
\vspace{0.5cm}
\noindent
{\bf 5. $BDjM}(B1$B$N>ZL@(B.} 

$f:A \rightarrow A$$B$O$M$8$l<LA|!"(B$p,q$$B$O8_$$$KAG$J@0?t!"(B$w_0 \in A$$B$O(B$f$$B$N(B$p/q-$$B<~4|E@$H$9$k!%(B$w_0$$B$O%P!<%3%U<~4|E@$G$J$$$H2>Dj$9$k!%%P!<%3%U<~4|E@$G$"$l$P>ZL@$O:Q$s$G$$$k$+$i$G$"$k!%(B

\vspace*{3cm}
\hspace*{6cm}     $B?^(B10

\vspace*{3cm}
$B<!$K(B$f$$B$O(B$p/q-$$B<~4|E@$r;}$D$+$i!"(B$p/q \in [\rho_0(f),\rho_1(f)]$$B$G$"$k$3$H$KCm0U$7$h$&!%(B$p/q \in (\rho_0(f),\rho_1(f))$$B$H2>Dj$G$-$k!%$H$$$&$N$O!"(B$i=0$$B$^$?$O(B$1$$B$H$7$F(B$\rho_i(f)=p/q$$B$J$i!"(B$f \mid_{\{ (x,i):x \in {\bf R} \}}$$B$O2sE>?t(B$p/q$$B$N1_HyJ,F1Aj<LA|$N;}$A>e$2$G$"$C$F!"(B$p/q-$$B%P!<%3%U<~4|E@$NB8:_$O1_<LA|$KBP$9$kDL>o$N5DO@$+$i=P$FMh$k$+$i$G$"$k(B([12]$B;2>H(B)$B!%$=$N$&$(!"3F(B$\alpha \in {\bf R}$$B$KBP$7$F(B$y_{\alpha} \in (0,1)$$B$G(B$\pi_1(f(\alpha,y_{\alpha}))=\alpha+p/q$$B$J$k$b$N$,B8:_$9$k$H2>Dj$G$-$k!%$=$G$J$$$H$9$k$H!"M?$($i$l$?<LA|(B$f$$B$KBP$7$F!"(B$f$$B$r(B$f_1:A_1 \rightarrow A_1$$B$K3HD%$G$-$k!%$3$3$G!"(B$A_1= \{ (x,y) \in {\bf R}^2: -1/2 \leq y \leq 3/2 \}$$B$G(B$f_1$$B$O>e$N>r7o$rK~$?$9$M$8$l<LA|$G$"$j!"(B$f$$B$H(B$f_1$$B$O(B$A$$B>e$G0lCW$9$k!%(B$\rho_0(f) < p/q < \rho_1(f)$$B$G$"$k$+$i!"(B$A_1$$B>e$G(B$f_1$$B$N(B$p/q-$$B%P!<%3%U<~4|E@$r8+$D$1$k$3$H$,$G$-$l$P!"<B:]$O$=$N50F;$O(B$A$$BFb$K$"$k$O$:$G$"$k(B(10$B?^;2>H(B)$B!%(B

\vspace{0.3cm}
\noindent
{\bf $B<gD%(B 1.} $B0J2<$N>r7o$rK~$?$9(B$C^2$$B%[%b%H%T!<(B$H:A \times [0,1] \rightarrow A$$B$,$"$k!%(B

(a) $H( \cdot, 0) = f( \cdot )$$B!%(B

(b) $B$9$Y$F$N(B$s \in [0,1]$$B$KBP$7$F!"(B$H(\cdot, s)$$B$O$M$8$l<LA|$G$"$k!%(B

(c) $B$9$Y$F$N(B$s \in [0,1]$$B$KBP$7$F!"(B$w_0$$B$O(B$H( \cdot, s)$$B$N(B$p/q-$$B<~4|E@$G$"$k!%(B

(d) $H( \cdot, 1)$$B$O(B$p/q-$$B%P!<%3%U<~4|E@$r;}$D!%(B

\vspace*{0.3cm}
\noindent
\underline{$B<gD%(B 1$B$N>ZL@(B.} $z_0 \in A$$B$r8GDj$7$F(B$A$$B$NFbIt$K4^$^$l$k$h$&$K$7!"$^$?(B
$$
\begin{array}{l}
 \{ \pi_1(z_0)+jp/q, j=0, \cdots,q-1 \} \cap 
( \{ \pi_1(w):w \in {\cal O} (f,w_0) \} \cup {\bf Z} ) = \emptyset,
\end{array}
$$
$B$*$h$S(B$\pi_1(f(z_0))=\pi_1(z_0) +p/q$$B$H$J$k$h$&$K$9$k!%(B$y_i \in [0,1]$$B$r8GDj$7$F!"(B$\pi_1(f(\pi_1(z_0) + ip/q,y_i))= \pi_1(z_0)+(i+1)p/q$$B$,(B$i=0, \cdots, q-1$$B$KBP$7$F@.$jN)$D$h$&$K$9$k(B($B$7$?$,$C$F!"(B$y_0=\pi_2(z_0)$)$B!%G$0U$N(B$\varepsilon > 0$$B$KBP$7$F!"(B1-$B%Q%i%a!<%?!<HyJ,F1Aj<LA|B2(B$h_{i,s}:A \rightarrow A,$ $i=1, \cdots, q,$ $s \in [0,1]$$B$rDj5A$7$F!"<!$rK~$?$9$h$&$K$G$-$k!%(B

(1) $B$9$Y$F$N(B$s \in [0,1],$ $i=1, \cdots, q$$B$KBP$7$F!"$^$?$9$Y$F$N(B$(x,y) \in A$$B$KBP$7$F!"(B$h_{i,s}(x+1,y)= h_{i,s}(x,y)+(1,0)$$B!%(B

(2) $B$9$Y$F$N(B$s \in [0,1],$ $i=1, \cdots, q$$B$KBP$7$F!"Bf(B$(h_{i,s}-{\rm $B91Ey<LA|(B})$ $\cap \{ (x,y) \in A: 0 \leq x < 1 \}$$B$O(B$\{ (x,y) \in A:$ $\mid x - (\pi_1( z_0 ) + i p/q - [ \pi_1( z_0 ) + i p/q ]) \mid \varepsilon \}$ ($B$3$3$G(B$[\cdot]$$B$O:GBg@0?t4X?t$rI=$o$9(B)$B$K4^$^$l!"$^$?(B$h_{i, \cdot} (\cdot)$$B$O(B$A \times [0,1]$$B>e$G(B$C^{\infty}$$B$G$"$k!%(B

(3) $B$9$Y$F$N(B$s \in [0,1],$ $i=1, \cdots, q$$B$KBP$7$F!"$^$?$9$Y$F$N(B$(x,y) \in A$$B$KBP$7$F!"(B$\pi_1(h_{i,s} (x,y)$ $) = x$$B$G$"$k!%(B

(4) $i=1, \cdots, q$$B$N$H$-(B$h_{i,0}={\rm $B91Ey<LA|(B}$$B$G$"$j!"$^$?(B
$$ \begin{array}{l}
  \pi_2(h_{i,1}(\pi_1(z_0) + ip/q - [\pi_1(z_0) + ip/q], 
  \pi_2(f(\pi_1(z_0) + (i-1)p/q, y_{i-1}))) \\ 
\hspace*{1cm} = \displaystyle 
\left\{  \begin{array}{crl}
 y_i & \hspace{0.5cm} {\rm if} & i=1, \cdots, q-1, \\
 y_0 & \hspace{0.5cm} {\rm if} & i=q.
 $B!!!!!!!!(B\end{array}  \right. 
 $B!!(B\end{array}
$$

\vspace*{0.3cm}
$\varepsilon$$B$r==J,>.$5$/A*$s$G(B
$$
\displaystyle 
\left( \cup^q_{i=1} \cup_{s \in [0,1]} $BBf(B(h_{i,s} - $B91Ey<LA|(B) \right) 
\cap ({\cal O} (f, w_0) \cup \{ (j,y) \in A: j \in {\bf Z} \} ) 
= \emptyset 
$$
$B$H$7!"<!$rDj5A$9$k(B:
$$ \begin{array}{c}
 H: A \times [0,1] \rightarrow A \\
H(z,s) = h_{q,s} \circ f \circ h_{q-1,s} \circ h_{q-2,s} \circ 
\cdots \circ h_{1,s} (z).
 \end{array}
$$
$B$3$N$H$-!"(B$H$$B$O>e$N@-<A(B(a)-(d)$B$r;}$D!%$H$/$K!"$9$Y$F$N(B$s \in [0,1]$$B$KBP$7$F!"(B$w_0$$B$O(B$H(\cdot,s)$$B$N(B$p/q$$B<~4|E@$G$"$k!%$H$$$&$N$O!"(B$f$$B$H(B$H(\cdot,s)$$B$O(B${\cal O} (f, w_0)$$B$N6aK5$G0lCW$7!"(B$H(\cdot,s)$$B$O$M$8$l<LA|$@$+$i$G$"$k!%(B$H(\cdot,s)$$B$,$M$8$l<LA|$J$N$O!"$M$8$l<LA|$H(B$x=$$B0lDj$NMUAX$rJ]B8$9$k<LA|$H$N9g@.$O$d$O$j$M$8$l<LA|$@$+$i$G$"$k!%<LA|(B$H(\cdot,1)$$B$O>e$N>r7o(B(4)$B$K$h$C$F(B$z_0$$B$r(B$p/q-$$B<~4|E@$H$7$F;}$A!"(B$z_0$$B$O(B$p/q-$$B%P!<%3%U<~4|E@$G$"$k!%$J$<$J$i!"G$0U$N(B$\zeta \in {\cal O} (H(\cdot,1), z_0)$$B$O$U$?$?$S>r7o(B(4)$B$K$h$C$F!"(B$\pi_1(H(\zeta,1)) = \pi_1(\zeta) + p/q$$B$rK~$?$9$+$i$G$"$k!%$f$($K!"(B$H$$B$O5a$a$k%[%b%H%T!<$G$"$C$F!"<gD%$N>ZL@$,40N;$7$?!%(B\hfill $\Box$

\vspace*{0.3cm}
$\Xi = \{ s \in [0,1] : {\rm $B$9$Y$F$N(B}s_1 \geq s{\rm $B$KBP$7$F!"(B} H(\cdot,s_1){\rm $B$O(B}p/q-{\rm $B%P!<%3%U<~4|E@$r;}$D(B} \}$$B$H$*$/!%(B$1 \in \Xi \not= \emptyset$$B$G$"$k$+$i!"E@(B$s_1 = \inf \{ s \in [0,1] : s \in \Xi \}$$B$,(B$\Xi$$B$NFbIt$K$"$k$3$H!"$D$^$j(B$\Xi$$B$,3+$+$DJD$G$"$k$3$H!"$r<($;$l$P!"(B$\Xi = [0,1]$$B$G$J$1$l$P$J$i$J$$!%$f$($K!"(B$H(\cdot,0)=f(\cdot)$$B$O(B$p/q-$$B%P!<%3%U<~4|E@$r;}$D$O$:$G$"$j!">ZL@$O40N;$9$k$3$H$K$J$k!%(B

\vspace*{0.3cm}
\noindent
{\bf $B<gD%(B 2.} $\Xi$$B$OJD$8$F$$$k!%(B

\vspace*{0.3cm}
\noindent
\underline{$B<gD%(B 2$B$N>ZL@(B}. $BJdBj(B1$B$+$iD>$A$K$7$?$,$&!%(B\hfill $\Box$

\vspace*{0.3cm}
\noindent
{\bf $B<gD%(B 3.} $\Xi=[0,1]$.

\vspace*{0.3cm}
\noindent
\underline{$B<gD%(B 3$B$N>ZL@(B}. $\Xi \not= [0,1]$$B$H2>Dj$9$k!%$9$k$H!"(B$s_1= \inf \{ s \in [0,1] : s \in \Xi \} > 0$$B$G$"$k!%<gD%(B2$B$h$j!"(B$s_1 \in \Xi$$B$G$"$k$+$i!"(B$H(\cdot,s_1)$$B$O(B$p/q-$$B%P!<%3%U<~4|E@(B$\zeta_0 \in A$$B$r;}$D!%9=@.K!$+$i!"(B$w_0$$B$O(B$H(\cdot,s_1)$$B$N(B$p/q-$$B<~4|E@$G$"$C$F!"(B$p/q-$$B%P!<%3%U<~4|E@$G$J$$!%$f$($K!"JdBj(B4$B$,E,MQ$G$-$F!"(B$H(\cdot,s_1)$$B$,JL$N(B$p/q-$$B%P!<%3%U<~4|E@(B$z_1 \in A$$B$G(B$z_1 \not\in {\cal O} (H(\cdot,s_1), z_1)$$B$J$k$b$N$r;}$D$3$H!"$5$i$K=EMW$J$3$H$K!"(B$s_1$$B$K==J,6a$$(B$s$$B$KBP$7$F!"JdBj(B4$B$N0BDj@-$NDD=R$+$i(B$H(\cdot,s)$$B$b(B$p/q-$$B%P!<%3%U<~4|E@$r;}$D$3$H$,<($;$k!%$f$($K!"(B$s_1$$B$O(B$\Xi$$B$NFbIt$K$"$k$3$H$K$J$j!"(B$s_1$$B$NDj5A$KL7=b$9$k!%$3$NL7=b$h$j!"(B$s_1=0$$B!"$D$^$j(B$\Xi=[0,1]$$B$,=P$F!"<gD%$N>ZL@$,40N;$7$?!%(B\hfill $\Box$

\vspace*{0.3cm}
\noindent
$B>e$GCm0U$7$?$h$&$K!"(B$0 \in \Xi$$B$O(B$H(\cdot,0)=f(\cdot)$$B$,(B$p/q-$$B%P!<%3%U<~4|E@$r;}$D$3$H$r0UL#$7!"DjM}(B1$B$N>ZL@$O40N;$7$?!%(B \hfill $\Box$
