3$N5i?t$r<0(B(23)$B$KBeF~$7$F<!$rF@$k!%(B
$$ \begin{array}{rcl}
 2a_n \sin (nb) \sin (n\psi) = 0, & {\rm for} & n \neq 1 \\
 2a_1 \sin b \sin \psi + 2b \sin \psi = 0.
   \end{array}
$$
$BITJQ%H!<%i%9$,(B$m_2$$B$K4X$7$F3j$i$+$G$"$k$3$H$+$i(B$n \neq 1$$B$KBP$7$F(B$a_n=0$$B$G$"$j!"$^$?(B$a_1=B/\sin b$$B$G$"$k$H$$$($k!%$7$?$,$C$F<!$rF@$k!%(B
$$
  h(\psi) = \displaystyle a_0 + \frac{B \cos \psi}{\sin b} + h.o.t.
$$

$B<!$K!"(B$\gamma$$B$H$O0[$J$k%[%b%/%j%K%C%/50F;$K1h$C$F$N(B$J$$B$NJQJ,$r7W;;$7$h$&!%(B5$B@a$G;XE&$7$?$h$&$K!"1_@)8B;0BNLdBj$K$O86E@(B$O$$B$KBP$7$FJL$NBP>NE*2#CGE*%[%b%/%j%K%C%/50F;$,$"$k!%$3$N%[%b%/%j%K%C%/50F;(B$\overline{\gamma}$$B$O$9$Y$F$N(B$-\infty<\phi<\infty$$B$K$o$?$C$F(B$x=\xi(\phi+\pi)$$B$*$h$S(B$y=\eta(\phi+\pi)$$B$K$h$C$F0lMM$K6a;w$G$-$k!%(B$\Delta J(\overline{\gamma})$$B$r$3$N%[%b%/%j%K%C%/50F;(B$\overline{\gamma}$$B$K1h$C$F$N(B$J$$B$NJQJ,$H$9$l$P!"<!$rF@$k!%(B
$$ \begin{array}{rl}
 \Delta J(\overline{\gamma}) & = \displaystyle \int_{-\infty}^{\pi}
\frac{-2m_2ex^4 \sin \phi \cos \psi}{1-x^4 \rho} \left( -1 + 
\frac{3}{(1-2x^2 \cos \phi + x^4)^{5/2}} \right) d \phi \\ 
& = \overline{A} \cos \psi_0 + \overline{B} \sin \psi_0 . 
   \end{array}                       \eqno(24)
$$
$B$3$3$G(B$x$$B$H(B$\psi$$B$NCM$O%[%b%/%j%K%C%/50F;(B$\overline{\gamma}$$B$K1h$C$F$H$j!"(B$\psi_0$$B$O(B$\phi=0$$B$N$H$-$N(B$\psi$$B$NCM$G$"$k!%$U$?$?$S(B$\overline{\gamma}$$B$N50F;$r6a;w$7$F<!$rF@$k!%(B
$$ \begin{array}{rl}
 \overline{A} = & \displaystyle \int_{-\infty}^0
\frac{2m_2e\xi^4 \sin \phi \cos (\psi - \psi_0)}{1+Jx^4} 
\left( -1 + 
\frac{3}{(1+2\xi^2 \cos \phi + \xi^4)^{5/2}} \right) d \phi \\ 
  & + o(m_2e), \\
 \overline{B} = & \displaystyle \int_{-\infty}^0
\frac{-2m_2e\xi^4 \sin \phi \sin (\psi - \psi_0)}{1+Jx^4} \left( -1 + 
\frac{3}{(1+2\xi^2 \cos \phi + \xi^4)^{5/2}} \right) d \phi \\ 
  & + o(m_2e). 
   \end{array}                       \eqno(25)
$$
$B>e$N@QJ,$O$H$b$K(B$J=\sqrt{2}$$B$N6a$/$G<B2r@OE*$G$"$k$3$H$r;XE&$7$F$*$/!%$5$i$K<0(B(15)$B$h$j!"(B$A(J) = \overline{A}(\sqrt{-1}J)$$B$*$h$S(B$B(J) = \overline{B}(\sqrt{-1}J)$$B$G$"$k!%(B$A(J)$$B$H(B$B(J)$$B$ODj?t%<%m$G$J$$$+$i(B$\overline{A}(J)$$B$H(B$\overline{B}(J)$$B$ODj?t%<%m$G$J$$!%(B

$B$$$^$d(B$C_{J^*}$$B$N0BDj$*$h$SIT0BDjB?MMBN$,2#CGE*$K8r$o$k$3$H$O4JC1$K$o$+$k!%A0@a$N(B$d,d_1$$B$*$h$S(B$d_2$$B$NDj5A$K$*$$$F!"(B$W^s(q)$$B$H(B$W^u(S)$$B$N2#CGE*8r:9$NA*Br$r8GDj$7$F%[%b%/%j%K%C%/E@(B$p$$B$+$i$N@\B3E@$H$7$?!%$3$N8r:9$r(B$p$$B$NBe$o$j$K(B$\overline{p}$$B$+$i@\B3$7$?E@$HA*$Y$P!"$?$H$($P(B$\overline{d},\overline{d}_1$$B$d(B$\overline{d}_2$$B$N$h$&$KF1MM$J<LA|$rF@$k$O$:$G$"$k!%(B$J=\overline{h}_1(\psi)$$B$r<LA|(B$\overline{d}_1^{-1}$$B$N$b$H$G$NITJQ6J@~(B$C_{J^*}$$B$NA|$H$9$k!%$3$N$H$-(B$\overline{h}_1(\psi)$$B$O<!$N$h$&$K6a;w$G$-$k!%(B
$$ \begin{array}{rl}
 \overline{h}_1(\psi) & = h(\psi-\overline{b}) + \overline{A} \cos \psi 
+ \overline{B} \sin \psi + O(m_2^2e) \\
 & = a_0 + \frac{\displaystyle B}{\displaystyle \sin b} 
\cos (\psi - \overline{b}) + \overline{A} \cos \psi + 
\overline{B} \sin \psi + O(m_2^2e).
   \end{array}
$$
$B$3$3$G(B$\overline{b}$$B$O%[%b%/%j%K%C%/50F;(B$\overline{\gamma}$$B$K1h$C$F$N(B$\psi$$B$NJQJ,$G$"$j!"(B
$$
  \overline{b} = \displaystyle 
\int_{-\infty}^{\pi} \frac{1}{1-x^4 \rho} d\phi = \displaystyle 
  \int_{-\infty}^0 \frac{1}{1+J\xi^4(\phi)} d\phi + O(m_2^2,e,m_3) , 
$$
$B$G$"$k!%F1MM$K!"(B$J=\overline{h}_2(\psi)$$B$r<LA|(B$\overline{d}_2^{-1}$$B$N$b$H$G$N6J@~(B$C_{J^*}$$B$NA|$G$"$k$H$9$k!%$9$k$H<!$,@.$jN)$D!%(B
$$
  \overline{h}_2(\psi) = a_0 + 
\frac{\displaystyle B}{\displaystyle \sin b} \cos (\psi + \overline{b}) 
+ \overline{A} \cos \psi - \overline{B} \sin \psi + O(m_2^2e).     
$$
$B4JC1$K$o$+$k$h$&$K!"(B2$BK\$N6J@~(B$J=\overline{h}_1(\psi)$$B$H(B$J=\overline{h}_2(\psi)$$B$,2#CGE*$K8r$o$l$P!"ITJQ6J@~(B$C_{J^*}$$B$N0BDj$*$h$SIT0BDjB?MMBN$O2#CGE*$K8r$o$k!%0lJ}!"(B$B \neq \overline{B} \sin b/\sin \overline{b}$$B$J$i!"(B2$B$D$N6J@~(B$\overline{h}_1(\psi)$$B$H(B$\overline{h}_2(\psi)$$B$O(B$e$$B$H(B$m_3$$B$,>.$5$$$H$-2#CGE*$K8r$o$k!%$7$+$7!"$9$G$K4Q;!$7$?$h$&$K!"(B$J$$B$,(B$\sqrt{2}$$B$K6a$$$H(B$\overline{B}$$B$H(B$\overline{b}$$B$O<B2r@OE*$G$"$j!"(B$B \sim c_2(J-\sqrt{2})^{-2}$$B$+$D(B$b \sim c_1 \log(C-\sqrt{2})$$B$G$"$k!%$7$?$,$C$F(B$\delta_2>\delta_1>0$$B$rA*$s$G!"$9$Y$F$N(B$J \in T = (\sqrt{2}+\delta_1,\sqrt{2}+\delta_2)$$B$KBP$7$F(B$B \neq \overline{B} \sin b/\sin \overline{b}$$B$K$G$-$k!%$7$?$,$C$F(B1$B<!6a;w$^$G$G!"(B$W^s(C_{J^*})$$B$H(B$W^u(C_{J^*})$$B$O2#CGE*$K8r$o$k!%(B

$B$3$l$GDjM}(B7.1$B$O>ZL@$5$l$?!%(B

\vspace{0.3cm}
\noindent
\underline{$BDjM}(B7.2$B$N>ZL@(B.} 

\vspace{0.3cm}
$BDjM}(B7.2$B$ODjM}(B7.1$B$N7O$G$"$k!%5<%"!<%N%k%I3H;6$K$*$1$k?6F02r$r8+$D$1$k$?$a$K5-9fNO3X$KAJ$($k!%$O$8$a$K(BArnold[2]$B$HF1$8$h$&$K(B\underline{$BK832=89g(B}(obstructing set)$B$rDj5A$9$k!%(B

$M$$B$r6u4V(B$X$$B$N3j$i$+$JItJ,B?MMBN$H$9$k!%=89g(B$\Omega$$B$,E@(B$x \in M$$B$K$*$$$FB?MMBN(B$M$$B$r(B\underline{$BK832$9$k(B}$B$H$O!"(B$x$$B$K$*$$$F(B$M$$B$H8r$o$k$I$NB?MMBN(B$N$$B$b(B$\Omega$$B$K$h$C$F8r:9$5$l$k$H$-$G$"$k!%Nc$H$7$F!"JD6J@~(B$M$$B$K4,$-IU$/124,(B$\Omega$$B$O6J@~$rK832$9$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BJdBj(B 8.1.} (1) $C_{J^*}$$B$r(B$S$$B>e$N<LA|(B$d$$B$NITJQ6J@~$H$9$k!%(B$U$$B$O(B$\Sigma^{\phi_0}$$B>e$N(B3$B<!85CGLL$G$"$C$F!"$"$kE@(B$q$$B$K$*$$$F(B$C_{J^*}$$B$N0BDjB?MMBN(B$W^s(C_{J^*})$$B$K2#CGE*$G$"$k$H$9$k!%(B$\Omega = \cup_{\phi \geq 0} U(\phi)$$B$O(B$U$$B$+$i=PH/$9$k$9$Y$F$N50F;$N=89g$G$"$k$H$9$k!%$3$N$H$-(B$C_{J^*}$$B$NIT0BDjB?MMBN(B$W^u(C_{J^*})$$B$NG$0U$NE@(B$q^*$$B$KBP$7$F!"(B$\Omega$$B$O(B$q^*$$B$K$*$$$F(B$W^u(C_{J^*})$$B$rK832$9$k!%(B

(2) $C_{J_1}, C_{J_2}, C_{J_3}, \ldots$$B$ODjM}(B7.1$B$NITJQ%H!<%i%9$G$"$C$F!"(B$i=1,2,3,\ldots$$B$KBP$7$F!"(B$C_{J_i}$$B$N0BDjB?MMBN$O(B$C_{J_{i+1}}$$B$NIT0BDjB?MMBN$H2#CGE*$K8r$o$j!"(B$C_{J_i}$$B$NIT0BDjB?MMBN$O(B$C_{J_{i+1}}$$B$N0BDjB?MMBN$H%]%"%s%+%lCGLL(B$\Sigma^{\phi_0}$$B$N$"$kE@$G8r$o$k$H$9$k!%(B$U$$B$O(B$\Sigma^{\phi_0}$$BFb$N(B3$B<!85CGLL$G$"$C$F$"$k(B$i$$B$KBP$7$F(B$C_{J_i}$$B$N0BDjB?MMBN$K2#CGE*$G$"$k$H$7!"(B$\Omega = \cup_{\phi \geq 0} U(\phi)$$B$O(B$U$$B$+$i=PH/$9$k$9$Y$F$N50F;$N=89g$G$"$k$H$9$k!%$3$N$H$-(B$\Omega$$B$O!"G$0U$N(B$j$$B$KBP$7$F(B$W^u(C_{J_j})$$B$I$NE@$K$*$$$F$b(B$W^u(C_{J_i})$$B$rK832$9$k!%(B

\vspace{0.3cm}
\noindent
\underline{$B>ZL@(B.} $BG$0U$NE@(B$q \in W^s(C_{J^*})$$B$KBP$7$F(B$q_1 \in C_{J^*} \subset S$$B$O!"(B$\phi \rightarrow \infty$$B$N$H$-(B$q(\phi) \rightarrow q_1$$B$J$kE@$G$"$k$H$9$k!%$^$:(B$q^* \in W^u(q_1)$$B$J$i(B$\Omega$$B$O(B$q^*$$B$K$*$$$F(B$W^u(C_{J^*})$$B$rK832$9$k$3$H$r<($=$&!%(B

$B>ZL@$O(B$S$$B$N6a$/$NN.$l$N6I=j9=B$$+$i=P$k!%DjM}(B2.1$B$N>ZL@(B(Robinson[16])$B$K$*$$$F!"$$$o$f$k%0%i%UJQ49(B(graph transform)$B$N5;=Q$,;H$o$l$?!%$=$l$K$h$k$H!"4pK\E*$K$O%0%i%UJQ49$N$b$H$G=89g(B$U$$B$,(B$C^{\infty}$$B0LAj$G=89g(B$S_1$$B$NIT0BDjB?MMBN$K<}B+$9$k!%$?$@$7(B$S_1 \subset S$$B$O(B$S$$B$NE@(B$q_1$$B$N>.$5$J6aK5$G$"$k!%$7$?$,$C$F!"G$0U$N(B$\varepsilon>0$$B$*$h$S(B$W^u(S_1)$$BFb$N(B$q^*$$B$N>.$5$J6aK5(B$U_1$$B$KBP$7$F!"@0?t(B$n$$B$,B8:_$7$F(B$\Pi^n(U)$$B$NCGJR$,(B$C^{\infty}$$B0LAj$G(B$U_1$$B$K(B$\varepsilon$$B6a@\$G$"$k!%$7$?$,$C$F(B$q^*$$B$K$*$$$F(B$W^u(S_1)$$B$K2#CGE*$JG$0U$NB?MMBN$O!"$"$k@0?t(B$n$$B$KBP$7$F(B$\Pi^n(U) \subset \Omega$$B$H2#CGE*$K8r$o$k$O$:$G$"$k!%$@$+$i(B$\Omega$$B$O(B$q^* \in W^u(q_1)$$B$K$*$$$F(B$W^u(C_{J^*})$$B$rK832$9$k!%(B

$B$5$F(B$\Omega$$B$,G$0U$N(B$q^* \in W^u(C_{J^*})$$B$KBP$7$F(B$W^u(C_{J^*})$$B$rK832$9$k$3$H$r<($=$&!%$O$8$a$KE@Ns(B$q_1,q_2, \ldots, q_k$$B$,$"$C$F!"(B$q \in W^s(q_1), d(q_1)=q_2, d(q_2)=q_3, \ldots, d(q_{k-1})=d(q_k)$$B$*$h$S(B$q^* \in W^u(q_k)$$B$G$"$k$H$9$k!%(B$q_1^*=d_1^{-1}(q_1)$$B$H$*$/!%$9$k$H(B$q_1^* \in W^u(q_1)$$B$G$"$j!">e$N7k2L$h$j!"(B$\Omega$$B$O(B$q_1^*$$B$K$*$$$F(B$W^u(C_{J^*})$$B$rK832$9$k!%$=$N>e!"<LA|(B$d$$B$NDj5A$h$j!"@0?t(B$n$$B$,$"$C$F(B$\Pi^n(U)$$B$O(B$W^s(q_2)$$B$K2#CGE*$G$"$k$+$i(B$U$$B$O(B$W^s(q_2)$$B$K2#CGE*$G$"$k!%F1$85DO@$K=>$($P!"(B$U$$B$,$9$Y$F$N(B$i=1,2, \ldots, k$$B$KBP$7$F(B$W^s(q_i)$$B$K2#CGE*$G$"$k$3$H$,$o$+$k!%(B$q^* \in W^u(q_k)$$B$J$i(B$\Omega$$B$O(B$q^*$$B$K$*$$$F(B$W^u(C_{J^*})$$B$rK832$9$k!%$3$s$I$OG$0U$N(B$U$$B$*$h$S(B$q^* \in W^u(C_{J^*})$$B$KBP$7$F!"ITJQ6J@~(B$C_{J^*}$$B$K@)8B$7$?<LA|(B$d$$B$NL5M}?t@-$h$j!"E@(B$q' \in U$$B$G(B$q' \in W^s(C_{J^*})$$B$J$k$b$N$,$"$C$F!"(B$C_{J^*}$$BFb$NE@Ns(B$q_1,q_2, \ldots, q_k$$B$G(B$q' \in W^s(q_1), d(q_1)=q_2, \ldots, d(q_{k-1})=q_k$$B$*$h$S(B$q^* \in W^u(q_k)$$B$rK~$?$9$b$N$r8+$D$1$k$3$H$,$G$-$k!%$7$?$,$C$F(B$U$$B$O(B$W^s(C_{J^*})$$B$N$I$NE@$K$*$$$F$b(B$W^s(C_{J^*})$$B$rK832$9$k!%$3$l$GJdBj$N:G=i$NItJ,$,>ZL@$5$l$?!%JdBj$NBhFs$NItJ,$bDjM}(B7.1$B$rMQ$$$FF1MM$K>ZL@$G$-$k!%(B\hfill $\Box$

\vspace{0.3cm}
$B$3$s$I$ODjM}(B7.2$B$N>ZL@$r$7$h$&!%$^$:<LA|(B$d$$B$NITJQ6J@~(B$C_{J^*}$$B$r8GDj$7!"(B$U$$B$O(B$W^s(C_{J^*})$$B$K2#CGE*$J(B3$B<!85CGLL$H$9$k!%A+0\:?(B$C_{J_1}, C_{J_2}, C_{J_3}, \ldots$$BFb$KG$0U$KL58BE@Ns(B$q_1, q_2, q_3, \ldots$$B$rA*$V!%>e$NJdBj$h$j!"(B$U$$B$O$"$kE@!"$?$H$($P(B$q_1^* \in W^u(q_1)$$B$K$*$$$F(B$q_1$$B$NIT0BDjB?MMBN$H2#CGE*$K8r$o$k!%(B$U_1$$B$O(B$U$$B$NFbIt$N(B$q_1^*$$B$N>.$5$J6aK5$H$9$k!%$U$?$?$S(B$U_1$$B$O$"$kE@!"$?$H$($P(B$q_2^*$$B$K$*$$$F(B$q_2$$B$N0BDjB?MMBN$H2#CGE*$K8r$o$k!%(B$U_{12}$$B$O(B$U_1$$B$NFbIt$N(B$q_2^*$$B$N>.$5$J6aK5$H$9$k!%$3$s$I$OCGLL(B$U_{12}$$B$HA+0\:?Fb$NE@(B$q_3$$B$r9M$($k$H(B$U_{12}$$B$H(B$W^s(q_3)$$B$N2#CGE*8rE@(B$q_3^*$$B$r8+$D$1$k$3$H$,$G$-$k!%$3$N$h$&$KB3$1$F$$$1$P!"(B$U$$BFb$K!"F~$l;R$K$J$C$?3+=89g$NNs(B
$$
 U \supset U_1 \supset U_{12} \supset U_{123} \supset \cdots,
$$
$B$rF@$k!%F~$l;R$K$J$C$?$3$N3+=89g$NNs$O6u$G$J$$8r$o$j$r;}$D!%(B$\tilde{q}$$B$r6&DLE@$H$9$k!%4JC1$K$o$+$k$h$&$K!"==J,$K>.$5$J=89gNs$rA*$Y$P!"(B$\tilde{q}$$B$N50F;$O?6F0E*$G$"$k!%$9$J$o$A(B$\tilde{q}$$B$+$i=PH/$9$k50F;$K4X$7$F!"(B$\phi \rightarrow \infty$$B$N$H$-<!$rF@$k!%(B
$$ \begin{array}{lcl}
 \limsup |x| > 0 & $B$*$h$S(B & \liminf |x| = 0.
   \end{array}
$$
$B$=$N>e!"E@Ns(B$q_1,q_2,q_3, \cdots$$B$r$&$^$/A*$V$3$H$K$h$j!"(B$\tilde{q}$$B$N50F;$N(B$\omega$$B6K8B=89g$,A+0\:?Fb$N$9$Y$F$NITJQ1_$r4^$`$h$&$K$G$-$k!%$7$?$,$C$F(B$\tilde{q}$$B$N50F;$O%+%*%9E*$J?6F0$b3H;6$b<($9!%(B  

$B$3$l$GDjM}(B7.2$B$N>ZL@$,40N;$7$?!%(B\hfill $\Box$

\vspace{0.3cm}
\noindent
\underline{$BDjM}(B6.1$B$N>ZL@(B.} 

\vspace{0.3cm}
$T_J^e$$B$O(B$\Omega_n$$B>e$NITJQ1_$G$"$C$F(B$J \in T = (\sqrt{2}+\delta_1, \sqrt{2}+\delta_2)$$B$G$"$k$H$9$k!%(B$e=m_3=0$$B$J$i(B$T_J^e$$B$NIT0BDjB?MMBN$O1_Fb$N=89g(B$S$$B$N0BDjB?MMBN$H8r$o$j!"$3$N8r$o$j$O2#CGE*$G$"$k!%$7$?$,$C$F(B$e>0$$B$,==J,>.$5$$$J$i!"ITJQ1_(B$T_J^e$$B$NIT0BDjB?MMBN$H=89g(B$S$$B$N0BDjB?MMBN$O$d$O$j1_Fb$G!"$?$H$($P(B$s_1$$BFb$G$G8r$o$k!%(B$s_1$$B$+$iN.$l$K=>$C$FA0$K?J$s$G=89g(B$S$$B$^$G9T$1$P!"6K8B$H$7$F(B$S$$B>e$N1_(B$s_1^*$$B$rF@$k!%F1MM$K(B$s_2$$B$r(B$T_J^e$$B$N0BDjB?MMBN$H=89g(B$S$$B$NIT0BDjB?MMBN$H$N8rE@$H$7!"(B$s_2^*$$B$r$=$N(B$\alpha$$B6K8B=89g$H$9$k!%6J@~(B$s_1^*$$B$H(B$s_2^*$$B$N0BDjB?MMBN$HIT0BDjB?MMBN$K4X$7<!$NJdBj$rF@$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BJdBj(B 8.2.}  $\delta_2>\delta_1>0, \varepsilon>0, e_0>0$$B$,B8:_$7!"$9$Y$F$N(B$0<e<e_0$$B$*$h$S(B$0 \leq m_3<\varepsilon$$B$KBP$7!"(B$J \in T = (\sqrt{2}+\delta_1, \sqrt{2}+\delta_2)$$B$J$i(B$s_1^*$$B$NIT0BDjB?MMBN$H(B$s_2^*$$B$N0BDjB?MMBN$O%]%"%s%+%lCGLL(B$\Sigma^{\phi_0}$$B$N$"$kE@$G2#CGE*$K8r$o$k!%(B

\vspace{0.3cm}
\noindent
\underline{$B>ZL@(B.} $BJQ?t(B$J$$B$H(B$\psi$$B$r(B$S$$B$N6I=j:BI8$H$7$F;H$&!%(B$J=h(\psi)$$B$r6J@~(B$s_1^*$$B$NI=<0$H$9$k!%$9$k$H(B$h(\psi)$$B$O3j$i$+$J<~4|4X?t$G$"$k!%BP>N@-$K$h$j!"6J@~(B$s_2^*$$B$O(B$J=h(-\psi)$$B$H=q$1$k!%(B$e=m_3=0$$B$N$H$-!"(B$W^u(s_1^*)$$B$O(B$\Sigma^{\phi=0}$$B>e$N1_(B$\{(x,y)=p;\phi=0 \}$$B$*$h$S(B$\Sigma^{\phi=\pi}$$B>e$N1_(B$\{(x,y)=\overline{p};\phi=\pi \}$$B$K$*$$$F(B$W^s(s_2^*)$$B$H8r$o$k!%$3$N8r:9$O$b$A$m$s=LB`$7$F$$$k!%$7$+$7=LB`$O(B$J$$B$K4X$7$F$N$_$G$"$C$F(B$\psi$$BJ}8~$G$"$k!%(B$e$$B$,>.$5$$$H$-8r:9$,Hs=LB`$G$"$k$3$H$r<($9$?$a$K$O!"(B$e$$B$,>.$5$$$H$-(B$J$$B$K4X$7$F(B$\psi$$BJ}8~$GHs=LB`$G$"$k$3$H$r<($;$P$h$$!%$7$?$,$C$F6J@~(B$\gamma$$B$*$h$S(B$\overline{\gamma}$$B$K1h$C$F$N6J@~(B$s_1^*$$B$*$h$S(B$s_2^*$$B$N(B$J$$B$K4X$9$kJQJ,$r9M$($k$@$1$G$h$$!%(B$C=h_1(\psi)$$B$r<LA|(B$d_1^{-1}$$B$N$b$H$G$N6J@~(B$J=h(\psi)$$B$NA|$H$9$k!%$3$N$H$-<!$,@.$jN)$D!%(B
$$
  h_1(\psi) = h(\psi-b) + A \cos \psi + B \sin \psi + h.o.t. 
$$
$B$3$3$G(B$A,B$$B$*$h$S(B$b$$B$O$3$N@a$N$O$8$a$KDj5A$7$?$b$N$HF1$8$G$"$k!%BP>N@-$K$h$j!"<LA|(B$d_2^{-1}$$B$N$b$H$G$N6J@~(B$s_2^*$$B$NA|$O<!$N<0$G5-=R$5$l$k!%(B
$$
  J = h_1(-\psi) = h(-\psi-b) + A \cos (-\psi) + B \sin (-\psi) + h.o.t.
$$
$B$3$l$i$N(B2$BK\$N6J@~$O(B$h'(-b) + B \neq 0$$B$J$i2#CGE*$K8r$o$k!%$3$l$,@.$jN)$?$J$$$H$7!"$9$Y$F$N(B$C \in J$$B$KBP$7$F(B$h'(-b) + B \equiv 0$$B$H2>Dj$9$k!%(B$W^u(s_1^*)$$B$H(B$W^s(s_2^*)$$B$NB>$N8rE@$r9M$(!"8r:9$N$$$/$D$+$O2#CGE*$G$"$k$3$H$r<($9!%(B

$p_1^* \in \Sigma^0$$B$O2#CGE*%[%b%/%j%K%C%/50F;$G$"$C$F!"(B$p_1^*$$B$N50F;$,(B$p$$B$N6a$/$r(B2$BEYDL2a$9$k$H$9$k(B(Fig.3$B;2>H(B)$B!%A0$K;XE&$7$?$h$&$K!"$3$N$h$&$J%[%b%/%j%K%C%/50F;$OL58B$K$"$k!%$5$i$K!"G$0U$N(B$\varepsilon>0$$B$KBP$7$F(B$p_1^*$$B$rA*$s$G(B$p_1^*$$B$N50F;$,(B$p$$B$N50F;$N(B$\varepsilon$$B6aK5Fb$K$"$k$h$&$K$G$-$k!%$3$3$G(B$d_1$$B$*$h$S(B$d_2$$B$K;w$?<LA|(B$d_{1,p_1^*}$$B$*$h$S(B$d_{2,p_1^*}$$B$r9M$($k!%$=$N:]!"(B$W^u(q)$$B$H(B$W^s(S)$$B$N2#CGE*8rE@$H$7$F(B$p$$B$NBe$o$j$K(B$p_1^*$$B$r;H$&!%(B$J=h_2(\psi)$$B$r<LA|(B$d_{1,p_1^*}^{-1}$$B$N$b$H$G$N6J@~(B$s_1^*$$B$NA|$H$9$k!%>e$N7W;;$HF1MM$K$7$F<!$rF@$k!%(B
$$
  J = h_2(\psi) = h(\psi-2b) + 2B \sin (\psi-b) + O(\varepsilon) + 
h.o.t.
$$
$B$^$?<LA|(B$d_{2,p_1^*}^{-1}$$B$N$b$H$G$N(B$s_2^*$$B$NA|$O<!<0$GI=$o$5$l$k!%(B
$$
  J = h_2(-\psi) = h(-\psi-2b) + 2B \sin (-\psi-b) + O(\varepsilon) + 
h.o.t.
$$
$B>e$N(B2$BK\$N6J@~$O(B$h'_2(0) \neq 0$$B$J$i!"$^$?F1$8$3$H$@$,(B
$$
  h'(-2b) + 2B \cos (-b) + O(\varepsilon) \neq 0,
$$
$B$J$i2#CGE*$K8r$o$k!%(B$\varepsilon$$B$OG$0U$K>.$5$/$G$-$k$+$i!"(B$h'(-2b) + 2B \cos (-b) \neq 0$$B$J$iJdBj$O>ZL@$5$l$?$3$H$K$J$k!%(B

$B>e$N$3$H$,@.$jN)$?$J$$!"$D$^$j$9$Y$F$N(B$J \in T=(\sqrt{2}+\delta_1,\sqrt{2}+\delta_2)$$B$KBP$7$F(B$h'(-2b) + 2B \cos (-b) \equiv 0$$B$H2>Dj$9$k!%$3$3$G86E@$N6a$/$NJL$NBP>N%[%b%/%j%K%C%/E@(B$p_2^*$$B$G!"(B$p_2^*$$B$N50F;$,(B$p$$B$N6a$/$r(B4$BEY(B  $BDL2a$7!"(B$p_2^*$$B$N50F;$O(B$p$$B$N50F;$N(B$\varepsilon$$B6aK5$K$"$k$h$&$J$b$N$r9M$($k!%>e$N5DO@$HF1MM$K$7$F!"<!$,@.$jN)$F$PJdBj$O>ZL@$5$l$?$3$H$K$J$k!%(B
$$
  h'(-4b) + 2B \cos (-3b) + 2B \cos (-b) \neq 0.
$$
$B$U$?$?$S!">e$,@5$7$/$J$$$H2>Dj$9$k!%$9$J$o$A<!$N$h$&$K2>Dj$9$k!%(B
$$
  h'(-4b) + 2B \cos (-3b) + 2B \cos (-b) \equiv 0.
$$
$B$3$N$h$&$J$d$jJ}$r(B$k=1,2,3, \ldots$$B$9$Y$F$KB3$1$l$P<!$rF@$k!%(B
$$
  h'(-2kb) - h'((2k-2)b)+ 2B \cos ((2k-1)b) \equiv 0.
$$
$b/2\pi$$B$,L5M}?t$J$i!"(B$h'(\psi)$$B$NO"B3@-$h$j<!$rF@$k!%(B
$$ \begin{array}{lcl}
 h'(\psi+b) - h'(\psi-b) + 2B \cos \psi = 0, & {\rm for \ all} & 
\psi \in S^1. 
   \end{array}     \eqno(26)
$$
$B6J@~(B$s_1^*$$B$H(B$s_2^*$$B$,3j$i$+$G$"$k$3$H$+$i!">e$N$3$H$,$9$Y$F$N(B$b$$B$KBP$7$F$b@.$jN)$D$3$H$,$o$+$k!%$3$s$I$O(B$h(\psi)$$B$r%U!<%j%(5i?t$KE83+$7$h$&!%(B
$$
 h(\psi) = \sum_{k=0}^{\infty} a_n \cos n\psi + b_n \sin n\psi.
$$
$B$3$N<0$r(B(26)$B$KBeF~$7!"(B$a_n$$B$H(B$b_n$$B$K4X$7$F2r$1$P!"$9$Y$F$N(B$n \neq 0,1$$B$KBP$7$F(B$a_n=b_n=0$$B$G$"$j!"(B$b_1=0, a_1=B/\sin b$$B$G$"$k$3$H$,$o$+$k!%(B($B$3$3$G$U$?$?$S6J@~(B$s_1^*$$B$,(B$m_3$$B$K4X$7$F3j$i$+$G$"$k$3$H$r;H$C$?(B.) $B$7$?$,$C$F6J@~(B$s_1^*$$B$O<!$N7A$r$H$k!%(B
$$
  J = h(\psi) = \displaystyle a_0 + \frac{B}{\sin b} \cos \psi.  
$$

$BDjM}(B7.1$B$N>ZL@$+$i!"0l<!6a;w$G$O(B$s_1^*=s_2^*$$B$,(B$S$$B>e$N<LA|(B$d$$B$NITJQ6J@~$G$"$k$3$H$,=P$k!%F1$85DO@$K$7$?$,$($P!"(B$s_1^*$$B$NIT0BDjB?MMBN$H(B$s_2^*$$B$N0BDjB?MMBN$,!"%]%"%s%+%lCGLL(B$\Sigma^{\phi_0}$$B$N$"$kE@$K$*$$$F2#CGE*$K8r$o$k$3$H$,<($;$k!%$3$l$GJdBj$,>ZL@$5$l$?!%(B\hfill $\Box$

\vspace{0.3cm}
$B$3$s$I$OJdBj$r;H$C$FDjM}(B6.1$B$N>ZL@$r407k$5$;$h$&!%(B

$BA0$K8+$?$h$&$K!"ITJQ1_(B$T_J^e$$B$NIT0BDjB?MMBN$O=89g(B$S$$B$N0BDjB?MMBN$H6J@~(B$s_1$$B$K$*$$$F2#CGE*$K8r$o$k!%$U$?$?$SDjM}(B2.1$B$N>ZL@$K;H$C$?%0%i%UJQ49$N5;=Q$K$7$?$,$($P!"%]%"%s%+%l<LA|$N7+$jJV$7$N$b$H$G$N(B$W^u(T_J^e)$$B$NA|$O(B$C^{\infty}$$B0LAj$G6J@~(B$s_1^*$$B$NIT0BDjB?MMBN$K<}B+$9$k!%F1MM$K!"%]%"%s%+%l<LA|$N5U$N7+$jJV$7$N$b$H$G$N(B$W^s(T_J^e)$$B$NA|$O(B$C^{\infty}$$B0LAj$G6J@~(B$s_2^*$$B$N0BDjB?MMBN$K<}B+$9$k!%>e$NJdBj$K$h$j!"(B$s_1^*$$B$NIT0BDjB?MMBN$H(B$s_2^*$$B$N0BDjB?MMBN$O%]%"%s%+%lCGLLFb$N$"$kE@$G2#CGE*$K8r$o$k$+$i!"ITJQ1_(B$T_J^e$$B$N0BDjB?MMBN$HIT0BDjB?MMBN$O%]%"%s%+%lCGLL(B$\Sigma^{\phi_0}$$BFb$N$"$kE@$G2#CGE*$K8r$o$k$H8@$($k!%$3$l$GDjM}$O>ZL@$5$l$?!%(B\hfill $\Box$

\vspace{0.5cm}
\noindent
{\bf References} 

\re
1. V.M. Alekseev, Quasirandom dynamical systems, I, II, III, {\it Math. USSR-Sb.} {\bf 5} (1968), 73-128; {\bf 6} (1968), 505-560; {\bf 7} (1969), 1-43.

\re
2. V.I. Arnold, Instability of dynamical systems with several degrees of freedom, {\it Dokl. Akad. Nauk SSSR} {\bf 156} (1964), 9.

\re
3. V.I. Arnold(Ed), {\it Dynamical Systems III}, Encyclopaedia of Mathematical Sciences, Vol.3, Springer-Verlag, New York, 1988.

\re
4. V.I. Arnold, Sur une propri\'{e}t\'{e} topologique des applications globalement canonique de la m\'{e}canique classique, {\it C. R. Acad. Sci. Paris} {\bf 26} (1965), 3719.

\re
5. B.V. Chirikov, A universal instability of many dimensional oscillator systems, {\it Phys. Rep.} {\bf 52} (1979), 265.

\re
6. R. Easton, Parabolic orbits for the planar three-body problem, {\it J. Differential Equations} {\bf 52} (1984), 116-134.

\re
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\re
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