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g(Ho)n(w)n(ev)n(er,)g(w)n(e)h(note)g(that)g(man)n(y)f(other)f(examples) h(are)g(certainly)f(accessible)g(as)83 1888 y(suggested)35 b(b)n(y)h([14)o(,)h(Sec.)g(4])g(\(for)f(instance,)f(it)i(w)n(ould)e(b)r (e)i(in)n(teresting)d(to)i(study)h(the)g(cases)e(of)i(the)g(nonlinear) 83 1996 y(Klein-Gordon)20 b(and)j(Sc)n(hr\177)-42 b(odinger)20 b(equations)j(where)g(the)h(scattering)f(theory)g(is)h(w)n(ell)f (de\014ned,)g(see)h([24)o(,)h(25)o(]\).)g(The)83 2104 y(notation)h Fq(h)p Fr(y)s Fq(i)e Fy(:=)659 2033 y Fm(p)p 742 2033 272 4 v 71 x Fy(1)18 b(+)g Fq(j)p Fr(y)s Fq(j)976 2080 y FA(2)1042 2104 y Fy(is)27 b(used)f(for)h(an)n(y)g Fr(y)f Fq(2)e Fp(R)1801 2074 y Fo(n)1853 2104 y Fy(.)83 2267 y Ft(Example)32 b(2.3)g Fy(\()p Fr(H)733 2279 y FA(0)770 2267 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))26 b(=)e Fr(h)p Fy(\()p Fr(p)p Fy(\)\))p Ft(.)42 b Fz(Consider)33 b(on)g Fr(M)g Fy(:=)24 b Fr(T)2121 2237 y Fl(\003)2158 2267 y Fp(R)2212 2237 y Fo(n)2288 2267 y Fq(')h Fp(R)2432 2237 y FA(2)p Fo(n)2550 2267 y Fz(the)32 b(c)-5 b(anonic)g(al)33 b(c)-5 b(o)g(or)g(dinates)32 b Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))p Fz(,)83 2375 y(with)29 b Fr(q)d Fq(\021)d Fy(\()p Fr(q)497 2345 y FA(1)535 2375 y Fr(;)14 b(:)g(:)g(:)f(;)h(q)759 2345 y Fo(n)805 2375 y Fy(\))30 b Fz(and)e Fr(p)23 b Fq(\021)f Fy(\()p Fr(p)1265 2387 y FA(1)1303 2375 y Fr(;)14 b(:)g(:)g(:)f(;)h(p)1530 2387 y Fo(n)1576 2375 y Fy(\))p Fz(,)29 b(and)g(the)f(c)-5 b(anonic)g(al)29 b(symple)-5 b(ctic)29 b(form)g Fr(!)d Fy(:=)3195 2313 y Fm(P)3283 2334 y Fo(n)3283 2400 y(j)s FA(=1)3415 2375 y Fy(d)p Fr(q)3502 2345 y Fo(j)3549 2375 y Fq(^)12 b Fy(d)p Fr(p)3706 2387 y Fo(j)3741 2375 y Fz(.)83 2483 y(T)-7 b(ake)33 b(a)f(pur)-5 b(ely)33 b(kinetic)g(Hamiltonian)g Fr(H)1520 2495 y FA(0)1556 2483 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))26 b(:=)e Fr(h)p Fy(\()p Fr(p)p Fy(\))34 b Fz(with)e Fr(h)25 b Fq(2)g Fj(C)2469 2453 y Fl(1)2539 2483 y Fy(\()p Fp(R)2625 2453 y Fo(n)2677 2483 y Fy(;)14 b Fp(R)q Fy(\))p Fz(,)39 b(and)33 b(let)f Fy(\010)3227 2495 y Fo(j)3261 2483 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))26 b(:=)e Fr(q)3623 2453 y Fo(j)3691 2483 y Fz(b)-5 b(e)83 2591 y(the)32 b(usual)g(p)-5 b(osition)33 b(functions.)g(Then)g Fr(')1505 2561 y FA(0)1505 2612 y Fo(t)1542 2591 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))25 b(=)1840 2524 y Fm(\000)1878 2591 y Fr(q)d Fy(+)d Fr(t)p Fy(\()p Fq(r)p Fr(h)p Fy(\)\()p Fr(p)p Fy(\))p Fr(;)14 b(p)2420 2524 y Fm(\001)2459 2591 y Fz(,)33 b Fq(r)p Fr(H)2656 2603 y FA(0)2717 2591 y Fy(=)24 b Fq(r)p Fr(h)p Fz(,)34 b(and)e(Assumption)i(2.1)83 2699 y(is)e(satis\014e)-5 b(d:)1116 2740 y Fm(\010)1165 2807 y Fq(f)p Fy(\010)1267 2819 y Fo(j)1301 2807 y Fr(;)14 b(H)1406 2819 y FA(0)1443 2807 y Fq(g)p Fr(;)g(H)1590 2819 y FA(0)1626 2740 y Fm(\011)1674 2807 y Fy(\()p Fr(q)s(;)g(p)p Fy(\))24 b(=)1970 2740 y Fm(\010)2018 2807 y Fy(\()p Fr(@)2093 2819 y Fo(j)2128 2807 y Fr(h)p Fy(\)\()p Fr(p)p Fy(\))p Fr(;)14 b(h)p Fy(\()p Fr(p)p Fy(\))2509 2740 y Fm(\011)2582 2807 y Fy(=)23 b(0)p Fr(:)83 2956 y Fz(F)-7 b(urthermor)i(e,)30 b(we)j(have)e 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Fz(with)41 b(the)g(c)-5 b(o)g(or)g(dinates)41 b Fr(q)i Fq(\021)38 b Fy(\()p Fr(q)3192 3197 y FA(1)3230 3227 y Fr(;)14 b(q)3304 3239 y Fl(?)3359 3227 y Fy(\))40 b Fq(2)f Fy(\012)i Fz(and)83 3344 y Fr(p)g Fq(\021)g Fy(\()p Fr(p)348 3356 y FA(1)385 3344 y Fr(;)14 b(p)465 3356 y Fl(?)520 3344 y Fy(\))42 b Fq(2)g Fp(R)745 3314 y Fo(n)p FA(+1)924 3344 y Fz(and)g(with)h(the)f(symple)-5 b(ctic)43 b(form)g Fr(!)h Fy(:=)2363 3282 y Fm(P)2451 3302 y Fo(n)p FA(+1)2451 3369 y Fo(j)s FA(=1)2595 3344 y Fy(d)p Fr(q)2682 3314 y Fo(j)2742 3344 y Fq(^)27 b Fy(d)p Fr(p)2914 3356 y Fo(j)2947 3344 y Fz(.)44 b(Then,)f(c)-5 b(onsider)43 b(the)83 3452 y(Hamiltonian)32 b(given)g(by)g(the)g(map)1147 3629 y Fr(H)1215 3641 y FA(0)1275 3629 y Fy(:)23 b Fr(M)32 b Fq(!)23 b Fp(R)q Fr(;)111 b Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))24 b Fq(7!)f(j)p Fr(p)p Fq(j)2129 3595 y FA(2)2166 3629 y Fr(=)p Fy(2)17 b(+)h Fr(v)2392 3641 y FA(0)2429 3562 y Fm(\000)2467 3629 y Fq(j)p Fr(q)2527 3641 y Fl(?)2583 3629 y Fq(j)2606 3595 y 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b(otential)39 b Fr(v)1089 4589 y FA(0)1126 4577 y Fz(,)h(which)h(r)-5 b(ep)g(els)40 b(the)g(p)-5 b(article)39 b(ne)-5 b(ar)40 b(the)g(b)-5 b(oundary)40 b(of)g Fy(\012)p Fz(.)h(The)f(motion)g(of)83 4685 y(the)c(p)-5 b(article)36 b(is)i(uniform)f(along)g(the)f Fr(q)1442 4655 y FA(1)1479 4685 y Fz(-axis)i(and)e(given)i(by)g(the)e(e)-5 b(quation)43 b Fy(\177)-48 b Fr(q)2788 4697 y Fl(?)2875 4685 y Fy(=)31 b Fq(\000)p Fy(2)p Fr(q)3115 4697 y Fl(?)3170 4685 y Fr(v)3215 4655 y Fl(0)3212 4706 y FA(0)3248 4618 y Fm(\000)3286 4685 y Fq(j)p Fr(q)3346 4697 y Fl(?)3402 4685 y Fq(j)3425 4655 y FA(2)3462 4618 y Fm(\001)3537 4685 y Fz(in)38 b(the)83 4793 y(tr)-5 b(ansverse)32 b(dir)-5 b(e)g(ction.)32 b(T)-7 b(o)32 b(show)g(the)f(c)-5 b(ompleteness)32 b(of)g(the)f(c)-5 b(orr)g(esp)g(onding)32 b(Hamiltonian)g(ve)-5 b(ctor)31 b(\014eld)825 5007 y Fr(X)893 5019 y Fo(H)946 5027 y Fg(0)982 5007 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))23 b(=)g Fr(p)1359 4951 y(@)p 1339 4988 89 4 v 1339 5064 a(@)5 b(q)1437 4887 y 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5421 y Fr(q)2529 5387 y FA(1)2567 5354 y Fm(\013)2606 5421 y Fr(:)968 b Fy(\(2.4\))1905 5670 y(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 83 307 a Fz(As)36 b(a)f(function)h Fy(\010)p Fz(,)g(we)g(take)e(the)h(longitudinal)f(c)-5 b(o)g(or)g(dinate)35 b Fy(\010\()p Fr(q)s(;)14 b(p)p Fy(\))29 b(:=)g Fr(q)2611 277 y FA(1)2649 307 y Fz(.)35 b(This)h(gives)g Fy(\()p Fq(r)p Fr(H)3311 319 y FA(0)3349 307 y Fy(\)\()p Fr(q)s(;)14 b(p)p Fy(\))29 b(=)g Fr(p)3731 319 y FA(1)83 415 y Fz(as)j(velo)-5 b(city)32 b(observable)f(and)g (implies)h(that)1001 528 y Fm(\010)1050 595 y Fq(f)p Fy(\010)p Fr(;)14 b(H)1257 607 y FA(0)1293 595 y Fq(g)p Fr(;)g(H)1440 607 y FA(0)1476 528 y Fm(\011)1524 595 y Fy(\()p Fr(q)s(;)g(p)p Fy(\))24 b(=)1820 528 y Fm(\010)1868 595 y Fr(p)1911 607 y FA(1)1948 595 y Fr(;)14 b Fq(j)p Fr(p)p Fq(j)2074 561 y FA(2)2111 595 y Fr(=)p Fy(2)k(+)g Fr(v)2338 607 y FA(0)2374 528 y Fm(\000)2413 595 y Fq(j)p Fr(q)2473 607 y Fl(?)2528 595 y Fq(j)2551 561 y FA(2)2588 528 y Fm(\001\011)2698 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1136 y Fz(endowe)-5 b(d)33 b(with)83 1244 y(the)e(R)n(iemannian)h(metric)g Fr(g)j Fz(given)e(by)728 1450 y Fr(g)768 1462 y Fo(q)804 1450 y Fy(\()p Fr(X)904 1462 y Fo(q)940 1450 y Fr(;)14 b(Y)1024 1462 y Fo(q)1060 1450 y Fy(\))23 b(:=)1400 1394 y(4)p 1236 1431 369 4 v 1236 1507 a(\(1)18 b Fq(\000)g(j)p Fr(q)s Fq(j)1497 1483 y FA(2)1535 1507 y Fy(\))1567 1483 y FA(2)1624 1450 y Fy(\()p Fr(X)1724 1462 y Fo(q)1778 1450 y Fq(\001)h Fr(Y)1867 1462 y Fo(q)1903 1450 y Fy(\))p Fr(;)106 b(q)26 b Fq(2)2215 1431 y Fy(\027)2205 1450 y Fr(B)2268 1462 y FA(1)2305 1450 y Fr(;)46 b(X)2442 1462 y Fo(q)2478 1450 y Fr(;)14 b(Y)2562 1462 y Fo(q)2621 1450 y Fq(2)23 b Fr(T)2748 1462 y Fo(q)2793 1431 y Fy(\027)2784 1450 y Fr(B)2847 1462 y FA(1)2907 1450 y Fq(')f Fp(R)3048 1416 y Fo(n)3100 1450 y Fr(:)83 1684 y Fz(L)-5 b(et)23 b Fr(T)289 1654 y Fl(\003)335 1666 y Fy(\027)325 1684 y Fr(B)388 1696 y FA(1)448 1684 y Fq(')536 1617 y Fm(\010)584 1684 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))24 b Fq(2)880 1666 y Fy(\027)870 1684 y Fr(B)933 1696 y FA(1)972 1684 y Fq(\002)r Fp(R)1092 1654 y Fo(n)1144 1617 y Fm(\011)1216 1684 y Fz(b)-5 b(e)24 b(the)e(c)-5 b(otangent)23 b(bund)5 b(le)23 b(on)2255 1666 y Fy(\027)2245 1684 y Fr(B)2308 1696 y FA(1)2368 1684 y Fz(with)g(symple)-5 b(ctic)25 b(form)e Fr(!)j Fy(:=)3369 1622 y Fm(P)3457 1643 y Fo(n)3457 1709 y(j)s FA(=1)3590 1684 y Fy(d)p Fr(q)3677 1654 y Fo(j)3713 1684 y Fq(^)83 1792 y Fy(d)p Fr(p)173 1804 y Fo(j)207 1792 y Fz(,)32 b(and)f(let)764 1963 y Fr(H)832 1975 y FA(0)891 1963 y Fy(:)24 b Fr(T)999 1929 y Fl(\003)1044 1944 y Fy(\027)1035 1963 y Fr(B)1098 1975 y FA(1)1158 1963 y Fq(!)f Fp(R)q Fr(;)111 b Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))24 b Fq(7!)1776 1907 y Fy(1)p 1776 1944 42 4 v 1776 2020 a(2)1903 1859 y Fo(n)1864 1884 y Fm(X)1841 2063 y Fo(j;k)q FA(=1)2021 1963 y Fr(g)2064 1929 y Fo(j)s(k)2135 1963 y Fy(\()p Fr(q)s Fy(\))p Fr(p)2282 1975 y Fo(j)2317 1963 y Fr(p)2360 1975 y Fo(k)2423 1963 y Fy(=)2521 1907 y(1)p 2521 1944 V 2521 2020 a(8)2581 1963 y Fq(j)p Fr(p)p Fq(j)2670 1929 y FA(2)2708 1896 y Fm(\000)2746 1963 y Fy(1)18 b Fq(\000)g(j)p Fr(q)s Fq(j)2975 1929 y FA(2)3012 1896 y Fm(\001)3050 1913 y FA(2)83 2197 y Fz(b)-5 b(e)31 b(the)g(kinetic)g(ener)-5 b(gy)32 b(Hamiltonian.)f(Then,)g(we)h(know)f (fr)-5 b(om)31 b([14,)g(Se)-5 b(c.)32 b(4.3\(D\)])g(that)d(the)h (Hamiltonian)83 2304 y(ve)-5 b(ctor)31 b(\014eld)g Fr(X)605 2316 y Fo(H)658 2324 y Fg(0)726 2304 y Fz(is)h(c)-5 b(omplete)32 b(on)f Fr(M)h Fy(:=)23 b Fr(T)1599 2274 y Fl(\003)1645 2286 y Fy(\027)1635 2304 y Fr(B)1698 2316 y FA(1)1753 2304 y Fq(n)18 b Fr(H)1887 2267 y Fl(\000)p FA(1)1881 2325 y(0)1975 2304 y Fy(\()p Fq(f)p Fy(0)p Fq(g)p Fy(\))k Fq(')2285 2286 y Fy(\027)2275 2304 y Fr(B)2338 2316 y FA(1)2393 2304 y Fq(\002)c Fp(R)2530 2274 y Fo(n)2601 2304 y Fq(n)g(f)p Fy(0)p Fq(g)30 b Fz(and)h(that)g(the)g(function)1086 2526 y Fy(\010)23 b(:)h Fr(M)31 b Fq(!)23 b Fp(R)q Fr(;)112 b Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))23 b Fq(7!)g Fy(tanh)2100 2489 y Fl(\000)p FA(1)2202 2409 y 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b(is)g(suc)n(h)f(that)h(the)g(w)n(a)n(v)n(e)f(maps)g(exist,)h(ha)n(v)n (e)f(common)f(ranges)83 3823 y(and)d(are)h(in)n(v)n(ertible)9 b(:)83 3989 y Ft(Assumption)30 b(2.6)h Fy(\(W)-7 b(a)n(v)n(e)28 b(maps\))p Ft(.)149 4159 y Fz(\(i\))43 b(The)31 b(p)-5 b(ointwise)33 b(limits)f Fr(W)1188 4171 y Fl(\006)1267 4159 y Fy(:=)22 b(lim)1494 4171 y Fo(t)p Fl(!\0061)1718 4159 y Fr(')1773 4171 y Fl(\000)p Fo(t)1870 4159 y Fq(\016)c Fr(')1985 4129 y FA(0)1985 4180 y Fo(t)2054 4159 y Fz(exist)32 b(on)g(some)g(op)-5 b(en)32 b(sets)g Fq(D)3066 4171 y Fl(\006)3145 4159 y Fq(\032)22 b Fr(M)9 b Fz(.)122 4329 y(\(ii\))43 b(The)31 b(maps)h Fr(W)783 4341 y Fl(\006)871 4329 y Fz(ar)-5 b(e)31 b(invertible,)h(with)f(inverses)i Fr(W)2065 4292 y Fl(\000)p FA(1)2053 4348 y Fl(\006)2177 4329 y Fy(:)23 b Fj(Ran)o Fy(\()p Fr(W)2468 4341 y Fl(\006)2524 4329 y Fy(\))g Fq(!)h(D)2750 4341 y Fl(\006)2805 4329 y Fz(.)95 4499 y(\(iii\))43 b(The)31 b(maps)h Fr(W)783 4511 y Fl(\006)871 4499 y Fz(have)f(c)-5 b(ommon)32 b(r)-5 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Fq(\016)18 b Fr(W)1037 5073 y Fl(\006)1093 4993 y Fm(\001)1131 5061 y Fy(\()p Fr(m)1238 5073 y Fl(\006)1293 5061 y Fy(\))24 b(=)1436 4993 y Fm(\000)1474 5061 y Fr(W)1551 5073 y Fl(\006)1626 5061 y Fq(\016)17 b Fr(')1740 5026 y FA(0)1740 5081 y Fo(t)1777 4993 y Fm(\001)1815 5061 y Fy(\()p Fr(m)1922 5073 y Fl(\006)1978 5061 y Fy(\))84 b(for)27 b(all)f Fr(t)d Fq(2)g Fp(R)35 b Fy(and)26 b Fr(m)2791 5073 y Fl(\006)2870 5061 y Fq(2)d(D)3012 5073 y Fl(\006)3068 5061 y Fr(:)506 b Fy(\(2.5\))83 5241 y(Due)27 b(to)h(p)r(oin)n(ts)e (\(ii\))h(and)f(\(iii\))h(of)h(Assumption)d(2.6)i(the)g(scattering)f (map)1637 5421 y Fr(S)h Fy(:=)c Fr(W)1915 5384 y Fl(\000)p FA(1)1903 5440 y(+)2022 5421 y Fq(\016)18 b Fr(W)2159 5433 y Fl(\000)1905 5670 y Fy(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 83 307 a Fy(is)33 b(w)n(ell)g(de\014ned)f(and)g (in)n(v)n(ertible)f(from)i Fq(D)1452 319 y Fl(\000)1540 307 y Fy(to)h Fq(D)1712 319 y FA(+)1767 307 y Fy(.)f(F)-7 b(urthermore,)31 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b(supp)r(ort)e(of)h(the)h(p)r(oten)n(tial)e(and)g(the)i (virial)d(condition)g(app)r(earing)g(in)i(the)g(case)g(\010\()p Fr(q)s(;)c(p)p Fy(\))24 b(=)f Fr(q)s Fy(,)c Fr(H)3204 4224 y FA(0)3241 4212 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))23 b(:=)g Fq(j)p Fr(p)p Fq(j)3648 4181 y FA(2)3685 4212 y Fr(=)p Fy(2)83 4320 y(and)f Fr(H)6 b Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))24 b(=)e Fq(j)p Fr(p)p Fq(j)698 4289 y FA(2)735 4320 y Fr(=)p Fy(2)10 b(+)g Fr(V)17 b Fy(\()p Fr(q)s Fy(\))25 b(on)d Fr(M)32 b Fy(=)23 b Fr(T)1469 4289 y Fl(\003)1505 4320 y Fp(R)1559 4289 y Fo(n)1611 4320 y Fy(.)h(As)f(in)g(Section)f(2.2,)h(w)n(e)g(alw)n(a)n(ys)f(assume)g(that) h(the)h(Hamiltonians)83 4427 y Fr(H)151 4439 y FA(0)216 4427 y Fy(and)i Fr(H)33 b Fy(ha)n(v)n(e)27 b(complete)f(\015o)n(ws)h Fq(f)p Fr(')1325 4397 y FA(0)1325 4448 y Fo(t)1361 4427 y Fq(g)1403 4439 y Fo(t)p Fl(2)p Fk(R)1545 4427 y Fy(and)f Fq(f)p Fr(')1803 4439 y Fo(t)1831 4427 y Fq(g)1873 4439 y Fo(t)p Fl(2)p Fk(R)1987 4427 y Fy(.)249 4535 y(W)-7 b(e)33 b(start)f(b)r(elo)n(w)g(with)g(a)g(result)g(on)f(the)i(domain)d (and)h(the)i(range)d(of)j(the)f(w)n(a)n(v)n(e)g(maps)f Fr(W)3217 4547 y Fl(\006)3273 4535 y Fy(.)i(F)-7 b(or)31 b(this,)h(w)n(e)83 4643 y(need)27 b(to)g(in)n(tro)r(duce)e(the)j(sets)f (of)h(\010-b)r(ounded)c(tra)5 b(jectories)692 4816 y Fr(B)759 4779 y Fl(\006)755 4839 y FA(\010)837 4816 y Fy(:=)948 4749 y Fm(\010)996 4816 y Fr(m)24 b Fq(2)f Fr(M)32 b Fq(j)23 b(9)9 b Fr(R)23 b Fq(\025)g Fy(0)k(suc)n(h)f(that) 1996 4746 y Fm(\014)1996 4796 y(\014)2024 4816 y Fy(\010)2084 4749 y Fm(\000)2122 4816 y Fr(')2177 4828 y Fl(\006)p Fo(t)2256 4816 y Fy(\()p Fr(m)p Fy(\))2395 4749 y Fm(\001)2434 4746 y(\014)2434 4796 y(\014)2484 4816 y Fq(\024)d Fr(R)28 b Fy(for)f(all)f Fr(t)d Fq(\025)g Fy(0)3088 4749 y Fm(\011)3136 4816 y Fr(:)83 4989 y Fy(The)i(sets)h Fr(B)480 4952 y Fl(\006)476 5012 y FA(\010)560 4989 y Fy(coincide)d(with)j(the)g(usual) d(sets)j(of)g(b)r(ounded)d(tra)5 b(jectories)23 b([10)o(,)j(Def.)g (2.1.1])e(if)i(the)f(map)g Fq(j)p Fy(\010)p Fq(j)e Fy(:)g Fr(M)32 b Fq(!)83 5097 y Fy([0)p Fr(;)14 b Fq(1)p Fy(\))31 b(is)g(prop)r(er.)e(If)j(it)f(is)g(not)g(the)g(case,)f(the)h(inclusion) d Fr(m)i Fq(2)g Fr(B)2230 5060 y Fl(\000)2226 5120 y FA(\010)2305 5097 y Fq(\\)22 b Fr(B)2449 5060 y FA(+)2445 5120 y(\010)2535 5097 y Fy(do)r(es)30 b(not)h(guaran)n(tee)d(that)j (the)h(orbit)83 5205 y Fq(f)p Fr(')180 5217 y Fo(t)208 5205 y Fy(\()p Fr(m)p Fy(\))p Fq(g)389 5217 y Fo(t)p Fl(2)p Fk(R)534 5205 y Fy(sta)n(ys)f(in)e(a)i(compact)e(subset)h(of)h Fr(M)39 b Fy(\(consider)29 b(for)h(instance)f(the)h(case)g Fr(M)37 b Fy(=)28 b Fr(T)3140 5175 y Fl(\003)3177 5205 y Fp(R)3231 5175 y FA(2)3274 5205 y Fy(,)j(\010\()p Fr(q)s(;)14 b(p)p Fy(\))29 b(=)f Fr(q)3731 5217 y FA(1)83 5313 y Fy(and)e Fr(H)6 b Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))24 b(:=)f Fq(j)p Fr(p)p Fq(j)726 5283 y FA(2)763 5313 y Fr(=)p Fy(2\).)249 5421 y(W)-7 b(e)28 b(also)e(need)h(the)g(follo)n (wing)e(assumption)g(on)h(the)i(p)r(oten)n(tial)9 b(:)1905 5670 y(7)p eop 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y FA(0)1109 1757 y Fr(;)k Fy(\010\))24 b(b)r(e)f(compact.)f(Then,)h(there)g(exists)g Fr(T)34 b(>)22 b Fy(0)h(suc)n(h)f(that)2989 1687 y Fm(\014)2989 1736 y(\014)3016 1757 y Fy(\010)3076 1690 y Fm(\000)3114 1757 y Fr(')3169 1727 y FA(0)3169 1778 y Fo(t)3206 1757 y Fy(\()p Fr(m)p Fy(\))3345 1690 y Fm(\001)3384 1687 y(\014)3384 1736 y(\014)3435 1757 y Fr(>)h(R)3586 1769 y Fo(V)3652 1757 y Fy(+)10 b(1)83 1865 y(for)34 b(all)f Fr(t)h(>)g(T)45 b Fy(and)33 b Fr(m)i Fq(2)f Fr(K)6 b Fy(.)34 b(In)g(particular,)d Fr(')1674 1835 y FA(0)1674 1886 y Fo(t)1711 1865 y Fy(\()p Fr(K)6 b Fy(\))23 b Fq(\\)g Fy(supp)o(\()p Fr(V)18 b Fy(\))34 b(=)g Fp(?)g Fy(for)g(all)f Fr(t)h(>)g(T)12 b Fy(.)33 b(Th)n(us,)g(one)g(has)h Fr(')3633 1835 y FA(0)3633 1886 y Fo(t)3703 1865 y Fy(=)83 1973 y Fr(')138 1943 y FA(0)138 1996 y Fo(t)p Fl(\000)p Fo(T)283 1973 y Fq(\016)18 b Fr(')398 1943 y FA(0)398 1996 y Fo(T)472 1973 y Fy(=)k Fr(')614 1985 y Fo(t)p Fl(\000)p Fo(T)759 1973 y Fq(\016)c Fr(')874 1943 y FA(0)874 1996 y Fo(T)953 1973 y Fy(on)26 b Fr(K)34 b Fy(for)27 b(all)f Fr(t)d(>)g(T)12 b Fy(,)26 b(whic)n(h)h(implies)1054 2163 y(lim)1034 2215 y Fo(t)p Fl(!1)1203 2163 y Fr(')1258 2175 y Fl(\000)p Fo(t)1356 2163 y Fq(\016)18 b Fr(')1471 2128 y FA(0)1471 2183 y Fo(t)1530 2163 y Fy(=)42 b(lim)1618 2215 y Fo(t)p Fl(!1)1787 2163 y Fr(')1842 2175 y Fl(\000)p Fo(t)1939 2163 y Fq(\016)18 b Fr(')2054 2175 y Fo(t)p Fl(\000)p Fo(T)2199 2163 y Fq(\016)g Fr(')2314 2128 y FA(0)2314 2183 y Fo(T)2388 2163 y Fy(=)k Fr(')2530 2175 y Fl(\000)p Fo(T)2651 2163 y Fq(\016)c Fr(')2766 2128 y FA(0)2766 2183 y Fo(T)83 2375 y Fy(on)39 b Fr(K)6 b Fy(.)39 b(Thence,)f Fr(W)752 2387 y FA(+)847 2375 y Fy(is)h(a)g(hamiltomorphism)34 b(on)39 b(an)n(y)f(compact)g(subset)h(of)g Fr(M)c Fq(n)26 b Fj(Crit)p Fy(\()p Fr(H)3130 2387 y FA(0)3167 2375 y Fr(;)14 b Fy(\010\),)40 b(and)e(th)n(us)g(a)83 2483 y (symplectomorphism)23 b(on)k Fr(M)g Fq(n)18 b Fj(Crit)p Fy(\()p Fr(H)1351 2495 y FA(0)1388 2483 y Fr(;)c Fy(\010\).)249 2591 y(Let)36 b 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y(then)32 b(lim)395 1609 y Fl(j)p Fo(t)p Fl(j!1)605 1528 y Fm(\000)643 1596 y Fq(j)p Fy(\010)p Fq(j)749 1565 y FA(2)809 1596 y Fq(\016)21 b Fr(')927 1608 y Fo(t)955 1528 y Fm(\001)993 1596 y Fy(\()p Fr(m)p Fy(\))33 b(=)e(+)p Fq(1)i Fy(for)f(all)g Fr(m)g Fq(2)g Fr(M)9 b Fy(,)33 b(and)e(so)h Fr(B)2373 1558 y Fl(\006)2369 1618 y FA(\010)2460 1596 y Fy(=)f Fp(?)p Fy(.)i(Accordingly)-7 b(,)30 b(it)j(is)f(su\016cien)n(t)f(to)83 1703 y(pro)n(v)n(e)26 b(that)489 1636 y Fm(\010\010)585 1703 y Fq(j)p Fy(\010)p Fq(j)691 1673 y FA(2)729 1703 y Fr(;)14 b(H)840 1636 y Fm(\011)888 1703 y Fr(;)g(H)999 1636 y Fm(\011)1071 1703 y Fr(>)24 b(\016)32 b Fy(on)27 b Fr(M)37 b Fy(to)29 b(get,)f(under)e(Assumptions)g(2.1)h(and)g(2.7,)h (the)g(completeness)d(of)j(the)83 1811 y(w)n(a)n(v)n(e)21 b(maps.)g(No)n(w,)h(a)g(direct)f(calculation)e(sho)n(ws)i(that)h(the)f (expression)f(for)2493 1744 y Fm(\010)q(\010)2590 1811 y Fq(j)p Fy(\010)p Fq(j)2696 1781 y FA(2)2734 1811 y Fr(;)14 b(H)2845 1744 y Fm(\011)2893 1811 y 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FA(0)2273 2319 y Fr(;)g(H)6 b Fy(\))27 b(for)f(whic)n(h)g(this)g(expression)e(is)i(b)r(ounded)83 2427 y(a)n(w)n(a)n(y)i(from)g(zero)f(is)h(rather)f(daun)n(ting.)f(Ho)n (w)n(ev)n(er,)i(if)h(one)e(assumes)g(that)i Fq(f)p Fy(\010)p Fr(;)14 b(V)j Fq(g)24 b Fy(=)h(0)j(\(as)g(in)g(the)h(standard)d(case)83 2535 y(where)34 b Fr(V)53 b Fy(dep)r(ends)32 b(only)i(on)g(the)g(p)r (osition)e(\010\()p Fr(q)s(;)14 b(p)p Fy(\))36 b(=)e Fr(q)s Fy(\),)h(then)f(F)-7 b(orm)n(ula)32 b(\(2.9\))i(reduces)f(to)h (the)h(m)n(uc)n(h)d(more)83 2643 y(sympathetic)26 b(equation)9 b(:)1182 2684 y Fm(\010)1230 2751 y Fy(\010)19 b Fq(\001)f(r)p Fr(H)1487 2763 y FA(0)1524 2751 y Fr(;)c(H)1635 2684 y Fm(\011)1706 2751 y Fy(=)23 b Fq(jr)p Fr(H)1954 2763 y FA(0)1991 2751 y Fq(j)2014 2717 y FA(2)2070 2751 y Fy(+)18 b(\010)g Fq(\001)2273 2684 y Fm(\010)2321 2751 y Fq(r)p Fr(H)2458 2763 y FA(0)2495 2751 y Fr(;)c(V)2598 2684 y Fm(\011)2646 2751 y Fr(:)83 2905 y Fy(Lemma)30 b(2.10)f(b)r(elo)n(w)h(pro)n(vides)e(conditions)f(under)i(whic)n(h)g (one)h(reco)n(v)n(ers)e(this)i(simpli\014ed)e(situation.)g(F)-7 b(or)30 b(it,)g(w)n(e)83 3013 y(need)d(the)g(follo)n(wing)9 b(:)83 3183 y Ft(Assumption)30 b(2.9.)106 b Fz(\(i\))43 b Fr(H)1078 3195 y FA(0)1151 3183 y Fz(is)36 b(b)-5 b(ounde)g(d)5 b(ly)36 b(pr)-5 b(eserve)g(d)36 b(by)g(the)f(\015ow)h(of)g Fr(H)6 b Fz(,)36 b Fy(i.e.)g Fz(for)f(e)-5 b(ach)36 b Fr(m)30 b Fq(2)g Fr(M)45 b Fz(ther)-5 b(e)291 3291 y(exists)32 b(a)g(c)-5 b(onstant)31 b Ff(c)1016 3303 y Fo(m)1103 3291 y Fq(\025)23 b Fy(0)31 b Fz(such)h(that)1649 3220 y Fm(\014)1649 3270 y(\014)1677 3291 y Fr(H)1745 3303 y FA(0)1782 3223 y Fm(\000)1820 3291 y Fr(')1875 3303 y Fo(t)1903 3291 y Fy(\()p Fr(m)p Fy(\))2042 3223 y Fm(\001)2100 3291 y Fq(\000)18 b Fr(H)2251 3303 y FA(0)2287 3291 y Fy(\()p Fr(m)p Fy(\))2426 3220 y Fm(\014)2426 3270 y(\014)2478 3291 y Fq(\024)23 b Ff(c)2615 3303 y Fo(m)2711 3291 y Fz(for)31 b(al)5 b(l)30 b Fr(t)23 b Fq(2)h Fp(R)q Fz(.)122 3463 y(\(ii\))43 b(Ther)-5 b(e)34 b(exists)h(an)f(incr)-5 b(e)g(asing)36 b(function)f Fr(\032)27 b Fy(:)h([0)p Fr(;)14 b Fq(1)p Fy(\))27 b Fq(!)g Fy([0)p Fr(;)14 b Fq(1)p Fy(\))34 b Fz(with)h Fy(lim)2742 3475 y Fo(R)p Fl(!1)2940 3463 y Fr(\032)p Fy(\()p Fr(R)q Fy(\))27 b(=)g(+)p Fq(1)34 b Fz(such)h(that)291 3571 y Fq(j)p Fy(\()p Fq(r)p Fr(H)483 3583 y FA(0)520 3571 y Fy(\)\()p Fr(m)p Fy(\))p Fq(j)24 b(\025)f Fr(R)32 b Fz(implies)g Fq(j)p Fr(H)1317 3583 y FA(0)1354 3571 y Fy(\()p Fr(m)p Fy(\))p Fq(j)24 b(\025)f Fr(\032)p Fy(\()p Fr(R)q Fy(\))32 b Fz(for)f(al)5 b(l)30 b Fr(m)24 b Fq(2)f Fr(M)41 b Fz(and)31 b Fr(R)24 b Fq(\025)e Fy(0)p Fz(.)249 3740 y Fy(If)k Fr(V)43 b Fy(is)25 b(b)r(ounded,)d(then)j(\(i\))g(holds)f(automatically)-7 b(,)22 b(since)2120 3670 y Fm(\014)2120 3719 y(\014)2148 3740 y Fr(H)2216 3752 y FA(0)2252 3673 y Fm(\000)2290 3740 y Fr(')2345 3752 y Fo(t)2373 3740 y Fy(\()p Fr(m)p Fy(\))2512 3673 y Fm(\001)2565 3740 y Fq(\000)14 b Fr(H)2712 3752 y FA(0)2748 3740 y Fy(\()p Fr(m)p Fy(\))2887 3670 y Fm(\014)2887 3719 y(\014)2939 3740 y Fy(=)3027 3670 y Fm(\014)3027 3719 y(\014)3054 3740 y Fr(V)3120 3673 y Fm(\000)3158 3740 y Fr(')3213 3752 y Fo(t)3241 3740 y Fy(\()p Fr(m)p Fy(\))3380 3673 y Fm(\001)3433 3740 y Fq(\000)g Fr(V)j Fy(\()p Fr(m)p Fy(\))3716 3670 y Fm(\014)3716 3719 y(\014)3745 3740 y Fr(:)83 3848 y Fy(On)27 b(the)g(other)g(hand,)f (\(i\))i(also)e(holds)g(for)h(some)g(un)n(b)r(ounded)d Fr(V)18 b Fy('s.)28 b(F)-7 b(or)26 b(example,)g(if)i Fr(H)2898 3860 y FA(0)2958 3848 y Fq(\025)23 b Fy(0)k(and)g Fr(V)45 b Fy(is)27 b(b)r(ounded)83 3956 y(from)g(b)r(elo)n(w)f(b)n(y)i Fr(V)681 3968 y FA(0)740 3956 y Fq(2)c Fp(R)q Fy(,)34 b(then)27 b(\(i\))g(is)g(v)n(eri\014ed)g(with)g Ff(c)1851 3968 y Fo(m)1938 3956 y Fy(=)c Fr(H)2094 3968 y FA(0)2131 3956 y Fy(\()p Fr(m)p Fy(\))c(+)f Fr(H)6 b Fy(\()p Fr(m)p Fy(\))20 b Fq(\000)e Fr(V)2736 3968 y FA(0)2772 3956 y Fy(.)83 4126 y Ft(Lemma)45 b(2.10)e Fy(\(Completeness)37 b(of)i(w)n(a)n(v)n(e)f(maps\))p Ft(.)47 b Fz(L)-5 b(et)43 b Fr(H)2067 4138 y FA(0)2147 4126 y Fz(and)g Fr(H)49 b Fz(satisfy)44 b(Assumptions)h(2.1)d(and)h(2.7.)83 4234 y(Supp)-5 b(ose)37 b(either)e(that)g Fq(f)p Fy(\010)p Fr(;)14 b(V)j Fq(g)31 b Fy(=)f(0)36 b Fz(or)g(that)f(Assumption)j(2.9)e (holds.)f(Assume)k(ther)-5 b(e)35 b(exists)j Fr(\016)c(>)d Fy(0)36 b Fz(such)83 4342 y(that)30 b Fq(f)p Fy(\010)18 b Fq(\001)h(r)p Fr(H)564 4354 y FA(0)601 4342 y Fr(;)14 b(H)6 b Fq(g)p Fy(\()p Fr(m)p Fy(\))23 b Fr(>)g(\016)35 b Fz(for)d(al)5 b(l)30 b Fr(m)24 b Fq(2)f Fr(M)9 b Fz(.)32 b(Then,)g Fr(B)1997 4304 y Fl(\006)1993 4364 y FA(\010)2075 4342 y Fy(=)22 b Fp(?)32 b Fz(and)f(the)g(maps)332 4527 y Fr(W)409 4539 y Fl(\006)488 4527 y Fy(:)23 b Fr(M)k Fq(n)18 b Fj(Crit)p Fy(\()p Fr(H)931 4539 y FA(0)969 4527 y Fr(;)c Fy(\010\))23 b Fq(!)g Fr(M)100 b Fz(and)91 b Fr(S)27 b Fy(=)c Fr(W)1894 4489 y Fl(\000)p FA(1)1882 4545 y(+)2001 4527 y Fq(\016)18 b Fr(W)2138 4539 y Fl(\000)2216 4527 y Fy(:)24 b Fr(M)j Fq(n)18 b Fj(Crit)p Fy(\()p Fr(H)2660 4539 y FA(0)2697 4527 y Fr(;)c Fy(\010\))23 b Fq(!)g Fr(M)k Fq(n)18 b Fj(Crit)p Fy(\()p Fr(H)3352 4539 y FA(0)3390 4527 y Fr(;)c Fy(\010\))83 4712 y Fz(ar)-5 b(e)37 b(wel)5 b(l)36 b(de\014ne)-5 b(d)37 b(symple)-5 b(ctomorphisms.)38 b(In)g(p)-5 b(articular,)35 b(Assumptions)k(2.6\(i\)-\(iii\))g(hold)c (on)i(the)g(sub-)83 4820 y(manifold)31 b Fr(M)d Fq(n)18 b Fj(Crit)p Fy(\()p Fr(H)850 4832 y FA(0)887 4820 y Fr(;)c Fy(\010\))p Fz(.)83 4989 y(Pr)-5 b(o)g(of.)42 b Fy(If)30 b Fq(f)p Fy(\010)p Fr(;)14 b(V)j Fq(g)25 b Fy(=)h(0,)i(then)h(the)g (claim)f(follo)n(ws)f(directly)h(from)h(the)g(observ)-5 b(ations)26 b(made)i(b)r(efore)h(Assumption)83 5097 y(2.9.)249 5205 y(So,)24 b(supp)r(ose)e(that)i(Assumption)e(2.9)h(holds.)g(Since)g Fq(f)p Fy(\010)12 b Fq(\001)g(r)p Fr(H)2197 5217 y FA(0)2233 5205 y Fr(;)i(H)6 b Fq(g)22 b Fr(>)g(\016)t Fy(,)j(w)n(e)f(ha)n(v)n(e) 2913 5172 y FA(d)p 2901 5186 62 4 v 2901 5234 a(d)p Fo(t)2972 5205 y Fy(\(\010)12 b Fq(\001)g(r)p Fr(H)3248 5217 y FA(0)3285 5205 y Fy(\))3317 5138 y Fm(\000)3355 5205 y Fr(')3410 5217 y Fo(t)3438 5205 y Fy(\()p Fr(m)p Fy(\))3577 5138 y Fm(\001)3639 5205 y Fr(>)23 b(\016)83 5313 y Fy(for)29 b(all)f Fr(t)e Fq(2)g Fp(R)37 b Fy(and)28 b Fr(m)e Fq(2)h Fr(M)9 b Fy(.)29 b(In)g(particular,)d(there)j(exist)g(for)g(all)f Fr(m)e Fq(2)h Fr(M)38 b Fy(and)28 b(all)g Fr(R)e Fq(\025)g Fy(0)j(times)g Fr(t)3301 5325 y Fl(\006)3382 5313 y Fq(2)d Fp(R)3517 5325 y Fl(\006)3608 5313 y Fy(suc)n(h)83 5421 y(that)e(either)493 5350 y Fm(\014)493 5400 y(\014)520 5421 y Fy(\010)580 5354 y Fm(\000)618 5421 y Fr(')673 5433 y Fo(t)697 5441 y Fe(\006)748 5421 y Fy(\()p Fr(m)p Fy(\))887 5354 y Fm(\001)926 5350 y(\014)926 5400 y(\014)977 5421 y Fr(>)f(R)i Fy(or)1251 5350 y Fm(\014)1251 5400 y(\014)1279 5421 y Fy(\()p Fq(r)p Fr(H)1448 5433 y FA(0)1485 5421 y Fy(\))1517 5354 y Fm(\000)1556 5421 y Fr(')1611 5433 y Fo(t)1635 5441 y Fe(\006)1686 5421 y Fy(\()p Fr(m)p Fy(\))1825 5354 y Fm(\001)1864 5350 y(\014)1864 5400 y(\014)1914 5421 y Fr(>)e(R)q Fy(.)h(Ho)n(w)n(ev)n(er,)g(w)n(e)g(kno)n (w)g(b)n(y)h(Assumption)d(2.9\(ii\))h(that)1905 5670 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 83 307 a Fy(if)157 237 y Fm(\014)157 286 y(\014)184 307 y Fy(\()p Fq(r)p Fr(H)353 319 y FA(0)391 307 y Fy(\))423 240 y Fm(\000)461 307 y Fr(')516 319 y Fo(t)540 327 y Fe(\006)591 307 y Fy(\()p Fr(m)p Fy(\))730 240 y Fm(\001)769 237 y(\014)769 286 y(\014)820 307 y Fr(>)22 b(R)j Fy(then)1182 237 y Fm(\014)1182 286 y(\014)1210 307 y Fr(H)1278 319 y FA(0)1315 240 y Fm(\000)1353 307 y Fr(')1408 319 y Fo(t)1432 327 y Fe(\006)1483 307 y Fy(\()p Fr(m)p Fy(\))1622 240 y Fm(\001)1661 237 y(\014)1661 286 y(\014)1712 307 y Fr(>)d(\032)p Fy(\()p Fr(R)q Fy(\).)j(W)-7 b(e)25 b(also)e(kno)n(w)h(from)g(Assumption)e(2.9\(i\))i(that)h(there) 83 415 y(exists)i Ff(c)362 427 y Fo(m)449 415 y Fq(\025)c Fy(0)k(suc)n(h)f(that)974 345 y Fm(\014)974 394 y(\014)1001 415 y Fr(H)1069 427 y FA(0)1106 348 y Fm(\000)1144 415 y Fr(')1199 427 y Fo(t)1223 435 y Fe(\006)1274 415 y Fy(\()p Fr(m)p Fy(\))1413 348 y Fm(\001)1471 415 y Fq(\000)18 b Fr(H)1622 427 y FA(0)1658 415 y Fy(\()p Fr(m)p Fy(\))1797 345 y Fm(\014)1797 394 y(\014)1849 415 y Fq(\024)23 b Ff(c)1986 427 y Fo(m)2050 415 y Fy(.)28 b(This)e(implies)f(that)557 577 y Fr(\032)p Fy(\()p Fr(R)q Fy(\))e Fr(<)839 506 y Fm(\014)839 556 y(\014)867 577 y Fr(H)935 589 y FA(0)971 509 y Fm(\000)1009 577 y Fr(')1064 589 y Fo(t)1088 597 y Fe(\006)1139 577 y Fy(\()p Fr(m)p Fy(\))1278 509 y Fm(\001)1317 506 y(\014)1317 556 y(\014)1368 577 y Fy(=)1456 506 y Fm(\014)1456 556 y(\014)1483 577 y Fr(H)1551 589 y FA(0)1588 509 y Fm(\000)1626 577 y Fr(')1681 589 y Fo(t)1705 597 y Fe(\006)1756 577 y Fy(\()p Fr(m)p Fy(\))1895 509 y Fm(\001)1953 577 y Fq(\000)18 b Fr(H)2104 589 y FA(0)2141 577 y Fy(\()p Fr(m)p Fy(\))h(+)f Fr(H)2450 589 y FA(0)2487 577 y Fy(\()p Fr(m)p Fy(\))2626 506 y Fm(\014)2626 556 y(\014)2678 577 y Fq(\024)k Ff(c)2814 589 y Fo(m)2897 577 y Fy(+)c Fq(j)p Fr(H)3071 589 y FA(0)3108 577 y Fy(\()p Fr(m)p Fy(\))p Fq(j)p Fr(;)83 738 y Fy(whic)n(h)j(is)g(a) g(con)n(tradiction)d(since)j Fr(\032)p Fy(\()p Fr(R)q Fy(\))h(is)f(not)g(b)r(ounded)f(for)h Fr(R)h Fy(big)f(enough.)e(Th)n (us,)i(w)n(e)h(m)n(ust)f(ha)n(v)n(e)f(the)i(follo)n(wing:)83 846 y(for)33 b(all)e Fr(m)i Fq(2)g Fr(M)42 b Fy(and)32 b(all)f Fr(R)i Fq(\025)f Fy(0)h(suc)n(h)e(that)i Fr(\032)p Fy(\()p Fr(R)q Fy(\))g Fq(\025)e Fr(C)1945 858 y Fo(m)2032 846 y Fy(+)22 b Fr(H)2187 858 y FA(0)2223 846 y Fy(\()p Fr(m)p Fy(\),)35 b(there)d(exist)h(times)f Fr(t)3099 858 y Fl(\006)3187 846 y Fq(2)g Fp(R)3328 858 y Fl(\006)3423 846 y Fy(suc)n(h)f(that)83 884 y Fm(\014)83 933 y(\014)111 954 y Fy(\010)171 887 y Fm(\000)209 954 y Fr(')264 966 y Fo(t)288 974 y Fe(\006)339 954 y Fy(\()p Fr(m)p Fy(\))478 887 y Fm(\001)517 884 y(\014)517 933 y(\014)575 954 y Fr(>)e(R)q Fy(.)j(In)f(particular,)e(w)n(e)j(ha)n(v)n(e)f(that)h(lim) 1929 967 y Fl(j)p Fo(t)p Fl(j!1)2140 887 y Fm(\000)2178 954 y Fq(j)p Fy(\010)p Fq(j)2284 924 y FA(2)2342 954 y Fq(\016)21 b Fr(')2460 966 y Fo(t)2488 887 y Fm(\001)2526 954 y Fy(\()p Fr(m)p Fy(\))32 b(=)e(+)p Fq(1)h Fy(for)h(all)e Fr(m)h Fq(2)g Fr(M)9 b Fy(,)31 b(whic)n(h)83 1062 y(implies)25 b(the)j(claim.)p 3708 1062 4 57 v 3712 1009 50 4 v 3712 1062 V 3761 1062 4 57 v 249 1231 a(Let)h Fr(U)k Fq(\032)24 b Fp(R)36 b Fy(b)r(e)28 b(an)g(op)r(en)g(set)g(suc)n(h)f(that)i Fr(H)1672 1193 y Fl(\000)p FA(1)1666 1251 y(0)1760 1231 y Fy(\()p Fr(U)9 b Fy(\))19 b Fq(\\)g Fj(Crit)p Fy(\()p Fr(H)2213 1243 y FA(0)2250 1231 y Fr(;)14 b Fy(\010\))25 b(=)f Fp(?)p Fy(.)29 b(Then,)e Fr(H)2925 1193 y Fl(\000)p FA(1)2919 1251 y(0)3013 1231 y Fy(\()p Fr(U)9 b Fy(\))29 b(is)f(a)g(submanifold)83 1338 y(of)d Fr(M)33 b Fy(preserv)n(ed)23 b(b)n(y)h(the)h(\015o)n(w)g(of)g Fr(H)1241 1350 y FA(0)1277 1338 y Fy(.)g(But)g(in)f(general,)f Fr(H)1963 1301 y Fl(\000)p FA(1)1957 1359 y(0)2051 1338 y Fy(\()p Fr(U)9 b Fy(\))25 b(is)f(not)g(preserv)n(ed)f(b)n(y)h(the)h(\015o)n(w)g(of)f Fr(H)6 b Fy(.)25 b(Ho)n(w)n(ev)n(er,)83 1446 y(if)g(Theorem)d(2.8)i (applies,)f(one)g(has)h Fr(H)18 b Fq(\016)13 b Fr(W)1447 1458 y Fl(\006)1525 1446 y Fy(=)23 b Fr(H)1681 1458 y FA(0)1742 1446 y Fy(and)h Fr(W)1978 1458 y Fl(\006)2034 1379 y Fm(\000)2072 1446 y Fr(H)2146 1409 y Fl(\000)p FA(1)2140 1467 y(0)2234 1446 y Fy(\()p Fr(U)9 b Fy(\))2363 1379 y Fm(\001)2424 1446 y Fy(=)22 b Fr(H)2585 1416 y Fl(\000)p FA(1)2673 1446 y Fy(\()p Fr(U)9 b Fy(\))k Fq(n)g Fr(B)2937 1409 y Fl(\006)2933 1469 y FA(\010)3015 1446 y Fy(is)24 b(also)f(a)i(submanifold)83 1554 y(of)j Fr(M)9 b Fy(.)27 b(Therefore,)f(the)h(follo)n(wing)e(corollary)g(is)i(a)g (consequence)e(of)j(Theorem)e(2.8)g(and)h(Lemma)g(2.10.)83 1705 y Ft(Corollary)k(2.11.)38 b Fz(L)-5 b(et)29 b Fr(H)962 1717 y FA(0)1029 1705 y Fz(and)g Fr(H)36 b Fz(satisfy)31 b(Assumptions)g(2.1)f(and)f(2.7.)h(Supp)-5 b(ose)30 b(either)f(that)f Fq(f)p Fy(\010)p Fr(;)14 b(V)j Fq(g)23 b Fy(=)g(0)83 1813 y Fz(or)31 b(that)f(Assumption)j(2.9)e(holds.)h(L)-5 b(et)31 b Fr(U)g Fq(\032)23 b Fp(R)39 b Fz(b)-5 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b(fr)-5 b(om)26 b Fp(R)2596 1033 y Fo(n)2655 1063 y Fq(\002)7 b Fp(R)2781 1033 y Fo(n)2840 1063 y Fq(n)g Fy(\()p Fq(r)p Fr(h)p Fy(\))3071 1033 y Fl(\000)p FA(1)3160 1063 y Fy(\()p Fq(f)p Fy(0)p Fq(g)p Fy(\))25 b Fz(to)g Fr(M)16 b Fq(n)7 b Fr(B)3687 1025 y Fl(\006)3683 1085 y FA(\010)3741 1063 y Fz(.)83 1171 y(F)-7 b(urthermor)i(e,)30 b(the)h(c)-5 b(ommutation)32 b Fq(f)p Fy(\010)p Fr(;)14 b(V)j Fq(g)23 b Fy(=)f(0)32 b Fz(implies)g(that)122 1294 y Fm(\010)170 1361 y Fy(\010)19 b Fq(\001)f(r)p Fr(H)427 1373 y FA(0)464 1361 y Fr(;)c(H)575 1294 y Fm(\011)623 1361 y Fy(\()p Fr(q)s(;)g(p)p Fy(\))24 b(=)f Fq(j)p Fy(\()p Fq(r)p Fr(h)p Fy(\)\()p Fr(p)p Fy(\))p Fq(j)1254 1327 y FA(2)1311 1361 y Fq(\000)18 b Fr(q)1434 1327 y Fd(T)1499 1361 y Fq(\001)h Fy(\()p Fj(Hess)13 b Fr(h)p Fy(\)\()p Fr(p)p Fy(\))c(\()p Fq(r)p Fr(V)20 b Fy(\)\()p Fr(q)s Fy(\))k Fq(\025)f(j)p Fy(\()p Fq(r)p Fr(h)p Fy(\)\()p Fr(p)p Fy(\))p Fq(j)2695 1327 y FA(2)2752 1361 y Fq(\000)18 b Fr(n)9 b Ff(c)2946 1373 y Fo(V)3064 1361 y Fy(max)3016 1416 y FA(1)p Fl(\024)p Fo(i;j)s Fl(\024)p Fo(n)3280 1291 y Fm(\014)3280 1340 y(\014)3307 1361 y Fy(\()p Fr(@)3382 1373 y Fo(i)3411 1361 y Fr(@)3454 1373 y Fo(j)3489 1361 y Fr(h)p Fy(\)\()p Fr(p)p Fy(\))3677 1291 y Fm(\014)3677 1340 y(\014)3706 1361 y Fr(;)83 1590 y Fz(wher)-5 b(e)33 b Fy(\()p Fj(Hess)14 b Fr(h)p Fy(\)\()p Fr(p)p Fy(\))34 b Fz(is)f(the)f(Hessian)j(matrix)d (of)h Fr(h)g Fz(at)g Fr(p)f Fz(and)h Ff(c)2228 1602 y Fo(V)2309 1590 y Fy(:=)25 b(sup)2548 1610 y Fo(q)r Fl(2)p Fk(R)2671 1594 y Fi(n)2723 1590 y Fq(j)p Fr(q)s Fq(j)20 b(\001)f(j)p Fy(\()p Fq(r)p Fr(V)f Fy(\)\()p Fr(q)s Fy(\))p Fq(j)p Fz(.)35 b(Ther)-5 b(efor)g(e,)33 b(if)83 1698 y(ther)-5 b(e)31 b(exist)h(c)-5 b(ontinuous)32 b(functions)h Fr(g)1389 1710 y FA(1)1426 1698 y Fr(;)14 b(g)1503 1710 y FA(2)1564 1698 y Fy(:)23 b Fj(Ran)o Fy(\()p Fr(h)p Fy(\))h Fq(!)f Fy([0)p Fr(;)14 b Fq(1)p Fy(\))32 b Fz(such)g(that)498 1889 y Fq(j)p Fy(\()p Fq(r)p Fr(h)p Fy(\)\()p Fr(p)p Fy(\))p Fq(j)833 1854 y FA(2)895 1889 y Fq(\025)22 b Fr(g)1022 1901 y FA(1)1060 1821 y Fm(\000)1098 1889 y Fr(h)p Fy(\()p Fr(p)p Fy(\))1254 1821 y Fm(\001)1384 1889 y Fz(and)153 b Fy(max)1631 1944 y FA(1)p Fl(\024)p Fo(i;j)s Fl(\024)p Fo(n)1895 1818 y Fm(\014)1895 1868 y(\014)1922 1889 y Fy(\()p Fr(@)1997 1901 y Fo(i)2026 1889 y Fr(@)2069 1901 y Fo(j)2104 1889 y Fr(h)p Fy(\)\()p Fr(p)p Fy(\))2292 1818 y Fm(\014)2292 1868 y(\014)2344 1889 y Fq(\024)23 b Fr(g)2472 1901 y FA(2)2509 1821 y Fm(\000)2547 1889 y Fr(h)p Fy(\()p Fr(p)p Fy(\))2703 1821 y Fm(\001)2833 1889 y Fz(for)32 b(al)5 b(l)30 b Fr(p)23 b Fq(2)g Fp(R)3301 1854 y Fo(n)83 2135 y Fz(\(which)32 b(o)-5 b(c)g(curs)33 b(for)e(instanc)-5 b(e)33 b(when)f Fr(h)p Fy(\()p Fr(p)p Fy(\))24 b(=)e Fq(j)p Fr(p)p Fq(j)1718 2105 y FA(2)1756 2135 y Fr(=)p Fy(2)30 b Fz(or)h Fr(h)p Fy(\()p Fr(p)p Fy(\))24 b(=)2253 2062 y Fm(p)p 2336 2062 223 4 v 73 x Fy(1)18 b(+)g Fr(p)2522 2111 y FA(2)2559 2135 y Fz(\),)32 b(then)g(the)f(op)-5 b(en)32 b(sets)1053 2325 y Fr(U)1109 2337 y Fo(\016)1168 2325 y Fy(:=)1279 2258 y Fm(\010)1327 2325 y Fr(x)23 b Fq(2)g Fp(R)31 b Fq(j)23 b Fr(g)1646 2337 y FA(1)1683 2325 y Fy(\()p Fr(x)p Fy(\))c Fq(\000)f Fr(n)9 b Ff(c)2007 2337 y Fo(V)2064 2325 y Fr(g)2104 2337 y FA(2)2141 2325 y Fy(\()p Fr(x)p Fy(\))23 b Fr(>)g(\016)2405 2258 y Fm(\011)2453 2325 y Fr(;)105 b(\016)28 b(>)22 b Fy(0)p Fr(;)83 2516 y Fz(satisfy)33 b(the)e(c)-5 b(onditions)32 b(\(i\))h(and)e(\(ii\))i(of)f(Cor)-5 b(ol)5 b(lary)30 b(2.11.)h(Thus,)h(the)f(maps)584 2707 y Fr(W)661 2719 y Fl(\006)740 2707 y Fy(:)23 b Fr(H)860 2669 y Fl(\000)p FA(1)854 2727 y(0)948 2707 y Fy(\()p Fr(U)1036 2719 y Fo(\016)1072 2707 y Fy(\))h Fq(!)f Fr(H)1308 2672 y Fl(\000)p FA(1)1396 2707 y Fy(\()p Fr(U)1484 2719 y Fo(\016)1520 2707 y Fy(\))92 b Fz(and)f Fr(S)27 b Fy(=)c Fr(W)2132 2669 y Fl(\000)p FA(1)2120 2725 y(+)2239 2707 y Fq(\016)18 b Fr(W)2376 2719 y Fl(\000)2454 2707 y Fy(:)23 b Fr(H)2574 2669 y Fl(\000)p FA(1)2568 2727 y(0)2662 2707 y Fy(\()p Fr(U)2750 2719 y Fo(\016)2787 2707 y Fy(\))g Fq(!)g Fr(H)3022 2669 y Fl(\000)p FA(1)3016 2727 y(0)3110 2707 y Fy(\()p Fr(U)3198 2719 y Fo(\016)3234 2707 y Fy(\))83 2897 y Fz(ar)-5 b(e)32 b(wel)5 b(l)31 b(de\014ne)-5 b(d)31 b(symple)-5 b(ctomorphisms,)33 b(and)e(Assumption)i(2.6)f(holds)e(on)i (the)f(submanifold)h Fr(H)3458 2860 y Fl(\000)p FA(1)3452 2917 y(0)3546 2897 y Fy(\()p Fr(U)3634 2909 y Fo(\016)3670 2897 y Fy(\))p Fz(.)83 3071 y Ft(Example)h(2.14)e Fy(\(P)n(article)d (in)f(a)i(tub)r(e,)f(con)n(tin)n(ued\))p Ft(.)39 b Fz(L)-5 b(et)33 b Fr(V)43 b Fq(2)25 b Fj(C)2240 3041 y Fl(1)2240 3092 y FA(c)2309 3071 y Fy(\(\012;)14 b Fp(R)r Fy(\))39 b Fz(b)-5 b(e)33 b(a)g Fj(C)2806 3041 y Fl(1)2908 3071 y Fz(function)g(with)g(c)-5 b(omp)g(act)83 3179 y(supp)g(ort)31 b(and)h(let)f Fr(H)e Fq(2)24 b Fj(C)927 3149 y Fl(1)997 3179 y Fy(\()p Fr(M)9 b Fy(;)14 b Fp(R)q Fy(\))38 b Fz(b)-5 b(e)33 b(the)e(p)-5 b(erturb)g(e)g(d)31 b(Hamiltonian)h(given)h(by)g Fr(H)6 b Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))24 b(:=)f Fr(H)3247 3191 y FA(0)3284 3179 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))19 b(+)f Fr(V)g Fy(\()p Fr(q)s Fy(\))p Fz(.)83 3287 y(Then,)32 b(Assumption)i(2.7)d(is)i(satis\014e)-5 b(d)32 b(and)g(we)h(know)f(fr) -5 b(om)33 b([1)o(,)g(Pr)-5 b(op.)32 b(2.1.20])f(\(onc)-5 b(e)33 b(mor)-5 b(e)32 b(applie)-5 b(d)31 b(with)83 3395 y(the)g(function)40 b Fy(\(2.4\))p Fz(\))32 b(that)f Fr(X)1080 3407 y Fo(H)1173 3395 y Fz(is)h(c)-5 b(omplete.)32 b(So,)g(The)-5 b(or)g(em)32 b(2.8)g(implies)g(that)f(the)g(wave)h(maps) g Fr(W)3505 3407 y Fl(\006)3593 3395 y Fz(exist)83 3503 y(and)27 b(ar)-5 b(e)27 b(symple)-5 b(ctomorphisms)28 b(fr)-5 b(om)27 b Fy(\012)9 b Fq(\002)1571 3436 y Fm(\000)1609 3503 y Fp(R)h Fq(n)f(f)p Fy(0)p Fq(g)g(\002)g Fp(R)1991 3473 y Fo(n)2043 3436 y Fm(\001)2108 3503 y Fz(to)27 b Fr(M)18 b Fq(n)9 b Fr(B)2426 3465 y Fl(\006)2422 3525 y FA(\010)2480 3503 y Fz(.)28 b(Now,)g(we)g(c)-5 b(annot)27 b(apply)f(Cor)-5 b(ol)5 b(lary)83 3611 y(2.11)31 b(to)g(obtain)g(the)g (identity)h(of)g(the)f(r)-5 b(anges)32 b(of)g Fr(W)1845 3623 y Fl(\006)1900 3611 y Fz(,)g(sinc)-5 b(e)34 b(the)d(virial)g (identity)1234 3734 y Fm(\010)1282 3801 y Fy(\010)19 b Fq(\001)f(r)p Fr(H)1539 3813 y FA(0)1576 3801 y Fr(;)c(H)1687 3734 y Fm(\011)1735 3801 y Fy(\()p Fr(q)s(;)g(p)p Fy(\))24 b(=)f Fr(p)2074 3767 y FA(2)2074 3822 y(1)2129 3801 y Fq(\000)18 b Fr(q)2252 3767 y FA(1)2290 3734 y Fm(\000)2328 3801 y Fr(@)2371 3813 y FA(1)2408 3801 y Fr(V)2474 3734 y Fm(\001)2512 3801 y Fy(\()p Fr(q)s Fy(\))83 3992 y Fz(do)-5 b(es)41 b(not)f(involve)g(observables)g(c)-5 b(omp)g(ar)g(able)40 b(with)h(the)e(fr)-5 b(e)g(e)41 b(ener)-5 b(gy)42 b Fr(H)2595 4004 y FA(0)2631 3992 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))p Fz(.)42 b(Inste)-5 b(ad,)41 b(we)g(set)f Fr(K)3587 4004 y Fo(V)3680 3992 y Fy(:=)83 4100 y(sup)209 4120 y Fo(q)r Fl(2)p FA(\012)350 4029 y Fm(\014)350 4079 y(\014)378 4100 y Fr(q)418 4070 y FA(1)455 4032 y Fm(\000)493 4100 y Fr(@)536 4112 y FA(1)574 4100 y Fr(V)640 4032 y Fm(\001)678 4100 y Fy(\()p Fr(q)s Fy(\))782 4029 y Fm(\014)782 4079 y(\014)842 4100 y Fz(and)32 b(de\014ne)f(the)h(op)-5 b(en)31 b(set)916 4290 y Fr(M)995 4302 y Fo(\016)1055 4290 y Fy(:=)1165 4223 y Fm(\010)1214 4290 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))23 b Fq(2)h Fr(M)31 b Fq(j)24 b(j)p Fy(\()p Fq(r)p Fr(H)1849 4302 y FA(0)1886 4290 y Fy(\)\()p Fr(q)s(;)14 b(p)p Fy(\))p Fq(j)2125 4256 y FA(2)2186 4290 y Fr(>)23 b(K)2344 4302 y Fo(V)2418 4290 y Fy(+)18 b Fr(\016)2542 4223 y Fm(\011)2590 4290 y Fr(;)106 b(\016)27 b(>)22 b Fy(0)p Fr(;)83 4481 y Fz(Then,)31 b Fr(')397 4450 y FA(0)397 4502 y Fo(t)465 4481 y Fz(is)g(a)g(di\013e)-5 b(omorphism)31 b(on)f Fr(M)1466 4493 y Fo(\016)1533 4481 y Fz(for)h(e)-5 b(ach)30 b Fr(t)23 b Fq(2)h Fp(R)38 b Fz(and)2269 4410 y Fm(\014)2269 4460 y(\014)2297 4414 y(\010)2345 4481 y Fy(\010)16 b Fq(\001)h(r)p Fr(H)2598 4493 y FA(0)2635 4481 y Fr(;)d(H)2746 4414 y Fm(\011)2794 4410 y(\014)2794 4460 y(\014)2845 4481 y Fr(>)22 b(\016)35 b Fz(on)c Fr(M)3210 4493 y Fo(\016)3246 4481 y Fz(.)g(So,)g(for)f(e)-5 b(ach)83 4589 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))24 b Fq(2)f Fr(M)448 4601 y Fo(\016)516 4589 y Fz(ther)-5 b(e)31 b(exists)i Fr(T)h(>)23 b Fy(0)31 b Fz(such)h(that)e Fr(W)1688 4601 y Fl(\000)1744 4589 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))23 b(=)2039 4521 y Fm(\000)2077 4589 y Fr(')2132 4601 y Fo(T)2201 4589 y Fq(\016)18 b Fr(')2316 4558 y FA(0)2316 4611 y Fl(\000)p Fo(T)2418 4521 y Fm(\001)2456 4589 y Fy(\()p Fr(q)s(;)c(p)p Fy(\))33 b Fz(and)752 4779 y Fy(lim)714 4837 y Fl(j)p Fo(t)p Fl(j!1)921 4712 y Fm(\000)959 4779 y Fq(j)p Fy(\010)p Fq(j)1065 4745 y FA(2)1121 4779 y Fq(\016)18 b Fr(')1236 4791 y Fo(t)1282 4779 y Fq(\016)g Fr(W)1419 4791 y Fl(\000)1475 4712 y Fm(\001)1513 4779 y Fy(\()p Fr(q)s(;)c(p)p Fy(\))23 b(=)71 b(lim)1808 4837 y Fl(j)p Fo(t)1851 4820 y Fe(0)1871 4837 y Fl(j!1)2034 4712 y Fm(\000)2072 4779 y Fq(j)p Fy(\010)p Fq(j)2178 4745 y FA(2)2234 4779 y Fq(\016)18 b Fr(')2349 4791 y Fo(t)2373 4775 y Fe(0)2416 4779 y Fq(\016)g Fr(')2531 4745 y FA(0)2531 4800 y Fl(\000)p Fo(T)2633 4712 y Fm(\001)2671 4779 y Fy(\()p Fr(q)s(;)c(p)p Fy(\))23 b(=)g(+)p Fq(1)p Fr(:)83 5024 y Fz(This)32 b(implies)h(that)d Fr(W)853 5036 y Fl(\000)908 5024 y Fy(\()p Fr(M)1019 5036 y Fo(\016)1056 5024 y Fy(\))23 b Fq(\032)g Fr(M)k Fq(n)1365 4957 y Fm(\010)1413 5024 y Fr(B)1480 4986 y Fl(\000)1476 5046 y FA(\010)1554 5024 y Fq([)19 b Fr(B)1695 4986 y FA(+)1691 5046 y(\010)1749 4957 y Fm(\011)1798 5024 y Fz(,)32 b(and)f(thus)1158 5214 y Fr(S)c Fy(:=)c Fr(W)1436 5177 y Fl(\000)p FA(1)1424 5233 y(+)1543 5214 y Fq(\016)18 b Fr(W)1680 5226 y Fl(\000)1758 5214 y Fy(:)24 b Fr(M)1884 5226 y Fo(\016)1943 5214 y Fq(!)2049 5147 y Fm(\000)2087 5214 y Fr(W)2176 5177 y Fl(\000)p FA(1)2164 5233 y(+)2283 5214 y Fq(\016)18 b Fr(W)2420 5226 y Fl(\000)2475 5147 y Fm(\001)2513 5214 y Fy(\()p Fr(M)2624 5226 y Fo(\016)2661 5214 y Fy(\))83 5405 y Fz(is)32 b(a)g(wel)5 b(l)31 b(de\014ne)-5 b(d)32 b(symple)-5 b(ctomorphism.)1884 5670 y Fy(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 83 307 a Ft(Example)45 b(2.15)g Fy(\(P)n(oincar)n(\023)-39 b(e)36 b(ball,)i(con)n(tin)n(ued\))p Ft(.)46 b Fz(L)-5 b(et)43 b Fr(V)62 b Fq(2)43 b Fj(C)2175 277 y Fl(1)2245 307 y Fy(\()2287 289 y(\027)2277 307 y Fr(B)2340 319 y FA(1)2377 307 y Fy(;)14 b Fp(R)q Fy(\))50 b Fz(with)44 b Fy(supp)n(\()p Fr(V)18 b Fy(\))44 b Fq(\032)f Fr(B)3273 319 y Fo(R)3322 327 y Fi(V)3417 307 y Fz(for)h(some)83 415 y Fr(R)146 427 y Fo(V)225 415 y Fq(2)23 b Fy([0)p Fr(;)14 b Fy(1\))24 b Fz(and)g(let)f Fr(H)29 b Fq(2)23 b Fj(C)1014 385 y Fl(1)1084 415 y Fy(\()p Fr(M)9 b Fy(;)14 b Fp(R)q Fy(\))30 b Fz(b)-5 b(e)25 b(the)e(p)-5 b(erturb)g(e)g(d)24 b(Hamiltonian)g(given)h(by)f Fr(H)6 b Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))24 b(:=)f Fr(H)3278 427 y FA(0)3314 415 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))s(+)s Fr(V)19 b Fy(\()p Fr(q)s Fy(\))p Fz(.)83 523 y(Then,)29 b(Assumption)g(2.7)f(is)h (satis\014e)-5 b(d)28 b(\(sinc)-5 b(e)30 b Fq(j)p Fy(\010)p Fq(j)23 b(\024)g Fy(tanh)2039 486 y Fl(\000)p FA(1)2127 523 y Fy(\(2)p Fr(R)2264 535 y Fo(V)2320 523 y Fy(\))28 b Fz(on)g Fy(supp)o(\()p Fr(V)18 b Fy(\))11 b Fq(\002)g Fp(R)2949 493 y Fo(n)3001 523 y Fz(\))28 b(and)g(we)h(know)g(fr)-5 b(om)83 631 y(Gor)g(don)c('s)34 b(The)-5 b(or)g(em)34 b([13)o(])h(that)d Fr(X)1266 643 y Fo(H)1361 631 y Fz(is)i(c)-5 b(omplete.)34 b(So,)g(The)-5 b(or)g(em)34 b(2.8)f(implies)i(that)d(the) h(wave)h(maps)g Fr(W)3712 643 y Fl(\006)83 739 y Fz(exist)c(and)g(ar)-5 b(e)30 b(symple)-5 b(ctomorphisms)30 b(fr)-5 b(om)1654 720 y Fy(\027)1644 739 y Fr(B)1707 751 y FA(1)1759 739 y Fq(\002)14 b Fp(R)1892 709 y Fo(n)1959 739 y Fq(n)h(f)p Fy(0)p Fq(g)28 b Fz(to)h Fr(M)24 b Fq(n)14 b Fr(B)2501 702 y Fl(\006)2497 761 y FA(\010)2556 739 y Fz(.)30 b(Now,)h(dir)-5 b(e)g(ct)30 b(c)-5 b(alculations)29 b(using)83 847 y(the)i(inclusion)i Fy(supp)n(\()p Fr(V)18 b Fy(\))24 b Fq(\032)e Fr(B)1088 859 y Fo(R)1137 867 y Fi(V)1220 847 y Fz(and)32 b(the)f(b)-5 b(ound)32 b Fq(j)p Fy(\010)p Fq(j)23 b(\024)f Fy(tanh)2183 810 y Fl(\000)p FA(1)2271 847 y Fy(\(2)p Fr(R)2408 859 y Fo(V)2463 847 y Fy(\))32 b Fz(on)g Fy(supp)o(\()p Fr(V)18 b Fy(\))h Fq(\002)f Fp(R)3114 817 y Fo(n)3198 847 y Fz(show)32 b(that)944 940 y Fm(\010)993 1007 y Fy(\010)18 b Fq(\001)h(r)p Fr(H)1250 1019 y FA(0)1287 1007 y Fr(;)14 b(H)1398 940 y Fm(\011)1469 1007 y Fy(=)22 b(2)p Fr(H)1666 1019 y FA(0)1721 1007 y Fy(+)1804 930 y Fm(p)p 1887 930 147 4 v 77 x Fy(2)p Fr(H)1997 1019 y FA(0)2047 940 y Fm(\010)2096 1007 y Fy(\010)p Fr(;)14 b(V)2259 940 y Fm(\011)2326 1007 y Fy(+)k(\010)2478 940 y Fm(\010)2526 930 y(p)p 2609 930 V 77 x Fy(2)p Fr(H)2719 1019 y FA(0)2756 1007 y Fr(;)c(V)2858 940 y Fm(\011)83 1167 y Fz(with)272 1097 y Fm(\014)272 1146 y(\014)300 1103 y Fq(p)p 369 1103 V 64 x Fy(2)p Fr(H)479 1179 y FA(0)529 1100 y Fm(\010)578 1167 y Fy(\010)p Fr(;)g(V)741 1100 y Fm(\011)789 1097 y(\014)789 1146 y(\014)844 1167 y Fz(and)1013 1097 y Fm(\014)1013 1146 y(\014)1041 1167 y Fy(\010)1110 1100 y Fm(\010)1159 1103 y Fq(p)p 1228 1103 V 64 x Fy(2)p Fr(H)1338 1179 y FA(0)1374 1167 y Fr(;)g(V)1477 1100 y Fm(\011)1526 1097 y(\014)1526 1146 y(\014)1581 1167 y Fz(b)-5 b(ounde)g(d)27 b(on)h Fr(M)9 b Fz(.)27 b(Ther)-5 b(efor)g(e,)28 b(ther)-5 b(e)26 b(exists)j(a)e(c)-5 b(onstant)27 b Fr(K)3560 1179 y Fo(V)3639 1167 y Fq(\025)c Fy(0)83 1275 y Fz(such)39 b(that)482 1208 y Fm(\010)530 1275 y Fy(\010)23 b Fq(\001)h(r)p Fr(H)797 1287 y FA(0)834 1275 y Fr(;)14 b(H)945 1208 y Fm(\011)1027 1275 y Fq(\025)34 b Fy(2)p Fr(H)1236 1287 y FA(0)1296 1275 y Fq(\000)23 b Fr(K)1454 1287 y Fo(V)1548 1275 y Fz(on)39 b Fr(M)9 b Fz(.)38 b(Mor)-5 b(e)g(over,)39 b(Assumption)h(2.9)e(is)i(satis\014e) -5 b(d)38 b(due)h(to)f(the)83 1383 y(b)-5 b(ounde)g(dness)32 b(of)f Fr(V)49 b Fz(and)31 b(the)f(identity)h Fq(r)p Fr(H)1569 1395 y FA(0)1629 1383 y Fy(=)1716 1319 y Fq(p)p 1785 1319 V 64 x Fy(2)p Fr(H)1895 1395 y FA(0)1932 1383 y Fz(.)g(So,)g(the)f(op)-5 b(en)31 b(intervals)g Fr(U)2919 1395 y Fo(\016)2978 1383 y Fy(:=)3088 1316 y Fm(\000)3136 1349 y Fo(K)3190 1357 y Fi(V)3238 1349 y FA(+)p Fo(\016)p 3136 1364 186 4 v 3212 1412 a FA(2)3331 1383 y Fr(;)14 b Fq(1)3451 1316 y Fm(\001)3489 1383 y Fz(,)32 b Fr(\016)27 b(>)22 b Fy(0)p Fz(,)83 1491 y(satisfy)33 b(the)e(c)-5 b(onditions)32 b(\(i\))h(and)e(\(ii\))i(of)f(Cor)-5 b(ol)5 b(lary)30 b(2.11,)h(and)g(the)g(maps)584 1651 y Fr(W)661 1663 y Fl(\006)740 1651 y Fy(:)23 b Fr(H)860 1614 y Fl(\000)p FA(1)854 1671 y(0)948 1651 y Fy(\()p Fr(U)1036 1663 y Fo(\016)1072 1651 y Fy(\))h Fq(!)f Fr(H)1308 1617 y Fl(\000)p FA(1)1396 1651 y Fy(\()p Fr(U)1484 1663 y Fo(\016)1520 1651 y Fy(\))92 b Fz(and)f Fr(S)27 b Fy(=)c Fr(W)2132 1614 y Fl(\000)p FA(1)2120 1670 y(+)2239 1651 y Fq(\016)18 b Fr(W)2376 1663 y Fl(\000)2454 1651 y Fy(:)23 b Fr(H)2574 1614 y Fl(\000)p FA(1)2568 1671 y(0)2662 1651 y Fy(\()p Fr(U)2750 1663 y Fo(\016)2787 1651 y Fy(\))g Fq(!)g Fr(H)3022 1614 y Fl(\000)p FA(1)3016 1671 y(0)3110 1651 y Fy(\()p Fr(U)3198 1663 y Fo(\016)3234 1651 y Fy(\))83 1811 y Fz(ar)-5 b(e)35 b(wel)5 b(l)35 b(de\014ne)-5 b(d)35 b(symple)-5 b(ctomorphisms.)36 b(In)g(p)-5 b(articular,)33 b(Assumption)k(2.6)e (holds)f(on)h(the)g(submanifold)83 1919 y Fr(H)157 1882 y Fl(\000)p FA(1)151 1939 y(0)245 1919 y Fy(\()p Fr(U)333 1931 y Fo(\016)369 1919 y Fy(\))p Fz(.)83 2196 y Fs(3)135 b(Time)45 b(dela)l(y)h(in)f(classical)h(scattering)g(theory)83 2391 y Fy(In)37 b(this)g(section,)f(w)n(e)h(consider)e(for)i(general)e (scattering)h(systems)g(\()p Fr(M)t(;)14 b(H)2512 2403 y FA(0)2549 2391 y Fr(;)g(H)6 b Fy(\))38 b(the)f(symmetrised)f(time)g (dela)n(y)83 2499 y(de\014ned)26 b(in)h(terms)f(of)h(so)5 b(journ)25 b(times)h(in)h(the)g(dilated)e(regions)g(\010)2176 2469 y Fl(\000)p FA(1)2264 2499 y Fy(\()p Fr(B)2359 2511 y Fo(r)2396 2499 y Fy(\).)i(Under)g(appropriate)c(assumptions,)h(w)n(e) 83 2607 y(pro)n(v)n(e)i(its)h(existence)f(and)h(relate)f(it)h(to)h(the) f(time)g(of)h(arriv)-5 b(al)25 b Fr(T)39 b Fy(de\014ned)26 b(in)h(\(2.3\))o(.)h(When)f(the)g(scattering)f(pro)r(cess)83 2715 y(preserv)n(es)i(the)h(norm)g(of)g(the)h(v)n(elo)r(cit)n(y)f (observ)-5 b(able)27 b Fq(r)p Fr(H)1906 2727 y FA(0)1943 2715 y Fy(,)j(w)n(e)g(sho)n(w)f(that)g(the)h(original)c (\(unsymmetrised\))i(time)83 2823 y(dela)n(y)i(also)f(exists)h(and)g (coincides)e(with)j(the)g(symmetrised)d(time)j(dela)n(y)-7 b(.)29 b(W)-7 b(e)32 b(refer)e(to)h([5)o(,)g(7],)g([6,)g(Sec.)g(I)r (I.B],)g([9,)83 2931 y(Sec.)24 b(4.1],)f([12)o(,)h(Sec.)g(I)r(I)r(I],)h ([20)o(,)f(Ch.)f(10],)g([26)o(,)h(27)o(])g(and)f([34)o(,)h(Sec.)g(3.4]) f(for)g(previous)e(w)n(orks)i(on)g(classical)e(time)i(dela)n(y)83 3039 y(for)k Fr(H)278 3051 y FA(0)315 3039 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))24 b(=)e Fq(j)p Fr(p)p Fq(j)699 3009 y FA(2)736 3039 y Fr(=)p Fy(2)27 b(and)f Fr(H)6 b Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))24 b(=)e Fq(j)p Fr(p)p Fq(j)1466 3009 y FA(2)1504 3039 y Fr(=)p Fy(2)17 b(+)h Fr(V)g Fy(\()p Fr(q)s Fy(\))29 b(on)d Fr(M)32 b Fy(=)23 b Fr(T)2262 3009 y Fl(\003)2298 3039 y Fp(R)2352 3009 y Fo(n)2404 3039 y Fy(.)249 3147 y(So,)k(let)1340 3284 y Fr(T)1401 3250 y FA(0)1389 3305 y Fo(r)1437 3284 y Fy(\()p 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3708 y Fl(\000)2373 3629 y Fm(\001)2411 3696 y Fy(\()p Fr(m)2518 3708 y Fl(\000)2574 3696 y Fy(\))83 3874 y(b)r(e)29 b(the)g(corresp)r(onding)c(so)5 b(journ)26 b(time)j(of)g(the)g(p)r (erturb)r(ed)f(tra)5 b(jectory)27 b(starting)h(from)g Fr(W)2983 3886 y Fl(\000)3039 3874 y Fy(\()p Fr(m)3146 3886 y Fl(\000)3202 3874 y Fy(\))h(at)g(time)g Fr(t)d Fy(=)f(0.)83 3982 y(The)32 b(free)h(so)5 b(journ)30 b(time)i Fr(T)985 3952 y FA(0)973 4002 y Fo(r)1021 3982 y Fy(\()p Fr(m)1128 3994 y Fl(\000)1184 3982 y Fy(\))h(is)f(\014nite)h(for)f(eac) n(h)f Fr(m)1955 3994 y Fl(\000)2043 3982 y Fq(2)h(D)2194 3994 y Fl(\000)2271 3982 y Fq(n)21 b Fj(Crit)p Fy(\()p Fr(H)2565 3994 y FA(0)2602 3982 y Fr(;)14 b Fy(\010\))33 b(due)f(to)h(Equation)e(\(2.1\).)i(The)83 4090 y(\014niteness)25 b(of)i(the)f(p)r(erturb)r(ed)f(so)5 b(journ)24 b(time)i Fr(T)1606 4102 y Fo(r)1642 4090 y Fy(\()p Fr(m)1749 4102 y Fl(\000)1805 4090 y Fy(\))h(is)f(sho)n(wn)f(in)h(Lemma)g(3.1)g(b)r (elo)n(w)f(under)g(some)h(additional)83 4198 y(assumptions.)c(Under)i (these)h(assumptions,)c(one)k(can)f(de\014ne)h(the)g(symmetrised)e (time)h(dela)n(y)g(in)g(\010)3250 4168 y Fl(\000)p FA(1)3338 4198 y Fy(\()p Fr(B)3433 4210 y Fo(r)3470 4198 y Fy(\))i(for)e(the)83 4306 y(scattering)i(system)h(\()p Fr(M)t(;)14 b(H)963 4318 y FA(0)1000 4306 y Fr(;)g(H)6 b Fy(\))27 b(with)h(starting)e(p)r (oin)n(t)g Fr(m)1964 4318 y Fl(\000)2043 4306 y Fy(:)989 4497 y Fr(\034)1026 4509 y Fo(r)1064 4497 y Fy(\()p Fr(m)1171 4509 y Fl(\000)1226 4497 y Fy(\))e(:=)e Fr(T)1441 4509 y Fo(r)1478 4497 y Fy(\()p Fr(m)1585 4509 y Fl(\000)1641 4497 y Fy(\))d Fq(\000)1785 4440 y Fy(1)p 1785 4478 42 4 v 1785 4554 a(2)1845 4429 y Fm(\010)1894 4497 y Fr(T)1955 4462 y FA(0)1943 4517 y Fo(r)1991 4497 y Fy(\()p Fr(m)2098 4509 y Fl(\000)2154 4497 y Fy(\))g(+)2288 4429 y Fm(\000)2326 4497 y Fr(T)2387 4462 y FA(0)2375 4517 y Fo(r)2441 4497 y Fq(\016)f Fr(S)2557 4429 y Fm(\001)2595 4497 y Fy(\()p Fr(m)2702 4509 y Fl(\000)2758 4497 y Fy(\))2790 4429 y Fm(\011)2839 4497 y Fr(:)83 4685 y Fy(The)26 b(time)g Fr(\034)479 4697 y Fo(r)516 4685 y Fy(\()p Fr(m)623 4697 y Fl(\000)679 4685 y Fy(\))g(can)g(b)r(e)g(in)n(terpreted)e(as)h(the)h (time)g(sp)r(en)n(t)g(the)g(p)r(erturb)r(ed)e(tra)5 b(jectory)2997 4617 y Fm(\010\000)3083 4685 y Fr(')3138 4697 y Fo(t)3182 4685 y Fq(\016)16 b Fr(W)3317 4697 y Fl(\000)3372 4617 y Fm(\001)3410 4685 y Fy(\()p Fr(m)3517 4697 y Fl(\000)3573 4685 y Fy(\))3605 4617 y Fm(\011)3654 4718 y Fo(t)p Fl(2)p Fk(R)83 4793 y Fy(within)k(\010)397 4762 y Fl(\000)p FA(1)486 4793 y Fy(\()p Fr(B)581 4805 y Fo(r)617 4793 y Fy(\))i(min)n(us)e(the)h(time)g(sp)r(en)n(t)f(b)n(y)h(the)h(corresp)r (onding)17 b(free)k(tra)5 b(jectory)20 b(\(b)r(efore)g(and)h(after)g (scattering\))83 4901 y(within)27 b(the)g(same)g(region.)249 5008 y(In)g(the)h(next)g(lemma,)e(w)n(e)h(use)g(the)h(auxiliary)d(time) 644 5204 y Fr(\034)691 5170 y FA(free)681 5225 y Fo(r)801 5204 y Fy(\()p Fr(m)908 5216 y Fl(\000)964 5204 y Fy(\))e(:=)1140 5148 y(1)p 1140 5185 V 1140 5261 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Fl(\000)p Fo(t)2903 5321 y Fm(\001)2941 5389 y Fy(\()p Fr(m)3048 5401 y Fl(\000)3104 5389 y Fy(\))3136 5321 y Fm(\011)3184 5389 y Fr(;)1884 5670 y Fy(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 83 307 a Fy(whic)n(h)31 b(is)h(\014nite)h(for)f (all)f Fr(m)962 319 y Fl(\000)1049 307 y Fq(2)h(D)1200 319 y Fl(\000)1287 307 y Fy(due)g(to)h(Theorem)d(2.2.)i(The)g (de\014nition)f(of)h Fr(\034)2774 277 y FA(free)2764 328 y Fo(r)2917 307 y Fy(is)g(inspired)e(b)n(y)i(a)g(similar)83 415 y(de\014nition)26 b(in)g(the)i(con)n(text)f(of)g(quan)n(tum)f (scattering)g(theory)h([29)o(,)g(Sec.)h(4].)83 583 y Ft(Lemma)42 b(3.1.)j Fz(L)-5 b(et)41 b Fr(H)858 595 y FA(0)935 583 y Fz(and)g Fr(H)46 b Fz(satisfy)c(Assumptions)g(2.1)f(and) f(2.6,)h(and)f(let)g Fr(m)2974 595 y Fl(\000)3068 583 y Fq(2)e(D)3225 595 y Fl(\000)3305 583 y Fq(n)24 b Fj(Crit)p Fy(\()p Fr(H)3602 595 y FA(0)3639 583 y Fr(;)14 b Fy(\010\))83 691 y Fz(satisfy)33 b Fr(S)5 b Fy(\()p Fr(m)524 703 y Fl(\000)579 691 y Fy(\))33 b Fr(=)-51 b 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y(\000)705 1165 y Fr(\037)760 1130 y FA(\010)760 1185 y Fo(r)830 1165 y Fq(\016)18 b Fr(W)967 1177 y FA(+)1041 1165 y Fq(\000)g Fr(\037)1179 1130 y FA(\010)1179 1185 y Fo(r)1231 1097 y Fm(\001\000)1307 1165 y Fy(\()p Fr(S)24 b Fq(\016)18 b Fr(')1529 1130 y FA(0)1529 1185 y Fo(t)1566 1165 y Fy(\)\()p Fr(m)1705 1177 y Fl(\000)1761 1165 y Fy(\))1793 1097 y Fm(\001)1831 1094 y(\014)1831 1144 y(\014)1882 1165 y Fq(\024)k Fr(g)2009 1177 y FA(+)2065 1165 y Fy(\()p Fr(t)p Fy(\))92 b Fz(for)31 b(al)5 b(l)44 b Fr(r)25 b(>)e Fy(0)31 b Fz(and)41 b Fr(t)23 b Fq(2)h Fp(R)3127 1177 y FA(+)3189 1165 y Fr(:)385 b Fy(\(3.3\))83 1317 y Fz(Then,)32 b Fr(T)392 1329 y Fo(r)429 1317 y Fy(\()p Fr(m)536 1329 y Fl(\000)592 1317 y Fy(\))g Fz(is)g(\014nite)g (for)f(e)-5 b(ach)32 b Fr(r)25 b(>)e Fy(0)p Fz(,)31 b(and)1363 1500 y Fy(lim)1339 1550 y Fo(r)r Fl(!1)1516 1433 y Fm(\010)1565 1500 y Fr(\034)1602 1512 y Fo(r)1639 1500 y Fy(\()p Fr(m)1746 1512 y Fl(\000)1802 1500 y Fy(\))19 b Fq(\000)f Fr(\034)1983 1466 y FA(free)1973 1521 y Fo(r)2093 1500 y Fy(\()p Fr(m)2200 1512 y Fl(\000)2256 1500 y Fy(\))2288 1433 y Fm(\011)2360 1500 y Fy(=)k(0)p Fr(:)249 1699 y Fy(Before)31 b(mo)n(ving)e(on)i(to)g (the)g(pro)r(of)f(of)i(the)f(lemma,)f(w)n(e)h(mak)n(e)g(a)g(digression) c(to)32 b(sho)n(w)e(that)h(the)g(conditions)83 1807 y(\(3.2\)-\(3.3\))c (are)f(automatically)f(v)n(eri\014ed)h(if)i(Assumption)d(2.7)i(holds)9 b(:)83 1975 y Ft(Remark)36 b(3.2.)42 b Fz(If)36 b(Assumptions)h(2.1,)e (2.6)g(and)g(2.7)g(hold,)f(then)h(we)h(know)g(fr)-5 b(om)36 b(the)f(pr)-5 b(o)g(of)35 b(of)g(The)-5 b(or)g(em)83 2083 y(2.8)35 b(that)e(ther)-5 b(e)34 b(exists)i Fr(T)j(>)28 b Fy(0)35 b Fz(such)g(that)1543 2016 y Fm(\000)1581 2083 y Fr(W)1658 2095 y FA(+)1735 2083 y Fq(\016)20 b Fr(S)1853 2016 y Fm(\001)1890 2083 y Fy(\()p Fr(m)1997 2095 y Fl(\000)2053 2083 y Fy(\))29 b(=)2207 2016 y Fm(\000)2245 2083 y Fr(')2300 2095 y Fl(\000)p Fo(T)2422 2083 y Fq(\016)21 b Fr(')2540 2052 y FA(0)2540 2105 y Fo(T)2611 2083 y Fq(\016)f 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b(qual)37 b(to)g(the)g(symmetrise)-5 b(d)38 b(time)g(delay)f(in)h(the)e (limit)83 5421 y Fr(r)25 b Fq(!)f(1)p Fz(.)32 b(We)f(r)-5 b(efer)32 b(to)f(the)g(Examples)h(3.7)f(and)g(3.8)h(b)-5 b(elow)32 b(for)f(an)h(il)5 b(lustr)-5 b(ation)30 b(of)i(this)f (observation.)1884 5670 y Fy(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 83 307 a Ft(Example)32 b(3.7)f Fy(\()p Fr(H)732 319 y FA(0)769 307 y Fy(\()p Fr(q)s(;)14 b(p)p Fy(\))24 b(=)g Fr(h)p Fy(\()p Fr(p)p Fy(\),)29 b(con)n(tin)n(ued\))p Ft(.)38 b Fz(We)32 b(know)h(that)d(Assumptions)k (2.1,)e(2.6)g(and)g(2.7)f(hold)g(on)83 415 y(the)38 b(manifold)h Fr(H)689 378 y Fl(\000)p FA(1)683 435 y(0)777 415 y Fy(\()p Fr(U)865 427 y Fo(\016)901 415 y Fy(\))p Fz(,)h(with)f Fr(S)h Fy(:)35 b Fr(H)1424 378 y Fl(\000)p FA(1)1418 435 y(0)1512 415 y Fy(\()p Fr(U)1600 427 y Fo(\016)1636 415 y Fy(\))h Fq(!)f Fr(H)1896 378 y Fl(\000)p FA(1)1890 435 y(0)1984 415 y Fy(\()p Fr(U)2072 427 y Fo(\016)2108 415 y Fy(\))40 b Fz(\(se)-5 b(e)40 b(Example)f(2.13\).)g(It)f(fol)5 b(lows)39 b(that)e(e)-5 b(ach)83 523 y Fy(\()p Fr(q)152 535 y Fl(\000)208 523 y Fr(;)14 b(p)288 493 y Fl(\000)342 523 y Fy(\))24 b Fq(2)f Fr(H)550 486 y Fl(\000)p FA(1)544 543 y(0)638 523 y Fy(\()p Fr(U)726 535 y Fo(\016)762 523 y Fy(\))30 b Fz(satisfy)g Fr(S)5 b Fy(\()p Fr(q)1224 535 y Fl(\000)1280 523 y Fr(;)14 b(p)1360 493 y Fl(\000)1414 523 y Fy(\))33 b Fr(=)-51 b Fq(2)23 b Fj(Crit)p Fy(\()p Fr(H)1779 535 y FA(0)1816 523 y Fr(;)14 b Fy(\010\))30 b Fz(and)37 b Fy(\(3.2\))p Fz(-)p Fy(\(3.3\))29 b Fz(\(se)-5 b(e)30 b(R)-5 b(emark)30 b(3.2\).)f(So,)h(The)-5 b(or)g(em)83 631 y(3.3)31 b(applies,)h(and)f(the)g(glob)-5 b(al)31 b(time)h(delay)f(exists)i(and)e(satis\014es)580 861 y Fy(lim)556 911 y Fo(r)r Fl(!1)733 861 y Fr(\034)770 873 y Fo(r)808 861 y Fy(\()p Fr(q)877 873 y Fl(\000)932 861 y Fr(;)14 b(p)1012 826 y Fl(\000)1067 861 y Fy(\))23 b(=)g Fr(T)12 b Fy(\()p Fr(q)1340 873 y Fl(\000)1395 861 y Fr(;)i(p)1475 826 y Fl(\000)1529 861 y Fy(\))19 b Fq(\000)f Fr(T)12 b Fy(\()p Fr(q)1793 873 y FA(+)1848 861 y Fr(;)i(p)1928 826 y FA(+)1983 861 y Fy(\))23 b(=)2135 805 y Fr(q)2172 817 y Fl(\000)2246 805 y Fq(\001)c Fy(\()p Fq(r)p Fr(h)p Fy(\)\()p Fr(p)2545 774 y Fl(\000)2601 805 y Fy(\))p 2135 842 498 4 v 2170 918 a Fq(j)p Fy(\()p Fq(r)p Fr(h)p Fy(\)\()p Fr(p)2450 894 y Fl(\000)2506 918 y Fy(\))p Fq(j)2561 894 y FA(2)2662 861 y Fq(\000)2755 805 y Fr(q)2792 817 y FA(+)2865 805 y Fq(\001)g Fy(\()p Fq(r)p Fr(h)p Fy(\)\()p Fr(p)3164 774 y FA(+)3220 805 y Fy(\))p 2755 842 V 2789 918 a Fq(j)p Fy(\()p Fq(r)p Fr(h)p Fy(\)\()p Fr(p)3069 894 y FA(+)3126 918 y Fy(\))p Fq(j)3181 894 y FA(2)3272 861 y Fr(;)83 1096 y Fz(wher)-5 b(e)38 b Fy(\()p Fr(q)410 1108 y FA(+)466 1096 y Fr(;)14 b(p)546 1066 y FA(+)600 1096 y Fy(\))34 b(:=)f Fr(S)5 b Fy(\()p Fr(q)912 1108 y Fl(\000)967 1096 y Fr(;)14 b(p)1047 1066 y Fl(\000)1102 1096 y Fy(\))p Fz(.)38 b(Now,)h(if)f(ther) -5 b(e)37 b(exists)i(a)e(di\013e)-5 b(omorphism)38 b Fr(h)2769 1108 y FA(0)2840 1096 y Fy(:)33 b(\(0)p Fr(;)14 b Fq(1)p Fy(\))33 b Fq(!)g Fj(Ran)p Fy(\()p Fr(h)3489 1108 y FA(0)3527 1096 y Fy(\))38 b Fz(such)83 1204 y(that)33 b Fr(h)p Fy(\()p Fr(p)p Fy(\))c(=)e Fr(h)594 1216 y FA(0)632 1204 y Fy(\()p Fr(p)707 1174 y FA(2)744 1204 y Fy(\))35 b Fz(for)g(al)5 b(l)33 b Fr(p)28 b Fq(2)g Fp(R)1295 1174 y Fo(n)1382 1204 y Fz(\(such)35 b(as)g(when)g Fr(h)p Fy(\()p Fr(p)p Fy(\))29 b(=)f Fr(p)2293 1174 y FA(2)2330 1204 y Fr(=)p Fy(2)p Fz(\),)34 b(then)g Fq(j)p Fy(\()p Fq(r)p Fr(h)p Fy(\)\()p Fr(p)p Fy(\))p Fq(j)3047 1174 y FA(2)3114 1204 y Fy(=)3207 1137 y Fm(\000)3245 1204 y Fr(f)29 b Fq(\016)20 b Fr(h)3426 1137 y Fm(\001)3464 1204 y Fy(\()p Fr(p)p Fy(\))35 b Fz(with)83 1329 y Fr(f)i Fq(2)30 b Fj(C)299 1298 y Fl(1)368 1261 y Fm(\000)420 1329 y Fj(Ran)o Fy(\()p Fr(h)637 1341 y FA(0)675 1329 y Fy(\))707 1261 y Fm(\001)781 1329 y Fz(given)36 b(by)g Fr(f)8 b Fy(\()p Fr(x)p Fy(\))29 b(:=)g(4)p Fr(h)1535 1291 y Fl(\000)p FA(1)1535 1349 y(0)1623 1329 y Fy(\()p Fr(x)p Fy(\))1735 1258 y Fm(\014)1735 1308 y(\014)1762 1329 y Fr(h)1811 1298 y Fl(0)1811 1349 y FA(0)1849 1261 y Fm(\000)1887 1329 y Fr(h)1936 1291 y Fl(\000)p FA(1)1936 1349 y(0)2025 1329 y Fy(\()p Fr(x)p Fy(\))2137 1261 y Fm(\001)2175 1258 y(\014)2175 1308 y(\014)2203 1279 y FA(2)2240 1329 y Fz(.)35 b(Ther)-5 b(efor)g(e,)36 b(one)f(has)g(for)g (any)g Fy(\()p Fr(q)3461 1341 y Fl(\000)3517 1329 y Fr(;)14 b(p)3597 1298 y Fl(\000)3652 1329 y Fy(\))29 b Fq(2)83 1437 y Fr(H)157 1399 y Fl(\000)p FA(1)151 1457 y(0)245 1437 y Fy(\()p Fr(U)333 1449 y Fo(\016)369 1437 y Fy(\))411 1628 y Fq(j)p Fy(\()p Fq(r)p Fr(H)603 1640 y FA(0)641 1628 y Fy(\)\()p Fr(q)742 1640 y Fl(\000)798 1628 y Fr(;)14 b(p)878 1593 y Fl(\000)932 1628 y Fy(\))p Fq(j)987 1593 y FA(2)1048 1628 y Fy(=)1136 1560 y Fm(\000)1174 1628 y Fr(f)27 b Fq(\016)18 b Fr(H)1370 1640 y FA(0)1406 1560 y Fm(\001)1444 1628 y Fy(\()p Fr(q)1513 1640 y Fl(\000)1569 1628 y Fr(;)c(p)1649 1593 y Fl(\000)1704 1628 y Fy(\))23 b(=)1847 1560 y Fm(\000)1885 1628 y Fr(f)k 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y Fl(\000)p FA(1)956 2514 y(+)1077 2495 y Fq(\016)20 b Fr(W)1216 2507 y Fl(\000)1272 2428 y Fm(\001)1310 2495 y Fy(\()p Fr(M)1421 2507 y Fo(\016)1458 2495 y Fy(\))36 b Fz(has)f(the)g(r)-5 b(e)g(quir)g(e)g(d)36 b(pr)-5 b(op)g(erties)35 b(\(se)-5 b(e)37 b(Example)e(2.14\).)g(It)h (fol)5 b(lows)83 2603 y(that)40 b(e)-5 b(ach)42 b Fy(\()p Fr(q)555 2615 y Fl(\000)611 2603 y Fr(;)14 b(p)691 2573 y Fl(\000)746 2603 y Fy(\))40 b Fq(2)g Fr(M)992 2615 y Fo(\016)1071 2603 y Fz(satisfy)i Fr(S)5 b Fy(\()p Fr(q)1483 2615 y Fl(\000)1539 2603 y Fr(;)14 b(p)1619 2573 y Fl(\000)1674 2603 y Fy(\))49 b Fr(=)-51 b Fq(2)40 b Fj(Crit)p Fy(\()p Fr(H)2072 2615 y FA(0)2110 2603 y Fr(;)14 b Fy(\010\))42 b Fz(and)49 b Fy(\(3.2\))p Fz(-)p Fy(\(3.3\))41 b Fz(\(se)-5 b(e)44 b(R)-5 b(emark)42 b(3.2\).)g(So,)83 2711 y(The)-5 b(or)g(em)32 b(3.3)f(applies)h(and)f(the)g(glob)-5 b(al)31 b(time)h(delay)f(in)h(the)f(tub)-5 b(e)32 b(exists)h(and)e(satis\014es) 878 2952 y Fy(lim)854 3002 y Fo(r)r Fl(!1)1032 2952 y 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Ft(Example)c(3.9)f Fy(\(P)n(oincar)n(\023)-39 b(e)21 b(ball,)h(con)n(tin)n(ued\))p Ft(.)34 b Fz(We)28 b(know)g(that)f(Assumptions)i(2.1,)f(2.6)f(and)h(2.7)f(hold)g(on)h(the) 83 3908 y(manifold)34 b Fr(H)529 3871 y Fl(\000)p FA(1)523 3928 y(0)617 3908 y Fy(\()p Fr(U)705 3920 y Fo(\016)741 3908 y Fy(\))p Fz(,)g(with)g Fr(S)e Fy(:)27 b Fr(H)1237 3871 y Fl(\000)p FA(1)1231 3928 y(0)1325 3908 y Fy(\()p Fr(U)1413 3920 y Fo(\016)1449 3908 y Fy(\))g Fq(!)g Fr(H)1692 3871 y Fl(\000)p FA(1)1686 3928 y(0)1780 3908 y Fy(\()p Fr(U)1868 3920 y Fo(\016)1904 3908 y Fy(\))34 b Fz(\(se)-5 b(e)36 b(Example)d(2.15\).)h(F)-7 b(urthermor)i(e,)33 b(one)h(has)g(for)83 4016 y(e)-5 b(ach)32 b Fy(\()p Fr(q)353 4028 y Fl(\000)408 4016 y Fr(;)14 b(p)488 3986 y Fl(\000)543 4016 y Fy(\))23 b Fq(2)h Fr(H)751 3979 y Fl(\000)p FA(1)745 4036 y(0)839 4016 y Fy(\()p Fr(U)927 4028 y Fo(\016)963 4016 y Fy(\))530 4207 y Fq(j)p Fy(\()p Fq(r)p Fr(H)722 4219 y FA(0)759 4207 y Fy(\)\()p Fr(q)860 4219 y Fl(\000)916 4207 y Fr(;)14 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Fo(E)1742 3138 y Fy(\)\()p Fr(\015)5 b Fy(\()p Fr(t)p Fy(\)\))p Fq(j)2003 3150 y Fo(t)p FA(=0)2140 3138 y Fy(=)2252 3082 y(d)p 2237 3119 V 2237 3195 a(d)p Fr(t)2333 3138 y(f)2382 3071 y Fm(\000)2421 3138 y Fy(\011)2486 3150 y Fo(E)2540 3138 y Fy(\()p Fr(m)p Fy(\))20 b(+)e Fr(t)9 b Fq(j)p Fy(\()p Fq(r)p Fr(H)3013 3150 y FA(0)3050 3138 y Fy(\)\()p Fr(m)p Fy(\))p Fq(j)3244 3104 y FA(2)3283 3071 y Fm(\001)3321 3138 y Fq(j)3344 3150 y Fo(t)p FA(=0)2140 3304 y Fy(=)22 b Fq(j)p Fy(\()p Fq(r)p Fr(H)2419 3316 y FA(0)2457 3304 y Fy(\)\()p Fr(m)p Fy(\))p Fq(j)2651 3270 y FA(2)2689 3304 y Fr(f)2738 3270 y Fl(0)2761 3237 y Fm(\000)2799 3304 y Fy(\011)2864 3316 y Fo(E)2919 3304 y Fy(\()p Fr(m)p Fy(\))3058 3237 y Fm(\001)2140 3483 y Fy(=)g Fq(j)p Fy(\()p Fq(r)p Fr(H)2419 3495 y FA(0)2457 3483 y Fy(\)\()p Fr(m)p Fy(\))p Fq(j)2651 3449 y FA(2)2714 3427 y Fr(@)p 2699 3464 79 4 v 2699 3540 a(@)5 b(t)2787 3388 y Fm(\014)2787 3438 y(\014)2787 3488 y(\014)2815 3542 y FA(\011)2866 3550 y Fi(E)2912 3542 y FA(\()p Fo(m)p FA(\))3028 3483 y Fy(\()p Fr(f)j Fy(\))p Fr(;)83 3716 y Fy(and)21 b(th)n(us)h(\(d\011) 562 3728 y Fo(E)616 3716 y Fy(\))648 3728 y Fo(m)713 3649 y Fm(\000)751 3716 y Fr(X)819 3728 y Fo(H)872 3736 y Fg(0)908 3716 y Fy(\()p Fr(m)p Fy(\))1047 3649 y Fm(\001)1109 3716 y Fy(=)h Fq(j)p Fy(\()p Fq(r)p Fr(H)1389 3728 y FA(0)1426 3716 y Fy(\)\()p Fr(m)p Fy(\))p Fq(j)1620 3686 y FA(2)1681 3683 y Fo(@)p 1668 3697 63 4 v 1668 3745 a(@)t(t)1741 3646 y Fm(\014)1741 3695 y(\014)1768 3749 y FA(\011)1819 3757 y Fi(E)1865 3749 y FA(\()p Fo(m)p FA(\))1981 3716 y Fy(.)g(Since)e Fq(j)p Fy(\()p Fq(r)p Fr(H)2431 3728 y FA(0)2469 3716 y Fy(\)\()p Fr(m)p Fy(\))p Fq(j)j(6)p Fy(=)f(0,)f(this)g(implies)e(that)i(\(d\011)3617 3728 y Fo(E)3672 3716 y Fy(\))3704 3728 y Fo(m)83 3847 y Fy(is)32 b(surjectiv)n(e,)f(and)h(so)g(\000)899 3859 y Fo(E)s(;\013)1048 3847 y Fq(\021)f Fy(\011)1209 3810 y Fl(\000)p FA(1)1209 3870 y Fo(E)1297 3847 y Fy(\()p Fq(f)p Fr(\013)p Fq(g)p Fy(\))h(is)g(a)h(regular)d(submanifold)f(of)k (\006)2617 3817 y FA(0)2617 3870 y Fo(E)2704 3847 y Fy(of)g(co)r (dimension)c(1)j(b)n(y)h(the)f(reg-)83 3955 y(ular)h(lev)n(el)h(set)h (theorem.)e(Moreo)n(v)n(er,)g(w)n(e)i(also)e(obtain)h(that)g Fr(X)2169 3967 y Fo(H)2222 3975 y Fg(0)2258 3955 y Fy(\()p Fr(m)p Fy(\))46 b Fr(=)-51 b Fq(2)35 b Fy(k)n(er)2649 3888 y Fm(\000)2687 3955 y Fy(\(d\011)2831 3967 y Fo(E)2886 3955 y Fy(\))2918 3967 y Fo(m)2982 3888 y Fm(\001)3020 3955 y Fy(,)g(whic)n(h)f(implies)e(that)83 4063 y Fr(X)151 4075 y Fo(H)204 4083 y Fg(0)240 4063 y Fy(\()p Fr(m)p Fy(\))h Fr(=)-51 b Fq(2)24 b Fr(T)531 4075 y Fo(m)594 4063 y Fy(\000)645 4075 y Fo(E)s(;\013)790 4063 y Fy(since)i(k)n(er) 1120 3996 y Fm(\000)1158 4063 y Fy(\(d\011)1302 4075 y Fo(E)1356 4063 y Fy(\))1388 4075 y Fo(m)1452 3996 y Fm(\001)1513 4063 y Fy(=)d Fr(T)1650 4075 y Fo(m)1713 4063 y Fy(\000)1764 4075 y Fo(E)s(;\013)1909 4063 y Fy(\(see)28 b(the)f(remark)f(after)i([1)o(,)g(Prop.)e(1.6.18]\).)249 4171 y(T)-7 b(o)27 b(pro)n(v)n(e)f(\(b\),)i(tak)n(e)f Fr(m)d Fq(2)f Fy(\006)1174 4140 y FA(0)1174 4193 y Fo(E)1257 4171 y Fy(and)j(observ)n(e)g(that)171 4399 y Fr(')226 4364 y FA(0)226 4419 y Fo(t)263 4399 y Fy(\()p Fr(m)p Fy(\))e Fq(2)f Fy(\000)555 4411 y Fo(E)s(;\013)719 4399 y Fq(\()-14 b(\))46 b Fy(\011)982 4411 y Fo(E)1036 4331 y Fm(\000)1074 4399 y Fr(')1129 4364 y FA(0)1129 4419 y Fo(t)1166 4399 y Fy(\()p Fr(m)p Fy(\))1305 4331 y Fm(\001)1367 4399 y Fy(=)23 b Fr(\013)46 b Fq(\()-14 b(\))47 b Fy(\011)1819 4411 y Fo(E)1873 4399 y Fy(\()p Fr(m)p Fy(\))20 b(+)e Fr(t)9 b Fq(j)p Fy(\()p Fq(r)p Fr(H)2346 4411 y FA(0)2383 4399 y Fy(\)\()p Fr(m)p Fy(\))p Fq(j)2577 4364 y FA(2)2639 4399 y Fy(=)23 b Fr(\013)46 b Fq(\()-14 b(\))46 b Fr(t)23 b Fy(=)3199 4343 y Fr(\013)c Fq(\000)f Fy(\011)3420 4355 y Fo(E)3475 4343 y Fy(\()p Fr(m)p Fy(\))p 3176 4380 462 4 v 3176 4456 a Fq(j)p Fy(\()p Fq(r)p Fr(H)3368 4468 y FA(0)3405 4456 y Fy(\)\()p Fr(m)p Fy(\))p Fq(j)3599 4432 y FA(2)3657 4399 y Fr(:)83 4647 y Fy(Th)n(us,)33 b(the)h(time)f Fr(t)700 4659 y FA(0)771 4647 y Fy(:=)938 4607 y FA(\011)989 4615 y Fi(E)1035 4607 y FA(\()p Fo(m)p FA(\))p Fl(\000)p Fo(\013)p 903 4628 374 4 v 903 4676 a Fl(j)p FA(\()p Fl(r)p Fo(H)1056 4684 y Fg(0)1087 4676 y FA(\)\()p Fo(m)p FA(\))p Fl(j)1244 4659 y Fg(2)1320 4647 y Fq(2)h Fp(R)41 b Fy(and)33 b(the)h(p)r(oin)n(t)f Fr(m)2121 4659 y FA(0)2192 4647 y Fy(:=)h Fr(')2369 4617 y FA(0)2369 4668 y Fl(\000)p Fo(t)2444 4676 y Fg(0)2480 4647 y Fy(\()p Fr(m)p Fy(\))h Fq(2)f Fy(\000)2794 4659 y Fo(E)s(;\013)2946 4647 y Fy(are)f(unique)g(and)g(satisfy)83 4767 y Fr(m)24 b Fy(=)e Fr(')324 4736 y FA(0)324 4787 y Fo(t)348 4795 y Fg(0)385 4767 y Fy(\()p Fr(m)492 4779 y FA(0)530 4767 y Fy(\).)p 3708 4767 4 57 v 3712 4714 50 4 v 3712 4767 V 3761 4767 4 57 v 249 4940 a(Lemma)27 b(4.1)g(implies)e(in)i(particular)d(that)j(the)h(submanifold)1117 5127 y(\000)1168 5139 y Fo(E)1246 5127 y Fy(:=)23 b(\000)1408 5139 y Fo(E)s(;)p FA(0)1538 5127 y Fq(\021)1626 5059 y Fm(\010)1674 5127 y Fr(m)h Fq(2)f Fy(\006)1911 5092 y FA(0)1911 5147 y Fo(E)1989 5127 y Fq(j)g Fy(\(\010)c Fq(\001)f(r)p Fr(H)2324 5139 y FA(0)2361 5127 y Fy(\)\()p Fr(m)p Fy(\))25 b(=)d(0)2686 5059 y Fm(\011)83 5313 y Fy(is)i(a)h(P)n(oincar)n(\023)-39 b(e)22 b(section)h(in)i(the)f(sense)g (of)i([8)o(,)f(Assumption)e(2.2].)h(The)h(rest)g(of)g(the)g (assumptions)c(of)k([8])g(are)f(v)n(eri\014ed)83 5421 y(in)j(the)g(follo)n(wing)e(lemma)9 b(:)1884 5670 y(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 83 307 a Ft(Lemma)31 b(4.2.)39 b Fz(Assume)33 b(that)c Fr(M)40 b Fz(is)32 b(exact)e(\(that)g(is,)i (with)f Fr(!)j Fz(exact\))d(and)g(satis\014es)g Fy(dim)o(\()p Fr(M)9 b Fy(\))23 b Fq(\025)g Fy(4)p Fz(.)31 b(Supp)-5 b(ose)83 415 y(also)31 b(that)65 585 y(\(H1\))43 b(Assumption)32 b(2.1)g(holds,)65 757 y(\(H2\))43 b Fr(V)49 b Fz(has)32 b(c)-5 b(omp)g(act)31 b(supp)-5 b(ort,)65 929 y(\(H3\))43 b Fq(f)p Fy(\010)p Fr(;)14 b(V)j Fq(g)22 b Fy(=)h(0)31 b Fz(or)h(Assumption)g(2.9)g(holds,)65 1102 y(\(H4\))43 b 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b(implies)g(the)i(noncompactness)d (of)k Fr(M)32 b Fy([21)o(,)24 b(Rem.)f(V.9.4].)g(Note)83 1863 y(also)j(that)h(Assumption)f(2.7)g(follo)n(ws)g(from)h(the)h (compactness)d(of)i(the)h(supp)r(ort)d(of)j Fr(V)18 b Fy(.)83 2036 y Fz(Pr)-5 b(o)g(of.)42 b Fy(The)32 b(h)n(yp)r(otheses)e (\(H1\)-\(H4\))j(imply)e(that)i(Corollary)c(2.11)i(applies.)f(Th)n(us,) h(eac)n(h)g Fr(E)36 b Fq(2)31 b Fr(U)41 b Fy(is)32 b(a)g(regular)83 2144 y(v)-5 b(alue)33 b(of)g Fr(H)472 2156 y FA(0)542 2144 y Fy(and)g Fr(H)39 b Fy(\(Assumption)31 b(2.1\(i\))i(of)g([8]\))h (and)e Fr(W)2042 2156 y FA(+)2098 2077 y Fm(\000)2136 2144 y Fr(H)2210 2106 y Fl(\000)p FA(1)2204 2164 y(0)2298 2144 y Fy(\()p Fq(f)p Fr(E)5 b Fq(g)p Fy(\))2512 2077 y Fm(\001)2582 2144 y Fy(=)32 b Fr(W)2756 2156 y Fl(\000)2811 2077 y Fm(\000)2850 2144 y Fr(H)2924 2106 y Fl(\000)p FA(1)2918 2164 y(0)3012 2144 y Fy(\()p Fq(f)p Fr(E)5 b Fq(g)p Fy(\))3226 2077 y Fm(\001)3296 2144 y Fy(=)32 b Fr(H)3467 2114 y Fl(\000)p FA(1)3555 2144 y Fy(\()p Fq(f)p 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