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\begin{flushright}
$BC+@n@6N4(B $BLu(B
\end{flushright}

\noindent
{\it Michigan Math. J.} {\bf 24} (1977), 21-31.

\begin{center}
{\Large
{\bf $B%]%"%s%+%l!&%P!<%3%U$NITF0E@DjM}$N>ZL@(B \\
Proof of the Poincar\'{e}-Birkhoff fixed point theorem}
}

\vspace{0.5cm}
M. Brown and W.D. Neumann
\end{center}

\vspace{0.3cm}
\noindent
{\bf 1. $B=x(B.} 

$B%]%"%s%+%l!&%P!<%3%U$NITF0E@DjM}(B($B%]%"%s%+%l$N:G8e$N4v2?3XDjM}$H$b8F$P$l$k(B)$B$N<gD%$K$h$l$P!"1_4D$N$$$o$f$kJ]B,$M$8$l<LA|$K$OITF0E@$,>/$J$/$H$b(B2$B$D$"$k!%$3$NDjM}$O%]%"%s%+%l(B[3]$B$,;`$ND>A0$KM=A[$H$7$FDj<02=$7!"FC<l$J>l9g$K$D$$$F>ZL@$7$?!%(B1913$BG/$K%8%g!<%8!&%P!<%3%U(B[1]$B$O>ZL@$rH/I=$7$?!%$?$@$7$3$N>ZL@$O(B1$B$D$NITF0E@$K4X$7$F$O@5$7$$$,!"(B2$B$DL\$NITF0E@$NB8:_$rF3$/$N$K(B1$B$DL\$NITF0E@$,;X?t(B0$B$r;}$D2DG=@-$r8+F($7$F$$$?!%$3$N4V0c$$$O(B1925$BG/$N<+?H$NO@J8(B[2]$B$G=$@5$5$l$?!%$3$NBhFs$NO@J8$G$O!VJ]B,!W$N2>Dj$r=c?h$K0LAjE*$J>r7o$KCV$-49$(!"!VF1Aj<LA|!W$r$b$C$H0lHL$N>u67$KCV$-49$($F!"DjM}$r0lHL2=$7$F>ZL@$7$?!%$H$3$m$,!"?t3X<T$NCf$K$O!"$3$N>ZL@$b4V0c$C$F$$$k$H<gD%$9$k?M$,$*$j!"$3$3?tG/!"#2$DL\$NITF0E@$NB8:_$N@5$7$$>ZL@$rF@$h$&$H$9$k$$$/$D$+$N9-HO0O$K$o$?$kEXNO$,$J$5$l$F$-$?!%(B

$B$3$NO@J8$G$O!"#1$DL\$NITF0E@$NB8:_$K4X$9$k%P!<%3%U$NM-L>$J$b$H$b$H$N>ZL@$rC1=c$K=$@5$7$F#2$D$NITF0E@$NB8:_$r<($9=iEyE*$J>ZL@$rM?$($k!%BhFs$NITF0E@$rF@$k$?$a$N=$@5$O!"%P!<%3%U$,(B1925$BG/$K0LAjHG$N>ZL@$N:]$K#1$D$NITF0E@$+$i#2$D$NITF0E@$rF@$k$N$K%9%1%C%A$7$?$b$N$HK\<AE*$KF1$8$G$"$k!%(B

$B$7$?$,$C$F!"$3$NO@J8$O$"$k0UL#$G2r@bE*$JO@J8$G$"$k!%>ZL@$r2DG=$J8B$j8+DL$7$h$/$9$k$?$a$K!"3F6-3&$r0lDj$N3QEY$@$12sE>$9$k1_4D$N$M$8$lF1Aj<LA|$H$$$&$b$C$H$b4JC1$J>u67$K@)8B$9$k!%:G=*@a$G;XE&$9$k$h$&$K!">ZL@$O$3$N@)8B$r<h$j5n$C$?>l9g$X$[$\0l8l0l8lK]Lu2DG=$G$"$k!%$^$?!">ZL@$O1_4D>e$NI8=`E*%j%e%Y!<%0B,EY0J30$N0lHL$NB,EY$X$b3HD%$G$-$k!%(B

$B>ZL@$O5?OG$K$5$i$5$l$F$$$k%P!<%3%U$N>ZL@$KHs>o$K6a$$$N$G!"I,MW0J>e$K>\$7$$5-=R$r9T$J$C$?!%$3$l$O$3$NO@J8$,2r@bE*$J@-3J$r;}$D$3$H$H50$r0l$K$7$F$$$k!%(B

\vspace{0.3cm}
\noindent
{\bf 2. $BDjM}$NDD=R(B}

$A = \{ P \in {\bf R}^2: 1 \leq || P || \leq 2 \}$$B$r1_4D$H$9$k!%J88%$G$OF1Aj<LA|(B$g: A \rightarrow A$$B$,IaDL!V$M$8$lF1Aj<LA|!W$H8F$P$l$k$N$O!"!V(B$A$$B$N(B2$B$D$N6-3&$rH?BP$N3QEYJ}8~$K2sE>$5$;$k!W$H$-$G$"$k!%$3$l$O[#Kf$G$"$k!%H?;~7W2s$j$N(B$\theta$$B$N2sE>$O;~7W2s$j$N(B$2 \pi - \theta$$B$N2sE>$HF1$8$G$"$k!%$3$N[#Kf$5$r2r>C$9$kDL>o$N$d$jJ}$O(B$A$$B$NIaJWHoJ$$K9T$/$3$H$G$"$k!%$9$J$o$A!"F1Aj<LA|(B$g: A \rightarrow A$$B$,(B\underline{$B$M$8$lF1Aj<LA|(B}$B$H8F$P$l$k$N$O!"(B$\partial A$$B>e$KITF0E@$r;}$?$:!"(B$A$$B$NIaJWHoJ$(B$\tilde{A} \tilde{=} S = \{(x,y) \in {\bf R}^2: 0 \leq y \leq 1 \}$$B$NF1Aj<LA|(B$\tilde{g}: \tilde{A} \rightarrow \tilde{A}$$B$K;}$A>e$,$j!"$7$+$b(B$\tilde{A}$$B$N#2$D$N6-3&@.J,$rH?BPJ}8~$KF0$+$9$H$-$G$"$k!%(B

$\tilde{g}$$B$r(B$S$$B$NF1Aj<LA|(B$h$$B$H8+$J$7!"$3$N(B$h$$B$r<+L@$J;EJ}$G(B${\bf R}^2$$BA4BN$K3HD%$9$k$3$H$K$h$C$F!"$3$l$+$i>ZL@$9$kDjM}$r<!$N$h$&$KDj<02=$G$-$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B.} $h: {\bf R}^2 \rightarrow {\bf R}^2$$B$OJ]B,$JF1Aj<LA|$G!"$"$k(B$r_1, r_2 > 0$$B$KBP$7(B
$$ \begin{array}{ll}
  h(x,y) = (x - r_1, y), & y \geq 1; \\
  h(x,y) = (x + r_2, y), & y \leq 0; \\
  h(x+2 \pi, y) = h(x,y) + ( 2 \pi, 0),
   \end{array}
$$
$B$rK~$?$9$H$9$k!%$3$N$H$-(B$h$$B$OF1$8<~4|B2$KB0$5$J$$#2$D$N0[$J$kITF0E@(B$F_1$$B$H(B$F_2$$B$r;}$D!%$D$^$j!"(B$F_1 - F_2$$B$O(B$(2 \pi, 0)$$B$N@0?tG\$G$O$J$$!%(B

\vspace{0.3cm}
\noindent
\underline{$BCm0U(B.} $B<~4|@-$N>r7o(B$h(x+2 \pi, y) = h(x,y) + ( 2 \pi, 0)$$B$O!"(B$S = \{(x,y): 0 \leq y \leq 1 \}$$B$X$N(B$h$$B$N@)8B$,!"HoJ$<LA|(B$\pi: S \rightarrow A,$ $\pi(x,y) = ((y+1) \cos x, (y+1) \sin x)$$B$K$h$k<LA|(B$g: A \rightarrow A$$B$N;}$A>e$2$G$"$k$H$$$&>r7o$K@53N$KEy$7$$!%(B

\vspace{0.3cm}
$B2f!9$O(B${\bf R}^2$$B$NI8=`E*LL@QB,EY(B$dxdy$$B$r9M$($F$$$k$+$i!"(B$A$$BFb$G$OHsI8=`E*LL@Q!"$D$^$j(B$rdr d \theta$$B$N$+$o$j$K(B$dr d\theta$$B$r9M$($F$$$k$h$&$K8+$($k$+$b$7$l$J$$!%$7$+$7!"6K:BI8$G(B$\phi(r, \theta) = ((r^2+2)/3, \theta)$$B$GM?$($i$l$kF1Aj<LA|(B$\phi: A \rightarrow A$$B$K$h$C$F0lJ}$NB,EY$O$b$&$R$H$D$NB,EY$NDj?tG\$K$&$D$k!%(B

$BDjM}$N>ZL@$r9T$J$&$?$a$K!"$^$:2sE>?t$H;X?t$K$D$$$F$$$/$D$+=R$Y$F$*$/I,MW$,$"$k!%$3$l$O>ZL@$KI,MW$JM#0l$N%H%]%m%8!<$G$"$k!%>ZL@$K$O$b$C$H$b4pK\E*$JHoJ$$N@-<A$7$+;H$o$J$$!%(B

\vspace{0.3cm}
\noindent
{\bf 3. $B2sE>?t$H;X?t(B}

\noindent
\underline{$BMQ8l(B}. $P$$B$H(B$Q$$B$,(B${\bf R}^2$$B$N0[$J$kE@$N$H$-!"(B$P$\underline{$B$+$i(B}$Q$\underline{$B$X$NJ}8~(B}$B$H$O(B$D(P,Q) = (Q - P)/ || Q - P ||$$B$N$3$H$G$"$k!%(B

\vspace{0.3cm}
$X \subset {\bf R}^2$$B$rItJ,=89g$H$7!"(B$h: X \rightarrow {\bf R}^2$$B$r(B$X$$B$+$i(B${\bf R}^2$$B$NCf$X$NF1Aj<LA|$GITF0E@$r;}$?$J$$$b$N$H$9$k!%$D$^$j!"$9$Y$F$N(B$P \in X$$B$KBP$7$F(B$h(P) \neq P$$B$H$9$k!%(B$C$$B$,(B$X$$BFb$NG$0U$N6J@~$N$H$-!"(B$h$\underline{$B$K4X$9$k(B}$C$\underline{$B$N;X?t(B}$B$r!"(B$P$$B$,6J@~(B$C$$B$K1h$C$FF0$/$H$-J}8~(B$D(P,h(P))$$B$,9T$J$&A42sE>$H$7$?$$!%$?$H$($P!"?^(B1$B$K$*$$$F$3$NJ}8~$O;~7W2s$j$K(B($BIiJ}8~$K(B)$1 + 1/2$$B2sE>$9$k!%$@$+$i;X?t$O(B$-(1+1/2)$$B$G$"$k!%(B

\vspace{3cm}
\hspace*{6cm} Figure 1

\newpage
$B$-$A$s$H$7$?Dj5A$r$9$k$?$a$K!"$O$8$a$K?7$7$$<LA|(B
$$
\overline{h}: X \rightarrow S^1 = \{ (x,y) \in {\bf R}^2: 
x^2 + y^2 = 1 \} 
$$
$B$r(B
$$
\overline{h}(P) = D(P, h(P)) = (h(P) - P) / || h(P) - P ||,
$$
$B$GDj5A$9$k!%$3$N$H$-G$0U$N6J@~(B$C: [a,b] \rightarrow X$$B$KBP$7$F!"(B$[a,b]$$B$OC1O"7k$G$"$k$+$i!"9g@.<LA|(B$\overline{h}C: [a,b] \rightarrow S^1$$B$r(B$S^1$$B$NIaJWHoJ$$X;}$A>e$2$k$3$H$,$G$-$k!%(B

\vspace{1cm}
\hspace*{6cm} $B?^$"$j(B

\vspace*{2.5cm}
$B$3$3$G(B$\pi$$B$OHoJ$<LA|(B$\pi(r) = (\cos (r), \sin(r))$$B$G$"$k!%;}$A>e$2(B$\tilde{C}$$B$O(B${\bf R}$$B$NHoJ$JQ49$NITDj@-$r=|$$$F0l0U$G$"$k!%$9$J$o$A!"(B$2 \pi$$B$N@0?tG\$r2C$($kITDj@-$,$"$k!%$f$($K(B$\tilde{C}(b) - \tilde{C}(a)$$B$O;}$A>e$2$K0M$i$J$$!%$@$+$i<!$N$h$&$KDj5A$G$-$k!%(B
$$
  Ind_h C = (\tilde{C}(b) - \tilde{C}(a))/2 \pi.
$$

\vspace{0.3cm}
\noindent
\underline{$B;X?t$N@-<A(B.}

1. $B>e$GF3F~$7$?6J@~(B$C$$B$^$?$OF1Aj<LA|(B$h$$B$N(B1-$B%Q%i%a!<%?B2$KBP$7$F(B$Ind_h C$$B$O%Q%i%a!<%?$H$H$b$KO"B3$KJQ2=$9$k!%(B

2. $C$$B$,E@(B$A$$B$+$iE@(B$B$$B$^$G1d$S$F$$$l$P!"(B$Ind_h C$$B$O(B$mod \ 1$$B$GJ}8~(B$D(A, h(A))$$B$H(B$D(B, h(B))$$B$N4V$N3QEY$N(B$(1/ 2 \pi)$$BG\$KEy$7$$!%(B

3. $C=C_1C_2$$B$,(B$C_1$$B$H(B$C_2$$B$rC<E@$G$D$J$$$@$b$N$J$i(B($B$9$J$o$A!"(B$a < c < b$$B$H$7$F(B$C_1 =C|[a,c]$$B$*$h$S(B$C_2 = C|[c,b]$)$B!"(B$Ind_h C = Ind_h C_1 + Ind_h C_2$$B$G$"$k!%(B$C$$B$r5UJ}8~$K$?$I$k6J@~$r(B$-C$$B$H$9$l$P!"(B$Ind_h(-C) = - Ind_h C$$B$G$"$k!%(B

4. $Ind_h C = Ind_{h^{-1}}(h(C))$.

\vspace{0.3cm}
$B:G=i$N@-<A$O%[%b%H%T!<;}$A>e$2$N@-<A$rHoJ$$XC1=c$K1~MQ$7$?$b$N$G$"$k!%$9$J$o$A!"<LA|(B$\overline{h}C: [a,b] \rightarrow S^1$$B$NG$0U$N%[%b%H%T!<$O(B$\tilde{C}$$B$N%[%b%H%T!<$K;}$A>e$,$k!%@-<A(B2$B$H(B3$B$O<+L@$G$"$k!%@-<A(B4$B$O4JC1$K7W;;$+$i=P$k!%;v<B!"(B$Ind_{h^{-1}}(h(C))$$B$r7W;;$9$k$N$K!"J}8~(B$D(h(P), P)$$B$N2sE>$KCmL\$9$k!%$3$l$O(B$\pi$$B$K(B$Ind_h C$$B$r7W;;$9$k$H$-$NJ}8~(B$D(P, h(P))$$B$r2C$($?$b$N$G$"$k!%$@$+$iF1$8J}8~$KF1$8NL$@$12sE>$9$k!%(B

$B:G8e$K!":#8e$N$?$a$K!">e$G=R$Y$?%[%b%H%T!<;}$A>e$2$N@-<A$NAjBPHG$K$D$$$F?($l$F$*$/!%C<E@(B$\overline{h}C(a)$$B$H(B$\overline{h}C(b)$$B$r8GDj$9$k<LA|(B$\overline{h}C$$B$NG$0U$N%[%b%H%T!<$O(B$\tilde{C}(a)$$B$H(B$\tilde{C}(b)$$B$r8GDj$9$k(B$\tilde{C}$$B$N%[%b%H%T!<$K;}$A>e$,$k!%$@$+$i(B$Ind_h C$$B$r7W;;$9$k$N$K!"C<E@$r8GDj$7$F$*$/8B$j!"$O$8$a$K(B$\overline{h}C$$B>e$G%[%b%H%T!<$r9T$J$C$F!"$=$l$r4JC12=$9$k$3$H$,5v$5$l$k!%(B

\newpage
%\vspace{0.3cm}
\noindent
{\bf 4. $BDjM}$N>ZL@(B}

$B$O$8$a$K5-K!$rF3F~$9$k!%<!$N$h$&$K$*$/!%(B
$$ \begin{array}{l}
 H_+ = \{ (x,y) \in {\bf R}^2: y \geq 1 \}, \\
 H_- = \{ (x,y) \in {\bf R}^2: y \leq 0 \}, \\
 S = \{ (x,y) \in {\bf R}^2: 0 \leq y \leq 1 \} .
   \end{array}
$$

$B$=$3$G(B$h: {\bf R}^2 \rightarrow {\bf R}^2$$B$rDjM}$N$b$N$9$k!%$9$J$o$A!"(B$h$$B$OLL@QJ]B8!"$^$?$"$k(B$r_1, r_2 > 0$$B$KBP$7$F(B
$$ \begin{array}{ll}
  h(x,y) = (x - r_1, y), & (x,y) \in H_+; \\
  h(x,y) = (x + r_2, y), & (x,y) \in H_-; \\
  h(x+2 \pi, y) = h(x,y) + ( 2 \pi, 0).
   \end{array}
$$

$F$$B$,(B$h$$B$NITF0E@$J$i!"$=$N$9$Y$F$N<~4|A|(B$F + (2k \pi, 0), k \in {\bf Z}$$B$b$=$&$G$"$k!%(B$h$$B$,$?$+$@$+#1$D$7$+$3$N$h$&$JITF0E@$N<~4|B2$r;}$?$J$$$H2>Dj$7$F0J2<$N$h$&$KL7=b$rF3$3$&!%(B$H_-$$B$+$i(B$H_+$$B$K1d$S$k#2K\$N6J@~(B$C$$B$H(B$C'$$B$G(B$h$$B$N$9$Y$F$NITF0E@$rHr$1$FDL$j!"(B$Ind_h C = 1/2 = - Ind_h C'$$B$J$k$b$N$r9=@.$7$h$&!%$3$l$O<!$NJdBj$KL7=b$9$k!%(B

\vspace{4.5cm}
\hspace*{5cm} Figure 2.

\vspace{4.3cm}
\noindent
{\bf $BJdBj(B.} $H_-$$B$+$i(B$H_+$$B$K1d$S!"(B$h$$B$N$I$NITF0E@$r$bDL$i$J$$G$0U$N6J@~(B$C$$B$KBP$7$F!"(B

(a) $Ind_h C \equiv 1/2 (mod \ 1)$;

(b) $Ind_h C$$B$O(B$C$$B$K0MB8$7$J$$!%(B

\vspace{0.3cm}
\noindent
\underline{$B>ZL@(B.} (a)$B$O;X?t$N@-<A(B2$B$+$iL@$i$+$G$"$k!%(B(b)$B$r>ZL@$9$k$?$a$K(BFix$(h)$$B$r(B$h$$B$NITF0E@=89g$H$7$h$&!%(BFix$(h) = \emptyset$$B$G$"$k$+(BFix$(h) = \{ F + (2 \pi k, 0): k \in {\bf Z} \}$$B$G$"$k!%$5$i$K(B$A_i \in H_-$$B$+$i(B$B_i \in H_+, i=1,2$$B$XAv$k(B$C_i$$B$O(B${\bf R}^2 - {\rm Fix}(h)$$BFb$N#2$D$N6J@~$G$"$k$H$9$k!%(B$H_+$$BFb$G(B$B_1$$B$+$i(B$B_2$$B$XAv$kG$0U$N6J@~(B$C_3$$B$*$h$S(B$H_-$$BFb$G(B$A_2$$B$+$i(B$A_1$$B$XAv$kG$0U$N6J@~(B$C_4$$B$r$H$j!"(B$C'$$B$rJD6J@~(B$C_1C_3(-C_2)C_4$$B$H$9$k(B(Figure 2)$B!%(B$D(P,h(P))$$B$O(B$H_+$$B$*$h$S(B$H_-$$BFb$G0lDj$G$"$k$+$i!"(B$Ind_hC' =Ind_hC_1 - Ind_hC_2$$B$G$"$k!%$@$+$i(B$Ind_hC_1 = Ind_hC_2$$B$r<($9$K$O(B$Ind_hC' =0$$B$r<($5$J$P$J$i$J$$!%$H$3$m$,4pK\72(B$\pi_1({\bf R}^2 - {\rm Fix}(h), A_1)$$B$O!"(B$A_1$$B$+$i=PH/$7$F6J@~(B$C_0$$B$K1h$C$FAv$j!"(B($B$b$7$"$k$H$7$F(B)$BITF0E@$N6a$/$KC#$7!"$3$NITF0E@$N$^$o$j$r0l2s$j$7!"(B$-C_0$$B$K1h$C$F(B$A_1$$B$KLa$kF;$K$h$C$F@8@.$5$l$k!%$f$($K(B$C'$$B$O$3$N$h$&$JF;$N9g@.$X$HJQ7A2DG=$@$+$i!"$3$N$h$&$JG$0U$NF;$K4X$7$F(B$Ind_h$$B$,%<%m$G$"$k$3$H$r<($;$P==J,$G$"$k(B(Figure 3$B;2>H(B)$B!%$H$3$m$,!"C10l$NITF0E@$r=d$k$3$N$h$&$JG$0U$NF;$O(BFigure 4$B$K<($5$l$k$h$&$JF;(B$C''$$B$KJQ7A2DG=$G$"$k!%(B$Ind_hC''$$B$X$N>e2<$N?eJ?@~$N4sM?$O$H$b$K%<%m$G$"$j!"#2K\$N?bD>$J@~J,$N4sM?$O(B$h$$B$N<~4|@-$K$h$C$F8_$$$KBG$A>C$79g$&!%$@$+$i(B$Ind_hC'' =0$$B$G$"$j!"JdBj$O>ZL@$5$l$?!%1_4D$N>l9g$K(BLefschetz$B$NITF0E@DjM}$NFCJL$J>l9g$r>ZL@$7$?$K$9$.$J$$$3$H$KFI<T$O5$$E$+$l$?$H;W$&!%(B

\vspace{3.5cm}
\hspace*{5cm} Figure 3.

\vspace{7.0cm}
\hspace*{5cm} Figure 4.

\vspace{3.5cm}
$BJX59>e!"ITF0E@$N<~4|B2$,$b$7$"$l$P!"$=$l$OD>@~(B$x \equiv 0 (mod \ 2 \pi)$$B>e$K$"$k$H2>Dj$9$k!%$3$l$O<+L@$J:BI8JQ49$G<B8=$G$-$k!%(B

$W$$B$O1tD>BS$N<~4|OB(B
$$
  W = \{ (x,y): $B$"$k(B k \in {\bf Z} $B$KBP$7$F(B 2k\pi + \pi/2 \leq x \leq 2k \pi + 3 \pi/2 \} ,
$$
$B$rI=$o$9$b$N$H$9$k!%$@$+$i(B$W$$B$OITF0E@$N$I$l$b4^$^$J$$!%(B$\varepsilon > 0$$B$rA*$s$G(B
$$ \begin{array}{ccc}
  || P - h(P) || > \varepsilon & {\rm for \ all} & P \in W,
   \end{array}
$$
$B$H$9$k!%$3$l$O2DG=$G$"$k!%$9$J$o$A<~4|@-$K$h$j!"%3%s%Q%/%H$JNN0h(B$\{ (x,y): \pi/2 \leq x \leq 3 \pi/2, 0 \leq y \leq 1 \}$$B$K$*$1$k(B$P$$B$KBP$7$F$N$_$3$N>r7o$rK~$?$;$P$h$$$3$H$OL@$i$+$G$"$k!%$^$?$3$NNN0h>e$G4X?t(B$\phi(P) = || P - h(P) ||$$B$OO"B3$+$D@5$G$"$k$+$i!"@5$N6K>.CM$r;}$D!%(B

$T:{\bf R}^2 \rightarrow {\bf R}^2$$B$r(B$T(x,y) = (x,y+(\varepsilon/2)(|\cos(x)| - \cos(x)))$$B$GDj5A$9$k!%(B$T$$B$OG$0U$N1tD>@~>e$GC1$J$k>eJ}$X$N0\F0$G$"$k$+$i(B(Figure 5)$B!"=iEy2r@O(B(calculus)$B$h$jLL@QJ]B8$G$"$k$3$H$r3NG'$7$FM_$7$$!%$5$i$K!"(B$T$$B$O(B$W$$B$NE@$N$_$rF0$+$7!"$7$+$b9b!9(B$\varepsilon$$B$7$+F0$+$5$J$$!%$f$($K!"(B$Th$$B$O(B$W$$BFb$KITF0E@$r;}$?$J$$!%$J$<$J$i!"(B$P \in W$$B$,(B$P=Th(P)$$B$J$i!"(B
$$
  \varepsilon \geq ||Th(P) - h(P)|| = ||P - h(P)|| > \varepsilon
$$
$B$H$J$C$F$7$^$&$+$i$G$"$k!%(B

\vspace{3.5cm}
\hspace*{5cm} Figure 5.

\vspace{4.5cm}
$B;D$j$N5DO@$NF;6Z$r$^$H$a$F$*$3$&!%E@(B$P_0 \in H_-$$B$,$"$C$F!"$"$k(B$n$$B$KBP$7$F(B$(Th)^n P_0 \in H_+$$B$G$"$k$3$H$r<($9!%$3$l$K$h$j!"(B$H_-$$B$+$i(B$H_+$$B$X$N(B$Th$$B$N!VN.@~!W$H$J$k6J@~(B$C$$B$r8+$D$1$k$3$H$,$G$-$k!%$9$J$o$A!"$3$N6J@~$O(B($BC<E@$N(B1$B$D$N6a$/$r=|$$$F(B)$Th$$B$K$h$j$=$l<+?H$K<L$5$l$k!%$H$3$m$,!";X?t$ON.@~$K1h$C$F$O4JC1$K7W;;$G$-$k!%(B$Ind_{Th}C$$B$,(B$1/2$$B$KHs>o$K6a$$$3$H$,$o$+$k!%(B$Th$$B$O(B$h$$B$K6a$$$+$i!"(B$Ind_hC = 1/2$$B$G$"$k$3$H$rF3$1$k!%4JC1$JBP>N@-$N5DO@$+$i!"(B$Ind_h C' = -1/2$$B$J$k$b$&#1$D$N6J@~(B$C'$$B$,B8:_$9$k$+$i!"5a$a$kL7=b$,F3$1$k!%(B

$B>e$G=R$Y$?E@(B$P_0$$B$,B8:_$9$k$3$H$r8+$k$?$a$K!"<!$N$h$&$K?J$`!%(B
$$ \begin{array}{cc}
  D_0 = H_- - (Th)^{-1}H_-, &  D_1 = (Th)D_0 = (Th)H_- - H_-, 
   \end{array}
$$
$B$HDj5A$9$k!%$"$k$$$O$b$C$H0lHL$K(B
$$ \begin{array}{ccc}
  D_i = (Th)^iD_0 & {\rm for} & i \in {\bf Z} ,
   \end{array}
$$
$B$HDj5A$9$k!%(B$D_1 \subset {\rm Int}(S) \cup H_+$$B$*$h$S(B$Th({\rm Int}(S) \cup H_+) \subset {\rm Int}(S) \cup H_+$$B$G$"$k$+$i!"<+L@$J5"G<K!$K$h$j!"$9$Y$F$N(B$i \geq 1$$B$KBP$7$F(B$D_i \subset {\rm Int}(S) \cup H_+$$B$G$"$k!%F1MM$K!"$9$Y$F$N(B$i \leq 0$$B$KBP$7$F(B$D_i \subset H_-$$B$G$"$k!%$H$/$K!"(B$i>0$$B$KBP$7$F(B$D_i \cap D_0 = \emptyset$$B$G$"$k$+$i!"$3$N<0$K(B$Th$$B$N$Y$-$r:nMQ$5$;$F(B$j \neq k$$B$J$i$$$D$b(B$D_j \cap D_k = \emptyset$$B$G$"$k$3$H$,$o$+$k!%(B

$B$5$F!"!VLL@Q!W$H$O!V(Brolled up$B!WLL(B
$$
  S^1 \times {\bf R} = {\bf R}^2/((x,y) \equiv (x+2\pi,y))
$$
$BFb$NLL@Q$G$"$k$H$9$k!%$3$N0UL#$G!"3F(B$D_i$$B$OF1$8LL@Q(B($B$9$J$o$A(B$2 \varepsilon$)$B$rM-$9$k!%(B$T$$B$b(B$h$$BLL@QJ]B8$G$"$j!"(B$Th$$B$b$=$&$@$+$i$G$"$k!%$@$+$i(B$i \geq 1$$B$N(B$D_i$$B$O$$$:$l$OLL@Q(B$2 \pi$$B$N(B$S$$B$rKd$a$D$/$7!"$h$C$F$$$:$l$O(B$H_+$$B$H8r$o$k!%(B

\vspace{1cm}
\hspace*{5cm} Figure 6.

\vspace{1.5cm}
$D_n \subset (Th)^n H_-$$B$G$"$k$+$i!"(B$n>0$$B$,B8:_$7$F(B$(Th)^n H_- \cap H_+ \neq \emptyset$$B$G$"$k$3$H$r<($7$?$3$H$K$J$k!%$=$N$h$&$J(B$n$$B$KBP$7$F!"(B$y$$B:BI8$,6KBg$NE@(B$P_n \in (Th)^n H_-$$B$rA*$V!%(B$P_n$$B$O0l0U$G$J$/$H$b!"B8:_$9$l$P$h$$!%$J$<$J$i!"<~4|@-$K$h$j!"%3%s%Q%/%H$JNN0h(B$Th^n(H_-) \cap \{ (x,y): y \geq 0, 0 \leq x \leq 2 \pi \}$$B$r9M$($k$@$1$G$h$/!"(B$y$$B$O$3$NNN0h$GO"B34X?t$G$"$k$+$i!"6KBg$rC#@.$9$k$+$i$G$"$k!%<!$rDj5A$7$h$&!%(B
$$ \begin{array}{ccc}
  P_i = (x_i,y_i) = (Th)^{i-n} P_n  &  {\rm for} & i \in {\bf Z} .
   \end{array}
$$
$B$@$+$i$9$Y$F$N(B$i$$B$KBP$7$F(B$P_{i+1} = Th(P_i)$$B$G$"$j!"(B$P_{-1},P_0 \in H_-$$B$*$h$S(B$P_n,P_{n+1} \in H_+$$B$G$"$k!%(B$C_0$$B$r(B$P_{-1}$$B$+$i(B$P_0$$B$X$ND>@~J,$H$7!"(B$i \in {\bf Z}$$B$KBP$7$F(B$C_i = (Th)^i C_0$$B$H$*$/!%(B$P_0$$B$O<B:]$O(B$y=0$$B>e$K$"$k$,!"$3$N$3$H$OI,MW$J$$(B(Figure 7)$B!%(B$C = C_0 C_1 \cdots C_{n-1}C_n$$B$H$*$1$P!"(B$Th(C) = C_1 C_2 \cdots C_n C_{n+1}$$B$G$"$k!%(B$C$$B$K4X$9$k<!$N;v<B$,I,MW$G$"$k!%(B

\vspace{4cm}
\hspace*{5cm} Figure 7.

\newpage
\noindent
{\bf $C$$B$N@-<A(B.}

1. $B6J@~(B$C C_{n+1} = C_0 \cdots C_{n+1}$$B$OFs=EE@$r;}$?$J$$!%(B

2. $C$$B$N$I$NE@$N(B$y$$B:BI8$b(B$P_{n+1}$$B$N(B$y$$B:BI8$h$jBg$-$/$J$i$J$$!%(B

3. $Th(C)$$B$N$I$NE@$N(B$y$$B:BI8$b(B$P_{-1}$$B$N(B$y$$B:BI8$h$j>.$5$/$J$i$J$$!%(B

\vspace{0.3cm}
$C \subset (Th)^n H_-$$B$G$"$k$3$H!"$*$h$S(B$P_n$$B$NA*$SJ}$h$j(B$(x,y) \in H_-$$B$KBP$7$F(B$y \leq y_n \leq y_{n+1}$$B$G$"$k$3$H$+$i@-<A(B2$B$OEvA3$G$"$k!%(B$Th(x,y) = (x',y')$$B$G(B$y \geq y_{-1}$$B$J$i(B$y' \geq y_{-1}$$B$G$"$k$3$H$r;H$C$F5"G<K!$r9T$J$($P!"@-<A(B3$B$O4JC1$K>ZL@$G$-$k!%@-<A(B1$B$r8+$k$?$a$K!"(B$i \neq j$$B$KBP$7$F#2K\$N6J@~(B$C_i$$B$H(B$C_j$$B$r9M$($k!%$3$l$i$O(B$|i-j|=1$$B$J$i6&DL$NC<E@$G8r$o$k$,!"$=$l0J30$NE@$G8r$o$k$H$9$k!%(B$Th$$B$NIi$NBg$-$J$Y$-$r:nMQ$5$;$F!"(B$i$$B$H(B$j$$B$rIi$K$G$-$k!%$H$3$m$,$3$l$O$*$+$7$$!%$J$<$J$i(B$i$$B$,@5$G$J$$$H$-(B$C_i$$B$O40A4$KBS(B
$$
  \{ (x,y): x_0 + (i-1)r_2 \leq x \leq x_0 + ir_2 \}
$$
$B$NCf$K4^$^$l!"C<E@$K$*$$$F$N$_$3$NBS$N6-3&$H8r$o$k$+$i$G$"$k(B($B$J$<$J$i!"$3$l$O(B$C_0$$B$K4X$7$F$O@5$7$/!"(B$H_-$$B$K$*$1$k(B$(Th)^{-1}$$B$N(B$x$$B@.J,$OC1$J$k(B$-r_2$$B$@$1$NJ?9T0\F0$@$+$i$G$"$k(B)$B!%(B$i \neq j$$B$KBP$7$F$3$l$i$NBS$O$?$+$@$+6-3&$K$*$$$F$7$+8r$o$i$J$$!%(B

\vspace{4cm}
\hspace*{5cm} Figure 8.

\vspace{4cm}
$B$$$^$d(B$Ind_{Th} C$$B$,7W;;$G$-$k!%9=@.K!$h$j!"(B
$$ \begin{array}{ll}
P_{n+1} = (x_{n+1}, y_{n+1}) = (x_n - r_1, y_n + \delta_1) ; &
         0 \leq \delta_1 \leq \varepsilon, \\
P_0 = (x_0, y_0) = (x_{-1} + r_2, y_{-1} + \delta_2), &
         0 \leq \delta_2 \leq \varepsilon .
   \end{array}
$$
$B$@$+$i(B$D(P_{-1}, P_0)$$B$H(B$D(P_n, P_{n+1})$$B$N4V$N3QEY$O(B
$$
  \theta = \pi - ({\rm arctan}(\delta_1/r_1) + 
{\rm arctan}(\delta_2/r_2) ) 
$$
$B$G$"$j(B(Figure 8$B;2>H(B)$B!"$=$l$f$(!"(B
$$ \begin{array}{ll}
  Ind_{Th} C \equiv \theta/2 \pi = 
1/2 - (1/2 \pi)({\rm arctan}(\delta_1/r_1) + 
{\rm arctan}(\delta_2/r_2))  & ({\rm mod} 1) .
   \end{array}
$$

$i=1,2$$B$KBP$7$F(B$0 \leq \delta_i \leq \varepsilon < r_i$$B$G$"$k$+$i(B$0 \leq {\rm arctan}(\delta_i/r_i) < \pi/4$$B$G$"$j!"$h$C$F(B$1/4 < \theta/2 \pi \leq 1/2$$B$G$"$k$3$H$r3NG'$7$FM_$7$$!%>e$N9gF14X78$,Ey<0$K$J$k$3$H$D$^$j!"(B$Ind_{Th} C = \theta/ 2 \pi$$B$G$"$k$3$H$r<($=$&!%D>4QE*$J5DO@$r%9%1%C%A$9$k!%(B

$Ind_{Th} C$$B$O(B$P$$B$,(B$C$$B$K1h$C$FF0$/$H$-$N(B$D(P,Q), Q=Th(P)$$B$NA42sE>3Q$G$"$k!%H/A[$O!"$O$8$a$K(B$P = P_{-1}$$B$r8GDj$7$F$*$$$F!"(B$Q$$B$r(B$Th(C)$$B$K1h$C$F(B$Th(P_{-1}) = P_0$$B$+$i(B$P_{n+1}$$B$^$GF0$+$7!"<!$K(B$Q$$B$r(B$P_{n+1}$$B$K8GDj$7$F$*$$$F!"(B$P$$B$r(B$C$$B$K1h$C$F(B$P_{-1}$$B$+$i(B$P_n$$B$^$GF0$+$9$H!"F1$8A42sE>$,F@$i$l$k$3$H$r3NG'$9$k$3$H$G$"$k!%$7$+$7!"$$$^$d!"(B$Q$$B$r(B$Th(C)$$B$K1h$C$FF0$+$7!"(B$P$$B$r(B$C$$B$K1h$C$FF0$+$9Be$o$j$K!"(B$Q$$B$r(B$P_0$$B$+$i(B$P_{n+1}$$B$^$GD>@~J,$K1h$C$FF0$+$7!"<!$K(B$P$$B$r(B$P_{-1}$$B$+$i(B$P_n$$B$^$GD>@~J,$K1h$C$FF0$+$9$3$H$,$G$-$k!%$3$N>u67$G$O!"4JC1$KJ,$+$k$h$&$K!"A42sE>$O<gD%DL$j$G$"$k(B(Figure 8$B;2>H(B)$B!%(B

$B>\:Y$J5DO@$N$?$a$K!"<LA|(B$[-1,0] \rightarrow {\bf R}^2$$B$K$h$C$F6J@~(B$C_0$$B$K%Q%i%a!<%?$rF3F~$7!"<!$K(B$-1 \leq t \leq n$$B$KBP$7$F(B$Th(P(t)) = P(t+1)$$B$G$"$k$H$7$F!"$3$l$r(B$CC_n$$B$N%Q%i%a!<%?2=(B$P: [-1,n+1] \rightarrow {\bf R}^2$$B$K3HD%$9$k!%$@$+$i(B$i=-1,0, \cdots, n+1$$B$KBP$7$F(B$P(i) = P_i$$B$G$"$k!%(B

$BDj5A$K$h$j!"(B$Ind_{Th} C$$B$O(B
$$ \begin{array}{cc}
\overline{P} :[-1,n] \rightarrow S^1, & 
\overline{P}(t) = D(P(t), P(t+1)),
   \end{array}
$$
$B$+$i7W;;$G$-$k!%Be$o$j$K(B
$$
  \overline{P}_0 = \left\{
\begin{array}{ll}
  \overline{P}(t), & -1 \leq t \leq n, \\
  \overline{P}(n), & n \leq t \leq 2n+1, 
\end{array}        \right.
$$
$B$GDj5A$5$l$k(B$\overline{P}_0: [-1,2n+1] \rightarrow S^1$$B$,;H$($k!%(B

$B$5$F!"<LA|$NO"B3B2(B$\overline{P}_{\lambda}: [-1, 2n+1] \rightarrow S^1, 0 \leq \lambda \leq n+2$$B$G!"$9$Y$FF1$8C<E@$rM-$7!":G8e$N<LA|(B$\overline{P}_{n+2}$$B$,(B$\pi - {\rm arctan}(\delta_1/r_1)$$B$rDL$C$F(B${\rm arctan}(\delta_2/r_2)$$B$+$i$NC1$J$kC1D43QEYA}2C$G$"$k$h$&$J$b$N$rDj5A$9$k!%$9$k$H2f!9$N<gD%$O#3@a$N:G8e$G=R$Y$?%[%b%H%T!<;}$A>e$2$K4X$9$kCm0U$+$iD>$A$KF@$i$l$k!%(B

$BB2(B$\overline{P}_{\lambda}$$B$r(B2$B$D$KJ,$1$FDj5A$9$k!%(B

\vspace{0.3cm}
\noindent
{\it Part 1:} $0 \leq \lambda \leq n+1$. \\
$$
  \overline{P}_{\lambda}(t) = \left\{
\begin{array}{ll}
  D(P(-1), P(t+1)), & -1 \leq t \leq \lambda -1; \\
  D(P(t-\lambda), P(t+1)), & \lambda-1 \leq t \leq n; \\
  D(P(t-\lambda), P(n+1)), & n \leq t \leq n + \lambda; \\
  D(P(n), P(n+1)), & n + \lambda \leq t \leq 2n + 1. 
\end{array}        \right.
$$
$B>e$G<($7$?G$0U$N(B$\lambda$$B$H(B$t$$B$KBP$7$F!"(B$-1 \leq t_0 < t_1 \leq n+1$$B$H$7$F!"(B$\overline{P}_{\lambda}$$B$O$D$M$K(B$D(P(t_0), P(t_1))$$B$N7A$r$7$F$*$j!"$f$($K$&$^$/Dj5A$5$l$F$*$j!"(B$CC_n$$B$,C1=c6J@~$G$"$k$+$i(B(C$B$N@-<A(B1)$B!"(B$P(t_0) \neq P(t_1)$$B$G$"$k!%(B

\vspace{0.3cm}
\noindent
{\it Part 2:} $n+1 \leq \lambda \leq n+2$. $P':[0,n+1] \rightarrow {\bf R}^2$$B$*$h$S(B$P'':[-1,n] \rightarrow {\bf R}^2$$B$r$=$l$>$l(B$P(0)$$B$+$i(B$P(n+1)$$B$^$G$*$h$S(B$P(-1)$$B$+$i(B$P(n)$$B$^$G$ND>@~J,$H$9$k!%(B$0 \leq \mu \leq 1$$B$KBP$7$F(B
$$
  \overline{P}_{n+1+\mu}(t) = \left\{
\begin{array}{ll}
  D(P(-1), (1-\mu)P(t+1)+\mu P'(t+1)) , & -1 \leq t \leq n; \\
  D((1-\mu)P(t-n-1)+\mu P''(t-n-1), P(n+1)), & n \leq t \leq 2n + 1 , 
\end{array}        \right.
$$
$B$rDj5A$9$k!%$3$l$,$&$^$/Dj5A$5$l$F$$$k$3$H$r8+$k$?$a$K!"(B$P \neq Q$$B$H$7$F1&JU$,$D$M$K(B$D(P,Q)$$B$N7A$r$7$F$$$k$3$H$r3NG'$9$Y$-$G$"$k!%(B$P = (1-\mu)P(t-n-1) + \mu P''(t-n-1)$$B$*$h$S(B$Q = P(n+1), n \leq t \leq 2n+1$$B$KBP$7$F!"(BC$B$N@-<A(B2$B$h$j!"(B$P$$B$O$D$M$K(B$Q$$B$h$j(B$y$$B:BI8$,>.$5$$!%$?$@$7Nc30$O(B$t = 2n+1$$B$^$?$O(B$\mu=0$$B$N$H$-$G$"$C$F!"$=$N$H$-$O!"(BC$B$N@-<A(B1$B$h$j(B
$$
  P = P(t-n-1) \neq P(n+1) = Q,
$$
$B$G$"$k!%F1MM$K!"(BC$B$N@-<A(B3$B$*$h$S(B1$B$h$j!"(B$P=P(-1)$$B$O7h$7$F(B
$$ \begin{array}{ccc}
  Q = (1-\mu)P(t+1) + \mu P'(t+1) & {\rm for} & -1 \leq t \leq n,
\end{array} 
$$
$B$KEy$7$/$J$i$J$$!%(B

$B;03QK!$N<+L@$J7W;;$K$h$C$F!"(B$\overline{P}_{n+2}(t), -1 \leq t \leq 2n+1$$B$O!"<gD%DL$j!"3QEY$rC1D4$K(B$\theta$$B$@$1A}2C$5$;$k$3$H$,$o$+$k!%$3$l$h$j(B
$$
  Ind_{Th} C = \theta/2 \pi = 1/2 - 
(1/2\pi)({\rm arctan}(\delta_1/r_1) + {\rm arctan}(\delta_2/r_2) ).
$$
$B$3$&$7$F(B$Ind_{Th} C$$B$N7W;;$,407k$7$?!%(B

$B<!$K(B$T_s: {\bf R}^2 \rightarrow {\bf R}^2$$B$r(B$T_s (x,y) = (x,y+(s \varepsilon/2) (|\cos (x)| - \cos(x)))$$B$GDj5A$9$k!%$h$C$F(B$T_0 = id$$B$*$h$S(B$T_1=T$$B$G$"$k!%$9$k$H!"(B$0 \leq s \leq 1$$B$KBP$7$F(B$Ind_{T_sh} C$$B$,Dj5A$5$l!"BP1~$9$k<gD%$,(B$Th = T_1 h$$B$KBP$7$F@.$jN)$D$N$HF1$8O@5r$G!"<!$,@.$jN)$D!%(B
$$ \begin{array}{cc}
  Ind_{T_sh} C \equiv 1/2 - 
(1/2\pi)({\rm arctan}(s\delta_1/r_1) + {\rm arctan}(s\delta_2/r_2)) & 
({\rm mod} 1) .
\end{array} 
$$
$B$H$3$m$,!"$3$N9gF14X78$O(B$s=1$$B$K4X$7$F$O<B$OEy<0$G$"$k$3$H$r8+$?!%$@$+$i(B$Ind$$B$NO"B3@-$K$h$j!"$9$Y$F$N(B$0 \leq s \leq 1$$B$KBP$7$F!"$3$l$OEy<0$G$"$k!%$H$/$K!"(B$s=0$$B$N>l9g!"K>$s$G$$$?DL$j!"(B$Ind_h C = 1/2$$B$rF@$k!%(B

$B5DO@A4BN$r(B$h$$B$NBe$o$j$K(B$h^{-1}$$B$r;H$$!"(B$T$$B$O$=$N$^$^$K$7$F7+$jJV$9$3$H$,$G$-$k!%$3$N$H$-$9$Y$F$O:8$H1&$,F~$lBX$o$j!"(B$Ind_{h^{-1}} C_0 = -1/2$$B$J$k6J@~(B$C_0$$B$,F@$i$l$k!%$9$k$H(B$Ind$$B$N@-<A(B4$B$h$j!"(B$C' = h^{-1} C_0$$B$KBP$7$F(B$Ind_h C' = -1/2$$B$G$"$k!%$3$&$7$FJdBj$KL7=b$9$k6J@~(B$C$$B$*$h$S(B$C'$$B$r8+$D$1$k$3$H$,$G$-$?!%(B

\vspace{0.5cm}
\noindent
{\bf 5. $B:G8e$N(BRemarks.}

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$B>ZL@$O$^$?1_4D>e$NI8=`E*$JLL@QB,EY0J30$NB,EY$K$bMF0W$K3HD%$G$-$k!%$7$+$7!"2f!9$NCN$k8B$j!"$3$NJ}8~$X$N:GNI$N7k2L$O!"$=$l$[$ID>@\E*$KF@$i$l$k$o$1$G$O$J$$$,!"%P!<%3%U$N0LAjHG$G$"$k$h$&$@!%$3$l$OLL@QJ]B8$r0J2<$N>r7o$KCV$-49$($F$$$k!%$9$J$o$A!"1_4D$N0lJ}$N6-3&$N3+6aK5(B$U$$B$G!"(B$U$$B$,(B$h(U)$$B$KHscGL)$JItJ,=89g$H$7$F4^$^$l$k$b$N$O$J$$$H$$$&>r7o$G$"$k!%%P!<%3%U$N>ZL@$O!"$3$NO@J8$G<($7$?$N$HF1MM$NJ}?K$K=>$C$F$$$k!%$7$+$7(B$h$$B$KBP$9$k!V6a;wE*N.@~!W(B$C$$B$*$h$S(B$C'$$B$G!"1_4D$NFbB&$H30B&$r7k$V$b$N$G!";X?t(B$+1/2$$B$*$h$S(B$-1/2$$B$r;}$D$b$N$r9=@.$9$k$N$O$O$k$+$KHyL/$JLdBj$G$"$k!%(B

$B>e$G5-=R$7$?5DO@$r>/$73HD%$9$k$@$1$G!"(B$h:A \rightarrow A$$B$N<~4|$,$A$g$&$I(B$n$$B$KEy$7$$<~4|E@$N=89g$N@.J,$N?t$N@53N$J2<8B$rF@$F!"%]%"%s%+%l!&%P!<%3%U$NDjM}$r3HD%$G$-$k!%$3$N2<8B$O%*%$%i!<4X?t(B$\phi(n)$$B$NDj?tG\$KA26a$9$k!%>\:Y$OBhFsCx<T$NO@J8$K:\$k!%(B

\vspace{0.5cm}
{\bf $B;29MJ88%(B}

\re
1. G.D. Birkhoff, Proof of Poincar\'{e}'s geometric theorem. 
{\it Trans. Amer. Math. Soc.} {\bf 14}(1913), 14-22.

\re
2. G.D. Birkhoff, An extension of Poincar\'{e}'s last geometric theorem. 
{\it Acta Math.} {\bf 47} (1925), 297-311.

\re
3. H. Poincar\'{e}, Sur un th\'{e}or$\grave{\rm e}$me de g\'{e}om\'{e}trie. 
{\it Rend. Circ. Mat. Palermo} {\bf 33} (1912), 375-407. 

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