
\vspace{0.3cm}
\noindent
{\bf $BJdBj(B 5.1.} $BBg$-$J(B$n$$B$r8GDj$9$k!%(B$\delta_2>\delta_1>0$$B$*$h$S(B
$\varepsilon>0$$B$,$"$C$F!"$9$Y$F$N(B$C \in J = (\sqrt{2}+\delta_1,\sqrt{2}+\delta_2)$$B$*$h$S(B$0<m_2 \leq \varepsilon_0$$B$KBP$7$F!"(B$\Omega_n$$B>e$KDj5A$5$l$k<LA|(B$\tilde{\Pi}$$B$O<B2r@OE*$M$8$l<LA|$G$"$k!%$b$C$H@53N$K8@$($P!"(B$f_n(C)$$B$,(B
$$ \begin{array}{ll}
  \tilde{\Pi}_n(J) = J, & \tilde{\Pi}_n(\psi) = \psi + f_n(J); 
   \end{array} 
$$
$B$J$k4X?t$G$"$l$P!"$3$N$H$-$9$Y$F$N(B$J \in T$$B$KBP$7$F(B$df_n(J)/dJ \neq 0$$B$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$B>ZL@(B.} (14)$B<0$h$j!"(B$\tilde{\Pi}_n(J) =J$$B$*$h$S(B
$$
  \tilde{\Pi}_n(\psi) = \int_0^{2n\pi} 
\frac{\displaystyle 1}{\displaystyle 1 + x^4J} d\phi + \psi,  
$$
$B$,$G$k!%$3$3$G(B$x$$B$O<~4|50F;(B$\gamma_n$$B$K1h$C$FI>2A$9$k!%(B

$B>e$N@QJ,$rI>2A$9$k$N$O!"(B$n$$B$,Bg$-$$$H$-$H$/$KFq$7$$!%$7$+$7!"(B$m_2$$B$,==J,>.$5$$$H$-(B$\gamma_n$$B$,%[%b%/%j%K%C%/%k!<%W$K6a$$$H$$$&;v<B$,;H$($k!%$3$NL\E*$N$?$a$K!"<!<0$G?7$7$$JQ?t$rF3F~$7$h$&!%(B
$$
  {\bf v} = (1 - x^2 \rho^2) {\bf s} - y \rho i {\bf s}.  \eqno(16)
$$
${\bf v}$$B$O(B$m_2=0$$B$J$i1?F0$NDj?t$G$"$j!"<!$N$h$&$K4JC1$K7W;;$G$-$k!%(B
$$
 \frac{ \displaystyle d {\bf v} }{ \displaystyle d \phi } = 
\frac{  \displaystyle -2m_2 x^2 \rho g_2(x,r,\phi) {\bf s} + 
         m_2( \rho g_1(x,r,\phi) + 
         yg_2(x,r,\phi)) i {\bf s}  }
     {\displaystyle 1 - x^4 \rho} + O(m_2^2).    \eqno(17)  
$$

$\alpha \in S^1$$B$r%Y%/%H%k(B${\bf v}$$B$H(B$x$$B<4$N4V$N3QEY$H$9$k!%$D$^$j(B${\bf v} = |{\bf v}| e^{\alpha i}$$B!%C1EaD>F~$K7W;;$9$k$H<!$,F@$i$l$k!%(B
$$ \begin{array}{rl}
\displaystyle 
 \frac{d \alpha}{d \phi}  = & \displaystyle 
\frac{2m_2 x^2 \rho^2 y g_2(x,r,\phi) + 
         m_2( 1 - x^2 \rho^2)(\rho g_1(x,r,\phi) + yg_2(x,r,\phi)) }
     {(y^2 \rho^2 + (1 - x^2 \rho^2)^2)(1 - x^4 \rho)} 
+ O(m_2^2) \\
 = & \displaystyle \frac{-m_2(1+x^2)x^4y \sin \phi + 
                     m_2(1-x^2 \rho^2) \rho x^4 (1 + 2x^2 \cos \phi) }
     {(y^2 \rho^2 + (1 - x^2 \rho^2)^2)(1 - x^4 \rho)} \\
 & \displaystyle - 
\frac{m_2x^4[(1+x^2 \rho^2)y \sin \phi + 
          (1-x^2 \rho^2)p(1-x^2 \cos \phi)]}
     {(1-2x^2 \cos \phi + x^4)^{3/2}(y^2 \rho^2 + 
          (1-x^2 \rho^2)^2)(1-x^4 \rho)} + O(m_2^2).      
   \end{array}                    \eqno(18)
$$

$m_2=0$$B$J$i(B$\alpha$$B$O1?F0$NDj?t$G$"$k!%(B$m_2$$B$,>.$5$$$H$-!"(B$\Delta \alpha(\gamma)$$B$O%[%b%/%j%K%C%/50F;(B$\gamma$$B$K1h$C$F$N(B$\alpha$$B$NJQJ,!"$9$J$o$A!"(B$\Delta \alpha(\gamma)$$B$O(B$\gamma$$B$K1h$C$F$N(B$m_2$$B$N0l<!$^$G$N(B$\alpha$$B$NJQ2=!"$H$9$k!%(B$\Delta \alpha(\gamma)$$B$O>e$N<0$r@QJ,$7$FF@$i$l$k!%(B
$$
 \Delta \alpha(\gamma) = \int_{-\infty}^{\infty} 
\frac{\displaystyle \left( \begin{array}{r} 
     2m_2 x^2 \rho^2 y g_2(x,r,\phi) + m_2(1-x^2\rho^2) \\
     \times (\rho g_1(x,r,\phi) + y g_2(x,r,\phi)) 
                          \end{array} \right) }
{\displaystyle (y^2\rho^2 + (1-x^2\rho^2)^2)(1-x^4\rho)} d \phi 
+ O(m_2^2).  
$$
$B$3$3$G(B$p=-J$$B$G$"$j!"(B$x$$B$H(B$y$$B$O%[%b%/%j%K%C%/50F;(B$\gamma$$B$K1h$C$FI>2A$9$k!%(B$m_2$$B$N(B1$B<!$^$G$J$i!"(BFig.1$B$K<($5$l$k$h$&$K(B$x$$B$H(B$y$$B$NCM$OFsBNLdBj$NJ|J*2r$K$h$C$F6a;w$G$-$k!%(B$x=\xi(t,\rho), y=\eta(t,\rho)$$B$OJ|J*2r$N50F;$G$"$C$F!"(B$\xi(t,\rho) = \xi(-t,\rho)$$B$+$D(B$\eta(t,\rho) = -\eta(-t,\rho)$$B$H$9$k!%$3$N$H$-!"<!$rF@$k!%(B
$$ \begin{array}{ll}
 \Delta \alpha(\gamma) = & \displaystyle \int_{-\infty}^{\infty} 
-m_2(1+\xi^2\rho^2)\xi^4\eta \sin (t-\theta) \\
 & \displaystyle 
+ m_2(1-\xi^2\rho^2)\rho \xi^4 (1+2\xi^2 \cos (t-\theta)) dt \\ 
 & \displaystyle + \int_{-\infty}^{\infty} 
\frac{m_2\xi^4(1+\xi^2\rho^2)\eta \sin (t-\theta)}
     {(1-2\xi^2 \cos (t-\theta)+\xi^4)^{3/2}} dt \\ 
 & \displaystyle - \int_{-\infty}^{\infty} 
\frac{m_2\xi^4(1-\xi^2\rho^2)\rho
                                      (1-\xi^2 \cos (t-\theta))}
     {(1-2\xi^2 \cos (t-\theta)+\xi^4)^{3/2}} dt 
         + O(m_2^2). 
   \end{array}              \eqno(19)
$$
$B$3$3$G(B$\theta$$B$O<!$N@QJ,$GM?$($i$l$k!%(B
$$
  \theta(t,\rho) = \int_0^t \xi^4(t,\rho)\rho dt.    \eqno(20)
$$

$\xi = \sqrt{2}|J|^{-1} - 2\sqrt{2}t^2/|J|^{-7}$$B$KCm0U$9$l$P!"(B(19)$B$N@QJ,$O(B$J=\sqrt{2}$$B$N$H$-HsM-3&$G$"$k$3$H$,$o$+$j!"$?$@$A$K(B$\delta_2>\delta_1>0$$B$,==J,>.$5$$$H$-$9$Y$F$N(B$J \in T = (\sqrt{2}+\delta_1, \sqrt{2}+\delta_2)$$B$KBP$7$F!"(B$\Delta \alpha(\gamma) \neq 0$$B$,=P$k!%$^$?4JC1$K$o$+$k$h$&$K!"(B$\delta_2>\delta_1>0$$B$r>.$5$/A*$s$G!"(B$\Delta \alpha(\gamma)$$B$N(B$J$$B$K4X$9$kHyJ,$,$9$Y$F$N(B$J \in T$$B$KBP$7$FHs%<%m$K$G$-$k!%(B

$B8e$N;29M$N$?$a$K!"$"$k(B$c_1 \neq 0$$B$KBP$7$F(B$\Delta \alpha(\gamma) \sim c_1 \log (J-\sqrt{2})$$B$G$"$k$3$H$r;XE&$7$F$*$/!%(B

$BJdBj$N>ZL@$KLa$m$&!%(B$\Delta \psi(\gamma_n)$$B$O(B$\gamma_n$$B$K1h$C$F$N(B$\psi$$B$NJQJ,$G$"$k$H$9$k!%$3$N$H$-(B$\tilde{\Pi}_n(\psi) = \psi + \Delta \psi(\gamma_n)$$B$*$h$S(B
$$ \begin{array}{ll}
 \Delta \psi(\gamma_n) = \displaystyle \int_0^{2n \pi}
\frac{1}{1-x^4\rho} d\phi = \displaystyle \int_0^{2n \pi}
\frac{x^4\rho}{1-x^4\rho} d\phi & {\rm mod} (2\pi), 
   \end{array}
$$
$B$G$"$k!%$3$3$G(B$x$$B$O<~4|50F;(B$\gamma_n$$B$K1h$C$FI>2A$9$k!%(B$\Delta \theta(\gamma_n)$$B$O<~4|50F;(B$\gamma_n$$B$K1h$C$F$N(B$\theta$$B$NJQJ,$G$"$k$H$9$k!%$9$k$H<!$,@.$jN)$D!%(B
$$
 \Delta \theta(\gamma) = \int_0^{2n \pi} 
\frac{\displaystyle x^4\rho}{\displaystyle 1-x^4\rho} d\phi. 
$$
$B$3$3$G$b(B$x$$B$O(B$\gamma_n$$B$K1h$C$FI>2A$9$k!%$7$?$,$C$F(B$\Delta \psi(\gamma_n) = \Delta \theta(\gamma_n)$$B$G$"$k!%(B$\gamma_n$$B$O%[%b%/%j%K%C%/50F;(B$\gamma$$B$K$h$C$F6a;w$G$-$k$+$i!"(B
$$
 \Delta \psi(\gamma_n) = \Delta \theta(\gamma_n) = 
\int_{-\infty}^{\infty} 
\frac{\displaystyle x^4\rho}{\displaystyle 1-x^4\rho} d\phi 
+ \varepsilon_n, 
$$
$B$G$"$k!%$3$3$G(B$n \rightarrow \infty$$B$N$H$-(B$\varepsilon_n \rightarrow 0$$B$G$"$j!"(B$x$$B$O%[%b%/%j%K%C%/50F;(B$\gamma$$B$K1h$C$FI>2A$9$k!%(B$\alpha$$B$NDj5A(B($B<0(B(16))$B$K$h$j!"(B$x=y=0$$B$J$i(B$\alpha=\theta$$B$G$"$k!%$f$($K<!$rF@$k!%(B$$
 \Delta \psi(\gamma_n) = \Delta \theta(\gamma_n) = 
 \Delta \alpha(\gamma) + \varepsilon_n + O(m_2^2). 
$$
$B$7$?$,$C$F(B$\delta_2>\delta_1>0$$B$*$h$S(B$m_2>0$$B$,==J,>.$5$/(B$n$$B$,==J,Bg$-$$$H$-!"(B$\Delta \psi(\gamma_n)(J)$$B$*$h$S(B$J$$B$K4X$9$k$=$NHyJ,$O$9$Y$F$N(B$J \in (\sqrt{2}+\delta_1, \sqrt{2}+\delta_2)$$B$KBP$7$F$H$b$KHs%<%m$G$"$k!%(B

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

\vspace{0.3cm}
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\vspace{0.5cm}
\noindent
{\bf 6. $B%"!<%N%k%I3H;6$HJ?LL;0BNLdBj$NHs2D@QJ,@-(B}

$B$3$N@a$G$O(B$m_3$$B$H(B$e$$B$,>.$5$$>l9g$N40A4$JJ?LL;0BNLdBj$r9M$($k!%$U$?$?$S(B$0<m_3 \ll e$$B$H2>Dj$9$k!%$O$8$a$KA0@a$NITJQ=89g(B$\Omega_n$$B$,(B$m_3$$B$H(B$e$$B$K$h$k@]F0$N$b$H$GJ]B8$5$l$k$3$H!"$5$i$K$3$NITJQ=89g>e$NITJQ%H!<%i%9$N$[$H$s$I$,@8$-;D$k$3$H$r<($=$&!%(B

\vspace{0.3cm}
\noindent
{\bf $BJdBj(B 6.1.} $e>0$$B$,==J,>.$5$1$l$P!"(B$\Omega_n$$B$O@]F0$N$b$H$GJ]B8$5$l!"6a$/$NITJQB?MMBN(B$\Omega_n^e$$B$K$J$k!%$=$N>e!"(B$\Omega_n^e$$B$K@)8B$7$?2#CG<LA|(B$\tilde{\Pi}_n$$B$OLL@QJ]B8HyJ,F1Aj$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$B>ZL@(B.} $\Omega_n$$B$,@8$-;D$k$3$H$r<($9$N$O4JC1$G$"$k!%$9$J$o$A!"ITJQB?MMBN(B$\Omega_n$$B$O@55,AP6J$G$"$j!"@55,AP6JITJQB?MMBN$O>.$5$J@]F0$N$b$H$G@8$-;D$k!%$@$+$i(B$\Omega_n$$B$O@]F0$r<u$1$F6a$/$NITJQB?MMBN(B$\Omega_n^e$$B$KJQ$o$k!%(B$\tilde{\Pi}_n$$B$,LL@QJ]B8$G$"$k$3$H$r<($9$?$a$K!"J?LL;0BNLdBj$O%O%_%k%H%s7O$G$"$k$3$H!"$7$?$,$C$F(B$\tilde{\Pi}_n$$B$O%7%s%W%l%/%F%#%C%/$G$"$j!"7O$KIU?o$9$kJDITJQHyJ,(B2-$B7A<0(B$\omega$$B$,$"$k$3$H$KCm0U$9$k!%$3$N7A<0$rITJQItJ,B?MMBN(B$\Omega_n^e$$B$K@)8B$7$?$b$N$r(B$\omega_n$$B$H$9$k!%(B$\omega_n$$B$O(B$\Omega_n$$B>e$X$N@)8B<LA|(B$\tilde{\Pi}_n$$B$N$b$H$GITJQ$G$"$k!%4JC1$K$o$+$k$h$&$K!"$3$N(B2-$B7A<0(B$\omega_n$$B$OHs=LB`$G$"$k!%$3$l$OHs=LB`$N7A<0$+$i$N@]F0$@$+$i$G$"$k!%$f$($K(B$\tilde{\Pi}_n$$B$O(B$\Omega_n^e$$B>e$NLL@QJ]B8<LA|$G$"$k!%;v<B!"7O$N%O%_%k%H%s@-$h$j!"(B$\tilde{\Pi}_n$$B$O(B$\Omega_n^e$$B>e$G40A4%7%s%W%l%/%F%#%C%/$G$"$k!%$3$l$GJdBj$,>ZL@$5$l$?!%(B

\vspace{0.3cm}
$B$3$3$G(BKAM(Kolmogorov-Arnold-Moser)$BM}O@$rITJQB?MMBN(B$\Omega_n^e$$B$KE,MQ$G$-$k!%(B$\Omega_n^e$$B>e$N6I=j:BI8$H$7$F(B$J$$B$H(B$\psi$$B$r0zB3$-;H$($P!"<LA|(B$\tilde{\Pi}_n$$B$O<!$N7A$r$H$k!%(B
$$ \begin{array}{ll}
 \tilde{\Pi}_n(J) = J + O(m_3,e), & 
   \tilde{\Pi}_n(\psi) = \psi + f_n(J) + O(m_3,e). 
   \end{array}
$$
$BG$0U$N<B?t(B$J \in T$$B$KBP$7$F!"Hs=LB`$HHs6&LD$N>r7o$,@.$jN)$F$P!"(B$\Omega_n$$B>e$NITJQ%H!<%i%9(B$T_J$$B$O!"(B$e>0$$B$H(B$m_3>0$$B$,==J,>.$5$$$H$7$F6a$/$NITJQ%H!<%i%9(B$T_J^e$$B$KJQ$o$j!"(B$T_J^e$$B$OHs@]F0%H!<%i%9(B$T_j$$B$HF1$8?6F0?t$r;}$D!%$=$N>e!"(B$\Omega_n$$B>e$NITJQ%H!<%i%9$NBgItJ,(B($B%j%e%Y!<%0B,EY$N0UL#$G(B)$B$O>.$5$J@]F0$N$b$H$G@8$-;D$k!%$7$+$7!"<~4|2r$NO"B3B2$r4^$`6&LD%H!<%i%9$O0lHL$K8r8_$K8=$o$l$kBJ1_$*$h$SAP6J<~4|2r$NM-8B=89g$K2u$l!"$3$l$i$N<~4|50F;$N0BDjB?MMBN$HIT0BDjB?MMBN$ODL>o2#CGE*$K8r$o$C$FJ#;($JNO3X$r$D$/$j!"!V%9%H%+%9%F%#%C%/AX!W(B(stochasitic layer)$B$H8F$P$l$k%+%*%9NN0h$r$D$/$k!%$3$l$i$N%9%H%+%9%F%#%C%/AX$OC`<!E*$JJ?6QA`:n$G<h$j=|$/$3$H$,$G$-$k$+$i!"$3$l$i$NAX$N8|$5$O@]F0%Q%i%a!<%?(B$e$$B$NG$0U$N$Y$-$h$j$b>.$5$$!%;v<B!"$3$l$i$N%9%H%+%9%F%#%C%/AX$O0lHL$K;X?t4X?tE*$K>.$5$$!%(B

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$B$$$^$d<!$NDjM}$r=R$Y$k$3$H$,$G$-$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B 6.1.} $B==J,Bg$-$J(B$n$$B$H==J,>.$5$J(B$m_2$$B$r8GDj$9$k!%(B$\delta_2>\delta_1>0$$B!"(B$e_0>0$$B$*$h$S(B$\varepsilon_1>0$$B$,B8:_$7$F<!$,@.$jN)$D!%(B$J^* \in T = (\sqrt{2} + \delta_1, \sqrt{2} + \delta_2)$$B$H$7!"ITJQ%H!<%i%9(B$T_{J^*}$$B$O@8$-;D$C$F(B$T_{J^*}^e$$B$K$J$k$H$9$k!%$3$N$H$-!"$9$Y$F$N(B$0<e<e_0$$B$*$h$S(B$0<m_3<\varepsilon_1$$B$KBP$7$F!"(B$T_{J^*}^e$$B$N0BDjB?MMBN(B$W^s(T_{J^*}^e)$$B$HIT0BDjB?MMBN(B$W^u(T_{J^*}^e)$$B$O$"$kE@$K$*$$$F8r$o$j!"$=$N8r:9$O2#CGE*$G$"$k!%$=$N>e!"(B$\varepsilon>0$$B$,B8:_$7$F(B$|J_1 - J_2| \leq \varepsilon$$B$*$h$S(B$J_1,J_2 \in T$$B$J$i(B$W^u(T_{J_1})$$B$O(B$W^s(T_{J_2})$$B$H2#CGE*$K8r$o$k!%(B

\vspace{0.3cm}
$B$3$l$i$NITJQ%H!<%i%9$N0BDj$*$h$SIT0BDjB?MMBN$,2#CGE*$K8r$o$k$3$H$N>ZL@$OITJQ=89g(B$S= \{ x=y=0, J \in {\bf R}, \psi \in S^1 \}$$B$N6a$/$NN.$l$N@-<A$K4p$E$$$F$*$j!"$^$?(B$p_n$$B$N50F;$,(B$p$$B$N50F;$K6a$$$H$$$&;v<B$K$b4p$E$$$F$$$k!%$3$NDjM}$N>ZL@$O8e$N@a$G9T$J$&!%(B

$BCN$i$l$F$$$k$b$N0J30$K(B$n$$BBNLdBj$K@QJ,$,$"$k$+$I$&$+$H$$$&LdBj$OE7BNNO3X$N8&5f$G=EMW$JLr3d$r2L$7$F$-$?!%J?LL;0BNLdBj$K$*$$$F!">e$G<($7$?$h$&$K%"!<%N%k%I3H;6$,B8:_$9$k$3$H$O!"$3$NLdBj$NNO3X$NJ#;($5$r<($7$F$$$k!%$3$l$O$^$?J?LL;0BNLdBj$K$3$l0J>e<B2r@OE*@QJ,$,$J$$$3$H$b<($7$F$$$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B 6.2.} $BJ?LL;0BNLdBj$O!"D>@~1?F0NL!"3Q1?F0NL$*$h$S%(%M%k%.!<@QJ,0J30$K<B2r@OE*$J4X?tE*$KFHN)$J@QJ,$,$J$$$H$$$&0UL#$GHs2D@QJ,$G$"$k!%(B

\vspace{0.3cm}
$B$3$NDjM}$O8EE5E*$J%]%"%s%+%l(B[15]$B$NDjM}$N3HD%$K$J$C$F$$$k!%%]%"%s%+%l$NDjM}$K$h$l$P!"J?LL;0BNLdBj$K$O<B2r@OE*$J@QJ,$G;0BN$N<ANL$K4X$7$F$b<B2r@OE*$J@QJ,$OCN$i$l$F$$$k$b$N0J30$K$J$$!%(B

\vspace{0.5cm}
\noindent
{\bf 7. $BL58B1s$K$*$1$k3H;6$H3H;6E*?6F02r(B}

$B$3$N@a$G$O<!$N=89g$r9M$($k!%(B
$$
  S = \{ x=y=0; J \in {\bf R}; \psi \in S^1; \phi=0 \}.
$$
$BJ*M}6u4V$K$*$$$F=89g(B$S$$B$O(B$m_3$$B$,B.EY%<%m$GL58B1s$K$"$k$3$H$KBP1~$9$k!%$o$l$o$l$O<g$H$7$F$3$N=89g$N6a$/$N2r!"$H$/$K$3$N=89g$N(B$J \in T$$B$J$kItJ,$N6a$/$N2r$K6=L#$r;}$C$F$$$k!%(B

$B$O$8$a$K(B$S$$B$N3FE@$O%]%"%s%+%l<LA|(B$\Pi$$B$NITF0E@$G$"$k$3$H$r;XE&$7$F$*$/!%DjM}(B2.1$B$K$h$j!"3FE@$O(B1$B<!850BDjB?MMBN$H(B1$B<!85IT0BDjB?MMBN$r;}$D!%N>J}$NB?MMBN$H$b(B$x>0$$B$G$O<B2r@OE*$G$"$k!%(B

$q$$B$r(B$S$$B$NE@$H$9$k!%(B$q$$B$NIT0BDjB?MMBN(B$W^u(q)$$B$O!"(B$x>0$$B$J$kCGLL(B$\Sigma^{\phi_0}|_{\phi_0=0}$$B>e$N(B1$B<!85<B2r@OB?MMBN$G$"$k!%=89g(B$S$$B$N0BDjB?MMBN(B$W^s(S)$$B$OM><!85(B1$B$G$"$k!%$7$?$,$C$F(B$W^u(q)$$B$O(B$W^s(S)$$B$H$"$kE@$G2#CGE*$K8r$o$jF@$k!%<B:]!"0JA0$K8+$?$h$&$K!"(B$m_3=e=0$$B$+$D(B$J \in T$$B$N$H$-$3$l$,@.$jN)$C$F$$$k!%(B$W^u(q)$$B$H(B$W^s(S)$$B$,2#CGE*$K8r$o$k$H$7!"(B$q^*$$B$,(B$W^u(q)$$B$H(B$W^s(S)$$B$N2#CGE*8rE@$G$"$k$H$9$k!%(B$q^{**}$$B$O(B$S$$B$NE@$G(B$q^* \in W^s(q^{**})$$B$J$k$b$N$H$9$k!%$3$3$G3+ItJ,=89g(B$D \subset S$$B$+$i(B$S$$B$X$N<LA|(B$d$$B$r(B$d(q) = q^{**}$$B$GDj5A$9$k!%(B$W^u(q)$$B$H(B$W^s(S)$$B$N2#CGE*8rE@$O$?$/$5$s$"$k$+$i!"<LA|(B$d$$B$O2#CGE*8rE@(B$q^*$$B$NA*$SJ}$K0MB8$9$k!%(B$q^*$$B$K$*$$$F8r:9$O2#CGE*$G$"$k$+$i!"(B$q^*$$B$,(B$q$$B$KO"B3E*$K0MB8$9$k$h$&$K8rE@(B$q^*$$B$rA*$V!%$3$l$K$h$j!"<LA|(B$d$$B$ODj5A$5$l$F$$$k$H$3$m$G$O$I$3$G$bO"B3$G2DHyJ,$G$"$k$3$H$,J]>Z$5$l$k!%(B

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\vspace{0.3cm}
\noindent
{\bf $BJdBj(B 7.1.} $D_0 \subset S$$B$r<!$N=89g$H$9$k!%(B
$$
  D_0 = \{ x=y=0; J \in T \subset (\sqrt{2}+\delta_1,\sqrt{2}+\delta_2);\psi \in S^1; \phi=0 \}.
$$

1. $B@5?t(B$\delta_2>\delta_1,\varepsilon_1,\varepsilon_2,$$B$*$h$S(B$\varepsilon_3$$B$,$"$C$F!"(B$J \in T, 0 < m_2 \leq \varepsilon_1, 0 \leq m_3 \leq \varepsilon_2$$B$*$h$S(B$0 \leq e \leq \varepsilon_3$$B$J$i!"<LA|(B$d$$B$O(B$D_0$$B>e$G$&$^$/Dj5A$5$l$F$$$k!%$9$J$o$A!"(B$D_0 \subset D$$B$G$"$k!%(B

2. $m_3=e=0$$B$J$i!"<LA|(B$d$$B$O(B$D_0$$B>e$N3j$i$+$J$M$8$l<LA|$G$"$k!%(B

3. $B<LA|(B$d$$B$O(B$D$$B>e$GLL@QJ]B8$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$B>ZL@(B.} Xia[21]$B$h$j!"(B$m_3=e=0$$B$N$H$-!"E@(B$q \in S$$B$N0BDjB?MMBN$HIT0BDjB?MMBN$O!"(B$J$$B$N$[$H$s$I$9$Y$F$NCM$*$h$S>.$5$J(B$m_2$$B$NCM$KBP$7$F(B$x-y$$BLL$G2#CGE*$K8r$o$k!%$H$/$K!"(B$\delta_2>\delta_1>0$$B$*$h$S(B$\varepsilon_1>0$$B$,B8:_$7$F!"(B$J \in (\sqrt{2}+\delta_1,\sqrt{2}+\delta_2)$$B$*$h$S(B$0<m_2 \leq \varepsilon_1$$B$J$i(B$W^s(q)$$B$H(B$W^u(q)$$B$O(B$x-y$$BLL$G2#CGE*$K8r$o$k!%$3$l$+$i(B$W^u(q)$$B$,(B$W^s(S)$$B$H2#CGE*$K8r$o$k$3$H$,=P$k!%$7$?$,$C$F(B$m_3$$B$H(B$e$$B$,>.$5$$$H$-!"$9$Y$F$N(B$J \in T$$B$KBP$7$F(B$W^u(q)$$B$H(B$W^s(S)$$B$,2#CGE*$K8r$o$k$3$H$,$o$+$k!%$3$l$GJdBj$N:G=i$NItJ,$,>ZL@$5$l$?!%(B

$m_3=e=0$$B$N$H$-<LA|(B$d$$B$O<!$N7A$r$H$k!%(B
$$ \begin{array}{lcl}
 d(J)=J & $B$*$h$S(B & d(\psi) = \displaystyle 
\psi + \int_{-\infty}^{\infty} \frac{1}{1+x^4J} d\phi. 
   \end{array}
$$
$B$3$3$G(B$x$$B$O2#CGE*%[%b%/%j%K%C%/50F;(B$\gamma$$B$K1h$C$FI>2A$9$k!%JdBj(B5.1$B$N>ZL@$K$h$l$P(B
$$
\displaystyle \int_{-\infty}^{\infty} 
\frac{1}{1+x^4J} d\phi = \Delta \alpha (\gamma) + O(m_2^2), 
$$
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$d$$B$,LL@QJ]B8$G$"$k$3$H$r<($9$?$a$K!"$U$?$?$S$3$NLdBj$N%O%_%k%H%s@-$KAJ$($M$P$J$i$J$$!%(B$\omega$$B$r7O$NHs=LB`JD%7%s%W%l%/%F%#%C%/7A<0$H$9$k!%(B$\omega$$B$O(B$\Sigma^0$$B>e$N<LA|(B$\Pi$$B$N$b$H$GJ]B8$9$k!%(B$\tilde{\omega}$$B$r(B$S$$B$X$N(B$\omega$$B$N@)8B$H$9$k!%(B$\tilde{\omega}$$B$OHs=LB`$G$"$k!%(B$d_* \tilde{\omega} = \tilde{\omega}$$B$r<($9I,MW$,$"$k!%(B$D^*$$B$O(B$d$$B$NDj5A$K$*$1$k$9$Y$F$N(B$q^*$$B$N=89g$H$9$k!%4JC1$K$o$+$k$h$&$K!"(B$D^*$$B$O(B$D$$B$KHyJ,F1Aj$G$"$k!%(B$d_1$$B$r(B$D^*$$B$+$i(B$D$$B$X$NHyJ,F1Aj$H$9$k!%$3$N$H$-(B$d_1$$B$O(B$n \rightarrow \infty$$B$N$H$-$N<LA|(B$\Pi^{-n}$$B$N6K8B$GM?$($i$l$k!%F1MM$K(B$D^*$$B$O(B$d(D)$$B$KHyJ,F1Aj$G$"$k!%(B$d_2:D^* \rightarrow d(D)$$B$O(B$n \rightarrow \infty$$B$N$H$-$N(B$\Pi^n$$B$N6K8B$GM?$($i$l$kHyJ,F1Aj<LA|$H$9$k!%(B$\Pi$$B$,%7%s%W%l%/%F%#%C%/7A<0$rJ]B8$9$k$+$i(B$d_1$$B$b(B$d_2$$B$b%7%s%W%l%/%F%#%C%/7A<0(B$\omega$$B$rJ]B8$9$k!%$7$?$,$C$F(B$d=(d_1)^{-1}d_2$$B$O(B$\omega$$B$rJ]B8$9$k!%$f$($K(B$d_*\tilde{\omega} = \tilde{\omega}$$B$G$"$k!%;v<B!"(B$d$$B$O40A4(B(exact)$B%7%s%W%l%/%F%#%C%/$G$"$k!%$3$l$GJdBj$,>ZL@$5$l$?!%(B\hfill $\Box$

\vspace{0.3cm}
$B$5$F(BKAM$BM}O@$,(B$D \subset S$$B>e$N<LA|(B$d$$B$KE,MQ$G$-$k!%(B$m_3=e=0$$B$J$i(B$D$$B>e(B$J=J^*$$B$J$k3F1_$OITJQ1_$G$"$k!%$3$l$i$NITJQ1_$N$&$A%G%#%*%U%!%s%H%9>r7o$rK~$?$92sE>?t$b$N$O>.$5$J(B$e$$B$H(B$m_3$$B$N$b$H$G(B$D_0$$B>e$K@8$-;D$k!%DjM}(B2.1$B$h$j3F@8$-;D$jITJQ1_(B$C_{J^*}$$B$O(B2$B<!850BDjB?MMBN(B$W^s(C_{J^*})$$B$H(B2$B<!85IT0BDjB?MMBN(B$W^u(C_{J^*})$$B$r(B$\Sigma^0$$B>e$N<LA|(B$\Pi$$B$KBP$7$F;}$D!%<LA|(B$d$$B$NDj5A$K$h$j$3$l$i(B2$B$D$NB?MMBN(B$W^s(C_{J^*})$$B$H(B$W^u(C_{J^*})$$B$O1_(B$d_1^{-1}(C_{J^*})$$BFb$G8r$o$k!%0J2<$N8r:9DjM}$O$3$l$i(B2$B$D$NB?MMBN$,<B:](B$\Sigma^0$$BFb$NB>$N$"$kE@$G8r$o$k$3$H$*$h$S8r:9$,(B$D_0$$BFb$N$9$Y$F$NITJQ1_(B$C_{J^*}$$B$KBP$7$F2#CGE*$G$"$k$3$H$r=R$Y$F$$$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B 7.1.} $\delta_2>\delta_1, \varepsilon_1, \varepsilon_2$$B$*$h$S(B$\varepsilon_3$$B$OJdBj(B7.1$B$N@5?t$H$7!"(B$C_{J^*}$$B$O(B$D_0$$BFb$G@8$-;D$kITJQ1_$H$9$k!%(B$\varepsilon_2$$B$H(B$\varepsilon_3$$B$,==J,>.$5$/A*$P$l!"(B$0 \leq m_3 \leq \varepsilon_2$$B$*$h$S(B$0 < e \leq \varepsilon_3$$B$J$i!"(B$W^s(C_{J^*})$$B$H(B$W^u(C_{J^*})$$B$O$"$kE@$G2#CGE*$K8r$o$k!%$=$N>e!"(B$\varepsilon>0$$B$,B8:_$7$F!"(B$J_1,J_2 \in T$$B$+$D(B$|J_1-J_2| \leq \varepsilon$$B$J$i(B$J_1$$B$N0BDjB?MMBN(B$W^s(C_{J_1})$$B$O(B$J_2$$B$NIT0BDjB?MMBN(B$W^u(C_{J_2})$$B$H2#CGE*$K8r$o$k!%(B

\vspace{0.3cm}
$B$3$3$G$b!"$3$NDjM}$N>ZL@$r8e$N@a$K1d4|$9$k!%>e$NDjM}$N5"7k$H$7$F!"%"!<%N%k%I3H;6$K;w$?!"$o$l$o$l$,!V5<%"!<%N%k%I3H;6!W(B(pseudo Arnold diffusion)$B$H8F$V!"$"$k<o$N3H;6$,J*M}6u4V$G$OL58B1s$KBP1~$9$k=89g(B$D_0$$B$N$^$o$j$K8=$o$l$k!%%"!<%N%k%I3H;6$K$*$$$F$O!"%+%*%9NO3X$r0z$-5/$3$94pK\E*5!9=$OA+0\%H!<%i%9$HA+0\:?$G$"$C$?$3$H$r;W$$$@$=$&!%>e$NDjM}$NITJQ1_$O%"!<%N%k%I3H;6$NA+0\%H!<%i%9$HF1$88z2L$r;}$D(B($B0J2<$NJdBj(B8.1$B;2>H(B)$B!%$3$N=LB`$7$?3H;6$+$i$NLLGr$$5"7k$H$7$F!"J?LL;0BNLdBj$N3H;6?6F02r$NN`$NB8:_$,=P$k!%(B$n$$BBNLdBj$N(B\underline{$B?6F02r(B}$B$H$O(B$t \rightarrow \infty$$B$N$H$-(B$\limsup \max r_{ij} = \infty$$B$+$D(B$\liminf \max r_{ij} < \infty$$B$J$k2r$N$3$H$G$"$k!%$?$@$7(B$r_{ij}$$B$O(B$i$$BHVL\$H(B$j$$BHVL\$NN3;R4V$N5wN%$G$"$k!%(B\underline{$B3H;6?6F02r(B}$B$H$OA+0\%H!<%i%9$N4V$r%+%*%9E*$K3H;6$9$k?6F02r$N$3$H$G$"$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B 7.2.} $BJ?LL;0BNLdBj$K$O?6F02r$,B8:_$9$k!%$o$l$o$l$N5-K!$G>\$7$/8@$($P!";0BNLdBj$K$O!"(B$t \rightarrow \infty$$B$N$H$-(B
$$ \begin{array}{lcl}
 \limsup |x| > 0 & $B$*$h$S(B & \liminf |x| = 0,
   \end{array}
$$
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\vspace{0.3cm}
$BF1MM$K(B$t$$B$,(B$\infty$$B$*$h$S(B$-\infty$$B$K6a$E$/$K$D$l$F?6F0$7!"%+%*%9E*$K3H;6$9$k2r$r9=@.$G$-$k$3$H$rCm0U$7$F$*$/!%$^$?Ja3M$*$h$SC&=P2r$bF1MM$J$d$jJ}$G9=@.$G$-$k!%(B


