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y(gets)30 b(t)m(w)m(o)g(disjoin)m(t)f(closed)h(orien)m(ted)f(curv)m (es)j(in)d Fo(M)10 b Fq(.)43 b(In)30 b(this)g(case,)h(they)g(ha)m(v)m (e)g(a)456 1016 y(w)m(ell)23 b(de\014ned)i(linking)d(n)m(um)m(b)s(er.) 41 b(The)25 b(asymptotic)e(linking)f(n)m(um)m(b)s(er)j Fo(\025)3079 1031 y Fm(X)q(;Y)3218 1016 y Fq(\()p Fo(x;)17 b(y)t Fq(\))456 1132 y(is)27 b(found)h(b)m(y)g(dividing)e(the)j (linking)c(n)m(um)m(b)s(er)j(of)g(the)g(t)m(w)m(o)g(closed)g(curv)m(es) i(b)m(y)f Fo(T)14 b(S)456 1248 y Fq(and)46 b(letting)f Fo(T)61 b Fq(and)46 b Fo(S)53 b Fq(gro)m(w)47 b(to)f(in\014nit)m(y)-8 b(.)85 b(The)48 b(a)m(v)m(erage)f(linking)e(n)m(um)m(b)s(er)456 1365 y(\003\()p Fo(X)r(;)17 b(Y)k Fq(\))32 b(is)g(de\014ned)i(to)e(b)s (e)h(the)g(in)m(tegral)e(of)h Fo(\025)2214 1380 y Fm(X)q(;Y)2354 1365 y Fq(\()p Fo(x;)17 b(y)t Fq(\))31 b(o)m(v)m(er)j Fo(M)f Fk(\002)23 b Fo(M)10 b Fq(.)555 1481 y(The)42 b(theorem)f(form)m(ulated)f(b)m(y)i(Arnold)e(yields)g(an)h(in)m (terpretation)f(of)h(the)456 1597 y(Hopf)47 b(in)m(v)-5 b(arian)m(t)47 b(of)g Fo(X)r(;)17 b(Y)70 b Fq(in)47 b(terms)h(of)f(the) i(asymptotic)e(linking)f(n)m(um)m(b)s(er,)456 1713 y(more)31 b(precisely)-8 b(,)32 b(the)h(Hopf)e(in)m(v)-5 b(arian)m(t)31 b(of)h Fo(X)r(;)17 b(Y)53 b Fq(equals)32 b(their)g(a)m(v)m(erage)g (linking)456 1830 y(n)m(um)m(b)s(er.)42 b(Recall)28 b(that)i(the)f (classical)f(Hopf)i(in)m(v)-5 b(arian)m(t)28 b(classifying)g(con)m(tin) m(uous)456 1946 y(maps)f Fo(S)772 1910 y Fn(3)838 1946 y Fk(!)h Fo(S)1032 1910 y Fn(2)1098 1946 y Fq(can)g(b)s(e)f(in)m (terpreted)h(as)g(linking)d(n)m(um)m(b)s(er,)j(cf.)g([BT)q(].)41 b(Supp)s(ose)456 2062 y Fo(f)53 b Fq(:)42 b Fo(S)692 2026 y Fn(3)774 2062 y Fk(!)g Fo(S)982 2026 y Fn(2)1062 2062 y Fq(is)f(a)g(di\013eren)m(tiable)f(represen)m(tativ)m(e)j(of)e (the)g(homotop)m(y)g(class)456 2178 y([)p Fo(f)11 b Fq(])46 b Fk(2)g Fo(\031)782 2193 y Fn(3)822 2178 y Fq(\()p Fo(S)926 2142 y Fn(2)965 2178 y Fq(\))e(and)f Fo(p;)17 b(q)50 b Fk(2)c Fo(S)1611 2142 y Fn(2)1694 2178 y Fq(are)d(regular)g(v)-5 b(alues)43 b(of)g Fo(f)11 b Fq(.)76 b(Then)45 b(the)e(Hopf)456 2294 y(in)m(v)-5 b(arian)m(t)31 b(of)h Fo(f)43 b Fq(is)32 b(equal)h(to)f(the)h(linking)d(n)m(um)m(b)s(er)j(of)f Fo(f)2557 2258 y Fi(\000)p Fn(1)2651 2294 y Fq(\()p Fo(p)p Fq(\))h(and)f Fo(f)3057 2258 y Fi(\000)p Fn(1)3151 2294 y Fq(\()p Fo(q)t Fq(\).)555 2411 y(The)g(v)m(ector)g(space)g(of)e (di\013eren)m(tial)f(forms)h(of)g(degree)i Fo(k)h Fq(will)c(b)s(e)i (denoted)g(b)m(y)456 2527 y(\012)526 2491 y Fm(k)569 2527 y Fq(,)i(in)f(case)h(of)g(compact)f(supp)s(ort)h(w)m(e)h(will)d (use)i(\012)2379 2491 y Fm(k)2379 2552 y(c)2455 2527 y Fq(.)44 b(Recall)32 b(that)g(if)g Fo(N)43 b Fq(is)32 b(an)456 2643 y(arbitrary)37 b(manifold,)h(a)g(double)h(form)e(on)i Fo(N)e Fk(\002)26 b Fo(M)50 b Fq(is)38 b(a)h(form)e(on)i Fo(M)49 b Fq(taking)456 2759 y(v)-5 b(alues)27 b(in)g(the)g(forms)g(on) g Fo(N)10 b Fq(.)43 b(Here,)29 b(one)f(can)f(in)m(terc)m(hange)i(the)e (roles)g(of)g Fo(N)38 b Fq(and)456 2876 y Fo(M)10 b Fq(,)39 b(a)e(double)g(form)f(on)i Fo(N)e Fk(\002)26 b Fo(M)48 b Fq(can)37 b(also)g(b)s(e)g(considered)i(as)e(a)g(form)g(on)g Fo(N)456 2992 y Fq(taking)31 b(v)-5 b(alues)33 b(in)f(the)h(forms)e(on) i Fo(M)10 b Fq(.)1530 3263 y(2.)55 b Fp(Linking)38 b(f)n(orms)555 3438 y Fq(In)f(this)g(section,)h(w)m(e)g(de\014ne)g(the)f(linking)e(n)m (um)m(b)s(er)i(and)g(the)h(linking)c(form.)456 3554 y(W)-8 b(e)42 b(giv)m(e)h(a)f(new)h(pro)s(of)e(for)h(the)h(existence)h(of)d(a) h(linking)f(form.)71 b(The)43 b(pro)s(of)456 3670 y(in)31 b([ArK])h(uses)h(argumen)m(ts)f(of)f(a)h(more)f(homological)d(t)m(yp)s (e)33 b(and)f(requires)h(that)456 3786 y Fo(M)46 b Fq(has)35 b(the)h(in)m(tegral)d(homology)h(of)h(a)g(3-sphere.)52 b(W)-8 b(e)35 b(use)i(the)e(theory)h(of)f(the)456 3903 y(P)m(oisson)e(equation)f(on)g(Riemannian)f(manifolds.)555 4019 y(The)36 b(linking)d(n)m(um)m(b)s(er)j(of)f(t)m(w)m(o)g(closed)h (orien)m(ted)f(disjoin)m(t)f(one-dimensional)456 4135 y(submanifolds)40 b Fo(N)1132 4150 y Fn(1)1171 4135 y Fo(;)17 b(N)1293 4150 y Fn(2)1376 4135 y Fk(\032)44 b Fo(M)52 b Fq(can)42 b(b)s(e)g(de\014ned)h(as)f(follo)m(ws.)70 b(F)-8 b(or)41 b Fo(i)j Fk(2)g(f)p Fq(1)p Fo(;)17 b Fq(2)p Fk(g)456 4251 y Fq(c)m(ho)s(ose)35 b(disjoin)m(t)f(tubular)g(neigh)m(b) s(orho)s(o)s(ds)g Fo(W)2211 4266 y Fm(i)2274 4251 y Fq(of)g Fo(N)2465 4266 y Fm(i)2528 4251 y Fq(in)g Fo(M)45 b Fq(and)35 b(represen)m(ta-)456 4368 y(tiv)m(es)i Fo(\027)736 4383 y Fm(i)799 4368 y Fk(2)e Fq(\012)970 4331 y Fn(2)970 4392 y Fm(c)1010 4368 y Fq(\()p Fo(W)1140 4383 y Fm(i)1168 4368 y Fq(\))i(of)f(the)h(P)m(oincar)m(\023)-46 b(e)37 b(duals)g(of)f Fo(N)2387 4383 y Fm(i)2450 4368 y Fk(\032)f Fo(W)2654 4383 y Fm(i)2719 4368 y Fq(.)56 b(The)38 b(extensions)456 4484 y(of)43 b Fo(\027)626 4499 y Fm(i)697 4484 y Fq(to)h(the)g(whole)f (of)g Fo(M)54 b Fq(b)m(y)45 b(zero)f(will)d(also)h(b)s(e)i(denoted)h(b) m(y)f Fo(\027)3047 4499 y Fm(i)3119 4484 y Fq(.)76 b(Since)456 4600 y Fo(H)545 4564 y Fn(2)584 4600 y Fq(\()p Fo(M)10 b Fq(\))28 b(=)f(0,)33 b(w)m(e)g(can)g(c)m(ho)s(ose)h(1-forms)d Fo(\015)2037 4615 y Fm(i)2097 4600 y Fq(suc)m(h)j(that)f Fo(d\015)2631 4615 y Fm(i)2686 4600 y Fq(=)27 b Fo(\027)2837 4615 y Fm(i)2898 4600 y Fq(.)456 4789 y Ft(De\014nition)36 b(1.)49 b Fq(The)33 b Fh(linking)i(numb)-5 b(er)32 b Fq(of)g Fo(N)2169 4804 y Fn(1)2209 4789 y Fo(;)17 b(N)2331 4804 y Fn(2)2403 4789 y Fq(is)33 b(denoted)g(b)m(y)h Fo(l)r(k)i Fq(and)d(de-)456 4905 y(\014ned)g(b)m(y)1415 5135 y Fo(l)r(k)s Fq(\()p Fo(N)1616 5150 y Fn(1)1656 5135 y Fo(;)17 b(N)1778 5150 y Fn(2)1817 5135 y Fq(\))28 b(=)1986 5000 y Fl(Z)2041 5225 y Fm(M)2137 5135 y Fo(\015)2188 5150 y Fn(1)2249 5135 y Fk(^)23 b Fo(\027)2386 5150 y Fn(2)2458 5135 y Fo(:)p eop %%Page: 3 3 3 2 bop 1131 235 a Fj(ON)33 b(THE)h(ASYMPTOTIC)f(LINKING)f(NUMBER)637 b(3)555 435 y Fq(The)30 b(linking)c(n)m(um)m(b)s(er)i(is)g(indep)s (enden)m(t)i(of)e(c)m(hoices.)43 b(The)29 b(pro)s(of)e(and)i(the)g(re-) 456 551 y(lation)24 b(of)h(De\014nition)g(1)h(to)f(other)i(p)s(ossible) e(de\014nitions)h(of)f(the)i(linking)d(n)m(um)m(b)s(er)456 667 y(can)32 b(b)s(e)h(found)g(in)f([BT].)456 841 y Ft(De\014nition)k (2.)49 b Fq(A)25 b Fh(linking)j(form)d Fq(on)h Fo(M)36 b Fq(is)25 b(a)h(double)f(form)f Fo(L)i Fq(on)g Fo(M)19 b Fk(\002)8 b Fo(M)37 b Fq(suc)m(h)456 957 y(that)29 b(for)g(an)m(y)h(t)m(w)m(o)g(disjoin)m(t)f(closed)g(orien)m(ted)h (one-dimensional)d(submanifolds)456 1074 y Fo(N)534 1089 y Fn(1)573 1074 y Fo(;)17 b(N)695 1089 y Fn(2)767 1074 y Fq(of)32 b Fo(M)43 b Fq(the)33 b(equalit)m(y)1464 1278 y Fo(l)r(k)s Fq(\()p Fo(N)1665 1293 y Fn(1)1705 1278 y Fo(;)17 b(N)1827 1293 y Fn(2)1867 1278 y Fq(\))27 b(=)2036 1142 y Fl(Z)2091 1368 y Fm(N)2147 1377 y Fg(1)2202 1142 y Fl(Z)2258 1368 y Fm(N)2314 1377 y Fg(2)2369 1278 y Fo(L)-1979 b Fq(\(1\))456 1495 y(holds.)456 1669 y Fh(Example)33 b(1.)49 b Fq(W)-8 b(e)32 b(w)m(an)m(t)g(to)f(describ)s(e)h(a)g(linking) d(form)h(on)i Ff(R)2709 1633 y Fn(3)2754 1669 y Fq(.)43 b(F)-8 b(or)31 b(this)g(exam-)456 1785 y(ple,)41 b(let)f Fk(\002)g Fq(b)s(e)g(the)h(usual)f(cross-pro)s(duct)g(of)g(t)m(w)m(o)h (v)m(ectors)g(in)f Ff(R)2943 1749 y Fn(3)2988 1785 y Fq(,)i(the)e(stan-)456 1901 y(dard)d(inner)g(pro)s(duct)g(on)g Ff(R)1518 1865 y Fn(3)1601 1901 y Fq(will)e(b)s(e)i(denoted)h(b)m(y)g Fk(h)f Fo(;)54 b Fk(i)p Fq(.)j(If)37 b Fo(V)57 b Fk(2)36 b Fo(T)3095 1916 y Fm(x)3139 1901 y Ff(R)3205 1865 y Fn(3)3287 1901 y Fq(and)456 2018 y Fo(W)41 b Fk(2)28 b Fo(T)740 2033 y Fm(y)782 2018 y Ff(R)848 1981 y Fn(3)893 2018 y Fq(,)33 b(w)m(e)g(de\014ne)h Fo(L)1444 2034 y Fe(R)1492 2015 y Fg(3)1564 2018 y Fq(b)m(y)1203 2237 y Fo(L)1269 2253 y Fe(R)1317 2234 y Fg(3)1356 2237 y Fq(\()p Fo(V)5 b(;)17 b(W)d Fq(\))27 b(=)1814 2169 y(1)p 1785 2214 108 4 v 1785 2305 a(4)p Fo(\031)1912 2169 y Fk(h)p Fo(V)5 b(;)17 b(W)36 b Fk(\002)22 b Fq(\()p Fo(x)h Fk(\000)g Fo(y)t Fq(\))p Fk(i)p 1912 2214 716 4 v 2086 2305 a(k)p Fo(x)g Fk(\000)f Fo(y)t Fk(k)2415 2276 y Fn(3)2670 2237 y Fo(:)-2241 b Fq(\(2\))456 2458 y(This)32 b(double)h(form)e(is)h (a)g(linking)f(form)g(on)i Ff(R)2157 2422 y Fn(3)2202 2458 y Fq(,)g(cf.)f([ArK].)555 2632 y(W)-8 b(e)29 b(c)m(ho)s(ose)h(a)e (metric)g(on)g Fo(M)40 b Fq(whic)m(h)29 b(w)m(e)g(will)e(use)i(in)f (sev)m(eral)i(constructions.)456 2748 y(Let)37 b Fk(\003)685 2763 y Fm(y)763 2748 y Fo(;)54 b(\016)887 2763 y Fm(y)966 2748 y Fq(resp)s(ectiv)m(ely)38 b Fo(d)1553 2763 y Fm(y)1632 2748 y Fq(denote)g(the)f(Ho)s(dge-)p Fk(\003)p Fq(-op)s(erator,)g(co)s (deriv)-5 b(ativ)m(e)456 2864 y(resp)s(ectiv)m(ely)27 b(deriv)-5 b(ativ)m(e)26 b(with)g(resp)s(ect)h(to)f(the)h(second)g (factor)f(in)f Fo(M)20 b Fk(\002)9 b Fo(M)h Fq(.)43 b(Let)456 2980 y(the)29 b(double)g(form)f Fo(g)33 b Fq(b)s(e)d(a)f(fundamen)m (tal)f(solution)g(for)h(\001)e(:)h(\012)2751 2944 y Fi(\003)2791 2980 y Fq(\()p Fo(M)10 b Fq(\))28 b Fk(!)g Fq(\012)3197 2944 y Fi(\003)3237 2980 y Fq(\()p Fo(M)10 b Fq(\).)456 3097 y(This)37 b(means)g(that)f(for)h Fo(\014)j Fk(2)c Fq(\012)1622 3060 y Fi(\003)1662 3097 y Fq(\()p Fo(M)10 b Fq(\),)39 b(the)e(form)f Fo(\020)42 b Fk(2)35 b Fq(\012)2571 3060 y Fi(\003)2611 3097 y Fq(\()p Fo(M)10 b Fq(\))38 b(de\014ned)g(b)m(y)g(the)456 3213 y(in)m(tegral)1343 3417 y Fo(\020)8 b Fq(\()p Fo(x)p Fq(\))27 b(=)1656 3281 y Fl(Z)1711 3507 y Fm(y)r Fi(2)p Fm(M)1891 3417 y Fo(\014)6 b Fq(\()p Fo(y)t Fq(\))21 b Fk(^)h(\003)2239 3432 y Fm(y)2281 3417 y Fo(g)t Fq(\()p Fo(x;)17 b(y)t Fq(\))456 3638 y(solv)m(es)35 b(the)h(P)m(oisson)f(equation)g(\001)p Fo(\020)k Fq(=)32 b Fo(\014)6 b Fq(.)50 b(A)35 b(fundamen)m(tal)f(solution)g(exists)h(if) 456 3754 y(there)d(are)h(no)f(non)m(trivial)e(harmonic)g(forms)i(on)g Fo(M)10 b Fq(.)44 b(As)32 b(the)h(manifold)c Fo(M)43 b Fq(has)456 3870 y(the)26 b(real)g(cohomology)e(of)i(a)g(3-sphere,)i (it)d(meets)i(this)f(condition)f(b)m(y)i(the)f(Ho)s(dge)456 3987 y(theorem.)54 b(F)-8 b(or)36 b(the)h(explicit)e(construction)h(of) g Fo(g)k Fq(cf.)c([deR)q(].)55 b(The)37 b(solution)e(of)456 4103 y(the)41 b(equation)f(\001)p Fo(\020)50 b Fq(=)41 b Fo(\014)47 b Fq(is)40 b(uniquely)h(determined)f(if)g Fo(\014)47 b Fq(has)41 b(degree)g(1)g(or)g(2.)456 4219 y(In)d(particular,)f(if)g Fo(\014)42 b Fk(2)36 b Fq(\012)1430 4183 y Fn(1)1470 4219 y Fq(\()p Fo(M)10 b Fq(\))39 b(and)e Fo(\020)45 b Fq(solv)m(es)39 b(\001)p Fo(\020)44 b Fq(=)36 b Fo(\014)6 b Fq(,)39 b(then)f Fo(d\020)44 b Fq(solv)m(es)39 b(the)456 4335 y(P)m(oisson)33 b(equation)f(with)g(the)h(inhomogeneous) f(part)g Fo(d\014)6 b Fq(.)43 b(The)34 b(equalit)m(y)924 4404 y Fl(Z)980 4629 y Fm(y)r Fi(2)p Fm(M)1159 4539 y Fo(d\014)6 b Fq(\()p Fo(y)t Fq(\))21 b Fk(^)h(\003)1558 4554 y Fm(y)1600 4539 y Fo(g)t Fq(\()p Fo(x;)17 b(y)t Fq(\))26 b(=)h Fo(d)2075 4404 y Fl(Z)2130 4629 y Fm(y)r Fi(2)p Fm(M)2310 4539 y Fo(\014)6 b Fq(\()p Fo(y)t Fq(\))21 b Fk(^)h(\003)2658 4554 y Fm(y)2699 4539 y Fo(g)t Fq(\()p Fo(x;)17 b(y)t Fq(\))-2521 b(\(3\))456 4760 y(follo)m(ws.)47 b(Both)35 b Fo(g)j Fq(and)c Fo(L)d Fq(=)g Fk(\003)1594 4775 y Fm(y)1635 4760 y Fo(d)1686 4775 y Fm(y)1727 4760 y Fo(g)38 b Fq(are)d(smo)s(oth)e(a)m(w)m(a)m(y)j(from)d(the)i(diagonal) d(in)456 4876 y Fo(M)37 b Fk(\002)26 b Fo(M)10 b Fq(.)62 b(The)39 b(singularities)d(of)i Fo(g)k Fq(resp)s(ectiv)m(ely)d Fo(L)g Fq(on)f(the)h(diagonal)d(are)j(of)456 4993 y(order)e(1)g(resp)s (ectiv)m(ely)i(2,)f(\(cf.)58 b([deR)q(],)39 b(p.)58 b(134\).)f(In)38 b(particular)d(b)s(oth)j(double)456 5109 y(forms)27 b(are)i(in)e Fo(L)1054 5073 y Fn(1)1094 5109 y Fq(.)42 b(The)29 b(pro)s(of)f(that)g Fo(L)h Fq(is)f(indeed)g(a)h(linking)d(form)h(relies)h(on)g(the)456 5225 y(follo)m(wing)h(fundamen)m(tal)j(prop)s(ert)m(y)h(of)g Fo(L)p Fq(.)p eop %%Page: 4 4 4 3 bop 456 235 a Fj(4)1263 b(T.)34 b(V)n(OGEL)456 435 y Ft(Lemma)j(2.)49 b Fh(F)-7 b(or)34 b(any)g(1-form)g Fo(!)39 b Fh(on)34 b Fo(M)46 b Fh(ther)-5 b(e)35 b(is)f(a)h(function)f Fo(h)h Fh(such)g(that)1053 509 y Fl(Z)1108 734 y Fm(y)r Fi(2)p Fm(M)1288 644 y Fo(d!)t Fq(\()p Fo(y)t Fq(\))20 b Fk(^)j(\003)1691 659 y Fm(y)1732 644 y Fo(d)1783 659 y Fm(y)1824 644 y Fo(g)t Fq(\()p Fo(x;)17 b(y)t Fq(\))26 b(=)i Fo(!)t Fq(\()p Fo(x)p Fq(\))21 b(+)h Fo(dh)p Fq(\()p Fo(x)p Fq(\))35 b Fo(:)-2391 b Fq(\(4\))456 891 y Fh(Pr)-5 b(o)g(of.)41 b Fq(Let)32 b(\()p Fo(M)h Fk(\002)22 b Fo(M)10 b Fq(\))1334 906 y Fm(")1403 891 y Fq(denote)33 b(the)g(complemen)m(t)e (of)h(an)h Fo(")p Fq(-neigh)m(b)s(orho)s(o)s(d)d(of)456 1009 y(the)i(diagonal)e(in)h Fo(M)h Fk(\002)22 b Fo(M)10 b Fq(.)44 b(Let)36 b(~)-52 b Fo(g)35 b Fq(denote)e(a)f(smo)s(oth)f (extension)i(of)f Fo(g)3115 925 y Fl(\014)3115 984 y(\014)3147 1048 y Fn(\()p Fm(M)7 b Fi(\002)p Fm(M)g Fn(\))3406 1056 y Fd(")456 1142 y Fq(to)32 b(the)h(whole)f(of)g Fo(M)h Fk(\002)23 b Fo(M)10 b Fq(.)44 b(Since)33 b Fo(\016)1830 1157 y Fm(y)1904 1142 y Fq(and)g Fo(d)2145 1157 y Fm(y)2218 1142 y Fq(are)g(formally)d(adjoin)m(t)h(w)m(e)j(ha)m(v)m(e)510 1225 y Fl(Z)565 1450 y Fm(y)r Fi(2)p Fm(M)745 1360 y Fo(d!)t Fq(\()p Fo(y)t Fq(\))20 b Fk(^)j(\003)1148 1375 y Fm(y)1189 1360 y Fo(d)1240 1375 y Fm(y)1281 1360 y Fo(g)t Fq(\()p Fo(x;)17 b(y)t Fq(\))26 b(=)1689 1225 y Fl(Z)1744 1450 y Fm(y)r Fi(2)p Fm(M)1924 1360 y Fo(d!)f Fk(^)e(\003)2200 1375 y Fm(y)2241 1360 y Fo(d)2292 1375 y Fm(y)2336 1360 y Fq(~)-52 b Fo(g)s Fq(\()p Fo(x;)17 b(y)t Fq(\))22 b(+)g Fo(o)p Fq(\()p Fo(")p Fq(\))1015 1636 y(=)1118 1501 y Fl(Z)1174 1726 y Fm(y)r Fi(2)p Fm(M)1353 1636 y Fo(\016)t(d!)t Fq(\()p Fo(y)t Fq(\))f Fk(^)h(\003)1803 1651 y Fm(y)1848 1636 y Fq(~)-53 b Fo(g)t Fq(\()p Fo(x;)17 b(y)t Fq(\))k(+)h Fo(o)p Fq(\()p Fo(")p Fq(\))1015 1912 y(=)1118 1776 y Fl(Z)1174 2002 y Fm(y)r Fi(2)p Fm(M)1353 1912 y Fq(\001)p Fo(!)k Fk(^)d(\003)1660 1927 y Fm(y)1701 1912 y Fo(g)t Fq(\()p Fo(x;)17 b(y)t Fq(\))k Fk(\000)2099 1776 y Fl(Z)2155 2002 y Fm(y)r Fi(2)p Fm(M)2334 1912 y Fo(d\016)t(!)t Fq(\()p Fo(y)t Fq(\))g Fk(^)h(\003)2784 1927 y Fm(y)2829 1912 y Fq(~)-53 b Fo(g)t Fq(\()p Fo(x;)17 b(y)t Fq(\))k(+)h Fo(o)p Fq(\()p Fo(")p Fq(\))1015 2197 y(=)27 b Fo(!)t Fq(\()p Fo(x)p Fq(\))22 b Fk(\000)h Fo(d)1504 2056 y Fl(\022)1576 2061 y(Z)1632 2287 y Fm(y)r Fi(2)p Fm(M)1811 2197 y Fo(\016)t(!)t Fq(\()p Fo(y)t Fq(\))e Fk(^)h(\003)2210 2212 y Fm(y)2252 2197 y Fo(g)t Fq(\()p Fo(x;)17 b(y)t Fq(\))2530 2056 y Fl(\023)2624 2197 y Fq(+)22 b Fo(o)p Fq(\()p Fo(")p Fq(\))32 b Fo(:)456 2429 y Fq(The)h(last)f(equalit)m(y)g(is)g(due)h(to)g(\(3\))f(and)g(to)h(the) g(fact)f(that)g Fo(g)k Fq(is)c(a)g(fundamen)m(tal)456 2545 y(solution)i(of)i(\001.)54 b(Let)36 b Fo(h)g Fq(b)s(e)g(the)g(in)m (tegral)f(in)g(the)h(second)i(summand.)53 b(Since)36 b Fo(")456 2661 y Fq(can)c(b)s(e)h(c)m(hosen)h(to)f(b)s(e)f (arbitrarily)e(small)g(w)m(e)k(ha)m(v)m(e)g(pro)m(v)m(ed)g(the)f (lemma.)p 3374 2661 4 66 v 3378 2599 59 4 v 3378 2661 V 3436 2661 4 66 v 456 2839 a Ft(Theorem)k(3.)49 b Fh(The)34 b(form)g Fo(L)28 b Fq(=)g Fk(\003)1749 2854 y Fm(y)1790 2839 y Fo(d)1841 2854 y Fm(y)1882 2839 y Fo(g)38 b Fh(is)d(a)g(linking) e(form)i(on)f Fo(M)10 b Fh(.)456 3016 y(Pr)-5 b(o)g(of.)41 b Fq(Let)53 b Fo(")61 b(>)h Fq(0.)104 b(F)-8 b(or)52 b Fo(i)62 b Fk(2)h(f)p Fq(1)p Fo(;)17 b Fq(2)p Fk(g)p Fq(,)57 b(let)52 b Fo(N)2358 3031 y Fm(i)2448 3016 y Fk(\032)63 b Fo(M)g Fq(b)s(e)53 b(disjoin)m(t)f(one-)456 3132 y(dimensional)26 b(submanifolds.)41 b(W)-8 b(e)29 b(denote)h(the)f Fo(")p Fk(\000)p Fq(ball)e(in)h Ff(R)2739 3096 y Fn(2)2813 3132 y Fq(b)m(y)i Fo(D)3029 3096 y Fn(2)3026 3157 y Fm(")3068 3132 y Fq(.)43 b(Cho)s(ose)456 3249 y(disjoin)m(t)32 b(tubular)h(neigh)m(b)s(orho)s(o)s(ds)g Fo(W)1896 3264 y Fm(i)1953 3221 y Fk(\030)1954 3253 y Fq(=)2060 3249 y Fo(N)2138 3264 y Fm(i)2189 3249 y Fk(\002)24 b Fo(D)2374 3213 y Fn(2)2371 3273 y Fm(")2446 3249 y Fq(of)34 b Fo(N)2637 3264 y Fm(i)2698 3249 y Fq(suc)m(h)h(that)f(the)g (dif-)456 3365 y(feomorphism)43 b(is)h(giv)m(en)h(b)m(y)h(the)g(geo)s (desic)f(exp)s(onen)m(tial)f(map)g(restricted)i(to)456 3481 y(the)c(normal)e(bundle)i(of)g Fo(N)1501 3496 y Fm(i)1571 3481 y Fq(in)g Fo(M)10 b Fq(.)72 b(Co)s(ordinates)42 b(on)g Fo(W)2690 3496 y Fn(1)2772 3481 y Fq(resp)s(ectiv)m(ely)h Fo(W)3405 3496 y Fn(2)456 3597 y Fq(compatible)33 b(with)j(this)f(pro)s (duct)h(decomp)s(osition)e(will)f(b)s(e)j(denoted)h(b)m(y)g(\()p Fo(p;)17 b(a)p Fq(\))456 3714 y(resp)s(ectiv)m(ely)37 b(\()p Fo(q)t(;)17 b(b)p Fq(\).)54 b(As)36 b(describ)s(ed)h(in)f([BT],) h(w)m(e)g(c)m(ho)s(ose)g(represen)m(tativ)m(es)i(of)456 3830 y(the)33 b(P)m(oincar)m(\023)-46 b(e)32 b(duals)h Fo(!)1340 3845 y Fm(i)1395 3830 y Fk(2)28 b Fq(\012)1559 3794 y Fn(2)1559 3854 y Fm(c)1599 3830 y Fq(\()p Fo(W)1729 3845 y Fm(i)1757 3830 y Fq(\))33 b(of)f Fo(N)2017 3845 y Fm(i)2078 3830 y Fq(suc)m(h)i(that)e(for)g(ev)m(ery)j Fo(p)27 b Fk(2)h Fo(W)3178 3845 y Fm(i)1517 3909 y Fl(Z)1573 4135 y Fi(f)p Fm(p)p Fi(g\002)p Fm(D)1794 4116 y Fg(2)1792 4151 y Fd(")1849 4045 y Fq(exp)1998 4004 y Fi(\003)2054 4045 y Fo(!)2115 4060 y Fm(i)2171 4045 y Fq(=)f(1)32 b Fo(:)-1926 b Fq(\(5\))456 4285 y(Since)36 b Fo(H)803 4249 y Fn(2)842 4285 y Fq(\()p Fo(M)10 b Fq(\))34 b(=)f(0)j(w)m(e)h (can)g(c)m(ho)s(ose)f(a)g(form)f Fo(\021)2259 4300 y Fn(1)2332 4285 y Fk(2)f Fq(\012)2502 4249 y Fn(1)2542 4285 y Fq(\()p Fo(M)10 b Fq(\))37 b(satisfying)e Fo(d\021)3296 4300 y Fn(1)3369 4285 y Fq(=)456 4401 y Fo(!)517 4416 y Fn(1)588 4401 y Fq(.)44 b(Then)33 b(b)m(y)h(Lemma)d(2)h(there)i(is)e (a)g(function)g Fo(h)h Fq(with)f(the)h(prop)s(ert)m(y)554 4611 y Fo(l)r(k)s Fq(\()p Fo(N)755 4626 y Fn(1)795 4611 y Fo(;)17 b(N)917 4626 y Fn(2)956 4611 y Fq(\))28 b(=)1114 4475 y Fl(Z)1170 4701 y Fm(M)1265 4611 y Fo(\021)1313 4626 y Fn(1)1375 4611 y Fk(^)22 b Fo(!)1524 4626 y Fn(2)1022 4882 y Fq(=)1114 4747 y Fl(Z)1170 4972 y Fm(x)p Fi(2)p Fm(W)1324 4981 y Fg(1)1378 4742 y Fl(\022)1452 4747 y(Z)1507 4972 y Fm(y)r Fi(2)p Fm(W)1658 4981 y Fg(2)1713 4882 y Fo(!)1774 4897 y Fn(1)1813 4882 y Fq(\()p Fo(y)t Fq(\))g Fk(^)g(\003)2101 4897 y Fm(y)2142 4882 y Fo(d)2193 4897 y Fm(y)2234 4882 y Fo(g)t Fq(\()p Fo(x;)17 b(y)t Fq(\))k Fk(\000)i Fo(dh)p Fq(\()p Fo(x)p Fq(\))2871 4742 y Fl(\023)2966 4882 y Fk(^)g Fo(!)3116 4897 y Fn(2)3155 4882 y Fq(\()p Fo(x)p Fq(\))33 b Fo(:)456 5109 y Fq(The)28 b(summand)f(con)m(taining)f Fo(dh)i Fq(is)f(zero)h(b)m(y)h(the)f(Stok)m(es's)h(theorem.)42 b(Since)28 b Fo(W)3405 5124 y Fn(1)456 5225 y Fq(and)i Fo(W)735 5240 y Fn(2)804 5225 y Fq(are)g(disjoin)m(t,)g(w)m(e)h(can)f (treat)g Fo(g)k Fq(lik)m(e)29 b(a)h(smo)s(oth)f(form)g(in)g(this)h(in)m (tegral.)p eop %%Page: 5 5 5 4 bop 1131 235 a Fj(ON)33 b(THE)h(ASYMPTOTIC)f(LINKING)f(NUMBER)637 b(5)456 435 y Fq(The)33 b(last)f(expression)i(is)e(therefore)h(equal)f (to)477 635 y Fl(Z)533 860 y Fn(\()p Fm(p;a)p Fn(\))p Fi(2)p Fm(W)794 869 y Fg(1)849 635 y Fl(Z)904 860 y Fn(\()p Fm(q)r(;b)p Fn(\))p Fi(2)p Fm(W)1156 869 y Fg(2)1212 770 y Fo(!)1273 785 y Fn(1)1312 770 y Fq(\(\()p Fo(p;)17 b(a)p Fq(\)\))22 b Fk(^)g Fo(!)1779 785 y Fn(2)1819 770 y Fq(\(\()p Fo(q)t(;)17 b(b)p Fq(\)\))k Fk(^)i Fq(\()p Fk(\003)2301 785 y Fm(y)2342 770 y Fo(d)2393 785 y Fm(y)2434 770 y Fo(g)t Fq(\(\()p Fo(p;)17 b Fq(0\))p Fo(;)g Fq(\()p Fo(q)t(;)g Fq(0\)\))j(+)i Fo(o)p Fq(\()p Fo(")p Fq(\)\))32 b Fo(:)456 1016 y Fq(Because)i(of)e(the)h(sp)s(ecial)e(c)m(hoice)i (\(5\))f(of)h(the)g(P)m(oincar)m(\023)-46 b(e)32 b(duals)1155 1241 y Fo(l)r(k)s Fq(\()p Fo(N)1356 1256 y Fn(1)1396 1241 y Fo(;)17 b(N)1518 1256 y Fn(2)1557 1241 y Fq(\))28 b(=)1726 1105 y Fl(Z)1781 1331 y Fm(p)p Fi(2)p Fm(N)1920 1340 y Fg(1)1975 1105 y Fl(Z)2031 1331 y Fm(q)r Fi(2)p Fm(N)2168 1340 y Fg(2)2223 1241 y Fk(\003)2273 1256 y Fm(y)2314 1241 y Fo(d)2365 1256 y Fm(y)2406 1241 y Fo(g)e Fq(+)c Fo(o)p Fq(\()p Fo(")p Fq(\))456 1477 y(follo)m(ws.)77 b(This)44 b(completes)g(the)h(pro)s(of)e(b)s(ecause)j Fo(")h(>)g Fq(0)d(can)h(b)s(e)f(arbitrarily)456 1593 y(small.)p 3374 1593 4 66 v 3378 1531 59 4 v 3378 1593 V 3436 1593 4 66 v 1293 1834 a(3.)55 b Fp(Systems)37 b(of)h(shor)-7 b(t)38 b(p)-7 b(a)g(ths)555 2009 y Fq(F)f(or)33 b(de\014ning)h(the)h(asymptotic)e(linking)f(n)m(um)m(b)s(er,)j(w)m(e)g (need)g(a)e(sp)s(eci\014c)i(w)m(a)m(y)456 2125 y(to)h(get)g(closed)g (curv)m(es)j(from)c(pieces)i(of)f(\015o)m(w)h(lines)e(of)h Fo(X)44 b Fq(or)36 b Fo(Y)21 b Fq(.)55 b(This)36 b(will)e(b)s(e)456 2241 y(ac)m(hiev)m(ed)24 b(b)m(y)g(closing)e(up)h(these)h(pieces)g(b)m (y)g(prescrib)s(ed)g(paths.)41 b(The)24 b(paths)f(used)456 2357 y(for)k(this)h(purp)s(ose)i(m)m(ust)e(ha)m(v)m(e)i(certain)e(prop) s(erties)g(describ)s(ed)h(in)f(De\014nition)f(3.)555 2474 y(As)h(this)f(de\014nition)g(is)g(tec)m(hnical)g(w)m(e)h(\014rst)g (explain)f(the)h(meaning)e(of)h(some)g(of)456 2590 y(the)32 b(requiremen)m(ts.)44 b(In)m(tegration)32 b(o)m(v)m(er)h(a)f(path)g(in) f(\006)i(is)f(p)s(ossible)f(b)m(y)i(\(ii\).)41 b(The)456 2706 y(con)m(tin)m(uit)m(y)i(condition)g(in)g(\(iii\))e(ensures)k(that) f(all)e(the)i(in)m(tegrals)e(considered)456 2822 y(are)d(measurable)f (functions)i(on)f Fo(M)e Fk(\002)27 b Fo(M)10 b Fq(,)42 b(the)e(conditions)e(\(6\),)j(\(7\))d(and)i(\(8\))456 2939 y(guaran)m(tee)h(that)g(the)h(system)g(of)f(short)g(paths)h(do)s (es)g(not)f(con)m(tribute)g(to)g(the)456 3055 y(asymptotic)32 b(linking)e(n)m(um)m(b)s(er.)45 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Fd(S)2248 5065 y Fm(y)r(;y)r Fn(\))2390 4975 y Fk(j)p Fo(L)p Fk(j)28 b Fq(=)f(0)-2236 b(\(8\))673 5225 y(exist)32 b(in)g(the)h Fo(L)1251 5189 y Fn(1)1291 5225 y Fq(-sense.)p eop %%Page: 6 6 6 5 bop 456 235 a Fj(6)1263 b(T.)34 b(V)n(OGEL)469 435 y Fq(\(iv\))49 b(The)33 b(sets)828 605 y Fk(S)888 620 y Fm(X)q(;)p Fn(\006)1050 605 y Fq(=)1154 525 y Fl(\010)1212 605 y Fq(\()p Fo(x;)17 b(y)t Fq(\))27 b Fk(2)h Fo(M)33 b Fk(\002)23 b Fo(M)1891 521 y Fl(\014)1891 580 y(\014)1924 605 y Fo(\036)1982 621 y Fn([0)p Fm(;T)10 b Fn(])2131 605 y Fo(x)23 b Fk(\\)f Fo(\033)t Fq(\()p Fo( )2457 620 y Fm(S)2509 605 y Fo(y)t(;)17 b(y)t Fq(\))25 b Fk(6)p Fq(=)j Fk(;)2874 525 y Fl(\011)831 766 y Fk(S)891 781 y Fn(\006)p Fm(;Y)1050 766 y Fq(=)1154 686 y Fl(\010)1212 766 y Fq(\()p Fo(x;)17 b(y)t Fq(\))27 b Fk(2)h Fo(M)33 b Fk(\002)23 b Fo(M)1891 682 y Fl(\014)1891 741 y(\014)1924 766 y Fo(\033)t Fq(\()p Fo(\036)2079 781 y Fm(T)2134 766 y Fo(x;)17 b(x)p Fq(\))23 b Fk(\\)f Fo( )2500 782 y Fn([0)p Fm(;S)t Fn(])2646 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y(c)m(ho)s(oses)45 b Fo(x)i Fq(=)1039 3125 y Fl(\000)1085 3206 y Fq(2)p Fo(e)1179 3170 y Fi(\000)p Fn(2)p Fm(=z)1337 3179 y Fg(0)1376 3206 y Fo(;)17 b(';)g(z)1573 3221 y Fn(0)1612 3125 y Fl(\001)1701 3206 y Fq(and)44 b Fo(z)1947 3221 y Fn(0)2031 3206 y Fq(small)d(enough,)47 b(pro)m(vided)d(that)g(the)456 3322 y(curv)m(e)34 b(is)e(close)g (enough)h(to)g(the)g Fo(z)t Fq(-axes.)555 3503 y(Using)39 b(our)g(de\014nition)f(of)h(a)g(system)h(of)f(short)g(paths)h(w)m(e)g (can)f(pro)m(v)m(e)i(their)456 3619 y(existence)50 b(while)d(the)i (remaining)d(parts)i(of)g(the)h(construction)g(remain)d(un-)456 3736 y(c)m(hanged.)61 b(This)39 b(closes)f(a)g(gap)g(in)g(the)g (original)d(pro)s(of)j(of)f(Arnold's)i(theorem)456 3852 y(found)e(in)f([Arn,)h(ArK].)56 b(Our)37 b(de\014nition)f(of)g(the)h (system)h(of)e(short)i(paths)f(has)456 3968 y(the)30 b(additional)c(adv)-5 b(an)m(tage)30 b(that)g(the)g(system)g(can)g(b)s (e)g(c)m(hosen)i(indep)s(enden)m(tly)456 4084 y(of)g Fo(X)r(;)17 b(Y)k Fq(,)33 b(cf.)f([ArK].)456 4266 y Ft(Theorem)37 b(5.)49 b Fh(L)-5 b(et)26 b Fq(\006)g Fh(b)-5 b(e)25 b(the)g(set)h(c)-5 b(onsisting)24 b(of)i(a)f(ge)-5 b(o)g(desic)24 b(of)i(minimal)e(length)456 4382 y(having)36 b(starting)h(p)-5 b(oint)36 b Fo(p)h Fh(and)g(end)f(p)-5 b(oint)37 b Fo(q)k Fh(for)36 b(any)h Fo(p;)17 b(q)36 b Fk(2)c Fo(M)10 b Fh(.Then)36 b Fq(\006)i Fh(is)f(a)456 4498 y(system)d(of)h(short)g(p)-5 b(aths.)456 4679 y(Pr)g(o)g(of.)41 b Fq(The)c(set)g(of)f(paths)h(\006)f (ful\014lls)f(the)h(conditions)g(\(i\),\(ii\))d(and)j(the)h(con)m(ti-) 456 4795 y(n)m(uit)m(y)28 b(condition)f(in)g(\(iii\))f(in)h (De\014nition)f(3)i(b)m(y)h(construction.)43 b(Let)28 b Fo(F)3017 4810 y Fm(T)6 b(;S)3163 4795 y Fq(denote)456 4912 y(the)33 b(function)1532 5041 y(1)p 1488 5085 137 4 v 1488 5177 a Fo(T)14 b(S)1651 4973 y Fl(Z)1706 5198 y Fm(\036)1748 5212 y Fg([0)p Fd(;T)8 b Fg(])1881 5198 y Fm(x)1942 4973 y Fl(Z)1997 5198 y Fm(\033)r Fn(\()p Fm(y)r(; )2169 5209 y Fd(S)2215 5198 y Fm(y)r Fn(\))2300 5108 y Fk(j)p Fo(L)p Fk(j)p eop %%Page: 7 7 7 6 bop 1131 235 a Fj(ON)33 b(THE)h(ASYMPTOTIC)f(LINKING)f(NUMBER)637 b(7)456 435 y Fq(on)38 b Fo(M)g Fk(\002)27 b Fo(M)10 b Fq(.)63 b(T)-8 b(o)39 b(pro)m(v)m(e)h(that)f(\006)g(has)g(the)g(prop) s(erties)g(\(iii\))d(in)i(De\014nition)g(3,)456 551 y(w)m(e)49 b(sho)m(w)h(\014rst)g(that)e(lim)1454 566 y Fm(T)6 b(;S)t Fi(!1)1729 551 y Fo(F)1792 566 y Fm(T)g(;S)1965 551 y Fq(=)55 b(0)48 b(holds)h(in)f(the)h Fo(L)2845 515 y Fn(1)2885 551 y Fq(-sense.)93 b(This)456 672 y(corresp)s(onds)39 b(to)f(\(6\).)61 b(F)-8 b(or)37 b Fo(y)k Fk(2)d Fo(M)49 b Fq(de\014ne)40 b Fo(S)2198 687 y Fn(1)2237 672 y Fq(\()p Fo(y)t Fq(\))d(=)2515 591 y Fl(\010)2573 672 y Fo(A)28 b Fk(2)g Fo(T)2825 687 y Fm(y)2867 672 y Fo(M)2971 587 y Fl(\014)2971 647 y(\014)3005 672 y Fk(k)p Fo(A)p Fk(k)f(\024)h Fq(1)3359 591 y Fl(\011)3417 672 y Fq(.)456 788 y(Let)1244 960 y Fo(f)11 b Fq(\()p Fo(x;)17 b(y)t Fq(\))26 b(=)72 b(max)1660 1026 y Fm(A)p Fi(2)p Fm(S)1803 1035 y Fg(1)1837 1026 y Fn(\()p Fm(y)r Fn(\))1946 879 y Fl(\010)2004 960 y Fk(j)p Fo(L)p Fq(\()p Fo(X)8 b Fq(\()p Fo(x)p Fq(\))p Fo(;)17 b(A)p Fq(\))p Fk(j)2539 879 y Fl(\011)2629 960 y Fo(:)-2200 b Fq(\(9\))456 1191 y(Let)37 b Fo(r)s Fq(\()p Fo(x;)17 b(y)t Fq(\))37 b(denote)h(the)g(distance)g(of)f(t)m(w)m(o)h(p) s(oin)m(ts)g Fo(x;)17 b(y)39 b Fk(2)e Fo(M)10 b Fq(.)59 b(By)38 b([deR)q(])f(\(p.)456 1307 y(122)f(and)h(134\),)g(there)h(is)f (a)f(smo)s(oth)h(double)f(form)g Fo(\013)i Fq(on)f Fo(M)f Fk(\002)25 b Fo(M)48 b Fq(suc)m(h)39 b(that)456 1424 y(for)32 b Fo(V)49 b Fk(2)28 b Fo(T)862 1439 y Fm(x)906 1424 y Fo(M)43 b Fq(and)33 b Fo(A)28 b Fk(2)g Fo(T)1485 1439 y Fm(y)1526 1424 y Fo(M)44 b Fq(w)m(e)33 b(ha)m(v)m(e)1525 1657 y Fo(L)p Fq(\()p Fo(V)5 b(;)17 b(A)p Fq(\))28 b(=)1988 1590 y Fo(\013)q Fq(\()p Fo(V)5 b(;)17 b(A)p Fq(\))p 1988 1635 318 4 v 1991 1726 a Fo(r)s Fq(\()p Fo(x;)g(y)t Fq(\))2265 1697 y Fn(2)2348 1657 y Fo(:)456 1889 y Fq(By)33 b(the)g(de\014nition)e(\(9\))h(of)h Fo(f)711 2109 y(f)11 b Fq(\()p Fo(x;)17 b(y)t Fq(\))27 b(=)1269 2042 y(1)p 1137 2087 313 4 v 1137 2178 a Fo(r)s Fq(\()p Fo(x;)17 b(y)t Fq(\))1411 2149 y Fn(2)1520 2109 y Fq(max)1476 2176 y Fm(A)p Fi(2)p Fm(S)1619 2185 y Fg(1)1654 2176 y Fn(\()p Fm(y)r Fn(\))1762 2029 y Fl(\010)1820 2109 y Fk(j)p Fo(\013)q Fq(\()p Fo(X)8 b Fq(\()p Fo(x)p Fq(\))p Fo(;)17 b(A)p Fq(\))p Fk(j)2352 2029 y Fl(\011)2437 2109 y Fq(=:)2709 2042 y(1)p 2578 2087 V 2578 2178 a Fo(r)s Fq(\()p Fo(x;)g(y)t Fq(\))2852 2149 y Fn(2)2910 2109 y Fq(^)-59 b Fo(\013)q Fq(\()p Fo(x;)17 b(y)t Fq(\))456 2336 y(and)31 b(since)g Fo(\013)g Fq(is)g(a)f(con)m(tin)m(uous)i (double)e(form,)40 b(^)-59 b Fo(\013)32 b Fq(is)e(a)h(con)m(tin)m(uous) g(function)f(on)456 2452 y Fo(M)25 b Fk(\002)15 b Fo(M)10 b Fq(.)44 b(Hence)1128 2372 y Fl(R)1175 2486 y Fm(x)p Fi(2)p Fm(M)1358 2452 y Fo(f)11 b Fq(\()p Fo(x;)17 b(y)t Fq(\))p Fo(\026)27 b Fq(is)i(b)s(ounded)g(ab)s(o)m(v)m(e)h(b)m(y)g(a)f (constan)m(t)h Fo(C)36 b Fq(whic)m(h)456 2574 y(do)s(es)24 b(not)g(dep)s(end)i(on)e Fo(y)t Fq(.)40 b(Let)24 b(v)m(ol\()p Fo(M)10 b Fq(\))24 b(denote)h(the)g(v)m(olume)f(of)f Fo(M)35 b Fq(with)24 b(resp)s(ect)456 2690 y(to)29 b(the)g(v)m(olume)g (form)f Fo(\026)p Fq(,)i(the)g(diameter)e(of)h Fo(M)40 b Fq(is)29 b(denoted)i(b)m(y)f(diam)n(\()p Fo(M)10 b Fq(\).)43 b(The)456 2806 y(geo)s(desics)d(are)g(parameterized)g(b)m(y)h Fo(s)g Fk(2)g Fq([0)p Fo(;)17 b Fq(1],)41 b(hence)h(their)d(v)m(elo)s (cit)m(y)h(v)m(ector)472 2922 y(_)-43 b Fo(\033)t Fq(\()p Fo( )616 2937 y Fm(S)667 2922 y Fo(y)t(;)17 b(y)t Fq(\)\()p Fo(s)p Fq(\))30 b(is)i(no)h(longer)e(than)i(diam)n(\()p Fo(M)10 b Fq(\).)44 b(By)33 b(F)-8 b(ubini's)32 b(theorem)g(w)m(e)i (\014nd)456 3147 y Fk(k)p Fo(F)569 3162 y Fm(T)6 b(;S)686 3147 y Fk(k)27 b Fq(=)867 3011 y Fl(Z)922 3237 y Fn(\()p Fm(x;y)r Fn(\))p Fi(2)p Fm(M)7 b Fi(\002)p Fm(M)1400 3079 y Fq(1)p 1356 3124 137 4 v 1356 3215 a Fo(T)14 b(S)1519 3011 y Fl(Z)1575 3237 y Fm(\036)1617 3251 y Fg([0)p Fd(;T)8 b Fg(])1750 3237 y Fm(x)1810 3011 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Fl(Z)1891 3645 y Fm(T)1847 3844 y Fn(0)1963 3619 y Fl(Z)2018 3844 y Fm(x)p Fi(2)p Fm(M)2200 3754 y Fo(f)2259 3674 y Fl(\000)2305 3754 y Fo(\036)2363 3769 y Fm(t)2392 3754 y Fo(x;)17 b(\033)t Fq(\()p Fo( )2651 3769 y Fm(S)2702 3754 y Fo(y)t(;)g(y)t Fq(\)\()p Fo(s)p Fq(\))3010 3674 y Fl(\001)3054 3754 y Fo(\026)32 b(dt)g(ds)h(\026)714 4026 y Fk(\024)873 3959 y Fq(1)p 829 4003 V 829 4094 a Fo(T)14 b(S)992 4026 y Fq(diam)n(\()p Fo(M)c Fq(\))p Fo(C)24 b Fq(v)m(ol\()p Fo(M)10 b Fq(\))p Fo(T)42 b Fq(=)2003 3959 y(1)p 1995 4003 66 4 v 1995 4094 a Fo(S)2087 4026 y Fq(diam)n(\()p Fo(M)10 b Fq(\))17 b(v)m(ol\()p Fo(M)10 b Fq(\))p Fo(C)40 b(:)456 4227 y Fq(The)d(last)e(expression)i(b)s (ecomes)g(arbitrarily)c(small)h(if)g Fo(S)40 b Fk(!)33 b(1)p Fq(.)53 b(This)37 b(pro)m(v)m(es)456 4344 y(\(6\))o(.)82 b(With)45 b(an)g(analogous)g(argumen)m(t)g(one)g(can)h(sho)m(w)i (\(7\).)82 b(T)-8 b(o)45 b(pro)m(v)m(e)i(\(8\))456 4460 y(observ)m(e)29 b(that)d(the)i(corresp)s(onding)f(in)m(tegral)f(in)g 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Fq(\(ii\))47 b Fh(L)-5 b(et)862 1340 y Fq(~)841 1366 y Fo(f)45 b Fh(b)-5 b(e)35 b(the)g(limit)f (function,)h(then)1251 1440 y Fl(Z)1306 1666 y Fm(M)7 b Fi(\002)p Fm(M)1553 1550 y Fq(~)1532 1576 y Fo(f)45 b(\026)22 b Fk(^)h Fo(\026)k Fq(=)1985 1440 y Fl(Z)2040 1666 y Fm(M)7 b Fi(\002)p Fm(M)2265 1576 y Fo(f)46 b(\026)22 b Fk(^)g Fo(\026)35 b(:)555 1805 y Fq(W)-8 b(e)52 b(omit)d(the)i(pro)s (of)f(whic)m(h)i(is)f(standard)g(using)g(metho)s(ds)g(from)f([Geo)o(]) 456 1921 y(and)37 b([DuS].)59 b(The)39 b(Birkho\013)e(ergo)s(dic)g (theorem)g(used)i(in)e([Arn,)h(ArK])g(implies)456 2037 y(that)31 b(the)g(limits)d(exist)k(in)e(the)i(p)s(oin)m(t)m(wise)f (sense.)45 b(\(Additionally)-8 b(,)28 b(in)j(the)h(theo-)456 2153 y(rem)21 b(of)g(Birkho\013)g(the)i(sequences)h(are)e(required)h (to)e(b)s(e)h(monotonous,)h(cf.)f([T)-8 b(em].\))456 2270 y(As)39 b(the)g(limits)d(in)i(De\014nition)f(3)h(are)h(supp)s (osed)h(to)e(exist)h(in)f(the)h Fo(L)3043 2233 y Fn(1)3083 2270 y Fk(\000)g Fq(sense,)456 2386 y(Theorem)32 b(6)h(is)f(suitable.) 612 2583 y(4.)55 b Fp(Asymptotic)38 b(linking)g(number)g(and)g(Arnolds) g(theorem)555 2758 y Fq(W)-8 b(e)31 b(no)m(w)g(de\014ne)g(the)g (asymptotic)e(linking)g(n)m(um)m(b)s(er.)43 b(T)-8 b(o)30 b(simplify)e(notation)456 2874 y(w)m(e)33 b(abbreviate)g Fo(\033)t Fq(\()p Fo(\036)1234 2889 y Fm(T)1289 2874 y Fo(x;)17 b(x)p Fq(\))33 b(b)m(y)g Fo(\033)1704 2889 y Fm(T)1792 2874 y Fq(and)g Fo(\033)t Fq(\()p Fo( )2142 2889 y Fm(S)2193 2874 y Fo(y)t(;)17 b(y)t Fq(\))30 b(b)m(y)k Fo(\033)2600 2889 y Fm(S)2651 2874 y Fq(.)456 3039 y Ft(De\014nition)i(4.)49 b Fq(The)37 b Fh(asymptotic)i(linking)e(numb)-5 b(er)37 b Fo(\025)2543 3054 y Fm(X)q(;Y)2719 3039 y Fq(is)f(the)h (limit)c(func-)456 3155 y(tion)847 3336 y Fo(\025)904 3351 y Fm(X)q(;Y)1044 3336 y Fq(\()p Fo(x;)17 b(y)t Fq(\))27 b(=)87 b(lim)1401 3398 y Fm(T)6 b(;S)t Fi(!1)1726 3268 y Fq(1)p 1682 3313 137 4 v 1682 3404 a Fo(T)14 b(S)1828 3336 y(l)r(k)s Fq(\()p Fo(\036)2009 3351 y Fn([0)p Fm(;T)c Fn(])2159 3336 y Fo(x)22 b Fk([)h Fo(\033)2380 3351 y Fm(T)2435 3336 y Fo(;)17 b( )2542 3351 y Fn([0)p Fm(;S)t Fn(])2687 3336 y Fo(y)25 b Fk([)e Fo(\033)2904 3351 y Fm(S)2955 3336 y Fq(\))33 b Fo(:)456 3544 y Fq(The)g Fh(aver)-5 b(age)34 b(linking)g(numb)-5 b(er)32 b Fq(\003\()p Fo(X)r(;)17 b(Y)k Fq(\))32 b(is)h(de\014ned)h(b)m(y)1311 3754 y(\003\()p Fo(X)r(;)17 b(Y)k Fq(\))28 b(=)1791 3618 y Fl(Z)1847 3844 y Fm(M)7 b Fi(\002)p Fm(M)2072 3754 y Fo(\025)2129 3769 y Fm(X)q(;Y)2301 3754 y Fo(\026)22 b Fk(^)g Fo(\026)33 b(:)456 3987 y Ft(Prop)s(osition)i(7.)49 b Fh(The)40 b(limit)g(in)g(De\014nition)g(4)g(exists)g(in)g(the)h Fo(L)2951 3951 y Fn(1)2990 3987 y Fh(-sense)f(and)456 4103 y(do)-5 b(es)42 b(not)i(dep)-5 b(end)42 b(on)h(the)h(system)f(of)h (short)f(p)-5 b(aths.)70 b(The)43 b(limit)g(function)h(is)456 4219 y(inte)-5 b(gr)g(able.)456 4384 y(Pr)g(o)g(of.)41 b Fq(By)32 b(prop)s(ert)m(y)h(\(iv\))e(of)h(the)g(short)g(path)g (system,)h(the)g(linking)c(n)m(um)m(b)s(ers)456 4501 y(are)39 b(almost)e(alw)m(a)m(ys)i(w)m(ell)g(de\014ned.)64 b(Using)38 b(the)i(linking)d(form)g(and)j(prop)s(ert)m(y)456 4617 y(\(iii\))29 b(in)j(De\014nition)f(3)h(w)m(e)i(\014nd)808 4826 y Fo(\025)865 4841 y Fm(X)q(;Y)1004 4826 y Fq(\()p Fo(x;)17 b(y)t Fq(\))27 b(=)87 b(lim)1362 4888 y Fm(T)6 b(;S)t Fi(!1)1687 4759 y Fq(1)p 1643 4803 V 1643 4895 a Fo(T)14 b(S)1806 4691 y Fl(Z)1861 4916 y Fm(\036)1903 4930 y Fg([0)p Fd(;T)8 b Fg(])2036 4916 y Fm(x)p Fi([)p Fm(\033)2163 4927 y Fd(T)2233 4691 y Fl(Z)2288 4916 y Fm( )2334 4930 y Fg([0)p Fd(;S)s Fg(])2463 4916 y Fm(y)r Fi([)p Fm(\033)2587 4927 y Fd(S)2653 4826 y Fo(L)-2263 b Fq(\(10\))1258 5135 y(=)87 b(lim)1362 5197 y Fm(T)6 b(;S)t Fi(!1)1687 5068 y Fq(1)p 1643 5112 V 1643 5204 a Fo(T)14 b(S)1806 5000 y Fl(Z)1905 5026 y Fm(T)1861 5225 y Fn(0)1977 5000 y Fl(Z)2077 5026 y Fm(S)2032 5225 y Fn(0)2144 5135 y Fo(L)2210 5054 y Fl(\000)2256 5135 y Fo(X)8 b Fq(\()p Fo(\036)2441 5150 y Fm(t)2470 5135 y Fo(x)p Fq(\))p Fo(;)17 b(Y)22 b Fq(\()p Fo( )2787 5150 y Fm(s)2824 5135 y Fo(y)t Fq(\))2914 5054 y Fl(\001)2991 5135 y Fo(ds)32 b(dt)g(:)p eop %%Page: 9 9 9 8 bop 1131 235 a Fj(ON)33 b(THE)h(ASYMPTOTIC)f(LINKING)f(NUMBER)637 b(9)456 435 y Fq(The)29 b(last)g(limit)c(exists)30 b(b)m(y)g(Theorem)f (6.)42 b(This)29 b(theorem)g(also)f(implies)e(that)j(the)456 551 y(limit)g(function)j(is)g Fo(L)1235 515 y Fn(1)1274 551 y Fq(,)h(just)g(lik)m(e)f(the)h(double)f(form)f Fo(L)i Fq(itself.)42 b(Hence)34 b(\003)e(is)g(w)m(ell)456 667 y(de\014ned.)p 3374 667 4 66 v 3378 605 59 4 v 3378 667 V 3436 667 4 66 v 456 888 a(W)-8 b(e)42 b(no)m(w)g(giv)m(e)g(the)g (de\014nition)f(of)g(the)h(Hopf)g(in)m(v)-5 b(arian)m(t)41 b(for)g(div)m(ergence{free)456 1004 y(v)m(ector)30 b(\014elds)g(on)f Fo(M)10 b Fq(.)43 b(Since)30 b(the)g(forms)f Fo(i)2014 1019 y Fm(X)2081 1004 y Fo(\026)g Fq(and)h Fo(i)2389 1019 y Fm(Y)2450 1004 y Fo(\026)f Fq(are)h(closed)f(and)h Fo(M)40 b Fq(has)456 1120 y(the)27 b(real)g(cohomology)e(of)i(the)h (three-sphere,)i(these)f(forms)d(are)i(exact.)42 b(W)-8 b(e)28 b(can)456 1237 y(therefore)j(c)m(ho)s(ose)h(1-forms)e Fo(\021)1568 1252 y Fm(X)1666 1237 y Fq(and)h Fo(\021)1902 1252 y Fm(Y)1994 1237 y Fq(with)g(the)g(prop)s(ert)m(y)h(that)f Fo(d\021)3087 1252 y Fm(X)3182 1237 y Fq(=)c Fo(i)3318 1252 y Fm(X)3386 1237 y Fo(\026)456 1353 y Fq(and)32 b Fo(d\021)744 1368 y Fm(Y)833 1353 y Fq(=)27 b Fo(i)969 1368 y Fm(Y)1030 1353 y Fo(\026)p Fq(.)456 1553 y Ft(De\014nition)36 b(5.)49 b Fq(The)28 b Fh(Hopf)i(invariant)c Fq(of)h Fo(X)r(;)17 b(Y)49 b Fq(is)26 b(de\014ned)j(to)e(b)s(e)g(the)h(in)m(tegral)1406 1802 y Fk(H)q Fq(\()p Fo(X)r(;)17 b(Y)22 b Fq(\))27 b(=)1904 1666 y Fl(Z)1959 1891 y Fm(M)2055 1802 y Fo(\021)2103 1817 y Fm(X)2193 1802 y Fk(^)22 b Fo(i)2314 1817 y Fm(Y)2375 1802 y Fo(\026)32 b(:)456 2053 y Ft(Prop)s(osition)j(8.)49 b Fh(The)41 b(Hopf)g(invariant)g(of)g Fo(X)r(;)17 b(Y)62 b Fh(do)-5 b(es)41 b(not)g(dep)-5 b(end)40 b(on)h(the)456 2169 y(choic)-5 b(e)42 b(of)g Fo(\021)928 2184 y Fm(X)996 2169 y Fh(.)69 b(F)-7 b(urthermor)i(e)42 b Fk(H)q Fq(\()p Fo(X)r(;)17 b(Y)k Fq(\))43 b(=)g Fk(H)q Fq(\()p Fo(Y)5 b(;)17 b(X)8 b Fq(\))42 b Fh(and)g(if)h Fo(Z)50 b Fh(is)43 b(another)456 2285 y(diver)-5 b(genc)g(e{fr)g(e)g(e)34 b(ve)-5 b(ctor)36 b(\014eld)g(on)g Fo(M)47 b Fh(we)36 b(have)f Fk(H)q Fq(\()p Fo(X)r(;)17 b(Y)45 b Fq(+)23 b Fo(Z)7 b Fq(\))30 b(=)g Fk(H)q Fq(\()p Fo(X)r(;)17 b(Y)k Fq(\))j(+)456 2401 y Fk(H)q Fq(\()p Fo(X)r(;)17 b(Z)7 b Fq(\))p Fh(.)456 2601 y(Pr)-5 b(o)g(of.)41 b Fq(Let)32 b Fo(\021)980 2565 y Fi(0)976 2626 y Fm(X)1076 2601 y Fq(b)s(e)g(another)g(1-form)e(with)i(the)g(prop)s(ert)m(y)h Fo(d\021)2766 2565 y Fi(0)2762 2626 y Fm(X)2857 2601 y Fq(=)28 b Fo(d\021)3060 2616 y Fm(X)3154 2601 y Fq(=)g Fo(i)3291 2616 y Fm(X)3359 2601 y Fo(\026)p Fq(.)456 2717 y(Hence)33 b Fo(\021)793 2732 y Fm(X)881 2717 y Fk(\000)22 b Fo(\021)1032 2681 y Fi(0)1028 2743 y Fm(X)1127 2717 y Fq(is)32 b(closed.)43 b(Since)32 b Fo(H)1893 2681 y Fn(1)1932 2717 y Fq(\()p Fo(M)10 b Fq(\))29 b(=)e(0)32 b(,)g(w)m(e)h(can)f(c)m(ho)s(ose)h(a)f(function)456 2834 y Fo(f)43 b Fq(on)32 b Fo(M)44 b Fq(suc)m(h)34 b(that)e Fo(d)-16 b(f)38 b Fq(=)27 b Fo(\021)1523 2849 y Fm(X)1613 2834 y Fk(\000)c Fo(\021)1765 2798 y Fi(0)1761 2859 y Fm(X)1828 2834 y Fq(.)44 b(By)33 b(the)g(Stok)m(es's)h(theorem)f(w)m(e) g(\014nd)788 2953 y Fl(Z)843 3178 y Fm(M)922 3088 y Fq(\()p Fo(\021)1008 3103 y Fm(X)1097 3088 y Fk(\000)23 b Fo(\021)1249 3047 y Fi(0)1245 3113 y Fm(X)1313 3088 y Fq(\))f Fk(^)g Fo(i)1494 3103 y Fm(Y)1555 3088 y Fo(\026)28 b Fq(=)1745 2953 y Fl(Z)1801 3178 y Fm(M)1896 3088 y Fo(d)-16 b(f)32 b Fk(^)23 b Fo(i)2133 3103 y Fm(Y)2194 3088 y Fo(\026)k Fq(=)2384 2953 y Fl(Z)2439 3178 y Fm(M)2535 3088 y Fo(d)p Fq(\()p Fo(f)11 b(i)2716 3103 y Fm(Y)2776 3088 y Fo(\026)p Fq(\))28 b(=)f(0)32 b Fo(:)456 3347 y Fq(Hence)38 b Fk(H)q Fq(\()p Fo(X)r(;)17 b(Y)k Fq(\))37 b(is)f(w)m(ell)g(de\014ned.)58 b(Cho)s(ose)37 b(a)g(1-form)e Fo(\021)2637 3362 y Fm(Y)2734 3347 y Fq(suc)m(h)k(that)d Fo(d\021)3273 3362 y Fm(Y)3369 3347 y Fq(=)456 3463 y Fo(i)489 3478 y Fm(Y)550 3463 y Fo(\026)c(:)g Fq(Since)1184 3664 y Fo(d)p Fq(\()p Fo(\021)1321 3679 y Fm(X)1410 3664 y Fk(^)23 b Fo(\021)1547 3679 y Fm(Y)1608 3664 y Fq(\))k(=)h Fo(i)1810 3679 y Fm(X)1877 3664 y Fo(\026)22 b Fk(^)h Fo(\021)2095 3679 y Fm(Y)2178 3664 y Fk(\000)f Fo(\021)2325 3679 y Fm(X)2415 3664 y Fk(^)h Fo(i)2537 3679 y Fm(Y)2598 3664 y Fo(\026)32 b(;)456 3866 y Fq(symmetry)f(follo)m(ws)f(from)g(the)i(Stok)m(es's)h(theorem)e (and)g(the)h(fact)f(that)g(2-forms)456 3983 y(comm)m(ute)45 b(with)h(di\013eren)m(tial)e(forms)h(of)g(an)m(y)i(degree.)85 b(The)47 b(last)e(statemen)m(t)456 4099 y(follo)m(ws)31 b(from)g Fo(i)1039 4114 y Fn(\()p Fm(Y)17 b Fn(+)p Fm(Z)5 b Fn(\))1263 4099 y Fo(\026)27 b Fq(=)g Fo(i)1485 4114 y Fm(Y)1546 4099 y Fo(\026)22 b Fq(+)g Fo(i)1758 4114 y Fm(Z)1815 4099 y Fo(\026)p Fq(.)p 3374 4099 V 3378 4036 59 4 v 3378 4099 V 3436 4099 4 66 v 456 4320 a(No)m(w)42 b(w)m(e)h(are)f(in)f(a)h(p)s(osition)e(to)i(state)g(and)g(pro)m(v)m(e)h (Arnolds)f(theorem.)71 b(The)456 4436 y(pro)s(of)24 b(is)h(analogous)f (to)h(the)h(pro)s(of)e(found)i(in)e([Arn)q(,)h(ArK],)i(but)e(the)h (asymptotic)456 4552 y(linking)k(n)m(um)m(b)s(er)k(is)e(no)h(longer)f (de\014ned)i(as)f(a)f(p)s(oin)m(t)m(wise)h(limit)c(but)k(as)g(a)g (limit)456 4668 y(existing)f(in)f(the)i Fo(L)1164 4632 y Fn(1)1204 4668 y Fk(\000)p Fq(sense.)456 4868 y Ft(Theorem)k(9.)49 b Fh(The)31 b(aver)-5 b(age)32 b(linking)f(numb)-5 b(er)32 b(of)g Fo(X)r(;)17 b(Y)54 b Fh(is)32 b(e)-5 b(qual)32 b(to)h(the)f(Hopf)456 4984 y(invariant)i(of)g Fo(X)r(;)17 b(Y)22 b Fh(,)34 b(i.e.)1496 5186 y Fq(\003\()p Fo(X)r(;)17 b(Y)k Fq(\))27 b(=)h Fk(H)q Fq(\()p Fo(X)r(;)17 b(Y)k Fq(\))35 b Fo(:)p eop %%Page: 10 10 10 9 bop 456 235 a Fj(10)1225 b(T.)34 b(V)n(OGEL)456 435 y Fh(Pr)-5 b(o)g(of.)41 b Fq(Recall)30 b(that)i Fo(d\021)1357 450 y Fm(X)1452 435 y Fq(=)27 b Fo(i)1588 450 y Fm(X)1656 435 y Fo(\026)32 b Fq(and)g Fo(L)c Fq(=)f Fk(\003)2183 450 y Fm(y)2224 435 y Fo(d)2275 450 y Fm(y)2316 435 y Fo(g)t Fq(.)43 b(Using)32 b(Lemma)e(2)i(w)m(e)h(\014nd)548 647 y Fk(H)q Fq(\()p Fo(X)r(;)17 b(Y)22 b Fq(\))27 b(=)1046 512 y Fl(Z)1101 737 y Fm(M)1197 647 y Fo(\021)1245 662 y Fm(X)1334 647 y Fk(^)c Fo(i)1456 662 y Fm(Y)1517 647 y Fo(\026)942 919 y Fq(=)1046 783 y Fl(Z)1101 1009 y Fm(x)p Fi(2)p Fm(M)1283 778 y Fl(\022)q(\022)1430 783 y(Z)1485 1009 y Fm(y)r Fi(2)p Fm(M)1665 919 y Fo(i)1698 934 y Fm(X)1766 919 y Fo(\026)p Fq(\()p Fo(y)t Fq(\))e Fk(^)h(\003)2112 934 y Fm(y)2154 919 y Fo(d)2205 934 y Fm(y)2246 919 y Fo(g)t Fq(\()p Fo(x;)17 b(y)t Fq(\))2524 778 y Fl(\023)2618 919 y Fk(\000)22 b Fo(dh)p Fq(\()p Fo(x)p Fq(\))2955 778 y Fl(\023)3051 919 y Fk(^)g Fo(i)3172 934 y Fm(Y)3234 919 y Fo(\026)32 b(:)456 1140 y Fq(Because)e Fo(i)858 1155 y Fm(Y)919 1140 y Fo(\026)e Fq(is)g(closed,)i(the)f (summand)f(con)m(taining)f Fo(dh)i Fq(do)s(es)g(not)f(con)m(tribute)456 1256 y(to)k(the)h(in)m(tegral)e(b)m(y)i(the)g(Stok)m(es's)i(theorem,)d (hence)1002 1460 y Fk(H)q Fq(\()p Fo(X)r(;)17 b(Y)k Fq(\))28 b(=)1499 1325 y Fl(Z)1555 1550 y Fm(x)p Fi(2)p Fm(M)1737 1325 y Fl(Z)1792 1550 y Fm(y)r Fi(2)p Fm(M)1972 1460 y Fo(L)p Fq(\()p Fo(X)r(;)17 b(Y)22 b Fq(\))32 b Fo(\026)p Fq(\()p Fo(x)p Fq(\))22 b Fk(^)h Fo(\026)p Fq(\()p Fo(y)t Fq(\))31 b Fo(:)-2442 b Fq(\(11\))456 1687 y(By)34 b(the)g(second)i (part)d(of)h(Theorem)g(6,)g(the)h(righ)m(t)e(hand)h(side)g(of)41 b(\(11\))33 b(is)g(equal)456 1803 y(to)463 1987 y Fl(Z)519 2213 y Fn(\()p Fm(x;y)r Fn(\))p Fi(2)p Fm(M)7 b Fi(\002)p Fm(M)943 1982 y Fl(\022)1075 2123 y Fq(lim)1016 2185 y Fm(T)f(;S)t Fi(!1)1341 2055 y Fq(1)p 1297 2100 137 4 v 1297 2191 a Fo(T)14 b(S)1460 1987 y Fl(Z)1560 2014 y Fm(T)1515 2213 y Fn(0)1631 1987 y Fl(Z)1731 2014 y Fm(S)1687 2213 y Fn(0)1798 2123 y Fo(L)1864 2042 y Fl(\000)1910 2123 y Fo(X)8 b Fq(\()p Fo(\036)2095 2138 y Fm(t)2124 2123 y Fo(x)p Fq(\))p Fo(;)17 b(Y)22 b Fq(\()p Fo( )2441 2138 y Fm(s)2478 2123 y Fo(y)t Fq(\))2568 2042 y Fl(\001)2645 2123 y Fo(ds)32 b(dt)2860 1982 y Fl(\023)2950 2123 y Fo(\026)p Fq(\()p Fo(x)p Fq(\))22 b Fk(^)h Fo(\026)p Fq(\()p Fo(y)t Fq(\))456 2353 y(and)32 b(this)h(is)f(equal)g(to)g (\003\()p Fo(X)r(;)17 b(Y)k Fq(\))33 b(b)m(y)i(\(10\))o(.)p 3374 2353 4 66 v 3378 2291 59 4 v 3378 2353 V 3436 2353 4 66 v 1660 2551 a Fp(References)456 2708 y Fs([Arn])41 b(V.)23 b(I.)g(Arnold,)g Fc(The)k(asymptotic)f(Hopf)g(invariant)h(and)e (its)g(applic)l(ations)p Fs(,)i(Pro)r(c.)21 b(Sum-)668 2808 y(mer)27 b(Sc)n(ho)r(ol)f(in)h(Di\013.)h(Equations)e(at)h (Dilizhan,)h(1973)d(\(1974\),)h(Erev)-5 b(an)26 b(\(in)i(Russian\);)668 2908 y(English)f(transl.:)36 b(Sel.)28 b(Math.)f(So)n(v.)g(5)g (\(1986\),)g(327-345.)456 3007 y([ArK])41 b(V.)19 b(I.)g(Arnold,)h(B.)f (A.)g(Khesin,)i Fb(T)-7 b(op)r(ological)17 b(Metho)r(ds)i(in)g(Hydro)r (dynamics)p Fs(,)h(Springer)668 3107 y(1998.)456 3207 y([BT])47 b(R.)28 b(Bott,)f(L.)h(W.)g(T)-7 b(u,)28 b Fb(Di\013eren)n(tial)f(F)-7 b(orms)27 b(in)h(Algebraic)f(T)-7 b(op)r(ology)p Fs(,)26 b(Springer)g(1982.)456 3306 y([deR])41 b(G.)28 b(de)g(Rham,)g Fb(Di\013eren)n(tiable)f(manifolds)p Fs(,)h(Springer)e(1984.)456 3406 y([DuS])42 b(N.)22 b(Dunford,)i(J.)e (T.)g(Sc)n(h)n(w)n(artz,)g Fb(Linear)f(op)r(erators)f(I)p Fs(,)i(In)n(terscience)f(Publishers)g(1958.)456 3505 y([FH])50 b(M.)24 b(F)-7 b(reedman,)25 b(Z.-X.)e(He,)j Fc(Diver)l(genc)l(e{fr)l(e)l(e)h(\014elds:)38 b(Ener)l(gy)27 b(and)g(asymptotic)g(cr)l(oss-)668 3605 y(ing)j(numb)l(er)p Fs(,)d(Ann.)h(of)g(Math.)f(134)g(\(1991\),)f(189-229.)456 3705 y([Geo])41 b(H.)28 b(O.)f(Georgii,)g Fb(Gibbs)g(Measures)g(and)g (Phase)g(T)-7 b(ransitions)p Fs(,)26 b(de)i(Gruyter)f(1988.)456 3804 y([Kre])40 b(U.)28 b(Krengel,)f Fb(Ergo)r(dic)f(theorems)p Fs(,)h(de)g(Gruyter)g(1985.)456 3904 y([Mof)6 b(])42 b(H.)c(K.)f(Mo\013at,)k Fc(The)f(de)l(gr)l(e)l(e)g(of)g(knotte)l(dness) e(of)i(tangle)l(d)g(vortex)f(lines)p Fs(,)i(J.)d(Fluid.)668 4004 y(Mec)n(h.)27 b(35)g(\(1969\),)f(117-129.)456 4103 y([T)-7 b(em])41 b(A.)30 b(A.)f(T)-7 b(emp)r(elman,)30 b Fc(Er)l(go)l(dic)j(the)l(or)l(ems)e(for)h(gener)l(al)g(dynamic)l(al)h (systems)p Fs(,)c(So)n(viet.)668 4203 y(Math.)f(Dokl.)f(8)g(\(1967\),)g (no.5,)g(1213-1216.)456 4302 y([W)-7 b(ol])41 b(L.)32 b(W)-7 b(oltjer,)33 b Fc(A)g(the)l(or)l(em)h(on)f(for)l(c)l(e-fr)l(e)l (e)i(magnetic)f(\014elds)p Fs(,)f(Pro)r(c.)e(Natn.)h(Acad.)f(Sci.)668 4402 y(44)26 b(\(1958\),)h(489-491.)555 4589 y Fr(Ma)-6 b(thema)g(tisches)46 b(Institut,)h(Universit)2103 4582 y(\177)2100 4589 y(at)d(M)2329 4582 y(\177)2326 4589 y(unchen,)j(Theresienstr.)e(39,)456 4689 y(80333)30 b(M)801 4682 y(\177)798 4689 y(unchen,)h(Germany)555 4788 y Fc(E-mail)g(addr)l (ess)7 b Fs(:)38 b Fa(thomas.vogel@mat)o(he)o(mat)o(ik)o(.un)o(i-)o(mu) o(enc)o(he)o(n.d)o(e)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF