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b(of)i(the)f(cen)m(ter)i(manifold)c(and) i(giv)m(e)h(an)g(estimate)g(of)g(its)f(action)g(and)g(of)h(its)f (energy)-8 b(.)41 b(These)75 5239 y(estimates)33 b(are)f(the)g(main)f (no)m(v)m(elties)i(compared)f(with)e([3)q(],)j(they)f(allo)m(w)g(in)m (teresting)f(new)g(applications.)75 5352 y(The)22 b(energy)g(is)g (close)h(to)g(zero)g(\(the)g(energy)f(of)h(the)f(\014xed)g(p)s(oin)m (t\))g(when)f(the)i(system)f(is)g(close)g(to)h(a)g(pro)s(duct)75 5465 y(system)37 b(and)g(the)g(homo)s(clinic)d(w)m(e)k(\014nd)d(should) g(b)s(e)i(seen)g(as)g(the)g(con)m(tin)m(uation,)i(when)d(a)i(coupling)d (is)75 5578 y(in)m(tro)s(duced,)45 b(of)e(the)g(orbit)f(homo)s(clinic)e (to)k(the)f(origin)e(that)j(existed)e(in)g(the)h(pro)s(duct)e(system.) 79 b(In)1890 5841 y(2)p eop %%Page: 3 3 3 2 bop 75 399 a Fs(this)33 b(sense)g(w)m(e)h(can)g(sa)m(y)h(that)f (the)g(homo)s(clinic)d(w)m(e)k(\014nd)d(is)h(the)g(closest)i(to)f(the)g (origin,)f(although)g(new)75 511 y(orbits,)39 b(longer)f(and)f(closer,) 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y(A)46 b(ma)5 b(jor)46 b(in)m(terest)g(of)g(homo)s(clinic)e (orbits)h(is)g(their)f(link)g(with)h(c)m(haotic)i(b)s(eha)m(vior.)86 b(The)46 b(orbit)75 2205 y(structure)33 b(near)g(a)h(transv)m(ersal)f (homo)s(clinic)e(orbit)i(to)h(a)g(h)m(yp)s(erb)s(olic)d(\014xed)h(p)s (oin)m(t)h(of)g(a)h(p)s(erio)s(dic)d(time-)75 2318 y(dep)s(enden)m(t)25 b(system)g(has)g(b)m(y)h(no)m(w)f(b)s(een)g(w)m(ell)g(describ)s(ed.)37 b(The)25 b(natural)f(analog)i(of)g(this)e(structure)h(exists)75 2431 y(in)i(an)h(autonomous)g(system)h(around)e(a)h(transv)m(ersal)g (homo)s(clinic)e(orbit)i(to)h(an)f(h)m(yp)s(erb)s(olic)d(\014xed)j(p)s (oin)m(t.)75 2544 y(It)f(should)e(b)s(e)i(noted)g(ho)m(w)m(ev)m(er)h (that)g(the)f(b)s(eha)m(vior)g(asso)s(ciated)g(with)f(homo)s(clinic)f (orbits)h(to)i(h)m(yp)s(erb)s(olic)75 2657 y(\014xed)d(p)s(oin)m(ts)g (of)i(autonomous)f(systems)g(is)g(not)g(as)g(w)m(ell)f(understo)s(o)s (d,)h(see)h([15)q(])f(and)g([8])h(for)f(some)g(results)75 2770 y(on)g(this)g(sub)5 b(ject.)39 b(One)26 b(of)g(the)h(in)m(terests) f(of)h(our)e(w)m(ork)i(is)e(that)i(the)g(homo)s(clinic)d(w)m(e)j (\014nd,)f(if)f(transv)m(ersal,)75 2883 y(lead)34 b(to)g(the)g(w)m(ell) f(describ)s(ed)f(case,)37 b Fl(i.e.)50 b Fs(to)35 b(a)f(Bernoulli)e (shift)h(with)f(p)s(ositiv)m(e)h(en)m(trop)m(y)-8 b(.)53 b(Consider)32 b(for)75 2995 y(example)d(a)g(classical)f(plane)h(p)s (endulum,)d(our)i(results)g(pro)m(vide)g(a)i(new)e(w)m(a)m(y)i(to)g (break)f(in)m(tegrabilt)m(y)g(and)75 3108 y(in)m(tro)s(duce)c(c)m (haotic)i(b)s(eha)m(vior.)39 b(Instead)25 b(of)i(considering)d(that)i (there)h(is)e(some)h(small)f(in\015uence)f(from)i(the)75 3221 y(exterior)32 b(\(a)h(time)f(dep)s(enden)m(t)f(p)s(erturbation\),) g(one)h(can)h(consider)d(that)j(the)f(bar)g(has)g(some)g(elasticit)m(y) -8 b(.)75 3334 y(In)32 b(this)g(case,)i(the)f(unstable)f(equilibrium)c (is)j(surrounded)g(b)m(y)h(unstable)g(oscillations.)46 b(W)-8 b(e)34 b(pro)m(v)m(e)f(that)75 3447 y(one)d(of)g(these)g 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b(exists)g(a)g(function)f Fp(G)1433 2142 y Fn(1)1542 2128 y Fs(on)h Fp(T)13 b(M)d Fs(,)34 b(a)g(n)m(um)m(b)s(er) e Fp(\013)f(>)f Fs(0,)k(a)g(compact)h(set)f Fp(K)j Fm(\032)29 b Fp(T)13 b(M)44 b Fs(and)75 2241 y(a)38 b(compact)h(set)f Fp(B)j Fm(\032)c Fp(K)31 b Fm(\002)25 b Fr(R)1160 2208 y Fo(2)1243 2241 y Fs(suc)m(h)37 b(that)h Fp(G)p Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))39 b(=)e Fp(G)2235 2255 y Fn(1)2310 2241 y Fs(\()p Fp(z)t Fs(\))h(outside)f Fp(B)5 b Fs(,)39 b(and)e Fp(G)3178 2255 y Fn(1)3253 2241 y Fs(\()p Fp(z)t Fs(\))h(=)f Fp(\013)p Fm(k)p Fp(z)t Fm(k)3709 2208 y Fo(2)75 2354 y Fs(outside)30 b Fp(K)7 b Fs(.)75 2467 y(As)31 b(a)h(consequence)g(of)g([HL],)g(the)g(tra)5 b(jectories)32 b(of)g Fp(L)f Fs(on)g Fp(M)g Fm(\002)21 b Fr(R)40 b Fs(are)31 b(the)h(pro)5 b(jections)31 b(of)h(the)f(in)m (tegral)75 2580 y(curv)m(es)f(of)f(a)h(v)m(ector-\014eld)g Fp(Y)1052 2594 y Fq(L)1134 2580 y Fs(on)f Fp(T)13 b(M)d Fs(,)30 b(that)g(is)f(conjugated)h(to)g(the)g(Hamiltonian)e(v)m(ector)j (\014eld)d Fp(X)3557 2594 y Fq(H)3654 2580 y Fs(on)75 2693 y Fp(T)141 2660 y Fn(\003)180 2693 y Fp(M)10 b Fs(,)31 b(where)f Fp(H)37 b Fs(is)29 b(the)i(\014b)s(erwise)d(dual)h(of)i Fp(L)473 2878 y(H)7 b Fs(\()p Fp(\020)g(;)15 b(q)s(;)g(p)p Fs(\))25 b(=)249 b(sup)964 2962 y Fq(z)s Fn(2)p Fq(\031)1090 2943 y Fh(\000)p Fg(1)1172 2962 y Fo(\()p Fq(\031)1242 2943 y Fh(\003)1279 2962 y Fo(\()p Fq(\020)5 b Fo(\)\))p Fq(;v)r Fn(2)p Ff(R)1548 2878 y Fm(h)p Fp(\020)i(;)15 b(z)t Fm(i)21 b Fs(+)f Fp(pv)j Fm(\000)d Fp(L)p Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))93 b(\()p Fp(\020)7 b(;)15 b(q)s(;)g(p)p Fs(\))26 b Fm(2)f Fp(T)2974 2840 y Fn(\003)3013 2878 y Fp(M)30 b Fm(\002)20 b Fr(R)3282 2840 y Fo(2)3327 2878 y Fp(:)75 3131 y Fs(See)27 b(Section)f(2)i(for)e(more)h(details.) 38 b(The)27 b(\015o)m(w)f(of)h Fp(Y)1823 3145 y Fq(L)1902 3131 y Fs(has)f(an)h(in)m(v)-5 b(arian)m(t)26 b(manifold,)f(the)i(cen)m (ter)h(manifold,)75 3244 y(of)f(equation)h Fp(z)h Fs(=)c Fp(z)753 3258 y Fo(0)793 3244 y Fs(.)39 b(The)27 b(cen)m(ter)h (manifold)e(is)g(\014lled)f(with)h(p)s(erio)s(dic)f(orbits,)i(whic)m(h) f(are)h(the)h(liftings)d(of)1465 3430 y Fp(O)1534 3444 y Fq(r)1573 3430 y Fs(\()p Fp(t)p Fs(\))g(=)g(\()p Fp(\022)1875 3444 y Fo(0)1915 3430 y Fp(;)15 b(r)j Fs(cos)q(\()p Fp(!)s(t)p Fs(\)\))p Fp(;)75 3615 y Fs(and)30 b(can)g(b)s(e)g(describ)s(ed)e(also) j(b)m(y)953 3801 y Fp(O)1022 3815 y Fq(r)1085 3801 y Fs(=)25 b Fm(f)p Fs(\()p Fp(z)1303 3815 y Fo(0)1344 3801 y Fp(;)15 b(q)s(;)g(v)s Fs(\))26 b Fm(2)f Fp(T)13 b(M)30 b Fm(\002)20 b Fr(R)1997 3763 y Fo(2)2057 3801 y Fp(=)15 b(v)2164 3763 y Fo(2)2225 3801 y Fs(+)20 b Fp(!)2376 3763 y Fo(2)2415 3801 y Fp(q)2459 3763 y Fo(2)2524 3801 y Fs(=)k Fp(!)2679 3763 y Fo(2)2719 3801 y Fp(r)2763 3763 y Fo(2)2802 3801 y Fm(g)p Fp(:)75 3986 y Fs(W)-8 b(e)35 b(are)g(lo)s(oking)e(for)h(orbits)f(homo)s(clinic)f(to)j Fp(O)1763 4000 y Fq(r)1801 3986 y Fs(,)g(i.e.)52 b(tra)5 b(jectories)35 b Fp(x)d Fs(=)f(\()p Fp(\022)s(;)15 b(q)s Fs(\))32 b(:)g Fr(R)40 b 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b(in)f(certain)i(instances)f(to)h(pro)m(v)m(e)h(that) f(the)302 3384 y(homo)s(clinic)24 b(w)m(e)j(\014nd)e(is)g(actually)h (close)h(to)g(the)g(saddle-cen)m(ter.)39 b(This)25 b(enables)h(new)f (applications.)302 3497 y(The)j(estimate)h(\(5\))g(is)e(also)h(new,)h (w)m(e)f(ha)m(v)m(e)i(to)f(relax)e(it)h(in)f([3)q(])h(to)h(lo)s(calize) e(the)i(homo)s(clinic)d(orbits.)302 3609 y(Our)f(b)s(elief)g(is)g(that) j(the)e(homo)s(clinic)e(w)m(e)j(obtain)f(is)g(the)g(closest)h(to)h(the) e(\014xed)g(p)s(oin)m(t)f(among)i(those)302 3722 y(whic)m(h)32 b(satisfy)h(\(5\).)49 b(The)33 b(price)f(for)h(these)g(estimates)h(is)e (that)i(w)m(e)f(obtain)g(only)f(one)h(homo)s(clinic)302 3835 y(orbit,)h(while)e(in\014nitely)e(man)m(y)k(are)g(found)e(in)h ([3].)51 b(It)34 b(should)d(b)s(e)i(p)s(ossible,)g(although)g(not)h (easy)-8 b(,)302 3948 y(to)31 b(carry)f(o)m(v)m(er)h(the)g(results)d (of)j(this)d(pap)s(er)h(to)i(the)f(setting)g(of)h([3],)g(and)e(the)h (results)f(of)h([3)q(])g(to)h(this)302 4061 y(setting.)186 4245 y(2.)46 b(As)30 b(a)g(consequence)h(of)f(the)g(h)m(yp)s(otheses)f ([HR1,2],)j(the)e(orbit)f Fp(O)2549 4259 y Fq(r)2617 4245 y Fs(is)g(h)m(yp)s(erb)s(olic)e(with)h(resp)s(ect)i(to)302 4358 y(its)j(energy)g(shell)e(and)h(the)h(\014xed)f(p)s(oin)m(t)g(\()p Fp(\022)1813 4372 y Fo(0)1853 4358 y Fp(;)15 b Fs(0\))34 b(is)e(of)h(saddle)e(cen)m(ter)j(t)m(yp)s(e,)g(with)e(2)p Fp(n)h Fs(h)m(yp)s(erb)s(olic)302 4471 y(dimensions)28 b(and)i(2)g(elliptic)e(dimensions)g(in)h(phase)h(space.)41 b(This)29 b(is)g(pro)m(v)m(ed)i(in)e(Section)h(6.)186 4654 y(3.)46 b(The)30 b(h)m(yp)s(othesis)f([HG3])j(can)f(also)f(b)s(e)g (written)907 4897 y Fp(L)p Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))27 b Fi(>)1389 4835 y Fs(1)p 1389 4876 46 4 v 1389 4959 a(2)1460 4769 y Fk(\022)1526 4897 y Fp(q)1580 4835 y(@)5 b(L)p 1580 4876 116 4 v 1589 4959 a(@)g(q)1705 4897 y Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))23 b(+)c Fp(v)2162 4835 y(@)5 b(L)p 2162 4876 V 2169 4959 a(@)g(v)2288 4897 y Fs(\()p Fp(z)t(;)15 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y(~)75 4026 y Fp(L)137 4040 y Fq(\017)199 4026 y Fs(has)29 b(b)s(een)g(constructed)h(to)h(obtain)e([HG1].)42 b(The)29 b(h)m(yp)s(othesis)f([HL])i(is)f(a)h(direct)f(consequence)h (of)g(\(6\))75 4139 y(and)j(\(7\))i(when)e Fp(\017)h Fs(is)f(small)f(enough.)51 b(It)34 b(is)f(not)i(harder)e(to)h(see)h (that)f([HG2])h(holds)e(with)g(the)h(constan)m(t)75 4252 y Fp(b=)p Fs(2)i(instead)e(of)i Fp(b)f Fs(for)g(su\016cien)m(tly)e (small)h Fp(\017)p Fs(.)55 b(W)-8 b(e)36 b(also)f(obtain)g(from)f (\(6\))j(and)d(\(7\))i(the)g(existence)f(of)h(a)75 4365 y(function)29 b Fp(c)p Fs(\()p Fp(\017)p Fs(\))d Fp(>)f Fs(0)31 b(with)e(lim)1108 4379 y Fq(\017)p Fn(!)p Fo(0)1262 4365 y Fp(c)p Fs(\()p Fp(\017)p Fs(\))d(=)f(0)31 b(suc)m(h)f(that)1451 4484 y Fk(\014)1451 4538 y(\014)1482 4561 y Fp(L)1544 4575 y Fq(\017)1596 4561 y Fm(\000)20 b Fp(L)1749 4484 y Fk(\014)1749 4538 y(\014)1805 4561 y Fi(6)25 b Fp(c)p Fs(\()p Fp(\017)p Fs(\))p Fp(d)2094 4524 y Fo(2)2134 4561 y Fs(\()p Fp(z)t(;)15 b(z)2297 4575 y 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b(z)3429 5023 y Fo(0)3469 5009 y Fs(\))p Fp(:)75 5283 y Fs(The)38 b(h)m(yp)s(othesis)f([HG3])i(is)f(th)m(us)g(satis\014ed)f (with)g(the)h(constan)m(t)i Fp(b=)p Fs(2)f(when)e Fp(\017)h Fs(is)g(small)f(enough.)64 b(W)-8 b(e)75 5396 y(moreo)m(v)m(er)32 b(ha)m(v)m(e)g(the)e(inequalit)m(y)830 5592 y Fp(U)892 5606 y Fq(\017)950 5592 y Fs(=)25 b Fp(U)30 b Fm(\000)20 b Fp(c)p Fs(\()p Fp(\017)p Fs(\))p Fp(d)1422 5555 y Fo(2)1462 5592 y Fs(\()p Fp(z)t(;)15 b(z)1625 5606 y Fo(0)1666 5592 y Fs(\))26 b Fi(6)1843 5569 y Fs(~)1823 5592 y Fp(G)1894 5606 y Fq(\017)1952 5592 y Fi(6)f Fp(W)33 b Fs(+)20 b Fp(c)p Fs(\()p Fp(\017)p Fs(\))p Fp(d)2451 5555 y Fo(2)2491 5592 y Fs(\()p Fp(z)t(;)15 b(z)2654 5606 y Fo(0)2695 5592 y Fs(\))26 b Fi(6)f Fp(W)2938 5606 y Fq(\017)2970 5592 y Fp(:)1890 5841 y Fs(8)p eop %%Page: 9 9 9 8 bop 75 399 a Fs(The)27 b(estimate)h(\(3\))h(com)m(bined)d(with)g (the)i(fact)g(that)g([HG2])h(is)e(satis\014ed)f(with)g(the)i(constan)m (t)g Fp(b=)p Fs(2)h(implies)1131 521 y Fk(\014)1131 575 y(\014)1161 598 y Fp(I)7 b Fs(\()p Fp(W)1329 612 y Fq(\017)1362 598 y Fs(\))21 b Fm(\000)e Fp(I)7 b Fs(\()p Fp(U)1652 612 y Fq(\017)1686 598 y Fs(\))1721 521 y Fk(\014)1721 575 y(\014)1797 598 y Fm(\000)-17 b(\000)c(\000)k(!)1854 656 y Fq(\017)p Fn(!)p Fo(0)2091 521 y Fk(\014)2091 575 y(\014)2122 598 y Fp(I)7 b Fs(\()p Fp(W)13 b Fs(\))20 b Fm(\000)g Fp(I)7 b Fs(\()p Fp(U)j Fs(\))2638 521 y Fk(\014)2638 575 y(\014)2669 598 y Fp(:)75 848 y Fs(It)34 b(is)f(p)s(ossible)e(to)j(apply)f(Theorem)g(1)h(to)1576 826 y(~)1566 848 y Fp(L)1628 862 y Fq(\017)1694 848 y Fs(when)e Fp(\017)i Fs(is)f(small)f(enough,)i(and)g(get)g(a)g(homo)s (clinic)e(orbit)75 961 y(to)f Fp(O)255 975 y Fq(r)324 961 y Fs(with)1248 1084 y Fp(a!)s(r)1400 1046 y Fo(2)1464 1084 y Fi(6)1590 1022 y Fp(!)p 1570 1063 101 4 v 1570 1146 a Fs(2)p Fp(\031)1681 1010 y Fk(\000)1722 1084 y Fp(I)7 b Fs(\()p Fp(W)1890 1098 y Fq(\017)1923 1084 y Fs(\))21 b Fm(\000)f Fp(I)7 b Fs(\()p Fp(U)2214 1098 y Fq(\017)2247 1084 y Fs(\))2282 1010 y Fk(\001)2349 1084 y Fi(6)25 b Fp(E)2512 1098 y Fo(0)2552 1084 y Fp(:)3679 1270 y Fi(\003)75 1513 y Fd(1.2)112 b(P)m(erturbation)36 b(from)h(pro)s(duct)h(systems)75 1685 y Fs(Let)31 b(us)f(\014rst)g (consider)g(the)h(case)g(where)f Fp(G)i Fs(do)s(es)e(not)h(dep)s(end)e (on)h(\()p Fp(q)s(;)15 b(v)s Fs(\).)43 b(W)-8 b(e)32 b(can)f(set)g Fp(U)36 b Fs(=)25 b Fp(G)i Fs(=)e Fp(W)43 b Fs(in)75 1798 y(the)31 b(notations)f(of)h(Theorem)f(1,)h(and)1178 2002 y Fp(Q)p Fs(\()p Fp(q)s(;)15 b(v)s Fs(\))26 b(=)f Fp(a)p Fs(\()p Fp(v)1703 1965 y Fo(2)1763 2002 y Fm(\000)20 b Fp(!)1914 1965 y Fo(2)1954 2002 y Fp(q)1998 1965 y Fo(2)2037 2002 y Fs(\))p Fp(;)107 b Fs(\()p Fp(q)s(;)15 b(v)s Fs(\))26 b Fm(2)f Fr(R)2577 1965 y Fo(2)2622 2002 y Fp(:)75 2206 y Fs(The)41 b(system)h Fp(L)f Fs(is)f(the)i(uncoupled)e (pro)s(duct)g(b)s(et)m(w)m(een)i(the)g(linear)e(oscillating)f (Lagrangian)j(system)75 2319 y Fp(Q)g Fs(on)h Fr(R)51 b Fs(and)42 b(the)h(Lagrangian)f(system)h Fp(U)52 b Fs(on)42 b Fp(M)10 b Fs(.)78 b(It)42 b(is)g(w)m(ell)f(kno)m(wn)h(that)i(if)d ([HU1,2])k(hold)c(the)75 2432 y(Lagrangian)27 b(system)g Fp(U)36 b Fs(has)27 b(an)g(orbit)f Fp(h)p Fs(\()p Fp(t)p Fs(\))h(homo)s(clinic)e(to)i Fp(\022)2231 2446 y Fo(0)2270 2432 y Fs(,)h(see)f(Section)g(5.)40 b(This)25 b(can)i(b)s(e)f(reco)m(v) m(ered)75 2545 y(from)33 b(Theorem)g(1.)49 b(The)33 b(h)m(yp)s(othesis) f([HG3])i(alw)m(a)m(ys)g(holds)e(in)g(this)g(case,)j(and)e(Theorem)g(1) g(giv)m(es)h(the)75 2658 y(existence)e(of)g(an)f(orbit)g(homo)s(clinic) e(to)j(\()p Fp(\022)1567 2672 y Fo(0)1606 2658 y Fp(;)15 b Fs(0\))33 b(for)e Fp(L)p Fs(,)h(whic)m(h)e(of)i(course)g(implies)c (the)k(existence)g(of)g(the)75 2771 y(homo)s(clinic)c(of)j Fp(U)10 b Fs(.)40 b(All)29 b(the)i(orbits)e Fp(O)1406 2785 y Fq(r)1475 2771 y Fs(ha)m(v)m(e)i(a)g(homo)s(clinic)d(for)i Fp(L)g Fs(in)g(this)f(case,)i(giv)m(en)g(b)m(y)1437 2975 y Fp(h)1489 2989 y Fq(r)1527 2975 y Fs(\()p Fp(t)p Fs(\))26 b(=)f(\()p Fp(h)p Fs(\()p Fp(t)p Fs(\))p Fp(;)15 b(r)k Fs(cos)q(\()p 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(coupling,)f(and)g(w)m(e)h(obtain)g(that)g(the)g(radius)e Fp(r)k Fs(tends)e(to)g(zero)h(when)e(the)h(coupling)75 3857 y(tends)34 b(to)i(0.)55 b(The)34 b(orbit)g(obtained)h(b)m(y)f (Theorem)h(1)g(can)h(b)s(e)e(considered)g(as)h(the)g(con)m(tin)m (uation)g(of)g(the)75 3970 y(orbit)41 b(homo)s(clinic)g(to)i(the)f (\014xed)g(p)s(oin)m(t)f(that)i(existed)f(in)f(the)i(uncoupled)d (system.)77 b(Moreo)m(v)m(er,)48 b(the)75 4083 y(h)m(yp)s(othesis)29 b([HG3])j(is)d(satis\014ed)h(when)f(the)i(coupling)d(is)i(small)f (since)g(it)h(can)h(b)s(e)f(written)1143 4275 y(1)p 1143 4316 46 4 v 1143 4399 a(2)1214 4209 y Fk(\022)1281 4337 y Fp(q)1335 4275 y(@)5 b(C)p 1335 4316 125 4 v 1349 4399 a(@)g(q)1490 4337 y Fs(+)20 b Fp(v)1638 4275 y(@)5 b(C)p 1638 4316 V 1650 4399 a(@)g(v)1773 4209 y Fk(\023)1860 4337 y Fm(\000)20 b Fp(C)31 b Fi(6)25 b Fp(U)30 b Fm(\000)20 b Fp(cd)2412 4299 y Fo(2)2452 4337 y Fs(\()p Fp(z)t(;)15 b(z)2615 4351 y Fo(0)2656 4337 y Fs(\))75 4591 y(if)34 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Fp(\017)p Fs(.)56 b(Shortly)-8 b(,)36 b(The)f(homo)s(clinic)e (orbit)i(to)h(the)g(\014xed)f(p)s(oin)m(t)f(that)i(existed)f(in)75 5320 y(the)26 b(uncoupled)e(system)i(is)f(turned)g(to)i(an)e(orbit)g (homo)s(clinic)f(to)j Fp(O)2405 5334 y Fq(r)2469 5320 y Fs(when)e(the)h(coupling)e(is)h(in)m(tro)s(duced,)75 5433 y(with)34 b Fp(r)j Fs(as)e(small)e(as)i(the)g(coupling)f(is)f (small,)i(and)f(this)g(homo)s(clinic)f(exists)h(as)h(long)g(as)g([HG3]) h(holds.)75 5546 y(Com)m(bining)28 b(this)h(with)g(the)i(metho)s(ds)f (of)g(the)h(preceding)e(subsection,)h(w)m(e)h(obtain)1890 5841 y(9)p eop %%Page: 10 10 10 9 bop 75 399 a Fj(Application)35 b(1)46 b Fl(L)-5 b(et)43 b(us)f(c)-5 b(onsider)43 b(a)g(smo)-5 b(oth)45 b(one)d(p)-5 b(ar)g(ameter)45 b(family)e Fp(L)2775 413 y Fq(\017)2850 399 y Fl(of)g(L)-5 b(agr)g(angian)44 b(systems)75 511 y(such)33 b(that)718 624 y Fp(L)780 638 y Fo(0)819 624 y Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))27 b(=)e Fp(a)1292 551 y Fk(\000)1334 624 y Fp(v)1381 587 y Fo(2)1441 624 y Fm(\000)20 b Fp(!)1592 587 y Fo(2)1631 624 y Fp(q)1675 587 y Fo(2)1715 551 y Fk(\001)1776 624 y Fs(+)g Fp(U)10 b Fs(\()p Fp(z)t Fs(\))p Fp(;)110 b Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))27 b Fm(2)e Fp(T)13 b(M)30 b Fm(\002)20 b Fr(R)29 b Fm(\002)20 b Fr(R)75 791 y Fl(satis\014es)26 b([HL])e(and)i([HU1,2].)39 b(Ther)-5 b(e)26 b(is)e(an)i Fp(\017)1665 805 y Fo(0)1729 791 y Fp(>)f Fs(0)h Fl(and)g(a)f(function) g Fp(e)p Fs(\()p Fp(\017)p Fs(\))h Fi(>)f Fs(0)h Fl(satisfying)f Fs(lim)3350 805 y Fq(\017)p Fn(!)p Fo(0)3504 791 y Fp(e)p Fs(\()p Fp(\017)p Fs(\))h(=)75 904 y(0)j Fl(such)f(that)h(for)g Fp(\017)c Fi(6)g Fp(\017)868 918 y Fo(0)936 904 y Fl(the)k(system)g Fp(L)1438 918 y Fq(\017)1498 904 y Fl(has)h(a)e(sadd)5 b(le-c)-5 b(enter)30 b(\014xe)-5 b(d)29 b(p)-5 b(oint)29 b Fp(p)p Fs(\()p Fp(\017)p Fs(\))g Fl(and)g(a)g(c)-5 b(enter)28 b(manifold)75 1017 y Fm(C)5 b Fs(\()p Fp(\017)p Fs(\))27 b Fl(interse)-5 b(cting)26 b(tr)-5 b(ansversal)5 b(ly)28 b(the)f(ener)-5 b(gy)26 b(level)g Fp(E)1964 984 y Fn(\000)p Fo(1)1959 1039 y Fq(\017)2058 943 y Fk(\000)2100 1017 y Fp(e)p Fs(\()p Fp(\017)p Fs(\))2249 943 y Fk(\001)2318 1017 y Fl(along)h(a)f(hyp)-5 b(erb)g(olic)28 b(p)-5 b(erio)g(dic)28 b(tr)-5 b(aje)g(ctory)75 1130 y(which)34 b(has)f(a)g(homo)-5 b(clinic)35 b(orbit.)2439 b Fi(\003)75 1373 y Fd(1.3)112 b(Singular)37 b(p)s(erturbation)75 1545 y Fs(The)30 b(case)h Fp(!)d Fm(\000)-15 b(!)25 b(1)30 b Fs(is)g(of)g(ph)m(ysical)f(in)m (terest.)41 b(Let)31 b(us)f(consider)f(a)i(system)673 1749 y Fp(L)735 1712 y Fq(!)785 1749 y Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))27 b(=)e Fp(a)p Fs(\()p Fp(v)1325 1712 y Fo(2)1386 1749 y Fm(\000)19 b Fp(!)1536 1712 y Fo(2)1576 1749 y Fp(q)1620 1712 y Fo(2)1659 1749 y Fs(\))i(+)f Fp(G)p Fs(\()p Fp(z)t(;)15 b(q)s Fs(\))p Fp(;)108 b Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))27 b Fm(2)e Fp(T)13 b(M)30 b Fm(\002)20 b Fr(R)28 b Fm(\002)20 b Fr(R)s Fp(;)75 1953 y Fs(and)30 b(set)1398 2066 y Fp(G)1469 2080 y Fo(0)1509 2066 y Fs(\()p Fp(z)t 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b(us)f(supp)s(ose)g(that)h(there)g(is)e (a)i(critical)f(p)s(oin)m(t)g Fp(z)2405 2699 y Fo(0)2473 2685 y Fs(=)d(\()p Fp(\022)2651 2699 y Fo(0)2690 2685 y Fp(;)15 b Fs(0\))31 b Fm(2)d Fp(T)13 b(M)43 b Fs(of)33 b Fp(G)3304 2699 y Fo(0)3376 2685 y Fs(suc)m(h)f(that)75 2798 y Fp(G)146 2812 y Fo(0)186 2798 y Fs(\()p Fp(z)263 2812 y Fo(0)303 2798 y Fs(\))26 b(=)f(0)30 b(and)75 2911 y Fj(HG2)35 b(:)41 b Fs(There)29 b(is)h(a)h Fp(b)25 b(>)g Fs(0)31 b(suc)m(h)f(that)h Fp(G)1535 2925 y Fo(0)1575 2911 y Fs(\()p Fp(z)t Fs(\))26 b Fi(>)f Fp(bd)1899 2878 y Fo(2)1939 2911 y Fs(\()p Fp(z)t(;)15 b(z)2102 2925 y Fo(0)2142 2911 y Fs(\))p Fp(:)75 3024 y Fs(The)33 b(p)s(oin)m(t)g Fp(z)548 3038 y Fo(0)619 3024 y Fs(=)e(\()p Fp(\022)799 3038 y Fo(0)838 3024 y Fp(;)15 b Fs(0\))35 b(is)e(then)h(a)g(h)m(yp)s (erb)s(olic)d(rest)j(p)s(oin)m(t)f(of)h(the)g(limit)d(\015o)m(w)j (\(the)h(\015o)m(w)f(of)g Fp(G)3495 3038 y Fo(0)3534 3024 y Fs(\))h(and)75 3137 y(there)30 b(is)e(an)i(orbit)e(of)i Fp(G)918 3151 y Fo(0)987 3137 y Fs(homo)s(clinic)d(to)j(this)f(\014xed) g(p)s(oin)m(t,)g(see)h(Section)f(5.)41 b(It)29 b(is)g(in)m(teresting)g (to)h(study)75 3249 y(the)c(limit)e(pro)s(cess)h(and)h(describ)s(e)e (what)i(remains)e(of)i(this)f(homo)s(clinic)f(orbit)h(in)f(the)j(total) f(\015o)m(w)g(for)g(large)75 3362 y(but)k(\014nite)f Fp(!)s Fs(.)40 b(W)-8 b(e)32 b(will)c(furthermore)h(assume)h(that)h Fp(G)g Fs(satis\014es)75 3475 y Fj(HG1)k(lo)s(c)h(:)k Fs(There)30 b(is)f(is)h(an)g Fp(\017)25 b(>)g Fs(0)31 b(suc)m(h)f(that)h Fp(G)p Fs(\()p Fp(z)1917 3489 y Fo(0)1958 3475 y Fp(;)15 b(q)s Fs(\))25 b(=)g(0)31 b(and)f Fp(dG)p Fs(\()p Fp(z)2646 3489 y Fo(0)2687 3475 y Fp(;)15 b(q)s Fs(\))26 b(=)f(0)30 b(when)g Fm(j)p Fp(q)s Fm(j)25 b Fi(6)g Fp(\017)p Fs(.)75 3588 y Fe(Example)35 b(:)42 b Fs(Let)33 b(us)d(consider)h(a)h(p)s(endulum,)c(in)j(the)h(plane)e(or) i(in)e(space,)j(where)e(the)h(bar)f(is)f(replaced)75 3701 y(b)m(y)i(a)h(sti\013)f(spring)f(whic)m(h)g(has)i(v)-5 b(ariable)31 b(length)h(but)g(remains)f(alw)m(a)m(ys)i(straigh)m(t,)h (see)f(\014gure)f(1,)i(page)f(2.)75 3814 y(The)d(Lagrangian)g(of)h (this)e(system)h(can)h(b)s(e)f(written)712 4018 y Fp(L)p Fs(\()p Fp(\022)s(;)913 3994 y Fs(_)895 4018 y Fp(\022)r(;)15 b(q)s(;)33 b Fs(_)-43 b Fp(q)t Fs(\))25 b(=)42 b(_)-42 b Fp(q)1309 3981 y Fo(2)1369 4018 y Fm(\000)20 b Fp(!)1520 3981 y Fo(2)1559 4018 y Fp(q)1603 3981 y Fo(2)1663 4018 y Fs(+)g(\()p Fp(l)1816 4032 y Fo(0)1876 4018 y Fs(+)g Fp(q)s Fs(\))2046 3981 y Fo(2)2103 3994 y Fs(_)2085 4018 y Fp(\022)2131 3981 y Fo(2)2190 4018 y Fs(+)g(\()p Fp(l)2343 4032 y Fo(0)2403 4018 y Fs(+)g Fp(q)s Fs(\)\(cos)c Fp(')p Fs(\()p Fp(\022)s Fs(\))21 b Fm(\000)f Fs(1\))75 4222 y(where)38 b Fp(\022)k Fm(2)d Fp(S)592 4189 y Fo(2)670 4222 y Fs(is)e(the)i(direction)f(of)h(the)g(spring,)g Fp(')p Fs(\()p Fp(\022)s Fs(\))g(is)f(the)h(angle)f(b)s(et)m(w)m(een)i (the)f(spring)e(and)h(the)75 4335 y(v)m(ertical)h(axis)f(p)s(oin)m (ting)f(up,)k(and)d Fp(l)1347 4349 y Fo(0)1412 4335 y Fs(+)26 b Fp(q)41 b Fs(is)d(the)h(length)g(of)g(the)g(spring,)g Fp(l)2756 4349 y Fo(0)2834 4335 y Fs(b)s(eing)f(its)g(length)g(in)g (the)75 4448 y(unstable)d(equilibrium)d(p)s(osition.)57 b(Let)37 b(us)f(call)f Fp(\022)1841 4462 y Fo(0)1917 4448 y Fs(the)i(v)m(ertical)f(direction)f(p)s(oin)m(ting)g(up,)i(that)g (is)f(the)75 4561 y(direction)24 b(of)i(the)f(unstable)g(equilibrium.) 34 b(It)26 b(is)e(not)i(hard)e(to)i(c)m(hec)m(k)i(that)e(b)s(oth)e(h)m (yp)s(otheses)i(ab)s(o)m(v)m(e)g(hold)75 4674 y(for)f(that)h(system.)39 b(There)25 b(is)g(an)g(unstable)f(in)m(v)-5 b(arian)m(t)24 b(manifold)g(\()p Fp(\022)s(;)2447 4650 y Fs(_)2430 4674 y Fp(\022)r Fs(\))h(=)g(\()p Fp(\022)2709 4688 y Fo(0)2749 4674 y Fp(;)15 b Fs(0\))26 b(\014lled)e(with)g(oscillations)75 4787 y(of)36 b(the)f(spring.)55 b(In)35 b(view)f(of)i(the)g (application)e(b)s(elo)m(w,)i(one)g(of)g(these)g(oscillations)e(ha)m(v) m(e)i(a)g(homo)s(clinic)75 4900 y(orbit)27 b(if)f(the)i(spring)d(is)i (sti\013)g(enough.)39 b(The)27 b(whole)g(structure,)h(cen)m(ter)g (manifold)e(and)g(homo)s(clinic)g(orbit,)75 5013 y(is)e(preserv)m(ed)h (b)m(y)g(a)g(small)f(p)s(erturbation.)37 b(The)24 b(homo)s(clinic)f(of) i(the)g(sti\013)g(elastic)g(p)s(endulum)c(can)k(b)s(e)g(seen)75 5126 y(as)37 b(the)g(con)m(tin)m(uation)f(of)h(the)g(homo)s(clinic)d (that)j(exists)f(in)f(the)i(rigid)e(p)s(endulum,)f(whic)m(h)i(is)f(the) i(limit)75 5239 y(system)28 b(when)f(the)h(sti\013ness)f(tends)g(to)i (in\014nit)m(y)-8 b(.)38 b(Note)29 b(that)f(the)h(energy)f(of)g(the)g (homo)s(clinic)d(orbit)i(do)s(es)75 5352 y(not)32 b(tend)g(to)h(zero)g (in)e(general)h(when)f(the)i(sti\013ness)e(tends)h(to)h(in\014nit)m(y)d (\(or)i(at)h(least)g(w)m(e)f(can)h(not)f(pro)m(v)m(e)75 5464 y(that)c(it)f(do)s(es\))g(although)g(the)g(length)g(of)g(the)g (spring)f(is)g(con)m(v)m(erging)i(to)g Fp(l)2594 5478 y Fo(0)2634 5464 y Fs(.)39 b(The)27 b(homo)s(clinic)e(ha)m(v)m(e)j (small)75 5577 y(but)i(fast)g(oscillations.)1867 5841 y(10)p eop %%Page: 11 11 11 10 bop 75 399 a Fj(Application)35 b(2)46 b Fl(The)h(p)-5 b(oint)47 b Fs(\()p Fp(z)1258 413 y Fo(0)1298 399 y Fp(;)15 b Fs(0)p Fp(;)g Fs(0\))48 b Fl(is)e(a)h(sadd)5 b(le-c)-5 b(enter)47 b(\014xe)-5 b(d)47 b(p)-5 b(oint)48 b(of)e Fp(L)2969 366 y Fq(!)3019 399 y Fl(.)83 b(It)46 b(has)h(a)g(c)-5 b(enter)75 511 y(manifold)36 b Fp(z)d Fs(=)c Fp(z)670 525 y Fo(0)710 511 y Fl(,)35 b(which)g(is)g(\014l)5 b(le)-5 b(d)35 b(with)g(the)g(p)-5 b(erio)g(dic)37 b(orbits)e Fp(O)2373 478 y Fq(!)2370 534 y(r)2452 511 y Fs(=)29 b Fm(f)p Fp(v)2644 478 y Fo(2)2705 511 y Fs(+)22 b Fp(!)2858 478 y Fo(2)2897 511 y Fp(q)2941 478 y Fo(2)3009 511 y Fs(=)29 b Fp(!)s Fm(g)p Fl(.)48 b(Ther)-5 b(e)35 b(is)f(an)75 624 y(ener)-5 b(gy)33 b Fp(E)429 638 y Fn(1)529 624 y Fi(>)25 b Fs(0)33 b Fl(such)g(that)211 810 y Fm(\017)46 b Fl(When)30 b Fp(!)i Fl(is)e(lar)-5 b(ge)30 b(enough)g(ther)-5 b(e)30 b(is)g(an)g(orbit)g Fp(h)1979 777 y Fq(!)2055 810 y Fs(=)2151 736 y Fk(\000)2193 810 y Fp(z)2239 777 y Fq(!)2289 810 y Fp(;)15 b(q)2373 777 y Fq(!)2424 810 y Fp(;)32 b Fs(_)-42 b Fp(q)2508 777 y Fq(!)2559 736 y Fk(\001)2630 810 y Fl(of)29 b Fp(L)2795 777 y Fq(!)2875 810 y Fl(homo)-5 b(clinic)31 b(to)f Fp(O)3505 777 y Fq(!)3502 832 y(r)3585 810 y Fl(with)1752 1090 y Fp(r)e Fi(6)1934 1029 y Fs(1)p 1927 1069 60 4 v 1927 1153 a Fp(!)1997 946 y Fk(r)p 2088 946 162 4 v 2098 1029 a Fp(E)2165 1043 y Fn(1)p 2098 1069 142 4 v 2145 1153 a Fp(a)2275 1090 y Fs(;)211 1357 y Fm(\017)46 b Fl(The)33 b(orbits)h Fp(h)795 1324 y Fq(!)878 1357 y Fl(c)-5 b(onver)g(ge)33 b(to)g Fp(M)43 b Fl(in)32 b(c)-5 b(on\014gur)g(ation)35 b(sp)-5 b(ac)g(e:)1690 1538 y Fm(k)p Fp(q)1779 1505 y Fq(!)1830 1538 y Fm(k)1875 1552 y Fn(1)1996 1538 y Fm(\000)-17 b(\000)d(\000)i(!)2027 1593 y Fq(!)r Fn(!1)2292 1538 y Fs(0;)211 1811 y Fm(\017)46 b Fl(The)33 b(function)g Fp(!)28 b Fm(\000)-15 b(!)1103 1738 y Fk(R)1178 1811 y Fp(d)1225 1778 y Fo(2)1265 1811 y Fs(\()p Fp(z)1346 1778 y Fq(!)1397 1811 y Fp(;)15 b(z)1479 1825 y Fo(0)1519 1811 y Fs(\))33 b Fl(is)g(b)-5 b(ounde)g(d;)211 1998 y Fm(\017)46 b Fl(F)-7 b(or)35 b(any)f(se)-5 b(quenc)g(e)34 b Fp(!)1076 2012 y Fq(n)1150 1998 y Fm(\000)-16 b(!)27 b(1)p Fl(,)34 b(ther)-5 b(e)35 b(is)e(a)i(subse)-5 b(quenc)g(e)33 b Fp(p)2427 2012 y Fq(n)2474 1998 y Fl(,)g(a)h(\014nite)g(numb)-5 b(er)34 b Fp(m)g Fl(of)g(orbits)g Fp(Z)3722 1965 y Fq(i)302 2111 y Fl(of)f Fp(L)471 2125 y Fo(0)543 2111 y Fl(homo)-5 b(clinic)34 b(to)g Fp(z)1150 2125 y Fo(0)1222 2111 y Fl(and)f Fp(m)g Fl(se)-5 b(quenc)g(es)32 b Fp(t)1950 2078 y Fq(i)1950 2133 y(p)2022 2111 y Fl(such)h(that)h Fs(lim)2541 2125 y Fq(p)p Fn(!1)2737 2037 y Fk(\000)2779 2111 y Fp(t)2812 2078 y Fq(i)p Fo(+1)2812 2133 y Fq(p)2950 2111 y Fm(\000)20 b Fp(t)3074 2078 y Fq(i)3074 2133 y(p)3114 2037 y Fk(\001)3180 2111 y Fs(=)25 b Fm(1)33 b Fl(and)1534 2375 y Fp(z)1580 2341 y Fq(!)1624 2349 y Fb(p)1664 2375 y Fs(\()p Fp(t)20 b Fm(\000)g Fp(t)1876 2341 y Fq(i)1876 2397 y(p)1916 2375 y Fs(\))2056 2308 y Fq(C)2111 2284 y Fg(1)2106 2331 y Fb(loc)1998 2374 y Fm(\000)-17 b(\000)c(\000)k(!) 2034 2429 y Fq(p)p Fn(!1)2293 2375 y Fp(Z)2362 2341 y Fq(i)2390 2375 y Fs(\()p Fp(t)p Fs(\))p Fp(:)211 2668 y Fm(\017)46 b Fl(If)39 b Fp(!)j Fl(is)e(lar)-5 b(ge)40 b(enough)f(and)i(\014xe)-5 b(d,)41 b(ther)-5 b(e)40 b(is)g(an)f Fp(\017)e(>)g Fs(0)j Fl(such)f(that)i(any)f(L)-5 b(agr)g(angian)41 b(system)3699 2645 y Fs(~)3688 2668 y Fp(L)302 2781 y Fl(satisfying)d Fm(k)767 2758 y Fs(~)756 2781 y Fp(L)24 b Fm(\000)f Fp(L)998 2748 y Fq(!)1048 2781 y Fm(k)1093 2801 y Fq(C)1148 2782 y Fg(3)1220 2781 y Fi(6)33 b Fp(\017)k Fl(also)i(has)f(a)f(sadd)5 b(le-c)-5 b(enter)38 b(\014xe)-5 b(d)38 b(p)-5 b(oint)38 b(with)g(a)g(c)-5 b(enter)37 b(manifold)302 2894 y(and)d(an)f(orbit)827 2870 y Fs(~)826 2894 y Fp(h)g Fl(homo)-5 b(clinic)34 b(to)g(this)f(c)-5 b(enter)33 b(manifold)h(and)g(such)f(that)2889 2871 y Fs(~)2868 2894 y Fp(E)6 b Fs(\()2977 2870 y(~)2976 2894 y Fp(h)p Fs(\))26 b Fi(6)f Fp(E)3252 2908 y Fn(1)3327 2894 y Fl(.)75 3079 y Fe(Remarks)33 b(:)186 3265 y Fs(1.)46 b(The)39 b(limit)d(con\014guration)i(space)i Fp(M)49 b Fs(=)39 b Fp(M)d Fm(\002)25 b(f)p Fs(0)p Fm(g)40 b Fs(is)e(not)h(in)m(v)-5 b(arian)m(t)38 b(for)h Fp(L)3046 3232 y Fq(!)3135 3265 y Fs(hence)g(the)g(\014xed)302 3378 y(p)s(oin)m(t)c(\()p Fp(z)622 3392 y Fo(0)662 3378 y Fp(;)15 b Fs(0)p Fp(;)g Fs(0\))38 b(do)s(es)e(not)g(ha)m(v)m(e)h(an)m (y)g(homo)s(clinic)c(orbit)j(in)e(general)i(\(its)g(stable)g(and)f (unstable)302 3491 y(manifold)29 b(ha)m(v)m(e)i(dimension)d Fp(n)i Fs(in)f(a)i(2)p Fp(n)20 b Fs(+)g(1-dimensional)28 b(energy)j(shell\).)186 3677 y(2.)46 b(The)34 b(energy)g Fp(E)859 3644 y Fq(!)909 3677 y Fs(\()p Fp(h)996 3644 y Fq(!)1047 3677 y Fs(\))g(is)f(b)s(ounded,)g(but)g(do)s(es)g(not)h (con)m(v)m(erge)i(to)f(zero,)g(or)f(at)g(least)g(w)m(e)h(can)f(not)302 3790 y(pro)m(v)m(e)i(that)g(it)f(do)s(es.)56 b(It)35 b(should)e(b)s(e)i(in)m(teresting)g(to)h(understand)d(whether)i(this)f (is)h(only)f(a)i(side)302 3903 y(e\013ect)c(due)e(to)h(our)f(approac)m (h,)h(or)f(whether)g(it)g(has)g(a)h(ph)m(ysical)e(meaning.)186 4090 y(3.)46 b(It)29 b(should)d(b)s(e)h(p)s(ossible,)g(when)g Fp(M)38 b Fs(is)28 b(not)g(simply)e(connected,)k(to)f(pro)m(v)m(e)g (that)g(the)f Fp(z)3266 4057 y Fq(!)3345 4090 y Fs(is)f(actually)302 4203 y(con)m(v)m(erging)32 b(to)f(a)f(single)f(homo)s(clinic)g(of)h Fp(L)1817 4217 y Fo(0)1856 4203 y Fs(.)186 4389 y(4.)46 b(The)29 b(h)m(yp)s(othesis)e(HG1)j(lo)s(c)f(is)f(an)h(unpleasan)m(t)f (restriction,)h(assumed)f(in)g(order)g(that)i(Theorem)f(1)302 4502 y(can)f(b)s(e)f(readily)g(applied.)37 b(It)28 b(is)f(not)h(hard)e (to)j(see)f(ho)m(w)m(ev)m(er)h(that)g(ev)m(en)f(without)f(this)f (assumption)302 4615 y(a)31 b(saddle-cen)m(ter)g(exists)f(in)g Fp(L)1347 4582 y Fq(!)1428 4615 y Fs(for)g(large)h Fp(!)s Fs(,)f(and)g(it)g(ma)m(y)i(b)s(e)e(p)s(ossible)e(using)h(the)i(tec)m (hniques)f(of)302 4728 y(Section)g(1.1)i(to)f(pro)m(v)m(e)g(that)g(the) g(phenomenon)e(describ)s(ed)f(in)h(the)i(application)e(still)f(o)s (ccurs.)186 4915 y(5.)46 b(Adding)29 b(more)h(than)g(one)g(degree)h(of) g(freedom)f(mak)m(es)h(things)e(m)m(uc)m(h)h(harder.)40 b(Ev)m(en)30 b(in)f(the)h(ideal)302 5028 y(case)39 b(where)e(a)h(cen)m (ter)h(manifold)c(foliated)i(b)m(y)h(quasi-p)s(erio)s(dic)c(tori)j(w)m (ould)g(exist,)i(there)f(w)m(ould)302 5141 y(remain)e(the)h(problem)e (that)i(the)g(in)m(tersection)f(b)s(et)m(w)m(een)h(the)g(cen)m(ter)h (manifold)c(and)i(an)h(energy)302 5254 y(shell)d(w)m(ould)f(con)m(tain) j(families)d(of)i(suc)m(h)g(quasi-p)s(erio)s(dic)c(tori,)37 b(in)c(con)m(trast)k(with)c(our)i(situation)302 5367 y(where)40 b(eac)m(h)h(p)s(erio)s(dic)c(orbit)i(is)g(the)h(in)m (tersection)g(b)s(et)m(w)m(een)g(its)f(energy)h(shell)f(and)g(the)h (cen)m(ter)302 5479 y(manifold.)d(Moreo)m(v)m(er,)27 b(this)22 b(ideal)g(case)i(is)f(not)g(as)h(rigid)d(as)j(our)e(case,)k (since)d(some)h(of)f(the)h(in)m(v)-5 b(arian)m(t)302 5592 y(tori)30 b(are)h(usually)d(destro)m(y)m(ed)j(b)m(y)g(a)f(p)s (erturbation.)1867 5841 y(11)p eop %%Page: 12 12 12 11 bop 186 399 a Fs(6.)46 b(The)30 b(classical)g(p)s(endulum)c(is)k (describ)s(ed)e(b)m(y)i(the)h(Lagrangian)1255 603 y Fp(G)1326 617 y Fo(0)1366 603 y Fs(\()p Fp(\022)s(;)1504 579 y Fs(_)1487 603 y Fp(\022)r Fs(\))25 b(=)g Fm(k)1751 579 y Fs(_)1733 603 y Fp(\022)s Fm(k)1824 565 y Fo(2)1884 603 y Fs(+)20 b(\(1)h Fm(\000)f Fs(cos)c Fp(\022)s Fs(\))p Fp(;)106 b(\022)27 b Fm(2)e Fp(S)2733 565 y Fo(1)2772 603 y Fp(:)302 807 y Fs(It)31 b(is)f(w)m(ell)f(kno)m(wn)h(that)i(in)m (tegrabilit)m(y)d(can)i(b)s(e)f(destro)m(y)m(ed)h(and)f(c)m(haotic)i(b) s(eha)m(vior)e(turned)g(on)g(b)m(y)302 920 y(a)h(time-dep)s(enden)m(t)e (small)g(p)s(erturbation.)39 b(Let)31 b(us)f(consider)f(a)i(system)942 1124 y Fp(L)1004 1087 y Fq(!)1054 1124 y Fs(\()p Fp(\022)s(;)1193 1100 y Fs(_)1175 1124 y Fp(\022)r(;)15 b(q)s(;)33 b Fs(_)-43 b Fp(q)t Fs(\))25 b(=)42 b(_)-42 b Fp(q)1589 1087 y Fo(2)1649 1124 y Fm(\000)20 b Fp(!)1800 1087 y Fo(2)1839 1124 y Fp(q)1883 1087 y Fo(2)1943 1124 y Fs(+)f Fp(G)p Fs(\()p 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(is)f(\014rst)h(necessary)g(to)75 1968 y(c)m(hange)28 b(the)f(Lagrangian)f(function)g(outside)f(a)i(neigh)m(b)s(orho)s(o)s(d) e(of)i Fp(q)h Fs(=)d(0.)39 b(W)-8 b(e)28 b(need)f(a)g(smo)s(oth)f (function)75 2081 y Fp(')k Fs(:)h([0)p Fp(;)15 b Fm(1)p Fs(])31 b Fm(\000)-16 b(!)30 b Fs([0)p Fp(;)15 b Fs(1])35 b(suc)m(h)e(that)h Fp(')p Fm(j)1360 2099 y Fo([0)p Fq(;)p Fo(1])1524 2081 y Fs(=)c(1)k(and)e Fp(')p Fm(j)1967 2099 y Fo([2)p Fq(;)p Fn(1)p Fo(\))2174 2081 y Fs(=)e(0)k(and)e(0)f Fi(>)e Fp(')2768 2048 y Fn(0)2822 2081 y Fi(>)h Fm(\000)p Fs(2.)49 b(Let)34 b(us)e(\014x)h Fp(\016)h(>)c Fs(0)75 2193 y(and)g(de\014ne)996 2306 y Fp(G)1067 2321 y Fq(\016)1106 2306 y Fs(\()p Fp(\022)s(;)15 b(q)s Fs(\))25 b(=)g Fp(')p Fs(\()p Fp(q)s(=\016)s Fs(\))p Fp(G)p Fs(\()p Fp(\022)s(;)15 b(q)s Fs(\))23 b(+)2073 2233 y Fk(\000)2115 2306 y Fs(1)e Fm(\000)e Fp(')p Fs(\()p Fp(q)s(=\016)s Fs(\))2532 2233 y Fk(\001)2576 2306 y Fp(G)2647 2320 y Fo(0)2687 2306 y Fs(\()p Fp(\022)s Fs(\))p Fp(:)75 2473 y Fs(It)35 b(is)f(clear)g (that)i Fp(G)768 2488 y Fq(\016)841 2473 y Fs(satis\014es)e(HG1)h(when) f Fp(\016)39 b Fs(is)33 b(small)h(enough.)53 b(T)-8 b(o)35 b(c)m(hec)m(k)h(the)f(other)g(h)m(yp)s(otheses)g(of)75 2586 y(Theorem)30 b(1)h(let)f(us)g(\014rst)f(notice)i(that)g(there)g (is)e(a)i(constan)m(t)h Fp(D)c(>)d Fs(0)30 b(suc)m(h)g(that)919 2821 y Fm(j)p Fp(G)21 b Fm(\000)f Fp(G)1198 2835 y Fo(0)1238 2821 y Fm(j)26 b Fi(6)f Fp(D)s(\016)s(d)1553 2784 y Fo(2)1593 2821 y Fs(\()p Fp(z)t(;)15 b(z)1756 2835 y Fo(0)1797 2821 y Fs(\))91 b(and)2175 2689 y Fk(\014)2175 2744 y(\014)2175 2798 y(\014)2175 2853 y(\014)2216 2760 y Fp(@)5 b(G)p 2216 2800 125 4 v 2230 2883 a(@)g(q)2351 2689 y Fk(\014)2351 2744 y(\014)2351 2798 y(\014)2351 2853 y(\014)2406 2821 y Fi(6)25 b Fp(D)s(d)2627 2784 y Fo(2)2667 2821 y Fs(\()p Fp(z)t(;)15 b(z)2830 2835 y Fo(0)2870 2821 y Fs(\))75 3083 y(when)28 b Fm(j)p Fp(q)s Fm(j)e Fi(6)f Fs(2)p Fp(\016)s Fs(.)42 b(It)29 b(follo)m(ws)g(from)f(the)i(\014rst)f(estimate)h(ab)s 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y Fo(1)799 2925 y Fs(\([0)p Fp(;)15 b(T)e Fs(])p Fp(;)i(M)10 b Fs(\))15 b Fp(=)g(x)p Fs(\(0\))38 b(=)33 b Fp(x)p Fs(\()p Fp(T)13 b Fs(\))p Fm(g)p Fp(:)36 b Fs(There)f(is)f(a)i(one)f(to)h(one)g (corresp)s(ondence)e(b)s(et)m(w)m(een)75 3038 y(tra)5 b(jectories)31 b Fp(x)g Fs(of)f Fp(L)g Fs(and)g(in)m(tegral)g(curv)m (es)h Fp(z)j Fs(of)d Fp(X)7 b Fs(,)31 b(giv)m(en)f(b)m(y)1184 3204 y Fp(x)c Fm(\000)-16 b(!)25 b Fp(z)30 b Fs(=)25 b Fp(\036)p Fs(\()p Fp(@)5 b(x)p Fs(\))26 b Fp(;)106 b(z)30 b Fm(\000)-15 b(!)25 b Fp(x)g Fs(=)g Fp(\031)2459 3166 y Fn(\003)2498 3204 y Fs(\()p Fp(z)t Fs(\))p Fp(:)75 3369 y Fs(As)i(a)g(consequence,)i(there)e(is)f(a)h(v)m(ector-\014eld)h Fp(Y)47 b Fs(on)26 b Fp(T)13 b(M)37 b Fs(suc)m(h)27 b(that)g Fp(x)g Fs(is)f(a)h(tra)5 b(jectory)29 b(of)e Fp(L)g Fs(if)e(and)i(only) 75 3482 y(if)i Fp(@)5 b(x)31 b Fs(is)e(an)i(in)m(tegral)f(curv)m(e)g (of)h Fp(Y)20 b Fs(,)30 b(and)g(w)m(e)h(ha)m(v)m(e)1411 3648 y Fp(Y)20 b Fs(\()p Fp(z)t Fs(\))27 b(=)d(\()p Fp(d\036)1858 3662 y Fq(z)1899 3648 y Fs(\))1934 3611 y Fn(\000)p Fo(1)2029 3648 y Fs(\()p Fp(X)7 b Fs(\()p Fp(\036)p Fs(\()p Fp(z)t Fs(\)\))p Fp(:)75 3814 y Fs(In)23 b(an)m(y)i(canonical)e(c)m(hart)i(\() p Fp(q)s(;)15 b(v)s Fs(\))25 b(of)g Fp(T)13 b(M)d Fs(,)25 b(the)f(tra)5 b(jectories)25 b(of)g Fp(L)e Fs(satisfy)h(the)g (Euler-Lagrange)g(equations)1275 3961 y Fp(d)p 1258 4002 81 4 v 1258 4085 a(dt)1358 3961 y(@)5 b(L)p 1358 4002 116 4 v 1365 4085 a(@)g(v)1483 4022 y Fs(\()p Fp(q)s Fs(\()p Fp(t)p Fs(\))p Fp(;)33 b Fs(_)-43 b Fp(q)t Fs(\()p Fp(t)p Fs(\)\))27 b(=)2020 3961 y Fp(@)5 b(L)p 2020 4002 V 2029 4085 a(@)g(q)2146 4022 y Fs(\()p Fp(q)s Fs(\()p Fp(t)p Fs(\))p Fp(;)33 b Fs(_)-43 b Fp(q)t Fs(\()p Fp(t)p Fs(\)\))p Fp(:)75 4227 y Fs(The)39 b(Hamiltonian)f(function)h Fp(H)46 b Fs(is)39 b(in)m(v)-5 b(arian)m(t)39 b(along)g(in)m(tegral)h (curv)m(es)g(of)f Fp(X)7 b Fs(,)43 b(and)c(the)h(energy)g Fp(E)45 b Fs(is)75 4340 y(in)m(v)-5 b(arian)m(t)34 b(along)g(in)m (tegral)h(curv)m(es)f(of)h Fp(Y)55 b Fs(hence)34 b Fp(E)5 b Fs(\()p Fp(@)g(x)p Fs(\))36 b(is)e(constan)m(t)h(if)f Fp(x)g Fs(is)g(a)h(tra)5 b(jectory)36 b(of)f Fp(L)p Fs(.)53 b(This)75 4453 y(construction)27 b(can)g(b)s(e)g(rev)m(ersed.)40 b(Let)28 b Fp(H)k Fs(:)25 b Fp(T)1650 4420 y Fn(\003)1689 4453 y Fp(N)36 b Fm(\000)-16 b(!)25 b Fr(R)36 b Fs(b)s(e)27 b(a)g(Hamiltonian)f(function.)38 b(If)27 b(the)h(mapping)1554 4618 y Fp( )h Fs(:)c Fp(T)13 b(N)35 b Fm(\000)-15 b(!)25 b Fp(T)13 b(N)1794 4756 y(z)30 b Fm(7\000)-15 b(!)25 b Fp(H)2114 4770 y Fq(v)2154 4756 y Fs(\()p Fp(z)t Fs(\))75 4922 y(is)k(a)i(di\013eomorphism,)d(whic)m(h)h(happ)s(ens)g(when)g Fp(H)37 b Fs(is)30 b(\014b)s(erwise)e(con)m(v)m(ex)k(and)e(prop)s(er,)f (w)m(e)i(de\014ne)1265 5088 y Fp(L)p Fs(\()p Fp(z)t Fs(\))26 b(=)f(\()p Fp(z)t(;)15 b( )1748 5050 y Fn(\000)p Fo(1)1844 5088 y Fs(\()p Fp(z)t Fs(\)\))21 b Fm(\000)f Fp(H)7 b Fs(\()p Fp( )2287 5050 y Fn(\000)p Fo(1)2382 5088 y Fs(\()p Fp(z)t Fs(\)\))p Fp(;)75 5254 y Fs(the)31 b(asso)s(ciated)h(mapping)e 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Fs(\))g(=)f(0)k(when)e Fp(x)e Fi(6)g Fp(d)j Fs(and)f Fp(f)2545 2652 y Fq(\013)2594 2638 y Fs(\()p Fp(x)p Fs(\))f(=)f Fp(\013x)2960 2605 y Fo(2)3034 2638 y Fs(when)i Fp(x)e Fi(>)g Fp(d)3507 2652 y Fo(1)3547 2638 y Fs(.)52 b(W)-8 b(e)75 2751 y(no)m(w)27 b(consider)f(a)h(function)f Fp(')g Fs(:)f Fr(R)1232 2718 y Fo(+)1322 2751 y Fm(\000)-15 b(!)25 b Fs([0)p Fp(;)15 b Fs(1])28 b(suc)m(h)f(that)h Fp(')p Fs(\()p Fp(x)p Fs(\))e(=)f(1)i (when)f Fp(x)f Fi(6)g Fp(d)2927 2765 y Fo(1)2994 2751 y Fs(and)h Fp(')p Fs(\()p Fp(x)p Fs(\))h(=)e(0)i(when)75 2864 y Fp(x)e Fi(>)g Fp(d)295 2878 y Fo(2)365 2864 y Fs(for)30 b(some)h Fp(d)779 2878 y Fo(2)844 2864 y Fp(>)25 b(d)987 2878 y Fo(1)1057 2864 y Fs(and)30 b(w)m(e)h(set)1099 3068 y Fp(G)1170 3082 y Fo(1)1210 3068 y Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))27 b(=)e Fp(')p Fs(\()p Fm(k)p Fp(z)t Fm(k)p Fs(\))p Fp(G)p Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))25 b(+)20 b Fp(f)2404 3082 y Fq(\013)2453 3068 y Fs(\()p Fm(k)p Fp(z)t Fm(k)2624 3030 y Fo(2)2665 3068 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Fs(\))31 b(on)g Fp(d)589 3625 y Fo(1)654 3611 y Fi(6)25 b Fm(k)p Fp(z)t Fm(k)h Fi(6)f Fp(d)1055 3625 y Fo(2)1095 3611 y Fp(:)30 b Fs(W)-8 b(e)32 b(\014rst)e(note)h(that)1278 3815 y Fp(d)1325 3778 y Fo(2)1325 3838 y Fq(v)1366 3815 y Fs(\()p Fp(G)1472 3829 y Fo(1)1533 3815 y Fs(+)20 b Fp(av)1719 3778 y Fo(2)1759 3815 y Fs(\))25 b(=)g Fp(')p Fs(\()p Fm(k)p Fp(z)t Fm(k)p Fs(\))p Fp(d)2227 3778 y Fo(2)2227 3838 y Fq(v)2271 3815 y Fp(G)20 b Fs(+)g(2)p Fp(a)75 4019 y Fs(is)37 b(p)s(ositiv)m(e)f(since)h Fp(d)794 3986 y Fo(2)794 4042 y Fq(v)836 4019 y Fp(G)25 b Fs(+)g(2)p Fp(a)38 b Fs(is)f(p)s(ositiv)m(e.)61 b(On)37 b(the)h(other)g(hand,)h (it)e(is)f(not)i(hard)f(to)h(see)h(that)f(giv)m(en)75 4132 y Fp(\014)h(>)33 b Fs(0)i(one)h(can)f(tak)m(e)i Fp(\013)f Fs(large)f(enough)g(so)h(that)f Fp(d)1898 4099 y Fo(2)1898 4155 y Fq(\027)1942 4132 y Fp(G)2013 4146 y Fo(1)2086 4132 y Fi(>)e Fp(\014)5 b(I)i(d)36 b Fs(on)f Fp(d)2554 4146 y Fo(1)2627 4132 y Fi(6)e Fm(k)p Fp(z)t Fm(k)i Fi(6)e Fp(d)3053 4146 y Fo(2)3093 4132 y Fp(:)i Fs(Since)f(the)i(cross)75 4245 y(deriv)-5 b(ativ)m(es)30 b Fp(d)575 4259 y Fq(\027)618 4245 y Fp(d)665 4259 y Fq(v)706 4245 y Fs(\()p Fp(G)812 4259 y Fo(1)873 4245 y Fs(+)20 b Fp(av)1059 4212 y Fo(2)1099 4245 y Fs(\))30 b(do)h(not)f(dep)s(end)f(on)h Fp(\013)p Fs(,)h(one)g(can)g(c)m(hec)m(k) g(that)g(the)g(Hessian)1217 4375 y Fk(\024)1309 4447 y Fp(d)1356 4414 y Fo(2)1356 4469 y Fq(\027)1399 4447 y Fs(\()p Fp(G)1505 4461 y Fo(1)1566 4447 y Fs(+)20 b Fp(av)1752 4414 y Fo(2)1792 4447 y Fs(\))127 b Fp(d)2001 4461 y Fq(v)2042 4447 y Fp(d)2089 4461 y Fq(\027)2133 4447 y Fs(\()p Fp(G)2239 4461 y Fo(1)2299 4447 y Fs(+)20 b Fp(av)2485 4414 y Fo(2)2525 4447 y Fs(\))1265 4560 y Fp(d)1312 4574 y Fq(\027)1355 4560 y Fp(d)1402 4574 y Fq(v)1443 4560 y Fs(\()p Fp(G)1549 4574 y Fo(1)1610 4560 y Fs(+)g Fp(av)1796 4527 y Fo(2)1836 4560 y Fs(\))128 b Fp(d)2046 4527 y Fo(2)2046 4582 y Fq(v)2087 4560 y Fs(\()p Fp(G)2193 4574 y Fo(1)2254 4560 y Fs(+)20 b Fp(av)2440 4527 y Fo(2)2480 4560 y Fs(\))2560 4375 y Fk(\025)75 4769 y Fs(is)32 b(p)s(ositiv)m(e)g(de\014nite)g(when)g Fp(\013)h Fs(is)f(large)h(enough.)48 b(The)32 b(function)g Fp(G)2459 4783 y Fo(1)2521 4769 y Fs(+)21 b Fp(av)2708 4736 y Fo(2)2781 4769 y Fs(is)32 b(th)m(us)h(\014b)s(erwise)d(con)m(v)m (ex,)75 4882 y(as)j(w)m(ell)f(as)h(the)g(function)f Fp(L)1070 4896 y Fo(1)1139 4882 y Fs(=)d Fp(a)p Fs(\()p Fp(v)1369 4849 y Fo(2)1431 4882 y Fm(\000)22 b Fp(!)1584 4849 y Fo(2)1623 4882 y Fp(q)1667 4849 y Fo(2)1706 4882 y Fs(\))h(+)e Fp(G)1927 4896 y Fo(1)1967 4882 y Fs(.)49 b(The)32 b(function)g Fp(G)2660 4896 y Fo(1)2733 4882 y Fs(satis\014es)g([HG1-3])j(with)d (the)75 4995 y(same)f(constan)m(t)g Fp(b)p Fs(.)41 b(Let)31 b(us)f(de\014ne)f(the)i(functions)1268 5199 y Fp(U)1330 5213 y Fo(1)1370 5199 y Fs(\()p Fp(z)t Fs(\))26 b(=)f Fp(')p Fs(\()p Fm(k)p Fp(z)t Fm(k)p Fs(\))p Fp(U)10 b Fs(\()p Fp(z)t Fs(\))23 b(+)d Fp(f)2220 5213 y Fq(\013)2269 5199 y Fs(\()p Fm(k)p Fp(z)t Fm(k)2440 5161 y Fo(2)2481 5199 y Fs(\))p Fp(;)1093 b Fs(\(9\))75 5403 y(and)30 b Fp(W)338 5417 y Fo(1)407 5403 y Fs(in)f(the)i(same)g(w)m(a)m(y)-8 b(,)32 b(so)e(that)1610 5516 y Fp(U)1672 5530 y Fo(1)1737 5516 y Fi(6)25 b Fp(G)1904 5530 y Fo(1)1969 5516 y Fi(6)g Fp(W)2151 5530 y Fo(1)2190 5516 y Fp(:)1867 5841 y Fs(17)p eop %%Page: 18 18 18 17 bop 75 399 a Fs(If)30 b Fp(\013)g Fs(has)g(b)s(een)g(c)m(hosen)g (large)h(enough,)f(the)g(functions)f Fp(U)2092 413 y Fo(1)2162 399 y Fs(and)g Fp(W)2424 413 y Fo(1)2494 399 y Fs(are)h(\014b)s(erwise)e(con)m(v)m(ex)k(and)d(satisfy)954 603 y Fp(U)1016 617 y Fo(1)1056 603 y Fs(\()p Fp(z)t Fs(\))d(=)f Fp(U)10 b Fs(\()p Fp(z)t Fs(\))31 b(and)f Fp(W)1776 617 y Fo(1)1815 603 y Fs(\()p Fp(z)t Fs(\))c(=)f Fp(W)13 b Fs(\()p Fp(z)t Fs(\))92 b(when)29 b Fp(z)g Fm(2)c Fp(K)2831 617 y Fo(0)75 807 y Fs(and)1149 920 y Fp(U)1211 934 y Fo(1)1250 920 y Fs(\()p Fp(z)t Fs(\))h(=)f Fp(W)1574 934 y Fo(1)1614 920 y Fs(\()p Fp(z)t Fs(\))h(=)f Fp(\013)p Fm(k)p Fp(z)t Fm(k)2046 882 y Fo(2)2178 920 y Fs(when)k Fp(z)h Fm(62)25 b Fp(K)q(:)75 1087 y Fs(If)d Fp(K)235 1101 y Fo(0)296 1087 y Fs(is)g(su\016cien)m(tly)e(large,)25 b(the)d(de\014nition)e(of)i Fp(I)29 b Fs(is)22 b Fp(I)7 b Fs(\()p Fp(U)j Fs(\))26 b(=)f Fp(I)7 b Fs(\()p Fp(U)2315 1101 y Fo(1)2355 1087 y Fs(\))22 b(and)g Fp(I)7 b Fs(\()p Fp(W)13 b Fs(\))25 b(=)g Fp(I)7 b Fs(\()p Fp(W)3086 1101 y Fo(1)3126 1087 y Fs(\))22 b(\(see)h(De\014nition)75 1200 y(2)31 b(of)f(Section)g(5\).)42 b(So)30 b(the)h(maximal)e(energy)i Fp(E)1731 1214 y Fo(0)1800 1200 y Fs(has)g(not)f(b)s(een)g(c)m(hanged.) 75 1313 y Fe(second)e(step)g(:)38 b Fs(W)-8 b(e)27 b(no)m(w)f(w)m(an)m (t)h(to)f(con)m(trol)h(the)f(b)s(eha)m(vior)f(for)g(large)h Fp(v)s Fs(.)40 b(Let)26 b Fp(B)2887 1327 y Fo(0)2952 1313 y Fs(b)s(e)f(a)h(compact)i(subset)75 1425 y(of)38 b Fp(T)13 b(M)34 b Fm(\002)25 b Fr(R)530 1392 y Fo(2)575 1425 y Fs(,)40 b(w)m(e)e(will)c(de\014ne)j(a)h(function)1696 1403 y(~)1675 1425 y Fp(G)g Fs(suc)m(h)f(that)2221 1403 y(~)2201 1425 y Fp(G)g Fs(=)g Fp(G)2488 1439 y Fo(1)2565 1425 y Fs(on)h Fp(B)2768 1439 y Fo(0)2845 1425 y Fs(and)3049 1403 y(~)3028 1425 y Fp(G)g Fs(=)f Fp(U)3307 1439 y Fo(1)3384 1425 y Fs(outside)g(a)75 1538 y(compact)e(subset)d Fp(B)j Fm(\033)30 b Fp(B)997 1552 y Fo(0)1036 1538 y Fs(.)50 b(T)-8 b(o)34 b(do)f(this)f(w)m(e)i(\014rst)f(observ)m(e)h(that)f Fp(U)2473 1552 y Fo(1)2535 1538 y Fm(\000)22 b Fp(G)2699 1552 y Fo(1)2772 1538 y Fs(is)33 b(b)s(ounded,)f(whic)m(h)g(easily)75 1651 y(implies)19 b(that)k Fp(d)611 1665 y Fq(v)652 1651 y Fp(G)723 1665 y Fo(1)785 1651 y Fs(is)e(b)s(ounded)e(since)j Fp(d)1492 1618 y Fo(2)1492 1674 y Fq(v)1533 1651 y Fp(G)1604 1665 y Fo(1)1669 1651 y Fi(>)j Fm(\000)p Fs(2)p Fp(a)p Fs(.)38 b(W)-8 b(e)23 b(no)m(w)f(tak)m(e)h(a)g(compactly)f(supp)s (orted)e(function)75 1764 y Fp( )34 b Fs(:)d Fr(R)284 1731 y Fo(2)360 1764 y Fm(\000)-15 b(!)31 b Fs([0)p Fp(;)15 b Fs(1])35 b(suc)m(h)e(that)i Fp( )s Fs(\()p Fp(q)s(;)15 b(v)s Fs(\))32 b(=)f(1)j(when)f(there)g(exists)h(a)g Fp(z)h Fm(2)30 b Fp(T)13 b(M)44 b Fs(with)32 b(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))33 b Fm(2)d Fp(B)3504 1778 y Fo(0)3544 1764 y Fs(,)35 b(and)75 1877 y(suc)m(h)30 b(that)h Fm(k)p Fp(d )s Fm(k)676 1897 y Fq(C)731 1878 y Fg(1)797 1877 y Fi(6)25 b Fp(\017)30 b Fs(and)g Fp(q)s(d)1228 1891 y Fq(q)1266 1877 y Fp( )24 b Fs(+)19 b Fp(v)s(d)1533 1891 y Fq(v)1575 1877 y Fp( )29 b Fi(6)c Fs(0.)41 b(The)30 b(function)936 2058 y(~)916 2081 y Fp(G)p Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))28 b(=)d Fp( )s Fs(\()p Fp(q)s(;)15 b(v)s Fs(\))p Fp(G)1732 2095 y Fo(1)1773 2081 y Fs(\()p Fp(z)t(;)g(q)s(;)g(v)s Fs(\))22 b(+)e(\(1)h Fm(\000)f Fp( )s Fs(\()p Fp(q)s(;)15 b(v)s Fs(\)\))p Fp(U)2725 2095 y Fo(1)2767 2081 y Fs(\()p Fp(z)t Fs(\))p Fp(;)75 2286 y Fs(satis\014es)30 b([HG4])i(with)d Fp(G)954 2300 y Fn(1)1054 2286 y Fs(=)c Fp(U)1212 2300 y Fo(1)1252 2286 y Fs(.)40 b(With)30 b(the)h(notation)f Fp(z)g Fs(=)25 b(\()p Fp(\022)s(;)15 b(\027)6 b Fs(\),)30 b(w)m(e)h(deriv)m(e)1075 2490 y Fp(d)1122 2452 y Fo(2)1122 2512 y Fq(\027)1186 2467 y Fs(~)1166 2490 y Fp(G)25 b Fs(=)g Fp( )s(d)1467 2452 y Fo(2)1467 2512 y Fq(\027)1511 2490 y Fp(G)1582 2504 y Fo(1)1643 2490 y Fs(+)19 b(\(1)i Fm(\000)f Fp( )s Fs(\))p Fp(d)2069 2452 y Fo(2)2069 2512 y Fq(\027)2114 2490 y Fp(U)2176 2504 y Fo(1)987 2641 y Fp(d)1034 2655 y Fq(v)1075 2641 y Fp(d)1122 2655 y Fq(\027)1186 2618 y Fs(~)1166 2641 y Fp(G)25 b Fs(=)g Fp( )s(d)1467 2655 y Fq(v)1509 2641 y Fp(d)1556 2655 y Fq(\027)1600 2641 y Fp(G)1671 2655 y Fo(1)1731 2641 y Fs(+)20 b Fp(d)1869 2655 y Fq(v)1910 2641 y Fp( )s Fs(\()p Fp(d)2054 2655 y Fq(\027)2098 2641 y Fp(G)2169 2655 y Fo(1)2229 2641 y Fm(\000)g Fp(d)2367 2655 y Fq(\027)2411 2641 y Fp(U)2473 2655 y Fo(1)2512 2641 y Fs(\))1078 2791 y Fp(d)1125 2754 y Fo(2)1125 2814 y Fq(v)1186 2768 y Fs(~)1166 2791 y Fp(G)25 b Fs(=)g Fp( )s(d)1467 2754 y Fo(2)1467 2814 y Fq(v)1509 2791 y Fp(G)1580 2805 y Fo(1)1640 2791 y Fs(+)20 b Fp(d)1778 2754 y Fo(2)1778 2814 y Fq(v)1819 2791 y Fp( )f Fs(\()p Fp(G)2003 2805 y Fo(1)2063 2791 y Fm(\000)h Fp(U)2216 2805 y Fo(1)2256 2791 y Fs(\))g(+)g(2)p Fp(d)2494 2805 y Fq(v)2536 2791 y Fp( )f(d)2661 2805 y Fq(v)2702 2791 y Fp(G)2773 2805 y Fo(1)2813 2791 y Fp(;)75 2996 y Fs(and)30 b(get)413 3113 y Fk(\015)413 3167 y(\015)413 3222 y(\015)463 3217 y Fp(d)510 3184 y Fo(2)510 3249 y(\()p Fq(v)r(;\027)t Fo(\))665 3217 y Fs(\()721 3194 y(~)700 3217 y Fp(G)21 b Fs(+)f Fp(av)978 3184 y Fo(2)1018 3217 y Fs(\))g Fm(\000)g Fp( )s(d)1273 3184 y Fo(2)1273 3249 y(\()p Fq(v)r(;\027)t Fo(\))1428 3217 y Fs(\()p Fp(G)1534 3231 y Fo(1)1595 3217 y Fs(+)g Fp(av)1781 3184 y Fo(2)1820 3217 y Fs(\))h Fm(\000)f Fs(\(1)h Fm(\000)f Fp( )s Fs(\))p Fp(d)2303 3184 y Fo(2)2303 3249 y(\()p Fq(v)r(;\027)t Fo(\))2458 3217 y Fs(\()p Fp(U)2555 3231 y Fo(1)2615 3217 y Fs(+)g Fp(av)2801 3184 y Fo(2)2841 3217 y Fs(\))2876 3113 y Fk(\015)2876 3167 y(\015)2876 3222 y(\015)2927 3281 y Fn(1)3047 3217 y Fm(\000)-17 b(\000)d(\000)i(!)3105 3275 y Fq(\017)p Fn(!)p Fo(0)3342 3217 y Fs(0)p Fp(;)75 3468 y Fs(whic)m(h)32 b(implies)f(that)871 3445 y(~)850 3468 y Fp(G)j Fs(is)e(\014b)s(erwise) f(con)m(v)m(ex)k(when)e Fp(\017)g Fs(is)g(small)f(enough.)49 b(The)33 b(function)3304 3445 y(~)3284 3468 y Fp(G)g Fs(moreo)m(v)m(er)75 3581 y(satis\014es)d([HG1-2])i(with)d(the)i(same)g (constan)m(t)g Fp(b)g Fs(and)f([HG3])h(follo)m(ws)f(from)g(:)525 3800 y(~)504 3823 y Fp(G)21 b Fm(\000)696 3762 y Fs(1)p 696 3802 46 4 v 696 3886 a(2)752 3750 y Fk(\000)794 3823 y Fp(q)s(d)885 3837 y Fq(q)943 3800 y Fs(~)923 3823 y Fp(G)f Fs(+)g Fp(v)s(d)1199 3837 y Fq(v)1261 3800 y Fs(~)1241 3823 y Fp(G)1312 3750 y Fk(\001)1379 3823 y Fs(=)p Fp( )1512 3722 y Fk(\020)1567 3823 y Fp(G)1638 3837 y Fo(1)1698 3823 y Fm(\000)1799 3762 y Fs(1)p 1799 3802 V 1799 3886 a(2)1854 3750 y Fk(\000)1896 3823 y Fp(q)s(d)1987 3837 y Fq(q)2025 3823 y Fp(G)2096 3837 y Fo(1)2156 3823 y Fs(+)g Fp(v)s(d)2341 3837 y Fq(v)2383 3823 y Fp(G)2454 3837 y Fo(1)2494 3750 y Fk(\001)2535 3722 y(\021)1470 4039 y Fs(+)g(\(1)h Fm(\000)f Fp( )s Fs(\))p Fp(U)1912 4053 y Fo(1)1973 4039 y Fs(+)2073 3977 y(1)p 2073 4018 V 2073 4101 a(2)2129 4039 y(\()p Fp(U)2226 4053 y Fo(1)2286 4039 y Fm(\000)g Fp(G)2448 4053 y Fo(1)2488 4039 y Fs(\))2523 3965 y Fk(\000)2565 4039 y Fp(q)s(d)2656 4053 y Fq(q)2694 4039 y Fp( )k Fs(+)c Fp(v)s(d)2962 4053 y Fq(v)3003 4039 y Fp( )3065 3965 y Fk(\001)1475 4215 y Fi(>)25 b Fp( )s(b)15 b(d)1734 4177 y Fo(2)1775 4215 y Fs(\()p Fp(z)t(;)g(z)1938 4229 y Fo(0)1978 4215 y Fs(\))21 b(+)f(\(1)h Fm(\000)f Fp( )s Fs(\))p Fp(b)15 b(d)2515 4177 y Fo(2)2555 4215 y Fs(\()p Fp(z)t(;)g(z)2718 4229 y Fo(0)2759 4215 y Fs(\))26 b(=)f Fp(b)15 b(d)3017 4177 y Fo(2)3057 4215 y Fs(\()p Fp(z)t(;)g(z)3220 4229 y Fo(0)3260 4215 y Fs(\))p Fp(:)75 4432 y Fs(The)32 b(function)632 4409 y(~)621 4432 y Fp(L)c Fs(=)g Fp(a)p Fs(\()p Fp(v)940 4399 y Fo(2)1002 4432 y Fm(\000)21 b Fp(!)1154 4399 y Fo(2)1193 4432 y Fp(q)1237 4399 y Fo(2)1277 4432 y Fs(\))g(+)1446 4409 y(~)1425 4432 y Fp(G)33 b Fs(satis\014es)f(all)f(the)h(h)m(yp)s(otheses)g(of)g (Theorem)g(1)g(with)f(the)i(same)75 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b(b)m(y)j(Bolotin)e([5)q(],)i(and)f(then)g(b)m (y)g(sev)m(eral)h(authors)f(\(see)h(e.g.)41 b([2])28 b(,[25)q(]\).)40 b(These)27 b(w)m(orks)h(ha)m(v)m(e)75 1053 y(also)k(b)s(een)f(extended)h(to)h(more)f(general)g(Hamiltonian)e (systems)i(in)f([16)q(])h(and)g([10)q(],)h([11)q(].)46 b(W)-8 b(e)33 b(just)e(giv)m(e)75 1166 y(here)j(a)g(presen)m(tation)g (of)g(these)g(results)f(that)i(will)c(b)s(e)i(useful)f(in)h(the)h(pro)s (of)f(of)h(Theorem)g(1.)52 b(W)-8 b(e)35 b(shall)75 1279 y(\014rst)30 b(study)f(Lagrangians)h Fp(U)41 b Fs(with)29 b(the)h(additional)e(h)m(yp)s(othesis)75 1392 y Fj(HU3)35 b(:)41 b Fs(There)30 b(exists)g(an)g Fp(\013)c(>)f Fs(0)30 b(suc)m(h)g(that)h Fp(U)10 b Fs(\()p Fp(z)t Fs(\))27 b(=)e Fp(\013)p Fm(k)p Fp(z)t Fm(k)2196 1359 y Fo(2)2267 1392 y Fs(outside)30 b(a)g(compact)i(set)f(of)g Fp(T)13 b(M)d Fs(.)75 1505 y(Although)30 b(w)m(e)h(are)g(in)m(terested)f (mainly)f(in)g(the)i(homo)s(clinic)d(orbit,)i(w)m(e)h(shall)e(use)h(a)h (v)-5 b(ariational)30 b(setting)75 1618 y(for)g Fp(T)13 b Fs(-p)s(erio)s(dic)28 b(orbits)h(of)i Fp(U)10 b Fs(:)40 b(Let)31 b(\003)1384 1632 y Fq(T)1470 1618 y Fs(b)s(e)f(the)g(manifold) e(of)j Fp(H)2313 1585 y Fo(1)2352 1618 y Fs(-lo)s(ops)1484 1815 y Fp(\015)f Fs(:)c Fp(S)1668 1829 y Fq(T)1748 1815 y Fs(=)f Fr(R)r Fp(=)q(T)13 b Fr(Z)27 b Fm(!)e Fp(M)5 b(;)75 2013 y Fs(the)31 b(action)f(functional)1357 2211 y Fm(U)1414 2225 y Fq(T)1494 2211 y Fs(:)c(\003)1608 2225 y Fq(T)1688 2211 y Fm(\000)-15 b(!)25 b Fr(R)1611 2421 y Fp(\015)30 b Fm(7\000)-15 b(!)1860 2297 y Fk(Z)1951 2323 y Fq(T)1910 2503 y Fo(0)2021 2421 y Fp(U)10 b Fs(\()p Fp(@)5 b(\015)g Fs(\()p Fp(t)p Fs(\)\))15 b Fp(dt)75 2659 y Fs(is)33 b(smo)s(oth)h(and)g(satis\014es)f(the)i(P)m (alais-Smale)e(condition.)51 b(W)-8 b(e)35 b(note)g Fp(H)41 b Fs(the)35 b(rational)e(cohomology)-8 b(.)54 b(A)75 2772 y(p)s(oin)m(ted)36 b(set)h(\()p Fp(S;)15 b(s)p Fs(\))38 b(is)e(a)h(set)g Fp(S)42 b Fs(with)36 b(a)h(distinguished)c(elemen)m(t) k Fp(s)f Fm(2)g Fp(S)5 b Fs(.)60 b(W)-8 b(e)38 b(will)c(use)j(the)g (notation)75 2885 y Fp(H)7 b Fs(\()p Fp(S;)15 b(s)p Fs(\))30 b(for)g(the)g(relativ)m(e)g(cohomology)h Fp(H)7 b Fs(\()p Fp(S;)15 b Fm(f)p Fp(s)p Fm(g)p Fs(\).)42 b(F)-8 b(or)31 b(an)m(y)f(closed)f(subset)h Fp(S)k Fs(of)c(\003)3097 2899 y Fq(T)3182 2885 y Fs(con)m(taining)g(the)75 2998 y(constan)m(t)i(lo)s(op)d Fp(\022)f Fm(\021)d Fp(\022)851 3012 y Fo(0)920 2998 y Fs(w)m(e)31 b(consider)e(the)i(morphism)1372 3196 y Fp(i)1403 3158 y Fn(\003)1403 3218 y Fq(S)1479 3196 y Fs(:)26 b Fp(H)7 b Fs(\(\003)1711 3210 y Fq(T)1766 3196 y Fp(;)15 b(\022)1849 3210 y Fo(0)1889 3196 y Fs(\))25 b Fm(\000)-15 b(!)25 b Fp(H)7 b Fs(\()p Fp(S;)15 b(\022)2378 3210 y Fo(0)2418 3196 y Fs(\))75 3394 y(asso)s(ciated)31 b(with)e(the)h(inclusion.)75 3576 y Fj(De\014nition)35 b(1)46 b Fl(We)28 b(c)-5 b(al)5 b(l)30 b Fs(\006)1052 3590 y Fq(T)1136 3576 y Fl(the)f(family)h(of)f(al)5 b(l)30 b(c)-5 b(omp)g(act)31 b(subsets)e Fp(\033)j Fl(of)d Fs(\003)2686 3590 y Fq(T)2770 3576 y Fl(c)-5 b(ontaining)30 b Fp(\022)3253 3590 y Fo(0)3321 3576 y Fl(and)g(having)75 3689 y(induc)-5 b(e)g(d)34 b(c)-5 b(ohomolo)g(gy,)35 b(i.e.)42 b(such)32 b(that)i Fp(i)1502 3656 y Fn(\003)1502 3711 y Fq(\033)1575 3689 y Fm(6)p Fs(=)25 b(0)p Fl(.)75 3871 y Fs(The)30 b(distinguished)c(lev)m(el)1499 3984 y Fp(I)1539 3998 y Fq(T)1594 3984 y Fs(\()p Fp(U)10 b Fs(\))26 b(=)64 b(inf)1858 4044 y Fq(\033)r Fn(2)p Fo(\006)1998 4055 y Fb(T)2062 3984 y Fs(sup)2109 4056 y Fq(\033)2214 3984 y Fm(U)2271 3998 y Fq(T)75 4187 y Fs(satis\014es:)75 4369 y Fj(Lemma)33 b(2)45 b Fl(Ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(onstant)34 b Fp(M)h(>)25 b Fs(0)33 b Fl(indep)-5 b(endent)35 b(of)d Fp(T)46 b Fl(such)32 b(that)i Fs(0)26 b Fp(<)f(I)3101 4383 y Fq(T)3156 4369 y Fs(\()p Fp(U)10 b Fs(\))26 b Fi(6)f Fp(M)5 b(:)75 4552 y Fe(Pr)n(oof)28 b(:)38 b Fs(T)-8 b(o)25 b(pro)m(v)m(e)h(that)g Fp(I)1055 4566 y Fq(T)1110 4552 y Fs(\()p Fp(U)10 b Fs(\))26 b Fp(>)f Fs(0,)i(w)m(e)e(tak)m(e)i(a) e(small)f(disk)g Fp(D)k Fm(2)c Fp(M)36 b Fs(cen)m(tered)26 b(at)f Fp(\022)3095 4566 y Fo(0)3135 4552 y Fs(,)h(and)e(let)h(\003) 3546 4566 y Fq(T)3602 4552 y Fs(\()p Fp(D)s Fs(\))75 4664 y(b)s(e)j(the)h(set)g(of)f Fp(H)676 4631 y Fo(1)744 4664 y Fs(lo)s(ops)g(in)f Fp(D)s Fs(.)40 b(It)29 b(is)e(not)i(hard)f (to)h(see)g(that)g(\003)2294 4678 y Fq(T)2349 4664 y Fs(\()p Fp(D)s Fs(\))g(is)f(con)m(tractible,)h(th)m(us)g Fp(i)3367 4631 y Fn(\003)3367 4696 y Fo(\003)3416 4707 y Fb(T)3464 4696 y Fo(\()p Fq(D)r Fo(\))3609 4664 y Fs(=)c(0)75 4789 y(and)33 b Fp(i)286 4756 y Fn(\003)286 4811 y Fq(\033)363 4789 y Fs(=)d(0)k(for)f(all)g Fp(\033)g Fm(\032)d Fs(\003)1064 4803 y Fq(T)1119 4789 y Fs(\()p Fp(D)s Fs(\))k(con)m(taining)f Fp(\022)1789 4803 y Fo(0)1828 4789 y Fs(.)50 b(F)-8 b(rom)34 b(this)e(follo)m(ws)h(that)h(all)e Fp(\033)h Fm(2)d Fs(\006)j(m)m(ust)h (con)m(tain)75 4902 y(a)h(curv)m(e)f(lea)m(ving)g Fp(D)s Fs(.)52 b(Suc)m(h)34 b(a)h(curv)m(e)f(has)g(its)g(action)h(b)s(ounded)d (a)m(w)m(a)m(y)k(from)e(0.)53 b(T)-8 b(o)34 b(pro)m(v)m(e)h(the)g (second)75 5015 y(inequalit)m(y)-8 b(,)29 b(let)i(us)e(in)m(tro)s(duce) h(the)g(set)h(of)g(lo)s(ops)e(starting)h(at)h Fp(\022)2297 5029 y Fo(0)1315 5212 y Fs(\003)1378 5175 y Fo(0)1378 5235 y Fq(T)1459 5212 y Fs(=)25 b Fm(f)p Fp(\022)s Fs(\()p Fp(t)p Fs(\))g Fm(2)g Fs(\003)1923 5226 y Fq(T)2004 5212 y Fp(=)g(\022)s Fs(\(0\))h(=)f Fp(\022)2400 5226 y Fo(0)2439 5212 y Fm(g)p Fp(:)75 5410 y Fs(W)-8 b(e)32 b(need)e(the)75 5592 y Fj(Lemma)j(3)45 b Fl(Ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(omp)g(act)35 b(subset)d Fp(K)g Fm(\032)25 b Fs(\003)2021 5559 y Fo(0)2021 5617 y(1)2093 5592 y Fl(such)33 b(that)h Fp(i)2517 5559 y Fn(\003)2517 5619 y Fq(K)2611 5592 y Fm(6)p Fs(=)25 b(0)p Fl(.)1867 5841 y Fs(19)p eop %%Page: 20 20 20 19 bop 75 399 a Fe(Pr)n(oof)33 b(of)g(Lemma)g(3:)40 b Fs(This)28 b(is)g(v)m(ery)i(classical,)f(and)g(w)m(e)h(shall)e(only)h (outline)f(the)i(pro)s(of.)39 b(If)30 b Fp(M)39 b Fs(is)29 b(not)75 511 y(simply)k(connected,)38 b(there)e(is)e(a)i(non)f(con)m (tractible)h(curv)m(e)f Fp(\015)k Fm(2)34 b Fs(\003)2418 478 y Fo(0)2418 536 y(1)2457 511 y Fs(.)56 b(W)-8 b(e)37 b(tak)m(e)g Fp(K)k Fs(=)33 b Fm(f)p Fp(\015)5 b(;)15 b(\022)3304 525 y Fo(0)3344 511 y Fm(g)p Fs(,)37 b(and)e(see)75 624 y(that)g Fp(i)307 591 y Fn(\003)307 651 y Fq(K)375 624 y Fs(\()p Fp(H)493 591 y Fo(0)533 624 y Fs(\(\003)631 638 y Fo(1)671 624 y Fp(;)15 b(\022)754 638 y Fo(0)793 624 y Fs(\)\))32 b Fm(6)p Fs(=)f(0.)52 b(Things)32 b(are)i(m)m(uc)m(h)g (harder)f(when)g Fp(M)44 b Fs(is)33 b(simply)e(connected.)53 b(Let)34 b(us)f(set)75 737 y Fp(C)f Fs(=)25 b Fp(C)340 704 y Fo(0)379 737 y Fs(\()p Fp(S)475 704 y Fo(1)514 737 y Fp(;)15 b(M)10 b Fs(\))22 b(and)f Fp(C)949 704 y Fo(0)1013 737 y Fs(=)k Fm(f)p Fp(\015)31 b Fm(2)24 b Fp(C)1389 704 y Fo(0)1428 737 y Fs(\()p Fp(S)1524 704 y Fo(1)1564 737 y Fp(;)15 b(M)10 b Fs(\))26 b Fp(=)g(\015)5 b Fs(\(0\))26 b(=)f Fp(\022)2166 751 y Fo(0)2206 737 y Fs(\),)e(the)e(inclusion)d Fp(i)2837 751 y Fo(\003)2886 760 y Fg(0)2950 737 y Fs(:)26 b(\(\003)3099 704 y Fo(0)3139 737 y Fp(;)15 b(\022)3222 751 y Fo(0)3261 737 y Fs(\))26 b Fm(\000)-16 b(!)26 b Fs(\(\003)p Fp(;)15 b(\022)3675 751 y Fo(0)3715 737 y Fs(\))75 850 y(is)30 b(homotop)m(y)i(equiv)-5 b(alen)m(t)31 b(to)g(the)h(inclusion)c Fp(ic)f Fs(:)f(\()p Fp(C)1929 817 y Fo(0)1969 850 y Fp(;)15 b(\022)2052 864 y Fo(0)2091 850 y Fs(\))27 b Fm(\000)-16 b(!)27 b Fs(\()p Fp(C)q(;)15 b(\022)2510 864 y Fo(0)2550 850 y Fs(\).)43 b(A)31 b(theorem)h(of)f(Sulliv)-5 b(an)28 b(giv)m(es)75 963 y(the)40 b(existence)f(of)h(in\014nitely)c(man)m(y)k(nonzero)f (rational)g(Betti)h(n)m(um)m(b)s(ers)e(of)i(the)f(space)h Fp(C)3309 930 y Fo(0)3348 963 y Fs(\()p Fp(S)3444 930 y Fo(1)3484 963 y Fp(;)15 b(M)10 b Fs(\))40 b(if)75 1076 y Fp(\031)127 1090 y Fo(1)166 1076 y Fs(\()p Fp(M)10 b Fs(\))26 b(=)f(0,)31 b(see)g([28)q(],)g(page)g(46.)42 b(Then,)30 b(w)m(e)g(consider)g(the)g(Serre)g(\014bration)1636 1261 y Fp(C)90 b Fm(\000)-16 b(!)83 b Fp(M)1656 1399 y(\015)88 b Fm(7\000)-16 b(!)83 b Fp(\015)5 b Fs(\(0\))75 1584 y(of)33 b(\014b)s(er)e Fp(C)465 1551 y Fo(0)536 1584 y Fs(to)i(pro)m(v)m(e)h(that)f Fp(ic)1166 1551 y Fn(\003)1238 1584 y Fs(is)f(nonzero,)h(hence)g Fp(i)1984 1551 y Fn(\003)1984 1611 y Fo(\003)2033 1620 y Fg(0)2104 1584 y Fs(is)f(nonzero.)47 b(W)-8 b(e)34 b(no)m(w)e(use)h(brok)m(en)f (geo)s(desics)75 1697 y(appro)m(ximation,)e(see)h([7],)g(to)g(\014nd)e (a)i(compact)g Fp(K)37 b Fs(represen)m(ting)30 b(this)f(cohomology)-8 b(.)654 b Fi(\003)75 1810 y Fe(Pr)n(oof)35 b(of)h(Lemma)f(2:)44 b Fs(F)-8 b(or)33 b(an)m(y)f Fp(T)41 b Fi(>)27 b Fs(1,)33 b(w)m(e)g(can)f(extend)g(lo)s(ops)f(in)g(\003)2643 1777 y Fo(0)2643 1834 y(1)2714 1810 y Fs(to)i([0)p Fp(;)15 b(T)e Fs(])33 b(b)m(y)f(\014xing)e(them)i(in)75 1923 y Fp(\022)118 1937 y Fo(0)187 1923 y Fs(outside)e([0)p Fp(;)15 b Fs(1],)32 b(this)e(de\014nes)f(the)i(injection)1079 2108 y Fp(j)1116 2122 y Fq(T)1197 2108 y Fs(:)25 b(\(\003)1345 2071 y Fo(0)1345 2131 y(1)1385 2108 y Fp(;)15 b(\022)1468 2122 y Fo(0)1508 2108 y Fs(\))83 b Fm(\000)-15 b(!)83 b Fs(\(\003)1954 2071 y Fo(0)1954 2131 y Fq(T)2009 2108 y Fp(;)15 b(\022)2092 2122 y Fo(0)2132 2108 y Fs(\))1395 2246 y Fp(\022)s Fs(\()p Fp(t)p Fs(\))82 b Fm(7\000)-15 b(!)83 b Fp(j)1893 2260 y Fq(T)1948 2246 y Fs(\()p Fp(\022)s Fs(\()p Fp(t)p Fs(\)\))26 b(=)f Fp(\022)s Fs(\(min)n(\(1)p Fp(;)15 b(t)p Fs(\)\))75 2431 y(whic)m(h)29 b(is)h(homotopic)g(to)h (the)g(di\013eomorphism)1240 2616 y(\(\003)1338 2579 y Fo(0)1338 2639 y(1)1378 2616 y Fp(;)15 b(\022)1461 2630 y Fo(0)1500 2616 y Fs(\))84 b Fm(\000)-16 b(!)83 b Fs(\(\003)1946 2579 y Fo(0)1946 2639 y Fq(T)2002 2616 y Fp(;)15 b(\022)2085 2630 y Fo(0)2124 2616 y Fs(\))1387 2754 y Fp(\022)s Fs(\()p Fp(t)p Fs(\))83 b Fm(7\000)-16 b(!)83 b Fp(S)1904 2768 y Fq(T)1959 2754 y Fs(\()p Fp(\022)s Fs(\()p Fp(t)p Fs(\)\))26 b(=)f Fp(\022)s Fs(\()p Fp(t=T)13 b Fs(\))p Fp(:)75 2939 y Fs(It)30 b(follo)m(ws)g(that)h Fp(j)706 2953 y Fq(T)761 2939 y Fs(\()p Fp(K)7 b Fs(\))26 b Fm(2)f Fs(\006)1093 2953 y Fq(T)1148 2939 y Fs(,)30 b(th)m(us)1387 3125 y Fp(I)1427 3139 y Fq(T)1482 3125 y Fs(\()p Fp(U)10 b Fs(\))26 b Fi(6)54 b Fs(sup)1745 3207 y Fq(j)1774 3218 y Fb(T)1823 3207 y Fo(\()p Fq(K)5 b Fo(\))1957 3125 y Fm(U)2014 3139 y Fq(T)2094 3125 y Fs(=)25 b(sup)2226 3203 y Fq(K)2342 3125 y Fm(U)2399 3139 y Fo(1)75 3377 y Fs(b)s(ecause)41 b(the)f(tra)5 b(jectory)42 b Fp(t)h Fm(7\000)-16 b(!)42 b Fp(\022)1330 3391 y Fo(0)1410 3377 y Fs(has)e(zero)i(action.)72 b(This)38 b(ends)i(the)h(pro)s(of)f(of)h(the)f(lemma)h(since)75 3489 y(sup)212 3511 y Fq(K)295 3489 y Fm(U)352 3503 y Fo(1)422 3489 y Fs(is)29 b(a)i(\014nite)f(n)m(um)m(b)s(er.)2530 b Fi(\003)75 3602 y Fs(There)35 b(is)f(a)h Fp(T)13 b Fs(-p)s(erio)s(dic)33 b(tra)5 b(jectory)36 b Fp(\015)1445 3616 y Fq(T)1536 3602 y Fs(suc)m(h)e(that)i Fm(U)2004 3616 y Fq(T)2059 3602 y Fs(\()p Fp(\015)2141 3616 y Fq(T)2196 3602 y Fs(\))e(=)f Fp(I)2409 3616 y Fq(T)2464 3602 y Fs(\()p Fp(U)10 b Fs(\))p Fp(:)36 b Fs(W)-8 b(e)36 b(shall)e(not)h(pro) m(v)m(e)h(it)f(since)75 3715 y(it)c(is)g(v)m(ery)h(classical,)g(and)f (in)m(v)m(olv)m(es)h(argumen)m(ts)g(simpler)e(than)h(those)i(of)f (Section)f(8.)46 b(Here)32 b(non-trivial)75 3828 y(means)e(that)h Fp(\015)597 3842 y Fq(T)678 3828 y Fm(6\021)25 b Fp(\022)817 3842 y Fo(0)856 3828 y Fs(.)40 b(W)-8 b(e)32 b(can)f(de\014ne)e(the)i (n)m(um)m(b)s(er)1492 4013 y Fp(I)7 b Fs(\()p Fp(U)j Fs(\))26 b(=)f(lim)15 b(inf)1811 4074 y Fq(T)10 b Fn(\000)-12 b(!1)2070 4013 y Fp(I)2110 4027 y Fq(T)2165 4013 y Fs(\()p Fp(U)10 b Fs(\))p Fp(:)75 4227 y Fs(There)30 b(m)m(ust)g(b)s(e)g(a)h (non)m(trivial)d(homo)s(clinic)g(orbit)i(to)h Fp(z)2010 4241 y Fo(0)2080 4227 y Fs(suc)m(h)f(that)1450 4323 y Fk(Z)1541 4350 y Fn(1)1500 4529 y(\0001)1645 4447 y Fp(U)10 b Fs(\()j(_)-38 b Fp(\015)6 b Fs(\()p Fp(t)p Fs(\)\))15 b Fp(dt)26 b Fi(6)f Fp(I)7 b Fs(\()p Fp(U)j Fs(\))p Fp(;)75 4680 y Fs(w)m(e)29 b(obtain)g(it)f(as)h(an)g(accum)m(ulation)f(p)s(oin) m(t)g(of)h(the)g(sequence)g Fp(\015)2284 4694 y Fq(T)2368 4680 y Fs(of)g(p)s(erio)s(dic)d(orbits,)j(compare)g(Section)75 4793 y(7.)41 b(The)30 b(follo)m(wing)f(prop)s(osition)f(is)h(useful)g (for)h(applications)75 4965 y Fj(Prop)s(osition)36 b(3)46 b Fl(The)33 b(function)g Fp(U)i Fm(7\000)-16 b(!)26 b Fp(I)7 b Fs(\()p Fp(U)j Fs(\))33 b Fl(is)f(incr)-5 b(e)g(asing)34 b(and)g(c)-5 b(ontinuous:)1405 5150 y Fp(U)35 b Fi(6)25 b Fp(W)37 b Fs(=)-15 b Fm(\))25 b Fp(I)7 b Fs(\()p Fp(U)j Fs(\))26 b Fi(6)f Fp(I)7 b Fs(\()p Fp(W)13 b Fs(\))1256 5253 y Fk(\014)1256 5308 y(\014)1256 5362 y(\014)1256 5417 y(\014)1287 5385 y Fs(1)20 b Fm(\000)1453 5324 y Fp(I)7 b Fs(\()p Fp(W)13 b Fs(\))p 1453 5364 217 4 v 1466 5447 a Fp(I)7 b Fs(\()p Fp(U)j Fs(\))1679 5253 y Fk(\014)1679 5308 y(\014)1679 5362 y(\014)1679 5417 y(\014)1735 5385 y Fi(6)25 b Fs(sup)1881 5457 y Fq(z)1993 5324 y Fm(j)p Fp(W)13 b Fs(\()p Fp(z)t Fs(\))21 b Fm(\000)f Fp(U)10 b Fs(\()p Fp(z)t Fs(\))p Fm(j)p 1993 5364 567 4 v 2086 5447 a Fp(b)15 b(d)2187 5421 y Fo(2)2227 5447 y Fs(\()p Fp(z)t(;)g(z)2390 5461 y Fo(0)2430 5447 y Fs(\))75 5592 y Fl(for)33 b(al)5 b(l)33 b Fp(U)43 b Fl(and)33 b Fp(W)45 b Fl(satisfying)34 b([HU1-3].)41 b(R)-5 b(e)g(c)g(al)5 b(l)34 b(that)g Fp(b)e Fl(is)h(the)g(c)-5 b(onstant)34 b(of)f([HU2].)1867 5841 y Fs(20)p eop %%Page: 21 21 21 20 bop 75 399 a Fe(Pr)n(oof)34 b(:)40 b Fs(The)30 b(monotonicit)m(y)h(is)e(clear,)i(w)m(e)f(shall)f(pro)m(v)m(e)i (regularit)m(y)-8 b(.)41 b(Let)31 b(us)e(set)1444 644 y Fp(\025)c Fs(=)g(sup)1669 716 y Fq(z)1780 583 y Fm(j)p Fp(W)13 b Fs(\()p Fp(z)t Fs(\))21 b Fm(\000)f Fp(U)10 b Fs(\()p Fp(z)t Fs(\))p Fm(j)p 1780 623 567 4 v 1810 706 a Fp(b)15 b(d)1911 680 y Fo(2)1951 706 y Fs(\()p Fp(\031)s Fs(\()p Fp(z)t Fs(\))p Fp(;)g(\022)2240 720 y Fo(0)2281 706 y Fs(\))2356 644 y Fp(;)75 896 y Fs(w)m(e)31 b(obtain)f(using)f([HU2])i(that)1382 1009 y(\(1)21 b Fm(\000)f Fp(\025)p Fs(\))p Fp(U)36 b Fi(6)25 b Fp(W)37 b Fi(6)25 b Fs(\(1)c(+)f Fp(\025)p Fs(\))p Fp(U:)75 1175 y Fs(This)29 b(yields)1129 1288 y(\(1)21 b Fm(\000)e Fp(\025)p Fs(\))p 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Fq(U)1070 2049 y Fs(is)g(prop)s(er)g(and)g (the)h(sets)g Fp(E)2043 2016 y Fq(e)2038 2075 y(U)2124 2049 y Fs(=)c Fm(f)p Fp(E)2335 2063 y Fq(U)2422 2049 y Fi(6)f Fp(e)p Fm(g)32 b Fs(are)h(compact.)46 b(Let)32 b Fm(E)3408 2063 y Fq(U)3499 2049 y Fs(b)s(e)f(the)75 2161 y(set)37 b(of)f(all)f(Lagrangians)i Fp(U)1045 2175 y Fo(1)1120 2161 y Fs(satisfying)e([HU1-3])k(and)c(suc)m(h)h(that)h Fp(U)2534 2175 y Fo(1)2609 2161 y Fs(=)e Fp(U)46 b Fs(on)36 b Fp(E)3027 2128 y Fq(e)3022 2188 y(U)3117 2161 y Fs(for)g(some)h Fp(e)e(>)g Fs(0.)75 2274 y(Elemen)m(ts)30 b(of)h Fm(E)620 2288 y Fq(U)709 2274 y Fs(can)g(b)s(e)e(constructed)i(b)m(y)f(the)h (metho)s(ds)f(of)g(Section)g(4.)75 2461 y Fj(De\014nition)35 b(2)46 b Fl(F)-7 b(or)34 b(al)5 b(l)33 b(L)-5 b(agr)g(angian)34 b(function)f Fp(U)43 b Fl(satisfying)33 b([HU1,2],)f(we)h(set)1647 2664 y Fp(I)7 b Fs(\()p Fp(U)j Fs(\))26 b(=)f Fp(I)7 b Fs(\()p Fp(U)2102 2678 y Fo(1)2142 2664 y Fs(\))75 2867 y Fl(for)32 b(any)h Fp(U)455 2881 y Fo(1)519 2867 y Fm(2)25 b(E)653 2881 y Fq(U)712 2867 y Fl(.)41 b(The)32 b(pr)-5 b(op)g(osition)36 b(3)c(holds)h(for)f Fp(U)42 b Fl(and)32 b Fp(W)44 b Fl(satisfying)33 b([HU1,2])e(with)h(this)h (extende)-5 b(d)75 2979 y(de\014nition)34 b(of)e Fp(I)7 b Fl(.)75 3166 y Fe(Pr)n(oof)44 b(:)60 b Fs(One)40 b(has)g(to)h(pro)m (v)m(e)g(that)f(the)h(n)m(um)m(b)s(er)e Fp(I)7 b Fs(\()p Fp(U)2101 3180 y Fo(1)2141 3166 y Fs(\))40 b(do)s(es)g(not)g(dep)s(end) f(on)h(the)g(c)m(hoice)h(of)g(the)75 3279 y(Lagrangian)22 b Fp(U)605 3293 y Fo(1)670 3279 y Fm(2)j(E)804 3293 y Fq(U)863 3279 y Fs(.)38 b(Let)22 b(us)g(tak)m(e)i(t)m(w)m(o)f (Lagrangians)f Fp(U)2107 3293 y Fo(1)2169 3279 y Fs(and)g Fp(U)2400 3293 y Fo(0)2461 3279 y Fs(in)f Fm(E)2607 3293 y Fq(U)2666 3279 y Fs(,)j(de\014ne)e Fp(U)3032 3293 y Fq(t)3087 3279 y Fs(=)j Fp(tU)3278 3293 y Fo(1)3321 3279 y Fs(+)t(\(1)t Fm(\000)t Fp(t)p Fs(\))p Fp(U)3685 3293 y Fo(0)3725 3279 y Fs(,)75 3392 y Fp(t)g Fm(2)g Fs([0)p Fp(;)15 b Fs(1],)30 b(and)c(let)h Fp(E)822 3406 y Fq(t)879 3392 y Fs(b)s(e)f(the)h(energy)g(function)f(asso)s(ciated)i(with)d Fp(U)2487 3406 y Fq(t)2517 3392 y Fs(.)40 b(There)26 b(is)g(an)h(energy)g Fp(e)f(>)f Fs(0)i(suc)m(h)75 3505 y(that)33 b Fp(U)336 3519 y Fq(t)365 3505 y Fs(\()p Fp(z)t Fs(\))c(=)f Fp(U)10 b Fs(\()p Fp(z)t Fs(\))33 b(for)e(all)g Fp(z)i Fm(2)27 b Fp(E)1333 3472 y Fq(e)1328 3532 y(U)1419 3505 y Fs(and)k(all)g Fp(t)d Fm(2)g Fs([0)p Fp(;)15 b Fs(1].)47 b(The)31 b(Lagrangians)h Fp(U)2891 3519 y Fq(t)2953 3505 y Fs(satisfy)f([HU1,2])j(with)75 3618 y(the)k(same)f(constan)m(t)i Fp(b)p Fs(,)g(and)e Fp(U)1196 3632 y Fq(t)1263 3618 y Fs(=)f Fp(\013)1428 3632 y Fq(t)1458 3618 y Fm(k)p Fp(z)t Fm(k)1594 3585 y Fo(2)1672 3618 y Fs(at)i(in\014nit)m(y)d(hence)j Fp(E)2437 3632 y Fq(t)2503 3618 y Fs(=)f Fp(\013)2669 3632 y Fq(t)2699 3618 y Fm(k)p Fp(z)t Fm(k)2835 3585 y Fo(2)2913 3618 y Fs(at)h(in\014nit)m(y)-8 b(.)60 b(Since)36 b Fp(\013)3695 3632 y Fq(t)3725 3618 y Fs(,)75 3730 y Fp(t)25 b Fm(2)g Fs([0)p Fp(;)15 b Fs(1])28 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Fq(r)1339 4038 y Fs(\()p Fp(t)p Fs(\))25 b(=)g(\(0)p Fp(;)15 b(r)k Fs(cos)q(\()p Fp(!)s(t)p Fs(\))p Fp(;)c Fm(\000)p Fp(!)s(r)j Fs(sin)o(\()p Fp(!)s(t)p Fs(\)\))p Fp(:)75 4242 y Fl(Each)34 b(of)f(these)h(p)-5 b(erio)g(dic)35 b(orbits)f(is)g(the)f(interse)-5 b(ction)35 b(b)-5 b(etwe)g(en)34 b(its)f(ener)-5 b(gy)34 b(shel)5 b(l)34 b(and)g(the)g(c)-5 b(enter)34 b(man-)75 4355 y(ifold,)i(and)g (is)e(hyp)-5 b(erb)g(olic)37 b(with)f(r)-5 b(esp)g(e)g(ct)36 b(to)f(its)g(ener)-5 b(gy)35 b(shel)5 b(l)36 b(\(but)e(not)i(with)f(r) -5 b(esp)g(e)g(ct)37 b(to)e(the)g(ful)5 b(l)35 b(phase)75 4468 y(sp)-5 b(ac)g(e\).)75 4656 y Fe(Pr)n(oof)39 b(:)50 b Fs(Let)36 b Fp(\036)d Fs(:)h Fp(T)13 b(M)33 b Fm(\002)23 b Fr(R)1133 4623 y Fo(2)1212 4656 y Fm(\000)-16 b(!)34 b Fp(T)1458 4623 y Fn(\003)1497 4656 y Fp(M)f Fm(\002)24 b Fr(R)1772 4623 y Fo(2)1853 4656 y Fs(b)s(e)34 b(the)i (di\013eomorphism)c(de\014ned)i(in)g(Section)h(2,)i(w)m(e)75 4769 y(ha)m(v)m(e)32 b Fp(\036)p Fs(\()p Fm(f)p Fs(0)p Fm(g)22 b(\002)e Fr(R)681 4736 y Fo(2)727 4769 y Fs(\))25 b(=)g Fm(f)p Fs(0)p Fm(g)d(\002)e Fr(R)1190 4736 y Fo(2)1266 4769 y Fs(and)30 b(the)g(Hamiltonian)f Fp(H)j Fs(=)25 b Fp(E)h Fm(\016)21 b Fp(\036)2539 4736 y Fn(\000)p Fo(1)2663 4769 y Fs(can)31 b(b)s(e)f(written)816 5011 y Fp(H)7 b Fs(\()p Fp(\022)s(;)15 b(\020)7 b(;)15 b(q)s(;)g(p)p Fs(\))25 b(=)1427 4950 y(1)p 1403 4990 94 4 v 1403 5074 a(4)p Fp(a)1507 5011 y(p)1553 4974 y Fo(2)1612 5011 y Fs(+)20 b Fp(a!)1811 4974 y Fo(2)1850 5011 y Fp(q)1894 4974 y Fo(2)1954 5011 y Fs(+)2055 4950 y(1)p 2055 4990 46 4 v 2055 5074 a(2)2110 5011 y(\()p Fp(g)2191 4973 y Fn(\000)p Fo(1)2188 5041 y Fq(\022)2286 5011 y Fp(\020)7 b(;)15 b(\020)7 b Fs(\))20 b(+)2586 4988 y(~)2566 5011 y Fp(R)q Fs(\()p Fp(\022)s(;)15 b(\020)7 b(;)15 b(q)s(;)g(p)p Fs(\))75 5257 y(where)353 5234 y(~)333 5257 y Fp(R)26 b Fs(=)f Fp(O)s Fs(\()p Fm(k)p Fp(\022)s Fm(k)767 5224 y Fo(2)818 5257 y Fs(+)11 b Fm(k)p Fp(\020)c Fm(k)1037 5224 y Fo(2)1077 5257 y Fs(\).)39 b(It)26 b(follo)m(ws)f(that)i(the)f (plane)f Fm(f)p Fs(0)p Fm(g)11 b(\002)g Fr(R)2436 5224 y Fo(2)2507 5257 y Fs(is)25 b(in)m(v)-5 b(arian)m(t)25 b(for)h(the)g(Hamiltonian)75 5370 y(\015o)m(w,)31 b(and)e(foliated)h(b) m(y)g(the)h(p)s(erio)s(dic)d(orbits)1162 5551 y(~)1142 5574 y Fp(O)1211 5588 y Fq(r)1249 5574 y Fs(\()p Fp(t)p Fs(\))e(=)f(\(0)p Fp(;)15 b Fs(0)p Fp(;)g(r)k Fs(cos)q(\()p Fp(!)s(t)p Fs(\))p Fp(;)c Fm(\000)p Fs(2)p Fp(a!)s(r)k Fs(sin)n(\()p Fp(!)s(t)p Fs(\)\))p Fp(;)1867 5841 y Fs(22)p eop %%Page: 23 23 23 22 bop 75 399 a Fs(w)m(e)28 b(apply)f Fp(\036)508 366 y Fn(\000)p Fo(1)630 399 y Fs(to)h(obtain)g(the)g(expression)e(of)i (the)g(asso)s(ciated)g(orbits)f(of)h Fp(Y)20 b Fs(.)40 b(W)-8 b(e)29 b(no)m(w)e(pro)m(v)m(e)i(h)m(yp)s(erb)s(ol-)75 511 y(icit)m(y)-8 b(.)41 b(The)30 b(h)m(yp)s(ersurface)769 706 y(\006)25 b(=)g Fm(f)p Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))28 b Fm(2)d Fp(T)13 b(M)30 b Fm(\002)20 b Fr(R)1736 668 y Fo(2)1797 706 y Fp(=)15 b(q)29 b(>)c Fs(0)30 b(and)g Fp(v)f Fs(=)c(0)p Fm(g)h Fs(=)f Fp(T)13 b(M)30 b Fm(\002)20 b Fr(R)2991 668 y Fo(+)2991 728 y Fn(\003)75 900 y Fs(is)26 b(transv)m(ersal)h(to)h(the)g(\015o)m(w)f(around)f Fm(f)p Fs(0)p Fm(g)14 b(\002)g Fr(R)1669 867 y Fo(+)1669 923 y Fn(\003)1734 900 y Fs(,)28 b(and)e(w)m(e)i(de\014ne)e(the)i(asso)s (ciated)g(P)m(oincar)m(\023)-43 b(e)28 b(return)e(map)75 1013 y(\010.)48 b(Let)34 b(us)e(\014x)g(a)i Fp(r)e(>)d Fs(0,)35 b(w)m(e)e(w)m(an)m(t)h(to)g(study)e(the)h(eigen)m(v)-5 b(alues)33 b(of)g(mo)s(dulus)e(1)i(of)g(the)g(linearized)e(map)75 1126 y Fp(d)p Fs(\010\(0)p Fp(;)15 b(r)s Fs(\).)62 b(Note)38 b(that)g(\010)970 1156 y Fn(jf)p Fo(0)p Fn(g\002)p Ff(R)1198 1126 y Fg(+)1198 1168 y Fh(\003)1289 1126 y Fs(=)f Fp(I)7 b(d)p Fs(,)39 b(th)m(us)e Fp(d)p Fs(\010\(0)p Fp(;)15 b(r)s Fs(\))2074 1144 y Fn(jf)p Fo(0)p Fn(g\002)p Ff(R)2344 1126 y Fs(=)36 b Fp(I)7 b(d)p Fs(.)61 b(It)37 b(follo)m(ws)f(that)i (for)f(all)f Fp(\017)g(>)h Fs(0)75 1249 y(there)31 b(is)e(a)i(fully)d (resonan)m(t)j(appro)m(ximation)e(\011)h(of)h Fp(d)p Fs(\010\(0)p Fp(;)15 b(r)s Fs(\))31 b(suc)m(h)f(that)1258 1444 y Fm(k)p Fs(\011\()p Fp(q)s(;)15 b(z)t Fs(\))22 b Fm(\000)e Fp(d)p Fs(\010\(0)p Fp(;)15 b(r)s Fs(\)\()p Fp(q)s(;)g(z)t Fs(\))p Fm(k)28 b Fi(6)d Fp(\017)p Fm(k)p Fp(z)t Fm(k)p Fp(:)75 1638 y Fs(By)30 b(fully)e(resonan)m(t,)i(w)m(e)g (mean)g(that)h(all)d(the)i(eigen)m(v)-5 b(alues)30 b(of)g(mo)s(dulus)d (1)j(of)g(\011)f(are)h(ro)s(ots)g(of)g(the)g(unit)m(y)-8 b(.)75 1751 y(W)g(e)45 b(can)f(moreo)m(v)m(er)i(tak)m(e)f Fp(\017)f Fs(small)e(enough)i(so)g(that)h(\011)e(and)h Fp(d)p Fs(\010\(0)p Fp(;)15 b(r)s Fs(\))45 b(ha)m(v)m(e)g(the)f(same)g (n)m(um)m(b)s(er)f(of)75 1864 y(eigen)m(v)-5 b(alues)39 b(of)h(mo)s(dulus)c(1.)68 b(Since)38 b(\011)1497 1894 y Fn(jf)p Fo(0)p Fn(g\002)p Ff(R)1725 1864 y Fg(+)1725 1906 y Fh(\003)1820 1864 y Fs(=)i Fp(I)7 b(d)40 b Fs(there)f(exists)g (a)h(neigh)m(b)s(orho)s(o)s(d)d(of)i(\(0)p Fp(;)15 b(r)s Fs(\))42 b Fm(2)d Fs(\006)75 1977 y(where)1028 2089 y Fm(j)p Fs(\010\()p Fp(z)t(;)15 b(q)s Fs(\))22 b Fm(\000)d Fp(d)p Fs(\010\(0)p Fp(;)c(r)s Fs(\)\()p Fp(z)t(;)g(q)26 b Fm(\000)19 b Fp(r)s Fs(\))i Fm(\000)e Fs(\(0)p Fp(;)c(r)s Fs(\))p Fm(j)27 b Fi(6)e Fp(\017)p Fm(k)p Fp(z)t Fm(k)2731 2052 y Fo(2)2772 2089 y Fp(:)75 2251 y Fs(As)44 b(a)g(consequence,)k (there)c(exists)f(a)h(function)f Fp(G)1907 2265 y Fo(1)1990 2251 y Fs(satisfying)g([HG1])i(and)e([HG2])i(with)d(a)i(smaller)75 2364 y(constan)m(t)32 b Fp(b)481 2378 y Fo(1)550 2364 y Fs(and)e(suc)m(h)g(that)h(P)m(oincar)m(\023)-43 b(e)32 b(map)e(\010)1770 2378 y Fo(1)1839 2364 y Fs(of)h(the)f(\015o)m(w)h (asso)s(ciated)g(to)1053 2559 y Fp(L)1115 2573 y Fo(1)1154 2559 y Fs(\()p Fp(\022)s(;)15 b(\027)q(;)g(q)s(;)g(v)s Fs(\))26 b(=)f Fp(a)1712 2485 y Fk(\000)1754 2559 y Fp(v)1801 2521 y Fo(2)1861 2559 y Fm(\000)20 b Fp(!)2012 2521 y Fo(2)2051 2559 y Fp(q)2095 2521 y Fo(2)2135 2485 y Fk(\001)2196 2559 y Fs(+)g Fp(G)2358 2573 y Fo(1)2398 2559 y Fs(\()p Fp(\022)s(;)15 b(\027)q(;)g(q)s(;)g(v)s Fs(\))75 2753 y(satis\014es)22 b(\010)468 2767 y Fo(1)507 2753 y Fs(\()p Fp(z)t(;)15 b(q)s Fs(\))26 b(=)f(\(0)p Fp(;)15 b(r)s Fs(\))t(+)t(\011\()p Fp(z)t(;)g(q)7 b Fm(\000)t Fp(r)s Fs(\))21 b(in)g(a)i(neigh)m(b)s(orho)s(o)s(d)c(of)j(\(0)p Fp(;)15 b(r)s Fs(\).)39 b(Let)23 b(us)e(consider)g(an)h(eigenspace)75 2866 y(of)38 b(\011)f(asso)s(ciated)g(with)f(a)i(pair)e(of)i(eigen)m(v) -5 b(alues)37 b(of)h(mo)s(dulus)c(one,)40 b(whic)m(h)c(are)i(therefore) g(ro)s(ot)f(of)h(the)75 2979 y(unit)m(y)-8 b(.)71 b(This)39 b(eigenspace)i(is)f(\014lled)e(with)h(p)s(erio)s(dic)f(p)s(oin)m(ts,)43 b(moreo)m(v)m(er)f(giv)m(en)e Fp(\016)46 b(>)c Fs(0)f(there)g(exists)g (a)75 3092 y(neigh)m(b)s(orho)s(o)s(d)28 b(of)k(0)f(in)e(the)j (eigenspace)f(suc)m(h)g(that)g(all)f(the)h(p)s(oin)m(ts)f(in)f(this)h (neigh)m(b)s(orho)s(o)s(d)f(ha)m(v)m(e)j(their)75 3205 y(\011-orbit)26 b(con)m(tained)h(in)e(the)i(zone)h(where)e(\010)1587 3219 y Fo(1)1626 3205 y Fs(\()p Fp(z)t(;)15 b(q)s Fs(\))27 b(=)e(\(0)p Fp(;)15 b(r)s Fs(\))e(+)g(\011\()p Fp(z)t(;)i(q)i Fm(\000)c Fp(r)s Fs(\),)28 b(and)e(suc)m(h)g(that)i(the)f(p)s(erio)s (dic)75 3317 y(orbits)37 b(of)h Fp(L)514 3331 y Fo(1)592 3317 y Fs(asso)s(ciated)g(with)f(these)i(\010)1554 3331 y Fo(1)1593 3317 y Fs(-orbits)e(are)i(con)m(tained)f(in)f Fp(B)2651 3332 y Fq(\016)2714 3317 y Fm(\002)25 b Fr(R)2870 3284 y Fo(2)2915 3317 y Fs(.)64 b(W)-8 b(e)40 b(no)m(w)e(apply)f(the)75 3430 y(lemma)g(5)g(b)s(elo)m(w)f(to)i Fp(L)901 3444 y Fo(1)977 3430 y Fs(and)f(obtain)f(that)i(the)f(p)s(erio)s(dic)d(orbits) i(w)m(e)i(just)e(constructed)h(m)m(ust)g(b)s(e)g(the)75 3543 y(trivial)28 b(ones,)j(corresp)s(onding)d(to)i(the)h(\014xed)e (space)i Fm(f)p Fs(0)p Fm(g)21 b(\002)e Fr(R)39 b Fs(of)30 b(\011.)40 b(As)30 b(a)h(consequence,)g(the)f(linearized)75 3656 y(P)m(oincar)m(\023)-43 b(e)36 b(map)e Fp(d)p Fs(\010\(0)p Fp(;)15 b(r)s Fs(\))35 b(can)g(ha)m(v)m(e)g(no)g(eigen)m(v)-5 b(alue)34 b(of)g(mo)s(dulus)e(1)j(except)g(the)g(one)f(asso)s(ciated)h (with)75 3769 y(this)29 b(\014xed)h(space.)2970 b Fi(\003)75 4061 y Fj(Lemma)33 b(5)45 b Fl(L)-5 b(et)33 b Fp(L)f Fl(b)-5 b(e)32 b(the)g(L)-5 b(agr)g(angian)35 b(function)d(\(1\))h (with)g(a)f(function)h Fp(G)f Fl(satisfying)h([HG1-2],)f(ther)-5 b(e)75 4174 y(is)33 b(a)g(two-p)-5 b(ar)g(ameters)36 b(family)d(of)g(p)-5 b(erio)g(dic)34 b(orbits)g(of)f Fp(L)1400 4369 y(O)s Fs(\()p Fp(t)p Fs(\))26 b(=)f(\()p Fp(\022)1775 4383 y Fo(0)1815 4369 y Fp(;)15 b(r)j Fs(cos\()p Fp(!)s(t)j Fs(+)e Fp(\036)p Fs(\)\))p Fp(;)75 4563 y Fl(and)34 b(ther)-5 b(e)33 b(exists)g(a)g Fp(\016)c(>)c Fs(0)33 b Fl(such)g(that)h(they)f(ar)-5 b(e)33 b(the)g(only)h(p)-5 b(erio)g(dic)34 b(orbits)g(satisfying)f Fp(@)5 b(x)25 b Fm(2)g Fp(B)3424 4578 y Fq(\016)3482 4563 y Fm(\002)20 b Fr(R)3633 4530 y Fo(2)3678 4563 y Fl(.)75 4742 y Fe(Pr)n(oof)34 b(:)42 b Fs(W)-8 b(e)32 b(w)m(ork)f(in)f(lo)s(cal)g(co)s(ordinates)h (as)g(describ)s(ed)e(ab)s(o)m(v)m(e.)44 b(The)30 b(tra)5 b(jectories)32 b(lying)e(in)f Fp(D)3534 4757 y Fq(\016)3593 4742 y Fm(\002)20 b Fr(R)75 4855 y Fs(satisfy)30 b(the)g(standard)g (Euler-Lagrange)g(equation)1671 5026 y Fp(d)p 1654 5066 81 4 v 1654 5150 a(dt)1755 5026 y(@)5 b(G)p 1755 5066 125 4 v 1765 5150 a(@)g(\027)1915 5087 y Fs(=)2020 5026 y Fp(@)g(G)p 2020 5066 V 2033 5150 a(@)g(\022)2155 5087 y Fm(\001)75 5307 y Fs(As)30 b(a)h(consequence)g(of)g([HG2)g(lo)s(c])f (there)h(is)e(a)i Fp(\016)e(>)c Fs(0)31 b(suc)m(h)f(that)1009 5423 y Fk(\034)1077 5552 y Fp(\027)q(;)1173 5490 y(@)5 b(G)p 1173 5531 V 1183 5614 a(@)g(\027)1308 5423 y Fk(\035)1401 5552 y Fi(>)1511 5490 y Fp(b)p 1507 5531 46 4 v 1507 5614 a Fs(2)1563 5552 y Fm(k)p Fp(\027)h Fm(k)1704 5514 y Fo(2)1835 5552 y Fs(and)2087 5423 y Fk(\034)2155 5552 y Fp(\022)s(;)2251 5490 y(@)f(G)p 2251 5531 125 4 v 2264 5614 a(@)g(\022)2386 5423 y Fk(\035)2479 5552 y Fi(>)2588 5490 y Fp(b)p 2585 5531 46 4 v 2585 5614 a Fs(2)2640 5552 y Fm(k)p Fp(\022)s Fm(k)2776 5514 y Fo(2)1867 5841 y Fs(23)p eop %%Page: 24 24 24 23 bop 75 399 a Fs(in)36 b Fp(B)257 414 y Fq(\016)320 399 y Fm(\002)25 b Fr(R)476 366 y Fo(2)521 399 y Fs(.)63 b(Let)39 b(us)e(no)m(w)g(consider)g(a)h(closed)g(tra)5 b(jectory)39 b(\()p Fp(\022)s Fs(\()p Fp(t)p Fs(\))p Fp(;)15 b(q)s Fs(\()p Fp(t)p Fs(\)\))39 b(suc)m(h)e(that)i(\()p Fp(\022)s(;)3253 375 y Fs(_)3236 399 y Fp(\022)r Fs(\))f Fm(2)f Fp(B)3521 414 y Fq(\016)3559 399 y Fs(,)j(the)75 511 y(equation)994 626 y Fk(Z)1100 621 y(\034)1168 749 y Fp(\022)s(;)1264 688 y(@)5 b(G)p 1264 729 125 4 v 1277 812 a(@)g(\022)1399 621 y Fk(\035)1492 749 y Fs(=)1563 626 y Fk(Z)1669 621 y(\034)1737 749 y Fp(\022)s(;)1849 688 y(d)p 1833 729 81 4 v 1833 812 a(dt)1933 688 y(@)g(G)p 1933 729 125 4 v 1943 812 a(@)g(\027)2068 621 y Fk(\035)2161 749 y Fs(=)25 b Fm(\000)2343 626 y Fk(Z)2449 621 y(\034)2535 726 y Fs(_)2517 749 y Fp(\022)r(;)2613 688 y(@)5 b(G)p 2613 729 V 2623 812 a(@)g(\027)2747 621 y Fk(\035)75 1012 y Fs(yields)1118 1214 y Fp(b)p 1115 1255 46 4 v 1115 1338 a Fs(2)1185 1152 y Fk(Z)1291 1275 y Fm(k)p Fp(\022)s Fm(k)1427 1238 y Fo(2)1492 1275 y Fi(6)1563 1152 y Fk(Z)1669 1147 y(\034)1737 1275 y Fp(\022)s(;)1849 1214 y(d)p 1833 1255 81 4 v 1833 1338 a(dt)1933 1214 y(@)g(G)p 1933 1255 125 4 v 1943 1338 a(@)g(\027)2068 1147 y Fk(\035)2161 1275 y Fi(6)25 b Fm(\000)2341 1214 y Fp(b)p 2338 1255 46 4 v 2338 1338 a Fs(2)2408 1152 y Fk(Z)2514 1275 y Fm(k)2577 1251 y Fs(_)2559 1275 y Fp(\022)s Fm(k)2650 1238 y Fo(2)2690 1275 y Fp(:)75 1535 y Fs(It)30 b(follo)m(ws)g(that)h Fm(k)p Fp(\022)s Fm(k)25 b(\021)g(k)989 1511 y Fs(_)971 1535 y Fp(\022)s Fm(k)h(\021)f Fs(0.)2425 b Fi(\003)75 1820 y Ft(7)135 b(Con)l(v)l(ergence)46 b(of)f(sequences)g(of)h(p)t(erio)t(dic)e(orbits.)75 2023 y Fs(In)32 b(this)f(section,)j(w)m(e)e(pro)m(v)m(e)i(the)e(con)m(v)m (ergence)j(of)e(go)s(o)s(d)f(sequences)h(of)g(p)s(erio)s(dic)d(orbits)h (to)i(homo)s(clinic)75 2136 y(orbits.)40 b(W)-8 b(e)31 b(\014rst)f(state)i(the)e(strong)h(minimizing)26 b(prop)s(ert)m(y)k(of) h(the)f(subspace)g Fp(z)g Fs(=)25 b Fp(z)3062 2150 y Fo(0)3101 2136 y Fs(.)75 2314 y Fj(Lemma)33 b(6)45 b Fl(A)n(ny)32 b Fp(T)13 b Fl(-p)-5 b(erio)g(dic)34 b(tr)-5 b(aje)g(ctory)35 b Fp(X)e Fs(=)25 b(\()p Fp(\022)s(;)15 b(q)s Fs(\))32 b Fl(of)h Fp(L)g Fl(satis\014es)1407 2568 y Fm(L)1470 2582 y Fq(T)1525 2568 y Fs(\()p Fp(X)7 b Fs(\))26 b Fi(>)f Fp(b)1853 2444 y Fk(Z)1944 2471 y Fq(T)1904 2651 y Fo(0)2015 2568 y Fp(d)p Fs(\()p Fp(@)5 b(\022)s(;)15 b(z)2278 2582 y Fo(0)2318 2568 y Fs(\))2353 2531 y Fo(2)2393 2568 y Fp(:)75 2807 y Fe(Pr)n(oof)34 b(:)40 b Fs(If)30 b Fp(X)j Fs(=)25 b(\()p Fp(\022)s(;)15 b(q)s Fs(\))30 b(is)g(a)g(tra)5 b(jectory)-8 b(,)33 b Fp(q)g Fs(m)m(ust)d(satisfy)g (the)h(\014rst)e(Euler-Lagrange)i(equation)1262 3051 y(2)p Fp(a)1370 2977 y Fk(\000)1419 3051 y Fs(\177)-52 b Fp(q)23 b Fs(+)d Fp(!)1627 3013 y Fo(2)1666 3051 y Fp(q)1710 2977 y Fk(\001)1777 3051 y Fs(=)1883 2989 y Fp(@)5 b(G)p 1883 3030 125 4 v 1897 3113 a(@)g(q)2038 3051 y Fm(\000)2155 2989 y Fp(d)p 2139 3030 81 4 v 2139 3113 a(dt)2244 2922 y Fk(\022)2321 2989 y Fp(@)g(G)p 2321 3030 125 4 v 2333 3113 a(@)g(v)2456 2922 y Fk(\023)2538 3051 y Fp(:)75 3289 y Fs(If)30 b Fp(X)38 b Fs(is)29 b(closed,)i(w)m(e)f (can)h(in)m(tegrate)h(b)m(y)e(parts)g(to)h(write)f(its)f(action)949 3519 y Fm(L)p Fs(\()p Fp(X)7 b Fs(\))26 b(=)1286 3395 y Fk(Z)1392 3519 y Fm(\000)p Fp(aq)1570 3445 y Fk(\000)1618 3519 y Fs(\177)-52 b Fp(q)23 b Fs(+)d Fp(!)1826 3481 y Fo(2)1865 3519 y Fp(q)1909 3445 y Fk(\001)1971 3519 y Fs(+)g Fp(G)p Fs(\()p Fp(@)5 b(\022)s(;)15 b(q)s(;)33 b Fs(_)-43 b Fp(q)t Fs(\))1190 3762 y(=)1286 3638 y Fk(Z)1392 3634 y(\022)1459 3762 y Fp(G)20 b Fm(\000)1651 3700 y Fs(1)p 1651 3741 46 4 v 1651 3824 a(2)1707 3762 y Fp(q)1761 3700 y(@)5 b(G)p 1761 3741 125 4 v 1775 3824 a(@)g(q)1895 3634 y Fk(\023)1982 3762 y Fs(+)2073 3638 y Fk(Z)2189 3700 y Fs(1)p 2189 3741 46 4 v 2189 3824 a(2)2245 3762 y Fp(q)2315 3700 y(d)p 2299 3741 81 4 v 2299 3824 a(dt)2404 3634 y Fk(\022)2481 3700 y Fp(@)g(G)p 2481 3741 125 4 v 2493 3824 a(@)g(v)2615 3634 y Fk(\023)2697 3762 y Fp(;)75 4025 y Fs(and)30 b(in)m(tegrating)g(b)m(y)g(part)h(again)f(the)h(last)f (term,)949 4281 y Fm(L)p Fs(\()p Fp(X)7 b Fs(\))26 b(=)1286 4157 y Fk(Z)1392 4281 y Fp(G)20 b Fm(\000)1584 4219 y Fs(1)p 1584 4260 46 4 v 1584 4343 a(2)1640 4281 y Fp(q)1694 4219 y(@)5 b(G)p 1694 4260 125 4 v 1708 4343 a(@)g(q)1848 4281 y Fm(\000)1949 4219 y Fs(1)p 1949 4260 46 4 v 1949 4343 a(2)2022 4281 y(_)-42 b Fp(q)2059 4219 y(@)5 b(R)p 2059 4260 124 4 v 2070 4343 a(@)g(v)2217 4281 y Fi(>)25 b Fp(b)2367 4157 y Fk(Z)2473 4281 y Fp(d)p Fs(\()p Fp(@)5 b(\022)s(;)15 b(z)2736 4295 y Fo(0)2776 4281 y Fs(\))2811 4243 y Fo(2)2851 4281 y Fp(:)3679 4511 y Fi(\003)75 4624 y Fs(This)24 b(lemma)i(roughly)e(implies)g(that)i(if)f(there)h(exists)g (a)g(sequence)h(of)f(p)s(erio)s(dic)d(orbits)i(of)h Fp(L)g Fs(of)g(un)m(b)s(ound-)75 4736 y(ed)39 b(p)s(erio)s(d)e(and)i(b)s (ounded)e(action,)42 b(there)e(m)m(ust)f(b)s(e)g(an)g(orbit)g(homo)s (clinic)e(to)j(the)f(cen)m(ter)i(manifold.)75 4849 y(Unfortunately)-8 b(,)36 b(there)f(is)e(no)i(con\014nemen)m(t)g(in)e(the)i Fp(q)j Fs(direction,)c(and)g(w)m(e)i(m)m(ust)e(ha)m(v)m(e)i(some)f (estimate)75 4962 y(of)30 b(the)g Fp(q)j Fs(part)d(of)g(the)h(p)s(erio) s(dic)c(orbits)i(in)g(order)g(to)i(b)s(e)f(able)f(to)i(pro)m(v)m(e)g (con)m(v)m(ergence.)43 b(As)30 b(explained)e(in)75 5075 y(the)k(sk)m(etc)m(h)i(of)e(pro)s(of,)g(w)m(e)g(m)m(ust)g(allo)m(w)f (the)i(parameter)f Fp(!)j Fs(to)d(v)-5 b(ary)d(.)46 b(Consider)31 b(no)m(w)h(a)g(sequence)g Fp(!)3598 5089 y Fq(n)3677 5075 y Fs(of)75 5188 y(pulsations,)j(with)f(a)h(limit)e Fp(!)s Fs(,)k(and)d(the)i(asso)s(ciated)f(Lagrangian)h(and)e(action)i Fp(L)2956 5202 y Fq(n)3038 5188 y Fs(and)e Fm(L)3282 5202 y Fq(n)3329 5188 y Fs(.)55 b(W)-8 b(e)37 b(ha)m(v)m(e)75 5301 y(the)31 b(follo)m(wing)d(con)m(v)m(ergence)33 b(prop)s(ert)m(y:) 75 5479 y Fj(Prop)s(osition)j(4)46 b Fl(If)d(ther)-5 b(e)45 b(exist)f(a)g(c)-5 b(onstant)45 b Fp(M)10 b Fl(,)47 b(a)d(r)-5 b(adius)45 b Fp(r)h Fl(and)f(a)f(se)-5 b(quenc)g(e)43 b Fp(X)3135 5493 y Fq(n)3228 5479 y Fs(=)i(\()p Fp(\022)3422 5493 y Fq(n)3469 5479 y Fp(;)15 b(q)3550 5493 y Fq(n)3597 5479 y Fs(\))44 b Fl(of)75 5592 y Fp(T)128 5606 y Fq(n)175 5592 y Fl(-p)-5 b(erio)g(dic)34 b(orbits)f(of)g Fp(L)971 5606 y Fq(n)1051 5592 y Fl(such)f(that)1867 5841 y Fs(24)p eop %%Page: 25 25 25 24 bop 211 399 a Fm(\017)46 b Fp(T)355 413 y Fq(n)428 399 y Fm(\000)-16 b(!)25 b(1)p Fl(,)211 586 y Fm(\017)46 b(L)365 600 y Fq(n)412 586 y Fs(\()p Fp(X)522 600 y Fq(n)570 586 y Fs(\))25 b Fi(6)g Fp(M)10 b Fl(,)211 774 y Fm(\017)46 b(k)p Fp(q)388 788 y Fq(n)435 774 y Fm(k)480 741 y Fo(2)520 774 y Fp(=T)618 788 y Fq(n)691 774 y Fm(\000)-15 b(!)25 b Fp(r)907 741 y Fo(2)946 774 y Fp(=)p Fs(2)p Fl(,)211 961 y Fm(\017)46 b Fp(\022)345 975 y Fq(n)417 961 y Fm(6\021)25 b Fp(\022)556 975 y Fo(0)595 961 y Fl(,)75 1149 y(then)33 b(ther)-5 b(e)34 b(exists)f(an)g(orbit)g Fp(X)1175 1163 y Fn(1)1275 1149 y Fs(=)25 b(\()p Fp(\022)1449 1163 y Fn(1)1524 1149 y Fp(;)15 b(q)1605 1163 y Fn(1)1680 1149 y Fs(\))32 b Fl(homo)-5 b(clinic)35 b(to)e Fp(O)2381 1163 y Fq(r)2452 1149 y Fl(and)g(such)g(that)851 1272 y Fk(Z)901 1478 y Ff(R)968 1396 y Fp(G)p Fs(\()p Fp(@)5 b(X)1202 1410 y Fn(1)1278 1396 y Fs(\))21 b Fm(\000)1435 1335 y Fs(1)p 1435 1375 46 4 v 1435 1458 a(2)1490 1396 y Fp(q)1531 1410 y Fn(1)1616 1335 y Fp(@)5 b(G)p 1616 1375 125 4 v 1630 1458 a(@)g(q)1750 1396 y Fs(\()p Fp(@)g(X)1913 1410 y Fn(1)1989 1396 y Fs(\))21 b Fm(\000)2145 1335 y Fs(1)p 2145 1375 46 4 v 2145 1458 a(2)2218 1396 y(_)-42 b Fp(q)2242 1410 y Fn(1)2326 1335 y Fp(@)5 b(G)p 2326 1375 125 4 v 2338 1458 a(@)g(v)2461 1396 y Fs(\()p Fp(@)g(X)2624 1410 y Fn(1)2700 1396 y Fs(\))25 b Fi(6)g Fp(M)5 b(:)75 1641 y Fe(Pr)n(oof)34 b(:)42 b Fs(Since)30 b Fp(\022)746 1655 y Fq(n)820 1641 y Fm(6\021)c Fp(\022)960 1655 y Fo(0)999 1641 y Fs(,)31 b(Lemma)h(5)f(implies)d(that)k Fp(@)5 b(\022)2060 1655 y Fq(n)2138 1641 y Fs(do)s(es)31 b(not)g(sta)m(y)h(in)e Fp(B)2875 1656 y Fq(\016)2913 1641 y Fs(.)43 b(W)-8 b(e)32 b(can)f(consider)f Fp(\022)3703 1655 y Fq(n)75 1754 y Fs(as)h(a)f(p)s(erio)s(dic)e(curv)m(e)j (de\014ned)e(on)h Fr(R)s Fs(,)36 b(and)30 b(b)m(y)g(c)m(hanging)h(time) f(origin,)f(w)m(e)i(can)f(require)f(that)1570 1958 y Fp(d)p Fs(\()p Fp(@)5 b(\022)1748 1972 y Fq(n)1796 1958 y Fs(\(0\))p Fp(;)15 b(z)1993 1972 y Fo(0)2034 1958 y Fs(\))26 b Fi(>)f Fp(\016)n(:)75 2163 y Fs(Since)k(the)i(sequence)g Fm(L)909 2177 y Fq(n)955 2163 y Fs(\()p Fp(X)1065 2177 y Fq(n)1113 2163 y Fs(\))g(is)e(b)s(ounded,)g(w)m(e)h(obtain)g(from)g (Lemma)h(6)f(that)h(the)g(sequence)1562 2308 y Fk(Z)1653 2335 y Fq(T)1694 2343 y Fb(n)1736 2335 y Fq(=)p Fo(2)1612 2515 y Fn(\000)p Fq(T)1708 2523 y Fb(n)1751 2515 y Fq(=)p Fo(2)1841 2432 y Fp(d)1888 2395 y Fo(2)1927 2432 y Fs(\()p Fp(@)5 b(\022)2058 2446 y Fq(n)2106 2432 y Fp(;)15 b(z)2188 2446 y Fo(0)2228 2432 y Fs(\))75 2694 y(is)29 b(b)s(ounded.)39 b(Asso)s(ciated)31 b(with)e(Lemma)h(1)h(this)e(yields)1256 2935 y Fp(E)1323 2949 y Fq(n)1370 2935 y Fs(\()p Fp(X)1480 2949 y Fq(n)1528 2935 y Fs(\))20 b Fm(\000)1710 2874 y Fp(a)p 1684 2914 101 4 v 1684 2997 a(T)1737 3011 y Fq(n)1810 2811 y Fk(Z)1932 2935 y Fs(_)-41 b Fp(q)1960 2897 y Fo(2)1957 2958 y Fq(n)2023 2935 y Fs(+)20 b Fp(!)2174 2897 y Fo(2)2171 2958 y Fq(n)2218 2935 y Fp(q)2262 2897 y Fo(2)2326 2935 y Fm(\000)-15 b(!)25 b Fs(0)p Fp(:)75 3176 y Fs(On)39 b(the)g(other)h(side,)h(w)m(e)f(obtain)f(using)f(the)i (Euler-Lagrange)g(equations)f(and)g(t)m(w)m(o)i(in)m(tegrations)e(b)m (y)75 3289 y(parts)30 b(that)829 3378 y Fp(a)p 803 3419 V 803 3502 a(T)856 3516 y Fq(n)928 3316 y Fk(Z)1051 3440 y Fs(_)-42 b Fp(q)1078 3402 y Fo(2)1075 3462 y Fq(n)1142 3440 y Fm(\000)20 b Fp(!)1293 3402 y Fo(2)1290 3462 y Fq(n)1337 3440 y Fp(q)1381 3402 y Fo(2)1445 3440 y Fs(=)1601 3378 y(1)p 1551 3419 146 4 v 1551 3502 a(2)p Fp(T)1649 3516 y Fq(n)1722 3316 y Fk(Z)1828 3440 y Fp(q)1869 3454 y Fq(n)1925 3378 y Fp(@)5 b(G)p 1925 3419 125 4 v 1939 3502 a(@)g(q)2060 3440 y Fs(\()p Fp(X)2170 3454 y Fq(n)2218 3440 y Fs(\))20 b(+)37 b(_)-42 b Fp(q)2405 3454 y Fq(n)2462 3378 y Fp(@)5 b(G)p 2462 3419 V 2474 3502 a(@)g(v)2596 3440 y Fs(\()p Fp(X)2706 3454 y Fq(n)2754 3440 y Fs(\))26 b Fm(\000)-16 b(!)25 b Fs(0)75 3650 y(b)s(ecause)30 b([HG1])i(and)e ([HG4])i(imply)1106 3897 y Fp(q)1160 3836 y(@)5 b(G)p 1160 3876 V 1174 3959 a(@)g(q)1294 3897 y Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))23 b(+)c Fp(v)1751 3836 y(@)5 b(G)p 1751 3876 V 1763 3959 a(@)g(v)1886 3897 y Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))28 b Fi(6)d Fp(C)7 b(d)2416 3860 y Fo(2)2455 3897 y Fs(\()p Fp(z)t(;)15 b(z)2618 3911 y Fo(0)2659 3897 y Fs(\))p Fp(:)75 4140 y Fs(Com)m(bining)28 b(these)j(equations)f(and)g(the)g(third)f(h)m(yp)s(othesis)g(yields)681 4394 y Fp(E)748 4408 y Fn(1)849 4394 y Fs(=)24 b(lim)14 b Fp(E)1153 4408 y Fq(n)1200 4394 y Fs(\()p Fp(X)1310 4408 y Fq(n)1358 4394 y Fs(\))25 b(=)g(2)15 b(lim)1752 4333 y Fp(a)p 1726 4374 101 4 v 1726 4457 a(T)1779 4471 y Fq(n)1836 4394 y Fm(k)j Fs(_)-43 b Fp(q)1922 4408 y Fq(n)1969 4394 y Fm(k)2014 4357 y Fo(2)2014 4417 y(2)2079 4394 y Fs(=)25 b(2)15 b(lim)2387 4333 y Fp(a!)2495 4300 y Fo(2)2492 4355 y Fq(n)p 2387 4374 152 4 v 2413 4457 a Fp(T)2466 4471 y Fq(n)2549 4394 y Fm(k)p Fp(q)2635 4408 y Fq(n)2682 4394 y Fm(k)2727 4357 y Fo(2)2727 4417 y(2)2792 4394 y Fs(=)25 b Fp(a!)2996 4357 y Fo(2)3035 4394 y Fp(r)3079 4357 y Fo(2)3118 4394 y Fp(:)75 4638 y Fs(Since)33 b Fp(E)383 4652 y Fq(n)430 4638 y Fs(\()p Fp(X)540 4652 y Fq(n)587 4638 y Fs(\))h(is)f(a)h(b)s(ounded)d(sequence) j(and)f(since)g Fp(@)5 b(X)2118 4652 y Fq(n)2199 4638 y Fs(is)33 b(an)g(in)m(tegral)h(curv)m(e)g(of)f Fp(Y)3163 4652 y Fq(n)3244 4638 y Fs(the)h(sequence)75 4751 y Fp(@)5 b(X)203 4765 y Fq(n)280 4751 y Fs(is)28 b Fp(C)442 4718 y Fo(1)481 4751 y Fs(-)i(b)s(ounded,)d(and)i(b)m(y)g(Ascoli's)g (Theorem)g(it)g(has)g(a)h(subsequence)e(con)m(v)m(erging)j(uniformly)26 b(on)75 4864 y(compact)e(sets)g(to)g(a)f(limit)1010 4841 y(~)984 4864 y Fp(X)1059 4878 y Fn(1)1157 4864 y Fs(that)h(is)e(an)h (in)m(tegral)g(curv)m(e)h(of)f Fp(Y)2259 4878 y Fq(n)2329 4864 y Fs(and)f(th)m(us)h(the)h(lifting)d(of)i(a)g Fp(L)p Fs(-tra)5 b(jectory)75 4977 y Fp(X)150 4991 y Fn(1)255 4977 y Fs(of)31 b(energy)g Fp(E)717 4991 y Fn(1)791 4977 y Fs(.)41 b(Recall)30 b(that)1562 5026 y Fk(Z)1653 5053 y Fq(T)1694 5061 y Fb(n)1736 5053 y Fq(=)p Fo(2)1612 5233 y Fn(\000)p Fq(T)1708 5241 y Fb(n)1751 5233 y Fq(=)p Fo(2)1841 5150 y Fp(d)1888 5113 y Fo(2)1927 5150 y Fs(\()p Fp(@)5 b(\022)2058 5164 y Fq(n)2106 5150 y Fp(;)15 b(z)2188 5164 y Fo(0)2228 5150 y Fs(\))75 5375 y(is)29 b(b)s(ounded.)39 b(It)30 b(follo)m(ws)g(that)1590 5385 y Fk(Z)1681 5412 y Fn(1)1640 5591 y(\0001)1785 5509 y Fp(d)1832 5471 y Fo(2)1872 5509 y Fs(\()p Fp(@)5 b(\022)2003 5523 y Fn(1)2078 5509 y Fp(;)15 b(z)2160 5523 y Fo(0)2200 5509 y Fs(\))1867 5841 y(25)p eop %%Page: 26 26 26 25 bop 75 399 a Fs(is)29 b(\014nite.)40 b(Since)29 b(the)i(curv)m(e)g Fp(@)5 b(\022)1167 413 y Fn(1)1272 399 y Fs(has)30 b(b)s(ounded)e(deriv)-5 b(ativ)m(e)30 b(this)f(yields)1627 603 y(lim)1579 659 y Fq(t)p Fn(!\0061)1816 603 y Fp(@)5 b(\022)s Fs(\()p Fp(t)p Fs(\))25 b(=)g Fp(z)2181 617 y Fo(0)2221 603 y Fp(:)75 838 y Fs(Using)30 b(once)h(more)f(the)h (lemma)f(1)g(w)m(e)h(get)h(that)1237 1042 y Fp(a)p Fs(\()18 b(_)-43 b Fp(q)1364 1004 y Fo(2)1361 1064 y Fn(1)1456 1042 y Fs(+)20 b Fp(!)1607 1004 y Fo(2)1646 1042 y Fp(q)1690 1004 y Fo(2)1687 1064 y Fn(1)1762 1042 y Fs(\))63 b Fm(\000)-16 b(!)1822 1098 y Fq(t)p Fn(!\0061)2069 1042 y Fp(E)2136 1056 y Fn(1)2236 1042 y Fs(=)25 b Fp(a!)2440 1004 y Fo(2)2479 1042 y Fp(r)2523 1004 y Fo(2)2562 1042 y Fp(:)75 1277 y Fs(This)k(is)g(the)i(de\014nition)d(w)m(e)i(ha)m(v)m(e)i(tak)m(en)g (for)e(a)g(homo)s(clinic)e(orbit.)40 b(The)30 b(last)g(inequalit)m(y)f (follo)m(ws)h(from)661 1417 y Fk(Z)752 1444 y Fq(T)793 1452 y Fb(n)835 1444 y Fq(=)p Fo(2)712 1624 y Fn(\000)p Fq(T)808 1632 y Fb(n)850 1624 y Fq(=)p Fo(2)940 1541 y Fp(G)p Fs(\()p Fp(@)5 b(X)1174 1555 y Fq(n)1222 1541 y Fs(\))21 b Fm(\000)1379 1480 y Fs(1)p 1379 1520 46 4 v 1379 1603 a(2)1434 1541 y Fp(q)1475 1555 y Fq(n)1532 1480 y Fp(@)5 b(G)p 1532 1520 125 4 v 1546 1603 a(@)g(q)1666 1541 y Fs(\()p Fp(@)g(X)1829 1555 y Fq(n)1877 1541 y Fs(\))21 b Fm(\000)2034 1480 y Fs(1)p 2034 1520 46 4 v 2034 1603 a(2)2106 1541 y(_)-42 b Fp(q)2130 1555 y Fq(n)2187 1480 y Fp(@)5 b(G)p 2187 1520 125 4 v 2199 1603 a(@)g(v)2321 1541 y Fs(\()p Fp(@)g(X)2484 1555 y Fq(n)2532 1541 y Fs(\))26 b(=)f Fm(L)p Fs(\()p Fp(X)2862 1555 y Fq(n)2909 1541 y Fs(\))h Fi(6)f Fp(M)75 1803 y Fs(since)30 b(the)g(in)m(tegrand)g(is)g(non-negativ)m(e.)2197 b Fi(\003)75 2089 y Ft(8)135 b(Existence)46 b(of)f(p)t(erio)t(dic)g (orbits.)75 2292 y Fs(Let)31 b(us)f(\014x)f(a)i(p)s(erio)s(d)d Fp(T)38 b Fs(=)25 b(2)p Fp(\031)s(\034)10 b(=!)s Fs(,)32 b Fp(\034)j Fm(2)25 b Fr(N)6 b Fp(:)37 b Fs(F)-8 b(or)31 b(an)m(y)g Fp(l)c Fm(2)e Fr(R)r Fs(,)37 b(the)31 b(functional)1138 2496 y Fm(Q)1212 2511 y Fq(l)1264 2496 y Fs(:)25 b Fp(C)1386 2459 y Fn(1)1379 2519 y Fq(T)1460 2496 y Fs(\()p Fr(R)s Fs(\))32 b Fm(\000)-15 b(!)25 b Fr(R)1441 2706 y Fp(x)p Fs(\()p Fp(t)p Fs(\))h Fm(7\000)-15 b(!)25 b Fp(a)1857 2583 y Fk(Z)1948 2609 y Fq(T)1908 2789 y Fo(0)2034 2706 y Fs(_)-41 b Fp(x)p Fs(\()p Fp(t)p Fs(\))2173 2669 y Fo(2)2233 2706 y Fm(\000)20 b Fp(l)2353 2669 y Fo(2)2393 2706 y Fp(!)2453 2669 y Fo(2)2492 2706 y Fp(x)p Fs(\()p Fp(t)p Fs(\))2647 2669 y Fo(2)75 2952 y Fs(can)31 b(b)s(e)e(computed)i (using)d(F)-8 b(ourier)30 b(expansion:)925 3228 y Fm(Q)999 3243 y Fq(l)1040 3073 y Fk( )1112 3142 y(X)1158 3339 y Fq(k)1258 3228 y Fp(q)1299 3243 y Fq(k)1342 3228 y Fp(e)1384 3190 y Fq(ik)r(!)r(t=\034)1597 3073 y Fk(!)1694 3228 y Fs(=)25 b Fp(aT)1919 3142 y Fk(X)1965 3339 y Fq(k)2065 3100 y Fk(\022)2142 3166 y Fp(k)2192 3133 y Fo(2)2232 3166 y Fp(!)2292 3133 y Fo(2)p 2142 3207 189 4 v 2192 3290 a Fp(\034)2242 3264 y Fo(2)2361 3228 y Fm(\000)20 b Fp(l)2481 3190 y Fo(2)2521 3228 y Fp(!)2581 3190 y Fo(2)2620 3100 y Fk(\023)2702 3228 y Fm(j)p Fp(q)2768 3243 y Fq(k)2810 3228 y Fm(j)2835 3190 y Fo(2)2875 3228 y Fp(:)75 3511 y Fs(It)30 b(follo)m(ws)g(that)h Fm(Q)743 3526 y Fq(l)799 3511 y Fs(can)g(b)s(e)f(extended)g(to)1407 3715 y Fp(E)1474 3729 y Fq(T)1554 3715 y Fs(=)25 b Fp(H)1733 3678 y Fo(1)1772 3715 y Fs(\()p Fp(S)1863 3729 y Fq(T)1944 3715 y Fs(=)g Fr(R)r Fp(=)q(T)13 b Fr(Z)p Fp(;)i Fr(R)t Fs(\))75 3920 y(as)45 b(a)f(con)m(tin)m(uous)g(quadratic)g(form.)82 b(It)44 b(has)g(a)h(t)m(w)m(o)g(dimensional)d(k)m(ernel)i(when)f Fp(l)50 b Fm(2)e Fr(Z)p Fp(=\034)10 b Fs(,)44 b(and)g(is)75 4032 y(non-degenerate)32 b(for)e(other)g(v)-5 b(alues)30 b(of)h Fp(l)r Fs(.)40 b(Let)31 b(us)f(set)1038 4237 y Fp(E)1110 4199 y Fo(+)1252 4237 y Fs(=)83 b Fm(f)p Fp(q)33 b Fs(suc)m(h)d(that)h Fp(q)1968 4252 y Fq(k)2036 4237 y Fs(=)25 b(0)30 b(when)g Fm(j)p Fp(k)s Fm(j)c Fi(6)f Fp(\034)10 b Fm(g)1038 4375 y Fp(E)1110 4337 y Fn(\000)1252 4375 y Fs(=)83 b Fm(f)p Fp(q)33 b Fs(suc)m(h)d(that)h Fp(q)1968 4390 y Fq(k)2036 4375 y Fs(=)25 b(0)30 b(when)g Fm(j)p Fp(k)s Fm(j)c Fp(>)f(\034)10 b Fm(g)p Fp(;)75 4579 y Fs(there)31 b(is)e(an)h(orthogonal)h(splitting)1591 4692 y Fp(E)1658 4706 y Fq(T)1739 4692 y Fs(=)25 b Fp(E)1907 4654 y Fo(+)1986 4692 y Fm(\010)20 b Fp(E)2149 4654 y Fn(\000)2208 4692 y Fp(;)75 4859 y Fs(suc)m(h)30 b(that)42 b Fm(\006Q)633 4874 y Fq(l)659 4859 y Fm(j)684 4885 y Fq(E)740 4867 y Fh(\006)826 4859 y Fs(is)30 b(p)s(ositiv)m(e)g (de\014nite)g(for)g(all)g Fp(l)e Fm(2)d Fs(\(1)p Fp(;)15 b Fs(1)22 b(+)e(1)p Fp(=\034)10 b Fs(\).)44 b(Notice)31 b(that)h Fp(E)3063 4826 y Fn(\000)3153 4859 y Fs(is)e(\014nite)f (dimen-)75 4971 y(sional,)g(whic)m(h)g(is)h(the)g(usual)f(feature)i(of) g(Lagrangian)f(form)m(ulations.)39 b(Let)31 b(us)f(de\014ne)f(the)i (functionals)961 5176 y Fm(G)g Fs(:)25 b Fp(A)1164 5190 y Fq(T)1227 5176 y Fs(=)o(\003)1360 5190 y Fq(T)1436 5176 y Fm(\002)20 b Fp(E)1594 5190 y Fq(T)1730 5176 y Fm(\000)-16 b(!)26 b Fr(R)1071 5386 y Fp(x)p Fs(\()p Fp(t)p Fs(\))q(=)o(\()p Fp(\022)s Fs(\()p Fp(t)p Fs(\))p Fp(;)15 b(q)s Fs(\()p Fp(t)p Fs(\)\))27 b Fm(7\000)-16 b(!)1892 5262 y Fk(Z)1982 5288 y Fq(T)1942 5468 y Fo(0)2053 5386 y Fp(G)p Fs(\()p Fp(@)5 b(\022)s Fs(\()p Fp(t)p Fs(\))p Fp(;)15 b(q)s Fs(\()p Fp(t)p Fs(\))p Fp(;)34 b Fs(_)-44 b Fp(q)5 b Fs(\()p Fp(t)p Fs(\)\))15 b Fp(dt:)1867 5841 y Fs(26)p eop %%Page: 27 27 27 26 bop 75 399 a Fs(and)932 628 y Fm(L)995 643 y Fq(l)1046 628 y Fs(:)25 b Fp(A)1164 642 y Fq(T)1227 628 y Fs(=)o(\003)1360 642 y Fq(T)1436 628 y Fm(\002)20 b Fp(E)1594 642 y Fq(T)1730 628 y Fm(\000)-16 b(!)26 b Fr(R)1071 765 y Fp(x)p Fs(\()p Fp(t)p Fs(\))q(=)o(\()p Fp(\022)s Fs(\()p Fp(t)p Fs(\))p Fp(;)15 b(q)s Fs(\()p Fp(t)p Fs(\)\))27 b Fm(7\000)-16 b(!L)1939 780 y Fq(l)1965 765 y Fs(\()p Fp(x)p Fs(\))26 b(=)f Fm(Q)2283 780 y Fq(l)2309 765 y Fs(\()p Fp(q)s Fs(\))c(+)f Fm(G)5 b Fs(\()p Fp(x)p Fs(\))p Fp(:)75 1108 y Fs(W)-8 b(e)32 b(also)e(de\014ne)g(the)g(pro)5 b(jection)30 b Fp(P)1323 1122 y Fo(\003)1402 1108 y Fs(:)25 b(\003)1515 1122 y Fq(T)1591 1108 y Fm(\002)20 b Fp(E)1749 1122 y Fq(T)1829 1108 y Fm(\000)-15 b(!)25 b Fs(\003)2064 1122 y Fq(T)2119 1108 y Fs(.)75 1295 y Fj(Lemma)33 b(7)45 b Fl(F)-7 b(or)32 b(any)e Fp(l)i Fl(in)e(the)h(interval)f Fs(\(1)p Fp(;)15 b Fs(1)p Fp(=\034)10 b Fs(\))p Fl(,)33 b(the)e(functional)g Fm(L)2522 1310 y Fq(l)2577 1295 y Fl(is)f Fp(C)2744 1262 y Fo(1)2813 1295 y Fl(and)h(satis\014es)g(the) g(Palais-)75 1408 y(Smale)38 b(c)-5 b(ondition.)57 b(The)37 b(critic)-5 b(al)38 b(p)-5 b(oints)38 b(of)g Fm(L)1743 1423 y Fq(l)1805 1408 y Fl(ar)-5 b(e)38 b(the)g Fp(T)13 b Fl(-p)-5 b(erio)g(dic)38 b(smo)-5 b(oth)39 b(tr)-5 b(aje)g(ctories)39 b(of)e(the)h(L)-5 b(a-)75 1521 y(gr)g(angian)592 1725 y Fp(L)654 1740 y Fq(l)680 1725 y Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))27 b(=)e Fp(a)1153 1652 y Fk(\000)1195 1725 y Fp(v)1242 1688 y Fo(2)1302 1725 y Fm(\000)20 b Fp(l)1422 1688 y Fo(2)1461 1725 y Fp(!)1521 1688 y Fo(2)1560 1725 y Fp(q)1604 1688 y Fo(2)1644 1652 y Fk(\001)1706 1725 y Fs(+)f Fp(G)p Fs(\()p Fp(z)t(;)c(q)s(;)g(v)s Fs(\))p Fp(;)111 b Fs(\()p Fp(z)t(;)15 b(q)s(;)g(v)s Fs(\))27 b Fm(2)e Fp(T)13 b(M)30 b Fm(\002)20 b Fr(R)29 b Fm(\002)20 b Fr(R)s Fp(:)75 1929 y Fe(Pr)n(oof)28 b(:)38 b Fs(W)-8 b(e)27 b(will)22 b(often)k(omit)f(the)g(subscript)f Fp(l)j Fs(in)d(the)h(follo)m(wing)f(pro)s(of.)38 b(Recall)25 b(that)h(\003)3222 1943 y Fq(T)3302 1929 y Fs(is)e(a)i(smo)s(oth)75 2042 y(manifold,)j(and)g(that)i(the)g(mappings)1318 2247 y(exp)1457 2261 y Fq(c)1517 2247 y Fs(:)25 b Fp(H)1650 2209 y Fo(1)1690 2247 y Fs(\()p Fm(O)1797 2261 y Fq(c)1832 2247 y Fs(\))h Fm(\000)-16 b(!)26 b Fp(H)2148 2209 y Fo(1)2187 2247 y Fs(\()p Fp(S)2278 2261 y Fq(T)2333 2247 y Fp(;)15 b(M)10 b Fs(\))1720 2384 y Fp(\030)t Fs(\()p Fp(t)p Fs(\))26 b Fm(7\000)-16 b(!)26 b Fs(exp)o(\()p Fp(\030)t Fs(\()p Fp(t)p Fs(\)\))75 2589 y(are)k(c)m(harts)h(of)f(this) e(manifold,)g(where)i Fp(c)25 b Fm(2)g Fp(C)1662 2556 y Fn(1)1736 2589 y Fs(\()p Fp(S)1827 2603 y Fq(T)1883 2589 y Fp(;)15 b(M)10 b Fs(\),)31 b Fm(O)2184 2603 y Fq(c)2248 2589 y Fs(is)e(a)i(su\016cien)m(tly)d(small)g(neigh)m(b)s (orho)s(o)s(d)g(of)75 2702 y(the)i(zero)g(section)g(in)e(the)h(bundle)e Fp(c)1321 2669 y Fn(\003)1361 2702 y Fp(T)13 b(M)39 b Fs(of)30 b(tangen)m(ts)h(v)m(ectors)g(of)e Fp(M)40 b Fs(along)29 b Fp(c)p Fs(,)h(and)f(exp)h(:)25 b Fp(T)13 b(M)35 b Fm(\000)-15 b(!)25 b Fp(M)75 2814 y Fs(is)k(the)h(exp)s(onen)m (tial)e(map)i(asso)s(ciated)g(with)e(some)i(spra)m(y)g(on)g Fp(M)10 b Fs(,)30 b(see)g([20)q(].)41 b(Let)30 b Fp(\026)2967 2828 y Fq(T)3047 2814 y Fs(:)c Fr(R)34 b Fm(\000)-16 b(!)25 b Fp(S)3416 2828 y Fq(T)3501 2814 y Fs(b)s(e)k(the)75 2927 y(natural)35 b(pro)5 b(jection,)38 b(the)e(induced)e(v)m(ector)k (bundle)c Fp(\026)2008 2894 y Fn(\003)2047 2927 y Fp(c)2086 2894 y Fn(\003)2126 2927 y Fp(T)13 b(M)46 b Fs(is)35 b(trivial)f(since)i(it)g(is)f(a)i(v)m(ector)g(bundle)75 3040 y(o)m(v)m(er)32 b Fr(R)r Fs(,)37 b(w)m(e)31 b(ha)m(v)m(e)g(the)g (comm)m(utativ)m(e)h(diagram)1138 3268 y Fr(R)d Fm(\002)20 b Fr(R)1375 3235 y Fq(n)1572 3216 y Fo(\010)1473 3268 y Fm(\000)-17 b(\000)c(\000)k(!)45 b Fp(\026)1822 3235 y Fn(\003)1862 3268 y Fp(c)1901 3235 y Fn(\003)1940 3268 y Fp(T)13 b(M)2259 3211 y Fo(~)-41 b Fq(\026)2150 3268 y Fm(\000)-17 b(\000)c(\000)k(!)45 b Fp(c)2483 3235 y Fn(\003)2523 3268 y Fp(T)13 b(M)1252 3339 y Fk(?)1252 3393 y(?)1252 3448 y(y)1906 3339 y(?)1906 3393 y(?)1906 3448 y(y)2535 3339 y(?)2535 3393 y(?)2535 3448 y(y)1250 3636 y Fr(R)1567 3584 y Fq(id)1473 3636 y Fm(\000)-17 b(\000)c(\000)k(!)181 b Fr(R)2253 3579 y Fq(\026)2150 3636 y Fm(\000)-17 b(\000)c(\000)k(!)99 b Fp(S)2554 3650 y Fq(T)2608 3636 y Fp(;)75 3816 y Fs(where)30 b(\010)g(is)f(a)i(v)m (ector)h(bundle)c(isomorphism)g(and)h(w)m(e)i(de\014ne)f(the)g(co)m(v)m (ering)1298 4020 y Fp(r)1339 4034 y Fq(c)1399 4020 y Fs(=)j(~)-53 b Fp(\026)20 b Fm(\016)h Fs(\010)k(:)g Fr(R)k Fm(\002)20 b Fr(R)2014 3983 y Fq(n)2092 4020 y Fm(\000)-15 b(!)25 b Fp(c)2303 3983 y Fn(\003)2343 4020 y Fp(T)13 b(M)5 b(:)75 4225 y Fs(A)30 b Fp(H)256 4192 y Fo(1)326 4225 y Fs(section)h Fp(\030)e Fs(:)c Fp(S)805 4239 y Fq(T)885 4225 y Fm(\000)-15 b(!)25 b Fp(c)1096 4192 y Fn(\003)1136 4225 y Fp(T)13 b(M)40 b Fs(has)30 b(a)h(unique)d(lifting) 2138 4201 y(~)2129 4225 y Fp(\030)h Fs(:)c Fr(R)34 b Fm(\000)-15 b(!)25 b Fr(R)2571 4192 y Fq(n)2654 4225 y Fs(suc)m(h)30 b(that)h(the)f(diagram)1476 4431 y Fr(R)f Fm(\002)20 b Fr(R)1713 4398 y Fq(n)1904 4380 y(r)1936 4388 y Fb(c)1811 4431 y Fm(\000)-17 b(\000)d(\000)j(!)45 b Fp(c)2145 4398 y Fn(\003)2185 4431 y Fp(T)13 b(M)1422 4607 y Fo(\()p Fq(id;)1536 4590 y Fo(~)1529 4607 y Fq(\030)s Fo(\))1591 4503 y Fk(x)1591 4557 y(?)1591 4612 y(?)2197 4503 y(x)2197 4557 y(?)2197 4612 y(?)2257 4601 y Fq(\030)1588 4787 y Fr(R)1915 4730 y Fq(\026)1811 4787 y Fm(\000)-17 b(\000)d(\000)j(!)111 b Fp(S)2228 4801 y Fq(T)75 4964 y Fs(comm)m(utes.)51 b(Let)33 b(us)g(tak)m(e)i(a)f(compact)h(neigh)m(b) s(orho)s(o)s(d)30 b Fp(U)2113 4978 y Fq(c)2182 4964 y Fs(of)j(the)h(origin)d(in)i Fr(R)2877 4931 y Fq(n)2963 4964 y Fs(suc)m(h)h(that)f Fr(R)e Fm(\002)22 b Fp(U)3614 4978 y Fq(c)3679 4964 y Fm(\032)75 5076 y Fp(r)119 5043 y Fn(\000)p Fo(1)116 5099 y Fq(c)213 5076 y Fs(\()p Fm(O)320 5090 y Fq(c)355 5076 y Fs(\),)31 b(and)f(supp)s(ose)f(without)g(loss)h (of)h(generalit)m(y)f(that)h Fm(O)2268 5090 y Fq(c)2328 5076 y Fs(=)25 b Fp(r)2465 5090 y Fq(c)2500 5076 y Fs(\()p Fr(R)k Fm(\002)20 b Fp(U)2774 5090 y Fq(c)2809 5076 y Fs(\).)41 b(The)30 b(mapping)1261 5281 y Fp(\032)25 b Fs(:)g Fp(H)1466 5243 y Fo(1)1506 5281 y Fs(\()p Fp(c)1580 5243 y Fn(\003)1620 5281 y Fp(T)13 b(M)d Fs(\))25 b Fm(\000)-15 b(!)25 b Fp(H)2099 5243 y Fo(1)2138 5281 y Fs(\([0)p Fp(;)15 b(T)e Fs(])p Fp(;)i Fr(R)2476 5243 y Fq(n)2529 5281 y Fs(\))1775 5432 y Fp(\030)29 b Fm(7\000)-15 b(!)2025 5408 y Fs(~)2016 5432 y Fp(\030)2056 5451 y Fn(j)p Fo([0)p Fq(;T)10 b Fo(])1867 5841 y Fs(27)p eop %%Page: 28 28 28 27 bop 75 399 a Fs(is)26 b(a)i(linear)d(isomorphism)f(on)m(to)k(its) f(image)g Fp(T)1687 376 y Fs(~)1663 399 y Fp(H)32 b Fm(\032)25 b Fp(H)1950 366 y Fo(1)1990 399 y Fs(\([0)p Fp(;)15 b(T)e Fs(])p Fp(;)i Fr(R)2327 366 y Fq(n)2380 399 y Fs(\).)40 b(W)-8 b(e)28 b(will)d(also)i(note)g Fp(\032)g Fs(the)h(mapping)75 511 y(\()p Fp(\032;)15 b(id)275 525 y Fq(E)327 536 y Fb(T)381 511 y Fs(\),)31 b(and)f(w)m(e)g(call)973 488 y(~)949 511 y Fp(H)37 b Fs(the)31 b(set)1385 488 y(~)1361 511 y Fp(H)h Fs(=)25 b Fp(T)1655 488 y Fs(~)1631 511 y Fp(H)i Fm(\\)20 b Fp(H)1898 478 y Fo(1)1937 511 y Fs(\([0)p Fp(;)15 b(T)e Fs(])p Fp(;)i(U)2275 525 y Fq(c)2312 511 y Fs(\).)41 b(Let)31 b(us)e(de\014ne)h(the)h(smo)s(oth)f(map)351 676 y(~)340 699 y Fp(L)402 713 y Fq(c)462 699 y Fs(:)25 b Fr(R)k Fm(\002)20 b Fp(U)751 713 y Fq(c)806 699 y Fm(\002)g Fr(R)957 662 y Fq(n)1030 699 y Fm(\002)g Fr(R)29 b Fm(\002)20 b Fr(R)34 b Fm(\000)-16 b(!)25 b Fr(R)917 851 y Fs(\()p Fp(t;)1035 827 y Fs(~)1025 851 y Fp(\030)5 b(;)15 b(\027)q(;)g(q)s(;)g (v)s Fs(\))27 b Fm(7\000)-16 b(!)25 b Fp(L)1622 778 y Fk(\000)1664 851 y Fs(exp)20 b Fm(\016)h Fp(r)1930 865 y Fq(c)1964 851 y Fs(\()p Fp(t;)2082 827 y Fs(~)2072 851 y Fp(\030)5 b Fs(\))15 b Fp(;)1636 1003 y(d)1683 1017 y Fo(1)1723 1003 y Fs(\(exp)20 b Fm(\016)h Fp(r)2024 1017 y Fq(c)2059 1003 y Fs(\)\()p Fp(t;)2212 979 y Fs(~)2202 1003 y Fp(\030)t Fs(\))p Fp(:)p Fs(1)h(+)e Fp(d)2511 1017 y Fo(2)2550 1003 y Fs(\(exp)h Fm(\016)f Fp(r)2851 1017 y Fq(c)2886 1003 y Fs(\)\()p Fp(t;)3040 979 y Fs(~)3029 1003 y Fp(\030)5 b Fs(\))p Fp(:\027)22 b(;)30 b(q)18 b(;)31 b(v)3418 929 y Fk(\001)3460 1003 y Fp(;)75 1191 y Fs(and)f(the)g(functional)755 1356 y(~)734 1379 y Fm(L)25 b Fs(:)g Fp(H)955 1342 y Fo(1)995 1379 y Fs(\([0)p Fp(;)15 b(T)e Fs(])p Fp(;)i(U)1333 1393 y Fq(c)1369 1379 y Fs(\))21 b Fm(\002)f Fp(E)1583 1393 y Fq(T)1663 1379 y Fm(\000)-15 b(!)25 b Fr(R)1232 1589 y Fs(\()1276 1565 y(~)1267 1589 y Fp(\030)t Fs(\()p Fp(t)p Fs(\))p Fp(;)15 b(q)s Fs(\()p Fp(t)p Fs(\)\))27 b Fm(7\000)-15 b(!)1835 1465 y Fk(Z)1926 1492 y Fq(T)1885 1671 y Fo(0)2007 1566 y Fs(~)1996 1589 y Fp(L)2058 1603 y Fq(c)2093 1589 y Fs(\()p Fp(t;)2210 1565 y Fs(~)2201 1589 y Fp(\030)t Fs(\()p Fp(t)p Fs(\))p Fp(;)2398 1565 y Fs(~)2388 1589 y Fp(\030)2432 1551 y Fn(0)2456 1589 y Fs(\()p Fp(t)p Fs(\))p Fp(;)15 b(q)s Fs(\()p Fp(t)p Fs(\))p Fp(;)34 b Fs(_)-44 b Fp(q)5 b Fs(\()p Fp(t)p Fs(\)\))15 b Fp(dt;)75 1839 y Fs(w)m(e)43 b(ha)m(v)m(e)g Fm(L)h Fs(=)686 1816 y(~)665 1839 y Fm(L)27 b(\016)i Fp(\032:)42 b Fs(One)g(can)g(c)m(hec)m(k)i(from)d([HG4])j(and) d(the)h(expression)f(of)3018 1816 y(~)3007 1839 y Fp(L)3069 1853 y Fq(c)3146 1839 y Fs(ab)s(o)m(v)m(e)i(that)g(the)75 1951 y(estimates)1366 2140 y Fm(j)1402 2117 y Fs(~)1391 2140 y Fp(L)1453 2154 y Fq(c)1488 2140 y Fs(\()p Fp(t;)1606 2116 y Fs(~)1596 2140 y Fp(\030)t(;)15 b(\027)q(;)g(q)s(;)g(v)s Fs(\))p Fm(j)27 b Fi(6)e Fp(C)7 b Fs(\(1)21 b(+)f Fp(q)2388 2102 y Fo(2)2447 2140 y Fs(+)g Fm(j)p Fp(\027)6 b Fm(j)2639 2102 y Fo(2)2699 2140 y Fs(+)20 b Fp(v)2837 2102 y Fo(2)2876 2140 y Fs(\))913 2207 y Fk(\014)913 2261 y(\014)913 2316 y(\014)913 2370 y(\014)913 2425 y(\014)944 2210 y( )1025 2304 y Fp(@)1090 2281 y Fs(~)1078 2304 y Fp(L)1140 2318 y Fq(c)p 1025 2345 150 4 v 1052 2440 a Fp(@)1114 2416 y Fs(~)1105 2440 y Fp(\030)1185 2366 y(;)1236 2304 y(@)1300 2281 y Fs(~)1289 2304 y Fp(L)1351 2318 y Fq(c)p 1236 2345 V 1262 2428 a Fp(@)5 b(q)1396 2210 y Fk(!)1483 2366 y Fs(\()p Fp(t;)1601 2342 y Fs(~)1591 2366 y Fp(\030)t(;)15 b(\027)q(;)g(q)s(;)g(v)s Fs(\))1927 2207 y Fk(\014)1927 2261 y(\014)1927 2316 y(\014)1927 2370 y(\014)1927 2425 y(\014)1984 2366 y Fi(6)25 b Fp(C)7 b Fs(\(1)21 b(+)f Fp(q)j Fs(+)c Fm(j)p Fp(\027)6 b Fm(j)2599 2328 y Fo(2)2659 2366 y Fs(+)20 b Fp(v)2797 2328 y Fo(2)2837 2366 y Fs(\))913 2513 y Fk(\014)913 2567 y(\014)913 2622 y(\014)913 2676 y(\014)913 2731 y(\014)944 2516 y( )1025 2610 y Fp(@)1090 2587 y Fs(~)1078 2610 y Fp(L)1140 2624 y Fq(c)p 1025 2651 V 1048 2734 a Fp(@)5 b(\027)1185 2672 y(;)1236 2610 y(@)1300 2587 y Fs(~)1289 2610 y Fp(L)1351 2624 y Fq(c)p 1236 2651 V 1260 2734 a Fp(@)g(v)1396 2516 y Fk(!)1483 2672 y Fs(\()p Fp(t;)1601 2648 y Fs(~)1591 2672 y Fp(\030)t(;)15 b(\027)q(;)g(q)s(;)g(v)s Fs(\))1927 2513 y Fk(\014)1927 2567 y(\014)1927 2622 y(\014)1927 2676 y(\014)1927 2731 y(\014)1984 2672 y Fi(6)25 b Fp(C)7 b Fs(\(1)21 b(+)f Fp(q)2388 2634 y Fo(2)2447 2672 y Fs(+)g Fm(j)p Fp(\027)6 b Fm(j)20 b Fs(+)g Fp(v)s Fs(\))75 2937 y(hold)26 b(on)i Fr(R)17 b Fm(\002)d Fp(U)619 2951 y Fq(c)674 2937 y Fm(\002)g Fr(R)819 2904 y Fq(n)886 2937 y Fm(\002)g Fr(R)k Fm(\002)c Fr(R)9 b Fs(.)45 b(These)28 b(gro)m(wth)g(conditions)e(imply)f(b)m(y)i (w)m(ell-kno)m(wn)f(results)h(\(see)h([22)q(]\))75 3050 y(that)294 3027 y(~)273 3050 y Fm(L)p Fs(,)j(and)g(th)m(us)g Fm(L)p Fs(,)g(are)h(con)m(tin)m(uously)e(di\013eren)m(tiable.)42 b(W)-8 b(e)32 b(also)g(ha)m(v)m(e)g(the)g(lo)s(cal)e(expression)g(of)i (the)75 3163 y(di\013eren)m(tial:)834 3316 y Fp(d)903 3293 y Fs(~)881 3316 y Fm(L)p Fs(\()988 3292 y(~)979 3316 y Fp(\030)t(;)15 b(q)s Fs(\))26 b(=)1264 3192 y Fk(Z)1355 3219 y Fq(T)1315 3399 y Fo(0)1435 3255 y Fp(@)1499 3232 y Fs(~)1488 3255 y Fp(L)1550 3269 y Fq(c)p 1435 3295 V 1462 3391 a Fp(@)1524 3367 y Fs(~)1515 3391 y Fp(\030)1595 3316 y(d)1651 3292 y Fs(~)1642 3316 y Fp(\030)f Fs(+)1808 3255 y Fp(@)1872 3232 y Fs(~)1861 3255 y Fp(L)1923 3269 y Fq(c)p 1808 3295 V 1830 3379 a Fp(@)5 b(\027)1967 3316 y Fs(\()p Fp(d)2058 3292 y Fs(~)2049 3316 y Fp(\030)g Fs(\))2129 3279 y Fn(0)2173 3316 y Fs(+)2274 3255 y Fp(@)2338 3232 y Fs(~)2327 3255 y Fp(L)2389 3269 y Fq(c)p 2274 3295 V 2300 3379 a Fp(@)g(q)2434 3316 y(dq)23 b Fs(+)2646 3255 y Fp(@)2710 3232 y Fs(~)2699 3255 y Fp(L)2761 3269 y Fq(c)p 2646 3295 V 2670 3379 a Fp(@)5 b(v)2806 3316 y Fs(\()p Fp(dq)s Fs(\))2967 3279 y Fn(0)75 3547 y Fs(and)27 b Fp(d)p Fm(L)p Fs(\()p Fp(\030)t(;)15 b(q)s Fs(\))26 b(=)f Fp(d)748 3524 y Fs(~)726 3547 y Fm(L)p Fs(\()833 3523 y(~)824 3547 y Fp(\030)t(;)15 b(q)s Fs(\))f Fm(\016)g Fp(\032)p Fs(.)42 b(Let)28 b(us)e(no)m(w)i(pro)m(v)m(e)g(that)g(the)g (P)m(alais-Smale)e(condition)g(is)h(satis\014ed.)39 b(W)-8 b(e)75 3660 y(tak)m(e)32 b(a)f(P)m(alais-Smale)e(sequence)i(\()p Fp(\022)1337 3674 y Fq(n)1384 3660 y Fp(;)15 b(q)1465 3674 y Fq(n)1512 3660 y Fs(\).)41 b(The)30 b(sequence)1305 3848 y Fm(L)1368 3863 y Fq(l)1393 3848 y Fs(\()p Fp(\022)1471 3862 y Fq(n)1518 3848 y Fp(;)15 b(q)1599 3862 y Fq(n)1646 3848 y Fs(\))26 b(=)f Fm(Q)1877 3863 y Fq(l)1903 3848 y Fs(\()p Fp(q)1979 3862 y Fq(n)2026 3848 y Fs(\))c(+)f Fm(G)5 b Fs(\()p Fp(\022)2310 3862 y Fq(n)2357 3848 y Fp(;)15 b(q)2438 3862 y Fq(n)2485 3848 y Fs(\))75 4036 y(is)23 b(b)s(ounded.)37 b(Since)23 b Fm(Q)870 4051 y Fq(l)920 4036 y Fs(is)g(a)h(non-degenerate)h(quadratic)f(form,)h(there) f(exists)g(an)g(op)s(erator)g Fp(A)3354 4051 y Fq(l)3406 4036 y Fs(:)h Fp(E)3523 4050 y Fq(T)3604 4036 y Fm(\000)-16 b(!)75 4149 y Fp(E)142 4163 y Fq(T)227 4149 y Fs(suc)m(h)30 b(that)1279 4262 y Fp(d)p Fm(Q)1400 4277 y Fq(l)1427 4262 y Fs(\()p Fp(q)s Fs(\))p Fp(:A)1634 4277 y Fq(l)1661 4262 y Fp(q)e Fs(=)d Fm(jQ)1925 4277 y Fq(l)1951 4262 y Fs(\()p Fp(q)s Fs(\))p Fm(j)h Fi(>)f Fp(C)7 b Fm(k)p Fp(q)s Fm(k)2418 4225 y Fo(2)2418 4290 y Fq(H)2481 4271 y Fg(1)2520 4262 y Fp(:)75 4421 y Fs(Let)31 b(us)f(no)m(w)g(write)g (using)e([HG4])k(and)e(that)1654 4344 y Fk(\015)1654 4398 y(\015)1704 4421 y Fp(d)p Fm(L)p Fs(\()p Fp(\022)1892 4435 y Fq(n)1939 4421 y Fp(;)15 b(q)2020 4435 y Fq(n)2067 4421 y Fs(\))2102 4344 y Fk(\015)2102 4398 y(\015)2179 4421 y Fs(=)24 b Fp(\017)2311 4435 y Fq(n)2384 4421 y Fm(\000)-16 b(!)25 b Fs(0)496 4609 y Fp(\017)533 4623 y Fq(n)580 4609 y Fm(k)p Fp(q)666 4623 y Fq(n)713 4609 y Fm(k)758 4629 y Fq(H)821 4610 y Fg(1)885 4609 y Fi(>)g Fp(d)p Fm(L)1091 4624 y Fq(l)1117 4609 y Fs(\()p Fp(\022)1195 4623 y Fq(n)1242 4609 y Fp(;)15 b(q)1323 4623 y Fq(n)1370 4609 y Fs(\)\(0)p Fp(;)g(A)1593 4624 y Fq(l)1621 4609 y Fp(q)1662 4623 y Fq(n)1708 4609 y Fs(\))26 b(=)f Fp(d)p Fm(Q)1986 4624 y Fq(l)2013 4609 y Fs(\()p Fp(q)2089 4623 y Fq(n)2135 4609 y Fs(\))p Fp(:A)2263 4624 y Fq(l)2290 4609 y Fp(q)2331 4623 y Fq(n)2398 4609 y Fs(+)20 b Fp(d)p 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Fq(n)2069 5142 y Fm(k)2114 5104 y Fo(2)2114 5170 y Fq(H)2177 5151 y Fg(1)2237 5142 y Fm(\000)20 b Fp(C)2400 5104 y Fn(0)o(0)2442 5142 y Fm(k)p Fp(q)2528 5156 y Fq(n)2575 5142 y Fm(k)2620 5162 y Fq(H)2683 5143 y Fg(1)2722 5142 y Fp(:)75 5330 y Fs(It)30 b(follo)m(ws)g(that)h(the)g(sequence)f Fm(k)p Fp(q)1288 5344 y Fq(n)1335 5330 y Fm(k)1380 5350 y Fq(H)1443 5331 y Fg(1)1513 5330 y Fs(is)f(b)s(ounded.)39 b(Plugging)29 b(this)g(in)m(to)i(the)f(action)710 5560 y Fp(C)i Fi(>)25 b Fm(L)966 5575 y Fq(l)991 5560 y Fs(\()p Fp(\022)1069 5574 y Fq(n)1116 5560 y Fp(;)15 b(q)1197 5574 y Fq(n)1244 5560 y Fs(\))26 b Fi(>)1401 5436 y Fk(Z)1507 5560 y Fp(G)21 b Fs(+)f Fm(Q)1764 5575 y Fq(l)1790 5560 y Fs(\()p Fp(q)1866 5574 y Fq(n)1913 5560 y Fs(\))25 b Fi(>)g Fp(b)2123 5436 y Fk(Z)2230 5560 y Fp(d)2277 5522 y Fo(2)2316 5560 y Fs(\()p Fp(@)5 b(\022)2447 5574 y Fq(n)2495 5560 y Fp(;)15 b(z)2577 5574 y Fo(0)2617 5560 y Fs(\))20 b Fm(\000)g Fp(C)7 b Fm(k)p Fp(q)2921 5574 y Fq(n)2968 5560 y Fm(k)3013 5522 y Fo(2)3013 5588 y Fq(H)3076 5569 y Fg(1)1867 5841 y Fs(28)p eop %%Page: 29 29 29 28 bop 75 399 a Fs(yields)26 b(that)523 325 y Fk(R)598 399 y Fm(k)p Fp(@)5 b(\022)739 413 y Fq(n)787 399 y Fm(k)832 366 y Fo(2)899 399 y Fs(is)27 b(also)h(b)s(ounded.)38 b(By)28 b(a)h(standard)e(application)f(of)i(the)g(theorem)g(of)g (Ascoli,)g(see)75 511 y([20)q(],)g(Lemma)g(1.4.4,)h(w)m(e)f(can)f (\014nd)f(a)h Fp(C)1447 478 y Fo(0)1486 511 y Fs(-con)m(v)m(ergen)m(t)j (subsequence)d(of)g Fp(\022)2622 525 y Fq(n)2668 511 y Fs(,)h(and)f(b)m(y)g(extracting)h(another)75 624 y(subsequence)33 b(w)m(e)g(can)h(obtain)e(that)i Fp(q)1424 638 y Fq(n)1504 624 y Fs(also)f(has)g(a)h(uniform)d(limit.)47 b(F)-8 b(rom)34 b(no)m(w)f(on,)h(w)m(e)g(will)c(supp)s(ose)75 737 y(that)1557 862 y(\()p Fp(\022)1635 876 y Fq(n)1682 862 y Fp(;)15 b(q)1763 876 y Fq(n)1810 862 y Fs(\))1899 810 y Fq(C)1954 787 y Fg(0)1871 862 y Fm(\000)-16 b(!)25 b Fs(\()p Fp(\022)s(;)15 b(q)s Fs(\))p Fp(:)75 1017 y Fs(It)27 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Fm(\002)19 b Fr(R)34 b Fm(\000)-15 b(!)25 b Fr(R)k Fm(\002)20 b Fp(U)2338 1549 y Fq(c)2393 1535 y Fm(\002)g Fr(R)2543 1498 y Fq(n)2617 1535 y Fm(\002)f Fr(R)29 b Fm(\002)20 b Fr(R)1456 1761 y Fs(\()p Fp(t;)1574 1737 y Fs(~)1564 1761 y Fp(\030)5 b(;)15 b(\027)q(;)g(q)s(;)g(v)s Fs(\))26 b Fm(7\000)-15 b(!)2099 1606 y Fk( )2171 1761 y Fp(t;)2254 1737 y Fs(~)2244 1761 y Fp(\030)t(;)2339 1700 y(@)2403 1677 y Fs(~)2392 1700 y Fp(L)2454 1714 y Fq(c)p 2339 1741 150 4 v 2362 1824 a Fp(@)5 b(\027)2498 1761 y(;)15 b(q)s(;)2633 1700 y(@)2697 1677 y Fs(~)2686 1700 y Fp(L)2748 1714 y Fq(c)p 2633 1741 V 2658 1824 a Fp(@)5 b(v)2793 1606 y Fk(!)2880 1761 y Fp(:)75 2034 y Fs(It)30 b(is)f(straigh)m(tforw) m(ard)g(from)g(the)h(explicit)f(expression)f(of)2145 2011 y(~)2134 2034 y Fp(L)2196 2048 y Fq(c)2260 2034 y Fs(and)h(from)h([HL])g(that)g(\012)3118 2048 y Fq(c)3182 2034 y Fs(is)f(a)h(di\013eomor-)75 2147 y(phism,)e(and)i(the)h (estimate)1232 2299 y(1)p 1219 2340 72 4 v 1219 2423 a Fp(C)1300 2360 y Fm(j)p Fp(X)7 b Fm(j)21 b(\000)f Fp(C)32 b Fi(6)25 b Fs(\012)1803 2374 y Fq(c)1837 2360 y Fs(\()p Fp(t;)15 b(X)7 b Fs(\))27 b Fi(6)e Fp(C)7 b Fs(\()p Fm(j)p Fp(X)g Fm(j)21 b Fs(+)f(1\))75 2566 y(is)29 b(a)i(consequence)g(of)g ([HG4].)42 b(A)30 b(theorem)h(of)g(Krasnoselskii)d(implies)f(that)k (the)g(mapping)1071 2747 y(\007)1142 2761 y Fq(c)1201 2747 y Fs(:)26 b Fp(L)1314 2709 y Fo(2)1353 2747 y Fs(\([0)p Fp(;)15 b(T)e Fs(])p Fp(;)i Fr(R)1691 2709 y Fo(2)p Fq(n)p Fo(+2)1869 2747 y Fs(\))26 b Fm(\000)-16 b(!)26 b Fp(L)2164 2709 y Fo(2)2203 2747 y Fs(\([0)p Fp(;)15 b(T)e Fs(])p Fp(;)i Fr(R)2540 2709 y Fo(2)q Fq(n)p Fo(+2)2719 2747 y Fs(\))1719 2884 y Fp(X)7 b Fs(\()p Fp(t)p Fs(\))26 b Fm(7\000)-16 b(!)26 b Fs(\012)2168 2898 y Fq(c)2202 2884 y Fs(\()p Fp(t;)15 b(X)7 b Fs(\()p Fp(t)p Fs(\)\))75 3064 y(is)29 b(a)i(homeomorphism.)39 b(It)31 b(is)e(not)i(hard)e(to)i (see)g(that)g(the)g(sequence)1110 3161 y Fk( )1192 3255 y Fp(@)1256 3232 y Fs(~)1245 3255 y Fp(L)1307 3269 y Fq(c)p 1192 3296 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1917 y Fq(c)1765 1903 y Fs(\))20 b Fm(\002)g Fp(E)1978 1917 y Fq(T)2033 1903 y Fs(.)1621 b Fi(\003)75 2016 y Fs(W)-8 b(e)33 b(no)m(w)f(ha)m(v)m(e)h (to)g(study)e(the)h(top)s(ology)g(of)g(the)g(functional.)44 b(Let)32 b(us)f(de\014ne)h(a)g(group)f(\000)h(of)g(admissible)75 2129 y(deformations)e(of)g Fp(A)790 2143 y Fq(T)845 2129 y Fs(:)75 2316 y Fj(De\014nition)35 b(3)46 b Fl(A)41 b(home)-5 b(omorphism)45 b Fp(h)c Fs(:)g Fp(A)1680 2330 y Fq(T)1776 2316 y Fm(\000)-15 b(!)40 b Fp(A)2031 2330 y Fq(T)2128 2316 y Fl(b)-5 b(elongs)42 b(to)f Fs(\000)g Fl(if)g(and)h(only)g(if)f(ther)-5 b(e)42 b(exist)g(a)75 2429 y(p)-5 b(ar)g(ameter)34 b Fp(l)27 b Fm(2)e Fs(\(1)p Fp(;)15 b Fs(1)20 b(+)d(1)p Fp(=\034)10 b Fs(\))32 b Fl(and)h(a)f(c)-5 b(ontinuous)32 b(isotopy)h Fp(k)c Fs(:)c([0)p Fp(;)15 b Fs(1])20 b Fm(\002)d Fp(A)2623 2443 y Fq(T)2703 2429 y Fm(\000)-15 b(!)25 b Fp(A)2943 2443 y Fq(T)3030 2429 y Fl(such)31 b(that)i Fp(k)3467 2443 y Fo(0)3532 2429 y Fs(=)25 b Fp(I)7 b(d)p Fl(,)75 2542 y Fp(k)122 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Fp(E)2345 3009 y Fo(+)2404 2973 y Fk(\001)2471 3047 y Fm(\032)25 b Fs(\003)p Fp(:)75 3251 y Fj(Lemma)33 b(8)i(\(In)m (tersection)f(prop)s(ert)m(y)i(\))45 b Fl(L)-5 b(et)32 b Fs(\006)f Fl(b)-5 b(e)31 b(the)h(family)h(of)f(c)-5 b(omp)g(act)34 b(subsets)d(of)h Fs(\003)3389 3265 y Fq(T)3476 3251 y Fl(de\014ne)-5 b(d)75 3364 y(in)33 b(Se)-5 b(ction)33 b(5,)g(De\014nition)f(1,)h(we)g(have)1316 3568 y Fp(\033)28 b Fm(2)d Fs(\006)32 b Fl(and)i Fp(h)26 b Fm(2)e Fs(\000)i(=)-15 b Fm(\))24 b Fp(h:\033)29 b Fm(2)c Fs(\006)p Fp(:)75 3772 y Fe(Pr)n(oof)34 b(:)40 b Fs(Compare)30 b([19)q(],)h(Prop)s (osition)e(1.)41 b(Let)31 b(us)e(consider)h(the)g(mapping)1035 3977 y Fp(T)1088 3991 y Fq(s)1151 3977 y Fs(:)25 b Fp(\033)f Fm(\002)19 b Fp(E)1439 3939 y Fn(\000)1524 3977 y Fm(\000)-15 b(!)25 b Fp(E)1768 3939 y Fn(\000)1298 4114 y Fs(\()p Fp(z)t(;)15 b(q)s Fs(\))26 b Fm(7\000)-15 b(!)25 b Fp(T)1749 4128 y Fq(s)1786 4114 y Fs(\()p Fp(z)t(;)15 b(q)s Fs(\))26 b(=)f Fp(q)e Fm(\000)d Fp(P)2334 4077 y Fn(\000)2413 4114 y Fm(\016)h Fp(k)2526 4128 y Fq(s)2563 4114 y Fs(\()p Fp(z)t(;)15 b(q)s Fs(\))p Fp(;)75 4319 y Fs(where)31 b Fp(P)410 4286 y Fn(\000)495 4319 y Fs(:)c(\003)610 4333 y Fq(T)686 4319 y Fm(\002)21 b Fp(E)845 4333 y Fq(T)926 4319 y Fm(\000)-15 b(!)27 b Fp(E)1172 4286 y Fn(\000)1262 4319 y Fs(is)j(the)i(pro)5 b(jection)31 b(asso)s(ciated)g(with)f(the)i (splitting)d Fp(E)3171 4333 y Fq(T)3252 4319 y Fs(=)e Fp(E)3422 4286 y Fo(+)3502 4319 y Fm(\010)20 b Fp(E)3665 4286 y Fn(\000)3725 4319 y Fs(,)75 4432 y(and)30 b Fp(k)299 4446 y Fq(s)366 4432 y Fs(is)g(the)g(homotop)m(y)h(b)s(et)m(w)m(een)g Fp(k)1436 4446 y Fo(0)1501 4432 y Fs(=)25 b Fp(I)7 b(d)31 b Fs(and)f Fp(k)1946 4446 y Fo(1)2011 4432 y Fs(=)25 b Fp(h)p Fs(.)41 b(Let)31 b(us)e(set)1178 4636 y Fp(F)1236 4650 y Fq(s)1298 4636 y Fs(=)c Fm(f)p Fs(\()p Fp(z)t(;)15 b(q)s Fs(\))27 b Fm(2)e Fp(\033)e Fm(\002)d Fp(E)1990 4598 y Fn(\000)2075 4636 y Fp(=)26 b(T)2199 4650 y Fq(s)2236 4636 y Fs(\()p Fp(z)t(;)15 b(q)s Fs(\))26 b(=)f Fp(q)s Fm(g)75 4840 y Fs(and)1186 4953 y Fp(I)1226 4967 y Fq(s)1288 4953 y Fs(=)g Fp(k)1431 4967 y Fq(s)1468 4953 y Fs(\()p Fp(\033)f Fm(\002)19 b Fp(E)1741 4915 y Fn(\000)1801 4953 y Fs(\))h Fm(\\)g Fs(\003)h Fm(\002)e Fp(E)2183 4915 y Fo(+)2268 4953 y Fs(=)25 b Fp(k)2411 4967 y Fq(s)2448 4953 y Fs(\()p Fp(F)2541 4967 y Fq(s)2579 4953 y Fs(\))p Fp(:)75 5120 y Fs(Both)31 b Fp(I)341 5134 y Fq(s)407 5120 y Fs(and)e Fp(F)641 5134 y Fq(s)708 5120 y Fs(con)m(tain)i(\()p Fp(\022)1107 5134 y Fo(0)1146 5120 y Fp(;)15 b Fs(0\).)42 b(Since)29 b Fm(Q)1644 5135 y Fq(l)1700 5120 y Fs(is)g(negativ)m(e)i (de\014nite)e(on)g Fp(E)2669 5087 y Fn(\000)2759 5120 y Fs(and)2935 5047 y Fk(R)3011 5120 y Fp(@)5 b(W)42 b Fs(is)29 b(b)s(ounded)f(on)75 5233 y Fp(\033)s Fs(,)g(there)f(is)e(a)i Fp(c)f(>)f Fs(0)i(suc)m(h)f(that)h Fm(Q)1272 5248 y Fq(l)1299 5233 y Fs(\()p Fp(q)s Fs(\))13 b(+)1510 5160 y Fk(R)1585 5233 y Fp(W)g Fs(\()p Fp(@)5 b(\022)s Fs(\))25 b Fp(<)g Fs(0)i(for)g(all)e Fp(\022)j Fm(2)c Fp(\033)30 b Fs(and)c Fp(q)i Fm(2)d Fp(E)2942 5200 y Fn(\000)3028 5233 y Fs(satisfying)g Fm(k)p Fp(q)s Fm(k)h Fi(>)f Fp(c)p Fs(.)75 5346 y(As)30 b(a)h(consequence,)g(the)g(mapping)e Fp(T)1407 5360 y Fq(s)1474 5346 y Fs(satis\014es)211 5533 y Fm(\017)46 b Fp(T)355 5547 y Fo(0)420 5533 y Fs(=)25 b(0,)1867 5841 y(30)p eop %%Page: 31 31 31 30 bop 211 399 a Fm(\017)46 b Fp(T)355 413 y Fq(s)392 399 y Fs(\()p Fp(\022)s(;)15 b(q)s Fs(\))26 b(=)f(0)30 b(for)g(all)g Fp(q)j Fs(suc)m(h)d(that)h Fm(k)p Fp(q)s Fm(k)26 b Fi(>)f Fp(c)30 b Fs(and)g(all)g Fp(s)p Fs(,)211 586 y Fm(\017)46 b Fp(T)355 600 y Fq(s)392 586 y Fs(\()p Fp(\022)470 600 y Fo(0)510 586 y Fp(;)15 b(q)s Fs(\))26 b(=)e(0)31 b(for)f(all)g Fp(q)j Fs(and)d(all)f Fp(s)p Fs(,)75 772 y(and)38 b(w)m(e)g(can)h(apply)e(Dold's)h(\014xed)f(p)s (oin)m(t)h(transfer,)i(see)e([14)r(])g(and)g([19)q(],)j(page)e(433,)i (that)e(asserts)g(the)75 885 y(injectivit)m(y)32 b(of)h(the)g(morphism) d Fp(P)1270 852 y Fn(\003)1257 911 y Fo(\003)1340 885 y Fs(:)g Fp(H)1478 852 y Fn(\003)1517 885 y Fs(\()p Fp(\033)n(;)15 b(\022)1685 899 y Fo(0)1725 885 y Fs(\))30 b Fm(\000)-16 b(!)30 b Fp(H)2049 852 y Fn(\003)2088 885 y Fs(\()p Fp(F)2181 899 y Fq(s)2219 885 y Fp(;)15 b Fs(\()p Fp(\022)2337 899 y Fo(0)2376 885 y Fp(;)g Fs(0\)\))p Fp(:)35 b Fs(W)-8 b(e)34 b(no)m(w)f(tak)m(e)h Fp(s)29 b Fs(=)g(1)k(and)g(ha)m(v)m(e)75 997 y(the)e(comm)m(utativ)m(e)g(diagram)1182 1220 y Fp(H)1265 1187 y Fn(\003)1304 1220 y Fs(\()p Fp(I)1379 1234 y Fo(1)1419 1220 y Fp(;)15 b Fs(\()p Fp(\022)1537 1234 y Fo(0)1577 1220 y Fp(;)g Fs(0\)\))1865 1169 y Fq(h)1906 1145 y Fh(\003)1779 1220 y Fm(\000)-17 b(\000)c(\000)k(!)45 b Fp(H)2156 1187 y Fn(\003)2196 1220 y Fs(\()p Fp(F)2289 1234 y Fo(1)2329 1220 y Fp(;)15 b Fs(\()p Fp(\022)2447 1234 y Fo(0)2487 1220 y Fp(;)g Fs(0\)\))1337 1385 y Fq(P)1392 1362 y Fh(\003)1382 1408 y Fg(\003)1427 1291 y Fk(x)1427 1346 y(?)1427 1401 y(?)2328 1291 y(x)2328 1346 y(?)2328 1401 y(?)2389 1385 y Fq(P)2444 1362 y Fh(\003)2434 1408 y Fg(\003)1236 1572 y Fp(H)1319 1539 y Fn(\003)1358 1572 y Fs(\()p Fp(h:\033)n(;)g(\022) 1603 1586 y Fo(0)1644 1572 y Fs(\))496 b Fp(H)2258 1539 y Fn(\003)2298 1572 y Fs(\()p Fp(\033)n(;)15 b(\022)2466 1586 y Fo(0)2506 1572 y Fs(\))1306 1737 y Fq(i)1330 1713 y Fh(\003)1330 1760 y Fb(h:\033)1427 1643 y Fk(x)1427 1698 y(?)1427 1752 y(?)2328 1643 y(x)2328 1698 y(?)2328 1752 y(?)2389 1740 y Fq(i)2413 1717 y Fh(\003)2413 1757 y Fb(\033)1268 1955 y Fp(H)1351 1922 y Fn(\003)1390 1955 y Fs(\(\003)p Fp(;)g(\022)1571 1969 y Fo(0)1611 1955 y Fs(\))1860 1889 y Fq(h)1901 1865 y Fh(\003)1901 1911 y Fg(\003)1779 1955 y Fm(\000)-17 b(\000)c(\000)k(!)128 b Fp(H)2239 1922 y Fn(\003)2279 1955 y Fs(\(\003)p Fp(;)15 b(\022)2460 1969 y Fo(0)2500 1955 y Fs(\))p Fp(;)75 2139 y Fs(where)30 b Fp(h)390 2106 y Fn(\003)390 2166 y Fo(\003)474 2139 y Fs(is)f(the)i(isomorphism)c(that)k(mak)m(es)g(the)g(follo)m (wing)e(diagram)h(comm)m(ute)1024 2371 y Fp(H)1107 2338 y Fn(\003)1146 2371 y Fs(\(\003)21 b Fm(\002)f Fp(E)5 b(;)15 b Fs(\()p Fp(\022)1546 2385 y Fo(0)1586 2371 y Fp(;)g Fs(0\)\))1874 2320 y Fq(h)1915 2296 y Fh(\003)1788 2371 y Fm(\000)-17 b(\000)c(\000)k(!)45 b Fp(H)2165 2338 y Fn(\003)2205 2371 y Fs(\(\003)21 b Fm(\002)e Fp(E)5 b(;)15 b Fs(\()p Fp(\022)2604 2385 y Fo(0)2645 2371 y Fp(;)g Fs(0\)\))1262 2536 y Fq(P)1317 2513 y Fh(\003)1307 2559 y Fg(\003)1353 2443 y Fk(x)1353 2497 y(?)1353 2552 y(?)2412 2443 y(x)2412 2497 y(?)2412 2552 y(?)2472 2536 y Fq(P)2527 2513 y Fh(\003)2517 2559 y Fg(\003)1194 2754 y Fp(H)1277 2721 y Fn(\003)1316 2754 y Fs(\(\003)p Fp(;)g(\022)1497 2768 y Fo(0)1537 2754 y Fs(\))1869 2688 y Fq(h)1910 2665 y Fh(\003)1910 2711 y Fg(\003)1788 2754 y Fm(\000)-17 b(\000)c(\000)k(!)203 b Fp(H)2323 2721 y Fn(\003)2362 2754 y Fs(\(\003)p Fp(;)15 b(\022)2543 2768 y Fo(0)2583 2754 y Fs(\))p Fp(:)75 2939 y Fs(Coming)35 b(bac)m(k)i(to)f(the)g (\014rst)g(diagram,)g(w)m(e)h(see)f(that)h Fp(i)2020 2906 y Fn(\003)2020 2966 y Fq(h:\033)2163 2939 y Fs(can)f(not)h(b)s(e)e (zero)i(b)s(ecause)e Fp(P)3242 2906 y Fn(\003)3229 2966 y Fo(\003)3307 2939 y Fm(\016)24 b Fp(i)3407 2906 y Fn(\003)3407 2961 y Fq(\033)3478 2939 y Fm(\016)h Fp(h)3600 2906 y Fn(\003)3600 2966 y Fo(\003)3689 2939 y Fs(is)75 3052 y(nonzero.)3271 b Fi(\003)75 3164 y Fs(F)-8 b(or)31 b(all)e Fp(G)i Fs(satisfying)e([HR1-4])j(and)e(all)f Fp(l)f Fm(2)c Fs(\(1)p Fp(;)15 b Fs(1)22 b(+)e(1)p Fp(=\034)10 b Fs(\))32 b(w)m(e)f(de\014ne)1288 3367 y Fp(c)1327 3329 y Fq(G)1327 3389 y(T)1387 3367 y Fs(\()p Fp(l)r Fs(\))26 b(=)40 b(inf)1608 3427 y Fq(\033)r Fn(2)p Fo(\006)1799 3367 y Fs(inf)1789 3428 y Fq(h)p Fn(2)p Fo(\000)1936 3367 y Fs(sup)14 b Fm(L)2151 3382 y Fq(l)2192 3290 y Fk(\014)2192 3344 y(\014)2222 3386 y Fq(h)p Fo(\()p Fq(\033)r Fn(\002)p Fq(E)2443 3367 y Fh(\000)2495 3386 y Fo(\))75 3602 y Fs(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(estimate:)75 3788 y Fj(Lemma)i(9)45 b Fl(If)33 b Fp(G)g Fl(satis\014es)g(\(2\))h(then)f(the)g(ine)-5 b(quality)1441 3991 y Fp(I)1481 4005 y Fq(T)1536 3991 y Fs(\()p Fp(U)10 b Fs(\))26 b Fi(6)f Fp(c)1839 3953 y Fq(G)1839 4013 y(T)1899 3991 y Fs(\()p Fp(l)r Fs(\))h Fi(6)f Fp(I)2160 4005 y Fq(T)2215 3991 y Fs(\()p Fp(W)13 b Fs(\))75 4193 y Fl(holds.)75 4379 y Fe(Pr)n(oof)43 b(:)119 b Fs(Since)38 b Fp(G)h Fm(\000)-15 b(!)39 b Fp(c)1132 4346 y Fq(G)1132 4406 y(T)1192 4379 y Fs(\()p Fp(l)r Fs(\))g(is)f(an)g(increasing)g(function)f(this)h(is)f(an)i(easy)g (consequence)h(of)f(the)75 4492 y(follo)m(wing)29 b(lemma.)75 4678 y Fj(Lemma)k(10)46 b Fl(F)-7 b(or)33 b(al)5 b(l)33 b Fp(U)43 b Fl(satisfying)33 b([HU1-3],)f(we)h(have)1621 4881 y Fp(c)1660 4843 y Fq(U)1660 4903 y(T)1720 4881 y Fs(\()p Fp(l)r Fs(\))26 b(=)f Fp(I)1981 4895 y Fq(T)2036 4881 y Fs(\()p Fp(U)10 b Fs(\))p Fp(:)75 5083 y Fe(Pr)n(oof)34 b(:)40 b Fs(Recall)30 b(that)1109 5285 y Fp(c)1148 5248 y Fq(U)1148 5308 y(T)1207 5285 y Fs(\()p Fp(l)r Fs(\))c(=)15 b(inf)1403 5346 y Fq(\033)r Fn(2)p Fo(\006)1595 5285 y Fs(inf)1584 5347 y Fq(h)p Fn(2)p Fo(\000)1922 5285 y Fs(sup)1741 5369 y Fo(\()p Fq(z)s(;x)p Fo(\))p Fn(2)p Fq(h)p Fo(\()p Fq(\033)r Fn(\002)p Fq(E)2159 5350 y Fh(\000)2211 5369 y Fo(\))2239 5285 y Fm(Q)2313 5300 y Fq(l)2339 5285 y Fs(\()p Fp(x)p Fs(\))21 b(+)f Fm(U)9 b Fs(\()p Fp(z)t Fs(\))1069 5507 y Fp(I)1109 5521 y Fq(T)1164 5507 y Fs(\()p Fp(U)h Fs(\))26 b(=)15 b(inf)1403 5567 y Fq(\033)r Fn(2)p Fo(\006)1584 5507 y Fs(sup)1590 5579 y Fq(z)s Fn(2)p Fq(\033)2239 5507 y Fm(U)9 b Fs(\()p Fp(z)t Fs(\))1867 5841 y(31)p eop %%Page: 32 32 32 31 bop 75 399 a Fs(W)-8 b(e)32 b(can)e(tak)m(e)i Fp(h)26 b Fs(=)f Fp(I)7 b(d)31 b Fs(in)e(the)h(de\014nition)e(of)j Fp(c)f Fs(to)h(obtain)833 603 y Fp(c)25 b Fi(6)40 b Fs(inf)993 663 y Fq(\033)r Fn(2)p Fo(\006)1308 603 y Fs(sup)1149 686 y Fo(\()p Fq(z)s(;q)1266 667 y Fh(\000)1318 686 y Fo(\))p Fn(2)p Fo(\003)p Fn(\002)p Fq(E)1552 667 y Fh(\000)1619 603 y Fm(Q)1693 618 y Fq(l)1719 603 y Fs(\()p Fp(q)1798 565 y Fn(\000)1857 603 y Fs(\))21 b(+)f Fm(U)9 b Fs(\()p Fp(z)t Fs(\))26 b(=)40 b(inf)2308 663 y Fq(\033)r Fn(2)p Fo(\006)2464 603 y Fs(sup)2466 681 y Fq(z)s Fn(2)p Fo(\003)2616 603 y Fm(U)9 b Fs(\()p Fp(z)t Fs(\))26 b(=)f Fp(I)7 b(:)75 875 y Fs(T)-8 b(o)31 b(obtain)f(the)g(other)h(inequalit)m(y)-8 b(,)29 b(w)m(e)i(apply)e(Lemma)i(8)f(and)g(get)1229 1079 y(sup)1048 1163 y Fo(\()p Fq(z)s(;x)p Fo(\))p Fn(2)p Fq(h)p Fo(\()p Fq(\033)r Fn(\002)p Fq(E)1466 1144 y Fh(\000)1519 1163 y Fo(\))1561 1079 y Fm(Q)1635 1094 y Fq(l)1661 1079 y Fs(\()p Fp(x)p Fs(\))21 b(+)f Fm(U)9 b Fs(\()p Fp(z)t Fs(\))26 b Fi(>)50 b Fs(sup)2199 1158 y Fq(z)s Fn(2)p Fq(h:\033)2400 1079 y Fm(U)9 b Fs(\()p Fp(z)t Fs(\))26 b Fi(>)f Fp(I)7 b(:)3679 1349 y Fi(\003)75 1462 y Fs(W)-8 b(e)32 b(are)e(no)m(w)h(in)e(a)i(p)s(osition)d(to)j(pro)m(v)m(e)g(Prop) s(osition)e(2.)75 1575 y Fe(Pr)n(oof)45 b(of)g(Pr)n(oposition)g(2:)130 b Fs(First,)43 b(notice)e(that)h(the)f(third)f(conclusion)f(is)h(a)i (consequence)g(of)75 1688 y(the)30 b(t)m(w)m(o)i(other)e(ones)h(since)e (the)h(only)g Fp(T)43 b Fs(p)s(erio)s(dic)27 b(solution)i(of)h Fp(L)2374 1703 y Fq(l)2430 1688 y Fs(satisfying)f Fp(\022)2878 1702 y Fq(T)2958 1688 y Fm(\021)c Fp(\022)3097 1702 y Fo(0)3166 1688 y Fs(is)k(the)i(constan)m(t)75 1801 y(curv)m(e)22 b(\()p Fp(\022)387 1815 y Fo(0)427 1801 y Fp(;)15 b Fs(0\),)25 b(and)c(has)h(zero)h(action,)h(whic)m(h)c(is)h(forbidden)f(b)m(y)i(the) g(second)g(conclusion)e(since)i Fp(c)3363 1815 y Fq(T)3418 1801 y Fs(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\)\))26 b Fi(>)75 1914 y Fp(I)115 1928 y Fq(T)170 1914 y Fs(\()p Fp(U)10 b Fs(\))31 b Fp(>)f Fs(0.)50 b(Let)34 b(us)e(no)m(w)i(c)m(ho)s (ose)g Fp(l)r Fs(\()p Fp(T)13 b Fs(\).)50 b(The)33 b(function)f Fp(l)g Fm(\000)-16 b(!)31 b Fp(c)2397 1928 y Fq(T)2452 1914 y Fs(\()p Fp(l)r Fs(\))j(is)e(non-increasing)g(th)m(us)h(almost)75 2027 y(ev)m(erywhere)e(di\013eren)m(tiable.)39 b(Moreo)m(v)m(er,)33 b(the)d(inequalit)m(y)1255 2168 y Fk(Z)1346 2194 y Fo(1+1)p Fq(=\034)1306 2374 y Fo(1)1565 2291 y Fp(c)1604 2254 y Fn(0)1604 2314 y Fq(T)1660 2291 y Fs(\()p Fp(l)r Fs(\))p Fp(dl)e Fi(>)d Fp(I)1997 2305 y Fq(T)2052 2291 y Fs(\()p Fp(U)10 b Fs(\))21 b Fm(\000)e Fp(I)2345 2305 y Fq(T)2400 2291 y Fs(\()p Fp(W)13 b Fs(\))75 2542 y(holds)29 b(and)h(w)m(e)g(can)h (c)m(ho)s(ose)h(an)e Fp(l)r Fs(\()p Fp(T)13 b Fs(\))30 b(in)g(the)g(in)m(terv)-5 b(al)30 b(\(1)p Fp(;)15 b Fs(1)p Fp(=\034)10 b Fs(\))32 b(suc)m(h)e(that)1181 2746 y Fp(c)1220 2708 y Fn(0)1269 2746 y Fs(=)25 b Fm(j)p Fp(c)1429 2708 y Fn(0)1429 2768 y Fq(T)1485 2746 y Fs(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\)\))p Fm(j)26 b Fi(6)f Fp(\034)10 b Fs(\()p Fp(I)1992 2760 y Fq(T)2047 2746 y Fs(\()p Fp(W)j Fs(\))21 b Fm(\000)f Fp(I)7 b(T)13 b Fs(\()p Fp(U)d Fs(\)\))p Fp(:)75 2950 y Fs(Let)31 b(us)f(set)g Fp(c)c Fs(=)f Fp(c)696 2964 y Fq(T)751 2950 y Fs(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\)\))32 b(and)e(recall)f(that)1313 3154 y Fm(L)1376 3169 y Fq(l)1402 3154 y Fs(\()p Fp(\022)s(;)15 b(q)s Fs(\))25 b(=)g Fm(L)p Fs(\()p Fp(\022)s(;)15 b(q)s Fs(\))20 b Fm(\000)g Fp(al)2174 3117 y Fo(2)2213 3154 y Fp(!)2273 3117 y Fo(2)2313 3154 y Fm(k)p Fp(q)s Fm(k)2447 3117 y Fo(2)2447 3177 y(2)2487 3154 y Fp(:)75 3358 y Fs(W)-8 b(e)32 b(shall)c(pro)m(v)m(e)j(that)g(there)g(exists)f(a)h(critical)e (p)s(oin)m(t)g Fp(X)2061 3372 y Fq(T)2142 3358 y Fs(=)c(\()p Fp(\022)2316 3372 y Fq(T)2371 3358 y Fp(;)15 b(q)2452 3372 y Fq(T)2507 3358 y Fs(\))31 b(of)f Fm(L)2739 3377 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))2901 3358 y Fs(suc)m(h)30 b(that)1520 3574 y(2)p Fp(a!)1673 3536 y Fo(2)1713 3574 y Fm(k)p Fp(q)1799 3588 y Fq(T)1854 3574 y Fm(k)1899 3536 y Fo(2)1899 3596 y(2)1964 3574 y Fi(6)25 b Fs(1)c(+)f Fp(c:)2281 3536 y Fn(0)75 3778 y Fs(Arguing)k(b)m(y)h(con)m(tradiction) g(w)m(e)h(assume)f(that)h(there)g(is)e(no)h(critical)f(p)s(oin)m(t)g (of)i Fp(L)2846 3796 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))3003 3778 y Fs(at)26 b(lev)m(el)f Fp(c)g Fs(satisfying)75 3891 y(2)p Fp(a!)228 3858 y Fo(2)268 3891 y Fm(k)p Fp(q)354 3905 y Fq(T)409 3891 y Fm(k)454 3858 y Fo(2)454 3915 y(2)528 3891 y Fi(6)34 b Fs(1)24 b(+)f Fp(c)835 3858 y Fn(0)859 3891 y Fs(.)57 b(W)-8 b(e)36 b(can)g(then)g(\014nd)e(using)g (a)i(standard)f(deformation)g(argumen)m(t)h(an)g Fp(\017)f Fs(in)g(the)75 4004 y(in)m(terv)-5 b(al)30 b(\(0)p Fp(;)15 b(c=)p Fs(2\))33 b(and)c(a)i(homeomorphism)e Fp(h)1708 4018 y Fo(0)1773 4004 y Fm(2)c Fs(\000)30 b(satisfying)1423 4208 y Fm(L)1486 4226 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))1617 4208 y Fs(\()p Fp(h)1704 4222 y Fo(0)1745 4208 y Fs(\()p Fp(X)d Fs(\)\))26 b Fi(6)f Fm(L)2117 4226 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))2249 4208 y Fs(\()p Fp(X)d Fs(\))75 4412 y(for)30 b(all)f Fp(X)k Fm(2)25 b Fp(A)602 4426 y Fq(T)657 4412 y Fs(,)31 b(and)f(suc)m(h)g(that)1503 4525 y Fm(L)1566 4543 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))1698 4525 y Fs(\()p Fp(h)1785 4539 y Fo(0)1825 4525 y Fs(\()p Fp(X)d Fs(\)\))27 b Fi(6)e Fp(c)20 b Fm(\000)g Fp(\017)75 4692 y Fs(for)30 b(all)f Fp(X)k Fs(=)25 b(\()p Fp(\022)s(;)15 b(q)s Fs(\))25 b Fm(2)g Fp(A)923 4706 y Fq(T)1009 4692 y Fs(satisfying)978 4896 y Fm(L)1041 4915 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))1173 4896 y Fs(\()p Fp(X)d Fs(\))26 b Fi(6)f Fp(c)c Fs(+)e Fp(\017)122 b Fs(and)e(2)p Fp(a!)2176 4859 y Fo(2)2216 4896 y Fm(k)p Fp(q)s Fm(k)2350 4859 y Fo(2)2350 4919 y(2)2416 4896 y Fi(6)24 b Fp(c)2550 4859 y Fn(0)2594 4896 y Fs(+)c(1)p Fp(=)p Fs(2)p Fp(:)75 5100 y Fs(Let)33 b Fp(l)267 5114 y Fq(n)347 5100 y Fs(b)s(e)f(an)h(increasing)f (sequence)h(con)m(v)m(erging)h(to)f Fp(l)r Fs(\()p Fp(T)13 b Fs(\),)34 b(and)e(let)h Fp(c)2553 5114 y Fq(n)2630 5100 y Fs(=)c Fp(c)2769 5114 y Fq(T)2824 5100 y Fs(\()p Fp(l)2886 5114 y Fq(n)2934 5100 y Fs(\))k(and)f Fm(L)3244 5114 y Fq(n)3320 5100 y Fs(=)d Fm(L)3483 5115 y Fq(l)3504 5123 y Fb(n)3551 5100 y Fs(.)48 b(W)-8 b(e)75 5213 y(can)31 b(c)m(ho)s(ose)g Fp(\033)582 5227 y Fq(n)654 5213 y Fm(2)25 b Fs(\006)30 b(and)g Fp(h)1065 5227 y Fq(n)1137 5213 y Fm(2)25 b Fs(\000)30 b(suc)m(h)g(that)1109 5417 y(sup)14 b Fm(L)1324 5431 y Fq(n)1386 5340 y Fk(\014)1386 5395 y(\014)1417 5437 y Fq(h)1458 5445 y Fb(n)1500 5437 y Fo(\()p Fq(\033)1567 5445 y Fb(n)1610 5437 y Fn(\002)p Fq(E)1721 5418 y Fh(\000)1772 5437 y Fo(\))1839 5417 y Fi(6)25 b Fp(c)1974 5431 y Fq(n)2042 5417 y Fs(+)20 b(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))21 b Fm(\000)e Fp(l)2471 5431 y Fq(n)2519 5417 y Fs(\))p Fp(=)p Fs(10)p Fp(:)1867 5841 y Fs(32)p eop %%Page: 33 33 33 32 bop 75 399 a Fs(When)30 b Fp(n)g Fs(is)f(large)i(enough)f(this)f (implies)305 603 y Fm(L)368 621 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))515 525 y Fk(\014)515 580 y(\014)545 622 y Fq(h)586 630 y Fb(n)628 622 y Fo(\()p Fq(\033)695 630 y Fb(n)739 622 y Fn(\002)p Fq(E)850 603 y Fh(\000)901 622 y Fo(\))968 603 y Fi(6)25 b Fm(L)1127 617 y Fq(n)1189 525 y Fk(\014)1189 580 y(\014)1219 622 y Fq(h)1260 630 y Fb(n)1302 622 y Fo(\()p Fq(\033)1369 630 y Fb(n)1412 622 y Fn(\002)p Fq(E)1523 603 y Fh(\000)1575 622 y Fo(\))1699 603 y Fi(6)83 b Fp(c)1892 617 y Fq(n)1960 603 y Fs(+)19 b(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))21 b Fm(\000)f Fp(l)2389 617 y Fq(n)2436 603 y Fs(\))p Fp(=)p Fs(10)1699 741 y Fi(6)83 b Fp(c)21 b Fs(+)f(\()p Fp(c)2078 703 y Fn(0)2122 741 y Fs(+)g(1)p Fp(=)p Fs(10\)\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))23 b Fm(\000)d Fp(l)2769 755 y Fq(n)2816 741 y Fs(\))g(+)g(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))21 b Fm(\000)f Fp(l)3301 755 y Fq(n)3348 741 y Fs(\))p Fp(=)p Fs(10)1699 878 y Fi(6)83 b Fp(c)21 b Fs(+)f(\()p Fp(c)2078 841 y Fn(0)2122 878 y Fs(+)g(1)p Fp(=)p Fs(5\)\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))22 b Fm(\000)e Fp(l)2723 892 y Fq(n)2770 878 y Fs(\))p Fp(:)75 1083 y Fs(T)-8 b(ak)m(e)32 b(a)e(lo)s(op)g Fp(X)j Fs(=)25 b(\()p Fp(\022)s(;)15 b(q)s Fs(\))25 b Fm(2)g Fp(h)1137 1097 y Fq(n)1184 1083 y Fs(\()p Fp(\033)1271 1097 y Fq(n)1339 1083 y Fm(\002)20 b Fp(E)1502 1050 y Fn(\000)1561 1083 y Fs(\),)31 b(either)1347 1287 y Fm(L)1410 1305 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))1541 1287 y Fs(\()p Fp(X)d Fs(\))27 b Fi(6)e Fp(c)20 b Fm(\000)g Fs(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))21 b Fm(\000)f Fp(l)2305 1301 y Fq(n)2352 1287 y Fs(\))p Fp(=)p Fs(5)75 1491 y(and)1265 1604 y Fm(L)1328 1622 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))1460 1604 y Fs(\()p Fp(h)1547 1618 y Fo(0)1587 1604 y Fs(\()p Fp(X)d Fs(\)\))27 b Fi(6)e Fp(c)20 b Fm(\000)g Fs(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))21 b Fm(\000)f Fp(l)2386 1618 y Fq(n)2433 1604 y Fs(\))p Fp(=)p Fs(5)75 1771 y(or)1411 1884 y Fm(L)1474 1902 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))1630 1884 y Fi(>)25 b Fp(c)c Fm(\000)f Fs(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))21 b Fm(\000)f Fp(l)2216 1898 y Fq(n)2263 1884 y Fs(\))p Fp(=)p Fs(5)p Fp(:)75 2051 y Fs(In)30 b(the)g(second)h(case,)176 2255 y(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))376 2217 y Fo(2)437 2255 y Fm(\000)19 b Fp(l)556 2217 y Fo(2)554 2277 y Fq(n)602 2255 y Fs(\))p Fp(a!)745 2217 y Fo(2)784 2255 y Fm(k)p Fp(q)s Fm(k)918 2217 y Fo(2)984 2255 y Fs(=)25 b Fm(L)1143 2269 y Fq(n)1189 2255 y Fs(\()p Fp(X)7 b Fs(\))22 b Fm(\000)e(L)1517 2273 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))1648 2255 y Fs(\()p Fp(X)d Fs(\))27 b Fi(6)d Fp(c)d Fs(+)f(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))21 b Fm(\000)f Fp(l)2412 2269 y Fq(n)2459 2255 y Fs(\)\()p Fp(c)2568 2217 y Fn(0)2612 2255 y Fs(+)g(1)p Fp(=)p Fs(5\))i Fm(\000)e Fp(c)h Fs(+)f(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))21 b Fm(\000)e Fp(l)3475 2269 y Fq(n)3523 2255 y Fs(\))p Fp(=)p Fs(5)984 2402 y Fi(6)25 b Fs(\()p Fp(c)1154 2365 y Fn(0)1198 2402 y Fs(+)20 b(1)p Fp(=)p Fs(2\)\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))22 b Fm(\000)e Fp(l)1799 2416 y Fq(n)1846 2402 y Fs(\))p Fp(;)75 2631 y Fs(th)m(us)591 2861 y(2)p Fp(a!)744 2823 y Fo(2)784 2861 y Fm(k)p Fp(q)s Fm(k)918 2823 y Fo(2)984 2861 y Fi(6)25 b Fs(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))21 b(+)f Fp(l)1419 2875 y Fq(n)1466 2861 y Fs(\))p Fp(a!)1609 2823 y Fo(2)1648 2861 y Fm(k)p Fp(q)s Fm(k)1782 2823 y Fo(2)1848 2861 y Fi(6)25 b Fp(c)1983 2823 y Fn(0)2027 2861 y Fs(+)19 b(1)p Fp(=)p Fs(2)p Fp(;)75 3065 y Fs(and)30 b(w)m(e)h(get)1491 3178 y Fm(L)1554 3196 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))1685 3178 y Fs(\()p Fp(h)1772 3192 y Fo(0)1812 3178 y Fs(\()p Fp(X)d Fs(\)\))27 b Fi(6)e Fp(c)20 b Fm(\000)g Fp(\017;)75 3345 y Fs(when)29 b Fp(n)h Fs(is)g(large)g(enough.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)h(seen)e (that)1031 3549 y Fm(L)1094 3567 y Fq(l)q Fo(\()p Fq(T)10 b Fo(\))1225 3549 y Fs(\()p Fp(h)1312 3563 y Fo(0)1373 3549 y Fm(\016)20 b Fp(h)1490 3563 y Fq(n)1538 3549 y Fs(\()p Fp(\033)1625 3563 y Fq(n)1692 3549 y Fm(\002)g Fp(E)1855 3511 y Fn(\000)1914 3549 y Fs(\)\))26 b Fi(6)f Fp(c)c Fm(\000)f Fs(\()p Fp(l)r Fs(\()p Fp(T)13 b Fs(\))21 b Fm(\000)f Fp(l)2596 3563 y Fq(n)2643 3549 y Fs(\))p Fp(=)p Fs(5)p Fp(;)75 3753 y Fs(whic)m(h)29 b(is)h(a)g(con)m (tradiction)h(since)e Fp(h)1330 3767 y Fo(0)1390 3753 y Fm(\016)21 b Fp(h)1508 3767 y Fq(n)1581 3753 y Fm(2)j Fs(\000.)1931 b Fi(\003)75 4152 y Ft(References)120 4355 y Fs([1])47 b(Arnold)29 b(V.)h(I.)h(:)40 b Fl(Mathematic)-5 b(al)35 b(metho)-5 b(ds)35 b(of)e(classic)-5 b(al)34 b(me)-5 b(chanics,)31 b Fs(Springer)e(\(1978\).)120 4543 y([2])47 b(Benci)29 b(V.)g(-)g(Giannoni)e(F.)i(:)40 b(Homo)s(clinic)28 b(orbits)f(on)i(compact)h(manifolds,)d(J.)i(Math.)h(Anal.)e(Appl.)262 4656 y Fj(157)j Fs(\(1991\),)i(568-576.)120 4843 y([3])47 b(Bernard)20 b(P)-8 b(.)21 b(:)35 b(Homo)s(clinic)19 b(orbitsin)f(families)g(of)j(h)m(yp)s(ersurfaces)e(with)g(h)m(yp)s(erb) s(olic)f(p)s(erio)s(dic)g(orbits.)262 4956 y(Preprin)m(t)29 b(CEREMADE.)120 5144 y([4])47 b(Bernard)22 b(P)-8 b(.)24 b(:)37 b(Rec)m(herc)m(he)25 b(v)-5 b(ariationnelle)21 b(d'orbites)h(homo)s(clines)f(dans)i(les)f(syst)m(\022)-43 b(emes)25 b(dynamiques)262 5257 y(hamiltoniens,)j(th)m(\022)-43 b(ese,)32 b(2000.)120 5445 y([5])47 b(Bolotin)27 b(S.V.)g(:)40 b(Libration)25 b(motions)i(of)h(natural)e(dynamical)g(systems,)i(V)-8 b(estnik)28 b(Mosk.)g(Univ.)e(Ser)262 5557 y(I)k(Mat.)i Fj(6)p Fs(\(1978\),72-77.)1867 5841 y(33)p eop %%Page: 34 34 34 33 bop 120 399 a Fs([6])47 b(Bolotin)23 b(S.V.)g(:The)g(existence)h (of)f(homo)s(clinic)e(motions,)k(V)-8 b(estnik)23 b(Mosk.)h(Univ.)e (Ser)h(I)g(Mat.)i(Mekh.)262 511 y Fj(6)p Fs(\(1983\),98-102.)120 693 y([7])47 b(Bott)32 b(R.)e(:)41 b(Lectures)31 b(on)f(Morse)h(theory) -8 b(,)31 b(old)f(and)f(new,)h(BAMS)h Fj(7)g Fs(\(1982\))i(,)d(No.)h (2,)g(331-358.)120 875 y([8])47 b(Bu\013oni)28 b(B.)i(-)g(S)m(\023)-43 b(er)m(\023)g(e)32 b(E.)d(:)40 b(A)30 b(global)f(condition)f(for)h (quasi-random)f(b)s(eha)m(vior)h(in)f(a)i(class)f(of)g(conser-)262 988 y(v)-5 b(ativ)m(e)31 b(systems,)g(Comm.)f(Pure)g(and)f(App.)h (Math.)h(,)g Fj(49)g Fs(\(1996\),)i(285-305.)120 1170 y([9])47 b(Cielebak)39 b(K.:)61 b(Pseudoholomorphic)38 b(curv)m(es)j(and)f(p)s(erio)s(dic)e(orbits)h(on)h(cotangen)m(t)j (bundles,)e(J.)262 1283 y(Math.)31 b(Pures)f(Appl.)f(\(a\))i Fj(73)p Fs(\(1994\),)j(251-278.)75 1465 y([10])47 b(Cielebak)28 b(K.)i(-)g(S)m(\023)-43 b(er)m(\023)g(e)31 b(E.)f(:)40 b(Pseudoholomorphic)27 b(curv)m(es)j(and)f(m)m(ultiplicit)m(y)d(of)k (homolcinic)e(orbits,)262 1578 y(Duk)m(e)j(Math.)g(J.)p Fj(77)p Fs(\(1995\),)j(483-518.)75 1760 y([11])47 b(Cielebak)42 b(K.)g(-)i(S)m(\023)-43 b(er)m(\023)g(e)44 b(E.)f(:)66 b(Pseudoholomorphic)41 b(curv)m(es)i(and)f(the)h(shado)m(wing)f(lemma,) k(Duk)m(e)262 1873 y(Math.)31 b(J.)p Fj(98)p Fs(\(1999\).)75 2055 y([12])47 b(Conley)34 b(C.)h(C.)g(:)50 b(On)35 b(the)g(ultimate)f (b)s(eha)m(vior)h(of)g(orbits)f(with)g(resp)s(ect)h(to)h(an)f(unstable) f(critical)262 2167 y(p)s(oin)m(t)29 b(I.)i(oscillating,)e(asymptotic)h (and)g(capture)h(orbits,)e(J.)i(Di\013.)f(Eq.)g Fj(5)p Fs(\(1969\),)k(136-158.)75 2349 y([13])47 b(Coti)32 b(Zelati)g(V.,)i (Ek)m(eland)e(I.,)i(and)e(Sere)h(E.)g(:)45 b(A)33 b(V)-8 b(ariational)32 b(Approac)m(h)h(to)h(Homo)s(clinic)d(Orbits)262 2462 y(in)e(Hamiltonian)g(Systems,)h(Math.)h(Ann.)f Fj(288)h Fs(\(1990\),)i(133-160.)75 2644 y([14])47 b(Dold)21 b(A.)i(:)36 b(The)21 b(\014xed)h(p)s(oin)m(t)e(transfer)i(of)g(\014bre)f (preserving)f(maps,)j(Math.)g(Z.)f Fj(148)g Fs(\(1976\),)27 b(215-244.)75 2826 y([15])47 b(Dev)-5 b(aney)33 b(R.)g(L.)f(:)45 b(Homo)s(clinic)31 b(orbits)h(in)f(hamiltonian)f(systems,)j(J.)g (Di\013.)f(Eq.)h Fj(21)g Fs(\(1976\),)i(431-)262 2939 y(438.)75 3121 y([16])47 b(F)-8 b(elmer)30 b(P)-8 b(.)31 b(:)41 b(Hetero)s(clinic)29 b(orbits)g(for)h(spatially)f(p)s(erio)s (dic)e(hamiltonian)h(systems,)j(Ann.)f(Inst.)g(H.)262 3234 y(P)m(oincar)m(\023)-43 b(e,)32 b(Anal.)e(Non)g(Lin)m(\023)-43 b(eaire)30 b Fj(8)h Fs(\(1991\),)i(477-497.)75 3416 y([17])47 b(F)-8 b(enic)m(hel)39 b(N.)h(:)60 b(P)m(ersistence)40 b(and)f(smo)s(othness)g(of)g(in)m(v)-5 b(arian)m(t)39 b(manifolds)f(for)h(\015o)m(ws,)j(Ind.)d(Univ.)262 3529 y(Math.)31 b(Jour.)f Fj(26)h Fs(\(1971\),)i(193-225.)75 3711 y([18])47 b(Hirsc)m(h)36 b(M.)g(W.)i(-)e(Pugh)g(C.)g(C.)g(-)h(Sh)m (ub)e(M.)i(:)53 b Fl(Invariant)39 b(Manifolds)p Fs(,)h(Lecture)c(Notes) i(in)d(Math.)262 3823 y Fj(583)c Fs(\(1977\),)i(Springer.)75 4005 y([19])47 b(Hofer)d(H.)g(-)g(Viterb)s(o)f(C.)g(:)67 b(The)43 b(W)-8 b(einstein)43 b(conjecture)i(in)d(cotangen)m(t)k (bundles)41 b(and)i(related)262 4118 y(results,)29 b(Ann.)h(Scuola)g (Normale)g(Sup)s(eriore)e(di)h(Pisa,)h(Serie)g(5,)p Fj(15)h Fs(\(1988\),)j(411-445.)75 4300 y([20])47 b(Klingen)m(b)s(erg)28 b(W.)j(:)p Fl(le)-5 b(ctur)g(es)34 b(on)g(close)-5 b(d)33 b(ge)-5 b(o)g(desics,)32 b Fs(Grundlehren)c(der)i(Math.)h(Wiss.)f Fj(230)h Fs(\(1978\),)262 4413 y(Springer.)75 4595 y([21])47 b(Lerman)22 b(L.)g(M.)h(:)37 b(Hamiltonian)21 b(systems)i(with)e(lo)s (ops)g(of)i(a)g(separatrix)f(of)h(a)g(saddle-cen)m(ter,)h(Selecta)262 4708 y(Math.)31 b(So)m(v.)g Fj(10)g Fs(\(1991\),)i(297-306.)75 4890 y([22])47 b(Ma)m(whin)36 b(J.)h(-)h(Willem)e(M.)h(:)55 b Fl(Critic)-5 b(al)40 b(p)-5 b(oint)40 b(the)-5 b(ory)41 b(and)f(Hamiltonian)h(systems,)e Fs(App.)e(Math.)262 5003 y(Sciences)30 b Fj(105)h Fs(\(1989\),)i(Springer.)75 5185 y([23])47 b(Mielk)m(e)38 b(A.)g(-)g(Holmes)f(P)-8 b(.)38 b(and)f(O'Reilly)f(O.)i(:)55 b(Cascades)38 b(of)g(homo)s(clinic) e(orbits)g(to,)41 b(and)c(c)m(haos)262 5298 y(near,)30 b(a)h(Hamiltonian)e(saddle-cen)m(ter,)i(J.)f(Dyn.)h(Di\013.)f(Eq.)g Fj(4)p Fs(\(1992\),)k(95-126.)75 5479 y([24])47 b(Rabino)m(witz)30 b(P)-8 b(.)32 b(H.)f(:)42 b Fl(Minimax)34 b(metho)-5 b(ds)35 b(in)e(critic)-5 b(al)34 b(p)-5 b(oint)35 b(the)-5 b(ory)35 b(with)f(epplic)-5 b(ations)35 b(to)f(di\013er-)262 5592 y(ential)f(e)-5 b(quations)p Fs(,)32 b(CBMS,)e Fj(65)p Fs(,)h(AMS,)g(Pro)m(vidence)f(\(1986\).)1867 5841 y(34)p eop %%Page: 35 35 35 34 bop 75 399 a Fs([25])47 b(Rabino)m(vitz)27 b(P)-8 b(.)28 b(:)39 b(P)m(erio)s(dic)26 b(and)h(hetero)s(clinic)e(orbits)i (for)g(a)h(p)s(erio)s(dic)c(Hamiltonian)i(system,)j(Ann.)262 511 y(Inst.)h(H.)h(P)m(oincar)m(\023)-43 b(e,)32 b(Anal.)e(Non)g(Lin)m (\023)-43 b(eaire)30 b Fj(6)h Fs(\(1989\),)i(331-346.)75 699 y([26])47 b(Ragazzo)28 b(C.)d(G.)h(:)38 b(Irregular)24 b(dynamics)g(and)h(homo)s(clinic)e(orbits)h(to)i(hamiltonian)d (saddle-cen)m(ters,)262 812 y(Comm.)30 b(Pure)g(Appl.)f(Math.,)i Fj(49)p Fs(\(1996\))j(1-37.)75 1000 y([27])47 b(Stru)m(w)m(e)30 b(M.)h(:)42 b Fl(V)-7 b(ariational)35 b(Metho)-5 b(ds)p Fs(,)32 b(Springer,)c(Second)i(Edition,)f(\(1996\).)75 1187 y([28])47 b(Sulliv)-5 b(an)26 b(D.)k(:)40 b(Di\013eren)m(tial)28 b(forms)h(and)g(the)g(top)s(ology)h(of)f(manifolds,)f(Pro)s(c.)h(In)m (t.)g(Conf.)g(on)g(Man-)262 1300 y(ifolds)f(and)i(Related)h(T)-8 b(opics)29 b(in)h(T)-8 b(op)s(ology)g(,)31 b(T)-8 b(oky)m(o)31 b(\(1973\),)j(37-49.)75 1513 y Fa(P)n(atric)n(k)26 b(BERNARD,)i (CEREMADE,)f(Univ)n(ersit)n(\023)-39 b(e)26 b(P)n(aris)g(Dauphine,)75 1626 y(Place)h(du)g(Mar)n(\023)-39 b(ec)n(hal)25 b(de)j(Lattre)f(de)h (T)-7 b(assign)n(y)g(,)26 b(75775)f(P)n(aris)h(Cedex)h(16.)75 1738 y(pb)r(ernard@clipp)r(er.ens.fr)1867 5841 y Fs(35)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0001220507302--