%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: Fds.dvi %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -O 0cm,2cm Fds %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2001.05.11:0637 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 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2496 y Fp(n)985 2489 y Fr(as)g(follo)o(ws.)34 b(First)20 b(construct)g(the)0 2553 y(set)87 2540 y(~)76 2553 y Fq(F)j Fr(of)17 b(all)f Fq(n)p Fr(-tuples)455 2540 y(~)445 2553 y Fq(f)k Fr(=)14 b(\()572 2539 y(~)560 2553 y Fq(f)589 2538 y Fk(1)609 2553 y Fq(;)643 2539 y Fr(~)631 2553 y Fq(f)660 2538 y Fk(2)680 2553 y Fq(;)8 b(:)g(:)g(:)f(;)803 2540 y Fr(~)789 2553 y Fq(f)818 2538 y Fp(n)842 2553 y Fr(\))p Fq(;)16 b Fr(whic)o(h)g(either)f(are)i(in)f Fq(F)7 b Fr(,)16 b(or)h(arise)f(from)0 2611 y(an)22 b(elemen)o(t)d(in)j Fq(F)28 b Fr(b)o(y)22 b(replacing)f(one)h(of)g(the)g(co)q(ordinates)g (b)o(y)g(a)g(0-lo)q(cal)h(function,)0 2669 y(that)g(is,)g(b)o(y)f(a)g (function)g(from)f Fq(L)653 2651 y Fp(i)653 2681 y Fk(0)696 2669 y Fr(for)h(some)f Fq(i)p Fr(.)39 b(No)o(w)22 b(de\014ne)g(the)g (graph)i(\010\()p Fq(F)7 b Fr(\))21 b(as)p eop %%Page: 6 6 6 5 bop 0 187 a Fl(6)316 b(REINHARD)13 b(LA)o(UBENBA)o(CHER)f(AND)h (BODO)g(P)m(AREIGIS)0 293 y Fr(follo)o(ws.)28 b(An)18 b(edge)h Fq(i)p Fr({)p 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Fo(\002)f(\001)e (\001)g(\001)k(\002)e Fq(L)1124 536 y Fp(n)1124 569 y Fk(1)1148 557 y Fr(\()p 1167 516 V Fq(G)p Fr(\))p Fq(;)0 632 y Fr(where)p 141 592 V 16 w Fq(G)17 b Fr(is)f(the)g(complem)o(en)o (t)d(of)k Fq(G)f Fr(in)g Fq(K)809 639 y Fp(n)833 632 y Fr(.)0 712 y(W)l(e)f(need)h(to)g(sho)o(w)g(that)g(\010)g(and)g(\011)g (are)g(inclusion)e(rev)o(ersing,)h(that)h Fq(F)k Fo(\032)14 b Fr(\011\010\()p Fq(F)7 b Fr(\),)14 b(and)0 774 y(that)j Fq(G)d Fo(\032)f Fr(\010\011\()p Fq(G)p Fr(\).)22 b(T)l(o)17 b(sho)o(w)g(the)f(\014rst)g(prop)q(ert)o(y)l(,)f(let)h Fq(F)k Fo(\032)14 b Fq(F)1195 756 y Fh(0)1222 774 y Fr(in)i Fo(F)5 b Fr(.)20 b(Then)1493 762 y(~)1482 774 y Fq(F)g Fo(\032)1599 762 y Fr(~)1586 774 y Fq(F)1625 760 y Fh(0)1636 774 y Fr(.)0 832 y(An)13 b(edge)h Fq(i)p Fr({)p Fq(j)i Fr(of)e Fq(K)357 839 y Fp(n)395 832 y Fr(is)f(in)g(\010\()p Fq(F)588 814 y Fh(0)599 832 y Fr(\))h(if)f(and)h(only)g Fq(f)900 814 y Fp(i)928 832 y Fr(and)g Fq(f)1049 814 y Fp(j)1081 832 y Fr(comm)o(ute)c(for)k(ev)o(ery)e(elemen)o(t)0 895 y Fq(f)24 b Fr(=)18 b(\()p Fq(f)152 876 y Fp(l)166 895 y Fr(\))g Fo(2)266 882 y Fr(~)255 895 y Fq(F)294 876 y Fh(0)305 895 y Fr(.)30 b(Since)490 882 y(~)479 895 y Fq(F)25 b Fo(\032)604 882 y Fr(~)593 895 y Fq(F)632 876 y Fh(0)643 895 y Fr(,)19 b Fq(i)p Fr({)p Fq(j)j Fr(is)d(also)h(con) o(tained)e(in)h(\010\()p Fq(F)7 b Fr(\).)29 b(If)18 b Fq(G)i Fo(\032)e Fq(G)1554 876 y Fh(0)1585 895 y Fr(are)0 955 y(subgraphs)k(of)f Fq(K)336 962 y Fp(n)360 955 y Fr(,)g(then)p 511 915 51 2 v 21 w Fq(G)549 941 y Fh(0)582 955 y Fo(\032)p 642 915 39 2 v 21 w Fq(G)q Fr(.)34 b(A)20 b(1-lo)q(cal)h(function)g(on)g(the)f(smaller)f(graph)i(is)0 1013 y(certainly)13 b(also)i(1-lo)q(cal)g(on)g(the)g(larger)f(graph.)22 b(This)14 b(sho)o(ws)i(that)f(the)f(corresp)q(ondence)0 1071 y(is)i(inclusion)g(rev)o(ersing.)0 1151 y(T)l(o)j(sho)o(w)f(that)h Fq(F)k Fo(\032)17 b Fr(\011\010\()p Fq(F)7 b Fr(\),)18 b(let)f(\()p Fq(f)717 1133 y Fk(1)737 1151 y Fq(;)8 b(:)g(:)g(:)f(;)h (f)875 1133 y Fp(n)899 1151 y Fr(\))17 b Fo(2)g Fq(F)7 b Fr(.)26 b(W)l(e)18 b(ha)o(v)o(e)g(to)g(sho)o(w)h(that)f Fq(f)1585 1133 y Fp(i)1617 1151 y Fo(2)0 1215 y Fq(L)33 1197 y Fp(i)33 1227 y Fk(1)53 1215 y Fr(\()p 72 1172 112 2 v(\010\()p Fq(F)7 b Fr(\))o(\),)14 b(i.e.)20 b(that)14 b Fq(f)446 1197 y Fp(i)460 1215 y Fr(\()p Fq(x)p Fr(\))g(do)q(es)h(not) f(dep)q(end)g(on)h(the)e Fq(j)s Fr(-th)i(co)q(ordinate)f Fq(x)1411 1222 y Fp(j)1443 1215 y Fr(of)g Fq(x)g Fr(if)f Fq(j)k Fr(is)0 1280 y(not)c(connected)f(to)h Fq(i)g Fr(in)f(the)g (graph)p 661 1237 V 14 w(\010\()p Fq(F)7 b Fr(\).)20 b(Let)12 b Fq(j)s Fr(\()p Fo(6)p Fr(=)i Fq(i)p Fr(\))e(b)q(e)h(a)g(v)o (ertex)f(in)p 1330 1237 V 12 w(\010\()p Fq(F)7 b Fr(\))12 b(suc)o(h)h(that)0 1344 y Fq(i)p Fr({)p Fq(j)18 b Fr(is)e(not)f(in)p 269 1301 V 15 w(\010\()p Fq(F)7 b Fr(\).)21 b(Let)513 1331 y(~)502 1344 y Fq(f)531 1326 y Fp(j)565 1344 y Fr(b)q(e)16 b(a)g(function)f(in)g Fq(L)950 1321 y Fp(j)950 1356 y Fk(0)970 1344 y Fr(.)21 b(Then)15 b(\()p Fq(f)1179 1326 y Fk(1)1199 1344 y Fq(;)8 b(:)g(:)g(:)g(;)1319 1331 y Fr(~)1309 1344 y Fq(f)1338 1326 y Fp(j)1356 1344 y Fq(;)g(:)g(:)g(:)g (;)g(f)1495 1326 y Fp(n)1518 1344 y Fr(\))14 b Fo(2)1609 1332 y Fr(~)1598 1344 y Fq(F)6 b Fr(.)0 1408 y(Since)20 b Fq(i)p Fr({)q Fq(j)k Fr(is)d(not)h(in)p 426 1365 V 21 w(\010\()p Fq(F)7 b Fr(\))o(,)23 b(it)e(is)g(in)g(\010\()p Fq(F)7 b Fr(\))21 b(hence)f Fq(f)1046 1390 y Fp(i)1075 1408 y Fo(\016)1125 1395 y Fr(~)1115 1408 y Fq(f)1144 1390 y Fp(j)1185 1408 y Fr(=)1256 1395 y(~)1246 1408 y Fq(f)1275 1390 y Fp(j)1308 1408 y Fo(\016)14 b Fq(f)1376 1390 y Fp(i)1412 1408 y Fr(for)22 b(all)f(four)0 1471 y(functions)221 1458 y(~)210 1471 y Fq(f)239 1453 y Fp(j)272 1471 y Fo(2)14 b Fq(L)352 1447 y Fp(j)352 1483 y Fk(0)372 1471 y Fr(.)21 b(No)o(w)358 1546 y(~)347 1559 y Fq(f)376 1538 y Fp(j)406 1559 y Fo(\016)11 b Fq(f)471 1538 y Fp(i)485 1559 y Fr(\()p Fq(x)p Fr(\))j(=)f(\()p Fq(x)663 1566 y Fk(1)683 1559 y Fq(;)8 b(:)g(:)g(:)f(;)h(f)821 1538 y Fp(i)816 1571 y(i)835 1559 y Fr(\()p Fq(x)p Fr(\))p Fq(;)g(:)g(:)g(:)f(;)1021 1546 y Fr(~)1010 1559 y Fq(f)1039 1535 y Fp(j)1034 1572 y(j)1058 1559 y Fr(\()p Fq(x)p Fr(\))p Fq(;)h(:)g(:)g(:)f(;)h(x)1261 1566 y Fp(n)1284 1559 y Fr(\))0 1642 y(and)172 1726 y Fq(f)201 1705 y Fp(i)227 1726 y Fo(\016)274 1713 y Fr(~)263 1726 y Fq(f)292 1705 y Fp(j)311 1726 y Fr(\()p Fq(x)p Fr(\))13 b(=)h(\()p Fq(x)489 1733 y Fk(1)508 1726 y Fq(;)8 b(:)g(:)g(:)g(;)g(f)647 1705 y Fp(i)642 1738 y(i)661 1726 y Fr(\()p Fq(x)708 1733 y Fk(1)727 1726 y Fq(;)g(:)g(:)g(:)g(;)j Fr(~)-27 b Fq(x)865 1733 y Fp(j)883 1726 y Fq(;)8 b(:)g(:)g(:)f(;)h(x)1020 1733 y Fp(n)1043 1726 y Fr(\))p Fq(;)g(:)g(:)g(:)f(;)1182 1713 y Fr(~)1171 1726 y Fq(f)1200 1702 y Fp(j)1195 1739 y(j)1219 1726 y Fr(\()p Fq(x)p Fr(\))p Fq(;)h(:)g(:)g(:)f(;)h(x)1422 1733 y Fp(n)1445 1726 y Fr(\))p Fq(;)0 1821 y Fr(where)19 b(~)-27 b Fq(x)169 1828 y Fp(j)201 1821 y Fr(=)263 1808 y(~)252 1821 y Fq(f)281 1797 y Fp(j)276 1834 y(j)300 1821 y Fr(\()p Fq(x)p Fr(\),)16 b(hence)f Fq(f)560 1803 y Fp(i)555 1833 y(i)575 1821 y Fr(\()p Fq(x)p Fr(\))g(and)i Fq(f)780 1803 y Fp(i)795 1821 y Fr(\()p Fq(x)p Fr(\))f(do)g(not)h(dep)q (end)f(on)h Fq(x)1296 1828 y Fp(j)1314 1821 y Fr(.)0 1903 y(No)o(w)f(w)o(e)g(sho)o(w)h(that)f Fq(G)f Fo(\032)e Fr(\010\011\()p Fq(G)p Fr(\).)22 b(First)16 b(observ)o(e)g(that)332 1986 y Ff(])322 2001 y Fr(\011\()p Fq(G)p Fr(\))e(=)g(\011\()p Fq(G)p Fr(\))g(=)g Fq(L)715 1980 y Fk(1)715 2013 y(1)735 2001 y Fr(\()p 754 1961 39 2 v Fq(G)p Fr(\))d Fo(\002)g Fq(L)905 1980 y Fk(2)905 2013 y(1)925 2001 y Fr(\()p 944 1961 V Fq(G)p Fr(\))h Fo(\002)e(\001)e(\001)g(\001)k(\002)e Fq(L)1214 1980 y Fp(n)1214 2013 y Fk(1)1238 2001 y Fr(\()p 1257 1961 V Fq(G)p Fr(\))p Fq(;)0 2087 y Fr(since)17 b Fq(L)154 2069 y Fp(i)154 2099 y Fk(0)189 2087 y Fo(\032)f Fq(L)277 2069 y Fp(i)277 2099 y Fk(1)297 2087 y Fr(\()p 316 2047 V Fq(G)p Fr(\).)25 b(Let)18 b Fq(i)p Fr({)p Fq(j)i Fr(b)q(e)e(in)f Fq(G)h Fr(and)g(let)e Fq(f)22 b Fo(2)16 b Fr(\011\()p Fq(G)p Fr(\).)25 b(W)l(e)17 b(ha)o(v)o(e)g(to)h (sho)o(w)g(that)0 2145 y Fq(f)29 2127 y Fp(i)55 2145 y Fo(\016)11 b Fq(f)120 2127 y Fp(j)152 2145 y Fr(=)j Fq(f)233 2127 y Fp(j)263 2145 y Fo(\016)d Fq(f)328 2127 y Fp(i)359 2145 y Fr(holds,)16 b(since)g(then)g Fq(i)p Fr({)p Fq(j)j Fr(is)e(an)f(edge)h(in)f(\010\011\()p Fq(G)p Fr(\).)22 b(No)o(w)16 b Fq(f)j Fo(2)c Fr(\011\()p Fq(G)p Fr(\))f(=)0 2205 y Fq(L)33 2187 y Fk(1)33 2218 y(1)53 2205 y Fr(\()p 72 2165 V Fq(G)p Fr(\))e Fo(\002)f Fq(L)224 2187 y Fk(2)224 2218 y(1)244 2205 y Fr(\()p 263 2165 V Fq(G)p Fr(\))g Fo(\002)h(\001)c(\001)g(\001)j(\002)g Fq(L)534 2187 y Fp(n)534 2218 y Fk(1)558 2205 y Fr(\()p 577 2165 V Fq(G)p Fr(\))17 b(implies)d(that)j Fq(f)952 2187 y Fp(i)947 2218 y(i)966 2205 y Fr(\()p Fq(x)p Fr(\))g(do)q(es)g (not)g(dep)q(end)g(on)g Fq(x)1512 2212 y Fp(j)1547 2205 y Fr(since)0 2269 y(the)h(edge)h Fq(i)p Fr({)p Fq(j)i Fr(is)e(not)g(in)p 482 2229 V 18 w Fq(G)g Fr(and,)g(similarly)l(,)d Fq(f)891 2245 y Fp(j)886 2282 y(j)910 2269 y Fr(\()p Fq(x)p Fr(\))i(do)q(es)h(not)h(dep)q(end)e(on)h Fq(x)1465 2276 y Fp(i)1479 2269 y Fr(.)28 b(Hence)0 2327 y(w)o(e)16 b(get)58 2412 y Fq(f)87 2391 y Fp(i)113 2412 y Fo(\016)11 b Fq(f)178 2391 y Fp(j)196 2412 y Fr(\()p Fq(x)p Fr(\))j(=)g Fq(f)357 2391 y Fp(i)371 2412 y Fr(\()p Fq(x)418 2419 y Fk(1)437 2412 y Fq(;)8 b(:)g(:)g(:)g(;)g(f)576 2388 y Fp(j)571 2425 y(j)594 2412 y Fr(\()p Fq(x)p Fr(\))p Fq(;)g(:)g(:)g(:)f(;)h(x)797 2419 y Fp(n)820 2412 y Fr(\))14 b(=)g(\()p Fq(x)952 2419 y Fk(1)971 2412 y Fq(;)8 b(:)g(:)g(:)g(;)g(f) 1110 2391 y Fp(i)1105 2424 y(i)1124 2412 y Fr(\()p Fq(x)p Fr(\))p Fq(;)g(:)g(:)g(:)f(;)h(f)1328 2388 y Fp(j)1323 2425 y(j)1347 2412 y Fr(\()p Fq(x)p Fr(\))p Fq(;)g(:)g(:)g(:)f(;)h(x) 1550 2419 y Fp(n)1573 2412 y Fr(\))0 2495 y(and,)16 b(similarly)l(,)51 2580 y Fq(f)80 2560 y Fp(j)110 2580 y Fo(\016)11 b Fq(f)175 2560 y Fp(i)190 2580 y Fr(\()p Fq(x)p Fr(\))i(=)h Fq(f)350 2560 y Fp(j)368 2580 y Fr(\()p Fq(x)415 2587 y Fk(1)435 2580 y Fq(;)8 b(:)g(:)g(:)f(;)h(f)573 2560 y Fp(i)568 2593 y(i)588 2580 y Fr(\()p Fq(x)p Fr(\))p Fq(;)g(:)g(:)g(:)f(;)h(x)791 2587 y Fp(n)814 2580 y Fr(\))13 b(=)h(\()p Fq(x)945 2587 y Fk(1)965 2580 y Fq(;)8 b(:)g(:)g(:)f(;)h(f)1103 2560 y Fp(i)1098 2593 y(i)1117 2580 y Fr(\()p Fq(x)p Fr(\))p Fq(;)g(:)g(:)g(:)f(;)h(f)1321 2557 y Fp(j)1316 2593 y(j)1340 2580 y Fr(\()p Fq(x)p Fr(\))p Fq(;)g(:)g(:)g(:)f(;)h(x)1543 2587 y Fp(n)1566 2580 y Fr(\))p Fq(:)0 2669 y Fr(Th)o(us,)16 b(w)o(e)g(ha)o(v)o(e)f Fq(f)350 2651 y Fp(i)376 2669 y Fo(\016)c Fq(f)441 2651 y Fp(j)473 2669 y Fr(=)j Fq(f)554 2651 y Fp(j)584 2669 y Fo(\016)d Fq(f)649 2651 y Fp(i)663 2669 y Fr(.)934 b Fe(\003)p eop %%Page: 7 7 7 6 bop 528 187 a Fl(FINITE)16 b(D)o(YNAMICAL)g(SYSTEMS)509 b(7)0 286 y Fr(W)l(e)22 b(illustrate)f(this)h(corresp)q(ondence)g(with) f(an)i(example.)36 b(Let)22 b Fq(n)i Fr(=)g(3,)f(and)f(let)g Fq(F)0 344 y Fr(consist)16 b(of)h(the)f(single)g(triple)f(of)h (functions)h Fq(f)i Fr(=)14 b(\()p Fq(f)975 326 y Fk(1)995 344 y Fq(;)8 b(f)1046 326 y Fk(2)1066 344 y Fq(;)g(f)1117 326 y Fk(3)1136 344 y Fr(\))17 b(with)469 443 y Fq(f)498 424 y Fk(1)518 443 y Fr(\()p Fq(x)565 450 y Fk(1)585 443 y Fq(;)8 b(x)635 450 y Fk(2)654 443 y Fq(;)g(x)704 450 y Fk(3)723 443 y Fr(\))42 b(=)14 b(\()p Fq(x)883 450 y Fk(2)902 443 y Fq(;)8 b(x)952 450 y Fk(2)971 443 y Fq(;)g(x)1021 450 y Fk(3)1041 443 y Fr(\))p Fq(;)469 501 y(f)498 483 y Fk(2)518 501 y Fr(\()p Fq(x)565 508 y Fk(1)585 501 y Fq(;)g(x)635 508 y Fk(2)654 501 y Fq(;)g(x)704 508 y Fk(3)723 501 y Fr(\))42 b(=)14 b(\()p Fq(x)883 508 y Fk(1)902 501 y Fq(;)8 b(x)952 508 y Fk(1)971 501 y Fq(;)g(x)1021 508 y Fk(3)1041 501 y Fr(\))p Fq(;)469 559 y(f)498 541 y Fk(3)518 559 y Fr(\()p Fq(x)565 566 y Fk(1)585 559 y Fq(;)g(x)635 566 y Fk(2)654 559 y Fq(;)g(x)704 566 y Fk(3)723 559 y Fr(\))42 b(=)14 b(\()p Fq(x)883 566 y Fk(1)902 559 y Fq(;)8 b(x)952 566 y Fk(2)971 559 y Fq(;)g(x)1021 566 y Fk(1)1052 559 y Fr(+)j Fq(x)1129 566 y Fk(3)1148 559 y Fr(\))p Fq(:)0 658 y Fr(Insp)q(ection)19 b(sho)o(ws)i(that)f(the)f(only)g(edge)h(of)g Fq(K)904 665 y Fk(3)943 658 y Fr(con)o(tained)g(in)f(\010\()p Fq(F)7 b Fr(\))19 b(is)g(2-3.)32 b(Hence)0 717 y(\010\()p Fq(F)7 b Fr(\))16 b(consists)h(of)g(a)g(graph)g(with)f(three)g(v)o (ertices)f(and)i(one)g(edge.)k(Hence)15 b(its)i(comple-)0 775 y(men)o(t)c(is)i(the)g(graph)h(with)g(three)e(v)o(ertices)g(and)h (t)o(w)o(o)g(edges)h(emanating)e(from)g(v)o(ertex)g(1.)0 833 y(Therefore,)k(\011\010\()p Fq(F)7 b Fr(\))18 b(consists)h(of)g (the)f(set)h(of)g(all)f(functions)g(\()p Fq(g)1194 815 y Fk(1)1214 833 y Fq(;)8 b(g)1261 815 y Fk(2)1281 833 y Fq(;)g(g)1328 815 y Fk(3)1347 833 y Fr(\))19 b(suc)o(h)f(that)h Fq(g)1630 815 y Fk(1)0 891 y Fr(is)c(arbitrary)l(,)f Fq(g)288 873 y Fk(2)323 891 y Fr(do)q(es)i(not)f(dep)q(end)g(on)g(the)g (third)f(v)m(ariable,)h(and)g Fq(g)1268 873 y Fk(3)1303 891 y Fr(do)q(es)h(not)f(dep)q(end)0 949 y(on)i(the)f(second)g(v)m (ariable.)0 1028 y Fy(Corollary)j(2.3.)h Fn(With)d(notation)h(as)f(in)h (the)g(ab)n(ove)g(the)n(or)n(em,)f(we)h(have)74 1127 y Fr(\(1\))691 1206 y(\010\011\()p Fq(G)p Fr(\))d(=)f Fq(G;)157 1295 y Fn(for)j(al)r(l)i(gr)n(aphs)d Fq(G)p Fn(.)74 1353 y Fr(\(2\))637 1431 y(\010\011\010\()p Fq(F)7 b Fr(\))14 b(=)g(\010\()p Fq(F)7 b Fr(\))p Fq(;)157 1520 y Fn(and,)18 b(in)g(p)n(articular,)580 1619 y Fr(\011\010\(\011\010\()p Fq(F)7 b Fr(\)\))14 b(=)g(\011\010\()p Fq(F)7 b Fr(\))p Fq(;)157 1718 y Fn(for)21 b(al)r(l)i(sets)f Fq(F)27 b Fo(2)22 b(F)5 b Fn(.)34 b(That)21 b(is,)h Fr(\011\010)g Fn(is)f(a)g(closur)n(e)h(op)n(er)n(ator)e(on)h(the)h(set)g(of)157 1776 y Fq(n)p Fn(-tuples)d(of)f(lo)n(c)n(al)f(functions)i(on)f Fq(k)819 1758 y Fp(n)842 1776 y Fn(.)0 1905 y(Pr)n(o)n(of.)h Fr(The)12 b(second)g(claim)d(is)j(a)g(standard)h(consequence)e(of)h (the)g(prop)q(erties)f(of)h(a)h(Galois)0 1964 y(corresp)q(ondence.)33 b(T)l(o)20 b(sho)o(w)h(the)f(\014rst)g(claim,)f(let)g Fq(i)p Fr({)p Fq(j)k Fr(b)q(e)d(an)h(edge)f(in)f(\010\011\()p Fq(G)p Fr(\),)i(and)0 2024 y(supp)q(ose)c(that)g(it)f(is)g(not)h(in)f Fq(G)p Fr(.)21 b(Then)c Fq(i)p Fr({)p Fq(j)f Fo(2)p 857 1984 39 2 v 15 w Fq(G)p Fr(.)21 b(Recall)15 b(that)507 2123 y(\011\()p Fq(G)p Fr(\))f(=)g Fq(L)720 2102 y Fk(1)720 2135 y(1)740 2123 y Fr(\()p 759 2083 V Fq(G)p Fr(\))d Fo(\002)g(\001)d(\001)g(\001)j(\002)g Fq(L)1029 2102 y Fp(n)1029 2135 y Fk(1)1053 2123 y Fr(\()p 1072 2083 V Fq(G)p Fr(\))p Fq(:)0 2222 y Fr(Then)19 b(\011\()p Fq(G)p Fr(\))f(con)o(tains)h(the)f(function)g Fq(f)23 b Fr(=)18 b(\()p Fq(f)886 2204 y Fk(1)906 2222 y Fq(;)8 b(:)g(:)g(:)f(;)h(f)1044 2204 y Fp(n)1068 2222 y Fr(\),)18 b(suc)o(h)g(that)h Fq(f)1368 2204 y Fp(p)1406 2222 y Fr(=)e Fq(id)i Fr(for)g(all)0 2280 y Fq(p)14 b Fo(6)p Fr(=)g Fq(i;)8 b(j)s Fr(,)15 b(and)251 2379 y Fq(f)280 2358 y Fp(i)294 2379 y Fr(\()p Fq(x)341 2386 y Fk(1)361 2379 y Fq(;)8 b(:)g(:)g(:)f(;)h(x)498 2386 y Fp(n)521 2379 y Fr(\))42 b(=)f(\()p Fq(x)708 2386 y Fk(1)727 2379 y Fq(;)8 b(:)g(:)g(:)g(;)g(x)865 2386 y Fp(i)p Fh(\000)p Fk(1)924 2379 y Fq(;)g(x)974 2386 y Fp(i)998 2379 y Fr(+)j Fq(x)1075 2386 y Fp(j)1093 2379 y Fq(;)d(x)1143 2386 y Fp(i)p Fk(+1)1202 2379 y Fq(;)g(:)g(:)g(:)f(;)h(x)1339 2386 y Fp(n)1363 2379 y Fr(\))p Fq(;)247 2454 y(f)276 2433 y Fp(j)294 2454 y Fr(\()p Fq(x)341 2461 y Fk(1)361 2454 y Fq(;)g(:)g(:)g(:)f(;)h(x)498 2461 y Fp(n)521 2454 y Fr(\))42 b(=)f(\()p Fq(x)708 2461 y Fk(1)727 2454 y Fq(;)8 b(:)g(:)g(:)g(;)g(x)865 2461 y Fp(j)r Fh(\000)p Fk(1)928 2454 y Fq(;)g Fr(0)p Fq(;)g(x)1024 2461 y Fp(j)r Fk(+1)1087 2454 y Fq(;)g(:)g(:)g(:)f(;)h(x)1224 2461 y Fp(n)1247 2454 y Fr(\))p Fq(;)0 2553 y Fr(that)17 b(is,)f Fq(f)198 2535 y Fp(i)213 2553 y Fr(,)g(resp.)23 b Fq(f)396 2535 y Fp(j)415 2553 y Fr(,)16 b(c)o(hanges)h(only)f(the)h Fq(i)p Fr(th,)f(resp.)23 b(the)16 b Fq(j)s Fr(th,)h(co)q(ordinate.)23 b(Observ)o(e)0 2611 y(no)o(w)17 b(that)h Fq(f)238 2593 y Fp(i)264 2611 y Fo(\016)12 b Fq(f)330 2593 y Fp(j)363 2611 y Fo(6)p Fr(=)k Fq(f)446 2593 y Fp(j)476 2611 y Fo(\016)11 b Fq(f)541 2593 y Fp(i)556 2611 y Fr(.)24 b(This)17 b(implies)d(that)k Fq(i)p Fr({)p Fq(j)i Fr(is)d(not)g(in)g (\010\011\()p Fq(G)p Fr(\),)g(whic)o(h)g(is)g(a)0 2669 y(con)o(tradiction.)1316 b Fe(\003)p eop %%Page: 8 8 8 7 bop 0 187 a Fl(8)316 b(REINHARD)13 b(LA)o(UBENBA)o(CHER)f(AND)h (BODO)g(P)m(AREIGIS)348 286 y Fr(3.)24 b Fs(Equiv)l(alence)16 b(Rela)m(tions)h(On)h(Systems)0 395 y Fr(In)e(this)g(section)g(w)o(e)g (consider)g(sev)o(eral)f(equiv)m(alence)g(relations)h(on)h(systems.)j (The)d(\014rst)0 453 y(one)23 b(corresp)q(onds)g(to)g(the)f(notion)h (of)g Fn(top)n(olo)n(gic)n(al)r(ly)h(c)n(onjugate)g Fr(discrete)d (dynamical)0 511 y(systems.)0 581 y Fy(De\014nition)h(3.1.)f Fr(Tw)o(o)g(systems)d Fq(f)s(;)8 b(g)22 b Fr(:)e Fq(k)828 563 y Fp(n)871 581 y Fo(\000)-8 b(!)19 b Fq(k)998 563 y Fp(n)1042 581 y Fr(are)h(called)f Fn(isomorphic)g Fr(or)h Fn(dy-)0 639 y(namic)n(al)r(ly)d(e)n(quivalent)f Fr(if)e(there)f (exists)h(a)h(bijectiv)o(e)c(function)j Fq(\036)g Fr(:)f Fq(k)1256 621 y Fp(n)1294 639 y Fo(\000)-9 b(!)14 b Fq(k)1415 621 y Fp(n)1453 639 y Fr(suc)o(h)g(that)0 698 y Fq(g)f Fo(\016)e Fq(\036)j Fr(=)g Fq(\036)d Fo(\016)g Fq(f)5 b Fr(.)0 789 y(It)22 b(is)h(easy)g(to)g(see)f(that)i(t)o(w)o(o)e (systems)g(are)h(isomorphic)e(if)h(and)i(only)e(if)h(they)f(ha)o(v)o(e) 0 847 y(isomorphic)f(state)h(spaces,)h(that)g(is,)f(the)g(function)g Fq(\036)g Fr(induces)f(an)i(isomorphism)d(of)0 905 y(directed)14 b(graphs.)22 b(This)15 b(de\014nition)g(of)g(dynamic)f(equiv)m(alence)f (has)j(the)f(prop)q(ert)o(y)g(that)0 963 y(p)q(o)o(w)o(ers)d(of)g Fq(f)17 b Fr(and)c(of)f Fq(g)i Fr(are)e(also)g(isomorphic,)f(since)g Fq(g)995 945 y Fp(s)1016 963 y Fo(\016)r Fq(\036)i Fr(=)h Fq(g)1162 945 y Fp(s)p Fh(\000)p Fk(1)1228 963 y Fo(\016)r Fq(\036)r Fo(\016)r Fq(f)k Fr(=)c Fo(\001)8 b(\001)g(\001)14 b Fr(=)g Fq(\036)r Fo(\016)r Fq(f)1618 945 y Fp(s)1636 963 y Fr(.)0 1022 y(So)j(the)f(dynamic)e(b)q(eha)o(vior)i(\(under)h (iteration\))e(of)i Fq(f)k Fr(and)c Fq(g)i Fr(is)d(the)g(same.)0 1092 y Fy(Lemma)g(3.2.)21 b Fn(If)c Fq(f)k Fr(:)15 b Fq(k)460 1074 y Fp(n)498 1092 y Fo(\000)-8 b(!)14 b Fq(k)620 1074 y Fp(n)662 1092 y Fn(is)k(a)g(system,)g(and)g Fq(\036)d Fr(:)g Fq(k)1128 1074 y Fp(n)1167 1092 y Fo(\000)-9 b(!)15 b Fq(k)1289 1074 y Fp(n)1330 1092 y Fn(is)j(an)h(invertible)0 1150 y(function,)g(then)f(the)g(systems)f Fq(f)23 b Fn(and)18 b Fq(\036)746 1132 y Fh(\000)p Fk(1)804 1150 y Fo(\016)11 b Fq(f)16 b Fo(\016)11 b Fq(\036)18 b Fn(ar)n(e)e(dynamic)n(al)r(ly)i (e)n(quivalent.)0 1241 y Fr(W)l(e)13 b(no)o(w)g(de\014ne)f(a)h(w)o(eak) o(er)f(equiv)m(alence)f(relation)h(on)i(the)e(whole)h(collection)e(of)i (systems)0 1299 y Fo(f)p Fq(f)29 b Fr(:)23 b Fq(k)142 1281 y Fp(n)189 1299 y Fo(\000)-31 b(!)24 b Fq(k)298 1281 y Fp(n)345 1299 y Fo(j)f Fq(n)g Fo(2)h Ff(N)p Fo(g)p Fr(,)g(whic)o(h)d(w)o(e)g(call)h Fn(stable)h(e)n(quivalenc)n(e)p Fr(.)41 b(Then)22 b(w)o(e)f(sho)o(w)0 1358 y(that)f(stable)f(equiv)m (alence)e(of)i(systems)f(corresp)q(onds)j(to)e(the)g(existence)e(of)j (a)f(digraph)0 1416 y(isomorphism)14 b(b)q(et)o(w)o(een)h(the)h(limit)e (cycles)h(in)h(the)g(resp)q(ectiv)o(e)e(state)j(spaces.)0 1486 y Fy(De\014nition)i(3.3.)i Fr(Let)d Fq(f)k Fr(:)16 b Fq(k)564 1468 y Fp(n)605 1486 y Fo(\000)-31 b(!)17 b Fq(k)707 1468 y Fp(n)748 1486 y Fr(b)q(e)h(a)h(system)d(with)i(state) g(space)g Fo(S)1418 1493 y Fp(f)1459 1486 y Fr(and)g(with)0 1544 y(the)c(sub-digraph)h Fo(L)382 1551 y Fp(f)420 1544 y Fr(of)f(limit)d(cycles.)20 b(Then)14 b Fq(x)f Fo(2)h Fq(k)986 1526 y Fp(n)1024 1544 y Fr(is)g(a)h(v)o(ertex)d(in)i Fo(L)1344 1551 y Fp(f)1381 1544 y Fr(if)g(and)h(only)f(if)0 1602 y(there)i(exists)g(a)h(p)q(ositiv)o(e)f(in)o(teger)g Fq(m)g Fr(suc)o(h)h(that)g Fq(f)949 1584 y Fp(m)983 1602 y Fr(\()p Fq(x)p Fr(\))d(=)h Fq(x)p Fr(.)22 b(Let)17 b Fq(m)f Fr(b)q(e)h(the)g(smallest)0 1660 y(in)o(teger)d(suc)o(h)h (that)g Fq(f)402 1642 y Fp(m)436 1660 y Fr(\()p Fq(x)p Fr(\))e(=)h Fq(x)h Fr(for)g(all)g Fq(x)e Fo(2)h(L)872 1667 y Fp(f)895 1660 y Fr(.)21 b(W)l(e)15 b(call)f Fq(m)h Fr(the)g Fn(or)n(der)e Fr(of)j(the)e(system)0 1718 y Fq(f)5 b Fr(,)16 b(denoted)g(Order\()p Fq(f)5 b Fr(\).)0 1788 y Fy(Lemma)16 b(3.4.)j Fn(The)f(inte)n(ger)g Fr(Order\()p Fq(f)5 b Fr(\))18 b Fn(exists.)0 1897 y(Pr)n(o)n(of.)h Fr(F)l(or)f(eac)o(h)g Fq(x)f Fo(2)g(L)479 1904 y Fp(f)520 1897 y Fr(there)h(is)g(an)g(in)o(teger)g Fq(m)974 1904 y Fp(x)1013 1897 y Fr(suc)o(h)g(that)h Fq(f)1262 1879 y Fp(m)1293 1883 y Fi(x)1315 1897 y Fr(\()p Fq(x)p Fr(\))e(=)g Fq(x)p Fr(.)26 b(Hence)0 1955 y(the)17 b(least)g(common)d(m)o(ultiple)g (of)j(all)g Fq(m)755 1962 y Fp(x)794 1955 y Fr(is)f(an)i(in)o(teger)e Fq(m)g Fr(suc)o(h)h(that)h Fq(f)1380 1937 y Fp(m)1413 1955 y Fr(\()p Fq(x)p Fr(\))d(=)g Fq(x)i Fr(for)0 2013 y(all)f Fq(x)p Fr(.)1501 b Fe(\003)0 2101 y Fy(De\014nition)19 b(3.5.)i Fr(Let)c Fq(f)22 b Fr(:)15 b Fq(k)562 2083 y Fp(r)597 2101 y Fo(\000)-8 b(!)15 b Fq(k)720 2083 y Fp(r)757 2101 y Fr(and)j Fq(g)g Fr(:)e Fq(k)951 2083 y Fp(m)1000 2101 y Fo(\000)-9 b(!)16 b Fq(k)1123 2083 y Fp(m)1174 2101 y Fr(b)q(e)i(t)o(w)o(o)f(systems.)24 b(Then)0 2159 y Fq(f)j Fr(and)22 b Fq(g)i Fr(are)d(called)g Fn(stably)i(e)n (quivalent)g Fr(if)e(there)g(exist)g(maps)g Fq(p)i Fr(:)g Fq(k)1345 2141 y Fp(r)1386 2159 y Fo(\000)-8 b(!)23 b Fq(k)1517 2141 y Fp(m)1571 2159 y Fr(and)0 2217 y Fq(q)e Fr(:)f Fq(k)104 2199 y Fp(m)157 2217 y Fo(\000)-9 b(!)20 b Fq(k)284 2199 y Fp(r)303 2217 y Fr(,)g(a)g(p)q(ositiv)o(e)f(in)o (teger)g Fq(s)h Fr(prime)e(to)i(lcm)n(\(Order)o(\()p Fq(f)5 b Fr(\))p Fq(;)j Fr(Order\()p Fq(g)r Fr(\)\),)20 b(and)h(a)0 2275 y(nonnegativ)o(e)16 b(in)o(teger)f Fq(n)p Fr(,)h(suc)o(h)g(that)h(the)f(diagram)541 2604 y Fq(k)568 2586 y Fp(r)778 2604 y Fq(k)805 2586 y Fp(m)p 602 2588 162 2 v 722 2587 a Fd(-)674 2571 y Fp(p)541 2361 y Fq(k)568 2343 y Fp(r)778 2361 y Fq(k)805 2343 y Fp(m)p 602 2344 V -114 w Fd(-)674 2327 y Fp(p)1029 2604 y Fq(k)1056 2586 y Fp(r)p 853 2588 V 973 2587 a Fd(-)925 2571 y Fp(q)1029 2361 y Fq(k)1056 2343 y Fp(r)p 853 2344 V -99 w Fd(-)925 2327 y Fp(q)p 564 2553 2 176 v 564 2553 a Fd(?)584 2473 y Fp(f)605 2462 y Fi(s)p 807 2553 V 808 2553 a Fd(?)828 2473 y Fp(g)846 2461 y Fi(s)p 1051 2553 V 1052 2553 a Fd(?)1071 2473 y Fp(f)1092 2462 y Fi(s)0 2669 y Fr(comm)o(utes,)d(and)k Fq(q)12 b Fo(\016)f Fq(p)j Fr(=)g Fq(f)527 2651 y Fp(n)551 2669 y Fr(,)i Fq(p)11 b Fo(\016)g Fq(q)16 b Fr(=)d Fq(g)766 2651 y Fp(n)790 2669 y Fr(.)p eop %%Page: 9 9 9 8 bop 528 187 a Fl(FINITE)16 b(D)o(YNAMICAL)g(SYSTEMS)509 b(9)0 286 y Fr(W)l(e)15 b(p)q(ostp)q(one)i(the)e(pro)q(of)h(that)g (stable)f(equiv)m(alence)f(is)h(an)h(equiv)m(alence)d(relation)i(un)o (til)0 344 y(after)h(the)g(follo)o(wing)g(theorem.)0 415 y Fy(De\014nition)h(3.6.)j Fr(Let)c Fq(f)480 422 y Fk(1)499 415 y Fq(;)8 b(f)545 422 y Fk(2)581 415 y Fr(b)q(e)16 b(systems)e(with)i(state)g(spaces)g Fo(S)1236 422 y Fp(f)1253 427 y Fi(i)1284 415 y Fr(and)g(sub-digraphs)0 473 y(of)d(limit)e(cycles)g Fo(L)336 480 y Fp(f)353 485 y Fi(i)369 473 y Fr(.)20 b(W)l(e)13 b(call)f Fq(f)594 480 y Fk(1)627 473 y Fr(and)i Fq(f)743 480 y Fk(2)776 473 y Fn(stably)h(isomorphic)d Fr(if)h(there)f(exists)h(a)g(digraph)0 531 y(isomorphism)h(b)q(et)o(w)o(een)h Fo(L)507 538 y Fp(f)524 543 y Fj(1)560 531 y Fr(and)i Fo(L)689 538 y Fp(f)706 543 y Fj(2)726 531 y Fr(.)0 624 y(It)f(is)g(clear)f(that)i (stable)f(isomorphism)e(is)i(an)h(equiv)m(alence)e(relation.)0 694 y Fy(Theorem)i(3.7.)j Fn(Two)e(systems)f Fq(f)23 b Fn(and)18 b Fq(g)i Fn(ar)n(e)d(stably)h(e)n(quivalent)i(if)d(and)h (only)g(if)g(they)0 753 y(ar)n(e)f(stably)h(isomorphic.)0 863 y(Pr)n(o)n(of.)h Fr(First)24 b(assume)f(that)h Fq(f)30 b Fr(and)24 b Fq(g)i Fr(are)e(stably)g(equiv)m(alen)o(t,)g(that)g(is,)i (there)d(are)0 921 y(maps)e Fq(p;)8 b(q)r Fr(,)22 b(and)g(a)f(p)q (ositiv)o(e)g(in)o(teger)f Fq(s)i Fr(prime)d(to)j(lcm)n(\(Order)o(\()p Fq(f)5 b Fr(\))p Fq(;)j Fr(Order\()p Fq(g)r Fr(\)\))22 b(and)g(a)0 979 y(non-negativ)o(e)15 b(in)o(teger)f Fq(n)p Fr(,)h(suc)o(h)g(that)h Fq(g)743 961 y Fp(s)761 979 y Fq(p)f Fr(=)e Fq(pf)904 961 y Fp(s)924 979 y Fr(,)i Fq(q)r(g)1002 961 y Fp(s)1033 979 y Fr(=)f Fq(f)1114 961 y Fp(s)1133 979 y Fq(q)r Fr(,)g(and)i Fq(q)r(p)e Fr(=)g Fq(f)1422 961 y Fp(n)1445 979 y Fr(,)h Fq(pq)h Fr(=)e Fq(g)1613 961 y Fp(n)1636 979 y Fr(.)0 1059 y(Let)h Fq(a)e Fr(:=)h(Order)o(\()p Fq(f)5 b Fr(\),)15 b Fq(b)f Fr(:=)f(Order\()p Fq(g)r Fr(\),)i(lcm)n(\()p Fq(a;)8 b(b)p Fr(\))13 b(=)g Fq(a)1001 1041 y Fh(0)1013 1059 y Fq(b)g Fr(=)h Fq(ab)1146 1041 y Fh(0)1172 1059 y Fr(for)h(some)f(in)o(tegers)g Fq(a)1571 1041 y Fh(0)1582 1059 y Fq(;)8 b(b)1625 1041 y Fh(0)1636 1059 y Fr(.)0 1117 y(Let)18 b Fq(r)q(s)12 b Fr(+)h Fq(t)e Fo(\001)h Fr(lcm)n(\()p Fq(a;)c(b)p Fr(\))16 b(=)g(1)j(for)f(some)e(p)q (ositiv)o(e)h(in)o(teger)g Fq(r)q Fr(.)27 b(W)l(e)17 b(ha)o(v)o(e)g Fq(x)f Fo(2)h(L)1484 1124 y Fp(f)1525 1117 y Fr(if)g(and)0 1176 y(only)f(if)g Fq(f)181 1158 y Fp(a)202 1176 y Fr(\()p Fq(x)p Fr(\))d(=)h Fq(x)p Fr(.)21 b(Similarly)l(,)13 b Fq(x)g Fo(2)i(L)736 1183 y Fp(g)772 1176 y Fr(if)h(and)h(only)f(if)f Fq(g)1088 1158 y Fp(b)1106 1176 y Fr(\()p Fq(x)p Fr(\))e(=)h Fq(x)p Fr(.)0 1256 y(Giv)o(en)h Fq(x)f Fo(2)g(L)264 1263 y Fp(f)287 1256 y Fr(,)i(w)o(e)f(ha)o(v)o(e)220 1345 y Fq(g)245 1325 y Fp(sa)280 1313 y Fc(0)292 1325 y Fp(b)309 1345 y Fq(p)p Fr(\()p Fq(x)p Fr(\))f(=)g(\()p Fq(g)509 1325 y Fp(s)527 1345 y Fr(\))546 1325 y Fp(a)565 1313 y Fc(0)576 1325 y Fp(b)593 1345 y Fq(p)p Fr(\()p Fq(x)p Fr(\))g(=)g Fq(p)p Fr(\()p Fq(f)821 1325 y Fp(s)840 1345 y Fr(\))859 1325 y Fp(ab)893 1313 y Fc(0)906 1345 y Fr(\()p Fq(x)p Fr(\))g(=)f Fq(p)p Fr(\()p Fq(f)1109 1325 y Fp(a)1131 1345 y Fr(\))1150 1325 y Fp(sb)1181 1313 y Fc(0)1195 1345 y Fr(\()p Fq(x)p Fr(\))g(=)h Fq(p)p Fr(\()p Fq(x)p Fr(\))p Fq(:)0 1429 y Fr(Hence)h Fq(p)p Fr(\()p Fq(x)p Fr(\))f Fo(2)g(L)330 1436 y Fp(g)350 1429 y Fr(,)i(so)h(that)g Fq(p)f Fr(induces)g(a)h(set)f (map)669 1512 y Fq(P)21 b Fr(:)13 b Fo(L)782 1519 y Fp(f)819 1512 y Fo(\000)-8 b(!)13 b(L)947 1519 y Fp(g)968 1512 y Fq(:)0 1596 y Fr(This)19 b(map)g(is)g(also)h(a)f(morphism)e(of)j (digraphs.)31 b(T)l(o)20 b(sho)o(w)g(this,)f(let)f Fq(x)h Fo(2)g(L)1455 1603 y Fp(f)1498 1596 y Fr(so)h(that)0 1654 y Fq(f)29 1636 y Fp(a)50 1654 y Fr(\()p Fq(x)p Fr(\))14 b(=)f Fq(x)p Fr(.)21 b(Then)133 1742 y Fq(P)7 b(f)e Fr(\()p Fq(x)p Fr(\))14 b(=)f Fq(pf)384 1721 y Fp(r)q(s)p Fk(+)p Fp(tab)491 1710 y Fc(0)506 1742 y Fr(\()p Fq(x)p Fr(\))g(=)h Fq(pf)690 1721 y Fp(r)q(s)726 1742 y Fr(\()p Fq(x)p Fr(\))g(=)f Fq(g)882 1721 y Fp(r)q(s)918 1742 y Fq(p)p Fr(\()p Fq(x)p Fr(\))h(=)g Fq(g)1099 1721 y Fp(r)q(s)p Fk(+)p Fp(ta)1191 1710 y Fc(0)1202 1721 y Fp(b)1219 1742 y Fq(p)p Fr(\()p Fq(x)p Fr(\))g(=)g Fq(g)r(P)7 b Fr(\()p Fq(x)p Fr(\))p Fq(;)0 1825 y Fr(since)22 b Fq(p)p Fr(\()p Fq(x)p Fr(\))k Fo(2)g(L)335 1832 y Fp(g)379 1825 y Fr(and)d Fq(g)505 1807 y Fp(b)523 1825 y Fr(\()p Fq(p)p Fr(\()p Fq(x)p Fr(\)\))i(=)h Fq(p)p Fr(\()p Fq(x)p Fr(\).)42 b(But)23 b Fq(P)7 b(f)31 b Fr(=)26 b Fq(g)r(P)k Fr(on)24 b Fo(L)1342 1832 y Fp(f)1388 1825 y Fr(implies)d(that)0 1883 y Fq(P)g Fr(:)13 b Fo(L)113 1890 y Fp(f)150 1883 y Fo(\000)-8 b(!)13 b(L)278 1890 y Fp(g)315 1883 y Fr(is)j(a)h(morphism)d(of)i (digraphs.)0 1963 y(Note)22 b(that)i Fq(f)266 1945 y Fp(n)312 1963 y Fr(is)f(a)g(bijection)f(on)h Fo(L)729 1970 y Fp(f)775 1963 y Fr(for)g(all)f Fq(n)p Fr(.)41 b(F)l(rom)22 b(the)h(de\014nition)f(of)h(stable)0 2021 y(equiv)m(alence)e(w)o(e)h(obtain)i(an)f Fq(n)g Fr(suc)o(h)f(that)i Fq(q)r(p)h Fr(=)g Fq(f)1021 2003 y Fp(n)1044 2021 y Fr(.)41 b(Hence)22 b Fq(P)32 b Fr(:)25 b Fo(L)1387 2028 y Fp(f)1435 2021 y Fo(\000)-9 b(!)25 b(L)1574 2028 y Fp(g)1617 2021 y Fr(is)0 2079 y(injectiv)o(e)12 b(and)j Fq(Q)e Fr(:)g Fo(L)398 2086 y Fp(g)433 2079 y Fo(\000)-9 b(!)14 b(L)561 2086 y Fp(f)598 2079 y Fr(is)g(surjectiv)o(e.)19 b(Similarly)11 b Fq(P)22 b Fr(is)14 b(surjectiv)o(e)e(so)j(that)g Fq(P)21 b Fr(is)0 2137 y(an)c(isomorphism)d(of)i(digraphs.)22 b(This)17 b(sho)o(ws)g(that)g Fq(f)k Fr(and)c Fq(g)h Fr(are)f(stably)f(isomorphic.)0 2217 y(Con)o(v)o(ersely)l(,)j(assume)g (that)i Fq(f)k Fr(and)c Fq(g)h Fr(are)e(stably)g(isomorphic,)f(with)h (a)g(digraph)h(iso-)0 2275 y(morphism)669 2338 y Fq(P)g Fr(:)13 b Fo(L)782 2345 y Fp(f)819 2338 y Fo(\000)-8 b(!)13 b(L)947 2345 y Fp(g)968 2338 y Fq(:)0 2411 y Fr(F)l(rom)f(eac)o (h)g(limit)e(cycle)h(in)h Fo(L)548 2418 y Fp(f)584 2411 y Fr(c)o(ho)q(ose)h(a)g(v)o(ertex)e(as)j(represen)o(tativ)o(e,)d(with)h Fo(f)p Fq(x)1457 2418 y Fk(1)1477 2411 y Fq(;)c(x)1527 2418 y Fk(2)1546 2411 y Fq(;)g(:)g(:)g(:)o Fo(g)0 2469 y Fr(the)14 b(full)f(set)h(of)g(represen)o(tativ)o(es.)19 b(Similarly)-5 b(,)11 b(c)o(ho)q(ose)k(represen)o(tativ)o(es)d Fo(f)p Fq(y)1389 2476 y Fk(1)1408 2469 y Fq(;)c(y)1454 2476 y Fk(2)1474 2469 y Fq(;)g(:)g(:)g(:)o Fo(g)14 b Fr(for)0 2528 y(the)h(limit)d(cycles)i(of)h Fo(L)425 2535 y Fp(g)446 2528 y Fr(,)f(suc)o(h)h(that)h Fq(P)7 b Fr(\()p Fq(x)773 2535 y Fp(i)787 2528 y Fr(\))13 b(=)h Fq(y)895 2535 y Fp(i)909 2528 y Fr(.)21 b(The)15 b(restriction)f(of)h Fq(P)23 b Fr(to)15 b(eac)o(h)g(limit)0 2586 y(cycle)g(giv)o(es)g(an)i (isomorphism)d(of)i(digraphs)190 2669 y Fq(p)214 2676 y Fp(i)243 2669 y Fr(:)d Fo(f)p Fq(x)323 2676 y Fp(i)337 2669 y Fq(;)8 b(f)d Fr(\()p Fq(x)435 2676 y Fp(i)449 2669 y Fr(\))p Fq(;)j(f)519 2649 y Fk(2)539 2669 y Fr(\()p Fq(x)586 2676 y Fp(i)600 2669 y Fr(\))p Fq(;)g(:)g(:)g(:)o Fo(g)13 b(\000)-8 b(!)13 b(f)p Fq(P)7 b Fr(\()p Fq(x)940 2676 y Fp(i)954 2669 y Fr(\))14 b(=)g Fq(y)1063 2676 y Fp(i)1077 2669 y Fq(;)8 b(g)r Fr(\()p Fq(y)1167 2676 y Fp(i)1181 2669 y Fr(\))p Fq(;)g(g)1247 2649 y Fk(2)1266 2669 y Fr(\()p Fq(y)1309 2676 y Fp(i)1323 2669 y Fr(\))p Fq(;)g(:)g(:)g(:)o Fo(g)p Fq(;)p eop %%Page: 10 10 10 9 bop 0 187 a Fl(10)297 b(REINHARD)13 b(LA)o(UBENBA)o(CHER)f(AND)h (BODO)g(P)m(AREIGIS)0 286 y Fr(with)632 346 y Fq(p)656 353 y Fp(i)670 346 y Fq(f)699 325 y Fp(t)715 346 y Fr(\()p Fq(x)762 353 y Fp(i)775 346 y Fr(\))h(=)g Fq(g)885 325 y Fp(t)900 346 y Fq(p)924 353 y Fp(i)938 346 y Fr(\()p Fq(x)985 353 y Fp(i)999 346 y Fr(\))0 416 y(for)j(all)e Fq(t)p Fr(.)0 495 y(W)l(e)20 b(no)o(w)g(construct)g(a)g(function)g Fq(p)h Fr(:)f Fq(k)756 477 y Fp(n)799 495 y Fo(\000)-8 b(!)20 b Fq(k)927 477 y Fp(m)980 495 y Fr(as)h(follo)o(ws.)32 b(Let)20 b Fq(x)g Fo(2)g(S)1456 502 y Fp(f)1479 495 y Fr(.)32 b(There)0 554 y(exists)16 b(a)h(unique)g(minim)o(al)d Fq(s)h Fo(2)g Ff(N)647 561 y Fk(0)684 554 y Fr(suc)o(h)i(that)g Fq(f)930 535 y Fp(s)949 554 y Fr(\()p Fq(x)p Fr(\))d Fo(2)h(L)1111 561 y Fp(f)1134 554 y Fr(.)23 b(Let)17 b Fq(r)h Fr(b)q(e)f(minim)o(al)d(suc)o(h)0 612 y(that)j Fq(f)135 594 y Fp(s)153 612 y Fr(\()p Fq(x)p Fr(\))d(=)g Fq(f)314 594 y Fp(r)333 612 y Fr(\()p Fq(x)380 619 y Fp(i)394 612 y Fr(\))i(for)h(a)f(unique)g(represen)o(tativ)o(e)e Fq(x)1044 619 y Fp(i)1058 612 y Fr(.)21 b(Then)16 b(de\014ne)635 692 y Fq(p)p Fr(\()p Fq(x)p Fr(\))e(=)g Fq(p)815 699 y Fp(i)829 692 y Fq(f)858 671 y Fp(r)q Fh(\000)p Fp(s)922 692 y Fr(\()p Fq(x)969 699 y Fp(i)982 692 y Fr(\))p Fq(;)0 772 y Fr(where)g Fq(f)168 754 y Fp(r)q Fh(\000)p Fp(s)231 772 y Fr(\()p Fq(x)278 779 y Fp(i)292 772 y Fr(\))g(is)g(to)g(b)q(e)g (tak)o(en)g(in)g Fo(L)713 779 y Fp(f)750 772 y Fr(if)g Fq(r)8 b Fo(\000)f Fq(s)13 b Fr(is)h(negativ)o(e.)20 b(Note)14 b(that)g Fq(f)20 b Fr(is)14 b(bijectiv)o(e)0 830 y(on)20 b Fo(L)105 837 y Fp(f)128 830 y Fr(,)g(so)g(that)g(negativ) o(e)e(exp)q(onen)o(ts)i Fq(r)14 b Fo(\000)f Fq(s)20 b Fr(mak)o(e)d(sense.)31 b(W)l(e)19 b(ha)o(v)o(e)g Fq(s)g Fr(=)h(0)f(if)g(and)0 888 y(only)d(if)g Fq(x)d Fo(2)h(L)274 895 y Fp(f)297 888 y Fr(.)22 b(Observ)o(e)15 b(that)598 970 y Fq(pf)5 b Fr(\()p Fq(x)p Fr(\))14 b(=)g Fq(p)807 977 y Fp(i)822 970 y Fq(f)851 949 y Fp(r)q Fh(\000)p Fp(s)p Fk(+1)959 970 y Fr(\()p Fq(x)1006 977 y Fp(i)1020 970 y Fr(\))p Fq(:)0 1050 y Fr(Let)i Fq(a)d Fr(=)h(Order\()p Fq(f)5 b Fr(\).)21 b(By)15 b(adding)i(a)f(suitable)f(m)o(ultiple)e Fq(ta)i Fr(of)h Fq(a)f Fr(to)h Fq(r)c Fo(\000)d Fq(s)16 b Fr(w)o(e)f(can)h(force)0 1108 y(the)i(exp)q(onen)o(t)f(of)i Fq(f)k Fr(in)17 b(the)h(de\014nition)g(of)g Fq(p)p Fr(\()p Fq(x)p Fr(\))g(to)g(b)q(e)g(p)q(ositiv)o(e.)26 b(Observ)o(e)17 b(that)h Fq(p)p Fr(\()p Fq(x)p Fr(\))0 1166 y(is)e(an)h(elemen)o(t)c (of)k Fo(L)387 1173 y Fp(f)410 1166 y Fr(.)0 1246 y(W)l(e)h(de\014ne)f Fq(q)i Fr(:)d Fq(k)326 1228 y Fp(m)376 1246 y Fo(\000)-8 b(!)17 b Fq(k)501 1228 y Fp(n)542 1246 y Fr(similarly)l(,)e(using)k (the)f(in)o(v)o(erse)e(of)i Fq(P)7 b Fr(.)27 b(W)l(e)18 b(need)f(to)i(v)o(erify)0 1304 y(that)h Fq(p)f Fr(and)h Fq(q)h Fr(satisfy)e(the)g(conditions)g(of)h(De\014nition)f(3.5,)g (making)f Fq(f)25 b Fr(and)20 b Fq(g)h Fr(stably)0 1362 y(equiv)m(alen)o(t.)0 1442 y(First)16 b(of)g(all,)g(for)g Fq(x)e Fo(2)g Fq(k)446 1423 y Fp(n)469 1442 y Fr(,)i(w)o(e)g(ha)o(v)o (e)f(that)i Fq(p)p Fr(\()p Fq(x)p Fr(\))d(=)g Fq(p)969 1449 y Fp(i)983 1442 y Fq(f)1012 1423 y Fp(j)1031 1442 y Fr(\()p Fq(x)1078 1449 y Fp(i)1092 1442 y Fr(\))i(for)g(some)g Fq(i;)8 b(j)s Fr(.)20 b(Then)409 1523 y Fq(g)r(p)p Fr(\()p Fq(x)p Fr(\))14 b(=)g Fq(g)r(p)639 1530 y Fp(i)654 1523 y Fq(f)683 1502 y Fp(j)701 1523 y Fr(\()p Fq(x)748 1530 y Fp(i)762 1523 y Fr(\))g(=)g Fq(p)871 1530 y Fp(i)885 1523 y Fq(f)914 1502 y Fp(j)933 1523 y Fq(f)5 b Fr(\()p Fq(x)1009 1530 y Fp(i)1023 1523 y Fr(\))14 b(=)g Fq(pf)5 b Fr(\()p Fq(x)p Fr(\))p Fq(:)0 1603 y Fr(A)16 b(similar)f(argumen)o(t) g(sho)o(ws)j(that)f Fq(f)5 b(q)16 b Fr(=)f Fq(q)r(g)r Fr(.)22 b(This)17 b(pro)o(v)o(es)f(that)h Fq(p)g Fr(and)g Fq(q)i Fr(satisfy)d(the)0 1661 y(\014rst)h(condition)f(of)g(a)h(stable) f(equiv)m(alence,)e(with)i Fq(s)e Fr(=)g(1.)0 1740 y(In)i(order)g(to)g (v)o(erify)f(the)h(second)g(condition,)f(w)o(e)h(need)g(to)g(\014nd)h (a)f(nonnegativ)o(e)g(in)o(teger)0 1798 y Fq(n)g Fr(suc)o(h)g(that)h Fq(q)r(p)d Fr(=)g Fq(f)404 1780 y Fp(n)444 1798 y Fr(and)j Fq(pq)e Fr(=)f Fq(g)677 1780 y Fp(n)701 1798 y Fr(.)21 b(W)l(e)16 b(ha)o(v)o(e)385 1878 y Fq(p)p Fr(\()p Fq(x)p Fr(\))e(=)f Fq(p)564 1885 y Fp(i)579 1878 y Fq(f)608 1858 y Fp(r)q Fh(\000)p Fp(s)671 1878 y Fr(\()p Fq(x)718 1885 y Fp(i)732 1878 y Fr(\))h(=)f Fq(g)841 1858 y Fp(r)q Fh(\000)p Fp(s)904 1878 y Fq(p)928 1885 y Fp(i)943 1878 y Fr(\()p Fq(x)990 1885 y Fp(i)1004 1878 y Fr(\))g(=)h Fq(g)1113 1858 y Fp(r)q Fh(\000)p Fp(s)1176 1878 y Fr(\()p Fq(y)1219 1885 y Fp(i)1233 1878 y Fr(\))p Fq(:)0 1958 y Fr(Then,)i(for)g(eac)o(h)g Fq(x)p Fr(,)376 2038 y Fq(q)r(p)p Fr(\()p Fq(x)p Fr(\))d(=)h Fq(q)577 2045 y Fp(i)590 2038 y Fq(g)615 2018 y Fp(r)q Fh(\000)p Fp(s)678 2038 y Fr(\()p Fq(y)721 2045 y Fp(i)735 2038 y Fr(\))g(=)g Fq(f)849 2018 y Fp(r)q Fh(\000)p Fp(s)912 2038 y Fq(q)934 2045 y Fp(i)948 2038 y Fr(\()p Fq(y)991 2045 y Fp(i)1004 2038 y Fr(\))g(=)g Fq(f)1118 2018 y Fp(r)q Fh(\000)p Fp(s)1181 2038 y Fr(\()p Fq(x)1228 2045 y Fp(i)1242 2038 y Fr(\))p Fq(:)0 2118 y Fr(Hence)558 2178 y Fq(f)587 2158 y Fp(s)606 2178 y Fq(q)r(p)p Fr(\()p Fq(x)p Fr(\))f(=)h Fq(f)814 2158 y Fp(r)833 2178 y Fr(\()p Fq(x)880 2185 y Fp(i)894 2178 y Fr(\))g(=)g Fq(f)1008 2158 y Fp(s)1026 2178 y Fr(\()p Fq(x)p Fr(\))0 2248 y(and)506 2308 y Fq(q)r(p)p Fr(\()p Fq(x)p Fr(\))g(=)f Fq(f)714 2287 y Fp(ta)p Fh(\000)p Fp(s)p Fk(+)p Fp(s)836 2308 y Fq(q)r(p)p Fr(\()p Fq(x)p Fr(\))g(=)h Fq(f)1044 2287 y Fp(ta)1078 2308 y Fr(\()p Fq(x)p Fr(\))0 2377 y(for)19 b(all)f(su\016cien)o(tly)e(large)j Fq(t)f Fr(\(suc)o(h)g(that)h Fq(ta)e Fo(\025)g Fq(s)p Fr(\).)28 b(T)l(ak)o(e)18 b(the)g(largest)h Fq(t)f Fr(o)q(ccuring)h (for)0 2436 y(all)g Fq(x)h Fo(2)g(S)202 2443 y Fp(f)244 2436 y Fr(\(and)h(all)e Fq(y)i Fo(2)f(S)561 2443 y Fp(g)581 2436 y Fr(\))g(and)h(de\014ne)e Fq(n)h Fr(=)g Fq(ta)p Fr(.)30 b(Then)20 b Fq(n)g Fr(satis\014es)g(the)g(second)0 2494 y(condition)c(for)h(stable)f(equiv)m(alence.)924 b Fe(\003)0 2600 y Fr(The)16 b(pro)q(of)i(of)e(this)g(theorem)f (implies)f(the)i(follo)o(wing)f(corollary)l(.)0 2669 y Fy(Corollary)k(3.8.)p eop %%Page: 11 11 11 10 bop 528 187 a Fl(FINITE)16 b(D)o(YNAMICAL)g(SYSTEMS)490 b(11)74 286 y Fr(\(1\))21 b Fn(Stable)e(e)n(quivalenc)n(e)h(is)e(an)f (e)n(quivalenc)n(e)j(r)n(elation.)74 344 y Fr(\(2\))h Fn(Using)h Fq(s)f Fr(=)f(1)i 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b(m)o(ultipli)o(cativ)n(e)c(group)20 b Fy(F)1609 2593 y Fh(\003)1609 2623 y Fk(2)1627 2614 y Fi(n)0 2669 y Fr(is)e(cyclic)f(of)h(order)h(2)399 2651 y Fp(n)435 2669 y Fo(\000)12 b Fr(1.)29 b(F)l(or)18 b(eac)o(h)g(divisor)g Fq(t)g Fr(of)h(2)1033 2651 y Fp(n)1069 2669 y Fo(\000)13 b Fr(1)18 b(there)g(exists)g(a)h(\(unique\))p eop %%Page: 12 12 12 11 bop 0 187 a Fl(12)297 b(REINHARD)13 b(LA)o(UBENBA)o(CHER)f(AND)h (BODO)g(P)m(AREIGIS)0 286 y Fr(subgroup)19 b(of)f Fy(F)305 268 y Fh(\003)305 299 y Fk(2)323 289 y Fi(n)364 286 y Fr(of)f(order)h Fq(t)p Fr(.)25 b(Let)18 b Fq(a)f Fr(b)q(e)h(a)g (generator)g(of)g(this)f(subgroup.)27 b(Then)17 b(the)0 344 y(system)678 405 y Fq(f)j Fr(:)13 b Fq(k)776 384 y Fp(n)813 405 y Fo(\000)-8 b(!)13 b Fq(k)934 384 y Fp(n)958 405 y Fq(;)0 475 y Fr(giv)o(en)g(b)o(y)g(m)o(ultiplication)e(b)o(y)j Fq(a)f Fr(is)h(linear)f(and)h(in)o(v)o(ertible.)k(F)l(urthermore,)12 b(Order\()p Fq(f)5 b Fr(\))14 b(=)0 533 y Fq(t)p Fr(.)0 612 y(F)l(or)19 b(an)o(y)g(linear)f(system)g 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