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Fj(\003)186 648 y Fp(j)252 624 y Fw(is)27 b(a)f(dimensionless)g (constan)m(t)i(satisfying)1880 584 y Fs(\026)-38 b Fp(\024)p 1850 600 95 4 v 1850 610 a Fj(p)p 1909 610 36 3 v 55 x Fs(2)1982 624 y Fr(<)27 b(\024)2141 588 y Fj(\003)2141 648 y Fp(j)2209 624 y Fr(<)g Fw(1\))g(and,)g(follo)m(wing)d(notations)i (common)92 744 y(in)32 b(celestial)f(mec)m(hanics,)i(de\014ne)1415 962 y Fr(\025)1472 920 y Fj(\003)1472 986 y Fp(j)1594 962 y Fw(=)83 b Fr(`)1794 977 y Fp(j)1853 962 y Fw(+)22 b Fr(g)1998 977 y Fp(j)2051 962 y Fr(;)1394 1140 y(H)1475 1155 y Fp(j)1594 1140 y Fw(=)83 b(\003)1821 1099 y Fj(\003)1821 1164 y Fp(j)1877 1140 y Fw(\(1)22 b Fq(\000)2085 1039 y Fi(q)p 2168 1039 256 4 v 101 x Fw(1)g Fq(\000)h Fr(e)2384 1105 y Fs(2)2384 1163 y Fp(j)2424 1140 y Fw(\))17 b Fr(;)1424 1325 y(\030)1472 1284 y Fj(\003)1467 1350 y Fp(j)1594 1325 y Fw(=)1753 1225 y Fi(q)p 1836 1225 167 4 v 100 x Fw(2)p Fr(H)1966 1340 y Fp(j)2019 1325 y Fw(cos)g Fr(g)2213 1340 y Fp(j)2266 1325 y Fr(;)1420 1511 y(\021)1472 1470 y Fj(\003)1468 1536 y Fp(j)1594 1511 y Fw(=)83 b Fq(\000)1830 1411 y Fi(q)p 1914 1411 V 1914 1511 a Fw(2)p Fr(H)2044 1526 y Fp(j)2096 1511 y Fw(sin)17 b Fr(g)2280 1526 y Fp(j)2333 1511 y Fr(:)92 1800 y Fw(Since)40 b(w)m(e)h(are)f(in)m (terested)h(in)f(small)e(eccen)m(tricities,)i(collisions)d(are)k(a)m(v) m(oided)f(b)m(y)h(requiring)e(that)92 1920 y(the)30 b(ma)5 b(jor)29 b(semi-axes)g Fr(a)1024 1935 y Fp(j)1088 1920 y Fw(=)f Fr(a)1243 1935 y Fp(j)1280 1920 y Fw(\(\003)1386 1884 y Fj(\003)1425 1920 y Fw(\))f(:=)h(\(\003)1727 1884 y Fj(\003)1727 1945 y Fp(j)1766 1920 y Fr(=\024)1871 1884 y Fj(\003)1871 1945 y Fp(j)1910 1920 y Fw(\))1948 1884 y Fs(2)2018 1920 y Fw(are)h(di\013eren)m(t)h(\(and)g(di\013eren)m (t)g(from)e(zero\).)i(W)-8 b(e,)92 2040 y(therefore,)33 b Ft(\014x,)g(once)g(and)g(for)f(all,)1070 2258 y Fw(0)27 b Fr(<)h(a)1301 2273 y Fs(min)1451 2258 y Fr(<)f(a)1605 2273 y Fs(max)1936 2258 y Fw(and)195 b(0)28 b Fr(<)f(\013)2530 2273 y Fs(max)2693 2258 y Fr(<)h 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Fw(at)f(distance)h(less)g(than)f Fr(r)k Fw(from)31 b Fr(A)p Fw(,)i(i.e.,)f(the)h(set)1463 3427 y Fr(A)1536 3442 y Fp(r)1601 3427 y Fw(:=)1754 3344 y Fi([)1732 3527 y Fp(I)5 b Fj(2)p Fp(A)1884 3427 y Fr(D)1968 3386 y Fp(n)1965 3451 y(r)2015 3427 y Fw(\()p Fr(I)j Fw(\))27 b Fq(\032)i Fh(C)2341 3386 y Fp(n)2426 3427 y Fr(;)1170 b Fw(\(1.3\))92 3720 y(where)34 b Fr(D)458 3684 y Fp(n)455 3745 y(r)505 3720 y Fw(\()p Fr(I)8 b Fw(\))33 b(denotes)i(the)f(\(op)s(en\))f(complex)g Fr(n)p Fw(-ball)e(of)i(radius)g Fr(r)j Fw(cen)m(tered)f(at)e Fr(I)41 b Fw(while)33 b(the)h Ft(real)92 3840 y Fr(n)p Fw(-ball)d(of)i(radius)h Fr(r)i Fw(cen)m(tered)f(at)f Fr(I)41 b Fw(will)31 b(b)s(e)j(denoted)h Fr(B)2221 3804 y Fp(n)2216 3865 y(r)2268 3840 y Fw(\()p Fr(I)8 b Fw(\);)33 b(complex)g(or)g(real)g Fr(n)p Fw(-balls)f(cen)m(tered)92 3961 y(at)i(the)i(origin)d(will)g(b)s(e)i(simply)e(denoted)j(b)m(y)g Fr(D)1892 3925 y Fp(n)1889 3985 y(r)1974 3961 y Fw(or,)f(resp)s(ectiv)m (ely)-8 b(,)36 b Fr(B)2755 3925 y Fp(n)2750 3985 y(r)2802 3961 y Fw(.)f(The)h(follo)m(wing)c(classical)92 4081 y(result)g(holds.)92 4307 y Fu(Theorem)37 b(1.1)g(\(Delauna)m(y)-9 b(,)38 b(P)m(oincar)n(\023)-54 b(e\))47 b Fg(Ther)-5 b(e)25 b(exist)f(p)-5 b(ositive)24 b(c)-5 b(onstants)31 b Fw(\026)-55 b Fr(")p Fg(,)3176 4280 y Fw(\026)3171 4307 y Fr(\016)t Fg(,)25 b Fr({)3304 4322 y Fs(max)3465 4307 y Fg(and)f Fr(e)3689 4322 y Fs(max)92 4427 y Fg(such)47 b(that)g(the)h(ab)-5 b(ove)46 b(intr)-5 b(o)g(duc)g(e)g(d)47 b(variables)f Fw(\(\003)1988 4391 y Fj(\003)2027 4427 y Fr(;)17 b(\025)2128 4391 y Fj(\003)2167 4427 y Fw(\))47 b Fg(ar)-5 b(e,)47 b(for)g Fw(0)j Fr(<)h(")f(<)57 b Fw(\026)-56 b Fr(")p Fg(,)47 b(standar)-5 b(d,)46 b(r)-5 b(e)g(al-)92 4547 y(analytic)37 b(symple)-5 b(ctic)38 b(variables)e(describing)h(al) 5 b(l)38 b(motions)f(of)g(the)h(sp)-5 b(atial)38 b(thr)-5 b(e)g(e-b)g(o)g(dy)37 b(pr)-5 b(oblem)37 b(in)92 4668 y(a)31 b Fr(O)s Fw(\()294 4642 y(\026)289 4668 y Fr(\016)s Fw(\))p Fg(-neighb)-5 b(orho)g(o)g(d)29 b(of)i(any)g(p)-5 b(oint)31 b Fw(\003)1588 4632 y Fj(\003)1588 4693 y Fs(0)1658 4668 y Fg(in)g(the)g(c)-5 b(omp)g(act)30 b(set)h Fw(\(1.2\))p Fg(,)g(for)g Fr(e)2911 4683 y Fp(j)2975 4668 y Fq(\024)d Fr(e)3125 4683 y Fs(max)3292 4668 y Fg(and)j(for)g(non-)92 4788 y(vanishing)43 b(mutual)i(inclinations)38 b Fw(^)-43 b Fr({)45 b Fg(not)g(bigger)f(than)g Fr({)2227 4803 y Fs(max)2363 4788 y Fg(;)h(the)g(Hamiltonian)f(governing)f(such)92 4909 y(motions)31 b(\(with)i(r)-5 b(esp)g(e)g(ct)32 b(to)g(the)h (standar)-5 b(d)32 b(symple)-5 b(ctic)31 b(form)2392 4842 y Fi(P)2480 4929 y Fp(i)2525 4909 y Fr(d)p Fw(\003)2644 4872 y Fj(\003)2644 4933 y Fp(i)2699 4909 y Fq(^)18 b Fr(d\025)2891 4872 y Fj(\003)2891 4933 y Fp(i)2946 4909 y Fw(+)3039 4842 y Fi(P)3143 4909 y Fr(d\021)3246 4872 y Fj(\003)3242 4933 y Fp(i)3302 4909 y Fq(^)f Fr(d\030)3484 4872 y Fj(\003)3479 4933 y Fp(i)3522 4909 y Fg(\))33 b(is)f(the)92 5029 y(r)-5 b(e)g(al-analytic)34 b(function)1286 5216 y Fq(\000)1373 5148 y Fw(1)p 1373 5192 49 4 v 1373 5284 a(2)1493 5108 y Fs(2)1450 5133 y Fi(X)1449 5315 y Fp(j)t Fs(=1)1625 5148 y Fr(\024)1681 5163 y Fp(j)p 1598 5192 147 4 v 1598 5287 a Fw(\003)1666 5253 y Fj(\003)1666 5310 y Fp(j)1705 5245 y Fs(2)1777 5216 y Fw(+)22 b Fr("F)14 b Fw(\(\003)2104 5175 y Fj(\003)2142 5216 y Fr(;)j(\025)2243 5175 y Fj(\003)2282 5216 y Fr(;)g(\021)2378 5175 y Fj(\003)2417 5216 y Fr(;)g(\030)2509 5175 y Fj(\003)2548 5216 y Fw(\))g Fr(;)993 b Fw(\(1.4\))1933 5775 y(5)p eop %%Page: 6 6 6 5 bop 92 222 a Fg(wher)-5 b(e)46 b Fr(\024)435 237 y Fp(j)522 222 y Fw(:=)675 126 y Fi(\020)735 173 y Fp(m)797 183 y Ff(j)p 735 199 95 4 v 766 257 a Fp(")840 126 y Fi(\021)890 149 y Fs(3)1121 183 y(1)p 939 199 399 4 v 939 257 a Fp(m)1001 234 y Fe(2)1001 280 y(0)1036 257 y Fs(\()p Fp(m)1125 266 y Fe(0)1160 257 y Fs(+)p Fp(m)1277 267 y Ff(j)1310 257 y Fs(\))1395 222 y Fg(ar)-5 b(e)47 b(p)-5 b(ositive)46 b(c)-5 b(onstant)47 b(of)g(or)-5 b(der)46 b(1)h(\()2882 183 y Fs(\026)-38 b Fp(\024)2920 160 y Fe(3)p 2879 199 76 4 v 2899 257 a Fs(2)3015 222 y Fr(<)50 b(\024)3197 237 y Fp(j)3284 222 y Fr(<)g Fw(1)p Fg(\))d(and)f Fr(F)92 367 y Fg(is)c(a)h(suitable)g(function)f(r)-5 b(e)g(al-analytic)42 b(in)h(a)f(neighb)-5 b(orho)g(o)g(d)41 b(of)i Fr(B)2621 331 y Fs(2)2619 382 y(\026)2616 400 y Fp(\016)2660 367 y Fw(\(\003)2766 331 y Fj(\003)2766 392 y Fs(0)2806 367 y Fw(\))28 b Fq(\002)g Fh(T)3040 331 y Fs(2)3111 367 y Fq(\002)g(f)p Fw(\(0)p Fr(;)17 b Fw(0\))p Fq(g)p Fg(.)42 b(Mor)-5 b(e)92 487 y(pr)g(e)g(cisely,)30 b(\014x)h Fw(\003)709 451 y Fj(\003)709 512 y Fs(0)779 487 y Fg(in)g(the)g(c)-5 b(omp)g(act)31 b(set)g Fw(\(1.2\))g Fg(and)f(\014x)h Fw(0)c Fr(<)h({)2333 502 y Fs(min)2482 487 y Fr(<)g({)2617 502 y Fs(max)2753 487 y Fg(,)j(then,)g(for)g(any)g Fw(0)c Fr(<)h(")f(<)34 b Fw(\026)-56 b Fr(")p Fg(,)92 608 y Fr({)123 623 y Fs(min)272 608 y Fq(\024)28 b(j)-6 b Fw(^)-43 b Fr({)p Fq(j)28 b(\024)g Fr({)628 623 y Fs(max)763 608 y Fg(,)35 b(the)g(function)g Fr(F)48 b Fg(c)-5 b(an)35 b(b)-5 b(e)34 b(taken)h(to)g(b)-5 b(e)34 b(r)-5 b(e)g(al-analytic)34 b(on)h(the)g(c)-5 b(omplex)33 b(set)1434 828 y Fq(I)1488 843 y Fp(\033)1528 852 y Fe(0)1590 828 y Fq(\002)22 b Fh(T)1752 787 y Fs(2)1752 853 y(2)p Fp(s)1820 862 y Fe(0)1885 828 y Fq(\002)g Fr(D)2068 787 y Fs(4)2065 853 y(2)p Fp(\032)2136 862 y Fe(0)2203 828 y Fq(\032)28 b Fh(C)2374 787 y Fs(8)2454 828 y Fr(;)92 1048 y Fg(wher)-5 b(e)34 b Fr(\033)422 1063 y Fs(0)462 1048 y Fr(;)17 b(s)552 1063 y Fs(0)591 1048 y Fr(;)g(\032)685 1063 y Fs(0)759 1048 y Fg(ar)-5 b(e)35 b(suitable)g(p)-5 b(ositive)34 b(numb)-5 b(ers)34 b(and)g(wher)-5 b(e)1043 1268 y Fq(I)35 b Fw(:=)28 b([\003)1358 1227 y Fj(\003)1358 1293 y Fs(01)1454 1268 y Fq(\000)23 b Fr(\016)n(;)17 b Fw(\003)1707 1227 y Fj(\003)1707 1293 y Fs(01)1804 1268 y Fw(+)22 b Fr(\016)t Fw(])g Fq(\002)h Fw([\003)2193 1227 y Fj(\003)2193 1293 y Fs(02)2289 1268 y Fq(\000)g Fr(\016)n(;)17 b Fw(\003)2542 1227 y Fj(\003)2542 1293 y Fs(02)2639 1268 y Fw(+)22 b Fr(\016)t Fw(])35 b Fr(;)92 1488 y Fg(for)f(some)g Fr(\016)e Fq(\025)681 1462 y Fw(\026)676 1488 y Fr(\016)t Fg(.)92 1679 y(F)-7 b(urthermor)i(e,)33 b(ther)-5 b(e)33 b(exists)h(a)f(r)-5 b(e)g(al-analytic,)33 b(symple)-5 b(ctic)34 b(change)e(of)i(variables)e Fw(\011)3240 1694 y Fs(W)3346 1679 y Fw(:)h(\()p Fr(I)8 b(;)17 b(';)g(p;)g(q)t Fw(\))92 1800 y Fq(7!)34 b Fw(\(\003)332 1763 y Fj(\003)371 1800 y Fr(;)17 b(\025)472 1763 y Fj(\003)511 1800 y Fr(;)g(\021)607 1763 y Fj(\003)646 1800 y Fr(;)g(\030)738 1763 y Fj(\003)777 1800 y Fw(\))35 b Fg(of)f(the)h(form)760 2067 y Fw(\003)828 2026 y Fj(\003)895 2067 y Fw(=)28 b Fr(I)42 b(;)216 b(\025)1384 2026 y Fj(\003)1451 2067 y Fw(=)28 b Fr(')22 b Fw(+)1746 2041 y(^)1739 2067 y Fr(`)p Fw(\()p Fr(I)8 b(;)17 b(p;)g(q)t Fw(\))34 b Fr(;)2368 1971 y Fi(\020)2427 1999 y Fr(\021)2479 1963 y Fj(\003)2429 2135 y Fr(\030)2477 2106 y Fj(\003)2528 1971 y Fi(\021)2605 2067 y Fw(=)28 b Fr(A)p Fw(\()p Fr(I)8 b Fw(\))2926 1971 y Fi(\020)2985 1999 y Fr(p)2986 2135 y(q)3044 1971 y Fi(\021)3128 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Fi(\020)1277 4933 y Fw(\026)1266 4958 y(\012)1336 4973 y Fs(2)1398 4958 y Fq(\000)1508 4933 y Fw(\026)1497 4958 y(\012)1567 4973 y Fs(1)1607 4862 y Fi(\021)1684 4958 y Fr(>)67 b Fw(inf)1788 5017 y Fp(I)5 b Fj(2I)1910 5025 y Ff(\033)1946 5040 y Fe(0)2001 4958 y Fq(j)2040 4933 y Fw(\026)2029 4958 y(\012)2099 4973 y Fs(2)2160 4958 y Fq(\000)2271 4933 y Fw(\026)2260 4958 y(\012)2330 4973 y Fs(1)2370 4958 y Fq(j)27 b Fr(>)44 b Fw(const)h Fr(>)28 b Fw(0)17 b Fr(:)614 b Fw(\(1.8\))1933 5775 y(6)p eop %%Page: 7 7 7 6 bop 92 206 a Fu(Remark)37 b(1.1)49 b Fw(\(i\))25 b(Ev)m(en)i(though)f(the)h(results)f(listed)f(in)h(this)f(theorem)h (are)g(classical,)f(the)h(analyti-)92 326 y(cal)h(form)m(ulation)d (presen)m(ted)30 b(here)e(\(together)g(with)f(a)h(detailed)e(pro)s(of)h (of)g(it\))g(seems)h(not)g(easy)g(to)g(b)s(e)92 447 y(found)k(in)f(the) h(literature.)e(An)i(e\013ort)g(to)g(remedy)g(to)f(this)h(breac)m(h)h (has)f(b)s(een)g(done)h(at)e(the)h(Bureau)92 567 y(des)37 b(longitudes)f(in)g(P)m(aris)h(in)f(the)h(late)f(80's)h(and)g(w)m(e)g (refer)g(the)h(in)m(terested)f(reader)h(to)e(the)h Ft(Notes)92 687 y(scien)m(ti\014ques)31 b(et)g(tec)m(hniques)h(du)f(B.D.L.)f Fw(b)m(y)h(A.)g(Chenciner)g(and)f(J.)h(Lask)-5 b(ar)30 b(\([Ch88],)h([L88]\).)f(F)-8 b(or)92 808 y(completeness,)33 b(w)m(e)h(presen)m(t)g(the)f(pro)s(of)f(of)g(Theorem)h(1.1)f(in)2373 772 y Fs(3)2444 808 y Fw(App)s(endix)h(C.)92 928 y(\(ii\))h(W)-8 b(e)36 b(sk)m(etc)m(h)i(here,)f(v)m(ery)h(brie\015y)-8 b(,)36 b(the)h(ideas)e(b)s(ey)m(ond)j(Theorem)e(1.1,)g(referring)f(for) g(details)g(to)92 1048 y(App)s(endix)29 b(C.)g(First)f(of)g(all,)f(b)m (y)i(elemen)m(tary)g(mec)m(hanics,)g(one)g(can)g(cast)g(the)g(three-b)s (o)s(dy)g(problem)92 1169 y(in)m(to)37 b(a)g(nine-degree-of-freedom)g (Hamiltonian)d(formalism.)g(Then,)39 b(reduction)f(of)f(the)h(cen)m (ter)h(of)92 1289 y(mass)27 b(mak)m(es)h(it)e(p)s(ossible)g(to)h(lo)m (w)m(er)g(the)h(n)m(um)m(b)s(er)f(of)g(degrees)i(of)d(freedom)h(to)g 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b(the)j(reduction)f(of)g(the)h(an-)92 2132 y(gular)43 b(momen)m(tum)f(\(also)h(kno)m(wn)j(in)d(the)i (literature)d(as)j(Jacobi's)f(\\reduction)f(of)h(the)h(no)s(des"\))92 2252 y(in)m(tro)s(duces)g(t)m(w)m(o)g(extra)g(in)m(tegrals)f(of)g (motion)f(-)h(called)g Ft(P)m(oincar)m(\023)-46 b(e)45 b(in)m(tegrals)e Fw(in)h(App)s(endix)h(C)g(-)92 2373 y(whic)m(h)34 b(allo)m(w)e(to)h(lo)m(w)m(er)h(the)g(degrees)h(of)f (freedom)f(bringing)f(the)i(system)h(in)m(to)e(its)g(\014nal)g(form)f (of)h(a)92 2493 y(four-degree-of-freedom)e(Hamiltonian)e(system.)92 2613 y(The)49 b(non-planarit)m(y)f(condition)41 b(^)-43 b Fr({)55 b Fq(6)p Fw(=)g(0)49 b(is)f(only)g(needed)j(in)d(order)h(to)f Ft(de\014ne)i Fw(the)f(osculating)92 2734 y(P)m(oincar)m(\023)-46 b(e)30 b(v)-5 b(ariables,)28 b(while)h(the)i(smallness)e(assumption)g (on)h(the)g(inclination)21 b(^)-43 b Fr({)30 b Fw(\(i.e.)f Fq(j)-6 b Fw(^)-43 b Fr({)p Fq(j)27 b Fr(<)h({)3555 2749 y Fs(max)3691 2734 y Fw(\))h(is)92 2854 y(related)21 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b(w)m(e)k(stress)g(that)f(the)g(function)f Fr(F)43 b Ft(do)s(es)29 b(dep)s(end)h(up)s(on)f(the)g(c)m(hosen)i Fr(O)s Fw(\(1\))p Fq(\000)p Ft(neigh)m(b)s(orho)s(o)s(d)92 3576 y(of)i Fw(\003)272 3540 y Fj(\003)272 3601 y Fs(0)345 3576 y Ft(but)h(ob)s(eys)h(uniform)d(b)s(ounds)i(in)f(the)h(whole)g (domain)e Fw(\(1.2\).)h(Suc)m(h)i(dep)s(endence)h(is)e(hidden)92 3697 y(in)46 b(the)i(dep)s(endence)i(of)d(the)g(p)s(erturbation)g (function)g(up)s(on)g(the)h(ab)s(o)m(v)m(e)g(men)m(tioned)f(P)m(oincar) m(\023)-46 b(e)92 3817 y(in)m(tegrals,)45 b(whic)m(h,)h(for)f(small)f (eccen)m(tricities,)i(are)g(related)f(to)h(the)g(m)m(utual)f (inclination)d(of)k(the)92 3938 y(osculating)31 b(ellipses.)92 4058 y(\(iii\))25 b(The)j(c)m(hange)h(of)e(v)-5 b(ariables)27 b(\(\003)1402 4022 y Fj(\003)1441 4058 y Fr(;)17 b(\025)1542 4022 y Fj(\003)1581 4058 y Fr(;)g(\021)1677 4022 y Fj(\003)1716 4058 y Fr(;)g(\030)1808 4022 y Fj(\003)1846 4058 y Fw(\))28 b Fq(7!)f Fw(\()p Fr(I)8 b(;)17 b(';)g(p;)g(q)t Fw(\))27 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y(App)r(endix)i(C)f(b)n(y)g(means)g(of)g(a)f(\\)p Fn(\003)p FC(":)g(the)i(\\P)n(oincar)n(\023)-39 b(e)27 b(v)-5 b(ariables")30 b(\(\003)p Fm(;)14 b(\025;)g(\021)s(;)g(\030)t FC(\))29 b Fn(2)h Fh(R)2760 4631 y Fo(2)2824 4662 y Fn(\002)20 b Fh(T)2972 4631 y Fo(2)3034 4662 y Fn(\002)g Fh(R)3185 4631 y Fo(4)3259 4662 y FC(are)31 b(particularly)92 4761 y(suited)e(to)g(treat)g(the)g Fx(planar)f FC(three-b)r(o)r(dy)h (problem,)g(while)g(the)h(\\osculating)d(P)n(oincar)n(\023)-39 b(e)26 b(v)-5 b(ariables")28 b(\(de\014ned)h(only)92 4861 y(for)g(non-v)-5 b(anishing)29 b(m)n(utual)h(inclinations\))f (\(\003)1605 4831 y Fy(\003)1643 4861 y Fm(;)14 b(\025)1728 4831 y Fy(\003)1767 4861 y Fm(;)g(\021)1848 4831 y Fy(\003)1886 4861 y Fm(;)g(\030)1963 4831 y Fy(\003)2001 4861 y FC(\))27 b Fn(2)g Fh(R)2208 4831 y Fo(2)2272 4861 y Fn(\002)19 b Fh(T)2419 4831 y Fo(2)2480 4861 y Fn(\002)g Fh(R)2630 4831 y Fo(4)2703 4861 y FC(are)29 b(more)g(con)n(v)n(enien)n(t)g(in)h (dealing)92 4960 y(with)25 b(the)h Fx(spatial)e FC(three-b)r(o)r(dy)h (problem;)g(the)g(w)n(ord)f(\\osculating")f(refers)h(to)h(the)h(fact)f (that)g(these)h(v)-5 b(ariables)23 b(\\liv)n(e")92 5060 y(on)k(the)h(t)n(w)n(o)f(planes)g(asso)r(ciated)f(to)i(the)g(t)n(w)n(o) f(osculating)f(ellipses.)204 5130 y Fo(5)241 5160 y FC(\\Symplectic)f (diagonalization")e(of)i(a)g(quadratic)g(Hamiltonian)g Fn(Q)p FC(\()p Fm(z)t FC(\),)g Fm(z)h Fn(2)e Fh(R)2754 5130 y Fo(2)q Fl(n)2839 5160 y FC(,)h(means)g(diagonalization,)e(b)n(y) 92 5259 y(a)31 b(linear)f(symplectic)h(map,)h(of)f(the)g(constan)n(t)g (matrix)g Fm(S)1931 5271 y Fo(2)p Fl(n)2009 5259 y Fn(Q)2077 5229 y Fy(00)2119 5259 y FC(,)g Fm(S)2224 5271 y Fo(2)p Fl(n)2334 5259 y FC(b)r(eing)g(the)h(standard)e(\(2)p Fm(n)21 b Fn(\002)f FC(2)p Fm(n)p FC(\))31 b(symplectic)1933 5775 y Fw(7)p eop %%Page: 8 8 8 7 bop 92 206 a Fw(\\secular)39 b(term")g(of)g(the)h(Hamiltonian)c (\(1.4\),)j(i.e.,)g(of)g(the)h(a)m(v)m(erage)g(o)m(v)m(er)h(the)f (angles)f Fr(\025)3478 170 y Fj(\003)3557 206 y Fw(of)g Fr("F)14 b Fw(,)92 326 y(namely)-8 b(,)41 b Fr("f)553 341 y Fs(1)592 326 y Fw(.)h(Under)g(the)h(assumption)e(that)35 b(^)-42 b Fr({)41 b Fw(is)h(small)d(enough,)k(suc)m(h)g(quadratic)f (part)f(turns)92 447 y(out)d(to)g(b)s(e)h Ft(p)s(ositiv)m(e)f (de\014nite)h Fw(and,)g(hence,)h Fq(\006)1859 369 y(p)p 1943 369 127 4 v 1943 447 a(\000)p Fw(1\(2)p Fr(")2213 421 y Fw(\026)2202 447 y(\012)2272 462 y Fp(j)2308 447 y Fw(\))f(are,)g(simply)-8 b(,)37 b(the)i(purely)g(imaginary)92 567 y(eigen)m(v)-5 b(alues)41 b(of)g(the)g(\(4)28 b Fq(\002)h Fw(4\)-matrix)39 b Fr(S)1627 582 y Fs(4)1666 567 y Fq(Q)1747 531 y Fj(00)1790 567 y Fw(,)i(where)h Fr(S)2208 582 y Fs(4)2289 567 y Fw(denotes,)h(here,)f(the)f(standard)h(\(4)28 b Fq(\002)g Fw(4\))92 687 y(symplectic)33 b(unit)m(y)g(matrix)f(and)h Fq(Q)c Fw(=)g Fq(Q)p Fw(\()p Fr(\021)1722 651 y Fj(\003)1761 687 y Fr(;)17 b(\030)1853 651 y Fj(\003)1892 687 y Fw(;)g(\003)2004 651 y Fj(\003)2043 687 y Fw(\))33 b(denotes)h(the)g(\(p)s(ositiv)m (e-de\014nite\))e(quadratic)92 808 y(part)39 b(of)h Fr(")475 737 y Fi(R)502 833 y Fd(T)554 814 y Fe(2)587 808 y Fr(F)14 b(d\025)772 772 y Fj(\003)851 808 y Fw(\(though)m(t)40 b(of)f(as)h(a)g(function)f(of)g(\()p Fr(\021)2193 772 y Fj(\003)2232 808 y Fr(;)17 b(\030)2324 772 y Fj(\003)2363 808 y Fw(\))40 b(and)g(parameterized)f(b)m(y)i(\003)3490 772 y Fj(\003)3529 808 y Fw(\).)f(This)92 928 y(diagonalization)g(pro)s (cedure,)46 b(already)e(kno)m(wn)h(to)g(W)-8 b(eierstrass,)45 b(requires,)g(in)f(the)h(case)g(of)g(the)92 1048 y(three-b)s(o)s(dy)33 b(problem,)g(certain)f(calculations,)g(in)m(v)m(olving)g(Laplace)h(co)s (e\016cien)m(ts)h(\(whic)m(h,)g(in)f(turn,)92 1169 y(are)25 b(simply)e(related)i(to)f(the)i(Gauss)f(h)m(yp)s(er-geometric)f (function\):)h(suc)m(h)h(calculations)d(are)i(sk)m(etc)m(hed)92 1289 y(in)32 b(App)s(endix)h(C)g(\(see,)g(also,)f([LR95])g(and)h ([R95]\).)92 1410 y(\(iv\))e(The)i(Hamiltonian)28 b Fq(H)k Fw(describ)s(es)h(a)f(nearly-in)m(tegrable,)e(prop)s(erly)h(degenerate) i(systems:)h(the)92 1530 y(in)m(tegrable)g(limit)e(\()p Fr(")f Fw(=)h(0\))j(dep)s(ends)i(only)e(on)g(the)g(t)m(w)m(o)h(action)e (v)-5 b(ariables)34 b Fr(I)2965 1545 y Fs(1)3005 1530 y Fw(,)h Fr(I)3110 1545 y Fs(2)3149 1530 y Fw(.)h(The)g(frequency)92 1663 y(v)m(ector)45 b(asso)s(ciated)f(to)f(the)i(in)m(tegrable)e(limit) d(is)1999 1567 y Fi(\020)2049 1663 y Fr(\024)2105 1678 y Fs(1)2145 1663 y Fr(=I)2245 1627 y Fs(3)2237 1687 y(1)2284 1663 y Fr(;)17 b(\024)2384 1678 y Fs(2)2423 1663 y Fr(=I)2523 1627 y Fs(3)2515 1687 y(2)2562 1567 y Fi(\021)2612 1663 y Fw(,)44 b(whic)m(h)g(is)g(a)f(v)m(ector)i(of)f(order)92 1783 y(one.)39 b(This)g(means)g(that)g(the)h(conjugated)f(angles)f Fr(')h Fw(ma)m(y)g(b)s(e)g(regarded)h(as)f Ft(fast)g(angles)g Fw(and,)g(in)92 1904 y(\\\014rst)27 b(appro)m(ximation",)e(the)j Fq(H)q Fw(-motions)d(are)i(go)m(v)m(erned)i(b)m(y)f(the)g(a)m(v)m (eraged)h(Hamiltonian)23 b Fr(h)11 b Fw(+)g Fr("f)3757 1919 y Fs(1)3797 1904 y Fw(:)92 2024 y(suc)m(h)44 b(Hamiltonian,)39 b(whic)m(h)k(in)f(case)h(of)g(the)g(spatial)e(three-b)s(o)s(dy)i (problem)e(is)h Ft(non)h(in)m(tegrable)p Fw(,)92 2144 y(is)36 b(sometimes)f(referred)i(to)f(as)h(\\secular)f(Hamiltonian)2191 2108 y Fs(6)2227 2144 y Fw(".)g(By)h(the)g(discussion)f(in)g(the)h (preceding)92 2265 y(item)k(\(iii\))f(and)j(from)f(\(1.7\),)h(it)f (follo)m(ws)f(that)i Fr(p)i Fw(=)h(0)f(=)g Fr(q)i Fw(is)42 b(an)h Ft(elliptic)d(equilibrium)g Fw(for)j(the)92 2385 y(Hamiltonian)29 b Fr(f)701 2400 y Fs(1)773 2385 y Fw(and)j(hence,)i (for)e(an)m(y)i Fr(I)8 b Fw(,)1448 2579 y Fq(f)p Fr(')27 b Fq(2)i Fh(T)1747 2538 y Fs(2)1789 2579 y Fq(g)22 b(\002)h(f)p Fr(p)k Fw(=)h(0)f(=)h Fr(q)t Fq(g)1155 b Fw(\(1.9\))92 2773 y(is)46 b(an)g Ft(elliptic)d(t)m(w)m(o-torus)k Fw(for)f(the)h(a)m (v)m(eraged)g(Hamiltonian)c Fr(h)31 b Fw(+)h Fr("f)2760 2788 y Fs(1)2845 2773 y Fw(run)47 b(b)m(y)g(the)g(linear)d(\015o)m(w)92 2894 y Fr(')28 b Fq(!)g Fr(')23 b Fw(+)f Fq(r)p Fw(\()p Fr(h)h Fw(+)f Fr("f)889 2909 y Fs(1)p Fp(;)p Fs(0)983 2894 y Fw(\))p Fr(t)p Fw(.)33 b Ft(These)i(are)e(the)h(secular)f (motions)f(that)g(w)m(e)j(shall)c(pro)m(v)m(e)k(to)d(p)s(ersist)h(for) 92 3014 y Fw(0)27 b Fr(<)h(")f Fq(\034)g Fw(1)33 b Ft(and)f(for)g(sp)s (ecial,)g(but)h(nearly-full)d(measure,)j(v)-5 b(alues)32 b(of)h Fr(I)8 b Fw(.)92 3284 y(W)-8 b(e)36 b(can,)h(no)m(w,)g(giv)m(e)g (a)f(precise)h(form)m(ulation)c(of)j(our)g(main)f(result.)i(Let)43 b(\026)-56 b Fr(")p Fw(,)2985 3257 y(\026)2980 3284 y Fr(\016)t Fw(,)37 b Fr({)3122 3299 y Fs(max)3257 3284 y Fw(,)g Fr(e)3366 3299 y Fs(max)3501 3284 y Fw(,)g(\003)3633 3248 y Fj(\003)3633 3308 y Fs(0)3672 3284 y Fw(,)f Fq(I)7 b Fw(,)92 3404 y Fr({)123 3419 y Fs(min)245 3404 y Fw(,)43 b Fr(\033)370 3419 y Fs(0)409 3404 y Fw(,)g Fr(s)525 3419 y Fs(0)565 3404 y Fw(,)g Fr(\032)685 3419 y Fs(0)767 3404 y Fw(b)s(e)g(as)g(in)f(Theorem)h(1.1)g(ab)s(o)m(v)m(e)h(and)e(let) 75 b(meas)2636 3419 y Fp(n)2726 3404 y Fw(denote)43 b(the)h Fr(n)p Fw(-dimensional)92 3524 y(Leb)s(esgue)34 b(measure.)92 3723 y Fu(Theorem)j(1.2)49 b Fg(Fix)34 b Fr(\034)39 b(>)28 b Fw(1)34 b Fg(and)h(pick)f(two)h(numb)-5 b(ers)34 b Fr(b)2204 3738 y Fp(i)2267 3723 y Fg(such)h(that)1055 3956 y Fw(0)27 b Fr(<)h(b)1276 3971 y Fs(1)1344 3956 y Fr(<)1457 3889 y Fw(1)p 1457 3933 49 4 v 1457 4025 a(2)1551 3956 y Fr(;)216 b Fw(0)27 b Fr(<)h(b)2015 3971 y Fs(2)2082 3956 y Fr(<)2186 3860 y Fi(\020)2245 3889 y Fw(1)p 2245 3933 V 2245 4025 a(2)2326 3956 y Fq(\000)23 b Fr(b)2467 3971 y Fs(1)2507 3860 y Fi(\021)2653 3889 y Fw(1)p 2566 3933 223 4 v 2566 4025 a Fr(\034)34 b Fw(+)22 b(1)2834 3956 y Fr(:)92 4183 y Fg(Then,)36 b(ther)-5 b(e)38 b(exist)f Fw(0)32 b Fr(<)g(")1088 4198 y Fs(0)1160 4183 y Fr(<)39 b Fw(\026)-56 b Fr(")37 b Fg(and)g Fr(C)i(>)33 b Fw(1)p Fg(,)k(such)g(that,)h(for)f(any)g Fw(0)c Fr(<)f(")g(<)g(")3102 4198 y Fs(0)3141 4183 y Fg(,)38 b(one)e(c)-5 b(an)37 b(\014nd)g(a)92 4304 y(Cantor)d(set)h Fq(I)630 4319 y Fj(\003)698 4304 y Fq(\032)28 b(I)7 b Fg(,)35 b(with)1494 4424 y Fw(meas)1706 4439 y Fs(2)1745 4424 y Fw(\()p Fq(I)30 b(n)22 b(I)1993 4439 y Fj(\003)2032 4424 y Fw(\))28 b Fq(\024)g Fr(C)7 b(")2326 4383 y Fp(b)2356 4392 y Fe(1)2430 4424 y Fr(;)1118 b Fw(\(1.10\))92 4587 y Fg(such)34 b(that)h(the)g(fol) 5 b(lowing)33 b(holds.)h(Ther)-5 b(e)34 b(exist)g(a)h(Lipschitz)f(home) -5 b(omorphism)32 b Fr(!)3153 4602 y Fj(\003)3220 4587 y Fw(:)c Fq(I)3329 4602 y Fj(\003)3396 4587 y Fq(!)g Fh(R)3589 4551 y Fs(2)3670 4587 y Fg(and)92 4708 y(a)34 b(Lipschitz)h(c)-5 b(ontinuous)34 b(family)h(of)f(tori)h(emb)-5 b(e)g(dding)286 4902 y Fr(\036)27 b Fw(:)56 b(\()p Fr(\022)s(;)17 b(J)9 b Fw(\))27 b Fq(2)h Fh(T)869 4861 y Fs(2)934 4902 y Fq(\002)23 b(I)1088 4917 y Fj(\003)1155 4902 y Fq(7!)1283 4806 y Fi(\020)1349 4902 y Fr(I)1392 4917 y Fp(\036)1438 4902 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(J)9 b Fw(\))p Fr(;)33 b(')1793 4917 y Fp(\036)1839 4902 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(J)9 b Fw(\))p Fr(;)33 b(p)2179 4917 y Fp(\036)2225 4902 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(J)9 b Fw(\))p Fr(;)34 b(q)2560 4917 y Fp(\036)2606 4902 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(J)9 b Fw(\))2854 4806 y Fi(\021)2931 4902 y Fq(2)28 b(I)i(\002)22 b Fh(T)3271 4861 y Fs(2)3336 4902 y Fq(\002)h Fr(B)3515 4861 y Fs(4)3510 4927 y Fp(\032)3546 4935 y Fc(\003)3603 4902 y Fr(;)p 92 5012 1493 4 v 92 5104 a FC(unit)28 b(matrix;)f Fn(Q)628 5074 y Fy(00)698 5104 y FC(denotes)g(the)h(Hessian)f(matrix)g(of)h(the)g(second)f(deriv) -5 b(ativ)n(es)26 b(of)i Fn(Q)p FC(.)204 5173 y Fo(6)241 5203 y FC(A)g(computer-assisted)d(KAM)j(theory)e(for)h(the)g(secular)f (Hamiltonian)h(of)g(the)h(spatial)f(three-b)r(o)r(dy)f(problem)h(is)92 5303 y(studied)h(in)g([LG00].)1933 5775 y Fw(8)p eop %%Page: 9 9 9 8 bop 92 206 a Fg(with)35 b Fr(\032)354 221 y Fj(\003)424 206 y Fw(:=)29 b Fr(C)7 b(")679 170 y Fp(b)709 179 y Fe(2)748 206 y Fg(,)36 b(such)f(that,)h(for)g(any)g Fr(J)j Fq(2)30 b(I)1854 221 y Fj(\003)1893 206 y Fg(,)36 b Fr(\036)p Fw(\()p Fh(T)2118 170 y Fs(2)2161 206 y Fr(;)17 b(J)9 b Fw(\))35 b Fg(is)h(a)g(r)-5 b(e)g(al-analytic)35 b(\(el)5 b(liptic\))35 b Fq(H)q Fg(-inva-)92 326 y(riant)27 b(torus,)h(on)f (which)f(the)h Fq(H)q(\000)p Fg(\015ow)h(is)f(analytic)-5 b(al)5 b(ly)27 b(c)-5 b(onjugate)g(d)27 b(to)h(the)f(line)-5 b(ar)27 b(\015ow)g Fr(\022)k Fq(7!)c Fr(\022)8 b Fw(+)d Fr(!)3719 341 y Fj(\003)3759 326 y Fr(t)p Fg(.)92 447 y(F)-7 b(urthermor)i(e,)39 b Fr(\036)p Fw(\()p Fq(\001)p Fr(;)17 b(J)9 b Fw(\))40 b Fg(is)g(r)-5 b(e)g(al-analytic)39 b(on)h Fh(T)1876 410 y Fs(2)1876 473 y Fp(s)1909 482 y Fe(0)1947 473 y Fp(=)p Fs(8)2062 447 y Fg(and)g(the)g(fol)5 b(lowing)39 b(b)-5 b(ounds)40 b(hold,)f(uniformly)92 567 y(on)34 b Fh(T)294 582 y Fp(s)327 591 y Fe(0)365 582 y Fp(=)p Fs(8)462 567 y Fq(\002)22 b(I)615 582 y Fj(\003)655 567 y Fg(:)1527 807 y Fq(j)p Fr(I)1598 822 y Fp(\036)1643 807 y Fw(\()p Fr(\022)s(;)17 b(J)9 b Fw(\))22 b Fq(\000)h Fr(J)9 b Fq(j)83 b(\024)g Fr(C)7 b(")2463 739 y Fe(1)p 2463 751 31 4 v 2463 792 a(2)2503 766 y Fs(+)p Fp(b)2588 775 y Fe(2)2662 807 y Fr(;)1210 952 y Fq(j)p Fr(p)1287 967 y Fp(\036)1333 952 y Fw(\()p Fr(\022)s(;)17 b(J)9 b Fw(\))p Fq(j)21 b Fw(+)h Fq(j)p Fr(q)1782 967 y Fp(\036)1828 952 y Fw(\()p Fr(\022)s(;)17 b(J)9 b Fw(\))p Fq(j)83 b(\024)g Fr(C)7 b(")2453 911 y Fp(b)2483 920 y Fe(2)2539 952 y Fr(;)1392 1098 y Fq(j)p Fr(!)1481 1113 y Fj(\003)1520 1098 y Fw(\()p Fr(J)i Fw(\))22 b Fq(\000)h(r)p Fr(h)p Fw(\()p Fr(J)9 b Fw(\))p Fq(j)83 b(\024)g Fr(C)7 b(")17 b(:)1078 b Fw(\(1.11\))92 1317 y Fg(A)n(lso,)34 b(ther)-5 b(e)35 b(exists)f(a)h(Lipschitz)f(c)-5 b(ontinuous)35 b(function)g Fw(\012)2292 1332 y Fj(\003)2359 1317 y Fw(:)28 b Fq(I)2468 1332 y Fj(\003)2535 1317 y Fq(\000)-16 b(!)27 b Fh(R)2789 1281 y Fs(2)2870 1317 y Fg(such)34 b(that)1388 1537 y Fq(j)p Fw(\012)1486 1552 y Fj(\003)1526 1537 y Fw(\()p Fr(J)9 b Fw(\))22 b Fq(\000)h Fr(")1844 1512 y Fw(\026)1833 1537 y(\012\()p Fr(J)9 b Fw(\))p Fq(j)27 b(\024)h Fr(C)7 b(")2325 1496 y Fs(1+)p Fp(b)2445 1505 y Fe(2)2500 1537 y 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Fw(~)-55 b Fr(u)o Fw(\))17 b Fr(;)135 b Fw(\(2.22\))p 92 5165 1493 4 v 171 5252 a Fo(13)241 5282 y FC(\010)301 5252 y Fy(0)352 5282 y FC(has)27 b(generating)f(function)i Fm(J)1285 5252 y Fy(0)1327 5282 y Fn(\001)19 b Fm(')1423 5252 y Fy(0)1465 5282 y FC(+)1548 5190 y Fi(\020)1597 5282 y Fm(v)1640 5252 y Fy(0)1682 5282 y FC(+)f Fm(p)1807 5252 y Fy(0)1830 5282 y FC(\()p Fm(J)1916 5252 y Fy(0)1940 5282 y Fm(;)c(")p FC(\))2048 5190 y Fi(\021)2116 5282 y Fn(\001)2157 5190 y Fi(\020)2207 5282 y Fm(q)2247 5252 y Fy(0)2289 5282 y Fn(\000)k Fm(q)2412 5252 y Fy(0)2435 5282 y FC(\()p Fm(J)2521 5252 y Fy(0)2545 5282 y Fm(;)c(")p FC(\))2653 5190 y Fi(\021)2702 5282 y FC(.)1909 5775 y Fw(15)p eop %%Page: 16 16 16 15 bop 92 206 a Fw(where)1364 326 y Fr(h)1420 341 y Fs(0)1460 326 y Fw(\()1521 301 y(~)1498 326 y Fr(J)9 b Fw(\))27 b(:=)h Fr(h)p Fw(\()1874 301 y(~)1851 326 y Fr(J)9 b Fw(\))22 b(+)g Fr(")s Fw(~)-52 b Fr(g)s Fw(\()2230 301 y(~)2206 326 y Fr(J)9 b(;)17 b Fw(0)p Fr(;)g Fw(0\))32 b Fr(;)1023 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Fs(1)2400 1901 y Fq(j)28 b Fr(>)44 b Fw(const)h Fr(>)27 b Fw(0)17 b Fr(:)536 b Fw(\(2.26\))92 2240 y Fu(Step)39 b(3)h(\(KAM\))33 b Fw(W)-8 b(e)35 b(are,)g(no)m(w,)g(in)e(p) s(osition)g(to)h(apply)g(a)g(KAM)h(result)f(in)g(order)h(to)f(\014nd)h Ft(t)m(w)m(o-)92 2360 y(dimensional)k(elliptic)g(tori)p Fw(.)h(The)i(KAM)g(Theorem)g(w)m(e)h(shall)d(use)j(is,)e(basically)-8 b(,)40 b(the)i(v)m(ersion)g(in)92 2481 y([P96])35 b(of)g(a)g(result)g (\014rst)g(pro)m(v)m(ed)i(b)m(y)f(H.)f(Eliasson)g(\([E88]\))g(and)g (S.B.)h(Kuksin)f(\([K88]\))g(ab)s(out)g(the)92 2601 y(conserv)-5 b(ation)37 b(of)h(lo)m(w)m(er)g(dimensional)d(in)m(v)-5 b(arian)m(t)36 b(elliptic)f(tori)2469 2565 y Fs(14)2543 2601 y Fw(.)j(T)-8 b(o)38 b(state)g(the)g(KAM)g(theorem,)92 2721 y(w)m(e)33 b(need)h(a)e(bit)g(of)g(preparation.)92 2913 y(Consider)37 b(a)g(Hamiltonian)d(system)k(with)f(symplectic)g(v) -5 b(ariables)36 b(\()p Fr(y)t(;)17 b(x;)g(v)t(;)g(u)p Fw(\))33 b Fq(2)k Fh(R)3228 2876 y Fp(n)3306 2913 y Fq(\002)26 b 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Fw(=)h(0)p Fq(g)17 b Fr(;)92 4999 y Fw(whic)m(h)45 b(is)g(in)m(v)-5 b(arian)m(t)43 b(for)h(the)i(Hamiltonian)41 b(v)m(ector)46 b(\014eld)f Fr(X)2440 5014 y Fp(N)2507 4999 y Fw(,)g(the)g(\015o)m(w)h(b)s(eing,)e(simply)-8 b(,)43 b Fr(x)49 b Fq(7!)92 5119 y Fr(x)22 b Fw(+)g Fr(!)t Fw(\()p Fr(\030)5 b Fw(\))p Fr(t)p Fw(.)p 92 5212 1493 4 v 171 5273 a Fo(14)241 5303 y FC(Actually)-7 b(,)28 b([K88])e(and)i([P96])e(co)n(v)n(er)g(also)g(in\014nite-dimensional)i (\(in)g(\()p Fm(v)s(;)14 b(u)p FC(\)\))28 b(cases.)1909 5775 y Fw(16)p eop %%Page: 17 17 17 16 bop 92 206 a Ft(Assume)33 b(that)g Fr(P)45 b Ft(is)33 b(real-analytic)c(on)398 424 y Fr(D)s Fw(\()t(\026)-53 b Fr(r)s(;)20 b Fw(\026)-52 b Fr(s)o Fw(\))28 b(:=)f Fq(f)p Fw(\()p Fr(y)t(;)17 b(x;)g(v)t(;)g(u)p Fw(\))26 b Fq(2)i Fh(C)1510 383 y Fs(2\()q Fp(n)p Fs(+)p Fp(m)p Fs(\))1815 424 y Fw(:)44 b Fq(j)p Fr(y)t Fq(j)26 b Fr(<)32 b Fw(\026)-53 b Fr(r)2171 383 y Fs(2)2226 424 y Fr(;)17 b(x)28 b Fq(2)g Fh(T)2510 383 y Fp(n)2513 449 y Fs(\026)-38 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b(\030)5 b Fw(\))27 b Fq(2)h Fh(T)1209 4572 y Fp(n)1282 4613 y Fq(\002)22 b Fw(\005)1454 4628 y Fj(\003)1522 4613 y Fq(\000)-17 b(!)1710 4517 y Fi(\020)1776 4613 y Fr(y)1824 4628 y Fs(\010)1879 4613 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(\030)5 b Fw(\))p Fr(;)32 b(x)2209 4628 y Fs(\010)2265 4613 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(\030)5 b Fw(\))p Fr(;)32 b(v)2587 4628 y Fs(\010)2642 4613 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(\030)5 b Fw(\))17 b Fr(;)g(u)2975 4628 y Fs(\010)3028 4613 y Fw(\()p Fr(\022)s Fw(;)g Fr(\030)5 b Fw(\))3261 4517 y Fi(\021)3326 4613 y Fr(;)809 4780 y Fw(\010\()p Fh(T)980 4739 y Fp(n)1053 4780 y Fq(\002)23 b Fw(\005)1226 4795 y Fj(\003)1265 4780 y Fw(\))28 b Fq(\032)g(fj)p Fr(y)t Fq(j)e Fr(<)32 b Fw(\026)-53 b Fr(r)1771 4739 y Fs(2)1810 4780 y Fq(g)22 b(\002)g Fh(T)2044 4739 y Fp(n)2117 4780 y Fq(\002)g(fj)p Fr(v)t Fq(j)f Fw(+)h Fq(j)p Fr(u)p Fq(j)27 b Fr(<)32 b Fw(\026)-53 b Fr(r)r Fq(g)17 b Fr(;)92 4999 y Fg(a)26 b(Lipschitz)g(home)-5 b(omorphism)24 b Fr(!)1348 5014 y Fj(\003)1413 4999 y Fg(on)i Fw(\005)1617 5014 y Fj(\003)1683 4999 y Fg(and)g(a)g(Lipschitz) g(c)-5 b(ontinuous)26 b(function)g Fw(\012)3271 5014 y Fj(\003)3337 4999 y Fg(on)g Fw(\005)3541 5014 y Fj(\003)3581 4999 y Fg(,)g(such)92 5119 y(that,)42 b(for)f(any)h Fr(\030)j Fq(2)c Fw(\005)952 5134 y Fj(\003)991 5119 y Fg(,)h Fw(\010\()p Fh(T)1234 5083 y Fp(n)1285 5119 y Fr(;)17 b(\030)5 b Fw(\))41 b Fg(is)g(a)h(r)-5 b(e)g(al-analytic)41 b(\(el)5 b(liptic\))41 b Fr(H)8 b Fg(-invariant)40 b Fr(n)p Fq(\000)p Fg(dimensional)p 92 5212 1493 4 v 171 5273 a Fo(15)269 5303 y FC(Denote)28 b(\012)23 b(:=)f(\(\012)839 5315 y Fo(1)877 5303 y Fm(;)14 b(:)g(:)g(:)f(;)h FC(\012)1121 5315 y Fl(m)1184 5303 y FC(\))1909 5775 y Fw(17)p eop %%Page: 18 18 18 17 bop 92 206 a Fg(torus,)40 b(on)f(which)g(the)h(\015ow)g Fr(X)1260 221 y Fp(H)1367 206 y Fg(is)g(analytic)-5 b(al)5 b(ly)39 b(c)-5 b(onjugate)g(d)39 b(to)h(the)g(line)-5 b(ar)40 b(\015ow)f Fr(\022)h Fq(7!)d Fr(\022)29 b Fw(+)d Fr(!)3720 221 y Fj(\003)3759 206 y Fr(t)p Fg(.)92 326 y(F)-7 b(urthermor)i(e,)34 b Fw(\010\()p Fq(\001)p Fr(;)17 b(\030)5 b Fw(\))34 b Fg(is)g(r)-5 b(e)g(al-analytic)34 b(on)h Fh(T)1846 290 y Fp(n)1849 353 y Fs(\026)-38 b Fp(s)s(=)p Fs(2)1991 326 y Fg(and)35 b(the)g(fol)5 b(lowing)33 b(b)-5 b(ounds)34 b(hold:)822 542 y Fw(1)p 803 586 87 4 v 807 678 a(\026)-53 b Fr(r)850 649 y Fs(2)899 609 y Fq(j)p Fr(y)975 624 y Fs(\010)1029 609 y Fq(j)22 b Fw(+)g Fq(j)p Fr(x)1260 624 y Fs(\010)1338 609 y Fq(\000)g Fw(Id)q Fq(j)f Fw(+)1684 542 y(1)p 1684 586 49 4 v 1689 678 a(\026)-53 b Fr(r)1743 609 y Fw(\()p Fq(j)p Fr(v)1856 624 y Fs(\010)1911 609 y Fq(j)22 b Fw(+)g Fq(j)p Fr(u)2143 624 y Fs(\010)2197 609 y Fq(j)p Fw(\))g(+)634 864 y(+)827 797 y Fr(\015)p 803 841 105 4 v 803 932 a(M)918 768 y Fi(\020)996 797 y Fw(1)p 977 841 87 4 v 981 932 a(\026)-53 b Fr(r)1024 904 y Fs(2)1073 864 y Fq(j)p Fr(y)1149 879 y Fs(\010)1204 864 y Fq(j)1232 817 y Fs(Lip)1232 888 y(\005)1285 896 y Fc(\003)1361 864 y Fw(+)22 b Fq(j)p Fr(x)1542 879 y Fs(\010)1597 864 y Fq(j)1625 817 y Fs(Lip)1625 888 y(\005)1678 896 y Fc(\003)1754 864 y Fw(+)1862 797 y(1)p 1862 841 49 4 v 1867 932 a(\026)-53 b Fr(r)1920 864 y Fw(\()p Fq(j)p Fr(v)2033 879 y Fs(\010)2088 864 y Fq(j)2116 817 y Fs(Lip)2116 888 y(\005)2169 896 y Fc(\003)2245 864 y Fw(+)22 b Fq(j)p Fr(u)2427 879 y Fs(\010)2481 864 y Fq(j)2509 817 y Fs(Lip)2509 888 y(\005)2562 896 y Fc(\003)2616 864 y Fw(\))2654 768 y Fi(\021)2731 864 y Fq(\024)28 b Fr(c)2905 785 y Fq(j)-17 b(j)g(j)p Fr(X)3036 800 y Fp(P)3094 785 y Fq(j)g(j)g(j)3147 815 y Fs(\026)-38 b Fp(r)r(;)s Fs(\026)g Fp(s)o(;\015)p 2905 841 390 4 v 3072 932 a Fr(\015)3321 864 y(;)227 b Fw(\(2.33\))793 1095 y Fq(j)p Fr(!)882 1110 y Fj(\003)943 1095 y Fq(\000)23 b Fr(!)t Fq(j)e Fw(+)1289 1028 y Fr(\015)p 1265 1072 105 4 v 1265 1164 a(M)1379 1095 y Fq(j)p Fr(!)1468 1110 y Fj(\003)1529 1095 y Fq(\000)i Fr(!)t Fq(j)1722 1048 y Fs(Lip)1722 1120 y(\005)1775 1128 y Fc(\003)1856 1095 y Fq(\024)28 b Fr(c)17 b Fq(j)-17 b(j)g(j)o Fr(X)2150 1110 y Fp(P)2209 1095 y Fq(j)g(j)g(j)2262 1124 y Fs(\026)-38 b Fp(r)r(;)s Fs(\026)g Fp(s)o(;\015)2426 1095 y Fr(;)1122 b Fw(\(2.34\))793 1307 y Fq(j)p Fw(\012)891 1322 y Fj(\003)953 1307 y Fq(\000)22 b Fw(\012)p Fq(j)h Fw(+)1305 1240 y Fr(\015)p 1281 1284 V 1281 1375 a(M)1395 1307 y Fq(j)p Fw(\012)1493 1322 y Fj(\003)1555 1307 y Fq(\000)g Fw(\012)p Fq(j)1753 1260 y Fs(Lip)1753 1331 y(\005)1806 1339 y Fc(\003)1887 1307 y Fq(\024)29 b Fr(c)17 b Fq(j)-17 b(j)g(j)o Fr(X)2182 1322 y Fp(P)2241 1307 y Fq(j)g(j)g(j)2294 1336 y Fs(\026)-38 b Fp(r)q(;)s Fs(\026)g Fp(s;\015)2457 1307 y Fr(;)1091 b Fw(\(2.35\))793 1519 y Fq(j)p Fr(!)882 1534 y Fj(\003)943 1519 y Fq(\001)22 b Fr(k)j Fw(+)d(\012)1237 1534 y Fj(\003)1299 1519 y Fq(\001)g Fr(`)p Fq(j)27 b(\025)1693 1451 y Fr(\015)p 1560 1496 322 4 v 1560 1587 a Fw(1)22 b(+)g Fq(j)p Fr(k)s Fq(j)1839 1558 y Fp(\034)1908 1519 y Fr(;)117 b Fq(8)p Fw(\()p Fr(k)s(;)17 b(`)p Fw(\))27 b Fq(6)p Fw(=)h(\(0)p Fr(;)17 b Fw(0\))g Fr(;)43 b Fq(j)p Fr(`)p Fq(j)27 b(\024)h Fw(2)17 b Fr(;)495 b Fw(\(2.36\))828 1756 y(meas)1040 1771 y Fp(n)1087 1756 y Fw(\(\005)22 b Fq(n)g Fw(\005)1365 1771 y Fj(\003)1404 1756 y Fw(\))28 b Fq(\024)g Fr(c)1668 1688 y(\015)p 1644 1732 105 4 v 1644 1824 a(M)1775 1756 y Fw(\()p Fr(LM)44 b Fw(diam)15 b(\005\))2356 1715 y Fp(n)p Fj(\000)p Fs(1)2510 1756 y Fr(:)1038 b Fw(\(2.37\))92 2015 y Fu(Remark)37 b(2.3)49 b Fw(This)23 b(Theorem)h(is)f(a)g(summary)g(\(in)f(the)i(\014nite)f (dimensional)e(case\))k(of)e(Theorems)h(A)92 2135 y(and)k(B)h(and)g (Corollary)d(C)j(of)g([P96],)f(to)g(whic)m(h)i(w)m(e)f(refer)g(for)f (the)h(pro)s(of)2750 2099 y Fs(16)2824 2135 y Fw(.)g(Notice)f(that)g (\(2.30\))g(and)92 2256 y(\(2.31\))e(here)h(pla)m(y)f(the)h(role)f(of)g (the)h(\\Non-degeneracy)h(Assumption)e(A")g(of)g([P96],)h(while)f (Assump-)92 2376 y(tions)31 b(B)g(and)g(C)h(of)f([P96])g(are)h (trivially)c(satis\014ed)k(in)f(the)g(\014nite)h(dimensional)c(case.)33 b(Assumptions)92 2496 y(\(2.30\))f(and)g(\(2.31\))g(imply)f(the)i (measure)g(estimate)e(\(2.37\),)h(as)h(brie\015y)g(sho)m(wn)h(in)e(App) s(endix)h(B.)92 2725 y(T)-8 b(o)32 b(apply)h(Prop)s(osition)d(2.2)j(to) f(our)g(case,)i(w)m(e)f(let)351 2945 y Fr(n)28 b Fw(=)g(2)k Fr(;)114 b(!)824 2960 y Fs(0)891 2945 y Fw(=)28 b Fr(h)1051 2904 y Fj(0)1051 2969 y Fs(0)1123 2945 y Fr(;)114 b Fw(\012)1334 2960 y Fp(i)1390 2945 y Fw(=)28 b Fr(")1551 2920 y Fw(~)1540 2945 y(\012)1610 2960 y Fp(i)1671 2945 y Fr(;)114 b(L)28 b Fw(=)44 b(const)50 b Fr(;)114 b(M)39 b Fw(=)44 b(const)50 b Fr(;)114 b(K)3188 2960 y Fs(0)3255 2945 y Fw(:=)28 b(16)p Fr(LM)43 b(;)351 3113 y(J)37 b Fw(=)28 b Fr(\030)k Fq(2)c Fw(\005)g(=)f Fr(U)38 b Fw(=)28 b Fq(I)1183 3128 y Fs(\026)-37 b Fp(\015)5 b(;\034)1317 3113 y Fr(;)114 b(e)p Fw(\()p Fr(\030)5 b Fw(\))27 b(=)h Fr(h)1814 3128 y Fs(0)1853 3113 y Fw(\()p Fr(\030)5 b Fw(\))32 b Fr(;)119 b Fw(\026)-54 b Fr(r)31 b Fw(=)c Fr(")2374 3072 y Fs(3)p Fp(=)p Fs(4)2516 3113 y Fr(;)119 b Fw(\026)-53 b Fr(s)27 b Fw(=)2846 3046 y Fr(s)p 2845 3090 49 4 v 2845 3181 a Fw(8)2936 3113 y Fr(;)351 3373 y(N)10 b Fw(\()p Fr(y)t(;)17 b(v)t(;)g(u)p Fw(;)g Fr(\030)5 b Fw(\))25 b(=)j Fr(e)p Fw(\()p Fr(\030)5 b Fw(\))21 b(+)h Fr(!)1332 3388 y Fs(0)1372 3373 y Fw(\()p Fr(\030)5 b Fw(\))21 b Fq(\001)h Fr(y)j Fw(+)1780 3266 y Fs(2)1738 3290 y Fi(X)1741 3473 y Fp(i)p Fs(=1)1885 3348 y Fw(~)1875 3373 y(\012)1945 3388 y Fp(i)1973 3373 y Fw(\()p Fr(\030)5 b Fw(\)\()p Fr(u)2191 3332 y Fs(2)2191 3398 y Fp(i)2252 3373 y Fw(+)22 b Fr(v)2401 3332 y Fs(2)2397 3398 y Fp(i)2440 3373 y Fw(\))32 b Fr(;)114 b Fw(\()p Fr(x;)17 b(v)t(;)g(u)p Fw(\))27 b(=)h(\()3166 3347 y(~)3146 3373 y Fr( )t(;)20 b Fw(~)-52 b Fr(v)s(;)23 b Fw(~)-55 b Fr(u)o Fw(\))33 b Fr(;)351 3607 y(H)8 b Fw(\()p Fr(y)t(;)17 b(x;)g(v)t(;)g(u)p Fw(;)g Fr(\030)5 b Fw(\))24 b(=)1111 3581 y(~)1082 3607 y Fq(H)q Fw(\()p Fr(J)31 b Fw(+)22 b Fr(y)t(;)1503 3580 y Fw(~)1484 3607 y Fr( )s(;)e Fw(~)-52 b Fr(v)s(;)23 b Fw(~)-55 b Fr(u)p Fw(\))32 b Fr(:)1734 b Fw(\(2.38\))92 3827 y(Recall)31 b(that)h Fr(r)e Fq(\030)777 3755 y(p)p 860 3755 46 4 v 72 x Fr(")o Fw(,)j(whic)m(h)g(is)f(m)m(uc)m(h)h(larger)f(than)37 b(\026)-54 b Fr(r)31 b Fw(=)c Fr(")2330 3791 y Fs(3)p Fp(=)p Fs(4)2440 3827 y Fw(,)32 b(so)h(that)1409 4047 y(\005)1485 4063 y Fs(\026)-38 b Fp(r)1516 4044 y Fe(2)1582 4047 y Fw(:=)1733 3964 y Fi([)1713 4148 y Fp(\030)s Fj(2)p Fs(\005)1863 4047 y Fr(D)1947 4063 y Fs(\026)g Fp(r)1978 4044 y Fe(2)2016 4047 y Fw(\()p Fr(\030)5 b Fw(\))27 b Fq(\032)i Fr(U)2339 4062 y Fp(r)r(=)p Fs(2)2480 4047 y Fr(;)92 4356 y Fw(and)44 b(also)k(\026)-54 b Fr(r)50 b Fq(\034)d Fr(\032=)p Fw(4)d(so)h(that)e(the)i(Hamiltonian)c Fr(H)51 b Fw(in)44 b(\(2.38\))f(is)h(real-analytic)d(for)j Fq(j)p Fr(y)t Fq(j)i Fr(<)51 b Fw(\026)-53 b Fr(r)3758 4320 y Fs(2)3797 4356 y Fw(,)92 4477 y Fq(j)p Fr(u)p Fq(j)21 b Fw(+)h Fq(j)p Fr(v)t Fq(j)k Fr(<)32 b Fw(\026)-53 b Fr(r)35 b Fw(and)e Fq(j)17 b Fw(Im)e Fr(x)p Fq(j)28 b Fr(<)k Fw(\026)-53 b Fr(s)p Fw(,)32 b(for)g(an)m(y)h Fr(\030)g Fq(2)28 b Fw(\005.)92 4668 y(Next,)33 b(w)m(e)h(observ)m(e)g (that)f(the)g(p)s(erturbation)e Fr(P)46 b Fw(ma)m(y)33 b(b)s(e)f(written)h(as)g Fr(P)41 b Fw(=)2921 4601 y Fi(P)3009 4689 y Fs(1)p Fj(\024)p Fp(k)r Fj(\024)p Fs(4)3249 4668 y Fr(P)3312 4683 y Fp(k)3354 4668 y Fw(,)33 b(with)1070 4888 y Fr(P)1133 4903 y Fs(1)1256 4888 y Fw(=)83 b Fr(h)1471 4903 y Fs(0)1510 4888 y Fw(\()p Fr(\030)27 b Fw(+)22 b Fr(y)t Fw(\))f Fq(\000)i Fr(h)1983 4903 y Fs(0)2022 4888 y Fw(\()p Fr(\030)5 b Fw(\))22 b Fq(\000)g Fr(!)2328 4903 y Fs(0)2367 4888 y Fw(\()p Fr(\030)5 b Fw(\))22 b Fq(\001)g Fr(y)d(;)p 92 4981 1493 4 v 171 5042 a Fo(16)241 5072 y FC(F)-7 b(or)27 b(comparison)e(purp)r(oses,)h(w)n(e)h(ha)n(v)n (e)f(k)n(ept)h(the)g(notations)g(as)f(close)g(as)h(p)r(ossible)f(to)h (the)h(notations)e(in)h([P96];)92 5172 y(notice,)d(ho)n(w)n(ev)n(er,)e (that)j(\(in)f(order)f(to)h(a)n(v)n(oid)f(confusion)h(with)h(other)e (parameters)g(in)n(tro)r(duced)h(in)h(our)e(pap)r(er\))h Fm(r)j FC(and)92 5271 y Fm(s)g FC(in)h([P96])e(are)h(denoted)h(here)j (\026)-46 b Fm(r)30 b FC(and)h(\026)-45 b Fm(s)p FC(.)1909 5775 y Fw(18)p eop %%Page: 19 19 19 18 bop 1070 265 a Fr(P)1133 280 y Fs(2)1256 265 y Fw(=)1457 157 y Fs(2)1415 182 y Fi(X)1418 364 y Fp(i)p Fs(=1)1551 169 y Fi(\020)1601 265 y Fw(\012)1671 280 y Fp(i)1699 265 y Fw(\()p Fr(\030)27 b Fw(+)22 b Fr(y)t Fw(\))f Fq(\000)i Fw(\012)2186 280 y Fp(i)2214 265 y Fw(\()p Fr(\030)5 b Fw(\))2338 169 y Fi(\021)2404 265 y Fw(\()p Fr(u)2498 224 y Fs(2)2498 290 y Fp(i)2559 265 y Fw(+)22 b Fr(v)2708 224 y Fs(2)2704 290 y Fp(i)2747 265 y Fw(\))17 b Fr(;)1070 481 y(P)1133 496 y Fs(3)1256 481 y Fw(=)83 b Fr(")s Fw(~)-52 b Fr(g)1508 496 y Fs(0)1546 481 y Fw(\()p Fr(\030)27 b Fw(+)22 b Fr(y)t(;)17 b(v)t(;)g(u)p Fw(\))g Fr(;)1070 626 y(P)1133 641 y Fs(4)1256 626 y Fw(=)83 b Fr(")1461 585 y Fs(3)1521 600 y Fw(~)1500 626 y Fr(f)1548 641 y Fs(0)1587 626 y Fw(\()p Fr(\030)27 b Fw(+)22 b Fr(y)t(;)17 b(x;)g(v)t(;)g(u)p Fw(\))g Fr(:)92 826 y Fw(By)33 b(\(2.26\))f(and)g(\(2.38\),)g(w)m(e)i(see)g(that)e 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2177 y Fi(\020)2488 2274 y Fr(")2534 2233 y Fs(3)p Fp(=)p Fs(2)2666 2274 y Fw(+)22 b Fr(")2810 2233 y Fs(7)p Fp(=)p Fs(4)2941 2274 y Fw(+)g Fr(")3085 2233 y Fs(3)p Fp(=)p Fs(2)3195 2177 y Fi(\021)3272 2274 y Fq(\024)1011 2466 y(\024)100 b Fw(const)17 b Fr(")1473 2425 y Fs(3)p Fp(=)p Fs(2)1599 2466 y Fr(;)92 2667 y Fw(so)32 b(that,)h(letting)e(\(sa)m (y\))1511 2787 y Fr(\015)h Fw(:=)c Fr(\015)1776 2802 y Fs(0)1815 2787 y Fr(=)p Fw(2)f(=)44 b(const)18 b Fr(")32 b(;)1170 b Fw(\(2.39\))92 2953 y(one)33 b(\014nds)1491 3036 y Fq(j)-17 b(j)g(j)p Fr(X)1622 3051 y Fp(P)1680 3036 y Fq(j)g(j)g(j)1733 3065 y Fs(\026)-38 b Fp(r)r(;)s Fs(\026)g Fp(s)o(;\015)p 1491 3091 390 4 v 1658 3183 a Fr(\015)1918 3114 y Fq(\024)45 b Fw(const)2262 3038 y Fq(p)p 2345 3038 46 4 v 76 x Fr(")17 b(:)1140 b Fw(\(2.40\))92 3325 y(Therefore,)26 b(the)f(assumptions)f(of)g(Prop)s(osition)f(2.2)h (are)h(ful\014lled)e(and)h(the)h(existence)i(of)d(the)h(elliptic)92 3446 y(quasi-p)s(erio)s(dic)d(orbits)h(follo)m(ws)g(from)g(Prop)s (osition)f(2.2:)i(the)h(parameterization)d Fr(\036)i Fw(in)f(Theorem)h(1.2)92 3566 y(of)32 b(the)h(elliptic)d(tori)h(is)h (giv)m(en)h(b)m(y)817 3670 y Fi(\020)883 3766 y Fr(I)926 3781 y Fp(\036)972 3766 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(J)9 b Fw(\))p Fr(;)33 b(')1327 3781 y Fp(\036)1373 3766 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(J)9 b Fw(\))p Fr(;)33 b(p)1713 3781 y Fp(\036)1760 3766 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(J)9 b Fw(\))p Fr(;)33 b(q)2094 3781 y Fp(\036)2140 3766 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(J)9 b Fw(\))2388 3670 y Fi(\021)817 3949 y Fw(:=)27 b(\011)22 b Fq(\016)g Fw(\010)1187 3908 y Fj(0)1233 3949 y Fq(\016)1317 3917 y Fi(e)1305 3949 y Fw(\010)1376 3853 y Fi(\020)1425 3949 y Fr(J)32 b Fw(+)22 b Fr(y)1657 3964 y Fs(\010)1711 3949 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(J)9 b Fw(\))p Fr(;)33 b(x)2057 3964 y Fs(\010)2113 3949 y Fw(\()p Fr(\022)s Fw(;)17 b Fr(J)9 b Fw(\))p Fr(;)33 b(v)2451 3964 y Fs(\010)2506 3949 y Fw(\()p 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b(in)g(the)h(other) g(v)-5 b(ariables,)31 b(\(2.33\))p Fq(\004)p Fw(\(2.36\),)i(\(2.16\),)f (\(2.21\))f(and)i(\(2.23\).)92 4982 y(Finally)-8 b(,)32 b(w)m(e)j(turn)g(to)f(the)h(measure)g(estimates)f(\(1.10\).)g(It)g (follo)m(ws)g(from)f(\(2.37\))h(and)g(\(2.39\))g(that)92 5103 y(the)f(2)p Fq(\000)p Fw(dimensional)d(elliptic)g(tori)h(are)i (describ)s(ed)g(b)m(y)h(a)e(set)h(of)f(parameters)h(\005)3079 5118 y Fj(\003)3118 5103 y Fw(,)g(with)1440 5303 y(meas)1652 5318 y Fs(2)1691 5303 y Fw(\(\005)22 b Fq(n)g Fw(\005)1969 5318 y Fj(\003)2009 5303 y Fw(\))28 b Fq(\024)g Fw(const)17 b Fr(")g(:)1066 b Fw(\(2.43\))1909 5775 y(19)p eop %%Page: 20 20 20 19 bop 92 206 a Fw(Th)m(us,)34 b(from)d(\(2.13\),)h(\(2.42\))g(and)h (\(2.43\),)f(there)h(follo)m(ws)803 405 y(meas)1014 420 y Fs(2)1054 405 y Fw(\()p Fq(I)d(n)21 b(I)1301 420 y Fj(\003)1341 405 y Fw(\))96 b Fq(\024)129 b Fw(meas)1893 420 y Fs(2)1932 405 y Fw(\()p Fq(I)30 b(n)22 b(I)2182 420 y Fs(\026)-37 b Fp(\015)t(;\034)2283 405 y Fw(\))22 b(+)54 b(meas)2685 420 y Fs(2)2725 405 y Fw(\()p Fq(I)2819 420 y Fs(\026)-37 b Fp(\015)t(;\034)2942 405 y Fq(n)22 b(I)3068 420 y Fj(\003)3108 405 y Fw(\))1462 550 y(:=)116 b(meas)1893 565 y Fs(2)1932 550 y Fw(\()p Fq(I)30 b(n)22 b(I)2182 565 y Fs(\026)-37 b Fp(\015)t(;\034)2283 550 y Fw(\))22 b(+)54 b(meas)2685 565 y Fs(2)2725 550 y Fw(\(\005)22 b Fq(n)g Fw(\005)3003 565 y Fj(\003)3043 550 y Fw(\))1475 743 y Fq(\024)96 b Fw(const)1887 646 y Fi(\020)1937 743 y Fr(")1993 674 y Fe(1)p 1993 686 31 4 v 1993 728 a(2)2033 702 y Fj(\000)p Fp(b)p Fs(\()p Fp(\034)8 b Fs(+1\))2306 646 y Fi(\020)2373 743 y Fw(log)2525 675 y(1)p 2525 720 49 4 v 2527 811 a Fr(")2584 646 y Fi(\021)2633 669 y Fp(\034)g Fs(+1)2789 743 y Fw(+)22 b Fr(")2933 646 y Fi(\021)1475 976 y Fq(\024)96 b Fw(const)17 b Fr(")1943 908 y Fe(1)p 1943 920 31 4 v 1943 961 a(2)1983 935 y Fj(\000)p Fp(b)p Fs(\()p Fp(\034)8 b Fs(+1\))2257 880 y Fi(\020)2323 976 y Fw(log)2476 909 y(1)p 2476 953 49 4 v 2478 1044 a Fr(")2534 880 y 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b Fr(p;)17 b(q)36 b Fw(or)c Fr(')p Fw(\).)g(Let)618 3653 y Fr(D)f Fq(\032)d Fh(R)901 3611 y Fp(n)986 3653 y Fr(;)115 b(E)6 b(;)17 b(F)41 b Fq(\032)28 b Fh(R)1525 3611 y Fp(m)1630 3653 y Fr(;)114 b(U)39 b Fw(:=)27 b Fr(D)e Fq(\002)e Fr(E)28 b Fq(\002)22 b Fr(F)47 b(;)114 b(W)41 b Fw(:=)28 b Fr(U)33 b Fq(\002)22 b Fh(T)3187 3611 y Fp(n)3270 3653 y Fr(;)92 3852 y Fw(and)32 b(let)g Fr(r)m(;)17 b(r)551 3867 y Fp(p)591 3852 y Fr(;)g(r)679 3867 y Fp(q)717 3852 y Fr(;)g(s)27 b(>)g Fw(0)33 b(and)f Fr(v)g Fw(:=)27 b(\()p Fr(r)m(;)17 b(r)1584 3867 y Fp(p)1624 3852 y Fr(;)g(r)1712 3867 y Fp(q)1750 3852 y Fw(\).)32 b(F)-8 b(or)32 b(a)g(function)1021 4051 y Fr(f)11 b Fw(\()p Fr(u;)17 b(')p Fw(\))26 b(=)1477 3968 y Fi(X)1450 4154 y Fp(k)r Fj(2)p Fd(Z)1588 4135 y Ff(n)1641 4051 y Fr(f)1689 4066 y Fp(k)1731 4051 y Fw(\()p Fr(u)p Fw(\))p Fr(e)1908 4010 y Fp(ik)r Fj(\001)p Fp(')2073 4051 y Fr(;)212 b(u)27 b Fw(:=)h(\()p Fr(I)8 b(;)17 b(p;)g(q)t Fw(\))31 b Fr(;)92 4328 y Fw(real-analytic)h(for)i (\()p Fr(u;)17 b(')p Fw(\))30 b Fq(2)i Fr(W)1266 4343 y Fp(v)r(;s)1393 4328 y Fw(:=)f Fr(U)1593 4343 y Fp(v)1658 4328 y Fq(\002)24 b Fh(T)1822 4292 y Fp(n)1822 4353 y(s)1873 4328 y Fw(,)34 b(with)h Fr(U)2225 4343 y Fp(v)2300 4328 y Fw(:=)c Fr(D)2515 4343 y Fp(r)2577 4328 y Fq(\002)24 b Fr(E)2750 4343 y Fp(r)2782 4351 y Ff(p)2846 4328 y Fq(\002)g Fr(F)3010 4343 y Fp(r)3042 4351 y Ff(q)3080 4328 y Fw(,)35 b(w)m(e)h(shall)d(use)j(the)92 4449 y(norm)315 4412 y Fs(17)1314 4569 y Fq(k)p Fr(f)11 b Fq(k)1473 4584 y Fp(v)r(;s)1593 4569 y Fw(:=)1751 4486 y Fi(X)1724 4672 y Fp(k)r Fj(2)p Fd(Z)1862 4654 y Ff(n)1928 4569 y Fw(sup)1915 4648 y Fp(u)p Fj(2)p Fp(U)2051 4656 y Ff(v)2105 4569 y Fq(j)p Fr(f)2181 4584 y Fp(k)2223 4569 y Fw(\()p Fr(u)p Fw(\))p Fq(j)p Fr(e)2428 4528 y Fj(j)p Fp(k)r Fj(j)p Fp(s)2574 4569 y Fr(:)92 4820 y Fw(Finally)-8 b(,)30 b(let)i(\003)g(a)g(sub-lattice)f(of)i Fh(Z)1423 4784 y Fp(n)1499 4820 y Fw(and,)g(if)f Fr(f)38 b Fw(=)1996 4754 y Fi(P)2100 4820 y Fr(f)2148 4835 y Fp(k)2191 4820 y Fr(e)2236 4784 y Fp(ik)r Fj(\001)p Fp(')2368 4820 y Fw(,)33 b(w)m(e)g(set)1004 5020 y Fr(T)1061 5035 y Fp(K)1130 5020 y Fr(f)38 b Fw(:=)1385 4936 y Fi(X)1347 5125 y Fj(j)p Fp(k)r Fj(j\024)p Fp(K)1560 5020 y Fr(f)1608 5035 y Fp(k)1651 5020 y Fr(e)1696 4978 y Fp(ik)r Fj(\001)p Fp(')1861 5020 y Fr(;)212 b(P)2163 5035 y Fs(\003)2216 5020 y Fr(f)38 b Fw(:=)2440 4936 y Fi(X)2433 5121 y Fp(k)r Fj(2)p Fs(\003)2584 5020 y Fr(f)2632 5035 y Fp(k)2675 5020 y Fr(e)2720 4978 y Fp(ik)r Fj(\001)p Fp(')2885 5020 y Fr(:)p 92 5212 1493 4 v 171 5273 a Fo(17)297 5303 y FC(If)28 b Fm(k)d Fn(2)f Fh(Z)596 5273 y Fl(n)666 5303 y Fn(j)p Fm(k)s Fn(j)f FC(:=)892 5241 y Fi(P)979 5261 y Fl(n)979 5328 y(i)p Fo(=1)1105 5303 y Fn(j)p Fm(k)1171 5315 y Fl(i)1199 5303 y Fn(j)p Fm(:)1909 5775 y Fw(20)p eop %%Page: 21 21 21 20 bop 92 206 a Fu(Prop)s(osition)35 b(A.1)49 b Fg(L)-5 b(et)50 b Fr(H)62 b Fw(:=)55 b Fr(h)p Fw(\()p Fr(I)8 b Fw(\))33 b(+)g Fr(f)11 b Fw(\()p Fr(u;)17 b(')p Fw(\))48 b Fg(b)-5 b(e)50 b(a)f(r)-5 b(e)g(al-analytic)49 b(Hamiltonian)g(on)g Fr(W)3701 221 y Fp(v)r(;s)3794 206 y Fg(.)92 326 y(Denoting)33 b Fr(!)e Fw(:=)d Fr(h)790 290 y Fj(0)848 326 y Fg(and)35 b Fr(c)1080 341 y Fp(m)1174 326 y Fw(:=)28 b Fr(e)p Fw(\(1)22 b(+)g Fr(em)p Fw(\))p Fr(=)p Fw(2)p Fg(,)34 b(we)h(supp)-5 b(ose)34 b(that)745 515 y Fq(j)p Fr(!)t Fw(\()p Fr(I)8 b Fw(\))21 b Fq(\001)h Fr(k)s Fq(j)27 b(\025)h Fr(\013)h(>)e Fw(0)35 b Fr(;)216 b Fq(8)17 b(j)p Fr(k)s Fq(j)27 b(\024)h Fr(K)42 b(;)116 b(k)43 b(=)-61 b Fq(2)28 b Fw(\003)35 b Fr(;)86 b Fq(8)17 b Fr(I)36 b Fq(2)28 b Fr(D)3071 530 y Fp(r)3144 515 y Fr(;)428 b Fw(\(A.1\))92 704 y Fr(K)7 b(s)27 b Fq(\025)i Fw(6)34 b Fg(and)747 932 y Fq(k)p Fr(f)11 b Fq(k)906 947 y Fp(v)r(;s)1026 932 y Fw(=:)27 b Fr(")h(<)1453 864 y(\013)q(d)p 1343 908 333 4 v 1343 1000 a Fw(2)1392 971 y Fs(7)1431 1000 y Fr(c)1473 1015 y Fp(m)1540 1000 y Fr(K)7 b(s)1885 932 y Fw(where)201 b Fr(d)27 b Fw(:=)h(min)o Fq(f)p Fr(r)s(s;)17 b(r)2937 947 y Fp(p)2975 932 y Fr(r)3019 947 y Fp(q)3057 932 y Fq(g)35 b Fr(:)430 b Fw(\(A.2\))92 1161 y Fg(Then,)34 b(ther)-5 b(e)34 b(exists)h(a)g(r)-5 b(e)g(al-analytic)34 b(symple)-5 b(ctic)34 b(tr)-5 b(ansformation)1169 1350 y Fw(\011)28 b(:)g(\()p Fr(u)1422 1309 y Fj(0)1444 1350 y Fr(;)17 b(f)1547 1309 y Fj(0)1570 1350 y Fw(\))28 b Fq(2)g Fr(W)1822 1365 y Fp(v)1856 1373 y Fc(\003)1892 1365 y Fp(;s)1945 1373 y Fc(\003)2012 1350 y Fq(\000)-16 b(!)27 b Fw(\()p Fr(u;)17 b(')p Fw(\))27 b Fq(2)h Fr(W)2653 1365 y Fp(v)r(;s)92 1540 y Fg(with)34 b Fr(v)350 1555 y Fj(\003)418 1540 y Fw(:=)27 b Fr(v)t(=)p Fw(2)p Fr(;)35 b(s)805 1555 y Fj(\003)872 1540 y Fw(:=)27 b Fr(s=)p Fw(6)p Fr(;)35 b Fg(such)f(that)1376 1729 y Fr(H)1457 1744 y Fj(\003)1524 1729 y Fw(:=)27 b Fr(H)j Fq(\016)22 b Fw(\011)27 b(=)h Fr(h)22 b Fw(+)g Fr(g)k Fw(+)c Fr(f)2439 1744 y Fj(\003)2513 1729 y Fr(;)1059 b Fw(\(A.3\))92 1918 y Fg(with)34 b Fr(g)39 b Fg(in)34 b(normal)g(form:)1539 2038 y Fr(g)d Fw(=)1727 1955 y Fi(X)1720 2140 y Fp(k)r Fj(2)p Fs(\003)1871 2038 y Fr(g)1918 2053 y Fp(k)1961 2038 y Fw(\()p Fr(u)2055 1997 y Fj(0)2077 2038 y Fw(\))p Fr(e)2160 1997 y Fp(ik)r Fj(\001)p Fp(')2289 1974 y Fc(0)2350 2038 y Fr(:)1222 b Fw(\(A.4\))92 2276 y Fg(Mor)-5 b(e)g(over,)38 b(we)g(have)f(that)i(\(denoting)e(the)i(pr)-5 b(oje)g(ction)37 b(of)h Fw(\011\()p Fr(I)2475 2240 y Fj(0)2498 2276 y Fr(;)17 b(p)2591 2240 y Fj(0)2614 2276 y Fr(;)g(q)2705 2240 y Fj(0)2728 2276 y Fr(;)g(')2836 2240 y Fj(0)2859 2276 y Fw(\))38 b Fg(onto)g(the)h Fr(I)8 b Fq(\000)p Fg(variables)92 2396 y(by)35 b Fr(I)8 b Fw(\()p Fr(I)359 2360 y Fj(0)382 2396 y Fr(;)17 b(p)475 2360 y Fj(0)498 2396 y Fr(;)g(q)589 2360 y Fj(0)611 2396 y Fr(;)g(')719 2360 y Fj(0)742 2396 y Fw(\))p Fg(,)35 b(etc.\),)1122 2645 y Fq(k)p Fr(g)25 b Fq(\000)e Fr(P)1407 2660 y Fs(\003)1460 2645 y Fr(T)1517 2660 y Fp(K)1585 2645 y Fr(f)11 b Fq(k)1694 2660 y Fp(v)1728 2668 y Fc(\003)1764 2660 y Fp(;s)1817 2668 y Fc(\003)1884 2645 y Fq(\024)1999 2577 y Fw(12)p 1999 2621 98 4 v 1999 2713 a(11)2152 2577 y(2)2201 2541 y Fs(7)2240 2577 y Fr(c)2282 2592 y Fp(m)2348 2577 y Fr(")p 2152 2621 243 4 v 2216 2713 a(\013)q(d)2439 2645 y(")27 b Fq(\024)2627 2577 y Fw(1)p 2627 2621 49 4 v 2627 2713 a(4)2686 2645 y Fr(")35 b(;)805 b Fw(\(A.5\))1171 2937 y Fq(k)p Fr(f)1269 2952 y Fj(\003)1308 2937 y Fq(k)1358 2952 y Fp(v)1392 2960 y Fc(\003)1428 2952 y Fp(;s)1481 2960 y Fc(\003)1548 2937 y Fq(\024)1663 2869 y Fw(2)1712 2833 y Fs(9)1751 2869 y Fr(c)1793 2884 y Fp(m)1860 2869 y Fr(")p 1663 2913 243 4 v 1728 3005 a(\013)q(d)1916 2937 y(e)1961 2895 y Fj(\000)p Fp(K)5 b(s=)p Fs(6)2187 2937 y Fr(")28 b Fq(\024)g Fr(e)2411 2895 y Fj(\000)p Fp(K)5 b(s=)p Fs(6)2637 2937 y Fr(")35 b(;)854 b Fw(\(A.6\))159 3132 y Fr(s)35 b Fq(j)p Fr(I)8 b Fw(\()p Fr(u)413 3091 y Fj(0)435 3132 y Fr(;)17 b(f)538 3091 y Fj(0)561 3132 y Fw(\))22 b Fq(\000)g Fr(I)771 3091 y Fj(0)794 3132 y Fq(j)p Fr(;)52 b(r)945 3147 y Fp(q)1018 3132 y Fq(j)p Fr(p)p Fw(\()p Fr(u)1189 3091 y Fj(0)1211 3132 y Fr(;)17 b(f)1314 3091 y Fj(0)1337 3132 y Fw(\))22 b Fq(\000)g Fr(p)1545 3091 y Fj(0)1569 3132 y Fq(j)p Fr(;)51 b(r)1719 3147 y Fp(p)1793 3132 y Fq(j)p Fr(q)t Fw(\()p Fr(u)1962 3091 y Fj(0)1985 3132 y Fr(;)17 b(f)2088 3091 y Fj(0)2110 3132 y Fw(\))22 b Fq(\000)h Fr(q)2317 3091 y Fj(0)2340 3132 y Fq(j)p Fr(;)51 b(r)38 b Fq(j)p Fr(')p Fw(\()p Fr(u)2714 3091 y Fj(0)2736 3132 y Fr(;)17 b(f)2839 3091 y Fj(0)2862 3132 y Fw(\))22 b Fq(\000)h Fr(f)3081 3091 y Fj(0)3104 3132 y Fq(j)k(\024)h Fw(9)p Fr("=\013)35 b(:)67 b Fw(\(A.7\))92 3325 y(The)37 b(pro)s(of)e(of)h(this)f(result)h (rests)i(up)s(on)e(a)f(series)i(of)f(tec)m(hnical)g(elemen)m(tary)g (lemmata,)d(whic)m(h)k(w)m(e)92 3445 y(no)m(w)c(pro)s(ceed)g(to)g (state:)92 3638 y Fu(Lemma)k(A.1)48 b Fg(L)-5 b(et)842 3602 y Fs(18)952 3638 y Fw(0)28 b Fr(<)f(\027)35 b(<)27 b(v)39 b Fg(and)34 b Fw(0)27 b Fr(<)h(\033)k(<)27 b(s;)35 b Fg(wher)-5 b(e)34 b Fr(\027)g Fw(:=)28 b(\()p Fr(\032;)17 b(\032)2741 3653 y Fp(p)2781 3638 y Fr(;)g(\032)2875 3653 y Fp(q)2913 3638 y Fw(\))p Fg(.)35 b(Then)556 3797 y Fi(X)510 3980 y Fs(1)p Fj(\024)p Fp(i)p Fj(\024)p Fp(n)739 3880 y Fq(k)815 3813 y Fr(@)5 b(f)p 799 3857 149 4 v 799 3948 a(@)g(')919 3963 y Fp(i)957 3880 y Fq(k)1007 3895 y Fp(v)r(;s)p Fj(\000)p Fp(\033)1225 3880 y Fq(\024)1368 3813 y Fw(1)p 1341 3857 105 4 v 1341 3948 a Fr(e\033)1455 3880 y Fq(k)p Fr(f)11 b Fq(k)1614 3895 y Fp(v)r(;s)1741 3880 y Fr(;)216 b Fw(max)2063 3938 y Fp(i)2182 3880 y Fq(k)2248 3813 y Fr(@)5 b(f)p 2242 3857 129 4 v 2242 3948 a(@)g(I)2341 3963 y Fp(i)2380 3880 y Fq(k)2430 3895 y Fp(r)r Fj(\000)p Fp(\032;r)2607 3903 y Ff(p)2642 3895 y Fp(;r)2694 3903 y Ff(q)2728 3895 y Fp(;s)2812 3880 y Fq(\024)2928 3813 y Fw(1)p 2927 3857 51 4 v 2927 3948 a Fr(\032)2988 3880 y Fq(k)p Fr(f)11 b Fq(k)3147 3895 y Fp(r)n(;r)3229 3903 y Ff(p)3263 3895 y Fp(;r)3315 3903 y Ff(q)3349 3895 y Fp(;s)462 4169 y Fw(max)542 4229 y Fp(l)660 4169 y Fq(k)728 4102 y Fr(@)5 b(f)p 720 4146 132 4 v 720 4237 a(@)g(p)825 4252 y Fp(l)862 4169 y Fq(k)912 4184 y Fp(r)n(;r)994 4192 y Ff(p)1029 4184 y Fj(\000)p Fp(\032)1120 4192 y Ff(p)1156 4184 y Fp(;r)1208 4192 y Ff(q)1242 4184 y Fp(;s)1326 4169 y Fq(\024)1462 4102 y Fw(1)p 1441 4146 90 4 v 1441 4237 a Fr(\032)1491 4252 y Fp(p)1541 4169 y Fq(k)p Fr(f)11 b Fq(k)1700 4184 y Fp(v)r(;s)1827 4169 y Fr(;)216 b Fw(max)2150 4229 y Fp(l)2269 4169 y Fq(k)2334 4102 y Fr(@)5 b(f)p 2329 4146 127 4 v 2329 4237 a(@)g(q)2428 4252 y Fp(l)2464 4169 y Fq(k)2514 4184 y Fp(r)n(;r)2596 4192 y Ff(p)2632 4184 y Fp(;r)2684 4192 y Ff(q)2718 4184 y Fj(\000)p Fp(\032)2809 4192 y Ff(q)2843 4184 y Fp(;s)2927 4169 y Fq(\024)3062 4102 y Fw(1)p 3042 4146 89 4 v 3042 4237 a Fr(\032)3092 4252 y Fp(q)3140 4169 y Fq(k)p Fr(f)11 b Fq(k)3299 4184 y Fp(v)r(;s)3427 4169 y Fr(:)92 4415 y Fw(This)33 b(Lemma)e(is)h(a)g (precise)i(v)m(ersion)f(of)f(classical)f(Cauc)m(h)m(y)k(estimates.)d(W) -8 b(e)33 b(omit)e(the)i(w)m(ell)f(kno)m(wn)92 4536 y(pro)s(of;)42 b(just)i(notice)e(that)h(the)h(estimates)f(relativ)m(e)f(to)h Fr(')p Fw(-deriv)-5 b(ativ)m(es)42 b(is)h(a)g(consequence)j(of)d(the)92 4656 y(c)m(hoice)33 b(of)f(the)h(\(F)-8 b(ourier\))31 b(norm.)p 1474 4656 41 83 v 92 4929 a(An)c(immediate)d(corollary)h(of)i (Lemma)e(A.1)i(is)f(the)i(follo)m(wing)c(Lemma)h(on)i(estimates)g(for)f (P)m(oisson's)92 5050 y(brac)m(k)m(ets)440 5014 y Fs(19)517 5050 y Fw(.)p 92 5109 1493 4 v 171 5170 a Fo(18)297 5200 y FC(If)i Fm(a)22 b FC(:=)h(\()p Fm(a)633 5212 y Fo(1)671 5200 y Fm(;)14 b(:)g(:)g(:)f(;)h(a)899 5212 y Fl(j)934 5200 y FC(\),)28 b Fm(b)23 b FC(:=)f(\()p Fm(b)1254 5212 y Fo(1)1292 5200 y Fm(;)14 b(:)g(:)g(:)f(;)h(b)1512 5212 y Fl(j)1547 5200 y FC(\))28 b Fn(2)23 b Fh(R)1751 5170 y Fl(j)1820 5200 y FC(then)28 b Fm(a)23 b(<)f(b)28 b FC(means)f Fm(a)2525 5212 y Fl(i)2575 5200 y Fm(<)c(b)2699 5212 y Fl(i)2754 5200 y FC(for)k(all)g(1)c Fn(\024)f Fm(i)h Fn(\024)g Fm(j:)171 5270 y Fo(19)241 5300 y Fn(f)p Fm(f)t(;)14 b(g)s Fn(g)22 b FC(:=)582 5238 y Fi(P)670 5325 y Fl(j)719 5300 y Fm(@)763 5312 y Fl(')807 5320 y Fb(j)842 5300 y Fm(f)36 b(@)963 5312 y Fl(I)992 5320 y Fb(j)1028 5300 y Fm(g)21 b Fn(\000)d Fm(@)1216 5312 y Fl(I)1245 5320 y Fb(j)1280 5300 y Fm(f)37 b(@)1402 5312 y Fl(')1446 5320 y Fb(j)1481 5300 y Fm(g)21 b FC(+)1625 5238 y Fi(P)1712 5325 y Fl(i)1754 5300 y Fm(@)1798 5312 y Fl(q)1828 5320 y Fb(i)1859 5300 y Fm(f)36 b(@)1980 5312 y Fl(p)2014 5320 y Fb(i)2045 5300 y Fm(g)21 b Fn(\000)d Fm(@)2233 5312 y Fl(p)2267 5320 y Fb(i)2297 5300 y Fm(f)37 b(@)2419 5312 y Fl(q)2449 5320 y Fb(i)2480 5300 y Fm(g)s FC(.)1909 5775 y Fw(21)p eop %%Page: 22 22 22 21 bop 92 206 a Fu(Lemma)37 b(A.2)48 b Fg(L)-5 b(et)36 b Fw(0)27 b Fr(<)h(v)e Fq(\000)c Fr(\027)34 b(<)e Fw(~)-53 b Fr(v)t(;)35 b Fg(wher)-5 b(e)37 b Fw(~)-52 b Fr(v)31 b Fw(:=)d(\()p Fr(r)2095 221 y Fs(0)2134 206 y Fr(;)21 b Fw(~)-53 b Fr(r)2222 221 y Fp(p)2261 206 y Fr(;)22 b Fw(~)-54 b Fr(r)2349 221 y Fp(q)2387 206 y Fw(\))p Fr(:)35 b Fg(Then,)478 454 y Fq(kf)p Fr(f)5 b(;)17 b(g)t Fq(gk)826 469 y Fp(v)r Fj(\000)p Fp(\027;s)p Fj(\000)p Fp(\033)1189 454 y Fq(\024)1349 308 y Fi(")1684 387 y Fw(1)p 1407 431 603 4 v 1407 523 a Fr(e)p Fw(\()p Fr(r)1534 538 y Fs(0)1596 523 y Fq(\000)23 b Fr(r)i Fw(+)d Fr(\032)p Fw(\))p Fr(\033)2042 454 y Fw(+)2408 387 y(1)p 2150 431 565 4 v 2150 523 a Fr(e)p Fw(\()t(~)-53 b Fr(s)22 b Fq(\000)h Fr(s)f Fw(+)g Fr(\033)t Fw(\))p Fr(\032)2746 454 y Fw(+)1189 737 y(+)1648 669 y Fr(m)p 1359 713 664 4 v 1359 805 a Fw(\()t(~)-53 b Fr(r)1441 820 y Fp(p)1503 805 y Fq(\000)22 b Fr(r)1646 820 y Fp(p)1708 805 y Fw(+)g Fr(\032)1856 820 y Fp(p)1896 805 y Fw(\))p Fr(\032)1984 820 y Fp(q)2054 737 y Fw(+)2450 669 y Fr(m)p 2162 713 661 4 v 2162 805 a Fw(\()t(~)-53 b Fr(r)2244 820 y Fp(q)2305 805 y Fq(\000)22 b Fr(r)2448 820 y Fp(q)2508 805 y Fw(+)g Fr(\032)2656 820 y Fp(q)2695 805 y Fw(\))p Fr(\032)2783 820 y Fp(p)2833 590 y Fi(#)2881 737 y Fq(k)p Fr(f)11 b Fq(k)3043 752 y Fs(~)-38 b Fp(v)r(;)s Fs(~)g Fp(s)3167 737 y Fq(k)p Fr(g)t Fq(k)3318 752 y Fp(v)r(;s)3410 737 y Fr(:)92 1055 y Fw(If)29 b Fr(\036)f Fw(is)h(a)g(Hamiltonian)c(function,)k(w)m(e)h (denote)g(b)m(y)g Fr(X)2080 1019 y Fp(t)2072 1080 y(\036)2147 1055 y Fw(the)f(Hamiltonian)d(\015o)m(w)j(of)g Fr(\036)g Fw(at)f(the)i(time)e Fr(t)p Fw(.)92 1176 y(If)i Fr(f)40 b Fw(and)31 b Fr(\036)e Fw(are)h(analytic,)f(expanding)i(in)e(T)-8 b(a)m(ylor)30 b(series)g(in)g(time)e(the)j(function)e Fr(f)f Fq(\016)17 b Fr(X)3354 1139 y Fp(t)3346 1200 y(\036)3392 1176 y Fw(,)30 b(one)g(gets)1522 1432 y Fr(f)j Fq(\016)22 b Fr(X)1764 1391 y Fp(t)1756 1456 y(\036)1830 1432 y Fw(=)1933 1349 y Fi(X)1977 1531 y Fp(j)2080 1364 y Fr(t)2115 1328 y Fp(j)p 2080 1408 74 4 v 2080 1500 a Fr(j)6 b Fw(!)2163 1432 y Fr(L)2229 1385 y Fp(j)2229 1457 y(\036)2275 1432 y Fr(f)43 b(;)92 1702 y Fw(where)34 b Fr(L)440 1717 y Fp(\036)486 1702 y Fr(f)k Fw(:=)28 b Fq(f)p Fr(f)5 b(;)17 b(\036)p Fq(g)p Fw(,)32 b Fr(L)1083 1666 y Fs(0)1083 1727 y Fp(\036)1157 1702 y Fw(:=)c Fr(I)8 b(d)32 b Fw(and)1521 2014 y Fr(L)1587 1967 y Fp(j)1587 2039 y(\036)1661 2014 y Fw(:=)1940 1864 y Fp(j)k Fs(times)1791 1921 y Fi(z)p 1828 1921 190 10 v 190 w(}|)p 2092 1921 V 190 w({)1791 2014 y Fr(L)1857 2029 y Fp(\036)1926 2014 y Fq(\016)22 b Fr(:)17 b(:)g(:)22 b Fq(\016)f Fr(L)2272 2029 y Fp(\036)2368 2014 y Fr(:)92 2200 y Fu(Lemma)37 b(A.3)48 b Fg(L)-5 b(et)36 b Fw(0)27 b Fr(<)h(\027)34 b(<)27 b(v)32 b Fq(\024)g Fw(~)-53 b Fr(v)26 b Fq(\000)c Fr(\027)6 b Fg(,)36 b Fw(0)27 b Fr(<)h(\033)j(<)d(s)f Fq(\024)33 b Fw(~)-53 b Fr(s)22 b Fq(\000)g Fr(\033)39 b Fg(and)1267 2422 y Fq(k)p Fr(\036)p Fq(k)1428 2437 y Fs(~)-38 b Fp(v)r(;)s Fs(~)g Fp(s)1545 2422 y Fr(<)28 b(G)f Fw(:=)1894 2355 y(2)p 1894 2399 49 4 v 1896 2491 a Fr(e)1953 2422 y Fw(\()2126 2355 y Fr(\032\032)2226 2370 y Fp(p)2266 2355 y Fr(\032)2316 2370 y Fp(q)2355 2355 y Fr(\033)p 2001 2399 539 4 v 2001 2491 a(\032)2051 2506 y Fp(p)2091 2491 y Fr(\032)2141 2506 y Fp(q)2201 2491 y Fw(+)22 b Fr(em\032\033)2549 2422 y Fw(\))35 b Fr(:)950 b Fw(\(A.8\))92 2659 y Fg(Then)1071 2833 y Fq(k)p Fr(f)32 b Fq(\016)22 b Fr(X)1362 2792 y Fs(1)1354 2858 y Fp(\036)1402 2833 y Fq(k)1452 2848 y Fp(v)r Fj(\000)p Fp(\027;s)p Fj(\000)p Fp(\033)1760 2833 y Fq(\024)1865 2687 y Fi( )1931 2833 y Fw(1)g Fq(\000)2111 2766 y(k)p Fr(\036)p Fq(k)2272 2781 y Fs(~)-38 b Fp(v)r(;)s Fs(~)g Fp(s)p 2111 2810 251 4 v 2198 2901 a Fr(G)2372 2687 y Fi(!)2437 2710 y Fj(\000)p Fs(1)2532 2833 y Fq(k)p Fr(f)11 b Fq(k)2691 2848 y Fp(v)r(;s)2818 2833 y Fr(:)92 3082 y Fu(Pro)s(of)45 b Fw(Fix)f Fr(h)49 b Fq(\025)h Fw(1)45 b(and)g(let)50 b(\026)-54 b Fr(\027)55 b Fw(:=)49 b Fr(\027)6 b(=h)p Fw(,)51 b(\026)-54 b Fr(\033)53 b Fw(:=)c Fr(\033)t(=h)c Fq(k)31 b(\001)f(k)2423 3097 y Fp(i)2500 3082 y Fw(:=)49 b Fq(k)31 b(\001)f(k)2841 3097 y Fp(v)r Fj(\000)p Fp(i)t Fs(\026)-39 b Fp(\027)5 b(;s)p Fj(\000)p Fp(i)t Fs(\026)-39 b Fp(\033)3174 3082 y Fw(,)45 b(for)g(all)e Fr(i)i Fw(with)92 3202 y(1)27 b Fq(\024)h Fr(i)g Fq(\024)g Fr(h)p Fw(.)33 b(W)-8 b(e)32 b(will)e(use)j(Lemma)e (A.2)h(with)g Fr(\027)i Fw(=)f(\026)-54 b Fr(\027)38 b Fw(and)33 b Fr(\033)e Fw(=)i(\026)-54 b Fr(\033)t Fw(.)32 b(Then,)i Fr(v)25 b Fq(\000)c Fr(i)6 b Fw(\026)-55 b Fr(\027)35 b Fw(=)27 b([)p Fr(v)f Fq(\000)21 b Fw(\()p Fr(i)h Fq(\000)g Fw(1\))6 b(\026)-55 b Fr(\027)6 b Fw(])92 3322 y(and)32 b Fr(s)22 b Fq(\000)h Fr(i)5 b Fw(\026)-54 b Fr(\033)32 b Fw(=)c([)p Fr(s)22 b Fq(\000)g Fw(\()p Fr(i)h Fq(\000)f Fw(1\))5 b(\026)-54 b Fr(\033)t Fw(])p Fr(:)33 b Fw(Hence,)202 3568 y Fq(k)p Fr(L)318 3527 y Fp(i)318 3593 y(\036)364 3568 y Fr(f)11 b Fq(k)473 3583 y Fp(i)528 3568 y Fw(=)28 b Fq(kf)p Fr(L)798 3527 y Fp(i)p Fj(\000)p Fs(1)798 3593 y Fp(\036)916 3568 y Fr(;)17 b(\036)p Fq(gk)1118 3583 y Fp(i)1228 3568 y Fq(\024)1389 3422 y Fi(")1741 3501 y Fw(1)p 1447 3545 636 4 v 1447 3636 a Fr(e)p Fw(\()p Fr(r)1574 3651 y Fs(0)1636 3636 y Fq(\000)23 b Fr(r)h Fw(+)e Fr(i)9 b Fw(\026)-58 b Fr(\032)q Fw(\))5 b(\026)-54 b Fr(\033)2115 3568 y Fw(+)2497 3501 y(1)p 2223 3545 598 4 v 2223 3636 a Fr(e)p Fw(\()t(~)h Fr(s)22 b Fq(\000)h Fr(s)f Fw(+)g Fr(i)5 b Fw(\026)-54 b Fr(\033)t Fw(\))9 b(\026)-58 b Fr(\032)1486 3850 y Fw(+)1845 3783 y Fr(m)p 1572 3827 631 4 v 1572 3919 a Fw(\()t(~)-53 b Fr(r)1654 3934 y Fp(p)1716 3919 y Fq(\000)23 b Fr(r)1860 3934 y Fp(p)1921 3919 y Fw(+)f Fr(i)t Fw(\026)-53 b Fr(\015)6 b Fw(\))t(\026)-53 b Fr(\024)2235 3850 y Fw(+)2614 3783 y Fr(m)p 2343 3827 628 4 v 2343 3919 a Fw(\()t(~)g Fr(r)2425 3934 y Fp(q)2485 3919 y Fq(\000)23 b Fr(r)2629 3934 y Fp(q)2689 3919 y Fw(+)f Fr(i)t Fw(\026)-53 b Fr(\024)p Fw(\))t(\026)g Fr(\015)2980 3704 y Fi(#)3028 3850 y Fq(k)p Fr(\036)p Fq(k)3189 3865 y Fs(~)-38 b Fp(v)s(;)s Fs(~)g Fp(s)3311 3850 y Fq(k)p Fr(L)3427 3809 y Fp(i)p Fj(\000)p Fs(1)3427 3875 y Fp(\036)3546 3850 y Fq(k)3596 3865 y Fp(i)p Fj(\000)p Fs(1)1228 4133 y Fq(\024)1389 3986 y Fi(")1500 4065 y Fw(2)p 1447 4109 155 4 v 1447 4201 a Fr(e)9 b Fw(\026)-58 b Fr(\032)6 b Fw(\026)-54 b Fr(\033)1634 4133 y Fw(+)1742 4065 y(2)p Fr(m)p 1742 4109 135 4 v 1757 4201 a Fw(\026)h Fr(\015)9 b Fw(\026)-53 b Fr(\024)1886 3986 y Fi(#)2025 4065 y Fw(1)p 1944 4109 210 4 v 1944 4201 a Fr(h)23 b Fw(+)f Fr(i)2164 4133 y Fq(k)p Fr(\036)p Fq(k)2325 4148 y Fs(~)-38 b Fp(v)r(;)s Fs(~)g Fp(s)2414 4133 y Fq(k)p Fr(L)2530 4091 y Fp(i)p Fj(\000)p Fs(1)2530 4158 y Fp(\036)2649 4133 y Fq(k)2699 4148 y Fp(i)p Fj(\000)p Fs(1)2850 4133 y Fr(:)92 4373 y Fw(Iterating)31 b Fr(h)i Fw(times)f(the)h(previous)g(estimate)f(w)m (e)h(obtain)635 4632 y Fq(k)p Fr(L)751 4591 y Fp(h)751 4657 y(\036)797 4632 y Fq(k)847 4647 y Fp(h)976 4632 y Fw(=)84 b Fq(k)p Fr(L)1252 4591 y Fp(h)1252 4657 y(\036)1298 4632 y Fq(k)1348 4647 y Fp(v)r Fj(\000)p Fp(\027;s)p Fj(\000)p Fp(\033)1656 4632 y Fq(\024)1761 4486 y Fi(")1872 4565 y Fw(2)p 1820 4609 155 4 v 1820 4701 a Fr(e)9 b Fw(\026)-58 b Fr(\032)5 b Fw(\026)-54 b Fr(\033)2006 4632 y Fw(+)2114 4565 y(2)p Fr(m)p 2114 4609 135 4 v 2129 4701 a Fw(\026)h Fr(\015)9 b Fw(\026)-53 b Fr(\024)2258 4486 y Fi(#)2307 4509 y Fp(h)2424 4565 y Fr(h)p Fw(!)p 2362 4609 208 4 v 2362 4701 a(\(2)p Fr(h)p Fw(\)!)2579 4632 y Fq(k)p Fr(\036)p Fq(k)2737 4591 y Fp(h)2740 4657 y Fs(~)-38 b Fp(v)r(;)s Fs(~)g Fp(s)2830 4632 y Fq(k)p Fr(f)11 b Fq(k)2989 4647 y Fp(v)r(;s)976 4934 y Fw(=)1136 4788 y Fi(")1247 4867 y Fw(2)p 1194 4911 155 4 v 1194 5003 a Fr(e\032\033)1381 4934 y Fw(+)1511 4867 y(2)p Fr(m)p 1489 4911 179 4 v 1489 5003 a(\032)1539 5018 y Fp(p)1579 5003 y Fr(\032)1629 5018 y Fp(q)1677 4788 y Fi(#)1725 4811 y Fp(h)1780 4867 y Fr(h)1836 4831 y Fs(2)p Fp(h)1916 4867 y Fr(h)p Fw(!)p 1780 4911 220 4 v 1786 5003 a(\(2)p Fr(h)p Fw(\)!)2010 4934 y Fq(k)p Fr(\036)p Fq(k)2168 4893 y Fp(h)2171 4959 y Fs(~)-38 b Fp(v)r(;)s Fs(~)g Fp(s)2260 4934 y Fq(k)p Fr(f)11 b Fq(k)2419 4949 y Fp(v)r(;s)975 5237 y Fq(\024)1136 5090 y Fi(")1250 5169 y Fr(e)p 1194 5213 159 4 v 1194 5305 a Fw(2)p Fr(\032\033)1384 5237 y Fw(+)1521 5169 y Fr(e)1566 5133 y Fs(2)1605 5169 y Fr(m)p 1492 5213 228 4 v 1492 5305 a Fw(2)p Fr(\032)1591 5320 y Fp(p)1631 5305 y Fr(\032)1681 5320 y Fp(q)1729 5090 y Fi(#)1778 5113 y Fp(h)1822 5237 y Fr(h)p Fw(!)p Fq(k)p Fr(\036)p Fq(k)2063 5195 y Fp(h)2066 5261 y Fs(~)-38 b Fp(v)s(;)s Fs(~)g Fp(s)2156 5237 y Fq(k)p Fr(f)11 b Fq(k)2315 5252 y Fp(v)r(;s)2435 5237 y Fw(=)2563 5169 y(1)p 2549 5213 77 4 v 2549 5305 a Fr(G)2636 5237 y(h)p Fw(!)p Fq(k)p Fr(\036)p Fq(k)2877 5195 y Fp(h)2880 5261 y Fs(~)-38 b Fp(v)r(;)s Fs(~)g Fp(s)2970 5237 y Fq(k)p Fr(f)11 b Fq(k)3129 5252 y Fp(v)r(;s)3254 5237 y Fr(:)1909 5775 y Fw(22)p eop %%Page: 23 23 23 22 bop 92 206 a Fw(Finally)-8 b(,)30 b(summing)g(o)m(v)m(er)k Fr(h)p Fw(,)f(w)m(e)g(get)490 449 y Fq(k)p Fr(f)f Fq(\016)22 b Fr(X)781 408 y Fs(1)773 474 y Fp(\036)820 449 y Fq(k)870 464 y Fp(v)r Fj(\000)p Fp(\027;s)p Fj(\000)p Fp(\033)1235 449 y Fw(=)84 b Fq(k)1462 366 y Fi(X)1501 550 y Fp(h)1625 382 y Fw(1)p 1607 426 84 4 v 1607 517 a Fr(h)p Fw(!)1701 449 y Fr(L)1767 408 y Fp(h)1767 474 y(\036)1813 449 y Fr(f)11 b Fq(k)1922 464 y Fp(v)r Fj(\000)p Fp(\027;s)p Fj(\000)p Fp(\033)2230 449 y Fq(\024)2335 366 y Fi(X)2375 550 y Fp(h)2499 382 y Fw(1)p 2482 426 V 2482 517 a Fr(h)p Fw(!)2575 449 y Fq(k)p Fr(L)2691 408 y Fp(h)2691 474 y(\036)2737 449 y Fr(f)g Fq(k)2846 464 y Fp(v)r Fj(\000)p Fp(\027;s)p Fj(\000)p Fp(\033)1234 758 y Fq(\024)84 b(k)p Fr(f)11 b Fq(k)1554 773 y Fp(v)r(;s)1663 675 y Fi(X)1702 859 y Fp(h)1846 690 y Fw(1)p 1809 735 122 4 v 1809 826 a Fr(G)1886 797 y Fp(h)1941 758 y Fq(k)p Fr(\036)p Fq(k)2099 717 y Fp(h)2102 783 y Fs(~)-38 b Fp(v)r(;)s Fs(~)g Fp(s)2219 758 y Fw(=)2323 612 y Fi( )2388 758 y Fw(1)22 b Fq(\000)2569 690 y(k)p Fr(\036)p Fq(k)2730 705 y Fs(~)-38 b Fp(v)r(;)s Fs(~)g Fp(s)p 2569 735 251 4 v 2656 826 a Fr(G)2829 612 y Fi(!)2895 635 y Fj(\000)p Fs(1)2989 758 y Fq(k)p Fr(f)11 b Fq(k)3148 773 y Fp(v)r(;s)3241 758 y Fr(:)p 3386 758 41 83 v 92 1035 a Fw(Next)36 b(lemma)d(is)i(an)g(immediate)d (consequence)39 b(of)34 b(Hamilton)f(equations)i(\(and)h(the)f(trivial) e(pro)s(of)92 1155 y(is)f(omitted\).)92 1365 y Fu(Lemma)37 b(A.4)48 b Fg(L)-5 b(et)36 b Fw(0)27 b Fr(<)h(v)e Fq(\000)c Fr(\027)34 b(<)28 b(v)t Fg(,)34 b Fw(0)28 b Fr(<)f(s)22 b Fq(\000)h Fr(\033)32 b(<)27 b(s)35 b Fg(and)f(supp)-5 b(ose)35 b(that)886 1622 y Fw(max)871 1680 y Fs(1)p Fj(\024)p Fp(i)p Fj(\024)p Fp(n)1099 1622 y Fq(k)1166 1554 y Fr(@)5 b(\036)p 1159 1598 129 4 v 1159 1690 a(@)g(I)1258 1705 y Fp(i)1297 1622 y Fq(k)1347 1637 y Fp(v)r(;s)1468 1622 y Fq(\024)33 b Fw(\026)-54 b Fr(\033)32 b Fq(\024)c Fr(\033)39 b(;)2140 1514 y Fp(n)2102 1539 y Fi(X)2105 1721 y Fp(i)p Fs(=1)2238 1622 y Fq(k)2315 1554 y Fr(@)5 b(\036)p 2298 1598 149 4 v 2298 1690 a(@)g(')2418 1705 y Fp(i)2457 1622 y Fq(k)2507 1637 y Fp(v)r(;s)2628 1622 y Fq(\024)37 b Fw(\026)-58 b Fr(\032)28 b Fq(\024)g Fr(\032)35 b(;)780 1899 y Fw(max)756 1959 y Fs(1)p Fj(\024)p Fp(l)q Fj(\024)p Fp(m)1002 1899 y Fq(k)1071 1832 y Fr(@)5 b(\036)p 1062 1876 132 4 v 1062 1967 a(@)g(p)1167 1982 y Fp(l)1204 1899 y Fq(k)1254 1914 y Fp(v)r(;s)1374 1899 y Fq(\024)48 b Fw(\026)-68 b Fr(\032)1530 1914 y Fp(q)1596 1899 y Fq(\024)28 b Fr(\032)1751 1914 y Fp(q)1824 1899 y Fr(;)240 b Fw(max)2067 1959 y Fs(1)p Fj(\024)p Fp(l)q Fj(\024)p Fp(m)2313 1899 y Fq(k)2379 1832 y Fr(@)5 b(\036)p 2373 1876 127 4 v 2373 1967 a(@)g(q)2472 1982 y Fp(l)2509 1899 y Fq(k)2559 1914 y Fp(v)r(;s)2679 1899 y Fq(\024)49 b Fw(\026)-69 b Fr(\032)2835 1914 y Fp(p)2902 1899 y Fq(\024)28 b Fr(\032)3057 1914 y Fp(p)3132 1899 y Fr(:)92 2156 y Fg(L)-5 b(et)41 b Fw(\()p Fr(u)360 2171 y Fs(0)399 2156 y Fr(;)17 b(')507 2171 y Fs(0)546 2156 y Fw(\))38 b(:=)h(\()p Fr(u)p Fw(\(0\))p Fr(;)17 b(')p Fw(\(0\)\))37 b Fq(2)i Fr(W)1488 2171 y Fp(v)r Fj(\000)p Fp(\027;s)p Fj(\000)p Fp(\033)1769 2156 y Fg(.)h(Then)g Fr(X)2188 2120 y Fp(t)2180 2181 y(\036)2227 2156 y Fw(\()p Fr(u)p Fw(\(0\))p Fr(;)17 b(')p Fw(\(0\)\))39 b(=)f(\()p Fr(u)p Fw(\()p Fr(t)p Fw(\))p Fr(;)17 b(')p Fw(\()p Fr(t)p Fw(\)\))40 b Fq(2)f Fr(W)3569 2171 y Fp(v)r(;s)3703 2156 y Fg(for)92 2276 y(al)5 b(l)34 b Fr(t)h Fg(with)g Fw(0)27 b Fq(\024)i Fr(t)f Fq(\024)g Fw(1)p Fg(.)34 b(Mor)-5 b(e)35 b(pr)-5 b(e)g(cisely,)608 2533 y Fq(j)p Fr(I)8 b Fw(\()p Fr(t)p Fw(\))22 b Fq(\000)g Fr(I)8 b Fw(\(0\))p Fq(j)27 b(\024)1294 2425 y Fp(n)1255 2450 y Fi(X)1258 2632 y Fp(i)p Fs(=1)1392 2533 y Fq(j)p Fr(I)1463 2548 y Fp(i)1491 2533 y Fw(\()p Fr(t)p Fw(\))22 b Fq(\000)h Fr(I)1767 2548 y Fp(i)1795 2533 y Fw(\(0\))p Fq(j)k(\024)37 b Fw(\026)-58 b Fr(\032)35 b(;)216 b Fw(max)2487 2590 y Fp(i)2606 2533 y Fq(j)p Fr(')2698 2548 y Fp(i)2726 2533 y Fw(\()p Fr(t)p Fw(\))22 b Fq(\000)h Fr(')3023 2548 y Fp(i)3051 2533 y Fw(\(0\))p Fq(j)k(\024)33 b Fw(\026)-54 b Fr(\033)39 b(;)632 2748 y Fw(max)608 2808 y Fs(1)p Fj(\024)p Fp(l)q Fj(\024)p Fp(m)854 2748 y Fq(j)p Fr(p)931 2763 y Fp(l)957 2748 y Fw(\()p Fr(t)p Fw(\))22 b Fq(\000)g Fr(p)1238 2763 y Fp(l)1264 2748 y Fw(\(0\))p Fq(j)27 b(\024)49 b Fw(\026)-69 b Fr(\032)1600 2763 y Fp(p)1674 2748 y Fr(;)709 b Fw(max)2386 2808 y Fs(1)p Fj(\024)p Fp(l)q Fj(\024)p Fp(m)2632 2748 y Fq(j)p Fr(q)2703 2763 y Fp(l)2728 2748 y Fw(\()p Fr(t)p Fw(\))23 b Fq(\000)f Fr(q)3004 2763 y Fp(l)3031 2748 y Fw(\(0\))p Fq(j)27 b(\024)48 b Fw(\026)-69 b Fr(\032)3366 2763 y Fp(q)3439 2748 y Fr(:)92 3001 y Fu(Lemma)37 b(A.5)48 b Fg(L)-5 b(et)45 b Fr(H)8 b Fw(\()p Fr(u;)17 b(')p Fw(\))44 b(:=)h Fr(h)p Fw(\()p Fr(I)8 b Fw(\))29 b(+)g Fr(g)t Fw(\()p Fr(u;)17 b(')p Fw(\))27 b(+)i Fr(f)11 b Fw(\()p Fr(u;)17 b(')p Fw(\))43 b Fg(r)-5 b(e)g(al-analytic)43 b(on)h Fr(W)3294 3016 y Fp(v)r(;s)3432 3001 y Fg(with)g Fr(g)k Fw(=)92 3055 y Fi(P)179 3142 y Fp(k)r Fj(2)p Fs(\003)335 3121 y Fr(g)382 3136 y Fp(k)424 3121 y Fw(\()p Fr(u)p Fw(\))p Fr(e)601 3085 y Fp(ik)r Fj(\001)p Fp(')733 3121 y Fr(:)35 b Fg(L)-5 b(et)35 b Fr(\027)f(<)28 b(v)t(=)p Fw(2)34 b Fg(and)g Fr(\033)e(<)27 b(s=)p Fw(2)p Fg(.)35 b(Supp)-5 b(ose)34 b(that)645 3325 y Fq(j)p Fr(!)t Fw(\()p Fr(I)8 b Fw(\))21 b Fq(\001)h Fr(k)s Fq(j)27 b(\025)i Fr(\013)f(>)g Fw(0)34 b Fr(;)415 b Fq(8)17 b(j)p Fr(k)s Fq(j)27 b(\024)i Fr(K)42 b(;)116 b(k)42 b(=)-60 b Fq(2)28 b Fw(\003)34 b Fr(;)87 b Fq(8)17 b Fr(I)35 b Fq(2)28 b Fr(D)3170 3340 y Fp(r)3243 3325 y Fr(;)329 b Fw(\(A.9\))92 3530 y Fg(and)881 3650 y Fq(k)p Fr(f)11 b Fq(k)1040 3665 y Fp(v)r(;s)1161 3650 y Fr(<)27 b(\013)q(\016)t(=c)1465 3665 y Fp(m)1730 3650 y Fw(where)201 b Fr(\016)32 b Fw(:=)27 b(min)o Fq(f)p Fr(\032\033)n(;)17 b(\032)2794 3665 y Fp(p)2834 3650 y Fr(\032)2884 3665 y Fp(q)2923 3650 y Fq(g)34 b Fr(:)516 b Fw(\(A.10\))92 3818 y Fg(Then,)34 b(ther)-5 b(e)34 b(exists)h(a)g(r)-5 b(e)g(al-analytic)34 b(symple)-5 b(ctic)34 b(tr)-5 b(ansformation)1098 4022 y Fw(\010)29 b(:)e(\()6 b(~)-55 b Fr(u;)32 b Fw(~)-64 b Fr(')o Fw(\))28 b Fq(2)g Fr(W)1704 4037 y Fp(v)r Fj(\000)p Fs(2)p Fp(\027;s)p Fj(\000)p Fs(2)p Fp(\033)2083 4022 y Fq(\000)-16 b(!)27 b Fw(\()p Fr(u;)17 b(')p Fw(\))27 b Fq(2)h Fr(W)2724 4037 y Fp(v)r(;s)92 4226 y Fg(such)34 b(that)987 4347 y Fr(H)29 b Fq(\016)22 b Fw(\010)28 b(=)g Fr(h)22 b Fw(+)g Fr(g)1594 4362 y Fs(+)1675 4347 y Fw(+)g Fr(f)1821 4362 y Fs(+)1915 4347 y Fr(;)216 b(g)2205 4362 y Fs(+)2286 4347 y Fq(\000)22 b Fr(g)31 b Fw(=)d Fr(P)2630 4362 y Fs(\003)2683 4347 y Fr(T)2740 4362 y 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Fw(,)33 b(w)m(e)g(ha)m(v)m(e)961 2756 y Fr(f)1009 2771 y Fs(+)1095 2756 y Fw(=)1199 2639 y Fi(Z)1282 2665 y Fs(1)1245 2827 y(0)1321 2756 y Fq(f)p Fw(\()p Fr(g)26 b Fw(+)c Fr(f)1628 2771 y Fp(t)1657 2756 y Fw(\))p Fr(;)17 b(\036)p Fq(g)22 b(\016)g Fr(X)2030 2715 y Fp(t)2022 2780 y(\036)2084 2756 y Fr(dt)g Fw(+)g(\()p Fr(f)33 b Fq(\000)23 b Fr(T)2566 2771 y Fp(K)2634 2756 y Fr(f)11 b Fw(\))22 b Fq(\016)g Fw(\010)33 b Fr(:)92 3010 y Fw(W)-8 b(e)49 b(can)g(estimate)f Fr(G)h Fw(in)f(\(A.8\))h(with)f Fr(G)55 b Fq(\025)h Fr(\016)t(=c)2049 3025 y Fp(m)2116 3010 y Fw(.)49 b(Then,)h Fq(8)17 b Fr(F)62 b Fw(and)49 b Fq(8)17 b Fr(t)49 b Fw(with)g(0)55 b Fq(\024)h Fr(t)f Fq(\024)h Fw(1)p Fr(;)92 3130 y Fw(substituting)35 b(~)-52 b Fr(v)31 b Fq(!)c Fr(v)t(;)33 b(v)e Fq(!)d Fr(v)d Fq(\000)e Fr(\027)q(;)33 b(\027)h Fq(!)27 b Fr(\027)q(;)33 b Fw(w)m(e)g(ha)m(v)m(e)h(\(b)m(y)g(Lemma)d(A.3\))h(that)1220 3350 y Fq(k)p Fr(F)k Fq(\016)22 b Fr(X)1530 3309 y Fp(t)1522 3375 y(\036)1568 3350 y Fq(k)1618 3365 y Fp(v)r Fj(\000)p Fs(2)p Fp(\027;s)p Fj(\000)p Fs(2)p Fp(\033)1997 3350 y Fq(\024)28 b Fr(C)7 b Fq(k)p Fr(F)14 b Fq(k)2356 3365 y Fp(v)r Fj(\000)p Fp(\027;s)p Fj(\000)p Fp(\033)2668 3350 y Fr(;)92 3570 y Fw(with)37 b Fr(C)43 b Fw(:=)35 b(\(1)25 b Fq(\000)h Fr(c)827 3585 y Fp(m)894 3570 y Fq(k)p Fr(\036)p Fq(k)1052 3585 y Fp(v)r(;s)1144 3570 y Fr(=\016)t Fw(\))1278 3534 y Fj(\000)p Fs(1)1372 3570 y Fw(.)38 b(Then,)g(c)m(ho)s(osing)f Fr(F)50 b Fw(:=)36 b Fq(f)p Fr(f)2474 3585 y Fp(t)2503 3570 y Fr(;)17 b(\036)p Fq(g)p Fr(;)37 b Fw(and)h(using)f(Lemma)f(A.2)h(w)m(e)92 3691 y(ha)m(v)m(e)422 3932 y Fq(k)489 3815 y Fi(Z)572 3841 y Fs(1)535 4004 y(0)611 3932 y Fq(f)p Fr(f)709 3947 y Fp(t)739 3932 y Fr(;)17 b(\036)p Fq(g)k(\016)h Fr(X)1073 3891 y Fp(t)1065 3957 y(\036)1128 3932 y Fr(dt)p Fq(k)1264 3947 y Fp(v)r Fj(\000)p Fs(2)p Fp(\027;s)p Fj(\000)p Fs(2)p Fp(\033)1698 3932 y Fq(\024)1858 3815 y Fi(Z)1941 3841 y Fs(1)1904 4004 y(0)1997 3932 y Fq(kf)p Fr(f)2145 3947 y Fp(t)2175 3932 y Fr(;)17 b(\036)p Fq(g)k(\016)h Fr(X)2509 3891 y Fp(t)2501 3957 y(\036)2547 3932 y Fq(k)2597 3947 y Fp(v)r Fj(\000)p Fs(2)p Fp(\027;s)p Fj(\000)p Fs(2)p Fp(\033)2965 3932 y Fr(dt)1698 4171 y Fq(\024)83 b Fr(C)1952 4053 y Fi(Z)2035 4080 y Fs(1)1998 4242 y(0)2091 4171 y Fq(kf)p Fr(f)2239 4186 y Fp(t)2268 4171 y Fr(;)17 b(\036)p Fq(gk)2470 4186 y Fp(v)r Fj(\000)p Fp(\027;s)p Fj(\000)p Fp(\033)2767 4171 y Fr(dt)1698 4409 y Fq(\024)83 b Fr(C)1952 4292 y Fi(Z)2035 4318 y Fs(1)1998 4480 y(0)2074 4409 y Fw(\()2175 4342 y(2)p 2122 4386 155 4 v 2122 4477 a Fr(e\032\033)2309 4409 y Fw(+)2439 4342 y(2)p Fr(m)p 2417 4386 179 4 v 2417 4477 a(\032)2467 4492 y Fp(p)2507 4477 y Fr(\032)2557 4492 y Fp(q)2605 4409 y Fw(\))p Fq(k)p Fr(f)2741 4424 y Fp(t)2771 4409 y Fq(k)2821 4424 y Fp(v)r(;s)2914 4409 y Fq(k)p Fr(\036)p Fq(k)3072 4424 y Fp(v)r(;s)3181 4409 y Fr(dt)1698 4672 y Fq(\024)g Fr(C)1945 4605 y(c)1987 4620 y Fp(m)p 1945 4649 109 4 v 1976 4740 a Fr(\016)2080 4555 y Fi(Z)2163 4581 y Fs(1)2126 4744 y(0)2219 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Fj(\000)p Fp(\027;s)p Fj(\000)p Fp(\033)3550 5189 y Fw(\(A.16\))1909 5775 y(24)p eop %%Page: 25 25 25 24 bop 92 206 a Fw(and)412 420 y Fq(k)p Fw(\()p Fr(f)33 b Fq(\000)23 b Fr(T)738 435 y Fp(K)806 420 y Fr(f)11 b Fw(\))22 b Fq(\016)g Fr(X)1086 379 y Fs(1)1078 445 y Fp(\036)1125 420 y Fq(k)1175 435 y Fp(v)r Fj(\000)p Fs(2)p Fp(\027;s)p Fj(\000)p Fs(2)p Fp(\033)1554 420 y Fq(\024)28 b Fr(C)7 b Fq(k)p Fr(f)33 b Fq(\000)22 b Fr(T)2023 435 y Fp(K)2092 420 y Fr(f)11 b Fq(k)2201 435 y Fp(v)r Fj(\000)p Fp(\027;s)p Fj(\000)p Fp(\033)2509 420 y Fq(\024)28 b Fr(C)23 b(e)2752 379 y Fj(\000)p Fp(K)5 b(\033)2919 420 y Fq(k)p Fr(f)11 b Fq(k)3078 435 y Fp(v)r(;s)3203 420 y Fr(:)320 b Fw(\(A.17\))92 634 y(Collecting)30 b(\(A.15\),)j (\(A.16\),)f(\(A.17\))g(w)m(e)i(ha)m(v)m(e)g(\(A.12\).)p 2312 634 41 83 v 92 851 a(W)-8 b(e)33 b(are)f(no)m(w)h(ready)h(for)e (the)92 1042 y Fu(Pro)s(of)39 b(of)h(Prop)s(osition)e(A.1)c Fw(Let)h Fr(")1570 1057 y Fs(0)1640 1042 y Fw(:=)c Fr(";)k(\027)1930 1057 y Fs(0)2000 1042 y Fw(:=)c Fr(v)t(=)p Fw(8)p Fr(;)j(\033)2399 1057 y Fs(0)2470 1042 y Fw(:=)d Fr(s=)p Fw(6)p Fr(;)k(\016)2853 1057 y Fs(0)2923 1042 y Fw(:=)c(min)o Fq(f)p Fr(\032)3320 1057 y Fs(0)3360 1042 y Fr(\033)3415 1057 y Fs(0)3454 1042 y Fr(;)17 b(\032)3548 1057 y Fp(p)p Fs(0)3623 1042 y Fr(\032)3673 1057 y Fp(q)r Fs(0)3747 1042 y Fq(g)p Fr(:)92 1163 y Fw(Supp)s(ose)33 b(that)653 1127 y Fs(20)1549 1283 y Fr(e)1594 1242 y Fj(\000)p Fp(K)5 b(s=)p Fs(6)1848 1283 y Fq(\024)28 b Fw(32)p Fr(c)2093 1298 y Fp(m)2159 1283 y Fr("=\013)q(d)1182 b Fw(\(A.18\))92 1455 y(Substituting)30 b Fr(\027)k Fq(!)28 b Fr(\027)907 1470 y Fs(0)946 1455 y Fr(;)k(\033)f Fq(!)d Fr(\033)1274 1470 y Fs(0)1314 1455 y Fr(;)j(\016)g Fq(!)d Fr(\016)1617 1470 y Fs(0)1656 1455 y Fr(;)k Fw(w)m(e)g(can)g(use)g(Lemma)e(A.5)h(since)h Fr(\016)3010 1470 y Fs(0)3077 1455 y Fq(\025)c Fr(d=)p Fw(64)j(and)g(\(A.2\))92 1575 y(implies)21 b(\(A.10\).)j(De\014ning)g Fr(W)1212 1590 y Fs(1)1279 1575 y Fw(:=)k Fr(W)1502 1590 y Fp(v)1536 1599 y Fe(1)1571 1590 y Fp(;s)1624 1599 y Fe(1)1686 1575 y Fw(with)c Fr(v)1947 1590 y Fs(1)2014 1575 y Fw(:=)j Fr(v)9 b Fq(\000)c Fw(2)p Fr(\027)2379 1590 y Fs(0)2447 1575 y Fw(=)27 b(3)p Fr(=)p Fw(4)p Fr(v)g Fw(and)e Fr(s)2999 1590 y Fs(1)3066 1575 y Fw(:=)i Fr(s)5 b Fq(\000)g Fw(2)p Fr(\033)3433 1590 y Fs(0)3501 1575 y Fw(=)28 b(2)p Fr(=)p Fw(3)p Fr(s;)92 1696 y Fw(w)m(e)23 b(obtain)e(an)i(analytic)e(symplectic)h(transformation)e(\010)2199 1711 y Fs(0)2267 1696 y Fw(:)27 b Fr(W)2413 1711 y Fs(1)2481 1696 y Fq(!)g Fr(W)2700 1711 y Fp(v)r(;s)2816 1696 y Fw(with)22 b Fr(H)9 b Fq(\016)q Fw(\010)3239 1711 y Fs(0)3306 1696 y Fw(=)28 b Fr(h)q Fw(+)q Fr(g)3591 1711 y Fs(0)3632 1696 y Fw(+)q Fr(f)3757 1711 y Fs(1)3797 1696 y Fr(;)92 1816 y Fw(where)34 b Fr(g)421 1831 y Fs(0)487 1816 y Fw(=)28 b Fr(P)654 1831 y Fs(\003)707 1816 y Fr(T)764 1831 y Fp(K)832 1816 y Fr(f)11 b Fw(.)33 b(Moreo)m(v)m(er)h(from)d (\(A.2\),)i(\(A.18\),)f(\(A.12\))g(w)m(e)i(obtain)142 2090 y Fq(k)p Fr(f)240 2105 y Fs(1)279 2090 y Fq(k)329 2105 y Fs(1)396 2090 y Fw(:=)27 b Fq(k)p Fr(f)624 2105 y Fs(1)664 2090 y Fq(k)714 2105 y Fp(v)748 2114 y Fe(1)783 2105 y Fp(;s)836 2114 y Fe(1)901 2090 y Fw(=:)h Fr(")1078 2105 y Fs(1)1145 2090 y Fq(\024)1250 1969 y Fi(\022)1311 2090 y Fw(1)16 b Fq(\000)1479 2022 y Fw(64)p Fr(c)1619 2037 y Fp(m)1685 2022 y Fr(")p 1479 2067 252 4 v 1548 2158 a(\013)q(d)1740 1969 y Fi(\023)1801 1992 y Fj(\000)p Fs(1)1896 1969 y Fi(\024)1950 2022 y Fw(64)p Fr(c)2090 2037 y Fp(m)2156 2022 y Fr(")p 1950 2067 V 2019 2158 a(\013)q(d)2227 2090 y Fw(+)g Fr(e)2364 2049 y Fj(\000)p Fp(K)5 b(s)2520 1969 y Fi(\025)2564 2090 y Fr(")27 b Fq(\024)2776 2022 y Fw(9)p 2752 2067 98 4 v 2752 2158 a(11)2899 2022 y(2)2948 1986 y Fs(7)2987 2022 y Fr(c)3029 2037 y Fp(m)3096 2022 y Fr(")p 2899 2067 243 4 v 2963 2158 a(\013)q(d)3151 2090 y(")h Fq(\024)3340 2022 y Fw(1)p 3340 2067 49 4 v 3340 2158 a(6)3398 2090 y Fr(")h(:)50 b Fw(\(A.19\))92 2353 y(Letting)31 b(\()p Fr(u;)17 b(')p Fw(\))27 b(=)h(\010)875 2368 y 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Fp(i)3282 3389 y Fw(:=)k Fr(H)38 b Fq(\016)31 b Fw(\010)3705 3404 y Fp(i)p Fj(\000)p Fs(1)92 3509 y Fw(=:)37 b Fr(h)26 b Fw(+)g Fr(g)463 3524 y Fp(i)p Fj(\000)p Fs(1)607 3509 y Fw(+)g Fr(f)757 3524 y Fp(i)824 3509 y Fw(on)38 b Fr(W)1057 3524 y Fp(i)1124 3509 y Fw(and)g Fr(")1365 3524 y Fp(i)1431 3509 y Fw(:=)g Fq(k)p Fr(f)1670 3524 y Fp(i)1698 3509 y Fq(k)1748 3524 y Fp(W)1815 3534 y Ff(i)1882 3509 y Fw(=:)g Fq(k)p Fr(f)2121 3524 y Fp(i)2150 3509 y Fq(k)2200 3524 y Fp(i)2228 3509 y Fr(:)g Fw(Observing)h(that)f Fr(W)3072 3524 y Fp(i)p Fs(+1)3228 3509 y Fq(\032)g Fr(W)3435 3524 y Fp(i)3463 3509 y Fw(,)g(w)m(e)i(can)92 3629 y(iterate)32 b(Lemma)f(A.5)h(with)h Fr(v)e Fq(!)c Fr(v)1405 3644 y Fp(i)1434 3629 y Fw(,)32 b Fr(s)c Fq(!)f Fr(s)1740 3644 y Fp(i)1768 3629 y Fw(,)33 b(after)f(v)m(erifying)g(b)m(y)i(induction)d(\(see)j(b)s(elo)m(w\))e (that)1403 3844 y Fr(")1449 3859 y Fp(i)1505 3844 y Fq(\024)c Fr(")1656 3859 y Fs(1)1728 3844 y Fr(;)212 b Fq(8)17 b Fw(1)27 b Fq(\024)h Fr(i)g Fq(\024)g Fr(L)33 b(:)1038 b Fw(\(A.22\))92 4058 y(In)30 b(fact,)h(in)e(order)i(to)e(apply)h (Lemma)f(A.5)i(w)m(e)g(ha)m(v)m(e)h(to)e(v)m(erify)g(\(A.10\),)g(whic)m (h)h(is)f(implied,)d(for)j(an)m(y)92 4178 y(1)d Fq(\024)h Fr(i)g Fq(\024)g Fr(L)p Fw(,)33 b(b)m(y)h(\(A.22\))e(and)h(the)g (estimate)1724 4142 y Fs(21)1369 4393 y Fr(")1415 4408 y Fp(i)1471 4393 y Fq(\024)28 b Fr(")1622 4408 y Fs(1)1689 4393 y Fq(\024)g Fw(2)1843 4352 y Fj(\000)p Fs(6)1937 4393 y Fw(\()p Fr(\013)q(d=)p Fw(64)p Fr(c)2278 4408 y Fp(m)2343 4393 y Fr(L)2409 4352 y Fs(2)2449 4393 y Fw(\))k Fr(;)1004 b Fw(\(A.23\))92 4607 y(whic)m(h)33 b(follo)m(ws)e(directly)h(from)f(\(A.19\),)i(\(A.2\))f(and)h(\(A.21\).) f(W)-8 b(e)33 b(observ)m(e)h(that)f Fq(8)17 b Fw(1)27 b Fq(\024)h Fr(i)g Fq(\024)g Fr(L)1124 4821 y Fq(k)p Fr(g)1221 4836 y Fp(i)1271 4821 y Fq(\000)22 b Fr(g)1417 4836 y Fp(i)p Fj(\000)p Fs(1)1535 4821 y Fq(k)1585 4836 y Fp(i)1641 4821 y Fw(=)28 b Fq(k)p Fr(P)1858 4836 y Fs(\003)1910 4821 y Fr(T)1967 4836 y Fp(K)2036 4821 y Fr(f)2084 4836 y Fp(i)2112 4821 y Fq(k)2162 4836 y Fp(i)2218 4821 y Fq(\024)g(k)p Fr(f)2421 4836 y Fp(i)2449 4821 y Fq(k)2499 4836 y Fp(i)2555 4821 y Fw(=)f Fr(")2704 4836 y Fp(i)2765 4821 y Fr(:)758 b Fw(\(A.24\))p 92 4913 1493 4 v 171 4974 a Fo(20)294 5004 y FC(The)26 b(case)g(in)h(whic)n(h)f (in)h(\(A.18\))f(holds)g(\\)p Fm(>)p FC(")f(is)h(ev)n(en)g(simpler.)g (In)h(fact)f(it)h(is)f(su\016cien)n(t)h(to)f(apply)g(Lemma)g(A.5)92 5104 y(with)g Fm(\027)j FC(:=)469 5071 y Fl(v)p 469 5085 36 4 v 470 5132 a Fo(4)515 5104 y FC(,)d Fm(\033)g FC(:=)759 5071 y Fl(s)p 758 5085 34 4 v 758 5132 a Fo(3)801 5104 y FC(,)g Fm(v)19 b Fn(\000)c FC(2)p Fm(\027)28 b FC(=)22 b Fm(v)1227 5116 y Fy(\003)1266 5104 y FC(,)k Fm(s)15 b Fn(\000)g FC(2)p Fm(\033)26 b FC(=)d Fm(s)1691 5116 y Fy(\003)1729 5104 y FC(,)j Fm(g)g FC(=)c(0)p Fm(;)k FC(ha)n(ving)f(\011)e(:=)f(\010,)27 b Fm(g)2636 5116 y Fo(+)2713 5104 y FC(=)c Fm(P)2854 5116 y Fo(\003)2904 5104 y Fm(T)2953 5116 y Fl(K)3016 5104 y Fm(f)9 b FC(,)26 b Fm(f)3156 5116 y Fy(\003)3217 5104 y FC(:=)d Fm(f)3369 5116 y Fo(+)3423 5104 y Fm(:)k FC(\(A.6\))f(and)92 5203 y(\(A.7\))i(follo)n(w)f(from)g(\(A.2\),)h(\(A.12\))g(and)f(from)g Fm(K)6 b(s)23 b Fn(\025)f FC(6)p Fm(;)28 b FC(as)f(it)h(is)f(easy)g(to) g(v)n(erify)-7 b(.)171 5273 y Fo(21)297 5303 y FC(W)g(e)27 b(observ)n(e)f(that)i Fm(\016)e Fn(\025)d Fm(d=)p FC(64)p Fm(:)1909 5775 y Fw(25)p eop %%Page: 26 26 26 25 bop 92 206 a Fw(W)-8 b(e)48 b(no)m(w)h(pro)m(v)m(e)h(\(A.22\).)d (In)i(order)f(to)g(estimate)g Fr(f)2119 221 y Fp(i)p Fs(+1)2237 206 y Fw(,)g(w)m(e)h(ev)-5 b(aluate)48 b Fr(g)2916 221 y Fp(i)p Fj(\000)p Fs(1)3088 206 y Fw(=)3212 139 y Fi(P)3300 166 y Fp(i)p Fj(\000)p Fs(1)3300 230 y Fp(j)t Fs(=0)3447 206 y Fw(~)-53 b Fr(g)3490 221 y Fp(j)3575 206 y Fw(where)95 326 y(~)h Fr(g)139 341 y Fs(0)227 326 y Fw(:=)50 b Fr(g)427 341 y Fs(0)512 326 y Fw(and)f(~)-53 b Fr(g)761 341 y Fp(j)847 326 y Fw(:=)45 b(\()p Fr(g)1080 341 y Fp(j)1147 326 y Fq(\000)32 b Fr(g)1303 341 y Fp(j)t Fj(\000)p Fs(1)1429 326 y Fw(\))45 b(are)h(de\014ned)h(on)e Fr(W)2277 341 y Fp(j)2314 326 y Fr(:)g Fw(Since)h(from)e(\(A.14\))h(w)m (e)i(ha)m(v)m(e)f Fq(k)p Fr(\036)3718 341 y Fp(i)3746 326 y Fq(k)3796 341 y Fp(i)92 447 y Fq(\024)28 b(k)p Fr(f)295 462 y Fp(i)323 447 y Fq(k)373 462 y Fp(i)401 447 y Fr(=\013)33 b Fw(=)27 b Fr(")694 462 y Fp(i)722 447 y Fr(=\013)q(;)32 b Fw(and)241 722 y Fq(kf)p Fr(g)388 737 y Fp(i)p Fj(\000)p Fs(1)506 722 y Fr(;)17 b(\036)608 737 y Fp(i)635 722 y Fq(gk)735 737 y Fp(v)769 747 y Ff(i)795 737 y Fj(\000)p Fp(\027;s)938 747 y Ff(i)964 737 y Fj(\000)p Fp(\033)1149 722 y Fq(\024)1313 615 y Fp(i)p Fj(\000)p Fs(1)1311 639 y Fi(X)1309 822 y Fp(j)t Fs(=0)1448 722 y Fq(kf)s Fw(~)-52 b Fr(g)1595 737 y Fp(j)1631 722 y Fr(;)17 b(\036)1733 737 y Fp(i)1761 722 y Fq(gk)1861 737 y Fp(v)1895 747 y Ff(i)1921 737 y Fj(\000)p Fp(\027;s)2064 747 y Ff(i)2090 737 y Fj(\000)p Fp(\033)1149 1036 y Fq(\024)1313 928 y Fp(i)p Fj(\000)p Fs(1)1311 953 y Fi(X)1309 1136 y Fp(j)t Fs(=0)1448 890 y Fi(")1795 969 y Fw(1)p 1507 1013 625 4 v 1507 1105 a Fr(e)p Fw(\()p Fr(r)1634 1120 y Fp(j)1693 1105 y Fq(\000)22 b Fr(r)1836 1120 y Fp(i)1887 1105 y Fw(+)g Fr(\032)p Fw(\))p Fr(\033)2164 1036 y Fw(+)2562 969 y(1)p 2272 1013 629 4 v 2272 1105 a Fr(e)p Fw(\()p Fr(s)2401 1120 y Fp(j)2460 1105 y Fq(\000)g Fr(s)2605 1120 y Fp(i)2656 1105 y Fw(+)g Fr(\033)t Fw(\))p Fr(\032)1439 1339 y Fw(+)1842 1272 y Fr(m)p 1525 1316 720 4 v 1525 1408 a Fw(\()p Fr(r)1607 1423 y Fp(pj)1701 1408 y Fq(\000)h Fr(r)1845 1423 y Fp(pi)1931 1408 y Fw(+)f Fr(\032)2079 1423 y Fp(p)2119 1408 y Fw(\))p Fr(\032)2207 1423 y Fp(q)2277 1339 y Fw(+)2701 1272 y Fr(m)p 2385 1316 717 4 v 2385 1408 a Fw(\()p Fr(r)2467 1423 y Fp(q)r(j)2560 1408 y Fq(\000)g Fr(r)2703 1423 y Fp(q)r(i)2787 1408 y Fw(+)g Fr(\032)2935 1423 y Fp(q)2974 1408 y Fw(\))p Fr(\032)3062 1423 y Fp(p)3112 1193 y Fi(#)3160 1339 y Fq(k)s Fw(~)-52 b Fr(g)3257 1354 y Fp(j)3293 1339 y Fq(k)3343 1354 y Fp(j)3379 1339 y Fq(k)p Fr(\036)3487 1354 y Fp(i)3515 1339 y Fq(k)3565 1354 y Fp(i)1149 1632 y Fq(\024)1319 1565 y Fw(2)p 1319 1609 49 4 v 1319 1701 a(3)1378 1486 y Fi(")1489 1565 y Fw(1)p 1436 1609 155 4 v 1436 1701 a Fr(e\032\033)1623 1632 y Fw(+)1778 1565 y Fr(m)p 1731 1609 179 4 v 1731 1701 a(\032)1781 1716 y Fp(p)1821 1701 y Fr(\032)1871 1716 y Fp(q)1919 1486 y Fi(#)1989 1524 y Fp(i)p Fj(\000)p Fs(1)1986 1549 y Fi(X)1984 1732 y Fp(j)t Fs(=1)2124 1632 y Fq(k)p Fr(P)2237 1647 y Fs(\003)2289 1632 y Fr(T)2346 1647 y Fp(K)2415 1632 y Fr(f)2463 1647 y Fp(j)2499 1632 y Fq(k)2549 1647 y Fp(j)2603 1565 y Fw(1)p 2596 1609 63 4 v 2596 1701 a Fr(\013)2668 1632 y(")2714 1647 y Fp(i)2764 1632 y Fw(+)2914 1565 y Fr(")p 2872 1609 130 4 v 2872 1701 a(\013)q(L)3011 1632 y(")3057 1647 y Fp(i)3085 1486 y Fi(")3196 1565 y Fw(1)p 3144 1609 155 4 v 3144 1701 a Fr(e\032\033)3330 1632 y Fw(+)3485 1565 y Fr(m)p 3438 1609 179 4 v 3438 1701 a(\032)3488 1716 y Fp(q)3527 1701 y Fr(\032)3577 1716 y Fp(p)3626 1486 y Fi(#)1149 1946 y Fq(\024)1319 1879 y Fw(64)p Fr(c)1459 1894 y Fp(m)1525 1879 y Fr(L)1591 1843 y Fs(2)1631 1879 y Fr(")1677 1894 y Fp(i)p 1319 1923 386 4 v 1455 2015 a Fr(\013)q(d)1715 1800 y Fi(")1773 1879 y Fw(2)p 1773 1923 49 4 v 1773 2015 a(3)1853 1838 y Fp(i)p Fj(\000)p Fs(1)1850 1863 y Fi(X)1849 2045 y Fp(j)t Fs(=1)1988 1946 y Fr(")2034 1961 y Fp(j)2092 1946 y Fw(+)2211 1879 y Fr(")p 2200 1923 67 4 v 2200 2015 a(L)2276 1800 y Fi(#)2353 1946 y Fq(\024)2468 1879 y Fw(64)p Fr(c)2608 1894 y Fp(m)2674 1879 y Fr(L)2740 1843 y Fs(2)2780 1879 y Fr(")2826 1894 y Fp(i)p 2468 1923 386 4 v 2604 2015 a Fr(\013)q(d)2863 1800 y Fi(")2912 1946 y Fr(")2958 1961 y Fs(1)3019 1946 y Fw(+)3137 1879 y Fr(")p 3127 1923 67 4 v 3127 2015 a(L)3203 1800 y Fi(#)3268 1946 y Fr(;)255 b Fw(\(A.25\))92 2227 y(where)37 b(w)m(e)h(ha)m(v)m(e)f(used)h(Lemma)c(A.2)j(and)f(the)h (fact)f(that)j(~)-52 b Fr(g)2340 2242 y Fp(j)2410 2227 y Fw(=)36 b Fr(P)2585 2242 y Fs(\003)2638 2227 y Fr(T)2695 2242 y Fp(K)2764 2227 y Fr(f)2812 2242 y Fp(j)2848 2227 y Fr(;)g Fw(considered)i(separately)92 2347 y(the)33 b(case)g Fr(j)h Fw(=)27 b(0)33 b(from)e Fr(j)j(>)27 b Fw(0)p Fr(;)33 b Fw(and)f(observ)m(ed)j(that,)d(if)g Fr(j)h(>)28 b Fw(0,)k(then)823 2550 y Fr(v)26 b Fq(\000)d Fr(v)1043 2565 y Fp(j)1101 2550 y Fw(+)f Fr(\027)90 b Fq(\025)83 b Fr(v)26 b Fq(\000)d Fr(v)1717 2565 y Fs(1)1779 2550 y Fw(+)f Fr(\027)34 b Fq(\025)28 b Fr(\027)h Fw(+)22 b Fr(v)t(=)p Fw(4)27 b(=)g(\(2)p Fr(L)22 b Fw(+)g(1\))p Fr(\027)39 b(;)825 2695 y(s)22 b Fq(\000)g Fr(s)1038 2710 y Fp(j)1097 2695 y Fw(+)g Fr(\033)87 b Fq(\025)c Fr(s)23 b Fq(\000)f Fr(s)1711 2710 y Fs(1)1773 2695 y Fw(+)g Fr(\033)31 b Fq(\025)e Fr(\033)d Fw(+)c Fr(s=)p Fw(3)27 b(=)g(\(4)p Fr(L=)p Fw(3)22 b(+)g(1\))p Fr(\033)36 b(:)92 2898 y Fw(Using)c(\(A.21\),)g(\(A.19\),)g(\(A.23\))h(and)f Fr(")1574 2913 y Fp(i)1630 2898 y Fq(\024)c Fr(")1781 2913 y Fs(1)1853 2898 y Fw(w)m(e)33 b(obtain)f(from)f(Lemma)g(A.5)142 3180 y Fr(")188 3195 y Fp(i)p Fs(+1)333 3180 y Fw(=)d Fq(k)p Fr(f)535 3195 y Fp(i)p Fs(+1)653 3180 y Fq(k)703 3195 y Fp(i)p Fs(+1)849 3180 y Fq(\024)954 3034 y Fi( )1020 3180 y Fw(1)13 b Fq(\000)1183 3113 y Fw(64)p Fr(c)1323 3128 y Fp(m)1389 3113 y Fr(L)1455 3076 y Fs(2)1494 3113 y Fr(")1540 3128 y Fp(i)p 1183 3157 386 4 v 1319 3248 a Fr(\013)q(d)1578 3034 y Fi(!)1644 3057 y Fj(\000)p Fs(1)1738 3034 y Fi(")1797 3113 y Fw(64)p Fr(c)1937 3128 y Fp(m)2003 3113 y Fr(L)2069 3076 y Fs(2)2109 3113 y Fr(")2155 3128 y Fp(i)p 1797 3157 V 1933 3248 a Fr(\013)q(d)2192 3180 y Fw(\()p Fr(")2276 3195 y Fp(i)2317 3180 y Fw(+)g Fr(")2452 3195 y Fs(1)2505 3180 y Fw(+)g Fr("=L)p Fw(\))g(+)g Fr(e)2940 3139 y Fj(\000)p Fp(K)5 b(s)3096 3034 y Fi(#)3145 3180 y Fr(")3191 3195 y Fp(i)3246 3180 y Fq(\024)3361 3113 y Fr(")3407 3128 y Fp(i)p 3361 3157 74 4 v 3374 3248 a Fw(4)3473 3180 y Fr(:)50 b Fw(\(A.26\))92 3445 y(Moreo)m(v)m(er,)34 b(from)d(\(A.19\),)i(there)g(follo)m(ws)314 3707 y Fq(k)p Fr(f)412 3722 y Fj(\003)452 3707 y Fq(k)502 3722 y Fp(v)536 3730 y Fc(\003)572 3722 y Fp(;s)625 3730 y Fc(\003)692 3707 y Fw(=)27 b Fr(")841 3722 y Fp(L)p Fs(+1)1011 3707 y Fq(\024)h Fw(4)1165 3666 y Fj(\000)p Fp(L)1272 3707 y Fr(")1318 3722 y Fs(1)1385 3707 y Fq(\024)1500 3640 y Fw(2)1549 3604 y Fs(9)1588 3640 y Fr(c)1630 3655 y Fp(m)1697 3640 y Fr(")p 1500 3684 243 4 v 1564 3776 a(\013)q(d)1752 3707 y Fw(4)1801 3666 y Fj(\000)p Fs(\()p Fp(L)p Fs(+1\))2053 3707 y Fr(")f Fq(\024)2241 3640 y Fw(2)2290 3604 y Fs(9)2330 3640 y Fr(c)2372 3655 y Fp(m)2438 3640 y Fr(")p 2241 3684 V 2306 3776 a(\013)q(d)2494 3707 y Fw(4)2543 3666 y Fj(\000)2648 3639 y Ff(K)s(s)p 2607 3651 164 4 v 2607 3692 a Fe(12)12 b(ln)e(2)2785 3707 y Fr(")28 b Fw(=)2972 3640 y(2)3021 3604 y Fs(9)3060 3640 y Fr(c)3102 3655 y Fp(m)3169 3640 y Fr(")p 2972 3684 243 4 v 3037 3776 a(\013)q(d)3225 3707 y(e)3270 3666 y Fj(\000)p Fp(K)5 b(s=)p Fs(6)3496 3707 y Fr(")32 b(;)92 3936 y Fw(and)629 4106 y Fq(k)p Fr(g)25 b Fq(\000)d Fr(g)897 4121 y Fs(0)937 4106 y Fq(k)987 4121 y Fp(v)1021 4129 y Fc(\003)1057 4121 y Fp(;s)1110 4129 y Fc(\003)1177 4106 y Fq(\024)1318 3998 y Fp(L)1282 4023 y Fi(X)1285 4205 y Fp(i)p Fs(=1)1419 4106 y Fq(k)p Fr(g)1516 4121 y Fp(i)1565 4106 y Fq(\000)h Fr(g)1712 4121 y Fp(i)p Fj(\000)p Fs(1)1830 4106 y Fq(k)1880 4121 y Fp(v)1914 4131 y Ff(i)1941 4121 y Fp(;s)1994 4131 y Ff(i)2051 4106 y Fq(\024)2192 3998 y Fp(L)2156 4023 y Fi(X)2159 4205 y Fp(i)p Fs(=1)2293 4106 y Fr(")2339 4121 y Fp(i)2394 4106 y Fq(\024)28 b Fw(4)2601 3998 y Fp(L)2565 4023 y Fi(X)2568 4205 y Fp(i)p Fs(=1)2685 4106 y Fw(\()2733 4038 y(1)p 2733 4082 49 4 v 2733 4174 a(4)2791 4106 y(\))2829 4065 y Fp(i)2857 4106 y Fr(")2903 4121 y Fs(1)2970 4106 y Fw(=)3084 4038 y(4)p 3084 4082 V 3084 4174 a(3)3142 4106 y Fr(")3188 4121 y Fs(1)3260 4106 y Fr(;)92 4343 y Fw(from)j(whic)m(h)i(\(A.5\))g(and)f(\(A.6\))h(follo)m(w.)92 4534 y(Let,)h(no)m(w,)g(\011)c(:=)g Fr(\036)823 4549 y Fs(0)885 4534 y Fq(\016)23 b(\001)17 b(\001)g(\001)k(\016)i Fr(\036)1228 4549 y Fp(L)1314 4534 y Fw(and)34 b(\()p Fr(u)1599 4498 y Fs(\()p Fp(i)p Fs(\))1681 4534 y Fr(;)17 b(')1789 4498 y Fs(\()p Fp(i)p Fs(\))1872 4534 y Fw(\))30 b Fq(2)g Fr(W)2128 4549 y Fp(i)2156 4534 y Fr(:)k Fw(Using)g(\(A.21\),) f(\(A.20\))h(and)g(Lemma)e(A.5)92 4655 y(w)m(e)h(ha)m(v)m(e)205 4907 y Fq(j)p Fr(I)d Fq(\000)22 b Fr(I)456 4866 y Fj(0)479 4907 y Fq(j)28 b Fw(=)f Fq(j)p Fr(I)717 4866 y Fs(\()p Fp(L)p Fs(+1\))936 4907 y Fq(\000)c Fr(I)1087 4866 y Fs(\(0\))1181 4907 y Fq(j)k(\024)h(j)p Fr(I)1420 4866 y Fs(\(1\))1536 4907 y Fq(\000)23 b Fr(I)1687 4866 y Fs(\(0\))1781 4907 y Fq(j)f Fw(+)1965 4799 y Fp(L)1929 4824 y Fi(X)1932 5006 y Fp(i)p Fs(=1)2066 4907 y Fq(j)p Fr(I)2145 4866 y Fs(\()p Fp(i)p Fs(+1\))2339 4907 y Fq(\000)h Fr(I)2490 4866 y Fs(\()p Fp(i)p Fs(\))2573 4907 y Fq(j)k(\024)2751 4840 y Fw(8)p Fr(")p 2743 4884 109 4 v 2743 4975 a(\013)q(s)2884 4907 y Fw(+)2992 4840 y(4)p Fr(L)p 2992 4884 116 4 v 2995 4975 a(\013)q(s)3170 4799 y Fp(L)3134 4824 y Fi(X)3137 5006 y Fp(i)p Fs(=1)3270 4907 y Fr(")3316 4922 y Fp(i)3372 4907 y Fq(\024)3517 4840 y Fw(9)p 3487 4884 109 4 v 3487 4975 a Fr(\013)q(s)3606 4907 y(")32 b(:)92 5188 y Fw(The)h(estimates)f (for)g Fq(j)p Fr(p)22 b Fq(\000)h Fr(p)1118 5152 y Fj(0)1141 5188 y Fq(j)p Fw(,)32 b Fq(j)p Fr(q)26 b Fq(\000)d Fr(q)1472 5152 y Fj(0)1495 5188 y Fq(j)32 b Fw(and)h Fq(j)p Fr(')21 b Fq(\000)i Fr(')2022 5152 y Fj(0)2045 5188 y Fq(j)32 b Fw(are)h(analogous.)p 2831 5188 41 83 v 1909 5775 a(26)p eop %%Page: 27 27 27 26 bop 92 206 a Fv(B)161 b(KAM)54 b(measure)e(estimates)92 425 y Fw(Here,)33 b(w)m(e)h(sho)m(w)f(ho)m(w)h(assumptions)e(\(2.30\))g (and)h(\(2.31\))e(imply)g(the)i(measure)g(estimate)f(\(2.37\).)92 616 y(Indeed,)i(the)f(set)g(\005)829 631 y Fj(\003)901 616 y Fw(is)f(obtained)g(as)h(\005)1593 631 y Fj(\003)1660 616 y Fw(=)1764 550 y Fi(T)1833 637 y Fp(\027)t Fj(2)p Fd(N)1990 616 y Fw(\005)2063 631 y Fp(\027)2107 616 y Fw(,)f(where)i(\005)2521 631 y Fs(0)2588 616 y Fw(:=)28 b(\005)33 b(and,)f(recursiv)m(ely)-8 b(,)1228 810 y(\005)1301 825 y Fp(\027)t Fs(+1)1462 810 y Fw(:=)28 b(\005)1666 825 y Fp(\027)1731 810 y Fq(n)1803 714 y Fi(\020)2069 727 y([)1879 912 y Fe(\()p Ff(k)q(;`)p Fe(\))p Fc(2)p Fa(Z)2087 898 y Ff(n)p Fe(+)p Ff(m)2224 912 y Fc(nf)p Fe(0)p Fc(g)1921 972 y(j)p Ff(`)p Fc(j\024)p Fe(2)p Ff(;)17 b Fc(j)p Ff(k)q Fc(j)p Ff(>K)2270 980 y(\027)2377 810 y Fq(R)2461 769 y Fp(\027)t Fs(+1)2461 835 y Fp(k)r(`)2594 714 y Fi(\021)2661 810 y Fr(;)92 1146 y Fw(with)32 b Fr(K)397 1161 y Fp(\027)468 1146 y Fw(:=)c Fr(K)682 1161 y Fs(0)738 1146 y Fw(2)787 1110 y Fp(\027)830 1146 y Fw(,)33 b(and)g Fq(R)1164 1105 y Fp(\027)t Fs(+1)1164 1171 y Fp(k)r(`)1330 1146 y Fw(is)g(a)f(suitable)g(\\resonan)m(t)h (set")h(to)e(b)s(e)h(discarded)g(at)g(the)g Fr(\027)3546 1110 y Fs(th)3650 1146 y Fw(step)92 1266 y(of)39 b(the)h(KAM)f (iteration)f(\(compare)h(with)g(the)h(Iterativ)m(e)f(Lemma)f(in)h (Section)g(4)h(of)e([P96]\).)i(The)92 1387 y(sets)33 b Fq(R)366 1346 y Fp(\027)t Fs(+1)366 1412 y Fp(k)r(`)533 1387 y Fw(satisfy)f(the)h(measure)g(estimate)608 1618 y(meas)820 1633 y Fp(n)867 1618 y Fw(\()p Fq(R)989 1577 y Fp(\027)t Fs(+1)989 1643 y Fp(k)r(l)1123 1618 y Fw(\))27 b Fq(\024)1396 1551 y Fr(\025)p 1303 1595 243 4 v 1303 1686 a Fq(j)p Fr(k)s Fq(j)1413 1658 y Fp(\034)8 b Fs(+1)1589 1618 y Fr(;)211 b(\025)28 b Fw(:=)44 b(const)18 b(\()p Fr(LM)10 b Fw(\))2545 1577 y Fp(n)2627 1551 y Fr(\015)p 2603 1595 105 4 v 2603 1686 a(M)2717 1618 y Fw(\()17 b(diam)e(\005\))3111 1577 y Fp(n)p Fj(\000)p Fs(1)3281 1618 y Fr(;)295 b Fw(\(B.1\))92 1856 y(for)32 b(an)m(y)h Fq(j)p Fr(k)s Fq(j)27 b(\025)h Fr(K)750 1871 y Fs(0)790 1856 y Fw(,)k Fr(\027)i Fq(\025)29 b Fw(0,)j Fq(j)p Fr(`)p Fq(j)27 b(\024)h Fw(2)k(\(see)i(Lemma)d(5)i(in)e([P96]\).)i(Therefore,) 748 2037 y(meas)960 2052 y Fp(n)1007 2037 y Fw(\(\005)1118 2052 y Fp(\027)t Fs(+1)1251 2037 y Fw(\))83 b Fq(\025)116 b Fw(meas)1777 2052 y Fp(n)1824 2037 y Fw(\(\005)1935 2052 y Fp(\027)1978 2037 y Fw(\))23 b Fq(\000)39 b Fw(const)17 b Fr(\025)2523 1954 y Fi(X)2468 2143 y Fj(j)p Fp(k)r Fj(j)p Fp(>K)2662 2151 y Ff(\027)2715 2037 y Fq(j)p Fr(k)s Fq(j)2825 1996 y Fj(\000)p Fs(\()p Fp(\034)8 b Fs(+1\))3095 2037 y Fq(\025)1372 2332 y(\025)116 b Fw(meas)1777 2347 y Fp(n)1824 2332 y Fw(\(\005)1935 2347 y Fp(\027)1978 2332 y Fw(\))23 b Fq(\000)39 b Fw(const)17 b Fr(\025)2597 2265 y Fw(1)p 2461 2309 322 4 v 2461 2401 a Fr(K)2551 2372 y Fp(\034)8 b Fj(\000)p Fp(n)p Fs(+1)2544 2425 y Fp(\027)2825 2332 y Fr(:)92 2564 y Fw(Iterating)31 b(this)i(relation)d (and)j(using)f(the)h(de\014nition)f(of)g Fr(K)2284 2579 y Fp(\027)2327 2564 y Fw(,)h(w)m(e)g(get)713 2763 y(meas)925 2778 y Fp(n)972 2763 y Fw(\(\005)1083 2778 y Fp(\027)t Fs(+1)1217 2763 y Fw(\))83 b Fq(\025)116 b Fw(meas)1743 2778 y Fp(n)1789 2763 y Fw(\(\005\))23 b Fq(\000)39 b Fw(const)2351 2696 y Fr(\015)p 2326 2740 105 4 v 2326 2832 a(M)2441 2763 y Fw(\()p Fr(LM)60 b Fw(diam)15 b(\005\))3038 2722 y Fp(n)p Fj(\000)p Fs(1)3208 2763 y Fr(;)92 2984 y Fw(whic)m(h)33 b(pro)m(v)m(es)h(\(2.37\).)p 1067 2984 41 83 v 92 3389 a Fv(C)161 b(The)74 b(Delauna)l(y-P)l(oincar)l(\023)-77 b(e)74 b(theory)f(of)i(the)e(planetary)370 3572 y(three-b)t(o)t(dy)52 b(problem)92 3791 y Fw(In)44 b(this)h(app)s(endix)f(w)m(e)h(discuss,)h (in)e(a)g(self-con)m(tained)g(w)m(a)m(y)-8 b(,)45 b(the)g(Hamiltonian)c (form)m(ulation)g(of)92 3911 y(the)g(planetary)f(\(non-planar\))f (three-b)s(o)s(dy)i(problem,)e(discussing,)i(in)f(particular,)f(the)h (classical)92 4031 y(Delauna)m(y-P)m(oincar)m(\022)-46 b(e)43 b(Theorem)i(1.1)f(and)h(its)f(pro)s(of.)g(The)h(app)s(endix)g (is)f(divided)g(in)g(t)m(w)m(o)h(parts)92 4152 y(dealing,)38 b(resp)s(ectiv)m(ely)-8 b(,)41 b(with)e(the)h(canonical)f(treatmen)m(t) g(of)h(the)g(t)m(w)m(o-b)s(o)s(dy)g(problem)e(and)i(with)92 4272 y(the)33 b(\(partial\))d(extension)j(of)f(suc)m(h)i(theory)g(to)e (the)h(three-b)s(o)s(dy)g(problem.)585 4509 y Fu(P)-9 b(AR)g(T)36 b(I)h(-)h(Canonical)f(v)-6 b(ariables)36 b(for)i(the)f(t)m(w)m(o-b)s(o)s(dy)h(problem)92 4669 y(I.1)f(In)m(tegration)f(of)i(the)f(Kepler)g(problem)92 4828 y Fw(Consider)c(t)m(w)m(o)h(b)s(o)s(dies)e(P)1054 4843 y Fs(0)1094 4828 y Fr(;)h Fw(P)1220 4843 y Fs(1)1293 4828 y Fw(of)g(masses)h(m)1809 4843 y Fs(0)1847 4828 y Fr(;)g Fw(m)1990 4843 y Fs(1)2061 4828 y Fw(and)f(spatial)f(p)s (osition)f Fr(u)2995 4792 y Fs(\(0\))3089 4828 y Fr(;)i(u)3205 4792 y Fs(\(1\))3327 4828 y Fq(2)c Fh(R)3488 4792 y Fs(3)3534 4828 y Fw(,)k(in)m(ter-)92 4948 y(acting)i(through)i(gra)m(vit)m(y)-8 b(,)36 b(with)h(gra)m(vitational)c(constan)m(t)k(1;)g(the)g (\(inertial\))d(frame)h Fh(R)3360 4912 y Fs(3)3442 4948 y Fw(is)h(c)m(hosen)92 5069 y(so)c(that)h(its)f(origin)e(coincides)j (with)f(the)h(cen)m(ter)h(of)e(mass.)h(Let)603 5268 y(M)28 b(:=)f(m)932 5283 y Fs(0)993 5268 y Fw(+)22 b(m)1173 5283 y Fs(1)1245 5268 y Fr(;)114 b Fw(m)27 b(:=)1635 5200 y(m)1717 5215 y Fs(0)1756 5200 y Fw(m)1838 5215 y Fs(1)p 1635 5245 242 4 v 1711 5336 a Fw(M)1919 5268 y Fr(;)115 b(x)28 b Fw(:=)f Fr(u)2330 5227 y Fs(\(1\))2446 5268 y Fq(\000)c Fr(u)2602 5227 y Fs(\(0\))2728 5268 y Fr(;)115 b(X)35 b Fw(:=)28 b(m)16 b(_)-43 b Fr(x)32 b(:)289 b Fw(\(C.1\))1909 5775 y(27)p eop %%Page: 28 28 28 27 bop 92 206 a Fw(Then,)34 b(the)f(motion)d(of)i(the)h(t)m(w)m(o)g (b)s(o)s(dies)g(is)f(go)m(v)m(erned)i(b)m(y)f(the)g(Hamiltonian)1380 469 y Fq(K)q Fw(\()p Fr(X)r(;)17 b(x)p Fw(\))28 b(=)1897 401 y(1)p 1856 445 131 4 v 1856 537 a(2m)1997 469 y Fq(j)p Fr(X)8 b Fq(j)2142 427 y Fs(2)2202 469 y Fq(\000)2312 401 y Fw(mM)p 2312 445 171 4 v 2342 537 a Fq(j)p Fr(x)p Fq(j)2509 469 y Fr(;)1066 b Fw(\(C.2\))92 749 y(with)32 b(\()p Fr(X)r(;)17 b(x)p Fw(\))28 b Fq(2)g Fh(R)760 713 y Fs(3)828 749 y Fq(\002)23 b Fh(R)994 713 y Fs(3)1072 749 y Fw(conjugate)33 b(v)-5 b(ariables)1887 713 y Fs(22)1960 749 y Fw(,)33 b(i.e.,)f(the)h(equations)g(of)f(motion)f(are)50 b(_)-44 b Fr(x)28 b Fw(=)g Fr(@)3652 764 y Fp(X)3720 749 y Fq(K)q Fw(,)131 844 y(_)92 869 y Fr(X)35 b Fw(=)28 b Fq(\000)p Fr(@)440 884 y Fp(x)485 869 y Fq(K)q Fw(.)92 1160 y(As)35 b(w)m(ell)f(kno)m(wn)i(suc)m(h)h(system)f(is)e(in)m (tegrable)g(and)h(for)f Fq(K)f Fr(<)f Fw(0)i(the)h(\()p Fr(x)p Fw(-pro)5 b(jection)35 b(of)f(the\))i(orbits)92 1280 y(are)c(ellipses.)g(More)h(precisely)-8 b(,)33 b(one)g(has)92 1549 y Fu(Prop)s(osition)i(C.1)49 b Fg(Fix)40 b Fw(\003)1161 1564 y Fj(\000)1257 1549 y Fr(>)e Fw(0)f Fr(>)g Fq(K)1646 1564 y Fs(0)1726 1549 y Fg(and)j(let)g Fw(\003)2131 1564 y Fs(+)2227 1549 y Fw(:=)2368 1452 y Fi(\020)2427 1509 y Fs(m)2487 1486 y Fe(3)2521 1509 y Fs(M)2586 1486 y Fe(2)p 2427 1525 193 4 v 2434 1583 a Fj(\000)p Fs(2)p Fj(K)2578 1592 y Fe(0)2630 1452 y Fi(\021)2690 1448 y Fe(1)p 2690 1460 31 4 v 2690 1501 a(2)2734 1549 y Fg(.)g(Then,)g(ther) -5 b(e)40 b(exist)49 b Fw(^)-58 b Fr(\032)38 b(>)f Fw(0)92 1686 y Fg(and)d(a)h(r)-5 b(e)g(al-analytic)34 b(symple)-5 b(ctic)34 b(tr)-5 b(ansformation)2019 1649 y Fs(23)171 1959 y Fw(\011)247 1974 y Fs(DP)380 1959 y Fw(:)435 1863 y Fi(\020)485 1959 y Fw(\(\003)p Fr(;)17 b(\021)t(;)g Fw(p\))p Fr(;)g Fw(\()p Fr(\025;)g(\030)5 b(;)17 b Fw(q\))1187 1863 y Fi(\021)1262 1959 y Fq(2)1356 1863 y Fi(\020)1405 1959 y Fw([\003)1500 1974 y Fj(\000)1559 1959 y Fr(;)g Fw(\003)1671 1974 y Fs(+)1730 1959 y Fw(])22 b Fq(\002)h Fr(B)1958 1918 y Fs(2)1959 1984 y(^)-41 b Fp(\032)1997 1863 y Fi(\021)2069 1959 y Fq(\002)2169 1863 y Fi(\020)2218 1959 y Fh(T)26 b Fq(\002)c Fr(B)2485 1918 y Fs(2)2486 1984 y(^)-41 b Fp(\032)2525 1863 y Fi(\021)2602 1959 y Fq(7!)27 b Fw(\()p Fr(X)r(;)17 b(x)p Fw(\))28 b Fq(2)h(fj)p Fr(x)p Fq(j)e(\025)3482 1892 y Fw(^)-58 b Fr(\032)3523 1856 y Fs(2)p 3413 1936 211 4 v 3413 2028 a Fw(m)3495 1999 y Fs(2)3534 2028 y Fw(M)3633 1959 y Fq(g)35 b Fr(;)92 2214 y Fg(c)-5 b(asting)34 b Fw(\(C.2\))h Fg(into)g(the)g(inte)-5 b(gr)g(able)34 b(Hamiltonian)g Fw(\()p Fq(\000)p Fw(m)2236 2178 y Fs(3)2275 2214 y Fw(M)2364 2178 y Fs(2)2404 2214 y Fw(\))p Fr(=)p Fw(\(2\003)2646 2178 y Fs(2)2684 2214 y Fw(\))p Fg(.)92 2443 y Fw(This)39 b(classical)e(prop)s(osition)g(is)h (due)h(to)g(P)m(oincar)m(\023)-46 b(e)38 b(\([P)m(oi1905],)g(c)m (hapter)i(I)s(I)s(I\))e(and)h(the)g(v)-5 b(ariables)92 2563 y(\(\003)p Fr(;)17 b(\021)t(;)g Fw(p)p Fr(;)g(\025;)g(\030)5 b(;)17 b Fw(q\))28 b(are,)i(usually)-8 b(,)29 b(called)g Fu(P)m(oincar)n(\023)-54 b(e)34 b(v)-6 b(ariables)p Fw(.)30 b(The)h(pro)s(of)e(of)h(Prop)s(osition)e(C.1)i(is)92 2684 y(particularly)g(in)m(teresting)j(from)e(the)i(ph)m(ysical)g(p)s (oin)m(t)f(of)g(view)h(and)g(rests)g(up)s(on)g(the)g(in)m(tro)s (duction)92 2804 y(of)25 b(three)h(di\013eren)m(t)f(\(famous\))g(c)m (hange)h(of)f(v)-5 b(ariables,)25 b(whic)m(h)h(w)m(e,)g(no)m(w,)g(pro)s (ceed)h(to)e(describ)s(e)h(brie\015y)92 2924 y(\(for)32 b(more)g(details,)f(see)j([Ch88]\).)92 3116 y(Recall)45 b(that)h Fr(`)p Fw(,)h Fr(\022)j Fw(and)d Fr(g)j Fw(denote,)d(resp)s (ectiv)m(ely)-8 b(,)48 b(the)f(mean)g(anomaly)-8 b(,)45 b(the)i(longitude)e(of)i(the)92 3236 y(\(ascending\))33 b(no)s(de)f(and)h(the)g(argumen)m(t)f(of)g(the)h(p)s(erihelion)e(\(see) i(Figure)f(2)g(ab)s(o)m(v)m(e\).)92 3406 y Fg(Step)k(1.)e Fw(The)h(system)g(is)e(set)i(in)e Fg(\\symple)-5 b(ctic")36 b(spheric)-5 b(al)35 b(p)-5 b(olar)35 b(variables)p Fw(:)e(namely)-8 b(,)33 b(w)m(e)j(consider)92 3527 y(the)g(symplectic)f(map)g(\011)1042 3542 y Fs(sp)r(c)1179 3527 y Fw(:)1239 3430 y Fi(\020)1289 3527 y Fw(\()p Fr(R)q(;)17 b Fw(\012)p Fr(;)g Fw(\010\))p Fr(;)g Fw(\()p Fr(r)m(;)g(!)t(;)g(')p Fw(\))2046 3430 y Fi(\021)2127 3527 y Fq(7!)32 b Fw(\()p Fr(X)r(;)17 b(x)p Fw(\))36 b(\(where)h Fr(r)f(>)c Fw(0,)k(0)d Fr(<)f(!)37 b(<)32 b(\031)40 b Fw(and)92 3664 y(0)27 b Fq(\024)h Fr(')g(<)f Fw(2)p Fr(\031)t Fw(\))33 b(giv)m(en)f(b)m(y)1003 3627 y Fs(24)237 4122 y Fw(\011)313 4137 y Fs(sp)r(c)445 4122 y Fw(:)500 3798 y Fi(8)500 3873 y(>)500 3898 y(>)500 3923 y(>)500 3948 y(>)500 3973 y(>)500 3998 y(>)500 4023 y(<)500 4172 y(>)500 4197 y(>)500 4222 y(>)500 4247 y(>)500 4272 y(>)500 4297 y(>)500 4322 y(:)615 3881 y Fr(x)670 3896 y Fs(1)738 3881 y Fw(=)27 b Fr(r)20 b Fw(sin)c Fr(!)52 b Fw(cos)18 b Fr(')615 4121 y(x)670 4136 y Fs(2)738 4121 y Fw(=)27 b Fr(r)20 b Fw(sin)c Fr(!)52 b Fw(sin)17 b Fr(')615 4362 y(x)670 4377 y Fs(3)738 4362 y Fw(=)27 b Fr(r)20 b Fw(cos)d Fr(!)1466 4122 y(;)114 b(X)36 b Fw(=)1827 3827 y Fi(0)1827 3973 y(B)1827 4023 y(B)1827 4072 y(B)1827 4122 y(B)1827 4172 y(B)1827 4225 y(@)1927 3837 y Fs(sin)11 b Fp(!)47 b Fs(cos)12 b Fp(')2385 3839 y Fs(cos)h Fp(!)h Fs(cos)e Fp(')p 2385 3858 317 4 v 2527 3916 a(r)2809 3882 y Fq(\000)2920 3839 y Fs(sin)f Fp(')p 2896 3858 191 4 v 2896 3916 a(r)j Fs(sin)e Fp(!)1931 4078 y Fs(sin)f Fp(!)47 b Fs(sin)11 b Fp(')2389 4079 y Fs(cos)h Fp(!)i Fs(sin)e Fp(')p 2389 4099 309 4 v 2527 4157 a(r)2877 4079 y Fs(cos)g Fp(')p 2858 4099 191 4 v 2858 4157 a(r)i Fs(sin)d Fp(!)2021 4319 y Fs(cos)h Fp(!)2423 4363 y Fq(\000)2510 4324 y Fs(sin)f Fp(!)p 2510 4340 145 4 v 2565 4397 a(r)2929 4363 y Fw(0)3114 3827 y Fi(1)3114 3973 y(C)3114 4023 y(C)3114 4072 y(C)3114 4122 y(C)3114 4172 y(C)3114 4225 y(A)3203 3827 y(0)3203 3973 y(B)3203 4023 y(B)3203 4072 y(B)3203 4122 y(B)3203 4172 y(B)3203 4225 y(@)3292 3882 y Fr(R)3294 4122 y Fw(\012)3294 4363 y(\010)3384 3827 y Fi(1)3384 3973 y(C)3384 4023 y(C)3384 4072 y(C)3384 4122 y(C)3384 4172 y(C)3384 4225 y(A)3602 4122 y Fw(\(C.3\))92 4575 y(and)32 b(consider)h(the)g(new)h (Hamiltonian)29 b Fq(K)1667 4590 y Fs(sp)r(c)1799 4575 y Fw(:=)e Fq(K)d(\016)e Fw(\011)2177 4590 y Fs(sp)r(c)2281 4575 y Fw(.)p 92 4671 1493 4 v 171 4733 a Fo(22)241 4763 y FC(Often,)28 b(in)g(this)f(app)r(endix,)h(upp)r(er-lo)n(w)n(er)e (case)g(letters)h(indicate)h(couples)f(of)g(standard)f(symplectic)i (conjugate)92 4862 y(momen)n(tum-co)r(ordinate)e(v)-5 b(ariables.)171 4932 y Fo(23)241 4962 y FC(Recall)25 b(that)g Fm(B)733 4932 y Fl(n)729 4982 y(r)779 4962 y FC(,)g Fm(D)898 4932 y Fl(n)896 4982 y(r)943 4962 y FC(,)g Fm(B)1058 4932 y Fl(n)1054 4982 y(r)1104 4962 y FC(\()p Fm(x)1183 4974 y Fo(0)1221 4962 y FC(\))g(and)g Fm(D)1508 4932 y Fl(n)1506 4982 y(r)1553 4962 y FC(\()p Fm(x)1632 4974 y Fo(0)1670 4962 y FC(\))h(denote,)f(resp)r(ectiv)n(ely)-7 b(,)24 b(the)i(real)e Fm(n)p FC(-ball)g(of)h(radius)g Fm(r)j FC(cen)n(tered)c(at)92 5062 y(the)29 b(origin,)e(the)i(complex)g Fm(n)p FC(-ball)f(of)g(radius)g Fm(r)j FC(cen)n(tered)d(at)h(the)g (origin,)f(the)h(real)e Fm(n)p FC(-ball)h(of)h(radius)f Fm(r)j FC(cen)n(tered)d(at)92 5161 y Fm(x)139 5173 y Fo(0)204 5161 y FC(and)g(the)f(complex)h Fm(n)p FC(-ball)f(of)g(radius) g Fm(r)j FC(cen)n(tered)d(at)h Fm(x)1963 5173 y Fo(0)2001 5161 y FC(.)171 5234 y Fo(24)241 5264 y FC(The)g(matrix)f(in)h(\(C.3\)) g(is)f(the)h(transp)r(ose)e(of)i(the)g(in)n(v)n(erse)e(of)i(the)g (Jacobian)2787 5231 y Fl(@)t(x)p 2702 5245 247 4 v 2702 5293 a(@)t Fo(\()p Fl(r)n(;!)r(;')p Fo(\))2959 5264 y FC(.)1909 5775 y Fw(28)p eop %%Page: 29 29 29 28 bop 92 206 a Fg(Step)48 b(2.)f Fw(Using)f(the)i(metho)s(d)e(of)h (Hamilton-Jacobi,)c(one)48 b(can)f(\014nd)h(a)f(symplectic)f(map)h (\011)3687 221 y Fs(D)3797 206 y Fw(:)92 230 y Fi(\020)141 326 y Fw(\()p Fr(L;)17 b(G;)g Fw(\002\))p Fr(;)g Fw(\()p Fr(`;)g(g)t(;)g(\022)s Fw(\))872 230 y Fi(\021)955 326 y Fq(7!)1089 230 y Fi(\020)1139 326 y Fw(\()p Fr(R)q(;)g Fw(\012)p Fr(;)g Fw(\010\))p Fr(;)g Fw(\()p Fr(r)m(;)g(!)t(;)g(')p Fw(\))1896 230 y Fi(\021)1981 326 y Fw(that)36 b(in)m(tegrates)h(the)g (system:)h(\011)3251 341 y Fs(D)3346 326 y Fw(is)e(the)h(sym-)92 447 y(plectic)32 b(transformation)e(with)i(generating)g(function)142 746 y Fr(S)6 b Fw(\()p Fr(L;)17 b(G;)g Fw(\002)p Fr(;)g(r)m(;)g(!)t(;)g (')p Fw(\))25 b(=)1022 629 y Fi(Z)1121 580 y(s)p 1204 580 1004 4 v 166 x Fq(\000)1291 678 y Fw(m)1373 650 y Fs(4)1412 678 y Fw(M)1501 650 y Fs(2)p 1291 722 250 4 v 1364 814 a Fr(L)1430 785 y Fs(2)1573 746 y Fw(+)1681 678 y(2m)1812 650 y Fs(2)1851 678 y Fw(M)p 1681 722 259 4 v 1787 814 a Fr(r)1972 746 y Fq(\000)2082 678 y Fr(G)2159 650 y Fs(2)p 2082 722 117 4 v 2097 814 a Fr(r)2144 785 y Fs(2)2236 746 y Fr(dr)15 b Fw(+)2434 629 y Fi(Z)2533 581 y(s)p 2616 581 499 4 v 165 x Fr(G)2693 717 y Fs(2)2755 746 y Fq(\000)2927 678 y Fw(\002)3003 650 y Fs(2)p 2864 722 241 4 v 2864 815 a Fw(sin)2984 775 y Fs(2)3040 815 y Fr(!)3142 746 y(d!)h Fw(+)c(\002)p Fr(')27 b(:)50 b Fw(\(C.4\))92 1034 y(The)40 b(v)-5 b(ariables)711 938 y Fi(\020)760 1034 y Fw(\()p Fr(L;)17 b(G;)g Fw(\002\))p Fr(;)g Fw(\()p Fr(`;)g(g)t(;)g(\022)s Fw(\))1491 938 y Fi(\021)1578 1034 y Fw(are)40 b(kno)m(wn)h(as)f Ft(Delauna)m(y)f(v)-5 b(ariables)p Fw(.)39 b(In)h(suc)m(h)h(v)-5 b(ariables,)92 1155 y(the)33 b(new)g(Hamiltonian)c(b)s(ecomes)1347 1409 y Fq(K)1423 1424 y Fs(D)1508 1409 y Fw(:=)f Fq(K)1715 1424 y Fs(sp)r(c)1841 1409 y Fq(\016)22 b Fw(\011)1989 1424 y Fs(D)2075 1409 y Fw(=)27 b Fq(\000)2265 1342 y Fw(m)2347 1305 y Fs(3)2387 1342 y Fw(M)2476 1305 y Fs(2)p 2265 1386 250 4 v 2313 1477 a Fw(2)p Fr(L)2428 1449 y Fs(2)2542 1409 y Fr(:)92 1654 y Ft(Let)42 b Fr(C)49 b Ft(b)s(e)42 b(the)g(angular)f(momen)m(tum)f(of)i(the)g(planet,)f(let)h Fr(a)g Ft(b)s(e)g(the)g(ma)5 b(jor)41 b(semi-axis)g(and)h(let)92 1775 y Fr(i)h Ft(b)s(e)h(its)e(inclination)p Fw(,)e(i.e.,)j(the)h (angle)e(b)s(et)m(w)m(een)j(a)e(\014xed)i(reference)f(plane)f(and)h (the)f(Keplerian)92 1895 y(ellipse)27 b(plane;)h(compare)g(Figure)g(2)g (\(later,)f(suc)m(h)j(reference)g(plane)f(will)d(b)s(e)i(tak)m(en)i(to) e(b)s(e)h(the)g(\\total)92 2016 y(angular)i(momen)m(tum)g(plane"\).)h (By)h(construction,)g(the)g(follo)m(wing)d(relations)h(hold)1011 2245 y Fr(G)c Fw(=)h Fq(j)p Fr(C)7 b Fq(j)17 b Fr(;)113 b Fw(\002)27 b(=)h Fr(G)33 b Fw(cos)17 b Fr(i)98 b Fw(and)g Fr(L)28 b Fw(=)g(m)2639 2156 y Fq(p)p 2721 2156 141 4 v 2721 2245 a Fw(M)p Fr(a)17 b(:)92 2465 y Fg(Step)27 b(3.)e Fw(T)-8 b(o)25 b(remo)m(v)m(e)h(singularities,)d(follo)m(wing)f (P)m(oincar)m(\023)-46 b(e,)25 b(w)m(e)h(pro)s(ceed)g(as)f(follo)m(ws.) f(First,)g(w)m(e)i(in)m(tro-)92 2586 y(duce)38 b Fg(Poinc)-5 b(ar)n(\023)-47 b(e)37 b(action-angle)h(variables)e Fw(b)m(y)i(means)f (of)f(the)i(linear)d(symplectic)i(transformation)92 2706 y(\011)168 2721 y Fs(P)216 2729 y Fe(aa)312 2706 y Fw(:)367 2610 y Fi(\020)417 2706 y Fw(\(\003)p Fr(;)17 b(H)r(;)g(Z)7 b Fw(\))p Fr(;)17 b Fw(\()p Fr(\025;)g(h;)g(\020)8 b Fw(\))1178 2610 y Fi(\021)1252 2706 y Fq(7!)1380 2610 y Fi(\020)1429 2706 y Fw(\()p Fr(L;)17 b(G;)g Fw(\002\))p Fr(;)g Fw(\()p Fr(`;)g(g)t(;)g(\022)s Fw(\))2160 2610 y Fi(\021)2240 2706 y Fw(giv)m(en)33 b(b)m(y)832 3013 y(\011)908 3028 y Fs(P)956 3036 y Fe(aa)1052 3013 y Fw(:)1205 2867 y Fi(\()1313 2952 y Fw(\003)28 b(=)f Fr(L)17 b(;)114 b(H)36 b Fw(=)27 b Fr(L)c Fq(\000)f Fr(G)17 b(;)114 b(Z)35 b Fw(=)27 b Fr(G)22 b Fq(\000)h Fw(\002)17 b Fr(;)1313 3073 y(\025)28 b Fw(=)f Fr(`)22 b Fw(+)g Fr(g)k Fw(+)c Fr(\022)e(;)114 b(h)28 b Fw(=)f Fq(\000)p Fr(g)f Fq(\000)d Fr(\022)d(;)114 b(\020)35 b Fw(=)27 b Fq(\000)p Fr(\022)20 b(:)3602 3013 y Fw(\(C.5\))92 3320 y(Then,)39 b(w)m(e)g(let)637 3284 y Fs(25)748 3320 y Fw(\011)824 3335 y Fs(P)913 3320 y Fw(:)977 3224 y Fi(\020)1026 3320 y Fw(\(\003)p Fr(;)17 b(\021)t(;)g Fw(p\))p Fr(;)g Fw(\()p Fr(\025;)g(\030)5 b(;)17 b Fw(q\))1728 3224 y Fi(\021)1812 3320 y Fq(7!)1948 3224 y Fi(\020)1998 3320 y Fw(\(\003)p Fr(;)g(H)r(;)g(Z)7 b Fw(\))p Fr(;)17 b Fw(\()p Fr(\025;)g(h;)g(\020)8 b Fw(\))2759 3224 y Fi(\021)2843 3320 y Fw(b)s(e)38 b(the)g(symplectic)g (map)92 3441 y(de\014ned)c(b)m(y)f(the)g(relations)823 3666 y Fr(H)i Fw(=)1053 3624 y Fp(\021)1090 3600 y Fe(2)1125 3624 y Fs(+)p Fp(\030)1214 3600 y Fe(2)p 1053 3643 196 4 v 1133 3701 a Fs(2)1290 3666 y Fr(;)1529 3582 y Fq(p)p 1612 3582 138 4 v 84 x Fw(2)p Fr(H)41 b Fw(cos)17 b Fr(h)28 b Fw(=)f Fr(\021)21 b(;)2425 3582 y Fq(p)p 2508 3582 V 84 x Fw(2)p Fr(H)40 b Fw(sin)16 b Fr(h)28 b Fw(=)g Fr(\030)21 b(;)823 3801 y(Z)34 b Fw(=)1038 3758 y Fp(p)1074 3735 y Fe(2)1108 3758 y Fs(+)p Fp(q)1197 3735 y Fe(2)p 1038 3778 194 4 v 1117 3835 a Fs(2)1274 3801 y Fr(;)1513 3717 y Fq(p)p 1596 3717 123 4 v 84 x Fw(2)p Fr(Z)40 b Fw(cos)17 b Fr(\020)35 b Fw(=)27 b(p)17 b Fr(;)2390 3717 y Fq(p)p 2473 3717 V 84 x Fw(2)p Fr(Z)40 b Fw(sin)16 b Fr(\020)35 b Fw(=)27 b(q)17 b Fr(:)3602 3727 y Fw(\(C.6\))92 4045 y(As)32 b(P)m(oincar)m(\023)-46 b(e)32 b(sho)m(w)m(ed)i(\(see)f(b) s(elo)m(w\),)e Fg(the)j(symple)-5 b(ctic)34 b(map)f Fw(\011)2423 4060 y Fs(DP)2557 4045 y Fw(:)2612 3948 y Fi(\020)2661 4045 y Fw(\(\003)p Fr(;)17 b(\021)t(;)g Fw(p\))p Fr(;)g Fw(\()p Fr(\025;)g(\030)5 b(;)17 b Fw(q\))3363 3948 y Fi(\021)3438 4045 y Fq(7!)28 b Fw(\()p Fr(X)r(;)17 b(x)p Fw(\))92 4165 y Fg(with)1329 4285 y Fw(\011)1405 4300 y Fs(DP)1539 4285 y Fw(:=)27 b(\011)1745 4300 y Fs(sp)r(c)1872 4285 y Fq(\016)22 b Fw(\011)2020 4300 y Fs(D)2100 4285 y Fq(\016)g Fw(\011)2248 4300 y Fs(P)2296 4308 y Fe(aa)2387 4285 y Fq(\016)g Fw(\011)2535 4300 y Fs(P)3602 4285 y Fw(\(C.7\))92 4475 y Fg(is)45 b(r)-5 b(e)g(al-analytic)45 b(in)h(a)f(neighb)-5 b(orho)g(o)g(d)44 b(of)1714 4379 y Fi(\020)1764 4475 y Fw([\003)1859 4490 y Fj(\000)1917 4475 y Fr(;)17 b Fw(\003)2029 4490 y Fs(+)2088 4475 y Fw(])30 b Fq(\002)h(f)p Fw(\(0)p Fr(;)17 b Fw(0\))p Fq(g)2571 4379 y Fi(\021)2649 4475 y Fq(\002)2757 4379 y Fi(\020)2807 4475 y Fh(T)33 b Fq(\002)e(f)p Fw(\(0)p Fr(;)17 b Fw(0\))p Fq(g)3329 4379 y Fi(\021)3377 4475 y Fw(,)45 b(\(and)g(the)92 4628 y(t)m(w)m(o-b)s(o)s(dy)33 b(Hamiltonian,)c(in)i(P)m(oincar)m(\023) -46 b(e)33 b(v)-5 b(ariables,)31 b(is)h Fq(K)24 b(\016)e Fw(\011)27 b(=)h Fq(\000)2619 4589 y Fs(m)2679 4565 y Fe(3)2713 4589 y Fs(M)2778 4565 y Fe(2)p 2619 4605 193 4 v 2656 4662 a Fs(2\003)2740 4643 y Fe(2)2822 4628 y Fw(\).)p 92 4838 1493 4 v 171 4899 a Fo(25)241 4930 y FC(Do)c(not)g(confuse)g(the)h(v)-5 b(ariables)23 b(\(p)p Fm(;)14 b FC(q\))24 b(here)g(with)g(the)h(v)-5 b(ariables)23 b(\()p Fm(p;)14 b(q)s FC(\))24 b(used)g(in)h(the)f(text)h(\(and,)f(in)g (particular,)92 5029 y(in)k(Theorem)e(1.2,)h(where)g(the)h(v)-5 b(ariables)27 b(\()p Fm(p;)14 b(q)s FC(\))27 b(corresp)r(ond)f(to)i (the)g(v)-5 b(ariables)26 b(\()p Fm(\021)2722 4999 y Fy(0)2746 5029 y Fm(;)14 b(\030)2823 4999 y Fy(0)2846 5029 y FC(\))28 b(in)n(tro)r(duced)g(b)r(elo)n(w\).)1909 5775 y Fw(29)p eop %%Page: 30 30 30 29 bop 92 206 a Fu(Remark)37 b(C.1)49 b Fw(\(i\))31 b(If)i(w)m(e)g(denote)g(\()p Fr(X)r(;)17 b(x)p Fw(\))29 b(=)e(\010)1884 221 y Fs(DP)1991 109 y Fi(\020)2040 206 y Fw(\(\003)p Fr(;)17 b(\021)t(;)g Fw(p\))p Fr(;)g Fw(\()p Fr(\025;)g(\030)5 b(;)17 b Fw(q\))2742 109 y Fi(\021)2822 206 y Fw(then)3011 170 y Fs(26)3119 206 y Fw(,)1624 510 y Fr(X)36 b Fw(=)1854 442 y(m)1936 406 y Fs(4)1975 442 y Fw(M)2064 406 y Fs(2)p 1854 487 250 4 v 1926 578 a Fw(\003)1994 549 y Fs(3)2125 442 y Fr(@)5 b(x)p 2124 487 114 4 v 2124 578 a(@)g(\025)2264 510 y(:)92 761 y Fw(\(ii\))23 b(Let)i(us)g(collect,)f(here,)i(some)f(imp)s(ortan)m(t)e (relations)g(among)h(the)h(ab)s(o)m(v)m(e)h(in)m(tro)s(duced)f(quan)m (tities.)92 881 y(Let,)38 b(as)g(usual,)f Fr(e)h Fw(denote)h(the)f (eccen)m(tricit)m(y)g(of)g(the)g(Keplerian)e(ellipse)h(and)g(let)h Fr(a)f Fw(and)h Fr(i)g Fw(denote)92 1002 y(the)33 b(ma)5 b(jor)31 b(semi-axis)g(and)i(the)g(inclination.)c(Then,)34 b(b)m(y)g(construction,)f(one)f(sees)j(that)1201 1222 y(\003)27 b(=)h(m)1482 1133 y Fq(p)p 1564 1133 141 4 v 1564 1222 a Fw(M)p Fr(a)17 b(;)1201 1281 y Fi(q)p 1284 1281 299 4 v 106 x Fr(\030)1332 1358 y Fs(2)1393 1387 y Fw(+)22 b Fr(\021)1543 1358 y Fs(2)1610 1387 y Fw(=)1713 1298 y Fq(p)p 1796 1298 68 4 v 89 x Fw(\003)16 b Fr(e)h Fw(\(1)22 b(+)g Fr(O)s Fw(\()p Fr(e)2310 1346 y Fs(2)2349 1387 y Fw(\)\))32 b Fr(;)1201 1467 y Fi(q)p 1284 1467 305 4 v 106 x Fw(p)1338 1544 y Fs(2)1399 1573 y Fw(+)22 b(q)1548 1544 y Fs(2)1616 1573 y Fw(=)1720 1484 y Fq(p)p 1803 1484 68 4 v 89 x Fw(\003)16 b Fr(i)h Fw(\(1)22 b(+)g Fr(O)s Fw(\()p Fr(e)2305 1532 y Fs(2)2344 1573 y Fw(\))g(+)g Fr(O)s Fw(\()p Fr(i)2651 1532 y Fs(2)2690 1573 y Fw(\)\))17 b Fr(:)792 b Fw(\(C.8\))92 1793 y(A)32 b(more)g(explicit)f(link)h(b)s (et)m(w)m(een)i Fr(H)8 b Fw(,)33 b(the)g(eccen)m(tricit)m(y)g(and)g (the)g(ma)5 b(jor)31 b(semi-axis)g(is)h(giv)m(en)h(b)m(y)1149 2067 y Fr(H)i Fw(=)28 b(\003)17 b(\(1)k Fq(\000)1662 1980 y(p)p 1745 1980 256 4 v 87 x Fw(1)h Fq(\000)g Fr(e)1960 2038 y Fs(2)2000 2067 y Fw(\))27 b(=)h(\003)2279 1999 y Fr(e)2324 1963 y Fs(2)p 2279 2043 85 4 v 2297 2135 a Fw(2)2374 2067 y(\(1)22 b(+)g Fr(O)s Fw(\()p Fr(e)2742 2026 y Fs(2)2781 2067 y Fw(\)\))32 b Fr(;)686 b Fw(\(C.9\))1149 2338 y Fr(e)p Fw(\()p Fr(H)r(;)17 b Fw(\003\))27 b(=)1596 2175 y Fi(s)p 1679 2175 504 4 v 1689 2271 a Fr(H)p 1689 2315 89 4 v 1699 2407 a Fw(\003)1804 2242 y Fi(\020)1854 2338 y Fw(2)22 b Fq(\000)2034 2271 y Fr(H)p 2034 2315 V 2044 2407 a Fw(\003)2133 2242 y Fi(\021)2199 2338 y Fr(:)1327 b Fw(\(C.10\))92 2598 y(Also,)32 b(if)f Fr(C)40 b Fw(is)32 b(the)h(angular)e(momen)m(tum)g(of)h(the)h(system,)h(one)f (infers)f(that)700 2818 y Fq(j)p Fr(C)7 b Fq(j)27 b Fw(=)h(\003)1049 2731 y Fq(p)p 1131 2731 256 4 v 1131 2818 a Fw(1)22 b Fq(\000)h Fr(e)1347 2789 y Fs(2)1414 2818 y Fw(=)28 b(\003\(1)21 b(+)h Fr(O)s Fw(\()p Fr(e)1953 2777 y Fs(2)1992 2818 y Fw(\)\))32 b Fr(;)1426 b Fw(\(C.11\))700 3022 y Fr(Z)35 b Fw(=)27 b Fq(j)p Fr(C)7 b Fq(j)17 b Fw(\(1)k Fq(\000)i Fw(cos)17 b Fr(i)p Fw(\))28 b(=)f(\003)1697 2936 y Fq(p)p 1780 2936 V 86 x Fw(1)22 b Fq(\000)g Fr(e)1995 2993 y Fs(2)2052 3022 y Fw(\(1)f Fq(\000)i Fw(cos)17 b Fr(i)p Fw(\))28 b(=)g Fq(j)p Fr(C)7 b Fq(j)2784 2955 y Fr(i)2817 2919 y Fs(2)p 2784 2999 73 4 v 2796 3091 a Fw(2)2867 3022 y(\(1)22 b(+)g Fr(O)s Fw(\()p Fr(i)3223 2981 y Fs(2)3262 3022 y Fw(\)\))17 b Fr(:)171 b Fw(\(C.12\))92 3340 y Fz(P)m(oincar)n(\023)-46 b(e's)32 b(argumen)m(t)f(\([P)m(oi1905]\))26 b FC(for)h(pro)n(ving)f(the)i(analyticit)n(y)e(of)i(\011)2604 3352 y Fo(DP)2730 3340 y FC(go)r(es)f(as)g(follo)n(ws.)92 3440 y(Let)h(us)g(de\014ne)g Fm(\013)d FC(:=)f Fm(g)d FC(+)d Fm(v)s FC(,)29 b Fm( )e FC(:=)c Fm(')d Fn(\000)e Fm(\022)r FC(,)29 b Fn(X)36 b FC(:=)23 b Fm(r)17 b FC(cos)o(\()p Fm(v)23 b Fn(\000)18 b Fm(`)p FC(\),)28 b Fn(Y)j FC(:=)24 b Fm(r)17 b FC(sin\()p Fm(v)22 b Fn(\000)c Fm(`)p FC(\).)28 b(Let)h(us)f(also)f(denote)h(b)n(y)g(\\)14 b(trig)f(")92 3539 y(sin)27 b Fx(or)g FC(cos)g(and)g(b)n(y)h Fn(Z)34 b FC(the)28 b(v)n(ector)e(\()p Fn(X)12 b Fm(;)i Fn(Y)7 b FC(\).)29 b(By)f(\(C.3\),)f(one)h(gets)f(an)g(analytic)g(expression)f (of)i Fm(x)g FC(in)g(terms)f(of)915 3670 y Fi(\020)965 3762 y Fn(Z)7 b Fm(;)37 b FC(cos)1203 3728 y Fo(2)1270 3706 y Fm(i)p 1264 3743 42 4 v 1264 3819 a FC(2)1343 3762 y(trig)14 b Fm(\025;)38 b FC(sin)1698 3727 y Fo(2)1765 3706 y Fm(i)p 1759 3743 V 1759 3819 a FC(2)1838 3762 y(trig)14 b(\()p Fm(\025)19 b FC(+)f(2)p Fm(\020)6 b FC(\))p Fm(;)38 b FC(sin)13 b Fm(i)28 b FC(trig)14 b(\()p Fm(\025)19 b FC(+)f Fm(\020)6 b FC(\))2913 3670 y Fi(\021)2977 3762 y Fm(:)593 b FC(\(C.13\))92 3986 y(By)38 b(geometric)g (considerations,)f Fm(r)16 b FC(cos)d Fm(v)45 b FC(=)c Fm(a)p FC(\(cos)14 b Fm(u)25 b Fn(\000)g Fm(e)p FC(\))39 b(and)g Fm(r)16 b FC(sin)e Fm(v)45 b FC(=)c Fm(a)2619 3917 y Fn(p)p 2688 3917 219 4 v 69 x FC(1)18 b Fn(\000)g Fm(e)2870 3962 y Fo(2)2921 3986 y FC(sin)c Fm(u)p FC(.)38 b(Th)n(us,)h(one)f(gets)g(an)92 4086 y(analytic)27 b(expression)f(of)h Fn(Z)35 b FC(of)27 b(the)h(form)1064 4269 y Fn(Z)h FC(=)23 b Fn(Z)7 b FC(\()p Fm(a;)28 b(e)1474 4234 y Fo(2)1511 4269 y Fm(;)41 b FC(trig)15 b(\()p Fm(u)j Fn(\000)g Fm(`)p FC(\))p Fm(;)27 b(e)2057 4234 y Fo(2)2108 4269 y FC(trig)15 b(\()p Fm(u)j FC(+)g Fm(`)p FC(\))p Fm(;)27 b(e)14 b FC(trig)g Fm(`)p FC(\))g Fm(:)741 b FC(\(C.14\))92 4451 y(By)21 b(geometric)g(considerations,)f Fm(u)7 b Fn(\000)g Fm(`)21 b FC(=)h Fm(e)14 b FC(sin)f Fm(u)p FC(,)22 b(from)f(whic)n(h)h Fm(u)7 b Fn(\000)g Fm(`)19 b FC(results)j(to)f(b)r(e)h(an)g(analytic)f (function)h(of)f Fm(e)14 b FC(trig)g Fm(`)p FC(.)92 4551 y(Hence,)30 b(b)n(y)h(standard)e(trigonometric)g(computations,)h Fm(e)1934 4521 y Fo(2)1985 4551 y FC(trig)14 b(\()p Fm(u)21 b FC(+)f Fm(`)p FC(\))30 b(is)h(pro)n(v)n(ed)e(to)h(b)r(e)h(an)f (analytic)g(function)h(of)92 4650 y Fm(e)14 b FC(trig)g Fm(`)p FC(.)92 4750 y(Th)n(us,)23 b(from)f(\(C.14\),)h(w)n(e)g(get)g (an)g(analytic)f(represen)n(tation)g Fn(Z)29 b FC(=)23 b Fn(Z)7 b FC(\()p Fm(a;)28 b(e)2412 4720 y Fo(2)2449 4750 y Fm(;)f(e)14 b FC(trig)g Fm(`)p FC(\).)23 b(By)g Fm(e)2974 4720 y Fo(2)3034 4750 y FC(=)g(\()p Fm(e)14 b FC(sin)f Fm(`)p FC(\))3389 4720 y Fo(2)3436 4750 y FC(+)d(\()p Fm(e)k FC(cos)e Fm(`)p FC(\))3787 4720 y Fo(2)92 4850 y FC(and)27 b(the)h(\014rst)f(of)h(\(C.8\),)g(it)g(follo)n (ws)e(that)i Fn(Z)i FC(=)23 b Fn(Z)7 b FC(\(\003)p Fm(;)27 b(e)14 b FC(trig)g Fm(`)p FC(\).)92 4949 y(F)-7 b(rom)27 b(\(C.5\),)h(\(C.6\))g(and)f(\(C.9\),)h Fn(Z)i FC(=)22 b Fn(Z)7 b FC(\(\003)p Fm(;)28 b(\025;)g(\021)s(;)g(\030)t FC(\).)p 92 5038 1493 4 v 171 5125 a Fo(26)241 5155 y FC(By)k(Hamilton)f(equations)g(one)h(sees)f(that)h(:)1701 5133 y(_)1689 5155 y Fm(\025)e FC(=)g Fm(@)1906 5167 y Fo(\003)1955 5063 y Fi(\020)2026 5155 y Fn(\000)2122 5123 y Fo(m)2177 5097 y Fk(3)2209 5123 y Fo(M)2269 5097 y Fk(2)p 2122 5137 180 4 v 2156 5184 a Fo(2\003)2234 5167 y Fk(2)2311 5063 y Fi(\021)2391 5155 y FC(=)2495 5123 y Fo(m)2550 5097 y Fk(3)2582 5123 y Fo(M)2642 5097 y Fk(2)p 2495 5137 V 2546 5184 a Fo(\003)2591 5167 y Fk(3)2684 5155 y FC(,)i(and)2922 5134 y(_)2905 5155 y(\003)d(=)3105 5133 y(_)3087 5155 y Fm(\030)34 b FC(=)45 b(_)-38 b Fm(\021)33 b FC(=)41 b(_)-35 b(p)30 b(=)40 b(_)-33 b(q)30 b(=)f(0.)92 5302 y(Th)n(us,)e(b)n(y)g(the)h(c)n(hain)f(rule,)h Fm(X)h FC(=)23 b(m)14 b(_)-37 b Fm(x)24 b FC(=)e(m\()p Fm(@)1550 5314 y Fl(\025)1594 5302 y Fm(x)p FC(\))1700 5280 y(_)1687 5302 y Fm(\025)i FC(=)1857 5269 y Fo(m)1912 5244 y Fk(4)1944 5269 y Fo(M)2004 5244 y Fk(2)p 1857 5283 V 1907 5331 a Fo(\003)1952 5314 y Fk(3)2057 5269 y Fl(@)t(x)p 2056 5283 79 4 v 2056 5331 a(@)t(\025)2144 5302 y FC(.)1909 5775 y Fw(30)p eop %%Page: 31 31 31 30 bop 92 206 a FC(Hence,)28 b(w)n(e)g(come)g(bac)n(k)g(to)g(the)h (expression)e(in)h(\(C.13\).)h(T)-7 b(o)28 b(complete)g(the)h(pro)r(of) e(of)i(the)g(analytic)e(dep)r(endence)i(of)92 305 y Fm(x)f FC(w.r.t.)f(the)h(P)n(oincar)n(\023)-39 b(e)25 b(v)-5 b(ariables,)26 b(w)n(e)h(need)h(to)g(\014nd)g(an)f(expression)f(of)i (sin)2563 273 y Fl(i)p 2558 287 34 4 v 2558 334 a Fo(2)2629 305 y FC(trig)14 b Fm(\020)35 b FC(and)41 b(trig)3178 273 y Fl(i)p 3173 287 V 3173 334 a Fo(2)3216 305 y FC(.)28 b(F)-7 b(rom)27 b(\(C.6\),)728 548 y(sin)861 491 y Fm(i)p 854 529 42 4 v 854 605 a FC(2)933 548 y(cos)13 b Fm(\020)30 b FC(=)1373 491 y Fm(p)p 1221 529 346 4 v 1221 612 a FC(2)1263 545 y Fn(p)p 1332 545 235 4 v 67 x FC(\003)18 b Fn(\000)g Fm(H)1591 548 y(;)97 b FC(sin)1843 491 y Fm(i)p 1836 529 42 4 v 1836 605 a FC(2)1916 548 y(sin)13 b Fm(\020)30 b FC(=)2347 491 y Fm(q)p 2194 529 346 4 v 2194 612 a FC(2)2236 545 y Fn(p)p 2305 545 235 4 v 67 x FC(\003)18 b Fn(\000)g Fm(H)2564 548 y(;)97 b(H)30 b FC(=)2880 491 y Fm(\030)2920 461 y Fo(2)2976 491 y FC(+)18 b Fm(\021)3103 461 y Fo(2)p 2880 529 261 4 v 2990 605 a FC(2)3164 548 y Fm(:)92 777 y FC(Using)31 b(again)f(trigonometric)g(relations,)g(one)h(sees)g(that)h(sin)13 b Fm(i)h FC(trig)g Fm(\020)38 b FC(is)31 b(an)h(analytic)e(function)i (of)g(\(\003)p Fm(;)14 b(\025;)g(\021)s(;)g(\030)t(;)g(p;)g(q)s FC(\).)92 877 y(Moreo)n(v)n(er,)25 b(b)n(y)i(\(C.6\),)h(\(C.11\))f(and) g(\(C.12\),)h(one)f(has)1463 1111 y(1)18 b Fn(\000)g FC(cos)13 b Fm(i)23 b FC(=)2014 1055 y Fm(p)2056 1025 y Fo(2)2112 1055 y FC(+)18 b Fm(q)2235 1025 y Fo(2)p 1880 1092 526 4 v 1880 1168 a FC(2\003)g Fn(\000)g FC(\()p Fm(\021)2157 1144 y Fo(2)2213 1168 y FC(+)g Fm(\030)2336 1144 y Fo(2)2374 1168 y FC(\))2430 1111 y Fm(;)92 1350 y FC(and)i(an)f(analytic)h(expression)e(of)34 b(trig)1314 1317 y Fl(i)p 1309 1331 34 4 v 1309 1378 a Fo(2)1372 1350 y FC(in)21 b(terms)e(of)h(\(\003)p Fm(;)14 b(\021)s(;)g(\030)t(;)g (p;)g(q)s FC(\))20 b(easily)g(follo)n(ws.)f(Hence,)h(one)g(obtains)f (an)h(analytic)92 1449 y(expression)26 b(of)h Fm(x)i FC(in)e(terms)h(of)f(\(\003)p Fm(;)14 b(\025;)g(\021)s(;)g(\030)t(;)g (p;)g(q)s FC(\).)92 1570 y(F)-7 b(rom)27 b(p)r(oin)n(t)h(\(i\))g(of)g (Remark)e(C.1,)i(one)f(\014nally)g(gets)g(that)h Fm(X)34 b FC(is)28 b(analytic)e(in)i(\(\003)p Fm(;)14 b(\025;)g(\021)s(;)g (\030)t(;)g(p;)g(q)s FC(\).)p 3273 1570 41 83 v 92 1887 a Fu(I.2)37 b(\\Osculating")f(P)m(oincar)n(\023)-54 b(e)37 b(v)-6 b(ariables)92 2056 y Fw(F)e(ollo)m(wing)29 b(P)m(oincar)m(\023) -46 b(e,)32 b(w)m(e)h(in)m(tro)s(duce)f(a)g(new)h(set)g(of)f (action-angle)e(v)-5 b(ariables)31 b(\(linearly)f(related)h(to)92 2189 y(the)i(Delauna)m(y)f(v)-5 b(ariables\),)1161 2093 y Fi(\020)1211 2189 y Fw(\(\003)1317 2153 y Fj(\003)1356 2189 y Fr(;)17 b(H)1489 2153 y Fj(\003)1528 2189 y Fr(;)g(Z)1646 2153 y Fj(\003)1685 2189 y Fw(\))p Fr(;)g Fw(\()p Fr(\025)1862 2153 y Fj(\003)1901 2189 y Fr(;)g(h)2001 2153 y Fj(\003)2040 2189 y Fr(;)g(\020)2135 2153 y Fj(\003)2173 2189 y Fw(\))2211 2093 y Fi(\021)2288 2189 y Fq(2)28 b Fw(\()p Fh(R)2486 2153 y Fs(3)2554 2189 y Fq(\002)22 b Fh(T)2716 2153 y Fs(3)2759 2189 y Fw(\),)33 b(b)m(y)g(letting)704 2494 y(\011)780 2509 y Fs(P)828 2490 y Fc(\003)828 2526 y Fe(aa)925 2494 y Fw(:)1077 2348 y Fi(\()1186 2433 y Fw(\003)1254 2397 y Fj(\003)1321 2433 y Fw(=)27 b(\003)17 b Fr(;)260 b(H)1885 2397 y Fj(\003)1952 2433 y Fw(=)28 b Fr(H)c(;)212 b(Z)2474 2397 y Fj(\003)2541 2433 y Fw(=)27 b Fr(Z)i Fq(\000)23 b Fw(\003)f(+)g Fr(H)i(;)1186 2554 y(\025)1243 2518 y Fj(\003)1310 2554 y Fw(=)j Fr(\025)22 b Fw(+)g Fr(\020)i(;)114 b(h)1854 2518 y Fj(\003)1922 2554 y Fw(=)27 b Fr(h)c Fq(\000)f Fr(\020)i(;)114 b(\020)2462 2518 y Fj(\003)2528 2554 y Fw(=)28 b Fr(\020)c(:)3553 2494 y Fw(\(C.15\))92 2769 y(The)42 b(ph)m(ysical)g(in)m(terpretation)f(of)g (these)i(v)-5 b(ariables)41 b(follo)m(ws)f(from)h(the)h(ab)s(o)m(v)m(e) h(construction.)f(In)92 2889 y(particular,)1208 3010 y Fr(\025)1265 2969 y Fj(\003)1332 3010 y Fw(=)28 b Fr(`)22 b Fw(+)g Fr(g)e(;)114 b(h)1861 2969 y Fj(\003)1928 3010 y Fw(=)28 b Fq(\000)p Fr(g)20 b(;)114 b(\020)2368 2969 y Fj(\003)2435 3010 y Fw(=)27 b Fq(\000)p Fr(\022)20 b(:)846 b Fw(\(C.16\))92 3183 y(Observ)m(e)31 b(that)e Ft(the)g(angles)g Fw(\()p Fr(\025)1220 3147 y Fj(\003)1260 3183 y Fr(;)17 b(h)1360 3147 y Fj(\003)1399 3183 y Fw(\))29 b Ft(are)g(de\014ned)i(in)d(the)i(orbital)d(plane)p Fw(:)i(for)f(this)h (reason,)h(w)m(e)g(shall)92 3316 y(call)20 b(the)i(set)g(of)g(v)-5 b(ariables)1052 3220 y Fi(\020)1102 3316 y Fw(\(\003)1208 3280 y Fj(\003)1247 3316 y Fr(;)17 b(H)1380 3280 y Fj(\003)1418 3316 y Fr(;)g(Z)1536 3280 y Fj(\003)1575 3316 y Fw(\))p Fr(;)g Fw(\()p Fr(\025)1752 3280 y Fj(\003)1791 3316 y Fr(;)g(h)1891 3280 y Fj(\003)1931 3316 y Fr(;)g(\020)2026 3280 y Fj(\003)2064 3316 y Fw(\))2102 3220 y Fi(\021)2173 3316 y Fu(osculating)24 b(action-angle)h(v)-6 b(ariables)3732 3280 y Fs(27)3806 3316 y Fw(.)92 3449 y(Notice)32 b(that,)g(b)m(y)i (\(C.11\),)1045 3667 y Fr(Z)1119 3626 y Fj(\003)1186 3667 y Fw(=)27 b Fr(Z)i Fq(\000)23 b Fw(\003)1553 3580 y Fq(p)p 1636 3580 256 4 v 87 x Fw(1)f Fq(\000)g Fr(e)1851 3638 y Fs(2)1918 3667 y Fw(=)28 b Fq(\000)p Fw(\003)22 b(+)g Fr(O)s Fw(\()p Fr(e)2448 3626 y Fs(2)2487 3667 y Fw(\))g(+)g Fr(O)s Fw(\()p Fr(i)2794 3626 y Fs(2)2833 3667 y Fw(\))92 3885 y(is)32 b(negativ)m(e)h(for)f(small)e(eccen)m (tricit)m(y)j(and)g(inclination.)92 4054 y(In)i(a)g(w)m(a)m(y)i (analogous)d(to)g(\(C.6\),)i(w)m(e)g(can)g(in)m(tro)s(duce)f (symplectic)g Fu(osculating)k(P)m(oincar)n(\023)-54 b(e)39 b(v)-6 b(ari-)92 4187 y(ables)p Fw(,)43 b(\011)481 4202 y Fs(P)529 4183 y Fc(\003)615 4187 y Fw(:)685 4091 y Fi(\020)735 4187 y Fw(\(\003)841 4151 y Fj(\003)880 4187 y Fr(;)17 b(\021)976 4151 y Fj(\003)1015 4187 y Fr(;)g Fw(p)1113 4151 y Fj(\003)1152 4187 y Fw(\))p Fr(;)g Fw(\()p Fr(\025)1329 4151 y Fj(\003)1368 4187 y Fr(;)g(\030)1460 4151 y Fj(\003)1499 4187 y Fr(;)g Fw(q)1594 4151 y Fj(\003)1633 4187 y Fw(\))1671 4091 y Fi(\021)1764 4187 y Fq(7!)1907 4091 y Fi(\020)1956 4187 y Fw(\(\003)2062 4151 y Fj(\003)2101 4187 y Fr(;)g(H)2234 4151 y Fj(\003)2273 4187 y Fr(;)g(Z)2391 4151 y Fj(\003)2430 4187 y Fw(\))p Fr(;)g Fw(\()p Fr(\025)2607 4151 y Fj(\003)2646 4187 y Fr(;)g(h)2746 4151 y Fj(\003)2785 4187 y Fr(;)g(\020)2880 4151 y Fj(\003)2919 4187 y Fw(\))2957 4091 y Fi(\021)3006 4187 y Fw(,)43 b(through)h(the)f(rela-)92 4307 y(tions)459 4566 y Fr(H)548 4525 y Fj(\003)614 4566 y Fw(=)728 4499 y Fr(\021)780 4463 y Fj(\003)819 4458 y Fs(2)880 4499 y Fw(+)22 b Fr(\030)1026 4463 y Fj(\003)1065 4456 y Fs(2)p 728 4543 378 4 v 892 4635 a Fw(2)1147 4566 y Fr(;)1386 4478 y Fq(p)p 1469 4478 177 4 v 88 x Fw(2)p Fr(H)1607 4538 y Fj(\003)1679 4566 y Fw(cos)17 b Fr(h)1882 4525 y Fj(\003)1949 4566 y Fw(=)28 b Fr(\021)2105 4525 y Fj(\003)2161 4566 y Fr(;)2400 4478 y Fq(p)p 2483 4478 V 88 x Fw(2)p Fr(H)2621 4538 y Fj(\003)2692 4566 y Fw(sin)17 b Fr(h)2885 4525 y Fj(\003)2952 4566 y Fw(=)28 b Fr(\030)3104 4525 y Fj(\003)3159 4566 y Fr(;)367 b Fw(\(C.17\))p 92 4702 1493 4 v 171 4763 a Fo(27)241 4794 y FC(Ob)n(viously)-7 b(,)24 b(in)h(the)h(t)n(w)n(o-b)r(o)r(dy)e(problem)h(the)g (\\osculating)f(plane")g(coincides)g(with)i(the)g(orbital)e(plane,)h (but)g(w)n(e)92 4893 y(shall)32 b(use)h(these)h(symplectic)f(v)-5 b(ariables)32 b(also)g(for)g(the)i(spatial)e(three-b)r(o)r(dy)h (problem,)g(where)f(the)i(t)n(w)n(o)e(t)n(w)n(o-b)r(o)r(dy)92 4993 y(systems)f(considered)f(\(star+planet)h Fm(P)1361 5005 y Fl(j)1396 4993 y FC(\))h(will)g(not)f(mo)n(v)n(e)g(on)g(\014xed) h(planes)f(and,)g(in)h(suc)n(h)f(a)g(case,)g(it)h(mak)n(es)e(only)92 5092 y(sense)24 b(to)g(sp)r(eak)g(ab)r(out)h(\\osculating)e(planes";)h (w)n(e)g(also)f(an)n(ticipate)i(that,)g(c)n(ho)r(osing)e(as)h (reference)f(plane)i(the)g(\\total)92 5192 y(angular)k(momen)n(tum)h (plane",)g Fm(\020)1164 5162 y Fy(\003)1233 5192 y FC(will)h(turn)f (out)h(to)f(b)r(e)h(a)f Fx(cyclic)g FC(v)-5 b(ariable)30 b(and,)g(hence,)g Fm(Z)3075 5162 y Fy(\003)3069 5214 y Fl(j)3144 5192 y FC(will)g(b)r(e)h(in)n(tegrals)e(of)92 5303 y(the)f(motions)f(\()p Fx(P)n(oincar)n(\023)-39 b(e)24 b(in)n(tegrals)p FC(\).)1909 5775 y Fw(31)p eop %%Page: 32 32 32 31 bop 364 257 a Fq(\000)p Fr(Z)515 216 y Fj(\003)582 257 y Fw(=)696 190 y(p)750 154 y Fj(\003)789 149 y Fs(2)851 190 y Fw(+)22 b(q)1000 154 y Fj(\003)1040 149 y Fs(2)p 696 234 384 4 v 863 325 a Fw(2)1122 257 y Fr(;)1361 172 y Fq(p)p 1444 172 240 4 v 85 x(\000)p Fw(2)p Fr(Z)1644 228 y Fj(\003)1716 257 y Fw(cos)c Fr(\020)1915 216 y Fj(\003)1981 257 y Fw(=)27 b(p)2138 216 y Fj(\003)2195 257 y Fr(;)2433 172 y Fq(p)p 2516 172 V 85 x(\000)p Fw(2)p Fr(Z)2716 228 y Fj(\003)2789 257 y Fw(sin)17 b Fr(\020)2977 216 y Fj(\003)3043 257 y Fw(=)27 b(q)3197 216 y Fj(\003)3254 257 y Fr(:)272 b Fw(\(C.18\))92 462 y(Notice)46 b(that)h(\(C.17\))f (and)h(\(C.18\))g(are)g(singular)e(for)h Fr(H)2289 426 y Fj(\003)2380 462 y Fw(=)52 b(0)47 b(and)f Fr(Z)2881 426 y Fj(\003)2973 462 y Fw(=)51 b(0,)c(resp)s(ectiv)m(ely)3721 426 y Fs(28)3797 462 y Fw(.)92 583 y(Ho)m(w)m(ev)m(er,)k(the)f (Hamiltonian)45 b(formalism)h(in)i(osculating)g(P)m(oincar)m(\023)-46 b(e)49 b(v)-5 b(ariables)48 b(is)h(analytic)e(for)92 703 y(p)146 667 y Fs(2)214 703 y Fw(+)28 b(q)369 667 y Fs(2)451 703 y Fr(>)43 b Fw(0)e(\(whic)m(h,)h(for)f(small)e(eccen)m (tricities)j(and)g(in)e(view)i(of)f(\(C.8\),)h(means)f Fg(for)i(non-zer)-5 b(o)92 824 y(inclinations)p Fw(\):)92 1060 y Fu(Prop)s(osition)35 b(C.2)49 b Fg(Ther)-5 b(e)46 b(exists)g Fw(0)j Fr(<)g(\032)1757 1075 y Fs(0)1846 1060 y Fr(<)1972 960 y Fi(q)p 2055 960 874 4 v 100 x Fw(2)17 b(min)n Fq(f)p Fw(\003)2401 1075 y Fj(\000)2459 1060 y Fr(;)g Fw(\003)2571 1075 y Fs(+)2652 1060 y Fq(\000)22 b Fw(\003)2819 1075 y Fj(\000)2878 1060 y Fq(g)p Fg(,)46 b(such)h(that)g(the)f(sym-)92 1214 y(ple)-5 b(ctic)41 b(tr)-5 b(ansformation)41 b Fw(\011)1148 1229 y Fs(DP)1249 1204 y Fc(\003)1330 1214 y Fw(:)1397 1118 y Fi(\020)1446 1214 y Fw(\(\003)1552 1178 y Fj(\003)1591 1214 y Fr(;)17 b(\021)1687 1178 y Fj(\003)1726 1214 y Fr(;)g Fw(p)1824 1178 y Fj(\003)1864 1214 y Fw(\))p Fr(;)g Fw(\()p Fr(\025)2041 1178 y Fj(\003)2080 1214 y Fr(;)g(\030)2172 1178 y Fj(\003)2210 1214 y Fr(;)g Fw(q)2305 1178 y Fj(\003)2345 1214 y Fw(\))2383 1118 y Fi(\021)2472 1214 y Fq(7!)40 b Fw(\()p Fr(X)r(;)17 b(x)p Fw(\))41 b Fq(2)f(fj)p Fr(x)p Fq(j)g(\025)3395 1171 y Fs(^)-41 b Fp(\032)3425 1148 y Fe(2)p 3345 1191 159 4 v 3345 1248 a Fs(m)3405 1229 y Fe(2)3439 1248 y Fs(M)3513 1214 y Fq(g)42 b Fg(given)92 1334 y(by)1307 1455 y Fw(\011)1383 1470 y Fs(DP)1484 1445 y Fc(\003)1552 1455 y Fw(=)28 b(\011)1732 1470 y Fs(sp)r(c)1858 1455 y Fq(\016)22 b Fw(\011)2006 1470 y Fs(D)2086 1455 y Fq(\016)g Fw(\011)2234 1470 y Fs(P)2282 1451 y Fc(\003)2282 1486 y Fe(aa)2373 1455 y Fq(\016)g Fw(\011)2521 1470 y Fs(P)2569 1451 y Fc(\003)92 1644 y Fg(is)34 b(r)-5 b(e)g(al-analytic)35 b(for)907 1548 y Fi(\020)956 1644 y Fw(\(\003)1062 1608 y Fj(\003)1101 1644 y Fr(;)17 b(\025)1202 1608 y Fj(\003)1241 1644 y Fw(\))p Fr(;)g Fw(\()p Fr(\021)1413 1608 y Fj(\003)1452 1644 y Fr(;)g(\030)1544 1608 y Fj(\003)1582 1644 y Fw(\))p Fr(;)g Fw(\(p)1756 1608 y Fj(\003)1796 1644 y Fr(;)g Fw(q)1891 1608 y Fj(\003)1930 1644 y Fw(\))1968 1548 y Fi(\021)2046 1644 y Fq(2)28 b(C)41 b Fg(wher)-5 b(e)35 b Fq(C)f Fw(=)27 b Fq(C)6 b Fw(\(\003)2862 1659 y Fj(\000)2921 1644 y Fr(;)17 b Fw(\003)3033 1659 y Fs(+)3092 1644 y Fr(;)g(\032)3186 1659 y Fs(0)3225 1644 y Fw(\))35 b Fg(is)g(the)g(\\c) -5 b(oni-)92 1764 y(c)g(al")34 b(r)-5 b(e)g(gion)34 b(de\014ne)-5 b(d)34 b(as)92 2028 y Fq(C)g Fw(:=)434 1945 y Fi([)318 2130 y Fe(\003)359 2145 y Fc(\000)412 2130 y(\024)465 2117 y Fe(~)460 2130 y(\003)p Fc(\024)p Fe(\003)590 2145 y(+)367 2196 y(0)p Fc(\024)6 b Fe(~)-36 b Ff(\032)q(<\032)558 2211 y Fe(0)668 1932 y Fi(\020)717 2028 y Fq(f)776 2003 y Fw(~)767 2028 y(\003)p Fq(g)21 b(\002)g Fh(T)1067 1932 y Fi(\021)1141 2028 y Fq(\002)g(fj)p Fw(\()p Fr(\021)1407 1987 y Fj(\003)1446 2028 y Fr(;)c(\030)1538 1987 y Fj(\003)1576 2028 y Fw(\))p Fq(j)28 b Fw(=)36 b(~)-58 b Fr(\032)p Fq(g)21 b(\002)1992 1932 y Fi(n)2048 2028 y Fw(\(p)2140 1987 y Fj(\003)2179 2028 y Fr(;)c Fw(q)2274 1987 y Fj(\003)2314 2028 y Fw(\))28 b(:)2478 2003 y(~)2469 2028 y(\003)20 b Fq(\000)2674 1961 y Fw(~)-58 b Fr(\032)2715 1924 y Fs(2)2777 1961 y Fw(+)22 b Fr(\032)2925 1924 y 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2353 436 4 v 1883 2445 a Fw(1)g Fq(\000)h Fr(e)17 b Fw(cos)g Fr(u)2345 2377 y(:)1181 b Fw(\(C.27\))92 2615 y(Hence,)33 b(recalling)c(\(C.10\)) j(and)g(the)g(expressions)h(of)e Fr(u)g Fw(and)h Fr(v)k Fw(ab)s(o)m(v)m(e,)c(one)g(readily)f(expresses)j Fr(v)i Fw(as)92 2735 y(a)c(function)f(of)h(\(\003)771 2699 y Fj(\003)810 2735 y Fr(;)17 b(\025)911 2699 y Fj(\003)950 2735 y Fr(;)g(H)1083 2699 y Fj(\003)1122 2735 y Fr(;)g(h)1222 2699 y Fj(\003)1261 2735 y Fw(\).)32 b(Making)g(use)h(of)f(\(C.16\),)g (one)g(infers)g(that)g Fr(w)3044 2699 y Fj(\003)3111 2735 y Fw(=)27 b Fr(v)e Fq(\000)d Fr(h)3441 2699 y Fj(\003)3480 2735 y Fw(.)33 b(Hence,)92 2856 y(from)38 b(the)h(ab)s(o)m(v)m(e)h (construction)g(and)f(\(C.17\),)h(w)m(e)g(obtain)e(a)h(represen)m (tation)h(of)f Fr(w)3269 2820 y Fj(\003)3347 2856 y Fw(in)f(terms)h(of) 92 2976 y(osculating)31 b(P)m(oincar)m(\023)-46 b(e)33 b(v)-5 b(ariables)31 b(\()p Fr(\025)1452 2940 y Fj(\003)1491 2976 y Fr(;)17 b(\021)1587 2940 y Fj(\003)1626 2976 y Fr(;)g(\030)1718 2940 y Fj(\003)1757 2976 y Fw(\).)523 3228 y Fu(P)-9 b(AR)g(T)36 b(I)s(I)h(-)g(Canonical)g(v)-6 b(ariables)37 b(for)h(the)f(three-b)s(o)s(dy)h(problem)92 3392 y(I)s(I.1)f(The)h(P)m(oincar)n(\023)-54 b(e)36 b(Hamiltonian)f(of) j(the)f(planetary)h(three-b)s(o)s(dy)g(problem)92 3556 y Fw(The)46 b(discussion)f(of)f(this)h(paragraph)f(follo)m(ws)g([Ch88]) h(and)g([L88].)f(W)-8 b(e)46 b(consider)f(three)g(b)s(o)s(dies)92 3677 y Fr(P)155 3692 y Fs(0)194 3677 y Fr(;)33 b(P)317 3692 y Fs(1)373 3677 y Fr(;)17 b(P)480 3692 y Fs(2)549 3677 y Fw(of)30 b(mass)g Fr(m)980 3692 y Fs(0)1019 3677 y Fr(;)k(m)1165 3692 y Fs(1)1221 3677 y Fr(;)17 b(m)1350 3692 y Fs(2)1419 3677 y Fw(in)m(teracting)29 b(through)h(gra)m(vit)m(y) g(\(with)g(constan)m(t)g(of)g(gra)m(vitation)92 3797 y(1\).)i(Assume)h(that)1452 3941 y(\026)-53 b Fr(\024)q(")27 b Fq(\024)1693 3873 y Fr(m)1778 3888 y Fs(1)p 1693 3917 125 4 v 1693 4009 a Fr(m)1778 4024 y Fs(0)1844 3941 y Fr(;)1915 3873 y(m)2000 3888 y Fs(2)p 1915 3917 V 1915 4009 a Fr(m)2000 4024 y Fs(0)2077 3941 y Fq(\024)h Fr(")f(<)h Fw(1)k Fr(;)1086 b Fw(\(C.28\))92 4152 y(for)22 b(a)g(\014xed)i (constan)m(t)g(0)j Fr(<)32 b Fw(\026)-53 b Fr(\024)28 b Fq(\024)g Fw(1)22 b(and)h(a)g(small)d(parameter)j Fr(")k(>)g Fw(0.)c(W)-8 b(e)23 b(consider)g(a)g(\(inertial\))d(frame)92 4272 y Fq(f)p Fr(k)193 4287 y Fs(1)232 4272 y Fr(;)d(k)327 4287 y Fs(2)366 4272 y Fr(;)g(k)461 4287 y Fs(3)499 4272 y Fq(g)35 b Fw(in)f Fh(R)766 4236 y Fs(3)846 4272 y Ft(with)h(origin)d (in)j(the)g(cen)m(ter)h(of)e(mass)h(of)g(the)g(system)h(and)f(with)f(v) m(ertical)g(axis)92 4393 y(parallel)g(to)j(the)h(\(conserv)m(ed\))i (total)c(angular)g(momen)m(tum)p Fw(.)g(Th)m(us,)j(if)d Fr(u)2834 4356 y Fs(\()p Fp(i)p Fs(\))2954 4393 y Fw(denotes)j(the)f(p) s(osition)92 4513 y(of)32 b Fr(P)266 4528 y Fp(i)294 4513 y Fw(,)g Fr(U)429 4477 y Fs(\()p Fp(i)p Fs(\))541 4513 y Fw(:=)27 b Fr(m)756 4528 y Fp(i)802 4513 y Fw(_)-44 b Fr(u)841 4528 y Fp(i)901 4513 y Fw(denotes)33 b(the)g(momen)m(tum)e (of)h Fr(P)2107 4528 y Fp(i)2168 4513 y Fw(and)g Fr(C)39 b Fw(is)32 b(the)h(total)e(angular)g(momen)m(tum,)1554 4778 y Fr(C)j Fw(=)1804 4670 y Fs(2)1762 4695 y Fi(X)1765 4878 y Fp(i)p Fs(=0)1898 4778 y Fr(u)1954 4737 y Fs(\()p Fp(i)p Fs(\))2059 4778 y Fq(\002)23 b Fr(U)2235 4737 y Fs(\()p Fp(i)p Fs(\))2335 4778 y Fr(;)1191 b Fw(\(C.29\))92 5042 y(our)32 b(c)m(hoices)i(imply)1364 5121 y Fs(2)1321 5146 y Fi(X)1324 5328 y Fp(i)p Fs(=0)1458 5229 y Fr(m)1543 5244 y Fp(i)1571 5229 y Fr(u)1627 5188 y Fs(\()p Fp(i)p Fs(\))1738 5229 y Fw(=)27 b(0)32 b Fr(;)2199 5161 y(C)p 2171 5205 133 4 v 2171 5297 a Fq(j)p Fr(C)7 b Fq(j)2341 5229 y Fw(=)28 b Fr(k)2496 5244 y Fs(3)2567 5229 y Fr(:)959 b Fw(\(C.30\))1909 5775 y(35)p eop %%Page: 36 36 36 35 bop 92 206 a Fw(Newton's)45 b(la)m(ws)f(imply)f(that)g(the)i (three-b)s(o)s(dy)f(problem)f(is)h(go)m(v)m(erned)i(b)m(y)f(the)f (nine-degree-of-)92 326 y(freedom)32 b(Hamiltonian)437 526 y(\026)408 551 y Fq(H)493 510 y Fs(tb)563 551 y Fw(\()p Fr(U)677 510 y Fs(\(1\))772 551 y Fr(;)17 b(U)892 510 y Fs(\(2\))987 551 y Fr(;)g(U)1107 510 y Fs(\(3\))1202 551 y Fr(;)g(u)1302 510 y Fs(\(1\))1395 551 y Fr(;)g(u)1495 510 y Fs(\(2\))1589 551 y Fr(;)g(u)1689 510 y Fs(\(3\))1782 551 y Fw(\))28 b(:=)2021 443 y Fs(2)1979 468 y Fi(X)1982 650 y Fp(i)p Fs(=0)2182 484 y Fw(1)p 2125 528 163 4 v 2125 620 a(2)p Fr(m)2259 635 y Fp(i)2297 551 y Fq(j)p Fr(U)2401 510 y Fs(\()p Fp(i)p Fs(\))2485 551 y Fq(j)2513 510 y Fs(2)2574 551 y Fq(\000)2760 468 y Fi(X)2674 651 y Fs(0)p Fj(\024)p Fp(i)g FC(0)e(and)g Fm(\014)43 b(>)37 b FC(0)f(are)g(t)n(w)n(o)g(\\rescaling)e (factors",)h(the)i(Hamiltonian)f(\015o)n(w)g(go)n(v)n(erned)e(b)n(y)92 5072 y(a)e(Hamiltonian)h(function)g Fm(h)p FC(\()p Fm(X)r(;)14 b(x)p FC(\))33 b(\(w.r.t.)h(the)f(symplectic)g(form)f Fm(dX)c Fn(^)23 b Fm(dx)p FC(\))34 b(coincides)e(with)h(the)g (Hamiltonian)92 5172 y(\015o)n(w)28 b(go)n(v)n(erned)e(b)n(y)i(the)h (Hamiltonian)1389 5139 y Fo(1)p 1363 5153 84 4 v 1363 5200 a Fl(\013\014)1457 5172 y Fm(h)p FC(\()p Fm(\013)1615 5151 y FC(~)1590 5172 y Fm(X)7 b(;)14 b(\014)c FC(~)-48 b Fm(x)q FC(\))29 b(\(w.r.t.)f(the)h(symplectic)g(form)f Fm(d)2935 5151 y FC(~)2911 5172 y Fm(X)e Fn(^)19 b Fm(d)5 b FC(~)-47 b Fm(x)q FC(\))29 b(with)g(initial)g(data)116 5275 y(~)92 5296 y Fm(X)6 b FC(\(0\))23 b(=)399 5264 y Fo(1)p 394 5278 44 4 v 394 5325 a Fl(\013)447 5296 y Fm(X)7 b FC(\(0\))27 b(and)33 b(~)-48 b Fm(x)q FC(\(0\))23 b(=)1095 5264 y Fo(1)p 1092 5278 41 4 v 1092 5325 a Fl(\014)1142 5296 y Fm(x)p FC(\(0\).)1909 5775 y Fw(36)p eop %%Page: 37 37 37 36 bop 92 206 a Fw(Recalling)35 b(\(C.28\),)i(it)g(follo)m(ws)f (that)i(m)1583 221 y Fp(i)1648 206 y Fw(and)f(M)1931 221 y Fp(i)1997 206 y Fw(are)h(b)s(ounded)g(and)g(b)s(ounded)g(a)m(w)m (a)m(y)h(from)e(zero)92 326 y(\(uniformly)30 b(in)i Fr(")p Fw(\):)1316 416 y(\026)-53 b Fr(\024)p 1228 460 226 4 v 1228 551 a Fw(1)21 b(+)26 b(\026)-53 b Fr(\024)1490 483 y Fq(\024)28 b Fw(m)1677 498 y Fp(i)1733 483 y Fq(\024)g Fw(1)k 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5257 y Fc(\003)1301 5302 y Fe(1)1342 5265 y Fr(x)1397 5224 y Fs(\(1\))1514 5265 y Fq(\001)54 b Fr(@)1647 5280 y Fp(\025)1688 5257 y Fc(\003)1688 5302 y Fe(2)1729 5265 y Fr(x)1784 5224 y Fs(\(2\))1895 5265 y Fr(d\025)2003 5224 y Fj(\003)2003 5289 y Fs(1)2059 5265 y Fr(d\025)2167 5224 y Fj(\003)2167 5289 y Fs(2)2233 5265 y Fw(=)28 b(0)k Fr(:)1108 b Fw(\(C.42\))1909 5775 y(37)p eop %%Page: 38 38 38 37 bop 92 206 a Fu(I)s(I.2)37 b(Reduction)f(of)i(the)f(angular)h (momen)m(tum)92 371 y Fw(The)32 b(conserv)-5 b(ation)32 b(of)f(the)h(total)e(angular)g(momen)m(tum)g(allo)m(ws)h(to)g(lo)m(w)m (er)h(b)m(y)g(t)m(w)m(o)g(more)f(units)h(the)92 492 y(n)m(um)m(b)s(er) 37 b(of)f(degrees)i(of)e(freedom)g(\(\\)p Ft(reduction)g(of)g(the)h (angular)f(momen)m(tum)p Fw("\).)f(Recall)g(that)h(w)m(e)92 612 y(are)c Fg(excluding)i(planar)g(motions)p Fw(,)e(i.e.,)g(motions)f (with)i(v)-5 b(anishing)31 b(m)m(utual)g(inclinations.)92 821 y Fu(Prop)s(osition)k(C.3)49 b Fg(The)39 b(function)g Fq(H)1591 785 y 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b(asso)s(ciated)h(to)g(the)h(planet)e Fr(P)1544 2870 y Fp(j)1612 2855 y Fw(\(i.e.,)h(the)g(plane)g(spanned)i(b)m(y)f (the)f(p)s(osition)f(and)h(v)m(elo)s(cit)m(y)92 2975 y(of)d Fr(P)262 2990 y Fp(j)327 2975 y Fw(at)h(time)e Fr(t)696 2990 y Fs(0)736 2975 y Fw(\):)i Fr(i)863 2990 y Fp(j)900 2975 y Fw(\()p Fr(t)973 2990 y Fs(0)1012 2975 y Fw(\))g(is)f(the)h(angle)f(b)s(et)m(w)m(een)j(the)f Fq(f)p Fr(k)2227 2990 y Fs(1)2266 2975 y Fr(;)17 b(k)2361 2990 y Fs(2)2400 2975 y Fq(g)p Fw(-plane)27 b(and)i(the)g(plane)g (spanned)h(b)m(y)92 3096 y(the)j(p)s(osition)f(and)h(v)m(elo)s(cit)m(y) g(of)g Fr(P)1359 3111 y Fp(j)1428 3096 y Fw(at)g(time)f Fr(t)1806 3111 y Fs(0)1845 3096 y Fw(.)i(If)f Fr(i)2037 3111 y Fs(1)2105 3096 y Fq(6)p Fw(=)c Fr(i)2243 3111 y Fs(2)2282 3096 y Fw(,)34 b(w)m(e)g(can)f(de\014ne)i Ft(the)e(line)f(of)h(the)g(no)s(des)92 3216 y Fw(as)28 b(the)h(in)m(tersection)f(of)g(the)h(t)m(w)m(o)g(osculating)e(planes;)i (let)e Fr(N)39 b Fq(6)p Fw(=)27 b Fr(P)2539 3231 y Fs(0)2607 3216 y 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(di\013ers)g(from)f(the)i(one)f(in)g([R95])g(for)f(a)h(scaling)f (factor)g(of)h(size)92 5088 y Fm(")131 5058 y Fo(3)168 5088 y Fm(m)241 5058 y Fo(5)241 5109 y(0)278 5088 y FC(.)1909 5775 y Fw(41)p eop %%Page: 42 42 42 41 bop 92 206 a Fw(Let)29 b(us)i(in)m(tro)s(duce)e(some)g(notations) g(\(whic)m(h)h(w)m(e)g(shall)f(k)m(eep)i(similar)26 b(to)j(those)h (used)h(in)e([R95]\):)g(let)92 326 y Fr(\013)d Fw(denote)g(the)g(ratio) e(of)h(the)h(planetary)f(semi-axis,)f Fr(\013)k Fw(:=)g Fr(a)2281 341 y Fs(1)2320 326 y Fr(=a)2420 341 y Fs(2)2460 326 y Fw(,)d(\(recall)f(that)h(b)m(y)i(our)e(assumptions)92 447 y Fr(\013)j Fq(\024)g Fr(\013)349 462 y Fs(max)512 447 y Fr(<)g Fw(1\);)k(let)1534 637 y Fr(h)27 b Fw(:=)1758 570 y Fr(m)1843 585 y Fs(1)p 1758 614 125 4 v 1758 705 a Fr(m)1843 720 y Fs(2)1909 486 y Fi(s)p 1992 486 390 4 v 2002 570 a Fr(m)2087 585 y Fs(0)2149 570 y Fw(+)22 b Fr(m)2332 585 y Fs(2)p 2002 614 370 4 v 2002 705 a Fr(m)2087 720 y Fs(0)2149 705 y Fw(+)g Fr(m)2332 720 y Fs(1)92 868 y 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a(\000)p Fw(1\(2)p Fr(")3216 1355 y Fw(\026)3205 1380 y(\012)3275 1395 y Fp(j)3311 1380 y Fw(\))h(denote)h(the)92 1501 y(eigen)m(v)-5 b(alues)32 b(of)g(the)h(matrix)e Fr(Q)i Fw(in)f(\(C.54\))g(and)h(let)1014 1778 y Fq(L)1083 1793 y Fp(i)1138 1778 y Fw(:=)1279 1710 y(\003)1347 1725 y Fs(1)p 1279 1755 108 4 v 1311 1846 a Fr(c)1407 1753 y Fw(\026)1396 1778 y(\012)1466 1793 y Fp(i)1527 1778 y Fr(;)212 b(c)27 b Fw(:=)h Fq(\000)2294 1710 y Fw(2)p Fr(m)2428 1725 y Fs(1)2467 1710 y Fr(m)2552 1674 y Fs(3)2552 1735 y(2)p 2053 1755 780 4 v 2053 1846 a Fr(")2099 1817 y Fs(4)2139 1846 y Fr(m)2224 1812 y Fs(3)2224 1868 y(0)2263 1846 y Fw(\()p Fr(m)2386 1861 y Fs(0)2448 1846 y Fw(+)22 b Fr(m)2631 1861 y Fs(2)2671 1846 y Fw(\))17 b(\003)2794 1812 y Fs(2)2794 1868 y(2)2875 1778 y Fr(;)651 b Fw(\(C.55\))92 2051 y(\(notice)41 b(that)h Fr(c)h Fw(is)e(a)h(quan)m(tit)m(y)h(of)f(order)g(one\).)g(Finally)-8 b(,)40 b(recall)h(the)h(w)m(ell)g(kno)m(wn)h(de\014nition)e(of)92 2151 y(Laplace)32 b(co)s(e\016cien)m(ts)h(for)f(0)c Fq(\024)g Fr(\013)g(<)g Fw(1:)294 2379 y Fm(b)330 2344 y Fo(\()p Fl(k)q Fo(\))330 2399 y Fl(s)422 2379 y FC(\()p Fm(\013)p FC(\))c(=)f(2)716 2322 y Fm(s)p FC(\()p Fm(s)c FC(+)f(1\))c Fm(:)g(:)g(:)f FC(\()p Fm(s)19 b FC(+)f Fm(k)j Fn(\000)d FC(1\))p 716 2360 804 4 v 1084 2436 a Fm(k)s FC(!)1544 2379 y Fm(\013)1597 2344 y Fl(k)1666 2286 y Fi(\020)1715 2379 y FC(1)g(+)1858 2300 y Fi(X)1862 2478 y Fl(`)p Fy(\025)p Fo(0)1978 2379 y FC(\()2020 2322 y Fm(s)c(:)g(:)g(:)g FC(\()p Fm(s)19 b FC(+)f Fm(`)p FC(\))p 2020 2360 404 4 v 2193 2436 a Fm(`)p FC(!)2460 2322 y(\()p Fm(s)h FC(+)f Fm(k)s FC(\))c Fm(:)g(:)g(:)f FC(\()p Fm(s)19 b FC(+)f Fm(k)j FC(+)d Fm(`)p FC(\))p 2457 2360 768 4 v 2457 2436 a(\()p Fm(k)k FC(+)c(1\))c Fm(:)g(:)g(:)f FC(\()p Fm(k)22 b FC(+)c(1)g(+)g Fm(`)p FC(\))3249 2379 y Fm(\013)3302 2344 y Fo(2\()p Fl(`)p Fo(+1\))3503 2379 y FC(\))3535 2286 y Fi(\021)3599 2379 y Fm(:)92 2701 y Fw(After)33 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4645 y(inf)23 b Fq(L)1249 4660 y Fp(i)1305 4645 y Fr(>)29 b Fw(\026)-51 b Fr(c)28 b(>)f Fw(0)33 b Fr(;)211 b Fw(inf)2036 4548 y Fi(\020)2086 4645 y Fq(L)2155 4660 y Fs(2)2216 4645 y Fq(\000)23 b(L)2385 4660 y Fs(1)2424 4548 y Fi(\021)2501 4645 y Fr(>)30 b Fw(\026)-51 b Fr(c)27 b(>)h Fw(0)17 b Fr(:)682 b Fw(\(C.57\))92 4863 y(Finally)-8 b(,)34 b(b)m(y)39 b(a)e(standard)h(argumen)m(t)f(going)f(bac)m(k)j(to)e(W)-8 b(eierstrass,)38 b(one)g(can)g(\014nd,)g(for)f(an)m(y)h Fg(\014xe)-5 b(d)92 4983 y Fw(\003)160 4947 y Fj(\003)199 4983 y Fw(,)32 b(a)h(linear)e(symplectic)h(transformation)1357 5152 y Fi(\020)1417 5181 y Fr(\021)1469 5145 y Fj(0)1419 5317 y Fr(\030)1467 5288 y Fj(0)1502 5152 y Fi(\021)1579 5248 y Fq(7!)1706 5152 y Fi(\020)1766 5181 y Fr(\021)1818 5145 y Fj(\003)1768 5317 y Fr(\030)1816 5288 y Fj(\003)1867 5152 y Fi(\021)1944 5248 y Fw(=)c Fr(A)p Fw(\(\003)2227 5207 y Fj(\003)2266 5248 y Fw(\))2321 5152 y Fi(\020)2380 5181 y Fr(\021)2432 5145 y Fj(0)2382 5317 y Fr(\030)2430 5288 y Fj(0)2465 5152 y Fi(\021)2531 5248 y Fr(;)995 b Fw(\(C.58\))1909 5775 y(42)p eop %%Page: 43 43 43 42 bop 92 206 a Fw(whic)m(h)33 b(sends)h Fr(f)691 170 y Fj(\003)680 230 y Fs(1)p Fp(;)p Fs(2)807 206 y Fw(in)m(to)645 462 y Fr(f)693 477 y Fs(1)p Fp(;)p Fs(2)815 462 y Fw(:=)988 394 y Fr(c)p 955 438 108 4 v 955 530 a Fw(\003)1023 545 y Fs(1)1089 365 y Fi(\020)1139 462 y Fq(L)1208 477 y Fs(1)1247 462 y Fw(\(\003)1353 420 y Fj(\003)1392 462 y Fw(\))22 b Fq(\001)g Fw(\(\()p Fr(\021)1630 420 y Fj(0)1626 486 y Fs(1)1665 462 y Fw(\))1703 420 y Fs(2)1765 462 y Fw(+)g(\()p Fr(\030)1949 420 y Fj(0)1944 486 y Fs(1)1983 462 y Fw(\))2021 420 y Fs(2)2060 462 y Fw(\))g(+)g Fq(L)2287 477 y Fs(2)2326 462 y Fw(\(\003)2432 420 y Fj(\003)2472 462 y Fw(\))g Fq(\001)f Fw(\(\()p Fr(\021)2709 420 y Fj(0)2705 486 y Fs(2)2745 462 y Fw(\))2783 420 y Fs(2)2844 462 y Fw(+)h(\()p Fr(\030)3028 420 y Fj(0)3023 486 y Fs(2)3062 462 y Fw(\))3100 420 y Fs(2)3140 462 y Fw(\))3178 365 y Fi(\021)3244 462 y Fr(:)282 b 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Fj(0)2771 1188 y Fw(\))g Fr(;)700 b Fw(\(C.60\))92 1408 y(for)32 b(a)g(suitable)g(function)1075 1382 y(^)1068 1408 y Fr(`)p Fw(.)92 1599 y(Letting)f(\()p Fr(I)8 b(;)17 b(')p Fw(\))27 b(:=)h(\(\003)933 1563 y Fj(0)956 1599 y Fr(;)17 b(\025)1057 1563 y Fj(0)1080 1599 y Fw(\),)33 b(\()p Fr(p;)17 b(q)t Fw(\))27 b(:=)g(\()p Fr(\021)1641 1563 y Fj(0)1664 1599 y Fr(;)17 b(\030)1756 1563 y Fj(0)1779 1599 y Fw(\),)32 b Fr(f)1924 1614 y Fs(1)1991 1599 y Fw(:=)c Fr(f)2181 1563 y Fj(\003)2170 1624 y Fs(1)2242 1599 y Fq(\016)22 b Fw(\011)2390 1614 y Fs(W)2468 1599 y Fw(,)32 b Fr(f)2575 1614 y Fs(2)2642 1599 y Fw(:=)c Fr(f)2832 1563 y Fj(\003)2821 1624 y Fs(2)2893 1599 y Fq(\016)22 b Fw(\011)3041 1614 y Fs(W)3119 1599 y Fw(,)1038 1820 y Fr(\033)1093 1835 y Fs(0)1161 1820 y Fw(:=)27 b Fr(\032)1341 1778 y Fj(\003)1381 1774 y Fs(2)1453 1820 y Fr(;)212 b(\016)32 b Fw(:=)27 b Fr(\014)h Fq(\000)23 b Fr(\032)2130 1778 y Fj(\003)2169 1774 y Fs(2)2241 1820 y Fr(;)212 b(\032)2530 1835 y Fs(0)2597 1820 y Fw(:=)28 b Fr(\032)2778 1778 y Fj(\003)2850 1820 y Fr(;)676 b Fw(\(C.61\))92 2040 y(one)33 b(recognizes)g(that)f(the)h(pro)s(of)f(of) g(the)h(Delauna)m(y-P)m(oincar)m(\023)-46 b(e)32 b(Theorem)h(1.1)f(is)g (completed.)253 2472 y Fv(References)92 2670 y FC([A63])g(Arnold,)g(V.) g(I.:)h Fx(Small)f(denominators)f(and)h(problems)g(of)g(stabilit)n(y)g (of)h(motion)f(in)g(classical)f(and)i(celestial)92 2770 y(mec)n(hanics)p FC(,)27 b(Usp)r(ehi)h(Mat.)f(Nauk)h Fz(18)f FC(\(1963\),)f(no.)i(6)f(\(114\),)g(91-192.)92 2894 y([Ch88])35 b(Chenciner,)h(A.:)g Fx(In)n(t)n(\023)-39 b(egration)34 b(du)i(probl)n(\022)-39 b(eme)34 b(de)j(Kepler)e(par)g (la)h(m)n(\023)-39 b(etho)r(de)35 b(de)h(Hamilton-Jacobi:)f(co)r(or-)92 2994 y(donn)n(\023)-39 b(ees)23 b(\\action-angles")e(de)k(Delauna)n(y)f (and)g(co)r(ordonn)n(\023)-39 b(ees)22 b(de)i(P)n(oincar)n(\023)-39 b(e)p FC(,)21 b(Notes)k(scien)n(ti\014ques)f(et)h(tec)n(hniques)f(du)92 3094 y(B.D.L.,)k Fz(S026)f FC(\(1988\).)92 3218 y([E88])j(Eliasson,)f (L.:)j Fx(P)n(erturbations)d(of)i(stable)g(in)n(v)-5 b(arian)n(t)30 b(tori)h(for)f(Hamiltonian)h(systems)p FC(,)g(Ann.)h(Scuola)e(Norm.)92 3318 y(Sup.)e(Pisa,)e(Cl.)i(Sci.,)g Fz(15)f FC(\(1988\),)g(115-147.)92 3442 y([F02])g(F)n(\023)-39 b(ejoz,)26 b(J.:)i Fx(Quasip)r(erio)r(dic)e(motions)h(in)h(the)g (planar)f(three-b)r(o)r(dy)g(problem)p FC(,)g(preprin)n(t)g(\(2002\).) 92 3567 y([H95])g(Herman)g(M.:)h Fx(Priv)-5 b(ate)27 b(comm)n(unication)p FC(,)f(\(1995\).)92 3691 y([JM66])31 b(Je\013erys,)g(W.)i(H.;)f(Moser,)f(J.:)h Fx(Quasi-p)r(erio)r(dic)f (solutions)g(for)h(the)g(three-b)r(o)r(dy)g(problem)p FC(,)g(Astronom.)f(J.)92 3791 y Fz(71)c FC(\(1966\))f(568-578.)92 3916 y([K88])41 b(Kuksin,)g(S.)i(B.:)f Fx(P)n(erturbation)e(theory)h (of)i(conditionally)e(p)r(erio)r(dic)h(solutions)f(of)h (in\014nite-dimensional)92 4015 y(Hamiltonian)25 b(systems)g(and)g(its) h(applications)f(to)g(the)h(Kortew)n(eg-de)d(V)-7 b(ries)25 b(equation)p FC(.)g(\(Russian\))h(Mat.)f(Sb.)h(\(N.S.\))92 4115 y Fz(136)31 b(\(178\))c FC(\(1988\),)f(no.)i(3,)f(396-412,)d(431;) j(translation)f(in)i(Math.)g(USSR-Sb.)g Fz(64)f FC(\(1989\),)f(no.)i (2,)f(397-413.)92 4239 y([L88])38 b(Lask)-5 b(ar,)38 b(J.:)h Fx(Les)g(v)-5 b(ariables)38 b(de)h(P)n(oincar)n(\023)-39 b(e)36 b(et)j(le)h(dev)n(elopp)r(emen)n(t)f(de)g(la)g(fonction)g(p)r (erturbatrice)p FC(,)f(Notes)92 4339 y(scien)n(ti\014ques)27 b(et)h(tec)n(hniques)f(du)h(B.D.L.,)g Fz(S028)f FC(\(1988\).)92 4463 y([LR95])40 b(Lask)-5 b(ar,)39 b(J.;)i(Robutel,)g(P)-7 b(.:)40 b Fx(Stabilit)n(y)h(of)g(the)g(planetary)f(three-b)r(o)r(dy)g (problem.)h(I.)g(Expansion)e(of)i(the)92 4563 y(planetary)26 b(Hamiltonian)p FC(,)i(Celestial)f(Mec)n(h.)g(Dynam.)h(Astronom.)f Fz(62)g FC(\(1995\),)f(no.)i(3,)f(193-217.)92 4688 y([LG00])36 b(Lo)r(catelli,)g(U.;)h(Giorgilli,)f(A.:)i Fx(In)n(v)-5 b(arian)n(t)35 b(tori)i(in)g(the)g(secular)e(motions)i(of)f(the)i (three-b)r(o)r(dy)e(planetary)92 4787 y(systems)p FC(,)27 b(Celestial)g(Mec)n(h.)g(Dynam.)h(Astronom.)f Fz(78)g FC(\(2000\),)g(no.)g(1-4,)f(47-74.)92 4912 y([M65])e(Melnik)n(o)n(v,)f (V.)i(K.:)g Fx(On)f(certain)g(cases)g(of)h(conserv)-5 b(ation)23 b(of)i(almost)f(p)r(erio)r(dic)g(motions)g(with)i(a)e(small) g(c)n(hange)92 5011 y(of)j(the)h(Hamiltonian)g(function)p FC(.)g(\(Russian\))g(Dokl.)f(Ak)-5 b(ad.)28 b(Nauk)f(SSSR)h Fz(165)f FC(\(1965\),)g(1245-1248.)92 5136 y([N77])i(Nekhoroshev)g(N.)h (N.:)h Fx(An)g(exp)r(onen)n(tial)e(estimate)h(of)g(the)h(time)g(of)f (stabilit)n(y)g(of)g(nearly-in)n(tegrable)d(Hamil-)92 5236 y(tonian)g(systems)p FC(,)g(I,)h(Usp.)g(Mat.)f(Nauk.)h Fz(32)f FC(\(1977\),)f(5-66;)g(Russ.)i(Math.)g(Surv.)f Fz(32)g FC(\(1977\),)f(1-65.)1909 5775 y Fw(43)p eop %%Page: 44 44 44 43 bop 92 206 a FC([P)n(oi1905])24 b(P)n(oincar)n(\023)-39 b(e,)25 b(H.:)j Fx(Le\030)-37 b(cons)27 b(de)g(M)n(\023)-39 b(ecanique)26 b(C)n(\023)-39 b(eleste)p FC(,)27 b(T)-7 b(ome)27 b(1,)g(Gauthier-Villars,)g(P)n(aris)e(\(1905\))3463 176 y Fo(32)3532 206 y FC(.)92 330 y([P93])j(P\177)-42 b(osc)n(hel)28 b(J.:)i Fx(Nekhoroshev)e(estimates)h(for)g(quasi-con)n (v)n(ex)e(Hamiltonian)j(Systems)f FC(Math.)h(Zeitsc)n(hrift)g Fz(213)p FC(,)92 430 y(\(1993\),)c(187-216.)92 554 y([P96])d(P\177)-42 b(osc)n(hel,)23 b(J.:)h Fx(A)h(KAM-theorem)f(for)g(some)g(nonlinear)f (PDEs)p FC(,)h(Ann.)h(Scuola)f(Norm.)g(Sup.)h(Pisa,)e(Cl.)i(Sci.,)g Fz(23)92 654 y FC(\(1996\),)h(119-148.)92 779 y([R95])38 b(Robutel,)g(P)-7 b(.:)39 b Fx(Stabilit)n(y)f(of)h(the)g(planetary)e (three-b)r(o)r(dy)h(problem.)g(I)r(I.)h(KAM)g(theory)f(and)g(existence) g(of)92 878 y(quasip)r(erio)r(dic)26 b(motions)p FC(,)i(Celestial)f (Mec)n(h.)g(Dynam.)h(Astronom.)f Fz(62)g FC(\(1995\),)f(no.)i(3,)f (219-261.)p 92 5211 1493 4 v 171 5272 a Fo(32)241 5303 y FC(Av)-5 b(ailable)28 b(on-line)f(at)g FB(http://gallica.bn)o(f.)o (fr)o(/sc)o(ri)o(pts)o(/C)o(on)o(sul)o(ta)o(tio)o(nT)o(out)o(.e)o(xe)o (?E=)o(0&)o(O=N)o(09)o(50)o(10)1909 5775 y Fw(44)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF