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$BC+@n@6N4(B $BLu(B
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\noindent
{\it Ergodic Theory and Dynamical Systems} {\bf 4} (1984), 585-603

\vspace{0.5cm}
{\large 
\noindent
\begin{center}
{\bf $B$M$8$l<LA|$N%^%6!<$NDjM}$N0LAjHG(B \\ 
A topological version of a theorem of Mather on twist maps}
\end{center}
}

\vspace{0.5cm}
\begin{center}
G.R.Hall \\

Department of Mathematics, Boston University \\
Boston, MA 02215, USA
\end{center}

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

$B1_4D$N$M$8$l<LA|(B(twist map)$B$K2sE>?t(B$p/q$$B$N<~4|E@$,$"$l$P2sE>?t(B$p/q$$B$N%P!<%3%U<~4|E@$b$"$k$3$H$r<($9!%%H%]%m%8!<E*$J<jK!$r;H$&!%$@$+$iLL@QJ]B8!"1_8r:9@-$H$$$C$?@-<A$r2>Dj$9$k$OI,MW$O$J$$!%<LA|$,LL@QJ]B8$J$i$3$3$G>ZL@$9$kDjM}$H%P!<%3%U$NITF0E@DjM}$H$+$i(BAubry-Mather$B$N:G6a$NDjM}$,=P$k!%$M$8$l<LA|$N:G>.<~4|$N<~4|50F;$O%P!<%3%U<~4|50F;$G$"$k$3$H$b<($9!%(B

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

$B1_4D$+$i$=$l<+?H$N>e$X$NHyJ,F1Aj<LA|(B$f$$B$,$M$8$l>r7o$rK~$?$9$H$O!"(B$f$$B$N$b$H$G$N$"$kE@$NA|$N3QEY:BI8$,!"$=$NE@$NF07B:BI8$,Bg$-$/$J$k$K$D$l$FA}$($k$H$-$G$"$k(B($B@53N$JDj5A$K4X$7$F$O#2@a;2>H(B)$B!%$3$N$h$&$J<LA|$O%]%"%s%+%l$,;0BNLdBj$H$N4X78$G:G=i$K8&5f$7$?!%$3$NJ8L.$G$O<LA|$O<+A3$JITJQB,EY$r;}$D!%%P!<%3%U(B[4]$B!"(B[5]$B$O$3$N$h$&$JLL@QJ]B8$M$8$l<LA|$N>l9g!"B?$/$N<~4|50F;$,$"$k$3$H$r<($7$?!%$?$@$7!"%P!<%3%U$NDjM}$+$i$O50F;$N@-<A$K4X$7$F2?$NF6;!$bF@$i$l$J$$!%:G6a!"(BAubry[2]$B$H(BMather[15]([14]$B$b;2>H(B)$B$OLL@QJ]B8$M$8$l<LA|(B$f$$B$K$OE@$N3QEY:BI8$N=g=x(B(angular ordering)$B$,J]B8$9$k$h$&$J50F;$,B8:_$9$k$3$H$r<($7$?!%$3$l$i$O%P!<%3%U<~4|50F;$H8F$P$l$k(B([14]$B$*$h$S#2@a;2>H(B)$B!%$=$N$&$(!"$+$l$i$OLL@QJ]B8$M$8$l<LA|$K!V=`<~4|50F;!W$,B8:_$9$k$3$H$r<($7$?!%(BMather$B$NMxMQ$7$?JQJ,K!$N5;=Q$OLL@QJ]B8$M$8$l<LA|$NB>$N@-<A$r8&5f$9$k$N$K$bLrN)$D(B($B$?$H$($P(B[16]$B;2>H(B)$B!%(B

$B$M$8$l<LA|$OLL@QJ]B8$G$J$$<LA|$H$7$F$b$7$P$7$P$"$i$o$l$k!%Nc$($P!"(BHopf$BJ,4t$r5/$3$7$?J?LL$N<LA|$NITF0E@$N6aK5$G$O$M$8$l>r7o$,K~$?$5$l$k!%$3$N$h$&$J;60o<LA|$G$OITJQB,EY$OB8:_$7$J$$!%(B

$B$3$N%l%]!<%H$G$O!"$"$k2sE>?t$N<~4|50F;$r;}$D1_4D$N$M$8$l<LA|$,!"F1$82sE>?t$N%P!<%3%U<~4|50F;$r;}$D$3$H$r<($9!%$3$l$OLL@QJ]B8$N2>Dj$,<~4|50F;$NB8:_$N2>Dj$KCV$-JQ$o$C$?$b$N$H9M$($F$h$$!%$@$+$i$3$NDjM}$O;60o<LA|$K$bE,MQ$G$-$k(B([9]$B!"(B[10]$B;2>H(B)$B!%LL@QJ]B8$M$8$l<LA|$,M?$($i$l$?$H$-!">e$G=R$Y$?%P!<%3%U$NDjM}$K$h$jB?$/$N<~4|50F;$NB8:_$,M?$($i$l!"$f$($K(BAubry-Mather$B$NDjM}$O$=$N7O$H$7$FF3$+$l$k!%(B

$B>ZL@$NH/A[$O$M$8$l>r7o$rMxMQ$7$F%P!<%3%U<~4|50F;$G$J$$<~4|50F;$,0LAjE*$KJ#;($G$"$k$3$H$r<($9$3$H$K$"$k!%$?$H$($P$M$8$l<LA|$r7|?b$7(B(suspend)$B!"Hs%P!<%3%U<~4|50F;$r8+$k$H!"$3$l$i$OHs<+L@$JAH$_I3$r9=@.$9$k!%$M$8$l>r7o$N2<$G$3$l$i$N50F;$NMm$^$jJ}$rMxMQ$9$l$P!"4v2?3XE*<jK!$^$?$OAH$_I3$H<~4|50F;$K4X$9$k(BMatsuoka[17]$B$NDjM}$rMQ$$$F?7$7$$<~4|E@$NB8:_$,=P$;$k!%$H$/$K$b$7$M$8$l<LA|$KF1$82sE>?t$r;}$D%P!<%3%U<~4|50F;$HHs%P!<%3%U<~4|50F;$,$"$l$P!"$=$N2sE>?t$r;}$D0[$J$C$?(B2$B$D$N%P!<%3%U<~4|50F;$,$"$k$O$:$G$"$k!%(B

$B#2@a$G$O!"O@J8$rDL$7$FMQ$$$k5-9f$rF3F~$9$k!%#3@a$G$O!"<g7k2L$r=R$Y$k!%$=$N>ZL@$O#4@a$NJdBj$rMQ$$$F#5@a$G9T$J$&!%#6@a$G$O!"$M$8$l<LA|$K4XO"$7$?$$$/$D$+$NDjM}$r=R$Y$k!%$3$l$i$NDjM}$G$O!"4pK\E*$K$OM?$($i$l$?$M$8$l<LA|$NB>$N$I$N<~4|50F;$N<~4|$h$j!J$:$C$H!K>.$5$$<~4|$r;}$D<~4|50F;$O%P!<%3%U<~4|50F;$G$J$1$l$P$J$i$J$$$3$H$r=R$Y$k!%(B

\vspace{0.3cm}
\noindent
{\bf 2. $BDj5A$H5-K!(B.}

$A= \{ (x,y) \in {\bf R}^2: 0 \leq y \leq 1 \}$$B$H$7!"(B $\pi_1 {\rm (resp.} \ \pi_2) : A \rightarrow {\bf R}: (x,y) \rightarrow x  {\rm (resp.} \ y)$$B$rDL>o$N<M1F$H$7!"3F(B$\alpha \in {\bf R}$$B$KBP$7$F<!$N$h$&$K$*$/!%(B
$$
  l_{\alpha} = \{ (\alpha,y) \in A: 0 \leq y \leq 1 \} . 
$$

\vspace*{0.2cm}
\noindent
\underline{$BDj5A(B}. $B<LA|(B$f:A \rightarrow A$$B$,(B\underline{$B$M$8$l<LA|(B}$B$H8F$P$l$k$N$O<!$N>r7o$,K~$?$5$l$k$H$-$G$"$k(B:

(1) $f$$B$O(B$C^2$$BHyJ,F1Aj<LA|$G$"$C$F!"6-3&@.J,$*$h$SJ}8~$rJ]$D(B. 

(2) $B$9$Y$F$N(B$(x,y) \in A$$B$KBP$7$F!"(B$f((x,y))+(1,0)=f((x+1,y))$. 

(3) $\delta>0$$B$,$"$C$F!"$9$Y$F$N(B$z \in A$$B$KBP$7$F!"(B$\partial(\pi_1 \circ f)(z)/\partial y> \delta$.

\vspace*{2.5cm}    
\hspace*{5cm}   $B#1?^(B

\vspace*{2.8cm}
\noindent
\underline{$BCm0U(B}. $B>r7o(B(2)$B$O(B$f$$B$,(B$A$$B$rIaJWHoJ$$H$9$k$h$&$J1_4D>e$N<LA|$N;}$A>e$2$K$J$C$F$$$k$3$H$r=R$Y$F$$$k!%>r7o(B(3)$B$O!V$M$8$l>r7o!W$G$"$k!%$3$l$N0UL#$9$k$H$3$m$O!"G$0U$N(B$\alpha, \beta \in {\bf R}$$B$KBP$7$F!"(B$f(l_{\alpha}) \cap l_{\beta}$$B$,9b!90lE@$G$"$k$3$H$G$"$k!J#1?^;2>H!K!%$3$l$O$H$-$K!VC1D4!W$M$8$l>r7o$H$b8F$P$l$k!%(B

\vspace*{0.3cm}    
\noindent
\underline{$BDj5A(B}. $B$M$8$l<LA|(B$f:A \rightarrow A$$B$N$b$H$G$NE@(B$z \in A$$B$N(B\underline{$B50F;(B}$B$O=89g(B${\cal O}(f,z)= \{ f^k(z)+(l,0):k,l \in {\bf Z} \}$$B$HDj5A$5$l$k!%(B

\vspace*{0.3cm}    
\noindent
\underline{$BDj5A(B}. $BE@(B$z \in A$$B$,$M$8$l<LA|(B$f:A \rightarrow A$$B$N(B$p/q-$\underline{$B<~4|E@(B}$B$H8F$P$l$k$N$O<!$,@.$jN)$D$H$-$G$"$k!%(B
$$
  f^q(z) - (p,0) =z . 
$$
$BF1MM$K!"E@(B$z \in A$$B$,$M$8$l<LA|(B$f$$B$N(B$p/q-$\underline{$B%P!<%3%U<~4|E@(B}$B$H8F$P$l$k$N$O(B$z$$B$,(B$p/q-$$B<~4|E@$G$"$C$F!"$7$+$bG$0U$N(B$z_1,z_2$ $\in {\cal O}(f,z)$$B$KBP$7$F<!$,@.$jN)$D$H$-$G$"$k!%(B
$$
  \pi_1(z_1) < \pi_1(z_2) \Rightarrow \pi_1(f(z_1)) < \pi_1(f(z_2)).
$$

\vspace{0.3cm}
\noindent
\underline{$BCm0U(B}. $f$$B$O1_4D<LA|$N;}$A>e$2$G$"$k$+$i!">e$GDj5A$5$l$?!V50F;!W$*$h$S(B$p/q-$$B<~4|E@$NDj5A$O1_4D>e$NDL>o$N50F;$*$h$S<~4|E@$N;}$A>e$2$KBP1~$7$F$$$k$3$H$KCm0U$7$h$&!%(B$p/q-$$B%P!<%3%U<~4|E@$H$O(B$p/q-$$B<~4|E@(B$z$$B$N$&$A!"(B${\cal O}(f,z)$$B$K@)8B$7$?(B$f$$B$,(B$x$$B:BI8$K4X$7$F!V=g=xJ]B8!W$K$J$C$F$$$k$h$&$J$b$N$G$"$k!%(B

\vspace*{0.3cm}    
\noindent
\underline{$BDj5A(B}. $f:A \rightarrow A$$B$,$M$8$l<LA|$G$"$C$F(B$z \in A$$B$J$i!"(B$z$$B$K$*$1$k(B$f$$B$N(B\underline{$B2sE>?t(B}$B$O!"B8:_$9$k$H$9$l$P<!<0$GM?$($i$l$k(B:
$$
   \rho(f,z) = \displaystyle 
\lim_{n \rightarrow \infty} \frac{1}{n}(\pi_1(f^n(z))).
$$

\vspace*{0.3cm}    
\noindent
\underline{$B5-K!(B}. $B<LA|(B$f \mid_{\{ (x,i): x \in {\bf R} \} }, \ i=0,1$$B$O1_HyJ,F1Aj<LA|$N;}$A>e$2$G$"$k$+$i!"<!$N$h$&$K=q$/(B:
$$
  \rho_0(f)=\rho(f(x,0)) \hspace{0.5cm} {\rm $B$*$h$S(B} \hspace{0.5cm} 
\rho_1(f)=\rho(f(x,1)).
$$
$B$?$@$7!"$3$l$i$NDj5A$O(B$x \in {\bf R}$$B$K0MB8$;$:!"6K8B$,B8:_$9$k(B([12]$B;2>H(B)$B!%(B

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\noindent
{\bf 3. $B<gDjM}$NDD=R(B.}

$B$3$NO@J8$G$O<!$NDjM}$r>ZL@$9$k!%(B

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\noindent
{\bf $BDjM}(B 1.} $f: A \rightarrow A$$B$,$M$8$l<LA|$G(B$f$$B$K(B$p/q$-$B<~4|E@$,$"$l$P!"(B$p/q-$$B%P!<%3%U<~4|E@$,$"$k!%(B

\vspace{0.3cm}
$B$3$NDjM}$O(B$f$$B$,LL@QJ]B8$G$"$m$&$H$J$+$m$&$H@.$jN)$D!%(B$f$$B$,LL@QJ]B8$J$i!"$3$l$H0J2<$K=R$Y$k#2$D$N7k2L$r7k$S$D$1$F(BAubry-Mather$B$NDjM}$rF@$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B 2}($B%P!<%3%U(B[4],[5]. Chenciner[9]$B$b8+$h(B). $f: A \rightarrow A$$B$,LL@QJ]B8$M$8$l<LA|$G(B$p/q \in [\rho_0(f),\rho_1(f)]$$B$J$i!"(B$f$$B$K(B$p/q$-$B<~4|E@$,$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$BCm0U(B}. $B%P!<%3%U$NDjM}(B($B%]%"%s%+%l$N:G8e$N4v2?3XDjM}$H$7$FCN$i$l$F$$$k(B)$B$O<B:]$O$b$C$H<e$$$M$8$l>r7o$G$b@.$jN)$D!%(B

\vspace{0.3cm}
\noindent
{\bf $BJdBj(B 0}(Katok[14]). $f: A \rightarrow A$$B$,$M$8$l<LA|$G!"(B$\{ p_n/q_n \}_{n \geq 0}$$B$,(B$n \rightarrow \infty$$B$N$H$-(B$p_n/q_n \rightarrow \alpha \not\in {\bf Q}$$B$J$kM-M}?tNs$G!"3F(B$n$$B$KBP$7$F(B$f$$B$K(B$(p_n/q_n)-$$B%P!<%3%U<~4|E@(B$z_n$$B$,$"$l$P!"(B$\{ z_n \}_{n \geq 0}$$B$NG$0U$N6K8BE@(B$z$$B$O(B$\rho(f,z)=\alpha$$B$rK~$?$9!%(B

\vspace{0.3cm}
\noindent
\underline{$BCm0U(B}. $B<B$N$H$3$m!"(B$\{ z_n \}_{n \geq 0}$$B$N6K8BE@(B$z$$B$N50F;$K4X$7$F$O!"$b$C$H$?$/$5$s$N$3$H$,@.$jN)$D!%$3$N50F;$K@)8B$7$?<LA|(B$f$$B$O(B$x$$B:BI8$K4X$7$F=g=xJ]B8$G$"$j!"50F;$O<~4|E*%j%W%7%C%D4X?t$N%0%i%U>e$K$"$k(B([15],[14],[13]$B;2>H(B)$B!%$3$l$iIU2CE*$J;v<B$O$M$8$l<LA|$N4v2?3X$N$_$+$i=P$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B 3}(Aubry[2]$B$*$h$S(BMather[15]). $f: A \rightarrow A$$B$,LL@QJ]B8$M$8$l<LA|$G(B$\alpha \in [\rho_0(f),\rho_1(f)]$$B$J$i!"(B$\rho(f,z_{\alpha})=\alpha$$B$J$k(B$z_{\alpha} \in A$$B$,$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{ $BDjM}(B 3$B$N>ZL@(B}. $B%P!<%3%U$NDjM}(B($BDjM}(B2)$B$K$h$j!"3FM-M}?t(B$p/q \in [\rho_0(f),\rho_1(f)]$$B$KBP$7$F!"(B$z_{p/q} \in A$$B$,$"$C$F(B$f$$B$N(B$p/q-$$B<~4|E@$G$"$k!%DjM}(B1$B$h$j!"$3$N$H$-(B$p/q-$$B%P!<%3%U<~4|E@$,$"$k$O$:$H$$$&$3$H$,$o$+$k!%JdBj(B0$B$r%P!<%3%U<~4|E@Ns$KE,MQ$9$l$P!"(B$[\rho_0(f),\rho_1(f)]$$BFb$NG$0U$NL5M}?t$KBP$7$FL5M}?t2sE>?t$NE@$,F@$i$l$k!%$3$l$GDjM}$N>ZL@$,40N;$7$?!%(B\hfill $\Box$

\vspace*{0.3cm}
($B$3$l$i!V%P!<%3%UE@!W$N50F;$N$=$N$[$+$N@-<A$K4X$7$F$O!">e$NCm0U$d(BMather[15]$B!"(BKatok[14]$B!"$*$h$S(BHerman[13]$B$NO@J8$r8+$h!%(B)

$BDjM}(B1$B$N>ZL@$O<!$N$h$&$K?J$`(B: $p/q-$$B<~4|E@$r;}$D$M$8$l<LA|(B$f: A \rightarrow A$$B$,$"$k$H$9$k!%$3$N$H$-!"(B$f$$B$,(B$p/q-$$B%P!<%3%U<~4|E@$r;}$D$M$8$l<LA|(B$g: A \rightarrow A$$B$K%[%b%H!<%W$G$"$k$3$H$r<($9!%$=$N:]!"%[%b%H%T!<$K$*$1$k3F<LA|$O(B$p/q-$$B<~4|E@$r;}$D$b$N$K$9$k!%$9$k$H!"(B$p/q-$$B%P!<%3%U<~4|E@$r;}$D<LA|$N=89g$O!"$3$N%[%b%H%T!<$N%Q%i%a!<%?!<Cf$G3+$+$DJD$N=89g$+$i@.$k$3$H$,$o$+$k!%JD$8$F$$$k$3$H$O%P!<%3%U<~4|E@$NDj5A$+$iMF0W$K=P$k$7(B($B2<$NJdBj(B1$B$r8+$h(B)$B!"3+$G$"$k$3$H$O<!@a$NITF0E@JdBj(B($BJdBj(B4)$B$+$i=P$k!%4pK\E*$K$O!"$3$NJdBj$O!"(B$p/q-$$B%P!<%3%U<~4|50F;$*$h$SJL$N(B$p/q-$$B<~4|50F;$r;}$D<LA|$,<B$O#2$D$N(B$p/q-$$B%P!<%3%U<~4|50F;$r;}$A!"$=$N$&$($3$NBhFs$N50F;$O>.$5$J@]F0$N$b$H$G@8$-;D$k$O$:$G$"$k$3$H$r=R$Y$F$$$k!%(B

$B<!@a$K$O#5@a$G=R$Y$kDjM}(B1$B$N>ZL@$KI,MW$J$$$/$D$+$NJdBj$,$"$k!%#6@a$G$OF1MM$N$7$+$b$b$C$H4JC1$J5;=Q$r;H$C$F!"0J2<$N$3$H$r<($9!%(B$f: A \rightarrow A$$B$,$M$8$l<LA|$G!"(B$f$$B$,0?$k8_$$$KAG$J@0?t(B$p,q$$B$KBP$7$F!">r7o(B
$$
  (\dag)$B!!(Bf{\rm $B$N$I$N(B} r/s-{\rm $B<~4|E@$b(B} s=q {\rm $B$^$?$O(B} s>(3/2)q
$$
$B$rK~$?$9$H$-!"(B$f$$B$N$I$N(B$p/q-$$B<~4|E@$b(B$p/q-$$B%P!<%3%U<~4|50F;$G$"$k!%(B$p/q,$ $\rho_0(f)$$B$*$h$S(B$\rho_1(f)$$B$K4X$9$k$"$k>r7o$rM?$($k$3$H$,$G$-$k!%$=$N>r7o$OK\<AE*$K$O<LA|$,Bg$-$/$M$8$l$F$$$J$$$3$H$r8@$&$N$@$,!"$=$l$+$i>e$N>r7o(B$(\dag)$$B$,=P$k!%LL@QJ]B8$M$8$l<LA|$KBP$7$F$O!"(B$(\dag)$$B$r<!$N>r7o$HF~$l49$($F!"DjM}$r2~A1$G$-$k!%(B
$$
  (\ddag)$B!!(Bf {\rm $B$N$I$N(B} r/s-{\rm $B<~4|E@$b(B} s \geq q {\rm $B$rK~$?$9(B}.
$$
$B$3$N$H$-$I$N(B$p/q-$$B<~4|50F;$b(B$p/q-$$B%P!<%3%U<~4|50F;$G$"$k!"$H$$$&F1$87kO@$,F@$i$l$k!%(B

\vspace*{0.3cm}
\noindent
\underline{$BCm0U(B}. 
A. Katok$B$,EA$($F$/$l$?$H$3$m$K$h$k$H!"(BD. Bernstein$B$O:G6a!"DjM}(B3$B$NJQ7AHG$N>ZL@$rF@$?!%H`$OLL@QJ]B8@-$NBe$o$j$K!V1_8r:9@-!W$r;H$C$?(B([3]$B;2>H(B)$B!%$^$?(BP. Carter[8]$B$OJ]B,@-$NBe$o$j$K1_8r:9@-$rMQ$$$FDjM}(B2$B$K;w$?7k2L$rF@$F$$$k!%(B
