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Fo(g)p Fq(.)50 1488 y(Ein)19 b(erstes)g(einfac)o(hes)f(ab)q(er)h(w)o(eitreic)o(hendes) e(Abz)1015 1490 y(\177)1015 1488 y(ahlprinzip)h(ist)g(in)h(dem)f (folgenden)h(Satz)0 1546 y(angegeb)q(en.)0 1621 y Fr(Satz)g(1.6.)h Fm(In)d(einem)i(end)r(lichen)h(schlichten)f(Gr)n(aphen)i Fp(X)d Fq(=)c(\()p Fp(V)s(;)8 b(E)s(;)g(\031)r Fq(\))16 b Fm(gilt)730 1675 y Ff(X)731 1781 y Fn(a)p Fc(2)p Fn(V)811 1723 y Fp(d)p Fq(\()p Fp(a)p Fq(\))d(=)h(2)p Fo(j)p Fp(E)s Fo(j)p Fp(:)0 1888 y Fm(Pr)n(o)n(of.)19 b Fq(Jede)d(Kan)o(te)g(in)g Fp(E)j Fq(hat)e(zw)o(ei)e(Endpunkte,)g(die)h(b)q(eide)g(in)g(der)f (link)o(en)g(Summe)e(b)q(er)1711 1890 y(\177)1710 1888 y(uc)o(k-)0 1946 y(sic)o(h)o(tigt)i(w)o(erden.)g(Sie)h(wird)g(also)h (links)e(genau)i(zw)o(eimal)d(gez)1149 1948 y(\177)1149 1946 y(ahlt.)514 b Fb(\003)0 2045 y Fr(F)-5 b(olgerung)13 b(1.7.)j Fm(Die)d(A)o(nzahl)g(der)f(Knoten)i(unger)n(aden)g(Gr)n(ades)d (in)i(einem)g(end)r(lichen)i(schlich-)0 2103 y(ten)j(Gr)n(aphen)f(ist)h (ger)n(ade.)0 2223 y(Pr)n(o)n(of.)h Fq(Die)13 b(Knoten)h(geraden)g (Grades)h(tragen)f(n)o(ur)f(gerade)h(Summanden)e(in)h(der)h(link)o(en)e (Sum-)0 2281 y(me)j(b)q(ei.)1604 b Fb(\003)50 2400 y Fq(Eb)q(enso)13 b(wie)e(b)q(ei)g(algebraisc)o(hen)h(Grundstrukturen)g (gibt)g(es)f(auc)o(h)h(b)q(ei)f(Graphen)i(Homomor-)0 2459 y(phismen,)f(also)i(Abbildungen,)f(die)g(die)g(b)q(esondere)h (Struktur)g(v)o(on)f(Graphen)i(b)q(er)1530 2461 y(\177)1529 2459 y(uc)o(ksic)o(h)o(tigen.)0 2533 y Fr(De\014nition)k(1.8.)i Fq(Seien)16 b Fp(X)21 b Fq(=)16 b(\()p Fp(V)s(;)8 b(E)s(;)g(\031)r Fq(\))17 b(und)h Fp(Y)27 b Fq(=)16 b(\()p Fp(V)1080 2515 y Fc(0)1092 2533 y Fp(;)8 b(E)1153 2515 y Fc(0)1165 2533 y Fp(;)g(\031)1217 2515 y Fc(0)1228 2533 y Fq(\))17 b(Graphen.)h(Ein)f Fm(Homomor-)0 2591 y(phismus)26 b Fp(f)k Fq(:)23 b Fp(X)29 b Fo(!)24 b Fp(Y)33 b Fq(b)q(esteh)o(t)22 b(aus)h(zw)o(ei)e (Abbildungen)h Fp(f)1194 2598 y Fn(V)1249 2591 y Fq(:)h Fp(V)36 b Fo(!)24 b Fp(V)1463 2573 y Fc(0)1475 2591 y Fp(;)8 b(f)1521 2598 y Fn(E)1574 2591 y Fq(:)24 b Fp(E)j Fo(!)d Fp(E)1788 2573 y Fc(0)0 2649 y Fq(mit)346 2720 y Fo(8)p Fp(k)14 b Fo(2)g Fp(E)s Fq([)p Fp(\031)r Fq(\()p Fp(k)r Fq(\))f(=)h Fo(f)p Fp(a;)8 b(b)p Fo(g)13 b Fq(=)-8 b Fo(\))13 b Fp(\031)929 2699 y Fc(0)940 2720 y Fp(f)964 2727 y Fn(E)994 2720 y Fq(\()p Fp(k)r Fq(\))h(=)f Fo(f)p Fp(f)1173 2727 y Fn(V)1204 2720 y Fq(\()p Fp(a)p Fq(\))p Fp(;)8 b(f)1314 2727 y Fn(V)1344 2720 y Fq(\()p Fp(b)p Fq(\))p Fo(g)p Fq(])p Fp(:)0 2801 y Fq(Es)19 b(w)o(erden)f(also)h (Knoten)g(auf)g(Knoten)g(und)g(Kan)o(ten)f(auf)h(Kan)o(ten)f(so)i (abgebildet,)d(da\031)j(ihre)0 2859 y(Inzidenz)15 b(erhalten)g(bleibt.) p eop %%Page: 4 4 4 3 bop 0 -49 a Fh(4)741 b(SEMINAR)17 b(SS04)50 50 y Fq(Ein)22 b Fm(Isomorphismus)j Fp(f)k Fq(:)24 b Fp(X)k Fo(!)c Fp(Y)33 b Fq(ist)22 b(ein)g(Homomorphism)n(us,)d(zu)j(dem)f(ein) h(w)o(eiterer)0 108 y(Homomorphism)n(us)16 b Fp(g)k Fq(:)e Fp(Y)28 b Fo(!)18 b Fp(X)23 b Fq(existiert)17 b(mit)g Fp(g)r(f)23 b Fq(=)18 b(id)1114 115 y Fn(X)1167 108 y Fq(und)h 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Fm(Komplement)i(von)i Fp(Y)29 b Fm(in)0 562 y Fp(X)t Fq(.)16 b(Das)g Fm(Komplement)22 b Fq(eines)15 b(sc)o(hlic)o(h)o(ten)e (Graphen)j Fp(X)i Fq(=)c(\()p Fp(V)s(;)8 b(E)s(;)g(\031)r Fq(\))15 b(ist)g(sein)g(Komplemen)o(t)d(in)0 620 y(der)k(V)l(erv)o (ollst)273 622 y(\177)273 620 y(andigung.)0 691 y Fr(Beispiele)g(1.10.) k Fq(1.)c(Das)h(Komplemen)o(t)c(v)o(on)p 329 885 4 119 v 331 886 119 4 v 447 885 4 119 v 331 885 a Fd(r)106 b(r)331 766 y(r)g(r)563 826 y Fq(in)113 b Fe(\000)735 808 y(\000)717 867 y(@)735 885 y(@)p 775 885 V 776 886 119 4 v 776 768 V 893 885 4 119 v 912 826 a(@)894 808 y(@)912 867 y(\000)894 885 y(\000)-160 b Fd(r)106 b(r)776 766 y(r)g(r)717 826 y(r)224 b(r)102 b Fq(ist)114 b Fe(\000)1250 808 y(\000)1233 867 y(@)1250 885 y(@)p 1292 768 119 4 v 1428 826 a(@)1410 808 y(@)1428 867 y(\000)1410 885 y(\000)-160 b Fd(r)106 b(r)1292 766 y(r)g(r)1233 826 y(r)224 b(r)50 971 y Fq(2.)16 b(Das)h(Komplemen)n(t)d(v)o(on)p 641 1164 4 119 v 642 1164 a Fe(\000)684 1122 y(\000)719 1087 y(\000)p 642 1165 119 4 v 759 1164 4 119 v 642 1164 a Fd(r)106 b(r)642 1046 y(r)g(r)874 1105 y Fq(ist)p 1040 1047 119 4 v 1116 1164 a Fe(@)1075 1122 y(@)1040 1087 y(@)1040 1164 y Fd(r)g(r)1040 1046 y(r)g(r)0 1264 y Fr(Satz)22 b(1.11.)f Fm(Zwei)g(schlichte)h(Gr)n(aphen)d(sind)h(genau)i(dann)f (isomorph,)d(wenn)k(ihr)n(e)e(Komple-)0 1322 y(mente)f(isomorph)d (sind.)0 1433 y(Pr)n(o)n(of.)j Fq(Da)12 b(das)f(dopp)q(elte)g (Komplemen)o(t)d(eines)i(Graphen)h(der)g(Graph)h(selbst)f(ist,)f(gen) 1597 1435 y(\177)1596 1433 y(ugt)h(es)g(aus)0 1491 y(der)19 b(Isomorphie)e Fp(X)403 1477 y Fo(\030)403 1493 y Fq(=)459 1491 y Fp(Y)30 b Fq(auf)20 b(die)e(Isomorphie)f Fp(X)981 1473 y Fc(0)1011 1477 y Fo(\030)1012 1493 y Fq(=)1068 1491 y Fp(Y)1107 1473 y Fc(0)1138 1491 y Fq(zu)h(sc)o(hlie\031en.)f (Wir)i(iden)o(ti\014zieren)0 1549 y(en)o(tlang)j Fp(\031)r Fq(,)g(also)h Fp(E)k Fo(\032)e Fp(V)h Fo(\016)16 b Fp(V)11 b Fq(.)22 b(Sei)g Fp(f)30 b Fq(:)24 b Fp(V)850 1556 y Fn(X)908 1549 y Fo(!)h Fp(V)1011 1556 y Fn(Y)1064 1549 y Fq(bijektiv)c(und)h Fp(f)5 b Fq(\()p Fp(E)1435 1556 y Fn(X)1470 1549 y Fq(\))24 b(=)h Fp(E)1612 1556 y Fn(Y)1642 1549 y Fq(,)d(w)o(ob)q(ei)0 1607 y Fp(f)5 b Fq(\()p Fp(E)84 1614 y Fn(X)118 1607 y Fq(\))14 b(=)g Fo(ff)p Fp(f)5 b Fq(\()p Fp(a)p Fq(\))p Fp(;)j(f)d Fq(\()p Fp(b)p Fq(\))p Fo(gjf)p Fp(a;)j(b)p Fo(g)k(2)i Fp(E)s Fo(g)p Fq(.)i(Dann)h(ist)f Fp(f)5 b Fq(\()p Fp(V)1043 1614 y Fn(X)1088 1607 y Fo(\016)11 b Fp(V)1152 1614 y Fn(X)1197 1607 y Fo(n)g Fp(E)1269 1614 y Fn(X)1303 1607 y Fq(\))j(=)g Fp(V)1416 1614 y Fn(Y)1458 1607 y Fo(\016)d Fp(V)1522 1614 y Fn(Y)1564 1607 y Fo(n)g Fp(E)1636 1614 y Fn(Y)1666 1607 y Fp(:)81 b Fb(\003)0 1697 y Fr(Beispiel)17 b(1.12.)410 1837 y Fe(\010)451 1816 y(\010)493 1795 y(\010)534 1775 y(\010)545 1769 y(\010)410 1858 y(H)451 1878 y(H)493 1899 y(H)534 1920 y(H)545 1925 y(H)604 1837 y(H)563 1816 y(H)521 1795 y(H)480 1775 y(H)469 1769 y(H)604 1858 y(\010)563 1878 y(\010)521 1899 y(\010)480 1920 y(\010)469 1925 y(\010)469 1926 y(\012)496 1884 y(\012)524 1843 y(\012)552 1801 y(\012)559 1790 y(\012)559 1926 y(J)532 1884 y(J)504 1843 y(J)476 1801 y(J)469 1790 y(J)p 467 1926 4 178 v 585 1926 V 410 1839 237 4 v 410 1837 a Fd(r)224 b(r)469 1926 y(r)106 b(r)469 1748 y(r)g(r)757 1823 y Fo(\030)758 1839 y Fq(=)907 1837 y Fe(\012)935 1795 y(\012)939 1790 y(\012)907 1878 y(J)935 1920 y(J)939 1925 y(J)1116 1837 y(J)1088 1795 y(J)1085 1790 y(J)1116 1878 y(\012)1088 1920 y(\012)1085 1925 y(\012)p 967 1927 119 4 v 967 1750 V 965 1926 4 178 v 1083 1926 V 907 1839 237 4 v 907 1837 a Fd(r)225 b(r)967 1926 y(r)106 b(r)967 1748 y(r)g(r)1241 1837 y Fq(,)16 b(denn)410 2099 y Fe(\012)437 2057 y(\012)441 2052 y(\012)410 2140 y(J)437 2182 y(J)441 2187 y(J)618 2099 y(J)591 2057 y(J)587 2052 y(J)618 2140 y(\012)591 2182 y(\012)587 2187 y(\012)p 469 2189 119 4 v 469 2012 V 410 2099 a Fd(r)224 b(r)469 2187 y(r)106 b(r)469 2010 y(r)g(r)757 2085 y Fo(\030)758 2101 y Fq(=)907 2099 y Fe(\010)949 2078 y(\010)990 2057 y(\010)1032 2036 y(\010)1043 2031 y(\010)907 2119 y(H)949 2140 y(H)990 2161 y(H)1032 2182 y(H)1043 2187 y(H)1102 2099 y(H)1061 2078 y(H)1019 2057 y(H)978 2036 y(H)967 2031 y(H)1102 2119 y(\010)1061 2140 y(\010)1019 2161 y(\010)978 2182 y(\010)967 2187 y(\010)-42 b(\012)994 2146 y(\012)1022 2105 y(\012)1050 2063 y(\012)1057 2052 y(\012)1057 2187 y(J)1029 2146 y(J)1002 2105 y(J)974 2063 y(J)967 2052 y(J)907 2099 y Fd(r)225 b(r)967 2187 y(r)106 b(r)967 2010 y(r)g(r)8 2252 y Fm(\177)0 2263 y(Ubung)23 b Fq(1.1)p Fm(.)96 b Fq(\(1\))21 b(Zeigen)15 b(Sie,)g(da\031)i(die)f(Graphen)p 710 2342 V 709 2458 4 119 v 710 2458 a Fe(\000)752 2417 y(\000)787 2382 y(\000)p 710 2460 119 4 v 827 2458 4 119 v 710 2458 a Fd(r)106 b(r)710 2340 y(r)g(r)940 2386 y Fo(\030)940 2401 y Fq(=)1090 2382 y Fe(@)1131 2423 y(@)1166 2458 y(@)p 1088 2458 V -118 w(\000)1131 2417 y(\000)1166 2382 y(\000)p 1090 2460 119 4 v 1206 2458 4 119 v 1090 2458 a Fd(r)g(r)1090 2340 y(r)g(r)157 2558 y Fq(isomorph)16 b(sind.)74 2616 y(\(2\))21 b(Sei)15 b Fp(X)k Fq(ein)14 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Fq(:)23 b Fp(X)28 b Fo(!)23 b Fp(X)k Fq(b)q(e)22 b(a)g(dynamical)e(system.)h (The)g Fm(state)j(sp)n(ac)n(e)p Fq(,)d Fm(phase)0 1954 y(sp)n(ac)n(e)p Fq(,)e(or)h Fm(phase)h(gr)n(aph)e Fq(is)h(the)f (digraph)i Fo(S)t Fq(\()p Fp(f)5 b Fq(\))20 b(with)g(v)o(ertex)e(set)i Fp(V)1297 1962 y Fc(S)r Fk(\()p Fn(f)t Fk(\))1391 1954 y Fq(:=)g Fp(X)k Fq(and)c(edge)g(set)0 2012 y Fp(E)36 2020 y Fc(S)r Fk(\()p Fn(f)t Fk(\))124 2012 y Fq(:=)13 b Fo(f)p Fq(\()p Fp(x;)8 b(f)d Fq(\()p Fp(x)p Fq(\)\))p Fo(j)p Fp(x)13 b Fo(2)h Fp(X)t Fo(g)p Fq(.)i(The)g(incidence)f(map)g Fp(\031)j Fq(is)e(the)g(em)o(b)q(edding)f Fp(E)1480 2020 y Fc(S)r Fk(\()p Fn(f)t Fk(\))1567 2012 y Fo(\022)f Fp(X)h Fo(\002)c Fp(X)t Fq(.)0 2081 y Fm(R)n(emark)21 b Fq(2.5)p Fm(.)f Fq(A)15 b(digraph)h Fo(D)h Fq(is)e(the)g(state)g(space)g Fo(S)t Fq(\()p Fp(f)5 b Fq(\))16 b(of)f(a)h(dynamical)d(system)h Fp(f)19 b Fq(:)13 b Fp(X)18 b Fo(!)c Fp(X)t Fq(,)0 2139 y(i\013)i(ev)o(ery)f(v)o(ertex)g Fp(x)e Fo(2)h Fp(V)451 2146 y Fc(D)498 2139 y 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b(2.11.)j(\(Sark)n(o)n(vskii's)d(Theorem)g (\(1964\)\))p Fm(:)g(L)n(et)g Fp(f)i Fq(:)14 b Fl(R)d Fo(!)i Fl(R)i Fm(b)n(e)j(a)f(c)n(ontinuous)0 2742 y(function)g(and)f (let)h Fp(f)k Fm(have)16 b(a)g(p)n(erio)n(dic)f(p)n(oint)g(of)h(or)n (der)e Fp(n)p Fm(.)i(Then)h Fp(f)k Fm(has)15 b(p)n(erio)n(dic)g(p)n (oints)g(of)h(any)0 2801 y(or)n(der)g Fp(m)h Fm(for)g(al)r(l)i Fp(n)14 b(>)g(m)j Fm(in)h(the)f(fol)r(lowing)j Fq(Sark)o(o)o(vskii)15 b(order)j Fm(of)f Fl(N)p Fm(:)0 2859 y Fq(3)d Fp(>)g Fq(5)g Fp(>)g Fq(7)8 b Fp(:)g(:)g(:)g Fq(2)k Fo(\001)f Fq(3)j Fp(>)g Fq(2)d Fo(\001)g Fq(5)j Fp(>)g Fq(2)d Fo(\001)g Fq(7)d Fp(:)g(:)g(:)h Fq(2)762 2841 y Fn(n)797 2859 y Fo(\001)h Fq(3)15 b Fp(>)e Fq(2)935 2841 y Fn(n)970 2859 y Fo(\001)e Fq(5)j Fp(>)g Fq(2)1109 2841 y Fn(n)1144 2859 y Fo(\001)d Fq(7)d Fp(:)g(:)g(:)g Fq(2)1291 2841 y Fk(3)1325 2859 y Fp(>)14 b Fq(2)1401 2841 y Fk(2)1435 2859 y Fp(>)g Fq(2)g Fp(>)g Fq(1)p Fp(:)p eop %%Page: 12 12 12 11 bop 0 -49 a Fh(12)722 b(SEMINAR)17 b(SS04)50 50 y Fq(W)l(e)d(only)g(sho)o(w)h(the)f(follo)o(wing)f(Corollary)i(whic)o (h)e(is)h(a)h(m)o(uc)o(h)d(w)o(eak)o(er)h(v)o(ersion)h(of)g(this)g (Theo-)0 108 y(rem.)0 177 y Fr(F)-5 b(olgerung)20 b(2.12.)h Fm(L)n(et)c Fp(f)k Fq(:)14 b Fl(R)e Fo(!)j Fl(R)g Fm(b)n(e)k(a)f(c)n (ontinuous)h(function)g(and)g(let)g Fp(f)k Fm(have)c(a)f(p)n(erio)n (dic)0 235 y(p)n(oint)f(of)h(or)n(der)e Fq(3)p Fm(.)i(Then)g Fp(f)k Fm(has)c(p)n(erio)n(dic)e(p)n(oints)h(of)h(any)f(or)n(der)f Fp(n)p Fm(.)0 342 y(Pr)n(o)n(of.)j Fq(Let)i Fo(f)p Fp(a;)8 b(b;)g(c)p Fo(g)20 b Fq(b)q(e)i(a)f(p)q(erio)q(dic)g(orbit)h(of)f Fp(f)5 b Fq(.)21 b(W)l(e)g(ma)o(y)f(assume)g(that)i Fp(a)g(<)g(b)g(<)g (c)p Fq(.)f(W)l(e)0 400 y(distinguish)g(t)o(w)o(o)g(cases:)g Fp(f)5 b Fq(\()p Fp(a)p Fq(\))22 b(=)g Fp(b)e Fq(and)i Fp(f)5 b Fq(\()p Fp(a)p Fq(\))22 b(=)g Fp(c)p Fq(.)e(First)h(supp)q (ose)h(that)g Fp(f)5 b Fq(\()p Fp(a)p Fq(\))22 b(=)g Fp(b)p Fq(.)e(Then)0 458 y Fp(f)5 b Fq(\()p Fp(b)p Fq(\))14 b(=)g Fp(c)i Fq(and)h Fp(f)5 b Fq(\()p Fp(c)p Fq(\))14 b(=)g Fp(a)p Fq(.)h(The)h(pro)q(of)i(for)e(the)g(case)g Fp(f)5 b Fq(\()p Fp(a)p Fq(\))14 b(=)g Fp(c)i Fq(is)g(similar.)50 537 y(Let)i Fp(I)161 544 y Fk(0)196 537 y Fq(=)f([)p Fp(a;)8 b(b)p Fq(])16 b(and)i Fp(I)482 544 y Fk(1)518 537 y Fq(=)e([)p Fp(b;)8 b(c)p Fq(].)16 b(The)i(In)o(termediate)d(V)l (alue)i(Theorem)f(implies)f(that)k Fp(I)1726 544 y Fk(1)1761 537 y Fo(\032)0 595 y Fp(f)5 b Fq(\()p Fp(I)70 602 y Fk(0)90 595 y Fq(\))16 b(and)i Fp(I)243 602 y Fk(0)274 595 y Fo([)12 b Fp(I)341 602 y Fk(1)375 595 y Fo(\032)j Fp(f)5 b Fq(\()p Fp(I)499 602 y Fk(1)518 595 y Fq(\).)17 b(Namely)l(,)d(let)i Fp(d"I)898 602 y Fk(1)932 595 y Fq(=)f([)p Fp(f)5 b Fq(\()p Fp(a)p Fq(\))p Fp(;)j(f)d Fq(\()p Fp(b)p Fq(\)].)16 b(Let)h Fp(g)r Fq(\()p Fp(x)p Fq(\))d(=)h Fp(f)5 b Fq(\()p Fp(x)p Fq(\))12 b Fo(\000)f Fp(d)p Fq(,)17 b(then)0 654 y(the)f(In)o(termediate)d(V)l(alue)j (Theorem)f(implies)f(that)j Fp(g)h Fq(has)f(a)g(zero)f(in)g Fp(I)1336 661 y Fk(0)1355 654 y Fq(.)g(That)h(is,)f Fp(I)1592 661 y Fk(1)1625 654 y Fo(\032)e Fp(f)5 b Fq(\()p Fp(I)1748 661 y Fk(0)1767 654 y Fq(\).)0 712 y(Then)16 b Fp(f)22 b Fq(has)17 b(a)g(\014xed)e(p)q(oin)o(t)i(in)f Fp(I)625 719 y Fk(0)655 712 y Fo([)11 b Fp(I)721 719 y Fk(1)741 712 y Fq(,)k(that)i(is,)f Fp(f)21 b Fq(has)c(a)g(p)q(erio)q(dic)f(p)q (oin)o(t)g(of)h(p)q(erio)q(d)f(1.)50 791 y(No)o(w)h(let)f Fp(n)f(>)f Fq(1)k(b)q(e)f(an)g(in)o(teger.)e(If)i Fp(n)e Fq(=)g(3)i(w)o(e)f(are)h(done.)g(T)l(o)g(sho)o(w)h(that)f Fp(f)22 b Fq(has)c(a)f(p)q(erio)q(dic)0 849 y(p)q(oin)o(t)f(of)h(p)q (erio)q(d)g Fp(n)f Fq(w)o(e)g(will)f(construct)h(a)h(nested)f(sequence) 571 929 y Fp(A)608 936 y Fn(n)645 929 y Fo(\032)d Fp(:)8 b(:)g(:)13 b Fo(\032)h Fp(A)858 936 y Fk(2)891 929 y Fo(\032)g Fp(A)981 936 y Fk(1)1014 929 y Fo(\032)f Fp(A)1103 936 y Fk(0)1136 929 y Fq(=)h Fp(I)1210 936 y Fk(1)0 1009 y Fq(of)j(closed)e(in)o(terv)m(als)h(with)g(the)g(follo)o(wing)g(prop)q (erties:)74 1100 y(\(1\))21 b Fp(f)5 b Fq(\()p Fp(A)242 1107 y Fn(k)264 1100 y Fq(\))13 b(=)h Fp(A)385 1107 y Fn(k)q Fc(\000)p Fk(1)467 1100 y Fq(for)j Fp(k)f Fq(=)d(1)p Fp(;)8 b(:::n)j Fo(\000)f Fq(2;)74 1158 y(\(2\))21 b Fp(f)186 1140 y Fn(k)208 1158 y Fq(\()p Fp(A)264 1165 y Fn(k)285 1158 y Fq(\))14 b(=)f Fp(I)391 1165 y Fk(1)427 1158 y Fq(for)j Fp(k)g Fq(=)e(1)p Fp(;)8 b(:::;)g(n)i Fo(\000)h Fq(2;)74 1216 y(\(3\))21 b Fp(f)186 1198 y Fn(n)p Fc(\000)p Fk(1)255 1216 y Fq(\()p Fp(A)311 1223 y Fn(n)p Fc(\000)p Fk(1)379 1216 y Fq(\))14 b(=)g Fp(I)486 1223 y Fk(0)505 1216 y Fq(;)74 1274 y(\(4\))21 b Fp(f)186 1256 y Fn(n)210 1274 y Fq(\()p Fp(A)266 1281 y Fn(n)289 1274 y Fq(\))14 b(=)g Fp(I)396 1281 y Fk(1)415 1274 y Fq(.)0 1364 y(First)h(w)o(e)g(sho)o(w)i(that,)e(if)g(suc)o(h)h(a)g (sequence)e(of)i(in)o(terv)m(als)f(exists,)g(then)g Fp(f)21 b Fq(has)c(a)f(p)q(erio)q(dic)f(p)q(oin)o(t)0 1422 y(of)k(p)q(erio)q(d) f Fp(n)p Fq(.)g(Since)g Fp(A)439 1429 y Fn(n)479 1422 y Fo(\032)g Fp(I)558 1429 y Fk(1)577 1422 y Fq(,)g(Prop)q(ert)o(y)g (\(4\))h(implies)c(that)k Fp(A)1209 1429 y Fn(n)1249 1422 y Fo(\032)f Fp(f)1335 1404 y Fn(n)1358 1422 y Fq(\()p Fp(A)1414 1429 y Fn(n)1437 1422 y Fq(\),)g(so)h(that)g Fp(f)1687 1404 y Fn(n)1729 1422 y Fq(has)0 1481 y(a)f(\014xed)e(p)q (oin)o(t)h(in)g Fp(A)383 1488 y Fn(n)406 1481 y Fq(,)g(whic)o(h)f(just) i(means)e(that)h Fp(f)23 b Fq(has)18 b(a)f(p)q(erio)q(dic)g(p)q(oin)o (t)g Fp(p)h Fq(in)f Fp(A)1560 1488 y Fn(n)1600 1481 y Fq(suc)o(h)g(that)0 1539 y Fp(f)29 1521 y Fn(n)53 1539 y Fq(\()p Fp(p)p Fq(\))d(=)g Fp(p)p Fq(.)i(It)g(remains)f(to)h(sho)o(w) h(that)g Fp(n)f Fq(is)g(the)g(p)q(erio)q(d)h(of)g Fp(p)p Fq(.)50 1618 y(Then)23 b(Prop)q(ert)o(y)g(\(2\))h(of)f(the)g(sequence)f (implies)f(that)i Fp(f)1158 1600 y Fn(i)1173 1618 y Fq(\()p Fp(p)p Fq(\))j Fo(2)g Fp(I)1342 1625 y Fk(1)1386 1618 y Fq(=)g([)p Fp(b;)8 b(c)p Fq(])22 b(for)h(all)g Fp(i)i Fq(=)0 1676 y(0)p Fp(;)8 b(:)g(:)g(:)g(;)g(n)i Fo(\000)f Fq(2)17 b(\(with)e(the)h(con)o(v)o(en)o(tion)e(that)i Fp(f)852 1658 y Fk(0)872 1676 y Fq(\()p Fp(x)p Fq(\))e(=)f Fp(x)p Fq(\),)i(and)i Fp(f)1203 1658 y Fn(n)p Fc(\000)p Fk(1)1271 1676 y Fq(\()p Fp(p)p Fq(\))e Fo(2)f Fp(I)1417 1683 y Fk(0)1450 1676 y Fq(=)g([)p Fp(a;)8 b(b)p Fq(].)13 b(First)i(w)o(e)0 1734 y(sho)o(w)k(that)g Fp(p)g Fq(is)g(not)g(equal)f (to)h(either)e Fp(b)i Fq(or)g Fp(c)p Fq(.)f(Supp)q(ose)h(that)g Fp(p)g Fq(=)f Fp(c)p Fq(.)g(Then)g Fp(f)5 b Fq(\()p Fp(p)p Fq(\))19 b(=)f Fp(a)23 b(=)-30 b Fo(2)18 b Fp(I)1767 1741 y Fk(1)1786 1734 y Fq(.)0 1793 y(The)d(conditions)g(on)g(the)g (sequence)e(imply)g(that)i Fp(f)949 1775 y Fn(n)p Fc(\000)p Fk(1)1018 1793 y Fq(\()p Fp(p)p Fq(\))g(is)g(the)g(only)f(iterate)g(of) h Fp(p)h Fq(that)f(is)g(not)0 1851 y(in)i Fp(I)80 1858 y Fk(1)100 1851 y Fq(,)g(hence)g Fp(n)f Fq(=)g(2.)i(But)f(this)h(con)o (tradicts)f(the)h(fact)g(that)g(the)f(p)q(erio)q(d)h(of)g Fp(c)g Fq(is)g(3.)f(So)h Fp(p)f Fo(6)p Fq(=)f Fp(c)p Fq(.)0 1909 y(No)o(w)g(supp)q(ose)h(that)g Fp(p)d Fq(=)g Fp(b)p Fq(.)h(Again,)h(the)f(only)h(iterate)g(of)g Fp(p)g Fq(not)h(in)f Fp(I)1302 1916 y Fk(1)1337 1909 y Fq(is)g(the)g(\()p Fp(n)10 b Fo(\000)h Fq(1\)st.)16 b(Since)0 1967 y Fp(f)29 1949 y Fk(2)49 1967 y Fq(\()p Fp(p)p Fq(\))k(=)g Fp(a)26 b(=)-30 b Fo(2)20 b Fp(I)310 1974 y Fk(1)329 1967 y Fq(,)g(this)g (implies)d(that)j Fp(n)g Fq(=)g(3.)g(But)f(this)h(con)o(tradicts)g(our) g(assumption)g(that)0 2025 y Fp(n)14 b Fo(6)p Fq(=)g(3.)50 2105 y(No)o(w)19 b(observ)o(e)g(that)h Fp(f)479 2087 y Fn(n)p Fc(\000)p Fk(1)548 2105 y Fq(\()p Fp(p)p Fq(\))g Fo(2)f Fp(I)704 2112 y Fk(0)743 2105 y Fq(=)g([)p Fp(a;)8 b(b)p Fq(],)17 b(whic)o(h)i(is)g(disjoin)o(t)g(from)f(the)i(in)o(terv)m (al)e(\()p Fp(b;)8 b(c)p Fq(\).)0 2163 y(This)14 b(implies)d(that)k Fp(f)405 2145 y Fn(n)p Fc(\000)p Fk(1)474 2163 y Fq(\()p Fp(p)p Fq(\))f Fo(6)p Fq(=)g Fp(p)p Fq(,)f(so)i(that)f Fp(p)h Fq(cannot)f(ha)o(v)o(e)f(p)q(erio)q(d)i Fp(n)6 b Fo(\000)g Fq(1.)14 b(If)g(the)f(p)q(erio)q(d)i(is)e(less)0 2221 y(than)i Fp(n)7 b Fo(\000)g Fq(1,)13 b(then)h(Prop)q(ert)o(y)g (\(2\))g(implies)d(that)k(the)f(orbit)g(of)g 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