
\vspace{0.3cm}
\noindent
{\bf $BJdBj(B4$B$N>ZL@(B.} $B>e$N$h$&$K$9$Y$F$N(B$z \in A$$B$KBP$7$F(B$g(z)=f^q(z)-(p,0)$$B$H$7!"(B$\phi:A \times {\bf R} \rightarrow A$$B$OJdBj(B3$B$K$h$j(B$g$$B$*$h$S(B$z_0 \in A$$B$N50F;$KF1H<$9$k(B1-$B%Q%i%a!<%?!<B2$G$"$k$H$9$k!%JX59$N$?$a0J2<$N5;=QE*5-9f$rF3F~$9$k!#(B

$\zeta,\eta \in A$$B$N(B\underline{$B=g=x$,68$&(B}(get out of order)$B$H$$$o$l$k$N$O!"$"$k(B$i, 0<i<q$$B$KBP$7$F(B$\pi_1(\zeta)<\pi_1(\eta)$$B$+$D(B$\pi_1(f^i(\zeta))>\pi_1(f^i(\eta))$$B$G$"$k$+!"$"$k(B$i, 0<i<q$$B$KBP$7$F(B$\pi_1(\zeta)>\pi_1(\eta)$$B$+$D(B$\pi_1(f^i(\zeta))<\pi_1(f^i(\eta))$$B$N$H$-$G$"$k!%(B

$B$?$@$A$K8@$($k$3$H$O!"(B$\zeta,\eta$$B$,(B$f$$B$N(B$p/q-$$B<~4|E@$G$"$l$P!"(B$\zeta,\eta$$B$N=g=x$,68$&$?$a$NI,MW==J,>r7o$O$9$Y$F$N(B$k \in {\bf Z}$$B$KBP$7$F(B$f^k(\zeta),f^k(\eta)$$B$N=g=x$,68$&$3$H$G$"$k!%(B

$\mu_0, \mu_1, \cdots, \mu_{q-1} \in {\cal O}(f,z_0)$$B$O<!$N$h$&$KJB$s$G$$$k$H$9$k(B.
$$
    0 \leq \pi_1(\mu_0)<\pi_1(\mu_1)< \cdots< \pi_1(\mu_{q-1})<1.
$$
$B$^$?(B$\nu_0, \cdots, \nu_{q-1} \in {\cal O}(f,w_0)$$B$O<!$N$h$&$KJB$s$G$$$k$H$9$k(B.
$$
    \pi_1(\mu_0) \leq \pi_1(\nu_0)< \pi_1(\nu_1)<\cdots< \pi_1(\nu_{q-1})< \pi_1(\mu_0+(1,0)).
$$
($BI,MW$J$i:BI8$r$9$3$7JQ$($F(B$\pi_1(\nu_i) \not= \pi_1(\nu_{i+1})$$B$H2>Dj$G$-$k(B.)

$B$5$FH7$N7j86M}$K$h$C$F#2$D$N>l9g$r;XDj$9$k!%(B

\vspace{0.3cm}
\noindent
\underline{$B%1!<%9(B1}. $i_0,$ $0 \leq i_0 <q$$B$,$"$C$F$9$Y$F$N(B$j=0,1, \cdots, q-1$$B$KBP$7$F(B$\pi_1(\nu_j) \not\in [\pi_1(\mu_{i_0}), \pi_1(\mu_{i_0+1})$ $)$ ($B$^$?$O(B$i_0=q-1$$B$N$H$-$O(B$\pi_1(\nu_i) \not\in [\pi_1(\mu_{q-1}), \pi_1(\mu_0)+1)$).

\noindent
\underline{$B%1!<%9(B2.} $B3F(B$i, 0 \leq i <q$$B$KBP$7$F(B$\pi_1(\nu_i) \in [\pi_1(\mu_i), \pi_1(\mu_{i+1}))$($B$*$h$S(B$\pi_1(\nu_{q-1}) \in [\pi_1(\mu_{q-1}), \pi_1(\mu_0)$ $+1)$).

\vspace{0.3cm}
$B%1!<%9(B1$B$,@.$jN)$D$J$i(B$\zeta_0=\mu_{i_0}$$B$*$h$S(B$\zeta_1=\mu_{i_0+1}$$B$H<h$k!%%1!<%9(B2$B$,@.$jN)$D$J$i(B$\zeta_0=\mu_0$$B$*$h$S(B$\zeta_1=\mu_1$$B$H<h$k!%(B

$B<!$K(B$w_0$$B$O%P!<%3%U<~4|E@$G$J$$$+$i(B$z \in {\cal O}(f,z_0)$$B$,$"$C$F(B$z,w_0$$B$O=g=x$,68$&$O$:$G$"$k$3$H$KCm0U$9$k!%(B$\overline{z} \in {\cal O} (f,z_0)$$B$O(B$\overline{z},w_0$$B$N=g=x$,68$&$h$&$JE@$G$"$C$F!"(B$z \in {\cal O}(f,z_0)$$B$G(B$z,w_0$$B$N=g=x$,68$&$J$i(B$z=\overline{z}$$B$^$?$O(B$\pi_1(z)<\pi_1(\overline{z})$$B$H$J$kE@$H$9$k!%F1MM$K!"(B$\underline{z} \in {\cal O} (f,z_0)$$B$O(B$\underline{z},w_0$$B$N=g=x$,68$&$h$&$JE@$G$"$C$F!"(B$z \in {\cal O}(f,z_0)$$B$G(B$z,w_0$$B$N=g=x$,68$&$J$i(B$\underline{z}=z$$B$^$?$O(B$\pi_1(\underline{z})<\pi_1(z)$$B$H$J$kE@$H$9$k(B($\overline{z}=\underline{z}$$B$N$3$H$bM-$jF@$k(B)$B!%(B$f^{r_1}(\overline{z})=\zeta_0+(s_1,0)$$B$*$h$S(B$f^{r_2}(\underline{z})=\zeta_1+(s_2,0)$$B$H$J$k$h$&$K(B$r_1,r_2,s_1,s_2 \in {\bf Z}$$B$r8GDj$9$k!%$9$k$H!"(B$\zeta_0+(s_1,0),$ $f^{r_1}(w_0)$$B$O=g=x$,68$$!"(B$\zeta_1+(s_2,0),$ $f^{r_2}(w_0)$$B$N=g=x$b68$&!%(B
$$
    \eta_0 = f^{r_1}(w_0) - (s_1,0), \ \ 
    \eta_1 = f^{r_2}(w_0) - (s_2,0) , 
$$
$B$H$*$1$P!"(B$\zeta_0,\eta_0$$B$N=g=x$,68$$!"(B$\zeta_1,\eta_1$$B$N=g=x$b68$&!%$5$F!"(B$\zeta_0,\eta_1$$B$N=g=x$O68$o$J$$!%$J$<$J$i!"$b$7=g=x$,68$&$H(B$\zeta_0+(s_2,0),$ $f^{r_2}(w_0)$$B$N=g=x$,68$$!"(B$f^{-r_2}(\zeta_0+(s_2,0)),$ $w_0$$B$N=g=x$b68$&$O$:$G$"$k!%$H$3$m$,(B$z_0$$B$O%P!<%3%U<~4|50F;$G$"$k$+$i(B$\pi_1(\zeta_0)<\pi_1(\zeta_1)$$B$+$i(B
$$
   \pi_1(f^{-r_2}(\zeta_0+(s_2,0))) < \pi_1(f^{-r_2}(\zeta_1+(s_2,0)) = 
\pi_1(\underline{z})
$$
$B$,=P$F!"(B$\underline{z}$$B$NA*$SJ}$HL7=b$9$k!%F1MM$K(B$\zeta_1,\eta_0$$B$N=g=x$O68$o$J$$!%$9$J$o$A!"(B$i=0, \cdots, q-1$$B$KBP$7$F!"(B$\pi_1(f^i(\zeta_0)) < \pi_1(f^i(\eta_1)$$B$*$h$S(B$\pi_1(f^i(\zeta_1)) > \pi_1(f^i(\eta_0)$$B$G$"$k!%(B
$B$3$N$H$-!"%1!<%9(B2$B$N>l9g$J$i<!$N#3$D$N$&$A$I$l$+$,@.$jN)$D!%(B

(i) $\pi_1(\eta_0) < \pi_1(\zeta_0)$$B$+$D(B$\pi_1(\zeta_1) \leq \pi_1(\eta_1)$$B!"$^$?$O(B

(ii) $\pi_1(\eta_0) < \pi_1(\zeta_0)$$B$+$D(B$\pi_1(\zeta_0) \leq \pi_1(\eta_1) < \pi_1(\zeta_1)$$B!"$^$?$O(B

(iii) $\pi_1(\zeta_0) \leq \pi_1(\eta_0) < \pi_1(\zeta_1)$$B$+$D(B$\pi_1(\zeta_1) < \pi_1(\eta_1)$.

\noindent
$B0lJ}%1!<%9#1$N>l9g$K$O>r7o(B(i)$B$,@.$jN)$D$O$:$G$"$k!%(B

$\pi_1(\eta_0)<\pi_1(\zeta_0)$$B$H$7!"$"$k(B$i, 0<i<q$$B$KBP$7$F(B$\pi_1(f^i(\eta_0))>\pi_1(f^i(\zeta_0))$$B$H$9$k!%(B$\pi_1(\phi(\eta_0,$ $(i-1)/q))<\pi_1(\zeta_0)$$B$*$h$S(B$\pi_1(\phi(\eta_0,i/q))$ $> \pi_1(\zeta_0)$$B$,@.$jN)$D:G>.$N(B$i>0$$B$r8GDj$9$k!%$M$8$l>r7o$K$h$j!"(B$t \in [0,1/q)$$B$+$D(B$\pi_1(\phi(\eta_0,i/q+t))=\pi_1(\zeta_0)$$B$J$i(B
$$
    \pi_2(\phi(\eta_0,(i-1)/q+t)) > \pi_2(\zeta_0),
$$
$B$G$"$k$3$H$,$o$+$k!%(B$\pi_1(\phi(\eta_0,(j-1)/q)) > \pi_1(\zeta_0)$$B$+$D(B
$\pi_1(\phi(\eta_0, j/q)) \leq \pi_1(\zeta_0)$$B$rK~$?$9:G>.$N(B$j>i$$B$r8GDj$9$k$H!"F1MM$K(B$t \in [0,1/q)$$B$*$h$S(B$\pi_1(\phi(\eta_0,(j-1)/q+t)=\pi_1(\zeta_0)$$B$N$H$-(B
$$
    \pi_2(\phi(\eta_0,(j-1)/q+t)) < \pi_2(\zeta_0),
$$
$B$G$"$k$3$H$,$o$+$k!%$3$N5DO@$r(B$i>q$$B$^$GB3$1$l$P!"(B$\phi(\eta_0,[0,1])$$B$,(B$A \sim \{ \zeta_0 \}$$B$K$*$$$F2D=L$G$J$$$3$H$,$o$+$k!%F1MM$K!"(B$\pi_1(\eta_0) \geq \pi_1(\zeta_0)$$B$N$H$-$bF1$8$3$H$,8@$($k!%$^$?!"F1$85DO@$r$9$l$P!"(B$\phi( \eta_1,$ $[0,1] )$$B$,(B$A \sim \{ \zeta_1 \}$$B$K$*$$$F2D=L$G$J$$$3$H$,=P$k!%>u67$OLO<0E*$K#5?^$KI=$o$7$F$*$$$?!%(B

\vspace*{2.5cm}
\hspace*{6.0cm}    $B?^(B5

\vspace*{2.8cm}
\noindent
{\bf $B<gD%(B.} $\pi_1(\zeta_0) < \pi_1(z_1) < \pi_1(\zeta_1)$$B$J$k(B$z_1 \in A$$B$,$"$C$F!"(B$z_1$$B$O(B$g$$B$NITF0E@$G$"$j!"%k!<%W(B$\phi(z_1,[0,1])$$B$O(B$A \sim \{ \zeta_0, \zeta_1 \}$$B$K$*$$$F2D=L$G$"$k!%(B

\vspace*{0.3cm}
\noindent
\underline{$BCm0U(B}. (1) $B$3$N<gD%$r>ZL@$9$k$K$O$$$/$D$+$N$d$jJ}$,$"$k!%(B5$B?^$K<($5$l$F$$$k$h$&$K!"(B$\zeta_0,\zeta_1,\eta_0$$B$*$h$S(B$\eta_1$$B$N50F;$OHs>o$K4JC1$J7A$NAH$_I3$r$J$7$F$$$k!%$@$+$i6JLL>e$N;~4V<~4|E*>oHyJ,J}Dx<0$NAH$_I3$H<~4|50F;$K4X$9$k(BMatsuoka[17]$B$NDjM}$N7W;;$*$h$S1~MQ$rDL$7$F<gD%$,>ZL@$G$-$k$N$O$=$l$[$I6C$/$Y$-$3$H$G$O$J$$!%0J2<$G<($9=iEyE*>ZL@$O(BMatsuoka$B$NDjM}$N$$$/$D$+$N4JC1$N>l9g$N4v2?3XE*$J8+J}$r<($9!%(B($B!V<h$j=|$-F@$k!W<~4|50F;$H(BNielsen$BM}O@$r4X78$E$1$k(BAsimov and Franks[1]$B$N:G6a$N;E;v$b;2>H$7$FM_$7$$(B.)

(2) $B$3$N>ZL@$K4X$7(BM.Handel$B$N<(:6$K46<U$9$k!%$H$/$KJdBj(B4$B$N0BDj@-$K4X$9$kDD=R$O4v2?3XE*$J%"%W%m!<%A$G$b<($;$?$O$:$G$"$k!%$3$l$K$h$C$F<!@a$N$$$/$D$+$N%9%F%C%W$r4JC1$K$9$k$3$H$,$G$-$?!%(B

\vspace*{0.3cm}
\noindent
\underline{$B<gD%$N>ZL@(B.} $B$^$:%1!<%9(B1$B$N>l9g$N>ZL@$r<($=$&!%%1!<%9(B2$B$OF1MM$N$b$C$H4JC1$J5DO@$G$G$-$k!%(B

\vspace{0.3cm}
\noindent
Case 1: ($\pi_1(\eta_0)<\pi_1(\zeta_0)$$B$+$D(B$\pi_1(\eta_1) \geq \pi_1(\zeta_1)$). $B0J2<$N5-K!$,M-MQ$G$"$k!%(B
$$
  \begin{array}{ll}
B= \{ z \in A: \pi_1(\zeta_0) \leq \pi_1(z) \leq \pi_1(\zeta_1) \}, 
 & \\
b^+_i = \{ (\pi_1(\zeta_i),y) \in A: y> \pi_2(\zeta_i) \}, \ \  i=0,1 
 &  \\
b^-_i = \{ (\pi_1(\zeta_i),y) \in A: y< \pi_2(\zeta_i) \}, \ \  i=0,1 
 & (6$B?^;2>H(B).
  \end{array}
$$

$B0J2<$NDj5A$bM-MQ$G$"$k(B:

$t \in [0,1]$$B$KBP$7$FE@(B$\zeta \in B$$B$G(B$\phi(\zeta,t) \in \phi(B,t) \cap B$$B$J$k$b$N$O!"(B$\phi(\zeta,[0,1])$$B$*$h$S(B$\phi(\zeta,t)$$B$H(B$\zeta$$B$r7k$V@~J,$+$i$J$k%k!<%W$,(B$A \sim \{ \zeta_0,\zeta_1 \}$$B$K$*$$$F2D=L$G$"$k$H$-!";~9o(B$t$$B$K(B$B$$B$G(B\underline{$BNm(B(null)$B$G$"$k(B}$B$H$$$&!%<!$N$h$&$K$*$/!%(B
$$
  S_t = \{ \phi(\zeta,t) \in B: \zeta \in B {\rm $B$O;~9o(B} t 
{\rm $B$K$*$$$F(B} B {\rm $B$K$*$$$F(B} $BNm(B \}.
$$

\vspace*{2.5cm}
\hspace*{6.0cm}    $B?^(B6

\vspace*{2.8cm}
\vspace*{0.3cm}
\noindent
\underline{$BCm0U(B}. $B$[$+$K$b$$$/$D$+(B$S_t$$B$NFCD'$E$1$,$"$k!%$?$H$($P!"(B$\hat{A}$$B$r(B$A \sim \{ \zeta_0, \zeta_1 \}$$B$NIaJWHoJ$$H$7!"(B$\hat{\phi}:\hat{A} \times {\bf R} \rightarrow \hat{A}$$B$O(B$\phi$$B$N;}$A>e$2$G!"(B$\hat{\phi}(\cdot,0)$$B$,(B$\hat{A}$$B>e$N91Ey<LA|$KEy$7$$$b$N$H$7!"(B$\hat{B} \subseteq \hat{A}$$B$O(B$B$$B$N$"$kFCJL$K8GDj$7$?;}$A>e$2$H$9$k$H!"E@(B$\zeta \in S_t$$B$O@53N$K(B$\hat{\phi}(\hat{B},t) \cap \hat{B}$$BFb$NE@$N<M1F$K$J$C$F$$$k!%(B

\vspace*{0.3cm}
$\zeta$$B$,(B$B$$B$K$*$$$FNm$@$H$7$F$b(B$\phi(\zeta,[0,t]) \subseteq B$$B$OI,$:$7$b=P$F$3$J$$!%$7$+$7!"(B$0 \leq i/q \leq t$$B$J$i$$$D$G$b(B$\phi(\zeta,i/q) \in B$$B$G$"$j!"$"$k(B$t' \in [0,t]$$B$*$h$S(B$i=0$$B$^$?$O(B$1$$B$KBP$7$F(B$\pi_1(\phi(\zeta,t'))=\pi_1(\zeta_i)$$B$J$i(B$\phi(\zeta,t') \in b^-_0 \cup b^+_1$$B$G$"$k!%$3$l$O$M$8$l>r7o$+$i=P$k!%(B

\vspace*{2.5cm}
\hspace*{5.0cm}    $B#7?^(B

\vspace*{2.5cm}
$BF1MM$K$M$8$l>r7o$h$j!"3F(B$t \in [0,1]$$B$KBP$7$F<!$rF@$k!%(B
$$
  S_t \cap (b^+_0 \cup b^-_1) = \emptyset .
$$
$B$^$?(B$\sigma:[0,1] \rightarrow B$$B$,(B$\sigma(0) \in b^-_0$$B!"(B$\sigma(1) \in b^+_0$$B$J$i!"3F(B$t \in [0,1]$$B$KBP$7$F(B$\phi(\sigma([0,1]),t) \cap S_t$$B$OJRJ}$NC<E@$,(B$b^-_0$$B$K$"$j$b$&0lJ}$NC<E@$,(B$b^+_1$$B$K$"$k$h$&$J8I$r4^$`!%:G8e$K!"(B$\partial S_t \cap {\rm Int}(B)$$B$O!"(B$J$$B$r(B$b^+_0$$B$^$?$O(B$b^-_1$$BFb$N6h4V$H$7$F!"(B$\phi(J,t)$$B$N7A$N8I$+$i@.$j!"$^$?<!$,@.$jN)$D$3$H$rCm0U$7$F$*$/!%(B
$$
 \partial S_t \cap \partial B \subseteq 
    b^-_0 \cup b^+_1 \cup \{ \zeta_0, \zeta_1 \}
    \cup \{ (x,j) \in B: j=0,1 \}.
$$
$B<!$K(B$\eta_0$$B$H(B$\eta_1$$B$NB8:_$*$h$S(B$\phi$$B$N$b$H$G$N$=$l$i$N50F;$N@-<A$r;H$C$F!"(B$t=1$$B$N$H$-(B$S_t$$B$,O"7k@.J,(B$T_t$$B$G<!$N@-<A$rK~$?$9$b$N$r4^$`$3$H$r<($=$&!%(B

$(\alpha)$ $\partial T_t \cap \{ \zeta_0,\zeta_1 \} = \emptyset$;

$(\beta)$ $\partial T_t$$B$O(B$\phi(J_0,t),$ $\phi(J_1,t)$$B$N7A$N8I$r4^$`!%$?$@$7(B$J_0 \subseteq b^+_0,$ $J_1 \subseteq b^-_1$$B$G$"$k!%$^$?(B$\phi(J_0,t)$$B$H(B$\phi(J_1,t)$$B$O(B$b^-_0 \cup \{ (x,0) \in B \}$$B$*$h$S(B$b^+_1 \cup \{ (x,1) \in B \}$$B$NE@$N$I$A$i$r$b4^$`(B($B#7?^;2>H(B)$B!%(B

\vspace*{7.5cm}
\hspace*{5.0cm}    $B#8?^(B

\vspace*{7.5cm}
$B=89g(B$g^{-1}(T_1)$$B$K4JC1$J;X?t$N5DO@$rE,MQ$9$l$P5a$a$kITF0E@(B$z_1 \in A$$B$,F@$i$l$k!%(B

$\phi(\eta_0,i_0/q),$ $\phi(\eta_1,i_1/q) \in B$$B$rK~$?$9:G>.$N(B$i_0,i_1$$B$r8GDj$7!"$^$?(B$\phi(\eta_0,j_0/q),$ $\phi(\eta_1,j_1/q) \not\in B$$B$rK~$?$9:G>.$N(B$j_0>i_0,$ $j_1>i_1$$B$r8GDj$9$k!%$O$8$a$K(B$j_0 \leq j_1$$B$H2>Dj$7!"0J2<$N>l9g$r9M$($k(B:

\noindent
\underline{$B%1!<%9(B $a$}. $B$9$Y$F$N(B$i=i_0,i_0+1, \cdots, j_0$$B$KBP$7$F!"(B$\pi_1(\phi(\eta_0,i/q)) < \pi_1(\phi(\eta_1,i/q)).$

\noindent
\underline{$B%1!<%9(B $b$}. $B$"$k(B$i,$ $i_0 \leq i < j_0$$B$KBP$7$F!"(B$\pi_1(\phi(\eta_0,i/q)) > \pi_1(\phi(\eta_1,i/q))$ ($B$9$J$o$A!"(B$\eta_0,\eta_1$$B$N50F;$,Mm$s$G$$$J$$$+!"Mm$s$G$$$k$+(B)$B!%(B

\noindent
\underline{$B%1!<%9(B $a$}. $\phi(\eta_0,[0,j_0/q])$$B$K(B$\phi(\eta_0,j_0/q)$$B$H(B$\eta_0$$B$r7k$V@~J,$r$D$J$2$F$G$-$k%k!<%W$O(B$A \sim \{ \zeta_0 \}$$B$K$*$$$F2D=L$G$J$$!%$f$($K!"(B$S_{j_0/q}$$B$O>e$N(B$(\beta)$$B$rK~$?$9@.J,(B$T_{j_0/q}$$B$r4^$`$O$:$G$"$j!"(B$\zeta_0 \not\in T_{j_0/q}$$B$G$"$k(B($B#8?^(Bb$B;2>H(B)$B!%(B

$BF1MM$K(B$\phi(\phi^{-1}(T_{j_0/q},j_0/q),j_1/q)$$B$O(B$(\alpha)$$B$H(B$(\beta)$$B$rK~$?$9@.J,(B$T_{j_1/q}$$B$r4^$`$O$:$G$"$j!"(B
$g(\phi^{-1}($ $T_{j_1/q},j_1/q))$$B$O$7$?$,$C$F(B$S_1$$B$N@.J,$N$&$A!"5a$a$F$$$kJ}$N@.J,$r4^$`$O$:$G$"$k(B($B#8?^(Bc$B;2>H(B)$B!%(B

$B<B:]!"$3$N>l9g(B$z \in g^{-1}(T),$ $i=0,\cdots,q$$B$KBP$7$F(B
$$
 \pi_1(\phi(\eta_0,i/q)) <\pi_1(\phi(z,i/q)) <
 \pi_1(\phi(\eta_1,i/q)),
$$
$B$G$"$k!%$@$+$i(B$g^{-1}(T)$$B$N$I$NE@$b(B$\eta_0$$B$^$?$O(B$\eta_1$$B$H=g=x$,68$o$J$$!%(B($B$3$N$3$H$O0J2<$G%1!<%9(B2$B$G;H$o$l$k(B.)

\vspace*{0.3cm}
\noindent
\underline{$B%1!<%9(B $b$}. $\phi(\eta_0,[0,j_0/q])$$B$K(B$\eta_0$$B$H(B$\phi(\eta_0,j_0/q)$$B$r7k$V@~J,$r$D$J$2$F$G$-$k%k!<%W$O(B$A \sim \{ \zeta_0 \}$$B$K$*$$$F2D=L$G$J$/!"$"$k(B$i,0<i < j_0$$B$KBP$7$F!"(B$\pi_1(\phi(\eta_0,i/q)) > \pi_1(\phi(\eta_1,i/q))$$B$G$"$k$3$H$+$i$o$+$k$H$*$j!"(B$S_{j_0/q}$$B$O>e$N(B$(\alpha)$$B$*$h$S(B$(\beta)$$B$rK~$?$9@.J,(B$T_{j_0/q}$$B$r4^$`$+(B($B$3$N>l9g!"(B$g(\phi^{-1}(T_{j_0/q},j_0/q))$$B$N@.J,$,5a$a$k=89g$G$"$k(B)$B!"(B$S_{j_0/q}$$B$O(B$\{ \zeta_1 \}$$B$H6&DLE@$r;}$?$J$$@.J,(B$T_{j_0/q}$$B$G!"(B$\partial T_{j_0/q}$$B$,8I(B$\phi(J_0,j_0/q)$$B$*$h$S(B$\phi(J_1,j_0/q)$$B$r4^$`$h$&$J$b$N!"$r4^$`!%$?$@$7!"$3$3$G(B$J_0,$ $J_1$$B$O$=$l$>$l(B$b^+_0,$ $b^-_1$$B$K4^$^$l$k6h4V$G$"$j!"(B$\phi(J_0,j_0/q)$$B$ON>C<$,(B$b^-_0$$B$K4^$^$l$k$,!"(B$\phi(J_1,j_0/q)$$B$ON>C<$,(B$b^-_0$$B$K4^$^$l$k$+!"0lC<$,(B$b^-_0 \sim \{ \zeta_0 \}$$BFb$K$"$j!"B>C<$,(B$b^+_1 \sim \{ \zeta_1 \}$$B$K$"$k$+$I$A$i$+$G$"$k(B($B#8?^(Bf$B;2>H(B)$B!%$3$l$i$N8I$NN>C<E@$,$H$b$K(B$b^-_0$$B$K$"$l$P!"N>C<E@$r(B$b^-_0$$B$KCV$$$?$^$^(B$B \sim \phi(\eta_1,j_0/q)$$BFb$G$O(B$b^-_0$$B$K%[%b%H!<%W$G$J$$!%$7$+$7$3$N$H$-(B$\phi(\phi^{-1}(T_{j_0/q}, j_0/q), j_1/q)$$B$O(B$(\alpha)$$B$*$h$S(B$(\beta)$$B$rK~$?$9O"7k@.J,(B$T_{j_1/q}$$B$r4^$`$O$:$G$"$k$+$i!"(B$g(\phi^{-1}(T_{j_1/q},j_1/q))$$B$O5a$a$k@.J,$r4^$`(B($B#8?^(Bg$B;2>H(B)$B!%(B$j_1<j_0$$B$N>l9g$N>ZL@$O!"$$$^=R$Y$?>l9g$HBP>NE*$G$"$k!%(B

\vspace*{2.5cm}
\hspace*{6.0cm}    $B?^(B9

\vspace*{2.8cm}
$B_1=g^{-1}(T_1)$$B$H$9$k!%$3$N$H$-!"(B$T_1$$B$N@-<A(B$(\alpha)$$B$*$h$S(B$(\beta)$$B$+$i$o$+$k$H$*$j!"(B$z \in A$$B$r(B$\partial B_1$$B$N$^$o$j$KF0$+$7$F%Y%/%H%k(B$z-g(z)$$B$H(B$x$$B<4$N$J$93QEY$NA4JQ2=$r7W;;$9$l$P!"7k2L$O(B($B2s$kJ}8~$r$I$&$H$k$+$K$K0MB8$7$F(B)$\pm 2 \pi$$B$H$J$k(B($B#9?^;2>H(B)$B!%$@$+$i!"(B$g$$B$OITF0E@(B$z_1 \in B_1 \cap T_1$$B$r;}$D$O$:$G$"$k!%$=$N>e!"(B$T_1 \subseteq S_1$$B$G$"$k$+$i!"(B$\phi(z_1,[0,1])$$B$O(B$A \sim \{ \zeta_0,\zeta_1 \}$$B$K$*$$$F2D=L$N$O$:$G$"$j!"(B$\zeta_0,$ $\zeta_1 \not\in T_1$$B$G$"$k$+$i!"(B$z_1 \not\in \{ \zeta_0, \zeta_1 \}$$B$G$"$k$3$H$,H=$k!%(B

\vspace*{0.3cm}
$z_1$$B$O(B$g$$B$NITF0E@$G$"$k$+$i!"(B$f$$B$N(B$p/q-$$B<~4|E@$N$O$:$G$"$k!%(B${\cal O}(f,$ $z_0)$$B$O(B$B \sim \{ \zeta_0, \zeta_1 \}$$B$KE@$r;}$?$J$$$+$i(B$z_1 \not\in {\cal O}(f,z_0)$$B$G$"$k!%:G8e$K!"(B$\phi(z_1,[0,1])$$B$O(B$A \sim \{ \zeta_0,\zeta_1 \}$$B$K$*$$$F2D=L$G$"$k$+$i!"(B$i=0, \cdots, q$$B$KBP$7$F(B
$$
   \pi_1(f^i(\zeta_0)) < \pi_1(f^i(z_1)) < 
   \pi_1(f^i(\zeta_1)), 
$$
$B$G$"$j!"$f$($K(B$z_1$$B$O(B$f$$B$N(B$p/q-$$B%P!<%3%U<~4|E@$G$J$1$l$P$J$i$J$$!%%1!<%9(B$a$$B$N>l9g$O>e$KCm0U$7$?$h$&$KF1$8$/(B
$\pi_1(f^i(\eta_0)) < \pi_1(f^i(z_1)) < \pi_1(f^i(\eta_1))$$B$rF@$k!%(B($B$3$l$O0J2<$G%1!<%9(B2$B$N>l9g$KMxMQ$9$k(B.) 

$BJdBj$N0BDj@-$NItJ,$rF@$k$?$a$K!"<!$N$3$H$KCm0U$9$k!%$9$J$o$A!"(B$\tilde{f}$$B$,(B$f$$B$K==J,6a$/!"(B$\tilde{g}( \cdot ) \equiv \tilde{f}( \cdot ) - (p,0)$$B$G$"$l$P!"(B$\tilde{g}:B_1 \rightarrow A$$B$N;X?t$O$d$O$j%<%m$G$J$$$+$i!"(B$\tilde{g}$$B$OITF0E@(B$\tilde{z}_1 \in \tilde{g}(B_1) \cap B_1$$B$r;}$D!%(B$T_1 \cap B_1$$B$O(B($A$$B$KAjBPE*$K(B)$B$$B$NFbIt$K4^$^$l$k$+$i!"$3$l$O(B$\tilde{g}(B_1) \cap B_1$$B$KBP$7$F$b@.N)$A!"$7$?$,$C$F!"(B$\tilde{f}$$B$,(B$f$$B$K==J,6a$$$H$-$K$O(B$i=0, \cdots, q$$B$KBP$7$F<!$rF@$k!%(B
$$
   \pi_1(\tilde{f}^i(\zeta_0)) < \pi_1(\tilde{f}^i(\tilde{z}_1)) < \pi_1(\tilde{f}^i(\zeta_0)). 
$$
$B$3$N$H$->e$HF1$8$h$&$K(B($B$?$H$((B$\zeta_0,$ $\zeta_1$$B$,I,$:$7$b(B$\tilde{f}$$B$N<~4|E@$K6a$/$J$/$H$b(B)$B!"(B$\tilde{z}_1$$B$,(B$\tilde{f}$$B$N(B$p/q-$$B%P!<%3%U<~4|E@$G$"$k$3$H$,=P$F!"%1!<%9(B1$B$N>l9g$N>ZL@$,40N;$7$?!%(B

\noindent
\underline{$B%1!<%9(B 2}. $B$3$N>l9g(B$i=0, \cdots, q$$B$KBP$7$F(B$\pi_1(f^i(\eta_0)) < \pi_1(f^i(\eta_1))$$B$G$"$k$3$H$rCN$C$F$$$k!%$f$($K!"(B$\pi_1(\zeta_0) \leq \pi_1(\eta_0) < \pi_1(\zeta_1)$$B$N$H$-JdBj(B$3'$$B$,E,MQ$G$-$F(B$g$$B$KF1H<$9$k(B$\tilde{\phi}:A \times {\bf R} \rightarrow A$$B$,F@$i$l$F!"(B$\tilde{\phi}(\eta_0,t)=\eta_0,$ $\tilde{\phi}(\zeta_1,t)=\zeta_1$$B$,$9$Y$F$N(B$t$$B$KBP$7$F@.$jN)$D!%$3$N$H$-!"(B$\tilde{\phi}(\zeta_0,[0,1])$$B$O(B$A \sim \{ \eta_0 \}$$B$K$*$$$F2D=L$G$J$$$,!"(B$\tilde{\phi}(\zeta_0,[0,1])$$B$H(B$\tilde{\phi}(\eta_1,[0,1])$$B$OMm$s$G$$$J$$!%$@$+$i!"(B$B=\{ z \in A : \pi_1(\eta_0) \leq \pi_1(z) \leq \pi_1(\zeta_1) \}$$B$H<h$C$F>e$N%1!<%9(B1a$B$rE,MQ$G$-!"(B$g$$B$NITF0E@(B$z_1$$B$rF@$k!%%1!<%9(B1a$B$+$i$o$+$k$h$&$K!"(B$\tilde{\phi}(z_1,[0,1])$$B$O(B
$\zeta_0,$ $\zeta_1,$ $\eta_0,$$B$*$h$S(B$\eta_1$$B$N(B$\tilde{\phi}$$B$K$h$k50F;$N$I$l$H$bMm$s$G$$$J$$!%>ZL@$N;D$j$OA0$HF1$8$G$"$k!%(B

$\pi_1(\zeta_0) < \pi_1(\eta_1) \leq \pi_1(\zeta_1)$$B$N>l9g$O$b$A$m$s>e$HBP>NE*$G$"$k$+$i!"<gD%$*$h$SJdBj$N>ZL@$O407k$7$?!%(B \hfill $\Box$
