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Fm(1)3289 4901 y Ft(\))p 3021 4946 307 4 v 3104 5037 a Fo(@)g(I)3203 5052 y Fm(2)0 5268 y Ft(for)32 b(eac)m(h)i(2)27 b Fn(\024)h Fo(i)g Fn(\024)g Fo(n)23 b Fn(\000)f Fo(s)g Ft(+)g(1)33 b(and)f Fo(u)c Ft(=)f(0)32 b(for)g Fo(I)1793 5283 y Fm(2)1861 5268 y Ft(=)27 b(0,)32 b(one)h(sees)h(that)462 5541 y Fo(!)523 5556 y Fl(i)551 5541 y Ft(\()p Fo(I)8 b Ft(\))27 b(=)818 5473 y Fo(@)5 b(H)955 5488 y Fm(0)996 5473 y Ft(\()p Fo(I)j Ft(\))p 818 5518 304 4 v 906 5609 a Fo(@)d(I)1005 5624 y Fl(i)1160 5541 y Ft(=)1264 5400 y Fi(\024)1358 5480 y Ft(\()p Fo(u)p Ft(\()p Fo(I)1533 5495 y Fm(2)1572 5480 y Fo(;)17 b(:)g(:)g(:)e(;)i(I)1833 5495 y Fl(n)p Fr(\000)p Fl(s)p Fm(+1)2058 5480 y Ft(\)\))2134 5444 y Fl(i)p Fr(\000)p Fm(1)2335 5480 y Ft(for)82 b(1)28 b Fn(\024)g Fo(i)g Fn(\024)g Fo(n)23 b Fn(\000)f Fo(s)g Ft(+)g(1)1358 5600 y Fo(I)1401 5615 y Fl(i)2335 5600 y Ft(for)82 b Fo(n)23 b Fn(\000)f Fo(s)h Ft(+)f(2)27 b Fn(\024)h Fo(i)g Fn(\024)g Fo(n:)3702 5541 y Ft(\(5\))1889 5895 y(5)p eop %%Page: 6 6 6 5 bop 0 171 a Ft(Th)m(us,)28 b(the)d(Hamilton)e(function)h(\(4\))g (is)h(R)s(\177)-51 b(ussmann)24 b(nondegenerate)i(and)f(the)h(image)d (of)h(its)h(frequency)0 320 y(map)32 b(is)g(of)g(dimension)f Fo(s)p Ft(.)106 529 y(This)36 b(example)f(\(for)g(the)h(particular)e (case)j(of)e Fo(s)d Ft(=)h(1,)j Fo(n)d Ft(=)g(3\))i(w)m(as)i(\014rst)f (presen)m(ted)i(in)d([25].)52 b(The)0 678 y(case)31 b(of)f Fo(s)e Ft(=)f(1)j(and)h(arbitrary)e Fo(n)h Ft(w)m(as)h(considered)h(in) d([123].)42 b(A)31 b(similar)c(example)i(\(for)h(an)m(y)h Fo(s)f Ft(and)h Fo(n)p Ft(\))0 827 y(is)h(con)m(tained)h(in)f([22,)16 b(122].)106 977 y(As)37 b(far)g(as)f(the)h Fs(su\016ciency)45 b Ft(of)36 b(the)h(R)s(\177)-51 b(ussmann)36 b(nondegeneracy)j (condition)c(is)h(concerned,)k(ana-)0 1126 y(lyticit)m(y)c(of)g Fo(H)534 1141 y Fm(0)610 1126 y Ft(cannot)h(b)s(e)g(relaxed)g(to)g Fo(C)1615 1090 y Fr(1)1689 1126 y Ft(-smo)s(othness)g(\(see)h(b)s(elo)m (w\).)56 b(F)-8 b(or)36 b(the)i Fs(ne)-5 b(c)g(essity)45 b Ft(of)36 b(the)0 1276 y(R)s(\177)-51 b(ussmann)41 b(nondegeneracy)j (condition,)f(on)e(the)i(other)e(hand,)k(analyticit)m(y)40 b(is)i(not)f(imp)s(ortan)m(t)f(at)0 1425 y(all.)69 b(If)42 b(the)h(unp)s(erturb)s(ed)g(Hamilton)c(function)i Fo(H)1971 1440 y Fm(0)2052 1425 y Ft(\(of)h(an)m(y)g(smo)s(othness)h(class\))f (do)s(es)g(not)g(meet)0 1575 y(the)g(R)s(\177)-51 b(ussmann)41 b(condition)f(then)i(there)g(are)f(arbitrarily)e(small)g(p)s (erturbations)i Fo(H)3199 1590 y Fm(1)3280 1575 y Ft(\(of)g(the)h(same) 0 1724 y(smo)s(othness)30 b(class\))f(that)h(remo)m(v)m(e)g Fs(al)5 b(l)39 b Ft(the)30 b(in)m(v)-5 b(arian)m(t)28 b Fo(n)p Ft(-tori)g(of)h(the)g(unp)s(erturb)s(ed)i(system)f([22,)16 b(122].)0 1873 y(The)32 b(corresp)s(onding)f(p)s(erturb)s(ed)g(systems) i(admit)c(no)i(in)m(v)-5 b(arian)m(t)29 b Fo(n)p Ft(-tori)h(\(not)g (just)i(no)e(in)m(v)-5 b(arian)m(t)30 b(tori)0 2023 y(carrying)i (quasi-p)s(erio)s(dic)f(motions)g(but)h(no)h(in)m(v)-5 b(arian)m(t)31 b(tori)g(at)i(all\).)0 2231 y Fu(Example)38 b Ft([22,)16 b(122])p Fu(.)45 b Ft(Let)34 b Fn(h)p Fo(\035)t(;)17 b(!)t Ft(\()p Fo(I)8 b Ft(\))p Fn(i)27 b(\021)j Ft(0)j(for)g(some)g(v)m (ector)i Fo(\035)e Fn(2)c Fk(R)2581 2195 y Fl(n)2657 2231 y Fn(n)23 b(f)p Ft(0)p Fn(g)33 b Ft(\(here)h(and)g(henceforth,)0 2381 y Fn(h\001)p Fo(;)17 b Fn(\001i)33 b Ft(denotes)j(the)f(standard)g (inner)g(pro)s(duct)g(in)f Fk(R)1946 2345 y Fl(N)2020 2381 y Ft(\).)49 b(One)35 b(can)g(c)m(ho)s(ose)h(a)e(matrix)f Fo(A)f Fn(2)g Ft(GL)o(\()p Fo(n;)17 b Fk(R)5 b Ft(\))0 2530 y(arbitrarily)39 b(close)j(to)f(the)h(iden)m(tit)m(y)f Fo(n)29 b Fn(\002)g Fo(n)42 b Ft(matrix)e(in)h(suc)m(h)i(a)e(w)m(a)m(y) i(that)e(the)i(v)m(ector)f Fo(A\035)j Ft(will)40 b(b)s(e)0 2680 y(prop)s(ortional)30 b(to)i(an)h(in)m(teger)f(v)m(ector)i Fo(k)c Fn(2)f Fk(Z)1676 2644 y Fl(n)1742 2680 y Fn(n)22 b(f)p Ft(0)p Fn(g)p Ft(.)43 b(No)m(w)33 b(set)1006 2921 y Fo(H)1087 2936 y Fm(1)1126 2921 y Ft(\()p Fo(I)8 b(;)17 b(')p Ft(\))27 b(=)h Fo(H)1573 2936 y Fm(0)1612 2921 y Ft(\()p Fo(A)1723 2880 y Fr(\000)p Fm(1)1817 2921 y Fo(I)8 b Ft(\))22 b Fn(\000)h Fo(H)2109 2936 y Fm(0)2148 2921 y Ft(\()p Fo(I)8 b Ft(\))22 b(+)g Fo(\017)17 b Ft(cos)q Fn(h)p Fo(k)s(;)g(')p Fn(i)880 b Ft(\(6\))0 3162 y(with)29 b Fo(\017)f Fn(2)h Fk(R)21 b Fn(n)16 b(f)p Ft(0)p Fn(g)35 b Ft(arbitrarily)28 b(small.)40 b(Of)29 b(course,)j(this)d(function)g Fo(H)2578 3177 y Fm(1)2647 3162 y Ft(is)h(de\014ned)h(in)e(\()p Fo(D)19 b Fn(\\)d Fo(AD)s Ft(\))g Fn(\002)g Fk(T)3774 3126 y Fl(n)0 3311 y Ft(rather)39 b(than)g(in)f Fo(D)29 b Fn(\002)e Fk(T)930 3275 y Fl(n)981 3311 y Ft(,)40 b(but)g(this)e(is)h (of)f(no)h(imp)s(ortance)f(if)g(one)h(uses)h(the)g(precise)f (de\014nition)f(of)0 3461 y(KAM-stabilit)m(y)29 b([122])h(w)m(e)h(w)m (ould)g(not)f(lik)m(e)g(to)g(dw)m(ell)g(up)s(on)h(here)g(\(this)f (de\014nition)g(tak)m(es)h(care)g(of)f(the)0 3610 y(b)s(eha)m(viour)35 b(of)f(the)i(systems)g(near)f(the)g(b)s(oundary)h(of)e Fo(D)s Ft(\).)50 b(Denote)35 b Fo(@)5 b(H)2720 3625 y Fm(0)2760 3610 y Ft(\()p Fo(A)2871 3574 y Fr(\000)p Fm(1)2966 3610 y Fo(I)j Ft(\))p Fo(=@)d(I)43 b Ft(b)m(y)d Fi(e)-59 b Fo(!)s Ft(\()p Fo(I)8 b Ft(\),)35 b(then)0 3760 y Fn(h)p Fo(A\035)t(;)20 b Fi(e)-58 b Fo(!)r Ft(\()p Fo(I)8 b Ft(\))p Fn(i)39 b Ft(=)g Fn(h)p Fo(\035)t(;)17 b(!)t Ft(\()p Fo(A)912 3723 y Fr(\000)p Fm(1)1004 3760 y Fo(I)8 b Ft(\))p Fn(i)39 b Ft(whence)i Fn(h)p Fo(k)s(;)21 b Fi(e)-59 b Fo(!)s Ft(\()p Fo(I)8 b Ft(\))p Fn(i)38 b(\021)i Ft(0.)64 b(The)40 b(Hamilton)c(function)j Fo(H)47 b Ft(=)39 b Fo(H)3538 3775 y Fm(0)3604 3760 y Ft(+)27 b Fo(H)3788 3775 y Fm(1)0 3909 y Ft(determines)33 b(the)g(equations)g(of)f(motion) 1285 4125 y(_)1262 4150 y Fo(I)j Ft(=)28 b Fo(\017k)20 b Ft(sin)p Fn(h)p Fo(k)s(;)d(')p Fn(i)p Fo(;)237 b Ft(_)-53 b Fo(')27 b Ft(=)32 b Fi(e)-59 b Fo(!)s Ft(\()p Fo(I)8 b Ft(\))p Fo(:)1137 b Ft(\(7\))0 4391 y(One)46 b(has)g Fo(d)p Fn(h)p Fo(k)s(;)17 b(')p Fn(i)p Fo(=dt)48 b Ft(=)h Fn(h)p Fo(k)s(;)21 b Fi(e)-59 b Fo(!)s Ft(\()p Fo(I)8 b Ft(\))p Fn(i)49 b(\021)h Ft(0.)82 b(Consequen)m(tly)-8 b(,)2369 4366 y(_)2346 4391 y Fo(I)53 b Ft(is)45 b(an)h(in)m(tegral)e (of)h(motion,)h(and)g(if)0 4541 y(sin)p Fn(h)p Fo(k)s(;)17 b(')p Ft(\(0\))p Fn(i)32 b Ft(=)h Fo(c)g Fn(6)p Ft(=)g(0)j(then)g Fo(I)8 b Ft(\()p Fo(t)p Ft(\))33 b(=)g Fo(I)8 b Ft(\(0\))24 b(+)g Fo(\017ctk)s Ft(.)54 b(Th)m(us,)39 b(system)d(\(7\))g(has)g(no)g (compact)f(in)m(v)-5 b(arian)m(t)0 4690 y(manifolds)30 b(con)m(taining)i(p)s(oin)m(ts)g(\()p Fo(I)8 b(;)17 b(')p Ft(\))32 b(with)h(sin)o Fn(h)p Fo(k)s(;)17 b(')p Fn(i)27 b(6)p Ft(=)h(0.)44 b(On)33 b(the)g(other)g(hand,)g(function)f Fo(H)3630 4705 y Fm(1)3702 4690 y Ft(\(6\))0 4840 y(can)h(b)s(e)g(made) f(arbitrarily)e(small:)41 b(of)32 b(course,)i(the)f(length)f(of)h(v)m (ector)g Fo(k)s Ft(,)g(generally)f(sp)s(eaking,)h(tends)0 4989 y(to)f(in\014nit)m(y)g(as)h Fo(A\035)j Ft(tends)d(to)g Fo(\035)i Ft(but)e(w)m(e)h(can)f(paralyse)f(this)h(b)m(y)g(a)f (suitable)g(c)m(hoice)h(of)f Fo(\017)p Ft(.)106 5197 y(Although)25 b(this)g(argumen)m(t)g(is)g(v)m(ery)i(simple,)f(the)g (necessit)m(y)h(of)e(the)h(R)s(\177)-51 b(ussmann)25 b(nondegeneracy)i(for)0 5347 y(KAM-stabilit)m(y)f(w)m(as)k(\014rst)e (observ)m(ed,)k(as)c(far)g(as)g(the)h(author)f(kno)m(ws,)j(no)d (earlier)f(than)h(in)g(1995)f([122].)106 5496 y(Th)m(us,)40 b(for)d(analytic)e(functions)i Fo(H)1421 5511 y Fm(0)1460 5496 y Ft(,)i(one)e(has)g(the)h(follo)m(wing)c(alternativ)m(e.)56 b(If)37 b(the)g(image)f(of)g(the)0 5646 y(gradien)m(t)46 b(map)g Fo(!)56 b Ft(=)c Fo(@)5 b(H)1009 5661 y Fm(0)1049 5646 y Fo(=@)g(I)55 b Ft(do)s(es)48 b(not)e(lie)g(in)g(an)m(y)h(h)m(yp) s(erplane)h(passing)f(through)g(the)g(origin)1889 5895 y(6)p eop %%Page: 7 7 7 6 bop 0 171 a Ft(of)37 b(the)h(frequency)i(space,)f(then)g(an)m(y)f (Hamiltonian)c(system)k(su\016cien)m(tly)g(close)g(to)f(the)h (completely)0 320 y(in)m(tegrable)28 b(system)h(with)g(Hamilton)d (function)i Fo(H)1873 335 y Fm(0)1941 320 y Ft(admits)g(man)m(y)g(in)m (v)-5 b(arian)m(t)28 b Fo(n)p 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b(that)e(this)f(system)j(has)e(a)g(family)0 5646 y Fn(f)p Fj(T)113 5661 y Fl(\030)186 5646 y Fn(\032)40 b Fj(M)p Fn(g)451 5661 y Fl(\030)528 5646 y Ft(of)g(in)m(v)-5 b(arian)m(t)38 b Fo(n)p Ft(-tori)g(lab)s(elled)g(b)m(y)i(index)g Fo(\030)45 b Fn(2)40 b Ft(\004)g Fn(\032)g Fk(R)2607 5610 y Fl(q)2691 5646 y Ft(and)f(carrying)h(conditionally)1865 5895 y(12)p eop %%Page: 13 13 13 12 bop 0 171 a Ft(p)s(erio)s(dic)30 b(motions)g(with)h(frequency)i (v)m(ectors)g Fo($)1813 186 y Fl(\030)1879 171 y Fn(2)28 b Fk(R)2039 134 y Fl(n)2092 171 y Ft(.)43 b(It)31 b(is)g(assumed)h (that)g(\004)f(is)g(a)g(set)h(of)f(p)s(ositiv)m(e)0 320 y(measure.)48 b(The)35 b(structure)g(of)f(\004)g(can)g(b)s(e)g(\\bad")g (\(e.g.,)g(\004)g(ma)m(y)g(b)s(e)g(Can)m(tor-lik)m(e\).)47 b(Eac)m(h)35 b(torus)f Fj(T)3694 335 y Fl(\030)3761 320 y Ft(is)0 469 y(the)f(image)e(of)h(an)g(em)m(b)s(edding)h Fk(T)1253 433 y Fl(n)1331 469 y Fn(!)27 b Fj(M)p Ft(.)106 619 y(Let)43 b Fj(C)f Ft(b)s(e)h(an)m(y)g(smo)s(othness)g(class)f(\()p 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