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$@C+@n@6N4(B $@Lu(B
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\noindent
{\it Contemporary Mathematics} {\bf 81} (1988), 111-116

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{\Large
{\bf $@%]%"%s%+%l!&%P!<%3%U$NDjM}$NJQ7AHG(B \\ 
A variation of the Poincar\'{e}-Birkhoff theorem} 
}

\vspace{0.3cm}
John Franks
\end{center}

\vspace{0.5cm}
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{\bf 0. $@=x(B}

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{\bf 1. $@:?2s5"@-$*$h$S40Hw%j%c%W!<%N%U4X?t(B}

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$$ \begin{array}{ccc}
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$$
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(1.7){\bf $@JdBj(B.} $@O"B34X?t(B$g: X \rightarrow [0,1]$$@$,$"$C$F!"(B$g^{-1}(0) = A$$@$+$D(B$g^{-1}(1)=A^*$$@$G$"$j!"(B$g$$@$O(B$X - (A \cup A^*)$$@Fb$NE@$N50F;>e$G695AC1D48:>/$G$"$k!%(B

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\underline{$@>ZL@(B.} $g_0: X \rightarrow [0,1]$$@$r<!<0$GDj5A$9$k!%(B
$$
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$$
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$@$G$"$j!"$7$?$,$C$F(B$g_1$$@$O(B$N$$@>e$GO"B3$G$"$k!%(B$\cup_{n = -\infty}^{\infty} f^n(N) = X - (A \cup A^*)$$@$G$"$k$+$i(B$g_1$$@$OO"B3$G$"$k!%:G8e$K(B
$$
   g(x) = \sum_{n=0}^{\infty} \frac{g_1(f^n(x))}{2^{n+1}},
$$
$@$H$*$$$F!"O"B34X?t(B$g:X \rightarrow [0,1]$$@$G(B$g^{-1}(0)=A, g^{-1}(1) = A^*$$@$J$k$b$N$rF@$k!%$^$?(B
$$
   g(f(x)) - g(x) =
\sum_{n=0}^{\infty} \frac{g_1(f^{n+1}(x)) - g_1(f^n(x))}{2^{n+1}}
$$
$@$G$"$k!%$3$l$O(B$x \not\in A \cup A^*$$@$N$H$-Ii$G$"$k!%$J$<$J$i!"$9$Y$F$N(B$y$$@$KBP$7$F(B$g_1(f(y)) \leq g_1(y)$$@$G$"$j!"(B$g_1$$@$O(B$x$$@$N50F;>e$G0lDj$G$J$$$+$i$G$"$k!%(B\hfill $\Box$

\vspace{0.3cm}
$@<!$NDjM}$OK\<AE*$K(B[C]$@$N7k2L$G$"$k!%@_Dj$rN.$l$+$iF1Aj<LA|$KJQ$($F$"$k!%(B

\vspace{0.3cm}
\noindent
(1.8){\bf $@DjM}(B.} $f: X \rightarrow X$$@$,%3%s%Q%/%H5wN%6u4V$NF1Aj<LA|$J$i!"(B$f$$@$KBP$9$k40Hw%j%c%W!<%N%U4X?t(B$g: X \rightarrow R$$@$,$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$@>ZL@(B.} (1.2)$@$h$j!"(B$f$$@$K$O2D;;8D$N%"%H%i%/%?!<$7$+$J$$!%(B(1.7)$@$K$h$j!"(B$g_n^{-1}(0) = A_n, g_n^{-1}(1)$ $= A_n^*$$@$J$k(B$g_n:X \rightarrow R$$@$r8+$D$1$k$3$H$,$G$-$k!%(B$g_n$$@$O(B$X - (A_n \cup A_n^*)$$@>e$G695AC1D48:>/$G$"$k!%(B$g:X \rightarrow R$$@$r(B
$$
   g(x) = \sum_{n=1}^{\infty} \frac{2g_n(x)}{3^n}
$$
$@$GDj5A$7$h$&!%$3$N5i?t$O0lMM$K<}B+$9$k$+$i(B$g(x)$$@$OO"B3$G$"$k!%L@$i$+$K!"(B$x \not\in {\bf R}(f)$$@$J$i(B$x \not\in (A_i \cup A_i^*)$$@$J$k(B$A_i$$@$,$"$k$+$i!"(B
$g(f(x)) < g(x)$$@$G$"$k!%(B

$@$^$?!"(B$x \in {\bf R}(f)$$@$J$i!"$I$N(B$n$$@$KBP$7$F$b(B$x \in (A_n \cup A_n^*)$$@$G$"$k$+$i!"$9$Y$F$N(B$n$$@$KBP$7$F(B$g_n(x) = 0$$@$^$?$O(B$1$$@$G$"$k!%$7$?$,$C$F(B$g(x)$$@$N(B3$@?JE83+$O(B$0$$@$H(B$2$$@$N$_M-$7!"$f$($K(B$C$$@$r%+%s%H!<%k$NCf1{#3J,$N#1(B(middle third)$@=89g$H$7$F(B$g(x) \in C$$@$G$"$k!%$@$+$i(B$g({\bf R}(f)) \subset C$$@$G$"$j!"(B$g({\bf R}(f))$$@$O%3%s%Q%/%H$G$$$?$k$H$3$mAB$G$"$k!%$3$l$ODj5A$N(B(3)$@$r>ZL@$7$F$$$k!%(B

$@:G8e$K!"(B$x,y \in {\bf R}(f)$$@$+$D(B$g(x)=g(y)$$@$J$i!"L@$i$+$K$9$Y$F$N(B$n$$@$KBP$7$F(B$g_n(x) = g_n(y)$$@$G$"$k!%$J$<$J$i(B$2g_n(x)$$@$O(B$g(x)$$@$N#3?JE83+$N(B$n$$@7eL\$N?t;z$G$"$j!">e$HF1MM!"$3$NE83+$,(B$0$$@$H(B$2$$@$7$+4^$^$J$$$?$a$NI,MW==J,>r7o$O(B$x \in A_n, y \in A_n^*$$@$^$?$O(B$x \in A_n^*, y \in A_n$$@$H$J$k$h$&$J(B$n$$@$,$R$H$D$b$J$$$3$H$G$"$k$+$i$G$"$k!%$@$+$i!"(B(1.5)$@$h$j!"(B$g(x) = g(y)$$@$H$J$k$N$O(B$x$$@$H(B$y$$@$,F1$8:?A+0\@.J,$KB0$9$k$H$-!"$7$+$b$=$N$H$-$N$_$G$"$k!%(B\hfill $\Box$


\vspace{0.5cm}
\noindent
{\bf 2. $@6-3&$M$8$l<LA|(B}

$@$3$N@a$G$O1_4D(B$A$$@$NF1Aj<LA|$G!"91Ey<LA|$K%[%b%H!<%W$+$D6-3&$M$8$l>r7o$rK~$?$9$b$N$r9M$($k!%(B

\vspace{0.3cm}
\noindent
(2.1){\bf $@Dj5A(B.} $@91Ey<LA|$K%[%b%H!<%W$JF1Aj<LA|(B$f: A \rightarrow A$$@$,(B\underline{$@6-3&$M$8$l>r7o(B}$@$rK~$?$9$H$O!"IaJWHoJ$6u4V(B$\tilde{A} = R \times I$$@$X$N(B$f$$@$N;}$A>e$2(B$\tilde{f}: \tilde{A} \rightarrow \tilde{A}$$@$G(B$\tilde{f}_1(x,0) < x$$@$+$D(B$\tilde{f}_1(x,1) > x$$@$J$k$b$N$,B8:_$9$k$H$-$G$"$k!%$?$@$7!"(B$\tilde{f}(x,s) = (\tilde{f}_1(x,s),\tilde{f}_2(x,s))$$@$G$"$k!%(B

\vspace{0.3cm}
$@ITF0E@$r8+$D$1$k$?$a$K80$H$J$kJdBj$O<!$N$b$N$G$"$k!%$3$l$OK\<AE*$K$O(B[F]$@$K$h$k!%(B

\vspace{0.3cm}
\noindent
(2.2){\bf $@JdBj(B.} $f: R^2 \rightarrow R^2$$@$,J}8~J]B8F1Aj<LA|$G$"$C$F!":?2s5"=89g(B${\bf R}(f)$$@$,6u$G$J$$$J$i!"G$0U$N(B$\varepsilon > 0$$@$KBP$7$F(B$\parallel f(x) - x \parallel < \varepsilon$$@$J$kE@(B$x \in R^2$$@$,$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$@>ZL@(B.} [F]$@$G$N5DO@$r4JC1$K;W$$$@$=$&!%7kO@$,8m$j$J$i!"$9$Y$F$N(B$x \in R^2$$@$KBP$7$F(B$\parallel f(x) - x \parallel \geq \varepsilon_0$$@$J$k(B$\varepsilon_0$$@$,$"$k!%(B[Ox]$@$N7k2L$K$h$l$P!"(B$\delta >0$$@$,$"$C$F!"(B$||x_i - y_i|| < \delta$$@$J$kAH(B$\{ (x_i,y_i) \}$$@$NG$0U$NM-8B=89g$KBP$7$F!"(B$x_i$$@$+$i(B$y_i$$@$X$N$=$l$>$l8_$$$KAG$J6hJ,E*$K@~7A$J8I(B$\gamma_i$$@$GD9$5(B$< \varepsilon_0 /2$$@$J$k$b$N$N=89g$,$"$k!%(B$z \in {\bf R}(f)$$@$J$i!"(B$z_1 = z, z_2, \ldots, z_n=z$$@$r(B$z$$@$+$i(B$z$$@$X$N(B$\delta-$$@:?$H$9$k!%(B$y_i = z_i, x_i=f(z_{i-1})$$@$H$*$1$P!"(B$f(z_{i-1})$$@$+$i(B$z_i$$@$X$N8_$$$KAG$J8I(B$\gamma_i$$@$GD9$5(B$<\varepsilon_0 /2$$@$J$k$b$N$,$"$k$3$H$,$o$+$k!%$3$l$i$N8I$N6aK5$G%$%=%H!<%W$9$k$3$H$K$h$j!"(B$f$$@$N@]F0(B$g$$@$G!"$9$Y$F$N(B$x$$@$KBP$7$F(B$g(z_{i-1})=g(z_i)$$@$*$h$S(B$||f(x) - g(x)|| < \varepsilon_0$$@$rK~$?$9$b$N$r:n$l$k!%(B

$@$5$F(B$g$$@$O<~4|E@(B$z$$@$r;}$D!%$f$($K!"(B[Br]$@$^$?$O(B[Fa]$@$N7k2L$h$j!"(B$g$$@$OITF0E@(B$p$$@$r;}$D!%$@$+$i(B$||f(p) - p|| \leq ||f(p) - g(p)|| + ||g(p) - p|| < \varepsilon_0$$@$H$J$k$,!"$3$l$OL7=b$G$"$k!%(B\hfill $\Box$

\vspace{0.3cm}
\noindent
(2.3){\bf $@L?Bj(B.} $f:A \rightarrow A$$@$O91Ey<LA|$K%[%b%H!<%W$JF1Aj<LA|$G!"6-3&$M$8$l>r7o$rK~$?$9$H$9$k!%$I$N(B$\varepsilon$$@$KBP$7$F$b(B$S^1 \times \{0 \}$$@$+$i(B$S^1 \times \{ 1 \}$$@$X$N(B$f$$@$N(B$\varepsilon-$$@:?$*$h$S(B$S^1 \times \{1 \}$$@$+$i(B$S^1 \times \{ 0 \}$$@$X$N(B$f$$@$N(B$\varepsilon-$$@:?$,$"$l$P!"(B$f$$@$OITF0E@$r>/$J$/$H$b#1$D;}$D!%(B

\vspace{0.3cm}
\noindent
\underline{$@>ZL@(B.} $f$$@$r(B$S^1 \times [-\delta, 1 + \delta]$$@$NF1Aj<LA|$K3HD%$7$F!"(B$f$$@$,(B$S^1 \times \{ - \delta \}$$@$*$h$S(B$S^1 \times \{ 1 + \delta \}$$@>e$N2sE>$G$"$k$h$&$K$7!"$^$?;}$A>e$2(B$\tilde{f}: R \times [-\delta, 1 + \delta] \rightarrow R \times [-\delta, 1 + \delta]$$@$,<!$rK~$?$9$h$&$K$G$-$k$3$H$r$^$:3NG'$7$F$[$7$$!%(B

(1) $t \in [-\delta,0] \cup [1,1+\delta]$$@$KBP$7$F!"(B$\tilde{f}(x,t) = (\tilde{f}_1(x,t),t)$.

(2) $t \in [-\delta,0]$$@$KBP$7$F!"(B$\tilde{f}_1(x,t) < x$

(3) $t \in [1,1+\delta]$$@$KBP$7$F!"(B$\tilde{f}_1(x,t) > x$$@!%(B

\vspace{0.3cm}
\noindent
$@$3$l$i$N@-<A$+$i4JC1$K%A%'%C%/$G$-$k$h$&$K!"(B$S^1 \times [-\delta,0]$$@$O(B$S^1 \times [1,1+\delta]$$@$HF1MM!"C10l$N:?A+0\@.J,$K4^$^$l$k!%(B$S^1 \times [-\delta,0] \cup [1,1+\delta]$$@Fb$K$OITF0E@$OM-$jF@$J$$$+$i!"(B$f$$@$,6-3&>e$G2sE>$G$"$k$h$&$J3HD%$5$l$?1_4D$KBP$7$F7k2L$r>ZL@$9$k$@$1$G$h$$!%$3$l$r$9$k$N$K!"1_4D$r(B$S^1 \times [-\delta,1+\delta]$$@$H$9$k$h$j$b$`$7$m(B$A = S^1 \times [0,1]$$@$H$9$k!%(B

$@;}$A>e$2(B$\tilde{f}:\tilde{A} \rightarrow \tilde{A} (\tilde{A} = R \times [0,1])$$@$G!"$"$k(B$r_1,r_2 >0$$@$KBP$7$F(B
$$ \begin{array}{ccc}
  \tilde{f}(x,1) = (x + r_1,1) & $@$*$h$S(B & 
\tilde{f}(x,0) = (x - r_2,0) 
   \end{array}
$$
$@$rK~$?$9$b$N$r9M$($h$&!%(B$R \times \{ 0 \}$$@$N$I$NE@$b(B$\tilde{f}$$@$K4X$7$F:?2s5"E*$G$"$k$3$H$r<($=$&!%$7$+$7$3$l$r$9$kA0$K!"$3$l$G>ZL@$,407k$9$k$3$H$r<($7$F$*$3$&!%(B$t>1$$@$KBP$7$F$O(B$\tilde{f}(x,t) = (x+r_1,t)$$@$H$*$-!"(B$t<0$$@$KBP$7$F$O(B$\tilde{f}(x,t) = (x-r_0,t)$$@$H$*$$$F(B$\tilde{f}$$@$r(B$R^2$$@$K3HD%$G$-$k!%$@$+$i(B(2.2)$@$K$h$j!"$I$N(B$\varepsilon>0$$@$KBP$7$F$b(B$||\tilde{f}(x,t) - (x,t)|| < \varepsilon$$@$J$k(B$(x,t)$$@$,$"$k!%$3$NITEy<0$O(B$f$$@$*$h$S%3%s%Q%/%H$J1_4D(B$A$$@$X$N(B$(x,t)$$@$N<M1F$KBP$7$F$b@.$jN)$D$+$i!"(B$f$$@$OITF0E@$r;}$D!%(B

$@;D$C$F$$$k$N$O!"(B$R \times \{ 0 \}$$@$NE@$,:?2s5"E*$G$"$k$3$H$r<($9$3$H$G$"$k!%(B$x_0 \in S^1$$@$H$9$k!%>e$NCm0U$h$j!"$I$N(B$\varepsilon$$@$KBP$7$F$b(B$(x_0,0)$$@$+$i(B$(x_0,1)$$@$X$N(B$\varepsilon-$$@:?$*$h$S(B$(x_0,1)$$@$+$i(B$(x_0,0)$$@$X$N(B$\varepsilon-$$@:?$,$"$k!%(B$(x,0)$$@$,(B$(x_0,0)$$@$N;}$A>e$2$J$i!"(B$A$$@>e$N(B$\varepsilon-$$@:?$r;}$A>e$2$F(B$\tilde{f}:\tilde{A} \rightarrow \tilde{A}$$@$N(B$\varepsilon-$$@:?$G!"$"$k(B$a \in Z$$@$KBP$7$F(B$(x,0)$$@$+$i(B$(x+a,1)$$@$X9T$/$b$N$,F@$i$l$k!%$9$Y$F$N(B$y \in R$$@$KBP$7$F(B$\tilde{f}(y,1) = (y+r_1,1)$$@$G$"$k$H$$$&$3$H$O!"G$0U$N==J,Bg$-$J(B$p \in Z$$@$KBP$7$F(B$(x,1)$$@$+$i(B$(x+p,1)$$@$X$N(B$\varepsilon-$$@:?$,$"$k$3$H$r0UL#$9$k!%F1MM$K!"$"$k(B$b \in Z$$@$KBP$7$F(B$(x,1)$$@$+$i(B$(x+b,0)$$@$X$N(B$\varepsilon-$$@:?$,$"$k!%(B$\tilde{f}(x,t) + (0,1) = \tilde{f}(x+1,t)$$@$,$3$l$i$N(B$\varepsilon-$$@:?$rJ?9T0\F0$5$;$k$H$$$&;v<B$rMQ$$$F!"(B$(x,0)$$@$+$i(B$(x+a,1)$$@$X!"(B$(x+a+p,1)$$@$X!"(B$(x+a+p+b,0)$$@$X$N(B$\varepsilon$-$@:?$r$D$J$2$k$3$H$,$G$-$k!%(B$p$$@$,==J,Bg$-$1$l$P!"(B$m = a+p+b >0$$@$G$"$k!%(B

$@$^$?(B$\tilde{f}(x,0) = (x-r_2,0)$$@$G$"$k$+$i!"$"$k@0?t(B$n>0$$@$KBP$7$F(B$(x,0)$$@$+$i(B$(x-n,0)$$@$X$N(B$\varepsilon-$$@:?$,$"$k$3$H$O$o$+$C$F$$$k!%$b$&0lEY$D$J$2$F!"(B$(x,0)$$@$+$i(B$(x+m,0)$$@$X!"(B$(x+2m,0)$$@$X!"(B$\cdots$$@!"(B$(x+mn,0)$$@$X!"(B$(x+mn-n,0)$$@$X(B$\cdots$$@!"(B$(x,0)$$@$X$N(B$\varepsilon-$$@:?$rF@$k$3$H$,$G$-$k!%$@$+$i(B$(x,0)$$@$O:?2s5"E*$G$"$k!%(B\hfill $\Box$

\vspace{0.3cm}
$@$$$^$d<g7k2L$N>ZL@$,$G$-$k!%(B

\vspace{0.3cm}
\noindent
(2.4){\bf $@DjM}(B.} $f:A \rightarrow A$$@$O91Ey<LA|$K%[%b%H!<%W$JF1Aj<LA|$G$"$C$F6-3&$M$8$l>r7o$rK~$?$9$H$9$k!%$3$N$H$-(B$f$$@$OITF0E@$r>/$J$/$H$b#1$D;}$D$+!"$"$k$$$O3j$i$+$KKd$a9~$^$l$?4pK\JD6J@~(B$C \subset A$$@$,$"$C$F(B$f(C) \cap C = \emptyset$$@$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$@>ZL@(B.} (2.3)$@$N>ZL@$N$H$-$HF1MM!"1_4D$r(B$A_0 = S^1 \times [-\delta, 1 + \delta]$$@$K3HBg$7!"(B$f$$@$r3HD%$7$F3F6-3&>e(B
$@$G$O2sE>$G$"$k$h$&$K$7!"1_(B$\{ S^1 \times \{ t \} | t \in [-\delta,0] \cup [1,1 + \delta] \}$$@$,ITJQ$G$"$C$F!"ITF0E@$r4^$^$J$$$h$&$K$9$k!%$3$N$H$-!"(B$A^+ = \{ (x,t) \in A_0 | t \in [1,1 + \delta] \}$$@$*$h$S(B$A^- = \{ (x,t) \in A_0 | t \in [-\delta, 0] \}$$@$J$i!"(B$A^+$$@$H(B$A^-$$@$O:?2s5"E*$G$"$C$F!"3F!9$O:?A+0\@.J,$NItJ,=89g$G$"$k(B($@$9$J$o$A!"(B$A^+$$@$NG$0U$N#2E@$N4V$K(B$\varepsilon-$$@:?$,$"$k$7!"(B$A^-$$@$K$D$$$F$bF1MM(B)$@!%(B

$A^+$$@$H(B$A^-$$@$,(B$f$$@$NF1$8:?A+0\@.J,$KB0$;$P!"(B(2.3)$@$K$h$j!"(B$f$$@$OITF0E@$r>/$J$/$H$b#1$D;}$D!%$@$+$i$3$l$,@.$jN)$?$J$$$H2>Dj$9$k!%(B$g:A_0 \rightarrow R$$@$O(B$f$$@$K4X$9$k40Hw%j%c%W!<%N%U4X?t$G$"$k$H$9$k!%$3$N$H$-(B$g$$@$O(B$A^+$$@$*$h$S(B$A^-$$@>e$G0lDj$G$"$j!"(B$g(A^+) \neq g(A^-)$$@$G$"$k!%(B$g$$@$NNW3&CM$G$J$$E@(B$c \in R$$@$G!"(B$g(A^+)$$@$H(B$g(A^-)$$@$N4V$K$"$k$b$N$rA*$V!%=89g(B$g^{-1}(c) \subset A$$@$O(B$A$$@$H(B$f(g^{-1}(c)) \cap g^{-1}(c) = \emptyset$$@$rJ,N%$9$k(B(separate)$@$H$$$&@-<A$r;}$D!%(B$g$$@$,(B$C^{\infty}$$@$K$J$k$h$&$K(B$g$$@$r9=@.$G$-$k(B([W]$@;2>H(B)$@!%$?$@$7!"$3$l$O2f!9$NL\E*$K$OI,MW$G$J$$!%;v<B!"(B$g_0$$@$,(B$g$$@$K==J,(B$C^0$$@6a@\$J(B$C^{\infty}$$@4X?t$G$"$k$H$9$l$P!"(B$g_0^{-1}(c)$$@$O(B$A$$@$H(B$f(g^{-1}_0(c)) \cap g^{-1}_0(c) = \emptyset$$@$rJ,N%$9$k!%(BSard$@$NDjM}$K$h$j!"(B$g_0$$@$N@5B'CM$r(B$c$$@$NG$0U$N6a$/$K$H$l$k!%@5B'CM(B$c_0$$@$r!"(B$g^{-1}_0(c_0)$$@$,(B$A$$@$H(B$f(g^{-1}_0(c_0)) \cap g^{-1}_0(c_0) = \emptyset$$@$rJ,N%$9$k$h$&$KA*$V!%(B$g^{-1}_0(c_0)$$@$N@.J,$O3j$i$+$KKd$a9~$^$l$?1_$G$"$j!"$9$/$J$/$H$b$=$N#1$D$O4pK\E*$G$"!%$J$<$J$i(B$g^{-1}_0(c_0)$$@$O(B$A$$@$rJ,N%$9$k$+$i!%(B$C$$@$r$=$N$h$&$J4pK\E*@.J,$H$9$k!%$9$k$H(B$f(C) \cap C = \emptyset$$@$G$"$k!%(B\hfill $\Box$

\vspace{0.5cm}
\noindent
{\bf References}

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

\re
[B2] G.D. Birkhoff, An extension of Poincar\'{e}'s last geometric theorem, {\it Acta Math.} {\bf 47} (1925), 297-311.

\re
[Br] M. Brown, A new proof of Brouwer's lemma on translation arcs, {\it Houston Journ. of Math.} {\bf 10} (1984), 35-41.

\re
[BN] M. Brown and W.D. Neumann, Proof of the Poincar\'{e}-Birkhoff fixed point theorem, {\it Michigan Math. J.} {\bf 24} (1977), 21-31.

\re
[Ca] P. Carter, An improvement of the Poincar\'{e}-Birkhoff fixed point theorem, {\it Trans. Amer. Math. Soc.} {\bf 269} (1982), 285-299.

\re
[C] C. Conley, Isolated invariant sets and Morse index, C.B.M.S. Regional Conference Series in Math. {\bf 38}, Amer. Math. Soc., Providence, RI, 1978.

\re
[Fa] A. Fathi, An orbit closing proof of Brouwer's lemma on translation arcs, {\it L'enseignement Math\'{e}matique} {\bf 33} (1987), 315-322.

\re
[F] J. Franks, Recurrence and fixed points of surface homeomorphisms, {\it Ergod. Th. \& Dynam. Sys.} {\bf 8} (1988), 99-107.

\re
[Ox] J. Oxtoby, Diameters of arcs and the gerrymandering problem, {\it Amer. Math. Monthly} {\bf 84} (1977), 155-162.

\re
[W] W. Wilson, Smoothing derivatives of functions and applications, {\it Trans. Amer. Math. Soc.} {\bf 139} (1969), 413-428.

\end{document}
