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\begin{document}

\begin{flushright}
$BC+@n@6N4(B $BLu(B
\end{flushright}

\noindent
{\it Ergod. Th. \& Dynam. Sys.} {\bf 4} (1984), 301-309

\begin{center}
{\large
{\bf $BITJQ1_$NHsB8:_(B \\
Non-existence of invariant circles} 
}

\vspace{1cm}
John N. Mather
\end{center}

\vspace{0.5cm}
\noindent
{\bf $BMWLs(B.} $B:9J,J}Dx<0(B
$$
  \Delta^2 x_n = \displaystyle \frac{k}{2 \pi} \sin (2 \pi x_n)
$$
$B$K$H$b$J$&NO3X7O$O?t?M$NCx<T$K$h$C$F?tCME*$KD4$Y$i$l$F$-$?(B. $B?tCME*>Z5r$r(B
$B$b$H$K$7$?H`$i$N7kO@$G$O(B, $B?t(B$k_0 \approx 0.97$$B$,B8:_$7$F(B, $|k| \leq k_0$$B$J$i(B
$B%[%b%H%T!<Hs<+L@$JITJQ1_$,B8:_$7(B, $|k| > k_0$$B$J$iB8:_$7$J$$(B. 
$B$3$N>.O@J8$G$O(B, $|k| > \frac{4}{3}$$B$J$iITJQ1_$,B8:_$7$J$$$3$H$N4JC1$+$D87L)$J(B
$B>ZL@$rM?$($k(B.  

\vspace{0.5cm}
\noindent
{\bf 1. $B=x(B.}

$B<LA|(B${\bf Z} \rightarrow {\bf R} : n \mapsto x_n$$B$N6u4V$r9M$($k(B. 
$B$o$l$o$l$,6=L#$r;}$D$N$O(B, $BJ}Dx<0(B
$\Delta^2 x_n = (k/2 \pi) \sin (2 \pi x_n)$$B$rK~$?$9<LA|$G$"$k(B. $B$3$3$G(B
$\Delta^2 x_n = x_{n+1} - 2 x_n + x_{n-1}$$B$G$"$k(B. $B0J2<$O$3$N:9J,J}Dx<0$N(B
$B2r$rNO3X7O$N50@W$H$7$FI=$9I8=`E*J}K!$G$"$k(B. $y_n = x_n - x_{n-1}$$B$HCV$/(B. 
$B:9J,J}Dx<0$N2r$KBP$7$F<!$rF@$k(B. 
$$ \begin{array}{l}
   y_{n+1} = x_{n+1} - x_n \\
   y_{n+1} - y_n = (k/2 \pi) \sin (2 \pi x_n) .  
   \end{array}
$$
$B$3$l$O<!$N$h$&$K$b=q$1$k(B.
$$ \begin{array}{l}
   x_{n+1} = x_n + y_n + (k/2 \pi) \sin (2 \pi x_n) \\
   y_{n+1} = y_n + (k/2 \pi) \sin (2 \pi x_n) .  
   \end{array}
$$
$B$3$l$+$iD4$Y$?$$LdBj$O(B, $B>e$NJ}Dx<0$,L58B1_E{$N<LA|$rDj5A$7$F$$$k$H9M$($k$H(B
$B$?$$$X$s4JC1$K@_Dj$G$-$k(B. $(x,y)$$B$rL58B1_E{(B
$T^1 \times {\bf R} = ({\bf R}/{\bf Z}) \times {\bf R}$$B$N:BI8$H$7(B, $x$$B$O(B
${\rm mod} \ 1$$B$GDj5A$5$l$k$H$9$k(B. 
$$
  f_k : T^1 \times {\bf R} \rightarrow T^1 \times {\bf R}
$$
$B$O(B$f_k(x,y) = (x', y')$$B$GDj5A$5$l$k<LA|$H$9$k(B. $B$?$@$7(B,
$$ \begin{array}{l}
   x' = x + y + (k/2 \pi) \sin (2 \pi x) \\
   y' = y + (k/2 \pi) \sin (2 \pi x) . 
   \end{array}
$$

\underline{$BITJQ1_(B}$B$H$O(B, $BItJ,=89g(B$X \subset T^1 \times {\bf R}$$B$G(B, 
$f_k X = X$$B$+$D(B$X$$B$,(B$T^1 = {\bf R}/{\bf Z}$$B$KF1Aj$J$k$b$N$G$"$k(B. 
$B1_$KF1Aj$JItJ,=89g(B$X \subset T^1 \times {\bf R}$$B$,%[%b%H!<%T!<Hs<+L@$G$"$k(B
$B$?$a$NI,MW==J,>r7o$O(B$T^1$$B$X$N<M1F$,<LA|EY(B$\pm 1$$B$J$k$3$H$G$"$k(B. $BD>4QE*$K$O(B, 
$B1_4D$r0l2s$0$k$j$H$^$o$k$3$H$r0UL#$9$k(B. 

$B<!$,<g7k2L$G$"$k(B. 

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B.} $|k| > \frac{4}{3}$$B$J$i(B, $f_k$$B$N$b$H$GITJQ$J%[%b%H%T!<Hs<+L@$J(B
$B1_$O(B$T^1 \times {\bf R}$$BFb$K$O$J$$(B. 

\vspace{0.3cm}
J. Greene$B$,$3$NLdBj$r65$($F$/$l$?(B. $B$o$?$7$OH`$H2?2s$+5DO@$7$?(B. $B$+$l$O(B$f_k$
$B$K$D$$$F$N>\$7$$?tCM8&5f$r9T$J$$(B[7], $B?t(B$k_0 \approx 0.97$$B$,$"$C$F(B, 
$|k| \leq k_0$$B$J$i%[%b%H!<%T!<Hs<+L@$JITJQ1_$,(B$f_k$$B$KB8:_$7(B, 
$|k| > k_0$$B$J$iB8:_$7$J$$$3$H$r8+$D$1$?(B. $B2??M$+$NJL$NCx<T$b(B, $B?tCM8&5f$K(B
$B$b$H$E$$$FF1$87kO@$KC#$7$?(B([2], [5], [8], [12]). $B$7$+$7$3$N7kO@$O$$$^$N$H$3$m(B
$B87L)$K>ZL@$5$l$F$$$J$$(B. 

$B$o$l$o$l$N7k2L$O(BG.D. Birkhoff$B$NDjM}(B([4, $\S$3]$B$*$h$S(B[3, $\S$44])$B$NHs>o$KC1=c$J(B
$B5"7k$G$"$k(B. M. Herman$B$O:G6a%P!<%3%U$N7k2L$*$h$S4XO"$9$k7k2L$K$D$$$F$b$C$H>\$7$$(B
$B@bL@$r9T$J$C$?(B[9]. 

$B$o$l$o$l$NJ}K!$O(B[10]$B$G;H$C$?$b$N$HF1$8$G$"$k(B. $B$7$+$7<LA|(B$f_k$$B$X$N:G6a$N6=L#$N(B
$B9b$^$j$+$i$9$l$P(B, $B$3$N7k2L$O=PHG$9$k$KCM$9$k$H;W$o$l$k(B.  

\vspace{0.5cm}
\noindent
{\bf 2. $B%P!<%3%U$NDjM}(B}

$B$3$N@a$G$O(B, $B$o$l$o$l$,;H$&$3$H$K$J$k%P!<%3%U$NDjM}$r=R$Y$k(B. $B%P!<%3%U$NDjM}$r(B
$B=R$Y$k$?$a$K$b(B, $B$o$l$o$l$N9M$($k>u67$K%P!<%3%U$N>ZL@$r3HD%$9$k$?$a$K$b(B, $B<!$N(B
$BDj5A$,I,MW$G$"$k(B. 

\vspace{0.3cm}
\noindent
\underline{$BDj5A(B}. $C^1$\underline{$BKd$a9~$_(B}$B6J@~(B
$\gamma : J \rightarrow T^1 \times {\bf R}$
$B$r9M$($k(B. $B$?$@$7(B, $J = (-\infty, a), (-\infty, a]$$B$^$?$O(B$(-\infty, \infty)$
$B$G$"$j(B, $B$^$?(B$a \in {\bf R}$$B$G$"$k(B. $t \rightarrow -\infty$$B$N$H$-(B
$\gamma(t)_2 \rightarrow -\infty$$B$H$9$k(B. $B$?$@$7(B, $x \in T^1, y \in {\bf R}$
$B$KBP$7$F(B$(x,y)_2 = y$$B$G$"$k(B. $\gamma': J \rightarrow {\bf R}^2$$B$O(B$\gamma$$B$N(B
$B0l3,HyJ,$G$"$k$H$9$k(B. 
\underline{$B1tD>@~$+$i$N(B}$\gamma'$\underline{$B$N$:$l(B}$B$H8F$V(B
$\delta_{\gamma}: J \rightarrow {\bf R}$$B$r(B, $B0J2<$N(B2$B$D$N@-<A$r;}$D0l0U$N(B
$BO"B34X?t$H$7$FDj5A$7$h$&(B. 

(a) $\delta_{\gamma}(t)$$B$O1tD>%Y%/%H%k(B$(\partial/\partial y)_{\gamma(t)}$
$B$+$i(B$\gamma'(t)$$B$X$N3QEY$K(B${\rm mod} \ 2 \pi$$B$G0lCW$9$k(B. $B$?$@$7(B, $BH?;~7W2s$j(B
$B$r@5$H$9$k(B. 

(b)  $B$9$Y$F$N(B$s \leq t$$B$KBP$7$F(B$\gamma(s)_2 \leq \gamma(t)_2$$B$J$k(B$t \in J$$B$,(B
$BM?$($i$l$?$H$-(B, $-\pi/2 \leq \delta_{\gamma}(t) \leq \pi/2$$B$G$"$k(B. 

\vspace{0.3cm}
\noindent 
3$B@a$G$O(B, W. Thurston$B$,<(:6$7$F$/$l$?J}K!$G(B$\delta_{\gamma}$$B$NB8:_$H0l0U@-(B
$B$N>ZL@$N35N,$r=R$Y$k(B. $\delta_{\gamma}$$B$NB8:_$H0l0U@-$O(B$\gamma$$B$,Kd$a9~$^$l$F(B
$B$$$k$3$H(B, $B$9$J$o$A(B$\gamma$$B$,0lBP0l$G(B$\gamma'$$B$,%<%m$K$J$i$J$$$3$H(B, $B$K0MB8$9$k(B. 
$B<!$bI,MW$G$"$k(B. 

\vspace{0.3cm}
\noindent
\underline{$BDj5A(B}. $g: T^1 \times {\bf R} \rightarrow T^1 \times {\bf R}$$B$O(B
$T^1 \times {\bf R}$$B$N3FC<$r$=$l<+?H$K<L$9(B$C^1$$BHyJ,F1Aj<LA|$G$"$k$H$9$k(B. 
$g$$B$,(B\underline{$B:8$K(B}($B$^$?$O(B\underline{$B1&$K(B})\underline{$B79$$$F$$$k(B}$B$H$O(B, 
$T^1 \times {\bf R}$$BFb$N3F1tD>@~(B$x=$B0lDj(B$$B$KBP$7$F(B, $B$=$N(B$g$$B$N$b$H$G$NA|$,(B
$B$$$?$k$H$3$m1tD>@~$+$i@5$N(B($B$^$?$OIi$N(B)$B$:$l$r;}$D$H$-$G$"$k(B. 

\vspace{0.3cm}
\noindent
\underline{$BNc(B 2.1}. $h: {\bf R} \rightarrow {\bf R}$$B$r<~4|(B1$B$N(B$C^1$$B<LA|$H$9$k(B. 
$g: T^1 \times {\bf R} \rightarrow  T^1 \times {\bf R}$$B$O(B$g(x, y) = (x', y')$
$B$GM?$($i$l$k$H$9$k(B. $B$?$@$7(B
 
$$ \begin{array}{l}
   x' = x + y + h(x) \\
   y' = y + h(x) . 
   \end{array}    \eqno(1)
$$
$B$3$N<LA|$O1&$K79$$$F$$$k(B. 

$B=x$G5DO@$7$?<LA|(B$f_k$$B$O$3$N<LA|(B$g$$B$NFCJL$N>l9g$G$"$j(B, 
$h(x) = (k/2 \pi) \sin (2 \pi x)$$B$KBP1~$9$k(B.  

\vspace{0.3cm}
\noindent
{\bf $B%P!<%3%U$NDjM}(B}([4, $\S$3], [3, $\S$44], $B$*$h$S(B[9, I$B>O(B]).  
$g: T^1 \times {\bf R} \rightarrow  T^1 \times {\bf R}$$B$r(B$C^1$$BHyJ,F1Aj<LA|$H(B
$B$9$k(B. $g$$B$OLL@Q7A<0(B$dxdy$$B$rJ]B8$7(B, $T^1 \times {\bf R}$$B$N3FC<$r$=$l<+?H$K<L$7(B, 
$BJ}8~$rJ]B8$7(B, ($B:8$^$?$O1&$K(B)$B79$$$F$$$k$H$9$k(B. $U$$B$O(B $T^1 \times {\bf R}$$B$N3+(B
$BItJ,=89g$G$"$C$F(B, $gU = U$, $U$$B$O(B$T^1 \times {\bf R}$$B$KF1Aj$G$"$j(B, 
$B$"$k(B$a < b, a, b \in {\bf R}$$B$KBP$7$F(B
$T^1 \times (-\infty, a] \subset U \subset T^1 \times (-\infty, b)$$B$G$"$k$H(B
$B$9$k(B. $B$3$N$H$-(B,  $T^1 \times {\bf R}$$BFb$N(B$U$$B$N6-3&(B(frontier)$B$O%j%W%7%C%D4X?t(B
$\mu: T^1 \rightarrow {\bf R}$$B$N%0%i%U$G$"$k(B. $B$9$J$o$A(B, 
$\overline{U} - U = \{ (x, \mu(x)): x \in T^1 \}$$B$G$"$k(B. 
 
\vspace{0.3cm}
\noindent
\underline{$BNc(B 2.2}. (1)$B$GM?$($i$l$k<LA|$O>e$NDjM}$G(B$g$$B$K2]$7$?>r7o$r$9$Y$F(B
$BK~$?$9(B. 

\vspace{0.3cm}
\noindent
\underline{$BNc(B 2.3}. $X \subset T^1 \times {\bf R}$$B$r%[%b%H%T!<Hs<+L@$J(B
$B1_$H$9$k(B.  $T^1 \times {\bf R}$$BFb$N(B$X$$B$NJd=89g$N@.J,$N$R$H$D$O(B, $B>e$NDjM}$G(B
$U$$B$K2]$7$?>r7o$rK~$?$9(B. $B$3$l$O%H%]%m%8!<$N8EE5E*7k2L$G$"$k(B. $B$3$l$O(B
Schoenflies$B$NDjM}$+$i$?$@$A$K=P$k(B. ($T^1 \times {\bf R}$$B$rJ?LL$K1_4D$H$7$F(B
$BKd$a9~$`(B.) 

\vspace{0.3cm}
\noindent
\underline{$BCm0U(B}. $B:G8e$NNc$h$j(B, $B%P!<%3%U$NDjM}$N7O$rF@$k(B. $B$9$J$o$A(B, 
$X$$B$,(B$T^1 \times {\bf R}$$B$N(B$g$$BITJQ$J%[%b%H%T!<Hs<+L@$J1_$J$i(B, $X$$B$O(B 
$B%j%W%7%C%D4X?t(B$\mu: T^1 \rightarrow {\bf R}$$B$N%0%i%U$G$"$k(B.  

\vspace{0.3cm}
$B>e$NDjM}$G=R$Y$?>r7o$rK~$?$9<LA|(B$g$$B$KH<$C$F(B, $BN.NL(B(flux)$B$H8F$P$l$k?tCM(B
$BITJQNL$,$"$k(B. $B$3$NITJQNL$,%<%m$G$J$1$l$P(B, $B%P!<%3%U$NDjM}$OL5FbMF$G$"$k$,(B
$B@.$jN)$D(B. 

\vspace{0.3cm}
\noindent
\underline{$BDj5A(B}. $g$$B$O%P!<%3%U$NDjM}$N>r7o$rK~$?$9$H$9$k(B. $BN.NL(B$C(g)$
$B$r<!$GDj5A$9$k(B. 
$$
 {\rm Area}(gU \backslash U) - {\rm Area}(U \backslash gU) .  
$$
$B$?$@$7(B$U$$B$O(B$T^1 \times {\bf R}$$BFb$NG$0U$N3+=89g$G$"$C$F(B, $B$"$k(B
$a < b, a, b \in {\bf R}$$B$KBP$7$F(B
$T^1 \times (-\infty, a] \subset U \subset T^1 \times (-\infty, b)$$B$G$"$k(B. 

\vspace{0.3cm}
$g$$B$,(B$T^1 \times {\bf R}$$B$NLL@QJ]B8F1Aj<LA|$G$"$C$F3FC<$r$=$l<+?H$K<L$9$H$$$&(B
$B;v<B$+$i(B, $C(g)$$B$,(B$U$$B$K0M$i$J$$$3$H$,H=$k(B. $B%P!<%3%U$NDjM}$NDD=RCf$G2]$7$?(B
$B>r7o$r(B$U$$B$,K~$?$;$P(B, $BL@$i$+$K(B, 
$B$=$l$r;H$C$F(B$C(g)$$B$,7W;;$G$-(B, $C(g)=0$$B$,F@$i$l$k(B. 
$B8@$$49$($k$H(B, $BN.NL$,%<%m$G$J$1$l$P(B, $B%P!<%3%U$NDjM}Cf$G2]$5$l$?>r7o$rK~$?$9(B$U$
$B$O$J$$(B. $B$3$N>l9g(B, $B%P!<%3%U$NDjM}$O@5$7$$$,L50UL#$G$"$k(B. 

\vspace{0.3cm}
\noindent
\underline{$BNc(B 2.4}. $g$$B$r(B(1)$B$GM?$($i$l$k<LA|$H$9$k(B. $B$9$k$H<!$,@.$jN)$D(B.
$$
   C(g) = \int_0^1 h(x) dx .
$$
$C(g)$$B$NDj5A$N$H$-$K;H$C$?=89g(B$U$$B$r(B$T^1 \times (-\infty, 0)$$B$H$9$l$P>e$N<0$O(B
$BM}2r$G$-$k(B. $B$H$/$K(B, $f_k$$B$,=x$GDj5A$5$l$?<LA|$J$i(B$C(f_k) = 0$$B$G$"$k(B. 

\vspace*{0.3cm}
\noindent
[4, \S 3]$B$*$h$S(B[3, \S 44]$B$K$*$1$kDjM}$NA0Ds$O$o$l$o$lHG$N%P!<%3%U$NDjM}$N(B
$BA0Ds$H<c430[$J$C$F$$$k(B. $B;v<B(B, $B$o$l$o$l$NHG$O$I$A$i$N5"7k$G$b$J$$(B. $B$?$@(B
$BF1MM$J;EJ}$G>ZL@$G$-$k(B. $B%P!<%3%U$NDjM}$N$o$l$o$lHG$KHs>o$K;w$?$b$N$,(B[9]$B$G(B
$B>\$7$/>ZL@$5$l$F$l$$$k(B. 4$B@a$G$O%P!<%3%U$N>ZL@$r<jC;$+$K354Q$9$k(B. 

\newpage
%\vspace{0.5cm}
\noindent
{\bf 3. $\delta_{\gamma}$$B$NB8:_$H0l0U@-(B}

$\gamma$$B$,(B2$B@a$NDj5A$G=R$Y$?>r7o$rK~$?$9$H$9$k(B. $t \rightarrow -\infty$$B$N$H$-(B
$\gamma(t)_2 \rightarrow -\infty$$B$G$"$k$+$i(B, $t_0 \in J$$B$,$"$C$F(B, 
$s \leq t_0$$B$J$i(B$\gamma(s)_2 \leq \gamma(t_0)_2$$B$G$"$k(B.
$BL@$i$+$K(B, $B0l0U$NO"B3$J(B$\delta_{\gamma}$$B$,$"$C$F(B, $t=t_0$$B$K(B(b)$B$rK~$?$7(B, 
$B$^$?$9$Y$F$N(B$t$$B$KBP$7$F(B(a)$B$rK~$?$9(B. 
$t_1 \in J$$B$,M?$($i$l$F$9$Y$F$N(B$s \leq t_1$$B$KBP$7$F(B
$\gamma(s)_2 \leq \gamma(t_1)_2$$B$rK~$?$9$H$-$K(B
$\pi/2 \leq \delta_{\gamma}(t_1) \leq \pi/2$$B$G$"$k$3$H$r<($;$P==J,$G$"$k(B.

$B$3$N>ZL@K!$O(BW.Thurston$B$,<(:6$7$F$/$l$?(B. $\gamma$$B$,Kd$a9~$^$l$F$$$k$3$H$r(B
$B;H$&$3$H$,K\<AE*$G$"$k(B. $B4JC1$N$?$a(B$t_0 < t_1$$B$H$9$k(B. $B;D$j$N>l9g$bF1MM$G$"$k(B.  

\vspace*{0.3cm}
\noindent
\underline{$B%9%F%C%W(B 1}. $t_0$$B$*$h$S(B$t_1$$B$N6aK5$G(B$\gamma$$B$,1tD>$G(B, 
$i=1,2$$B$KBP$7$F(B$\gamma(s)_2 \leq \gamma(t_i)_2$$B$N>l9g$KLdBj$r5"Ce$5$;$k(B.  
$t_0$$B$*$h$S(B$t_1$$B$NG$0U$K>.$5$J6aK5$G(B$\gamma$$B$rJQ7A$9$l$P$3$l$,<B8=$G$-$k(B. 
$B$3$N>l9g(B, $B>ZL@$9$Y$-$3$H$O(B$\delta_{\gamma}(t_1)=0$$B$G$"$k(B.

\vspace*{0.3cm}
\noindent
\underline{$B%9%F%C%W(B 2}. $\gamma$$B$O(B $t_0$$B$N6aK5$G1tD>$G$"$C$F(B, 
$s \leq t_0$$B$N$H$-(B$\gamma(s)_2 \leq \gamma(t_0)_2$$B$G$"$k$+$i(B, 
$T^1 \times [\gamma(t_0)_2 - \varepsilon, \infty)$$B$rE@$4$H$K8GDj$7$F(B
$T^1 \times {\bf R}$$B$r%$%=%H!<%W$7(B, $\gamma|(-\infty, t_0+\varepsilon)$
$B$,1tD>$K$J$k$h$&$K(B$\gamma$$B$rJQ7A$G$-$k(B. $B$3$3$G(B$\varepsilon$$B$O>.$5$J(B
$B@5?t$G$"$k(B. $B$3$l$,$G$-$k$3$H$OHyJ,%H%]%m%8!<8&5f<T$K$O$h$/CN$i$l$F$*$j(B, 
$BD>4QE*$K$OL@$i$+$G$"$k(B. $B$9$k$HJQ7A$r<u$1$?(B$\gamma$$B$,(B
$\delta_{\gamma}(t_1) = 0$$B$rK~$?$9$N$O(B, $B$b$H$N(B$\gamma$$B$,$=$l$rK~$?$9$H$-(B, 
$B$^$?$=$N$H$-$N$_$G$"$k(B. 

\vspace*{0.3cm}
\noindent
\underline{$B%9%F%C%W(B 3}.
$B$$$^$d(B$\gamma$$B$O(B$(-\infty, t_0+\varepsilon]$$B>e$*$h$S(B$t_1$$B$N6aK5$G1tD>$G$"$j(B, 
$B$^$?(B$s \leq t_1$$B$KBP$7$F(B$\gamma(s)_2 \leq \gamma(t_1)_2$$B$G$"$k(B.
$B$3$N$H$-<!$,@.$jN)$D(B.
$$
  \gamma(t_0)_2 < \gamma(t_0+\varepsilon)_2 \leq \gamma(t_1)_2 .
$$
$y$$B:BI8$rJ]B8$9$k(B$T^1 \times {\bf R}$$B$N%$%=%H%T!<$r8+$D$1$F(B, 
$T^1 \times [\gamma(t_1)_2 - \varepsilon, \infty)$$B>e$G(B$x$$B:BI8$K4X$70lDj(B
$BB.EY$N2sE>$G$"$j(B, $B$^$?(B
$T^1 \times (-\infty, \gamma(t_0)_2 + \varepsilon]$$B>e$G91Ey<LA|$G$"$j(B, 
$B%$%=%H%T!<$N=*C<$G(B$\gamma(t_0)_1 = \gamma(t_1)_1$$B$G$"$k(B
$B$h$&$K$G$-$k(B. $BL@$i$+$KJQ7A8e$N(B$\gamma$$B$,(B$\delta_{\gamma}(t_1) = 0$$B$r(B
$BK~$?$9$N$O(B, $B$b$H$N(B$\gamma$$B$,$=$l$rK~$?$9$H$-(B, $B$^$?$=$N$H$-$N$_$G$"$k(B. 

\vspace*{0.3cm}
\noindent
\underline{$B%9%F%C%W(B 4}.
$B$$$^$d(B$\gamma$$B$O(B$(-\infty, t_0+\varepsilon]$$B>e$*$h$S(B$t_1$$B$N6aK5$G(B 
$B1tD>$G$"$j(B, $s \leq t_1$$B$KBP$7$F(B$\gamma(s)_2 \leq \gamma(t_1)_2$$B$G$"$j(B, 
$B$^$?(B$\gamma(t_0)_1 = \gamma(t_1)_1$$B$G$"$k(B. $T^1 \times {\bf R}$$B>e$K(B
$B1_(B$C_1, C_2$$B$,$"$C$F<!$rK~$?$9(B.
$$ \begin{array}{l}
  C_1 \cap {\rm image} \ \gamma = \gamma(t_0+\varepsilon) , \\
  C_2 \cap {\rm image} \ \gamma = \gamma(t_1-\varepsilon) ,
   \end{array}
$$
$B$*$h$S(B
$$
   C_1 \cap C_2 = \emptyset .
$$
$T^1 \times {\bf R}$$B$N%$%=%H%T!<$G(B, $C_1$$B$H(B$C_2$$B$K64$^$l$?1_4D$N30$G$O(B
$B91Ey<LA|$K$J$C$F$$$F(B, $B%$%=%H%T!<$N=*C<$G$O(B$\gamma$$B$,(B
$(-\infty, t_1+\varepsilon]$$B>e$G1tD>$K$J$k$b$N$r8+$D$1$k$3$H$,$G$-$k(B. 
$B$3$s$I$b(B, $B$3$l$,$G$-$k$3$H$OHyJ,%H%]%m%8!<8&5f<T$K$O$h$/CN$i$l$F$*$j(B, 
$BD>4QE*$K$OL@$i$+$G$"$k(B.
$BJQ7A8e$N(B$\gamma$$B$O(B$\delta_{\gamma}(t_1) = 0$$B$rK~$?$7(B, $B$b$H$N(B$\gamma$$B$b(B
$BF1$8J}Dx<0$rK~$?$9(B. \hfill $\Box$

\vspace{0.5cm}
\noindent
{\bf 4. $B%P!<%3%U$NDjM}$N>ZL@(B}

$B%P!<%3%U$NH/A[$K$[$\=>$$$J$,$i<jC;$+$K%9%1%C%A$7$h$&(B. $U$$BFb$NE@(B$x$$B$,(B
\underline{$B@5$K(B}($B$^$?$O(B\underline{$BIi$K(B})\underline{$BE~C#2DG=(B}$B$H$OKd$a9~$_6J@~(B
$\gamma: (-\infty,a] \rightarrow U$$B$,$"$C$F(B$t \rightarrow -\infty$$B$N$H$-$K(B
$\gamma(t)_2 \rightarrow -\infty$$B$rK~$?$7(B, $B1tD>@~$+$i@5$K(B($B$^$?$OIi$K(B)
$B$K$:$l$F$*$j(B, $\gamma(a) = x$$B$rK~$?$9$H$-$G$"$k(B. $W_+$($B$^$?$O(B$W_-$)
$B$r(B$U$$BFb$N@5$K(B($B$^$?$OIi$K(B)$BE~C#2DG=$JE@$N=89g$H$9$k(B. 

$B%P!<%3%U$NDjM}$r(B$g$$B$,1&$K79$$$F$$$k>l9g$K>ZL@$9$k(B. $g$$B$,:8$K79$$$F$$$k(B
$B>l9g$O(B$g$$B$r(B$g^{-1}$$B$KCV$-49$($l$P$3$N>l9g$K5"Ce$9$k(B.

$g$$B$,1&$K79$$$F$$$k>l9g(B, $B4JC1$K%A%'%C%/$G$-$k$h$&$K(B
$g(\overline{W}_- \cap U) \subset W_-$$B$*$h$S(B
$g^{-1}(\overline{W}_+ \cap U) \subset W_+$$B$G$"$k(B. $B$3$l$+$i(B
$W_- = U = W_+$$B$,<!$N$3$H$+$i$7$?$,$&(B. $B$9$J$o$A(B, 
$g$$B$,LL@QJ]B8$G$"$j(B, $a < b$$B$,$"$C$F<!$rK~$?$9$N$G$"$k(B. 
$$
 T^1 \times (-\infty, a] \subset W_- \cap U \cap W_+ \subset 
W_- \cup U \cup W_+ \subset T^1 \times (-\infty, b] .
$$
$W_- = U = W_+$$B$J$k;v<B$+$i(B, $B3F(B$x \in T^1$$B$KBP$7$F(B$a_x \in {\bf R}$$B$,$"$C$F(B
$B<!$rK~$?$9(B.
$$
  U \cap (x \times {\bf R}) = x \times (-\infty, a_x) . 
$$
$B$3$l$N>ZL@$O(B3$B@a$N>ZL@$K;w$F$$$k(B. $B$3$l$O%P!<%3%U$NO@J8$K$b(B
Herman$B$NO@J8(B[9]$B$K$b8@5Z$7$F$"$k(B. 

$u: {\bf R} \rightarrow {\bf R}$$B$,(B$C^1$$B4X?t$N$H$-(B
$\phi_u: T^1 \times {\bf R} \rightarrow T^1 \times {\bf R}$$B$rHyJ,F1Aj<LA|(B
$$
  \phi_u(x,y) = (x+u(y), y)
$$
$B$H$9$k(B. $g_u = \phi_u g \phi_u^{-1}$$B$HCV$/(B. $B$9$k$H(B$\phi_u U$$B$O(B$g_u$
$B$K4X$7$FITJQ$G$"$k(B. $u$$B$,%[%$%C%H%K!<$N(B$C^1$$B0LAj$G(B$0$$B$K==J,6a$1$l$P(B, 
$g_u$$B$O(B$g$$B$HF1$8J}8~$K79$$$F$$$k(B. $B$f$($K(B$\phi_u U$$B$O<!$N>r7o$rK~$?$9(B. 
$B$9$J$o$A(B, $B3F(B$x \in T^1$$B$KBP$7$F(B$a_{ux} \in {\bf R}$$B$,$"$C$F(B 
$$
  \phi_u U \cap (x \times {\bf R}) = x \times (-\infty, a_{ux}) .
$$
$B$3$l$O(B$0$$B$N==J,>.$5$J(B$C^1$$B6aK5Fb$NG$0U$N(B$u$$B$KBP$7$F@.$jN)$D$+$i(B
$U$$B$N6-3&(B(frontier)$B$O%j%W%7%C%D4X?t$N%0%i%U$G$"$j(B, $B%P!<%3%U$NDjM}$N(B
$B<gD%DL$j$G$"$k(B. \hfill $\Box$

\vspace{0.5cm}
\noindent
{\bf 5. $BJl4X?t(B}

$g$$B$rNc(B2.1$B$N<LA|$H$9$k(B. $BNc(B2.4$B$h$j(B, $BN.NL(B$C(g)$$B$O(B$\int_0^1 h(x)dx$$B$G$"$k(B. 
$B%P!<%3%U$NDjM}$O(B$C(g) \neq 0$$B$N$H$-L5FbMF$G@5$7$$$+$i(B$\int_0^1 h(x)dx =0$
$B$HCV$/(B. $H$$B$r(B$h$$B$NITDj@QJ,$H$9$k(B. $\int_0^1 h(x)dx =0$$B$G$"$k$+$i(B$H$$B$O(B
$B<~4|(B1$B$G<~4|E*$G$"$k(B.

$B8EE5NO3X$G(B$g$$B$NJl4X?t$H$7$FCN$i$l$k$b$N$r<!<0$GDj5A$9$k(B. 
$$
  G(x, x') = \displaystyle - \frac{(x'-x)^2}{2} - H(x) .
$$
$BNc(B(2.1)$B$NJ}Dx<0(B(1)$B$O<!<0$HF1CM$G$"$k(B.
$$ \begin{array}{l}
  y = \displaystyle \frac{\partial G(x,x')}{\partial x} \\
  y' = \displaystyle - \frac{\partial G(x,x')}{\partial x'} . 
   \end{array}     \eqno(2)
$$
$g$$B$O%[%b%H%T!<Hs<+L@$J1_(B$X$$B$r;}$D$H$9$k(B. $B%P!<%3%U$NDjM}$*$h$SNc(B2.3$B$KB3$/(B
$BCm0U$h$j(B, $X$$B$O%j%W%7%C%D4X?t(B$\mu: T^1 \rightarrow {\bf R}$$B$N%0%i%U$G$"$k(B. 
$B$7$?$,$C$FF1Aj<LA|(B$g_0: T^1 \rightarrow {\bf R}$$B$,$"$C$F<!$rK~$?$9(B. 
$$
  g(x, \mu(x)) = (g_0(x), \mu(g_0(x))). 
$$
$g$$B$O(B$T^1 \times {\bf R}$$B$N3FC<$r$=$l<+?H$K<L$9$+$i(B, $X$$B$NJd=89g$N3F@.J,$r(B
$B$=$l<+?H$K<L$9(B. $B$=$N>e(B, $g$$B$OJ}8~J]B8$G$"$k$+$i(B, $g_0$$B$OJ}8~J]B8$G$"$k(B. 
$BJ}Dx<0(B(2)$B$h$j<!$rF@$k(B. 
$$
\displaystyle
\frac{\partial}{\partial x} [G(\overline{x},x) + G(x,x')] = 0 . 
\eqno(3)
$$
$B$?$@$7(B$\overline{x} = g_0^{-1}(x), x' = g_0(x)$$B$G$"$k(B. 

\vspace{0.5cm}
\noindent
{\bf 5. $BDjM}$N>ZL@(B}

$BA0@a$N5-K!$rB3$$$FMQ$$$k(B. $g_0 = \mu_0^{-1}g \mu_0$, 
$\mu_0 = (1,\mu): T^1 \rightarrow {\rm graph} \mu$$B$G(B$\mu_0$$B$H(B$g$$B$O%j%W%7%C%D$G(B
$B$"$k$+$i(B, $g_0$$B$O%j%W%7%C%D$G$"$k(B. $BF1MM$K(B
$g_0^{-1} = \mu_0^{-1} g^{-1} \mu_0$$B$O%j%W%7%C%D$G$"$k(B. $B%j%W%7%C%D4X?t$O(B
$B@dBPO"B3$G$"$k$+$i(B, $B%k%Y!<%0M}O@$N8EE5E*DjM}(B, $B$9$J$o$A@dBPO"B3$J4X?t$O$[$H$s$I(B
$B$$$?$k$H$3$mHyJ,2DG=$G$"$j(B, $B$=$NHyJ,$NITDj@QJ,$O(B($BDj?t$NITDj@-$r=|$$$F(B)$B$b$H$N(B
$B4X?t$G$"$k(B, $B$rE,MQ$G$-$k(B. $B$3$NDjM}$O(B[13,11.7]$B$G=R$Y$i$l>ZL@$5$l$F$$$k(B. 

$B$f$($K(B(3)$B$rHyJ,$G$-$F<!$rF@$k(B.
$$
\displaystyle
\frac{dg_0^{-1}(x)}{dx} + \frac{dg_0(x)}{dx} = 2 + h'(x).  \eqno(4)
$$
$B$3$3$G(B$h'$$B$O(B$h$$B$NHyJ,$G$"$k(B. $BJ}Dx<0(B(4)$B$O(B$dg_0(x)/dx$$B$H(B$dg_0^{-1}(x)/dx$$B$,(B
$BDj5A$5$l$F$$$k$H$3$m$G$O$$$D$b@.$jN)$D(B. $B$@$+$i$[$H$s$I$$$?$k$H$3$m@.$jN)$D(B. 

$L$$B$r(B$g_0$$B$H(B$g_0^{-1}$$B$N%j%W%7%C%DDj?t$NBg$-$$J}$H$9$k(B. $B$9$J$o$A(B
$$
  L = \displaystyle \max \left\{ \sup \frac{|g_0(x) - g_0(x')|}{|x-x'|}, 
\sup \frac{|g_0^{-1}(x) - g_0^{-1}(x')|}{|x-x'|} 
 \right\} .
$$
$g_0$$B$H(B$g_0^{-1}$$B$OJ}8~J]B8$@$+$i(B, $dg_0(x)/dx$$B$H(B$dg_0^{-1}(x)/dx$$B$OHsIi$G(B
$B$"$k(B. $L$$B$NDj5A$+$i$9$k$H<!$,=P$k(B. 
$$ \begin{array}{l}
 L^{-1} \leq \displaystyle \frac{dg_0(x)}{dx} \leq L, \\
 L^{-1} \leq \displaystyle \frac{dg_0^{-1}(x)}{dx} \leq L.
   \end{array}   \eqno(5)
$$
$m = \min h', M = \max h'$$B$HCV$/(B. (4)$B$H(B(5)$B$h$j<!$rF@$k(B. 
$$
  2L^{-1} \leq 2 + m.     \eqno(6)
$$
$B$H$/$K(B
$$
  m > -2 .     \eqno(7)
$$
(7)$B$O(B$g$$B$,%[%b%H%T!<Hs<+L@$JITJQ1_$r5v$9$H$$$&2>Dj$N$_$+$iF3$+$l$?$+$i(B, 
$B$3$l$O%[%b%H%T!<Hs<+L@$JITJQ1_$NB8:_$N$?$a$NI,MW>r7o$G$"$k(B. $BJL$NI,MW>r7o$r(B
$B0J2<$N$h$&$KF@$k$3$H$,$G$-$k(B. 

$L$$B$NDj5A$*$h$S(B[13,11.7]$B$NM}O@$h$j<!$rF@$k(B. 
$$
  L = \displaystyle \max \left\{ {\rm ess.sup.} \frac{dg_0(x)}{dx},  
{\rm ess.sup.} \frac{dg_0^{-1}(x)}{dx} \right\} .  \eqno(8)
$$
$B$3$3$G(Bess.sup.$B$OB,EYO@E*$J0UL#$G$N!VK\<AE*>e8B!W(B(essential supremum)$B$N$3$H$G(B
$B$"$k(B.  $dg_0(x)/dx, dg_0^{-1}(x)/dx$$B$O%j%W%7%C%D4X?t$NHyJ,$G$"$k$+$i(B, $B3F<+$N(B
$BK\<AE*>e8B$O3F<+$,Dj5A$5$l$F$$$k=89g$N>e$N>e8B$HF1$8$G$"$k(B. 

(4), (5), (8)$B$h$j<!$rF@$k(B. 
$$
   L^{-1} + L \leq 2 + M .     \eqno(9)
$$
$B4X?t(B$L^{-1}+L$$B$O(B$L \geq 1$$B$N$H$-C1D4A}2C4X?t$G$"$k$+$i(B, (6)$B$H(B(9)$B$h$j<!$,(B
$BF@$i$l$k(B. 
$$
\displaystyle \frac{2+m}{2} + \frac{2}{2+m} \leq 2 + M .    \eqno(10)
$$
$B$3$3$G(B$h$$B$,<~4|E*$G$"$k$3$H$h$j(B$m \leq 0$$B$G$"$k$3$H$rMxMQ$7$?(B. 

$g = f_k$$B$N>l9g(B, $h(x) = (k/2 \pi) \sin (2 \pi x)$$B$G$"$k$+$i(B, $M=|k|, m = -|k|$
$B$G$"$k(B. $BITEy<0(B(7)$B$H(B(9)$B$O<!$r0UL#$9$k(B. 
$$
    |k| \leq \frac{4}{3} .  \eqno(11)
$$
(11)$B$O%[%b%H%T!<Hs<+L@$J1_$NB8:_$N2>Dj$N2<$KF3$+$l$?$+$i(B, $B>ZL@$,407k$7$?(B.  
\hfill $\Box$

\vspace{0.5cm}
\noindent
{\bf 6. $BJdB-E*Cm0U(B}

(1) $B>e$NK\J8$O%W%l%W%j%s%H(B[11]$B$K:3:Y$JJQ99$r2C$($?$b$N$G$"$k(B. 
$B$=$l$r=q$$$F0JMh(B, Fathi[6]$B$O%P!<%3%U$NDjM}$N(BHerman$BHG(B[9]$B$NJL$N>\$7$$>ZL@$r(B
$BM?$($?(B. 

(2) Aubry, Le Daeron$B$*$h$S(BAndr\'{e}$B$OITJQ1_$NB8:_$NI,MWH=Dj>r7o$r>ZL@$7$?(B[1]. 
$B$+$l$i$N7k2L$HD9$$?tCM7W;;$O$3$NO@J8$N<g7k2L$r4^$`$h$&$K8+$($k(B. 

(3) $g$$B$rNc(B2.1$B$GM?$($i$l$k%?%$%W$N<LA|$H$9$k(B. $BJ}Dx<0(B(3)$B$O<!<0$KF1CM$G$"$k(B. 
$$
   g_0^{-1}(x) + g_0(x) = 2x + h(x) .
$$ 
$B5U$K<LA|(B$g_0: {\bf R} \rightarrow {\bf R}$$B$,$"$C$F695AC1D4$G$"$j(B, 
$g_0(x+1) = g_0(x)+1$$B$H>e$N<0$rK~$?$;$P(B, $g$$B$O%[%b%H%T!<Hs<+L@$JITJQ1_$G(B
$B%j%W%7%C%D4X?t(B
$$
  \mu(x) = G_1(x, g_0(x))
$$
$B$N%0%i%U$H$J$C$F$$$k$b$N$r;}$D(B. $B$?$@$7(B$G_1 = \partial G(x,x')/\partial x$
$B$G$"$k(B. $B$3$N4Q;!$O(BM.Herman$B$K$h$k(B. 
$BH`$O(B1981$BG/2F$N%V%i%8%k$G$N9V5A$G(B, $g$$B$,(B$C^{3-\varepsilon}$$B5i(B 
$B$GL5M}?t2sE>?t$NHsA+0\E*ITJQ1_$r;}$DCmL\$9$Y$-Nc$r9=@.$7$?(B. ([9],III$B>O;2>H(B.)

(4) $B>e$G<($7$?>ZL@$K$h$l$P(B, $g$$B$,%[%b%H%T!<Hs<+L@$JITJQ1_$r;}$F$P(B(7)$B$H(B(10)$B$,(B
$BK~$?$5$l$k(B. 1983$BG/2F$NBg3X1!%3!<%9$G(B, $g$$B$,%[%b%H%T!<Hs<+L@$JITJQ1_$r;}$D(B
$B$H$$$&A0Ds$+$i(B$m$$B$H(B$M$$B$K4X$7$F$3$l0J>e$N>r7o$,=P$;$k$+$H$$$&2]Bj$r=P$7$?(B. 
Rafael Llave$B$O$3$N2]Bj$KEz$($?(B. $B$9$J$o$A(B, (7)$B$H(B(10)$B$rK~$?$9G$0U$N(B$m \leq 0$
$B$*$h$SG$0U$N(B$M \geq 0$$B$KBP$7$F(B$g$$B$,B8:_$7$F%[%b%H%T!<Hs<+L@$JITJQ1_$r;}$A(B, 
$B$^$?(B$m = \min h, M = \max h$$B$rK~$?$9$3$H$r<($7$?(B. $BH`$N>ZL@$OJd0d$K:\$;$F(B
$B$*$$$?(B.

(5) $B%l%U%'%j!<$O(B6$B@a$NITEy<0$,(B[9, II$B$*$h$S(BIII$B>O(B]$B$K4^$^$l$k$3$H$r;XE&$7(B, 
$B$^$?L\E*$O0c$C$F$$$?$,Jd0d$NNc$,(B[9, III.9]$B$K4^$^$l$k$3$H$r;XE&$7$?(B. 
$B%l%U%'%j!<$N%3%a%s%H$K$h$l$P(B, $B$3$l$i$NITEy<0$dNc$O(BM.Herman$B$,(B1980$BG/0JMh(B, 
Palaiseau, Zurich, Rio$BEy$K$*$$$F$$$m$$$m$J9V5A$G5DO@$7$F$-$?$b$N$G$"$k(B. 

\vspace{0.5cm}
\begin{center}
{\bf $BJd0d(B} \\
R.Llave
\end{center}

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B}. $m < 0 < M$$B$O(B(7)$B$H(B(9)$B$rK~$?$9(B2$B$D$N<B?t$H$9$k(B. $B$3$N$H$-(B$C^{\infty}$
$B4X?t(B$g_0, h: {\bf R} \rightarrow {\bf R}$$B$,$"$C$F0J2<$rK~$?$9(B. $g_0$$B$OC1D4A}2C(B
$B$G$"$j(B, $g_0(x+1) = g_0(x) +1$,
$$
  g_0^{-1}(x) + g_0(x) = 2x + h(x), \eqno(A1)
$$
$B$*$h$S(B
$$ \begin{array}{ll}
  \max h'(x) = M, & \min h'(x) = m .
   \end{array}  \eqno(A2)
$$

\vspace{0.3cm}
\noindent
\underline{$B>ZL@(B}. $-2 < m < 0$$B$G$"$k$+$i(B$l$$B$,$"$C$F<!$rK~$?$9(B.
$$
\displaystyle
\frac{2}{m+2} l + \frac{m+2}{2} (1-l) = 1 .
$$ 
$g_0$$B$O(B$[0,1]$$B$GDj5A$5$l$?6hJ,@~7A4X?t$G$"$C$F0J2<$rK~$?$9$H$9$k(B.
$$ \begin{array}{ll}
  g_0(0) = 0; & \\
  g'_0(x) = 2/(2+m), & x \in [0,l/2] \cup [1-l/2,1]; \\
  g'_0(x) = (2+m)/2, & x \in (l/2, 1-l/2) .
   \end{array}
$$
$B$3$N$H$-(B$g_0(1)=1$$B$G$"$k$+$i(B$g_0$$B$r(B${\bf R}$$BA4BN$K3HD%$7$F(B
$g_0(x+1) = g_0(x) + 1$$B$rK~$?$9$h$&$K$G$-$k(B. $h$$B$O(B(A1)$B$GDj5A$5$l$k$H$9$k(B.
$B$9$k$H<!$,@.$jN)$D(B.
$$ \begin{array}{ll}
  m = \min h'(x), & M_1 = \max h'(x) .   
   \end{array}   \eqno(A3)
$$
$B$3$3$G(B$M_1$$B$O<!<0$GDj5A$5$l$k(B. 
$$
  2/(2+m) + (2+m)/2 = 2 + M_1 . 
$$
1$B?^$r8+$h(B.

\vspace{2cm}
\hspace*{6cm} $B?^(B1

\vspace{1cm}

\newpage
$g_0$$B$N3Q$r4]$a$FL58B2sHyJ,2DG=$J4X?t(B$g_0$$B$H(B$h$($B$3$l$b(B(A1)$B$GDj5A$5$l$k(B)$B$rF@$F(B
(A3)$B$rK~$?$9$h$&$K$G$-$k(B. $B:G8e$K(B, $BG$0U$K>.$5$J6h4V$G(B$g_0$$B$rJQ7A$7$F(B, $B$=$3$G(B
$B$=$NHyJ,$rA}2C$5$;(B, $\max h'(x) = M$$B$H$7(B, $B$^$?DjM}$N$=$NB>$N>r7o$r$9$Y$F(B
$BK~$?$9$h$&$K$G$-$k(B. $B$H$$$&$N$O(B(10)$B$K$h$j(B$M \geq M_1$$B$@$+$i$G$"$k(B. 
\hfill $\Box$

\vspace{0.5cm}
\begin{center}
{\bf $B;29MJ88%(B}
\end{center}

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\re
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[3] G.D. Birkhoff. Surface transformations and their dynamical applications. 
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\re
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\re
[7] J.M. Greene. A method for determining a stochastic transition. 
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\re
[8] R.H.G. Hellemann \& T. Bountis. Periodic solutions of arbitrary period, 
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(ed. G. Casati and J. Ford). Springer Verlag: New York(1979), 353-.

\re
[9] M. Herman. Introduction $\grave{\rm a}$ l'\'{e}tude des courbes 
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{\it Asterisque} {\bf 103-104} (1983).

\re
[10] J. Mather. Glancing billiards. {\it Ergod. Th. \& Dyn. Sys.} {\bf 2} 
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\re
[11] J. Mather. Non-existence of invariant circles. Preprint (1982).

\re
[12] G.E. Powell \& J.C. Percival. A spectral entropy method for 
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\re
[13] E.C. Titchmarsh. {\it The Theory of Functions}, (2nd edition). 
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