This is a multi-part message in MIME format. ---------------9908301058650 Content-Type: text/plain; name="99-317.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-317.keywords" sormal forms, symmetry, nonlinear dynamics, dynamical systems ---------------9908301058650 Content-Type: application/postscript; name="CLG.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="CLG.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: CLG.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips CLG %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1999.08.30:1848 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 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b(is)e(written)h(lo)r(cally)f(in)h(terms)f (of)g(a)h(\(formal\))f(p)r(o)n(w)n(er)f(series)h(expansion)g(as)1494 1669 y(_)-37 b Fl(x)51 b Fo(=)f Fl(f)9 b Fo(\()p Fl(x)p Fo(\))52 b(=)2048 1565 y Fj(1)2022 1590 y Fi(X)2021 1769 y Fk(k)q Fh(=0)2170 1669 y Fl(f)2211 1681 y Fk(k)2251 1669 y Fo(\()p Fl(x)p Fo(\))29 b Fl(;)859 b Fo(\(1\))515 1931 y(where)25 b Fl(x)e Fg(2)h Fp(R)974 1900 y Fk(n)1018 1931 y Fo(,)i Fl(f)32 b Fo(:)23 b Fp(R)1258 1900 y Fk(n)1326 1931 y Fg(!)g Fo(T)p Fp(R)1564 1900 y Fk(n)1632 1931 y Fg(\031)f Fp(R)1791 1900 y Fk(n)1836 1931 y Fo(,)k(and)f Fl(f)2085 1943 y Fk(k)2126 1931 y Fo(\()p Fl(x)p Fo(\))h(is)g (homogeneous)d(of)j(degree)e Fl(k)17 b Fo(+)c(1)515 2030 y(in)26 b Fl(x)p Fo(,)g(i.e.)37 b Fl(f)890 2042 y Fk(k)930 2030 y Fo(\()p Fl(ax)p Fo(\))50 b(=)e Fl(a)1292 2000 y Fh(\()p Fk(k)q Fh(+1\))1483 2030 y Fl(f)1524 2042 y Fk(k)1564 2030 y Fo(\()p Fl(x)p Fo(\).)37 b(Our)25 b(results)g(will)h (apply)g(to)f(the)h(case)f Fl(f)3003 2042 y Fh(0)3040 2030 y Fo(\()p Fl(x)p Fo(\))i(is)e(not)515 2130 y(iden)n(tically)33 b(zero.)55 b(W)-7 b(e)34 b(can)g(equiv)-5 b(alen)n(tly)33 b(consider)g(the)h(v)n(ector)f(\014eld)h(\(VF\))h Fl(X)40 b Fg(\021)33 b Fl(X)3336 2142 y Fk(f)515 2229 y Fo(giv)n(en)27 b(in)g(the)h Fl(x)g Fo(co)r(ordinates)f(b)n(y)1598 2451 y Fl(X)1667 2463 y Fk(f)1760 2451 y Fo(:=)51 b Fl(f)1949 2416 y Fk(i)1976 2451 y Fo(\()p Fl(x)p Fo(\))2135 2395 y Fl(@)p 2112 2432 124 4 v 2112 2508 a(@)5 b(x)2208 2484 y Fk(i)2273 2451 y Fl(:)977 b Fo(\(2\))639 2657 y Fn(R)l(emark)27 b(1.)36 b Fo(Notice)24 b(that)h Fl(f)1522 2669 y Fk(k)1586 2657 y Fo(is)f(homogeneous)e(of)i(degree)f Fl(k)14 b Fo(+)d(1)24 b(rather)e(than)j Fl(k)s Fo(;)g(this)515 2756 y(notation)g(is)g(justi\014ed)i(on)e(the)h(one)g(hand)f(b)n(y)g (the)i(action)e(of)g Fl(f)2486 2768 y Fk(k)2552 2756 y Fo(under)h(the)g(fundamen)n(tal)515 2856 y(Lie-P)n(oisson)32 b(brac)n(k)n(et)1228 2826 y Fh(1)1299 2856 y Fg(f)p Fl(:;)14 b(:)p Fg(g)34 b Fo(de\014ned)i(b)n(y)f Fg(f)p Fl(';)14 b( )s Fg(g)2149 2826 y Fk(i)2211 2856 y Fo(:=)35 b(\()p Fl(')2420 2826 y Fk(k)2485 2856 y Fg(\001)24 b Fl(@)2576 2868 y Fk(k)2617 2856 y Fo(\))p Fl( )2706 2826 y Fk(i)2757 2856 y Fg(\000)f Fo(\()p Fl( )2934 2826 y Fk(k)2999 2856 y Fg(\001)g Fl(@)3089 2868 y Fk(k)3130 2856 y Fo(\))p Fl(')3216 2826 y Fk(i)3245 2856 y Fo(,)37 b(so)515 2956 y(that)32 b Fg(f)p Fl(f)782 2968 y Fk(k)822 2956 y Fl(;)14 b(:)p Fg(g)30 b Fo(:)h Fl(V)1056 2968 y Fk(m)1151 2956 y Fg(!)f Fl(V)1312 2968 y Fk(m)p Fh(+)p Fk(k)1464 2956 y Fo(,)j(where)f Fl(V)1813 2968 y Fk(m)1909 2956 y Fo(is)g(the)g(v)n (ector)f(space)h(of)g(v)n(ector)f(p)r(olynomial)515 3055 y(homogeneous)24 b(of)i(order)f Fl(m)15 b Fo(+)f(1)26 b(in)g(the)h Fl(x)p Fo(;)g(and)f(on)f(the)i(other)e(hand)h(b)n(y)g(the) g(fact)g(that)h(if)515 3155 y(w)n(e)c(rescale)f(co)r(ordinates)g(b)n(y) i Fl(x)f Fo(=)g Fl("\030)k Fo(\(so)c(that)h Fl(\030)k Fo(is)23 b(of)h(order)e(one)h(in)h(the)g(neigh)n(b)r(ourho)r(o)r(d)515 3254 y(of)35 b(the)g(origin)f(of)h(order)f Fl(")g Fo(w)n(e)h(will)g(ha) n(v)n(e)f(to)h(consider\),)h(then)g(\(1\))f(will)g(read)f(as)3222 3233 y(_)3204 3254 y Fl(\030)74 b Fo(=)515 3292 y Fi(P)602 3312 y Fj(1)602 3379 y Fk(k)q Fh(=0)755 3354 y Fl(")794 3324 y Fk(k)848 3354 y Fl(f)889 3366 y Fk(k)930 3354 y Fo(\()p Fl(\030)t Fo(\).)38 b Fg(\014)515 3653 y Fp(2.)68 b(Systems)39 b(equiv)-5 b(arian)m(t)42 b(under)f(simple)d(Lie)i (algebras)h(and)g(classical)f(Lie)515 3753 y(groups.)639 3952 y Fo(W)-7 b(e)37 b(will)f(then)h(consider)e(a)g(simple)i(Lie)f (algebra)e Fg(G)41 b Fo(and)36 b(the)h(\(simply)f(connected,)515 4051 y(compact)c(and)g(actually)h(orthogonal)1730 4021 y Fh(2)1764 4051 y Fo(\))g(simple)g(Lie)g(group)f Fl(G)h Fo(obtained)f(from)g(this)h(b)n(y)515 4151 y(exp)r(onen)n(tiation;)28 b(w)n(e)g(consider)f(for)h(these)g(the)h(fundamen)n(tal)g (\(de\014ning\))g(matrix)e(repre-)515 4251 y(sen)n(tation)20 b(in)h Fp(R)1028 4221 y Fk(n)1073 4251 y Fo(;)i(the)e(represen)n (tation)f(of)g Fl(G)i Fo(in)f Fp(R)2128 4221 y Fk(n)2194 4251 y Fo(will)g(b)r(e)g(giv)n(en)f(b)n(y)h Fl(T)34 b Fo(=)23 b Fg(f)p Fl(T)3031 4263 y Fk(g)3068 4251 y Fl(;)14 b(g)26 b Fg(2)d Fl(G)p Fg(g)p Fo(,)515 4350 y(and)30 b(the)g(Lie)g(algebra)e Fg(G)36 b Fo(will)30 b(act)g(via)f(matrices)g Fl(S)5 b Fo(;)31 b(w)n(e)f(will)g(denote)g(a)g(basis)f(of)h Fg(G)36 b Fo(\(i.e.)515 4450 y(of)h Fg(f)p Fl(S)5 b Fg(g)p Fo(,)39 b(whic)n(h)e(is)g(also)f(the)i(tangen)n(t)f(space)g(to)g Fl(T)48 b Fo(in)38 b Fl(I)46 b Fo(=)39 b Fl(T)2584 4462 y Fk(e)2619 4450 y Fo(\))f(b)n(y)f Fg(f)p Fl(H)2925 4465 y Fh(\(1\))3014 4450 y Fl(;)14 b(:::;)g(H)3226 4465 y Fh(\()p Fk(r)r Fh(\))3314 4450 y Fg(g)p Fo(.)515 4550 y(Equiv)-5 b(alen)n(tly)e(,)25 b Fg(G)31 b Fo(is)26 b(realized)f(b)n(y) g(v)n(ector)g(\014elds)h Fg(f)p Fl(Y)2137 4562 y Fh(1)2174 4550 y Fl(;)14 b(:::;)g(Y)2365 4562 y Fk(r)2401 4550 y Fg(g)26 b Fo(giv)n(en)f(in)h Fl(x)g Fo(co)r(ordinates)e(b)n(y)515 4649 y Fl(Y)563 4661 y Fk(h)629 4649 y Fo(=)f Fl(')771 4619 y Fk(i)771 4673 y(h)814 4649 y Fo(\()p Fl(x)p Fo(\)\()p Fl(@)5 b(=@)g(x)1144 4619 y Fk(i)1172 4649 y Fo(\))28 b(with)g Fl(')1475 4619 y Fk(i)1475 4673 y(h)1519 4649 y Fo(\()p Fl(x)p Fo(\))c(=)e(\()p Fl(H)1842 4664 y Fh(\()p Fk(h)p Fh(\))1938 4649 y Fo(\))1970 4619 y Fk(i)1970 4673 y(k)2011 4649 y Fl(x)2058 4619 y Fk(k)2099 4649 y Fo(.)p 515 4732 1146 4 v 607 4786 a Ff(1)642 4809 y Fe(The)i(homological)f(op)r(erator)h Fd(L)f Fe(is)g(de\014ned)i(in)f (terms)e(of)i(this)f(as)h Fd(L)p Fe(\()p Fc(:)p Fe(\))19 b(:=)h Fd(f)p Fc(f)2735 4818 y Ff(0)2769 4809 y Fc(;)12 b(:)p Fd(g)p Fe(.)607 4865 y Ff(2)642 4888 y Fe(W)-6 b(e)23 b(recall)f(that)i(for)e(eac)n(h)h(simple)e(Lie)i(algebra)f Fd(G)27 b Fe(there)c(exists)g(suc)n(h)g(a)f(group)h(admitting)f Fd(G)k Fe(as)d(the)515 4967 y(group)h(algebra)g(\(Ado's)g(theorem\).) 1926 5255 y Fo(2)p eop %%Page: 3 3 3 2 bop 639 523 a Fo(A)37 b(\(p)r(ossibly)f(formal\))f(dynamical)g (system)h(of)g(the)h(form)e(\(1\))i([v)n(ector)d(\014eld)j(of)f(the)515 623 y(form)24 b(\(2\),)i(p)r(o)n(w)n(er)e(series)g(of)h(the)h(form)e (in)i(the)f(r.h.s.)36 b(of)25 b(\(1\)])g(is)g Fl(G)p Fo(-equiv)-5 b(arian)n(t)24 b(if)i Fg(8)p Fl(g)e Fg(2)g Fl(G)515 722 y Fo(and)33 b Fg(8)p Fl(x)g Fg(2)h Fp(R)970 692 y Fk(n)1015 722 y Fo(,)i Fl(f)9 b Fo(\()p Fl(T)1205 734 y Fk(g)1243 722 y Fl(x)p Fo(\))34 b(=)f Fl(T)1503 734 y Fk(g)1541 722 y Fl(f)9 b Fo(\()p Fl(x)p Fo(\))34 b(\(w)n(e)g(use)g(here)f(linearit)n(y)g(of)h(the)g(action)f(and)h(also) 515 822 y(implicitely)27 b(the)h(standard)e(euclidean)h(connection)f (on)h Fp(R)2376 792 y Fk(n)2448 822 y Fo(to)g(iden)n(tify)h(tangen)n(t) e(spaces)515 922 y(at)h Fl(x)h Fo(and)g(at)f Fl(T)1003 934 y Fk(g)1041 922 y Fl(x)p Fo(\);)i(this)f(implies)g(in)f(particular) g(that)h(it)g(is)f Fg(G)5 b Fo(-symmetric,)27 b(i.e.)37 b(that)1302 1104 y Fg(f)p Fl(f)9 b Fo(\()p Fl(x)p Fo(\))p Fl(;)14 b(')1596 1116 y Fk(h)1640 1104 y Fo(\()p Fl(x)p Fo(\))p Fg(g)51 b Fo(=)f(0)111 b Fg(8)p Fl(h)22 b Fo(=)g(1)p Fl(;)14 b(:::;)g(r)30 b(:)682 b Fo(\(3\))639 1287 y Fn(R)l(emark)36 b(2.)58 b Fo(Notice)34 b(that)g(for)g(groups)f(obtained)h(b)n(y)g (direct)g(exp)r(onen)n(tiation)g(of)g Fg(G)515 1386 y Fo(and)24 b(th)n(us)h(simply)g(connected)g({)f(suc)n(h)h(as)f(e.g.)35 b Fl(S)5 b(O)r Fo(\()p Fl(n)p Fo(\))26 b(or)e Fl(S)5 b(U)k Fo(\()p Fl(n)p Fo(\))25 b({)f(the)h Fl(G)p Fo(-equiv)-5 b(ariance)515 1486 y(is)30 b(implied)g(b)n(y)g Fg(G)5 b Fo(-symmetry)-7 b(,)31 b(while)f(for)f(a)h(m)n(ultiply)h(connected)f (group)f({)g(suc)n(h)h(as)f(e.g.)515 1586 y Fl(O)r Fo(\()p Fl(n)p Fo(\))j(or)f Fl(U)9 b Fo(\()p Fl(n)p Fo(\))31 b({)g(w)n(e)g(should)g(still)h(c)n(hec)n(k)e(that)i Fl(f)40 b Fo(satis\014es)30 b(the)i(equiv)-5 b(ariance)30 b(relation)515 1685 y(with)f(resp)r(ect)f(to)h(one)f(elemen)n(t)h(of)f(eac)n(h)g (connected)g(comp)r(onen)n(t)h(of)f Fl(G)h Fo(\(i.e.)41 b(for)28 b Fl(G=G)3319 1655 y Fh(0)3356 1685 y Fo(,)515 1785 y(with)33 b Fl(G)774 1755 y Fh(0)845 1785 y Fo(the)h(connected)f (comp)r(onen)n(t)g(of)g(the)g(iden)n(tit)n(y)h(in)f Fl(G)p Fo(\).)55 b(Th)n(us,)34 b(for)f(all)f(the)i Fg(G)5 b Fo(-)515 1885 y(symmetric)24 b(NF)h(giv)n(en)e(b)r(elo)n(w,)i(one)f (should)g(still)g(add)g(the)h(restriction)e(to)h(terms)g(allo)n(w)n(ed) 515 1984 y(b)n(y)j(the)h(equiv)-5 b(ariance)26 b(condition)i(under)f (suc)n(h)g(a)h(discrete)f(group.)35 b Fg(\014)639 2084 y Fn(R)l(emark)30 b(3.)39 b Fo(A)29 b(situation)f(whic)n(h)f(is)h(also) f(of)h(ph)n(ysical)f(in)n(terest)h(is)g(the)g(one)g(in)g(whic)n(h)515 2183 y(the)20 b(system)f(is)g(Hamiltonian;)j(or)c(also)h(it)h(enjo)n (ys,)g(b)r(esides)f(the)h(symmetry)f(corresp)r(onding)515 2283 y(to)25 b(the)i(connected)e(Lie)h(group)f(generated)g(b)n(y)g Fg(G)5 b Fo(,)27 b(a)e Fn(r)l(eversing)k(symmetry)d Fo([4,)g(20)o(,)g (21)o(],)g(or)515 2383 y(ev)n(en)h(more)h(generally)e(a)i Fl(k)s Fo(-symmetry)f([19].)38 b(It)29 b(is)f(w)n(ell)g(kno)n(wn)f (that)i(in)f(these)g(case)g(the)515 2482 y(NF)f(will)g(also)f(b)r(e,)h (resp)r(ectiv)n(ely)-7 b(,)27 b(Hamiltonian)f([1,)h(2)o(,)g(17)o(])g (or)f(rev)n(ersible)f([4,)i(12)o(,)g(20)o(,)g(21)o(],)515 2582 y(or)f Fl(k)s Fo(-symmetric)h([19)o(].)37 b Fg(\014)639 2682 y Fo(W)-7 b(e)20 b(will)g(consider)f(the)h(classical)f(groups)f (corresp)r(onding)g(to)h(the)i(simple)e(Lie)h(algebras)515 2781 y Fl(A)577 2793 y Fk(n)622 2781 y Fo(,)26 b Fl(B)734 2793 y Fk(n)780 2781 y Fo(,)g Fl(C)888 2793 y Fk(n)933 2781 y Fo(,)g Fl(D)1051 2793 y Fk(n)1096 2781 y Fo(,)h(and)e(also)g (the)h(exceptional)f(groups)f(corresp)r(onding)g(to)h(the)h(algebras) 515 2881 y Fl(E)576 2893 y Fh(6)613 2881 y Fl(;)14 b(E)711 2893 y Fh(7)749 2881 y Fl(;)g(E)847 2893 y Fh(8)884 2881 y Fl(;)g(F)974 2893 y Fh(4)1012 2881 y Fl(;)g(G)1114 2893 y Fh(2)1151 2881 y Fo(,)36 b(acting)e(through)g(their)h(fundamen)n (tal)f Fn(r)l(e)l(al)h Fo(represen)n(tation)e(\(th)n(us)515 2980 y(for)27 b Fl(S)5 b(U)k Fo(\()p Fl(m)p Fo(\))27 b(groups)g(this)g(will)h(b)r(e)g(2)p Fl(m)p Fo(-dimensional\).)639 3080 y(All)36 b(these)f(group)e(represen)n(tations)g(share)h(a)g (common)h(prop)r(ert)n(y:)50 b(all)35 b(p)r(olynomial)515 3180 y(dynamical)k(systems)h(whic)n(h)g(are)f Fg(G)5 b Fo(-equiv)-5 b(arian)n(t)39 b(\(equiv)-5 b(alen)n(tly)e(,)43 b Fl(G)p Fo(-equiv)-5 b(arian)n(t\))39 b(are)515 3279 y(necessarily)26 b(written)i(in)f(the)h(form)1554 3524 y(_)-38 b Fl(x)52 b Fo(=)1797 3420 y Fk(s)1753 3445 y Fi(X)1754 3621 y Fk(p)p Fh(=0)1900 3524 y Fl(\013)1953 3536 y Fk(p)1992 3524 y Fo(\()p Fl(r)2063 3490 y Fh(2)2101 3524 y Fo(\))14 b Fl(K)2218 3536 y Fk(p)2256 3524 y Fl(x)28 b(;)919 b Fo(\(4\))515 3800 y(where)27 b Fl(r)794 3770 y Fh(2)855 3800 y Fo(=)942 3738 y Fi(P)1030 3759 y Fk(n)1030 3825 y(i)p Fh(=1)1142 3800 y Fo(\()p Fl(x)1221 3770 y Fk(i)1249 3800 y Fo(\))1281 3770 y Fh(2)1319 3800 y Fo(,)g(the)h Fl(\013)1565 3812 y Fk(p)1632 3800 y Fo(are)e(p)r(o)n(w)n(er)h(series)f (in)i Fl(r)2371 3770 y Fh(2)2409 3800 y Fo(,)g(and)f(the)h Fl(K)2835 3812 y Fk(p)2901 3800 y Fo(are)e(a)h(basis)g(of)515 3900 y(matrices)g(comm)n(uting)g(with)h Fl(T)12 b Fo(.)639 4000 y(This)31 b(result)g(can)f(b)r(e)i(obtained)e(in)i(sev)n(eral)d(w) n(a)n(ys)g(exploiting)i(their)g(prop)r(erties,)f(see)515 4099 y([16)o(].)58 b(One)34 b(of)h(the)g(w)n(a)n(ys)e(of)i(pro)n(ving)e (it)i(uses)f(simply)h(the)g(fact)f(that)h(all)g(these)f(group)515 4199 y(represen)n(tations)23 b(are)i(orthogonal)e(\(isometric\))j(and)f (are)g(transitiv)n(e)f(on)h(the)h(unit)h(sphere)515 4299 y Fl(S)571 4268 y Fk(n)p Fj(\000)p Fh(1)746 4299 y Fg(\032)44 b Fp(R)927 4268 y Fk(n)1013 4299 y Fo(of)d(their)g(carrier)e(space.)76 b(Equiv)-5 b(alen)n(tly)e(,)43 b(all)e(the)g Fl(H)2771 4311 y Fk(i)2839 4299 y Fo(are)f(orthogonal)515 4398 y(matrices,)33 b(and)g(the)h(v)n(ectors)d Fl(\021)1524 4368 y Fk(i)1584 4398 y Fo(=)h Fl(H)1750 4413 y Fh(\()p Fk(i)p Fh(\))1829 4398 y Fl(x)i Fo(span)e(the)i(tangen)n(t)e(space)g (to)h(the)h(sphere)e(of)515 4498 y(radius)26 b Fg(j)p Fl(x)p Fg(j)j Fo(in)f Fl(x)p Fo(;)g(see)f([16)o(])h(for)f(more)g (detail.)639 4597 y(Coming)j(bac)n(k)g(to)g(\(4\))h(this)g(means,)g(in) g(other)f(w)n(ords,)g(that)h(the)g Fl(K)2820 4609 y Fk(p)2888 4597 y Fo(span)f(the)h(cen-)515 4697 y(tralizer)f Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))31 b(of)h Fl(T)43 b Fo(in)32 b Fl(GL)p Fo(\()p Fl(n;)14 b Fp(R)p Fo(\);)34 b(w)n(e)d(will)h(tak)n(e) f Fl(K)2246 4709 y Fh(0)2313 4697 y Fo(=)e Fl(I)7 b Fo(,)33 b(and)f Fl(s)21 b Fo(+)g(1)29 b Fg(\021)h Fl(c)p Fo(\()p Fl(T)12 b Fo(\))31 b(is)h(the)515 4797 y(dimension)27 b(of)h Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\).)639 4896 y(It)24 b(happ)r(ens)g(that)f Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))23 b(can)g(assume)g(only)g(a)g(v)n(ery)g(limited)h(set)f(of)h(forms,)g (and)f(corre-)515 4996 y(sp)r(ondingly)h Fl(c)p Fo(\()p Fl(T)12 b Fo(\))22 b Fg(2)i Fp(N)g Fo(a)g(v)n(ery)f(limited)i(set)f(of) g(v)-5 b(alues;)25 b(these)f(are)g(completely)g(classi\014ed)1926 5255 y(3)p eop %%Page: 4 4 4 3 bop 515 523 a Fo(b)n(y)24 b(the)h(real)e(v)n(ersion)g(of)h(Sc)n(h)n (ur)g(Lemma,)h(whic)n(h)f(w)n(e)g(quote)g(here)g(in)h(full)g(generalit) n(y)d(from)515 623 y(Kirillo)n(v)k([18)o(]:)639 722 y Fp(Sc)m(h)m(ur)d(Lemma.)32 b Fn(Supp)l(ose)22 b(that)f(the)h(dimension) h(of)f(the)g(irr)l(e)l(ducible)h(r)l(epr)l(esentation)515 822 y Fl(T)35 b Fn(in)24 b(a)g(line)l(ar)h(sp)l(ac)l(e)f(over)h(the)f (\014eld)g Fp(K)g Fn(is)g(no)g(lar)l(ger)h(than)f(c)l(ountable.)37 b(Then,)26 b(if)f Fp(K)e Fo(=)g Fp(C)p Fn(,)515 922 y Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))22 b Fg(')h Fp(C)p Fn(;)30 b(if)g Fp(K)23 b Fo(=)g Fp(R)p Fn(,)30 b(then)f Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))29 b Fn(is)h(isomorphic)j(to)c(either)i Fp(R)e Fn(or)h Fp(C)g Fn(or)g Fp(H)p Fn(.)639 1021 y Fp(Corollary)-8 b(.)64 b Fn(The)39 b(intertwining)f(numb)l(er)g Fl(c)p Fo(\()p Fl(T)12 b Fo(\))38 b(=)g(dim[)p Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\)])37 b Fn(is)i(e)l(qual)f(to)h(1)f(for)515 1121 y Fp(K)23 b Fo(=)f Fp(C)30 b Fn(and)g(c)l(an)g(take)g(the)g (values)g(1)g(or)g(2)h(or)f(4)g(if)g Fp(K)23 b Fo(=)g Fp(R)p Fn(.)639 1220 y Fo(A)31 b(represen)n(tation)e(o)n(v)n(er)f Fp(R)i Fo(is)h(called)f(of)g(real,)g(complex)g(or)g(quaternionic)f(t)n (yp)r(e)i(ac-)515 1320 y(cording)26 b(to)i(the)g(form)f(of)g Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e([18)o (]:)639 1420 y Fp(Prop)s(osition.)33 b Fn(L)l(et)23 b Fl(T)1371 1432 y Fb(C)1452 1420 y Fn(b)l(e)h(the)g(c)l(omplexi\014c)l (ation)g(of)h(the)e(r)l(e)l(al)h(irr)l(e)l(ducible)h(r)l(epr)l(esen-) 515 1519 y(tation)e Fl(T)12 b Fn(.)35 b(If)23 b Fl(T)34 b Fn(is)23 b(of)h(r)l(e)l(al)f(or)g(c)l(omplex)h(or)f(quaternionic)g (typ)l(e,)i(then)e Fl(T)2760 1531 y Fb(C)2840 1519 y Fn(is,)i(r)l(esp)l(e)l(ctively,)515 1619 y(irr)l(e)l(ducible)i(or)g (the)f(sum)f(of)i(two)g(ine)l(quivalent)f(irr)l(e)l(ducible)i(r)l(epr)l (esentations)e(or)g(the)h(sum)515 1719 y(of)j(two)g(e)l(quivalent)g (irr)l(e)l(ducible)h(r)l(epr)l(esentations.)639 1818 y(R)l(emark)g(4.)41 b Fo(As)28 b(w)n(ell)h(kno)n(wn,)f(and)g(as)g (readily)g(seen)g(from)h(the)g(ab)r(o)n(v)n(e)e(form)n(ulation,)515 1918 y(when)f Fp(K)c Fo(=)h Fp(C)j Fo({)f(i.e.)36 b(for)26 b(complex)f(represen)n(tations)f({)h(the)h(Sc)n(h)n(ur)f(lemma)g(tells) h Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))22 b Fg(\031)515 2017 y Fp(R)p Fo(,)30 b(i.e.)43 b Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))29 b(is)h(trivial;)g(the)g(real)f(case)g(is)g(th)n(us)h(more)f (in)n(teresting.)43 b(It)30 b(is)f(not)h(to)g(us)f(to)515 2117 y(c)n(ho)r(ose)24 b(what)i(kind)f(of)h(represen)n(tation,)e(o)n(v) n(er)g(the)i(reals)e(or)h(o)n(v)n(er)f(the)h(complex)h(n)n(um)n(b)r (ers)515 2217 y(should)21 b(b)r(e)g(considered:)33 b(this)22 b(dep)r(ends)f(on)h(the)f(problems)g(w)n(e)g(are)f(considering,)h(and)g (\014rst)515 2316 y(of)26 b(all)g(on)g(the)h(nature)f(of)g(the)h(v)-5 b(ariables)25 b(in)n(v)n(olv)n(ed.)35 b(Ho)n(w)n(ev)n(er,)25 b(an)n(y)g(represen)n(tation)g(o)n(v)n(er)515 2416 y Fp(C)30 b Fo(can)g(b)r(e)h(written)g(as)f(a)g(represen)n(tation)e(o)n (v)n(er)h Fp(R)p Fo(,)i(so)f(that)h(w)n(e)f(can)g(in)g(full)i (generalit)n(y)515 2516 y(just)c(deal)f(with)h(the)g(latter.)37 b Fg(\014)639 2615 y Fn(R)l(emark)42 b(5.)77 b Fo(The)41 b(\(p)r(olynomial\))g(VFs)g(of)g(the)g(form)f(\(4\))h(are)f(sometimes)g (called)515 2715 y Fn(quasiline)l(ar)p Fo(,)25 b(corresp)r(onding)c(to) h(a)h(mo)r(dule)g(o)n(v)n(er)e(the)i(set)f Fg(I)2409 2727 y Fj(G)2480 2715 y Fo(of)h(\(p)r(olynomial\))f(in)n(v)-5 b(arian)n(t)515 2814 y(functions)28 b(for)f Fg(G)34 b Fo([7)o(],)28 b(whic)n(h)g(in)h(this)f(case)f(\(due)h(to)g(transitivit) n(y)f(o)n(v)n(er)f(the)j(unit)f(sphere\))515 2914 y(is)37 b(simply)g(giv)n(en)f(b)n(y)h(functions)h(of)f Fl(r)42 b Fo(=)c(\()p Fl(x)1967 2884 y Fh(2)1967 2935 y(1)2030 2914 y Fo(+)25 b Fl(:::)g Fo(+)f Fl(x)2350 2884 y Fh(2)2350 2935 y Fk(n)2396 2914 y Fo(\))2428 2884 y Fh(1)p Fk(=)p Fh(2)2532 2914 y Fo(:)56 b(w)n(e)37 b(denote)g(the)h(set)f(of)515 3014 y(quasilinear)32 b(\(p)r(olynomial\))h(equiv)-5 b(arian)n(t)32 b(VFs)i(as)f Fg(Q)2234 3026 y Fj(G)2314 3014 y Fo(:=)f Fg(I)2479 3026 y Fj(G)2549 3014 y Fg(\012)22 b(M)2736 3026 y Fj(G)2783 3014 y Fo(,)35 b(where)e Fg(M)3187 3026 y Fj(G)3268 3014 y Fo(are)515 3113 y(the)h(linear)f(equiv)-5 b(arian)n(t)33 b(VFs,)j Fg(M)1647 3125 y Fj(G)1728 3113 y Fo(=)d Fg(f)p Fl(M)9 b(x)47 b Fo(:)h([)p Fl(M)t(;)14 b(H)2337 3125 y Fk(i)2364 3113 y Fo(])34 b(=)f(0)h Fg(8)p Fl(i)e Fo(=)h(1)p Fl(;)14 b(:::;)g(r)r Fg(g)p Fo(.)56 b(Notice)515 3213 y(that)41 b(dealing)g(with)h(normal)f(forms)g(w)n(e)g (will)g(ha)n(v)n(e)g(to)g(consider)g(p)r(olynomial)f(v)n(ector)515 3313 y(\014elds,)27 b(hence)h(the)f(dep)r(endence)h(on)f Fl(r)1722 3282 y Fh(2)1788 3313 y Fo(instead)g(than)h Fl(r)i Fo(in)e(\(4\).)37 b(The)27 b(ab)r(o)n(v)n(e)f(men)n(tioned)515 3412 y(results)f(ensuring)h(\(4\))g(applies)g(can)f(then)i(b)r(e)g (stated)f(b)n(y)f(sa)n(ying)g(that)h(in)h(the)f(case)g(of)g(the)515 3512 y(de\014ning)f(represen)n(tation)f(of)i(simple)f(Lie)h(algebras)d (\(and)j(groups\),)f(and)g(more)g(generally)515 3611 y(of)j(an)n(y)g(linear)f(represen)n(tation)g(whic)n(h)h(is)g(transitiv) n(e)g(on)g(the)h(unit)g(sphere)e(of)i(its)f(carrier)515 3711 y(space)34 b(\(or)g(equiv)-5 b(alen)n(tly)35 b(satis\014es)f(the)i (other)e(conditions)h(discussed)f(in)i([16)o(]\),)h(all)e(the)515 3811 y(equiv)-5 b(arian)n(t)22 b(v)n(ector)h(\014elds)g(are)g (quasilinear,)g(and)h(all)f(the)h(p)r(olynomial)f(equiv)-5 b(arian)n(t)23 b(VFs)515 3910 y(are)28 b(p)r(olynomial)g(equiv)-5 b(arian)n(t)28 b(quasilinear)f(VFs.)41 b(W)-7 b(e)30 b(will)f(actually)f(use)h(only)g(the)g(fact)515 4010 y(that)f(\(4\))f(represen)n(ts)g(the)g(most)h(general)e(equiv)-5 b(arian)n(t)27 b(VF)h(in)g(these)f(cases.)36 b Fg(\014)639 4110 y Fn(R)l(emark)i(6.)63 b Fo(It)36 b(should)g(b)r(e)g(men)n(tioned) g(that)h(\(4\))f(ensures)f(the)h(DS)h(\(VF\))g(can)f(b)r(e)515 4209 y(written)24 b(in)g(terms)g(of)g(generalized)f(gradien)n(ts)f ([16])i(in)g(terms)g(of)g(\()p Fl(s)11 b Fo(+)g(1\))24 b(scalar)f(functions)515 4309 y Fg(fH)627 4321 y Fh(0)664 4309 y Fo(\()p Fl(r)735 4279 y Fh(2)773 4309 y Fo(\))p Fl(;)14 b(:::;)g Fg(H)1018 4321 y Fk(s)1054 4309 y Fo(\()p Fl(r)1125 4279 y Fh(2)1163 4309 y Fo(\))p Fg(g)34 b Fo(as)48 b(_)-38 b Fl(x)1426 4279 y Fk(i)1522 4309 y Fo(=)1654 4247 y Fi(P)1742 4267 y Fk(s)1742 4334 y(p)p Fh(=0)1878 4309 y Fo(\()p Fl(K)1981 4324 y Fh(\()p Fk(p)p Fh(\))2071 4309 y Fo(\))2103 4279 y Fk(ij)2176 4309 y Fg(r)2245 4321 y Fk(j)2294 4309 y Fg(H)2364 4321 y Fk(p)2402 4309 y Fo(.)56 b(As)34 b Fl(s)23 b Fo(+)f(1)34 b(is)g(nev)n(er)f(higher)515 4408 y(than)23 b Fl(n)p Fo(,)g(it)h(can)e(b)r(e)h(con)n(v)n(enien)n(t)f (to)h(deal)f(with)i(these)f(scalar)e(functions)i(rather)f(than)h(with) 515 4508 y(the)28 b(corresp)r(onding)d(DS)k(or)d(VF,)i(whic)n(h)g(are)f (giv)n(en)f(b)n(y)i Fl(n)p Fo(-dimensional)e(v)n(ectors.)36 b Fg(\014)515 4741 y Fp(3.)41 b(The)32 b(Sc)m(h)m(ur)i (classi\014cation.)639 4907 y Fo(W)-7 b(e)29 b(do)e(no)n(w)g(fo)r(cus)h (on)g(groups)e(rather)h(than)h(algebras)e(\(for)i(ease)f(of)h (language\))e(and)515 5006 y(use)20 b(the)h(classi\014cation)d(pro)n (vided)i(b)n(y)g(Sc)n(h)n(ur)f(Lemma,)j(considering)d(separately)g(the) h(three)1926 5255 y(4)p eop %%Page: 5 5 5 4 bop 515 523 a Fo(p)r(ossible)23 b(cases.)34 b(F)-7 b(or)23 b(ease)g(of)g(notation,)h(w)n(e)f(will)g(write)h Fl(\014)2347 535 y Fk(p)2408 523 y Fo(:=)f Fl(\013)2572 535 y Fk(p)2610 523 y Fo(\(0\),)i(so)d(that)i(the)g(linear)515 623 y(part)j(of)g(\(4\))h(reads)f Fl(f)1182 635 y Fh(0)1219 623 y Fo(\()p Fl(x)p Fo(\))51 b(=)g Fl(Ax)g Fo(=)1772 560 y Fi(P)1860 581 y Fk(s)1860 648 y(p)p Fh(=0)2010 623 y Fl(\014)2057 635 y Fk(p)2109 623 y Fl(K)2180 635 y Fk(p)2232 623 y Fl(x)p Fo(.)778 772 y Fp(3.1.)41 b(Case)31 b(A:)h Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))36 b Fg(')g Fp(R.)639 922 y Fo(In)27 b(this)h(case)e Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))26 b(corresp)r(onds)f(to)i(m)n(ultiples)g(of)g(the)h(iden)n (tit)n(y)-7 b(,)27 b(i.e.)37 b Fl(s)23 b Fo(=)f(0)27 b(and)g(\(4\))515 1021 y(reduces)g(to)1683 1121 y(_)-38 b Fl(x)51 b Fo(=)g Fl(\013)1935 1133 y Fh(0)1972 1121 y Fo(\()p Fl(r)2043 1087 y Fh(2)2081 1121 y Fo(\))14 b Fl(x)29 b Fo(;)1047 b(\(5\))515 1270 y(if)24 b Fl(\014)634 1282 y Fh(0)694 1270 y Fo(=)f(0)g(\(see)h(ab)r(o)n(v)n(e\),)f(the)h (linear)f(part)g(of)h(\(5\))f(v)-5 b(anishes,)24 b(and)g(w)n(e)f (cannot)g(do)h(an)n(ything)515 1370 y(\(at)32 b(least)f(with)i(the)f (standard)f(P)n(oincar)n(\023)-39 b(e)28 b(pro)r(cedure;)33 b(see)f(b)r(elo)n(w\))g(ab)r(out)f(normalizing)515 1469 y(it;)38 b(for)d Fl(\014)813 1481 y Fh(0)885 1469 y Fg(6)p Fo(=)f(0,)i(all)f(the)g(eigen)n(v)-5 b(alues)33 b Fg(f)p Fl(\025)1884 1481 y Fh(1)1922 1469 y Fl(;)14 b(:::;)g(\025)2113 1481 y Fk(n)2158 1469 y Fg(g)34 b Fo(are)g(equal)g(to)h Fl(\014)2762 1481 y Fh(0)2834 1469 y Fg(2)g Fp(R)p Fo(,)h(and)f(th)n (us)515 1569 y(no)26 b(resonance)e(is)j(presen)n(t,)f(and)g(moreo)n(v)n (er)d(the)k(sp)r(ectrum)g(of)f Fl(A)h Fo(b)r(elongs)e(to)h(a)g(P)n (oincar)n(\023)-39 b(e)515 1669 y(domain:)38 b(th)n(us)28 b(w)n(e)g(kno)n(w)f([1,)h(2,)g(5)o(])h(that)f(the)h(system)f(can)g(b)r (e)h(brough)n(t)e(to)h(its)g(P)n(oincar)n(\023)-39 b(e)515 1768 y(normal)28 b(form,)g(whic)n(h)h(coincides)f(with)i(its)e(linear)h (part)42 b(_)-37 b Fl(x)25 b Fo(=)g Fl(Ax)p Fo(,)30 b(b)n(y)e(a)h (transformation)515 1868 y(whic)n(h)e(is)h(con)n(v)n(ergen)n(t)d(in)j (a)f(ball)h(of)f(radius)g Fl(")g Fo(around)g(the)h(origin,)e(with)i Fl(")23 b(>)g Fo(0.)639 1968 y(Notice)41 b(that)g(in)g(this)g(case)e (\(see)i(remark)e(6\))h(w)n(e)h(ha)n(v)n(e)e(a)h(gradien)n(t)g(system,) j(and)515 2067 y(w)n(e)34 b(can)h(equiv)-5 b(alen)n(tly)34 b(Birkho\013)g(normalize)g(the)h(function)h(\(p)r(oten)n(tial\))f Fg(H)2962 2079 y Fh(0)2999 2067 y Fo(\()p Fl(\032)p Fo(\),)j(where)515 2167 y Fl(\032)i Fo(=)f Fl(r)741 2137 y Fh(2)779 2167 y Fo(,)h(reducing)d(it)i(\(in)f(normalized)f(co)r(ordinates)i Fi(b)-49 b Fl(x)p Fo(\))38 b(to)g(its)g(linear)f(part)g(in)43 b Fi(b)-51 b Fl(\032)p Fo(;)43 b(in)515 2267 y(terms)23 b(of)g(p)r(oten)n(tials,)h(this)f(means)g(reducing)g Fg(H)2045 2279 y Fh(0)2106 2267 y Fo(to)g(its)g(quadratic)f(part)h (\(in)h(normalized)515 2366 y(co)r(ordinates\).)639 2466 y(The)33 b(description)f(of)h(nonzero)f(solutions)g(to)g(\(5\))h(is)g (immediate:)48 b Fl(x)p Fo(\()p Fl(t)p Fo(\))33 b(=)e Fl(\032)p Fo(\()p Fl(t)p Fo(\))p Fl(x)p Fo(\(0\),)515 2565 y(where)c Fl(\032)p Fo(\()p Fl(t)p Fo(\))h(is)f(a)h(scalar)e (function)i(solution)f(to)44 b(_)-40 b Fl(\032)23 b Fo(=)g(2)p Fl(\032\013)2341 2577 y Fh(0)2378 2565 y Fo(\()p Fl(\032)p Fo(\))28 b(with)g Fl(\032)p Fo(\(0\))23 b(=)g Fg(j)p Fl(x)p Fo(\(0\))p Fg(j)3161 2535 y Fh(2)639 2665 y Fo(An)28 b(example)g(of)f(o)r(ccurrence)f(of)i(case)f(A)h(is)f(pro)n(vided)g(b)n (y)g Fg(G)h Fo(=)23 b Fl(so)p Fo(\(3\))g Fg(\021)g Fl(B)3033 2677 y Fh(1)3070 2665 y Fo(.)778 2814 y Fp(3.2.)41 b(Case)31 b(B:)g Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))37 b Fg(')f Fp(C.)639 2974 y Fo(No)n(w,)29 b(according)e(to)i(Sc)n(h)n(ur)f(Lemma,)h Fl(T)37 b Fo(=)25 b Fl(T)2109 2986 y Fh(0)2165 2974 y Fg(\010)2263 2953 y Fi(e)2249 2974 y Fl(T)2298 2986 y Fh(0)2363 2974 y Fo(as)k(a)f(complex)h(represen)n(tation)515 3074 y(\(notice)c(this)h(implies)g Fl(n)d Fo(=)g(2)p Fl(m)p Fo(\);)j(using)f(co)r(ordinates)f(corresp)r(onding)g(to)h(this)h (decomp)r(o-)515 3174 y(sition)h(of)h Fl(T)12 b Fo(,)27 b(\(4\))g(reads)1431 3356 y(_)-38 b Fl(x)51 b Fo(=)g Fl(\013)1683 3368 y Fh(0)1720 3356 y Fo(\()p Fl(r)1791 3322 y Fh(2)1829 3356 y Fo(\))14 b Fl(x)47 b Fo(+)f Fl(\013)2133 3368 y Fh(1)2170 3356 y Fo(\()p Fl(r)2241 3322 y Fh(2)2279 3356 y Fo(\))14 b Fl(J)8 b(x)29 b(;)795 b Fo(\(6\))515 3539 y(where)28 b Fl(J)36 b Fo(is)28 b(the)h(standard)f(simplectic)g (matrix)g(in)h(dimension)f Fl(n)c Fo(=)g(2)p Fl(m)p Fo(,)29 b(written)f(\(with)515 3638 y Fl(I)34 b Fo(the)28 b Fl(m)p Fo(-dimensional)f(iden)n(tit)n(y)h(matrix\))f(as)1645 3871 y Fl(J)59 b Fo(=)1865 3754 y Fi(\022)1941 3821 y Fo(0)83 b Fg(\000)p Fl(I)1940 3921 y(I)137 b Fo(0)2188 3754 y Fi(\023)639 4099 y Fo(The)33 b(linear)g(part)f(is)h(giv)n(en)f (b)n(y)h Fl(A)65 b Fo(=)g Fl(\014)1973 4111 y Fh(0)2010 4099 y Fl(I)43 b Fo(+)35 b Fl(\014)2236 4111 y Fh(1)2273 4099 y Fl(J)8 b Fo(;)36 b(if)e Fl(\014)2515 4111 y Fh(0)2584 4099 y Fo(=)d Fl(\014)2727 4111 y Fh(1)2797 4099 y Fo(=)g(0,)j(it)g(v) -5 b(anishes)515 4198 y(and)30 b(w)n(e)f(cannot)h(do)g(an)n(ything)f (to)h(normalize)f(\(6\);)i(for)f Fl(A)d Fg(6)p Fo(=)g(0,)j(w)n(e)g (should)g(distinguish)515 4298 y(among)c(v)-5 b(arious)27 b(sub)r(cases.)639 4398 y Fn(Case)k(B1.)p 639 4411 331 4 v 33 w Fo(If)i Fl(\014)1136 4410 y Fh(0)1204 4398 y Fg(6)p Fo(=)d(0)i(and)g Fl(\014)1586 4410 y Fh(1)1653 4398 y Fo(=)f(0,)h(w)n(e)g(are)f(in)i(the)f(same)g(situation)g(as)f(in) h(case)g(A)515 4497 y(considered)26 b(ab)r(o)n(v)n(e:)35 b(i.e.)i(all)27 b(the)h(eigen)n(v)-5 b(alues)26 b(of)i Fl(A)f Fo(are)g(real)f(and)h(equal)g(to)g Fl(\014)3030 4509 y Fh(0)3068 4497 y Fo(,)g(so)g(they)515 4597 y(b)r(elong)g(to)g(a) g(P)n(oincar)n(\023)-39 b(e)24 b(domain)j(and)h(no)f(resonances)e(are)i (presen)n(t;)g(the)g(system)h(can)f(b)r(e)515 4697 y(normalized)h(b)n (y)g(a)g(con)n(v)n(ergen)n(t)f(transformation)g(\(in)j(a)e(ball)h(of)f (radius)g Fl(")d(>)g Fo(0\),)k(and)g(the)515 4796 y(normal)d(form)i(is) f(linear,)53 b(_)-49 b Fl(w)26 b Fo(=)c Fl(Aw)31 b Fo(in)d(normalized)e (co)r(ordinates.)639 4896 y Fn(Case)31 b(B2.)p 639 4909 V 33 w Fo(If)i Fl(\014)1136 4908 y Fh(0)1204 4896 y Fg(6)p Fo(=)d(0)h(and)h Fl(\014)1585 4908 y Fh(1)1653 4896 y Fg(6)p Fo(=)e(0)i(as)f(w)n(ell,)i(the)f Fl(n)f Fo(=)f(2)p Fl(m)h Fo(eigen)n(v)-5 b(alues)31 b(of)h Fl(A)h Fo(split)515 4995 y(in)38 b(t)n(w)n(o)f(groups:)56 b Fl(m)38 b Fo(eigen)n(v)-5 b(alues)37 b(are)g(equal)g(to)h Fl(\025)2199 5007 y Fk(j)2275 4995 y Fo(=)i Fl(\014)2427 5007 y Fh(0)2489 4995 y Fo(+)25 b Fl(i\014)2655 5007 y Fh(1)2692 4995 y Fo(,)41 b(and)d Fl(m)g Fo(are)f(equal)1926 5255 y(5)p eop %%Page: 6 6 6 5 bop 515 523 a Fo(to)36 b Fl(\025)673 535 y Fk(j)747 523 y Fo(=)i Fl(\014)897 535 y Fh(0)959 523 y Fg(\000)24 b Fl(i\014)1124 535 y Fh(1)1161 523 y Fo(.)64 b(In)37 b(this)g(case)f(the)h(eigen)n(v)-5 b(alues)35 b(do)i(still)g(b)r(elong) f(to)h(a)f(P)n(oincar)n(\023)-39 b(e)515 623 y(domain,)26 b(so)g(that)h(the)g(normalizing)e(transformation)g(is)i(guaran)n(teed)e (to)h(b)r(e)h(con)n(v)n(ergen)n(t)515 722 y(in)k(a)f(ball)h(of)f (radius)g Fl(")e(>)g Fo(0,)j(and)g(they)g(cannot)f(giv)n(e)g(origin)g (to)g(resonances)f([since)i(an)n(y)515 822 y(com)n(bination)h(of)g(the) h Fl(\025)1288 834 y Fk(i)1349 822 y Fo(with)g(in)n(teger)f(co)r (e\016cien)n(ts)g Fl(m)2321 834 y Fk(i)2382 822 y Fo(satisfying)2757 760 y Fi(P)2844 847 y Fk(i)2886 822 y Fl(m)2959 834 y Fk(i)3018 822 y Fo(=)f Fl(m)g(>)g Fo(1,)515 922 y(i.e.)667 859 y Fi(P)755 946 y Fk(i)796 922 y Fl(m)869 934 y Fk(i)897 922 y Fl(\025)945 934 y Fk(i)973 922 y Fo(,)g(has)g(real)e(part)i (equal)f(to)g Fl(m\014)1973 934 y Fh(0)2039 922 y Fg(6)p Fo(=)d(Re\()p Fl(\025)2309 934 y Fk(j)2345 922 y Fo(\)],)32 b(and)f(th)n(us)f(the)h(NF)h(is)e(again)515 1021 y(linear,)53 b(_)-49 b Fl(w)25 b Fo(=)e Fl(Aw)31 b Fo(in)c(normalized)g(co)r (ordinates.)639 1121 y Fn(Case)k(B3.)p 639 1134 331 4 v 34 w Fo(If)h Fl(\014)1136 1133 y Fh(0)1204 1121 y Fo(=)e(0)i(but)h Fl(\014)1577 1133 y Fh(1)1645 1121 y Fg(6)p Fo(=)d(0,)j(the)f(eigen)n (v)-5 b(alues)31 b(of)i Fl(A)f Fo(do)g(again)f(split)h(in)h(t)n(w)n(o) 515 1220 y(groups,)c(with)h Fl(m)g Fo(eigen)n(v)-5 b(alues)28 b Fl(\025)1582 1232 y Fk(j)1645 1220 y Fo(=)e Fl(i\014)1812 1232 y Fh(1)1876 1220 y Fg(\021)g Fl(i!)s Fo(,)k(and)f Fl(m)h Fo(eigen)n(v)-5 b(alues)29 b Fl(\025)2850 1232 y Fk(j)2912 1220 y Fo(=)d Fg(\000)p Fl(i!)s Fo(.)43 b(No)n(w)515 1320 y(the)35 b(eigen)n(v)-5 b(alues)34 b(do)g Fn(not)h Fo(b)r(elong)f(to)h(a)f(P)n(oincar)n(\023)-39 b(e)32 b(domain,)k(and)f(moreo)n(v)n(er)d(they)j(giv)n(e)515 1420 y(origin)26 b(to)i(an)f(in\014nite)h(n)n(um)n(b)r(er)g(of)f (resonances.)639 1519 y(It)h(is)f(easy)g(to)g(c)n(hec)n(k)f(that)i(in)f (this)h(case)e(\(6\))i(is)f(already)f(in)i(NF,)f(and)h(con)n(v)n (ersely)d(the)515 1619 y(most)34 b(general)g(NF)h(with)g(linear)f(part) g Fl(A)h Fo(=)g Fl(\014)2035 1631 y Fh(1)2072 1619 y Fl(J)43 b Fo(\()p Fl(\014)2240 1631 y Fh(1)2312 1619 y Fg(6)p Fo(=)34 b(0\))h(is)g(giv)n(en)e(exactly)h(b)n(y)h(\(6\).)515 1719 y(Th)n(us,)27 b(no)g(\(standard\))h(P)n(oincar)n(\023)-39 b(e-Dulac)24 b(normalization)i(is)h(p)r(ossible)g(here.)639 1818 y(Notice)32 b(that)h(here)f(the)g(linear)g(part)46 b(_)-38 b Fl(x)32 b Fo(=)e Fl(Ax)j Fo(represen)n(ts)e(a)g(Hamiltonian)h (system,)515 1918 y(while)21 b(the)g(full)h(system)e(\(6\))h(is)g(in)g (general)f({)g(i.e.)35 b(unless)21 b Fl(\013)2337 1930 y Fh(0)2374 1918 y Fo(\()p Fl(r)2445 1888 y Fh(2)2484 1918 y Fo(\))i Fg(\021)f Fo(0)f({)f(non)h(Hamiltonian;)515 2017 y(the)28 b(NF)g(is)f(th)n(us)h(also)e(in)i(general)f (non-Hamiltonian.)639 2117 y(In)j(this)f(case)g(w)n(e)g(can)g(apply)g (to)g(the)g(system)g(the)h(\\further)f(normalization")e(pro)r(ce-)515 2217 y(dure)19 b([13)o(];)j(the)e(formal)f(computations)f(are)h (analogous)e(\(and)i(therefore)g(are)f(not)h(rep)r(eated)515 2316 y(here\))32 b(to)h(those)g(for)f Fl(m)g Fo(=)f(1,)j(whic)n(h)f(in) g(this)g(con)n(text)f(can)h(b)r(e)g(called)f(the)i Fl(S)5 b(O)r Fo(\(2\))33 b(case)515 2416 y([13)o(],)40 b(and)e(lead)f(to)h (the)g(same)g(result:)57 b(if)38 b Fl(\013)1971 2428 y Fh(0)2008 2416 y Fo(\()p Fl(r)2079 2386 y Fh(2)2117 2416 y Fo(\))h(is)e(not)h(iden)n(tically)g(zero,)h(w)n(e)e(write)515 2516 y Fl(\013)568 2528 y Fh(0)605 2516 y Fo(\()p Fl(r)676 2485 y Fh(2)714 2516 y Fo(\))24 b(=)857 2453 y Fi(P)945 2540 y Fk(k)1000 2516 y Fl(a)1044 2528 y Fk(k)1084 2516 y Fl(r)1123 2485 y Fh(2)p Fk(k)1198 2516 y Fo(,)i Fl(\013)1300 2528 y Fh(1)1337 2516 y Fo(\()p Fl(r)1408 2485 y Fh(2)1446 2516 y Fo(\))e(=)1589 2453 y Fi(P)1677 2540 y Fk(k)1732 2516 y Fl(b)1768 2528 y Fk(k)1808 2516 y Fl(r)1847 2485 y Fh(2)p Fk(k)1947 2516 y Fo(and)g(denote)h(b)n(y)g Fl(\026;)14 b(\027)30 b Fo(the)25 b(smallest)f(nonzero)515 2615 y Fl(k)34 b Fo(for)d(whic)n(h)h Fl(a)1009 2627 y Fk(k)1081 2615 y Fo(and)g(resp)r(ectiv)n(ely)e Fl(b)1739 2627 y Fk(k)1812 2615 y Fo(are)g(nonzero)g(\(w)n(e)i(assume)f(b)r(oth)h (exist\).)49 b(Then)515 2715 y([13)o(],)639 2814 y Fp(Lemma.)35 b Fn(The)c(system)e(c)l(an)h(b)l(e)g(r)l(e)l(duc)l(e)l(d)g(by)g(a)h(r)l (e)l(cursive)f(se)l(quenc)l(e)f(of)i(\(in)f(gener)l(al,)515 2914 y(only)g(formal\))h(changes)g(of)g(c)l(o)l(or)l(dinates)f(to)g (the)g(form)1127 3097 y Fo(_)-38 b Fl(x)54 b Fo(=)e Fl(\014)1377 3109 y Fh(1)1414 3097 y Fl(J)8 b(x)49 b Fo(+)1676 3029 y Fi(\002)1711 3097 y Fo(\()p Fl(c)1779 3109 y Fh(1)1816 3097 y Fl(r)1855 3062 y Fh(2)p Fk(\026)1952 3097 y Fo(+)18 b Fl(c)2071 3109 y Fh(2)2109 3097 y Fl(r)2148 3062 y Fh(4)p Fk(\026)2226 3097 y Fo(\))p Fl(I)39 b Fo(+)32 b Fl(c)2466 3109 y Fh(3)2503 3097 y Fl(r)2542 3062 y Fh(2)p Fk(\027)2618 3097 y Fl(J)2672 3029 y Fi(\003)2734 3097 y Fl(x)515 3279 y Fn(with)e Fl(c)731 3291 y Fk(i)782 3279 y Fg(2)23 b Fp(R)29 b Fn(and)i Fl(c)1159 3291 y Fh(1)1219 3279 y Fo(=)23 b Fl(a)1351 3291 y Fk(\026)1395 3279 y Fn(;)30 b(if)h Fl(\027)d(>)23 b(\026)p Fn(,)30 b(then)f Fl(c)2013 3291 y Fh(3)2074 3279 y Fo(=)22 b(0)p Fn(.)639 3379 y Fo(It)28 b(should)f(b)r(e)h(stressed)f(that)g(the)h (con)n(v)n(ergence)d(of)j(the)g(renormalizing)d(transforma-)515 3479 y(tion)32 b(has)g(not)g(b)r(een)h(studied)g(y)n(et)f(in)g(general) f(\(see)h(ho)n(w)n(ev)n(er)f([14)o(])h(for)g(partial)f(results\),)515 3578 y(but)25 b(from)g(the)h(explicit)f(form)n(ulae)f(giving)g(the)i (transformation)d(in)j(this)f(case)f([13)o(])i(\(when)515 3678 y(b)r(oth)k Fl(\026)f Fo(and)h Fl(\027)k Fo(exists,)c(i.e.)43 b(in)29 b(particular)g(for)g(non-hamiltonian)f(systems\))h(it)h(app)r (ears)515 3778 y(that)e(no)f(small)g(denominator)g(can)g(ho)n(w)n(ev)n (er)e(o)r(ccurr)i(in)h(it.)639 3877 y Fn(R)l(emark)g(7.)38 b Fo(In)26 b(the)g(hamiltonian)f(case)g(all)h(the)g Fl(a)2225 3889 y Fk(k)2292 3877 y Fo(are)e(zero,)i(and)f(the)i(renormaliza-)515 3977 y(tion)33 b(pro)r(cedure)f(as)h(describ)r(ed)g(in)h([13)o(])f(is)g (ine\013ectiv)n(e,)i(lea)n(ving)d(the)i(standard)e(normal)515 4076 y(form)d(unc)n(hanged)f([14)o(];)j(ho)n(w)n(ev)n(er,)d(the)h(same) g(result)g(is)h(kno)n(wn)e(to)i(hold)f(from)g(di\013eren)n(t)515 4176 y(considerations,)34 b(and)g(is)g(actually)g(a)g(classical)f (result)h(for)f Fl(n)i Fo(=)e(2,)j(i.e.)57 b(one)34 b(degree)f(of)515 4276 y(freedom)27 b([22)o(],)h(recen)n(tly)f(generalized)f(to)h(higher) g(dimension)h([11)o(].)37 b Fg(\014)639 4375 y Fn(R)l(emark)31 b(8.)40 b Fo(Notice)28 b(that)h(in)f(all)g(the)h(B)f(cases,)g Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))28 b(is)g(ab)r(elian;)g(this)h(is)f (connected)515 4475 y(to)f(the)g(fact)g(that)h(no)f(P)n(oincar)n(\023) -39 b(e-Dulac)23 b(normalization)j(is)h(p)r(ossible)f(in)i(the)f(sub)r (case)g(B3,)515 4575 y(nor)c(an)n(y)h(\\further)f(normalization")f(for) i(hamiltonian)g(systems)g(falling)g(in)g(this)h(sub)r(case.)515 4674 y Fg(\014)639 4774 y Fn(R)l(emark)31 b(9.)39 b Fo(The)29 b(in)n(terpretation)e(in)h(terms)g(of)h(\\p)r(oten)n(tials")e(\(or)g (\\Hamiltonians"\))515 4873 y Fg(H)585 4885 y Fk(i)655 4873 y Fo(is)43 b(immediate:)67 b(the)44 b(system)e(\(6\))h(can)f(b)r (e)i(written)f(in)g(terms)f(of)h(t)n(w)n(o)f(of)g(them,)529 4973 y(_)-37 b Fl(x)24 b Fo(=)e Fg(rH)812 4985 y Fh(0)850 4973 y Fo(\()p Fl(r)921 4943 y Fh(2)959 4973 y Fo(\))d(+)f Fl(J)8 b Fg(rH)1286 4985 y Fh(1)1324 4973 y Fo(\()p Fl(r)1395 4943 y Fh(2)1433 4973 y Fo(\);)28 b(in)g(case)f(B1)g(w)n(e)h(can)f(alw) n(a)n(ys)f(normalize)h(so)g(that)h Fg(H)3189 4985 y Fh(1)3250 4973 y Fg(\021)22 b Fo(0)1926 5255 y(6)p eop %%Page: 7 7 7 6 bop 515 523 a Fo(in)31 b(normalized)e(co)r(ordinates,)h(in)h(case)e (B2)h(w)n(e)h(can)f(alw)n(a)n(ys)e(obtain)j Fg(H)2800 535 y Fh(1)2865 523 y Fo(=)c Fl(c)p Fg(H)3063 535 y Fh(0)3131 523 y Fo(in)k(nor-)515 623 y(malized)25 b(co)r(ordinates,)g(and)h (\014nally)f(in)h(case)f(B3)g(w)n(e)h(get)f(no)h(simpli\014cation.)36 b(By)25 b(further)515 722 y(normalization,)g(in)j(the)f(last)g(case)f (B3)h(w)n(e)f(can)h(\(pro)n(vided)f(b)r(oth)i Fl(\026)f Fo(and)g Fl(\027)32 b Fo(exist\))c(reduce)515 822 y(the)23 b(Hamiltonians)g(to)f(the)i(form)e Fg(H)1655 834 y Fh(0)1716 822 y Fo(=)g(\()p Fl(c)1871 834 y Fh(1)1909 822 y Fl(=)p Fo(\(2)p Fl(\026)9 b Fo(+)g(2\)\)\()p Fl(r)2335 792 y Fh(2)2372 822 y Fo(\))2404 792 y Fk(\026)p Fh(+1)2542 822 y Fo(+)g(\()p Fl(c)2684 834 y Fh(2)2721 822 y Fl(=)p Fo(\(4)p Fl(\026)g Fo(+)g(2\)\)\()p Fl(r)3147 792 y Fh(2)3185 822 y Fo(\))3217 792 y Fh(2)p Fk(\026)p Fh(+1)515 922 y Fo(and)27 b Fg(H)746 934 y Fh(1)807 922 y Fo(=)22 b(\()p Fl(c)962 934 y Fh(3)1000 922 y Fl(=)p Fo(\(2)p Fl(\027)h Fo(+)18 b(2\)\)\()p Fl(r)1440 891 y Fh(2)1478 922 y Fo(\))1510 891 y Fk(\027)t Fh(+1)1636 922 y Fo(.)37 b Fg(\014)639 1021 y Fo(An)28 b(example)g(of)f(case)g(B)g(is)h(pro)n(vided)e(b)n(y)i Fg(G)g Fo(=)23 b Fl(so)p Fo(\(2\))g Fg(\021)g Fl(D)2528 1033 y Fh(1)2565 1021 y Fo(.)778 1171 y Fp(3.3.)41 b(Case)31 b(C:)g Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))37 b Fg(')f Fp(H.)639 1320 y Fo(This)k(case)e(is)h(actually)g(quite)g(rare;)44 b(in)c(particular,)g(the)g(only)f(simple)g(group)g(for)515 1420 y(whic)n(h)31 b(the)h(fundamen)n(tal)f(real)f(represen)n(tation)g (is)h(of)h(this)f(t)n(yp)r(e,)i(i.e.)48 b(quaternionic,)31 b(is)515 1519 y Fl(G)e Fo(=)f Fl(S)5 b(U)k Fo(\(2\))30 b(\(with)i Fg(G)i Fo(=)28 b Fl(su)p Fo(\(2\))g Fg(\021)g Fl(A)1737 1531 y Fh(1)1775 1519 y Fo(\):)44 b(that)31 b(is,)h(the)f(quaternion)f(group)g(itself.)47 b(This)515 1619 y(sp)r(ecial)37 b(case)f(is)h(not)h(only)f(needed)g(to)g(complete) h(our)e(classi\014cation,)j(but)f(also)e(quite)515 1719 y(imp)r(ortan)n(t)27 b(mathematically)g(and)h(ph)n(ysically)-7 b(.)639 1818 y(Examples)20 b(of)i(higher)e(represen)n(tations)f(of)i (other)g(classical)f(groups)g(whic)n(h)h(are)f(also)g(of)515 1918 y(quaternionic)j(t)n(yp)r(e)h(exist)f([9],)i(but)f(are)f(of)g (quite)i(high)e(dimension)h(and)g(seem)f(to)h(ha)n(v)n(e)e(no)515 2017 y(sp)r(ecial)29 b(\(including)h(ph)n(ysical\))g(relev)-5 b(ance,)29 b(b)r(eside)h(b)r(eing)g(not)g(relev)-5 b(an)n(t)29 b(to)h(our)f(presen)n(t)515 2117 y(classi\014cation.)35 b(I)28 b(will)g(therefore)e(directly)i(discuss)f(the)h Fl(S)5 b(U)k Fo(\(2\))27 b(case.)639 2217 y(The)h(basis)f(matrices)g Fg(f)p Fl(H)1458 2229 y Fh(1)1494 2217 y Fl(;)14 b(H)1600 2229 y Fh(2)1637 2217 y Fl(;)g(H)1743 2229 y Fh(3)1781 2217 y Fg(g)27 b Fo(for)g Fg(G)h Fo(=)23 b Fl(su)p Fo(\(2\))k(will)h(b) r(e)g(tak)n(en)f(to)g(b)r(e)907 2540 y Fl(H)976 2552 y Fh(1)1036 2540 y Fo(=)1124 2349 y Fi(0)1124 2495 y(B)1124 2548 y(@)1242 2391 y Fo(0)147 b(0)115 b(1)83 b(0)1242 2491 y(0)147 b(0)115 b(0)83 b(1)1210 2590 y Fg(\000)p Fo(1)114 b(0)h(0)83 b(0)1242 2690 y(0)115 b Fg(\000)p Fo(1)82 b(0)h(0)1768 2349 y Fi(1)1768 2495 y(C)1768 2548 y(A)1882 2540 y Fl(;)42 b(H)2016 2552 y Fh(2)2076 2540 y Fo(=)2164 2349 y Fi(0)2164 2495 y(B)2164 2548 y(@)2250 2391 y Fo(0)115 b(0)g(0)82 b Fg(\000)p Fo(1)2250 2491 y(0)115 b(0)g(1)g(0)2250 2590 y(0)83 b Fg(\000)p Fo(1)f(0)115 b(0)2250 2690 y(1)g(0)g(0)g(0)2808 2349 y Fi(1)2808 2495 y(C)2808 2548 y(A)2923 2540 y Fl(;)1447 3014 y(H)1516 3026 y Fh(3)1577 3014 y Fo(=)1664 2822 y Fi(0)1664 2968 y(B)1664 3021 y(@)1751 2864 y Fo(0)82 b Fg(\000)p Fo(1)115 b(0)g(0)1751 2964 y(1)g(0)147 b(0)115 b(0)1751 3064 y(0)g(0)147 b(0)115 b(1)1751 3163 y(0)g(0)g Fg(\000)p Fo(1)82 b(0)2309 2822 y Fi(1)2309 2968 y(C)2309 3021 y(A)2423 3014 y Fl(:)639 3304 y Fo(W)-7 b(e)36 b(ha)n(v)n(e)e(no)n(w)h(four)g(basic)g(matrices)f Fl(K)1978 3316 y Fk(p)2016 3304 y Fo(,)j(with)f Fl(K)2344 3316 y Fh(0)2417 3304 y Fo(=)f Fl(I)7 b Fo(,)38 b(spanning)d Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\).)59 b(The)515 3404 y(three)23 b(matrices)g Fl(K)1123 3416 y Fk(\013)1193 3404 y Fo(\(greek)g(indices) g(run)g(from)h(1)f(to)g(3,)h(latin)g(ones)f(from)g(0)g(to)g(3\),)h (satisfy)515 3504 y Fl(K)592 3473 y Fh(2)586 3524 y Fk(\013)657 3504 y Fo(=)h Fg(\000)p Fl(I)35 b Fo(and)29 b(the)g(quaternionic)f (relations)f Fl(K)2078 3516 y Fk(\013)2139 3504 y Fl(K)2210 3516 y Fk(\014)2308 3504 y Fo(=)53 b Fl(\017)2460 3516 y Fk(\013\014)s(\015)2586 3504 y Fl(K)2657 3516 y Fk(\015)2699 3504 y Fo(,)29 b(and)g(therefore)f(the)515 3603 y Fl(su)p Fo(\(2\))i(comm)n(utation)g(and)h(an)n(ticomm)n(utation)f(relations)f ([)p Fl(K)2494 3615 y Fk(\013)2541 3603 y Fl(;)14 b(K)2649 3615 y Fk(\014)2694 3603 y Fo(])59 b(=)g(2)p Fl(\017)2976 3615 y Fk(\013\014)s(\015)3101 3603 y Fl(K)3172 3615 y Fk(\015)3245 3603 y Fo(and)515 3703 y Fg(f)p Fl(K)628 3715 y Fk(i)654 3703 y Fl(;)14 b(K)762 3715 y Fk(j)797 3703 y Fg(g)34 b Fo(=)f Fg(\000)p Fo(2)p Fl(\016)1115 3715 y Fk(ij)1173 3703 y Fo(.)57 b(Th)n(us,)36 b(they)e(span)g(another) g Fl(su)p Fo(\(2\))g(algebra,)g(not)g(equiv)-5 b(alen)n(t)34 b(to)515 3802 y(the)c(one)g(spanned)f(b)n(y)h(the)g Fl(H)1472 3814 y Fk(\013)1550 3802 y Fo(\(they)g(corresp)r(ond)f(to)g(di\013eren) n(t)i(orien)n(tations)d(of)i(space)515 3902 y(and)d(are)g(dual)g(to)h (eac)n(h)f(other)g([15)o(]\).)639 4002 y(The)h(matrices)f Fl(K)1214 4014 y Fk(\013)1288 4002 y Fo(can)g(b)r(e)h(tak)n(en)f(to)h (b)r(e)905 4317 y Fl(K)976 4329 y Fh(1)1036 4317 y Fo(=)1124 4125 y Fi(0)1124 4271 y(B)1124 4325 y(@)1242 4168 y Fo(0)115 b(1)g(0)g(0)1210 4267 y Fg(\000)p Fo(1)82 b(0)115 b(0)g(0)1242 4367 y(0)g(0)g(0)g(1)1242 4467 y(0)g(0)83 b Fg(\000)p Fo(1)f(0)1768 4125 y Fi(1)1768 4271 y(C)1768 4325 y(A)1882 4317 y Fl(;)42 b(K)2018 4329 y Fh(2)2078 4317 y Fo(=)2165 4125 y Fi(0)2165 4271 y(B)2165 4325 y(@)2284 4168 y Fo(0)147 b(0)115 b(0)83 b(1)2284 4267 y(0)147 b(0)115 b(1)83 b(0)2284 4367 y(0)115 b Fg(\000)p Fo(1)82 b(0)h(0)2252 4467 y Fg(\000)p Fo(1)114 b(0)h(0)83 b(0)2810 4125 y Fi(1)2810 4271 y(C)2810 4325 y(A)2924 4317 y Fl(;)1447 4790 y(K)1518 4802 y Fh(3)1578 4790 y Fo(=)1665 4599 y Fi(0)1665 4745 y(B)1665 4798 y(@)1784 4641 y Fo(0)115 b(0)82 b(1)115 b(0)1784 4741 y(0)g(0)82 b(0)h Fg(\000)p Fo(1)1752 4840 y Fg(\000)p Fo(1)f(0)g(0)115 b(0)1784 4940 y(0)g(1)82 b(0)115 b(0)2310 4599 y Fi(1)2310 4745 y(C)2310 4798 y(A)2424 4790 y Fl(:)1926 5255 y Fo(7)p eop %%Page: 8 8 8 7 bop 639 523 a Fo(Th)n(us,)38 b(the)e(most)f(general)g Fl(S)5 b(U)k Fo(\(2\)-equiv)-5 b(arian)n(t)33 b(DS)k(\(VF\))f (expressed)f(as)g(a)g(p)r(o)n(w)n(er)515 623 y(series)26 b(is)1547 780 y(_)-38 b Fl(x)52 b Fo(=)1789 677 y Fh(3)1746 702 y Fi(X)1747 877 y Fk(p)p Fh(=0)1894 780 y Fl(\013)1947 792 y Fk(p)1985 780 y Fo(\()p Fl(r)2056 746 y Fh(2)2094 780 y Fo(\))14 b Fl(K)2211 792 y Fk(p)2263 780 y Fl(x)28 b Fo(;)912 b(\(7\))515 1024 y(the)33 b(linear)e(part)h(of)h(this)g(is)f (giv)n(en)g(b)n(y)g Fl(A)64 b Fo(=)2035 962 y Fi(P)2122 982 y Fh(3)2122 1049 y Fk(p)p Fh(=0)2272 1024 y Fl(\014)2319 1036 y Fk(p)2358 1024 y Fl(K)2429 1036 y Fk(p)2467 1024 y Fo(,)33 b(and)g(w)n(e)f(will)h(write)f Fl(!)66 b Fo(=)515 1075 y Fi(p)p 598 1075 469 4 v 72 x Fl(\014)649 1119 y Fh(2)645 1169 y(1)705 1147 y Fo(+)18 b Fl(\014)839 1119 y Fh(2)835 1169 y(2)895 1147 y Fo(+)g Fl(\014)1029 1119 y Fh(2)1025 1169 y(3)1066 1147 y Fo(.)69 b(It)38 b(is)g(easily)f(c)n(hec)n(k)n(ed,)i(e.g.)68 b(b)n(y)38 b(explicit)g(computation,)i(that)f(the)515 1247 y(eigen)n(v)-5 b(alues)26 b(of)i Fl(A)g Fo(are)e(equal)h(to)h Fl(\014)1635 1259 y Fh(0)1690 1247 y Fg(\006)18 b Fl(i!)s Fo(.)639 1347 y Fn(R)l(emark)39 b(10.)69 b Fo(By)37 b(a)h(rotation)e(in)i Fp(R)1861 1316 y Fh(4)1936 1347 y Fo({)f(whic)n(h)h(can)f(actually)g(b) r(e)i(realized)d(using)515 1446 y(the)g(matrices)g Fl(K)1079 1458 y Fk(\013)1162 1446 y Fo(themselv)n(es,)i(see)e(section)g(4)f({)h (c)n(hanging)f(the)i(co)r(ordinates)e Fl(x)h Fo(in)n(to)515 1546 y(co)r(ordinates)30 b Fi(e)-49 b Fl(x)q Fo(,)29 b(this)h(can)f(b)r(e)g(reduced)g(to)g(a)g(com)n(bination)f(of)h Fl(K)2615 1558 y Fh(0)2681 1546 y Fo(and)g(one)g(of)g(the)g Fl(K)3309 1558 y Fk(\013)3356 1546 y Fo(,)515 1645 y(sa)n(y)e Fl(A)54 b Fo(=)f Fl(\014)940 1657 y Fh(0)977 1645 y Fl(K)1048 1657 y Fh(0)1118 1645 y Fo(+)33 b Fl(\014)1263 1657 y Fh(1)1300 1645 y Fl(K)1371 1657 y Fh(1)1408 1645 y Fo(,)c(where)1710 1624 y Fi(e)1701 1645 y Fl(\014)1748 1657 y Fh(0)1810 1645 y Fo(=)24 b Fl(\014)1946 1657 y Fh(0)2012 1645 y Fo(and)2184 1624 y Fi(e)2174 1645 y Fl(\014)2221 1657 y Fh(1)2283 1645 y Fo(=)h Fl(!)s Fo(;)k(the)g(general)e(expression)g (of)515 1745 y(\(7\))g(in)h(the)g(new)g(co)r(ordinates)e(will)i(b)r(e)g (of)g(the)g(same)f(kind.)37 b Fg(\014)639 1845 y Fo(As)f(usual,)g(if)g (all)f(the)h Fl(\014)1424 1857 y Fk(p)1497 1845 y Fo(are)f(equal)f(to)i (zero,)g Fl(A)g Fo(=)f(0)g(and)g(there)g(is)g(no)g(P)n(oincar)n(\023) -39 b(e)515 1944 y(normalization)27 b(to)j(b)r(e)f(considered;)g(w)n(e) g(will)h(th)n(us)f(assume)g Fl(A)d Fg(6)p Fo(=)f(0)k(and)g(consider)f (again)515 2044 y(three)f(di\013eren)n(t)h(sub)r(cases)f(dep)r(ending)h (on)f(the)h(actual)f(form)g(of)h Fl(A)p Fo(.)639 2144 y Fn(Case)j(C1.)p 639 2157 331 4 v 33 w Fo(If)i Fl(\014)1137 2156 y Fh(0)1204 2144 y Fg(6)p Fo(=)d(0)h(and)h Fl(!)h Fo(=)d(0,)j(w)n(e)e(are)g(in)h(the)g(same)g(situation)g(as)f(in)h (cases)f(A)515 2243 y(and)c(B1)g(considered)f(ab)r(o)n(v)n(e:)35 b(i.e.)i(all)27 b(the)h(eigen)n(v)-5 b(alues)26 b(of)h Fl(A)h Fo(are)e(real)h(and)g(equal)g(to)g Fl(\014)3319 2255 y Fh(0)3356 2243 y Fo(,)515 2343 y(so)e(they)g(b)r(elong)h(to)f(a) g(P)n(oincar)n(\023)-39 b(e)23 b(domain)i(and)g(no)g(resonances)f(are)h (presen)n(t;)g(the)h(system)515 2442 y(can)31 b(b)r(e)h(normalized)e(b) n(y)h(a)g(con)n(v)n(ergen)n(t)e(transformation)h(\(in)i(a)f(ball)g(of)g (radius)g Fl(")e(>)g Fo(0\),)515 2542 y(and)e(the)h(normal)f(form)g(is) g(linear,)54 b(_)-50 b Fl(w)26 b Fo(=)d Fl(Aw)30 b Fo(in)e(normalized)f (co)r(ordinates.)639 2642 y Fn(Case)k(C2.)p 639 2655 V 35 w Fo(If)i Fl(\014)1139 2654 y Fh(0)1209 2642 y Fg(6)p Fo(=)f(0)h(and)h Fl(!)h Fg(6)p Fo(=)d(0)h(as)g(w)n(ell,)h(the)g Fl(n)f Fo(=)f(2)p Fl(m)h Fo(eigen)n(v)-5 b(alues)32 b(of)h Fl(A)h Fo(split)515 2741 y(in)g(t)n(w)n(o)e(groups:)48 b Fl(m)34 b Fo(eigen)n(v)-5 b(alues)32 b(are)h(equal)g(to)g Fl(\025)2160 2753 y Fk(j)2229 2741 y Fo(=)f Fl(\014)2373 2753 y Fh(0)2433 2741 y Fo(+)22 b Fl(i!)s Fo(,)34 b(and)g Fl(m)f Fo(are)g(equal)g(to)515 2841 y Fl(\025)563 2853 y Fk(j)621 2841 y Fo(=)23 b Fl(\014)756 2853 y Fh(0)809 2841 y Fg(\000)16 b Fl(i!)s Fo(.)36 b(In)27 b(this)g(case)e(the)i (eigen)n(v)-5 b(alues)25 b(do)i(still)f(b)r(elong)h(to)f(a)g(P)n (oincar)n(\023)-39 b(e)23 b(domain,)515 2941 y(so)i(that)h(the)h (normalizing)e(transformation)f(is)i(guaran)n(teed)e(to)i(b)r(e)h(con)n (v)n(ergen)n(t)c(in)k(a)e(ball)515 3040 y(of)k(radius)f Fl(")e(>)f Fo(0,)k(and)g(they)h(cannot)e(giv)n(e)h(origin)f(to)h (resonances,)f(and)h(th)n(us)g(the)g(NF)h(is)515 3140 y(again)c(linear,)53 b(_)-49 b Fl(w)26 b Fo(=)c Fl(Aw)31 b Fo(in)d(normalized)e(co)r(ordinates.)639 3239 y Fn(Case)31 b(C3.)p 639 3252 V 35 w Fo(If)j Fl(\014)1140 3251 y Fh(0)1210 3239 y Fo(=)f(0)g(but)h Fl(!)i Fg(6)p Fo(=)d(0,)h(the)g(eigen)n(v)-5 b(alues)33 b(of)g Fl(A)h Fo(do)g(again)e(split)i(in)g(t)n(w)n(o)515 3339 y(groups,)g(with)h Fl(m)f Fo(eigen)n(v)-5 b(alues)33 b Fl(\025)1601 3351 y Fk(j)1671 3339 y Fo(=)h Fl(i!)s Fo(,)h(and)f Fl(m)h Fo(eigen)n(v)-5 b(alues)33 b Fl(\025)2672 3351 y Fk(j)2741 3339 y Fo(=)h Fg(\000)p Fl(i!)s Fo(.)56 b(No)n(w)34 b(the)515 3439 y(eigen)n(v)-5 b(alues)26 b(do)g(not)h(b)r(elong)g(to)g(a)f(P)n(oincar)n(\023)-39 b(e)24 b(domain,)j(and)g(moreo)n(v)n(er)d(they)j(giv)n(e)f(origin)515 3538 y(to)h(an)h(in\014nite)g(n)n(um)n(b)r(er)f(of)h(resonances.)639 3638 y(Ho)n(w)n(ev)n(er,)d(no)n(w)g Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))25 b(is)g Fn(not)h Fo(ab)r(elian,)g(so)f(the)h(system)g (is)f(in)i(general)d Fn(not)h Fo(already)g(in)515 3738 y(NF.)19 b(T)-7 b(o)19 b(b)r(e)g(more)g(sp)r(eci\014c,)h(it)g(is)f(b)r (etter)g(to)g(use)g(the)g(p)r(ossibilit)n(y)g(to)g(assume)f Fl(A)23 b Fo(=)g Fl(\014)3123 3750 y Fh(0)3160 3738 y Fl(I)8 b Fo(+)q Fl(!)s(J)515 3837 y Fo(with)22 b Fl(J)31 b Fo(=)22 b Fl(K)933 3849 y Fk(\013)1002 3837 y Fo(\(e.g.)35 b Fl(J)c Fo(=)22 b Fl(K)1429 3849 y Fh(1)1466 3837 y Fo(\),)h(see)e(remark)f(10;)j(equiv)-5 b(alen)n(tly)e(,)23 b(consider)d Fl(A)j Fo(=)g Fl(\014)3090 3849 y Fh(0)3127 3837 y Fl(I)13 b Fo(+)6 b Fl(!)s(J)i Fo(,)515 3937 y(where)27 b Fl(J)k Fo(=)920 3875 y Fi(P)1007 3962 y Fk(\013)1054 3937 y Fo(\()p Fl(\014)1133 3949 y Fk(\013)1181 3937 y Fl(=!)s Fo(\))p Fl(K)1381 3949 y Fk(\013)1427 3937 y Fo(.)37 b(In)28 b(the)g(sub)r(case)f(w)n(e)g(are)g(considering,)f Fl(\014)2802 3949 y Fh(0)2862 3937 y Fo(=)d(0.)639 4036 y(No)n(w,)i Fl(A)911 4006 y Fh(+)990 4036 y Fo(=)d Fg(\000)p Fl(A)p Fo(,)k(and)e(the)i(most)e(general)g(matrix)g(in)h Fg(C)5 b Fo(\()p Fl(T)12 b Fo(\))24 b(whic)n(h)h(comm)n(utes)g(with)515 4136 y(this)i Fl(A)h Fo(is)f(a)g(linear)f(com)n(bination)h(of)g Fl(I)34 b Fo(and)27 b Fl(J)8 b Fo(.)37 b(F)-7 b(rom)27 b(this)g(it)h(follo)n(ws)e(at)h(once)g(that)h(the)515 4236 y(NF)g(for)f(\(7\))h({)f(with)h Fl(\014)1225 4248 y Fh(0)1285 4236 y Fo(=)23 b(0)k(and)h Fl(!)d Fg(6)p Fo(=)e(0)k({)g(is)h(of)f(the)h(form)1285 4468 y(_)-50 b Fl(w)54 b Fo(=)c Fl(!)17 b(J)8 b(w)49 b Fo(+)1854 4364 y Fj(1)1828 4389 y Fi(X)1827 4567 y Fk(k)q Fh(=1)1976 4468 y Fl(r)2015 4433 y Fh(2)p Fk(k)2103 4468 y Fo([)p Fl(a)2170 4480 y Fk(k)2211 4468 y Fl(I)25 b Fo(+)18 b Fl(b)2391 4480 y Fk(k)2432 4468 y Fl(J)8 b Fo(])14 b Fl(w)30 b(:)638 b Fo(\(8\))639 4707 y(Notice)38 b(that)g(again)f(here)h (the)g(linear)f(part)52 b(_)-37 b Fl(x)41 b Fo(=)e Fl(Ax)g Fo(represen)n(ts)e(a)g(Hamiltonian)515 4807 y(system,)i(while)f(the)g (full)g(system)f(\(7\))h(is)f(in)g(general)f({)h(i.e.)67 b(unless)37 b Fl(\013)2844 4819 y Fh(0)2881 4807 y Fo(\()p Fl(r)2952 4777 y Fh(2)2991 4807 y Fo(\))i Fg(\021)g Fo(0)e(and)515 4907 y Fl(\013)568 4919 y Fk(\014)613 4907 y Fo(\()p Fl(r)684 4876 y Fh(2)722 4907 y Fo(\))28 b(=)f Fl(c)910 4919 y Fk(\014)955 4907 y Fl(P)12 b Fo(\()p Fl(r)1091 4876 y Fh(2)1129 4907 y Fo(\))31 b(for)f(a)g(single)g(scalar)e (function)j Fl(P)43 b Fo({)30 b(non)g(Hamiltonian;)h(the)g(NF)g(is)515 5006 y(also)26 b(in)i(general)e(non-Hamiltonian,)h(unless)h(all)f(the)h Fl(a)2274 5018 y Fk(k)2342 5006 y Fo(v)-5 b(anish.)1926 5255 y(8)p eop %%Page: 9 9 9 8 bop 639 523 a Fo(Again)32 b(if)g(the)g Fl(a)1157 535 y Fk(k)1230 523 y Fo(are)f(not)g(all)h(zero,)g(w)n(e)f(can)h (\\further)f(normalize")f(this)i(NF;)h(once)515 623 y(again)28 b(the)h(formal)g(computations)g(are)f(exactly)h(the)g(same)g(as)f(in)i (the)g Fl(S)5 b(O)r Fo(\(2\))29 b(case)g([13)o(],)515 722 y(and)40 b(w)n(e)h(arriv)n(e)e(at)i(the)g(same)f(result,)k(i.e.)77 b(the)41 b(statemen)n(t)g(of)g(the)g(ab)r(o)n(v)n(e)e(Lemma.)515 822 y(Notice)31 b(that)h(in)g(this)h(case)d(the)i(initial)g(system)g (has)f(\014rst)h(to)f(b)r(e)h(put)h(in)f(standard)e(NF,)515 922 y(and)i(that)h(the)h(con)n(v)n(ergence)c(of)j(the)g(normalizing)e (transformation)g(for)h(this)h(\014rst)g(step)515 1021 y(is)39 b(not)g(guaran)n(teed)f(a)h(priori)f(\(see)i(section)f(4)g(b)r (elo)n(w\),)j(so)d(that)g(the)h(con)n(v)n(ergence)d(of)515 1121 y(the)f(renormalizing)e(transformation)h(do)r(es)h(not)g(in)g (itself)h(guaran)n(tee)d(that)i(the)h(initial)515 1220 y(system)31 b(is)g(actually)g(conjugated)g(to)h(the)g(renormalized)d (form)j(in)f(an)n(y)g(neigh)n(b)r(ourho)r(o)r(d)515 1320 y(of)c(the)h(origin.)639 1420 y Fn(R)l(emark)e(11.)37 b Fo(The)23 b(in)n(terpretation)g(in)g(terms)g(of)g(\\p)r(oten)n (tials")g(\(or)f(\\Hamiltonians"\))515 1519 y Fg(H)585 1531 y Fk(i)643 1519 y Fo(is)30 b(immediate:)43 b(the)30 b(system)g(\(8\))h(can)f(b)r(e)g(written)h(in)f(terms)g(of)h(four)f(of) g(them,)46 b(_)-38 b Fl(x)28 b Fo(=)515 1557 y Fi(P)602 1644 y Fk(i)644 1619 y Fl(K)715 1631 y Fk(i)742 1619 y Fg(rH)881 1631 y Fk(i)909 1619 y Fo(\()p Fl(r)980 1589 y Fh(2)1018 1619 y Fo(\);)37 b(in)d(case)e(C1)h(w)n(e)g(can)g(alw)n(a)n (ys)f(normalize)g(so)h(that)h Fg(H)2835 1631 y Fh(0)2905 1619 y Fo(=)e(\()p Fl(\014)3081 1631 y Fh(0)3119 1619 y Fl(=)p Fo(2\))p Fg(j)p Fl(w)r Fg(j)3342 1589 y Fh(2)515 1719 y Fo(and)g Fg(H)751 1731 y Fk(\013)829 1719 y Fg(\021)e Fo(0,)j(in)g(case)e(C2)h(w)n(e)g(can)g(alw)n(a)n(ys)e(obtain)i Fg(H)2325 1731 y Fh(0)2393 1719 y Fo(=)e(\()p Fl(\014)2567 1731 y Fh(0)2605 1719 y Fl(=)p Fo(2\))p Fg(j)p Fl(w)r Fg(j)2828 1688 y Fh(2)2865 1719 y Fo(,)k Fg(H)2992 1731 y Fk(\013)3070 1719 y Fo(=)c Fl(c)3201 1731 y Fk(\013)3249 1719 y Fg(H)3319 1731 y Fh(0)3356 1719 y Fo(,)515 1818 y(while)e(in)f(case)g(C3)g(w)n(e)g(are)g(guaran)n(teed)f(to)h(get)h (this)g(situation)f(only)g(formally)-7 b(.)36 b Fg(\014)515 2117 y Fp(4.)41 b(Conclusions.)639 2316 y Fo(W)-7 b(e)41 b(ha)n(v)n(e)e(considered)h(P)n(oincar)n(\023)-39 b(e-Dulac)37 b(normal)i(forms)h(for)g(dynamical)g(systems)515 2416 y(\(v)n(ector)27 b(\014elds,)j(formal)e(p)r(o)n(w)n(er)f(series\))h(in) h Fp(R)1966 2386 y Fk(n)2040 2416 y Fo(in)g(the)g(presence)f(of)h(a)f (symmetry)g(group)515 2516 y Fl(G)i Fo(with)h(Lie)f(algebra)f Fg(G)35 b Fo(when)c Fl(G)f Fo(is)g(a)g(simple)g(Lie)g(groups)f (\(actually)h(an)n(y)f(group)h(with)515 2615 y(transitiv)n(e)d(action)g (on)h Fl(S)1306 2585 y Fk(n)p Fj(\000)p Fh(1)1459 2615 y Fg(\032)c Fp(R)1620 2585 y Fk(n)1664 2615 y Fo(\))29 b(and)e(th)n(us)h Fg(G)34 b Fo(a)27 b(simple)h(Lie)g(algebra)e(of)i (the)h Fl(A)18 b Fg(\000)h Fl(G)515 2715 y Fo(t)n(yp)r(es,)32 b(acting)f(through)f(its)i(fundamen)n(tal)f(\(de\014ning\))h(real)e (represen)n(tation.)47 b(W)-7 b(e)31 b(ha)n(v)n(e)515 2814 y(used)26 b(the)g(fact)h(that)f(in)g(this)h(case)e(the)i(equiv)-5 b(arian)n(t)25 b(dynamical)g(systems)h(\(v)n(ector)f(\014elds,)515 2914 y(formal)i(p)r(o)n(w)n(er)f(series\))h(are)f(of)i(the)g(form)f (\(4\).)639 3014 y(W)-7 b(e)35 b(ha)n(v)n(e)f(completely)h(describ)r (ed)f(the)i(resulting)e(normal)g(form,)i(whic)n(h)f(moreo)n(v)n(er)515 3113 y(are)25 b(guaran)n(teed)f(to)h(b)r(e)i(obtained)e(b)n(y)h(a)f (transformation)f(whic)n(h)i(is)g(con)n(v)n(ergen)n(t)e(in)i(some)515 3213 y(neigh)n(b)r(ourho)r(o)r(d)c(of)h(the)g(origin;)h(in)f(some)g (cases)f(the)h(symmetry)g(guaran)n(tees)e(the)i(system)515 3313 y(is)33 b(necessarily)e(in)j(normal)e(form)h(without)g(the)h(need) f(of)g(an)n(y)f(transformation.)52 b(In)34 b(the)515 3412 y(cases)29 b(B3)g(and)h(C3)g(the)g(normal)f(form)h(con)n(tains)f (an)h(in\014nite)h(n)n(um)n(b)r(er)e(of)h(terms,)h(but)g(it)515 3512 y(is)25 b(p)r(ossible)h(to)f(\\further)g(normalize")f(them)i (\(pass)g(to)f(\\P)n(oincar)n(\023)-39 b(e)22 b(renormalized)i(forms") 515 3611 y([13)o(]\))30 b(obtaining)f(a)g(m)n(uc)n(h)h(simpler)f (expression,)g(except)h(when)g(w)n(e)f(are)g(in)h(case)f(B3)g(and)515 3711 y(the)34 b(system)g(is)g(Hamiltonian;)k(this)c(case)f(is)h(co)n(v) n(ered)f(b)n(y)h(other)f(approac)n(hes,)h(and)g(the)515 3811 y(same)27 b(result)g(applies)g([11)o(,)h(22)o(].)639 3910 y(W)-7 b(e)32 b(ha)n(v)n(e)f(also)g(argued,)h(on)f(the)h(basis)f (of)h(explicit)g(computations)g(presen)n(ted)f(else-)515 4010 y(where)c([13)o(])h(that)g(the)g(\\renormalizing")d (transformations)g(needed)j(in)g(case)f(B3)g(and)h(C3)515 4110 y(can)f(b)r(e)h(c)n(hosen)f(to)g(b)r(e)h(con)n(v)n(ergen)n(t)d(in) j(a)f(neigh)n(b)r(ourho)r(o)r(d)g(of)h(the)g(origin.)639 4209 y(Some)g(limitations)f(of)h(our)f(discussion)f(are)h(as)g(follo)n (ws:)639 4309 y(1\))22 b(W)-7 b(e)22 b(are)f(not)g(able)h(to)f(giv)n(e) g(conditions)g(guaran)n(teeing)f(the)i(con)n(v)n(ergence)d(\(or)i(div)n (er-)515 4408 y(gence\))i(of)g(the)h(standard)f(normalizing)f (transformation)g(in)h(the)h(case)f(C3.)35 b(W)-7 b(e)24 b(ha)n(v)n(e)e(also)515 4508 y(not)27 b(studied)g(in)g(detail)g(the)h (subtleties)f(of)g(the)g(renormalizing)e(pro)r(cedure)h(for)g(Hamilto-) 515 4608 y(nian)h(systems)g(of)h(t)n(yp)r(e)g(C3.)639 4707 y(2\))j(W)-7 b(e)31 b(ha)n(v)n(e)e(considered)h(only)g(systems)g (with)h(nonzero)f(linear)g(part)g Fl(A)p Fo(,)i(as)e(for)g(the)515 4807 y(fully)e(degenerate)f(case)g Fl(A)d Fo(=)f(0)k(no)h(P)n(oincar)n (\023)-39 b(e-Birkho\013)24 b(normalization)i(is)i(p)r(ossible;)g(the) 515 4907 y(tec)n(hnique)c(of)h(reduction)f(to)g(\\renormalized)e (forms")h(w)n(ould)h(also)g(apply)g(for)g Fl(A)f Fo(=)g(0,)h(but)1926 5255 y(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fo(the)32 b(con)n(v)n(ergence)d(issue)i(in)g(this)h (case)f(is)g(completely)g(op)r(en)h(and)f(an)n(y)g(result)g(w)n(ould)g (b)r(e)515 623 y(only)c(utterly)h(formal;)e(th)n(us)i(w)n(e)f(ha)n(v)n (e)g(not)g(discussed)h(this)f(case.)639 722 y(3\))h(W)-7 b(e)29 b(ha)n(v)n(e)e(only)h(considered)f(the)h(simple)h(\(connected\)) f(compact)g(Lie)g(groups)f(ob-)515 822 y(tained)h(b)n(y)f(direct)h(exp) r(onen)n(tiation)f(of)g(simple)h(Lie)g(algebras,)e(lea)n(ving)g(to)i (the)g(reader)e(to)515 922 y(apply)36 b(the)g(necessary)f(restrictions) g(on)h(the)g(normal)f(forms)h(w)n(e)g(obtained)g(for)f(groups)515 1021 y(ha)n(ving)26 b(sev)n(eral)g(connected)i(comp)r(onen)n(t.)639 1121 y(4\))35 b(Also,)h(w)n(e)e(ha)n(v)n(e)g(not)h(discussed)f (reducible)h(represen)n(tations,)f(lea)n(ving)g(again)f(to)515 1220 y(the)d(reader)f(to)h(com)n(bine)f(sev)n(eral)g(copies)g(of)h(the) h(fundamen)n(tal)f(represen)n(tation)e(and)i(to)515 1320 y(com)n(bine)21 b(suc)n(h)h(a)f(symmetry)g(with)i(discrete)e (symmetries)g(relating)g(di\013eren)n(t)h(irreducible)515 1420 y(blo)r(c)n(ks)27 b(where)g(appropriate.)639 1519 y(5\))g(Finally)-7 b(,)27 b(as)g(announced)f(in)i(the)f(title)h(w)n(e)f (only)f(considered)g(compact)h(simple)g(Lie)515 1619 y(groups,)f(i.e.)37 b(w)n(e)27 b(ha)n(v)n(e)g(not)g(discussed)g (non-compact)g(simple)h(Lie)f(groups.)515 1918 y Fp(Ac)m(kno)m (wledgemen)m(ts)639 2117 y Fo(I)k(w)n(ould)f(lik)n(e)g(to)h(thank)g(m)n (y)f(friends)g(Giampaolo)g(Cicogna,)g(Miguel)g(Angel)h(Rodr)-9 b(\023)-32 b(\020-)515 2217 y(guez)25 b(and)h(Sebastian)g(W)-7 b(alc)n(her)25 b(for)h(useful)g(discussions)f(and)h(for)g(correcting)e (a)i(previous)515 2316 y(v)n(ersion)g(of)h(the)h(pap)r(er.)639 2416 y(This)f(w)n(ork)e(w)n(as)g(p)r(erformed)h(while)h(I)f(w)n(as)g (visiting)g(the)h(Mathematics)f(Departmen)n(t)515 2516 y(of)k(the)g(\\Univ)n(ersidad)e(Carlos)h(I)r(I)r(I)h(de)g(Madrid";)g(I) g(thank)g(the)g(Director,)g(Luis)g(Alb)r(erto)515 2615 y(Ib)r(ort,)d(for)g(his)h(kind)g(in)n(vitation.)1905 5255 y(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fa(References)556 705 y Fo([1])41 b(V.I.)i(Arnold,)j Fn(Ge)l(ometric)l(al)f(metho)l(ds)f(in)g(the)f(the)l (ory)h(of)h(or)l(dinary)g(di\013er)l(ential)685 805 y(e)l(quations)p Fo(,)28 b(Springer)f(1983,)f(1988)556 971 y([2])41 b(V.I.)26 b(Arnold)e(and)h(Y)-7 b(u.S.)25 b(Il'y)n(ashenk)n(o,)f Fn(Or)l(dinary)j(di\013er)l(ential)i(e)l(quations)p Fo(,)d(in)f(\\Dy-) 685 1070 y(namical)20 b(Systems)h(I")f(\(D.V.)i(Anoso)n(v)d(and)h(V.I.) h(Arnold)g(eds.\),)h Fn(E.M.S.)g Fp(1)p Fo(,)g(Springer)685 1170 y(1988)556 1336 y([3])41 b(G.R.)36 b(Belitsky)-7 b(,)37 b(\\Equiv)-5 b(alence)33 b(and)i(normal)f(forms)h(of)g(germs)f (of)h(smo)r(oth)g(map-)685 1435 y(pings",)24 b Fn(R)n(uss.)i(Math.)h (Surveys)d Fp(33)f Fo(\(1978\),)g(107;)h(and)f(\\Normal)g(forms)g (relativ)n(e)f(to)685 1535 y(the)28 b(\014ltering)g(action)f(of)g(a)g (group",)f Fn(T)-6 b(r)l(ans.)30 b(Mosc)l(ow)i(Math.)f(So)l(c.)d Fp(40)f Fo(\(1981\),)f(1)556 1701 y([4])41 b(H.W.)d(Bro)r(er,)f(G.B.)g (Huitema)f(and)h(M.B.)f(Sevryuk,)i(\\Quasi)d(p)r(erio)r(dic)i(motions) 685 1801 y(in)31 b(families)g(of)f(dynamical)g(systems:)43 b(order)29 b(amidst)h(c)n(haos",)g Fn(L)l(e)l(ct.)i(Notes)h(Math.)685 1900 y Fp(1645)p Fo(,)27 b(Springer)g(1996)556 2066 y([5])41 b(A.D.)h(Bruno,)g Fn(L)l(o)l(c)l(al)g(metho)l(ds)g(in)g(nonline)l(ar)g (di\013er)l(ential)g(e)l(quations)p Fo(,)i(Springer)685 2166 y(1989;)37 b(see)e(also)f(\\Analytical)g(forms)h(of)g(di\013eren)n (tial)g(equations)f(I)h(&)g(I)r(I",)g Fn(T)-6 b(r)l(ans.)685 2266 y(Mosc)l(ow)31 b(Math.)h(So)l(c.)c Fp(25)f Fo(\(1971\),)f(131)g(&) i Fp(26)f Fo(\(1971\),)f(199)556 2432 y([6])41 b(G.)h(Cicogna,)h (\\Symmetries)e(of)g(dynamical)f(systems)h(and)g(con)n(v)n(ergen)n(t)e (normal)685 2531 y(forms",)26 b Fn(J.)i(Phys.)h(A)d Fp(28)f Fo(\(1995\),)g(L179-L182;)e(\\On)i(the)i(con)n(v)n(ergence)c(of)j(the)g (nor-)685 2631 y(malizing)37 b(transformation)f(in)i(the)g(presence)f (of)g(symmetries",)i Fn(J.)g(Math.)i(A)n(nal.)685 2731 y(Appl.)d Fp(199)e Fo(\(1996\),)h(243-255;)h(\\Con)n(v)n(ergen)n(t)c (normal)h(forms)h(of)g(symmetric)g(dy-)685 2830 y(namical)k(systems",)j Fn(J.)f(Phys.)h(A)d Fp(30)g Fo(\(1997\),)i(6021-6028;)h(\\Multiple-p)r (erio)r(dic)685 2930 y(bifurcation)32 b(and)f(resonance)f(in)i (dynamical)f(systems",)h Fn(Nuovo)i(Cimento)g(B)e Fp(113)685 3029 y Fo(\(1998\),)27 b(1425-1430)556 3196 y([7])41 b(G.)33 b(Cicogna)e(and)h(G.)h(Gaeta,)f(\\Nonlinear)f(symmetries)h(and) g(normal)f(forms",)i(in)685 3295 y Fn(Symmetry)24 b(and)g(p)l(erturb)l (ation)g(the)l(ory)g(II)g({)g(SPT98)p Fo(,)f(A.)f(Degasp)r(eris)e(and)g (G.)i(Gaeta)685 3395 y(eds.,)28 b(W)-7 b(orld)27 b(Scien)n(ti\014c,)h (1999)556 3561 y([8])41 b(G.)28 b(Cicogna)e(and)i(G.)f(Gaeta,)g Fn(Symmetry)j(and)g(p)l(erturb)l(ation)f(the)l(ory)i(in)e(nonline)l(ar) 685 3660 y(dynamics)p Fo(,)g(to)f(b)r(e)g(published)g(b)n(y)f(Springer) g(\(Lecture)g(Notes)h(in)g(Ph)n(ysics\),)e(1999)556 3827 y([9])41 b(G.)27 b(Cicogna)d(and)i(G.)g(Gaeta,)g(\\Bifurcation,)f (symmetries)h(and)f(maximal)h(isotrop)n(y)685 3926 y(subgroups",)g Fn(Nuovo)k(Cim.)h(B)d Fp(102)f Fo(\(1988\),)f(451)515 4092 y([10])40 b(C.)23 b(Elphic)n(k,)g(E.)g(Tirap)r(egui,)g(M.E.)f (Brac)n(het,)h(P)-7 b(.)22 b(Coullet)h(and)g(G.)g(Io)r(oss,)f(\\A)h (simple)685 4192 y(global)j(c)n(haracterization)f(for)h(normal)g(forms) g(of)h(singular)f(v)n(ector)f(\014elds",)i Fn(Physic)l(a)685 4291 y(D)h Fp(29)f Fo(\(1987\),)f(95)515 4457 y([11])40 b(E.)28 b(F)-7 b(orest)27 b(and)h(D.)h(Murra)n(y)-7 b(,)26 b(\\F)-7 b(reedom)27 b(in)h(minimal)g(normal)f(forms",)g Fn(Physic)l(a)32 b(D)685 4557 y Fp(74)c Fo(\(1994\),)e(181)515 4723 y([12])40 b(G.)26 b(Gaeta,)e(\\Normal)g(forms)g(for)g(rev)n (ersible)g(dynamical)g(systems",)g Fn(Int.)j(J.)g(The)l(or.)685 4823 y(Phys.)j Fp(33)d Fo(\(1994\),)f(1917)1905 5255 y(11)p eop %%Page: 12 12 12 11 bop 515 523 a Fo([13])40 b(G.)e(Gaeta,)h(\\Reduction)e(of)g(P)n (oincar)n(\023)-39 b(e)35 b(normal)h(forms",)i Fn(L)l(ett.)h(Math.)h (Phys.)f Fp(42)685 623 y Fo(\(1997\),)h(103;)j(\\P)n(oincar)n(\023)-39 b(e)34 b(renormalized)j(forms",)j(preprin)n(t)e Fn(mp-ar)l(c)i(98-739)p Fo(,)j(to)685 722 y(app)r(ear)27 b(in)h Fn(A)n(nn.)h(I.H.P.)515 888 y Fo([14])40 b(G.)31 b(Gaeta,)f(\\On)g(P)n(oincar)n(\023)-39 b(e)27 b(renormalized)i(forms";)h(in)h Fn(Symmetry)h(and)g(p)l(erturb)l (a-)685 988 y(tion)e(the)l(ory)h(II)f({)g(SPT98)p Fo(,)g(A.)e(Degasp)r (eris)f(and)g(G.)h(Gaeta)g(eds.,)g(to)f(b)r(e)i(published)685 1088 y(b)n(y)f(W)-7 b(orld)27 b(Scien)n(ti\014c,)h(1999)515 1254 y([15])40 b(G.)24 b(Gaeta)f(and)h(P)-7 b(.)23 b(Morando,)g(\\An)g (h)n(yp)r(erKahler)f(generalization)g(of)h(Hamiltonian)685 1353 y(Mec)n(hanics",)k(in)h(preparation)515 1519 y([16])40 b(G.)30 b(Gaeta)e(and)h(P)-7 b(.)29 b(Rossi,)g(\\Gradien)n(t)f(prop)r (ert)n(y)g(of)h(standard)f(represen)n(tation)g(for)685 1619 y(classical)i(orthogonal)g(groups",)g Fn(Nuovo)k(Cim.)h(B)d Fp(93)f Fo(\(1986\),)g(66;)h(G.)g(Gaeta,)g(\\A)685 1719 y(w)n(eak)n(er)i(condition)h(for)f(gradien)n(t)g(prop)r(ert)n(y)h(of)g (group)f(represen)n(tations",)h Fn(Nuovo)685 1818 y(Cim.)d(B)c Fp(96)f Fo(\(1986\),)g(147;)g(\\T)-7 b(ransitivit)n(y)26 b(and)i(gradien)n(t)e(prop)r(ert)n(y)h(for)g(real)g(group)685 1918 y(represen)n(tations",)f Fn(Nuovo)k(Cim.)h(B)d Fp(99)f Fo(\(1987\),)f(61)515 2084 y([17])40 b(G.)29 b(Io)r(oss)d(and)i(M.)g (Adelmey)n(er,)g Fn(T)-6 b(opics)31 b(in)f(bifur)l(c)l(ation)h(the)l (ory)g(and)f(applic)l(ations)p Fo(,)685 2183 y(W)-7 b(orld)28 b(Scien)n(ti\014c)g(1992)515 2350 y([18])40 b(A.A.)29 b(Kirillo)n(v,)d Fn(Elements)k(of)g(the)g(the)l(ory)g(of)h(r)l(epr)l (esentations)p Fo(,)d(Springer)f(1984)515 2516 y([19])40 b(J.)28 b(Lam)n(b,)f(\\Lo)r(cal)g(bifurcations)g(in)h Fl(k)s Fo(-symmetric)f(dynamical)g(systems",)g Fn(Nonlin-)685 2615 y(e)l(arity)i Fp(9)e Fo(\(1996\),)g(537)515 2781 y([20])40 b(J.)30 b(Lam)n(b)g(and)f(J.)h(Rob)r(erts,)g(\\Time)g(rev)n (ersal)e(symmetry)h(in)h(dynamical)g(systems:)685 2881 y(a)e(surv)n(ey",)d Fn(Physic)l(a)32 b(D)27 b Fp(112)g Fo(\(1998\),)g(1)515 3047 y([21])40 b(M.B.)e(Sevryuk,)h(\\Rev)n (ersible)c(systems",)k Fn(L)l(e)l(ct.)f(Notes)h(Math.)f Fp(1211)p Fo(,)h(Springer,)685 3147 y(1986)515 3313 y([22])h(K.)28 b(Siegel)f(and)h(J.K.)f(Moser,)f Fn(L)l(e)l(ctur)l(es)j(on)h(Celestial) i(Me)l(chanics)p Fo(,)d(Springer)e(1971;)685 3412 y(reprin)n(ted)g(in)h (\\Classics)e(in)i(Mathematics",)f(Springer)f(1995)515 3578 y([23])40 b(K.)20 b(Y)-7 b(osida,)22 b Fn(L)l(e)l(ctur)l(es)f(on)j (di\013er)l(ential)g(and)f(inte)l(gr)l(al)g(e)l(quations)p Fo(,)g(In)n(terscience)c(1960;)685 3678 y(reprin)n(ted)27 b(b)n(y)h(Do)n(v)n(er)e(1991)515 3844 y([24])40 b(S.)34 b(W)-7 b(alc)n(her,)33 b(\\On)f(di\013eren)n(tial)h(equations)f(in)h (normal)f(form",)h Fn(Math.)j(A)n(nn.)d Fp(291)685 3944 y Fo(\(1991\),)27 b(293-314)515 4110 y([25])40 b(S.)32 b(W)-7 b(alc)n(her,)32 b(\\On)f(con)n(v)n(ergen)n(t)e(normal)h(form)i (transformations)d(in)j(the)g(presence)685 4209 y(of)42 b(symmetry",)i(Preprin)n(t)c(1999,)j(to)f(app)r(ear)e(in)i Fn(J.)g(Math.)i(A)n(nal.)f(Appl.)p Fo(;)50 b(\\On)685 4309 y(transformation)29 b(in)n(to)h(normal)f(form",)h Fn(J.)i(Math.)h(A)n(nal.)f(Appl.)g Fp(180)d Fo(\(1993\),)h(617-)685 4408 y(632)1905 5255 y(12)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9908301058650--