
\vspace{0.3cm}
\noindent
{\bf 4. $B$$$/$D$+$NJdBj(B.}

$B$3$N@a$G$ODjM}#1$N>ZL@$KM-MQ$JJdBj$r$$$/$D$+>ZL@$9$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BJdBj(B 1.} $f_n: A \rightarrow A, n=1,2, \cdots$$B$O$M$8$l<LA|Ns$G$"$k$H$7!"$"$k8GDj$7$?8_$$$KAG$J@0?t(B$p,$ $q$$B$KBP$7$F!"3F(B$f_n$$B$O(B$p/q-$$B%P!<%3%U<~4|E@(B$z_n \in A$$B$r;}$D$H$9$k!%>e8B%N%k%`0LAj$G(B$f_n$$B$,$M$8$l<LA|(B $f_0: A \rightarrow A$$B$K<}B+$7(B($B$9$J$o$A!"(B$\parallel \cdot \parallel$$B$rDL>o$N(B${\bf R}^2$$B%N%k%`$H$7$F!"(B$n \rightarrow \infty$$B$N$H$-(Bsup$_{z \in A} \parallel f_n(z) - f_0(z) \parallel \rightarrow 0$)$B!"(B$n \rightarrow \infty$$B$N$H$-(B$z_n$$B$,(B$z_0 \in A$$B$K<}B+$9$k$J$i!"(B$z_0$$B$O(B$f_0$$B$N(B$p/q-$$B%P!<%3%U<~4|E@$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$BJdBj#1$N>ZL@(B}. $B$9$Y$F$N(B$n=1,2, \cdots$$B$KBP$7$F(B
$f_n^q(z_n)-(p,0)=z_n$$B$J$N$G!"(B$f_0^q(z_0)-(p,0)=z_0$$B$G$"$k!%(B
($B$=$N$&$(!"(B$p,$ $q$$B$O8_$$$KAG$@$+$i(B$s<q$$B$J$kG$0U$N(B$r,$ $s$$B$KBP$7$F(B$f_0^s(z_0)-(r,0) \not= z_0$$B$G$"$k(B.) 
$B$f$($K(B$z_0$$B$O(B$f_0$$B$N(B$p/q-$$B<~4|E@$G$"$k!%(B

$z_0$$B$,(B$f_0$$B$N(B$p/q-$$B%P!<%3%U<~4|E@$G$"$k$3$H$r<($9$?$a$K(B$k_1,$ $k_2,$ $l_1,$ $l_2 \in {\bf Z}$$B$r8GDj$9$k!%$3$N$H$-!"(B$n=1,2, \cdots$$B$KBP$7$F<!$,@.$jN)$D!%(B
$$
 \begin{array}{l}
 \pi_1(f_n^{k_1}(z_n))+l_1 < \pi_1(f_n^{k_2}(z_n))+l_2 \Rightarrow 
 \pi_1(f_n^{k_1+1}(z_n))+l_1 < \pi_1(f_n^{k_2+1}(z_n))+l_2.
 \end{array}
$$
$B$H$3$m$,$3$N$H$-O"B3@-$K$h$j!"(B$f_n$$B$H(B$z_n$$B$r$=$l$>$l(B$f_0$$B$H(B$z_0$$B$KCV$-49$(!"BhFs$NITEy<0$K!V(B$\leq$$B!W$rF~$l$F$bF1$8<gD%$,@.$jN)$D!%$7$+$7!"(B
$$
 \pi_1(f_0^{k_1}(z_0))+l_1 < \pi_1(f_0^{k_2}(z_0))+l_2,  \eqno(*)
$$
$B$*$h$S(B
$$
 \pi_1(f_0^{k_1+1}(z_0))+l_1 = \pi_1(f_0^{k_2+1}(z_0))+l_2,
$$
$B$H2>Dj$7$F$_$k!%$3$N$H$-$M$8$l>r7o$+$i(B$\pi_2(f_0^{k_1+1}$ $(z_0)) >$ $\pi_2(f_0^{k_2+1}$ $(z_0))$$B$rF@$k(B($B#2?^;2>H(B)$B!%$@$+$i:F$S$M$8$l>r7o$h$j(B
$\pi_1(f_0^{k_1+2}(z_0))+l_1 >$ $\pi_1(f_0^{k_2+2}(z_0))+l_2$$B$rF@$k!%$H$3$m$,!"$3$l$O(B$n$$B$,==J,Bg$-$$$H$-(B
$$
\pi_1(f_n^{k_1+2}(z_n))+l_1 > \pi_1(f_n^{k_2+2}(z_n))+l_2,
$$
$B$r0UL#$7!"0lJ}(B$(*)$$B<0$O(B$n$$B$,==J,Bg$-$$$H$-(B
$$
\pi_1(f_n^{k_1}(z_n))+l_1 < \pi_1(f_n^{k_2}(z_n))+l_2,
$$
$B$r0UL#$9$k$,!"$3$l$O(B$z_n$$B$,(B$f_n$$B$N(B$p/q-$$B%P!<%3%U<~4|E@$G$"$k$3$H$KL7=b$9$k!%$@$+$i(B$z_0$$B$O(B$f_0$$B$N(B$p/q-$$B%P!<%3%U<~4|E@$G$J$1$l$P$J$i$:!">ZL@$,407k$7$?!%!!(B\hfill $\Box$

\vspace*{2cm}
\hspace*{5cm}   $B?^(B2

\vspace*{2cm}
$B<!$NJdBj$K$h$l$P!"Cf4V$K$"$k<LA|$b$M$8$l<LA|$K$J$k$h$&$K!"M?$($i$l$?$M$8$l<LA|$r7|?b$G$-$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BJdBj(B 2.} $B$M$8$l<LA|(B $f: A \rightarrow A$$B$,M?$($i$l$?$H$-!"0J2<$N>r7o$rK~$?$9(B$C^1$$B<LA|(B$\phi : A \times {\bf R} \rightarrow A$$B$,B8:_$9$k(B.

(1) $\phi ( \cdot, 0)=$ $A$$B>e$N91Ey<LA|!"(B$\phi ( \cdot, 1)=f(\cdot)$;

(2) $B$9$Y$F$N(B$t \in [0,1]$$B$*$h$S$9$Y$F$N(B$n \in {\bf Z}$$B$KBP$7$F!"(B
$\phi ( \cdot, t+n)=\phi(f^n(\cdot),t)$;

(3) $B$9$Y$F$N(B$t \in [0,1]$$B$KBP$7$F!"(B$\phi ( \cdot, t)$$B$O$M$8$l<LA|$G$"$k(B;

(4) $B$9$Y$F$N(B$z \in A$$B$*$h$S$9$Y$F$N(B$t \in {\bf R}$$B$KBP$7$F!"(B
$\phi ( z+(1,0), t)= \phi(z,t)+(1,0)$.

\vspace{0.3cm}
\noindent
\underline{$BJdBj(B2$B$N>ZL@(B}. $A \times [0,1]$$B>e$G(B(1)$B!"(B(3)$B$*$h$S(B(4)$B$rK~$?$9$h$&$K(B$\phi$$B$rDj5A$9$l$P$h$$!%$H$$$&$N$O!"<!$K(B(2)$B$r;H$C$FDj5A$r(B${\bf R}$$BA4BN$K9-$2$l$P$h$$$+$i$G$"$k!%(B

$f_0: {\bf R} \rightarrow {\bf R}$$B$O(B$f_0(x)=\pi_1 (f(x,0))$$B$GM?$($i$l$k$H$7!"(B$G: {\bf R}^2 \rightarrow {\bf R}^2$$B$O$9$Y$F$N(B$z \in A$$B$KBP$7<!<0$GM?$($i$l$k$H$9$k!%(B
$$
   G(z)=Df(f^{-1}(z))(0,1).
$$
$B$?$@$7(B$Df(w)$$B$O(B$w \in A$$B$K$*$1$k(B$f$$B$NHyJ,(B($B9TNs(B)$B$G$"$k!%$3$N$H$-(B$f$$B$O<LA|(B$f_0$$B$*$h$S%Y%/%H%k>l(B$G$$B$K$h$C$F<!$N$h$&$KDj$^$k!%(B$\psi$$B$ODj5A0h$r(B$A \times {\bf R}$$BFb$K;}$D6I=jN.$l$G=i4|CMLdBj(B
$$$B!!(B\left\{ $B!!(B\begin{array}{l}
\displaystyle 
 \frac{{\rm d}\psi}{{\rm d}t}(z,t)=G(\psi(z,t)), \\
 \psi(z,0)=z, \ \ \  {\rm for \ all \ } z \in A, 
$B!!!!(B     \end{array}  \right.
$$
$B$GDj5A$5$l$F$$$k$H$9$k!%$3$N$H$-$9$Y$F$N(B$z \in A$$B$KBP$7$F(B$f$$B$O<!<0$GM?$($i$l$k!%(B
$$
        f(z) = \psi ((f_0(\pi_1(z)),0),\pi_2(z)).$B!!!!!!(B\eqno(**)
$$
$f$$B$N5a$a$k%[%b%H%T!<$OMF0W$K<!$N$h$&$K9=@.$G$-$k!%(B

(a) ${\bf R}$$B>e$G1_HyJ,F1Aj<LA|$N;}$A>e$2$K$J$C$F$$$kHyJ,F1Aj<LA|$rDL$C$F(B$f_0$$B$r91Ey<LA|$KJQ7A$9$k(B;

(b) $B%Y%/%H%k>l$N3j$i$+$J#1(B-$B%Q%i%a!<%?!<B2(B$G_s: A \rightarrow {\bf R}^2$$B$G%Q%i%a!<%?!<$,(B$s \in [0,1]$$B$G$"$k$b$N$G!"(B$G_0$$B$,Dj?t(B$(0,1)$$B$KEy$7$/!"(B$G_1$$B$,(B$G$$B$KEy$7$$$b$N$r$D$/$k!%>r7o(B(3)$B$rJ]>Z$9$k$?$a$K!"(B$G_s(z)$$B$H(B$x$$B<4$N3QEY$,$D$M$K(B$-\pi/2$$B$H(B$\pi/2$$B$N4V$K$"$C$F!"(B$s$$B$H$H$b$KA}2C$9$k$h$&$K(B$G_s$$B$r9=@.$9$k!%>r7o(B(4)$B$N$?$a$K!"$9$Y$F$N(B$z \in A,$ $s \in [0,1]$$B$KBP$7$F(B$G_s(z+(1,0))=G_s(z)$$B$G$"$k$3$H$rMW@A$9$k!%(B

$B<LA|(B$\phi(\cdot, s)$$B$O%Y%/%H%k>l(B$G_s$$B$r(B$(**)$$B$N7A$NJ}Dx<0$K$*$1$k>e$N(B(a)$B$rK~$?$9HyJ,F1Aj<LA|$G;XDj$5$l$k=i4|>r7o$G2r$$$FM?$($i$l$k(B(?)$B!%$b$A$m$s(B$\phi$$B$r(B$A \times {\bf R}$$B>e$NBg0hE*$J3j$i$+$J<LA|$K3HD%$9$k$?$a$K$O!"1_<LA|$NB2$H(B(a)$B$*$h$S(B(b)$B$GM?$($i$l$k%Y%/%H%k>l$r(B$s=0$$B$*$h$S(B$s=1$$B$N6a$/$G!V$D$J$2$k(B(match up)$B!WI,MW$,$"$k!%>\:Y$OFI<T$K$^$+$;$k!%(B\hfill $\Box$

\vspace*{0.3cm}
$BDjM}(B1$B$N>ZL@$N80$H$J$k$N$O!"$3$l$i$NN.$l$N$b$H$G$N(B$p/q-$$B%P!<%3%U<~4|E@$N50F;$N0LAjE*@-<A!"$9$J$o$A!"$3$l$i$,!VMm$s$G!W$$$J$$;v<B$G$"$k!%$3$l$O<!$NJdBj$G@:L)2=$5$l$k!%(B

$f: A \rightarrow A$$B$O$M$8$l<LA|$G(B$p,q$$B$O8_$$$KAG$N@0?t$H$9$k!%(B$g:A \rightarrow A$$B$O$9$Y$F$N(B$z \in A$$B$KBP$7$F<!$GDj5A$5$l$k$H$9$k!%(B
$$
               g(z)=f^q(z)-(p,0).
$$
$B$3$N$H$-(B$g$$B$NITF0E@$O@53N$K(B$f$$B$N(B$p/q-$$B<~4|E@$KBP1~$9$k!%(B$f$$B$,(B$p/q-$$B%P!<%3%U<~4|E@(B$z_0 \in A$$B$r;}$D$H$9$k!%$9$k$H(B

\vspace{0.3cm}
\noindent
{\bf $BJdBj(B 3.} $C^1$$B<LA|(B$\phi: A \times {\bf R} \rightarrow A$$B$,$"$C$F<!$rK~$?$9(B:

(i) $B$9$Y$F$N(B$z \in A$$B$*$h$S$9$Y$F$N(B$t \in {\bf R}$$B$KBP$7$F!"(B
$\phi(z+(1,0),t)=\phi(z,t)+(1,0);$

(ii) $\phi( \cdot,0)=$B91Ey<LA|(B, \ \ \phi( \cdot,1)=g(\cdot)$;

(iii) $B$9$Y$F$N(B$t \in [0,1]$$B$*$h$S$9$Y$F$N(B$n \in {\bf Z}$$B$KBP$7$F(B
$\phi( \cdot, t+n)=\phi(g^n(\cdot),t);$

(iv) $B$9$Y$F$N(B$t \in [0,1/q]$$B$*$h$S(B$i=0, \cdots, q-1$$B$KBP$7$F!"(B$\phi(\phi(\cdot,i/q)^{-1},i/q+t)$$B$O$M$8$l<LA|$G$"$k(B;

(v) $i=1, \cdots,q$$B$*$h$SG$0U$N(B$z,w \in A$$B$KBP$7$F!"(B$\pi_1(\phi(z,i/q)) <\pi_1(\phi(w,i/q))$$B$G$"$k$?$a$NI,MW==J,>r7o$O(B$\pi_1(f^i(z))<\pi_1(f^i(w))$$B$G$"$k(B;

(vi) $B$9$Y$F$N(B$\zeta \in {\cal O}(f,z_0)$$B$*$h$S$9$Y$F$N(B$t \in {\bf R}$$B$KBP$7$F!"(B$\phi(\zeta,t)=\zeta$.

\vspace{0.3cm}
\noindent
\underline{$BJdBj(B3$B$N>ZL@(B}. $\phi_1 : A \times {\bf R} \rightarrow A$$B$OJdBj(B2$B$K$h$C$F(B$f$$B$KF1H<$9$k(B1-$B%Q%i%a!<%?!<B2$G$"$k$H$9$k!%<LA|(B$\phi_1( \cdot,t)$$B$H(B$x=$$B0lDj$NMUAX$rJ]B8$9$k<LA|$H$r9g@.$9$k$3$H$K$h$j(B($B$9$J$o$A!"%Y%/%H%k>l(B$G_t$$B$GM?$($i$l$k%Y%/%H%k$N%N%k%`$rD4@0$9$k$3$H$K$h$j(B)$B!"<!$N$h$&$K2>Dj$9$k$3$H$,$G$-$k!%$9$J$o$A!"(B$\zeta_1,\zeta_2 \in {\cal O}(f,z_0)$$B$,(B$\pi_1(\zeta_1) < \pi_1(\zeta_2)$$B$J$i(B
$$
  \pi_1(\phi_1(\zeta_1,t)) < \pi_1(\phi_1(\zeta_2,t)) \hspace{0.5cm} 
{\rm for \ all \ } \hspace{0.5cm} t \in [0,1].
$$
$B$H$3$m$,$3$N$H$-(B$\phi_2:A \times [0,1] \rightarrow A$$B$,$9$Y$F$N(B$z \in A$$B$*$h$S$9$Y$F$N(B$t \in [0,1]$$B$KBP$7$F(B
$$
    \phi_2(z,t) = \phi_1(z,qt) - t(p,0),
$$
$B$K$h$C$FDj5A$5$l$F$$$k$H$9$k$H!"(B$\phi_2$$B$O>r7o(B(i)$B!"(B(ii)$B!"(B(iv)$B$*$h$S(B(v)$B$rK~$?$9$3$H$,$o$+$j!"$5$i$K(B$\zeta_1,\zeta_2 \in {\cal O}(f,z_0),$ $\pi_1(\zeta_1) < \pi_1(\zeta_2)$$B$J$i$9$Y$F$N(B$t \in [0,1]$$B$KBP$7$F(B$\pi_1(\phi_2(\zeta_1,t)) < \pi_1(\phi_2(\zeta_2,t))$$B$G$"$k$3$H$,$o$+$k!%$f$($K(B$\phi_2$$B$rB2(B$\phi:A \times [0,1] \rightarrow A$$B$KJQ7A$7$F(B(i)$B!"(B(ii)$B!"(B(iv)$B!"(B(v)$B$*$h$S(B(vi)$B$rK~$?$9$h$&$K$7!">r7o(B(iii)$B$K$h$C$F(B$\phi$$B$r(B${\bf R}$$B>e$N<LA|$K3HD%$G$-$k!%$3$l$,5a$`$Y$-<LA|(B$\phi$$B$G$"$k$+$i>ZL@$,40N;$7$?!%(B\hfill $\Box$

\vspace*{0.3cm}
\noindent
\underline{$BCm0U(B}. (1) $BHs8x<0$K$O(B$\zeta \in {\cal O}(f,z_0)$$B$KBP$7$F(B$\phi_2(\zeta,t)$$B$GM?$($i$l$kAH$_I3$O!V$^$C$9$0?-$P$;$k!W$+$i<+L@$G$"$k$H8@$($k(B(3$B?^;2>H(B)$B!%(B

\vspace*{3.5cm}
\hspace*{6cm}  $B?^(B3

\vspace{3.5cm}
(2) $z_0$$B$,(B$p/q-$$B%P!<%3%U<~4|E@$G$"$k$H$$$&2>Dj$O>e$NJdBj$GI,MW$G$"$k!%<B:]!"(B$z_0$$B$,(B$p/q-$$B<~4|E@$G$"$C$F(B$p/q-$$B%P!<%3%U<~4|E@$G$J$$$H$9$k$H!">e$NJdBj$O@.$jN)$?$J$$!%$3$l$r8+$k$?$a$K$O!"(B$z_0$$B$,(B$p/q-$$B%P!<%3%U<~4|E@$G$J$$$H$-!"(B$\zeta_1,\zeta_2 \in {\cal O}(f,z_0)$$B$G(B$\pi_1(\zeta_1)<\pi_1(\zeta_2)$$B$+$D(B$\pi_1(f(\zeta_1)) > \pi_1(f(\zeta_2))$$B$J$k$b$N$,$"$k$3$H$KCm0U$9$l$P$h$$!%$H$3$m$,<!$NJdBj$N>ZL@$G8+$k$h$&$K!"$3$N$3$H$+$i!"(B$\phi_1$$B$N$b$H$G$N(B$\zeta_1$$B$H(B$\zeta_2$$B$N50F;$OHs<+L@$K!VMm$_!W$"$C$F$*$j(B($B#4?^;2>H(B)$B!"F1;~$K$^$C$9$0?-$P$9$o$1$K$O$$$+$J$$!%(B

(3) $B:G8e$K!"JdBj#3$N>ZL@$O%P!<%3%U<~4|50F;$NE@$,=g=x$h$/JB$s$G$$$k$3$H$N$_$K0MB8$7$F$$$k$3$H$rCm0U$7$F$*$/!%F1$8>ZL@K!$G<!$,<($;$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BJdBj(B 3$'$.} $f: A \rightarrow A$$B$*$h$S(B$g: A \rightarrow A$$B$r>e$HF1MM$H$7!"(B$\zeta_1, \zeta_2$$B$O(B$f$$B$N(B$p/q-$$B<~4|E@$G$"$C$F!"$9$Y$F$N(B$i=0, \cdots, q$$B$KBP$7$F(B$\pi_1(f^i(\zeta_1))<\pi_1(f^i(\zeta_2))$$B$rK~$?$;$P!"<LA|(B$\phi: A \times {\bf R} \rightarrow A$$B$,$"$C$FJdBj(B3$B$N(B(ii)-(v)$B$rK~$?$7$5$i$K<!$rK~$?$9!%(B

(vi$'$) $B$9$Y$F$N(B$t \in {\bf R}$$B$KBP$7$F!"(B
$\phi(\zeta_i,t)=\zeta_i, \ i=1,2.$

\vspace{0.3cm}
\noindent
\underline{$BCm0U(B}. $B$3$NJdBj$N>l9g!">r7o(B(i)$B$D$^$j(B$x$$BJ}8~$N<~4|@-$rMW@A$9$k$3$H$O$G$-$J$$$3$H$KCm0U$7$h$&!%$H$$$&$N$O!"(B$\zeta_1$$B$O(B$\zeta_2$$B$H$OMm$s$G$$$J$/$F$b(B$\zeta_2-(1,0)$$B$H$OMm$s$G$$$k$+$b$7$l$J$$!%$@$+$i!"$3$N>l9g!V$^$C$9$0?-$P$9!W$3$H$,$G$-$k$N$O1_4D$NCf$G$O$J$/!"HoJ$6u4V(B$A$$B$NCf$G$J$N$G$"$k!%(B

\vspace*{2.5cm}
\hspace*{5cm}     $B#4?^(B

\vspace{2.8cm}
\noindent
{\bf $BJdBj(B3$'$$B$N>ZL@(B.} $BJdBj(B3$B$N>ZL@$HF1$8!%(B\hfill $\Box$

\vspace{0.3cm}
$B$3$N@a$N:G8e$NJdBj$ODjM}(B1$B$GMW5a$5$l$F$$$k!VITF0E@!WJdBj$G$"$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BJdBj(B 4.} $f: A \rightarrow A$$B$r$M$8$l<LA|$H$7!"(B$p,q$$B$O8_$$$KAG$J@0?t$H$9$k!%(B$f$$B$O(B$p/q-$$B%P!<%3%U<~4|E@(B$z_0 \in A$$B$H%P!<%3%U<~4|E@$G$J$$(B$p/q-$$B<~4|E@(B$w_0 \in A$$B$r;}$D$H$9$k!%$3$N$H$-(B$f$$B$OJL$N(B$p/q-$$B%P!<%3%U<~4|E@(B$z_1 \in A$$B$G(B$z_1 \not\in {\cal O} (f,z_0)$$B$J$k$b$N$r;}$D!%$=$N>e!"(B$\tilde{f} : A \rightarrow A$$B$,$M$8$l<LA|$G(B$\sup_{z \in A} \parallel f(z) - \tilde{f}(z) \parallel$$B$,==J,>.$5$1$l$P(B($B$?$@$7!"(B$\parallel \cdot \parallel$$B$O(B${\bf R}^2$$B$NDL>o$N%N%k%`(B)$B!"(B$\tilde{f}$$B$O(B$p/q-$$B%P!<%3%U<~4|E@$r;}$D!%(B

\vspace{0.3cm}
\noindent
\underline{$BCm0U(B}. $B$3$NDjM}$OLL@QJ]B8$M$8$l<LA|$K4X$9$k%P!<%3%U$NDjM}(B(3$B@a$NDjM}(B2$B$O$3$l$N7O$G$"$k(B)$B$r0E<($9$k!%$=$NDjM}$H$O!"$M$8$l<LA|(B $f$$B$KBP$7$F(B$0 \in [\rho_0(f),\rho_1(f)]$$B$J$i$P(B$f$$B$O#2$D$NITF0E@$r;}$D!"$H$$$&$b$N$G$"$k!%@8@.E*$J>l9g$K(B$f$$B$,#2$D$NITF0E@$r;}$D$Y$-$H$$$&$3$H$O;X?t$K4X$9$k5DO@$+$iMF0W$K=P$k$,!"ITF0E@$,=LB`$7$F$$$k>l9g$G$b!"(B$f$$B$O(B2$B$D$NITF0E@$r;}$?$M$P$J$i$J$$(B([4],[5]$B$*$h$S(B[7]$B;2>H(B)$B!%6/D4$7$?$$$N$O!"JdBj(B4$B$G$o$l$o$l$OLL@QJ]B8@-$b$I$s$J1_8r:9@-$b2>Dj$7$F$$$J$$$3$H$G$"$k!%(B
