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1853 y(men)m(t)k(of)h(Greene's)g(criterion)e(for)h (families)f(in)g(the)h(univ)m(ersalit)m(y)f(class.)873 2105 y(2.)46 b Fw(Some)33 b(preliminar)-6 b(y)33 b(ba)n(ck)n(gr)n(ound) g(and)g(not)-6 b(a)g(tion)555 2267 y Fx(In)24 b(this)f(section,)j(w)m (e)f(collect)g(some)g(rather)g(w)m(ell)e(kno)m(wn)h(results.)38 b(All)23 b(of)i(them)f(will)456 2375 y(b)s(e)i(undoubtedly)e(w)m(ell)h (kno)m(wn)h(to)i(exp)s(erts.)39 b(Besides)26 b(serving)g(as)g(a)h(quic) m(k)g(reference,)456 2483 y(this)41 b(will)f(serv)m(e)j(us)e(to)j (establish)c(our)i(notation.)77 b(A)43 b(general)f(reference)h(for)f (this)456 2590 y(bac)m(kground)30 b(material)g(is)f([dlL01].)456 2800 y(2.1.)46 b FA(Twist)f(maps.)g Fx(An)39 b(area{preserving)g(t)m (wist)g(map)g(of)g(the)h(cylinder)d(is)i(a)g(C)3405 2767 y Fu(1)456 2908 y Fx(di\013eomorphism)32 b Fr(T)46 b Fx(:)33 b Fq(T)1316 2875 y Fu(1)1381 2908 y Fp(\002)23 b Fq(R)43 b Fx(to)36 b(itself,)g(whic)m(h)d(preserv)m(es)i(area,)j (orien)m(tation)d(and)456 3016 y(the)30 b(top)s(ological)g(ends)g(of)g Fq(T)1444 2983 y Fu(1)1506 3016 y Fp(\002)20 b Fq(R)39 b Fx(and)29 b(whic)m(h)g(satis\014es)h(the)h(t)m(wist)f(condition:)456 3228 y(\(1\))1254 3167 y Fr(@)11 b Fx(\026)-51 b Fr(x)p 1254 3207 106 4 v 1256 3291 a(@)5 b(y)1395 3228 y(>)25 b Fx(0)101 b(where)g(\()6 b(\026)-51 b Fr(x;)22 b Fx(\026)-52 b Fr(y)t Fx(\))25 b(=)g Fr(T)13 b Fx(\()p Fr(x;)i(y)s Fx(\))51 b Fr(:)456 3446 y Fx(W)-8 b(e)39 b(also)e(assume)h(that)g(the) g(net)g(\015ux)e(of)i Fr(T)51 b Fx(\(the)38 b(orien)m(ted)g(area)g(b)s (et)m(w)m(een)h(a)f(non-)456 3554 y(con)m(tractible)26 b(circle)f(and)g(its)g(image)h(under)f Fr(T)13 b Fx(,)26 b(b)m(y)g(the)g(area)h(preserv)-5 b(ation)25 b(of)h Fr(T)13 b Fx(,)26 b(this)456 3662 y(orien)m(ted)36 b(area)h(is)e(indep)s(enden) m(t)f(of)i(the)h(circle)e(c)m(hosen\))j(is)d(zero.)59 b(Otherwise,)37 b(it)e(is)456 3770 y(clear)30 b(that)h(there)g(cannot)g (b)s(e)e(an)m(y)i(non-trivial)d(in)m(v)-5 b(arian)m(t)30 b(circles.)555 3878 y(One)d(example)f(that)i(w)m(e)f(will)d(use)j(v)m (ery)g(often)g(is)f(the)h(generalized)f(standard)g(map,)456 3986 y(giv)m(en)k(b)m(y:)1706 4232 y(\026)-51 b Fr(y)28 b Fx(=)d Fr(y)e Fx(+)d Fr(f)10 b Fx(\()p Fr(x)p Fx(\))1701 4367 y(\026)-51 b Fr(x)26 b Fx(=)f Fr(x)20 b Fx(+)26 b(\026)-51 b Fr(y)456 4307 y Fx(\(2\))456 4542 y(where)33 b Fr(f)10 b Fx(\()p Fr(x)22 b Fx(+)g(1\))32 b(=)f Fr(f)10 b Fx(\()p Fr(x)p Fx(\).)52 b(The)33 b(standard)g(map)h(corresp)s(onds)e (to)j(taking:)48 b Fr(f)10 b Fx(\()p Fr(x)p Fx(\))31 b(=)484 4619 y Fo(\025)p 466 4634 79 4 v 466 4686 a Fu(2)p Fo(\031)569 4654 y Fx(sin)n(\(2)p Fr(\031)s(x)p Fx(\).)555 4784 y(In)j(\(2\))q(,)j(the)e(net)g(\015ux)e(is)1452 4711 y Fn(R)1513 4737 y Fu(2)p Fo(\031)1495 4816 y Fu(0)1610 4784 y Fr(f)10 b Fx(\()p Fr(x)p Fx(\))p Fr(dx)p Fx(,)37 b(so)e(that)g(the)g(condition)f(of)h(zero)g(net)g(\015ux)456 4892 y(corresp)s(onds)29 b(to)i(the)f(in)m(tegral)g(v)-5 b(anishing.)555 5000 y(Giv)m(en)34 b(suc)m(h)f(map)h(it)f(is)g(often)h (con)m(v)m(enien)m(t)h(to)f(consider)f(a)h(lift)e Fr(F)44 b Fx(:)31 b Fq(R)3027 4967 y Fu(2)3103 5000 y Fp(!)g Fq(R)3284 4967 y Fu(2)3364 5000 y Fx(to)456 5108 y(the)f(univ)m(ersal)f (co)m(v)m(er.)43 b(Th)m(us,)29 b(w)m(e)i(de\014ne)f(a)h(lift)e(of)h Fr(T)13 b Fx(:)41 b(Let)31 b Fr(p)25 b Fx(:)h Fq(R)i Fp(\002)20 b Fq(R)34 b Fp(!)26 b Fq(T)3103 5075 y Fu(1)3164 5108 y Fp(\002)20 b Fq(R)39 b Fx(b)s(e)456 5216 y(the)c(co)m(v)m(ering) h(map)f Fr(p)p Fx(\()p Fr(x;)15 b(y)s Fx(\))34 b(=)f(\(\()p Fr(x)p Fx(\(mo)s(d)q(1\)\))p Fr(;)15 b(y)s Fx(\).)57 b(Then)34 b(a)i(lift)e Fr(F)46 b Fx(:)34 b Fq(R)2927 5183 y Fu(2)3006 5216 y Fp(!)f Fq(R)3190 5183 y Fu(2)3270 5216 y Fx(of)j Fr(T)p eop %%Page: 4 4 4 3 bop 456 251 a Fu(4)872 b(R.)23 b(DE)g(LA)h(LLA)-8 b(VE)23 b(AND)g(A.)g(OL)-8 b(VERA)456 450 y Fx(is)31 b(a)h(con)m(tin)m(uous)g(map)g(suc)m(h)f(that)i Fr(T)h Fp(\016)22 b Fr(p)28 b Fx(=)f Fr(p)22 b Fp(\016)f Fr(F)13 b Fx(.)46 b(Let)33 b Fr(R)c Fx(:)f Fq(R)2670 417 y Fu(2)2743 450 y Fp(!)g Fq(R)2922 417 y Fu(2)2999 450 y Fx(b)s(e)k(the)g(unit)456 558 y(translation)d Fr(R)q Fx(\()p Fr(x;)15 b(y)s Fx(\))26 b(=)f(\()p Fr(x)20 b Fp(\000)g Fx(1)p Fr(;)15 b(y)s Fx(\),)32 b(then)e Fr(F)43 b Fx(comm)m(utes)32 b(with)d Fr(R)q Fx(.)456 836 y(2.2.)46 b FA(Rev)m(ersible)29 b(mappings)e(and)g (critical)h(lines.)46 b Fx(W)-8 b(e)25 b(also)f(recall)f([Gre79)r(])h (and)456 944 y([D)m(V58)r(])36 b(that)h(man)m(y)f(maps)f Fr(F)49 b Fx(of)36 b(in)m(terest)g({)g(e.g.)59 b(\(2\))37 b({)f(ha)m(v)m(e)h(the)f(prop)s(ert)m(y)g(that)456 1052 y(they)d(can)g(b)s(e)f(written)h(as)g Fr(F)43 b Fx(=)29 b Fr(T)35 b Fp(\016)22 b Fr(S)38 b Fx(with)32 b Fr(T)2120 1019 y Fu(2)2189 1052 y Fx(=)d Fr(S)2350 1019 y Fu(2)2419 1052 y Fx(=)g(Id)k(\(equiv)-5 b(alen)m(tly)32 b Fr(F)3250 1019 y Fm(\000)p Fu(1)3374 1052 y Fx(=)456 1160 y Fr(T)h Fp(\016)20 b Fr(F)34 b Fp(\016)20 b Fr(T)830 1127 y Fm(\000)p Fu(1)924 1160 y Fx(\))555 1268 y(F)-8 b(or)31 b(suc)m(h)f(maps,)h(the)f (lines)f Fp(L)c Fx(=)g Fp(f)p Fr(x)p Fp(j)p Fr(T)13 b Fx(\()p Fr(x)p Fx(\))26 b(=)f Fr(x)p Fp(g)31 b Fx(are)f(called)g(the)h (critical)e(lines.)555 1376 y(The)d(imp)s(ortance)g(of)g(the)h (critical)e(lines)g(is)g(that)i(it)f(is)g(easy)h(to)g(sho)m(w)f(that)h (if)f Fr(x)f Fp(2)g(L)456 1484 y Fx(is)g(suc)m(h)i(that)g Fr(F)1009 1451 y Fo(N)1076 1484 y Fx(\()p Fr(x)p Fx(\))f Fp(2)f(L)p Fx(,)i(it)g(is)e(easy)j(to)f(sho)m(w)f(that)i Fr(F)2390 1451 y Fu(2)p Fo(N)2492 1484 y Fx(\()p Fr(x)p Fx(\))e(=)f Fr(x)p Fx(.)40 b(Hence,)28 b(one)f(can)456 1592 y(searc)m(h)k(for)f(p)s(erio)s(dic)d(orbits)j(b)m(y)g (one-dimensional)e(searc)m(hes.)555 1700 y(It)38 b(is)g(moreo)m(v)m(er) h(found)e(empirically)e(that)k(these)g(orbits)e(lying)f(in)h(critical)g (lines)456 1808 y(are)e(m)m(uc)m(h)h(more)g(orderly)e(and)h(ha)m(v)m(e) h(b)s(etter)g(prop)s(erties.)54 b(A)36 b(systematic)g(study)e(of)456 1915 y(critical)29 b(lines)f(is)i(in)f([PnJL87].)456 2194 y(2.3.)46 b FA(Bifurcation)39 b(surfaces.)47 b Fx(One)32 b(w)m(a)m(y)i(to)g(describ)s(e)e(bifurcations)f(that)j(is)e(v)m(ery)456 2301 y(con)m(v)m(enien)m(t)f(for)f(renormalization)e(is)h(the)i (discussion)c(in)i(terms)h(of)g(bifurcation)e(sur-)456 2409 y(faces.)41 b(\(See)31 b(e.g.)42 b([CH82)q(])30 b(p.)41 b(42,)31 b([AR67)q(].)41 b(\))555 2517 y(These)c(bifurcation)e (surfaces)h(are)h(co)s(dimension)e(one)i(surfaces)f(in)g(the)h(space)g (of)456 2625 y(mappings.)50 b(When)34 b(a)h(map)f(b)s(elongs)f(to)i (the)g(bifurcation)d(surfaces,)j(it)f(p)s(ossesses)g(a)456 2733 y(p)s(erio)s(dic)27 b(orbit)j(whic)m(h)f(exp)s(eriences)g(a)i (bifurcation.)555 2841 y(If)23 b(w)m(e)g(sp)s(ecify)e(the)i(orbit)f (and)g(the)h(t)m(yp)s(e)g(of)g(bifurcation)e(taking)i(place,)h (bifurcation)456 2949 y(surfaces)30 b(are)h(smo)s(oth)f(manifolds)e(in) h(the)i(space)f(of)h(mappings)e(considered.)555 3057 y(F)-8 b(or)45 b(example,)i(the)d(set)g(of)g(maps)f(exp)s(eriencing)f (a)i(saddle-no)s(de)f(in)f(a)i(certain)456 3165 y(neigh)m(b)s(orho)s(o) s(d)35 b(can)j(b)s(e)g(describ)s(ed)e(as)i(the)g(set)h(of)f(maps)f(for) h(whic)m(h)f(the)h(graph)f(is)456 3273 y(tangen)m(t)d(to)f(the)g (diagonal.)47 b(By)33 b(the)g(transv)m(ersalit)m(y)g(theorem)g(\(See)g (Theorem)g(18.2)456 3381 y(of)c([AR67)q(],)g(this)f(sho)m(ws)h(that)g (the)g(set)g(of)g(mappings)e(exp)s(eriencing)g(a)i(bifurcation)e(is)456 3489 y(lo)s(cally)h(a)j(smo)s(oth)f(manifold.)555 3597 y(In)g(our)h(case,)h(w)m(e)g(will)c(consider)i(the)h(bifurcation)e (manifolds)g(\000)2802 3611 y Fo(n)2849 3597 y Fx(,)i(whic)m(h)f(are)h (the)456 3705 y(sets)i(of)h(area)g(preserving)e(maps)h(for)h(whic)m(h)e (a)i(sp)s(eci\014ed)e(p)s(erio)s(dic)e(orbit)j({)h(selected)456 3812 y(b)m(y)h(the)h(index)e Fr(n)h Fx({)h(has)f(one)h(eigen)m(v)-5 b(alue)36 b(1)g(\(b)m(y)f(the)h(area)h(preserv)-5 b(ation,)36 b(then)f(the)456 3920 y(t)m(w)m(o)c(eigen)m(v)-5 b(alues)31 b(should)d(b)s(e)i(1\).)555 4028 y(These)25 b(bifurcating)f(maps)h(can) h(b)s(e)e(describ)s(ed)g(as)h(the)h(maps)f(in)f(whic)m(h)g(the)i (graphs)456 4136 y(of)k(an)g(iterate)h(has)f(a)h(non-transv)m(ersal)e (in)m(tersection)h(with)f(the)h(diagonal)g(in)f(a)h(sp)s(ec-)456 4244 y(i\014ed)f(neigh)m(b)s(orho)s(o)s(d.)555 4352 y(F)-8 b(or)30 b(n)m(umerical)f(applications,)f(it)h(will)e(also)i(b)s(e)g (con)m(v)m(enien)m(t)i(to)f(consider)e(the)i(sets)456 4460 y(\000)513 4474 y Fo(n)559 4460 y Fx(\()p Fr(\032)641 4474 y Fo(n)689 4460 y Fx(\),)h(in)e(whic)m(h)g(the)i(trace)g(of)g(the) g(deriv)-5 b(ativ)m(e)30 b(at)h(an)f(sp)s(eci\014ed)f(p)s(erio)s(dic)f (orbit)h(is)456 4568 y(sp)s(eci\014ed)35 b(to)j(b)s(e)e(a)h(v)-5 b(alue)37 b(di\013eren)m(t)f(from)g(2)i({)f(equiv)-5 b(alen)m(tly)d(,)38 b(the)f(eigen)m(v)-5 b(alues)37 b(are)456 4676 y(sp)s(eci\014ed)25 b(to)j(b)s(e)e(a)h(n)m(um)m(b)s(er)f (di\013eren)m(t)g(from)h(1)g({.)40 b(In)27 b(suc)m(h)f(a)i(case,)g(the) g(fact)g(that)f(the)456 4784 y(set)g(of)g(mappings)e(is)g(a)i (submanifold)d(of)j(co)s(dimension)d(1)j(is)f(an)g(easy)i(consequence)f (of)456 4892 y(the)i(implicit)e(function)h(theorem.)41 b(The)29 b(p)s(erio)s(dic)d(orbit)j(p)s(ersists)e(and)i(the)h(eigen)m (v)-5 b(al-)456 5000 y(ues)26 b(at)i(the)f(p)s(erio)s(dic)e(orbit)h (can)h(b)s(e)f(used)h(as)g(a)g(functional.)38 b(Setting)27 b(this)f(functional)456 5108 y(at)34 b(an)f(sp)s(eci\014ed)f(lev)m(el)h (de\014nes)g(a)h(smo)s(oth)f(manifold)f(in)g(the)i(space)g(of)g(maps)f (since,)456 5216 y(clearly)c(the)i(gradien)m(t)f(of)h(the)f(functional) f(is)g(not)i(zero.)p eop %%Page: 5 5 5 4 bop 1213 251 a Fu(OBSTR)n(UCTION)30 b(AND)g(RENORMALIZA)-6 b(TION)722 b(5)555 450 y Fx(Other)35 b(manifolds)e(that)j(w)m(e)f(will) e(consider)h(are)i(the)f(manifolds)e(in)h(the)h(space)h(of)456 558 y(maps)21 b(asso)s(ciated)h(to)g(in)m(tersections)f(of)h(the)g (stable)f(and)g(unstable)f(manifolds)g(of)h(sp)s(ec-)456 666 y(i\014ed)34 b(p)s(erio)s(dic)g(orbits)h(in)f(a)j(sp)s(eci\014ed)d (segmen)m(t.)59 b(F)-8 b(or)37 b(example,)g(w)m(e)g(will)c(consider)456 774 y(\006)522 788 y Fo(n)602 774 y Fx(whic)m(h)g(is)g(the)h(set)h(of)f (maps)g(in)f(whic)m(h)f(t)m(w)m(o)k(segmen)m(ts)f(of)f(stable)g(and)g (unstable)456 882 y(manifolds)28 b(of)k(t)m(w)m(o)g(orbits)e({)h (selected)h(b)m(y)f(the)g(index)f Fr(n)g Fx({)h(ha)m(v)m(e)i(a)e (transv)m(erse)g(in)m(ter-)456 990 y(section)d(and)g(also)h(the)g(sets) f(\003)1506 1004 y Fo(n)1554 990 y Fx(\()p Fr(\032)1636 1004 y Fo(n)1683 990 y Fx(\))h(in)e(whic)m(h)g(the)i(stable)f(and)g (unstable)g(manifolds)456 1098 y(in)m(tersect)j(enclosing)e(a)i(lob)s (e)e(of)i(area)g Fr(\032)p Fx(.)555 1206 y(The)36 b(fact)h(that)f(the)h (sets)f(\006)1547 1220 y Fo(n)1594 1206 y Fr(;)15 b Fx(\003)1697 1220 y Fo(n)1744 1206 y Fx(\()p Fr(\032)1826 1220 y Fo(n)1874 1206 y Fx(\))36 b(are)h(smo)s(oth)e(manifolds)f(of)j(co)s(dimension)456 1314 y(1)f(in)g(the)g(space)h(of)g(smo)s(oth)f(area)i(preserving)d (mappings)f(follo)m(ws)i(b)s(ecause)g(of)h(the)456 1421 y(results)21 b(men)m(tioned)h(in)f(Section)h(2.5)i(sho)m(w)e(that)h (the)g(dep)s(endence)e(of)i(the)f(observ)-5 b(ables)456 1529 y(de\014ning)28 b(the)j(manifold)d(with)h(resp)s(ect)h(to)h(the)g (map)f(has)g(a)h(non-zero)g(gradien)m(t.)456 1743 y(2.4.)46 b FA(The)35 b(obstruction)i(criterion.)46 b Fx(W)-8 b(e)31 b(recall)f(that)i(the)f(obstruction)f(criterion)456 1850 y(reads)g(\(See)h([OS87a)q(],)f([OS87b].\))456 2027 y FA(Theorem)h(2.1.)40 b Ft(L)-5 b(et)31 b Fr(T)43 b Ft(b)-5 b(e)31 b(a)f Fr(C)1608 1994 y Fu(1)1678 2027 y Ft(ar)-5 b(e)g(a)32 b(pr)-5 b(eserving)31 b(twist)h(map)f(of)g Fq(T)2882 1994 y Fu(1)2939 2027 y Fp(\002)15 b Fq(R)39 b Ft(to)31 b(itself.)555 2135 y(Assume)f(that)h Fr(T)42 b Ft(has)31 b(hyp)-5 b(erb)g(olic)31 b(p)-5 b(erio)g(dic)32 b(orbits)e Fr(p)2383 2149 y Fu(1)2422 2135 y Fr(;)15 b(p)2508 2149 y Fu(2)2577 2135 y Ft(of)30 b(typ)-5 b(e)31 b Fr(m)2949 2149 y Fu(1)2988 2135 y Fr(=n)3088 2149 y Fu(1)3128 2135 y Ft(,)f Fr(m)3266 2149 y Fu(2)3305 2135 y Fr(=n)3405 2149 y Fu(2)456 2243 y Ft(r)-5 b(esp)g(e)g(ctively)35 b(\()f Fr(m)1083 2257 y Fu(1)1122 2243 y Fr(;)15 b(m)1242 2257 y Fu(2)1281 2243 y Fr(;)g(n)1376 2257 y Fu(1)1416 2243 y Fr(;)g(n)1511 2257 y Fu(2)1577 2243 y Fp(2)27 b Fq(N)7 b Ft(\))40 b(with)34 b Fr(m)2080 2257 y Fu(1)2119 2243 y Fr(=n)2219 2257 y Fu(1)2286 2243 y Fr(>)27 b(m)2464 2257 y Fu(2)2503 2243 y Fr(=n)2603 2257 y Fu(2)2643 2243 y Ft(.)45 b(Assume,)33 b(mor)-5 b(e)g(over)456 2351 y(that)39 b(the)f(stable)g(and)h(unstable)g(manifolds)h(interse)-5 b(ct)38 b(tr)-5 b(ansversal)5 b(ly.)60 b(Then,)39 b(ther)-5 b(e)456 2459 y(is)34 b(no)h(homotopic)-5 b(al)5 b(ly)38 b(non-trivial)d(invariant)g(cir)-5 b(cle)35 b(with)g(a)g(r)-5 b(otation)37 b(numb)-5 b(er)34 b Fr(\032)29 b Fp(2)456 2567 y Fx([)p Fr(m)561 2581 y Fu(1)600 2567 y Fr(=n)700 2581 y Fu(1)740 2567 y Fr(;)15 b(m)860 2581 y Fu(2)899 2567 y Fr(=n)999 2581 y Fu(2)1039 2567 y Fx(])p Ft(.)555 2743 y Fx(The)20 b(pro)s(of)g(of)g(Theorem)g(2.1)i(dev)m(elop)s(ed)d (in)g([OS87b])i(is)e(based)h(on)g(the)h(observ)-5 b(ation)456 2851 y(that,)49 b(for)44 b(a)h(t)m(wist)f(mapping,)j(the)e(existence)g (of)f(an)h(homotopically)e(non-trivial)456 2959 y(in)m(v)-5 b(arian)m(t)35 b(circle)f(with)h(rotation)h(n)m(um)m(b)s(er)e Fr(\032)i Fx(is)f(incompatible)f(with)g(the)i(existence)456 3067 y(of)30 b(an)g(orbit)g(whic)m(h)f(is)g(asymptotic)i(in)e(the)i (future)e(to)i Fr(p)2417 3081 y Fu(1)2487 3067 y Fx(and)f(in)f(the)h (past)h(to)g Fr(p)3281 3081 y Fu(2)3320 3067 y Fx(.)456 3244 y FA(Remark)48 b(2.2.)h Fx(W)-8 b(e)45 b(also)d(note)i(that,)j(if) 42 b(the)h(in)m(tersection)g(of)g(these)g(stable)g(and)456 3351 y(unstable)31 b(manifolds)f(w)m(ere)i(transv)m(ersal,)h(then)f (there)h(is)e(a)h(horsesho)s(e)g(b)s(et)m(w)m(een)h(the)456 3459 y(orbits.)39 b(Using)30 b(the)g(sym)m(b)s(olic)f(dynamics)g(of)i (the)f(horsesho)s(e,)h(one)f(can)h(pro)s(duce)e(h)m(y-)456 3567 y(p)s(erb)s(olic)24 b(orbits)i(with)f(rotation)j(n)m(um)m(b)s(er)d Fr(\032)i Fx(for)g(an)m(y)g Fr(\032)e Fp(2)g Fx([)p Fr(m)2573 3581 y Fu(1)2613 3567 y Fr(=n)2713 3581 y Fu(1)2752 3567 y Fr(;)15 b(m)2872 3581 y Fu(2)2912 3567 y Fr(=n)3012 3581 y Fu(2)3051 3567 y Fx(].)40 b(Clearly)-8 b(,)456 3675 y(for)29 b(t)m(wist)h(mappings)e(the)i(existence)h(of)f(a)g(h)m (yp)s(erb)s(olic)d(orbit)i(of)i(rotation)f(n)m(um)m(b)s(er)e Fr(\032)456 3783 y Fx(is)h(incompatible)g(with)h(the)h(existence)g(of)g (an)g(in)m(v)-5 b(arian)m(t)30 b(circle)g(of)g(rotation)i(n)m(um)m(b)s (er)456 3891 y Fr(\032)p Fx(.)456 4068 y FA(Remark)48 b(2.3.)i Fx(Another)43 b(implemen)m(tation)e(of)i(the)h(obstruction)e (criterion)g(is)g(in)456 4176 y([OS93].)f(There,)30 b(it)f(is)g(sho)m (wn)g(that)i(if)e(a)h(map)g(comes)g(close)g(to)h(ha)m(ving)f(a)g (tangen)m(tial)456 4284 y(in)m(tersection,)k(then,)g(it)e(has)h(p)s (erio)s(dic)e(orbits)h(whic)m(h)g(are)i(not)f(w)m(ell)g(ordered)f (\(some-)456 4391 y(times)40 b(called)h(non-Birkho\013)f(orbits\).)73 b(Then,)43 b(a)f(criterion)e(of)i([BH87)q(])g({)f(see)h(also)456 4499 y([LC00])31 b({)g(sho)m(ws)f(that)h(there)f(is)g(no)g(in)m(v)-5 b(arian)m(t)29 b(circle.)456 4676 y(2.5.)46 b FA(Dep)s(endence)35 b(of)g(stable)f(manifolds)h(on)f(parameters.)45 b Fx(One)29 b(imp)s(ortan)m(t)456 4784 y(ingredien)m(t)24 b(in)h(our)g(form)m (ulation)g(is)g(the)h(observ)-5 b(ation)26 b(that)g(the)g(in)m(v)-5 b(arian)m(t)26 b(manifolds)456 4892 y(of)d(a)h(p)s(erio)s(dic)e(p)s (oin)m(t)g(dep)s(end)g(in)h(an)g(analytic)g(w)m(a)m(y)i(on)f(the)f (map.)39 b(This)21 b(result)i(can)h(b)s(e)456 5000 y(found)c(in)h ([Mey75)r(])i(and)e(a)i(more)f(mo)s(dern)f(v)m(ersion)h(\(including)d (more)j(general)g(results,)456 5108 y(explicit)34 b(form)m(ulas)i(for)g (the)h(deriv)-5 b(ativ)m(es)35 b(and)h(a)h(discussion)d(of)i(\014nite)g (regularities\))456 5216 y(can)30 b(b)s(e)g(found)f(in)g([CFdlL02b)o (].)p eop %%Page: 6 6 6 5 bop 456 251 a Fu(6)872 b(R.)23 b(DE)g(LA)h(LLA)-8 b(VE)23 b(AND)g(A.)g(OL)-8 b(VERA)456 450 y FA(Lemma)37 b(2.4.)44 b Ft(L)-5 b(et)37 b Fr(T)1251 464 y Fu(0)1327 450 y Ft(b)-5 b(e)36 b(an)h(analytic)g(mapping)h(of)e(a)h(neighb)-5 b(orho)g(o)g(d)40 b(of)c Fx(0)h Ft(in)f Fq(R)3398 417 y Fo(d)456 558 y Ft(with)d Fx(0)g Ft(a)g(hyp)-5 b(erb)g(olic)35 b(\014xe)-5 b(d)33 b(p)-5 b(oint.)555 666 y(Assume)32 b(that)h(the)g(dimension)g(of)g(the)f(stable)h(sp)-5 b(ac)g(e)33 b(of)g Fr(D)s(T)2650 680 y Fu(0)2689 666 y Fx(\(0\))g Ft(is)f Fr(s)p Ft(.)41 b(Denote)32 b(by)456 774 y Fr(B)530 741 y Fo(s)598 774 y Ft(the)h(unit)g(b)-5 b(al)5 b(l)33 b(in)g Fq(C)1278 741 y Fo(s)1321 774 y Ft(.)555 882 y(L)-5 b(et)33 b Fr(U)42 b Ft(b)-5 b(e)32 b(a)h(su\016ciently)f(smal)5 b(l)34 b(neighb)-5 b(orho)g(o)g(d)35 b(of)e Fr(T)2405 896 y Fu(0)2477 882 y Ft(in)f Fp(X)13 b Ft(,)32 b(a)h(Banach)g(sp)-5 b(ac)g(e)34 b(of)456 990 y(analytic)39 b(functions)g(\(endowe)-5 b(d)40 b(with)f(the)f(norm)i (of)e(the)h(supr)-5 b(emum)39 b(in)f(a)h(c)-5 b(omplex)456 1098 y(extension\).)555 1206 y(Then,)33 b(it)f(is)g(p)-5 b(ossible)33 b(to)g(\014nd)f(a)h(analytic)g(mapping)g Fp(W)g Fx(:)25 b Fr(X)i Fp(\002)18 b Fr(B)2851 1173 y Fo(s)2913 1206 y Fp(!)25 b Fr(M)42 b Ft(in)32 b(such)456 1314 y(a)h(way)g(that)671 1452 y Fp(\017)42 b Fr(p)804 1466 y Fo(T)884 1452 y Fp(\021)25 b(W)7 b Fx(\()p Fr(T)h(;)15 b Fx(0\))34 b Ft(is)f(a)g(hyp)-5 b(erb)g(olic)35 b(\014xe)-5 b(d)33 b(p)-5 b(oint)34 b(of)f(the)g(map)h Fr(T)13 b Ft(.)671 1560 y Fp(\017)42 b Fr(W)13 b Fx(\()p Fr(T)8 b(;)15 b(B)1067 1527 y Fo(s)1104 1560 y Fx(\))32 b Ft(is)h(c)-5 b(ontaine)g(d)35 b(in)d(the)h(stable)g(manifold)i(of)e Fr(p)2721 1574 y Fo(T)2775 1560 y Ft(.)671 1668 y Fp(\017)42 b Ft(The)33 b(mapping)h Fr(x)25 b Fp(!)h(W)7 b Fx(\()p Fr(T)h(;)15 b(x)p Fx(\))33 b Ft(is)g(a)g(lo)-5 b(c)g(al)34 b(di\013e)-5 b(omorphism.)555 1843 y Fx(Of)29 b(course,)i(analogous)f (results)e(hold)h(for)g(the)h(unstable)f(manifold)f(as)i(w)m(ell)f(as)h (for)456 1951 y(p)s(erio)s(dic)39 b(p)s(oin)m(ts)i(rather)h(than)g (just)f(\014xed)g(p)s(oin)m(ts.)75 b(W)-8 b(e)43 b(refer)f(to)h(the)f (references)456 2059 y(ab)s(o)m(v)m(e)31 b(for)f(pro)s(ofs)g(and)g (more)g(precisions.)555 2167 y(The)25 b(imp)s(ortance)f(for)g(us)h(of)g (Lemma)g(2.4)h(is)e(that)h(w)m(e)h(can)f(follo)m(w)f(the)h(dep)s (endence)456 2275 y(on)30 b(parameters)h(of)f(compact)i(subsets)d(of)i (the)f(stable)h(and)e(unstable)h(manifolds.)555 2383 y(Giv)m(en)d(t)m(w)m(o)g(h)m(yp)s(erb)s(olic)d(p)s(erio)s(dic)g(p)s (oin)m(ts,)i(w)m(e)h(\014x)f(t)m(w)m(o)i(closed)e(balls)f(of)i(the)g (stable)456 2491 y(manifold)e(of)i(the)g(\014rst)f(\014xed)h(p)s(oin)m (t)f(and)g(the)i(unstable)e(of)h(the)g(second)g(one.)40 b(The)27 b(set)456 2599 y(of)h(mappings)f(for)i(whic)m(h)e(the)i (stable)f(and)g(unstable)f(manifolds)g(in)m(tersect)i(transv)m(er-)456 2707 y(sally)34 b(in)h(the)i(in)m(terior)e(of)h(these)h(balls)d(will)g (con)m(tain)i(an)g(op)s(en)g(set)g(in)f(the)h(space)h(of)456 2814 y(mappings.)555 2922 y(The)h(b)s(oundary)e(of)j(this)e(set)i(of)f (transv)m(ersal)g(in)m(tersections)g(in)f(a)h(sp)s(eci\014ed)f(ball)456 3030 y(will)26 b(con)m(tain)k Fp(T)23 b Fx(,)29 b(the)h(set)g(of)f (maps)g(for)g(whic)m(h)f(the)i(stable)f(and)f(unstable)h(manifolds)456 3138 y(ha)m(v)m(e)g(a)g(tangency)g(in)e(the)i(in)m(terior)e(of)h(the)h (ball.)38 b(The)28 b(set)h Fp(T)51 b Fx(will)25 b(b)s(e)j(a)h(co)s (dimension)456 3246 y(one)h(manifold)f(in)g(the)h(space)h(of)g (mappings.)456 3421 y FA(Remark)f(2.5.)40 b Fx(W)-8 b(e)28 b(emphasize)f(that)h(w)m(e)g(sp)s(ecify)d(that)j(the)g(in)m (tersections)f(o)s(ccur)g(in)456 3529 y(a)37 b(ball)f(in)f(the)j(in)m (trinsic)c(top)s(ology)k(of)f(the)g(manifold.)59 b(This)35 b(will)g(allo)m(w)h(us)h(to)h(use)456 3637 y(implicit)27 b(function)i(theorems)i(in)e(in\014nite)f(dimensional)f(spaces)k(etc.) 555 3745 y(Di\013eren)m(t)g(c)m(hoice)g(of)g(balls)e(will)f(lead)i(to)h (di\013eren)m(t)f(p)s(ossibilities)c(of)31 b(applying)d(the)456 3853 y(obstruction)21 b(criterion.)37 b(This)21 b(is)h(p)s(erfectly)g (reasonable)g(since)g(the)h(obstruction)f(giv)m(es)456 3961 y(a)30 b(su\016cien)m(t)g(condition)f(for)h(non-existence)h(of)f (circles.)555 4069 y(When)39 b(w)m(e)g(ha)m(v)m(e)h(an)f(in)m (tersection)f(in)g(a)h(ball,)g(w)m(e)h(ha)m(v)m(e)g(in\014nitely)35 b(man)m(y)k(other)456 4177 y(in)m(tersections)31 b(of)g(the)h(stable)f (manifolds,)e(but)i(these)h(in)m(tersections)f(will)d(tak)m(e)33 b(place)456 4285 y(outside)j(of)h(the)g(balls.)58 b(In)36 b(particular,)h(w)m(e)g(will)d(obtain)j(secondary)f(in)m(tersections.) 456 4393 y(As)e(it)g(is)g(will)e(kno)m(wn,)j(the)g(dynamics)e(in)h(the) g(maps)g(with)g(a)h(tangency)g({)g(or)g(those)456 4501 y(close)i(to)g(that)h({)f(will)d(b)s(e)j(extremely)f(complicated)h (\(this)f(is)g(exploited)g(in)g([OS93].)456 4609 y(Nev)m(ertheless,)31 b(all)e(these)i(complications)e(happ)s(en)g(outside)g(of)i(the)g(sp)s (eci\014ed)d(balls.)555 4784 y(Giv)m(en)36 b(one)g(parameter)h(family)d (of)i(maps)g Fr(T)2126 4799 y Fo(\025)2207 4784 y Fx(whic)m(h)f (crosses)h(transv)m(ersally)f(the)456 4892 y(manifold)19 b Fp(T)43 b Fx(will)19 b(ha)m(v)m(e)j(a)g(critical)e(v)-5 b(alue)20 b Fr(\025)1918 4906 y Fu(0)1979 4892 y Fx(suc)m(h)h(that)h (for)f Fr(\016)j Fx(small)c(enough)h Fr(\025)k Fp(2)g Fx([)p Fr(\025)3332 4906 y Fu(0)3374 4892 y Fp(\000)456 5000 y Fr(\016)n(;)15 b(\025)587 5014 y Fu(0)627 5000 y Fx(\))37 b(there)g(is)f(transv)m(erse)h(in)m(tersection)f(of)h(the)g (stable)f(and)g(unstable)g(manifolds)456 5108 y(in)d(the)i(sp)s (eci\014ed)e(ball.)52 b(When)34 b Fr(\025)f Fp(2)f Fx(\()p Fr(\025)1859 5122 y Fu(0)1899 5108 y Fr(;)15 b(\025)1992 5122 y Fu(0)2055 5108 y Fx(+)22 b Fr(\016)s Fx(])36 b(there)f(is)f(no)g (in)m(tersection)h(in)e(the)456 5216 y(sp)s(eci\014ed)28 b(ball.)p eop %%Page: 7 7 7 6 bop 1213 251 a Fu(OBSTR)n(UCTION)30 b(AND)g(RENORMALIZA)-6 b(TION)722 b(7)555 450 y Fx(Again,)35 b(as)f(in)f(Remark)h(2.5)h(w)m(e) f(emphasize)g(that)g(there)h(is)e(no)g(claim)g(ab)s(out)h(the)456 558 y(global)e(b)s(eha)m(vior)h(of)h(the)f(map)g Fr(T)1625 573 y Fo(\025)1704 558 y Fx(in)f Fr(\025)f Fp(2)f Fx(\()p Fr(\025)2076 572 y Fu(0)2116 558 y Fr(;)15 b(\025)2209 572 y Fu(0)2271 558 y Fx(+)22 b Fr(\016)s Fx(].)51 b(The)33 b(only)f(thing)h(that)h(w)m(e)456 666 y(are)c(claiming)f(is)g(that)i(w) m(e)g(do)f(not)h(ha)m(v)m(e)h(an)m(y)e(in)m(tersection)h(in)e(the)h(sp) s(eci\014ed)f(balls.)555 774 y(It)j(is)f(heuristically)e(useful)h(to)i (think)e(that)j(w)m(e)f(are)g(discussing)d(lo)m(w)m(er)j(b)s(ounds)d (on)456 882 y(the)40 b(complexit)m(y)g(of)h(the)g(maps.)70 b(On)39 b(one)i(side)f(of)g Fp(T)63 b Fx(w)m(e)41 b(assert)g(that)g (there)f(is)g(a)456 990 y(transv)m(erse)34 b(in)m(tersection)g(of)g (the)g(stable)g(and)f(unstable)g(manifolds.)49 b(On)33 b(the)h(other)456 1098 y(side)h(of)i(the)f(manifold)f Fp(T)22 b Fx(,)38 b(our)e(b)s(ound)f(on)h(the)h(complexit)m(y)f(is)f (trivial)g(and)h(w)m(e)g(do)456 1206 y(not)30 b(claim)g(an)m(ything.) 456 1400 y(2.6.)46 b FA(F)-9 b(orm)m(ulations)45 b(of)g (renormalization)g(group.)h Fx(In)38 b(this)g(section,)k(w)m(e)d(will) 456 1508 y(discuss)30 b(t)m(w)m(o)j(form)m(ulations)e(of)i(the)f (renormalization)f(group)g(that)i(ha)m(v)m(e)h(b)s(een)d(used)456 1616 y(b)s(oth)e(for)h(n)m(umerical)f(exp)s(erimen)m(ts)h(and)f(for)i (some)f(theorems.)555 1724 y(There)25 b(are)h(other)f(form)m(ulations)f (of)h(renormalization)f(group)h(for)g(t)m(wist)g(mappings)456 1831 y(\(see)f([CJ02)q(])g(for)f(a)h(discussion)d(of)j(man)m(y)g(of)f (them\))h(but)f(w)m(e)h(p)s(oin)m(t)f(that)h(in)f(the)h(form)m(u-)456 1939 y(lation)e(of)g(the)h(conjectures)h(and)e(in)f(the)i(deriv)-5 b(ation)22 b(of)h(the)f(dynamical)g(consequences)456 2047 y(w)m(e)30 b(will)d(only)j(use)f(a)i(few)f(prop)s(erties)e(of)i (the)g(renormalization)f(groups.)40 b(Hence,)31 b(the)456 2155 y(results)36 b(here)g(can)i(b)s(e)e(v)m(eri\014ed)h(for)f(other)i (form)m(ulations)e(This)f(prop)s(erties)h(Section)456 2263 y(2.6.3.)555 2371 y(This)23 b(is)h(natural)f(since)h(the)h(main)f (consequence)h(\(the)g(optimalit)m(y)f(of)h(the)f(obstruc-)456 2479 y(tion)29 b(criterion\))h(can)h(b)s(e)e(form)m(ulated)h(indep)s (enden)m(tly)e(of)i(renormalization)f(theory)-8 b(.)456 2673 y(2.6.1.)47 b Ft(MacKay)31 b(R)-5 b(enormalization)34 b(op)-5 b(er)g(ator)33 b(and)e(c)-5 b(ommuting)32 b(p)-5 b(airs.)47 b Fx(In)27 b(this)f(sec-)456 2781 y(tion)34 b(w)m(e)i(describ)s(e)d(brie\015y)g(the)j(form)m(ulation)e(of)h (renormalization)f(group)g(included)456 2889 y(in)26 b([McK82)r(].)40 b(W)-8 b(e)28 b(do)g(not)g(aim)f(to)h(giv)m(e)g(a)g (full)e(description)f(of)j(the)g(theory)-8 b(.)40 b(W)-8 b(e)29 b(only)456 2997 y(co)m(v)m(er)c(the)f(de\014nitions)d(that)k (will)c(allo)m(w)i(us)g(to)i(form)m(ulate)f(the)g(conjectures)g (precisely)-8 b(.)555 3105 y(The)36 b(basic)f(idea)g(of)h(the)h (renormalization)d(group)h(describ)s(ed)f(b)s(elo)m(w)h(is)g(that)i(w)m (e)456 3213 y(select)31 b(an)f(small)f(neigh)m(b)s(orho)s(o)s(d)f(of)i (a)h(p)s(oin)m(t)e(in)g(the)i(circle)e(and)h(lo)s(ok)g(at)h(the)f (return)456 3321 y(map)k(at)h(this)e(neigh)m(b)s(orho)s(o)s(d.)50 b(By)34 b(scaling)g(this)f(neigh)m(b)s(orho)s(o)s(d)f(to)j(a)f(full)e (size,)k(w)m(e)456 3429 y(obtain)41 b(a)i(new)e(t)m(wist)h(map)g({)h (the)f(renormalized)f(map)h({)g(of)g(the)h(ann)m(ulus.)74 b(The)456 3537 y(n)m(um)m(b)s(er)28 b(of)i(iterates)g(that)g(tak)m(es)i (to)e(return)f(to)h(the)g(neigh)m(b)s(orho)s(o)s(d)d(is)i(giv)m(en)h(b) m(y)f(the)456 3645 y(\014rst)21 b(term)g(in)g(the)h(con)m(tin)m(ued)f (fraction)g(of)h(the)g(con)m(tin)m(ued)g(expansion)e(of)i(the)g (rotation)456 3753 y(n)m(um)m(b)s(er.)81 b(If)43 b(w)m(e)i(p)s(erform)e (the)h(renormalization)f(around)g(an)h(orbit)g(of)g(rotation)456 3860 y(n)m(um)m(b)s(er)27 b Fr(!)h Fx(=)d([)p Fr(m;)15 b(a)1159 3874 y Fu(1)1199 3860 y Fr(;)g(a)1287 3874 y Fu(2)1327 3860 y Fr(;)g(:)g(:)g(:)r Fx(],)29 b(the)g(renormalized)f (map)h(will)d(ha)m(v)m(e)k(at)g(the)f(origin)e(an)456 3968 y(orbit)37 b(of)i(rotation)h(n)m(um)m(b)s(er)d Fr(!)1555 3935 y Fm(0)1617 3968 y Fx(=)i([)p Fr(a)1800 3982 y Fu(1)1840 3968 y Fr(;)15 b(a)1928 3982 y Fu(2)1968 3968 y Fr(;)g(:)g(:)g(:)r Fx(].)66 b(The)38 b(orbit)g(of)h(the)g(renormalized)456 4076 y(map)c(will)e(b)s(e)i(an)h(in)m(v)-5 b(arian)m(t)35 b(circle)g(with)g(rotation)h(n)m(um)m(b)s(er)e Fr(!)2694 4043 y Fm(0)2753 4076 y Fx(if)h(and)g(only)g(if)g(the)456 4184 y(orbit)29 b(of)i(the)f(the)h(original)d(map)i(is)g(a)g(circle)g (of)h(rotation)f(n)m(um)m(b)s(er)f Fr(!)s Fx(.)555 4292 y(More)45 b(precisely)-8 b(,)47 b([McK82)r(])e(de\014nes)e(the)i (renormalization)e(op)s(erator)h Fp(N)3196 4306 y Fo(m)3263 4292 y Fx(,)k(for)456 4400 y Fr(m)25 b Fp(2)g Fq(Z)p Fx(,)h(acting)k(on)h(the)f(pair)f(\()p Fr(U;)15 b(T)e Fx(\))31 b(of)g(maps,)f(b)m(y)456 4617 y(\(3\))769 b Fp(N)1415 4631 y Fo(m)1507 4617 y Fx(=)1603 4489 y Fn(\032)1712 4562 y Fr(U)1784 4529 y Fm(0)1833 4562 y Fx(=)221 b Fr(B)5 b(T)13 b(B)2339 4529 y Fm(\000)p Fu(1)2483 4562 y Fr(;)1712 4670 y(T)1778 4637 y Fm(0)1827 4670 y Fx(=)89 b Fr(B)5 b(T)2127 4637 y Fo(m)2193 4670 y Fr(U)10 b(B)2339 4637 y Fm(\000)p Fu(1)2483 4670 y Fr(:)456 4829 y Fx(where)28 b Fr(B)k Fx(is)c(a)h(co)s(ordinate)f(c)m(hange)i(that)f(has)f(to)h(b)s (e)f(c)m(hosen)h(so)f(that)h(the)g(renormal-)456 4937 y(ization)h(has)g(a)g(\014xed)g(p)s(oin)m(t.)456 5108 y FA(Remark)42 b(2.6.)j Fx(It)38 b(is)e(more)h(natural)f(from)g(the)i (mathematical)f(p)s(oin)m(t)f(of)h(view)g(to)456 5216 y(c)m(hose)c Fr(B)38 b Fx(dep)s(ending)30 b(on)j(the)g(map)f(so)h(that) h(the)f(renormalized)e(set)i(of)g(comm)m(uting)p eop %%Page: 8 8 8 7 bop 456 251 a Fu(8)872 b(R.)23 b(DE)g(LA)h(LLA)-8 b(VE)23 b(AND)g(A.)g(OL)-8 b(VERA)456 450 y Fx(pairs)23 b(satis\014es)h(some)g(normalization.)38 b(Nev)m(ertheless,)26 b(one)f(often)g(\014nds)d(descriptions)456 558 y(of)30 b(the)h(renormalization)e(op)s(erator)h(as)h(including)c(a)k(\014xed)e Fr(B)5 b Fx(.)555 666 y(The)43 b(analysis)g(of)h(the)g(t)m(w)m(o)h (renormalization)d(op)s(erators)i(obtained)f(is)g(sligh)m(tly)456 774 y(di\013eren)m(t)29 b(b)s(ecause,)i(if)e(w)m(e)i(tak)m(e)h Fr(B)j Fx(dep)s(ending)28 b(on)i(the)h(map,)f(the)h(deriv)-5 b(ativ)m(e)30 b(of)g(the)456 882 y(renormalization)j(has)h(a)g (\014nite)g(rank)f(term)i(extra)g(compared)f(with)f(the)i(deriv)-5 b(ativ)m(e)456 990 y(of)30 b(the)h(renormalization)e(if)g(w)m(e)i (consider)e Fr(B)35 b Fx(as)c(\014xed.)555 1098 y(As)d(w)m(e)h(will)d (see,)j(the)g(analysis)d(that)j(w)m(e)g(will)c(describ)s(e)i(w)m(orks)h (for)g(b)s(oth)g(v)m(ersions.)456 1206 y(In)h(the)i(discussion)d(of)j (n)m(umerical)e(exp)s(erimen)m(ts,)h(w)m(e)h(will)d(see)j(that)g(w)m(e) g(imp)s(ose)f(cer-)456 1314 y(tain)22 b(normalizations,)i(hence)g(it)f (is)f(more)h(appropriate)g(to)h(consider)e Fr(B)28 b Fx(as)c(dep)s(ending)456 1421 y(on)29 b(the)h(map,)f(but)g(since)g(w)m (e)g(will)e(b)s(e)i(w)m(orking)g(close)g(to)i(the)e(\014xed)g(p)s(oin)m (ts)f(this)h(do)s(es)456 1529 y(not)36 b(mak)m(e)h(a)g(big)e (di\013erence)h(in)f(the)h(n)m(umerics.)58 b(In)35 b(n)m(umerical)g (applications,)h(one)456 1637 y(often)26 b(computes)g Fr(B)k Fx(in)25 b(a)h(preliminary)d(run)h(b)m(y)i(imp)s(osing)e(map)h (dep)s(enden)m(t)g(normal-)456 1745 y(izations)k(and,)h(then,)h(in)e (other)h(runs)f(use)h(the)h(computed)f(map.)555 1909 y(Let)j Fp(f)p Fr(p)811 1923 y Fo(n)858 1909 y Fr(;)15 b(q)939 1923 y Fo(n)986 1909 y Fp(g)33 b Fx(b)s(e)e(the)i(con)m(v)m (ergen)m(ts)h(of)e Fr(!)j Fx(suc)m(h)d(that)h(its)f(con)m(tin)m(ued)g (fraction)g(ex-)456 2017 y(pansion)22 b(is)h Fr(p)914 2031 y Fo(n)960 2017 y Fr(=q)1046 2031 y Fo(n)1119 2017 y Fx(=)i([)p Fr(m)1320 2031 y Fu(0)1359 2017 y Fr(;)15 b(m)1479 2031 y Fu(1)1519 2017 y Fr(;)g(m)1639 2031 y Fu(2)1678 2017 y Fr(;)g(:)g(:)g(:)i(;)e(m)1960 2031 y Fo(n)2007 2017 y Fx(])24 b(and)f(lim)2353 2031 y Fo(n)p Fm(!1)2556 2017 y Fr(p)2602 2031 y Fo(n)2649 2017 y Fr(=q)2735 2031 y Fo(n)2807 2017 y Fx(=)i Fr(!)s Fx(.)38 b(The)24 b(renor-)456 2125 y(malization)29 b(of)i(the)f(in)m(v)-5 b(arian)m(t)30 b(circle)f(with)g(rotation)i(n)m(um)m(b)s(er)e Fr(!)k Fx(is)d(de\014ned)f(b)m(y)456 2273 y(\(4\))180 b Fp(N)826 2287 y Fo(m)888 2295 y Fl(n)950 2273 y Fr(:)15 b(:)g(:)h Fp(N)1146 2287 y Fo(m)1208 2296 y Fs(0)1247 2273 y Fx(\()p Fr(F)s(;)f(R)q Fx(\))26 b(=)f(\()p Fr(B)1714 2287 y Fo(n)1761 2273 y Fr(F)1832 2235 y Fo(q)1864 2244 y Fl(n)p Fk(\000)p Fs(1)1989 2273 y Fr(R)2059 2235 y Fo(p)2095 2244 y Fl(n)p Fk(\000)p Fs(1)2219 2273 y Fr(B)2293 2235 y Fm(\000)p Fu(1)2288 2295 y Fo(n)2387 2273 y Fr(;)15 b(B)2496 2287 y Fo(n)2543 2273 y Fr(F)2614 2235 y Fo(q)2646 2243 y Fl(n)2693 2273 y Fr(R)2763 2235 y Fo(p)2799 2243 y Fl(n)2845 2273 y Fr(B)2919 2235 y Fm(\000)p Fu(1)2914 2295 y Fo(n)3012 2273 y Fx(\))77 b Fr(:)456 2421 y Fx(The)44 b(univ)m(ersal)f(map)h Fr(F)1343 2388 y Fm(\003)1427 2421 y Fx(is)g(de\014ned)f(as)i(the)g(limit)e(of)i(this)e(pro)s(cess,) 48 b(suc)m(h)d(that)456 2529 y Fp(N)531 2543 y Fo(m)593 2551 y Fl(n)654 2529 y Fr(:)15 b(:)g(:)i Fp(N)851 2543 y Fo(m)913 2552 y Fs(0)951 2529 y Fx(\()p Fr(F)s(;)e(R)q Fx(\))29 b(con)m(v)m(erges)h(to)f(a)f(\014xed)f(p)s(oin)m(t)g(\()p Fr(U)2367 2496 y Fm(\003)2407 2529 y Fr(;)15 b(T)2513 2496 y Fm(\003)2553 2529 y Fx(\),)29 b(with)d Fr(T)2912 2496 y Fm(\003)2977 2529 y Fx(=)f Fr(F)3144 2496 y Fm(\003)3183 2529 y Fx(,)k Fr(U)3309 2496 y Fm(\003)3374 2529 y Fx(=)456 2637 y Fr(B)5 b(F)601 2604 y Fm(\003)640 2637 y Fr(B)714 2604 y Fm(\000)p Fu(1)807 2637 y Fx(,)31 b(where)f Fr(B)5 b Fx(\()p Fr(y)s(;)15 b(x)p Fx(\))26 b(=)f(\()p Fr(\013y)s(;)15 b(\014)5 b(x)p Fx(\).)555 2745 y(Giv)m(en)41 b(a)h(one)f(parameter)g (family)f(of)h(maps)g Fr(F)2248 2759 y Fo(\026)2295 2745 y Fx(,)i(w)m(e)f(renormalize)e(this)g(family)456 2853 y(with)29 b(a)h(reparameterization)h(of)g Fr(\026)p Fx(,)f(then)g(the)g (one)h(parameter)g(families)456 3001 y(\(5\))1047 b Fr(B)1687 3015 y Fo(n)1733 3001 y Fr(F)1804 2957 y Fo(q)1836 2965 y Fl(n)1791 3035 y Fo(\026\016)1866 3016 y Fk(\000)p Fl(n)1963 3001 y Fr(R)2033 2964 y Fo(p)2069 2972 y Fl(n)2115 3001 y Fr(B)2189 2964 y Fm(\000)p Fu(1)2184 3024 y Fo(n)456 3161 y Fx(con)m(v)m(erge)32 b(to)f(a)g(univ)m(ersal)e(one)h(parameter)h (family)e Fr(F)2349 3128 y Fm(\003)2336 3183 y Fo(\026)2388 3161 y Fx(.)555 3273 y(In)c([McK82)q(])h(one)f(can)h(\014nd)d(v)m(ery)j (strong)f(n)m(umerical)f(evidence)h(that)h(the)f(op)s(erator)456 3381 y Fp(N)531 3395 y Fu(1)605 3381 y Fx(has)36 b(a)g(\014xed)f(p)s (oin)m(t)g(whic)m(h)f(is)h(h)m(yp)s(erb)s(olic)e(with)i(an)g(unstable)g (manifold)e(of)j(di-)456 3489 y(mension)25 b(1)i(and)f(a)h(stable)g (manifold)d(of)j(co)s(dimension)e(1.)39 b(This)25 b(\014xed)h(p)s(oin)m (t)g(con)m(trols)456 3597 y(the)h(disapp)s(earance)f(of)h(all)f(the)i (the)f(in)m(v)-5 b(arian)m(t)27 b(circles)f(whose)h(rotation)h(n)m(um)m (b)s(er)e(has)456 3705 y(a)k(con)m(tin)m(ued)g(fraction)g(whic)m(h)f (consists)h(of)g(1)h(except)g(for)f(a)h(\014nite)e(n)m(um)m(b)s(er)g (of)h(terms)456 3812 y(\(these)h(n)m(um)m(b)s(ers)e(are)h(commonly)g (called)g Ft(\\noble")37 b Fx(n)m(um)m(b)s(ers\).)555 3920 y(T)-8 b(o)32 b(the)f(b)s(est)g(of)g(our)g(kno)m(wledge,)h(in)e (the)h(literature)f(there)i(are)f(no)g(direct)g(calcu-)456 4028 y(lations)36 b(of)h(the)h(\014xed)f(p)s(oin)m(ts)f(for)h(op)s (erators)g Fp(N)2167 4042 y Fo(n)2214 4028 y Fx(.)61 b(Nev)m(ertheless,)40 b(there)e(is)e(strong)456 4136 y(indirect)f(evidence)i(of)h(their)f(existence.)62 b(It)38 b(has)f(b)s(een)f(rep)s(eatedly)h(observ)m(ed)h(that)456 4244 y(the)g(destruction)f(of)h(in)m(v)-5 b(arian)m(t)37 b(circles)g(of)h(the)h(corresp)s(onding)c(rotation)k(n)m(um)m(b)s(ers) 456 4352 y(exhibits)g(scaling)i(prop)s(erties)f(and)h(that)h(the)g (scaling)f(co)s(e\016cien)m(ts)h(are)g(univ)m(ersal.)456 4460 y(Indeed)29 b(in)g(Section)h(5)h(w)m(e)g(will)d(presen)m(t)i(some) h(of)f(this)g(evidence.)555 4568 y(The)42 b(precise)g(mathematical)h (form)m(ulation)e(of)i(the)g(renormalization)e(group)h(in)456 4676 y(terms)22 b(of)g(comm)m(uting)h(pairs)e(has)h(some)h(tec)m (hnical)f(di\016culties.)35 b(Notably)-8 b(,)25 b(it)d(is)g(some-)456 4784 y(what)k(in)m(v)m(olv)m(ed)g(to)h(mak)m(e)g(precise)f(what)g(is)f (the)h(nature)g(of)h(the)f(space)h(of)f(comm)m(uting)456 4892 y(pairs.)48 b(F)-8 b(rom)34 b(the)f(n)m(umerical)f(p)s(oin)m(t)h (of)g(view,)h(one)f(has)g(to)i(imp)s(ose)d(the)h(constrain)m(t)456 5000 y(of)h(area)h(preserv)-5 b(ation)34 b(as)g(w)m(ell)f(as)i(comm)m (utation.)53 b(These)34 b(incon)m(v)m(eniences)f(can)i(b)s(e)456 5108 y(solv)m(ed.)63 b(In)37 b(particular,)h([Sti97)q(])g(con)m(tains)g (a)g(precise)f(mathematical)i(form)m(ulation)456 5216 y(and)33 b(pro)m(v)m(es)i(the)f(existence)g(of)h(a)f(non-trivial)e (function)h(whic)m(h)f(is)h(a)i(\014xed)e(p)s(oin)m(t)g(of)p eop %%Page: 9 9 9 8 bop 1213 251 a Fu(OBSTR)n(UCTION)30 b(AND)g(RENORMALIZA)-6 b(TION)722 b(9)456 450 y Fx(a)35 b(p)s(o)m(w)m(er)g(of)g(the)g (renormalization)f(group)g(\(the)i(pro)s(of)e(in)f([Sti97)q(])i(do)s (es)f(not)i(estab-)456 558 y(lish)f(prop)s(erties)h(suc)m(h)h(as)h (area)g(preserving.)61 b(Numerical)36 b(evidence)i(suggests)g(v)m(ery) 456 666 y(strongly)26 b(that)i(this)e(ob)5 b(ject)29 b(is)d(a)i(\014xed)e(p)s(oin)m(t)h(and)f(that)i(it)f(satis\014es)g(the) g(other)h(prop-)456 774 y(erties.\))456 938 y(2.6.2.)47 b Ft(The)37 b(Ko)-5 b(ch)37 b(formulation)h(or)f(r)-5 b(enormalization)40 b(gr)-5 b(oup.)46 b Fx(In)34 b(the)h(form)m (ulation)456 1046 y(of)j([Ko)s(c99)r(],)j(the)d(mappings)f(considered)g (are)i(the)g(time)f(one)h(maps)f(of)h(a)g(p)s(erio)s(dic)456 1154 y(Hamiltonian)33 b(system)h(or)h(as)g(sections)g(of)f(a)h(v)m (ector)i(\014eld.)52 b(The)34 b(results)f(should)g(b)s(e)456 1262 y(the)j(same)g(as)g(the)g(consideration)f(of)h(maps)f(since)g(all) g(maps)g(can)i(b)s(e)e(generated)i(in)456 1370 y(this)30 b(w)m(a)m(y)-8 b(.)47 b(\(F)-8 b(or)33 b(mappings)d(of)i(class)f Fr(C)1853 1337 y Fo(r)1891 1370 y Fx(,)h Fr(r)e Fp(2)e Fq(N)33 b Fp([)21 b(f1g)p Fx(,)33 b(this)e(is)g(done)h(in)e([Mos86)r (],)456 1478 y(in)f(the)h(analytic)g(case)i([KP94].\))555 1586 y(F)-8 b(or)44 b(simplicit)m(y)c(w)m(e)j(just)g(describ)s(e)e(the) i(situation)f(of)h(one)g(degree)g(of)g(freedom)456 1694 y(systems)30 b(since)f(it)h(is)f(the)h(only)f(situation)g(where)h (there)g(are)g(n)m(umerics)f(suggesting)h(a)456 1802 y(non-trivial)e(\014xed)i(p)s(oin)m(t.)555 1910 y(As)35 b(w)m(e)h(will)d(see,)k(the)f(transformation)e(in)m(v)m(olv)m(es)h(sev) m(eral)h(c)m(hoices.)56 b(All)34 b(of)h(them)456 2017 y(ha)m(v)m(e)23 b(the)f(same)g(trivial)e(\014xed)i(p)s(oin)m(t)f(\(See) h(Theorem)g(1.6)h(in)e([Ko)s(c99)q(].\))38 b(Nev)m(ertheless,)456 2125 y(it)26 b(is)h(p)s(ossible)e(that)i(eac)m(h)i(of)e(these)h (transformations)f(ha)m(v)m(e)h(di\013eren)m(t)f(prop)s(erties)f(far) 456 2233 y(from)k(the)g(trivial)f(\014xed)g(p)s(oin)m(t.)555 2341 y(Giv)m(en)38 b(a)h(Hamiltonian)e(of)h(the)h(form)f Fr(H)7 b Fx(\()p Fr(p;)15 b(q)s Fx(\),)41 b(where)d Fr(p)g Fx(are)g(real)g(v)-5 b(ariables,)40 b Fr(q)456 2449 y Fx(angle)26 b(v)-5 b(ariables,)26 b Fr(p;)15 b(q)30 b Fx(are)d(symplectically)e(conjugate)j(v)-5 b(ariables,)26 b(The)g(de\014nition)e(of)456 2557 y(the)j(renormalization)f(in)g([Ko)s (c99)q(])h(\(1.11\))j(p.)39 b(481)28 b(is)e(obtained)h(b)m(y)g(setting) g(basically)1497 2752 y Fp(R)p Fx(\()p Fr(H)7 b Fx(\))26 b(=)1864 2690 y Fr(\022)p 1859 2731 55 4 v 1859 2814 a(\026)1924 2752 y(H)h Fp(\016)21 b(U)2150 2766 y Fo(H)2237 2752 y Fp(\016)g Fr(T)2356 2766 y Fo(\026)456 2942 y Fx(where)671 3070 y Fp(\017)42 b Fr(T)811 3084 y Fo(\026)888 3070 y Fx(is)30 b(a)h(transformation)e(of)i(the)f(form)g Fr(T)2203 3084 y Fo(\026)2250 3070 y Fx(\()p Fr(p;)15 b(q)s Fx(\))26 b(=)f(\()p Fr(\026)p Fx(\()p Fr(T)2763 3037 y Fm(\003)2803 3070 y Fx(\))2838 3037 y Fm(\000)p Fu(1)2932 3070 y Fr(p;)15 b(T)e(q)s Fx(\).)858 3178 y(That)30 b(is)f Fr(T)1229 3192 y Fo(\026)1306 3178 y Fx(consists)h(on)g(p)s (erforming)e(a)j(transformation)f Fr(T)38 b Fp(2)24 b Fr(S)5 b(L)p Fx(\(2)p Fr(;)15 b Fq(Z)p Fx(\))758 3285 y(that)29 b(mak)m(es)f(one)g(co)s(ordinate)f(angle)h(to)g(b)s(e)f(v)m (ery)h(close)g(to)g(a)g(resonan)m(t)g(com-)758 3393 y(bination.)60 b(Of)37 b(course,)j(to)e(k)m(eep)g(the)g(symplectic)e(c)m(haracter,)41 b(w)m(e)d(need)g(to)758 3501 y(p)s(erform)29 b(another)i (transformation)f(on)g(the)g(action)h(v)-5 b(ariables.)858 3609 y(The)22 b(parameter)i Fr(\026)e Fx(is)g(an)h(scaling)f(parameter) h(that)g(corresp)s(onds)f(to)h(lo)s(ok-)758 3717 y(ing)41 b(just)g(at)h(an)f(smaller)f(region)h(in)f(the)h(action)h(v)-5 b(ariable.)73 b(This)39 b(scaling)758 3825 y(mak)m(es)23 b(the)f(transformation)f(analyticit)m(y)h(impro)m(ving)e(and,)j(hence)f (compact.)671 3933 y Fp(\017)42 b Fx(The)23 b(transformation)g Fp(U)1608 3947 y Fo(H)1698 3933 y Fx(is)f(a)i(canonical)f (transformation)f(that)i(eliminates)758 4041 y(non-resonan)m(t)31 b(mo)s(des.)555 4168 y(The)g(full)e(transformation)i(requires)f(to)i (add)e(some)i(extra)g(transformations)f(that)456 4276 y(just)e(preserv)m(e)i(some)g(normalizations)456 4441 y(2.6.3.)47 b Ft(Assumptions)30 b(on)f(r)-5 b(enormalization)31 b(gr)-5 b(oup)30 b(formulation.)47 b Fx(The)26 b(form)m(ulation)456 4549 y(of)i(Conjectures)h(3.3,)h(3.5,)h(3.6)f(and)e(the)h(pro)s(of)f (of)g(their)g(consequences)h(can)g(b)s(e)g(done)456 4656 y(largely)g(indep)s(enden)m(t)f(of)j(the)g(renormalization)e(group)g (formalism)g(that)i(w)m(e)g(use.)555 4764 y(There)f(are)g(v)m(ery)h (few)e(prop)s(erties)g(of)h(the)g(renormalization)f(op)s(erator)h Fp(R)g Fx(that)h(w)m(e)456 4872 y(use.)555 4980 y(On)f(the)g(analytic)g (side,)g(w)m(e)h(will)c(assume:)671 5108 y Fp(\017)42 b Fx(It)d(is)f(p)s(ossible)f(to)i(de\014ne)g(\014xed)f(p)s(oin)m(ts,)i (their)e(linearization,)h(stable)g(and)758 5216 y(unstable)29 b(manifolds.)p eop %%Page: 10 10 10 9 bop 456 251 a Fu(10)837 b(R.)23 b(DE)g(LA)h(LLA)-8 b(VE)23 b(AND)g(A.)g(OL)-8 b(VERA)671 450 y Fp(\017)42 b Fx(There)31 b(are)h Fr(\025)f Fx(lemmas)g(for)h(the)f(stable)g (manifold)f(under)f(preimages)j(of)f(the)758 558 y(renormalization)25 b(and)h(for)g(the)h(unstable)e(manifold)g(under)g(iteration)h(of)g(the) 758 666 y(renormalization.)555 856 y(F)-8 b(or)32 b(the)f(MacKa)m(y)h (renormalization)e(group,)h(a)g(rather)f(satisfactory)i(mathemat-)456 964 y(ical)37 b(description)f(is)i(in)e([Sti97)q(].)64 b(The)38 b(pap)s(er)f([Ko)s(c99)q(])i(includes)c(a)k(rather)f(precise) 456 1072 y(mathematical)30 b(description.)555 1180 y(Besides)24 b(these)h(rather)e(analytical)h(and)f(tec)m(hnical)h(prop)s(erties,)g (w)m(e)h(use)f(only)f(a)i(few)456 1288 y(geometric)31 b(prop)s(erties.)555 1395 y(W)-8 b(e)30 b(denote)g(b)m(y)e Fr(p)1172 1409 y Fo(n)1219 1395 y Fr(=q)1305 1409 y Fo(n)1381 1395 y Fx(the)h(con)m(v)m(ergen)m(ts)i(of)e(an)f(irrational)f(n)m(um)m (b)s(er.)39 b(In)28 b(the)h(case)456 1503 y(of)36 b(the)h(golden)e (mean,)k(the)d(con)m(v)m(ergen)m(ts)j(will)33 b(b)s(e)j Fr(p)2309 1517 y Fo(n)2391 1503 y Fx(=)f Fr(F)2555 1517 y Fo(n)2602 1503 y Fx(,)j Fr(q)2706 1517 y Fo(n)2788 1503 y Fx(=)d Fr(F)2952 1517 y Fo(n)p Fu(+1)3126 1503 y Fx(with)g Fr(F)3397 1517 y Fo(n)456 1611 y Fx(the)30 b(Fib)s(onacci)g(n)m(um)m(b)s(ers.)671 1801 y Fp(\017)42 b Fx(If)f(the)h(map)f Fr(G)h Fx(has)g(a)g(p)s(erio)s(dic)d(orbit)h Fr(o)i Fx(of)g(t)m(yp)s(e)f Fr(p)2670 1815 y Fo(n)p Fu(+1)2807 1801 y Fr(=q)2893 1815 y Fo(n)p Fu(+1)3030 1801 y Fx(,)k(then)c(the)758 1909 y(renormalized)29 b(map)h Fp(R)p Fx(\()p Fr(G)p Fx(\))i(has)e(an)g(orbit)g Fr(o)2306 1876 y Fm(0)2359 1909 y Fx(of)h(t)m(yp)s(e)f Fr(p)2713 1923 y Fo(n)2760 1909 y Fr(=q)2846 1923 y Fo(n)2893 1909 y Fx(.)671 2017 y Fp(\017)42 b Fx(The)g(orbit)g Fr(o)1236 1984 y Fm(0)1302 2017 y Fx(is)g(h)m(yp)s(erb)s(olic)d(or)k(critical)f(if)f(and)h(only)g (if)f(the)i(orbit)f Fr(o)g Fx(is)758 2125 y(h)m(yp)s(erb)s(olic)28 b(or)i(critical.)671 2233 y Fp(\017)42 b Fx(If)j(the)h(map)f Fr(G)g Fx(has)g(t)m(w)m(o)i(h)m(yp)s(erb)s(olic)c(orbits)h Fr(o)p Fx(,)49 b Fr(p)c Fx(of)h(t)m(yp)s(e)f Fr(p)3059 2247 y Fo(n)p Fu(+1)3196 2233 y Fr(=q)3282 2247 y Fo(n)p Fu(+1)3419 2233 y Fx(,)758 2341 y Fr(p)804 2355 y Fo(n)851 2341 y Fr(=q)937 2355 y Fo(n)1015 2341 y Fx(resp)s(ectiv)m(ely)-8 b(,)31 b(whose)f(lo)s(cal)g(stable)g(and)h(unstable)e(manifolds)f(in)m (ter-)758 2449 y(sect,)34 b(then,)e(the)g(corresp)s(onding)e(orbits)h Fr(o)2250 2416 y Fm(0)2273 2449 y Fx(,)h Fr(p)2376 2416 y Fm(0)2431 2449 y Fx(of)g(the)g(renormalized)f(maps)758 2557 y(ha)m(v)m(e)h(lo)s(cal)e(stable)g(manifolds)e(that)j(in)m (tersect.)671 2676 y Fp(\017)42 b Fx(The)34 b(in)m(tersection)g(of)h Fr(W)1648 2628 y Fo(u;)p Fx(lo)s(c)n Fo(;)p Fu(\()p Fo(G)p Fu(\))1635 2688 y Fo(o)1955 2676 y Fx(,)g Fr(W)2114 2628 y Fo(s;)p Fx(lo)s(c)o Fo(;)p Fu(\()p Fo(G)p Fu(\))2101 2688 y Fo(p)2447 2676 y Fx(is)f(transv)m(ersal)g(\(resp.)52 b(tan-)758 2820 y(gen)m(tial\))30 b(if)d(and)i(only)e(if)h(the)h(in)m (tersection)g Fr(W)2365 2772 y Fo(u;)p Fx(lo)s(c)n Fo(;)p Fu(\()p Fm(R)p Fu(\()p Fo(G)p Fu(\)\))2352 2850 y Fo(o)2386 2831 y Fk(0)2787 2820 y Fx(,)g Fr(W)2940 2772 y Fo(s;)p Fx(lo)s(c)o Fo(;)p Fu(\()p Fm(R)p Fu(\()p Fo(G)p Fu(\)\))2927 2850 y Fo(p)2963 2831 y Fk(0)3383 2820 y Fx(is)758 2939 y(transv)m(ersal)h(\(resp.)41 b(tangen)m(tial\).)671 3047 y Fp(\017)h Fx(The)25 b(map)f Fr(G)h Fx(has)f(a)h(golden)g(mean)f (non-trivial)f(in)m(v)-5 b(arian)m(t)24 b(circle)g(if)f(and)h(only)758 3155 y(if)30 b(the)g(map)g Fp(R)p Fx(\()p Fr(G)p Fx(\))i(do)s(es.)555 3345 y(The)21 b(v)m(eri\014cation)g(that)h(b)s(oth)f(the)g(MacKa)m(y)i (and)e(the)h(Ko)s(c)m(h)f(form)m(ulation)f(of)i(renor-)456 3453 y(malization)29 b(groups)h(satisfy)g(the)g(ab)s(o)m(v)m(e)i(prop)s (erties)c(is)i(quite)g(straigh)m(tforw)m(ard.)555 3561 y(The)k(ab)s(o)m(v)m(e)h(geometric)h(prop)s(erties)c(seem)j(also)f(to)h (b)s(e)f(v)m(eri\014ed)f(b)m(y)h(other)h(renor-)456 3669 y(malization)24 b(group)g(form)m(ulations)g(that)i(ha)m(v)m(e)g(b)s (een)e(prop)s(osed)g(and)g(that)i(w)m(e)g(will)c(not)456 3777 y(discuss.)555 3885 y(Analogous)k(prop)s(erties)e(can)i(b)s(e)e (form)m(ulated)h(for)h(renormalization)e(groups)h(based)456 3993 y(on)30 b(other)g(rotation)h(n)m(um)m(b)s(ers.)456 4227 y FA(Remark)36 b(2.7.)43 b Fx(W)-8 b(e)34 b(note)f(that)g(it)f(is) f(not)i(in)e(principle)e(clear)k(at)g(all)e(that)i(di\013eren)m(t)456 4334 y(renormalization)23 b(groups)i(will)e(ha)m(v)m(e)j(the)f(same)h (non-trivial)d(\014xed)i(p)s(oin)m(ts)f(and)g(m)m(uc)m(h)456 4442 y(less)k(that)h(they)g(will)e(ha)m(v)m(e)j(the)f(same)g(stable)g (manifolds.)38 b(Of)28 b(course,)i(this)d(follo)m(ws)h(if)456 4550 y(one)e(can)h(giv)m(e)g(a)g(dynamical)f(c)m(haracterization)h(of)g (these)g(stable)g(manifolds)d(whic)m(h)h(is)456 4658 y(indep)s(enden)m(t)i(of)i(the)h(renormalization)d(group.)40 b(One)29 b(consequence)h(of)f(Conjectures)456 4766 y(3.3)34 b(and)e(3.5)i(is)e(that)i(the)f(in)m(v)-5 b(arian)m(t)32 b(manifolds)f(of)i(these)g(renormalization)f(groups)456 4874 y(can)27 b(b)s(e)f(c)m(haracterized)j(as)e(the)g(b)s(oundaries)e (of)i(the)g(maps)g(with)f(a)h(smo)s(oth)g(in)m(v)-5 b(arian)m(t)456 4982 y(circle)29 b(of)i(the)f(sp)s(eci\014ed)f(rotation)i(n)m(um)m(b)s (er.)555 5090 y(W)-8 b(e)35 b(think)d(that)i(it)e(w)m(ould)h(b)s(e)f (in)m(teresting)h(to)h(giv)m(e)g(a)g(more)f(direct)g(pro)s(of)g(of)g (the)456 5198 y(iden)m(tit)m(y)c(of)i(in)m(v)-5 b(arian)m(t)30 b(manifolds)e(of)i(di\013eren)m(t)g(renormalization)f(groups.)p eop %%Page: 11 11 11 10 bop 1213 251 a Fu(OBSTR)n(UCTION)30 b(AND)g(RENORMALIZA)-6 b(TION)687 b(11)946 450 y Fx(3.)46 b Fw(Precise)34 b(f)n(ormula)-6 b(tion)32 b(of)i(the)g(conjectures)555 612 y Fx(The)41 b(main)g(results)f(of)i(this)f(section)h(are)g(the)f(follo)m(wing)g (Lemma)g(3.1)i(and)e(the)456 720 y(form)m(ulation)24 b(of)i(Conjectures)g(3.3,)i(3.5,)h(3.6.)40 b(Later,)28 b(in)c(Section)i(4)g(w)m(e)h(will)c(establish)456 828 y(some)30 b(consequences)g(of)g(the)g(conjectures.)42 b(Some)30 b(of)g(the)g(quan)m(titativ)m(e)g(predictions)456 936 y(of)d(the)h(conjectures)g(will)d(b)s(e)i(v)m(eri\014ed)g(n)m (umerically)f(to)i(high)e(accuracy)j(in)d(Section)i(5.)456 1044 y(Other)h(v)-5 b(arian)m(ts)31 b(of)f(the)h(conjectures)g(will)c (b)s(e)j(form)m(ulated)g(in)f(Section)h(5.4)555 1152 y(The)39 b(follo)m(wing)g(Lemma)h(3.1)g(sho)m(ws)g(that)g(w)m(e)g(can)g (de\014ne)f(manifolds)f(of)i(maps)456 1260 y(with)33 b(a)i(tangency)h(in)e(a)h(prescrib)s(ed)e(segmen)m(t.)55 b(The)34 b(maps)h(with)e(a)j(tangency)f(in)f(a)456 1367 y(sp)s(eci\014ed)23 b(segmen)m(t)j(are)g(co)s(dimension)d(one)j (manifold)d(in)h(a)h(space)h(of)f(mappings)f(with)456 1475 y(w)m(e)30 b(will)e(denote)j(b)m(y)f Fp(T)23 b Fx(.)555 1583 y(On)f(one)h(side)f(of)h(this)e(submanifold)f Fp(T)i Fx(,)j(w)m(e)e(kno)m(w)g(that)g(there)g(will)d(b)s(e)i(a)h(transv)m (erse)456 1691 y(homo)s(clinic)38 b(in)m(tersection.)71 b(Hence,)44 b(w)m(e)d(will)c(b)s(e)j(able)g(to)h(apply)e(the)i (obstruction)456 1799 y(criterion)27 b(and)h(conclude)g(that)h(there)f (are)h(no)g(in)m(v)-5 b(arian)m(t)27 b(circles.)39 b(On)28 b(the)h(other)f(side)456 1907 y(of)i(the)h(submanifold,)c(w)m(e)k(will) d(not)i(claim)g(an)m(ything.)555 2015 y(The)24 b(gist)g(of)h(the)g (conjectures)g(is)e(that)i(the)g(unstable)e(manifold)f Fp(T)47 b Fx(at)25 b(the)g(non)m(triv-)456 2123 y(ial)e(\014xed)h(p)s (oin)m(t)g(has)h(a)g(transv)m(erse)g(in)m(tersection)g(with)e(some)j (bifurcation)c(manifolds.)456 2231 y(See)31 b(Conjecture)h(3.3)g(more)g (precise)f(form)m(ulation)f(of)h(the)h(main)e(conjecture)j(as)e(w)m (ell)456 2339 y(as)c(Conjectures)f(3.5)i(and)e(3.6)i(for)f(a)g(more)g (detailed)f(conjectures)h(that)g(lead)g(to)g(more)456 2447 y(consequences.)555 2555 y(Then,)i(it)f(will)e(follo)m(w)i(b)m(y)g (the)h Fr(\025)p Fx(-lemma)g(that)g Fp(N)2274 2522 y Fm(\000)p Fo(n)2376 2555 y Fx(\()p Fp(T)23 b Fx(\))29 b(will)d(accum)m(ulate)k(on)e(the)456 2663 y(stable)39 b(manifold)f(of)i(the)f(\014xed)g(p)s(oin)m(t.)68 b(W)-8 b(e)41 b(will)c(sho)m(w)j(that)g(if)f(a)h(family)e(crosses)456 2771 y Fp(N)544 2738 y Fm(\000)p Fo(n)645 2771 y Fx(\()p Fp(T)23 b Fx(\),)41 b(then,)g(high)c(p)s(erio)s(d)f(orbits)h(ha)m(v)m (e)j(hetero)s(clinic)c(tangencies.)66 b(This)36 b(also)456 2878 y(implies)i(that)43 b(there)e(is)g(no)g(in)m(v)-5 b(arian)m(t)41 b(circle)g(with)f(golden)h(mean)h(rotation)g(n)m(um-)456 2986 y(b)s(er.)g(Therefore,)32 b(b)m(y)f(the)g(accum)m(ulation,)h(of)f Fp(N)2153 2953 y Fm(\000)p Fo(n)2255 2986 y Fx(\()p Fp(T)23 b Fx(\),)32 b(w)m(e)g(see)g(that,)g(if)e Fr(T)3098 3001 y Fo(\025)3175 2986 y Fx(crosses)456 3094 y(transv)m(ersally)f(the)h (stable)f(manifold)f(of)i(the)g(\014xed)g(p)s(oin)m(t)f(at)h Fr(\025)c Fx(=)f Fr(\025)2838 3108 y Fu(0)2877 3094 y Fx(,)30 b(then,)g(for)g(an)m(y)456 3202 y Fr(\016)35 b(>)c Fx(0,)36 b(w)m(e)f(can)f(\014nd)f Fr(\025)f Fp(2)f Fx(\()p Fr(\025)1498 3216 y Fu(0)1538 3202 y Fr(;)15 b(\025)1631 3216 y Fu(0)1693 3202 y Fx(+)23 b Fr(\016)s Fx(\))35 b(so)g(that)f(the)h(map)f Fr(T)2636 3217 y Fo(\025)2715 3202 y Fx(do)s(es)g(not)h(ha)m(v)m(e)g(an)m(y)456 3310 y(in)m(v)-5 b(arian)m(t)29 b(circle)h(with)f(rotation)i(n)m(um)m(b)s (er)e(golden)h(mean.)555 3418 y(Actually)-8 b(,)24 b(a)f(more)g(quan)m (titativ)m(e)g(statemen)m(t)h(ab)s(out)e(the)h(in)m(tersections)f(will) e(imply)456 3526 y(the)36 b(sligh)m(tly)f(stronger)i(result)f(that)h (there)g(is)e(a)i Fr(\016)2225 3540 y Fu(0)2301 3526 y Fr(>)e Fx(0)i(suc)m(h)f(that)i Fr(\025)d Fp(2)h Fx(\()p Fr(\025)3177 3540 y Fu(0)3216 3526 y Fr(;)15 b(\025)3309 3540 y Fu(0)3374 3526 y Fx(+)456 3634 y Fr(\016)496 3648 y Fu(0)535 3634 y Fx(\),)34 b(then,)g Fr(T)918 3649 y Fo(\025)996 3634 y Fx(do)s(es)f(not)g(ha)m(v)m(e)h(a)f(non-trivial)e (in)m(v)-5 b(arian)m(t)32 b(circle)g(with)g(golden)g(mean)456 3742 y(rotation)e(n)m(um)m(b)s(er.)555 3850 y(Hence,)g(putting)c (together)k(the)e(results,)f(w)m(e)h(obtain)g(that)g(for)g(a)g(family)e Fr(F)3141 3865 y Fo(\025)3215 3850 y Fx(in)g(the)456 3958 y(univ)m(ersalit)m(y)33 b(class,)j(there)f(exists)f(an)h(analytic) f(golden)h(mean)f(in)m(v)-5 b(arian)m(t)34 b(circle)h(for)456 4066 y Fr(\025)29 b Fp(2)f Fx(\(0)p Fr(;)15 b(\025)800 4033 y Fm(\003)841 4066 y Fx(\),)34 b(there)f(is)e(a)i(circle)f(with)f (a)i(\014nite)f(and)g(univ)m(ersal)f(degree)i(of)g(regularit)m(y)456 4174 y(for)j Fr(\025)g Fx(=)g Fr(\025)850 4141 y Fm(\003)927 4174 y Fx(and)g(for)h Fr(\025)f(>)g(\025)1505 4141 y Fm(\003)1581 4174 y Fx(there)h(is)f(a)h(h)m(yp)s(erb)s(olic)e(Can)m (tor)i(set)g(with)f(rotation)456 4282 y(n)m(um)m(b)s(er)29 b(golden)h(mean.)456 4547 y(3.1.)46 b FA(T)-9 b(angencies)27 b(of)e(stable)g(and)h(unstable)f(manifolds)g(of)h(p)s(erio)s(dic)g(p)s (oin)m(ts.)456 4655 y Fx(The)j(\014rst)h(ingredien)m(t)f(of)i(the)f (form)m(ulation)f(of)i(our)f(conjectures)h(is)e(the)i(follo)m(wing.)456 4847 y FA(Lemma)25 b(3.1.)36 b Ft(L)-5 b(et)27 b Fr(F)40 b Ft(b)-5 b(e)26 b(a)h Fr(C)1516 4814 y Fo(!)1592 4847 y Ft(ar)-5 b(e)g(a)29 b(pr)-5 b(eserving)27 b(di\013e)-5 b(omorphism)30 b(of)c Fq(T)3002 4814 y Fu(1)3050 4847 y Fp(\002)7 b Fq(R)35 b Ft(\(r)-5 b(esp.)456 4955 y(ar)g(e)g(a)34 b(pr)-5 b(eserving)33 b(r)-5 b(eversible\).)43 b(Assume)32 b(that)656 5108 y Fx(i\))41 b Fr(F)e Ft(has)26 b(two)g(monotone)h(hyp) -5 b(erb)g(olic)28 b(p)-5 b(erio)g(dic)27 b(orbits)f Fr(Z)q(;)15 b(Z)2752 5075 y Fm(0)2800 5108 y Ft(of)26 b(r)-5 b(otation)28 b(num-)758 5216 y(b)-5 b(er)33 b Fr(m=n)p Ft(,)f Fr(m)1228 5183 y Fm(0)1252 5216 y Fr(=n)1352 5183 y Fm(0)1407 5216 y Ft(r)-5 b(esp)g(e)g(ctively.)p eop %%Page: 12 12 12 11 bop 456 251 a Fu(12)837 b(R.)23 b(DE)g(LA)h(LLA)-8 b(VE)23 b(AND)g(A.)g(OL)-8 b(VERA)631 466 y Fx(ii\))40 b Ft(The)47 b(lo)-5 b(c)g(al)49 b(invariant)f(manifolds)g Fr(W)2111 418 y Fo(S;L;)p Fu(\()p Fo(F)10 b Fu(\))2098 495 y Fo(Z)2354 466 y Ft(,)50 b Fr(W)2531 418 y Fo(U)o(;L)2646 395 y Fk(0)2667 418 y Fo(;)p Fu(\()p Fo(F)10 b Fu(\))2518 496 y Fo(Z)2571 477 y Fk(0)2847 466 y Ft(ar)-5 b(e)48 b(r)-5 b(e)g(gular)48 b(for)758 574 y(some)34 b Fr(L;)15 b(L)1155 541 y Fm(0)1203 574 y Fr(>)25 b Fx(0)p Ft(.)606 699 y Fx(iii\))39 b Fr(W)857 651 y Fo(S;L;)p Fu(\()p Fo(F)10 b Fu(\))844 728 y Fo(Z)1100 699 y Ft(,)32 b Fr(W)1259 651 y Fo(U)o(;L)1374 628 y Fk(0)1396 651 y Fo(;)p Fu(\()p Fo(F)10 b Fu(\))1246 728 y Fo(Z)1561 699 y Ft(have)33 b(exactly)g(one)f(tangency)h(in)f(the)g(interior)h(of)g(the)758 807 y(se)-5 b(gment)34 b(c)-5 b(onsider)g(e)g(d.)44 b(This)33 b(tangency)g(is)f(quadr)-5 b(atic.)456 934 y(\(In)31 b(the)h(c)-5 b(ase)31 b(that)i Fr(F)44 b Ft(is)31 b(a)g(r)-5 b(eversible)32 b(map,)g(we)g(assume)g(furthermor)-5 b(e)33 b(that)f Fr(Z)g Fp(2)25 b(L)p Ft(,)456 1042 y Fr(Z)525 1009 y Fm(0)573 1042 y Fp(2)g(L)722 1009 y Fm(0)777 1042 y Ft(with)34 b Fp(L)p Fr(;)15 b Fp(L)1141 1009 y Fm(0)1196 1042 y Ft(critic)-5 b(al)34 b(lines)e(for)i Fr(F)13 b Ft(.\))555 1150 y(Then)35 b(we)g(c)-5 b(an)36 b(\014nd)f(a)g(neighb)-5 b(orho)g(o)g(d)39 b Fr(U)44 b Ft(of)35 b Fr(F)48 b Ft(in)35 b(the)g(sp)-5 b(ac)g(e)36 b(of)f(ar)-5 b(e)g(a)36 b(pr)-5 b(eserving)456 1258 y(\(r)g(esp.)54 b(r)-5 b(eversible)37 b(ar)-5 b(e)g(a)38 b(pr)-5 b(eserving)37 b(maps\),)i Fx(\001)32 b Fr(>)g Fx(0)37 b Ft(and)g(neighb)-5 b(orho)g(o)g(ds)41 b(of)36 b Fr(V)5 b(;)15 b(V)3421 1225 y Fm(0)456 1366 y Ft(of)32 b Fr(Z)q(;)15 b(Z)734 1333 y Fm(0)791 1366 y Ft(in)32 b Fq(T)961 1333 y Fm(0)1006 1366 y Fp(\002)20 b Fq(R)41 b Ft(\(r)-5 b(esp.)43 b(in)33 b Fp(L)p Fr(;)15 b Fp(L)1739 1333 y Fm(0)1762 1366 y Ft(\))33 b(such)g(that:)555 1474 y(F)-7 b(or)34 b(every)e Fr(G)26 b Fp(2)f Fr(U)636 1600 y Fx(a\))42 b Ft(In)37 b Fr(V)56 b Ft(ther)-5 b(e)37 b(is)f(one)h(and)g(only)g(one)g(monotone)h(hyp)-5 b(erb)g(olic)38 b(p)-5 b(erio)g(dic)38 b(p)-5 b(oint)758 1708 y Fr(Z)820 1722 y Fo(G)909 1708 y Ft(of)30 b Fr(G)g Ft(of)f(r)-5 b(otation)32 b(numb)-5 b(er)30 b Fr(m=n)p Ft(.)41 b(The)29 b(mapping)i Fr(G)26 b Fp(!)f Fr(Z)2950 1722 y Fo(G)3039 1708 y Ft(is)k(analytic)758 1816 y(\(similarly)35 b(for)e Fr(Z)1385 1783 y Fm(0)1378 1843 y Fo(G)1437 1816 y Ft(\).)631 1949 y Fx(b\))41 b Fr(W)857 1901 y Fo(S;L)p Fm(\000)p Fu(\001)p Fo(;)p Fu(\()p Fo(G)p Fu(\))844 1978 y Fo(Z)892 1989 y Fl(G)1214 1949 y Ft(,)33 b Fr(W)1374 1901 y Fo(U)o(;L)p Fm(\000)p Fu(\001)p Fo(;)p Fu(\()p Fo(G)p Fu(\))1361 1983 y Fo(Z)1414 1960 y Fk(0)1409 2006 y Fl(G)1768 1949 y Ft(ar)-5 b(e)33 b(r)-5 b(e)g(gular.)456 2096 y(The)32 b(mappings)1166 2258 y Fr(G)25 b Fp(!)h Fr(W)1478 2210 y Fo(S;L)p Fm(\000)p Fu(\001)p Fo(;)p Fu(\()p Fo(G)p Fu(\))1465 2286 y Fo(Z)1513 2297 y Fl(G)1928 2258 y Fr(;)108 b(G)26 b Fp(!)f Fr(W)2373 2210 y Fo(U)o(;L)p Fm(\000)p Fu(\001)p Fo(;)p Fu(\()p Fo(G)p Fu(\))2360 2292 y Fo(Z)2413 2269 y Fk(0)2408 2315 y Fl(G)456 2424 y Ft(ar)-5 b(e)31 b(analytic)h(when)f(we)f(give)g(the)h(sp)-5 b(ac)g(e)32 b(of)f(se)-5 b(gments)31 b(the)g(Banach)g(manifold)i(struc-)456 2532 y(tur)-5 b(e)34 b(indic)-5 b(ate)g(d)37 b(b)-5 b(efor)g(e.)47 b(Mor)-5 b(e)g(over,)36 b(ther)-5 b(e)36 b(is)e(a)h(c)-5 b(o)g(dimension)36 b(1)f(Banach)h(manifold)456 2640 y Fx(\006)c Ft(c)-5 b(ontaine)g(d)34 b(in)f Fr(U)42 b Ft(such)33 b(that:)641 2767 y Fx(c\))42 b Ft(If)33 b Fr(G)26 b Fp(2)e Fx(\006)p Ft(,)32 b(then)1357 2933 y Fr(W)1456 2885 y Fo(S;L)p Fm(\000)p Fu(\001)p Fo(;)p Fu(\()p Fo(G)p Fu(\))1443 2962 y Fo(Z)1491 2973 y Fl(G)1878 2933 y Ft(and)66 b Fr(W)2186 2885 y Fo(S;L)p Fm(\000)p Fu(\001)p Fo(;)p Fu(\()p Fo(G)p Fu(\))2173 2967 y Fo(Z)2226 2944 y Fk(0)2221 2990 y Fl(G)758 3099 y Ft(have)33 b(a)g(quadr)-5 b(atic)35 b(tangency.)631 3207 y Fx(d\))41 b Ft(if)e Fr(F)911 3222 y Fo(\025)996 3207 y Ft(is)g(a)h(family)g(such)f(that)i Fr(F)1933 3221 y Fu(0)2010 3207 y Fp(2)36 b Fx(\006)p Ft(,)2242 3134 y Fn(\000)2314 3171 y Fo(d)p 2293 3186 78 4 v 2293 3239 a(d\025)2381 3207 y Fr(F)2452 3134 y Fn(\001)2494 3243 y Fo(\025)2576 3207 y Fp(6)p Fx(=)h(0)j Ft(tr)-5 b(ansversal)41 b(to)f Fx(\006)p Ft(.)758 3315 y(Then)986 3461 y Fr(W)1085 3413 y Fo(S;L)p Fm(\000)p Fu(\001)p Fo(;)p Fu(\()p Fo(F)1402 3425 y Fl(\025)1441 3413 y Fu(\))1072 3489 y Fo(Z)1120 3500 y Fl(F)1159 3518 y(\025)1498 3461 y Fj(t)24 b Fr(W)1682 3413 y Fo(U)o(;L)p Fm(\000)p Fu(\001)p Fo(;)p Fu(\()p Fo(F)2003 3425 y Fl(\025)2043 3413 y Fu(\))1669 3495 y Fo(Z)1722 3472 y Fk(0)1717 3518 y Fl(F)1756 3535 y(\025)2139 3461 y Ft(for)66 b Fr(\025)25 b Fp(2)g Fr(I)2521 3475 y Fu(1)2586 3461 y Fx(=)g(\(0)p Fr(;)15 b Fx(\001\))758 3644 y Ft(and)962 3790 y Fr(W)1061 3742 y Fo(S;L)p Fm(\000)p Fu(\001)p Fo(;)p Fu(\()p Fo(F)1378 3754 y Fl(\025)1417 3742 y Fu(\))1048 3819 y Fo(Z)1096 3830 y Fl(F)1135 3847 y(\025)1473 3790 y Fp(6)p Fj(t)25 b Fr(W)1658 3742 y Fo(U)o(;L)p Fm(\000)p Fu(\001)p Fo(;)p Fu(\()p Fo(F)1979 3754 y Fl(\025)2019 3742 y Fu(\))1645 3824 y Fo(Z)1698 3801 y Fk(0)1693 3847 y Fl(F)1732 3864 y(\025)2115 3790 y Ft(for)33 b Fr(\025)26 b Fx(=)f Fr(I)2475 3804 y Fu(2)2539 3790 y Fx(=)g(\()p Fp(\000)p Fx(\001)p Fr(;)15 b Fx(0\))758 3979 y Ft(or)34 b(we)e(have)i(the)f(similar)g(c)-5 b(onclusion)34 b(for)f Fr(I)2307 3993 y Fu(1)2372 3979 y Fx(=)25 b(\()p Fp(\000)p Fx(\001)p Fr(;)15 b Fx(0\))p Ft(,)33 b Fr(I)2871 3993 y Fu(2)2936 3979 y Fx(=)25 b(\(0)p Fr(;)15 b Fx(\001\))p Ft(.)555 4106 y(Which)41 b(of)g(the)g(two)h (alternatives)g(ab)-5 b(ove)42 b(takes)f(plac)-5 b(e)42 b(dep)-5 b(ends)42 b(on)f(the)g(sign)g(of)456 4135 y Fn(D)511 4236 y Fx(\010)577 4250 y Fo(F)622 4259 y Fs(0)660 4236 y Fr(;)700 4135 y Fn(\020)785 4200 y Fo(d)p 765 4215 V 765 4267 a(d\025)852 4236 y Fr(F)910 4251 y Fo(\025)956 4135 y Fn(\021)1010 4299 y Fo(\025)p Fu(=0)1146 4135 y Fn(E)1233 4236 y Ft(wher)-5 b(e)33 b Fx(\010)1555 4250 y Fo(F)1600 4259 y Fs(0)1669 4236 y Ft(is)f(a)g(c)-5 b(ontinuous)33 b(functional)f(\(which)h(we)f(wil)5 b(l)32 b(de-)456 4371 y(scrib)-5 b(e)32 b(later)i(somewhat)h(explicitly.)456 4568 y(Pr)-5 b(o)g(of.)70 b Fx(P)m(art)30 b(a\))g(is)f(just)g(the)g (implicit)d(function)i(theorem)i(for)f(h)m(yp)s(erb)s(olic)e(p)s(erio)s (dic)456 4676 y(\014xed)h(p)s(oin)m(ts.)40 b(P)m(art)30 b(b\))f(is)g(the)h(analytic)f(dep)s(endence)f(of)i(the)g(stable)f (manifold)f(that)456 4784 y(can)i(b)s(e)g(found)f(in)g([Mey75)r(],)i(a) g(more)f(mo)s(dern)f(treatmen)m(t)j(in)d([CFdlL02a].)555 4892 y(P)m(art)k(of)g(the)f(results)f(of)i([CFdlL02a])g(are)f(rather)g (explicit)f(form)m(ulas)h(for)g(the)g(de-)456 5000 y(riv)-5 b(ativ)m(e)26 b(of)h(stable)f(manifolds)e(with)i(resp)s(ect)g(to)i(the) f(map.)39 b(They)26 b(can)h(b)s(e)f(expressed)456 5108 y(as)i(sums)f(along)h(the)h(p)s(erio)s(dic)c(orbit.)39 b(\(Sometimes)28 b(this)f(sums)g(are)i(called)e(Melnik)m(o)m(v)456 5216 y(functions\).)p eop %%Page: 13 13 13 12 bop 1213 251 a Fu(OBSTR)n(UCTION)30 b(AND)g(RENORMALIZA)-6 b(TION)687 b(13)555 450 y Fx(In)37 b(particular,)g(this)g(sho)m(ws)g (that)h(if)e(one)i(has)f(a)h(tangen)m(t)g(in)m(tersection)f(and)g(the) 456 558 y(\014rst)k(order)h(p)s(erturbation)e(do)s(es)i(not)h(v)-5 b(anish,)44 b(then)e(the)g(family)f(starts)i(crossing.)456 666 y(This)28 b(establishes)h(d\).)1976 774 y Fj(\003)1329 b Fi(2)555 920 y Fx(Let)31 b Fr(F)776 934 y Fo(n)854 920 y Fx(denote)g(the)f(sequence)h(of)g(Fib)s(onacci)e(n)m(um)m(b)s (ers.)555 1028 y(Of)f(particular)f(in)m(terest)i(for)g(us)f(will)e(b)s (e)i(the)h(manifolds)e(\006)2602 1042 y Fo(n;B)2753 1028 y Fx(whic)m(h)h(corresp)s(ond)456 1135 y(to)g(the)g(tangen)m(t)g(in)m (tersection)g(of)f(the)h(stable)f(manifold)f(of)h(a)h(Birkho\013)f (orbit)g(\(mono-)456 1243 y(tone)d(p)s(erio)s(dic)d(orbit\))i(of)h(t)m (yp)s(e)g Fr(F)1597 1257 y Fo(n)1644 1243 y Fr(=F)1747 1257 y Fo(n)p Fu(+1)1909 1243 y Fx(and)f(the)h(unstable)f(manifold)e (of)j(a)g(Birkho\013)456 1351 y(orbit)29 b(of)i(t)m(yp)s(e)f Fr(F)1044 1365 y Fo(n)p Fu(+1)1182 1351 y Fr(=F)1285 1365 y Fo(n)p Fu(+2)1453 1351 y Fx(taking)g(place)h(in)e(a)i(ball)d Fr(B)5 b Fx(.)555 1459 y(As)25 b(argued)f(b)s(efore,)h(the)g(manifolds) d(\006)1892 1473 y Fo(n;B)2039 1459 y Fx(are)i(co)s(dimension)e(1)j (manifolds.)37 b(These)456 1567 y(manifolds)26 b(lo)s(cally)h(separate) i(the)g(space)g(of)g(maps)f(in)f(t)m(w)m(o)j(comp)s(onen)m(ts.)40 b(The)28 b(maps)456 1675 y(in)g(one)i(comp)s(onen)m(t)f({)h(whic)m(h)e (w)m(e)i(will)d(denote)j(\006)2186 1637 y Fu(+)2186 1704 y Fo(n;B)2338 1675 y Fx({)g(p)s(ossess)f(a)h(transv)m(erse)f(in)m(ter-) 456 1793 y(section)24 b(of)g(the)g(stable)f(manifolds)f(of)i(the)g (orbit)f(of)h(t)m(yp)s(e)g Fr(F)2478 1807 y Fo(n)2525 1793 y Fr(=F)2628 1807 y Fo(n)p Fu(+1)2790 1793 y Fx(and)g(the)g (unstable)456 1901 y(manifold)34 b(of)j(orbit)f(of)h(t)m(yp)s(e)g Fr(F)1556 1915 y Fo(n)p Fu(+1)1693 1901 y Fr(=F)1796 1915 y Fo(n)p Fu(+2)1971 1901 y Fx(The)f(maps)h(in)e(the)i(other)g (comp)s(onen)m(t)g({)456 2010 y(whic)m(h)d(w)m(e)h(will)e(denote)i (\006)1399 1972 y Fm(\000)1399 2039 y Fo(n;B)1557 2010 y Fx({)h(do)f(not)g(ha)m(v)m(e)i(suc)m(h)e(a)g(transv)m(erse)h(in)m (tersection)f(in)456 2123 y(the)j(ball)f Fr(B)42 b Fx({)d(ev)m(en)g(if) e(they)i(ma)m(y)f(ha)m(v)m(e)i(it)e(in)f(a)h(larger)g(ball.)63 b(W)-8 b(e)39 b(note)g(that)g(the)456 2231 y(obstruction)27 b(criterion)f(applies)h(to)h(the)h(maps)e(in)g(\006)2265 2193 y Fu(+)2265 2260 y Fo(n;B)2415 2231 y Fx(and)h(for)f(them,)i(w)m (e)f(can)h(con-)456 2344 y(clude)34 b(that)j(they)f(do)g(not)g(ha)m(v)m (e)h(an)f(in)m(v)-5 b(arian)m(t)35 b(circle)g(of)h(rotation)g(n)m(um)m (b)s(er)f(golden)456 2452 y(mean.)40 b(F)-8 b(or)32 b(the)e(maps)g(in)f (\006)1463 2414 y Fm(\000)1463 2481 y Fo(n;B)1586 2452 y Fx(,)h(w)m(e)h(are)g(not)g(claiming)d(an)m(ything.)555 2565 y(As)j(it)f(is)g(also)h(standard,)f(in)g(the)h(study)f(of)h (destruction)f(of)h(in)m(v)-5 b(arian)m(t)30 b(circles,)g(w)m(e)456 2673 y(de\014ne)j(the)h(manifolds)e(\000)1355 2687 y Fo(n)1436 2673 y Fx(to)j(b)s(e)f(the)g(set)h(of)f(p)s(oin)m(ts)f(for)h (whic)m(h)f(the)i(orbit)e(of)h(t)m(yp)s(e)456 2781 y Fr(F)514 2795 y Fo(n)561 2781 y Fr(=F)664 2795 y Fo(n)p Fu(+1)833 2781 y Fx(in)29 b(the)h(critical)g(line)f(are)i(suc)m(h)f (that)h(they)f(ha)m(v)m(e)i(an)f(eigen)m(v)-5 b(alue)30 b(1.)42 b(These)456 2889 y(bifurcation)28 b(surfaces)i(pla)m(y)m(ed)h (a)f(v)m(ery)h(imp)s(ortan)m(t)f(role)g(in)f(the)h(study)g(in)f([Gre79) q(].)555 2997 y(It)37 b(is)f(imp)s(ortan)m(t)f(to)j(note)f(that)g(all)f (the)h(form)m(ulations)e(of)i(the)g(renormalization)456 3105 y(group)e(ha)m(v)m(e)i(the)f(prop)s(ert)m(y)f(that)h(if)f(a)h(map) f(has)g(a)h(hetero)s(clinic)e(transv)m(erse)i(in)m(ter-)456 3213 y(section)e(b)s(et)m(w)m(een)h(an)g(orbit)e(of)i(t)m(yp)s(e)g Fr(F)1850 3227 y Fo(n)1897 3213 y Fr(=F)2000 3227 y Fo(n)p Fu(+1)2173 3213 y Fx(and)f(another)g(of)h(t)m(yp)s(e)g Fr(F)3066 3227 y Fo(n)p Fu(+1)3203 3213 y Fr(=F)3306 3227 y Fo(n)p Fu(+2)456 3321 y Fx(o)s(ccurring)30 b(in)h(a)h(sp)s (eci\014ed)e(ball,)h(then,)h(the)g(renormalized)e(mapping)h(has)g(a)h (hetero-)456 3429 y(clinic)27 b(transv)m(erse)j(in)m(tersection)f(b)s (et)m(w)m(een)h(orbit)f(of)g(t)m(yp)s(e)h Fr(F)2545 3443 y Fo(n)p Fm(\000)p Fu(1)2682 3429 y Fr(=F)2785 3443 y Fo(n)2863 3429 y Fx(and)f(another)g(of)456 3537 y(t)m(yp)s(e)c Fr(F)713 3551 y Fo(n)761 3537 y Fr(=F)864 3551 y Fo(n)p Fu(+1)1028 3537 y Fx(o)s(ccurring)f(in)g(another)i(ball)e(whic)m(h)h (is)g(just)g(as)h(scaled)f(v)m(ersion)g(of)h(the)456 3644 y(previous)i(one.)555 3752 y(Therefore,)j(as)f(an)g(immediate)g (consequence,)h(w)m(e)g(ha)m(v)m(e:)456 3894 y FA(Prop)s(osition)37 b(3.2.)42 b Ft(With)34 b(the)g(ab)-5 b(ove)34 b(de\014nitions)g(of)g (tangency)f(manifolds)i Fx(\006)e Ft(and)456 4002 y(the)g(r)-5 b(enormalization)36 b(gr)-5 b(oup)34 b Fp(N)13 b Ft(,)32 b(we)h(have)1576 4134 y Fp(N)13 b Fx(\(\006)1765 4148 y Fo(n;B)1888 4134 y Fx(\))26 b Fp(\032)f Fx(\006)2111 4148 y Fo(n)p Fm(\000)p Fu(1)p Fo(;B)1576 4274 y Fp(N)13 b Fx(\(\000)1756 4288 y Fo(n)1803 4274 y Fx(\))26 b Fp(\032)f Fx(\000)2017 4288 y Fo(n)p Fm(\000)p Fu(1)456 4415 y Fx(3.2.)46 b FA(F)-9 b(orm)m(ulation)35 b(of)g(the)f(conjectures.)456 4556 y(Conjecture)j(3.3.)43 b Ft(L)-5 b(et)35 b Fp(N)47 b Ft(b)-5 b(e)35 b(the)g(r)-5 b(enormalization)38 b(gr)-5 b(oup)36 b(for)f(golden)g(me)-5 b(an)36 b(in-)456 4664 y(variant)d(cir)-5 b(cles.)555 4772 y(L)g(et)33 b Fr(T)778 4739 y Fm(\003)850 4772 y Ft(b)-5 b(e)32 b(a)h(non-trivial)h(\014xe)-5 b(d)33 b(p)-5 b(oint)34 b(for)g(the)f(r)-5 b(enormalization.)555 4880 y(The)44 b(op)-5 b(er)g(ator)47 b Fp(N)57 b Ft(at)45 b Fr(T)1440 4847 y Fm(\003)1523 4880 y Ft(is)e(hyp)-5 b(erb)g(olic.)78 b(It)44 b(has)h(a)f(one)g(dimensional)i(unsta-)456 4988 y(ble)g(manifold)i Fr(W)1098 4955 y Fo(u)1085 5015 y(T)1136 4996 y Fk(\003)1222 4988 y Ft(and)f(a)g(c)-5 b(o)g(dimension)48 b Fx(1)f Ft(stable)g(manifold)h Fr(W)2882 4955 y Fo(s)2869 5015 y(T)2920 4996 y Fk(\003)2960 4988 y Ft(.)83 b(Sinc)-5 b(e)46 b(the)456 5096 y(manifold)32 b Fr(W)929 5063 y Fo(u)916 5123 y(T)967 5104 y Fk(\003)1037 5096 y Ft(is)e(one-dimensional,)j Fr(W)1936 5063 y Fo(u)1923 5123 y(T)1974 5104 y Fk(\003)2029 5096 y Fp(\000)15 b Fr(T)2181 5063 y Fm(\003)2250 5096 y Ft(has)31 b(two)h(c)-5 b(onne)g(cte)g(d)32 b(c)-5 b(omp)g(onents)456 5216 y Fr(W)555 5171 y Fo(u;)p Fu(+)542 5244 y Fo(T)593 5225 y Fk(\003)674 5216 y Ft(,)p Fr(W)801 5171 y Fo(u;)p Fm(\000)788 5244 y Fo(T)839 5225 y Fk(\003)919 5216 y Ft(.)p eop %%Page: 14 14 14 13 bop 456 251 a Fu(14)837 b(R.)23 b(DE)g(LA)h(LLA)-8 b(VE)23 b(AND)g(A.)g(OL)-8 b(VERA)601 451 y Fx(\(1\))42 b Ft(F)-7 b(or)43 b(some)f Fr(n)1233 465 y Fu(0)1313 451 y Fp(2)e Fq(N)54 b Ft(and)42 b(some)g(b)-5 b(al)5 b(l)42 b Fr(B)5 b Ft(,)43 b(the)e(manifold)i Fr(W)2911 407 y Fo(u;)p Fu(+)2898 480 y Fo(T)2949 461 y Fk(\003)3072 451 y Ft(interse)-5 b(cts)758 559 y(tr)g(ansversal)5 b(ly)35 b(a)e(manifold)i Fx(\006)1812 573 y Fo(n)1855 582 y Fs(0)1889 573 y Fo(;B)2002 559 y Ft(as)e(describ)-5 b(e)g(d)34 b(ab)-5 b(ove.)601 679 y Fx(\(2\))42 b Ft(The)33 b(manifold)i Fr(W)1420 635 y Fo(u;)p Fm(\000)1407 708 y Fo(T)1458 689 y Fk(\003)1571 679 y Ft(interse)-5 b(cts)33 b(tr)-5 b(ansversal)5 b(ly)36 b(the)d(manifold)h Fx(\000)3093 693 y Fo(n)3136 702 y Fs(0)3174 679 y Ft(.)601 800 y Fx(\(3\))42 b Ft(The)j(manifold)g Fr(W)1442 756 y Fo(u;)p Fm(\000)1429 829 y Fo(T)1480 810 y Fk(\003)1605 800 y Ft(is)f(c)-5 b(ontaine)g(d)46 b(in)e(the)g(b)-5 b(asin)45 b(of)f(attr)-5 b(action)47 b(of)d(the)758 908 y(trivial)34 b(\014xe)-5 b(d)33 b(p)-5 b(oint.)456 1119 y FA(Remark)36 b(3.4.)44 b Fx(Note)33 b(that)g(the)g(conjecture)g(in)m(v)m(olv)m(es)g (an)f(arbitrary)f(c)m(hoice)j(of)e(the)456 1227 y(sets)37 b Fr(B)42 b Fx(where)36 b(the)i(tangencies)f(tak)m(e)i(place.)61 b(As)37 b(w)m(e)g(will)e(see,)k(the)f(consequences)456 1335 y(happ)s(en)27 b(pro)m(vided)h(one)i(suitable)e(c)m(hoice)i(can)g (b)s(e)f(made.)40 b(In)29 b(the)g(n)m(umerical)f(section)456 1443 y(5,)33 b(w)m(e)g(ha)m(v)m(e)h(detailed)e(the)g(c)m(hoices)h(that) h(w)m(e)f(ha)m(v)m(e)g(made)g(and)f(that)h(seem)g(to)g(w)m(ork.)456 1551 y(F)-8 b(rom)43 b(the)g(n)m(umerical)e(p)s(oin)m(t)g(of)i(view,)i (it)e(do)s(es)f(not)h(seem)g(a)g(hard)e(task)j(to)f(\014nd)456 1659 y(c)m(hoices)31 b(of)f(the)h(balls)d(that)j(w)m(ork.)555 1767 y(Moreo)m(v)m(er,)f(it)c(can)g(b)s(e)g(v)m(eri\014ed)g(that)g (these)h(c)m(hoices)g(do)g(not)f(a\013ect)i(the)f(n)m(umerical)456 1875 y(results)k(appreciably)-8 b(.)48 b(Of)32 b(course,)i(part)f(of)g (the)g(predictions)e(of)i(the)h(conjectures)f(is)456 1983 y(that)38 b(the)h(results)e(obtained)h(with)f(an)m(y)h(of)h(these) g(c)m(hoices)g(will)c(accum)m(ulate)k(at)g(an)456 2091 y(exp)s(onen)m(tial)29 b(rate)i(\(with)f(a)g(rather)h(small)d(constan)m (t\).)555 2302 y(T)-8 b(aking)43 b(in)m(to)g(accoun)m(t)i(Prop)s (osition)c(\(3.2\))k(and)d(applying)f(the)j Fr(\025)p Fx(-Lemma,)j(w)m(e)456 2410 y(obtain)32 b(that,)j(\000)1024 2424 y Fo(n)1104 2410 y Fx(accum)m(ulate)f(on)f Fr(W)1812 2377 y Fo(s)1799 2437 y(T)1850 2418 y Fk(\003)1890 2410 y Fx(.)49 b(Similarly)-8 b(,)31 b(\006)2438 2424 y Fo(n;B)2554 2432 y Fl(n)2633 2410 y Fx(accum)m(ulate)j(on)f Fr(W)3341 2377 y Fo(s)3328 2437 y(T)3379 2418 y Fk(\003)3419 2410 y Fx(.)456 2521 y(Both)h(accum)m(ulations)g(happ)s(en)f(in)g(the)h Fr(C)1950 2488 y Fu(1)2023 2521 y Fx({)g(or)h(ev)m(en)f(stronger)h({)f (sense)g(and)g(at)h(a)456 2629 y(exp)s(onen)m(tial)22 b(rate)i(\(See)g([CEL80],)i([AK00)q(])d(Theorem)g(1.9)i(for)e(a)h(pro)s (of)e(of)i(a)f Fr(\025)p Fx(-lemma)456 2737 y(adapted)30 b(to)h(this)e(situation.\))555 2845 y(In)23 b(particular,)g(for)g(a)g (one)h(dimensional)c(family)i Fr(T)2278 2860 y Fo(\025)2346 2845 y Fx(su\016cien)m(tly)g Fr(C)2866 2812 y Fu(1)2928 2845 y Fx(close)h(to)h Fr(W)3341 2812 y Fo(u)3328 2872 y(T)3379 2853 y Fk(\003)3419 2845 y Fx(,)456 2953 y(it)30 b(will)d(also)k(cross)f(the)h(the)f(manifolds)e(\006)1915 2967 y Fo(n)1992 2953 y Fx(transv)m(ersally)-8 b(.)555 3061 y(Since)35 b(on)g(one)h(side)e(of)i(eac)m(h)g(of)g(the)f(\006)1942 3075 y Fo(n)2024 3061 y Fx(w)m(e)h(can)g(apply)e(the)h(obstruction)g (crite-)456 3169 y(rion,)g(w)m(e)g(conclude)f(that)i(a)f(family)f (family)f Fr(T)2090 3184 y Fo(\025)2170 3169 y Fx(as)i(ab)s(o)m(v)m(e)h (con)m(tains)g(a)f(sequence)g(of)456 3277 y(semin)m(terv)-5 b(als)33 b Fr(I)1016 3291 y Fo(n)1095 3277 y Fx(=)f(\()p Fr(\025)1286 3291 y Fo(n)1333 3277 y Fr(;)15 b(\025)1426 3291 y Fo(n)1497 3277 y Fx(+)22 b Fr(\016)1630 3291 y Fo(n)1678 3277 y Fx(\))35 b(suc)m(h)f(that)h Fr(\025)d Fp(2)g Fr(I)2376 3291 y Fo(n)2457 3277 y Fx(implies)g(that)j Fr(T)3024 3292 y Fo(\025)3104 3277 y Fx(do)s(es)f(not)456 3385 y(ha)m(v)m(e)d(an)f(in)m(v)-5 b(arian)m(t)30 b(circle)g(with)f(a)h (golden)g(mean)h(rotation.)555 3492 y(F)-8 b(urthermore,)30 b Fr(\025)1160 3506 y Fo(n)1232 3492 y Fp(!)c Fr(\025)1402 3506 y Fm(\003)1471 3492 y Fx(where)j Fr(\025)1786 3506 y Fm(\003)1855 3492 y Fx(is)g(determined)f(b)m(y)i Fr(T)2599 3507 y Fo(\025)2640 3515 y Fk(\003)2705 3492 y Fp(2)25 b Fr(W)2890 3459 y Fo(s)2877 3519 y(T)2928 3501 y Fk(\003)2968 3492 y Fx(.)40 b(Note)31 b(that,)456 3600 y(according)39 b(to)i([Ko)s(c02a)r(],)h Fr(T)1461 3615 y Fo(\025)1502 3623 y Fk(\003)1582 3600 y Fx(p)s(ossesses)e(an)f(in)m(v)-5 b(arian)m(t)39 b(circle)g({)h(alb)s(eit)f(not)h(v)m(ery)456 3708 y(regular,)35 b(see)h(also)f([Sti93)q(])g({.)55 b(F)-8 b(or)36 b Fr(\025)d Fp(2)g Fx(\()p Fr(\025)1983 3722 y Fm(\003)2046 3708 y Fp(\000)24 b Fr(\016)n(;)15 b(\025)2272 3722 y Fm(\003)2312 3708 y Fx(\),)37 b(w)m(e)f(will)c(ha)m (v)m(e)37 b(that)e Fp(N)3228 3675 y Fo(n)3275 3708 y Fx(\()p Fr(T)3363 3723 y Fo(\025)3409 3708 y Fx(\))456 3816 y(will)25 b(con)m(v)m(erge)30 b(to)f(the)f(trivial)e(\014xed)h(p)s (oin)m(t)g(and,)i(therefore,)g(according)f(to)g([Ko)s(c02a)r(])456 3924 y(ha)m(v)m(e)j(an)f(in)m(v)-5 b(arian)m(t)30 b(circle)g(of)g (golden)g(mean)g(rotation)h(n)m(um)m(b)s(er.)555 4032 y(W)-8 b(e)39 b(see)f(therefore)f(that)h Fr(T)1518 4047 y Fo(\025)1559 4055 y Fk(\003)1637 4032 y Fx(b)s(elongs)e(to)i(the)g(b) s(oundary)d(of)i(the)h(sets)f(of)h(maps)456 4140 y(that)31 b(ha)m(v)m(e)g(an)f(analytic)g(in)m(v)-5 b(arian)m(t)30 b(circle)g(with)f(golden)h(mean)g(rotation)h(n)m(um)m(b)s(er.)555 4248 y(The)22 b(follo)m(wing)e(Figure)h(3.2)i(depicts)e(the)h(geometry) i({)e(in)f(the)h(in\014nite)d(dimensional)456 4356 y(space)31 b(of)f(mappings)f({)h(describ)s(ed)f(in)g(Conjecture)h(3.3.)555 4464 y(Note)40 b(that)g(Conjecture)e(3.3)i(implies)c(that)j(in)f(a)h (semi-in)m(terv)-5 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%A8F8A87D7DF82727525227F852527D277DF852527D277DF87DF85227A852 %7D277D527DF87D5227277DF87DF8A8FFFF7D7DFD09FFA8A87DFD07FFA8A8 %52A852A827A8A8A852FF52FF7D7D52A8F8A8A8A87D7D7DA87DA87DA8FF7D %52A852FF272752FF52A852FF7DA8FD057DA87DA852FFA8FFFFA87DFD1AFF %52A827A8F8FF527DF8A827277D7D277D7D7D52A8F87D7D7DF87DF87DF87D %527DF8A852A8277D277DF87D527D7DA8277DF8FFFFFF7D52FD1FFF527D52 %7D52A8277D277D7DA852A827A8A8A87DA827FF52A8527D527D7DA87D7D27 %7D7DA827A8FF7D277D52A8FD27FF7D7D2727F852527DF87DF8A852A8F87D %F8A8277D277DF852527D5252F85227A82752A852F827527D52FD2AFF277D %27A87DA852FF7DA852FF7DA87D7D7DA852A87DFF7D7D7DA852FF7DA87DA8 %7DA852A87DA8FD2AFF7DFFFF27A8277D27A8277DF8A852527DA87D7D27A8 %5252F8A8277D277DF87D527D7D7DF87D7DFD2AFF52FD05FF7D7D52A852A8 %277D52FF527D7D27A87D52FF277D7DA827A87DA852A852A852A8FD2AFF52 %FD08FFF8A8F85252FF5252277D527DF8FF2752527D277DF8A85252F87D27 %7DA8FD28FF7DA85252A8FD09FFA8A87DA852A87D7D52FF7DA87DA87DA852 %FFA87D52FF7DA852FD2AFF27277DFD0EFFA8F87DF852277D527D277D27A8 %A852277D27FF277DFD2BFF27FD14FFA87DA87D7D277DA87DF87DA8A8F87D %A8FD23FFA87DFD06FF7DFD04FFA852FD42FF5252FD05FF52FD04FF52A8FD %43FF7D27FD04FF7DFFFFA827A8FD45FFA8F87DFFFF52FF5227A8FD47FFA8 %52FFFF52FF2752FD4BFF7D27FD4EFFF8F852FD4DFF7D27FD4BFF52F8FFFF %FF52A8FD48FF5252A8FFFFFF7DF8FD46FFA8277DFD05FFA8A827FD44FFA8 %52A8FD08FFA8F8A8FD43FFA8FD0AFFA827FD2AFFFF %%EndData %%EndComments %%BeginProlog %%BeginResource: procset Adobe_level2_AI5 1.2 0 %%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation) %%Version: 1.2 0 %%CreationDate: (04/10/93) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) userdict /Adobe_level2_AI5 26 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { (AI8_CMYK_CustomColor) 6 packedarray } bind def /findrgbcustomcolor { (AI8_RGB_CustomColor) 5 packedarray } bind def /setcustomcolor { exch aload pop dup (AI8_CMYK_CustomColor) eq { pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } { dup (AI8_RGB_CustomColor) eq { pop pop 3 { 1 exch sub 3 index mul 1 exch sub 3 1 roll } repeat 4 -1 roll pop setrgbcolor } { pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } ifelse } ifelse } def } if /setAIseparationgray { false setoverprint 0 setgray /setseparationgray where{ pop setseparationgray }{ /setcolorspace where{ pop [/Separation (All) /DeviceCMYK {dup dup dup}] setcolorspace 1 exch sub setcolor }{ setgray }ifelse }ifelse } def /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking %%EndResource %%BeginResource: procset Adobe_typography_AI5 1.0 1 %%Title: (Typography Operators) %%Version: 1.0 1 %%CreationDate:(6/10/1996) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_typography_AI5 68 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /havefont { systemdict /languagelevel known { /Font resourcestatus dup { exch pop exch pop } if } { systemdict /FontDirectory get 1 index known { pop true } { systemdict /fileposition known { dup length 6 add exch Ss 6 250 getinterval cvs pop Ss exch 0 exch getinterval status { pop pop pop pop true } { false } ifelse } { pop false } ifelse } ifelse } ifelse } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def /subststring { exch 2 index exch search { exch pop exch dup () eq { pop exch concatstring } { 3 -1 roll exch concatstring concatstring } ifelse exch pop true } { pop pop false } ifelse } def /concatstring { 1 index length 1 index length 1 index add string dup 0 5 index putinterval dup 2 index 4 index putinterval 4 1 roll pop pop pop } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def 2 index havefont { 3 index 255 string cvs dup (_Symbol_) eq { pop 2 index findfont } { 1 index 0 eq { dup length 1 sub 1 exch getinterval cvn findfont } { pop 2 index findfont } ifelse } ifelse } { dup 1 eq { 2 index 64 string cvs dup (-90pv-RKSJ-) (-83pv-RKSJ-) subststring { exch pop dup havefont { findfont false } { pop true } ifelse } { pop dup (-90ms-RKSJ-) (-Ext-RKSJ-) subststring { exch pop dup havefont { findfont false } { pop true } ifelse } { pop pop true } ifelse } ifelse { 1 index 1 eq { /Ryumin-Light-Ext-RKSJ-V havefont {/Ryumin-Light-Ext-RKSJ-V} {/Courier} ifelse } { /Ryumin-Light-83pv-RKSJ-H havefont {/Ryumin-Light-83pv-RKSJ-H} {/Courier} ifelse } ifelse findfont [1 0 0.5 1 0 0] makefont } if } { /Courier findfont } ifelse } ifelse _wv type /arraytype eq { _wv makeblendedfont } if dup length 10 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontScript exch def /FontDirection exch def /FontRequest exch def /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { W B } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { W F } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { W S } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat _shift aload pop _lineorientation 1 eq { exch } if translate _scale aload pop _lineorientation 1 eq _yokoorientation 1 eq or { exch } if scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { 1 index type /nametype eq { dup 0.75 mul 1 index 0.25 mul neg } if /_fontDescent exch ddef /_fontAscent exch ddef /_fontSize exch ddef /_fontRotateAdjust _fontAscent _fontDescent add 2 div neg ddef /_fontHeight _fontSize ddef findfont _fontSize scalefont setfont } def /Tl { pop neg 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { 0 exch _shift astore pop currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { count 1 eq { 100 } if 100 div exch 100 div exch _scale astore pop iTm } def /TA { pop } def /Tq { pop } def /Tg { pop } def /TG { pop } def /Tv { /_lineorientation exch ddef } def /TV { /_charorientation exch ddef } def /Ty { dup /_yokoorientation exch ddef 1 sub neg Tv } def /TY { pop } def /T~ { Tx } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { _fontSize mul 1000 div _lineorientation 0 eq { neg 0 } { 0 exch } ifelse rmoveto pop } def /TK { 2 npop } def /T* { _leading aload pop _lineorientation 0 ne { exch } if Td } def /T*- { _leading aload pop _lineorientation 0 ne { exch } if exch neg exch neg Td } def /T- { _ax neg 0 rmoveto _lineorientation 1 eq _charorientation 0 eq and { 1 TV _hyphen Tx 0 TV } { _hyphen Tx } ifelse } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ findfont _fontSize scalefont setfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking % /X^ { currentfont 5 1 roll dup havefont { findfont _fontSize scalefont setfont } { pop exch } ifelse 2 index 0 eq { Tx } { Tj } ifelse pop pop setfont } def /T^ /X^ load def %%EndResource %%BeginProcSet: Adobe_ColorImage_AI6 1.3 0 userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 53 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 41 dict def } if Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /_newproc null def /_proc1 null def /_proc2 null def /sourcearray 4 array def /_ptispace null def /_ptiname null def /_pti0 0 def /_pti1 0 def /_ptiproc null def /_ptiscale 0 def /_pticomps 0 def /_ptibuf 0 string def /_gtigray 0 def /_cticmyk null def /_rtirgb null def /XIEnable true def /XIType 0 def /XIEncoding 0 def /XICompression 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIRowBytes 0 def /XIFile null def /XIBuffer1 null def /XIBuffer2 null def /XIBuffer3 null def /XIDataProc null def /XIColorSpace /DeviceGray def /XIColorValues 0 def /XIPlateList false def end /ci6colorimage /colorimage where {/colorimage get}{null} ifelse def /ci6image systemdict /image get def /ci6curtransfer systemdict /currenttransfer get def /ci6curoverprint /currentoverprint where {/currentoverprint get}{{_of}} ifelse def /ci6foureq { 4 index ne { pop pop pop false }{ 4 index ne { pop pop false }{ 4 index ne { pop false }{ 4 index eq } ifelse } ifelse } ifelse } def /ci6testplate { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 ci6foureq { /plateindex 0 def }{ 0 1 0 0 ci6foureq { /plateindex 1 def }{ 0 0 1 0 ci6foureq { /plateindex 2 def }{ 0 0 0 1 ci6foureq { /plateindex 3 def }{ 0 0 0 0 ci6foureq { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /ci6concatprocs { /packedarray where { pop dup type /packedarraytype eq 2 index type /packedarraytype eq or }{ false } ifelse { /_proc2 exch cvlit def /_proc1 exch cvlit def _proc1 aload pop _proc2 aload pop _proc1 length _proc2 length add packedarray cvx }{ /_proc2 exch cvlit def /_proc1 exch cvlit def /_newproc _proc1 length _proc2 length add array def _newproc 0 _proc1 putinterval _newproc _proc1 length _proc2 putinterval _newproc cvx } ifelse } def /ci6istint { type /arraytype eq } def /ci6isspot { dup type /arraytype eq { dup length 1 sub get /Separation eq }{ pop false } ifelse } def /ci6spotname { dup ci6isspot {dup length 2 sub get}{pop ()} ifelse } def /ci6altspace { aload pop pop pop ci6colormake } def /ci6numcomps { dup /DeviceGray eq { pop 1 }{ dup /DeviceRGB eq { pop 3 }{ /DeviceCMYK eq { 4 }{ 1 } ifelse } ifelse } ifelse } def /ci6marksplate { dup /DeviceGray eq { pop plateindex 3 eq }{ dup /DeviceRGB eq { pop plateindex 5 ne }{ dup /DeviceCMYK eq { pop plateindex 5 ne }{ dup ci6isspot { /findcmykcustomcolor where { pop dup length 2 sub get 0.1 0.1 0.1 0.1 5 -1 roll findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop plateindex 5 ne } ifelse }{ pop plateindex 5 ne } ifelse } ifelse } ifelse } ifelse } def /ci6colormake { dup ci6numcomps exch 1 index 2 add 1 roll dup 1 eq {pop}{array astore} ifelse exch } def /ci6colorexpand { dup ci6spotname exch dup ci6istint { ci6altspace exch 4 1 roll }{ 1 3 1 roll } ifelse } def /ci6colortint { dup /DeviceGray eq { 3 1 roll 1 exch sub mul 1 exch sub exch }{ dup /DeviceRGB eq { 3 1 roll {1 exch sub 1 index mul 1 exch sub exch} forall pop 3 array astore exch }{ dup /DeviceCMYK eq { 3 1 roll {1 index mul exch} forall pop 4 array astore exch }{ 3 1 roll mul exch } ifelse } ifelse } ifelse } def /ci6colortocmyk { dup /DeviceGray eq { pop 1 exch sub 0 0 0 4 -1 roll 4 array astore }{ dup /DeviceRGB eq { pop aload pop _rgbtocmyk 4 array astore }{ dup /DeviceCMYK eq { pop }{ ci6altspace ci6colortint ci6colortocmyk } ifelse } ifelse } ifelse } def /ci6makeimagedict { 7 dict begin /ImageType 1 def /Decode exch def /DataSource exch def /ImageMatrix exch def /BitsPerComponent exch def /Height exch def /Width exch def currentdict end } def /ci6stringinvert { 0 1 2 index length 1 sub { dup 2 index exch get 255 exch sub 2 index 3 1 roll put } for } def /ci6stringknockout { 0 1 2 index length 1 sub { 255 2 index 3 1 roll put } for } def /ci6stringapply { 0 1 4 index length 1 sub { dup 4 index exch get 3 index 3 1 roll 3 index exec } for pop exch pop } def /ci6walkrgbstring { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval {} forall 5 index exec 3 index } for 5 {pop} repeat } def /ci6walkcmykstring { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval {} forall 6 index exec 3 index } for 5 { pop } repeat } def /ci6putrgbtograystr { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /ci6putcmyktograystr { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /ci6rgbtograyproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 dup 3 1 roll /ci6putrgbtograystr load exch ci6walkrgbstring end } def /ci6cmyktograyproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 dup 3 1 roll /ci6putcmyktograystr load exch ci6walkcmykstring end } def /ci6separatecmykproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop end } def /ci6compositeimage { dup 1 eq { pop pop image }{ /ci6colorimage load null ne { ci6colorimage }{ 3 1 roll pop sourcearray 0 3 -1 roll put 3 eq {/ci6rgbtograyproc}{/ci6cmyktograyproc} ifelse load image } ifelse } ifelse } def /ci6knockoutimage { gsave 0 ci6curtransfer exec 1 ci6curtransfer exec eq { 0 ci6curtransfer exec 0.5 lt }{ 0 ci6curtransfer exec 1 ci6curtransfer exec gt } ifelse {{pop 0}}{{pop 1}} ifelse systemdict /settransfer get exec ci6compositeimage grestore } def /ci6drawimage { ci6testplate -1 eq { pop ci6compositeimage }{ dup type /arraytype eq { dup length plateindex gt {plateindex get}{pop false} ifelse }{ { true }{ dup 1 eq {plateindex 3 eq}{plateindex 3 le} ifelse } ifelse } ifelse { dup 1 eq { pop pop ci6image }{ dup 3 eq { ci6compositeimage }{ pop pop sourcearray 0 3 -1 roll put /ci6separatecmykproc load ci6image } ifelse } ifelse }{ ci6curoverprint { 7 {pop} repeat }{ ci6knockoutimage } ifelse } ifelse } ifelse } def /ci6proctintimage { /_ptispace exch store /_ptiname exch store /_pti1 exch store /_pti0 exch store /_ptiproc exch store /_pticomps _ptispace ci6numcomps store /_ptiscale _pti1 _pti0 sub store level2? { _ptiname length 0 gt version cvr 2012 ge and { [/Separation _ptiname _ptispace {_ptiproc}] setcolorspace [_pti0 _pti1] ci6makeimagedict ci6image }{ [/Indexed _ptispace 255 {255 div _ptiscale mul _pti0 add _ptiproc}] setcolorspace [0 255] ci6makeimagedict ci6image } ifelse }{ _pticomps 1 eq { { dup { 255 div _ptiscale mul _pti0 add _ptiproc 255 mul cvi put } ci6stringapply } ci6concatprocs ci6image }{ { dup length _pticomps mul dup _ptibuf length ne {/_ptibuf exch string store}{pop} ifelse _ptibuf { exch _pticomps mul exch 255 div _ptiscale mul _pti0 add _ptiproc _pticomps 2 add -2 roll _pticomps 1 sub -1 0 { 1 index add 2 index exch 5 -1 roll 255 mul cvi put } for pop pop } ci6stringapply } ci6concatprocs false _pticomps /ci6colorimage load null eq {7 {pop} repeat}{ci6colorimage} ifelse } ifelse } ifelse } def /ci6graytintimage { /_gtigray 5 -1 roll store {1 _gtigray sub mul 1 exch sub} 4 1 roll /DeviceGray ci6proctintimage } def /ci6cmyktintimage { /_cticmyk 5 -1 roll store {_cticmyk {1 index mul exch} forall pop} 4 1 roll /DeviceCMYK ci6proctintimage } def /ci6rgbtintimage { /_rtirgb 5 -1 roll store {_rtirgb {1 exch sub 1 index mul 1 exch sub exch} forall pop} 4 1 roll /DeviceRGB ci6proctintimage } def /ci6tintimage { ci6testplate -1 eq { ci6colorexpand 3 -1 roll 5 -1 roll {0}{0 exch} ifelse 4 2 roll dup /DeviceGray eq { pop ci6graytintimage }{ dup /DeviceRGB eq { pop ci6rgbtintimage }{ pop ci6cmyktintimage } ifelse } ifelse }{ dup ci6marksplate { plateindex 5 lt { ci6colortocmyk plateindex get dup 0 eq ci6curoverprint and { 7 {pop} repeat }{ 1 exch sub exch {1 0}{0 1} ifelse () ci6graytintimage } ifelse }{ pop exch {0}{0 exch} ifelse 0 3 1 roll () ci6graytintimage } ifelse }{ ci6curoverprint { 8 {pop} repeat }{ pop pop pop {pop 1} 0 1 () /DeviceGray ci6proctintimage } ifelse } ifelse } ifelse } def /XINullImage { } def /XIImageMask { XIImageWidth XIImageHeight false [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load imagemask } def /XIImageTint { XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load XIType 3 eq XIColorValues XIColorSpace ci6tintimage } def /XIImage { XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load false XIChannelCount XIPlateList ci6drawimage } def /XG { pop pop } def /XF { 13 {pop} repeat } def /Xh { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def /XIImageHeight exch def /XIImageWidth exch def /XIImageMatrix exch def 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale /_lp /null ddef _fc /_lp /imagemask ddef end } def /XH { Adobe_ColorImage_AI6_Vars begin grestore end } def /XIEnable { Adobe_ColorImage_AI6_Vars /XIEnable 3 -1 roll put } def /XC { Adobe_ColorImage_AI6_Vars begin ci6colormake /XIColorSpace exch def /XIColorValues exch def end } def /XIPlates { Adobe_ColorImage_AI6_Vars begin /XIPlateList exch def end } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def cvi dup 256 idiv /XICompression exch store 256 mod /XIEncoding exch store pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi }{ XIImageWidth XIChannelCount mul } ifelse /XIRowBytes exch def XIEnable { /XIBuffer3 XIImageWidth string def XICompression 0 eq { /XIBuffer1 XIRowBytes string def XIEncoding 0 eq { {currentfile XIBuffer1 readhexstring pop} }{ {currentfile XIBuffer1 readstring pop} } ifelse }{ /XIBuffer1 256 string def /XIBuffer2 XIRowBytes string def {currentfile XIBuffer1 readline pop (%) anchorsearch {pop} if} /ASCII85Decode filter /DCTDecode filter /XIFile exch def {XIFile XIBuffer2 readstring pop} } ifelse /XIDataProc exch def XIType 1 ne { 0 setgray } if XIType 1 eq { XIImageMask }{ XIType 2 eq XIType 3 eq or { XIImageTint }{ XIImage } ifelse } ifelse }{ XINullImage } ifelse /XIPlateList false def grestore end } def end %%EndProcSet %%BeginResource: procset Adobe_Illustrator_AI5 1.3 0 %%Title: (Adobe Illustrator (R) Version 8.0 Full Prolog) %%Version: 1.3 0 %%CreationDate: (3/7/1994) () %%Copyright: ((C) 1987-1998 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 112 dict dup begin put /_?cmyk false def /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_lineorientation 0 def /_charorientation 0 def /_yokoorientation 0 def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_shift [0 0] def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fontSize 0 def /_fontAscent 0 def /_fontDescent 0 def /_fontHeight 0 def /_fontRotateAdjust 0 def /Ss 256 string def Ss 0 (fonts/) putinterval /_cnt 0 def /_scale [1 1] def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_hfname 100 string def /_hffound false def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_rgbf 3 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_rgbs 3 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /_lobyte 0 def /_hibyte 0 def /_cproc null def /_cscript 0 def /_hvax 0 def /_hvay 0 def /_hvwb 0 def /_hvcx 0 def /_hvcy 0 def /_bitfont null def /_bitlobyte 0 def /_bithibyte 0 def /_bitkey null def /_bitdata null def /_bitindex 0 def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 100 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin /_aicmykps where {pop /_?cmyk _aicmykps def}if discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /hswj { dup stringwidth 3 2 roll { _hvwb eq { exch _hvcx add exch _hvcy add } if exch _hvax add exch _hvay add } cforall } def /vswj { 0 0 3 -1 roll { dup 255 le _charorientation 1 eq and { dup cstring stringwidth 5 2 roll _hvwb eq { exch _hvcy sub exch _hvcx sub } if exch _hvay sub exch _hvax sub 4 -1 roll sub exch 3 -1 roll sub exch } { _hvwb eq { exch _hvcy sub exch _hvcx sub } if exch _hvay sub exch _hvax sub _fontHeight sub } ifelse } cforall } def /swj { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef _lineorientation 0 eq { hswj } { vswj } ifelse } def /sw { 0 0 0 6 3 roll swj } def /vjss { 4 1 roll { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto gsave false charpath currentpoint 5 index setmatrix stroke grestore _fontRotateAdjust sub moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto 90 rotate } { currentpoint _fontHeight sub 5 index sub 3 index _sp eq { 9 index sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto gsave 2 index false charpath 6 index setmatrix stroke grestore moveto pop pop } ifelse } cforall 6 npop } def /hjss { 4 1 roll { dup cstring gsave false charpath currentpoint 5 index setmatrix stroke grestore moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto } cforall 6 npop } def /jss { _lineorientation 0 eq { hjss } { vjss } ifelse } def /ss { 0 0 0 7 3 roll jss } def /vjsp { 4 1 roll { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto false charpath currentpoint _fontRotateAdjust sub moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto 90 rotate } { currentpoint _fontHeight sub 5 index sub 3 index _sp eq { 9 index sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto 2 index false charpath moveto pop pop } ifelse } cforall 6 npop } def /hjsp { 4 1 roll { dup cstring false charpath _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto } cforall 6 npop } def /jsp { matrix currentmatrix _lineorientation 0 eq {hjsp} {vjsp} ifelse } def /sp { matrix currentmatrix 0 0 0 7 3 roll _lineorientation 0 eq {hjsp} {vjsp} ifelse } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /_rgbtocmyk { 3 { 1 exch sub 3 1 roll } repeat 3 copy 1 4 1 roll 3 { 3 index 2 copy gt { exch } if pop 4 1 roll } repeat pop pop pop 4 1 roll 3 { 3 index sub 3 1 roll } repeat 4 -1 roll } def /setrgbfill { _rgbf astore pop /_fc { _lp /fill ne { _of setoverprint _rgbf aload pop setrgbcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /setrgbstroke { _rgbs astore pop /_sc { _lp /stroke ne { _os setoverprint _rgbs aload pop setrgbcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /Xa { _?cmyk { 3 npop k }{ setrgbfill 4 npop } ifelse } def /XA { _?cmyk { 3 npop K }{ setrgbstroke 4 npop } ifelse } def /Xs { /_gf exch ddef 5 npop /_fc { _lp /fill ne { _of setoverprint _gf setAIseparationgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /XS { /_gs exch ddef 5 npop /_sc { _lp /stroke ne { _os setoverprint _gs setAIseparationgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /Xx { exch /_gf exch ddef 0 eq { findcmykcustomcolor }{ _?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse { 4 1 roll 3 npop findcmykcustomcolor }{ 8 -4 roll 4 npop findrgbcustomcolor } ifelse } ifelse /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /XX { exch /_gs exch ddef 0 eq { findcmykcustomcolor }{ _?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse { 4 1 roll 3 npop findcmykcustomcolor }{ 8 -4 roll 4 npop findrgbcustomcolor } ifelse } ifelse /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /XK { 3 -1 roll pop 0 eq { 1 exch sub 3 {dup 3 1 roll mul 5 1 roll} repeat mul 4 1 roll K } { 1 exch sub 4 1 roll 3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat 4 -1 roll pop XA } ifelse } def /Xk { 3 -1 roll pop 0 eq { 1 exch sub 3 {dup 3 1 roll mul 5 1 roll} repeat mul 4 1 roll k } { 1 exch sub 4 1 roll 3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat 4 -1 roll pop Xa } ifelse } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /Xt { pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer {readline} stopped { % assume error was due to overfilling the buffer }{ not { stop } if endString eq { cleartomark stop } if }ifelse } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer {readline} stopped { % assume error was due to overfilling the buffer }{ not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse }ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 6 npop 7 2 roll 5 npop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 4 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setrgbcolor { 3 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end /XP { 4 npop } bind def /XD { pop } bind def end setpacking %%EndResource %%BeginResource: procset Adobe_cshow 2.0 8 %%Title: (Writing System Operators) %%Version: 2.0 8 %%CreationDate: (1/23/89) () %%Copyright: ((C) 1992-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_cshow 14 dict dup begin put /initialize { Adobe_cshow begin Adobe_cshow { dup xcheck { bind } if pop pop } forall end Adobe_cshow begin } def /terminate { currentdict Adobe_cshow eq { end } if } def /cforall { /_lobyte 0 ddef /_hibyte 0 ddef /_cproc exch ddef /_cscript currentfont /FontScript known { currentfont /FontScript get } { -1 } ifelse ddef { /_lobyte exch ddef _hibyte 0 eq _cscript 1 eq _lobyte 129 ge _lobyte 159 le and _lobyte 224 ge _lobyte 252 le and or and _cscript 2 eq _lobyte 161 ge _lobyte 254 le and and _cscript 3 eq _lobyte 161 ge _lobyte 254 le and and _cscript 25 eq _lobyte 161 ge _lobyte 254 le and and _cscript -1 eq or or or or and { /_hibyte _lobyte ddef } { _hibyte 256 mul _lobyte add _cproc /_hibyte 0 ddef } ifelse } forall } def /cstring { dup 256 lt { (s) dup 0 4 3 roll put } { dup 256 idiv exch 256 mod (hl) dup dup 0 6 5 roll put 1 4 3 roll put } ifelse } def /clength { 0 exch { 256 lt { 1 } { 2 } ifelse add } cforall } def /hawidthshow { { dup cstring show _hvax _hvay rmoveto _hvwb eq { _hvcx _hvcy rmoveto } if } cforall } def /vawidthshow { { dup 255 le _charorientation 1 eq and { -90 rotate 0 _fontRotateAdjust rmoveto cstring _hvcx _hvcy _hvwb _hvax _hvay 6 -1 roll awidthshow 0 _fontRotateAdjust neg rmoveto 90 rotate } { currentpoint _fontHeight sub exch _hvay sub exch _hvax sub 2 index _hvwb eq { exch _hvcy sub exch _hvcx sub } if 3 2 roll cstring dup stringwidth pop 2 div neg _fontAscent neg rmoveto show moveto } ifelse } cforall } def /hvawidthshow { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef _lineorientation 0 eq { hawidthshow } { vawidthshow } ifelse } def /hvwidthshow { 0 0 3 -1 roll hvawidthshow } def /hvashow { 0 0 0 6 -3 roll hvawidthshow } def /hvshow { 0 0 0 0 0 6 -1 roll hvawidthshow } def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_shading_AI8 1.0 0 %%Title: (Adobe Illustrator 8 Shading Procset) %%Version: 1.0 0 %%CreationDate: (12/17/97) () %%Copyright: ((C) 1987-1997 Adobe Systems Incorporated All Rights Reserved) userdict /defaultpacking currentpacking put true setpacking userdict /Adobe_shading_AI8 10 dict dup begin put /initialize { Adobe_shading_AI8 begin Adobe_shading_AI8 bdprocs Mesh /initialize get exec } def /terminate { currentdict Adobe_shading_AI8 eq { end } if } def /bdprocs { { dup xcheck 1 index type /arraytype eq and { bind } if pop pop } forall } def /X! {pop} def /X# {pop pop} def /Mesh 40 dict def Mesh begin /initialize { Mesh bdprocs Mesh begin /emulate? /AI8MeshEmulation where { pop AI8MeshEmulation }{ systemdict /shfill known not } ifelse def end } def /bd { shadingdict begin } def /paint { emulate? { end }{ /_lp /none ddef _fc /_lp /none ddef /AIColorSpace AIColorSpace tocolorspace store /ColorSpace AIColorSpace topsspace store version_ge_3010.106 not systemdict /setsmoothness known and { 0.0001 setsmoothness } if composite? { /DataSource getdatasrc def Matrix concat currentdict end shfill }{ AIColorSpace makesmarks AIPlateList markingplate and not isoverprint and { end }{ /ColorSpace /DeviceGray store /Decode [0 1 0 1 0 1] store /DataSource getplatesrc def Matrix concat currentdict end shfill } ifelse } ifelse } ifelse } def /shadingdict 12 dict def shadingdict begin /ShadingType 6 def /BitsPerCoordinate 16 def /BitsPerComponent 8 def /BitsPerFlag 8 def end /datafile null def /databuf 256 string def /dataptr 0 def /srcspace null def /srcchannels 0 def /dstchannels 0 def /dstplate 0 def /srctodstcolor null def /getplatesrc { /srcspace AIColorSpace store /srcchannels AIColorSpace getnchannels store /dstchannels 1 store /dstplate getplateindex store /srctodstcolor srcspace makesmarks { dstplate 4 eq { {1 exch sub} }{ {srcspace tocmyk 3 dstplate sub index 1 exch sub 5 1 roll 4 {pop} repeat} } ifelse }{ {srcchannels {pop} repeat 1} } ifelse store /datafile getdatasrc store /rdpatch168 load DataLength () /SubFileDecode filter } def /getdatasrc { /rdcmntline load /ASCII85Decode filter } def /rdpatch168 { /dataptr 0 store 49 rdcount 4 { dup {pop srcchannels getint8} if dup {pop srctodstcolor dstchannels putint8 true} if } repeat {databuf 0 dataptr getinterval}{()} ifelse } def /rdpatch3216 { /dataptr 0 store 97 rdcount 4 { dup {pop srcchannels getint16} if dup {pop srctodstcolor dstchannels putint16 true} if } repeat {databuf 0 dataptr getinterval}{()} ifelse } def /rdcount { dup 0 gt { datafile databuf dataptr 4 -1 roll getinterval readstring exch length dataptr add /dataptr exch store }{ true } ifelse } def /getint8 { mark true 3 -1 roll { dup {pop datafile read} if dup {pop 255 div true} if } repeat { counttomark 1 add -1 roll pop true }{ cleartomark false } ifelse } def /putint8 { dup dataptr add /dataptr exch store dataptr exch { 1 sub exch 255 mul cvi databuf 2 index 3 -1 roll put } repeat pop } def /getint16 { mark true 3 -1 roll { dup {pop datafile read} if dup {pop 256 mul datafile read} if dup {pop add 65535 div true} if } repeat { counttomark 1 add -1 roll pop true }{ cleartomark false } ifelse } def /putint16 { dup 2 mul dataptr add /dataptr exch store dataptr exch { 2 sub exch 65535 mul cvi dup 256 idiv databuf 3 index 3 -1 roll put 256 mod databuf 2 index 1 add 3 -1 roll put } repeat pop } def /srcbuf 256 string def /rdcmntline { currentfile srcbuf readline pop (%) anchorsearch {pop} if } def /getplateindex { 0 [cyan? magenta? yellow? black? customColor?] {{exit} if 1 add} forall } def /aicsarray 4 array def /aicsaltvals 4 array def /aicsaltcolr aicsaltvals def /tocolorspace { dup type /arraytype eq { mark exch aload pop aicsarray 0 3 -1 roll put aicsarray 1 3 -1 roll put dup aicsarray 2 3 -1 roll put gettintxform aicsarray 3 3 -1 roll put counttomark aicsaltvals 0 3 -1 roll getinterval /aicsaltcolr exch store aicsaltcolr astore pop pop aicsarray } if } def /subtintxform {aicsaltcolr {1 index mul exch} forall pop} def /addtintxform {aicsaltcolr {1 sub 1 index mul 1 add exch} forall pop} def /gettintxform { /DeviceRGB eq {/addtintxform}{/subtintxform} ifelse load } def /getnchannels { dup type /arraytype eq {0 get} if colorspacedict exch get begin Channels end } def /makesmarks { composite? { pop true }{ dup dup type /arraytype eq {0 get} if colorspacedict exch get begin MarksPlate end } ifelse } def /markingplate { composite? { pop true }{ dup type /arraytype eq { dup length getplateindex gt {getplateindex get}{pop false} ifelse } if } ifelse } def /tocmyk { dup dup type /arraytype eq {0 get} if colorspacedict exch get begin ToCMYK end } def /topsspace { dup dup type /arraytype eq {0 get} if colorspacedict exch get begin ToPSSpace end } def /colorspacedict 5 dict dup begin /DeviceGray 4 dict dup begin /Channels 1 def /MarksPlate {pop black?} def /ToCMYK {pop 1 exch sub 0 0 0 4 -1 roll} def /ToPSSpace {} def end def /DeviceRGB 4 dict dup begin /Channels 3 def /MarksPlate {pop isCMYKSep?} def /ToCMYK {pop _rgbtocmyk} def /ToPSSpace {} def end def /DeviceCMYK 4 dict dup begin /Channels 4 def /MarksPlate {pop isCMYKSep?} def /ToCMYK {pop} def /ToPSSpace {} def end def /Separation 4 dict dup begin /Channels 1 def /MarksPlate { /findcmykcustomcolor where { pop dup 1 exch ToCMYK 5 -1 roll 1 get findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop false } ifelse } def /ToCMYK { dup 2 get mark exch 4 2 roll 3 get exec counttomark -1 roll tocmyk 5 -1 roll pop } def /ToPSSpace {} def end def /Process 4 dict dup begin /Channels 1 def /MarksPlate { isCMYKSep? { 1 exch ToCMYK 4 array astore getplateindex get 0 ne }{ pop false } ifelse } def /ToCMYK { dup 2 get mark exch 4 2 roll 3 get exec counttomark -1 roll tocmyk 5 -1 roll pop } def /ToPSSpace { 4 array copy dup 0 /Separation put } def end def end def /isoverprint { /currentoverprint where {pop currentoverprint}{_of} ifelse } def /version_ge_3010.106 { version {cvr} stopped { pop false }{ 3010.106 ge } ifelse } def end end defaultpacking setpacking %%EndResource %%EndProlog %%BeginSetup %%IncludeFont: Symbol %%IncludeFont: Times-Italic userdict /_useSmoothShade false put userdict /_aicmykps true put userdict /_forceToCMYK true put Adobe_level2_AI5 /initialize get exec Adobe_cshow /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_shading_AI8 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron TE %AI55J_Tsume: None %AI3_BeginEncoding: _Symbol Symbol [/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /universal /numbersign /existential /percent /ampersand /suchthat /parenleft /parenright /asteriskmath /plus /comma /minus /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /congruent /Alpha /Beta /Chi /Delta /Epsilon /Phi /Gamma /Eta /Iota /theta1 /Kappa /Lambda /Mu /Nu /Omicron /Pi /Theta /Rho /Sigma /Tau /Upsilon /sigma1 /Omega /Xi /Psi /Zeta /bracketleft /therefore /bracketright /perpendicular /underscore /radicalex /alpha /beta /chi /delta /epsilon /phi /gamma /eta /iota /phi1 /kappa /lambda /mu /nu /omicron /pi /theta /rho /sigma /tau /upsilon /omega1 /omega /xi /psi /zeta /braceleft /bar /braceright /similar /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /Upsilon1 /minute /lessequal /fraction /infinity /florin /club /diamond /heart /spade /arrowboth /arrowleft /arrowup /arrowright /arrowdown /degree /plusminus /second /greaterequal /multiply /proportional /partialdiff /bullet /divide /notequal /equivalence /approxequal /ellipsis /arrowvertex /arrowhorizex /carriagereturn /aleph /Ifraktur /Rfraktur /weierstrass /circlemultiply /circleplus /emptyset /intersection /union /propersuperset /reflexsuperset /notsubset /propersubset /reflexsubset /element /notelement /angle /gradient /registerserif /copyrightserif /trademarkserif /product /radical /dotmath /logicalnot /logicaland /logicalor /arrowdblboth /arrowdblleft /arrowdblup /arrowdblright /arrowdbldown /lozenge /angleleft /registersans /copyrightsans /trademarksans /summation /parenlefttp /parenleftex /parenleftbt /bracketlefttp /bracketleftex /bracketleftbt /bracelefttp /braceleftmid /braceleftbt /braceex /.notdef /angleright /integral /integraltp /integralex /integralbt /parenrighttp /parenrightex /parenrightbt /bracketrighttp /bracketrightex /bracketrightbt /bracerighttp /bracerightmid /bracerightbt /.notdef /_Symbol/Symbol 0 0 0 TZ %AI3_EndEncoding AdobeType %AI55J_Tsume: None %AI3_BeginEncoding: _Times-Italic Times-Italic [/_Times-Italic/Times-Italic 0 0 1 TZ %AI3_EndEncoding AdobeType [161/degree 173/notequal 176/infinity/plusminus/lessequal/greaterequal 181/mu/partialdiff/summation/product/pi/integral 189/Omega 195/radical 197/approxequal 198/Delta 214/divide/lozenge 240/apple /_Symbol_/Symbol 0 0 0 TZ %AI5_Begin_NonPrinting Np %AI3_BeginPattern: (Brick) (Brick) 0 0 72 72 [ %AI3_Tile (0 O 0 R 0.3 0.85 0.85 0 k 0.3 0.85 0.85 0 K ) @ ( %AI6_BeginPatternLayer 800 Ar 0 J 0 j 1 w 4 M []0 d %AI3_Note: 0 D 0 XR 0 0 m 0 72 L 72 72 L 72 0 L 0 0 L f %AI6_EndPatternLayer ) & (0 O 0 R 1 g 1 G ) @ ( %AI6_BeginPatternLayer 800 Ar 0 J 0 j 0.3 w 4 M []0 d %AI3_Note: 0 D 0 XR 0 68.4097 m 72 68.4097 l S 0 61.209 m 72 61.209 L S 0 54.0088 m 72 54.0088 L S 0 46.8076 m 72 46.8076 L S 0 39.6084 m 72 39.6084 L S 0 32.4072 m 72 32.4072 L S 0 25.207 m 72 25.207 L S 0 18.0059 m 72 18.0059 L S 0 10.8057 m 72 10.8057 L S 0 3.6064 m 72 3.6064 L S 68.4102 68.4097 m 68.4102 61.2217 l S 54.0098 68.4097 m 54.0098 61.2217 L S 39.6094 68.4097 m 39.6094 61.2217 L S 25.21 68.4097 m 25.21 61.2217 L S 10.8105 68.4097 m 10.8105 61.2217 L S 68.4102 53.9717 m 68.4102 46.7842 l S 54.0098 53.9717 m 54.0098 46.7842 L S 39.6094 53.9717 m 39.6094 46.7842 L S 25.21 53.9717 m 25.21 46.7842 L S 10.8105 53.9717 m 10.8105 46.7842 L S 68.4102 39.5967 m 68.4102 32.4092 l S 54.0098 39.5967 m 54.0098 32.4092 L S 39.6094 39.5967 m 39.6094 32.4092 L S 25.21 39.5967 m 25.21 32.4092 L S 10.8105 39.5967 m 10.8105 32.4092 L S 68.4102 25.2217 m 68.4102 18.0342 l S 54.0098 25.2217 m 54.0098 18.0342 L S 39.6094 25.2217 m 39.6094 18.0342 L S 25.21 25.2217 m 25.21 18.0342 L S 10.8105 25.2217 m 10.8105 18.0342 L S 68.4102 10.7842 m 68.4102 3.5967 l S 54.0098 10.7842 m 54.0098 3.5967 L S 39.6094 10.7842 m 39.6094 3.5967 L S 25.21 10.7842 m 25.21 3.5967 L S 10.8105 10.7842 m 10.8105 3.5967 L S 61.1973 3.5967 m 61.1973 0 L S 46.7969 3.5967 m 46.7969 0 L S 32.3965 3.5967 m 32.3965 0 L S 17.9971 3.5967 m 17.9971 0 L S 3.5967 3.5967 m 3.5967 0 l S 61.1973 18.0342 m 61.1973 10.8467 L S 46.7969 18.0342 m 46.7969 10.8467 L S 32.3965 18.0342 m 32.3965 10.8467 L S 17.9971 18.0342 m 17.9971 10.8467 L S 3.5967 18.0342 m 3.5967 10.8467 l S 61.1973 32.4092 m 61.1973 25.2217 L S 46.7969 32.4092 m 46.7969 25.2217 L S 17.9971 32.4092 m 17.9971 25.2217 L S 3.5967 32.4092 m 3.5967 25.2217 l S 61.1973 46.7842 m 61.1973 39.5967 L S 46.7969 46.7842 m 46.7969 39.5967 L S 32.3965 46.7842 m 32.3965 39.5967 L S 17.9971 46.7842 m 17.9971 39.5967 L S 3.5967 46.7842 m 3.5967 39.5967 l S 61.1973 61.2217 m 61.1973 54.0347 L S 46.7969 61.2217 m 46.7969 54.0347 L S 32.3965 61.2217 m 32.3965 54.0347 L S 17.9971 61.2217 m 17.9971 54.0347 L S 3.5967 61.2217 m 3.5967 54.0347 l S 61.1973 71.959 m 61.1973 68.4717 L S 46.7969 71.959 m 46.7969 68.4717 L S 32.3965 71.959 m 32.3965 68.4717 L S 17.9971 71.959 m 17.9971 68.4717 L S 3.5967 71.959 m 3.5967 68.4717 l S 32.3965 32.4092 m 32.3965 25.2217 L S %AI6_EndPatternLayer ) & ] E %AI3_EndPattern %AI3_BeginPattern: (Confetti) (Confetti) 4.85 3.617 76.85 75.617 [ %AI3_Tile (0 O 0 R 1 g 1 G ) @ ( %AI6_BeginPatternLayer 800 Ar 0 J 0 j 1 w 4 M []0 d %AI3_Note: 0 D 0 XR 4.85 3.617 m 4.85 75.617 L 76.85 75.617 L 76.85 3.617 L 4.85 3.617 L f %AI6_EndPatternLayer ) & (0 O 0 R 0 g 0 G ) @ ( %AI6_BeginPatternLayer 800 Ar 0 J 0 j 0.3 w 4 M []0 d %AI3_Note: 0 D 0 XR 10.6 64.867 m 7.85 62.867 l S 9.1 8.617 m 6.85 6.867 l S 78.1 68.617 m 74.85 67.867 l S 76.85 56.867 m 74.35 55.117 l S 79.6 51.617 m 76.6 51.617 l S 76.35 44.117 m 73.6 45.867 l S 78.6 35.867 m 76.6 34.367 l S 76.1 23.867 m 73.35 26.117 l S 78.1 12.867 m 73.85 13.617 l S 68.35 14.617 m 66.1 12.867 l S 76.6 30.617 m 73.6 30.617 l S 62.85 58.117 m 60.956 60.941 l S 32.85 59.617 m 31.196 62.181 l S 47.891 64.061 m 49.744 66.742 l S 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31.6 35.617 m 27.85 36.367 l S 36.35 26.117 m 34.35 24.617 l S 33.85 14.117 m 31.1 16.367 l S 37.1 9.867 m 35.1 11.117 l S 34.35 20.867 m 31.35 20.867 l S 44.6 56.617 m 42.1 54.867 l S 47.35 51.367 m 44.35 51.367 l S 44.1 43.867 m 41.35 45.617 l S 43.35 33.117 m 42.6 30.617 l S 43.85 23.617 m 41.1 25.867 l S 44.35 15.617 m 42.35 16.867 l S 67.823 31.1 m 64.823 31.1 l S 27.1 32.617 m 29.6 30.867 l S 31.85 55.117 m 34.85 55.117 l S 19.6 40.867 m 22.1 39.117 l S 16.85 35.617 m 19.85 35.617 l S 20.1 28.117 m 22.85 29.867 l S 52.1 42.617 m 54.484 44.178 l S 52.437 50.146 m 54.821 48.325 l S 59.572 54.133 m 59.35 51.117 l S 50.185 10.055 m 53.234 9.928 l S 51.187 15.896 m 53.571 14.075 l S 58.322 19.883 m 59.445 16.823 l S 53.1 32.117 m 50.6 30.367 l S 52.85 24.617 m 49.6 25.617 l S 61.85 9.117 m 59.1 10.867 l S 69.35 34.617 m 66.6 36.367 l S 67.1 23.617 m 65.1 22.117 l S 24.435 46.055 m 27.484 45.928 l S 25.437 51.896 m 27.821 50.075 l S 62.6 47.117 m 65.321 46.575 l S 19.85 19.867 m 20.35 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Y=0 K=100) Pc 0 0.1 1 0 k (C=0 M=10 Y=100 K=0) Pc 0 0.5 0 0 k (C=0 M=50 Y=0 K=0) Pc 0 0.5 1 0 k (C=0 M=50 Y=100 K=0) Pc 1 0.55 1 0 k (C=100 M=55 Y=100 K=0) Pc 1 0.9 0.1 0 k (C=100 M=90 Y=10 K=0) Pc 0.15 1 1 0 k (C=15 M=100 Y=100 K=0) Pc 0.45 0.9 0 0 k (C=45 M=90 Y=0 K=0) Pc 0.5 0.4 0.3 0 k (C=50 M=40 Y=30 K=0) Pc 0.5 0.85 1 0 k (C=50 M=85 Y=100 K=0) Pc 0.75 0.05 1 0 k (C=75 M=5 Y=100 K=0) Pc 0.75 0.9 0 0 k (C=75 M=90 Y=0 K=0) Pc 0.8 0.05 0 0 k (C=80 M=5 Y=0 K=0) Pc Bb 2 (Black, White) -7885 8587 0 0 1 0 0 1 0 0 Bg 0 BB (Black, White) Pc Bb 2 (Chrome) -7885 8587 0 0 1 0 0 1 0 0 Bg 0 BB (Chrome) Pc Bb 2 (Rainbow) -7885 8587 0 0 1 0 0 1 0 0 Bg 0 BB (Rainbow) Pc Bb 0 0 0 0 Bh 2 (Yellow & Orange Radial) -7885 8587 0 0 1 0 0 1 0 0 Bg 0 BB (Yellow & Orange Radial) Pc (Brick) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p (Brick) Pc (Confetti) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p (Confetti) Pc (Leaves - Fall ) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p (Leaves - Fall ) Pc (Stripes) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p (Stripes) Pc PB %AI5_EndPalette %AI5_Begin_NonPrinting Np %AI8_BeginPluginObject (Adobe Scatter Brush Tool) (Dog Tracks) (1 /New Pattern 41/ 1 0 0 0 1 / 0 1 1 0 1 1 0 0 0 0 -90 -90 0 1 0) . %AI8_EndPluginObject %AI8_BeginPluginObject (Adobe Scatter Brush Tool) (Fall Leaf) (1 /New Pattern 34/ 1 0.0745 0.9 0.9019 0.18 / 0 0.602793 1 1 0.1 1 1 -) - (1 1 1 -180 180 1 0 0) . %AI8_EndPluginObject %AI8_BeginPluginObject (Adobe Scatter Brush Tool) (Ladybug) (1 /New Pattern 10/ 5 0.898039 0 0 / 0 1 1 0 0.803911 1.2 1 -1.55 1.55 ) - (1 -180 180 1 0 0) . %AI8_EndPluginObject %AI8_BeginPluginObject (Adobe Scatter Brush Tool) (Push Pin) (1 /New Pattern 36/ 1 0.025 0.1 0.475 0 / 0 1 1 0 0.401676 1 1 -1.06145) - ( 1.06 1 -180 180 1 0 0) . %AI8_EndPluginObject %AI8_BeginPluginObject (Adobe Scatter Brush Tool) (Strawberry) (1 /New Pattern 37/ 1 0 0 0 1 / 0 0.803911 1 1 0.803911 1 1 -0.5 0.5 1 ) - (-75 75.419 1 0 0) . %AI8_EndPluginObject %AI8_BeginPluginObject (Adobe Scatter Brush Tool) (Twinkle Star ) (1 /New Pattern 2/ 0 1 / 1 0.5 1 1 0.25 1 1 -0.5 0.5 1 0 0 0 0 0) . %AI8_EndPluginObject %AI8_BeginPluginObject (Adobe PatternOnPath Brush Tool) (Double Lines) (1 / New Pattern 62/ New Pattern 63/ New Pattern 64/ / / 1 1 0.14 0.09 ) - (0 / 1 0 1 0 1 0 0 0) . %AI8_EndPluginObject %AI8_BeginPluginObject (Adobe PatternOnPath Brush Tool) (Laurel) (1 / New Pattern 65/ New Pattern 42/ New Pattern 67/ / New Pattern 69/ ) - (1 0 0.55 1 0.3 / 1 0 1 0 1 0 0 0) . %AI8_EndPluginObject %AI8_BeginPluginObject (Adobe PatternOnPath Brush Tool) (Rope ) (1 / New Pattern 1/ / / New Pattern 3/ New Pattern 6/ 5 0 0 0 / 1 0 1 ) - (0 1 0 0 0) . %AI8_EndPluginObject %AI8_BeginPluginObject (Adobe ArtOnPath Brush Tool) (Arrow) (1 / New Pattern 45/ / / / / 5 0.898039 0 0 / 2 0 1 0 1 0 0 0) . %AI8_EndPluginObject %AI8_BeginPluginObject (Adobe ArtOnPath Brush Tool) (Marker) (1 / New Pattern 8/ / / / / 0 0 / 1 1 1 0 1 0 0 0) . %AI8_EndPluginObject %AI8_BeginPluginObject (Adobe ArtOnPath Brush Tool) (Paintbrush) (1 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PatternOnPath Brush Tool/ Laurel/ Adobe Pa) - (tternOnPath Brush Tool/ Rope /) . %AI8_EndPluginObject %AI5_End_NonPrinting-- %AI5_Begin_NonPrinting Np %AI8_PluginGroupInfo (Adobe Path Blends) (Adobe Blends Plugin) (Live Blends) %AI8_PluginGroupInfo (Adobe PatternOnPath Brush Tool) (Adobe Pattern Brush Plugin) (Art Brush Tool) %AI8_PluginGroupInfo (Adobe ArtOnPath Brush Tool) (Adobe Art Brush Plugin) (Art Brush Tool) %AI8_PluginGroupInfo (Adobe Calligraphic Brush Tool) (Undo New Calligraphic Brush) (Calligraphic Brush Tool) %AI8_PluginGroupInfo (Adobe Scatter Brush Tool) (Adobe Scatter Brush Plugin) (Scatter Brush Tool) %AI5_End_NonPrinting-- %%EndSetup %AI5_BeginLayer 1 1 1 1 0 0 1 0 79 128 255 0 50 Lb (Layer 1) Ln 0 A 1 Ap 0 R 0 0 0 1 K 800 Ar 0 J 0 j 2 w 4 M []0 d %AI3_Note: 0 D 0 XR 468.9697 285.6572 m 425.2305 282.4473 382.7764 287.2227 340.6006 300.1426 c 286.5449 315.9941 237.7876 339.1826 180.3242 340.2285 c 170.9746 339.6836 160.9424 341.2959 151.4326 339.7529 c 148.2632 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w 457.2207 721.041 m 440.6895 737.2827 l S 0.3713 w 458.7119 716.1123 m 439.9805 734.5146 l S 0.348 w 477.3867 707.9014 m 440.0811 744.5503 l S 0.3888 w 472.1826 715.7979 m 453.0039 734.6387 l S 0.3145 w 484.3994 714.936 m 464.7422 734.2476 l S U Q U 0 A 0 R 0 0 0 1 K 800 Ar 0 J 0 j 2 w 4 M []0 d %AI3_Note: 0 D 0 XR 308.3701 619.9702 m 329.2744 533.04 298.71 412.9287 288.4834 344.75 c 287.4834 339.75 286.4834 335.75 285.4834 330.75 c 275.4834 254.75 270.4834 180.75 272.4834 104.75 c S LB %AI5_EndLayer-- %%PageTrailer gsave annotatepage grestore showpage %%Trailer Adobe_Illustrator_AI5 /terminate get exec Adobe_shading_AI8 /terminate get exec Adobe_ColorImage_AI6 /terminate get exec Adobe_typography_AI5 /terminate get exec Adobe_cshow /terminate get exec Adobe_level2_AI5 /terminate get exec %%EOF %%EndDocument @endspecial 2705 2767 a currentpoint grestore moveto 2705 2767 a 891 2975 a Fw(Figure)34 b(1.)46 b Fx(Pictorial)29 b(represen)m(tation)i(of)f(Conjecture)h(3.3) 555 3195 y Ft(In)i(a)g(neighb)-5 b(orho)g(o)g(d)37 b(of)c Fr(W)1500 3162 y Fo(u)1487 3222 y(T)1538 3203 y Fk(\003)1578 3195 y Ft(,)f(the)i(unstable)f(manifold)i(of)e(the)g(\014xe)-5 b(d)34 b(p)-5 b(oint)34 b(under)456 3303 y(r)-5 b(enormalization,)36 b(we)c(have)h(for)h(some)f Fr(n)1915 3317 y Fu(0)1979 3303 y Fp(2)25 b Fq(N)45 b Ft(and)34 b Fr(B)j Ft(a)c(b)-5 b(al)5 b(l,)1375 3519 y Fx(\006)1441 3533 y Fo(n)1484 3542 y Fs(0)1518 3533 y Fu(+1)p Fo(;B)1681 3542 y Fl(n)1719 3557 y Fs(0)1754 3542 y(+1)1859 3519 y Fp(\\)20 b Fr(W)2039 3481 y Fo(u)2026 3541 y(T)2077 3523 y Fk(\003)2142 3519 y Fp(2)25 b Fx(\006)2294 3481 y Fu(+)2294 3548 y Fo(n)2337 3557 y Fs(0)2371 3548 y Fo(;B)2444 3556 y Fl(n)2482 3571 y Fs(0)555 3694 y Fx(Note)35 b(that)g(Conjecture)f(3.3)h(implies)c (that)j(a)g(segmen)m(t)h(of)f Fr(W)2747 3661 y Fo(u)2734 3721 y(T)2785 3702 y Fk(\003)2859 3694 y Fx(is)f(con)m(tained)h(in)456 3807 y(\006)522 3769 y Fu(+)522 3836 y Fo(n;B)644 3807 y Fx(.)77 b(Conjecture)43 b(3.5)g(mak)m(es)h(the)e(quan)m(titativ)m(e)i (assertion)e(that)h(this)e(in)m(terv)-5 b(al)456 3920 y(extends)30 b(to)h(reac)m(h)g(the)g(in)m(tersection)f(of)g(the)h(next) g(lev)m(el.)555 4028 y(If)37 b(Conjecture)g(3.5)h(w)m(ere)g(true,)h (then)d(w)m(e)i(see)g(that)f(the)h(di\013eren)m(t)e(obstructions)456 4136 y Ft(\\play)i(tag")p Fx(.)58 b(That)36 b(is,)h(b)s(efore)f(the)g (obstruction)f(due)h(to)g(\006)2592 4150 y Fo(n)2675 4136 y Fx(ceases)h(to)g(apply)-8 b(,)37 b(w)m(e)456 4244 y(can)26 b(start)g(applying)d(the)j(obstruction)f(due)g(to)h(\006)2164 4258 y Fo(n)p Fu(+1)2301 4244 y Fx(.)39 b(In)25 b(this)f(w)m(a)m(y)-8 b(,)29 b(w)m(e)d(can)g(see)g(that)456 4352 y(the)j(in)m(terv)-5 b(als)29 b Fr(I)1015 4366 y Fo(n)1092 4352 y Fx(describ)s(ed)e(ab)s(o)m (v)m(e)k(o)m(v)m(erlap)f(and)g(that)g(therefore)g Fp([)2880 4366 y Fo(n)2927 4352 y Fr(I)2967 4366 y Fo(n)3043 4352 y Fx(con)m(tains)g(a)456 4460 y(full)e(in)m(terv)-5 b(al)29 b(\()p Fr(\025)1031 4474 y Fm(\003)1071 4460 y Fr(;)15 b(\025)1164 4474 y Fm(\003)1224 4460 y Fx(+)20 b Fr(\016)s Fx(\))32 b(for)e(some)g Fr(\016)g(>)25 b Fx(0.)555 4568 y(It)30 b(seems)g(from)g(n)m(umerical)e(exp)s(erimen)m(ts)h(that)h(the) g Fr(\016)k Fx(ab)s(o)m(v)m(e)d(can)f(b)s(e)f(reasonably)456 4676 y(big.)45 b(Indeed,)32 b(if)f(Conjecture)h(3.5)i(w)m(ere)e(true,)h (w)m(e)g(could)e(just)h(select)h(it)e(b)m(y)h Fr(T)3185 4691 y Fo(\025)3226 4672 y Fk(\003)3263 4691 y Fu(+)p Fo(\016)3384 4676 y Fp(2)456 4784 y Fx(\006)522 4798 y Fo(n)565 4807 y Fs(0)603 4784 y Fx(.)555 4892 y(Once)g(w)m(e)h(ha)m (v)m(e)h(used)d(the)h(com)m(bination)g(of)g(the)h(obstruction)e(and)h (renormaliza-)456 5000 y(tion)24 b(to)j(exclude)d(existence)i(of)g(in)m (v)-5 b(arian)m(t)24 b(circles)h(in)f(the)h(initial)e(in)m(terv)-5 b(al)25 b(\()p Fr(\025)3112 4967 y Fm(\003)3151 5000 y Fr(;)15 b(\025)3244 4967 y Fm(\003)3294 5000 y Fx(+)10 b Fr(\016)s Fx(])456 5108 y(it)38 b(b)s(ecomes)i(p)s(ossible)c(to)k (use)f(other)g(criteria)g(to)h(exclude)e(existence)i(of)f(in)m(v)-5 b(arian)m(t)456 5216 y(circles)28 b(in)g(the)i(in)m(terv)-5 b(als)28 b([)p Fr(\025)1430 5183 y Fm(\003)1489 5216 y Fx(+)18 b Fr(\016)n(;)d Fp(1)p Fx(\).)41 b(F)-8 b(or)31 b(example,)e(in)f(the)i(n)m(umerical)e(studies)g(of)p eop %%Page: 16 16 16 15 bop 456 251 a Fu(16)837 b(R.)23 b(DE)g(LA)h(LLA)-8 b(VE)23 b(AND)g(A.)g(OL)-8 b(VERA)456 450 y Fx(the)30 b(standard)g(family)-8 b(,)30 b(whic)m(h)f(w)m(e)i(will)d(discuss)h (later,)i(it)f(seems)h(that)g(one)g(can)g(tak)m(e)456 558 y Fr(\016)g Fp(\031)c Fx(1.)46 b(There)31 b(exist)g(pro)s(ofs)g (that)i(the)f(standard)f(family)f(do)s(es)h(not)h(ha)m(v)m(e)h(in)m(v) -5 b(arian)m(t)456 666 y(circles)36 b(for)g Fr(\025)g(>)g Fx(4)p Fr(=)p Fx(3)i({)g(see)f([Mat88)r(])h({)f(and)f (computer-assisted)h(pro)s(ofs)f(that)h(get)456 774 y(m)m(uc)m(h)e (closer)f(to)i(the)f(n)m(umerically)e(conjectured)i(v)-5 b(alue)34 b(can)h(b)s(e)f(found)g(in)g([MP85)q(],)456 882 y([Jun91)o(].)555 990 y(Hence,)k(assuming)c(Conjecture)h(3.5)h({)g (as)f(w)m(ell)g(as)g(some)h(v)m(ery)f(mild)e(b)s(ounds)g(on)456 1098 y(the)38 b(v)-5 b(alue)38 b(of)g Fr(n)1029 1112 y Fu(0)1107 1098 y Fx({)g(and)g(that)h(the)f(standard)g(family)f (crosses)h(transv)m(ersally)g Fr(W)3367 1065 y Fo(u)3354 1125 y(T)3405 1106 y Fk(\003)456 1206 y Fx(su\016cien)m(tly)g(close)i (to)h Fr(T)1334 1173 y Fm(\003)1373 1206 y Fx(,)h(w)m(e)f(w)m(ould)d (conclude)i(that,)j(for)c(the)h(standard)f(family)-8 b(,)456 1314 y(there)30 b(is)g(a)g(golden)g(mean)h(in)m(v)-5 b(arian)m(t)29 b(circle)h(precisely)f(for)h Fr(\025)25 b Fp(2)g Fx([)p Fp(\000)p Fr(\025)2829 1328 y Fm(\003)2869 1314 y Fr(;)15 b(\025)2962 1328 y Fm(\003)3001 1314 y Fx(].)555 1421 y(When)33 b Fr(\025)e Fp(2)e Fx(\()p Fp(\000)p Fr(\025)1156 1435 y Fm(\003)1196 1421 y Fr(;)15 b(\025)1289 1435 y Fm(\003)1329 1421 y Fx(\),)34 b(the)g(circle)f(is)f(analytic.)49 b(When)33 b Fr(\025)e Fx(=)f Fp(\006)p Fr(\025)2885 1435 y Fm(\003)2924 1421 y Fx(,)k(the)g(circle)e(is)456 1529 y(\014nitely)f(di\013eren)m(tiable)g(according)i(to)g([Ko)s(c02a)r(].) 49 b(The)32 b(pap)s(er)g([Sti93])h(con)m(tains)g(an)456 1637 y(argumen)m(t)24 b(sho)m(wing)g(that)h(some)g(prop)s(erties)e(of)h (the)h(\014xed)f(p)s(oin)m(t)f(of)i(renormalization)456 1745 y(imply)g(that)k(the)f(critical)f(circle)g(is)g(not)h(more)g (regular)g(than)f(a)i(certain)f(v)m(ery)g(explicit)456 1853 y(n)m(um)m(b)s(er.)39 b(It)30 b(lo)s(oks)g(lik)m(e)g(the)g (argumen)m(t)h(in)e([Ko)s(c02a)r(])h(could)g(also)g(mak)m(e)h(similar)d (ex-)456 1961 y(plicit)22 b(n)m(umerical)g(predictions)h(or)h(that)h (the)f(argumen)m(t)h(of)f([Sti93])g(could)g(b)s(e)f(adapted)456 2069 y(to)31 b(the)f(renormalization)f(formalism)g(in)g([Ko)s(c02a)r (].)555 2177 y(Recen)m(t)45 b(w)m(ork)e(of)g(the)g(authors)g(with)f(N.) h(P)m(etro)m(v)i(extending)d(the)i(metho)s(ds)e(of)456 2285 y([dlLN02])26 b(measures)g(directly)f(the)h(regularit)m(y)g(of)g (the)h(critical)e(in)m(v)-5 b(arian)m(t)26 b(circles)f(and)456 2393 y(obtains)k(a)i(go)s(o)s(d)f(agreemen)m(t)i(with)d(the)i(n)m (umerical)e(v)-5 b(alues)30 b(of)g([Sti93].)555 2501 y(W)-8 b(e)35 b(certainly)e(b)s(eliev)m(e)f(that)j(Conjecture)e(3.5)i (is)e(true)g(and)g(w)m(e)h(will)d(presen)m(t)j(n)m(u-)456 2609 y(merical)25 b(evidence)h(for)f(its)h(v)-5 b(alidit)m(y)24 b(in)h(Section)h(5.)40 b(Clearly)-8 b(,)26 b(Conjecture)g(3.5)h (implies)456 2717 y(Conjecture)j(3.3.)555 2825 y(W)-8 b(e)36 b(will)c(end)h(this)h(section)g(b)m(y)h(form)m(ulating)e (Conjecture)i(3.6,)h(whic)m(h)e(is)f(clearly)456 2932 y(implied)28 b(b)m(y)j(Conjecture)g(3.3.)45 b(The)31 b(reason)g(for)g(form)m(ulating)f(suc)m(h)h(a)h(w)m(eak)g(conjec-)456 3040 y(ture)g(is)f(that)i(it)f(implies)d(man)m(y)j(of)h(the)f (consequences)h(of)g(the)f(sharp)s(er)f(conjectures)456 3148 y(but)i(it)g(will)e(b)s(e)i(easier)g(to)i(v)m(erify)e(in)f(a)i (computer)g(assisted)f(pro)s(of.)50 b(Certainly)-8 b(,)33 b(it)g(is)456 3256 y(so)d(w)m(eak)h(that)g(it)f(is)g(hard)f(to)i(b)s (eliev)m(e)f(it)g(could)f(fail.)456 3581 y FA(Conjecture)23 b(3.6.)35 b Ft(In)24 b(the)g(se)-5 b(gment)25 b Fr(W)1867 3536 y Fo(u;)p Fu(+)1854 3609 y Fo(T)1905 3590 y Fk(\003)2009 3581 y Ft(ther)-5 b(e)25 b(is)f(a)g(map)i Fr(T)2631 3596 y Fo(\025)2672 3577 y Fk(\003)2736 3581 y Ft(with)f(a)f(heter)-5 b(o)g(clinic)456 3689 y(tr)g(ansverse)30 b(c)-5 b(onne)g(ction)31 b(b)-5 b(etwe)g(en)30 b(an)f(orbit)h(of)g(typ)-5 b(e)30 b Fr(F)2352 3703 y Fo(n)2399 3689 y Fr(=F)2502 3703 y Fo(n)p Fu(+1)2669 3689 y Ft(and)g(an)g(orbit)g(of)f(typ)-5 b(e)456 3796 y Fr(F)514 3810 y Fo(n)p Fm(\000)p Fu(1)651 3796 y Fr(=F)754 3810 y Fo(n)802 3796 y Ft(.)555 4121 y Fx(Note)27 b(that)e(the)h(existence)f(of)g(a)h(transv)m(erse)f (hetero)s(clinic)f(connection)h(for)g Fr(T)3207 4136 y Fo(\025)3248 4117 y Fk(\003)3313 4121 y Fx(im-)456 4229 y(plies)d(the)j(existence)g(of)f(a)h(similar)d(connection)j(for)f (all)f(the)i(maps)f(in)f(a)i(neigh)m(b)s(orho)s(o)s(d)456 4337 y(of)30 b Fp(U)39 b Fx(of)31 b Fr(T)812 4352 y Fo(\025)853 4333 y Fk(\003)893 4337 y Fx(.)555 4445 y(Using)h(the)g(partial)f (linearization)f(established)h(in)g([CEL80])i(or)f(the)g Fr(\025)p Fx(-lemma)g(of)456 4553 y([AK00)q(],)27 b(w)m(e)f(ha)m(v)m(e) h(that)g(giv)m(en)e(an)m(y)i(neigh)m(b)s(orho)s(o)s(d)c Fp(V)33 b Fx(of)26 b(a)g(p)s(oin)m(t)f Fr(T)38 b Fp(2)25 b Fr(W)3049 4520 y Fo(s)3036 4579 y(T)3087 4561 y Fk(\003)3127 4553 y Fx(,)i(w)m(e)f(can)456 4667 y(\014nd)i(a)j Fr(k)e Fp(2)24 b Fq(N)43 b Fx(suc)m(h)30 b(that)h Fp(N)1460 4634 y Fm(\000)p Fo(k)1558 4667 y Fx(\()p Fp(V)7 b Fx(\))21 b Fp(\\)f(U)34 b(6)p Fx(=)25 b Fp(;)p Fx(.)555 4782 y(By)d(the)f(same)h (argumen)m(t)g(as)g(in)e(Prop)s(osition)f(3.2,)25 b(w)m(e)c(ha)m(v)m(e) i(that)f(if)2903 4759 y(~)2885 4782 y Fr(T)38 b Fp(2)25 b(N)3150 4749 y Fm(\000)p Fo(k)3248 4782 y Fx(\()p Fp(V)7 b Fx(\))r Fp(\\)456 4892 y(U)i Fx(,)34 b Fr(T)43 b Fx(=)30 b Fp(N)866 4859 y Fo(k)909 4892 y Fx(\()p Fp(N)1032 4859 y Fm(\000)p Fo(k)1130 4892 y Fx(\()p Fr(T)13 b Fx(\)\))34 b(has)f(a)h(transv)m(erse)g(hetero)s(clinic)d(in)m(tersection)i(b)s(et) m(w)m(een)h(an)456 5000 y(orbit)29 b(of)i(t)m(yp)s(e)f Fr(F)1044 5015 y Fo(n)p Fu(+)p Fo(k)1185 5000 y Fr(=F)1288 5015 y Fo(n)p Fu(+)p Fo(k)r Fu(+1)1550 5000 y Fx(and)g(an)g(orbit)f(of) i(t)m(yp)s(e)f Fr(F)2441 5015 y Fo(n)p Fu(+)p Fo(k)r Fm(\000)p Fu(1)2672 5000 y Fr(=F)2775 5015 y Fo(n)p Fu(+)p Fo(k)2917 5000 y Fx(.)555 5108 y(Therefore,)d(Conjecture)e(3.6)i (implies)22 b(that)27 b Fr(W)2183 5075 y Fo(s)2170 5135 y(T)2221 5116 y Fk(\003)2286 5108 y Fx(is)d(part)i(of)f(the)h(b)s (oundary)d(of)j(the)456 5216 y(p)s(oin)m(ts)j(with)g(an)h(analytic)g (in)m(v)-5 b(arian)m(t)30 b(circle)f(of)i(rotation)g(golden)e(mean.)p eop %%Page: 17 17 17 16 bop 1213 251 a Fu(OBSTR)n(UCTION)30 b(AND)g(RENORMALIZA)-6 b(TION)687 b(17)978 450 y Fx(4.)46 b Fw(Some)33 b(consequences)e(of)j (the)f(conjectures)456 612 y FA(Lemma)h(4.1.)43 b Ft(Under)34 b(Conje)-5 b(ctur)g(e)35 b(3.3)g(given)e(a)i(universal)f(family)h(of)f (twist)h(maps)456 722 y Fr(T)509 737 y Fo(\025)554 722 y Ft(,)g(then)h(\(by)f(switching)g(p)-5 b(erhaps)38 b(to)1842 699 y Fx(~)1824 722 y Fr(T)1877 737 y Fo(\025)1952 722 y Fp(\021)30 b Fr(T)2106 737 y Fm(\000)p Fo(\025)2241 722 y Ft(to)35 b(ensur)-5 b(e)36 b(a)f(pr)-5 b(op)g(er)37 b(orientation)456 830 y(of)32 b(the)h(p)-5 b(ar)g(ameter\))36 b(we)d(c)-5 b(an)33 b(\014nd)g Fr(\025)1714 844 y Fu(0)1754 830 y Ft(,)f Fr(\016)d(>)c Fx(0)33 b Ft(such)g(that:)671 956 y Fp(\017)42 b Ft(When)35 b Fr(\025)1076 970 y Fu(0)1138 956 y Fp(\000)21 b Fr(\016)34 b(<)29 b(\025)h(<)f(\025)1640 970 y Fu(0)1679 956 y Ft(,)36 b(then)f Fr(T)2000 971 y Fo(\025)2080 956 y Ft(has)h(a)g(smo)-5 b(oth)37 b(invariant)f(cir)-5 b(cle)35 b(with)758 1064 y(r)-5 b(otation)35 b(golden)f(me)-5 b(an.)671 1172 y Fp(\017)42 b Ft(When)e Fr(\025)1081 1186 y Fu(0)1158 1172 y Fr(<)e(\025)g(<)f(\025)1519 1186 y Fu(0)1584 1172 y Fx(+)25 b Fr(\016)s Ft(,)42 b(for)e(al)5 b(l)40 b Fr(N)49 b Ft(su\016ciently)40 b(lar)-5 b(ge,)42 b(ther)-5 b(e)41 b(ar)-5 b(e)40 b(hy-)758 1280 y(p)-5 b(erb)g(olic)33 b(orbits)f(of)f(typ)-5 b(e)32 b Fr(F)1693 1294 y Fo(N)1761 1280 y Fr(=F)1864 1294 y Fo(N)7 b Fu(+1)2022 1280 y Ft(,)32 b Fr(F)2140 1294 y Fo(N)7 b Fu(+1)2298 1280 y Fr(=F)2401 1294 y Fo(N)g Fu(+2)2559 1280 y Ft(,)31 b(such)g(that)i(their)f(stable)758 1388 y(and)i(unstable)f(manifolds)i (interse)-5 b(ct)33 b(tr)-5 b(ansversal)5 b(ly.)456 1535 y FA(Remark)50 b(4.2.)g Fx(Actually)-8 b(,)49 b(the)c(ab)s(o)m(v)m(e)h (results)d(can)i(b)s(e)f(made)h(somewhat)g(more)456 1643 y(precise)33 b(in)g(the)i(case)g(that)g(the)f(map)g(is)f(rev)m (ersible,)h(and,)h(as)g(it)e(is)h(w)m(ell)f(kno)m(wn,)i(w)m(e)456 1750 y(can)29 b(pro)m(v)m(e)i(that)f(the)f(h)m(yp)s(erb)s(olic)e (orbits)i(with)f(the)i(transv)m(ersal)f(in)m(tersection)g(lie)f(on)456 1858 y(the)23 b(critical)g(lines.)36 b(See)24 b(Subsection)e(2.2)j(for) e(more)h(details)e(and)h(de\014nitions.)36 b(F)-8 b(or)24 b(the)456 1966 y(momen)m(t,)33 b(w)m(e)g(just)f(p)s(oin)m(t)f(out)i (that)f(the)h(fact)g(that)g(the)f(orbits)g(lie)f(on)h(critical)f(lines) 456 2074 y(reduces)24 b(the)g(n)m(umerical)f(calculation)h(of)h(the)f (p)s(erio)s(dic)e(orbits)h(to)j(a)f(one-dimensional)456 2182 y(searc)m(h.)456 2329 y FA(Remark)45 b(4.3.)i Fx(When)40 b(that)g(the)h(map)e(is)g(rev)m(ersible,)i(then)f(the)g(p)s(erio)s(dic) d(orbits)456 2437 y(that)31 b(satisfy)g(\(7\))g(can)g(b)s(e)f(made)g (to)h(lie)e(on)i(the)f(critical)f(lines.)555 2583 y(Note)35 b(that)f(a)f(consequence)h(of)g(Lemma)f(4.1)i(is)d(that,)j(as)e(so)s (on)g(as)h(w)m(e)g(are)f(in)f(the)456 2691 y(second)d(alternativ)m(e)h (of)g(Lemma)f(4.1,)j(w)m(e)e(ha)m(v)m(e)g(that)g(for)g(the)f(v)-5 b(alues)29 b(of)h(parameters)456 2799 y(that)23 b(there)h(is)e(not)i (an)f(in)m(v)-5 b(arian)m(t)22 b(circle)h(with)f(golden)h(mean)g (rotation)h(n)m(um)m(b)s(er,)f(then,)456 2907 y(w)m(e)30 b(can)h(\014nd)e(a)i(h)m(yp)s(erb)s(olic)c(orbit)j(with)f(rotation)i(n) m(um)m(b)s(er)e(golden)h(mean.)555 3015 y(An)c(argumen)m(t)g(in)f ([FdlL92b],)i(sho)m(ws)f(that,)h(when)e(Lemma)h(4.1)h(applies,)f(w)m(e) g(ha)m(v)m(e)456 3123 y(that)38 b(for)g Fr(\025)g Fp(2)f Fx(\()p Fr(\025)1084 3137 y Fu(0)1124 3123 y Fr(;)15 b(\025)26 b Fx(+)e Fr(\016)s Fx(\))39 b(there)g(exists)e(a)h(sequence)h (of)f(p)s(oin)m(ts)f Fp(f)p Fr(p)2905 3137 y Fo(N)2972 3123 y Fp(g)h Fx(whic)m(h)f(are)456 3231 y(p)s(erio)s(dic)27 b(of)k(t)m(yp)s(e)f Fr(F)1170 3245 y Fo(N)1238 3231 y Fr(=F)1341 3245 y Fo(N)7 b Fu(+1)1530 3231 y Fx(for)30 b(the)g(map)g Fr(T)2080 3246 y Fo(\025)2156 3231 y Fx(and)g(suc)m(h)g (that)456 3394 y(\(6\))539 b(lim)1071 3455 y Fo(N)7 b Fm(!1)1290 3394 y Fp(j)15 b Fx(T)-8 b(r\()p Fr(D)s(T)1603 3345 y Fo(F)1648 3356 y Fl(N)5 b Fs(+1)1590 3424 y Fo(\025)1786 3394 y Fx(\()p Fr(p)1867 3408 y Fo(N)1935 3394 y Fx(\)\))21 b Fp(\000)f Fx(2)p Fp(j)2187 3357 y Fu(1)p Fo(=F)2302 3368 y Fl(N)5 b Fs(+1)2467 3394 y Fx(=)25 b Fr(\026)2618 3409 y Fo(\025)2688 3394 y Fr(>)g Fx(0)555 3578 y(The)i(ab)s(o)m(v)m(e) h(result)f(is)f(closely)h(related)g(to)h(Greene's)g(criterion)e([Gre79) q(])i(Assertion)456 3686 y(VI.)i(that)h(an)f(in)m(v)-5 b(arian)m(t)30 b(circle)g(exists)g(if)f(and)h(only)f(if)h(the)g(limit)e (in)h(\(6\))q(,)i(\(7\))h(is)d(zero.)555 3794 y(Hence,)g(under)d (Conjecture)h(3.5)i(w)m(e)e(ha)m(v)m(e)i(that,)g(for)e(univ)m(ersal)e (families,)h(w)m(e)i(ha)m(v)m(e)456 3902 y(that)j(if)e(there)h(is)g(no) g(circle,)g(then)g(the)h(mean)f(residue)f(is)g(p)s(ositiv)m(e.)555 4010 y(The)k(other)h(side)f(of)h(the)g(implication)d(in)h([Gre79)r(])i (is)e(established)g(rigorously)g(in-)456 4118 y(dep)s(enden)m(tly)c(of) h(renormalization.)40 b(In)29 b(the)g(pap)s(ers)g([FdlL92b],)h([Mac92)r (])g(\(See)g(also)456 4226 y([McK82)q(])35 b(1.3.2.4\))j(w)m(e)d(can)h (\014nd)d(pro)s(ofs)h(when)g(there)h(is)f(an)h(smo)s(oth)f(KAM)h (torus,)456 4334 y(w)m(e)30 b(ha)m(v)m(e)456 4479 y(\(7\))649 b(lim)1181 4540 y Fo(N)7 b Fm(!1)1401 4479 y Fp(j)15 b Fx(T)-8 b(r\()p Fr(D)s(T)1714 4430 y Fo(F)1759 4441 y Fl(N)5 b Fs(+1)1701 4509 y Fo(\025)1897 4479 y Fx(\()p Fr(p)1978 4493 y Fo(N)2045 4479 y Fx(\)\))21 b Fp(\000)f Fx(2)p Fp(j)2297 4442 y Fu(1)p Fo(=F)2412 4453 y Fl(N)5 b Fs(+1)2577 4479 y Fx(=)25 b(0)555 4658 y(Hence,)32 b(w)m(e)f(ha)m(v)m(e)g(that)456 4805 y FA(Lemma)36 b(4.4.)44 b Ft(If)36 b(Conje)-5 b(ctur)g(e)36 b(3.5)h(is)e(true,)i(Gr)-5 b(e)g(ene's)36 b(criterion)h(applies)g(for)f(uni-)456 4913 y(versal)k(functions.)64 b(Mor)-5 b(e)40 b(explicitly,)i(Ther)-5 b(e)40 b(is)g(an)g(invariant)h(cir)-5 b(cle)40 b(if)g(and)g(only)456 5021 y(if)456 5179 y Fx(\(8\))649 b(lim)1181 5240 y Fo(N)7 b Fm(!1)1401 5179 y Fp(j)15 b Fx(T)-8 b(r\()p Fr(D)s(T)1714 5130 y Fo(F)1759 5141 y Fl(N)5 b Fs(+1)1701 5209 y Fo(\025)1897 5179 y Fx(\()p Fr(p)1978 5193 y Fo(N)2045 5179 y Fx(\)\))21 b Fp(\000)f Fx(2)p Fp(j)2297 5142 y Fu(1)p Fo(=F)2412 5153 y Fl(N)5 b Fs(+1)2577 5179 y Fx(=)25 b(0)p eop %%Page: 18 18 18 17 bop 456 251 a Fu(18)837 b(R.)23 b(DE)g(LA)h(LLA)-8 b(VE)23 b(AND)g(A.)g(OL)-8 b(VERA)555 450 y Fx(W)g(e)35 b(will)30 b(also)j(establish)f(another)h(consequence)h(of)f(Conjecture) g(3.3)h(is)e(that)i(the)456 558 y(long)c(made)h(observ)-5 b(ation)31 b(that)g(the)h(stable)e(manifold)f(of)i(the)h(\014xed)e(p)s (oin)m(t)g(is)g(indeed)456 666 y(the)g(b)s(oundary)f(b)s(et)m(w)m(een)i (the)f(existence)h(and)f(non-existence)g(of)h(in)m(v)-5 b(arian)m(t)29 b(circles.)555 774 y(Therefore,)j(w)m(e)h(could)e(think) f(of)i(Conjecture)g(3.3)h(as)f(a)g(k)m(eystone)h(that)g(holds)d(to-)456 882 y(gether)e(the)g(renormalization)f(and)h(the)g(obstruction)f (criteria.)39 b(It)28 b(allo)m(ws)g(to)g(use)g(the)456 990 y(obstruction)k(criteria)g(to)i(sho)m(w)f(that)g(renormalization)f (is)g(optimal)g(and)g(renormal-)456 1098 y(ization)e(to)h(sho)m(w)f (that)h(obstruction)e(is)h(optimal.)456 1376 y FA(Remark)d(4.5.)37 b Fx(The)24 b(renormalization)f(group)h(has)g(b)s(een)g(studied)e (mainly)h(in)g(spaces)456 1484 y(of)30 b(analytic)g(functions.)555 1592 y(Hence,)36 b(it)d(is)g(p)s(erhaps)f(in)m(teresting)h(to)i (discuss)d(the)i(prosp)s(ects)f(of)h(adapting)f(the)456 1700 y(ideas)44 b(to)h(other)g(spaces.)84 b(One)44 b(motiv)-5 b(ation)45 b(for)f(this)g(study)g(is)f(that,)49 b(according)456 1808 y(to)35 b(the)g(study)f(here,)i(this)e(manifold)e(is)i(the)h(v)-5 b(alidit)m(y)33 b(among)j(stable)e(and)h(unstable)456 1915 y(manifolds.)555 2023 y(One)26 b(\014rst)f(obstacle)i(that)f(has)g (to)h(b)s(e)e(o)m(v)m(ercome)j(is)d(whether)h(the)g(renormalization)456 2131 y(op)s(erators)38 b(de\014ne)g(an)h(di\013eren)m(tiable)d(op)s (erator)j(in)e(the)i(spaces)g(of)g(\014nite)e(di\013eren-)456 2239 y(tiable)c(functions.)51 b(The)33 b(renormalization)g(of)h([McK82) r(])g(has)g(the)g(di\016cult)m(y)f(that)i(it)456 2347 y(in)m(v)m(olv)m(es)23 b(comp)s(osition,)g(whic)m(h)f(is)h(not)g(a)h (di\013eren)m(tiable)d(op)s(erator)j(in)d(spaces)j(of)f(\014nite)456 2455 y(di\013eren)m(tiable)29 b(functions)h(\(See)i(e.g.)45 b([dlLO99)o(].\))f(The)31 b(di\013eren)m(tiabilit)m(y)e(prop)s(erties) 456 2563 y(of)h(the)h(op)s(erator)f(in)f([Ko)s(c99)r(])h(ha)m(v)m(e)i (not)e(b)s(een)g(in)m(v)m(estigated.)555 2671 y(On)d(the)g(other)h (hand,)f(w)m(e)g(note)h(that)g(the)g(theory)f(of)g(stable)g(manifolds)e (for)j(renor-)456 2779 y(malization)e(group)h(op)s(erators)g(in)g (spaces)g(of)h(\014nitely)d(di\013eren)m(tiable)h(functions)g(ha)m(v)m (e)456 2887 y(b)s(een)j(in)m(v)m(estigated)i(in)e([Da)m(v96)s(],)i ([CE99].)555 2995 y(Noting)25 b(that)f(the)h(analytic)e(top)s(ology)i (is)e(stronger)h(than)g(the)h(\014nite)e(di\013eren)m(tiabil-)456 3103 y(it)m(y)30 b(one)h(sho)m(ws)f(that)i(the)f(b)s(oundary)d(in)h (analytic)i(spaces)g(is)e(part)i(of)g(the)f(b)s(oundary)456 3211 y(in)f(the)h(spaces)h(of)g(\014nitely)d(di\013eren)m(tiable)h (mappings.)555 3319 y(This)g(observ)-5 b(ation)30 b(can)h(b)s(e)f(made) h(signi\014can)m(tly)e(more)h(precise.)41 b(The)30 b(KAM)h(the-)456 3426 y(ory)26 b(sho)m(ws)h(that)g(if)e(some)i(map)g(has)f(an)g (analytic)h(in)m(v)-5 b(arian)m(t)25 b(circle,)i(then)g(in)e(a)i (neigh-)456 3534 y(b)s(orho)s(o)s(d)j(in)h(the)h(space)h(of)f(smo)s (oth)g(mappings)e(w)m(e)j(ha)m(v)m(e)g(also)f(an)g(in)m(v)-5 b(arian)m(t)32 b(circle.)456 3642 y(Hence,)38 b(the)f(in)m(terior)e(in) f(the)j(analytic)e(top)s(ology)i(is)e(con)m(tained)i(in)d(the)j(in)m (terior)e(in)456 3750 y(a)f(smo)s(oth)f(top)s(ology)-8 b(.)52 b(Con)m(v)m(ersely)-8 b(,)36 b(under)c(the)i(h)m(yp)s(othesis)e (that)j(Conjectures)e(3.3)456 3858 y(and)24 b(3.5)j(are)e(true,)i(w)m (e)e(obtain)g(that)h(the)f(places)g(where)g(the)h(KAM)f(tori)g(with)f (a)h(giv)m(en)456 3966 y(rotation)k(n)m(um)m(b)s(er)f(do)h(not)h(exist) f(is)f(c)m(haracterized)i(b)m(y)f(the)h(existence)g(of)f(transv)m(erse) 456 4074 y(hetero)s(clinic)f(in)m(tersections.)41 b(Those)30 b(p)s(ersist)e(under)h(small)g(smo)s(oth)h(p)s(erturbations.)456 4352 y FA(Remark)c(4.6.)36 b Fx(Note)25 b(that)f(Conjectures)f(3.3)i (and)d(3.5)j(can)e(b)s(e)g(v)m(eri\014ed)g(with)f(a)h(\014nite)456 4460 y(calculation.)555 4568 y(It)i(su\016ces)g(to)h(kno)m(w)f(with)f (a)i(\014nite)e(precision)f(the)j(stable)e(manifold)f(and)i(to)h(v)m (erify)456 4676 y(that)31 b(it)e(in)m(tersects)i(transv)m(ersally)f (the)g(manifold)e(\006.)555 4784 y(Similar)i(pro)s(ofs)h(ha)m(v)m(e)j (b)s(een)d(obtained)h(in)f(the)i(literature)e(for)h(similar)e (problems.)456 4892 y(F)-8 b(or)44 b(example)g(in)f([EW87)q(])h(one)h (can)f(\014nd)e(a)j(rigorous)e(computer)h(assisted)f(pro)s(of)456 5000 y(whic)m(h)27 b(establishes)g(that)i(the)g(stable)f(manifold)e (for)i(the)h(renormalization)e(of)i(p)s(erio)s(d)456 5108 y(doubling)j(maps)j(crosses)h(transv)m(ersally)e(the)i(p)s(erio)s (d)d(doubling)g(bifurcation)g(mani-)456 5216 y(fold.)p eop %%Page: 19 19 19 18 bop 1213 251 a Fu(OBSTR)n(UCTION)30 b(AND)g(RENORMALIZA)-6 b(TION)687 b(19)555 450 y Fx(F)-8 b(or)26 b(the)g(case)h(considered)d (here,)i(if)f(one)g(starts)h(from)f(the)h(\014xed)f(p)s(oin)m(t)f (established)456 558 y(in)e([Ko)s(c02b])i(and)e(one)i(w)m(an)m(ts)g(to) g(implemen)m(t)e(a)h(strategy)i(similar)20 b(to)k(that)g(of)g([EW87)q (])456 666 y(it)i(will)e(b)s(e)i(necessary)h(to)h(implemen)m(t)d(op)s (erations)h(acting)i(on)e(families)f(of)i(mappings.)555 774 y(F)-8 b(or)37 b(the)f(case)i(at)e(hand,)h(it)f(seems)g(that)h(a)f (v)m(eri\014cation)g(with)f(the)h(manifold)e(\006)3405 788 y Fu(0)456 882 y Fx(could)d(require)f(a)j(rather)f(extensiv)m(e)g (calculation.)45 b(On)31 b(the)i(other)f(hand,)g(the)g(v)m(eri\014-)456 990 y(cation)d(for)h(\006)932 1004 y Fo(n)975 1013 y Fs(0)1042 990 y Fx({)g(with)e Fr(n)1378 1004 y Fu(0)1447 990 y Fx(p)s(erhaps)g(around)g(6)i({)g(do)s(es)f(not)h(seem)g(to)g (require)e(more)456 1098 y(than)i(one)g(or)h(t)m(w)m(o)g(terms)g(in)e (the)h(expansion.)555 1206 y(W)-8 b(e)35 b(also)e(note)g(that)h(the)f (bulk)f(of)h(the)g(consequences)h(of)f(the)h(conjectures)g({)f(e.g.)456 1314 y(that)38 b Fr(W)759 1281 y Fo(u)746 1346 y(T)797 1327 y Fs(0)873 1314 y Fx(is)f(part)h(of)g(the)g(b)s(oundary)e(b)s(et)m (w)m(een)i(the)g(maps)g(with)e(an)i(analytic)f(in-)456 1421 y(v)-5 b(arian)m(t)30 b(tori)f(with)g(rotation)i(golden)e(mean)h (and)g(those)h(with)d(a)j(h)m(yp)s(erb)s(olic)c(Aubry-)456 1529 y(Mather)32 b(set)g(with)f(rotation)h(golden)f(mean)h({)h(can)f(b) s(e)f(established)f(b)m(y)i(establishing)456 1637 y(the)40 b(m)m(uc)m(h)g(w)m(eak)m(er)h(Conjecture)f(3.6)i(\(later)e(on,)j(w)m(e) d(in)m(tro)s(duce)f(the)h(ev)m(en)h(w)m(eak)m(er)456 1745 y(Conjecture)f(5.2\).)72 b(The)40 b(v)m(eri\014cation)g(of)h(this) e(conjecture)i(seems)g(to)g(us)e(to)i(b)s(e)f(of)456 1853 y(comparable)g(di\016cult)m(y)f(with)g(the)i(w)m(ork)f(already)h (done)f(in)f([Ko)s(c02b)q(].)71 b(The)40 b(only)456 1961 y(ingredien)m(t)24 b(not)h(already)h(presen)m(t)f(in)f([Ko)s(c02b)q(])i (the)f(v)m(eri\014cation)g(that)i(p)s(oin)m(ts)d(in)g(the)456 2069 y(unstable)32 b(manifold)f(of)i(the)g(\014xed)g(p)s(oin)m(ts)f (admits)g(p)s(erio)s(dic)f(orbits)h(with)g(transv)m(er-)456 2177 y(sal)i(in)m(tersections.)52 b(Note)36 b(that)f(the)g(existence)g (of)g(transv)m(ersal)f(in)m(tersection)g(of)h(the)456 2285 y(p)s(erio)s(dic)30 b(orbits)i(is)g(not)h(a)g(delicate)g(prop)s (ert)m(y)-8 b(,)34 b(hence,)g(only)e(a)h(mo)s(derate)g(accuracy)456 2393 y(for)d(the)g(p)s(oin)m(ts)g(in)f(the)h(unstable)f(manifold.)555 2501 y(The)f(w)m(ork)g(of)g([Ko)s(c02b])g(already)g(giv)m(es)g(go)s(o)s (d)g(estimates)g(of)g(the)g(unstable)f(eigen-)456 2609 y(v)-5 b(alue)30 b(and)h(estimates)g(on)g(the)g(second)g(deriv)-5 b(ativ)m(e.)43 b(Hence,)32 b(one)f(obtains)f(a)i(reason-)456 2717 y(able)e(appro)m(ximation)f(for)h(the)h(unstable)e(deriv)-5 b(ativ)m(e.)555 2825 y(There)33 b(are)g(already)g(in)f(the)h (literature)f(sev)m(eral)i(implemen)m(tations)e(of)h(the)g(study)456 2932 y(of)41 b(the)h(in)m(v)-5 b(arian)m(t)40 b(manifolds)f(whic)m(h)h (pro)m(vide)g(computer)i(assisted)e(pro)s(ofs)h(of)g(the)456 3040 y(transv)m(erse)c(in)m(tersection)f(of)g(the)h(manifolds.)57 b(F)-8 b(or)37 b(example)f([FR81)r(],)i([F)m(G99)s(].)59 b(\(A)456 3148 y(more)30 b(systematic)i(theory)f(of)f(these)i(metho)s (ds)d(is)h(dev)m(elop)s(ed)g(in)g([CFdlL02a].\))42 b(The)456 3256 y(maps)35 b(considered)g(in)f(these)i(references)g(are)g(p)s (olynomials,)f(but)g(the)h(metho)s(ds)f(de-)456 3364 y(v)m(elop)s(ed)29 b(for)i(the)f(non-rigorous)f(computations)i(in)e (this)g(pap)s(er)g(are)i(v)m(ery)g(similar)d(to)456 3472 y(the)i(calculations)g(that)h(could)e(b)s(e)h(the)g(basis)g(of)g(a)h (computer)f(assisted)g(pro)s(of.)555 3642 y(Note)h(also)f(that,)g(ev)m (en)h(if,)e(for)g(the)h(sak)m(e)h(of)e(simplicit)m(y)-8 b(,)28 b(w)m(e)i(ha)m(v)m(e)h(form)m(ulated)e(the)456 3750 y(conjectures)36 b(and)g(their)f(consequences)i(only)e(for)h(the)g (golden)g(mean)g(renormaliza-)456 3858 y(tion,)31 b(they)h(are)g(quite) g(trivial)d(to)k(adapt)f(for)f(other)h(\014xed)f(p)s(oin)m(ts)g (corresp)s(onding)f(to)456 3966 y(other)j(rotation)g(n)m(um)m(b)s(ers)f (of)h(p)s(erio)s(dic)e(con)m(tin)m(ued)h(n)m(um)m(b)s(er)g(expansion.) 48 b(W)-8 b(e)34 b(lea)m(v)m(e)456 4074 y(the)c(straigh)m(tforw)m(ard)g (form)m(ulation)f(to)j(the)e(in)m(terested)h(reader.)1320 4298 y(5.)80 b Fw(Numerical)33 b(experiments.)555 4460 y Fx(In)27 b(this)f(section)i(w)m(e)g(presen)m(t)f(some)h(n)m(umerical) e(evidences)h(that)h(pro)m(vide)f(supp)s(ort)456 4568 y(for)j(the)g(conjectures)h(form)m(ulated)f(in)f(in)g(Section)h(3.)555 4676 y(Roughly)21 b(sp)s(eaking,)h(w)m(e)h(will)c(presen)m(t)i(n)m (umerical)g(evidence)g(v)m(erifying)f(the)i(scaling)456 4784 y(relations)j(predicted)h(b)m(y)g(the)h(conjectures.)40 b(W)-8 b(e)28 b(will)c(pro)m(vide)i(detailed)f(calculations)456 4892 y(of)37 b(scaling)g(relations)g(related)h(to)g(the)g(scaling)f (relations)f(depicted)h(in)g(Figure)g(3.2.)456 5000 y(Indeed,)24 b(w)m(e)g(will)d(see)j(that)g(there)g(are)g(quite)f(quan)m(titativ)m(e) h(predictions)e(in)g(Figure)h(3.2)456 5108 y({)e(or)g(in)f(the)h(mo)s (di\014cation)f(depicted)g(in)g(Figure)h(5.4.)38 b(W)-8 b(e)23 b(note)f(that)f(the)h(quan)m(titativ)m(e)456 5216 y(implications)33 b(will)h(b)s(e)h(v)m(eri\014ed)g(with)g(sev)m(eral)i (\014gures)e(of)i(precision)d(and)i(they)g(will)p eop %%Page: 20 20 20 19 bop 456 251 a Fu(20)837 b(R.)23 b(DE)g(LA)h(LLA)-8 b(VE)23 b(AND)g(A.)g(OL)-8 b(VERA)456 450 y Fx(agree)28 b(with)d(precision)g(in)g(sev)m(eral)i(mappings.)38 b(This)25 b(pro)m(vides)g(with)h(strong)h({)g(alb)s(eit)456 558 y(circumstan)m(tial)f({)i(evidence)f(of)h(the)f(truth)g(of)g(the)h (conjectures.)40 b(W)-8 b(e)29 b(think)d(that)i(the)456 666 y(v)m(eri\014cations)f(presen)m(ted)h(here,)g(will)d(pro)m(vide)i (motiv)-5 b(ation)27 b(for)h(a)g(computer)g(assisted)456 774 y(pro)s(of)h(that,)h(as)g(w)m(e)h(discussed)c(in)i(Remark)h(4.6)h (seems)f(within)d(striking)h(distance)h(of)456 882 y(mo)s(dern)g(tec)m (hnology)-8 b(.)456 1089 y(5.1.)46 b FA(Description)41 b(of)f(the)g(exp)s(erimen)m(ts.)k Fx(All)33 b(our)i(exp)s(erimen)m(ts)e (will)f(b)s(e)i(p)s(er-)456 1197 y(formed)23 b(on)g(maps)g(of)h(the)f (form)h(\(2\))g(where)f(the)h(map)f Fr(f)33 b Fx(will)21 b(dep)s(end)h(on)h(a)h(parameter)456 1305 y Fr(K)37 b Fx(in)29 b(the)h(form)g Fr(f)10 b Fx(\()p Fr(x)p Fx(\))25 b(=)g Fr(K)7 b(f)1474 1319 y Fu(1)1513 1305 y Fx(\()p Fr(x)p Fx(\).)555 1413 y(W)-8 b(e)45 b(will)40 b(consider)j(the)g (scaling)f(b)s(eha)m(vior)h(not)g(only)g(of)g(golden)g(mean)g(circles) 456 1521 y(but)e(also)g(of)h(other)g(n)m(um)m(b)s(ers)e(with)h(relativ) m(ely)g(simple)e({)k(in)d(particular)g(p)s(erio)s(dic)456 1629 y({)i(con)m(tin)m(ued)g(fraction)g(expansions)e(suc)m(h)i(as)h([2) p Fr(;)15 b Fx(2)p Fr(;)g Fx(2)p Fr(;)g(:)g(:)g(:)s Fx(],)46 b([1)p Fr(;)15 b Fx(2)p Fr(;)g Fx(1)p Fr(;)g Fx(2)p Fr(;)g(:)g(:)g(:)5 b Fx(].)76 b(These)456 1737 y(n)m(um)m(b)s(ers)29 b(will)e(b)s(e)j (constan)m(t)i(t)m(yp)s(e,)e(in)g(particular)e(Diophan)m(tine.)555 1845 y(Giv)m(en)i(one)g(of)g(these)g(n)m(um)m(b)s(ers)f Fr(!)s Fx(,)h(w)m(e)g(consider)f(the)h(sequence)g Fr(p)2866 1859 y Fo(i)2894 1845 y Fr(=q)2980 1859 y Fo(i)3038 1845 y Fx(of)g(appro)m(x-)456 1953 y(iman)m(ts)e({)g(expressed)g(in)f(lo)m (w)m(est)j(common)e(denominator)g({)h(and)e(obtain)h(a)h(sequence)456 2061 y Fr(x)508 2075 y Fo(i)566 2061 y Fx(of)i(for)f(h)m(yp)s(erb)s (olic)d(orbits)j(of)g(t)m(yp)s(e)h Fr(p)1866 2075 y Fo(i)1894 2061 y Fr(=q)1980 2075 y Fo(i)2038 2061 y Fx(along)g(the)f(critical)g (lines.)555 2168 y(Giv)m(en)f(orbits)f Fr(x)1126 2182 y Fo(i)1183 2168 y Fx(and)g Fr(x)1410 2182 y Fo(i)p Fu(+1)1557 2168 y Fx(w)m(e)h(obtain)f(segmen)m(ts)i(of)f(their)f(stable)g(and)h (unstable)456 2276 y(manifolds)f(\(See)j(Section)f(5.2.2\).)555 2384 y(If)g(these)h(manifolds)d(in)m(tersect)j(transv)m(ersally)f(in)f (the)i(sp)s(eci\014ed)d(segmen)m(t)k(w)m(e)f(can)456 2492 y(compute)k(e\016cien)m(tly)g(the)g(area)h(of)f(the)g(lob)s(e)f (enclosed)g(b)m(y)h(t)m(w)m(o)i(consecutiv)m(e)e(in)m(ter-)456 2600 y(sections)30 b(\(See)h(Section)f(5.2.3\).)555 2708 y(See)40 b(Figure)f(2.)69 b(Note)41 b(that,)i(b)m(y)d(the)f(preserv)-5 b(ation)39 b(of)h(area,)j(the)d(v)-5 b(alue)39 b(of)h(the)456 2816 y(area)i(of)h(the)f(lob)s(e)f(is)g(indep)s(enden)m(t)f(of)i(the)g (iterate)h(and)f(it)f(is)g(also)h(an)g(in)m(v)-5 b(arian)m(t)456 2924 y(under)27 b(symplectic)i(c)m(hanges)h(of)f(v)-5 b(ariables.)39 b(Moreo)m(v)m(er,)32 b(as)d(w)m(e)h(will)d(see)i(it)g (is)f(related)456 3032 y(to)40 b(Melnik)m(o)m(v)g(functions.)68 b(Hence,)43 b(the)d(area)h(of)f(the)g(lob)s(es)f(has)h(b)s(een)f(often) h(used)456 3140 y(as)g(a)h(go)s(o)s(d)f(measure)h(of)f(the)h(size)f(of) h(the)g(in)m(tersections)f(of)g(stable)h(and)f(unstable)456 3248 y(manifolds.)555 3356 y(W)-8 b(e,)40 b(th)m(us)c(obtain)g(a)g (function)f Fr(A)1751 3370 y Fo(i)1780 3356 y Fx(\()p Fr(K)7 b Fx(\))37 b(that)g(giv)m(es)f(the)h(v)-5 b(alue)36 b(of)g(the)h(area)g(as)g(a)456 3464 y(function)25 b(of)h(the)h (parameter)g(of)g(the)f(map.)39 b(The)26 b(smo)s(oth)h(dep)s(endence)e (of)i(manifolds)456 3572 y(on)33 b(parameters)g(implies)d(that)k(the)f (function)f(will)e(b)s(e)j(de\014ned)e(and)i(smo)s(oth)g(for)f(an)456 3679 y(op)s(en)f(set)h(of)g(parameters.)45 b(The)31 b(pap)s(er)g([VT91) q(])h(and)f([Aub92])h(sho)m(w)g(that)g(for)g(high)456 3787 y(enough)i Fr(K)42 b Fx(there)35 b(are)g(transv)m(erse)g(homo)s (clinic)e(in)m(tersections)h(and)g(that)i(therefore)456 3895 y(the)30 b(functions)f Fr(A)1072 3909 y Fo(i)1131 3895 y Fx(are)i(de\014ned)e({)h(at)i(least)e({)h(in)e(domains)g(of)i (the)f(form)g([)p Fr(K)3120 3862 y Fm(\003)3113 3921 y Fo(i)3159 3895 y Fr(;)15 b Fp(1)p Fx(\).)555 4003 y(More)25 b(generally)-8 b(,)24 b(w)m(e)g(note)h(that)f(the)g(set)g(of)f(maps)h (for)f(whic)m(h)f Fr(A)2761 4017 y Fo(i)2813 4003 y Fx(tak)m(es)j(an)f (sp)s(eci\014c)456 4111 y(v)-5 b(alue)29 b(con)m(tains)i(a)g(co)s (dimension)d(1)j(manifold)d(in)h(the)i(space)g(of)f(maps.)555 4219 y(W)-8 b(e)36 b(denote)f(b)m(y)g(\003)1207 4175 y Fo(\032)1207 4246 y(i)1282 4219 y Fx(the)f(manifold)f(in)g(the)i (space)g(of)g(maps)f(where)g(the)h(function)456 4327 y Fr(A)524 4341 y Fo(i)582 4327 y Fx(tak)m(es)d(the)f(v)-5 b(alue)30 b Fr(\032)p Fx(.)555 4435 y(Since)20 b(the)i(MacKa)m(y)g (renormalization)e(group)h(just)f(in)m(v)m(olv)m(es)h(c)m(hanges)h(of)g (v)-5 b(ariables)456 4543 y({)29 b(that)g(do)g(not)g(c)m(hange)h(the)f Fr(A)1532 4557 y Fo(i)1589 4543 y Fx({)g(and)g(a)g(scaling)f(that)h(c)m (hanges)h Fr(A)2808 4557 y Fo(i)2865 4543 y Fx(b)m(y)f(m)m(ultiplying) 456 4651 y(it)k(b)m(y)i(a)f(\014xed)g(factor)h Fr(\016)j Fx(\()d Fr(\016)g Fp(\031)d Fx(4)p Fr(:)p Fx(3391438)38 b(in)33 b(the)h(case)i(of)e(the)h(golden)f(mean.)52 b(See)456 4759 y(later)30 b(for)g(description)f(of)h(other)h(scaling)e(v)-5 b(alues.\))555 4867 y(Therefore,)25 b(the)e(renormalization)e(op)s (erator)j Fp(R)f Fx(has)f(the)h(prop)s(ert)m(y)g(that)g Fp(R)p Fx(\(\003)3272 4822 y Fo(\032)3272 4894 y(i)3313 4867 y Fx(\))j(=)456 4992 y(\003)519 4948 y Fo(\016)r(\032)519 5020 y(i)p Fm(\000)p Fu(1)637 4992 y Fx(.)555 5108 y(W)-8 b(e)40 b(note)g(that)f(if)f(Conjecture)h(3.3)g(holds,)h(the)f (manifolds)e(\003)2776 5063 y Fo(\032)2776 5135 y(i)2855 5108 y Fx(are)i(in)m(terlea)m(v)m(ed)456 5216 y(with)21 b(the)i(manifolds)e(\006)1275 5230 y Fo(i)1325 5216 y Fx(of)i(homo)s(clinic)e(tangencies.)39 b(See)23 b(Figure)f(5.4)i(for)f (a)g(pictorial)p eop %%Page: 21 21 21 20 bop 1213 251 a Fu(OBSTR)n(UCTION)30 b(AND)g(RENORMALIZA)-6 b(TION)687 b(21)456 453 y Fx(description)31 b(of)j(the)g(situation.)50 b(Hence,)35 b(b)m(y)f(the)g Fr(\025)p Fx(-lemma,)h(\003)2674 420 y Fo(A)2727 429 y Fs(0)2761 420 y Fo(\016)2794 397 y Fl(n)2674 475 y Fo(n)2872 453 y Fx(=)c Fp(R)3051 420 y Fm(\000)p Fo(n)3153 453 y Fx(\(\003)3251 412 y Fo(A)3304 421 y Fs(0)3251 479 y Fu(0)3374 453 y Fx(=)456 561 y Fp(R)533 528 y Fm(\000)p Fo(n)665 561 y Fx(will)d(accum)m(ulate)j(on)f (the)h(unstable)e(manifold)f(of)j(the)f(\014xed)g(p)s(oin)m(t.)555 669 y(Our)k(exp)s(erimen)m(t,)h(therefore)g(is)e(to)i(\014nd)e(the)i(n) m(um)m(b)s(ers)e Fr(K)2641 683 y Fo(i)2704 669 y Fx(for)i(whic)m(h)e (the)i(map)456 782 y Fr(T)509 796 y Fo(K)569 806 y Fl(i)629 782 y Fx(in)m(tersects)c(transv)m(ersally)e(the)i(manifold)d(\003)2162 740 y Fo(A)2215 750 y Fl(i)2162 809 y Fo(i)2276 782 y Fx(where)i Fr(A)2607 796 y Fo(i)2660 782 y Fx(=)25 b Fr(A)2824 796 y Fu(0)2864 782 y Fr(\016)2907 749 y Fo(i)2936 782 y Fx(.)555 890 y(If)g(the)h(scaling)e Fr(\016)30 b Fx(is)24 b(c)m(hosen)i(appropriately)e({)i(but)e(the)i(c)m(hoice)g (is)f(not)g(that)h(critical)456 998 y({)35 b(the)h Fr(K)775 1012 y Fo(i)839 998 y Fx(will)c(con)m(v)m(erge)38 b(to)e(the)f(v)-5 b(alue)35 b(of)h Fr(K)2095 1012 y Fo(c)2165 998 y Fx(for)f(whic)m(h)f (the)h(family)f Fr(T)3071 1012 y Fo(K)3175 998 y Fx(crosses)456 1105 y(the)c(stable)g(manifold)f(of)h(the)h(\014xed)e(p)s(oin)m(t.)555 1213 y(W)-8 b(e)32 b(note)f(that)g(one)f(of)h(the)g(predictions)d(of)i (Conjecture)h(3.3)g(is)f(that)1617 1378 y Fr(K)1694 1392 y Fo(i)1748 1378 y Fp(\031)25 b Fr(K)1921 1392 y Fo(c)1976 1378 y Fx(+)20 b Fr(D)s(\033)2200 1340 y Fm(\000)p Fo(i)456 1542 y Fx(where)29 b Fr(\033)f(>)d Fx(1)31 b(is)e(the)h(unstable)e (eigen)m(v)-5 b(alue)30 b(of)g(the)g(deriv)-5 b(ativ)m(e)30 b(of)g(the)g(renormaliza-)456 1650 y(tion)d(at)h(the)g(non-trivial)d (\014xed)i(p)s(oin)m(t.)39 b(In)27 b(particular,)g(the)h(rate)g(of)g (accum)m(ulation)f(is)456 1758 y(univ)m(ersal)32 b(and,)j(it)f(is)g (the)g(same)h(as)g(that)g(of)f(accum)m(ulation)h(of)f(the)h(parameters) g(at)456 1865 y(whic)m(h)27 b(the)h(p)s(erio)s(dic)d(orbits)j(lo)s(ose) g(stabilit)m(y)-8 b(.)39 b(The)27 b(n)m(um)m(b)s(er)g Fr(D)s Fx(,)i(of)f(course,)h(dep)s(ends)456 1973 y(on)h(the)g(family)-8 b(.)555 2081 y(This)38 b(will)f(b)s(e)i(the)g(main)g(goal)h(of)f(our)g (n)m(umerics.)67 b(T)-8 b(o)40 b(v)m(erify)f(the)g(exp)s(onen)m(tial) 456 2189 y(con)m(v)m(ergence)k(of)d(the)h Fr(K)1326 2203 y Fo(i)1355 2189 y Fx(,)i(and)d(the)h(univ)m(ersalit)m(y)e(of)i(the)g (rate.)72 b(W)-8 b(e)42 b(will)c(do)i(this)456 2297 y(v)m (eri\014cation)f(for)i(sev)m(eral)f(rotation)h(n)m(um)m(b)s(ers)e(with) g(easy)i(con)m(tin)m(ued)f(fraction)g(ex-)456 2405 y(pansions.)555 2513 y(W)-8 b(e)39 b(note)f(that)f(the)h(pap)s(ers)e([OS87a)q (],[OS87b])i(con)m(tain)f(direct)g(calculations)f(of)456 2621 y(the)i(parameter)g(where)f(the)i(tangen)m(t)g(in)m(tersections)e (happ)s(en.)62 b(In)37 b(this)g(pap)s(er,)i(w)m(e)456 2729 y(ha)m(v)m(e)29 b(c)m(hosen)f(to)g(do)g(calculations)f(for)g(a)h (\014xed)f(v)-5 b(alue)27 b(of)h(the)g(area)g(of)g(the)g(lob)s(e.)39 b(This)456 2837 y(seems)21 b(to)i(b)s(e)e(more)g(e\016cien)m(t)h(n)m (umerically)-8 b(.)36 b(In)21 b(general,)j(it)d(is)f(m)m(uc)m(h)i(more) g(e\016cien)m(t)g(to)456 2945 y(determine)h(the)i(place)g(where)f(a)h (function)e(reac)m(hes)j(a)f(regular)f(p)s(oin)m(t)f(than)i(the)f(p)s (oin)m(ts)456 3053 y(where)30 b(a)h(function)f(reac)m(hes)i(a)f (critical)f(p)s(oin)m(t.)42 b(Precisely)30 b(at)h(the)g(critical)f(p)s (oin)m(t)g(the)456 3161 y(v)-5 b(alues)29 b(of)i(the)f(functions)f(the) i(dep)s(endence)e(on)i(the)f(parameters)h(is)e(w)m(eak)m(est.)555 3269 y(F)-8 b(or)26 b(similar)c(reasons)j([Gre79)q(],)h(one)f(prefers)f (to)h(\014nd)f(the)h(v)-5 b(alues)24 b(of)g(parameter)i(for)456 3376 y(whic)m(h)j(the)h(p)s(erio)s(dic)e(orbits)h(ha)m(v)m(e)i(a)g (\014xed)f(residue)e(of)j(1)p Fr(=)p Fx(2)g(rather)g(than)f(the)g(v)-5 b(alues)456 3484 y(for)30 b(whic)m(h)f(the)h(in)m(tersection)h(is)e (zero.)555 3592 y(This)e(do)s(es)h(not)h(a\013ect)h(the)e(scaling)g (results)f(since)h(the)g(corresp)s(onding)f(manifolds)456 3700 y(in)m(terlea)m(v)m(e.)41 b(See)31 b(Figure)f(5.4.)456 3899 y(5.2.)46 b FA(Some)35 b(indication)h(on)f(the)g(algorithms)g (used)g(in)h(the)f(computations.)456 4007 y Fx(All)25 b(our)g(computations)i(ha)m(v)m(e)g(b)s(een)f(carried)f(out)i(in)e (generalized)h(standard)f(maps)h(of)456 4115 y(the)k(form)g(\(2\))q(.) 555 4223 y(Man)m(y)f(of)f(them)g(ha)m(v)m(e)i(b)s(een)d(done)h(in)f (the)h(standard)f(map)h(whic)m(h)f(corresp)s(onds)g(to)456 4429 y(\(9\))995 b Fr(f)10 b Fx(\()p Fr(x)p Fx(\))25 b(=)1882 4367 y Fr(K)p 1874 4408 101 4 v 1874 4491 a Fx(2)p Fr(\031)1999 4429 y Fx(sin)o(\(2)p Fr(\031)s(x)p Fx(\))555 4624 y(Nev)m(ertheless,)41 b(w)m(e)d(note)g(that)h(the)f (function)e(\(9\))j(is)e(v)m(ery)h(sp)s(ecial.)62 b(In)37 b(particu-)456 4732 y(lar,)d(it)g(con)m(tains)g(only)f(one)i(harmonic.) 51 b(Since)33 b(one)h(of)h(the)f(imp)s(ortan)m(t)f(predictions)456 4839 y(of)39 b(renormalization)e(group)i(is)f(the)h(univ)m(ersalit)m(y) e(of)i(the)g(results,)h(to)g(con\014rm)e(the)456 4947 y(univ)m(ersalit)m(y)28 b(w)m(e)j(ha)m(v)m(e)h(also)e(used)456 5153 y(\(10\))603 b Fr(f)10 b Fx(\()p Fr(x)p Fx(\))25 b(=)1536 5092 y Fr(K)p 1527 5132 V 1527 5216 a Fx(2)p Fr(\031)1653 5153 y Fx(\(sin)o(\(2)p Fr(\031)s(x)p Fx(\))c(+)f(0)p Fr(:)p Fx(05)15 b(sin\(6)p Fr(\031)s(x)p Fx(\)\))p eop %%Page: 22 22 22 21 bop 456 251 a Fu(22)837 b(R.)23 b(DE)g(LA)h(LLA)-8 b(VE)23 b(AND)g(A.)g(OL)-8 b(VERA)456 450 y FA(Remark)24 b(5.1.)35 b Fx(It)22 b(w)m(ould)f(certainly)f(seem)j(desirable)c(to)k (extend)f(the)g(calculations)f(to)456 558 y(functions)29 b(with)h(more)h(harmonics)f(or)h(so)g(that)h(the)f(di\013eren)m(t)g (harmonics)f(are)h(more)456 666 y(dominan)m(t.)65 b(One)39 b(imp)s(ortan)m(t)f(question)g(that)h(remains)f(unsettled)g(is)g(to)i (obtain)e(a)456 774 y(description)30 b(of)j(the)g(domain)f(of)h(univ)m (ersalit)m(y)e(describ)s(ed)g(b)m(y)h(the)h(non-trivial)e(\014xed)456 882 y(p)s(oin)m(t.)56 b(Ev)m(en)36 b(if)f(for)g(some)i(one-dimensional) c(renormalizations)i(there)h(are)g(global)456 990 y(theories)e(for)g (circle)f(maps)h(and)f(for)h(unimo)s(dal)e(maps,)j(there)f(are)h (indications)d(that)456 1098 y(the)44 b(renormalization)g(group)g (description)e(for)i(circle)g(maps)g(is)g(only)g(lo)s(cal.)82 b(See)456 1206 y([FdlL92a])31 b(for)f(some)h(evidence)f(and)g(for)g (references)g(to)h(previous)e(literature.)456 1390 y(5.2.1.)47 b Ft(Calculation)29 b(of)f(p)-5 b(erio)g(dic)29 b(orbits.)46 b Fx(F)-8 b(or)26 b(a)f(\014xed)f(v)-5 b(alue)24 b(of)h Fr(K)7 b Fx(,)26 b(w)m(e)f(compute)g(the)456 1498 y(h)m(yp)s(erb)s (olic)19 b(p)s(erio)s(dic)h(orbits)i(of)g(an)g(sp)s(eci\014ed)f(t)m(yp) s(e)i(b)m(y)f(using)f(the)i(metho)s(d)e(describ)s(ed)456 1606 y(in)29 b([Gre79)q(].)555 1714 y(W)-8 b(e)29 b(recall)d(that)i (maps)f(of)g(the)h(form)f(\(2\))h(are)g(rev)m(ersible)d(\(see)k(2.2\).) 41 b(That)27 b(is,)g(they)456 1822 y(can)f(b)s(e)f(written)g(as)h(the)g (pro)s(duct)f(of)h(t)m(w)m(o)h(in)m(v)m(olutions.)38 b Fr(F)g Fx(=)25 b Fr(T)f Fp(\016)11 b Fr(S)31 b Fx(with)25 b Fr(S)16 b Fp(\016)11 b Fr(S)31 b Fx(=)25 b(Id)o(,)456 1930 y Fr(T)33 b Fp(\016)20 b Fr(T)38 b Fx(=)25 b(Id.)456 2256 y(\(11\))398 b Fr(S)30 b Fx(:)1151 2128 y Fn(\032)1271 2201 y Fx(\026)-51 b Fr(y)85 b Fx(=)25 b Fr(y)e Fp(\000)d Fr(f)10 b Fx(\()p Fr(x)p Fx(\))1266 2309 y(\026)-51 b Fr(x)83 b Fx(=)25 b Fp(\000)p Fr(x)2071 2256 y(T)38 b Fx(:)2212 2128 y Fn(\032)2332 2201 y Fx(\026)-51 b Fr(y)86 b Fx(=)25 b Fr(y)2328 2309 y Fx(\026)-51 b Fr(x)83 b Fx(=)25 b Fp(\000)p Fr(x)20 b Fx(+)g Fr(y)555 2491 y Fx(The)g(maps)g Fr(S)25 b Fx(and)20 b Fr(R)h Fx(ha)m(v)m(e)h(an)e(in)m (v)-5 b(arian)m(t)20 b(sets)h(whic)m(h)e(corresp)s(ond)g(to)i(four)f (straigh)m(t)456 2599 y(lines)34 b Fr(x)h Fx(=)f(0,)39 b Fr(x)c Fx(=)f(1)p Fr(=)p Fx(2,)39 b Fr(x)c Fx(=)g Fr(y)s(=)p Fx(2)i(and)e Fr(x)g Fx(=)g(\()p Fr(y)27 b Fx(+)d(1\))p Fr(=)p Fx(2,)39 b(they)d(are)h(called)e(critical)456 2707 y(lines.)i(It)27 b(is)f(w)m(ell)f(kno)m(wn)h(that)i(an)m(y)f(rev)m (ersible)e(map)h Fr(F)40 b Fx(has)26 b(a)h(set)h(of)e(p)s(erio)s(dic)e (orbits)456 2815 y(of)34 b(t)m(yp)s(e)g Fr(p=q)j Fx(suc)m(h)c(that)i(t) m(w)m(o)g(p)s(oin)m(ts)e(of)h(their)f(orbits)g(b)s(elong)g(to)i(the)f (critical)f(lines)456 2923 y([Gre79)q(].)60 b(This)35 b(prop)s(ert)m(y)h(allo)m(w)g(us)g(to)h(searc)m(h)g(p)s(erio)s(dic)d (orbits)i(only)g(in)f(the)i(four)456 3031 y(critical)31 b(lines.)44 b(Therefore)31 b(w)m(e)i(use)f(a)g(one)h(dimensional)c(ro)s (ot)j(\014nder)f(to)i(obtain)e(the)456 3139 y(p)s(erio)s(dic)22 b(orbits,)j(in)e(that)j(w)m(a)m(y)g(w)m(e)f(can)g(\014nd)e(this)h(kind) f(of)i(orbit)f(with)g(high)f(accuracy)456 3247 y(and)29 b(for)i(a)f(large)h(p)s(erio)s(d)d(of)i(the)h(orbit.)456 3489 y(5.2.2.)47 b Ft(L)-5 b(o)g(c)g(al)38 b(stable)f(and)g(unstable)g (manifolds.)47 b Fx(An)m(y)34 b(h)m(yp)s(erb)s(olic)e(p)s(oin)m(t)h(of) h(a)h(t)m(w)m(o-)456 3597 y(dimensional)d(area)k(preserving)d(map)i (has)f(t)m(w)m(o)j(one)e(dimensional)d(in)m(v)-5 b(arian)m(t)34 b(mani-)456 3705 y(folds,)29 b(one)i(stable)f(and)g(one)g(unstable.)555 3812 y(A)22 b(\014rst)f(linear)f(appro)m(ximation)h(of)h(these)g (manifolds)e(is)h(giv)m(en)h(b)m(y)f(the)h(eigen)m(v)m(ectors)456 3920 y(of)39 b(the)h(linear)e(map.)68 b(A)39 b(p)s(ossible)e(algorithm) i(is)g(to)h(select)g(a)g(segmen)m(t)h(along)e(the)456 4028 y(eigendirections)e(and)h(then)g(iterate)h(it.)65 b(This)36 b(metho)s(d)i(is)g(v)m(ery)h(inaccurate)f(when)456 4136 y(the)32 b(eigen)m(v)-5 b(alues)32 b(of)h(the)g(h)m(yp)s(erb)s (olic)c(p)s(oin)m(ts)i(are)i(close)g(to)g(1,)h(whic)m(h)d(is)g(the)i (case)g(of)456 4244 y(most)d(in)m(terest)h(to)g(us.)555 4352 y(Hence,)e(the)e(metho)s(d)f(that)h(w)m(e)g(will)d(dev)m(elop)j (is)e(to)j(compute)f(v)m(ery)g(accurate)h(lo)s(cal)456 4460 y(appro)m(ximations)33 b(of)j(the)f(stable)g(and)f(unstable)g (manifolds)f(b)m(y)i(computing)f(a)i(high)456 4568 y(order)19 b(p)s(olynomial)f(appro)m(ximation.)36 b(This)19 b(will)e(giv)m(e)k (enough)f(accuracy)i(so)e(that)h(only)456 4676 y(a)g(few)g(iterations)g (are)h(needed)f(to)h(reac)m(h)g(the)f(desired)f(places)h(where)g(the)h (in)m(tersections)456 4784 y(tak)m(e)32 b(place.)555 4892 y(The)20 b(standard)f(theory)i(of)f(in)m(v)-5 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5263 l 5350 5257 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P10 [8 0 0 -16 322.00 344.40] PATmp PATsp ef gr PATusp % Polyline 45.000 slw n 900 7800 m 901 7799 l 902 7798 l 904 7796 l 908 7793 l 913 7787 l 920 7781 l 929 7772 l 941 7761 l 954 7748 l 970 7733 l 988 7716 l 1008 7697 l 1031 7676 l 1055 7653 l 1082 7629 l 1110 7603 l 1140 7575 l 1172 7547 l 1205 7517 l 1240 7486 l 1276 7454 l 1313 7421 l 1352 7388 l 1392 7353 l 1434 7318 l 1477 7282 l 1522 7245 l 1568 7208 l 1616 7169 l 1667 7129 l 1719 7088 l 1775 7045 l 1833 7001 l 1893 6956 l 1957 6909 l 2024 6861 l 2094 6811 l 2166 6759 l 2242 6707 l 2320 6654 l 2400 6600 l 2474 6551 l 2546 6504 l 2618 6457 l 2687 6413 l 2752 6371 l 2814 6331 l 2872 6295 l 2925 6260 l 2975 6229 l 3019 6200 l 3059 6174 l 3095 6151 l 3127 6130 l 3155 6111 l 3180 6094 l 3202 6079 l 3221 6065 l 3238 6053 l 3253 6042 l 3266 6032 l 3278 6023 l 3289 6015 l 3300 6006 l 3311 5998 l 3322 5990 l 3335 5981 l 3348 5972 l 3364 5962 l 3381 5952 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4906 5405 l 4945 5411 l 4979 5416 l 5011 5420 l 5041 5425 l 5068 5428 l 5093 5432 l 5116 5435 l 5139 5438 l 5160 5441 l 5181 5444 l 5202 5446 l 5224 5449 l 5246 5452 l 5270 5454 l 5295 5457 l 5321 5460 l 5351 5463 l 5382 5467 l 5417 5470 l 5455 5474 l 5496 5479 l 5542 5483 l 5591 5488 l 5644 5493 l 5702 5498 l 5764 5504 l 5830 5510 l 5900 5516 l 5974 5522 l 6050 5528 l 6130 5534 l 6211 5540 l 6293 5545 l 6375 5550 l 6476 5555 l 6573 5560 l 6663 5564 l 6747 5567 l 6823 5569 l 6892 5571 l 6952 5573 l 7004 5574 l 7048 5575 l 7085 5576 l 7116 5577 l 7141 5578 l 7160 5579 l 7175 5579 l 7186 5580 l 7194 5580 l 7199 5581 l 7203 5581 l 7206 5581 l 7209 5581 l 7212 5581 l 7217 5581 l 7223 5580 l 7232 5579 l 7244 5577 l 7260 5575 l 7281 5572 l 7306 5569 l 7337 5564 l 7374 5559 l 7418 5553 l 7467 5546 l 7523 5537 l 7584 5527 l 7652 5516 l 7723 5504 l 7798 5490 l 7875 5475 l 7952 5459 l 8027 5441 l 8100 5423 l 8171 5405 l 8239 5386 l 8305 5368 l 8367 5349 l 8426 5330 l 8483 5311 l 8538 5292 l 8591 5273 l 8641 5254 l 8690 5235 l 8737 5216 l 8782 5198 l 8827 5179 l 8869 5160 l 8911 5142 l 8951 5123 l 8989 5106 l 9026 5088 l 9061 5071 l 9095 5055 l 9126 5040 l 9155 5025 l 9181 5012 l 9205 5000 l 9226 4989 l 9244 4979 l 9259 4971 l 9272 4965 l 9282 4960 l 9289 4956 l 9294 4953 l 9298 4951 l 9299 4950 l 9300 4950 l gs col0 s gr [] 0 sd /Times-Roman ff 450.00 scf sf 1050 4875 m gs 1 -1 sc (x) col0 sh gr /Times-Roman ff 270.00 scf sf 1275 5025 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 450.00 scf sf 1275 7875 m gs 1 -1 sc (x) col0 sh gr /Times-Roman ff 450.00 scf sf 2625 6975 m gs 1 -1 sc (W \(x \)) col0 sh gr /Times-Roman ff 270.00 scf sf 1500 8025 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 210.00 scf sf 4500 8055 m gs 1 -1 sc ( ) col0 sh gr /Times-Roman ff 210.00 scf sf 2925 7125 m gs 1 -1 sc (loc) col0 sh gr /Times-Roman ff 210.00 scf sf 3075 6675 m gs 1 -1 sc (u) col0 sh gr /Times-Roman ff 210.00 scf sf 3525 7050 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 450.00 scf sf 1950 4425 m gs 1 -1 sc (W \(x \)) col0 sh gr /Times-Roman ff 210.00 scf sf 2250 4575 m gs 1 -1 sc (loc) col0 sh gr /Times-Roman ff 210.00 scf sf 2850 4575 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 210.00 scf sf 2400 4200 m gs 1 -1 sc (s) col0 sh gr $F2psEnd rs end showpage %%EndDocument @endspecial 2877 1586 a currentpoint grestore moveto 2877 1586 a 724 1794 a Fw(Figure)34 b(2.)46 b Fx(Hetero)s(clinic)29 b(in)m(tersection)h(of)h(t)m(w)m(o)g(h) m(yp)s(erb)s(olic)d(p)s(oin)m(ts.)555 2023 y(W)-8 b(e)33 b(will)28 b(tak)m(e)33 b(p)s(olynomial)28 b(appro)m(ximations)i Fr(y)s Fx(\()p Fr(x)p Fx(\))d(=)2490 1955 y Fn(P)2586 1981 y Fo(N)2586 2050 y(i)p Fu(=1)2720 2023 y Fr(a)2768 2037 y Fo(i)2796 2023 y Fr(x)2848 1990 y Fo(i)2907 2023 y Fx(where)k(the)g(co-)456 2131 y(e\016cien)m(ts)i Fr(a)878 2145 y Fo(i)940 2131 y Fx(are)g(selected)h(so)f(that)h(the)g(graph)e (of)i(the)f(function)f Fr(y)s Fx(\()p Fr(x)p Fx(\))i(is)e(in)m(v)-5 b(arian)m(t)456 2241 y(up)26 b(to)i(order)e Fr(x)978 2208 y Fo(N)7 b Fu(+1)1136 2241 y Fx(.)39 b(It)28 b(is)e(p)s(ossible)f (to)j(see)f(that)h(the)g(p)s(olynomials)c(obtained)j(in)e(this)456 2349 y(w)m(a)m(y)37 b(will)c(b)s(e)j(the)g(T)-8 b(a)m(ylor)37 b(p)s(olynomials)c(of)k(the)f(function)f(whose)h(graph)g(giv)m(es)g (the)456 2457 y(in)m(v)-5 b(arian)m(t)29 b(manifold.)38 b(Hence,)32 b(as)e Fr(N)40 b Fx(gro)m(ws,)31 b(the)f(p)s(olynomial)e (ab)s(o)m(v)m(e)j(will)c(con)m(v)m(erge)456 2564 y(in)32 b(a)j(neigh)m(b)s(orho)s(o)s(d)c({)k(whic)m(h)d(in)h(practice)h(is)f (rather)h(large)g({.)52 b(Hence,)36 b(the)e(pro)s(ce-)456 2672 y(dure)23 b(will)g(pro)m(vide)g(with)h(an)h(extremely)g(accurate)h (appro)m(ximation)e(to)h(the)g(in)m(v)-5 b(arian)m(t)456 2780 y(manifolds.)555 2888 y(Let)41 b Fr(F)799 2855 y Fo(q)878 2888 y Fx(b)s(e)f(the)h Fr(q)s Fx(-th)g(iteration)f(of)h(our)f (map,)j(where)d Fr(q)j Fx(is)d(the)h(p)s(erio)s(d)d(of)j(the)456 2996 y(h)m(yp)s(erb)s(olic)27 b(p)s(erio)s(dic)h(p)s(oin)m(t,)i(w)m(e)h (can)f(represen)m(t)h(the)f(map)g(in)f(this)h(form,)1281 3142 y Fr(F)1352 3105 y Fo(q)1390 3142 y Fx(\()p Fr(x;)15 b(y)s Fx(\))26 b(=)f(\()p Fr(G)1828 3156 y Fo(q)1867 3142 y Fx(\()p Fr(x;)15 b(y)s Fx(\))p Fr(;)g(H)2193 3156 y Fo(q)2232 3142 y Fx(\()p Fr(x;)g(y)s Fx(\)\))117 b Fr(:)456 3288 y Fx(where)35 b Fr(G)795 3302 y Fo(q)834 3288 y Fx(\()p Fr(x;)15 b(y)s Fx(\))37 b(and)f Fr(H)1340 3302 y Fo(q)1377 3288 y Fx(\()p Fr(x;)15 b(y)s Fx(\))38 b(are)e(p)s(olynomials)e(of)i(order)g Fr(N)10 b Fx(.)59 b(These)36 b(functions)456 3396 y(are)30 b(obtained)g(using)f(the)i (recursion:)1276 3535 y Fr(G)1347 3549 y Fo(i)p Fu(+1)1491 3535 y Fx(=)83 b Fr(G)1716 3549 y Fu(0)1771 3535 y Fx(\()q Fr(G)1878 3549 y Fo(i)1906 3535 y Fx(\()p Fr(x;)15 b(y)s Fx(\))p Fr(;)g(H)2232 3549 y Fo(i)2261 3535 y Fx(\()p Fr(x;)g(y)s Fx(\)\))92 b Fr(;)1272 3643 y(H)1348 3657 y Fo(i)p Fu(+1)1491 3643 y Fx(=)83 b Fr(H)1721 3657 y Fu(0)1775 3643 y Fx(\()q Fr(G)1882 3657 y Fo(i)1910 3643 y Fx(\()p Fr(x;)15 b(y)s Fx(\))p 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Level 2 Emulation) %%Version: 1.2 0 %%CreationDate: (04/10/93) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) userdict /Adobe_level2_AI5 26 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { (AI8_CMYK_CustomColor) 6 packedarray } bind def /findrgbcustomcolor { (AI8_RGB_CustomColor) 5 packedarray } bind def /setcustomcolor { exch aload pop dup (AI8_CMYK_CustomColor) eq { pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } { dup (AI8_RGB_CustomColor) eq { pop pop 3 { 1 exch sub 3 index mul 1 exch sub 3 1 roll } repeat 4 -1 roll pop setrgbcolor } { pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } ifelse } ifelse } def } if /setAIseparationgray { false setoverprint 0 setgray /setseparationgray where{ pop setseparationgray }{ /setcolorspace where{ pop [/Separation (All) /DeviceCMYK {dup dup dup}] setcolorspace 1 exch sub setcolor }{ setgray }ifelse }ifelse } def /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking %%EndResource %%BeginResource: procset Adobe_typography_AI5 1.0 1 %%Title: (Typography Operators) %%Version: 1.0 1 %%CreationDate:(6/10/1996) () %%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_typography_AI5 68 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /havefont { systemdict /languagelevel known { /Font resourcestatus dup { exch pop exch pop } if } { systemdict /FontDirectory get 1 index known { pop true } { systemdict /fileposition known { dup length 6 add exch Ss 6 250 getinterval cvs pop Ss exch 0 exch getinterval status { pop pop pop pop true } { false } ifelse } { pop false } ifelse } ifelse } ifelse } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def /subststring { exch 2 index exch search { exch pop exch dup () eq { pop exch concatstring } { 3 -1 roll exch concatstring concatstring } ifelse exch pop true } { pop pop false } ifelse } def /concatstring { 1 index length 1 index length 1 index add string dup 0 5 index putinterval dup 2 index 4 index putinterval 4 1 roll pop pop pop } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def 2 index havefont { 3 index 255 string cvs dup (_Symbol_) eq { pop 2 index findfont } { 1 index 0 eq { dup length 1 sub 1 exch getinterval cvn findfont } { pop 2 index findfont } ifelse } ifelse } { dup 1 eq { 2 index 64 string cvs dup (-90pv-RKSJ-) (-83pv-RKSJ-) subststring { exch pop dup havefont { findfont false } { pop true } ifelse } { pop dup (-90ms-RKSJ-) (-Ext-RKSJ-) subststring { exch pop dup havefont { findfont false } { pop true } ifelse } { pop pop true } ifelse } ifelse { 1 index 1 eq { /Ryumin-Light-Ext-RKSJ-V havefont {/Ryumin-Light-Ext-RKSJ-V} {/Courier} ifelse } { /Ryumin-Light-83pv-RKSJ-H havefont {/Ryumin-Light-83pv-RKSJ-H} {/Courier} ifelse } ifelse findfont [1 0 0.5 1 0 0] makefont } if } { /Courier findfont } ifelse } ifelse _wv type /arraytype eq { _wv makeblendedfont } if dup length 10 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontScript exch def /FontDirection exch def /FontRequest exch def /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { W B } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { W F } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { W S } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat _shift aload pop _lineorientation 1 eq { exch } if translate _scale aload pop _lineorientation 1 eq _yokoorientation 1 eq or { exch } if scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { 1 index type /nametype eq { dup 0.75 mul 1 index 0.25 mul neg } if /_fontDescent exch ddef /_fontAscent exch ddef /_fontSize exch ddef /_fontRotateAdjust _fontAscent _fontDescent add 2 div neg ddef /_fontHeight _fontSize ddef findfont _fontSize scalefont setfont } def /Tl { pop neg 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { 0 exch _shift astore pop currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { count 1 eq { 100 } if 100 div exch 100 div exch _scale astore pop iTm } def /TA { pop } def /Tq { pop } def /Tg { pop } def /TG { pop } def /Tv { /_lineorientation exch ddef } def /TV { /_charorientation exch ddef } def /Ty { dup /_yokoorientation exch ddef 1 sub neg Tv } def /TY { pop } def /T~ { Tx } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { _fontSize mul 1000 div _lineorientation 0 eq { neg 0 } { 0 exch } ifelse rmoveto pop } def /TK { 2 npop } def /T* { _leading aload pop _lineorientation 0 ne { exch } if Td } def /T*- { _leading aload pop _lineorientation 0 ne { exch } if exch neg exch neg Td } def /T- { _ax neg 0 rmoveto _lineorientation 1 eq _charorientation 0 eq and { 1 TV _hyphen Tx 0 TV } { _hyphen Tx } ifelse } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ findfont _fontSize scalefont setfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking % /X^ { currentfont 5 1 roll dup havefont { findfont _fontSize scalefont setfont } { pop exch } ifelse 2 index 0 eq { Tx } { Tj } ifelse pop pop setfont } def /T^ /X^ load def %%EndResource %%BeginProcSet: Adobe_ColorImage_AI6 1.3 0 userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 53 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 41 dict def } if Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /_newproc null def /_proc1 null def /_proc2 null def /sourcearray 4 array def /_ptispace null def /_ptiname null def /_pti0 0 def /_pti1 0 def /_ptiproc null def /_ptiscale 0 def /_pticomps 0 def /_ptibuf 0 string def /_gtigray 0 def /_cticmyk null def /_rtirgb null def /XIEnable true def /XIType 0 def /XIEncoding 0 def /XICompression 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIRowBytes 0 def /XIFile null def /XIBuffer1 null def /XIBuffer2 null def /XIBuffer3 null def /XIDataProc null def /XIColorSpace /DeviceGray def /XIColorValues 0 def /XIPlateList false def end /ci6colorimage /colorimage where {/colorimage get}{null} ifelse def /ci6image systemdict /image get def /ci6curtransfer systemdict /currenttransfer get def /ci6curoverprint /currentoverprint where {/currentoverprint get}{{_of}} ifelse def /ci6foureq { 4 index ne { pop pop pop false }{ 4 index ne { pop pop false }{ 4 index ne { pop false }{ 4 index eq } ifelse } ifelse } ifelse } def /ci6testplate { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 ci6foureq { /plateindex 0 def }{ 0 1 0 0 ci6foureq { /plateindex 1 def }{ 0 0 1 0 ci6foureq { /plateindex 2 def }{ 0 0 0 1 ci6foureq { /plateindex 3 def }{ 0 0 0 0 ci6foureq { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /ci6concatprocs { /packedarray where { pop dup type /packedarraytype eq 2 index type /packedarraytype eq or }{ false } ifelse { /_proc2 exch cvlit def /_proc1 exch cvlit def _proc1 aload pop _proc2 aload pop _proc1 length _proc2 length add packedarray cvx }{ /_proc2 exch cvlit def /_proc1 exch cvlit def /_newproc _proc1 length _proc2 length add array def _newproc 0 _proc1 putinterval _newproc _proc1 length _proc2 putinterval _newproc cvx } ifelse } def /ci6istint { type /arraytype eq } def /ci6isspot { dup type /arraytype eq { dup length 1 sub get /Separation eq }{ pop false } ifelse } def /ci6spotname { dup ci6isspot {dup length 2 sub get}{pop ()} ifelse } def /ci6altspace { aload pop pop pop ci6colormake } def /ci6numcomps { dup /DeviceGray eq { pop 1 }{ dup /DeviceRGB eq { pop 3 }{ /DeviceCMYK eq { 4 }{ 1 } ifelse } ifelse } ifelse } def /ci6marksplate { dup /DeviceGray eq { pop plateindex 3 eq }{ dup /DeviceRGB eq { pop plateindex 5 ne }{ dup /DeviceCMYK eq { pop plateindex 5 ne }{ dup ci6isspot { /findcmykcustomcolor where { pop dup length 2 sub get 0.1 0.1 0.1 0.1 5 -1 roll findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop plateindex 5 ne } ifelse }{ pop plateindex 5 ne } ifelse } ifelse } ifelse } ifelse } def /ci6colormake { dup ci6numcomps exch 1 index 2 add 1 roll dup 1 eq {pop}{array astore} ifelse exch } def /ci6colorexpand { dup ci6spotname exch dup ci6istint { ci6altspace exch 4 1 roll }{ 1 3 1 roll } ifelse } def /ci6colortint { dup /DeviceGray eq { 3 1 roll 1 exch sub mul 1 exch sub exch }{ dup /DeviceRGB eq { 3 1 roll {1 exch sub 1 index mul 1 exch sub exch} forall pop 3 array astore exch }{ dup /DeviceCMYK eq { 3 1 roll {1 index mul exch} forall pop 4 array astore exch }{ 3 1 roll mul exch } ifelse } ifelse } ifelse } def /ci6colortocmyk { dup /DeviceGray eq { pop 1 exch sub 0 0 0 4 -1 roll 4 array astore }{ dup /DeviceRGB eq { pop aload pop _rgbtocmyk 4 array astore }{ dup /DeviceCMYK eq { pop }{ ci6altspace ci6colortint ci6colortocmyk } ifelse } ifelse } ifelse } def /ci6makeimagedict { 7 dict begin /ImageType 1 def /Decode exch def /DataSource exch def /ImageMatrix exch def /BitsPerComponent exch def /Height exch def /Width exch def currentdict end } def /ci6stringinvert { 0 1 2 index length 1 sub { dup 2 index exch get 255 exch sub 2 index 3 1 roll put } for } def /ci6stringknockout { 0 1 2 index length 1 sub { 255 2 index 3 1 roll put } for } def /ci6stringapply { 0 1 4 index length 1 sub { dup 4 index exch get 3 index 3 1 roll 3 index exec } for pop exch pop } def /ci6walkrgbstring { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval {} forall 5 index exec 3 index } for 5 {pop} repeat } def /ci6walkcmykstring { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval {} forall 6 index exec 3 index } for 5 { pop } repeat } def /ci6putrgbtograystr { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /ci6putcmyktograystr { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /ci6rgbtograyproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 dup 3 1 roll /ci6putrgbtograystr load exch ci6walkrgbstring end } def /ci6cmyktograyproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 dup 3 1 roll /ci6putcmyktograystr load exch ci6walkcmykstring end } def /ci6separatecmykproc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec XIBuffer3 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop end } def /ci6compositeimage { dup 1 eq { pop pop image }{ /ci6colorimage load null ne { ci6colorimage }{ 3 1 roll pop sourcearray 0 3 -1 roll put 3 eq {/ci6rgbtograyproc}{/ci6cmyktograyproc} ifelse load image } ifelse } ifelse } def /ci6knockoutimage { gsave 0 ci6curtransfer exec 1 ci6curtransfer exec eq { 0 ci6curtransfer exec 0.5 lt }{ 0 ci6curtransfer exec 1 ci6curtransfer exec gt } ifelse {{pop 0}}{{pop 1}} ifelse systemdict /settransfer get exec ci6compositeimage grestore } def /ci6drawimage { ci6testplate -1 eq { pop ci6compositeimage }{ dup type /arraytype eq { dup length plateindex gt {plateindex get}{pop false} ifelse }{ { true }{ dup 1 eq {plateindex 3 eq}{plateindex 3 le} ifelse } ifelse } ifelse { dup 1 eq { pop pop ci6image }{ dup 3 eq { ci6compositeimage }{ pop pop sourcearray 0 3 -1 roll put /ci6separatecmykproc load ci6image } ifelse } ifelse }{ ci6curoverprint { 7 {pop} repeat }{ ci6knockoutimage } ifelse } ifelse } ifelse } def /ci6proctintimage { /_ptispace exch store /_ptiname exch store /_pti1 exch store /_pti0 exch store /_ptiproc exch store /_pticomps _ptispace ci6numcomps store /_ptiscale _pti1 _pti0 sub store level2? { _ptiname length 0 gt version cvr 2012 ge and { [/Separation _ptiname _ptispace {_ptiproc}] setcolorspace [_pti0 _pti1] ci6makeimagedict ci6image }{ [/Indexed _ptispace 255 {255 div _ptiscale mul _pti0 add _ptiproc}] setcolorspace [0 255] ci6makeimagedict ci6image } ifelse }{ _pticomps 1 eq { { dup { 255 div _ptiscale mul _pti0 add _ptiproc 255 mul cvi put } ci6stringapply } ci6concatprocs ci6image }{ { dup length _pticomps mul dup _ptibuf length ne {/_ptibuf exch string store}{pop} ifelse _ptibuf { exch _pticomps mul exch 255 div _ptiscale mul _pti0 add _ptiproc _pticomps 2 add -2 roll _pticomps 1 sub -1 0 { 1 index add 2 index exch 5 -1 roll 255 mul cvi put } for pop pop } ci6stringapply } ci6concatprocs false _pticomps /ci6colorimage load null eq {7 {pop} repeat}{ci6colorimage} ifelse } ifelse } ifelse } def /ci6graytintimage { /_gtigray 5 -1 roll store {1 _gtigray sub mul 1 exch sub} 4 1 roll /DeviceGray ci6proctintimage } def /ci6cmyktintimage { /_cticmyk 5 -1 roll store {_cticmyk {1 index mul exch} forall pop} 4 1 roll /DeviceCMYK ci6proctintimage } def /ci6rgbtintimage { /_rtirgb 5 -1 roll store {_rtirgb {1 exch sub 1 index mul 1 exch sub exch} forall pop} 4 1 roll /DeviceRGB ci6proctintimage } def /ci6tintimage { ci6testplate -1 eq { ci6colorexpand 3 -1 roll 5 -1 roll {0}{0 exch} ifelse 4 2 roll dup /DeviceGray eq { pop ci6graytintimage }{ dup /DeviceRGB eq { pop ci6rgbtintimage }{ pop ci6cmyktintimage } ifelse } ifelse }{ dup ci6marksplate { plateindex 5 lt { ci6colortocmyk plateindex get dup 0 eq ci6curoverprint and { 7 {pop} repeat }{ 1 exch sub exch {1 0}{0 1} ifelse () ci6graytintimage } ifelse }{ pop exch {0}{0 exch} ifelse 0 3 1 roll () ci6graytintimage } ifelse }{ ci6curoverprint { 8 {pop} repeat }{ pop pop pop {pop 1} 0 1 () /DeviceGray ci6proctintimage } ifelse } ifelse } ifelse } def /XINullImage { } def /XIImageMask { XIImageWidth XIImageHeight false [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load imagemask } def /XIImageTint { XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load XIType 3 eq XIColorValues XIColorSpace ci6tintimage } def /XIImage { XIImageWidth XIImageHeight XIBitsPerPixel [XIImageWidth 0 0 XIImageHeight neg 0 0] /XIDataProc load false XIChannelCount XIPlateList ci6drawimage } def /XG { pop pop } def /XF { 13 {pop} repeat } def /Xh { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def /XIImageHeight exch def /XIImageWidth exch def /XIImageMatrix exch def 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale /_lp /null ddef _fc /_lp /imagemask ddef end } def /XH { Adobe_ColorImage_AI6_Vars begin grestore end } def /XIEnable { Adobe_ColorImage_AI6_Vars /XIEnable 3 -1 roll put } def /XC { Adobe_ColorImage_AI6_Vars begin ci6colormake /XIColorSpace exch def /XIColorValues exch def end } def /XIPlates { Adobe_ColorImage_AI6_Vars begin /XIPlateList exch def end } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIType exch def cvi dup 256 idiv /XICompression exch store 256 mod /XIEncoding exch store pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi }{ XIImageWidth XIChannelCount mul } ifelse /XIRowBytes exch def XIEnable { /XIBuffer3 XIImageWidth string def XICompression 0 eq { /XIBuffer1 XIRowBytes string def XIEncoding 0 eq { {currentfile XIBuffer1 readhexstring pop} }{ {currentfile XIBuffer1 readstring pop} } ifelse }{ /XIBuffer1 256 string def /XIBuffer2 XIRowBytes string def {currentfile XIBuffer1 readline pop (%) anchorsearch {pop} if} /ASCII85Decode filter /DCTDecode filter /XIFile exch def {XIFile XIBuffer2 readstring pop} } ifelse /XIDataProc exch def XIType 1 ne { 0 setgray } if XIType 1 eq { XIImageMask }{ XIType 2 eq XIType 3 eq or { XIImageTint }{ XIImage } ifelse } ifelse }{ XINullImage } ifelse /XIPlateList false def grestore end } def end %%EndProcSet %%BeginResource: procset Adobe_Illustrator_AI5 1.3 0 %%Title: (Adobe Illustrator (R) Version 8.0 Full Prolog) %%Version: 1.3 0 %%CreationDate: (3/7/1994) () %%Copyright: ((C) 1987-1998 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 112 dict dup begin put /_?cmyk false def /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_lineorientation 0 def /_charorientation 0 def /_yokoorientation 0 def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_shift [0 0] def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fontSize 0 def /_fontAscent 0 def /_fontDescent 0 def /_fontHeight 0 def /_fontRotateAdjust 0 def /Ss 256 string def Ss 0 (fonts/) putinterval /_cnt 0 def /_scale [1 1] def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_hfname 100 string def /_hffound false def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_rgbf 3 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_rgbs 3 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /_lobyte 0 def /_hibyte 0 def /_cproc null def /_cscript 0 def /_hvax 0 def /_hvay 0 def /_hvwb 0 def /_hvcx 0 def /_hvcy 0 def /_bitfont null def /_bitlobyte 0 def /_bithibyte 0 def /_bitkey null def /_bitdata null def /_bitindex 0 def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 100 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin /_aicmykps where {pop /_?cmyk _aicmykps def}if discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /hswj { dup stringwidth 3 2 roll { _hvwb eq { exch _hvcx add exch _hvcy add } if exch _hvax add exch _hvay add } cforall } def /vswj { 0 0 3 -1 roll { dup 255 le _charorientation 1 eq and { dup cstring stringwidth 5 2 roll _hvwb eq { exch _hvcy sub exch _hvcx sub } if exch _hvay sub exch _hvax sub 4 -1 roll sub exch 3 -1 roll sub exch } { _hvwb eq { exch _hvcy sub exch _hvcx sub } if exch _hvay sub exch _hvax sub _fontHeight sub } ifelse } cforall } def /swj { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef _lineorientation 0 eq { hswj } { vswj } ifelse } def /sw { 0 0 0 6 3 roll swj } def /vjss { 4 1 roll { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto gsave false charpath currentpoint 5 index setmatrix stroke grestore _fontRotateAdjust sub moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto 90 rotate } { currentpoint _fontHeight sub 5 index sub 3 index _sp eq { 9 index sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto gsave 2 index false charpath 6 index setmatrix stroke grestore moveto pop pop } ifelse } cforall 6 npop } def /hjss { 4 1 roll { dup cstring gsave false charpath currentpoint 5 index setmatrix stroke grestore moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto } cforall 6 npop } def /jss { _lineorientation 0 eq { hjss } { vjss } ifelse } def /ss { 0 0 0 7 3 roll jss } def /vjsp { 4 1 roll { dup cstring dup length 1 eq _charorientation 1 eq and { -90 rotate currentpoint _fontRotateAdjust add moveto false charpath currentpoint _fontRotateAdjust sub moveto _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto 90 rotate } { currentpoint _fontHeight sub 5 index sub 3 index _sp eq { 9 index sub } if currentpoint exch 4 index stringwidth pop 2 div sub exch _fontAscent sub moveto 2 index false charpath moveto pop pop } ifelse } cforall 6 npop } def /hjsp { 4 1 roll { dup cstring false charpath _sp eq { 5 index 5 index rmoveto } if 2 copy rmoveto } cforall 6 npop } def /jsp { matrix currentmatrix _lineorientation 0 eq {hjsp} {vjsp} ifelse } def /sp { matrix currentmatrix 0 0 0 7 3 roll _lineorientation 0 eq {hjsp} {vjsp} ifelse } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /_rgbtocmyk { 3 { 1 exch sub 3 1 roll } repeat 3 copy 1 4 1 roll 3 { 3 index 2 copy gt { exch } if pop 4 1 roll } repeat pop pop pop 4 1 roll 3 { 3 index sub 3 1 roll } repeat 4 -1 roll } def /setrgbfill { _rgbf astore pop /_fc { _lp /fill ne { _of setoverprint _rgbf aload pop setrgbcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /setrgbstroke { _rgbs astore pop /_sc { _lp /stroke ne { _os setoverprint _rgbs aload pop setrgbcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /Xa { _?cmyk { 3 npop k }{ setrgbfill 4 npop } ifelse } def /XA { _?cmyk { 3 npop K }{ setrgbstroke 4 npop } ifelse } def /Xs { /_gf exch ddef 5 npop /_fc { _lp /fill ne { _of setoverprint _gf setAIseparationgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /XS { /_gs exch ddef 5 npop /_sc { _lp /stroke ne { _os setoverprint _gs setAIseparationgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /Xx { exch /_gf exch ddef 0 eq { findcmykcustomcolor }{ _?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse { 4 1 roll 3 npop findcmykcustomcolor }{ 8 -4 roll 4 npop findrgbcustomcolor } ifelse } ifelse /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /XX { exch /_gs exch ddef 0 eq { findcmykcustomcolor }{ _?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse { 4 1 roll 3 npop findcmykcustomcolor }{ 8 -4 roll 4 npop findrgbcustomcolor } ifelse } ifelse /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc hvashow } ddef /_pjsf { _fc hvawidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /XK { 3 -1 roll pop 0 eq { 1 exch sub 3 {dup 3 1 roll mul 5 1 roll} repeat mul 4 1 roll K } { 1 exch sub 4 1 roll 3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat 4 -1 roll pop XA } ifelse } def /Xk { 3 -1 roll pop 0 eq { 1 exch sub 3 {dup 3 1 roll mul 5 1 roll} repeat mul 4 1 roll k } { 1 exch sub 4 1 roll 3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat 4 -1 roll pop Xa } ifelse } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /Xt { pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer {readline} stopped { % assume error was due to overfilling the buffer }{ not { stop } if endString eq { cleartomark stop } if }ifelse } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer {readline} stopped { % assume error was due to overfilling the buffer }{ not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse }ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 6 npop 7 2 roll 5 npop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 4 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setrgbcolor { 3 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end /XP { 4 npop } bind def /XD { pop } bind def end setpacking %%EndResource %%BeginResource: procset Adobe_cshow 2.0 8 %%Title: (Writing System Operators) %%Version: 2.0 8 %%CreationDate: (1/23/89) () %%Copyright: ((C) 1992-1996 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_cshow 14 dict dup begin put /initialize { Adobe_cshow begin Adobe_cshow { dup xcheck { bind } if pop pop } forall end Adobe_cshow begin } def /terminate { currentdict Adobe_cshow eq { end } if } def /cforall { /_lobyte 0 ddef /_hibyte 0 ddef /_cproc exch ddef /_cscript currentfont /FontScript known { currentfont /FontScript get } { -1 } ifelse ddef { /_lobyte exch ddef _hibyte 0 eq _cscript 1 eq _lobyte 129 ge _lobyte 159 le and _lobyte 224 ge _lobyte 252 le and or and _cscript 2 eq _lobyte 161 ge _lobyte 254 le and and _cscript 3 eq _lobyte 161 ge _lobyte 254 le and and _cscript 25 eq _lobyte 161 ge _lobyte 254 le and and _cscript -1 eq or or or or and { /_hibyte _lobyte ddef } { _hibyte 256 mul _lobyte add _cproc /_hibyte 0 ddef } ifelse } forall } def /cstring { dup 256 lt { (s) dup 0 4 3 roll put } { dup 256 idiv exch 256 mod (hl) dup dup 0 6 5 roll put 1 4 3 roll put } ifelse } def /clength { 0 exch { 256 lt { 1 } { 2 } ifelse add } cforall } def /hawidthshow { { dup cstring show _hvax _hvay rmoveto _hvwb eq { _hvcx _hvcy rmoveto } if } cforall } def /vawidthshow { { dup 255 le _charorientation 1 eq and { -90 rotate 0 _fontRotateAdjust rmoveto cstring _hvcx _hvcy _hvwb _hvax _hvay 6 -1 roll awidthshow 0 _fontRotateAdjust neg rmoveto 90 rotate } { currentpoint _fontHeight sub exch _hvay sub exch _hvax sub 2 index _hvwb eq { exch _hvcy sub exch _hvcx sub } if 3 2 roll cstring dup stringwidth pop 2 div neg _fontAscent neg rmoveto show moveto } ifelse } cforall } def /hvawidthshow { 6 1 roll /_hvay exch ddef /_hvax exch ddef /_hvwb exch ddef /_hvcy exch ddef /_hvcx exch ddef _lineorientation 0 eq { hawidthshow } { vawidthshow } ifelse } def /hvwidthshow { 0 0 3 -1 roll hvawidthshow } def /hvashow { 0 0 0 6 -3 roll hvawidthshow } def /hvshow { 0 0 0 0 0 6 -1 roll hvawidthshow } def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_shading_AI8 1.0 0 %%Title: (Adobe Illustrator 8 Shading Procset) %%Version: 1.0 0 %%CreationDate: (12/17/97) () %%Copyright: ((C) 1987-1997 Adobe Systems Incorporated All Rights Reserved) userdict /defaultpacking currentpacking put true setpacking userdict /Adobe_shading_AI8 10 dict dup begin put /initialize { Adobe_shading_AI8 begin Adobe_shading_AI8 bdprocs Mesh /initialize get exec } def /terminate { currentdict Adobe_shading_AI8 eq { end } if } def /bdprocs { { dup xcheck 1 index type /arraytype eq and { bind } if pop pop } forall } def /X! {pop} def /X# {pop pop} def /Mesh 40 dict def Mesh begin /initialize { Mesh bdprocs Mesh begin /emulate? /AI8MeshEmulation where { pop AI8MeshEmulation }{ systemdict /shfill known not } ifelse def end } def /bd { shadingdict begin } def /paint { emulate? { end }{ /_lp /none ddef _fc /_lp /none ddef /AIColorSpace AIColorSpace tocolorspace store /ColorSpace AIColorSpace topsspace store version_ge_3010.106 not systemdict /setsmoothness known and { 0.0001 setsmoothness } if composite? { /DataSource getdatasrc def Matrix concat currentdict end shfill }{ AIColorSpace makesmarks AIPlateList markingplate and not isoverprint and { end }{ /ColorSpace /DeviceGray store /Decode [0 1 0 1 0 1] store /DataSource getplatesrc def Matrix concat currentdict end shfill } ifelse } ifelse } ifelse } def /shadingdict 12 dict def shadingdict begin /ShadingType 6 def /BitsPerCoordinate 16 def /BitsPerComponent 8 def /BitsPerFlag 8 def end /datafile null def /databuf 256 string def /dataptr 0 def /srcspace null def /srcchannels 0 def /dstchannels 0 def /dstplate 0 def /srctodstcolor null def /getplatesrc { /srcspace AIColorSpace store /srcchannels AIColorSpace getnchannels store /dstchannels 1 store /dstplate getplateindex store /srctodstcolor srcspace makesmarks { dstplate 4 eq { {1 exch sub} }{ {srcspace tocmyk 3 dstplate sub index 1 exch sub 5 1 roll 4 {pop} repeat} } ifelse }{ {srcchannels {pop} repeat 1} } ifelse store /datafile getdatasrc store /rdpatch168 load DataLength () /SubFileDecode filter } def /getdatasrc { /rdcmntline load /ASCII85Decode filter } def /rdpatch168 { /dataptr 0 store 49 rdcount 4 { dup {pop srcchannels getint8} if dup {pop srctodstcolor dstchannels putint8 true} if } repeat {databuf 0 dataptr getinterval}{()} ifelse } def /rdpatch3216 { /dataptr 0 store 97 rdcount 4 { dup {pop srcchannels getint16} if dup {pop srctodstcolor dstchannels putint16 true} if } repeat {databuf 0 dataptr getinterval}{()} ifelse } def /rdcount { dup 0 gt { datafile databuf dataptr 4 -1 roll getinterval readstring exch length dataptr add /dataptr exch store }{ true } ifelse } def /getint8 { mark true 3 -1 roll { dup {pop datafile read} if dup {pop 255 div true} if } repeat { counttomark 1 add -1 roll pop true }{ cleartomark false } ifelse } def /putint8 { dup dataptr add /dataptr exch store dataptr exch { 1 sub exch 255 mul cvi databuf 2 index 3 -1 roll put } repeat pop } def /getint16 { mark true 3 -1 roll { dup {pop datafile read} if dup {pop 256 mul datafile read} if dup {pop add 65535 div true} if } repeat { counttomark 1 add -1 roll pop true }{ cleartomark false } ifelse } def /putint16 { dup 2 mul dataptr add /dataptr exch store dataptr exch { 2 sub exch 65535 mul cvi dup 256 idiv databuf 3 index 3 -1 roll put 256 mod databuf 2 index 1 add 3 -1 roll put } repeat pop } def /srcbuf 256 string def /rdcmntline { currentfile srcbuf readline pop (%) anchorsearch {pop} if } def /getplateindex { 0 [cyan? magenta? yellow? black? customColor?] {{exit} if 1 add} forall } def /aicsarray 4 array def /aicsaltvals 4 array def /aicsaltcolr aicsaltvals def /tocolorspace { dup type /arraytype eq { mark exch aload pop aicsarray 0 3 -1 roll put aicsarray 1 3 -1 roll put dup aicsarray 2 3 -1 roll put gettintxform aicsarray 3 3 -1 roll put counttomark aicsaltvals 0 3 -1 roll getinterval /aicsaltcolr exch store aicsaltcolr astore pop pop aicsarray } if } def /subtintxform {aicsaltcolr {1 index mul exch} forall pop} def /addtintxform {aicsaltcolr {1 sub 1 index mul 1 add exch} forall pop} def /gettintxform { /DeviceRGB eq {/addtintxform}{/subtintxform} ifelse load } def /getnchannels { dup type /arraytype eq {0 get} if colorspacedict exch get begin Channels end } def /makesmarks { composite? { pop true }{ dup dup type /arraytype eq {0 get} if colorspacedict exch get begin MarksPlate end } ifelse } def /markingplate { composite? { pop true }{ dup type /arraytype eq { dup length getplateindex gt {getplateindex get}{pop false} ifelse } if } ifelse } def /tocmyk { dup dup type /arraytype eq {0 get} if colorspacedict exch get begin ToCMYK end } def /topsspace { dup dup type /arraytype eq {0 get} if colorspacedict exch get begin ToPSSpace end } def /colorspacedict 5 dict dup begin /DeviceGray 4 dict dup begin /Channels 1 def /MarksPlate {pop black?} def /ToCMYK {pop 1 exch sub 0 0 0 4 -1 roll} def /ToPSSpace {} def end def /DeviceRGB 4 dict dup begin /Channels 3 def /MarksPlate {pop isCMYKSep?} def /ToCMYK {pop _rgbtocmyk} def /ToPSSpace {} def end def /DeviceCMYK 4 dict dup begin /Channels 4 def /MarksPlate {pop isCMYKSep?} def /ToCMYK {pop} def /ToPSSpace {} def end def /Separation 4 dict dup begin /Channels 1 def /MarksPlate { /findcmykcustomcolor where { pop dup 1 exch ToCMYK 5 -1 roll 1 get findcmykcustomcolor 1 setcustomcolor systemdict /currentgray get exec 1 ne }{ pop false } ifelse } def /ToCMYK { dup 2 get mark exch 4 2 roll 3 get exec counttomark -1 roll tocmyk 5 -1 roll pop } def /ToPSSpace {} def end def /Process 4 dict dup begin /Channels 1 def /MarksPlate { isCMYKSep? { 1 exch ToCMYK 4 array astore getplateindex get 0 ne }{ pop false } ifelse } def /ToCMYK { dup 2 get mark exch 4 2 roll 3 get exec counttomark -1 roll tocmyk 5 -1 roll pop } def /ToPSSpace { 4 array copy dup 0 /Separation put } def end def end def /isoverprint { /currentoverprint where {pop currentoverprint}{_of} ifelse } def /version_ge_3010.106 { version {cvr} stopped { pop false }{ 3010.106 ge } ifelse } def end end defaultpacking setpacking %%EndResource %%EndProlog %%BeginSetup %%IncludeFont: Symbol %%IncludeFont: Times-Italic %%IncludeFont: Times-Roman userdict /_useSmoothShade false put userdict /_aicmykps true put userdict /_forceToCMYK true put Adobe_level2_AI5 /initialize get exec Adobe_cshow /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_shading_AI8 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron TE %AI55J_Tsume: None %AI3_BeginEncoding: _Symbol Symbol [/.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /universal /numbersign /existential /percent /ampersand /suchthat /parenleft /parenright /asteriskmath /plus /comma /minus /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /congruent /Alpha /Beta /Chi /Delta /Epsilon /Phi /Gamma /Eta /Iota /theta1 /Kappa /Lambda /Mu /Nu /Omicron /Pi /Theta /Rho /Sigma /Tau /Upsilon /sigma1 /Omega /Xi /Psi /Zeta /bracketleft /therefore /bracketright /perpendicular /underscore /radicalex /alpha /beta /chi /delta /epsilon /phi 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b(able)33 b Fr(\025)2159 3383 y Fo(i)2188 3369 y Fr(=\025)2286 3383 y Fo(i)p Fu(+1)2439 3369 y Fx(de\014ne)h(a)h(co)s(dimension)d(one) 456 3477 y(foliation.)37 b(W)-8 b(e)27 b(denote)f(the)g(lea)m(v)m(es)h (of)e(suc)m(h)h(a)g(foliation)e(b)m(y)h(\003)2598 3444 y Fo(i)2598 3505 y(\014)2671 3477 y Fx(=)g Fp(f)p Fr(F)13 b Fp(j)p Fr(\025)2961 3491 y Fo(i)2990 3477 y Fr(=\025)3088 3491 y Fo(i)p Fu(+1)3207 3477 y Fx(\()p Fr(F)g Fx(\))26 b(=)456 3594 y Fr(\014)5 b Fp(g)p Fx(.)555 3702 y(Because)43 b(of)e(the)g(in)m(v)-5 b(ariance)40 b(of)h Fr(\025)1802 3716 y Fo(i)1830 3702 y Fr(=\025)1928 3716 y Fo(i)p Fu(+1)2088 3702 y Fx(under)e(renormalization,)k(the)e(lea)m(v)m(es)456 3810 y(of)h(the)h(foliation)e(\003)1171 3777 y Fo(i)1241 3810 y Fx(are)i(mapp)s(ed)e(b)m(y)h(renormalization)f(in)m(to)i(the)f (lea)m(v)m(es)i(of)e(the)456 3918 y(foliation)29 b(\003)875 3885 y Fo(i)p Fm(\000)p Fu(1)555 4026 y Fx(The)24 b(n)m(umerical)f(exp) s(erimen)m(ts)g(suggest)i(that)f(this)f(foliation)g(is)g(transv)m 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setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 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1960 2132 l 1970 2073 l cp clip n 1500 2025 m 1950 2100 l gs col0 s gr gr [] 0 sd % arrowhead n 1837 2051 m 1950 2100 l 1827 2110 l col0 s % Polyline [15 45] 45 sd gs clippath 1827 2184 m 1950 2175 l 1846 2241 l 1974 2199 l 1955 2142 l cp clip n 1500 2325 m 1950 2175 l gs col0 s gr gr [] 0 sd % arrowhead n 1827 2184 m 1950 2175 l 1846 2241 l col0 s % Polyline [15 45] 45 sd gs clippath 1839 2304 m 1950 2250 l 1877 2350 l 1981 2263 l 1942 2217 l cp clip n 1500 2625 m 1950 2250 l gs col0 s gr gr [] 0 sd % arrowhead n 1839 2304 m 1950 2250 l 1877 2350 l col0 s % Polyline [15 45] 45 sd gs clippath 1848 2396 m 1950 2325 l 1894 2435 l 1982 2333 l 1937 2294 l cp clip n 1425 2925 m 1950 2325 l gs col0 s gr gr [] 0 sd % arrowhead n 1848 2396 m 1950 2325 l 1894 2435 l col0 s % Polyline [15 45] 45 sd gs clippath 1857 2556 m 1950 2475 l 1906 2591 l 1983 2480 l 1934 2446 l cp clip n 1425 3225 m 1950 2475 l gs col0 s gr gr [] 0 sd % arrowhead n 1857 2556 m 1950 2475 l 1906 2591 l col0 s % Polyline [15 45] 45 sd gs clippath 1939 2639 m 2025 2550 l 1990 2669 l 2058 2552 l 2007 2522 l cp clip n 1500 3450 m 2025 2550 l gs col0 s gr gr [] 0 sd % arrowhead n 1939 2639 m 2025 2550 l 1990 2669 l col0 s /Times-Roman-iso ff 360.00 scf sf 600 4125 m gs 1 -1 sc 90.0 rot (log\(|\(A /A \) - 4.339143|\)) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 750 3075 m gs 1 -1 sc 90.0 rot (i) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 750 2700 m gs 1 -1 sc 90.0 rot (i+1) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 10575 7725 m gs 1 -1 sc (log\(q \)) col0 sh gr /Times-Roman-iso ff 210.00 scf sf 11400 7875 m gs 1 -1 sc (i) col0 sh gr /Times-Roman-iso ff 240.00 scf sf 1350 675 m gs 1 -1 sc (a) col0 sh gr /Times-Roman-iso ff 240.00 scf sf 1350 960 m gs 1 -1 sc (b) col0 sh gr /Times-Roman-iso ff 240.00 scf sf 1350 1245 m gs 1 -1 sc (c) col0 sh gr /Times-Roman-iso ff 240.00 scf sf 1350 1530 m gs 1 -1 sc (d) col0 sh gr /Times-Roman-iso ff 240.00 scf sf 1350 1815 m gs 1 -1 sc (e) col0 sh gr /Times-Roman-iso ff 240.00 scf sf 1350 2100 m gs 1 -1 sc (f) col0 sh gr /Times-Roman-iso ff 240.00 scf sf 1350 2385 m gs 1 -1 sc (g) col0 sh gr /Times-Roman-iso ff 240.00 scf sf 1350 2670 m gs 1 -1 sc (h) col0 sh gr /Times-Roman-iso ff 240.00 scf sf 1350 2955 m gs 1 -1 sc (i) col0 sh gr /Times-Roman-iso ff 240.00 scf sf 1350 3240 m gs 1 -1 sc (j) col0 sh gr /Times-Roman-iso ff 240.00 scf sf 1350 3525 m gs 1 -1 sc (k) col0 sh gr $F2psEnd rs showpage %%EndDocument @endspecial 3375 2167 a currentpoint grestore moveto 3375 2167 a 754 2338 a Fw(Figure)26 b(6.)45 b Fx(Similar)19 b(to)k(Figure)e(5.)38 b(In)21 b(this)g(case,)k(the)c(p)s(erturbation)754 2446 y(function)30 b(is:)39 b Fp(\000)1325 2410 y Fo(K)p 1318 2425 79 4 v 1318 2477 a Fu(2)p Fo(\031)1406 2446 y Fx(\(sin)o(\(2)p Fr(\031)s(x)p Fx(\))22 b(+)d(0)p Fr(:)p Fx(05)c(sin)q(\(6)p Fr(\031)s(x)p Fx(\)\).)456 2948 y Fr(n)27 b Fx(=)g(2)p Fr(;)15 b Fx(3)p Fr(;)g(:)g(:)g(:)j Fx(.)45 b(The)32 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Fx(169)34 b(and)c Fr(p)1316 4690 y Fu(5)1355 4676 y Fr(=q)1441 4690 y Fu(5)1507 4676 y Fx(=)c(169)p Fr(=)p Fx(408,)34 b(w)m(e)d(lo)s(ok)m(ed)g(for)g(these)g(h)m(yp)s(erb)s(olic)d (orbits)456 4784 y(suc)m(h)22 b(that)i(the)f(ratio)g(of)g(their)f (eigen)m(v)-5 b(alues)22 b(w)m(as)i(close)f(enough)g(to)g(0)p Fr(:)p Fx(349,)k(in)21 b(this)h(case)456 4892 y Fr(K)533 4906 y Fu(4)606 4892 y Fx(=)33 b(0)p Fr(:)p Fx(962535)39 b(and)34 b(the)i(corresp)s(onding)d(lob)s(e's)i(area)h(w)m(as)g Fr(A)2732 4906 y Fu(4)2805 4892 y Fx(=)d(1)p Fr(:)p Fx(369)26 b Fp(\002)d Fx(10)3324 4859 y Fm(\000)p Fu(9)3419 4892 y Fx(.)456 5000 y(Then,)32 b(our)g(\014rst)g(estimation)h(of)f(the)h (scale)g(factor)h(w)m(as)f(14)p Fr(:)p Fx(6092.)51 b(T)-8 b(able)32 b(T2)h(sho)m(ws)456 5108 y(the)d(v)-5 b(alue)31 b(of)f(the)h(sequence)g(of)g(h)m(yp)s(erb)s(olic)d(orbits)i(whose)g (rotation)h(n)m(um)m(b)s(ers)f(con-)456 5216 y(v)m(erge)h(to)g Fr(\015)851 5230 y Fu(2)891 5216 y Fx(.)p eop %%Page: 30 30 30 29 bop 456 251 a Fu(30)837 b(R.)23 b(DE)g(LA)h(LLA)-8 b(VE)23 b(AND)g(A.)g(OL)-8 b(VERA)p 555 370 2796 4 v 553 470 4 100 v 696 440 a Fh(p)738 452 y Fg(i)765 440 y Fh(=q)844 452 y Fg(i)p 1010 470 V 1089 440 a Fh(p)1131 452 y Fg(i)p Ff(+1)1243 440 y Fh(=q)1322 452 y Fg(i)p Ff(+1)p 1508 470 V 1870 440 a Fh(K)1941 452 y Fg(i)p 2326 470 V 2369 440 a Fh(\025)2417 452 y Fg(i)2445 440 y Fh(=\025)2535 452 y Fg(i)p Ff(+1)p 2687 470 V 2975 440 a Fh(A)3037 452 y Fg(i)p 3349 470 V 555 473 2796 4 v 553 573 4 100 v 701 543 a Fe(5)p Fh(=)p Fe(12)p 1010 573 V 288 w(12)p Fh(=)p Fe(29)p 1508 573 V 226 w(1)p Fh(:)p Fe(032676)15 b Fd(\006)j(\002)p Fe(10)2157 513 y Fc(\000)p Ff(7)p 2326 573 V 2372 543 a Fe(0)p Fh(:)p Fe(35045)p 2687 573 V 141 w(4)p Fh(:)p Fe(263)f Fd(\002)h Fe(10)3163 513 y Fc(\000)p Ff(6)p 3349 573 V 555 576 2796 4 v 553 676 4 100 v 680 646 a Fe(12)p Fh(=)p Fe(29)p 1010 676 V 267 w(29)p Fh(=)p Fe(70)p 1508 676 V 226 w(0)p Fh(:)p Fe(987999)d Fd(\006)j(\002)p Fe(10)2157 616 y Fc(\000)p Ff(7)p 2326 676 V 2414 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Fh(:)p Fe(33608)p 2687 985 V 104 w(9)p Fh(:)p Fe(3826)e Fd(\002)i Fe(10)3167 925 y Fc(\000)p Ff(11)p 3349 985 V 555 988 2796 4 v 553 1088 4 100 v 638 1058 a Fe(408)p Fh(=)p Fe(985)p 1010 1088 V 163 w(985)p Fh(=)p Fe(2378)p 1508 1088 V 120 w(0)p Fh(:)p Fe(95830144)d Fd(\006)j(\002)p Fe(10)2199 1028 y Fc(\000)p Ff(9)p 2326 1088 V 2372 1058 a Fe(0)p Fh(:)p Fe(33520)p 2687 1088 V 104 w(6)p Fh(:)p Fe(4264)e Fd(\002)i Fe(10)3167 1028 y Fc(\000)p Ff(12)p 3349 1088 V 555 1091 2796 4 v 553 1191 4 100 v 617 1161 a Fe(985)p Fh(=)p Fe(2378)p 1010 1191 V 121 w(2378)p Fh(=)p Fe(5741)p 1508 1191 V 141 w(0)p Fh(:)p Fe(957796)d Fd(\006)j(\002)p Fe(10)2157 1131 y Fc(\000)p Ff(7)p 2326 1191 V 2414 1161 a Fe(0)p Fh(:)p Fe(333)p 2687 1191 V 146 w(4)p Fh(:)p Fe(4016)e Fd(\002)i Fe(10)3167 1131 y Fc(\000)p Ff(13)p 3349 1191 V 555 1194 2796 4 v 553 1294 4 100 v 597 1264 a Fe(2378)p Fh(=)p Fe(5741)p 1010 1294 V 78 w(5741)p Fh(=)p Fe(13860)p 1508 1294 V 78 w(0)p Fh(:)p Fe(95759028)d Fd(\006)j(\002)p Fe(10)2199 1234 y Fc(\000)p Ff(9)p 2326 1294 V 2393 1264 a Fe(0)p Fh(:)p Fe(3304)p 2687 1294 V 104 w(3)p Fh(:)p Fe(01485)e Fd(\002)i Fe(10)3188 1234 y Fc(\000)p Ff(14)p 3349 1294 V 555 1297 2796 4 v 1785 1363 a Fe(T)-7 b(able)27 b(T2)555 1542 y Fx(The)g(v)-5 b(alue)26 b(of)h(the)g(scale)g(factor)h(that)f(w)m(e)h(could)e (estimate)h(from)f(this)g(exp)s(erimen)m(t)456 1650 y(w)m(as)k Fr(\016)670 1664 y Fo(\015)706 1673 y Fs(2)771 1650 y Fp(\030)25 b Fx(14)p Fr(:)p Fx(60,)33 b(this)c(factor)i(is)f(the)g (limit)e(of)j Fr(A)2204 1664 y Fo(i)2232 1650 y Fr(=)-5 b(A)2340 1664 y Fo(i)p Fu(+1)2459 1650 y Fx(.)555 1758 y(Our)31 b(next)h(exp)s(erimen)m(t)g(w)m(as)g(done)g(with)f(the)h(v)-5 b(alue)32 b(of)g(the)h(rotation)f(n)m(um)m(b)s(er)f(of)456 1866 y(the)20 b(critical)g(circle)g Fr(\015)1171 1880 y Fu(3)1236 1866 y Fx(=)25 b(\()1367 1791 y Fp(p)p 1443 1791 91 4 v 75 x Fx(13)q Fp(\000)q Fx(2\))p Fr(=)p Fx(2.)39 b(The)20 b(v)-5 b(alue)20 b(of)h(the)g(rational)f(appro)m(ximan)m(ts)g (of)456 1974 y Fr(\015)503 1988 y Fu(3)565 1974 y Fx(w)m(as)k(obtained) f(from)f(the)i(truncation)f(of)g(its)g(con)m(tin)m(ued)g(fraction)g (expansion)f Fr(\015)3309 1988 y Fu(3)3374 1974 y Fx(=)456 2082 y([3)p Fr(;)15 b Fx(3)p Fr(;)g Fx(3)p Fr(;)g Fx(3)p Fr(;)g(:)g(:)g(:)20 b Fx(].)39 b(The)24 b(initial)e(orbits)i(w)m(as)i (h)m(yp)s(erb)s(olic)c(orbits)i(of)h(t)m(yp)s(e)g Fr(p)2931 2096 y Fu(1)2970 2082 y Fr(=q)3056 2096 y Fu(1)3121 2082 y Fx(=)g(10)p Fr(=)p Fx(33)456 2190 y(and)40 b Fr(p)689 2204 y Fu(2)728 2190 y Fr(=q)814 2204 y Fu(2)895 2190 y Fx(=)i(33)p Fr(=)p Fx(109,)k(the)41 b(v)-5 b(alue)40 b(of)h(the)f(p)s(erturbation)f(parameter)i(w)m(as)g Fr(K)3292 2204 y Fu(1)3374 2190 y Fx(=)456 2298 y(0)p Fr(:)p Fx(906678,)50 b(the)43 b(corresp)s(onding)e(v)-5 b(alue)43 b(of)g(the)h(eigen)m(v)-5 b(alue's)43 b(ratio)g(w)m(as)h Fr(\025)3150 2312 y Fu(1)3189 2298 y Fr(=\025)3287 2312 y Fu(2)3374 2298 y Fx(=)456 2406 y(0)p Fr(:)p Fx(38178)26 b(and)e(the)g(lob)s(e's)f(area)i(w)m(as)f Fr(A)1774 2420 y Fu(1)1839 2406 y Fx(=)h(5)7 b Fp(\002)g Fx(10)2155 2373 y Fm(\000)p Fu(9)2251 2406 y Fx(.)39 b(T)-8 b(able)23 b(T3)h(sho)m(ws)g(the)g(sequence)456 2514 y(of)f(v)-5 b(alues)22 b(of)h(the)g(appro)m(ximan)m(ts)g(to)g Fr(\015)1772 2528 y Fu(3)1812 2514 y Fx(,)h(these)g(v)-5 b(alues)22 b(w)m(as)h(computed)g(with)f(the)h(help)456 2621 y(of)j(our)g(conjecture,)i(the)f(next)f(h)m(yp)s(erb)s(olic)e (orbits)h(w)m(ere)i(computed)f(\014xing)f(the)i(v)-5 b(alue)456 2729 y(of)41 b Fr(\025)623 2743 y Fo(i)651 2729 y Fr(=\025)749 2743 y Fo(i)p Fu(+1)910 2729 y Fp(\030)i Fx(0)p Fr(:)p Fx(3815.)74 b(The)41 b(scale)g(factor)h(obtained)e(from)g (this)g(pro)s(cedure)g(w)m(as)456 2837 y Fr(\016)496 2851 y Fo(\015)532 2860 y Fs(3)596 2837 y Fx(=)25 b(40)p Fr(:)p Fx(12.)p 555 2946 2713 4 v 553 3046 4 100 v 696 3016 a Fh(p)738 3028 y Fg(i)765 3016 y Fh(=q)844 3028 y Fg(i)p 1010 3046 V 1089 3016 a Fh(p)1131 3028 y Fg(i)p Ff(+1)1243 3016 y Fh(=q)1322 3028 y Fg(i)p Ff(+1)p 1508 3046 V 1849 3016 a Fh(K)1920 3028 y Fg(i)p 2285 3046 V 2328 3016 a Fh(\025)2376 3028 y Fg(i)2404 3016 y Fh(=\025)2494 3028 y Fg(i)p Ff(+1)p 2645 3046 V 2912 3016 a Fh(A)2974 3028 y Fg(i)p 3266 3046 V 555 3049 2713 4 v 553 3149 4 100 v 680 3119 a Fe(10)p Fh(=)p Fe(33)p 1010 3149 V 246 w(33)p Fh(=)p Fe(109)p 1508 3149 V 184 w(0)p Fh(:)p Fe(906678)15 b Fd(\006)j(\002)p Fe(10)2136 3089 y Fc(\000)p Ff(7)p 2285 3149 V 2331 3119 a Fe(0)p Fh(:)p Fe(38178)p 2645 3149 V 194 w(5)g Fd(\002)g Fe(10)3027 3089 y Fc(\000)p Ff(9)p 3266 3149 V 555 3152 2713 4 v 553 3252 4 100 v 659 3222 a Fe(33)p Fh(=)p Fe(109)p 1010 3252 V 205 w(109)p Fh(=)p Fe(360)p 1508 3252 V 162 w(0)p Fh(:)p Fe(895563)d Fd(\006)j(\002)p Fe(10)2136 3192 y Fc(\000)p Ff(7)p 2285 3252 V 2331 3222 a Fe(0)p Fh(:)p Fe(38172)p 2645 3252 V 83 w(1)p Fh(:)p Fe(2462)e Fd(\002)i Fe(10)3105 3192 y Fc(\000)p Ff(10)p 3266 3252 V 555 3255 2713 4 v 553 3355 4 100 v 638 3325 a Fe(109)p Fh(=)p Fe(360)p 1010 3355 V 163 w(360)p Fh(=)p Fe(1189)p 1508 3355 V 141 w(0)p Fh(:)p Fe(892266)d Fd(\006)j(\002)p Fe(10)2136 3295 y Fc(\000)p Ff(7)p 2285 3355 V 2351 3325 a Fe(0)p Fh(:)p Fe(3817)p 2645 3355 V 105 w(3)p Fh(:)p Fe(1064)e Fd(\002)i Fe(10)3105 3295 y Fc(\000)p Ff(12)p 3266 3355 V 555 3358 2713 4 v 553 3458 4 100 v 617 3428 a Fe(360)p Fh(=)p Fe(1189)p 1010 3458 V 121 w(1189)p Fh(=)p Fe(3927)p 1508 3458 V 120 w(0)p Fh(:)p Fe(891283)d Fd(\006)j(\002)p Fe(10)2136 3398 y Fc(\000)p Ff(7)p 2285 3458 V 2351 3428 a Fe(0)p Fh(:)p Fe(3812)p 2645 3458 V 105 w(7)p Fh(:)p Fe(7429)e Fd(\002)i Fe(10)3105 3398 y Fc(\000)p Ff(14)p 3266 3458 V 555 3461 2713 4 v 553 3561 4 100 v 597 3531 a Fe(1189)p Fh(=)p Fe(3927)p 1010 3561 V 78 w(3927)p Fh(=)p Fe(12970)p 1508 3561 V 78 w(0)p Fh(:)p Fe(8909888)d Fd(\006)j(\002)p Fe(10)2157 3501 y Fc(\000)p Ff(8)p 2285 3561 V 2351 3531 a Fe(0)p Fh(:)p Fe(3815)p 2645 3561 V 125 w(1)p Fh(:)p Fe(929)f Fd(\002)h Fe(10)3084 3501 y Fc(\000)p Ff(15)p 3266 3561 V 555 3564 2713 4 v 1785 3630 a Fe(T)-7 b(able)27 b(T3)555 3809 y Fx(Finally)-8 b(,)24 b(w)m(e)h(ha)m(v)m(e)g(computed)f(the)h(renormalization)e(pro)s (cess)g(of)i(in)m(v)-5 b(arian)m(t)23 b(circles)456 3917 y(whose)30 b(rotation)h(n)m(um)m(b)s(ers)e(ha)m(v)m(e)i(the)g(form)f([) p 2046 3867 216 4 v Fr(a)2094 3931 y Fu(1)2134 3917 y Fr(;)15 b(a)2222 3931 y Fu(2)2262 3917 y Fx(])30 b(and)g([)p 2519 3867 344 4 v Fr(a)2567 3931 y Fu(1)2607 3917 y Fr(;)15 b(a)2695 3931 y Fu(2)2735 3917 y Fr(;)g(a)2823 3931 y Fu(3)2863 3917 y Fx(].)41 b(W)-8 b(e)31 b(used)f(the)456 4025 y(same)k(n)m(umerical)e(metho)s(d)h(to)h(compute)g(the)g(corresp)s (onding)e(lob)s(e's)g(areas.)52 b(These)456 4133 y(results)29 b(are)i(going)f(to)h(b)s(e)f(rep)s(orted)f(in)h(our)f(next)i(pap)s(er.) 1685 4522 y Fw(References)456 4668 y Fz([AK00])182 b(Juan)20 b(J.)h(Abad)e(and)h(Hans)f(Ko)r(c)n(h.)i(Renormalization)f(and)g(p)r (erio)r(dic)h(orbits)f(for)h(Hamil-)874 4759 y(tonian)26 b(\015o)n(ws.)g Fv(Comm.)h(Math.)g(Phys.)p Fz(,)g(212\(2\):371{394,)j (2000.)456 4850 y([AR67])185 b(Ralph)33 b(Abraham)g(and)h(Jo)r(el)h (Robbin.)f Fv(T)-6 b(r)l(ansversal)38 b(mappings)d(and)h(\015ows)p Fz(.)f(W.)f(A.)874 4942 y(Benjamin,)26 b(Inc.,)f(New)h(Y)-6 b(ork-Amsterdam,)23 b(1967.)456 5033 y([Aub92])156 b(Serge)32 b(J.)h(Aubry)-6 b(.)30 b(The)i(concept)g(of)g(an)n(ti-in)n(tegrabilit)n (y:)48 b(de\014nition,)33 b(theorems)f(and)874 5124 y(applications)27 b(to)g(the)f(standard)g(map.)g(In)g Fv(Twist)i(mappings)h(and)f(their)h (applic)l(ations)p Fz(,)874 5216 y(v)n(olume)24 b(44)j(of)f Fv(IMA)h(V)-6 b(ol.)27 b(Math.)g(Appl.)p Fz(,)f(pages)g(7{54.)i (Springer,)e(New)g(Y)-6 b(ork,)25 b(1992.)p eop %%Page: 31 31 31 30 bop 1213 251 a Fu(OBSTR)n(UCTION)30 b(AND)g(RENORMALIZA)-6 b(TION)687 b(31)456 450 y Fz([BH87])188 b(Philip)37 b(L.)g(Bo)n(yland)g (and)f(Glen)h(Ric)n(hard)g(Hall.)g(In)n(v)l(arian)n(t)f(circles)i(and)f (the)f(order)874 541 y(structure)29 b(of)i(p)r(erio)r(dic)f(orbits)h (in)f(monotone)f(t)n(wist)h(maps.)g Fv(T)-6 b(op)l(olo)l(gy)p Fz(,)32 b(26\(1\):21{35,)874 633 y(1987.)456 724 y([CE99])193 b(Eleonora)23 b(Catsigeras)h(and)e(Heb)r(er)f(Enric)n(h.)h(P)n (ersistence)h(of)g(the)e(F)-6 b(eigen)n(baum)21 b(attrac-)874 815 y(tor)26 b(in)f(one-parameter)g(families.)j Fv(Comm.)e(Math.)i (Phys.)p Fz(,)e(207\(3\):621{640,)k(1999.)456 907 y([CEL80])145 b(P)-6 b(.)19 b(Collet,)k(J.-P)-6 b(.)19 b(Ec)n(kmann,)h(and)f(O.)g(E.) h(Lanford,)h(I)r(I)r(I.)e(Univ)n(ersal)h(prop)r(erties)g(of)g(maps)874 998 y(on)25 b(an)h(in)n(terv)l(al.)g Fv(Comm.)g(Math.)i(Phys.)p Fz(,)e(76\(3\):211{254,)k(1980.)456 1089 y([CFdlL02a])45 b(Xa)n(vier)26 b(Cabr)n(\023)-36 b(e,)28 b(Ernest)f(F)-6 b(on)n(tic)n(h,)27 b(and)g(Rafael)g(de)g(la)g(Lla)n(v)n(e.)h(The)f (parameterization)874 1181 y(metho)r(d)d(for)j(in)n(v)l(arian)n(t)e (manifolds)i(I.)f Fv(Ind.)h(Univ.)g(Mat.)g(Jour.)p Fz(,)g(2002.)h(MP)p 3126 1181 24 4 v 28 w(AR)n(C)d(02-)874 1272 y(510.)456 1363 y([CFdlL02b])40 b(Xa)n(vier)26 b(Cabr)n(\023)-36 b(e,)28 b(Ernest)f(F)-6 b(on)n(tic)n(h,)27 b(and)g(Rafael)g(de)g(la)g (Lla)n(v)n(e.)h(The)f(parameterization)874 1455 y(metho)r(d)34 b(for)j(in)n(v)l(arian)n(t)f(manifolds)g(II.)f Fv(Ind.)i(Univ.)f(Mat.)h (Jour.)p Fz(,)i(2002.)f(MP)p 3254 1455 V 28 w(AR)n(C)874 1546 y(02-511.)456 1637 y([CH82])187 b(Sh)n(ui)24 b(Nee)h(Cho)n(w)h (and)f(Jac)n(k)h(K.)f(Hale.)h Fv(Metho)l(ds)j(of)e(bifur)l(c)l(ation)h (the)l(ory)p Fz(,)f(v)n(olume)d(251)874 1729 y(of)d Fv(Grund)t(lehr)l (en)k(der)e(Mathematischen)i(Wissenschaften)g([F)-6 b(undamental)23 b(Principles)874 1820 y(of)k(Mathematic)l(al)h(Scienc)l(e])p Fz(.)f(Springer-V)-6 b(erlag,)26 b(New)g(Y)-6 b(ork,)25 b(1982.)456 1911 y([CJ02])206 b(C.)28 b(Chandre)f(and)g(H.)g(R.)g (Jauslin.)i(Renormalization-group)e(analysis)i(for)f(the)f(tran-)874 2003 y(sition)f(to)g(c)n(haos)g(in)g(Hamiltonian)g(systems.)f Fv(Phys.)j(R)l(ep.)p Fz(,)e(365\(1\):1{64,)j(2002.)456 2094 y([Da)n(v96])164 b(A.)43 b(M.)h(Da)n(vie.)g(P)n(erio)r(d)g (doubling)f(for)h Fb(C)2177 2062 y Fs(2+)p Fl(\017)2328 2094 y Fz(mappings.)f Fv(Comm.)g(Math.)h(Phys.)p Fz(,)874 2185 y(176\(2\):261{272,)30 b(1996.)456 2277 y([dlL01])188 b(Rafael)39 b(de)f(la)h(Lla)n(v)n(e.)g(A)f(tutorial)h(on)g(KAM)e (theory)-6 b(.)38 b(In)g Fv(Smo)l(oth)i(er)l(go)l(dic)h(the)l(ory)874 2368 y(and)30 b(its)h(applic)l(ations)g(\(Se)l(attle,)i(W)-6 b(A,)29 b(1999\))p Fz(,)i(pages)f(175{292.)h(Amer.)d(Math.)h(So)r(c.,) 874 2459 y(Pro)n(vidence,)d(RI,)f(2001.)456 2551 y([dlLN02])130 b(R.)21 b(de)h(la)h(Lla)n(v)n(e)f(and)g(P)n(etro)n(v)g(N.)f(Smo)r (othnes)h(of)g(conjugacies)i(of)f(critical)h(circle)f(maps:)874 2642 y(an)i(empirical)h(study)-6 b(.)25 b Fv(Exp.)i(Math.)p Fz(,)g(11:219{241,)i(2002.)456 2733 y([dlLO99])128 b(R.)19 b(de)h(la)h(Lla)n(v)n(e)f(and)f(R.)h(Oba)n(y)n(a.)g(Regularit)n(y)g(of) h(the)e(comp)r(osition)i(op)r(erator)g(in)e(spaces)874 2825 y(of)26 b(H\177)-38 b(older)26 b(functions.)g Fv(Discr)l(ete)j (Contin.)f(Dynam.)f(Systems)p Fz(,)g(5\(1\):157{184,)j(1999.)456 2916 y([D)n(V58])185 b(Ren)n(\023)-36 b(e)25 b(De)g(V)-6 b(ogelaere.)27 b(On)d(the)h(structure)h(of)g(symmetric)d(p)r(erio)r (dic)k(solutions)f(of)h(con-)874 3007 y(serv)l(ativ)n(e)21 b(systems,)h(with)g(applications.)h(In)e Fv(Contributions)k(to)f(the)h (the)l(ory)g(of)e(nonlin-)874 3099 y(e)l(ar)i(oscil)t(lations,)h(V)-6 b(ol.)24 b(IV)p Fz(,)f(Annals)g(of)g(Mathematics)h(Studies,)f(no)g(41,) h(pages)g(53{84.)874 3190 y(Princeton)i(Univ)n(ersit)n(y)e(Press,)j (Princeton,)g(N.J.,)g(1958.)456 3281 y([EW87])169 b(Jean-Pierre)18 b(Ec)n(kmann)e(and)g(P)n(eter)i(Witt)n(w)n(er.)g(A)e(complete)h(pro)r (of)h(of)g(the)f(Feigen)n(baum)874 3372 y(conjectures.)27 b Fv(J.)g(Statist.)h(Phys.)p Fz(,)f(46\(3-4\):455{475,)j(1987.)456 3464 y([F)-6 b(at89])188 b(Alb)r(ert)23 b(F)-6 b(athi.)24 b(Expansiv)n(eness,)h(h)n(yp)r(erb)r(olicit)n(y)e(and)g(Hausdor\013)h (dimension.)f Fv(Comm.)874 3555 y(Math.)k(Phys.)p Fz(,)g (126\(2\):249{262,)j(1989.)456 3646 y([FdlL92a])100 b(C.)18 b(F)-6 b(alcolini)19 b(and)e(R.)h(de)f(la)h(Lla)n(v)n(e.)g(Numerical)f (calculation)i(of)g(domains)e(of)h(analyticit)n(y)874 3738 y(for)29 b(p)r(erturbation)g(theories)h(in)f(the)g(presence)g(of)g (small)h(divisors.)g Fv(J.)g(Statist.)i(Phys.)p Fz(,)874 3829 y(67\(3-4\):645{666,)e(1992.)456 3920 y([FdlL92b])95 b(C.)32 b(F)-6 b(alcolini)34 b(and)d(R.)h(de)f(la)i(Lla)n(v)n(e.)f(A)f (rigorous)i(partial)g(justi\014cation)g(of)f(Greene's)874 4012 y(criterion.)27 b Fv(J.)g(Statist.)h(Phys.)p Fz(,)f (67\(3-4\):609{643,)j(1992.)456 4103 y([F)n(G99])192 b(John)28 b(Erik)g(F)-6 b(orn\032ss)30 b(and)d(Estela)j(A.)e(Ga)n(v)n (osto.)i(T)-6 b(angencies)29 b(for)g(real)g(and)f(complex)874 4194 y(H)n(\023)-36 b(enon)25 b(maps:)34 b(an)25 b(analytic)h(metho)r (d.)f Fv(Exp)l(eriment.)k(Math.)p Fz(,)d(8\(3\):253{260,)j(1999.)456 4286 y([FR81])193 b(V)-6 b(alter)32 b(F)-6 b(rancesc)n(hini)32 b(and)f(Lucio)i(Russo.)f(Stable)g(and)g(unstable)g(manifolds)g(of)h (the)874 4377 y(H)n(\023)-36 b(enon)25 b(mapping.)g Fv(J.)i(Statist.)i (Phys.)p Fz(,)d(25\(4\):757{769,)k(1981.)456 4468 y([Gor85])172 b(Daniel)36 b(L.)g(Goro\013.)h(Hyp)r(erb)r(olic)f(sets)g(for)h(t)n (wist)f(maps.)g Fv(Er)l(go)l(dic)i(The)l(ory)g(Dynam.)874 4560 y(Systems)p Fz(,)27 b(5\(3\):337{339,)j(1985.)456 4651 y([Gre79])176 b(J.)25 b(M.)g(Greene.)g(A)f(metho)r(d)f(for)j (determining)d(a)i(sto)r(c)n(hastic)h(transition.)f Fv(Jour.)i(Math.) 874 4742 y(Phys.)p Fz(,)f(20:1183{1201,)31 b(1979.)456 4834 y([Her])254 b(M.)35 b(Herman.)e(Some)h(op)r(en)g(problems)g(in)g (dynamical)g(systems.)g(In)g Fv(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)874 4925 y(the)24 b(International)h(Congr)l(ess)h(of)d(Mathematicians,)i(V) -6 b(ol.)23 b(II)g(\(Berlin,)h(1998\))p Fz(,)f(v)n(olume)874 5016 y(1998,)k(pages)g(797{808)h(\(electronic\).)456 5108 y([Jun91])175 b(I.)19 b(Jungreis.)h(A)f(metho)r(d)e(for)j(pro)n (ving)f(that)g(monotone)f(t)n(wist)i(maps)e(ha)n(v)n(e)h(no)g(in)n(v)l (arian)n(t)874 5199 y(circles.)27 b Fv(Er)l(go)l(dic)i(The)l(ory)f (Dynamic)l(al)g(Systems)p Fz(,)f(11\(1\):79{84,)i(1991.)p eop %%Page: 32 32 32 31 bop 456 251 a Fu(32)837 b(R.)23 b(DE)g(LA)h(LLA)-8 b(VE)23 b(AND)g(A.)g(OL)-8 b(VERA)456 450 y Fz([Ko)r(c99])166 b(H.)37 b(Ko)r(c)n(h.)h(A)f(renormalization)i(group)f(for)h (Hamiltonians,)i(with)d(applications)h(to)874 541 y(KAM)25 b(tori.)i Fv(Er)l(go)l(dic)h(The)l(ory)h(Dynam.)e(Systems)p Fz(,)g(19:1{47,)i(1999.)456 633 y([Ko)r(c02a])128 b(Hans)24 b(Ko)r(c)n(h.)g(On)f(the)h(renormalization)h(of)g(Hamiltonian)e(\015o)n (ws,)j(and)d(critical)j(in)n(v)l(ari-)874 724 y(an)n(t)f(tori.)i Fv(Discr)l(ete)i(Contin.)e(Dyn.)g(Syst.)p Fz(,)g(8\(3\):633{646,)i (2002.)456 815 y([Ko)r(c02b])123 b(Hans)22 b(Ko)r(c)n(h.)i(A)e (renormalization)i(group)f(\014xed)f(p)r(oin)n(t)h(asso)r(ciated)i (with)e(the)f(breakup)874 907 y(of)k(golden)g(in)n(v)l(arian)n(t)g (tori,)g(2002.)i(MP)p 2021 907 24 4 v 28 w(AR)n(C)d(#)h(02-175.)456 998 y([KP94])188 b(Sergei)33 b(Kuksin)f(and)g(J)r(\177)-41 b(urgen)34 b(P\177)-38 b(osc)n(hel.)33 b(On)f(the)g(inclusion)i(of)f (analytic)g(symplectic)874 1089 y(maps)26 b(in)h(analytic)h (Hamiltonian)f(\015o)n(ws)h(and)f(its)g(applications.)i(In)e Fv(Seminar)i(on)g(Dy-)874 1181 y(namic)l(al)g(Systems)i(\(St.)g (Petersbur)l(g,)h(1991\))p Fz(,)f(v)n(olume)26 b(12)j(of)g Fv(Pr)l(o)l(gr.)i(Nonline)l(ar)f(Dif-)874 1272 y(fer)l(ential)e (Equations)g(Appl.)p Fz(,)e(pages)g(96{116.)j(Birkh\177)-38 b(auser,)26 b(Basel,)i(1994.)456 1363 y([LC00])197 b(P)n(atrice)33 b(Le)g(Calv)n(ez.)h Fv(Dynamic)l(al)g(pr)l(op)l(erties)i(of)e(di\013e)l (omorphisms)h(of)f(the)h(annulus)874 1455 y(and)d(of)g(the)h(torus)p Fz(,)g(v)n(olume)d(4)h(of)g Fv(SMF/AMS)h(T)-6 b(exts)34 b(and)e(Mono)l(gr)l(aphs)p Fz(.)h(American)874 1546 y(Mathematical)21 b(So)r(ciet)n(y)-6 b(,)21 b(Pro)n(vidence,)h(RI,)d(2000.)k(T)-6 b(ranslated)21 b(from)f(the)f(1991)j(F)-6 b(renc)n(h)874 1637 y(original)27 b(b)n(y)e(Philipp)r(e)h(Mazaud.)456 1729 y([Mac92])158 b(R.)25 b(S.)h(MacKa)n(y)-6 b(.)26 b(Greene's)g(residue)g(criterion.)h Fv(Nonline)l(arity)p Fz(,)g(5\(1\):161{187,)i(1992.)456 1820 y([Mat88])162 b(J.)21 b(N.)f(Mather.)h(Destruction)g(of)g(in)n(v)l(arian)n(t)f (circles.)i Fv(Er)l(go)l(dic)i(The)l(ory)g(Dynamic)l(al)e(Sys-)874 1911 y(tems)p Fz(,)k(8\(Charles)i(Conley)e(Memorial)g(Issue\):199{214,) k(1988.)456 2003 y([McK82])136 b(R.)24 b(S.)f(McKa)n(y)-6 b(.)24 b Fv(R)l(enormalisation)j(in)f(A)n(r)l(e)l(a)h(Pr)l(eserving)h (Maps)p Fz(.)c(PhD)g(thesis,)h(Prince-)874 2094 y(ton)g(Univ)n(ersit)n (y)-6 b(,)25 b(1982.)456 2185 y([Mey75])155 b(K.)31 b(R.)h(Mey)n(er.)g (The)g(implicit)f(function)h(theorem)f(and)h(analytic)g(di\013eren)n (tial)g(equa-)874 2277 y(tions.)c(In)g Fv(Dynamic)l(al)h (systems|Warwick)j(1974)e(\(Pr)l(o)l(c.)g(Symp)l(os.)g(Appl.)f(T)-6 b(op)l(olo)l(gy)874 2368 y(and)29 b(Dynamic)l(al)f(Systems,)i(Univ.)e (Warwick,)h(Coventry,)h(1973/1974;)g(pr)l(esente)l(d)i(to)874 2459 y(E.)20 b(C.)g(Ze)l(eman)i(on)f(his)g(\014ftieth)g(birthday\))p Fz(,)h(pages)e(191{208.)h(Lecture)d(Notes)h(in)f(Math.,)874 2551 y(V)-6 b(ol.)26 b(468.)h(Springer,)f(Berlin,)g(1975.)456 2642 y([Mos86])162 b(J)r(\177)-41 b(urgen)29 b(Moser.)h(Monotone)f(t)n (wist)g(mappings)e(and)h(the)g(calculus)h(of)g(v)l(ariations.)h Fv(Er-)874 2733 y(go)l(dic)e(The)l(ory)g(Dynam.)g(Systems)p Fz(,)f(6\(3\):401{413,)j(1986.)456 2825 y([MP85])178 b(R.)19 b(S.)h(MacKa)n(y)f(and)h(I.)f(C.)h(P)n(erciv)l(al.)h(Con)n(v)n (erse)f(KAM:)f(theory)h(and)f(practice.)h Fv(Comm.)874 2916 y(Math.)27 b(Phys.)p Fz(,)g(98\(4\):469{512,)j(1985.)456 3007 y([Olv01])178 b(Arturo)23 b(Olv)n(era.)h(Estimation)g(of)h(the)e (amplitude)g(of)i(resonance)f(in)g(the)g(general)h(stan-)874 3099 y(dard)g(map.)g Fv(Exp)l(eriment.)k(Math.)p Fz(,)d (10\(3\):401{418,)k(2001.)456 3190 y([OS87a])159 b(Arturo)21 b(Olv)n(era)g(and)h(Carles)h(Sim\023)-38 b(o.)21 b(The)h(obstruction)g (metho)r(d)e(and)h(some)h(n)n(umerical)874 3281 y(exp)r(erimen)n(ts)35 b(related)i(to)g(the)f(standard)g(map.)g(In)g Fv(Perio)l(dic)h (solutions)i(of)e(Hamil-)874 3372 y(tonian)d(systems)h(and)f(r)l(elate) l(d)h(topics)f(\(Il)f(Cio)l(c)l(c)l(o,)i(1986\))p Fz(,)h(pages)d (235{244.)i(Reidel,)874 3464 y(Dordrec)n(h)n(t,)25 b(1987.)456 3555 y([OS87b])154 b(Arturo)28 b(Olv)n(era)h(and)g(Carles)i(Sim\023)-38 b(o.)28 b(An)h(obstruction)g(metho)r(d)f(for)i(the)e(destruction)874 3646 y(of)e(in)n(v)l(arian)n(t)f(curv)n(es.)h Fv(Phys.)i(D)p Fz(,)e(26\(1-3\):181{192,)k(1987.)456 3738 y([OS93])197 b(Arturo)33 b(Olv)n(era)h(and)g(Carles)i(Sim\023)-38 b(o.)34 b(Elliptic)h(non-birkho\013)e(p)r(erio)r(dic)i(in)f(the)g(t)n (wist)874 3829 y(map.)25 b Fv(Bifur)l(c)l(ation)j(and)g(Chaos)p Fz(,)e(3\(1\):165{185,)k(1993.)456 3920 y([PnJL87])118 b(E.)18 b(Pi)r(~)-41 b(na)20 b(and)e(L.)h(Jimenez)f(Lara.)h(On)f(the)f (symmetry)f(lines)j(of)g(the)f(standard)g(mapping.)874 4012 y Fv(Physic)l(a)p Fz(,)27 b(26D:369{378,)i(1987.)456 4103 y([Sim90])171 b(C.)36 b(Sim\023)-38 b(o.)34 b(Analytical)i(and)f (n)n(umerical)f(computation)h(of)g(in)n(v)l(arian)n(t)g(manifolds.)h (In)874 4194 y(Benest)23 b(D.)g(and)g(C.)g(F)-6 b(ro)r(esc)n(hl)n(\023) -36 b(e,)25 b(editors,)g Fv(Mo)l(dern)h(Metho)l(ds)g(in)f(Celestial)g (Me)l(chanics)p Fz(.)874 4286 y(Editions)h(F)-6 b(ron)n(ti)n(\022)-36 b(eres,)27 b(P)n(aris,)g(1990.)456 4377 y([Sta88])189 b(J.)27 b(Stark.)g(An)e(exhaustiv)n(e)i(criterion)g(for)h(the)e (nonexistence)h(of)g(in)n(v)l(arian)n(t)f(circles)i(for)874 4468 y(area-preserving)e(t)n(wist)g(maps.)g Fv(Comm.)g(Math.)i(Phys.)p Fz(,)e(117\(2\):177{189,)31 b(1988.)456 4560 y([Sti93])206 b(Andreas)18 b(Stirnemann.)f(Renormalization)j(for)f(golden)g(circles.) h Fv(Comm.)h(Math.)g(Phys.)p Fz(,)874 4651 y(152\(2\):369{431,)30 b(1993.)456 4742 y([Sti97])206 b(Andreas)33 b(Stirnemann.)f(T)-6 b(o)n(w)n(ards)34 b(an)f(existence)g(pro)r(of)i(of)f(MacKa)n(y's)g (\014xed)e(p)r(oin)n(t.)874 4834 y Fv(Comm.)26 b(Math.)i(Phys.)p Fz(,)e(188\(3\):723{735,)31 b(1997.)456 4925 y([VT91])187 b(J.)29 b(J.)g(P)-6 b(.)29 b(V)-6 b(eerman)27 b(and)h(F.)h(M.)h(T)-6 b(angerman.)28 b(In)n(tersection)h(prop)r(erties)g(of)h(in)n(v)l(arian) n(t)874 5016 y(manifolds)c(in)f(certain)i(t)n(wist)f(maps.)f Fv(Comm.)i(Math.)g(Phys.)p Fz(,)g(139\(2\):245{265,)j(1991.)p eop %%Page: 33 33 33 32 bop 1213 251 a Fu(OBSTR)n(UCTION)30 b(AND)g(RENORMALIZA)-6 b(TION)687 b(33)555 450 y Fy(Dep)-5 b(ar)g(tment)35 b(of)i(Ma)-5 b(thema)g(tics,)37 b(The)e(University)f(of)j(Texas)e(a)-5 b(t)36 b(A)n(ustin,)h(A)n(ustin,)456 541 y(TX)27 b(78712-1082)555 633 y Fv(E-mail)g(addr)l(ess)6 b Fz(:)37 b Fa(llave@math.utexas.edu)555 788 y Fy(I)r(IMAS)29 b({)f(UNAM,)g(01000)h(M)1513 782 y(\023)1513 788 y(exico)f(D.F.,)g(Mexico)555 879 y Fv(E-mail)f(addr)l (ess)6 b Fz(:)37 b Fa(aoc@mym.iimas.unam.mx)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF