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\begin{flushright}
$@C+@n@6N4(B $@Lu(B
\end{flushright}

\noindent
{\it Ergod. Th. \& Dynam. Sys.} {\bf 2} (1982), 185-194

\begin{center}
{\large
{\bf $@%P!<%3%U$*$h$S%^%6!<$N$M$8$l<LA|DjM}$K4X$9$kCm0U(B \\
Some remarks on Birkhoff and Mather twist map theorems} 
}

\vspace{1cm}
A.Katok
\end{center}

\vspace{0.5cm}
\noindent
{\bf $@MWLs(B.} 

$@$M$8$l<LA|$N=`<~4|50F;$NB8:_$K4X$9$k(BJ.Mather[1]$@$N:G6a$N7k2L$O!"<~4|50F;$K4X$9$k%P!<%3%U(B(G.D.Birkhoff)$@$N8EE5E*DjM}$r$&$^$/=$@5$7$FF3$-=P$;$k!%(BMather$@$N5DO@$r4v2?3X$K4JC12=$7$?$b$N$r;H$C$F%P!<%3%U$NDjM}$r>ZL@$9$k!%(BMather$@$NITJQ=89g$NJL$N@-<A$K$D$$$F$b5DO@$9$k!%(B

\vspace{0.5cm}
\noindent
{\bf 1. $@5-K!(B.}
$$
   A = S^1 \times [0,1] = \{ (\phi,r), \phi \in {\bf R}/{\bf Z}, 
0 \leq r \leq 1 \} , 
$$
$@$rI8=`E*1_4D$H$7!"(B
$$
$@!!(BS = {\bf R} \times [0,1] = 
\{ (x,y), x \in {\bf R}, 0 \leq y \leq 1 \} , 
$$
$@$r$=$NIaJWHoJ$$H$7!"(B$T: S \rightarrow S$$@$rC10LJ?9T0\F0(B
$$
   T(x,y) = (x+1, y),
$$
$@$H$9$k!%(B

$@G$0U$NF1Aj<LA|(B$f: A \rightarrow A$$@$KBP$7$F!"$=$N(B$S$$@$X$N;}$A>e$2$O(B$T$$@$N%Y%-$NITDj@-$r=|$$$FDj5A$5$l$k!%5U$K!"(B$F:S \rightarrow S$$@$G!"(B$F$$@$,(B$T$$@$H2D49$J$i(B$F$$@$O(B$A$$@$N$"$kF1Aj<LA|$N;}$A>e$2$G$"$k!%(B$F$$@$r<!$N$h$&$J:BI8$G=q$3$&!%(B
$$
    F(x,y) = (F_1(x,y), F_2(x,y)).
$$

$@F1Aj<LA|(B$f: A \rightarrow A$$@$,(B\underline{$@$M$8$lF1Aj<LA|(B}($@$^$?$O(B\underline{$@$M$8$l<LA|(B})$@$H8F$P$l$k$N$O!"$=$l$,J}8~J]B8$G!"(B$A$$@$N6-3&$rJ]B8$7!"$^$?(B$f$$@$N;}$A>e$2(B$F$$@$*$h$SG$0U$N(B$x \in {\bf R}$$@$KBP$7$F4X?t(B$F_1(x,y)$$@$,(B$y$$@$N695AC1D44X?t$G$"$k$H$-$G$"$k!%(B

$@L@$i$+$K!"0[$J$k(B$x$$@$KBP$9$k4X?t(B$F_1(x,y)$$@$O$9$Y$FA}2C$G$"$k$+8:>/$G$"$k$+$I$A$i$+$G$"$k!%BP1~$7$F1&$M$8$l<LA|$H$+:8$M$8$l<LA|$H$$$&$3$H$K$9$k!%(B$f$$@$,1&$M$8$l<LA|$J$i!"(B$f^{-1}$$@$O:8$M$8$l<LA|$G$"$k!%5U$b@.$jN)$D!%$O$C$-$j$5$;$k$?$a!"(B$f$$@$O1&$M$8$l<LA|$G$"$k$H$7$h$&!%(B

$@$M$8$l<LA|(B$f$$@$K<!$NBP>]$r7k$S$D$1$k!%(B

$@$^$:(B\underline{$@$M$8$l6h4V(B}$[\alpha_0(f),\alpha_1(f)]$$@!%$3$3$G(B$\alpha_0(f)$$@$H(B$\alpha_1(f)$$@$O(B$F$$@$r(B${\bf R} \times \{ 0 \}$$@$*$h$S(B${\bf R} \times \{ 1 \}$$@$K$=$l$>$l@)8B$7$?$H$-$N2sE>?t$G$"$k!%$M$8$l6h4V$O@0?tJ?9T0\F0$NITDj@-$r=|$$$FDj5A$5$l$k!%(B

$@<!$K(B\underline{$@$M$8$l(Bmodulus}$\omega_f(r)$$@!%$3$l$O(B$0 \leq r \leq 1$$@$KBP$7$F!"<!$GDj5A$5$l$k!%(B
$$
  \omega_f(r) = \min_{0 \leq x \leq 1} \min_{0 \leq y \leq 1 - r} 
(F_1(x, y+r) - F_1(x,y), \hat{F}_1(x,y) - \hat{F}_1(x, y+r)) . 
$$
$@$3$3$G(B$\hat{F}_1$$@$O(B$f^{-1}$$@$N;}$A>e$2$NBh0l:BI8$G$"$k!%(B

$f$$@$,$M$8$l<LA|$G!"2C$($F(B$f$$@$H(B$f^{-1}$$@$,%j%W%7%C%D<LA|$G!"$"$k(B$K$$@$KBP$7$F(B$\omega(r) \geq Kr$$@$J$i!"(B$f$$@$r(B\underline{$@%j%W%7%C%D$M$8$l<LA|(B}$@$H8F$V!%$?$H$($P!"(B$\partial F_1 / \partial y > 0$$@$J$kG$0U$N(B$C^1$$@HyJ,F1Aj(B$f$$@$O%j%W%7%C%D$M$8$l<LA|$G$"$k!%(B

\vspace{0.5cm}
\noindent
{\bf 2. $@%P!<%3%U<~4|50F;(B}

$f^qz=z$$@$J$i!"(B$z$$@$NG$0U$N;}$A>e$2(B$w$$@$KBP$7$F(B$F^q w =T^p w$$@$,@.$jN)$D!%$?$@$7(B$p$$@$O(B$q$$@$N@0?tG\$NITDj@-$rM-$9$k!%$@$+$iJ,?t(B$p/q$$@$O@0?tJ,$@$1ITDj@-$rM-$9$k!%$3$NJ,?t$r<~4|E@(B$z$$@$N(B\underline{$@2sE>?t(B}$@$H8F$\$&!%(B

$@E@(B$z$$@$,(B$(p,q)$\underline{$@7?%P!<%3%UE@(B}$@$H8F$P$l$k$N$O!"(B$z$$@$N;}$A>e$2(B$w$$@$KBP$7$F!"<LA|(B
$$
 \theta = ( \psi, \rho ) : {\bf Z} \rightarrow S 
$$
$@$,$"$C$F<!$,K~$?$5$l$k$H$-$G$"$k!%(B
$$ \begin{array}{c}
 \theta(0) = w , \\ 
\psi $@$O695AC1D44X?t(B , \\ 
\theta(n+q) = T \theta(n) , \\ 
\theta(n+p) = F \theta(n) . \\ 
   \end{array}  
$$
$(p,q)$$@7?%P!<%3%UE@$N50F;$r(B$(p,q)$\underline{$@7?%P!<%3%U<~4|50F;(B}$@$H8F$V!%(B

$@L@$i$+$K!"G$0U$N(B$(p,q)$$@7?%P!<%3%UE@$O2sE>?t(B$p/q$$@$N<~4|E@$G$"$k!%$=$N>e!"$3$N$h$&$JE@$KBP$7$F(B
$$ \begin{array}{ll}
$@!!!!!!!!(Bf^nz=(\phi_n, r_n), & n=0,1,...,q-1
   \end{array}
$$
$@$H=q$1$P!"(B$\phi_n$$@$O$9$Y$F0[$J$j!"1_<~>e$N(B$2 \pi p/q$$@$:$D$N2sE>$K$h$kA|$HF1$8=g=x$GJB$s$G$$$k!%$7$+$7!"$@$+$i$H8@$C$F$3$NFs$D$N@-<A$@$1$+$i$O(B$z$$@$,(B$(p,q)$$@7?%P!<%3%UE@$G$"$k$H$O8@$($J$$!%$3$H$K!"0J2<$NL?Bj(B1$@$N<gD%$O$3$NFs$D$N@-<A$+$i$O=P$F$3$J$$!%(B

$\beta(r)$$@$r(B$f$$@$H(B$f^{-1}$$@$N6&DL$NO"B3N((B(modulus of continuity)$@$H$9$k!%(B$\beta$$@$OFL4X?t$G$"$k$H2>Dj$9$k!%(B

\vspace{0.3cm}
\noindent
{\bf $@L?Bj(B1.} $f$$@$r$M$8$l<LA|$H$9$k!%$3$N$H$-1z$JC1D44X?t(B$\gamma(r)$$@$G0J2<$N>r7o$rK~$?$9$b$N$,B8:_$9$k!%(B$\gamma(r)$$@$O(B$r$$@$,@5$N$H$-@5$G$"$j!"(B$\omega_f$$@$H(B$\beta$$@$N$_$K0MB8$7!"3F%P!<%3%U<~4|50F;$KBP$7$F(B
$$
       \mid \phi_n - \phi_m \mid < r,
$$
$@$J$i$P(B
$$
       \mid r_n -r_m \mid < \gamma(r),
$$
$@$,@.$jN)$D!%(B

$@$H$/$K(B$f$$@$,%j%W%7%C%D$M$8$l<LA|$N$H$-!"$"$k(B$L$$@$r<h$C$F(B$\gamma(r)$$@$r(B$Lr$$@$H$G$-$k!%(B

\vspace*{0.3cm}
\noindent
\underline{$@>ZL@(B}. $@IaJWHoJ$LL>e$G9M$($k!%(B$\psi$$@$OA}2C4X?t$G$"$k$H$7!"(B$m,$ $n(m<n)$$@$r@0?t$H$9$k!%$h$C$F(B$\psi(m) < \psi(n)$$@$G$"$k!%(B$\rho(n) < \rho(m)$$@$H2>Dj$7!"(B
$$
$@!!!!!!!!(BF(\psi(n),\rho(m)) = (\tilde{\psi},\tilde{\rho})
$$
$@$H=q$/!%(B
$$
$@!!(BF(\psi(n),\rho(n))=(\psi(n+p),\rho(n+p))             \eqno(1)
$$
$@$G$"$j!"$^$?(B$\rho(n) < \rho(m)$$@$@$+$i!"$M$8$l>r7o$h$j(B
$$
$@!!(B\tilde{\psi} > \psi (n+p)+\omega_f (\rho (m) - \rho (n))  \eqno(2)
$$
$@$rF@$k!%0lJ}(B
$$
$@!!(B\psi (n+p) > \psi (m+p) > \tilde{\psi} - \beta (\psi (n) - \psi (m) ) \eqno(3)
$$
$@$G$"$k!%(B(1),(2),(3)$@<0$h$j(B
$$
    \omega_f(\rho (m) - \rho (n)) < \beta (\psi (n) - \psi (m))
$$
$@$rF@$k!%$@$+$i!"(B$\gamma = \tilde{\omega}^{-1} \circ \beta$$@$H$*$1$k!%$?$@$7!"$3$3$G(B$\tilde{\omega}$$@$O(B$r > 0$$@$N$H$-(B
$$
   0 < \tilde{\omega}(r) \leq \omega_f(r)
$$
$@$rK~$?$9695AC1D4FL4X?t$G$"$k!%(B

$\rho(n) > \rho(m)$$@$N>l9g$O(B$f$$@$NBe$o$j$K(B$f^{-1}$$@$rMQ$$$FF1$85DO@$r$9$l$P$h$$!%(B

$@%j%W%7%C%D$N>l9g$O$"$kDj?t(B$M$$@$rMQ$$$F(B$\beta(r)=Mr$$@$H<h$l!"(B$\tilde{\omega}(r)=Kr$$@$G$"$k$+$i!"(B$\gamma(r)=MK^{-1}r$$@$G$"$k!%(B


\vspace{0.5cm}
\noindent
{\bf 3. $@%^%6!<=89g(B(Mather sets)}

$f$$@ITJQ$JJD=89g(B$E \subset A$$@$,<!$NFs$D$N>r7o$rK~$?$9$H$-!"(B$E$$@$r(BMather$@=89g$H8F$V!%(B

(i) $E$$@$O3F6h4V(B$ \{ \phi \} \times [0,1]$$@$H9b!90lE@$G8r$o$k!%$9$J$o$A!"(B$\Phi$$@$,(B$S^1$$@$NJD=89g(B$K$$@>e$GDj5A$5$l$?O"B34X?t$G(B$[0,1]$$@$KCM$r$H$k$b$N$H$7$F!"(B$E= {\rm graph} \Phi$$@$H=q$1$k!%(B

(ii) $F$$@$O(B$E$$@$NHoJ$>e$N=g=x$rJ]$D!%(B

$f|_E$$@$,(B$E$$@>e$N<+A3$J1_=g=x$rJ]B8$9$k$3$H!"$^$?=g=xJ]B8$NF1Aj<LA|$rDL$7$F!"1_$NF1Aj<LA|$r=89g(B$K$$@$X@)8B$7$?$b$N$K0LAj6&Lr(B(topologically conjugate)$@$G$"$k$3$H$OMF0W$K$o$+$k!%$3$H$K!"2sE>?t(B$\rho (E)$$@$O@0?t$NITDj@-$rM-$9$k!%(B

Mather$@=89g$N$I$NJDItJ,=89g$b(BMather$@=89g$G$"$k!%$H$/$K!"$3$N$h$&$J=89g$O$I$l$b6K>.=89g$r4^$`!%$7$?$,$C$F!"1_F1Aj<LA|$NI8=`E*$J%]%"%s%+%l!&%@%s%8%g%"M}O@$h$j#3<oN`$N6K>.%^%6!<=89g$N$"$k$3$H$,$o$+$k!%$9$J$o$A!"(B

$\rho (E)$$@$,M-M}?t$GNc$($P(B$p/q$$@$KEy$7$$$H$-!"(B$E$$@$O(B$(p,q)$$@7?$N%P!<%3%U<~4|50F;$G$"$k!%(B

$\rho (E)$$@$,L5M}?t$N$H$-(B$E$$@$O1_$=$N$b$N$G$"$C$F!"(B$f_E$$@$O2sE>(B$\rho (E)$$@$K6&Lr$G$"$k$+!"(B

$E$$@$O%+%s%H!<%k=89g$G$"$C$F!"(B$f_E$$@$O!"$$$o$f$k%@%s%8%g%"$NH?Nc$N0l$D$KBP$9$k6K>.=89g$KF1CM$G$"$k!%(B

\vspace{0.3cm}
\noindent
{\bf $@L?Bj(B2.} $@G$0U$N(BMather$@=89g(B$E$$@$KBP$7$F!"4X?t(B$\Phi$$@$NO"B3N((B$\gamma$$@$OL?Bj(B1$@$G=R$Y$i$l$?@-<A$rM-$9$k!%(B

\vspace{0.3cm}
$@>ZL@$OL?Bj(B1$@$N>ZL@$NC1$J$k7+$jJV$7$G$"$k!%$H$$$&$N$O!"$3$N>ZL@$K:]$7$F!"50F;$N<~4|@-$O=EMW$G$J$/!"IaJWHoJ$LL>e$G$N=g=x$NJ]B8$@$1$,LdBj$H$J$k$+$i$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$@Cm0U(B}. $@%P!<%3%U$NDjM}$+$i%^%6!<$NDjM}$rF3$/$N$KL?Bj(B2$@$OI,MW$J$$!%(B

\vspace{0.3cm}
$@%3%s%Q%/%H$J5wN%6u4V(B$X$$@$N$9$Y$F$NJDItJ,=89g$+$i$J$k6u4V>e$N%O%&%9%I%k%U0LAj$NDj5A$r;W$$5/$3$=$&!%(B$U_{\varepsilon} (E)$$@$O(B$E \subset X$$@$N3+(B$\varepsilon-$$@6aK5$H$9$k!%%O%&%9%I%k%U0LAj$G$N(B$E$$@$N6aK5$N4p$O=89gC#(B
$$
$@!!(BV_{\varepsilon} (E) = \{ F \subset X: F \subset U_{\varepsilon}(E), E \subset U_{\varepsilon} (F) \}$@!!(B
$$
$@$+$i$J$k!%(B

\vspace{0.3cm}
\noindent
{\bf $@L?Bj(B3.} (a) $@$M$8$lF1Aj<LA|$N%^%6!<=89g$N$9$Y$F$N=89g$O%O%&%9%I%k%U0LAj$GJD$8$F$$$k!%(B

(b) $@%^%6!<=89g(BE$@$N2sE>?t(B$\rho (E)$$@$O%O%&%9%I%k%U0LAj$GO"B3$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$@>ZL@(B}. $\Phi_n:K_n \rightarrow [0,1]$$@$H$7$F!"(B$E_n={\rm graph} \Phi_n$$@$O(B$f$$@$N%^%6!<=89g$NNs$G$"$C$F!"%O%&%9%I%k%U0LAj$G=89g(B$E$$@$K<}B+$9$k$H$9$k!%(B
$$
$@!!(B  z=(\phi^0, r^0) \in E, z_n = (\phi^n, r^n) \in E_n, \ 
z_n \rightarrow z
$$
$@$H$7!">.$5$$?t(B$\varepsilon >0$$@$r8GDj$9$k!%L?Bj(B2$@$h$j(B
$$
\begin{array}{c}
\displaystyle 
$@!!!!(BE_n \cap [ \phi^n - \varepsilon ; \phi^n + \varepsilon ] \times S^1 \subset \{ ( \phi, r): \phi^n - \varepsilon \leq \phi \leq \phi^n + \varepsilon, \\
\displaystyle 
    r^n - \gamma (\mid r -r^n \mid) \leq r \leq r^n + 
\gamma ( \mid r-r^n \mid ) \} , 
\end{array}
$$
$@$G$"$j!"%O%&%9%I%k%U0LAj$G$N<}B+@-$K$h$C$F(B
$$
\begin{array}{c}
$@!!(BE \cap [ \phi^0 - \varepsilon ; \phi^0 + \varepsilon ] \times S^1
 \subset \{ ( \phi, r) : \phi^0 - \varepsilon \leq \phi \leq \phi^0 + \varepsilon, \\
  r^0 - \gamma ( \mid r-r^0 \mid ) \leq r \leq r^0 + 
\gamma ( \mid r-r^0 \mid ) \} , 
\end{array}
$$
$@$,J]>Z$5$l$k!%$3$N4X78$+$i=89g(BE$@$KBP$9$k>r7o(B(i) $@$,$G$k!%(B$E_n$$@$KBP$9$k$3$N>r7o$H<}B+@-$+$i$?$@$A$K>r7o(B(ii) $@$,$G$k!%$3$l$G(B(a) $@$,>ZL@$5$l$?!%(B

$@<!$K!"(B$\rho$$@$rNs(B$\rho (E_n)$$@$NG$0U$N6K8BE@$H$9$k!%Nc$($P!"(B$\rho (E_{n_k}) \rightarrow \rho$$@$H$9$k!%<!$N$h$&$K=q$3$&!%(B
$$
$@!!(Bf^m(z_{n_k})=( \phi^k_m,r^k_m), f^m(z)=(\phi_m, r_m) . 
$$
$\rho (E)= \rho$$@$r<($9$?$a$K$O!"1_<~>e$N(B$\phi_m$$@$N=g=x$,!"$"$kE@$N(B$2 \pi \rho$$@$:$D$N2sE>$NA|$N=g=x$HF1$8$G$"$k$3$H$r<($;$P$h$$!%G$0U$K8GDj$7$?(B$N$$@$KBP$7$F!"$&$^$/(B$k(N)$$@$r$H$l$P!"(B$k \geq k(N)$$@$KBP$7$F$O!":G=i$N(B$N$$@8D$N(B$2 \pi \rho (E_{n_k})$$@$:$D$N2sE>A|$N=g=x$,(B$2 \pi \rho$$@$:$D$N2sE>A|$N=g=x$HF1$8$K$G$-$k!%=89g(B$E_{n_k}$$@$KBP$9$k@-<A(B(ii)$@$K$h$C$F!"(B$\phi^k_m$$@$N=g=x$O(B$2 \pi \rho (E_{n_k})$$@$:$D$N2sE>A|$N=g=x$HF1$8$G$"$k!%$H$3$m$,!"3QEY(B$\phi_m,$ $m=0, \cdots,N-1$$@$O$9$Y$FCM$,8_$$$K0[$J$k!%$@$+$i!"==J,Bg$-$J(B$k$$@$KBP$7$F!"(B$\phi^k_m$$@$H(B$\phi_m$$@$N=g=x$O(B$m=0, \cdots, N-1$$@$G0lCW$9$k!%(B$N$$@$O$$$/$i$G$bBg$-$/$H$l$k$+$i!"(B$\rho (E)= \rho$$@$G$"$k!%$h$C$F(B(b)$@$b>ZL@$5$l$?!%(B \hfill $\Box$

\vspace{0.3cm}
$@$3$NL?Bj$H%O%&%9%I%k%U0LAj$N%3%s%Q%/%H@-$h$j!"$?$@$A$K$D$.$,=P$k!%(B

\vspace*{0.3cm}
\noindent
{\bf $@7O#1(B.} $@$9$Y$F$N%^%6!<=89g$NOB$OJD(B$f-$$@ITJQ=89g$G$"$k!%(B

\vspace*{0.3cm}
$@$3$N=89g$r(B$M(f)$$@$HI=$9!%$3$N=89g$N@-<A$N$$$/$D$+$K$D$$$F8e$K5DO@$9$k!%(B

\vspace*{0.3cm}
\noindent
{\bf $@7O(B2.} $f$$@$,$M$8$l6h4VFb$NG$0U$NM-M}?t(B$p/q$$@$KBP$7$F(B$(p,q)$$@7?$N%P!<%3%U50F;$r;}$F$P!"(B$f$$@$O$M$8$l6h4VFb$NG$0U$NL5M}?t$KBP$7$F6K>.%^%6!<=89g$r;}$D!%(B

\vspace*{0.3cm}
$@7O(B2$@$rLL@QJ]B8$M$8$lF1Aj<LA|$N>l9g$K<gD%$7$F$$$k$N$,%^%6!<$NDjM}(B([1])$@$NFbMF$G$"$k!%$@$+$i!"2f!9$O!"$3$NDjM}$r%P!<%3%U<~4|50F;$NB8:_$NLdBj$X5"Ce$5$;$?$3$H$K$J$k!%(B

\vspace*{0.3cm}
\noindent
\underline{$@LdBj(B}. $@<~4|50F;$O(B$M(f)$$@Fb$GcGL)$+(B? 

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

\noindent
{\bf $@DjM}(B.} $f$$@$O3+=89g>e$G@5$JB,EY$rJ]B8$9$k$M$8$lF1Aj<LA|$H$7!"(B$p/q$$@$O$M$8$l6h4V$KB0$9$k$H$9$k!%$3$N$H$-!"(B$f$$@$O(B$(p,q)$$@7?$N%P!<%3%U<~4|50F;$r;}$D!%(B

$@2C$($F!"(B$f$$@$,(B$C^1$$@HyJ,F1Aj<LA|$G$"$C$F!"@5$N$J$a$i$+$JL)EY$GM?$($i$l$kB,EY$rJ]B8$9$k$J$i!"F1$82>Dj$N2<$G(B$f$$@$O(B2$@8D$N0[$J$k(B$(p,q)$$@7?$N%P!<%3%U<~4|50F;$r;}$D!%$3$N(B2$@8D$N50F;$O%^%6!<=89g$r9=@.$9$k!%(B

\vspace*{0.3cm}
\noindent
\underline{$@>ZL@(B}. $@5-K!$NJX59$N$?$a$K!"1_4D$NBe$o$j$K!"$=$NIaJWHoJ$LL>e$G9M$($k!%(B $F_1(x,0)=g_0(x)$$@$*$h$S(B$F_1(x,1)=g_1(x)$$@$H=q$/!%0J2<$N$h$&$JHs8:>/<LA|(B$\phi :{\bf Z} \rightarrow {\bf R}$$@$r$H$j!"$3$N<LA|A4BN$N6u4V$r9M$($k!%$=$N<LA|$H$O(B
$$
$@!!!!!!(B\phi (n+q) = \phi (n)+1,
$$
$@$*$h$S(B
$$
$@!!(Bg_0(\phi (n) ) \leq \phi (n+p) \leq g_1( \phi (n)).$@!!!!!!!!(B\eqno(5)
$$
$@$5$i$K$I$N(B$k$$@$KBP$7$F$b!"$3$N$h$&$J<LA|(B$\phi (n)$$@$H(B$\phi (n)+k$$@$rF10l;k$9$k!%0x;R6u4V(B(factor space)$@$r(B$\Phi_{p,q}$$@$GI=$9!%(B

$\Phi_{p,q}$$@>e$N<+A3$J%3%s%Q%/%H0LAj$O(B
$$
$@!!(B\Phi_{p,q} \rightarrow {\bf R}^q/{\bf Z} \simeq {\bf R}^{q-1} \times 
{\bf R}/{\bf Z}$@!!(B    
$$
$@$J$kKd$a9~$_$+$iF@$i$l$k!%$3$NKd$a9~$_$K$h$C$F!"<LA|(B$\phi$$@$K%Y%/%H%k(B
$(\phi (0), \cdots, \phi (q-1))$$@$,3d$jEv$F$i$l$k!%(B${\bf R}^q$$@$K$*$1$k0x;R2=$OBP3Q@~>e$N@0?t0\F0(B
$$
$@!!!!!!(B(x_1, \cdots, x_q) \rightarrow (x_1+n, \cdots, x_q+n), n \in {\bf Z}
$$
$@$GDj5A$5$l$k!%(B

$\Phi_{p,q}$$@$K$*$1$kF10l;k$,$3$N0x;R2=$KBP1~$9$k$3$H$OMF0W$K$o$+$k!%%3%s%Q%/%H@-$O(B(5)$@$+$i$G$k!%(B

$\Phi_{p,q}$$@$,6u$G$J$$$3$H$OMF0W$K$o$+$k!%$=$N>e!"$I$N(B$x \in {\bf R}$$@$KBP$7$F$b(B$\phi (0)=x$$@$J$k(B$\phi \in \Phi_{p,q}$$@$,$"$k!%<B:]!"(B$g_t(x),$ $0 \leq t \leq 1$$@$O(B$g_0$$@$H(B$g_1$$@$r7k$V(B${\bf R}$$@$NF1Aj<LA|$NC1D4$JB2$G$"$k$H$9$k!%(B$\alpha_0(f) < p/q$$@$+$D(B$\alpha_1(f) > p/q$$@$G$"$k$+$i!"(B
$$ \begin{array}{cc}
         g^q_0(x)-x < p, & g^q_1(x)-x > p
   \end{array}
$$
$@$rF@!"$h$C$F$"$k(B$t \in [0,1]$$@$KBP$7$F!"(B$g^q_t(x)=x+p$$@$G$"$k!%$3$N$H$-(B$\phi (n)=g^n_t(x)$$@$GDj5A$5$l$k(B$\phi$$@$O(B$\Phi_{p,q}$$@$KB0$9$k!%(B

$x,x' \in {\bf R}$$@$+$D(B$g_0(x) \leq x \leq g_1(x)$$@$H$9$k!%$M$8$l@-$K$h$j!"6h4V(B$I = \{ x \} \times [0,1]$$@$NA|$O6h4V(B$I' = \{ x' \} \times [0,1]$$@$H$A$g$&$I0lE@!"Nc$($P(B$(x',h(x,x'))$$@!"$G8r$o$k!%(B$S$$@$NDlJU6-3&(B$S_0$$@!"6h4V(B$I'$$@!"$*$h$S6J@~(B$F(I)$$@$K$h$C$F0O$^$l$k!V;03Q7A!W$r(B$T(x,x')$$@$H=q$3$&!%(B

$\phi \in \Phi_{p,q}$$@$KBP$7$F<!$rDj5A$9$k!%(B
$$
$@!!!!!!!!(BL_{p,q}( \phi )= \displaystyle \sum^{q-1}_{n=0} \mu 
(T( \phi (n), \phi (n+p))) . 
$$
$@$3$3$G!"(B$\mu$$@$O(B$f$$@$N$b$H$GITJQ$+$D3+=89g>e$G@5$NB,EY$N(B$S$$@$X$N;}$A>e$2$G$"$k!%(B

\vspace*{1cm}
\hspace*{6cm} $@?^(B1

\vspace{2.5cm}
$@$5$i$K$$$/$D$+5-9f$,I,MW$G$"$k!%$9$J$o$A(B
$$
\begin{array}{l}
    h_1(n)=h( \phi (n-p), \phi (n)), \\
    \psi_1(n)=( \phi (n), h_1(n)), \\
    \psi_2(n)= F^{-1} \psi_1(n+p) \stackrel{\rm def}{=} 
( \phi (n), h_2(n)). 
\end{array}
$$
$@$3$N$H$-(B
$$
$@!!!!!!!!(Bh_1(n)=h_2(n) ,           \eqno(6)
$$
$@$J$i<!$,@.$jN)$D!%(B
$$
$@!!!!!!!!!!(B F \psi_1(n)= \psi_1(n+p) . 
$$
$@$@$+$i$I$N(B$n$$@$KBP$7$F$b(B(6)$@$,K~$?$5$l$l$P!"<LA|(B$\psi_1:{\bf Z} \rightarrow {\bf R}$$@$O2sE>?t(B$p/q$$@$N<~4|50F;$rDj5A$9$k!%<B$O!"$3$l$O(B$(p,q)$$@7?$N%P!<%3%U50F;$G$"$k!%$H$$$&$N$O!"$=$&$G$J$$$H!"(B$\phi$$@$O695AC1D4$G$J$/$J$j!"(B$\phi (n)= \phi (n+1)$$@$+$D(B$h_1(n) \not= h_1(n+1)$$@$J$k(B$n$$@$,B8:_$9$k$3$H$K$J$k!%(B$\phi (n-p) \leq \phi (n-p+1)$$@$G$"$k$+$i!"$M$8$l@-$K$h$j!"(B$h_1(n) > h_1(n+1)$$@$,F@$i$l$k$,!"$=$N$H$-$K$O:F$S$M$8$l@-$K$h$j!"(B$\phi (n+p) > \phi (n+p+1)$$@$,F@$i$l!"(B$\phi$$@$,Hs8:>/$G$J$/$J$k!%$3$&$7$F!"DjM}$NBh0l$N<gD%$O(B$\Phi_{p,q}$$@$N%3%s%Q%/%H@-$*$h$S<!$NJdBj$+$i=P$k!%(B

\vspace{0.3cm}
\noindent
{\bf $@JdBj(B.} $@HF4X?t(B$L_{p,q}$$@$,(B$\phi$$@$K$*$$$F6I=jE*6K>.$K$J$C$F$$$l$P!"$3$N(B$\phi$$@$KBP$7$F(B
$$
    h_1(n)=h_2(n) , 
$$
$@$,$9$Y$F$N(B$n$$@$KBP$7$F@.$jN)$D!%(B

\vspace{0.3cm}
\noindent
\underline{$@>ZL@(B}. $@$"$k(B$n$$@$KBP$7$F(B
$$
$@!!(B\phi (n-1) < \phi (n) < \phi (n+1) \ {\rm $@$+$D(B} \ 
h_1(n) \not=h_2(n),
$$
$@$,@.$jN)$D>l9g$r$^$:9M$($h$&!%$3$N$H$-(B
$$
$@!!!!!!!!(B1 \geq h_1(n) > h_2(n) \geq 0 ,       \eqno(7)
$$
$@$^$?$O(B
$$
$@!!!!!!!!(B1 \geq h_2(n) > h_1(n) \geq 0 ,        \eqno(8)
$$
$@$N$I$A$i$+$,@.$jN)$D!%Bh0l$N>l9g$r9M$($k!%(B
$$
$@!!!!(BF( \phi (n), h_2(n))=( \phi (n+p), h_1(n+p)) , 
$$
$@$G$"$k$+$i!"(B(7)$@$h$j(B
$$
$@!!!!!!!!!!(Bh_1(n+p) < 1.$@!!(B     \eqno(9)
$$
$\varepsilon > 0$$@$H$7$F<!$N4X?t$rDj5A$7$h$&!%(B
$$ \begin{array}{rl} 
  \tilde{\phi}_{\varepsilon}(m) = \left\{ \right. & 
       \begin{array}{ll}
$@!!!!!!(B  \phi(m),   &  {\rm if} \ m \neq n( \bmod \ q), \\
$@!!!!!!(B  \phi(m) - \varepsilon,   &  {\rm if} \ m = n( \bmod \ q).
       \end{array} 
   \end{array} 
$$
$\varepsilon$$@$r==J,>.$5$/$H$l$P!"(B(7)$@$H(B(9) $@$K$h$C$F!"(B$\tilde{\phi}_{\varepsilon} \in \Phi_{p,q}$$@$,J]>Z$5$l$k!%<!$N$h$&$KCV$/!%(B
$$
\begin{array}{ll}
$@!!(BI_0= \{ \phi (n-p) \} \times [0,1], & I_1= \{ \phi (n) \} \times 
[0,1], \\
$@!!(BI_2= \{ \phi (n+p) \} \times [0,1], & \tilde{I}_1= \{ \tilde{\phi} (n) \} \times 
[0,1] .
\end{array}
$$
$A_1$$@$r(B$I_1$$@!"(B$FI_0$$@!"(B$\tilde{I}_1$$@!"$*$h$S(B$S_0$$@$K0O$^$l$?NN0h!"(B$A_2$$@$r(B$FI_1$$@!"(B$I_2$$@!"(B$F \tilde{I}_1$$@!"$*$h$S(B$S_0$$@$K0O$^$l$?NN0h$H$9$k!%(B

\vspace{1.0cm}
\hspace*{6.0cm} $@?^(B2

\vspace{4.0cm}
$\varepsilon$$@$,==J,>.$5$1$l$P!"(B$F^{-1} A_2 \subset A_1$$@$G$"$k$+$i!"(B$\mu$$@$N(B$F$$@ITJQ@-$K$h$j(B
$$
    \mu (A_1) > \mu (F^{-1} A_2)= \mu (A_2).
$$
$@$3$N$H$-<!$rF@$k!%(B
$$
\begin{array}{rl} 
L_{p,q}( \phi )- L_{p,q}( \tilde{\phi}_{\varepsilon}) = &
\mu (T( \phi (n), \phi (n+p)))+ \mu (T( \phi (n-p), \phi (n))) \\
&  - \mu (T( \phi (n)- \varepsilon, \phi (n+p))) - \mu(T( \phi (n-p), 
\phi (n)- \varepsilon)) \\
= & \mu (A_1)- \mu (A_2) > 0 . 
\end{array}
$$
$@$3$l$O(B$\phi$$@$,6I=j6K>.$G$J$$$3$H$r<($7$F$$$k!%ITEy<0(B(8)$@<0$,K~$?$5$l$k>l9g$b(B$\tilde{\phi}_{\varepsilon} (n)$$@$r(B$\phi (n)+ \varepsilon$$@$GDj5A$7$FF1MM$K07$($k(B(Fig.3$@;2>H(B)$@!%$3$N$H$-$O(B$A_1 \subset F^{-1}(A_2)$$@$G$"$k!%(B

$@$b$C$H0lHL$N>l9g$,;D$C$F$$$k!%$9$J$o$A!"$?$H$($P(B
$$
$@!!(B\phi (n-1) < \phi (n)= \phi (n+1)= \cdots = 
\phi (n+k) < \phi(n+k+1) , 
$$
$@$N>l9g$G$"$k!%$3$N$H$-(B
$$
$@!!(B1 \geq h_1(n) \geq h_1(n+1) \geq \cdots \geq h_1(n+k) \geq 0 , 
$$
$@$*$h$S(B
$$
  1 \geq h_2(n+k) \geq h_2(n+k-1) \geq \cdots \geq h_2(n) \geq 0 , 
$$
$@$G$"$k$+$i!"(B
$$
   1 \geq h_1(n) > h_2(n) \geq 0 ,    \eqno(10)
$$
$@$^$?$O(B
$$
$@!!!!(B1 \geq h_2(n+k) > h_1(n+k) \geq 0 ,   \eqno(11)
$$
$@$^$?$O(B
$$ \begin{array}{ll}
$@!!!!(Bh_1(n+l)=h_2(n+l), & l=0, \cdots, k , 
   \end{array}
$$
$@$G$"$k!%(B

(10)$@!"(B(11)$@$N>l9g$O$=$l$>$l(B(7)$@!"(B(8)$@$HF1MM$K9M$($l$P$h$$!%$3$l$,$G$-$k$N$O!"Bh0l$N>l9g$K(B$\phi (n)$$@$,:8$K$:$l!"BhFs$N>l9g$K(B$\phi (n+k)$$@$,1&$K$:$l$F(B$\phi$$@$NC1D4@-$,@]F0$N$"$H$G$bJ]$?$l$k$+$i$G$"$k!%$3$l$GJdBj$N>ZL@$,=*$o$k!%(B\hfill $\Box$ 

\vspace{1.0cm}
\hspace*{6.0cm} $@?^(B3

\vspace{4.0cm}
$@A0$K=R$Y$?%^%6!<$NDjM}$N>ZL@(B(3$@@a(B)$@$OK\@a$NDjM}$N#2HVL\$N<gD%$H$OFHN)$G$"$k$+$i!"$=$NItJ,$N>ZL@$N35N,$N$_$r=R$Y$h$&!%(B

$@$J$a$i$+$J>l9g!"(B$L_{p,q}$$@$O(B${\bf T}^q$$@$X$NKd$a9~$_$K$h$C$FM6F3$5$l$k$J$a$i$+$J9=B$$K4X$7$FHyJ,2DG=$G$"$k!%JdBj$N5DO@$NL58B>.(B($@$7$?$,$C$F4JC1(B)$@HG$K$h$l$P!"(B$\Phi_{p,q}$$@$NFbIt$K$"$k(B$L_{p,q}$$@$N$I$N(B\underline{$@NW3&(B}$@E@$b%P!<%3%U<~4|50F;$r@8@.$9$k$O$:$G$"$k!%(B

$@$3$N$H$-6u4V(B$\Phi_{p,q}$$@>e$GHF4X?t(B$L_{p,q}$$@$r6K>.$K$9$k<LA|(B$\varphi$$@$r<h$j!"(B(5)$@$rK~$?$7!"$^$?$9$Y$F$N(B$n$$@$KBP$7$F>r7o(B
$$
  \phi(n) \leq \psi(n) \leq \phi(n+1)
$$
$@$rK~$?$9<LA|(B$\psi : {\bf Z} \rightarrow {\bf R}$$@A4BN$N6u4V(B$\Phi_{p,q}^{\phi}$$@$r9M$($k!%<LA|(B$\phi$$@$H(B$\tilde{\phi}, \tilde{\phi}(n) = \phi(n+1)$$@$O(B$\Phi_{p,q}^{\phi}$$@>e$N(B$L_{p,q}$$@$NNW3&E@(B($@6K>.(B)$@$G$"$k!%(B

$@4X?t(B$L_{p,q}$$@$N8{G[N.$,(B$\Phi_{p,q}^{\phi}$$@$r$=$l<+?H$N>e$K<L$9$3$H$r<($9$3$H$K$h$C$F!"6K>.$G$J$/!"$7$?$,$C$F(B$\phi$$@$H$b(B$\tilde{\phi}$$@$H$b0[$J$kNW3&E@(B$\psi$$@$NB8:_$rJ]>Z$G$-$k!%$3$N<LA|(B$\psi$$@$OBhFs%P!<%3%U<~4|50F;$r7h$a$k!%(B

$@3j$i$+$5$N2>Dj$O<e$a$k$3$H$,$G$-$k!%$7$+$7ITJQB,EY$,O"B3$JL)EY$GM?$($i$l$k$3$H$O5DO@$K$H$C$F7hDjE*$K=EMW$G$"$k!%$3$N<o$N5DO@$r$b$C$H$b0lHL$K5DO@$9$k$K$O$I$&$7$?$i$h$$$N$+$o$?$7$OCN$i$J$$!%(B

[2]$@$G%P!<%3%U$O;X?t$N5DO@$G2sE>?t(B$p/q$$@$NBhFs$N<~4|50F;$rF@$?$,!"$3$l$,$J$<1_=g=x$rJ]B8$9$k$N$+$OL@$i$+$G$J$$!%(B


\vspace{0.5cm}
\noindent
{\bf 5. $@J|J*$*$h$SAP6J50F;(B}

$@HsM-M}2sE>?t$N$I$N(BMather$@=89g$b0l0U$N(B$f-$$@ITJQ3NN(B,EY(B$\mu_E$$@$r;}$C$F$$$k(B(carry)$@!%(B$f$$@$,(B$C^1$$@HyJ,F1Aj$J$i!"(B2$@$D$N%j%c%W!<%N%UFC@-;X?t(B[3]$@$rDj5A$9$k$3$H$,$G$-!"$3$l$i$O(B$\mu_E$$@$N0UL#$G$[$H$s$I$$$?$k$H$3$m0lDj$G$"$k!%(B$\chi_1^E \leq \chi_2^E$$@$r$3$l$i;X?t$NK\<AE*CM$H$9$k!%(B$f$$@$,3j$i$+$JB,EY$rJ]B8$9$k$J$i!"(B$\chi_1^E + \chi_2^E = 0$$@$G$"$j!"$@$+$iHsIi$N;X?t(B$\chi^E$$@$R$H$D$HHs@5$N;X?t(B$-\chi^E$$@$,$"$k!%(B

$\chi^E > 0$$@$J$i(BMather$@=89g(B$E$$@$rAP6JE*$H8F$S!"(B$\chi^E = 0$$@$J$iJ|J*E*$H8F$V!#(B

\vspace{0.3cm}
\noindent
\underline{$@LdBj(B}. $@AP6JE*(BMather$@=89g(B$E$$@$r;}$D(B$C^1(C^2, C^{\infty},$ $@<B2r@OE*(B)$@LL@QJ]B8$M$8$lHyJ,F1Aj$O$"$k$+!)!%(B

\vspace{0.3cm}
$@%^%6!<=89g$,(B\underline{$@$9$Y$F(B}$@AP6JE*$G$"$k$h$&$J6hJ,E*$K2r@OE*$J$M$8$lHyJ,F1Aj<LA|$NNc$OM-L>$J%9%?%8%"%`7?$NNc$b4^$a$F(BBunimovich$@$N%S%j%d!<%I$+$iF@$i$l$k(B[4]$@!%$7$+$7!"$3$NNc$G$O!"%^%6!<=89g$OFC0[E@$H8r$o$C$F$$$k!%(B

\vspace{0.3cm}
\noindent
{\bf $@L?Bj(B 4.} $f$$@$O(B$C^{1+\varepsilon}$$@$M$8$lHyJ,F1Aj$G!"%j%e%Y!<%0B,EY$KF1CM$JB,EY(B$\mu$$@$rJ]B8$9$k$H$9$k!%$3$N$H$-(B$f$$@$NAP6JE*(BMather$@=89g$NA4%j%e%Y!<%0B,EY$O(B$0$$@$G$"$k!%(B

\vspace*{0.3cm}
\noindent
\underline{$@>ZL@(B}. $@%^%6!<=89g$K$OB0$9$k$1$l$I$b6K>.%^%6!<=89g$KB0$5$J$$E@$O$I$l$b2s5"E*$G$J$$!%$9$J$o$A!"$3$NE@$O<+J,$N(B$\omega$$@6K8B=89g$KB0$5$J$$!%%]%"%s%+%l$N2s5"DjM}$K$h$j!"$3$N$h$&$JE@$NA4(B$\mu$$@B,EY$O%<%m$G$"$k!%$@$+$i6K>.%^%6!<=89g$r9M$($k$@$1$G==J,$G$"$k!%$3$l$i$N=89g$OJD$+$D8_$$$KAG$G$"$k!%$@$+$i(B$M(f)$$@$N8D!9$N%^%6!<=89g$X$NJ,3d$O2DB,$G$"$k!%$3$N$h$&$J=89g$OO"B34X?t$N%0%i%U$N0lIt$H$7$F$I$l$b%j%e%Y!<%0B,EY$,%<%m$G$"$k!%$@$+$i=89g(B$M(f)$$@>e$N(B$\mu$$@$N$I$N%(%k%4!<%I@.J,$bB,EY$,%<%m$G$"$k!%AP6J%^%6!<=89g$NOB$r(B$M_h(f)$$@$HI=$o$=$&!%(B$f$$@$O$3$N=89g>e$GHs%<%m$N%j%c%W!<%N%U;X?t$r;}$D!%(BPesin$@$NM}O@(B[3]$@$h$j!"Hs%<%m$N;X?t$r;}$D=89g>e$N%(%k%4!<%I@.J,$O$I$l$b@5$NB,EY$r;}$D!%$@$+$i(B$\mu(M_h(f)) = 0$$@$G$"$k!%(B\hfill $\Box$

\vspace{0.5cm}
\noindent
{\bf 6. $@7k8l(B.} 

$@=`<~4|50F;$K4X$9$k%^%6!<$NDjM}$O8EE5J,Ln$K$*$1$kBg$-$J7k2L$G$"$k!%$3$l$O!"<~4|50F;$K4X$9$k%]%"%s%+%l$*$h$S%P!<%3%U$N7k2L!"%[%b%/%j%K%C%/50F;$N6a$/$N50F;9=B$$K4X$9$k%P!<%3%U$d%9%a!<%k$N7k2L!"$*$h$SITJQ6J@~$N;}B3$K4X$9$k%3%k%b%4%m%U!"%"!<%N%k%I$*$h$S%b!<%6!<$N7k2L$KHf8*$7$&$k!%$H$/$K%^%6!<$NDjM}$O$J$/$J$C$?ITJQ6J@~$K$D$$$F4J7i$J@bL@$rM?$(!"$^$?ITJQ6J@~$N>C$(J}$r<($7$F$$$k!%(B

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\re
[1] J.N. Mather. Existence of quasi-periodic orbits for twist homeomorphisms. {\it Topology}. (To appear.)

\re
[2] G.D. Birkhoff, on the periodic motions of dynamical systems. In {\it G. D. Birkhoff collected Mathematical Papers}, Vol. 2, pp. 333-353, Amer. Math. Soc., 1950. 

\re
[3] Ja. B. Pesin. Characteristic Lyapunov exponents and smooth ergodic theory. {\it Russ. Math. Surveys} {\bf 32} (1977), 55-112. 

\re
[4] L.A. Bunimovich. On the ergodic properties of nowhere dispersing billiards. {\it Comm. Math. Phys.} {\bf 65} (1979), 295-312. 

\re
[5] G.D. Birkhoff. An extension of Poincar\'{e}'s last geometric theorem. In {\it G. D. Birkhoff collected Mathematical Papers}, Vol. 2, pp. 252-266, Amer. Math. Soc., 1950. 

\re
[6] A. Katok \& R. Spatzier. Problem list of special session on differential geometry and ergodic theory in Amherst, October 1981. (prepint)

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