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5 2111 1140 fsq 5 1330 1846 fsq 5 2073 1218 fsq 5 1228 1868 fsq 5 2021 1297 fsq 5 1138 1890 fsq 5 1970 1364 fsq 5 2162 446 fsq 5 1023 1902 fsq 5 1906 1442 fsq 5 2175 546 fsq 5 921 1913 fsq 5 1855 1510 fsq 5 2200 636 fsq 5 818 1935 fsq 5 1791 1588 fsq 5 2200 737 fsq 5 703 1935 fsq 5 1714 1655 fsq 5 2200 838 fsq 5 588 1935 fsq 5 1637 1711 fsq 5 2175 927 fsq 5 486 1913 fsq 5 1548 1745 fsq 5 2149 1017 fsq 5 1458 1790 fsq 5 2124 1106 fsq 5 1356 1834 fsq 5 2085 1196 fsq 5 1254 1868 fsq 5 2034 1286 fsq 5 1151 1890 fsq 5 1983 1364 fsq 5 2175 434 fsq 5 1049 1913 fsq 5 1945 1442 fsq 5 2200 524 fsq 5 946 1935 fsq 5 1881 1510 fsq 5 2200 625 fsq 5 831 1935 fsq 5 1804 1577 fsq 5 2200 726 fsq 5 716 1935 fsq 5 1714 1633 fsq 5 2175 815 fsq 5 614 1913 fsq 5 1637 1678 fsq 5 2162 905 fsq 5 499 1902 fsq 5 1548 1734 fsq 5 2149 1006 fsq 5 1471 1778 fsq 5 2124 1084 fsq 5 1382 1823 fsq 5 2098 1174 fsq 5 1292 1857 fsq 5 2060 1252 fsq 5 1202 1879 fsq 5 2009 1330 fsq 5 1087 1902 fsq 5 1957 1409 fsq 5 2188 490 fsq 5 985 1924 fsq 5 1893 1476 fsq 5 2188 591 fsq 5 870 1924 fsq 5 1817 1543 fsq 5 2188 692 fsq 5 755 1924 fsq 5 1740 1610 fsq 5 2188 793 fsq 5 639 1924 fsq 5 1650 1666 fsq 5 2162 882 fsq 5 524 1902 fsq 5 1573 1711 fsq 5 2149 972 fsq 5 1484 1767 fsq 5 2111 1062 fsq 5 1382 1801 fsq 5 2073 1151 fsq 5 1279 1834 fsq 5 2034 1241 fsq 5 1177 1868 fsq 5 1983 1319 fsq 5 1087 1879 fsq 5 1932 1386 fsq 5 2149 490 fsq 5 972 1890 fsq 5 1868 1465 fsq 5 2162 580 fsq 5 870 1902 fsq 5 1804 1521 fsq 5 2162 670 fsq 5 767 1902 fsq 5 1740 1577 fsq 5 2162 759 fsq 5 665 1902 fsq 5 1676 1644 fsq 5 2162 849 fsq 5 563 1902 fsq 5 1586 1689 fsq 5 2136 950 fsq 5 1497 1734 fsq 5 2098 1028 fsq 5 1407 1767 fsq 5 2073 1106 fsq 5 1318 1801 fsq 5 2034 1196 fsq 5 1215 1834 fsq 5 1996 1274 fsq 5 1126 1857 fsq 5 1945 1353 fsq 5 2136 446 fsq 5 1023 1879 fsq 5 1893 1420 fsq 5 2162 535 fsq 5 921 1902 fsq 5 1829 1498 fsq 5 2162 636 fsq 5 806 1902 fsq 5 1753 1566 fsq 5 2162 737 fsq 5 691 1902 fsq 5 1676 1622 fsq 5 2136 826 fsq 5 588 1879 fsq 5 1599 1666 fsq 5 2124 916 fsq 5 1509 1722 fsq 5 2111 1017 fsq 5 1420 1767 fsq 5 2085 1106 fsq 5 1330 1812 fsq 5 2047 1185 fsq 5 1228 1834 fsq 5 1996 1263 fsq 5 1138 1857 fsq 5 1945 1330 fsq 5 1036 1857 fsq 5 1881 1398 fsq 5 2124 524 fsq 5 921 1868 fsq 5 1804 1465 fsq 5 2124 625 fsq 5 806 1868 fsq 5 1727 1532 fsq 5 2111 726 fsq 5 691 1857 fsq 5 1650 1588 fsq 5 2111 815 fsq 5 588 1857 fsq 5 1586 1644 fsq 5 2111 905 fsq 5 486 1857 fsq 5 1509 1689 fsq 5 2085 983 fsq 5 1420 1734 fsq 5 2060 1073 fsq 5 1343 1778 fsq 5 2034 1151 fsq 5 1241 1801 fsq 5 1983 1230 fsq 5 1151 1823 fsq 5 1932 1297 fsq 5 1062 1846 fsq 5 1881 1364 fsq 5 2098 490 fsq 5 959 1846 fsq 5 1817 1431 fsq 5 2111 591 fsq 5 844 1857 fsq 5 1753 1498 fsq 5 2111 681 fsq 5 742 1857 fsq 5 1689 1554 fsq 5 2111 770 fsq 5 639 1857 fsq 5 1612 1610 fsq 5 2098 860 fsq 5 537 1846 fsq 5 1535 1655 fsq 5 2085 938 fsq 5 1471 1711 fsq 5 2073 1028 fsq 5 1382 1745 fsq 5 2047 1106 fsq 5 1292 1790 fsq 5 2009 1185 fsq 5 1202 1823 fsq 5 1983 1274 fsq 5 1113 1846 fsq 5 1932 1342 fsq 5 2124 446 fsq 5 1010 1868 fsq 5 1881 1420 fsq 5 2149 535 fsq 5 921 1890 fsq 5 1829 1487 fsq 5 2149 625 fsq 5 818 1890 fsq 5 1765 1543 fsq 5 2149 714 fsq 5 716 1890 fsq 5 1689 1599 fsq 5 2149 804 fsq 5 614 1890 fsq 5 1625 1655 fsq 5 2136 894 fsq 5 511 1879 fsq 5 1548 1711 fsq 5 2124 983 fsq 5 1458 1756 fsq 5 2098 1073 fsq 5 1369 1790 fsq 5 2060 1151 fsq 5 1279 1823 fsq 5 2021 1230 fsq 5 1190 1846 fsq 5 1970 1297 fsq 5 1074 1857 fsq 5 1919 1375 fsq 5 2136 479 fsq 5 985 1879 fsq 5 1868 1442 fsq 5 2149 569 fsq 5 882 1890 fsq 5 1804 1510 fsq 5 2149 658 fsq 5 780 1890 fsq 5 1727 1566 fsq 5 2136 759 fsq 5 665 1879 fsq 5 1650 1622 fsq 5 2136 849 fsq 5 563 1879 fsq 5 1573 1666 fsq 5 2111 927 fsq 5 1497 1711 fsq 5 2098 1006 fsq 5 1420 1745 fsq 5 2060 1084 fsq 5 1330 1778 fsq 5 2021 1162 fsq 5 1241 1812 fsq 5 1996 1241 fsq 5 1138 1834 fsq 5 1932 1319 fsq 5 1036 1846 fsq 5 1881 1386 fsq 5 2124 513 fsq 5 934 1868 fsq 5 1817 1465 fsq 5 2136 614 fsq 5 831 1879 fsq 5 1753 1521 fsq 5 2124 692 fsq 5 729 1868 fsq 5 1689 1577 fsq 5 2124 782 fsq 5 627 1868 fsq 5 1612 1633 fsq 5 2111 882 fsq 5 511 1857 fsq 5 1535 1689 fsq 5 2111 972 fsq 5 1445 1734 fsq 5 2073 1050 fsq 5 1356 1767 fsq 5 2047 1140 fsq 5 1279 1812 fsq 5 2021 1218 fsq 5 1190 1834 fsq 5 1970 1286 fsq 5 1087 1857 fsq 5 1919 1364 fsq 5 2136 468 fsq 5 998 1879 fsq 5 1868 1431 fsq 5 2136 558 fsq 5 895 1879 fsq 5 1804 1498 fsq 5 2149 647 fsq 5 793 1890 fsq 5 1740 1554 fsq 5 2149 737 fsq 5 691 1890 fsq 5 1676 1610 fsq 5 2136 815 fsq 5 601 1879 fsq 5 1599 1655 fsq 5 2124 905 fsq 5 486 1868 fsq 5 1509 1700 fsq 5 2085 994 fsq 5 1407 1734 fsq 5 2047 1073 fsq 5 1318 1767 fsq 5 2009 1162 fsq 5 1215 1801 fsq 5 1970 1241 fsq 5 1126 1834 fsq 5 1932 1330 fsq 5 2124 434 fsq 5 1023 1868 fsq 5 1881 1409 fsq 5 2136 535 fsq 5 921 1879 fsq 5 1829 1476 fsq 5 2149 614 fsq 5 831 1890 fsq 5 1765 1532 fsq 5 2149 703 fsq 5 729 1890 fsq 5 1701 1588 fsq 5 2136 793 fsq 5 627 1879 fsq 5 1625 1644 fsq 5 2136 882 fsq 5 524 1879 fsq 5 1548 1689 fsq 5 2111 961 fsq 5 1458 1734 fsq 5 2085 1050 fsq 5 1382 1778 fsq 5 2060 1129 fsq 5 1292 1801 fsq 5 2009 1196 fsq 5 1190 1823 fsq 5 1970 1274 fsq 5 1100 1857 fsq 5 1919 1353 fsq 5 2111 457 fsq 5 998 1857 fsq 5 1855 1420 fsq 5 2124 558 fsq 5 882 1868 fsq 5 1791 1487 fsq 5 2124 647 fsq 5 780 1868 fsq 5 1727 1543 fsq 5 2136 737 fsq 5 691 1879 fsq 5 1663 1599 fsq 5 2124 826 fsq 5 575 1868 fsq 5 1573 1655 fsq 5 2098 916 fsq 5 1497 1700 fsq 5 2085 1006 fsq 5 1407 1756 fsq 5 2060 1095 fsq 5 1318 1778 fsq 5 2021 1174 fsq 5 1228 1812 fsq 5 1983 1252 fsq 5 1138 1846 fsq 5 1945 1319 fsq 5 1049 1857 fsq 5 1893 1386 fsq 5 2136 502 fsq 5 959 1879 fsq 5 1842 1454 fsq 5 2136 591 fsq 5 857 1879 fsq 5 1778 1521 fsq 5 2149 692 fsq 5 742 1890 fsq 5 1714 1588 fsq 5 2149 782 fsq 5 639 1890 fsq 5 1637 1644 fsq 5 2136 871 fsq 5 537 1879 fsq 5 1561 1689 fsq 5 2111 950 fsq 5 1471 1734 fsq 5 2098 1050 fsq 5 1394 1778 fsq 5 2073 1129 fsq 5 1305 1812 fsq 5 2034 1207 fsq 5 1215 1846 fsq 5 1996 1286 fsq 5 1113 1868 fsq 5 1945 1364 fsq 5 2136 457 fsq 5 1010 1879 fsq 5 1881 1431 fsq 5 2149 546 fsq 5 908 1890 fsq 5 1817 1487 fsq 5 2149 636 fsq 5 806 1890 fsq 5 1753 1554 fsq 5 2149 726 fsq 5 703 1890 fsq 5 1689 1610 fsq 5 2149 815 fsq 5 601 1890 fsq 5 1612 1655 fsq 5 2124 894 fsq 5 499 1868 fsq 5 1522 1700 fsq 5 2098 983 fsq 5 1445 1745 fsq 5 2073 1062 fsq 5 1343 1767 fsq 5 2034 1140 fsq 5 1254 1801 fsq 5 1983 1218 fsq 5 1164 1823 fsq 5 1945 1297 fsq 5 1062 1834 fsq 5 1881 1353 fsq 5 2098 479 fsq 5 972 1846 fsq 5 1829 1420 fsq 5 2111 569 fsq 5 870 1857 fsq 5 1765 1487 fsq 5 2111 658 fsq 5 767 1857 fsq 5 1701 1543 fsq 5 2111 748 fsq 5 665 1857 fsq 5 1625 1599 fsq 5 2098 849 fsq 5 550 1846 fsq 5 1548 1655 fsq 5 2098 938 fsq 5 1471 1700 fsq 5 2073 1017 fsq 5 1394 1745 fsq 5 2047 1095 fsq 5 1305 1790 fsq 5 2021 1185 fsq 5 1215 1812 fsq 5 1970 1252 fsq 5 1113 1834 fsq 5 1919 1330 fsq 5 2111 434 fsq /Times-Italic-ISOLatin1 findfont 90 scalefont setfont end showpage %%Trailer %%EndDocument @endspecial 94 2031 a Fr(Figure)29 b(1:)41 b(Magni\014cation)30 b(of)g(an)g(orbit)f(of)h(the)h(discrete)e(mapping)g(\(2\),)i(with)e Fl(\013)c Fr(=)g(1)p Fl(=)p Fr(3.)42 b(The)30 b(exact)94 2144 y(motions)45 b(\(1\))h(are)g(quasi-p)s(erio)s(dic)c(with)i (irrational)f(rotation)j(n)m(um)m(b)s(er)e Fl(\027)56 b Fr(=)49 b(cos)3069 2111 y Fk(\000)p Fj(1)3163 2144 y Fr(\(1)p Fl(=)p Fr(6\))p Fl(=)p Fr(2)p Fl(\031)57 b Fr(=)94 2256 y(0)p Fl(:)p Fr(2233)p Fl(:::)p Fr(.)89 b(The)44 b(discrete)h(orbit)g(is)f(p)s(erio)s(dic)f(with)h(p)s(erio)s (d)f(1952,)51 b(and)44 b(it)h(constitutes)h(a)f(fuzzy)94 2369 y(represen)m(tation)30 b(of)h(the)g(\011-in)m(v)-5 b(arian)m(t)29 b(ellipses)f(3)p Fl(x)1837 2336 y Fj(2)1897 2369 y Fr(+)20 b Fl(xy)j Fi(\000)d Fr(3)p Fl(y)2292 2336 y Fj(2)2357 2369 y Fr(=)25 b Fl(const)p Fr(.)94 2643 y(discrete)30 b(setting,)g(and)f(it)h(is)f(hop)s(ed)f(that)j(the)f (constructs)g(dev)m(elop)s(ed)f(here)h(ma)m(y)h(\014nd)d(applications) 94 2784 y(b)s(ey)m(ond)i(the)g(sp)s(eci\014c)f(mo)s(del)h(considered.) 235 2925 y(W)-8 b(e)27 b(consider)e(a)h(linear)f(area-preserving)g(map) h(of)g(the)g(plane,)g(describing)e(elliptic-t)m(yp)s(e)g(motions)892 3156 y(\011)h(:)g Fh(R)1098 3118 y Fj(2)1169 3156 y Fi(7!)g Fh(R)1345 3118 y Fj(2)1572 3156 y Fr(\()p Fl(x;)15 b(y)s Fr(\))26 b Fi(7!)f Fr(\()p Fl(\013x)c Fi(\000)f Fl(y)s(;)15 b(x)p Fr(\))182 b Fi(j)p Fl(\013)p Fi(j)27 b Fl(<)e Fr(2)p Fl(:)682 b Fr(\(1\))94 3386 y(A)45 b(uniform)e(discretization)h(of)i (the)f(phase)g(space)h Fh(R)2015 3353 y Fj(2)2105 3386 y Fr(to)g(the)f(lattice)h Fh(Z)2760 3353 y Fj(2)2840 3386 y Fr(can)g(b)s(e)e(ac)m(hiev)m(ed)i(b)m(y)94 3527 y(in)m(tro)s(ducing)28 b(the)j(mapping)763 3758 y(\010)25 b(:)41 b Fh(Z)985 3720 y Fj(2)1045 3758 y Fi(7!)25 b Fh(Z)1226 3720 y Fj(2)1443 3758 y Fr(\010)g(:)41 b(\()p Fl(x;)15 b(y)s Fr(\))41 b Fi(7!)f Fr(\()p Fi(b)p Fl(\013x)p Fi(c)22 b(\000)e Fl(y)s(;)30 b(x)p Fr(\))182 b Fi(j)p Fl(\013)p Fi(j)27 b Fl(<)e Fr(2)553 b(\(2\))94 3988 y(where)46 b Fi(b\001c)i Fr(denotes)f(the)f(\015o)s(or)h(function)e(\(the)i (largest)g(in)m(teger)g(not)g(exceeding)f(its)g(argumen)m(t\))94 4130 y([30)q(,)d(20,)g(21)q(].)76 b(One)42 b(v)m(eri\014es)g(that)h (\010)e(is)h(in)m(v)m(ertible.)74 b(The)42 b(map)g(\010)g(is)f(here)h (regarded)g(as)h(a)f(dis-)94 4271 y(crete)e(appro)m(ximation)d(of)i (\011:)57 b(the)38 b(discretization)g(length)g(is)g(\014xed,)i(and)e (the)g(limit)f(of)i(v)-5 b(anishing)94 4412 y(discretization)31 b(corresp)s(onds)f(to)j(motions)f(at)g(in\014nit)m(y)-8 b(.)44 b(A)32 b(t)m(ypical)f(orbit)h(for)f(the)h(parameter)h(v)-5 b(alue)94 4553 y Fl(\013)26 b Fr(=)f(1)p Fl(=)p Fr(3)g(is)e(sho)m(wn)h (in)f(Figure)g(1.)39 b(Suc)m(h)24 b(orbit)f(is)g(p)s(erio)s(dic)f(\(as) i(apparen)m(tly)g(are)h(all)e(the)h(orbits)f(of)h(this)94 4694 y(system)32 b(|see)h(section)g(6)f(b)s(elo)m(w\),)h(and)f(it)g (consists)f(of)i(a)g(cloud)e(of)h(p)s(oin)m(ts)g(distributed)d (irregularly)94 4835 y(along)h(an)h(in)m(v)-5 b(arian)m(t)29 b(ellipses)f(of)j(the)f(mapping)f(\011.)235 4976 y(The)23 b(rounding)f(pro)s(cedure)h(\(2\))i(is)d(c)m(hosen)j(so)f(as)g(to)h (mak)m(e)f(the)g(arithmetic)g(natural.)37 b(The)23 b(results)1843 5225 y(2)p eop %%Page: 3 3 3 2 bop 94 573 a Fr(presen)m(ted)34 b(here)g(extend)h(easily)e(to)i (the)f(case)i(of)e(rounding)e(to)j(the)f(nearest)h(in)m(teger,)h (although)e(w)m(e)94 714 y(shall)29 b(not)h(consider)g(the)g(latter)h (problem.)235 855 y(W)-8 b(e)42 b(consider)d(a)i(dense)g(set)g(of)f Fg(irr)-5 b(ational)53 b Fr(winding)38 b(n)m(um)m(b)s(ers)h Fl(\027)46 b Fr(\(where)40 b Fl(\013)58 b Fr(=)f(2)15 b(cos)r(\(2)p Fl(\031)s(\027)6 b Fr(\)\),)94 996 y(corresp)s(onding)28 b(to)k Fg(r)-5 b(ational)42 b Fr(v)-5 b(alues)30 b(of)g Fl(\013)h Fr(of)g(the)g(form)f Fl(q)s(=p)2209 963 y Ff(n)2256 996 y Fr(,)g(where)g Fl(p)h Fr(is)e(a)i Fg(prime,)g Fl(q)i Fr(is)d(relativ)m(ely)94 1137 y(prime)h(to)i Fl(p)p Fr(,)g(and)e Fi(j)p Fl(q)s Fi(j)e Fl(<)f Fr(2)p Fl(p)1062 1104 y Ff(n)1109 1137 y Fr(.)47 b(T)-8 b(o)33 b(see)g(that)f(a)h(rational)f Fl(\013)g Fr(yields)f(an)h(irrational)f(rotation)h(n)m(um)m(b)s(er,)94 1279 y(w)m(e)e(note)h(that)g(when)e Fl(\027)35 b Fr(is)29 b(rational,)h(then)f Fl(\013)i Fr(is)e(t)m(wice)h(the)g(real)g(part)g (of)g(a)g(primitiv)m(e)e(ro)s(ot)i(of)h(unit)m(y)-8 b(,)94 1420 y(and)33 b(standard)h(theory)g(sho)m(ws)g(that)g Fl(\013)h Fr(m)m(ust)f(b)s(e)f(either)h(an)g(in)m(teger)g(\(in)f (\014nitely)g(man)m(y)h(cases\),)i(or)94 1561 y(an)31 b(algebraic)f(in)m(teger)h(of)g(degree)g(greater)h(than)f(1)g(\(see,)h (e.g.)g([23)q(],)f(c)m(hapter)h(2\).)42 b(As)31 b Fl(\013)g Fr(is)f(neither,)g Fl(\027)94 1702 y Fr(is)f(irrational.)235 1843 y(T)-8 b(o)35 b(illustrate)e(the)i(arithmetical)f(rationale)g(b)s (ehind)e(our)i(c)m(hoice)h(of)g(parameters,)h(w)m(e)f(consider)94 1984 y(the)f(ring)e Fh(Z)509 1998 y Ff(p)578 1984 y Fr(of)i Fl(p)p Fr(-adic)f(in)m(tegers,)i(whic)m(h)e(ma)m(y)h(b)s(e)f(represen)m (ted)h(as)g(the)g(set)g(of)g(expressions)e(of)i(the)94 2125 y(t)m(yp)s(e)792 2267 y Fl(\037)25 b Fr(=)g Fl(b)1009 2281 y Fj(0)1069 2267 y Fr(+)20 b Fl(b)1199 2281 y Fj(1)1238 2267 y Fl(p)g Fr(+)g Fl(b)1434 2281 y Fj(2)1473 2267 y Fl(p)1519 2229 y Fj(2)1579 2267 y Fr(+)g Fi(\001)15 b(\001)g(\001)431 b Fl(b)2245 2282 y Ff(k)2313 2267 y Fi(2)25 b(f)p Fr(0)p Fl(;)15 b(:)g(:)g(:)i(;)e(p)21 b Fi(\000)e Fr(1)p Fi(g)583 b Fr(\(3\))94 2460 y(whic)m(h)21 b(con)m(v)m(erge)k(with)c(resp)s(ect)i(to)g(the)g(non-ac)m(himedean)f (absolute)h(v)-5 b(alue)22 b Fi(j)5 b(\001)g(j)2790 2474 y Ff(p)2852 2460 y Fr(\(one)23 b(has)f Fi(j)p Fl(\037)p Fi(j)3307 2474 y Ff(p)3373 2460 y Fr(=)j Fl(p)3515 2427 y Fk(\000)p Ff(k)3612 2460 y Fr(,)94 2601 y(where)j Fl(k)k Fr(is)c(the)h(\014rst)f(non-zero)h(co)s(e\016cien)m(t)h(in)d(the)i (expansion)f(\(3\))i(|see,)f(e.g.,)i([13)q(],)f(part)e(I)s(I\).)h(The) 94 2742 y(main)d(result)f(of)i(this)f(pap)s(er)g(is)f(the)i(follo)m (wing)f(theorem,)i(whic)m(h)d(establishes)h(a)h(connection)g(b)s(et)m (w)m(een)94 2883 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[1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash 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closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Times-Roman) findfont 180 scalefont setfont LTb 1335 450 M -43 0 V -45 0 R (-2) Rshow 1335 2410 M -43 0 V -45 0 R (2) Rshow 1335 3390 M -43 0 V -45 0 R (4) Rshow 1335 4370 M -43 0 V -45 0 R (6) Rshow LT2 1335 450 M 0 4410 V LTb 1335 1430 M 0 -43 V -30 -110 R (0 ) Cshow LT2 2382 450 M 0 4410 V LTb 2382 1430 M 0 -43 V -30 -110 R (2 ) Cshow LT2 3428 450 M 0 4410 V LTb 3428 1430 M 0 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y(Firstly)-8 b(,)32 b(w)m(e)g(establish)f(a)h(corresp)s(ondence)g (b)s(et)m(w)m(een)h Fh(Z)2053 912 y Fj(2)2120 945 y Fr(and)e Fh(Z)p Fr([)p Fl(\025)p Fr(])m(,)h(iden)m(tifying)e Fh(Z)3039 912 y Fj(2)3106 945 y Fr(with)h Fl(P)3386 912 y Ff(n)3433 945 y Fr(,)i(and)94 1086 y(the)43 b(lattice)g Fi(M)662 1101 y Ff(k)748 1086 y Fr(with)e Fl(P)1038 1053 y Fj(\()p Ff(k)r Fj(+1\))p Ff(n)1311 1086 y Fr(\(prop)s(osition)g(4.1\).)80 b(Secondly)-8 b(,)45 b(w)m(e)e(em)m(b)s(ed)g(homomorphically)94 1227 y(the)e(ring)e Fh(Z)p Fr([)p Fl(\025)p Fr(])d(in)m(to)41 b Fh(Z)922 1241 y Ff(p)957 1227 y Fr(,)i(b)m(y)e(means)f(of)h(the)g (prime)d(ideal)i Fl(P)13 b Fr(.)71 b(Finally)-8 b(,)41 b(w)m(e)g(construct)g(the)g(lo)s(cal)94 1369 y(represen)m(tation)c(of)h (the)f(round-o\013)g(mapping,)h(and)f(extend)g(it)g(to)h(the)g(whole)e (of)i Fh(Z)3082 1383 y Ff(p)3154 1369 y Fr(\(prop)s(osition)94 1510 y(4.2\).)235 1651 y(W)-8 b(e)32 b(require)d(sev)m(eral)h (mappings.)39 b(The)30 b(\014rst)g(mapping)f(em)m(b)s(eds)g Fh(Z)2578 1618 y Fj(2)2644 1651 y Fr(in)m(to)h Fh(Z)p Fr([)p Fl(\025)p Fr(].)1025 1845 y Fi(L)1088 1859 y Fj(1)1152 1845 y Fr(:)c Fh(Z)1268 1807 y Fj(2)1328 1845 y Fi(7!)f Fh(Z)p Fr([)p Fl(\025)p Fr(])360 b(\()p Fl(x;)15 b(y)s Fr(\))26 b Fi(7!)f Fl(p)2370 1807 y Ff(n)2417 1845 y Fl(x)20 b Fi(\000)g Fl(\025y)s(:)770 b Fr(\(20\))94 2039 y(The)28 b(second)g(mapping)f(creates)j(a)f(homomorphic)e(image)h Fi(R)h Fr(of)f Fh(Z)p Fr([)p Fl(\025)p Fr(])d(in)i Fh(Z)2716 2053 y Ff(p)2751 2039 y Fr(,)i(b)m(y)f(iden)m(tifying)e Fl(\025)j Fr(with)94 2180 y(its)h(lo)s(cal)f(image)i Fl(\022)1003 2321 y Fi(L)1066 2335 y Fj(2)1131 2321 y Fr(:)25 b Fh(Z)p Fr([)p Fl(\025)p Fr(])47 b Fi(7!)j Fh(Z)1602 2335 y Ff(p)2001 2321 y Fl(x)20 b Fr(+)g Fl(\025y)53 b Fi(7!)e Fl(x)20 b Fr(+)g Fl(\022)d(y)751 b Fr(\(21\))94 2497 y(Comp)s(osing)29 b(the)i(ab)s(o)m(v)m(e)h(t)m(w)m(o)f(mappings)e (and)h(scaling,)g(w)m(e)h(obtain)f(an)h(em)m(b)s(edding)e(of)h(our)h (discrete)94 2638 y(phase)f(space)h(in)m(to)f(the)h Fl(p)p Fr(-adics)703 2842 y Fi(L)25 b Fr(:)g Fh(Z)906 2804 y Fj(2)967 2842 y Fi(7!)g Fh(Z)1148 2856 y Ff(p)1547 2842 y Fr(\()p Fl(x;)15 b(y)s Fr(\))26 b Fi(7!)1933 2780 y Fr(1)p 1909 2821 93 4 v 1909 2904 a Fl(p)1955 2878 y Ff(n)2027 2842 y Fi(L)2090 2856 y Fj(2)2129 2842 y Fr(\()p Fi(L)2227 2856 y Fj(1)2267 2842 y Fr(\()p Fl(x;)15 b(y)s Fr(\)\))41 b(=)f Fl(x)20 b Fi(\000)2861 2780 y Fl(\022)p 2837 2821 V 2837 2904 a(p)2883 2878 y Ff(n)2955 2842 y Fl(y)s(:)448 b Fr(\(22\))94 3046 y(The)30 b(image)g(of)h(the)g (round-o\013)e(phase)h(space)1533 3240 y Fi(Z)j Fr(=)25 b Fi(L)p Fr(\()p Fh(Z)1891 3203 y Fj(2)1926 3240 y Fr(\))41 b Fi(\032)25 b Fh(Z)2163 3254 y Ff(p)3476 3240 y Fr(\(23\))94 3434 y(is)31 b(an)h(additiv)m(e)f(subgroup)g(of)h(the)g Fl(p)p Fr(-adic)g(in)m(tegers,)h(whic)m(h)d(is)i(in)m(v)-5 b(arian)m(t)31 b(under)f(m)m(ultiplication)f(b)m(y)94 3575 y(elemen)m(ts)i(of)f(the)h(ring)e Fi(R)c Fr(=)g Fi(L)1174 3589 y Fj(2)1213 3575 y Fr(\()p Fh(Z)p 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4694 y Fg(wher)-5 b(e)33 b Fi(j)19 b(\001)g(j)463 4708 y Ff(p)535 4694 y Fg(is)32 b(the)g Fl(p)p Fg(-adic)g(absolute)h(value,) f(de\014nes)g(a)g(non-ar)-5 b(chime)g(de)g(an)35 b(norm)f(on)e Fh(Z)3183 4661 y Fj(2)3218 4694 y Fg(,)g(such)g(that)94 4835 y Fi(k)p Fl(z)d Fi(\000)24 b Fl(z)351 4802 y Fk(0)375 4835 y Fi(k)420 4849 y Ff(p)496 4835 y Fr(=)35 b Fl(p)648 4802 y Fk(\000)p Ff(k)r(n)827 4835 y Fg(if)j(and)h(only)g(the)g(c)-5 b(o)g(des)40 b(of)e Fl(z)43 b Fg(and)c Fl(z)2131 4802 y Fk(0)2193 4835 y Fg(have)g(pr)-5 b(e)g(cisely)40 b(the)f(\014rst)g Fl(k)i Fg(symb)-5 b(ols)40 b(in)94 4976 y(c)-5 b(ommon.)1820 5225 y Fr(11)p eop %%Page: 12 12 12 11 bop 94 573 a Fe(Pro)s(of:)92 b Fl(i)p Fr(\))51 b(W)-8 b(e)27 b(b)s(egin)d(to)j(relate)f(the)f(lattices)h Fi(M)1901 588 y Ff(k)1970 573 y Fr(to)g(the)g(algebraic)f(in)m(teger)h Fl(\025)p Fr(,)h(b)m(y)e(sho)m(wing)g(that)94 714 y(if)k Fl(a)225 729 y Ff(k)298 714 y Fr(and)h Fl(b)514 729 y Ff(k)587 714 y Fr(are)h(as)f(in)g(\(8\),)h(then)1192 921 y Fl(\025)1245 883 y Ff(k)1313 921 y Fr(=)25 b Fl(a)1457 936 y Ff(k)1514 921 y Fl(\025)c Fr(+)f Fl(b)1718 936 y Ff(k)1775 921 y Fl(p)1821 883 y Ff(n)2050 921 y Fl(k)28 b Fr(=)d(1)p Fl(;)15 b Fr(2)p Fl(;)g(:)g(:)g(:)j(:)937 b Fr(\(25\))94 1128 y(Indeed)29 b(for)i Fl(k)d Fr(=)d(1)31 b(w)m(e)f(ha)m(v)m(e)i Fl(\025)25 b Fr(=)g Fl(a)1341 1142 y Fj(1)1396 1128 y Fl(\025)20 b Fr(+)g Fl(b)1599 1142 y Fj(1)1654 1128 y Fl(p)1700 1095 y Ff(n)1746 1128 y Fr(,)31 b(as)f Fl(a)1961 1142 y Fj(1)2026 1128 y Fr(=)25 b(1)31 b(and)f Fl(b)2414 1142 y Fj(1)2478 1128 y Fr(=)25 b(0.)41 b(F)-8 b(urthermore,)31 b(using)e(\(14\))94 1269 y(w)m(e)i(\014nd)715 1476 y Fl(\025)768 1438 y Ff(k)r Fj(+1)999 1476 y Fr(=)98 b Fl(\025)15 b(\025)1289 1438 y Ff(k)1357 1476 y Fr(=)25 b Fl(\025)p Fr(\()p Fl(a)1589 1491 y Ff(k)1632 1476 y Fl(\025)20 b Fr(+)g Fl(b)1835 1491 y Ff(k)1893 1476 y Fl(p)1939 1438 y Ff(n)1986 1476 y Fr(\))25 b(=)g Fl(a)2190 1491 y Ff(k)2233 1476 y Fl(\025)2286 1438 y Fj(2)2346 1476 y Fr(+)19 b Fl(b)2475 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714 y Fi(j)20 b(\001)g(j)209 728 y Ff(p)279 714 y Fr(has)29 b(the)i(same)f(prop)s(ert)m(y)f(and)h Fi(L)g Fr(is)f(a)h(monomorphism)e(of)i Fh(Z)p Fr(-mo)s(dules.)35 b(If)29 b Fl(z)35 b Fr(and)29 b Fl(z)3248 681 y Fk(0)3302 714 y Fr(ha)m(v)m(e)i(the)94 855 y(same)k(co)s(de,)g(then)f(from)g (prop)s(ositions)e(2.2)j(and)f(4.1)h Fl(i)p Fr(\))g(w)m(e)f(ha)m(v)m(e) i(that)f Fi(L)2710 869 y Fj(1)2749 855 y Fr(\()p Fl(z)27 b Fi(\000)c Fl(z)2993 822 y Fk(0)3016 855 y Fr(\))35 b(is)e(divisible)d(b)m(y)94 996 y Fl(P)165 963 y Fj(\()p Ff(k)r Fj(+1\))p Ff(n)421 996 y Fr(and)24 b(no)h(larger)g(p)s(o)m(w)m (er)g(of)g Fl(P)13 b Fr(.)39 b(It)26 b(follo)m(ws)e(that)i Fi(L)p Fr(\()p Fl(z)14 b Fi(\000)c Fl(z)2319 963 y Fk(0)2342 996 y Fr(\))25 b(is)f(divisible)e(b)m(y)j Fl(p)3007 963 y Ff(k)r(n)3117 996 y Fr(and)f(no)i(larger)94 1137 y(p)s(o)m(w)m(er)k (of)g Fl(p)p Fr(,)g(whence)g Fi(jL)p Fr(\()p Fl(z)24 b Fi(\000)18 b Fl(z)1205 1104 y Fk(0)1229 1137 y Fr(\))p Fi(j)1289 1151 y Ff(p)1355 1137 y Fr(=)25 b 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y(4.1)f Fl(i)p Fr(\).)42 b(Th)m(us)662 3502 y(\010)728 3464 y Fk(\003)767 3502 y Fr(\()p Fl(\037)p Fr(\))84 b(=)f(\()p Fi(L)20 b(\016)h Fr(\010\)\()p Fl(x;)15 b(y)s Fr(\))41 b(=)f Fi(L)1857 3373 y Fd(\022\026)1987 3440 y Fl(q)18 b(x)p 1987 3481 111 4 v 1996 3564 a(p)2042 3538 y Ff(n)2108 3373 y Fd(\027)2181 3502 y Fi(\000)i Fl(y)s(;)15 b(x)2412 3373 y Fd(\023)978 3753 y Fr(=)1132 3625 y Fd(\026)1195 3691 y Fl(q)j(x)p 1195 3732 V 1204 3815 a(p)1250 3789 y Ff(n)1316 3625 y Fd(\027)1389 3753 y Fi(\000)i Fl(y)j Fi(\000)1672 3691 y Fl(\022)p 1649 3732 93 4 v 1649 3815 a(p)1695 3789 y Ff(n)1766 3753 y Fl(x)41 b Fr(=)2004 3691 y(1)p 1980 3732 V 1980 3815 a Fl(p)2026 3789 y Ff(n)2083 3652 y Fd(\020)2137 3753 y Fl(x)15 b Fr(\()p Fl(q)23 b Fi(\000)d Fl(\022)s Fr(\))g Fi(\000)g Fl(p)2632 3715 y Ff(n)2694 3753 y Fl(y)j Fi(\000)d Fl(c)p Fr(\()p Fl(x)p Fr(\))3014 3652 y Fd(\021)978 4015 y Fr(=)1165 3954 y(1)p 1142 3994 V 1142 4078 a Fl(p)1188 4051 y Ff(n)1244 3914 y Fd(\020)1299 4015 y Fl(x)p 1351 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Fk(\003)1629 4614 y Fr(\()p Fl(\037)p Fr(\))41 b(=)f Fl(\033)1963 4577 y Ff(n)2010 4614 y Fr(\()p 2045 4540 V Fl(\022)r(\037)p Fr(\))p Fl(:)94 4835 y Fr(No)m(w,)k(if)c Fl(\037)490 4802 y Fj(\()p Ff(k)r Fj(\))630 4835 y Fi(!)j Fl(\037)e Fr(is)e(a)j(Cauc)m(h)m(y)f(sequence)g(in)f Fi(Z)7 b Fr(,)43 b(so)e(is)f Fl(\033)2310 4802 y Ff(n)2357 4835 y Fr(\()p 2392 4761 V Fl(\022)s(\037)2495 4802 y Fj(\()p Ff(k)r Fj(\))2592 4835 y Fr(\),)k(and)d(w)m(e)g(can)g(extend)g(the)94 4976 y(mapping)29 b(\010)534 4943 y Fk(\003)603 4976 y Fr(to)i(the)g(whole)e(of)i Fh(Z)1297 4990 y Ff(p)1332 4976 y Fr(.)41 b(This)28 b(pro)m(v)m(es)j(equation)g(\(28\).)1820 5225 y(13)p eop %%Page: 14 14 14 13 bop 3564 503 74 4 v 3564 570 4 67 v 3634 570 V 3564 573 74 4 v 235 814 a Fr(W)-8 b(e)45 b(note)g(that)g(the)f (restriction)f(of)i(the)f(lo)s(cal)f(mapping)g(to)i Fi(Z)51 b Fr(is)43 b(in)m(v)m(ertible)g(\(whereas)h(the)94 955 y(extended)29 b(mapping)d(is)i(not\).)41 b(The)28 b(in)m(v)m(erse)g(is) 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b(erio)g(dic)25 b(p)-5 b(oint)25 b Fl(\037)e Fg(c)-5 b(orr)g(esp)g(onding)26 b(to)e(the)g Fl(T)13 b Fg(-p)-5 b(erio)g(dic)24 b(c)-5 b(o)g(de)25 b Fl(C)31 b Fr(=)25 b(\()p 3300 4544 466 4 v Fl(c)3339 4608 y Fj(0)3380 4594 y Fl(;)15 b(:)g(:)g(:)h(;)f(c)3620 4608 y Ff(T)10 b Fk(\000)p Fj(1)3766 4594 y Fr(\))94 4735 y Fg(takes)33 b(the)g(form)1407 4893 y Fl(\037)25 b Fr(=)1681 4832 y(1)p 1595 4872 217 4 v 1595 4956 a Fl(m)p Fr(\()p Fl(T)13 b Fr(\))1852 4765 y Fd(\022)1919 4893 y Fl(x)20 b Fi(\000)2115 4832 y Fl(\022)p 2092 4872 93 4 v 2092 4956 a(p)2138 4929 y Ff(n)2209 4893 y Fl(y)2257 4765 y Fd(\023)3476 4893 y Fr(\(32\))1820 5225 y(15)p eop %%Page: 16 16 16 15 bop 94 573 a Fg(wher)-5 b(e)34 b Fl(x)e Fg(and)i Fl(y)h Fg(ar)-5 b(e)34 b(inte)-5 b(gers)33 b(given)f(by)329 858 y Fl(x)25 b Fr(=)502 745 y Ff(T)10 b Fk(\000)p Fj(1)507 772 y Fd(X)511 966 y Ff(r)r Fj(=0)658 858 y Fl(c)697 872 y Ff(r)751 858 y Fl(U)813 872 y Ff(T)c(;)11 b(r)1178 858 y Fl(y)28 b Fr(=)1347 745 y Ff(T)10 b Fk(\000)p 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Fl(c)2218 4760 y Ff(r)2272 4746 y Fl(p)2318 4709 y Ff(nr)2398 4746 y Fr(\()p Fl(a)2481 4760 y Ff(T)g Fk(\000)p Ff(r)2645 4746 y Fi(\000)20 b Fl(\022)s(a)2830 4760 y Ff(T)10 b Fk(\000)p Ff(r)r Fk(\000)p Fj(1)3063 4746 y Fr(\))1820 5225 y(16)p eop %%Page: 17 17 17 16 bop 94 573 a Fr(is)27 b(a)i Fl(p)p Fr(-adic)f(in)m(teger.)40 b(The)28 b(righ)m(tmost)h(sum)e(w)m(as)i(deriv)m(ed)e(using)g(the)h(lo) s(cal)g(v)m(ersion)g(of)g(form)m(ula)g(\(25\))94 714 y(\(whic)m(h)34 b(is)h(v)-5 b(alid)33 b(since)i Fi(L)1006 728 y Fj(2)1080 714 y Fr(is)f(a)i(ring)e(homomorphism\))f(and)i (de\014ning)e Fl(a)2709 728 y Fj(0)2782 714 y Fr(=)g(0.)55 b(Multiplying)32 b(the)94 855 y(n)m(umerator)e(and)g(denominator)g(of)g (\(35\))i(b)m(y)e Fl(p)1711 822 y Ff(nT)1829 855 y Fi(\000)20 b Fl(\022)1966 822 y Ff(T)2020 855 y Fr(,)31 b(w)m(e)g(obtain)195 1117 y Fl(\037)25 b Fr(=)468 1056 y(1)p 383 1096 217 4 v 383 1180 a Fl(m)p Fr(\()p Fl(T)13 b Fr(\))634 1056 y(\()p Fl(p)715 1023 y Ff(nT)833 1056 y Fi(\000)20 b 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b(conjecture)h(that)f(the)h(system)f(is)f(globally)g(p)s(erio) s(dic)e(is)94 1086 y(supp)s(orted)d(b)m(y)i(extensiv)m(e)g(n)m (umerical)f(evidence)h([30)q(].)55 b(Ho)m(w)m(ev)m(er,)38 b(no)d(pro)s(of)f(of)h(global)g(stabilit)m(y)e(is)94 1227 y(kno)m(wn)d(to)h(the)g(authors)f(for)g(an)m(y)g(in)m(v)m(ertible) f(mo)s(del)g(of)i(linear)e(irrational)f(rotations.)235 1369 y(Stabilit)m(y)h(pro)s(ofs)h(for)h(planar)e(rotations)i(ha)m(v)m (e)h(b)s(een)e(pro)s(duced)f(for)i Fg(non-invertible)37 b Fr(discretiza-)94 1510 y(tions)f(of)h(irrational)d(rotations,)39 b(for)d(one)h(case)h(of)e(in)m(v)m(ertible)f(discretization)h(of)g Fg(r)-5 b(ational)49 b Fr(rotation)94 1651 y([21)q(],)31 b(and)f(\014nally)e(for)i(in)m(v)m(ertible)f(discretizations)g(of)i(t)m (wist)f(mappings)e([8)q(].)235 1792 y(The)36 b(completeness)g(of)h(the) f(co)s(de)h(yields)e(the)h(full)e(set)j(of)g(p)s(erio)s(dic)c(p)s(oin)m (ts,)k(as)g Fl(p)p Fr(-adic)f(in)m(teger)94 1933 y Fl(\037)h Fr(of)h(the)g(form)f(\(32\).)64 b(The)37 b(di\016culties)e(in)i(the)g (study)g(of)h(the)g(p)s(erio)s(dic)d(orbits)h(of)i(the)g(round-o\013)94 2074 y(map)28 b(\010)h(are)g(cen)m(tered)h(around)d(the)i(problem)e(of) i(deciding)e(whether)h(or)h(not)g(suc)m(h)f Fl(\037)h Fr(b)s(elongs)f(to)h(the)94 2215 y(submo)s(dule)22 b Fi(Z)31 b Fr(of)25 b Fh(Z)806 2229 y Ff(p)841 2215 y Fr(,)h(whic)m(h)e(is)f(the)i(case)h(when)d Fl(x)i Fr(and)f Fl(y)j Fr(are)e(divisible)c(b)m(y)j Fl(m)p Fr(\()p Fl(T)13 b Fr(\).)39 b(This)23 b(divisibilit)m(y)94 2357 y(question)e(is)h (delicate,)h(and)f(will)d(b)s(e)j(dealt)g(with)f(elsewhere.)37 b(Structurally)20 b(similar)g(problems)g(arise)i(in)94 2498 y(the)28 b(arc)m(himedean)g(case,)i(for)e(Bernoulli)e(shifts)h (and)g(h)m(yp)s(erb)s(olic)f(toral)i(automorphisms,)f(where)h(the)94 2639 y(lattices)23 b(con)m(taining)g(all)f(orbits)g(of)i(a)f(giv)m(en)h (p)s(erio)s(d)c(are)k(easily)e(determined,)i(but)e(one)i(do)s(es)f(not) g(kno)m(w)94 2780 y(a)31 b(priori)e(whether)h(a)h(giv)m(en)g (sublattice)f(con)m(tains)h(an)m(y)-8 b(.)43 b(This)29 b(class)i(of)g(problems)e(are)i(asso)s(ciated)g(to)94 2921 y(the)e(emergence)i(of)e(non-p)s(olynomial)d(time)j(algorithms,)g (and)g(of)g(statistical)g(b)s(eha)m(viour)f(in)g(discrete)94 3062 y(and)i(quan)m(tum)g(systems)g(\(see)h([9)q(])g(and)e(references)i (therein\).)235 3203 y(Nev)m(ertheless,)g(the)g(expansiv)m(e)f(prop)s (ert)m(y)f(of)i(the)f(em)m(b)s(edding)f(system)h(justi\014es)f(in)m (tuitiv)m(ely)f(the)94 3344 y(observ)m(ed)j(statistical)g(prop)s (erties)e(of)i(the)g(round-o\013)f(errors,)h(and)f(one)h(w)m(ould)f (exp)s(ect)h(the)g(accum)m(u-)94 3486 y(lation)24 b(of)i(suc)m(h)f (errors)g(to)h(giv)m(e)f(rise)g(to)h(a)f(gaussian)g(pro)s(cess.)38 b(T)-8 b(o)26 b(this)e(exten)m(t)j(w)m(e)f(note)g(that)f(a)h(recen)m(t) 94 3627 y(result)h(of)h(Vladimiro)m(v)d([28)r(],)j(who)g(pro)m(v)m(ed)g (a)g(cen)m(tral)g(limit)d(theorem)k(for)e(round-o\013)g(errors)g(in)g (linear)94 3768 y(systems.)41 b(Our)30 b(mo)s(del)f(ho)m(w)m(ev)m(er,)j (is)e(not)h(directly)e(applicable)f(to)k(that)f(case,)h(since)e(the)g (system)h(\(1\))94 3909 y(has)f(rational)g(en)m(tries.)1820 5225 y(18)p eop %%Page: 19 19 19 18 bop 94 573 a Fm(References)139 804 y Fr([1])47 b(S.)29 b(Alb)s(ev)m(erio,)f(A.)i(Khrennik)m(o)m(v,)e(B.)i(Tirozzi)e (and)h(D)g(De)h(Smedt,)g Fl(p)p Fr(-adic)e(dynamical)g(systems,)281 945 y Fb(Theoretical)i(and)g(Mathematical)h(Ph)m(ysics)p Fr(,)f Fe(114)p Fr(,)h(\(1998\),)i(276{287.)139 1158 y([2])47 b(D.)22 b(K.)g(Arro)m(wsmith)e(and)h(F.)h(Viv)-5 b(aldi,)22 b(Some)g Fl(p)p Fr(-adic)f(represen)m(tations)h(of)g(the)f (Smale)g(horsesho)s(e,)281 1299 y Fb(Ph)m(ys.)30 b(Lett.)i(A)e Fe(176)h Fr(\(1993\))i(292{294.)139 1511 y([3])47 b(D.)41 b(K.)g(Arro)m(wsmith)e(and)h(F.)i(Viv)-5 b(aldi,)41 b(Geometry)h(of)f Fl(p)p Fr(-adic)f(Siegel)g(discs,)i Fb(Ph)m(ysica)f(D)g Fe(71)281 1652 y Fr(\(1994\))33 b(222{236.)139 1865 y([4])47 b(R.)30 b(Benedetto,)i(F)-8 b(atou)32 b(comp)s(onen)m(ts)e(in)f Fl(p)p Fr(-adic)g(dynamics,)g(PhD)h(Thesis,)f(Bro)m(wn)h(Univ)m(ersit)m (y)281 2006 y(\(1998\).)139 2219 y([5])47 b(R.)21 b(Benedetto,)26 b Fl(p)p Fr(-adic)21 b(dynamics)f(and)g(Sulliv)-5 b(an's)19 b(No)i(W)-8 b(andering)21 b(Domain)h(Theorem,)h(\(1998\))281 2360 y Fb(Comp)s(osition)28 b(Mathematica)k Fr(at)f(press.)139 2572 y([6])47 b(R.)24 b(Benedetto,)j(Hyp)s(erb)s(olic)21 b(maps)i(in)g Fl(p)p Fr(-adic)g(dynamics,)h(preprin)m(t,)f(Univ)m (ersit)m(y)g(of)h(Ro)s(c)m(hester,)281 2713 y(\(1998\))33 b Fb(Ergo)s(dic)c(Theory)h(and)g(Dynamical)g(Systems)g Fr(at)h(press.)139 2926 y([7])47 b(M.)37 b(Blank,)g(P)m(athologies)g (generated)g(b)m(y)f(round-o\013)g(in)f(dynamical)f(systems,)k Fb(Ph)m(ysica)e(D)h Fe(78)281 3067 y Fr(\(1994\))c(93{114.)139 3280 y([8])47 b(M.)40 b(Blank,)i(T.)d(Kr)s(\177)-48 b(uger)39 b(and)g(L.)h(Pust)m(ylnik)m(o)m(v,)h(A)f(KAM)g(t)m(yp)s(e)f(theorem)i (for)e(systems)h(with)281 3421 y(round-o\013)30 b(errors,)g(preprin)m (t)e(\(1997\).)139 3633 y([9])47 b(B.V.)36 b(Chirik)m(o)m(v)e(and)h(F.) h(Viv)-5 b(aldi,)35 b(An)g(algorithmic)f(view)h(of)h(pseudo)s(c)m (haos,)g Fb(Ph)m(ysica)g(D)f Fe(129)281 3774 y Fr(\(1999\))e(223{235.) 94 3987 y([10])47 b(P)-8 b(.)33 b(Diamond,)f(P)-8 b(.)32 b(Klo)s(eden,)g(V.)h(Kozy)m(akin)f(and)g(A.)g(P)m(okro)m(vskii,)h (Boundedness)e(and)g(dissipa-)281 4128 y(tivit)m(y)c(of)h(truncated)f (rotations)h(on)f(uniform)f(planar)g(lattices,)j(Preprin)m(t)d(TR)h (M95/06,)k(Sc)m(ho)s(ol)281 4269 y(of)25 b(Computing)f(and)h (Mathematics,)i(Deaking)f(Univ)m(ersit)m(y)-8 b(,)26 b(Geelong,)i(Vic.,)e(Australia)f(\(1995\).)94 4482 y([11])47 b(H.)32 b(Cohn,)f Fb(A)h(second)g(course)g(in)e(n)m(um)m(b)s(er)g (theory)p Fr(,)j(John)e(Wiley)g(&)g(Sons,)g(New)h(Y)-8 b(ork)33 b(\(1962\).)281 4623 y(\(Reprin)m(ted)c(as)i Fb(Adv)-5 b(anced)30 b(n)m(um)m(b)s(er)f(theory)p Fr(,)i(Do)m(v)m(er,)i (New)d(Y)-8 b(ork)31 b(\(1980\).\))94 4835 y([12])47 b(H.)25 b(Cohn,)g Fb(In)m(tro)s(duction)e(to)i(the)g(construction)f(of) h(class)g(\014elds)p Fr(,)f(Cam)m(bridge)g(Univ)m(ersit)m(y)f(Press,) 281 4976 y(New)30 b(Y)-8 b(ork)31 b(\(1985\).)1820 5225 y(19)p eop %%Page: 20 20 20 19 bop 94 573 a Fr([13])47 b(H.)31 b(Hasse,)g Fb(Num)m(b)s(er)e (theory)-8 b(,)31 b Fr(Springer-V)-8 b(erlag,)30 b(Berlin)f(\(1980\).) 94 789 y([14])47 b(M.)40 b(R.)f(Herman)g(and)f(J-C.)h(Y)-8 b(o)s(ccoz,)44 b(Generalization)39 b(of)g(some)h(theorem)g(of)f(small)f (divisors)281 930 y(to)h(non-arc)m(himedean)f(\014elds,)h Fb(Geometric)h(Dynamics,)g Fr(LNM)f(1007,)j(Springer-V)-8 b(erlag,)40 b(New)281 1071 y(Y)-8 b(ork,)31 b(\(1983\),)i(408{447.)94 1287 y([15])47 b(L.)33 b(Hsia,)i(A)e(w)m(eak)i(N)m(\023)-43 b(eron)34 b(mo)s(del)e(with)h(applications)e(to)k Fl(p)p Fr(-adic)e(dynamical)f(systems,)i Fb(Com-)281 1428 y(p)s(ositio)29 b(Math.)i Fe(100)p Fr(,)g(\(1996\))i(277{304.)94 1644 y([16])47 b(L.)k(Hsia,)56 b(Closure)50 b(of)i(p)s(erio)s(dic)c(p)s(oin) m(ts)i(o)m(v)m(er)j(a)e(non-arc)m(himedean)g(\014eld,)k(preprin)m(t,)g (De-)281 1785 y(partmen)m(t)j(of)f(Mathematics,)66 b(National)57 b(Cen)m(tral)h(Univ)m(ersit)m(y)-8 b(,)64 b(T)-8 b(aiw)m(an)57 b(\(1999\))j(\(e-mail:)281 1926 y Fa(hsia@math.ncu.edu.tw)p Fr(\).)94 2142 y([17])47 b(V.)42 b(Kozy)m(akin,)i(On)c(\014niteness)g (of)i(tra)5 b(jectories)42 b(for)f(one)h(mapping)d(asso)s(ciated)j (with)e(quasi-)281 2283 y(in)m(v)m(ersion)46 b(of)i(rotation)g(mapping) e(on)h(in)m(teger)h(planar)f(lattice,)52 b(Pro)s(ceedings)47 b(of)h(the)g(15th)281 2424 y(IMA)m(CS)35 b(W)-8 b(orld)36 b(Congress)f(on)h(Scien)m(ti\014c)f(Computation,)h(Mo)s(delling)e(and)h (Applied)e(Mathe-)281 2565 y(matics,)24 b(August)d(24-29,)26 b(1997,)f(Berlin,)d(German)m(y)-8 b(,)25 b(V)-8 b(olume)22 b(I:)f(Computational)g(Mathematics,)281 2707 y(Wissensc)m(haft)30 b(und)f(T)-8 b(ec)m(hnik)30 b(V)-8 b(erlag,)31 b(Berlin,)e(\(1997\).)k (39{44.)94 2922 y([18])47 b(H-C.)29 b(Li,)f Fl(p)p Fr(-adic)g (dynamical)f(systems)i(and)f(formal)g(groups,)g Fb(Comp)s(ositio)f (Mathematica)j Fe(104)281 3064 y Fr(\(1996\))j(41{54.)94 3279 y([19])47 b(H-C.)d(Li,)i Fl(p)p Fr(-adic)d(p)s(erio)s(dic)e(p)s (oin)m(ts)h(and)h(Sen's)g(theorem,)k Fb(J.)c(Num)m(b)s(er)g(Theory)g Fe(56)h Fr(\(1996\))281 3421 y(309{318.)94 3636 y([20])j(J.)39 b(H.)g(Lo)m(w)m(enstein,)j(S.)d(Hatjisp)m(yros)g(and)f(F.)i(Viv)-5 b(aldi,)40 b(Quasi-p)s(erio)s(dicit)m(y)-8 b(,)38 b(global)h(stabilit)m (y)281 3778 y(and)30 b(scaling)f(in)g(a)i(mo)s(del)e(of)i(Hamiltonian)d (round-o\013,)i Fb(Chaos)h Fe(7)f Fr(\(1997\))j(49{66.)94 3993 y([21])47 b(J.)37 b(H.)g(Lo)m(w)m(enstein,)h(and)f(F.)g(Viv)-5 b(aldi,)36 b(Anomalous)h(transp)s(ort)f(in)f(a)i(mo)s(del)f(of)h (Hamiltonian)281 4134 y(round-o\013,)30 b Fb(Nonlinearit)m(y)f Fe(5)h Fr(\(1998\))j(1321{1350.)94 4350 y([22])47 b(J.)30 b(Lubin,)e(Non-arc)m(himedean)j(dynamical)e(systems,)h Fb(Comp.)g(Maths.)h Fe(94)g Fr(\(1994\))i(321{346.)94 4566 y([23])47 b(D.)31 b(A.)g(Marcus,)f Fb(Num)m(b)s(er)g(\014elds,)f Fr(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork)31 b(\(1977\).)94 4782 y([24])47 b(W.)33 b(Narkiewicz,)g(P)m(olynomial)f(cycles)g(in)g (algebraic)g(n)m(um)m(b)s(er)g(\014elds,)g Fb(Coll.)f(Math.)j Fe(58)f Fr(\(1989\))281 4923 y(151{155.)1820 5225 y(20)p eop %%Page: 21 21 21 20 bop 94 573 a Fr([25])47 b(T.)30 b(P)m(ezda,)h(P)m(olynomial)e (cycles)i(in)e(certain)h(lo)s(cal)f(domains,)h Fb(Acta)h(Arith.)e Fe(LXVI)h Fr(\(1994\))j(11{)281 714 y(22.)94 930 y([26])47 b(J-P)-8 b(.)31 b(Serre,)f Fb(A)g(course)h(in)e(arithmetic,)h Fr(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork)31 b(\(1973\).)94 1146 y([27])47 b(E.)35 b(Thiran,)g(D.)h(V)-8 b(erstegen,)39 b(and)c(J.)g(W)-8 b(ey)m(ers,)38 b Fl(p)p Fr(-adic)d(dynamics,)h Fb(J.)f(Stat.)i(Ph)m(ys.)e Fe(54)h Fr(\(1989\))281 1287 y(893{913.)94 1503 y([28])47 b(I.)23 b(Vladimiro)m(v,)g(Quan)m(tized)g (linear)f(systems)h(on)g(in)m(teger)h(lattices:)37 b(frequency-based)23 b(approac)m(h)281 1644 y(I)30 b(&)g(I)s(I,)g(preprin)m(t,)f(Deakin)h (Univ)m(ersit)m(y)-8 b(,)30 b(Geelong,)i(Victoria.)e(\(1996\).)94 1860 y([29])47 b(F.)31 b(Viv)-5 b(aldi,)28 b(Dynamics)i(o)m(v)m(er)i (irreducible)27 b(p)s(olynomials,)h Fb(Nonlinearit)m(y)h Fe(5)h Fr(\(1992\))j(941{960.)94 2076 y([30])47 b(F.)27 b(Viv)-5 b(aldi,)25 b(P)m(erio)s(dicit)m(y)g(and)h(transp)s(ort)f(from) h(round-o\013)g(errors,)h Fb(Exp)s(erimen)m(tal)d(Mathemat-)281 2217 y(ics)30 b Fe(3)g Fr(\(1994\))j(303{315.)94 2433 y([31])47 b(C.)37 b(W)-8 b(o)s(o)s(dco)s(c)m(k)39 b(and)e(N.)g(Smart,)i Fl(p)p Fr(-adic)e(c)m(haos)i(and)e(random)f(n)m(um)m(b)s(er)g (generation,)k Fb(Exp)s(eri-)281 2574 y(men)m(tal)30 b(Mathematics)i Fe(7)e Fr(\(1998\))j(333{342.)94 2790 y([32])47 b(M.)35 b(E.)h(Ziev)m(e,)g(Cycles)f(of)g(P)m(olynomial)f (Mappings,)h(Ph.D.)g(thesis,)h(Univ)m(ersit)m(y)e(of)i(California)281 2931 y(at)31 b(Berk)m(eley)-8 b(,)32 b(\(1996\).)1820 5225 y(21)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF