
\vspace{0.5cm}
\noindent
{\bf 6. $B$$$/$D$+$N4XO"$9$kDjM}(B.} 

$B$3$N@a$G$O!"<!$r<($9!%(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B 4.} $f:A \rightarrow A$$B$O$M$8$l<LA|!"(B$p,q$$B$O8_$$$KAG$J@0?t$H$9$k!%(B$f$$B$N$I$N(B$r/s-$$B<~4|E@$b(B$s=q$$B$^$?$O(B$s > 3q/2$$B$rK~$?$;$P!"(B$f$$B$N$I$N(B$p/q-$$B<~4|E@$b(B$p/q-$$B%P!<%3%U<~4|E@$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$BCm0U(B}. $B$3$NDjM}$,8@$C$F$$$k$N$O!"$M$8$l<LA|$N$"$k<~4|50F;$N<~4|$,$9$Y$F$NBg$-$J<~4|$h$j>.$5$1$l$P!"$3$N50F;$,%P!<%3%U<~4|50F;$G$"$k$3$H$G$"$k!%2DG=$J<~4|$O$$$D$b6-3&$N2sE>?t$N4V$K4^$^$l$F$$$k$+$i!"DjM}$N2>Dj$,@.N)$9$k$?$a$N>r7o$r0J2<$N$h$&$KM?$($k$3$H$,$G$-$k!%(B

${\cal F}_n$$B$O(B$n$$B<!(B(order)$B$N(BFarey$B?tNs$G$"$k$H$9$k!%$9$J$o$A!"(B${\cal F}_n$$B$O(B$0$$B$H(B$1$$B$N$"$$$@$N4{LsJ,?tNs$GJ,Jl$,(B$n$$B0J2<$N$b$N$r>.$5$$=g$K$J$i$Y$?$b$N$G$"$k!%$@$+$i(B([11]$B;2>H(B)$B!"(B
$$
  {\cal F}_1 = \displaystyle 
      \left\{ \frac{0}{1} , \frac{1}{1} \right\},  
  {\cal F}_2 = \displaystyle 
      \left\{ \frac{0}{1} , \frac{1}{2}, \frac{1}{1} \right\},  
  {\cal F}_3 = \displaystyle 
      \left\{ \frac{0}{1} , \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, 
         \frac{1}{1} \right\}, \cdots.  
$$

\vspace{0.3cm}
\noindent
{\bf $B7O(B.} $f:A \rightarrow A$$B$O$M$8$l<LA|!"(B$p,q$$B$O8_$$$KAG$J@0?t!"(B$\rho_0(f),$ $p/q$ $\rho_1(f)$$B$O$"$k(B$n$$B$KBP$7$F(B${\cal F}_n$$B$NAjB3$/MWAG$G$"$k$H$9$k!%:GDc<!$N7A$G(B$\rho_0(f)=r_0/s_0,$ $\rho_1(f)=r_1/s_1$$B$G$"$j!"(B$s_0 > 3q/2,$ $s_1 > 3q/2$$B$J$i!"(B$f$$B$N$9$Y$F$N(B$p/q-$$B<~4|E@$O(B($B$b$7$"$l$P(B)$p/q-$$B%P!<%3%U<~4|E@$G$"$k!%(B

\vspace{0.3cm}
\noindent
\underline{$B7O$N>ZL@(B}. $i>0$$B$KBP$7$F(B$r/s \in {\cal F}_{n+i} \sim {\cal F}_{n+i-1}$$B$G$"$j!"(B$r_0/s_0 < r/s < p/q$$B$H$9$k!%$9$k$H!"(B[11]$B$NDjM}(B29-31$B$h$j!"(B$r/s=(a_1+a_2)/(b_1+b_2)$$B$G$"$k!%$?$@$7!"(B$a_1/b_1,$ $a_2/b_2$$B$O(B${\cal F}_{n+i-1}$$B$NAjB3$/MWAG$G$"$C$F!"(B$a_1+a_2,$ $b_1+b_2$$B$O8_$$$KAG$G$"$k!%$H$3$m$,$=$N$H$-(B$a_1/b_1,$ $a_2/b_2$ $\in [r_0/s_0,p/q]$$B$G$"$k$+$i(B$a_1/b_1 \in {\cal F}_n$$B$G$"$k$?$a$NI,MW==J,>r7o$O(B$a_1/b_1=r_0/s_0$$B$G$"$j!"(B$a_2/b_2 \in {\cal F}_n$$B$G$"$k$?$a$NI,MW==J,>r7o$O(B$a_2/b_2=p/q$$B$G$"$k!%$@$+$i!"(B$b_1 \geq q$$B$*$h$S(B$b_2 \geq q$$B$G$"$C$F(B$s \geq 2q$$B$H$J$k!%F1MM$K!"(B$p/q < r/s < r_1/s_1$$B$N$H$-(B$s \geq 2q$$B$H$J$j!"(B$f,p,q$$B$ODjM}(B4$B$N2>Dj$rK~$?$9$+$i7O$N>ZL@$,40N;$7$?!%(B\hfill $\Box$

\vspace{0.3cm}
\noindent
\underline{$BCm0U(B}. $BK\<AE*$K$O!">e$NDjM}$H7O$,8@$C$F$$$k$N$O!"(B$f: A \rightarrow A$$B$,$M$8$l<LA|$G(B$f$$B$,$=$l$[$I!V$M$8$l$F!W$$$J$$$H$-!":G>.<~4|$N<~4|50F;$O%P!<%3%U<~4|50F;$G$J$1$l$P$J$i$J$$$3$H$G$"$k!%DjM}(B4$B$N>r7o$,:G>e$N$b$N$G$"$k$3$H$r<($9Nc$rCN$i$J$$$,!"(B$f$$B$,LL@QJ]B8$J$i!"DjM}$O$:$C$H2~A1$G$-$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B 5.} $f:A \rightarrow A$$B$OLL@QJ]B8$M$8$l<LA|!"(B$p,q$$B$O8_$$$KAG$J@0?t$H$9$k!%(B$f$$B$N$I$N(B$r/s-$$B<~4|E@$b(B$s \geq q$$B$rK~$?$;$P!"(B$f$$B$N$I$N(B$p/q-$$B<~4|E@$b(B
$p/q-$$B%P!<%3%U<~4|E@$G$"$k!%(B

\vspace{0.3cm}
$BDjM}(B4$B$N>ZL@$ODjM}(B1$B$N>ZL@$H;w$?H/A[$G!"$7$+$b$b$C$H5;=QE*$K4JC1$K$G$-$k!%DjM}(B5$B$N>ZL@$O(B[6]$B$N(BSharkovskii$B$NDjM}$X$N!V%0%i%UM}O@E*!W%"%W%m!<%A$KGyA3$H4X78$7$F$$$k!%(B

\vspace{0.3cm}
\noindent
\underline{$BDjM}(B4$B$N>ZL@(B}. $f:A \rightarrow A$$B$H(B$p,q$$B$rDjM}$K$*$1$k$h$&$K8GDj$9$k!%(B$z_0 \in A$$B$O(B$f$$B$N(B$p/q$-$B<~4|E@$G$"$C$F(B$p/q$-$B%P!<%3%U<~4|E@$G$J$$$H2>Dj$9$k!%$3$N$H$-(B$\zeta_1,\zeta_2 \in {\cal O} (f,z_0)$$B$r8GDj$7!"$"$k(B$j, 1<j \leq q/2+1$$B$KBP$7$F(B
$\pi_1(\zeta_1) < \pi_1(\zeta_2),$ $\pi_1(f(\zeta_1)) > \pi_1(f(\zeta_2))$$B$*$h$S(B$\pi_1(f^j(\zeta_1)) < \pi_1(f^j(\zeta_2))$$B$H$J$k$h$&$K$9$k!%(B
$B$3$N$H$-@0?t(B$r$$B$,B8:_$7$F!"$"$k(B$k, j \leq k \leq 3q/2, k \not= q$$B$KBP$7$F(B$f^k(\zeta_2)=\zeta_1+(r,0)$$B$G$"$k!%(B

\vspace*{4cm}
\hspace{5cm} 11$B?^(B

\vspace*{4cm}
$B$5$F==J,Bg$-$J(B$N>0$$B$r8GDj$7$F!"(B$i=0, \cdots, 2q$$B$KBP$7$F(B$\pi_1(f^i(\pi_1(\zeta_2),1)) < \pi_1(f^i(\pi_1(\zeta_) + N, 0))$$B$H$J$k$h$&$K$7!"(B
$B= \{ (x,y) : \pi_1(\zeta_1) \leq x \leq \pi_1(\zeta_1)+N \}$$B$H$*$/!%(B$g: A \rightarrow A$$B$O$9$Y$F$N(B$z \in A$$B$KBP$7$F(B$g(z)=f^k(z) - (r,0)$$B$GDj5A$5$l$F$$$k$b$N$H$9$k!%$3$N$H$-(B$g(\zeta_2)=\zeta_1$$B$G$"$k!%(B
$$
  J_1 = \{ ( \pi_1(\zeta_1),y): $B$9$Y$F$N(By_1 \geq y$B$KBP$7$F(B 
g(\pi_1(\zeta_1),y_1) \in B \}
$$
$B$*$h$S(B
$$ 
  J_2 = \{ ( \pi_1(\zeta_1)+N,y): 
$B$9$Y$F$N(By_1 \leq y$B$KBP$7$F(B g(\pi_1(\zeta_1)+N,y_1) \in B \} 
$$
$B$HCV$/!%$3$N$H$-(B$g(B) \cap B$$B$N@.J,(B$T$$B$G6-3&$,(B$g(J_1) \cup g(J_2)$$B$r4^$`$b$N$OITF0E@$r;}$D$O$:$G$"$k!%$3$N$3$H$O!"(B$z$$B$,(B$g^{-1}(T)$$B$N6-3&$rF0$$$?$H$-$N%Y%/%H%k(B$z-g(z)$$B$N3QEY$NJQ2=$r7W;;$9$l$P4JC1$K$o$+$k!%$3$NJQ2=$OF0$/J}8~$K0MB8$7$F(B$\pm 2 \pi$$B$G$"$k(B($B?^(B11$B;2>H(B).

$B$H$3$m$,(B$g$$B$N$3$NITF0E@$O(B$f$$B$N(B$r/k-$$B<~4|E@$KBP1~$9$k!%$f$($K(B$f$$B$,(B$s_1 \not= q, s_1 \leq 3q/2$$B$J$k(B$r/s_1$-$B<~4|E@$r;}$?$J$1$l$P!"(B$f$$B$N$I$N(B$p/q$-$B<~4|E@$b(B$p/q$-$B%P!<%3%U<~4|E@$G$J$1$l$P$J$i$:!">ZL@$,40N;$7$?!%(B\hfill $\Box$

\vspace{0.3cm}
\noindent
\underline{$BDjM}(B5$B$N>ZL@(B}. $BLL@QJ]B8$M$8$l<LA|(B$f:A \rightarrow A$$B!"8_$$$KAG$J@0?t(B$p,q$$B$*$h$S(B$f$$B$N(B$p/q$-$B<~4|E@(B$z_0$$B$r8GDj$9$k!%(B
$$
  \{ \zeta_0, \cdots, \zeta_{q-1} \} = 
  {\cal O} (f,z_0) \cap \{ z \in A : \pi_1(z) \in [0,1) \}
$$
$B$H$7!"(B$\pi_1(\zeta_0) \leq \pi_1(\zeta_1) \leq \cdots \leq \pi_1(\zeta_{q-1})$$B$HHV9f$E$1$k!%I,MW$J$iJQ?t$r$9$3$7JQ$($k$3$H$K$h$j!"(B$i=0, \cdots, q-1$$B$KBP$7$F(B$\pi_1(\zeta_i) \not= \pi_1(\zeta_{i+1})$$B$H2>Dj$G$-$k!%(B$\zeta_0, \cdots, \zeta_{q-1}$$B$r@a$H$9$kJ}8~IU$-%0%i%U$r<!$N$h$&$KDj5A$9$k!%(B

$\zeta_i$$B$+$i(B$\zeta_j$$B$X$NJU$,$"$k$?$a$NI,MW==J,>r7o$O(B

(a) $B$"$k(B$s \in {\bf Z}$$B$KBP$7$F(B
$\pi_1(f(\pi_1(\zeta_i),0) \in (\pi_1(\zeta_{j-1}) +s, \pi_1(\zeta_j)+s]$$B$G$"$k$+!"(B

(b) $B$"$k(B$s \in {\bf Z}$$B$KBP$7$F(B
$\pi_1(f(\pi_1(\zeta_i),1) \in (\pi_1(\zeta_j)+s, \pi_1(\zeta_{j+1})+s)$$B$G$"$k!%(B

$B$f$($K3F(B$\zeta_i$$B$O!"<+J,$+$i=P$kJU$r>/$J$/$H$b#1$D;}$A!"$=$N>e(B$z_0$$B$,(B$p/q$-$B%P!<%3%U<~4|E@$G$J$1$l$P<+J,$+$i=P$kJU$r#2$D;}$D(B$\zeta_i$$B$,$"$k!%$J$<$J$i!"$"$k(B$\zeta_i, \zeta_j$$B$KBP$7$F(B$\pi_1(\zeta_i)<\pi_1(\zeta_j)$$B$G$"$k$,(B
$\pi_1(f(\zeta_i))>\pi_1(f(\zeta_j))$$B$G$"$C$F(B($B?^(B12$B;2>H(B)$B!"<!$,@.$jN)$D$+$i$G$"$k!%(B
$$
  \pi_1(f(\pi_1(\zeta_i),0)) < \pi_1(f(\zeta_j)) <
  \pi_1(f(\zeta_i)) < \pi_1(f(\pi_1(\zeta_i),1))
$$

\vspace{2cm}
\hspace{5cm} 12$B?^(B

\vspace{2cm}
$B$3$N>l9g!"$G$-$?%0%i%U$OD9$5(B$q$$BL$K~$N%k!<%W$rM-$9$k!%$H$3$m$,!"$3$l$O(B$s<q$$B$J$kM-M}?t(B$r/s \in [\rho_0(f),\rho_1(f)]$$B$NB8:_$r0UL#$9$k!%%P!<%3%U$NDjM}(B($BDjM}(B2)$B$K$h$l$P(B($B%P!<%3%U$,;XE&$7$?$h$&$K!"DjM}(B2$B$,@.N)$9$k$?$a$K$O(B$f:A \rightarrow A$$B$O%<%m$G$J$$L)EY$NB,EY$rJ]B8$9$l$P$h$$(B([4]$B;2>H(B)$B!%$@$+$i>e$NJQ?tJQ49$K$O2?$bLdBj$,$J$$(B)$B!"(B$f$$B$O(B$r/s$-$B<~4|E@$r;}$D$O$:$G$"$k!%$f$($K(B$f:A \rightarrow A$$B$,(B$s<q$$B$J$k(B$r/s$-$B<~4|E@$r;}$?$J$1$l$P!"(B$f$$B$N$I$N(B$p/q$-$B<~4|E@$b(B$p/q$-$B%P!<%3%U<~4|E@$G$"$k!%(B\hfill $\Box$

\vspace{0.5cm}
\noindent
{\bf $B<U<-(B.} $B$3$N;E;v$N4V!"OC$r?IJz6/$/J9$$$F$/$l!"<(:6$dNe$^$7$r$/$l$??M!9$K46<U$9$k!%$H$/$K(BC. Conley, E. Mansfield, D. Terman$B$K46<U$9$k!%(BM. Handel$B$K$OFCJL$K46<U$9$k!%H`$OJdBj(B4$B$r4JC12=$73HD%$9$k$?$a$N<(:6$r$/$l$?!%$*$+$2$G5;=QE*$J:$Fq$,=|$+$l$?!%$^$?H`$O(B6$B@a$NDjM}$N2DG=@-$r<(:6$7$F$/$l$?!%(B
$B:G8e$KJdBj(B$3'$$B$N4V0c$$$r;XE&$7$?(BDavid Bernstein$B$K46<U$9$k!%(B

$BJF9qN&73$N7@LsHV9f(BDAAG 29-80-C-0041$B$*$h$S(BNSF$B$NGn;N2]Dx=$N;<T%U%'%m%&%7%C%W$N1g=u$r<u$1$?!%(B

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

\re
[1] D. Asimov \& J. Franks. Unremovable closed orbits. Preprint.

\re
[2] S. Aubry. Theory of the devil's staircase. {\it Seminar on the Riemann Problem and Complete Integrability}, 1978/1979, (Ed. D.O.Chudrovsky), Lecture Notes in Math. 925 Springer, Berlin.

\re
[3] D. Bernstein. Birkhoff periodic orbits for twist maps with graph 
intersection property. {\it Ergod. Th. \& Dynam. Sys.} To appear.

\re
[4] G. D. Birkhoff. Proof of Poincar\'{e}'s geometric theorem. In 
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Dover, New York, 1968, p.673.

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[5] G. D. Birkhoff. An extension of Poincar\'{e}'s last geometric theorem. In {\it George David Birkhoff: Collected Mathematical Papers}, Vol.2, Dover, New York, 1968, p.252.

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[6] L. Block, J. Guckenheimer, M. Misiurewicz \& L.-S.Young. Periodic points and topological entropy of one dimensional maps. preprint.

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

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

\re
[9] A. Chenciner. Sur un enonc\'{e} dissipatif du Th\'{e}orem\`{e} 
geometric de Poincar\'{e}-Birkhoff. {\it C.R. Acad. Sc. Paris}, 
{\bf 294} (1982), 243-245.

\re
[10] A. Chenciner. Bifurcation de points fixes elliptiques. preprint.

\re
[11] G. H. Hardy \& E. M. Wright. {\it An Introduction to the Theory of Numbers}, (fifth edition), Oxford University Press, Oxford, 1979.

\re
[12] M. Herman. Sur la conjugaison differentiable des diffeomorphisms 
de cercle a des rotations. {\it Publ. Math. I.H.E.S.} {\bf 49}, 1979.

\re
[13] M. Herman. Introduction a l'etude des courbes invariantes par les diffeomorphisms de l'anneau, Vol.1. Preprint.

\re
[14] A. Kotok. Some remarks on Birkhoff and Mather twist map Theorems. 
{\it Ergod. Th. \& Dynam Sys.} {\bf 2} (1982), 185-192.

\re
[15] J. N. Mather. Existence of quasi-periodic orbits for twist homeomorphisms. {\it Topology}(1982).

\re
[16] J. N. Mather. A criterion for the non-existence of invariant circles. 
Preprint.

\re
[17] T. Matsuoka. The number and linking of periodic solutions of periodic systems. {\it Invent. Math.} {\bf 70} (1983), 319-340.
