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y(an)n(t)27 b Fr(H)5 b Ft(\()p Fr(X)r(;)11 b(Y)k Ft(\))27 b Fq(2)h Fp(R)i Ft(for)d(pairs)g(\()p Fr(X)r(;)11 b(Y)k Ft(\))27 b(of)g(div)n(ergence-free)i(v)n(ector)f(\014elds)f(on) 308 2043 y(a)h(three-dimensional)e(homology)g(sphere)h Fr(M)7 b Ft(.)45 b(Arnold)28 b([1])g(in)n(terpreted)e(this)308 2122 y(Hopf)20 b(in)n(v)l(arian)n(t)g(as)f(the)g(a)n(v)n(erage)i(o)n(v) n(er)e Fr(M)e Fq(\002)10 b Fr(M)27 b Ft(of)20 b(the)f(long-time)g (asymptotic)308 2200 y(linking)25 b(n)n(um)n(b)r(ers)e(of)h(pairs)h(of) f(\015o)n(w)g(lines)h(for)g Fr(X)k Ft(and)24 b Fr(Y)15 b Ft(.)37 b(In)24 b(order)g(to)g(mak)n(e)308 2279 y(this)k(in)n (terpretation)e(precise,)k(one)e(has)f(to)g(\014nd)g(a)g(coheren)n(t)h (w)n(a)n(y)f(of)h(closing)308 2358 y(long)i(pieces)h(of)f(\015o)n(w)f (lines)i(to)e(form)g(lo)r(ops)g(so)h(that)f(one)h(can)g(ev)l(aluate)g (the)308 2436 y(corresp)r(onding)g(linking)g(n)n(um)n(b)r(ers)f(and)h (compute)f(their)h(long-time)g(asymp-)308 2515 y(totics.)i(Arnold's)23 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a Fn(Date)t Fv(:)18 b(April)i(8,)f(2002;)f(MSC)h (2000:)25 b(57R30,)19 b(37C10,)g(37C40,)f(37C85.)376 3317 y(The)23 b(\014rst)g(author)h(gratefully)h(ac)n(kno)n(wledges)g (supp)r(ort)e(from)h(the)f Fn(V)l(olkswagenStiftung)308 3384 y Fv(and)28 b(the)f(hospitalit)n(y)j(of)d(Harv)m(ard)g(Univ)n (ersit)n(y)-5 b(.)52 b(The)27 b(second)h(author)g(is)g(supp)r(orted)f (b)n(y)308 3451 y(the)17 b Fn(DF)o(G)i(Gr)m(aduiertenkol)s(le)m(g)e (\\Mathematik)g(im)h(Ber)m(eich)g(ihr)m(er)f(We)m(chselwirkung)h(mit)g (der)308 3519 y(Physik".)1307 3586 y Fm(1)p eop %%Page: 2 2 2 1 bop 308 159 a Fm(2)586 b(D.)18 b(K)o(OTSCHICK)e(AND)h(T.)g(V)o (OGEL)308 294 y Ft(order)26 b(to)g(de\014ne)g(linking)g(n)n(um)n(b)r (ers.)41 b(In)26 b(this)g(pap)r(er)g(w)n(e)g(de\014ne)g(the)g(a)n(v)n (erage)308 373 y(asymptotic)19 b(linking)h(n)n(um)n(b)r(er)f(of)h(a)g (div)n(ergence-free)h(v)n(ector)g(\014eld)f(and)f(a)h(mea-)308 451 y(sured)f(co)r(dimension)g(t)n(w)n(o)g(foliation.)30 b(W)-6 b(e)20 b(close)h(up)e(pieces)h(of)g(\015o)n(w)f(lines)h(of)g (the)308 530 y(v)n(ector)g(\014eld)g(as)g(in)f([14)q(],)i(but)d(w)n(e)i (do)g(not)f(parameterise)g(or)g(close)i(the)f(lea)n(v)n(es)h(of)308 609 y(the)d(foliation.)29 b(In)n(tuitiv)n(ely)-6 b(,)19 b(parameterisation)d(and)h(\\closing")i(is)f(done)g(b)n(y)f(the)308 687 y(in)n(v)l(arian)n(t)26 b(measure.)42 b(Thinking)25 b(of)i(a)f(div)n(ergence-free)h(\015o)n(w)f(as)g(a)h(\(singular\))308 766 y(one-dimensional)i(foliation)h(with)f(a)g(holonom)n(y-in)n(v)l (arian)n(t)f(transv)n(erse)h(mea-)308 845 y(sure,)c(w)n(e)f(reco)n(v)n (er)g(the)g(construction)f(of)h([1,)g(14].)36 b(A)24 b(di\013eren)n(t)f(generalisation)308 923 y(of)f(Arnold's)h (construction)e(has)g(b)r(een)h(prop)r(osed)f(b)n(y)h(Rivi)n(\022)-31 b(ere)22 b([12)q(].)376 1002 y(In)34 b(section)i(3)f(w)n(e)g(shall)h (consider)f(the)g(linking)h(of)f(the)g(\015o)n(w)f(lines)i(of)g(a)308 1081 y(div)n(ergence-free)19 b(v)n(ector)f(\014eld)g(with)f(a)h(n)n (ull-homologous)e(closed)j(orien)n(ted)e(sub-)308 1159 y(manifold)g Fr(N)24 b Ft(of)17 b(co)r(dimension)f(t)n(w)n(o.)28 b(On)17 b(three-manifolds,)g(this)g(situation)g(w)n(as)308 1238 y(already)28 b(considered)g(b)n(y)f(Arnold)h([1)q(],)h(but)e(our)h (approac)n(h)f(is)h(di\013eren)n(t.)46 b(F)-6 b(or)308 1317 y(manifolds)23 b(with)g(v)l(anishing)g(\014rst)f(Betti)i(n)n(um)n (b)r(er)d(w)n(e)j(pro)n(v)n(e)f(the)g(existence)i(of)308 1395 y(suitable)18 b(systems)f(of)g(short)g(paths)g(whic)n(h)g(one)h (can)g(use)f(to)h(close)g(the)g(\015o)n(w)f(lines)308 1474 y(of)j(the)g(v)n(ector)g(\014eld.)29 b(The)20 b(resulting)f(a)n(v) n(erage)i(asymptotic)d(linking)i(n)n(um)n(b)r(er)e(is)308 1553 y(giv)n(en)h(b)n(y)g(a)f(Hopf-t)n(yp)r(e)h(in)n(tegral.)29 b(It)18 b(is)h(then)f(clear)h(ho)n(w)g(to)f(generalise)i(further)308 1631 y(and)25 b(replace)h(the)g(closed)g(submanifold)e Fr(N)33 b Ft(b)n(y)25 b(a)g(measured)g(foliation)h(\()p Fq(F)7 b Fr(;)k(\027)t Ft(\))308 1710 y(of)30 b(co)r(dimension)e(t)n(w) n(o,)j(whose)e(Ruelle{Sulliv)l(an)h(cycle)h(is)e(n)n(ull-homologous.) 308 1789 y(This)c(is)h(carried)f(out)g(in)g(section)g(4.)40 b(The)25 b(discussion)g(there)g(is)h(motiv)l(ated)e(in)308 1867 y(part)d(b)n(y)g(the)g(w)n(ork)f(of)i(Arnold,)f(Khesin)h([8],)h (No)n(vik)n(o)n(v)f(and)e(others)h(on)g(higher-)308 1946 y(dimensional)30 b(generalisations)g(of)h(Arnold's)f(construction)f (describ)r(ed)h(in)g([2])308 2024 y(\(Chapter)21 b(I)r(I)r(I,)g(7.B\).) 376 2103 y(In)15 b(section)i(5)f(w)n(e)g(apply)f(our)h(construction)f (to)h(giv)n(e)g(an)g(in)n(terpretation)f(of)h(the)308 2182 y(four-dimensional)h(Go)r(dbillon-V)-6 b(ey)18 b(in)n(v)l(arian)n (t)g(discussed)f(in)h([9])g(as)g(an)f(a)n(v)n(erage)308 2260 y(asymptotic)22 b(linking)i(n)n(um)n(b)r(er)d(of)j(a)f(v)n(ector)h (\014eld)f(and)g(a)g(measured)g(co)r(dimen-)308 2339 y(sion)28 b(t)n(w)n(o)f(foliation.)47 b(F)-6 b(or)28 b(this)f(it)h(is)g(imp)r(ortan)n(t)d(that)i(the)g(constructions)g(of) 308 2418 y(section)c(4)f(w)n(ork)f(for)h(singular)g(foliations.)308 2533 y Fl(Ac)n(kno)n(wledgemen)n(ts:)33 b Ft(W)-6 b(e)25 b(are)f(grateful)f(to)h(B.)g(Khesin)g(for)f(useful)h(con)n(v)n(ersa-) 308 2612 y(tions,)c(for)e(Remark)f(19,)i(and)f(for)h(sending)f(us)g(a)h (cop)n(y)f(of)h([8];)i(and)c(to)i(S.)f(Hurder)308 2690 y(for)25 b(pro)r(dding)f(us)g(to)h(b)r(e)g(more)f(careful)h(in)g(our)g (treatmen)n(t)e(of)i(singular)g(folia-)308 2769 y(tions,)d(and)g(for)g (p)r(oin)n(ting)f(us)h(to)f(the)h(w)n(ork)f(of)i(Moussu.)1044 2929 y(2.)34 b Fs(Preliminaries)376 3047 y Ft(Let)25 b Fr(M)568 3022 y Fk(n)625 3047 y Ft(b)r(e)h(a)f(smo)r(oth)f(closed)i (orien)n(ted)f Fr(n)p Ft(-manifold.)40 b(Consider)24 b(smo)r(oth)308 3126 y(closed)g(orien)n(ted)f(submanifolds)f Fr(N)1220 3101 y Fk(k)1213 3142 y Fo(1)1272 3126 y Ft(and)h Fr(N)1462 3101 y Fk(l)1455 3142 y Fo(2)1505 3126 y Ft(of)g Fr(M)7 b Ft(,)24 b(with)e Fr(k)d Ft(+)d Fr(l)22 b Ft(=)f Fr(n)16 b Fq(\000)g Ft(1.)34 b(If)308 3204 y Fr(N)361 3214 y Fo(1)411 3204 y Ft(and)23 b Fr(N)594 3214 y Fo(2)643 3204 y Ft(are)h(n)n(ull-homologous)e(o)n(v)n(er)h Fp(R)p Ft(,)j(they)c(ha)n(v)n(e)i(a)f(w)n(ell-de\014ned)h(link-)308 3283 y(ing)f(n)n(um)n(b)r(er)e(de\014ned)h(as)h(follo)n(ws.)32 b(Let)23 b Fr(S)1374 3293 y Fo(1)1423 3283 y Ft(b)r(e)f(a)h(real)g (orien)n(ted)f(\()p Fr(k)c Ft(+)e(1\)-c)n(hain)308 3362 y(with)24 b Fr(@)t(S)541 3372 y Fo(1)589 3362 y Ft(=)f Fr(N)716 3372 y Fo(1)743 3362 y Ft(,)i(and)e(de\014ne)h(lk\()p Fr(N)1241 3372 y Fo(1)1268 3362 y Fr(;)11 b(N)1350 3372 y Fo(2)1377 3362 y Ft(\))24 b(to)g(b)r(e)g(the)g(algebraic)g(in)n (tersection)308 3440 y(n)n(um)n(b)r(er)h(of)i Fr(S)674 3450 y Fo(1)727 3440 y Ft(and)f Fr(N)913 3450 y Fo(2)940 3440 y Ft(,)i(whic)n(h)f(b)n(y)f(assumption)f(ha)n(v)n(e)i(complemen)n (tary)e(di-)308 3519 y(mensions.)43 b(This)26 b(linking)h(n)n(um)n(b)r (er)e(is)i(alw)n(a)n(ys)g(rational.)44 b(If)26 b Fr(N)1950 3529 y Fo(1)2003 3519 y Ft(and)g Fr(N)2189 3529 y Fo(2)2243 3519 y Ft(are)p eop %%Page: 3 3 3 2 bop 677 159 a Fm(LINKING)23 b(NUMBERS)f(OF)h(MEASURED)g(F)o(OLIA)l (TIONS)342 b(3)308 294 y Ft(n)n(ull-homologous)23 b(o)n(v)n(er)h Fp(Z)-6 b Ft(,)21 b(then)i(their)g(linking)h(n)n(um)n(b)r(er)e(is)i(in) n(tegral.)36 b(In)23 b(an)n(y)308 373 y(case,)g(w)n(e)g(ha)n(v)n(e)f (lk\()p Fr(N)848 383 y Fo(1)874 373 y Fr(;)11 b(N)956 383 y Fo(2)983 373 y Ft(\))19 b(=)g(\()p Fq(\000)p Ft(1\))1235 348 y Fo(\()p Fk(k)q Fo(+1\)\()p Fk(l)q Fo(+1\))1474 373 y Ft(lk)q(\()p Fr(N)1607 383 y Fo(2)1633 373 y Fr(;)11 b(N)1715 383 y Fo(1)1742 373 y Ft(\).)376 451 y(There)21 b(is)g(a)g(classical)i(expression)e(for)g(these)h(linking)f(n)n(um)n(b) r(ers)e(as)i(in)n(tegrals)308 530 y(of)d(certain)g(di\013eren)n(tial)f (forms.)27 b(If)18 b Fr(W)1250 540 y Fk(i)1287 530 y Ft(is)g(a)f(tubular)f(neigh)n(b)r(ourho)r(o)r(d)g(of)i Fr(N)2200 540 y Fk(i)2219 530 y Ft(,)h(w)n(e)308 609 y(can)d(c)n(ho)r(ose)f(a)h(closed)g(form)e Fr(\021)1044 619 y Fk(i)1078 609 y Ft(on)h Fr(W)1225 619 y Fk(i)1259 609 y Ft(represen)n(ting)g(the)g(compactly)f(supp)r(orted)308 687 y(P)n(oincar)n(\023)-31 b(e)23 b(dual)f(of)g Fr(N)854 697 y Fk(i)895 687 y Ft(in)h Fr(W)1035 697 y Fk(i)1054 687 y Ft(,)g(and)e(then)h(extend)f(this)h(form)f(b)n(y)h(zero)h(to)f (all)h(of)308 766 y Fr(M)7 b Ft(.)28 b(The)15 b(assumption)g(that)g([)p Fr(N)1106 776 y Fo(1)1132 766 y Ft(])i(v)l(anishes)f(in)g(real)g (homology)f(means)g(that)g(the)308 845 y(extension)23 b(of)h Fr(\021)711 855 y Fo(1)760 845 y Ft(to)f Fr(M)7 b Ft(,)24 b(also)g(denoted)e Fr(\021)1372 855 y Fo(1)1399 845 y Ft(,)h(is)h(exact.)34 b(Let)23 b Fr(\013)1873 855 y Fo(1)1923 845 y Ft(b)r(e)g(a)g(primitiv)n(e)308 923 y(and)f(de\014ne)308 1074 y(\(1\))560 b Fr(H)5 b Ft(\()p Fr(N)1092 1084 y Fo(1)1118 1074 y Fr(;)11 b(N)1200 1084 y Fo(2)1227 1074 y Ft(\))18 b(=)1341 982 y Fj(Z)1379 1135 y Fk(M)1443 1074 y Fr(\013)1485 1084 y Fo(1)1527 1074 y Fq(^)d Fr(\021)1620 1084 y Fo(2)1668 1074 y Fr(:)308 1243 y Ft(If)22 b(w)n(e)g(c)n(hose)h(a)f(di\013eren)n(t)f(primitiv)n(e) h Fr(\013)1286 1219 y Fi(0)1286 1260 y Fo(1)1334 1243 y Ft(for)g Fr(\021)1468 1253 y Fo(1)1495 1243 y Ft(,)g(then)661 1319 y Fj(Z)698 1471 y Fk(M)763 1410 y Fr(\013)805 1382 y Fi(0)805 1427 y Fo(1)846 1410 y Fq(^)15 b Fr(\021)939 1420 y Fo(2)981 1410 y Fq(\000)1048 1319 y Fj(Z)1086 1471 y Fk(M)1150 1410 y Fr(\013)1192 1420 y Fo(1)1234 1410 y Fq(^)g Fr(\021)1327 1420 y Fo(2)1372 1410 y Ft(=)1442 1319 y Fj(Z)1479 1471 y Fk(M)1533 1410 y Ft(\()p Fr(\013)1601 1382 y Fi(0)1601 1427 y Fo(1)1642 1410 y Fq(\000)g Fr(\013)1751 1420 y Fo(1)1778 1410 y Ft(\))g Fq(^)g Fr(\021)1912 1420 y Fo(2)1960 1410 y Fr(:)308 1579 y Ft(This)28 b(v)l(anishes)g(b)r (ecause)g Fr(\013)1022 1555 y Fi(0)1022 1596 y Fo(1)1067 1579 y Fq(\000)20 b Fr(\013)28 b Ft(is)g(closed)g(and)f Fr(\021)1648 1589 y Fo(2)1702 1579 y Ft(is)h(exact)g(as)g Fr(N)2094 1589 y Fo(2)2148 1579 y Ft(is)g(also)308 1658 y(assumed)23 b(to)h(b)r(e)g(n)n(ull-homologous.)34 b(If)25 b(w)n(e)f(mak)n(e)f(a)h(di\013eren)n(t)g(c)n(hoice,)i Fr(\021)2185 1634 y Fi(0)2183 1675 y Fo(2)2209 1658 y Ft(,)f(for)308 1737 y Fr(\021)341 1747 y Fo(2)368 1737 y Ft(,)d(then)f Fr(\021)593 1712 y Fi(0)591 1753 y Fo(2)632 1737 y Fq(\000)16 b Fr(\021)733 1747 y Fo(2)778 1737 y Ft(=)j Fr(d\014)t Ft(,)k(for)e(a)h Fr(\014)k Ft(with)c(supp)r(ort)e (in)i Fr(W)1716 1747 y Fo(2)1743 1737 y Ft(.)30 b(Then)308 1812 y Fj(Z)346 1965 y Fk(M)410 1904 y Fr(\013)452 1914 y Fo(1)492 1904 y Fq(^)13 b Fr(\021)585 1876 y Fi(0)583 1921 y Fo(2)623 1904 y Fq(\000)688 1812 y Fj(Z)726 1965 y Fk(M)790 1904 y Fr(\013)832 1914 y Fo(1)872 1904 y Fq(^)g Fr(\021)963 1914 y Fo(2)1008 1904 y Ft(=)1078 1812 y Fj(Z)1116 1965 y Fk(M)1180 1904 y Fr(\013)1222 1914 y Fo(1)1262 1904 y Fq(^)g Fr(d\014)23 b Ft(=)c Fq(\006)1547 1812 y Fj(Z)1585 1965 y Fk(M)1650 1904 y Fr(d\013)1726 1914 y Fo(1)1766 1904 y Fq(^)13 b Fr(\014)23 b Ft(=)c Fq(\006)2017 1812 y Fj(Z)2055 1965 y Fk(M)2120 1904 y Fr(\021)2153 1914 y Fo(1)2192 1904 y Fq(^)13 b Fr(\014)25 b(:)308 2070 y Ft(No)n(w)20 b(the)f(righ)n(t)f(hand)g(side)i(v)l (anishes)f(b)r(ecause)h Fr(\021)1546 2080 y Fo(1)1591 2070 y Ft(and)f Fr(\014)k Ft(ha)n(v)n(e)c(supp)r(ort)f(in)h Fr(W)2304 2080 y Fo(1)308 2149 y Ft(and)j Fr(W)499 2159 y Fo(2)548 2149 y Ft(resp)r(ectiv)n(ely)-6 b(,)23 b(whic)n(h)f(can)g(b) r(e)g(c)n(hosen)g(to)g(b)r(e)g(disjoin)n(t.)376 2228 y(Th)n(us)14 b(w)n(e)h(ha)n(v)n(e)g(sho)n(wn)g(that)f Fr(H)5 b Ft(\()p Fr(N)1240 2238 y Fo(1)1266 2228 y Fr(;)11 b(N)1348 2238 y Fo(2)1375 2228 y Ft(\))k(is)g(indep)r(enden)n(t)f(of)i (c)n(hoices,)i(except)308 2306 y(p)r(ossibly)28 b(for)f(the)h(c)n (hoice)h(of)e Fr(\021)1109 2316 y Fo(1)1136 2306 y Ft(.)46 b(Ho)n(w)n(ev)n(er,)31 b(w)n(e)d(can)f(in)n(terc)n(hange)h(the)f(roles) 308 2385 y(of)j Fr(\021)424 2395 y Fo(1)480 2385 y Ft(and)f Fr(\021)649 2395 y Fo(2)705 2385 y Ft(b)r(ecause)h(w)n(e)f(ha)n(v)n(e)h Fr(H)5 b Ft(\()p Fr(N)1360 2395 y Fo(1)1386 2385 y Fr(;)11 b(N)1468 2395 y Fo(2)1495 2385 y Ft(\))32 b(=)g(\()p Fq(\000)p Ft(1\))1773 2361 y Fo(\()p Fk(k)q Fo(+1\)\()p Fk(l)q Fo(+1\))2012 2385 y Fr(H)5 b Ft(\()p Fr(N)2151 2395 y Fo(2)2178 2385 y Fr(;)11 b(N)2260 2395 y Fo(1)2287 2385 y Ft(\).)308 2464 y(Th)n(us)21 b Fr(H)5 b Ft(\()p Fr(N)614 2474 y Fo(1)640 2464 y Fr(;)11 b(N)722 2474 y Fo(2)749 2464 y Ft(\))21 b(is)h(indep)r(enden)n(t)e(of)h(all)h(c)n (hoices)h(and)d(only)i(dep)r(ends)e(on)h(the)308 2542 y(submanifolds)i Fr(N)762 2552 y Fo(1)813 2542 y Ft(and)h Fr(N)997 2552 y Fo(2)1023 2542 y Ft(,)i(as)e(suggested)g(b)n(y)g(the)g (notation.)36 b(By)24 b(making)f(a)308 2621 y(con)n(v)n(enien)n(t)g(c)n (hoice)g(for)f Fr(\021)965 2631 y Fo(2)1013 2621 y Ft(as)g(in)g([3],)h (one)f(sees)h(that)308 2788 y(\(2\))1032 2697 y Fj(Z)1069 2849 y Fk(M)1134 2788 y Fr(\013)1176 2798 y Fo(1)1218 2788 y Fq(^)15 b Fr(\021)1311 2798 y Fo(2)1356 2788 y Ft(=)1426 2697 y Fj(Z)1463 2849 y Fk(N)1501 2855 y Fh(2)1538 2788 y Fr(\013)1580 2798 y Fo(1)308 2965 y Ft(and)25 b(that)g(these)h(in)n(tegrals)f(equal)h(the)g(linking)f(n)n(um)n(b)r (er)f(lk\()p Fr(N)1922 2975 y Fo(1)1949 2965 y Fr(;)11 b(N)2031 2975 y Fo(2)2058 2965 y Ft(\).)40 b(An)26 b(in-)308 3043 y(stance)18 b(of)g(this)f(equalit)n(y)h(is)g(the)f(di\013eren)n (tial)h(form)e(in)n(terpretation)g(of)i(the)f(Hopf)308 3122 y(in)n(v)l(arian)n(t,)h(and)d(w)n(e)h(shall)g(refer)g(to)g(either) 1362 3068 y Fj(R)1394 3146 y Fk(M)1459 3122 y Fr(\013)1501 3132 y Fo(1)1530 3122 y Fq(^)s Fr(\021)1611 3132 y Fo(2)1653 3122 y Ft(or)1727 3068 y Fj(R)1759 3146 y Fk(N)1797 3152 y Fh(2)1834 3122 y Fr(\013)1876 3132 y Fo(1)1919 3122 y Ft(as)g(a)g(Hopf-t)n(yp)r(e)308 3204 y(in)n(tegral.)376 3283 y(The)j(linking)h(n)n(um)n(b)r(er)e(of)i(t)n(w)n(o)f(disjoin)n(t)h (closed)g(orien)n(ted)f(submanifolds)g(can)308 3362 y(b)r(e)31 b(expressed)g(through)e(so-called)i(linking)g(forms)f(\(see)h([11,)h (14]\).)55 b(A)32 b Fg(link-)308 3440 y(ing)g(form)e Fr(L)h Ft(on)f(an)g Fr(n)p Ft(-manifold)g Fr(M)37 b Ft(is)31 b(a)f(double)g(form)f(on)h Fr(M)e Fq(\002)21 b Fr(M)37 b Ft(with)308 3519 y(the)23 b(prop)r(ert)n(y)e(that)h(whenev)n(er)g (the)g(linking)h(n)n(um)n(b)r(er)e(of)i(t)n(w)n(o)f(orien)n(ted)h (closed)p eop %%Page: 4 4 4 3 bop 308 159 a Fm(4)586 b(D.)18 b(K)o(OTSCHICK)e(AND)h(T.)g(V)o (OGEL)308 294 y Ft(submanifolds)k Fr(N)760 304 y Fo(1)808 294 y Ft(and)h Fr(N)990 304 y Fo(2)1038 294 y Ft(is)h(w)n (ell-de\014ned,)f(w)n(e)g(ha)n(v)n(e)973 446 y(lk\()p Fr(N)1105 456 y Fo(1)1131 446 y Fr(;)11 b(N)1213 456 y Fo(2)1241 446 y Ft(\))18 b(=)1355 355 y Fj(Z)1393 507 y Fk(N)1431 513 y Fh(1)1468 355 y Fj(Z)1505 507 y Fk(N)1543 513 y Fh(2)1581 446 y Fr(L)k(:)308 608 y Ft(A)d(linking)g(form)e(can)h (b)r(e)h(constructed)e(as)i(follo)n(ws.)29 b(W)-6 b(e)19 b(c)n(ho)r(ose)g(a)f(Riemannian)308 686 y(metric)25 b(on)h Fr(M)7 b Ft(,)27 b(and)e(de\014ne)g Fr(H)31 b Ft(to)25 b(b)r(e)g(the)h(pro)t(jection)f(op)r(erator)g(mapping)f(a)308 765 y(di\013eren)n(tial)k(form)f(to)h(its)g(harmonic)e(part.)47 b(Then,)29 b(for)e(ev)n(ery)i(degree)f Fr(i)p Ft(,)h(the)308 844 y(Green's)22 b(op)r(erator)1018 936 y Fr(G)7 b Ft(:)23 b(\012)1166 909 y Fk(i)1185 936 y Ft(\()p Fr(M)7 b Ft(\))18 b Fq(\000)-11 b(!)19 b Ft(\()p Fq(H)1537 909 y Fk(i)1556 936 y Ft(\))1582 909 y Fi(?)308 1043 y Ft(is)j(de\014ned)e(to)g(map)g (a)h(di\013eren)n(tial)f(form)g Fr(\013)h Ft(of)g(degree)h Fr(i)e Ft(to)g(the)h(unique)f Fr(i)p Ft(-form)308 1122 y Fr(!)25 b Ft(that)c(is)h(p)r(erp)r(endicular)f(to)h(all)h(harmonic)d (forms)h(and)h(solv)n(es)g(the)g(equation)1066 1242 y(\001)p Fr(!)g Ft(=)d Fr(\013)c Fq(\000)h Fr(H)5 b Ft(\()p Fr(\013)p Ft(\))21 b Fr(:)308 1362 y Ft(In)32 b(other)g(w)n(ords,)j Fr(G)d Ft(is)h(c)n(haracterised)g(b)n(y)f(the)h(prop)r(erties)e Fr(H)5 b(G)37 b Ft(=)g(0)32 b(and)308 1441 y(\001)p Fr(G)19 b Ft(=)g(Id)11 b Fq(\000)p Fr(H)5 b Ft(.)30 b(It)21 b(can)h(b)r(e)g (written)f(in)h(the)g(form)852 1586 y Fr(G)p Ft(\()p Fr(\013)p Ft(\)\()p Fr(x)p Ft(\))c(=)1176 1494 y Fj(Z)1213 1647 y Fk(y)r Fi(2)p Fk(M)1335 1586 y Fr(\013)p Ft(\()p Fr(y)r Ft(\))d Fq(^)g(\003)1572 1596 y Fk(y)1600 1586 y Fr(g)r Ft(\()p Fr(x;)c(y)r Ft(\))308 1753 y(for)28 b(a)g(suitable)g(double)g(form)f Fr(g)r Ft(\()p Fr(x;)11 b(y)r Ft(\))28 b(on)f Fr(M)f Fq(\002)20 b Fr(M)34 b Ft(whic)n(h)28 b(is)h(smo)r(oth)d(a)n(w)n(a)n(y)308 1832 y(from)h(the)g(diagonal)g (and)g(has)g(a)g(p)r(ole)h(of)g(order)e Fr(n)19 b Fq(\000)g Ft(2)28 b(along)f(the)g(diagonal,)308 1910 y(cf.)f([11])f Fq(x)p Ft(31.)37 b(Here)25 b Fq(\003)871 1920 y Fk(y)923 1910 y Ft(denotes)f(the)g(Ho)r(dge)h(star)f(op)r(erator)f(with)h(resp)r (ect)g(to)308 1989 y(the)e(second)g(factor)g(of)g Fr(M)g Fq(\002)15 b Fr(M)7 b Ft(.)376 2068 y(W)-6 b(e)26 b(denote)f(b)n(y)g (\()p Fq(\000)p Ft(1\))941 2043 y Fk(\017)989 2068 y Ft(the)g(linear)h(op)r(erator)e(on)i(double)f(forms)f(acting)i(on)308 2146 y(decomp)r(osable)21 b(double)h(forms)e Fr(!)r Ft(\()p Fr(x)p Ft(\))14 b Fq(\001)h Fr(\021)r Ft(\()p Fr(y)r Ft(\))21 b(with)g Fr(\021)g Fq(2)e Ft(\012)1761 2122 y Fk(s)1785 2146 y Ft(\()p Fr(M)7 b Ft(\))21 b(b)n(y)h(m)n(ultiplica-) 308 2227 y(tion)i(with)f(\()p Fq(\000)p Ft(1\))735 2203 y Fo(\()p Fk(n)p Fi(\000)p Fk(s)p Fo(\))p Fk(s)885 2227 y Ft(.)36 b(In)23 b(later)h(sections)h(w)n(e)f(will)h(b)r(e)e (concerned)i(only)e(with)308 2306 y(the)f(case)h Fr(s)c Ft(=)g Fr(n)c Fq(\000)h Ft(1.)30 b(\(This)21 b(is)i(due)f(to)f(the)h (fact)g(that)f(the)h(P)n(oincar)n(\023)-31 b(e)23 b(duals)f(of)308 2384 y(one-dimensional)f(submanifolds)f(of)h Fr(M)28 b Ft(ha)n(v)n(e)21 b(degree)h Fr(n)14 b Fq(\000)f Ft(1.\))29 b(W)-6 b(e)22 b(de\014ne)f(the)308 2463 y(double)h(form)f Fr(L)h Ft(on)f Fr(M)h Fq(\002)16 b Fr(M)29 b Ft(b)n(y)308 2583 y(\(3\))513 b Fr(L)p Ft(\()p Fr(x;)11 b(y)r Ft(\))19 b(=)g(\()p Fq(\000)p Ft(1\))1330 2556 y Fk(\017)1367 2583 y Fq(\003)1401 2593 y Fk(y)1443 2583 y Fr(d)1477 2593 y Fk(y)1506 2583 y Fr(g)r Ft(\()p Fr(x;)11 b(y)r Ft(\))22 b Fr(:)308 2704 y Fw(Prop)r(osition)28 b(1.)g Fg(The)e(double)f(form)f Fr(L)p Ft(\()p Fr(x;)11 b(y)r Ft(\))25 b Fg(is)g(a)g(linking)h(form.)34 b(Denoting)308 2782 y(by)c Fr(r)h Fg(the)f(R)n(iemannian)g(distanc)m(e)f(function,)k Fr(L)c Fg(has)g(a)g(singularity)h(of)g(or)m(der)308 2861 y Ft(\()p Fr(r)r Ft(\()p Fr(x;)11 b(y)r Ft(\)\))545 2837 y Fo(1)p Fi(\000)p Fk(n)661 2861 y Fg(along)24 b(the)g(diagonal)f(in)h Fr(M)f Fq(\002)15 b Fr(M)31 b Fg(and)24 b(is)g(smo)m(oth)e(elsewher)m (e.)32 b(It)308 2940 y(has)c(the)g(fol)s(lowing)g(additional)f(pr)m(op) m(erty:)38 b(for)28 b(every)h Fr(i)p Fg(-form)e Fr(\013)h Fg(ther)m(e)g(exists)308 3018 y(an)c Ft(\()p Fr(i)14 b Fq(\000)h Ft(1\))p Fg(-form)23 b Fr(h)g Fg(such)h(that)308 3174 y Ft(\(4\))607 3082 y Fj(Z)645 3235 y Fk(y)r Fi(2)p Fk(M)767 3174 y Fr(L)p Ft(\()p Fr(x;)11 b(y)r Ft(\))k Fq(^)g Fr(d\013)p Ft(\()p Fr(y)r Ft(\))k(=)g Fr(\013)p Ft(\()p Fr(x)p Ft(\))14 b Fq(\000)i Fr(H)5 b Ft(\()p Fr(\013)p Ft(\)\()p Fr(x)p Ft(\))13 b(+)i Fr(dh)p Ft(\()p Fr(x)p Ft(\))23 b Fr(:)308 3347 y Fg(Pr)m(o)m(of.)k Ft(By)g(the)h (de\014nition)f(of)g Fr(G)p Ft(,)j(and)d(b)r(ecause)h Fr(G)f Ft(comm)n(utes)f(with)h(\001,)j(w)n(e)308 3426 y(ha)n(v)n(e)833 3519 y Fr(G)p Ft(\()p Fr(d)945 3491 y Fi(\003)972 3519 y Fr(d\013)p Ft(\))19 b(=)g Fr(\013)d Fq(\000)f Fr(H)5 b Ft(\()p Fr(\013)p Ft(\))15 b Fq(\000)g Fr(G)p Ft(\()p Fr(dd)1670 3491 y Fi(\003)1697 3519 y Fr(\013)p Ft(\))22 b Fr(:)p eop %%Page: 5 5 5 4 bop 677 159 a Fm(LINKING)23 b(NUMBERS)f(OF)h(MEASURED)g(F)o(OLIA)l (TIONS)342 b(5)308 294 y Ft(As)23 b Fr(G)f Ft(comm)n(utes)e(with)h Fr(d)i Ft(w)n(e)f(can)g(set)g Fr(h)d Ft(=)g Fq(\000)p Fr(G)p Ft(\()p Fr(d)1609 270 y Fi(\003)1636 294 y Fr(\013)p Ft(\))j(to)g(obtain)387 348 y Fj(Z)424 500 y Fk(y)r Fi(2)p Fk(M)546 439 y Fr(d\013)p Ft(\()p Fr(y)r Ft(\))16 b Fq(^)f(\003)818 449 y Fk(y)845 439 y Fr(d)879 449 y Fk(y)908 439 y Fr(g)r Ft(\()p Fr(x;)c(y)r Ft(\))18 b(=)h Fr(G)p Ft(\()p Fr(d)1295 411 y Fi(\003)1322 439 y Fr(d\013)p Ft(\))g(=)g Fr(\013)p Ft(\()p Fr(x)p Ft(\))c Fq(\000)g Fr(H)5 b Ft(\()p Fr(\013)p Ft(\)\()p Fr(x)p Ft(\))13 b(+)j Fr(dh)p Ft(\()p Fr(x)p Ft(\))21 b Fr(:)308 592 y Ft(If)33 b(w)n(e)f(c)n(hange)h(the)f(order)g (of)h(the)f(factors)g(in)h(the)f(in)n(tegrand,)j(w)n(e)d(ha)n(v)n(e)h (to)308 671 y(m)n(ultiply)21 b(b)n(y)h(\()p Fq(\000)p Ft(1\))801 646 y Fk(\017)845 671 y Ft(and)f(w)n(e)h(obtain)g(\(4\))o(.) 376 750 y(That)e Fr(L)h Ft(is)g(a)g(linking)g(form)f(can)h(b)r(e)g(sho) n(wn)g(as)g(in)g(the)f(pro)r(of)h(of)g(Theorem)f(3)308 828 y(in)25 b([14].)37 b(W)-6 b(e)26 b(brie\015y)e(indicate)g(the)h (argumen)n(t.)35 b(Cho)r(osing)23 b Fr(W)1916 838 y Fk(i)1936 828 y Ft(,)i Fr(\021)2012 838 y Fk(i)2055 828 y Ft(and)f Fr(\013)2228 838 y Fk(i)2272 828 y Ft(as)308 907 y(ab)r(o)n(v)n(e,)f (and)e(using)i(\(4\))o(,)f(w)n(e)h(ha)n(v)n(e)385 1052 y Fr(l)q(k)r Ft(\()p Fr(N)521 1062 y Fo(1)548 1052 y Fr(;)11 b(N)630 1062 y Fo(2)657 1052 y Ft(\))19 b(=)772 960 y Fj(Z)809 1113 y Fk(M)874 1052 y Fr(\013)916 1062 y Fo(1)958 1052 y Fq(^)c Fr(\021)1051 1062 y Fo(2)497 1236 y Ft(=)559 1144 y Fj(Z)597 1297 y Fk(x)p Fi(2)p Fk(M)720 1141 y Fj(\022)770 1236 y Fr(H)5 b Ft(\()p Fr(\013)898 1246 y Fo(1)924 1236 y Ft(\)\()p Fr(x)p Ft(\))14 b Fq(\000)h Fr(dh)p Ft(\()p Fr(x)p Ft(\))f(+)1364 1144 y Fj(Z)1401 1297 y Fk(y)r Fi(2)p Fk(M)1523 1236 y Fr(L)p Ft(\()p Fr(x;)d(y)r Ft(\))k Fq(^)g Fr(\021)1829 1246 y Fo(1)1855 1236 y Ft(\()p Fr(y)r Ft(\))1941 1141 y Fj(\023)2005 1236 y Fq(^)g Fr(\021)2098 1246 y Fo(2)2125 1236 y Ft(\()p Fr(x)p Ft(\))20 b Fr(:)308 1389 y Ft(The)i(in)n(tegrals)832 1411 y Fj(Z)870 1564 y Fk(M)934 1503 y Fr(H)5 b Ft(\()p Fr(\013)1062 1513 y Fo(1)1089 1503 y Ft(\))14 b Fq(^)h Fr(\021)1222 1513 y Fo(2)1315 1503 y Ft(and)1498 1411 y Fj(Z)1536 1564 y Fk(M)1600 1503 y Fr(dh)h Fq(^)e Fr(\021)1780 1513 y Fo(2)308 1633 y Ft(v)l(anish)25 b(b)n(y)g(Stok)n(es's)g(theorem) e(b)r(ecause)i Fr(\021)1402 1643 y Fo(2)1453 1633 y Ft(is)h(exact.)38 b(As)26 b Fr(\021)1865 1643 y Fk(i)1908 1633 y Ft(has)f(supp)r(ort)e (in)308 1712 y Fr(W)370 1722 y Fk(i)390 1712 y Ft(,)f(w)n(e)g(obtain) 562 1859 y Fr(l)q(k)r Ft(\()p Fr(N)698 1869 y Fo(1)725 1859 y Fr(;)11 b(N)807 1869 y Fo(2)834 1859 y Ft(\))18 b(=)941 1767 y Fj(Z)978 1920 y Fk(x)p Fi(2)p Fk(W)1082 1926 y Fh(2)1120 1764 y Fj(\022)1169 1767 y(Z)1207 1920 y Fk(y)r Fi(2)p Fk(W)1310 1926 y Fh(1)1346 1859 y Fr(L)p Ft(\()p Fr(x;)11 b(y)r Ft(\))k Fq(^)g Fr(\021)1652 1869 y Fo(1)1678 1859 y Ft(\()p Fr(y)r Ft(\))1764 1764 y Fj(\023)1829 1859 y Fq(^)g Fr(\021)1922 1869 y Fo(2)1948 1859 y Ft(\()p Fr(x)p Ft(\))21 b Fr(:)308 2012 y Ft(No)n(w)i(the)f Fr(W)635 2022 y Fk(i)677 2012 y Ft(can)g(b)r(e)h(tak)n(en)f(to)g(b)r(e)g (disjoin)n(t)h(and)f(making)f(a)h(go)r(o)r(d)g(c)n(hoice)i(for)308 2091 y(the)e Fr(\021)455 2101 y Fk(i)496 2091 y Ft(the)f(ab)r(o)n(v)n (e)h(in)n(tegral)h(reduces)f(to)1030 2140 y Fj(Z)1067 2293 y Fk(x)p Fi(2)p Fk(N)1164 2299 y Fh(2)1201 2140 y Fj(Z)1239 2293 y Fk(y)r Fi(2)p Fk(N)1335 2299 y Fh(1)1371 2232 y Fr(L)p Ft(\()p Fr(x;)11 b(y)r Ft(\))22 b Fr(:)308 2385 y Ft(The)d(claim)h(ab)r(out)e(the)h(order)g(of)h(the)f(singularit) n(y)g(along)g(the)h(diagonal)f(follo)n(ws)308 2464 y(from)i(what)g(w)n (e)h(said)g(ab)r(o)n(v)n(e)g(ab)r(out)f(the)h(singularit)n(y)g(of)g Fr(g)r Ft(.)472 b Ff(\003)376 2580 y Ft(W)-6 b(e)22 b(shall)h(also)f (need)g(to)g(use)g(the)g(mean)f(ergo)r(dic)h(theorem)e(as)j(in)f([14].) 308 2674 y Fw(Theorem)31 b(2)g Ft(\([6]\))p Fw(.)g Fg(L)m(et)d Fr(f)35 b Fg(b)m(e)29 b(an)f Fr(L)1294 2650 y Fo(1)1321 2674 y Fg(-function)h(on)f(the)g(c)m(omp)m(act)f(manifold)308 2753 y Fr(M)7 b Fg(.)52 b(L)m(et)31 b Fr(\036)611 2763 y Fk(t)662 2753 y Fg(b)m(e)g(a)f(di\013er)m(entiable)i(\015ow)e(on)g Fr(M)38 b Fg(pr)m(eserving)31 b(a)f(given)i(volume)308 2832 y(form)24 b Fr(\026)p Fg(.)409 2926 y Ft(\(1\))j Fg(The)d(limit)912 3063 y Ft(~)898 3081 y Fr(f)7 b Ft(\()p Fr(x)p Ft(\))17 b(=)30 b(lim)1115 3121 y Fk(t)p Fi(!1)1246 3035 y Ft(1)p 1246 3065 34 3 v 1251 3127 a Fr(t)1297 2989 y Fj(Z)1365 3007 y Fk(t)1335 3142 y(s)p Fo(=0)1432 3081 y Fr(f)7 b Ft(\()p Fr(\036)1537 3091 y Fk(s)1561 3081 y Ft(\()p Fr(x)p Ft(\)\))p Fr(ds)521 3232 y Fg(exists)24 b(in)g(the)g Fr(L)937 3207 y Fo(1)964 3232 y Fg(-sense)f(and)g(is)h(an) g(inte)m(gr)m(able)f(function.)409 3313 y Ft(\(2\))k Fg(The)d(inte)m(gr)m(al)f(of)983 3295 y Ft(~)969 3313 y Fr(f)30 b Fg(satis\014es)1073 3366 y Fj(Z)1110 3519 y Fk(M)1189 3440 y Ft(~)1175 3458 y Fr(f)6 b(\026)19 b Ft(=)1343 3366 y Fj(Z)1380 3519 y Fk(M)1445 3458 y Fr(f)7 b(\026)23 b(:)p eop %%Page: 6 6 6 5 bop 308 159 a Fm(6)586 b(D.)18 b(K)o(OTSCHICK)e(AND)h(T.)g(V)o (OGEL)516 294 y Ft(3.)34 b Fs(Linking)27 b(numbers)e(between)g(vector)g (fields)g(and)1137 373 y(submanif)o(olds)376 491 y Ft(Let)i Fr(M)570 466 y Fk(n)629 491 y Ft(b)r(e)g(a)g(smo)r(oth)e(closed)j (orien)n(ted)f Fr(n)p Ft(-manifold,)h(with)f Fr(n)g Fq(\025)h Ft(3.)45 b(W)-6 b(e)308 569 y(\014x)19 b(once)g(and)g(for)g(all)g(a)g (Riemannian)f(metric)g(and)h(the)g(corresp)r(onding)f(linking)308 648 y(form)j Fr(L)h Ft(as)g(in)g(Prop)r(osition)f(1.)376 727 y(Instead)i(of)h(considering)h(the)e(linking)i(b)r(et)n(w)n(een)f (t)n(w)n(o)f(closed)i(orien)n(ted)f(sub-)308 805 y(manifolds)g Fr(N)663 815 y Fo(1)713 805 y Ft(and)g Fr(N)897 815 y Fo(2)923 805 y Ft(,)h(w)n(e)g(shall)f(replace)h Fr(N)1501 815 y Fo(1)1551 805 y Ft(b)n(y)f(lo)r(ops)g(formed)f(b)n(y)h(closing) 308 884 y(up)e(the)g(\015o)n(w)g(lines)h(of)f(a)g(div)n(ergence-free)i (v)n(ector)e(\014eld)g Fr(X)5 b Ft(,)23 b(and)f(consider)g(ho)n(w)308 963 y(these)f(lo)r(ops)f(link)g(with)g(an)g(orien)n(ted)g(submanifold)f Fr(N)1696 938 y Fk(n)p Fi(\000)p Fo(2)1809 963 y Ft(pla)n(ying)i(the)f (role)h(of)308 1041 y Fr(N)361 1051 y Fo(2)410 1041 y Ft(ab)r(o)n(v)n(e.)32 b(As)23 b(b)r(efore,)g(w)n(e)f(assume)g(that)f Fr(N)27 b Fq(\032)20 b Fr(M)29 b Ft(is)23 b(n)n(ull-homologous)f(o)n(v) n(er)308 1120 y Fp(R)p Ft(.)41 b(T)-6 b(o)25 b(ensure)f(that)g(all)i (the)f(lo)r(ops)g(are)g(n)n(ull-homologous,)f(w)n(e)i(assume)e(that)308 1199 y(the)e(\014rst)f(Betti)h(n)n(um)n(b)r(er)e(of)i Fr(M)29 b Ft(v)l(anishes.)376 1277 y(Let)16 b Fr(\026)f Ft(b)r(e)g(a)h(v)n(olume)f(form)f(on)i Fr(M)7 b Ft(,)17 b(and)e Fr(X)20 b Ft(a)c(v)n(ector)g(\014eld)f(that)g(is)h(div)n (ergence-)308 1356 y(free)22 b(with)f(resp)r(ect)g(to)g Fr(\026)p Ft(,)h(i.)g(e.)g(suc)n(h)f(that)g Fr(L)1432 1366 y Fk(X)1477 1356 y Fr(\026)e Ft(=)g(0.)29 b(Then)21 b(the)g(\()p Fr(n)14 b Fq(\000)g Ft(1\)-form)308 1435 y Fr(\021)26 b Ft(=)e Fr(i)465 1445 y Fk(X)510 1435 y Fr(\026)g Ft(is)h(closed,)i(and)d(our)g(assumption)f(that)h(the)g (\014rst)g(Betti)h(n)n(um)n(b)r(er)e(of)308 1513 y Fr(M)34 b Ft(v)l(anishes)27 b(implies,)h(via)f(P)n(oincar)n(\023)-31 b(e)27 b(dualit)n(y)-6 b(,)28 b(that)e Fr(\021)j Ft(m)n(ust)c(b)r(e)i (exact.)43 b(Let)308 1592 y Fr(\013)28 b Ft(b)r(e)g(a)g(primitiv)n(e.) 46 b(Then,)29 b(generalising)g(\(2\),)g(w)n(e)f(can)g(de\014ne)f(a)h (Hopf-t)n(yp)r(e)308 1671 y(in)n(tegral)22 b(for)g Fr(X)27 b Ft(and)22 b Fr(N)29 b Ft(b)n(y)21 b(setting)1058 1811 y Fr(H)5 b Ft(\()p Fr(X)r(;)11 b(N)c Ft(\))18 b(=)1404 1719 y Fj(Z)1441 1871 y Fk(N)1498 1811 y Fr(\013)23 b(:)308 1956 y Ft(If)f(w)n(e)g(c)n(ho)r(ose)h(a)f(di\013eren)n(t)f(primitiv)n (e)h Fr(\013)1321 1931 y Fi(0)1359 1956 y Ft(for)g Fr(\021)e Ft(=)f Fr(i)1606 1966 y Fk(X)1652 1956 y Fr(\026)p Ft(,)j(then)951 2004 y Fj(Z)988 2157 y Fk(N)1045 2096 y Fr(\013)1087 2068 y Fi(0)1118 2096 y Fq(\000)1186 2004 y Fj(Z)1223 2157 y Fk(N)1280 2096 y Fr(\013)d Ft(=)1411 2004 y Fj(Z)1448 2157 y Fk(N)1505 2096 y Fr(\013)1547 2068 y Fi(0)1578 2096 y Fq(\000)d Fr(\013)308 2241 y Ft(v)l(anishes)23 b(b)r(ecause)f Fr(\013)854 2217 y Fi(0)886 2241 y Fq(\000)15 b Fr(\013)23 b Ft(is)g(closed)g(and)e Fr(N)29 b Ft(is)23 b(assumed)e(to)h(b)r(e)g(homologous)308 2320 y(to)g(zero.)30 b(Th)n(us)21 b Fr(H)5 b Ft(\()p Fr(X)r(;)11 b(N)c Ft(\))21 b(is)i(w)n(ell-de\014ned.)376 2399 y(W)-6 b(e)17 b(w)n(an)n(t)e(to)h (in)n(terpret)g(this)g(in)n(tegral)g(as)g(an)g(a)n(v)n(erage)h(of)g (asymptotic)e(linking)308 2477 y(n)n(um)n(b)r(ers)20 b(of)i(\015o)n(w)f(lines)i(of)e Fr(X)27 b Ft(with)21 b Fr(N)7 b Ft(.)30 b(T)-6 b(o)21 b(do)h(so,)g(w)n(e)g(\014x)f(once)h (and)f(for)g(all)i(a)308 2556 y(\\system)h(of)h(short)e(paths")h (connecting)h(an)n(y)f(pair)g(of)h(p)r(oin)n(ts)f Fr(p;)11 b(q)27 b Fq(2)c Fr(M)7 b Ft(.)37 b(The)308 2634 y(\015o)n(w)22 b(of)g Fr(X)27 b Ft(will)c(b)r(e)f(denoted)f(b)n(y)h Fr(\036)1198 2644 y Fk(t)1218 2634 y Ft(.)308 2728 y Fw(De\014nition)d(3.)j Ft(A)16 b(set)g(\006)f(of)g(piecewise)i (di\013eren)n(tiable)e(paths)f(in)i Fr(M)22 b Ft(is)15 b Fg(a)j(system)308 2807 y(of)24 b(short)f(p)m(aths)d Ft(if)j(it)f(has)f(the)h(follo)n(wing)h(prop)r(erties:)409 2900 y(\(1\))k(F)-6 b(or)21 b(an)n(y)f(t)n(w)n(o)h(p)r(oin)n(ts)f Fr(p;)11 b(q)22 b Fq(2)d Fr(M)27 b Ft(there)21 b(is)g(exactly)g(one)g (path)e Fr(\033)r Ft(\()p Fr(p;)11 b(q)r Ft(\))20 b Fq(2)521 2979 y Ft(\006)i(starting)f(at)h Fr(p)g Ft(and)f(ending)h(at)f Fr(q)r Ft(.)409 3057 y(\(2\))27 b(The)f(paths)f(dep)r(end)g(con)n(tin)n (uously)g(on)h(their)g(endp)r(oin)n(ts)f(almost)g(ev-)521 3136 y(erywhere.)409 3214 y(\(3\))i(The)22 b(limit)308 3358 y(\(5\))450 b(lim)833 3399 y Fk(t)p Fi(!1)964 3313 y Ft(1)p 964 3343 34 3 v 969 3404 a Fr(t)1015 3266 y Fj(Z)1052 3419 y Fk(y)r Fi(2)p Fk(\033)r Fo(\()p Fk(\036)1186 3425 y Fe(t)1204 3419 y Fo(\()p Fk(x)p Fo(\))p Fk(;x)p Fo(\))1341 3266 y Fj(Z)1378 3419 y Fk(p)p Fi(2)p Fk(N)1491 3358 y Fr(L)p Ft(\()p Fr(p;)11 b(y)r Ft(\))19 b(=)g(0)521 3519 y(exists)j(in)g(the)g Fr(L)939 3494 y Fo(1)966 3519 y Ft(-sense.)p eop %%Page: 7 7 7 6 bop 677 159 a Fm(LINKING)23 b(NUMBERS)f(OF)h(MEASURED)g(F)o(OLIA)l (TIONS)342 b(7)308 294 y Fw(Theorem)22 b(4.)k Fg(A)21 b(system)g(of)g(short)f(p)m(aths)g(exists.)30 b(It)22 b(c)m(an)e(b)m(e)i(chosen)f(indep)m(en-)308 373 y(dently)j(of)f(the)h (ve)m(ctor)g(\014eld)f Fr(X)5 b Fg(.)308 615 y(Pr)m(o)m(of.)27 b Ft(Let)d Fr(C)5 b Ft(\()p Fr(y)r Ft(\))22 b(=)864 561 y Fj(R)896 639 y Fk(p)p Fi(2)p Fk(N)1008 615 y Fr(L)p Ft(\()p Fr(p;)11 b(y)r Ft(\).)36 b(This)24 b(is)g(a)g(w)n (ell-de\014ned)g Fr(L)1941 591 y Fo(1)1968 615 y Ft(-form)f(on)g Fr(M)7 b Ft(.)308 700 y(Since)26 b Fr(M)33 b Ft(is)25 b(compact)g(it)h(can)f(b)r(e)g(co)n(v)n(ered)h(b)n(y)g(a)f(\014nite)g (n)n(um)n(b)r(er)f(of)i(geo)r(desic)308 779 y(balls)d Fr(U)508 789 y Fk(j)532 779 y Ft(,)g Fr(j)g Ft(=)c(1)p Fr(;)11 b(:)g(:)g(:)i(;)e(r)r Ft(.)31 b(In)21 b(eac)n(h)i(geo)r(desic)g (ball)f(w)n(e)h(\014x)e(a)i(basep)r(oin)n(t)e Fr(u)2154 789 y Fk(j)2197 779 y Fq(2)e Fr(U)2306 789 y Fk(j)308 857 y Ft(suc)n(h)j(that)f(the)h(follo)n(wing)h(conditions)e(are)h (satis\014ed.)409 1003 y(\(1\))27 b(F)-6 b(or)31 b(ev)n(ery)f(pair)g Fr(k)r(;)11 b(j)36 b Ft(there)29 b(is)i(a)f(path)g Fr(\015)1613 1013 y Fk(k)q(j)1694 1003 y Ft(parametrised)f(b)n(y)g([0)p Fr(;)11 b Ft(1])521 1081 y(joining)22 b Fr(u)777 1091 y Fk(k)828 1081 y Ft(and)f Fr(u)994 1091 y Fk(j)1040 1081 y Ft(suc)n(h)h(that)f(the)h(in)n(tegral)308 1343 y(\(6\))753 1251 y Fj(Z)791 1404 y Fk(y)r Fi(2)p Fk(\015)874 1412 y Fe(k)q(j)931 1343 y Fq(j)p Fr(C)5 b Ft(\()p Fr(y)r Ft(\))p Fq(j)41 b Ft(:=)1258 1251 y Fj(Z)1325 1269 y Fo(1)1295 1404 y(0)1363 1265 y Fj(\014)1363 1306 y(\014)1363 1346 y(\014)1386 1343 y Fr(i)7 b Fe(@)r(\015)1462 1355 y(k)q(j)p 1415 1365 92 3 v 1439 1393 a(@)r(s)1516 1343 y Fr(C)e Ft(\()p Fr(\015)1628 1353 y Fk(k)q(j)1679 1343 y Ft(\()p Fr(s)p Ft(\)\))1788 1265 y Fj(\014)1788 1306 y(\014)1788 1346 y(\014)1820 1343 y Fr(ds)521 1604 y Ft(is)23 b(\014nite.)409 1683 y(\(2\))k(Let)e Fr(\033)r Ft(\()p Fr(x;)11 b(u)813 1693 y Fk(j)837 1683 y Ft(\))25 b(denote)f(the)h(unique)f(geo)r(desic)i(in)f Fr(U)1824 1693 y Fk(j)1874 1683 y Ft(b)r(et)n(w)n(een)f Fr(x)g Fq(2)f Fr(U)2306 1693 y Fk(j)521 1762 y Ft(and)e Fr(u)687 1772 y Fk(j)712 1762 y Ft(.)29 b(F)-6 b(or)23 b(all)f Fr(j)k Ft(the)c(in)n(tegral)308 2012 y(\(7\))953 1920 y Fj(Z)990 2072 y Fk(x)p Fi(2)p Fk(U)1081 2079 y Fe(j)1118 1920 y Fj(Z)1155 2072 y Fk(y)r Fi(2)p Fk(\033)r Fo(\()p Fk(x;u)1329 2079 y Fe(j)1349 2072 y Fo(\))1382 2012 y Fq(j)p Fr(C)5 b Ft(\()p Fr(y)r Ft(\))p Fq(j)p Fr(\026)p Ft(\()p Fr(x)p Ft(\))521 2272 y(is)23 b(\014nite.)308 2418 y(The)c(second)g(condition)g(is)g(satis\014ed)g(if)h(all)g(the)e Fr(u)1547 2428 y Fk(j)1591 2418 y Ft(are)h(outside)g(of)g Fr(N)7 b Ft(.)28 b(Since)20 b Fr(N)308 2496 y Ft(has)k(co)r(dimension)g (t)n(w)n(o,)g(b)r(oth)f(conditions)h(hold)g(for)g(a)g(generic)h(c)n (hoice)h(of)e(the)308 2575 y Fr(u)346 2585 y Fk(j)371 2575 y Ft(.)376 2654 y(F)-6 b(or)27 b(all)h Fr(x)f Fq(2)g Fr(M)34 b Ft(\014x)27 b(a)g(n)n(um)n(b)r(er)e Fr(n)p Ft(\()p Fr(x)p Ft(\))h(suc)n(h)h(that)g Fr(x)f Fq(2)i Fr(U)1875 2664 y Fk(n)p Fo(\()p Fk(x)p Fo(\))1998 2654 y Ft(and)e Fr(n)p Ft(\()p Fr(x)p Ft(\))g(is)308 2732 y(lo)r(cally)32 b(constan)n(t)e(on)g(a)h(dense)g(op)r(en)f(subset)g(of) h Fr(M)7 b Ft(.)55 b(Let)31 b Fr(p;)11 b(q)37 b Fq(2)c Fr(M)7 b Ft(.)56 b(W)-6 b(e)308 2811 y(de\014ne)20 b(a)h(piecewise)h (di\013eren)n(tiable)e(path)g Fr(\033)r Ft(\()p Fr(p;)11 b(q)r Ft(\))21 b(joining)g Fr(p)f Ft(and)g Fr(q)j Ft(as)e(follo)n(ws.) 308 2890 y(The)k(\014rst)f(segmen)n(t)g(of)h Fr(\033)r Ft(\()p Fr(p;)11 b(q)r Ft(\))25 b(is)h(the)e(unique)h(geo)r(desic)g(in) g Fr(U)1922 2900 y Fk(n)p Fo(\()p Fk(p)p Fo(\))2040 2890 y Ft(b)r(et)n(w)n(een)g Fr(p)308 2968 y Ft(and)g Fr(u)478 2979 y Fk(n)p Fo(\()p Fk(p)p Fo(\))571 2968 y Ft(,)i(the)e(second)g (segmen)n(t)g(is)h Fr(\015)1312 2979 y Fk(n)p Fo(\()p Fk(p)p Fo(\))p Fk(;n)p Fo(\()p Fk(q)r Fo(\))1533 2968 y Ft(and)f(the)g(third)g(is)h(the)f(unique)308 3047 y(geo)r(desic)20 b(in)f Fr(U)683 3057 y Fk(n)p Fo(\()p Fk(q)r Fo(\))793 3047 y Ft(starting)f(at)h Fr(u)1152 3057 y Fk(n)p Fo(\()p Fk(q)r Fo(\))1262 3047 y Ft(and)f(ending)h(at)f Fr(q)r Ft(.)29 b(Note)20 b(that)d(for)i Fr(x)f Ft(with)308 3126 y Fr(n)p Ft(\()p Fr(x)p Ft(\))g(=)h Fr(j)t Ft(,)i(the)e(short)g(path)g (b)r(et)n(w)n(een)g Fr(x)g Ft(and)g Fr(u)1498 3136 y Fk(j)1542 3126 y Ft(is)h(the)g(unique)f(geo)r(desic)h(in)g Fr(U)2306 3136 y Fk(j)308 3204 y Ft(joining)i Fr(x)g Ft(and)f Fr(u)752 3214 y Fk(j)777 3204 y Ft(,)h(this)g(justi\014es)g (the)f(notation)g Fr(\033)j Ft(for)e(b)r(oth)f(ob)t(jects.)376 3283 y(W)-6 b(e)32 b(de\014ne)f(\006)g(to)g(b)r(e)g(the)g(set)g(of)h (paths)e(obtained)h(this)g(w)n(a)n(y)-6 b(.)58 b(F)-6 b(or)31 b(eac)n(h)308 3362 y Fr(p;)11 b(q)30 b Fq(2)c Fr(M)33 b Ft(there)26 b(is)h(a)f(unique)g(path)f(with)h(starting)f(p)r (oin)n(t)h Fr(p)h Ft(and)e(end)h(p)r(oin)n(t)308 3440 y Fr(q)r Ft(.)52 b(By)29 b(construction,)i(the)e(paths)f(dep)r(end)h (in)g(a)g(con)n(tin)n(uous)g(w)n(a)n(y)g(on)g(their)308 3519 y(starting)21 b(and)h(end)f(p)r(oin)n(ts)h(on)f(an)h(op)r(en)g (dense)g(subset)f(of)h Fr(M)g Fq(\002)16 b Fr(M)7 b Ft(.)p eop %%Page: 8 8 8 7 bop 308 159 a Fm(8)586 b(D.)18 b(K)o(OTSCHICK)e(AND)h(T.)g(V)o (OGEL)376 294 y Ft(Dividing)22 b(the)g(paths)f(in)n(to)h(their)g (di\013eren)n(tiable)g(pieces)h(w)n(e)f(obtain)393 351 y Fj(Z)431 503 y Fk(x)p Fi(2)p Fk(M)554 345 y Fj(\014)554 385 y(\014)554 426 y(\014)554 466 y(\014)577 351 y(Z)614 503 y Fk(y)r Fi(2)p Fk(\033)r Fo(\()p Fk(\036)748 509 y Fe(t)766 503 y Fo(\()p Fk(x)p Fo(\))p Fk(;x)p Fo(\))903 442 y Fr(C)5 b Ft(\()p Fr(y)r Ft(\))1041 345 y Fj(\014)1041 385 y(\014)1041 426 y(\014)1041 466 y(\014)1074 442 y Fr(\026)p Ft(\()p Fr(x)p Ft(\))18 b Fq(\024)1293 351 y Fj(Z)1330 503 y Fk(x)p Fi(2)p Fk(M)1454 351 y Fj(Z)1491 503 y Fk(y)r Fi(2)p Fk(\033)r Fo(\()p Fk(\036)1625 509 y Fe(t)1643 503 y Fo(\()p Fk(x)p Fo(\))p Fk(;u)1749 513 y Fe(n)p Fh(\()p Fe(\036)1816 522 y(t)1834 513 y Fh(\()p Fe(x)p Fh(\)\))1909 503 y Fo(\))1942 442 y Fq(j)p Fr(C)5 b Ft(\()p Fr(y)r Ft(\))p Fq(j)p Fr(\026)p Ft(\()p Fr(x)p Ft(\))1284 639 y(+)1350 547 y Fj(Z)1388 700 y Fk(x)p Fi(2)p Fk(M)1511 547 y Fj(Z)1549 700 y Fk(y)r Fi(2)p Fk(\015)1632 710 y Fe(n)p Fh(\()p Fe(\036)1699 719 y(t)1716 710 y Fh(\()p Fe(x)p Fh(\)\))p Fe(;n)p Fh(\()p Fe(x)p Fh(\))1900 639 y Fq(j)p Fr(C)g Ft(\()p Fr(y)r Ft(\))p Fq(j)p Fr(\026)p Ft(\()p Fr(x)p Ft(\))1284 836 y(+)1350 744 y Fj(Z)1388 897 y Fk(x)p Fi(2)p Fk(M)1511 744 y Fj(Z)1549 897 y Fk(y)r Fi(2)p Fk(\033)r Fo(\()p Fk(u)1683 907 y Fe(n)p Fh(\()p Fe(x)p Fh(\))1766 897 y Fk(;x)p Fo(\))1838 836 y Fq(j)p Fr(C)g Ft(\()p Fr(y)r Ft(\))p Fq(j)p Fr(\026)p Ft(\()p Fr(x)p Ft(\))20 b Fr(:)308 999 y Ft(The)29 b(\014rst)e(and)h (the)g(third)g(summand)d(on)k(the)f(righ)n(t)g(hand)f(side)i(are)g(in)g (fact)308 1078 y(equal.)45 b(T)-6 b(o)27 b(see)g(this,)i(apply)d(the)g (v)n(olume-preserving)g(transformation)f Fr(x)h Fq(7!)308 1157 y Fr(\036)347 1167 y Fi(\000)p Fk(t)405 1157 y Ft(\()p Fr(x)p Ft(\))17 b(to)h(the)g(\014rst)g(summand.)25 b(In)18 b(particular)g(these)h(t)n(w)n(o)f(summands)d(do)j(not)308 1235 y(dep)r(end)j(on)h Fr(t)p Ft(.)30 b(The)21 b(second)h(summand)d (is)k(b)r(ounded)d(ab)r(o)n(v)n(e)i(b)n(y)865 1404 y(max)901 1445 y Fk(i;j)999 1288 y Fj( )1052 1312 y(Z)1090 1464 y Fk(y)r Fi(2)p Fk(\015)1173 1471 y Fe(ij)1223 1404 y Fq(j)p Fr(C)5 b Ft(\()p Fr(y)r Ft(\))p Fq(j)1399 1288 y Fj(!)1485 1312 y(Z)1522 1464 y Fk(x)p Fi(2)p Fk(M)1645 1404 y Fr(\026)p Ft(\()p Fr(x)p Ft(\))308 1572 y(and)22 b(the)f(third)g(summand)e(is)k(b)r(ounded)d(ab)r(o)n(v)n(e)i(b)n(y)920 1649 y Fk(r)883 1669 y Fj(X)893 1811 y Fk(i)p Fo(=1)991 1641 y Fj(Z)1029 1794 y Fk(x)p Fi(2)p Fk(U)1120 1801 y Fe(i)1152 1641 y Fj(Z)1189 1794 y Fk(y)r Fi(2)p Fk(\033)r Fo(\()p Fk(u)1323 1801 y Fe(i)1339 1794 y Fk(;x)p Fo(\))1412 1733 y Fq(j)p Fr(C)5 b Ft(\()p Fr(y)r Ft(\))p Fq(j)p Fr(\026)p Ft(\()p Fr(x)p Ft(\))20 b Fr(:)308 1902 y Ft(Th)n(us)i(\006)f (meets)h(the)f(third)g(condition)h(of)g(De\014nition)g(3.)544 b Ff(\003)376 2019 y Ft(If)23 b Fr(x)g Ft(is)g(a)g(p)r(oin)n(t)g(in)g Fr(M)30 b Ft(and)23 b Fr(t)d Fq(2)h Fp(R)p Ft(,)26 b(denote)c(b)n(y)h Fr(\036)p Ft(\()p Fr(x;)11 b(t)p Ft(\))23 b(the)g(\015o)n(w)f(line)i (of)g Fr(X)308 2098 y Ft(generated)j(b)n(y)g Fr(x)g Ft(in)g(the)g (time-in)n(terv)l(al)g([0)p Fr(;)11 b(t)p Ft(].)46 b(W)-6 b(e)28 b(shall)g(denote)f(b)n(y)g Fr(\015)t Ft(\()p Fr(x;)11 b(t)p Ft(\))308 2177 y(the)27 b(closed)g(lo)r(op)f(obtained)g(b)n(y)g (connecting)h(the)f(endp)r(oin)n(ts)g Fr(x)g Ft(and)g Fr(\036)2142 2187 y Fk(t)2162 2177 y Ft(\()p Fr(x)p Ft(\))g(of)308 2255 y Fr(\036)p Ft(\()p Fr(x;)11 b(t)p Ft(\))22 b(b)n(y)f(the)h(path)f Fr(\033)r Ft(\()p Fr(\036)976 2265 y Fk(t)996 2255 y Ft(\()p Fr(x)p Ft(\))p Fr(;)11 b(x)p Ft(\).)376 2334 y(The)21 b(follo)n(wing)i(is)f(true)g(for)g(dimension)f(reasons:)308 2428 y Fw(Lemma)34 b(5.)e Fg(L)m(et)f Fr(t)h Fq(2)g Fp(R)h Fg(b)m(e)e(\014xe)m(d.)53 b(Then)31 b(for)f Fr(\026)p Fg(-almost)g(al)s(l)g Fr(x)i Fq(2)g Fr(M)7 b Fg(,)33 b(the)308 2507 y(pie)m(c)m(ewise)24 b(di\013er)m(entiable)h(curve)g Fr(\015)t Ft(\()p Fr(x;)11 b(t)p Ft(\))23 b Fg(is)g(emb)m(e)m(dde)m(d)g (in)h Fr(M)e Fq(n)15 b Fr(N)7 b Fg(.)376 2602 y Ft(Giv)n(en)22 b(this)g(Lemma)e(and)i(our)f(assumptions)f(that)h(the)h(\014rst)f (Betti)h(n)n(um)n(b)r(er)308 2680 y(of)d Fr(M)25 b Ft(v)l(anishes)19 b(and)f(that)f Fr(N)26 b Ft(is)19 b(n)n(ull-homologous)e(o)n(v)n(er)i Fp(R)p Ft(,)i(w)n(e)d(can)h(de\014ne)f(the)308 2759 y(linking)k(n)n(um) n(b)r(er)f(lk\()p Fr(\015)t Ft(\()p Fr(x;)11 b(t)p Ft(\))p Fr(;)g(N)c Ft(\))21 b(for)h(almost)f(all)h Fr(x)g Ft(and)f Fr(t)p Ft(.)30 b(W)-6 b(e)22 b(then)g(ha)n(v)n(e:)308 2853 y Fw(Prop)r(osition)k(6.)i Fg(The)c(limit)887 2993 y Ft(lk\()p Fr(x;)11 b(N)c Ft(\))18 b(=)30 b(lim)1207 3033 y Fk(t)p Fi(!1)1338 2947 y Ft(1)p 1338 2977 34 3 v 1343 3039 a Fr(t)1378 2993 y Ft(lk\()p Fr(\015)t Ft(\()p Fr(x;)11 b(t)p Ft(\))p Fr(;)g(N)c Ft(\))308 3131 y Fg(exists)29 b(in)g(the)f Fr(L)738 3106 y Fo(1)765 3131 y Fg(-sense.)45 b(It)29 b(is)g(an)f(inte)m(gr)m(able)g(function)i(on)e Fr(M)35 b Fg(which)28 b(do)m(es)308 3209 y(not)c(dep)m(end)f(on)g(the)h (chosen)f(system)h(of)g(short)e(p)m(aths.)308 3327 y(Pr)m(o)m(of.)27 b Ft(By)22 b(Prop)r(osition)f(1)h(w)n(e)g(ha)n(v)n(e)615 3469 y(lim)605 3510 y Fk(t)p Fi(!1)736 3424 y Ft(1)p 736 3454 V 741 3515 a Fr(t)775 3469 y Ft(lk\()p Fr(\015)t Ft(\()p Fr(x;)11 b(t)p Ft(\))p Fr(;)g(N)c Ft(\))18 b(=)30 b(lim)1238 3510 y Fk(t)p Fi(!1)1369 3424 y Ft(1)p 1369 3454 V 1374 3515 a Fr(t)1420 3377 y Fj(Z)1457 3530 y Fk(y)r Fi(2)p Fk(\015)s Fo(\()p Fk(x;t)p Fo(\))1650 3377 y Fj(Z)1688 3530 y Fk(p)p Fi(2)p Fk(N)1800 3469 y Fr(L)p Ft(\()p Fr(p;)11 b(y)r Ft(\))23 b Fr(:)p eop %%Page: 9 9 9 8 bop 677 159 a Fm(LINKING)23 b(NUMBERS)f(OF)h(MEASURED)g(F)o(OLIA)l (TIONS)342 b(9)308 294 y Ft(Using)21 b(the)e(third)h(prop)r(ert)n(y)e (of)i(the)g(system)f(of)h(short)f(paths,)h(w)n(e)g(\014nd)f(that)g(the) 308 373 y(righ)n(t)j(hand)f(side)h(equals)880 524 y(lim)869 564 y Fk(t)p Fi(!1)1000 478 y Ft(1)p 1000 508 34 3 v 1005 570 a Fr(t)1051 432 y Fj(Z)1118 450 y Fk(t)1088 585 y(s)p Fo(=0)1185 432 y Fj(Z)1223 585 y Fk(p)p Fi(2)p Fk(N)1336 524 y Fr(i)1359 534 y Fk(X)1404 524 y Fr(L)p Ft(\()p Fr(p;)11 b(\036)1576 534 y Fk(s)1601 524 y Fr(x)p Ft(\))p Fr(ds)21 b(:)308 680 y Ft(Since)f Fr(L)p Ft(\()p Fr(p;)11 b Fq(\001)p Ft(\))19 b(is)h(an)f(in)n(tegrable)g(form)g(on)g Fr(M)26 b Ft(for)19 b(ev)n(ery)h Fr(p)e Fq(2)h Fr(M)7 b Ft(,)20 b(w)n(e)g(can)f(apply)308 759 y(the)27 b(mean)f(ergo)r(dic)h (theorem,)h(Theorem)e(2.)45 b(Hence)28 b(the)f(limit)g(exists.)45 b(It)26 b(is)308 837 y(clearly)d(indep)r(enden)n(t)e(of)h(the)g(system) f(of)h(short)f(paths.)556 b Ff(\003)376 953 y Ft(Using)18 b(this,)g(w)n(e)g(can)g(\014nally)f(de\014ne)h(the)f(a)n(v)n(erage)h (asymptotic)e(linking)i(n)n(um-)308 1032 y(b)r(er)k(of)g(the)g(v)n (ector)g(\014eld)g Fr(X)27 b Ft(with)21 b(the)h(submanifold)e Fr(N)7 b Ft(,)23 b(b)n(y)e(setting)943 1173 y(lk\()p Fr(X)r(;)11 b(N)c Ft(\))18 b(=)1282 1081 y Fj(Z)1320 1234 y Fk(M)1384 1173 y Ft(lk)q(\()p Fr(x;)11 b(N)c Ft(\))p Fr(\026)21 b(:)308 1320 y Ft(The)32 b(analog)g(of)g(the)f(theorem)g (pro)n(v)n(ed)g(in)h([1,)h(14])f(for)f(v)n(ector)h(\014elds)g(on)g(3-) 308 1399 y(manifolds)22 b(is:)308 1493 y Fw(Theorem)27 b(7.)i Fg(L)m(et)c Fr(M)32 b Fg(b)m(e)26 b(a)f(close)m(d)f(oriente)m(d) h Fr(n)p Fg(-manifold)g(with)g Fr(b)2034 1503 y Fo(1)2061 1493 y Ft(\()p Fr(M)7 b Ft(\))20 b(=)i(0)p Fg(.)308 1572 y(L)m(et)30 b Fr(X)35 b Fg(b)m(e)30 b(a)g(diver)m(genc)m(e-fr)m(e)m(e)h (ve)m(ctor)e(\014eld)h(on)g Fr(M)7 b Fg(,)32 b(and)d Fr(N)37 b Fq(\032)31 b Fr(M)37 b Fg(a)29 b(close)m(d)308 1650 y(oriente)m(d)e(submanifold)h(of)f(c)m(o)m(dimension)g Ft(2)g Fg(which)g(is)g(nul)s(l-homolo)m(gous)g(over)308 1729 y Fp(R)p Fg(.)32 b(Then)21 b(the)g(aver)m(age)g(asymptotic)f (linking)i(numb)m(er)g(of)e(the)h(orbits)g(of)g Fr(X)26 b Fg(with)308 1807 y Fr(N)31 b Fg(exists)24 b(and)f(e)m(quals)h(a)f (Hopf-typ)m(e)g(inte)m(gr)m(al:)1000 1917 y Ft(lk)q(\()p Fr(X)r(;)11 b(N)c Ft(\))18 b(=)h Fr(H)5 b Ft(\()p Fr(X)r(;)11 b(N)c Ft(\))22 b Fr(:)308 2033 y Fg(Pr)m(o)m(of.)27 b Ft(W)-6 b(e)23 b(ha)n(v)n(e)f(seen)g(in)g(the)g(pro)r(of)f(of)h(Prop)r (osition)g(6)g(that)713 2183 y(lk\()p Fr(x;)11 b(N)c Ft(\))18 b(=)k(lim)1026 2224 y Fk(t)p Fi(!1)1156 2138 y Ft(1)p 1156 2168 V 1161 2230 a Fr(t)1207 2092 y Fj(Z)1275 2109 y Fk(t)1245 2244 y(s)p Fo(=0)1342 2092 y Fj(Z)1379 2244 y Fk(p)p Fi(2)p Fk(N)1492 2183 y Fr(i)1515 2193 y Fk(X)1560 2183 y Fr(L)p Ft(\()p Fr(p;)11 b(\036)1732 2193 y Fk(s)1757 2183 y Fr(x)p Ft(\))p Fr(ds)22 b(:)308 2336 y Ft(By)g(the)g(second)g(part)f(of)h(the)f(mean)h(ergo)r(dic)g (theorem,)f(Theorem)f(2,)j(w)n(e)f(\014nd)524 2487 y(lk\()p Fr(X)r(;)11 b(N)c Ft(\))18 b(=)855 2395 y Fj(Z)893 2548 y Fk(x)p Fi(2)p Fk(M)1016 2392 y Fj(\022)1076 2487 y Ft(lim)1066 2527 y Fk(t)p Fi(!1)1196 2441 y Ft(1)p 1196 2471 V 1201 2533 a Fr(t)1247 2395 y Fj(Z)1315 2413 y Fk(t)1285 2548 y(s)p Fo(=0)1382 2395 y Fj(Z)1419 2548 y Fk(p)p Fi(2)p Fk(N)1532 2487 y Fr(i)1555 2497 y Fk(X)1600 2487 y Fr(L)p Ft(\()p Fr(p;)11 b(\036)1772 2497 y Fk(s)1797 2487 y Fr(x)p Ft(\))p Fr(ds)1926 2392 y Fj(\023)1987 2487 y Fr(\026)p Ft(\()p Fr(x)p Ft(\))793 2673 y(=)855 2582 y Fj(Z)893 2734 y Fk(x)p Fi(2)p Fk(M)1016 2582 y Fj(Z)1054 2734 y Fk(p)p Fi(2)p Fk(N)1166 2673 y Fr(i)1189 2684 y Fk(X)t Fo(\()p Fk(x)p Fo(\))1298 2673 y Fr(L)p Ft(\()p Fr(p;)g(x)p Ft(\))k Fq(^)g Fr(\026)p Ft(\()p Fr(x)p Ft(\))793 2860 y(=)855 2768 y Fj(Z)893 2921 y Fk(x)p Fi(2)p Fk(M)1016 2768 y Fj(Z)1054 2921 y Fk(p)p Fi(2)p Fk(N)1166 2860 y Fr(L)p Ft(\()p Fr(p;)c(x)p Ft(\))k Fq(^)g Fr(i)1461 2871 y Fk(X)t Fo(\()p Fk(x)p Fo(\))1570 2860 y Fr(\026)p Ft(\()p Fr(x)p Ft(\))20 b Fr(;)308 3013 y Ft(where)f Fr(i)519 3023 y Fk(X)564 3013 y Fr(\026)f Ft(=)h Fr(d\013)p Ft(.)30 b(W)-6 b(e)19 b(no)n(w)f(apply)h(Prop)r (osition)f(1)g(to)h(the)f(last)h(in)n(tegral.)29 b(The)308 3091 y(in)n(tegral)23 b(of)g(the)f(harmonic)g(term)f(in)i(Prop)r (osition)e(1)i(o)n(v)n(er)g Fr(N)29 b Ft(is)23 b(zero)g(b)r(ecause)308 3170 y Fr(N)34 b Ft(is)27 b(n)n(ull-homologous,)g(and)e(b)n(y)i(Stok)n (es's)f(theorem)g(the)g(exact)g(term)g(pla)n(ys)308 3248 y(no)c(role.)30 b(Hence)23 b(w)n(e)f(obtain)g(the)f(desired)h(result) 888 3376 y(lk)q(\()p Fr(X)r(;)11 b(N)c Ft(\))18 b(=)1228 3285 y Fj(Z)1265 3437 y Fk(N)1322 3376 y Fr(\013)h Ft(=)g Fr(H)5 b Ft(\()p Fr(X)r(;)11 b(N)c Ft(\))21 b Fr(:)2278 3519 y Ff(\003)p eop %%Page: 10 10 10 9 bop 308 159 a Fm(10)560 b(D.)18 b(K)o(OTSCHICK)e(AND)h(T.)g(V)o (OGEL)376 294 y Ft(T)-6 b(o)22 b(end)g(this)f(section)i(w)n(e)f (consider)g(some)f(geometric)h(examples.)308 380 y Fg(Example)h Ft(8)p Fg(.)g Ft(Let)16 b(\()p Fr(M)s(;)11 b(!)r Ft(\))16 b(b)r(e)g(a)f(symplectic)h(manifold)f(of)h(dimension)e(2)p Fr(n)p Ft(.)28 b(Then)308 459 y Fr(!)351 434 y Fk(n)407 459 y Ft(is)23 b(a)g(v)n(olume)f(form)g(on)h Fr(M)7 b Ft(.)32 b(Consider)22 b(a)h(smo)r(oth)f(function)g Fr(H)28 b Ft(on)23 b Fr(M)29 b Ft(and)308 538 y(the)h(corresp)r(onding)g (Hamiltonian)g(v)n(ector)g(\014eld)h Fr(X)1664 548 y Fk(H)1709 538 y Ft(.)55 b(This)30 b(is)h(the)f(unique)308 616 y(v)n(ector)25 b(\014eld)f(on)h Fr(M)31 b Ft(satisfying)25 b Fr(dH)j Ft(=)23 b Fr(i)1355 626 y Fk(X)1394 634 y Fe(H)1436 616 y Fr(!)r Ft(.)38 b(Then)24 b Fr(X)1765 626 y Fk(H)1835 616 y Ft(is)g(div)n(ergence-free)308 695 y(with)e(resp)r(ect)g(to)f Fr(!)808 670 y Fk(n)863 695 y Ft(since)727 803 y Fr(L)772 813 y Fk(X)811 821 y Fe(H)853 803 y Fr(!)896 776 y Fk(n)947 803 y Ft(=)e Fr(di)1074 813 y Fk(X)1113 821 y Fe(H)1155 803 y Fr(!)1198 776 y Fk(n)1249 803 y Ft(=)g Fr(d)p Ft(\()p Fr(ndH)i Fq(^)15 b Fr(!)1631 776 y Fk(n)p Fi(\000)p Fo(1)1724 803 y Ft(\))j(=)i(0)i Fr(:)308 912 y Ft(Moreo)n(v)n(er,)h(the)f(\(2)p Fr(n)15 b Fq(\000)g Ft(2\){form)21 b Fr(nH)5 b(!)1300 888 y Fk(n)p Fi(\000)p Fo(1)1416 912 y Ft(satis\014es)493 1021 y Fr(d)p Ft(\()p Fr(nH)g(!)695 993 y Fk(n)p Fi(\000)p Fo(1)789 1021 y Ft(\))18 b(=)h Fr(ndH)i Fq(^)15 b Fr(!)1155 993 y Fk(n)p Fi(\000)p Fo(1)1267 1021 y Ft(=)k Fr(n)p Ft(\()p Fr(i)1425 1031 y Fk(X)1464 1039 y Fe(H)1506 1021 y Fr(!)r Ft(\))c Fq(^)g Fr(!)1693 993 y Fk(n)p Fi(\000)p Fo(1)1805 1021 y Ft(=)k Fr(i)1898 1031 y Fk(X)1937 1039 y Fe(H)1979 1021 y Ft(\()p Fr(!)2048 993 y Fk(n)2080 1021 y Ft(\))i Fr(:)308 1130 y Ft(Let)28 b Fr(N)34 b Ft(b)r(e)27 b(a)g(n)n(ull-homologous)g(submanifold)f(of)h(co)r (dimension)g(2)g(in)g Fr(M)7 b Ft(.)46 b(If)308 1208 y Fr(b)336 1218 y Fo(1)363 1208 y Ft(\()p Fr(M)7 b Ft(\))18 b(=)h(0,)j(the)g(linking)g(n)n(um)n(b)r(er)e(of)j Fr(X)k Ft(with)21 b Fr(N)29 b Ft(is)22 b(giv)n(en)h(b)n(y)308 1353 y(\(8\))566 b(lk\()p Fr(X)r(;)11 b(N)c Ft(\))18 b(=)h Fr(n)1348 1261 y Fj(Z)1386 1413 y Fk(N)1443 1353 y Fr(H)5 b(!)1546 1325 y Fk(n)p Fi(\000)p Fo(1)1661 1353 y Fr(:)308 1496 y Ft(F)-6 b(or)23 b(a)g(\014xed)f(Hamiltonian)g(v)n (ector)h(\014eld)g(the)f(function)h(generating)f(it)h(is)g(w)n(ell-)308 1575 y(de\014ned)35 b(only)g(up)f(to)h(the)g(addition)g(of)g(lo)r (cally)h(constan)n(t)e(functions.)69 b(By)308 1653 y(Stok)n(es's)29 b(theorem)e(this)h(am)n(biguit)n(y)f(in)i(the)f(c)n(hoice)i(of)e Fr(H)33 b Ft(do)r(es)c(not)f(c)n(hange)308 1732 y(the)22 b(v)l(alue)g(of)h(the)e(in)n(tegral)i(in)f(\(8\))f(since)i Fr(N)29 b Ft(is)22 b(homologous)f(to)g(zero.)308 1818 y Fg(Example)33 b Ft(9)p Fg(.)e Ft(Consider)26 b(a)h(manifold)f(of)h (dimension)f(2)p Fr(n)19 b Ft(+)g(1)27 b(with)f(a)h(con)n(tact)308 1897 y(form)22 b Fr(\013)f Fq(2)f Ft(\012)641 1872 y Fo(1)668 1897 y Ft(\()p Fr(M)7 b Ft(\).)31 b(This)23 b(means)f(that)g Fr(\013)16 b Fq(^)g Ft(\()p Fr(d\013)p Ft(\))1586 1872 y Fk(n)1640 1897 y Ft(is)24 b(a)f(v)n(olume)f(form)g (on)g Fr(M)7 b Ft(.)308 1975 y(There)21 b(is)g(a)g(unique)g(v)n(ector)f (\014eld)h Fr(X)5 b Ft(,)22 b(called)g(the)e(Reeb)h(v)n(ector)g (\014eld,)h(with)e(the)308 2054 y(prop)r(erties)i Fr(\013)p Ft(\()p Fr(X)5 b Ft(\))18 b(=)h(1)j(and)g Fr(i)1068 2064 y Fk(X)1113 2054 y Fr(d\013)d Ft(=)g(0.)30 b(Because)628 2162 y Fr(L)673 2172 y Fk(X)718 2162 y Ft(\()p Fr(\013)15 b Fq(^)g Ft(\()p Fr(d\013)p Ft(\))989 2135 y Fk(n)1021 2162 y Ft(\))j(=)h Fr(di)1192 2172 y Fk(X)1238 2162 y Ft(\()p Fr(\013)c Fq(^)g Ft(\()p Fr(d\013)p Ft(\))1509 2135 y Fk(n)1541 2162 y Ft(\))j(=)h Fr(d)p Ft(\()p Fr(d\013)p Ft(\))1817 2135 y Fk(n)1868 2162 y Ft(=)g(0)j Fr(;)308 2271 y Ft(the)17 b(Reeb)g(v)n(ector)f(\014eld)h(is)h(div)n (ergence{free)g(with)e(resp)r(ect)h(to)f(the)h(v)n(olume)f(form)308 2350 y Fr(\013)q Fq(^)p Ft(\()p Fr(d\013)p Ft(\))524 2325 y Fk(n)556 2350 y Ft(.)27 b(The)14 b(\(2)p Fr(n)p Fq(\000)p Ft(1\){form)h Fr(\013)q Fq(^)p Ft(\()p Fr(d\013)p Ft(\))1337 2325 y Fk(n)p Fi(\000)p Fo(1)1444 2350 y Ft(is)g(a)g (primitiv)n(e)f(of)h Fr(i)1917 2360 y Fk(X)1962 2350 y Ft(\()p Fr(\013)q Fq(^)p Ft(\()p Fr(d\013)p Ft(\))2204 2325 y Fk(n)2235 2350 y Ft(\))j(=)308 2428 y(\()p Fr(d\013)p Ft(\))436 2404 y Fk(n)468 2428 y Ft(.)29 b(Th)n(us,)21 b(if)f Fr(b)786 2438 y Fo(1)813 2428 y Ft(\()p Fr(M)7 b Ft(\))18 b(=)h(0,)i(the)f(linking)h(n)n(um)n(b)r(er)d(of)j(the)f (Reeb)g(v)n(ector)h(\014eld)308 2507 y Fr(X)27 b Ft(with)22 b(a)g(n)n(ull-homologous)f(submanifold)f Fr(N)29 b Ft(of)22 b(co)r(dimension)g(t)n(w)n(o)f(is)914 2647 y(lk\()p Fr(X)r(;)11 b(N)c Ft(\))18 b(=)1253 2556 y Fj(Z)1290 2708 y Fk(N)1347 2647 y Fr(\013)d Fq(^)g Ft(\()p Fr(d\013)p Ft(\))1592 2620 y Fk(n)p Fi(\000)p Fo(1)1707 2647 y Fr(:)308 2791 y Ft(In)22 b(particular)f(it)h(is)h(nonzero)e(if)i Fr(N)29 b Ft(is)22 b(a)g(con)n(tact)g(submanifold.)308 2877 y Fg(Example)29 b Ft(10)p Fg(.)f Ft(Consider)21 b Fr(M)26 b Ft(=)19 b Fr(S)1175 2852 y Fo(3)1216 2877 y Fq(\002)d Fr(S)1329 2852 y Fo(3)1373 2877 y Fq(\032)k Fp(C)1489 2852 y Fo(2)1535 2877 y Fq(\002)15 b Fp(C)1647 2852 y Fo(2)1700 2877 y Ft(and)21 b(the)h(map)771 2985 y Fr(f)15 b Ft(:)22 b Fr(S)904 2957 y Fo(3)945 2985 y Fq(\002)16 b Fr(S)1058 2957 y Fo(3)1103 2985 y Fq(\000)-11 b(!)19 b Fp(C)921 3088 y Ft(\()p Fr(z)s(;)11 b(w)r Ft(\))18 b Fq(7\000)-11 b(!)19 b(h)p Fr(z)s(;)11 b(w)r Fq(i)20 b Ft(=)f Fr(z)1515 3098 y Fo(0)1554 3088 y Ft(\026)-46 b Fr(w)1588 3098 y Fo(0)1630 3088 y Ft(+)15 b Fr(z)1727 3098 y Fo(1)1767 3088 y Ft(\026)-46 b Fr(w)1801 3098 y Fo(1)1849 3088 y Fr(:)308 3203 y Ft(As)31 b(0)f(is)g(a)g(regular)g(v) l(alue)g(of)g Fr(f)37 b Ft(the)29 b(preimage)h Fr(N)39 b Ft(=)32 b Fr(f)1804 3178 y Fi(\000)p Fo(1)1868 3203 y Ft(\(0\))d(is)i(a)e(smo)r(oth)308 3282 y(submanifold)21 b(of)h Fr(M)29 b Ft(of)22 b(dimension)f(four.)29 b(The)22 b(map)878 3390 y Fr(S)923 3362 y Fo(3)965 3390 y Fq(\002)15 b Fr(S)1077 3362 y Fo(1)1122 3390 y Fq(\000)-11 b(!)19 b Fr(N)788 3492 y Ft(\(\()p Fr(z)871 3502 y Fo(0)897 3492 y Fr(;)11 b(z)957 3502 y Fo(1)984 3492 y Ft(\))p Fr(;)g(\025)p Ft(\))18 b Fq(7\000)-11 b(!)19 b Ft(\(\()p Fr(z)1332 3502 y Fo(0)1358 3492 y Fr(;)11 b(z)1418 3502 y Fo(1)1445 3492 y Ft(\))p Fr(;)g Ft(\()p Fr(\025)t Ft(\026)-37 b Fr(z)1596 3502 y Fo(1)1622 3492 y Fr(;)11 b Fq(\000)p Fr(\025)t Ft(\026)-37 b Fr(z)1773 3502 y Fo(0)1799 3492 y Ft(\)\))p eop %%Page: 11 11 11 10 bop 677 159 a Fm(LINKING)23 b(NUMBERS)f(OF)h(MEASURED)g(F)o(OLIA) l(TIONS)316 b(11)308 294 y Ft(is)20 b(a)f(di\013eomorphism.)26 b(Since)20 b Fr(H)1138 304 y Fo(4)1164 294 y Ft(\()p Fr(M)7 b Ft(\))18 b(=)h(0,)i Fr(N)26 b Ft(is)20 b(n)n(ull-homologous.) 27 b(W)-6 b(e)20 b(no)n(w)308 373 y(construct)h(a)h(submanifold)f(in)h Fr(M)28 b Ft(whose)22 b(b)r(oundary)e(is)i Fr(N)7 b Ft(.)30 b(Consider)21 b(the)h(set)308 451 y([0)p Fr(;)11 b Fq(1)p Ft(\))26 b Fq(\032)f Fp(C)42 b Ft(of)26 b(nonnegativ)n(e)f(real)h(n)n (um)n(b)r(ers.)38 b(Let)26 b(\()p Fr(z)s(;)11 b(w)r Ft(\))25 b(b)r(e)g(a)h(p)r(oin)n(t)e(in)i Fr(M)308 530 y Ft(that)21 b(is)i(mapp)r(ed)d(b)n(y)i Fr(f)28 b Ft(to)22 b(a)g(p)r(ositiv)n(e)g (real)h(n)n(um)n(b)r(er)d Fr(c)p Ft(.)30 b(Then)1003 630 y Fr(d)p 991 660 59 3 v 991 722 a(dt)1056 578 y Fj(\014)1056 618 y(\014)1056 659 y(\014)1056 699 y(\014)1078 743 y Fk(t)p Fo(=0)1171 676 y Fr(f)7 b Ft(\()p Fr(e)1268 648 y Fk(it)1303 676 y Fr(z)s(;)k(w)r Ft(\))19 b(=)g Fr(c@)1593 686 y Fk(y)1643 676 y Fr(:)308 824 y Ft(T)-6 b(ogether)25 b(with)g(the)f(fact)h(that)g(0)g(is)g(a)g(regular)g(v)l(alue)g(of)g Fr(f)7 b Ft(,)26 b(this)f(sho)n(ws)g(that)308 903 y Fr(f)30 b Ft(is)23 b(transv)n(ersal)g(to)g([0)p Fr(;)11 b Fq(1)p Ft(\).)33 b(Th)n(us)23 b Fr(f)1289 878 y Fi(\000)p Fo(1)1352 903 y Ft(\([0)p Fr(;)11 b Fq(1)p Ft(\)\))23 b(is)h(a)f(\014v)n (e-dimensional)f(sub-)308 982 y(manifold)f(of)i Fr(M)29 b Ft(whose)21 b(b)r(oundary)f(is)j Fr(N)7 b Ft(.)376 1060 y(Denote)25 b(b)n(y)f(pr)761 1076 y Fo(1)813 1060 y Ft(the)g(pro)t(jection)g(of)h Fr(M)32 b Ft(on)n(to)24 b(the)g(\014rst)g(factor)g(and)g(b)n(y)h(pr)2304 1076 y Fo(2)308 1139 y Ft(the)i(pro)t(jection)g(on)n(to)f(the)h(second)g (factor.)44 b(W)-6 b(e)28 b(de\014ne)f(the)f(v)n(ector)h(\014eld)g Fr(H)2304 1149 y Fo(1)308 1218 y Ft(on)j Fr(M)37 b Ft(as)31 b(the)f(unique)f(v)n(ector)i(\014eld)f(with)f(the)h(prop)r(ert)n(y)f (that)g(pr)2073 1233 y Fo(1)p Fi(\003)2123 1218 y Ft(\()p Fr(H)2204 1228 y Fo(1)2230 1218 y Ft(\))h(is)308 1296 y(the)f(Hopf)h(v)n(ector)f(\014eld)h(on)f Fr(S)1098 1272 y Fo(3)1153 1296 y Ft(and)g(pr)1351 1312 y Fo(2)p Fi(\003)1402 1296 y Ft(\()p Fr(H)1483 1306 y Fo(1)1509 1296 y Ft(\))i(=)g(0.)52 b(The)29 b(v)n(ector)g(\014eld)g Fr(H)2304 1306 y Fo(2)308 1375 y Ft(is)i(de\014ned)e(similarly)h(with)f(the)h(roles)g(of)h(pr) 1484 1391 y Fo(1)1541 1375 y Ft(and)e(pr)1739 1391 y Fo(2)1796 1375 y Ft(in)n(terc)n(hanged.)53 b(F)-6 b(or)308 1453 y(arbitrary)26 b(constan)n(ts)g Fr(a)h Ft(and)f Fr(b)h Ft(the)g(v)n(ector)g(\014eld)g Fr(aH)1701 1463 y Fo(1)1745 1453 y Ft(+)19 b Fr(bH)1898 1463 y Fo(2)1951 1453 y Ft(preserv)n(es)27 b(the)308 1532 y(canonical)19 b(v)n(olume)f(form)g(on)g Fr(M)25 b Ft(=)19 b Fr(S)1263 1508 y Fo(3)1297 1532 y Fq(\002)8 b Fr(S)1402 1508 y Fo(3)1429 1532 y Ft(.)28 b(The)18 b(\015o)n(w)g(of)h(this)f(v)n(ector)h (\014eld)f(at)308 1611 y(time)k Fr(t)h Ft(maps)e(\()p Fr(z)s(;)11 b(w)r Ft(\))22 b(to)h(\()p Fr(e)1004 1586 y Fk(iat)1064 1611 y Fr(z)s(;)11 b(e)1158 1586 y Fk(ibt)1215 1611 y Fr(w)r Ft(\).)31 b(If)23 b(the)f(di\013erence)h Fr(a)15 b Fq(\000)h Fr(b)22 b Ft(is)h(a)g(rational)308 1689 y(m)n(ultiple)c(of)g(2)p Fr(\031)r Ft(,)h(then)f(all)g(\015o)n(w)g (lines)g(are)g(closed,)i(otherwise)d(all)i(\015o)n(w)e(lines)i(are)308 1768 y(op)r(en.)376 1847 y(W)-6 b(e)24 b(consider)f(no)n(w)h(the)f (\015o)n(w)g(line)h(with)f(starting)f(p)r(oin)n(t)h(\()p Fr(z)s(;)11 b(w)r Ft(\).)34 b(A)n(t)24 b(time)f Fr(t)308 1925 y Ft(w)n(e)f(\014nd)922 2006 y Fq(h)p Fr(z)s Ft(\()p Fr(t)p Ft(\))p Fr(;)11 b(w)r Ft(\()p Fr(t)p Ft(\))p Fq(i)17 b Ft(=)i Fr(e)1356 1978 y Fk(i)p Fo(\()p Fk(a)p Fi(\000)p Fk(b)p Fo(\))p Fk(t)1512 2006 y Fq(h)p Fr(z)s(;)11 b(w)r Fq(i)23 b Fr(:)308 2100 y Ft(If)c Fr(a)f Fq(6)p Ft(=)h Fr(b)f Ft(this)h(means)e(that)g(ev)n(ery)i(\015o)n(w)f(line)h(emerging) e(from)h(a)g(p)r(oin)n(t)g(of)g Fr(M)d Fq(n)8 b Fr(N)308 2178 y Ft(in)n(tersects)31 b Fr(f)649 2154 y Fi(\000)p Fo(1)713 2178 y Ft(\([0)p Fr(;)11 b Fq(1)p Ft(\)\))31 b(transv)n(ersely)g(during)e(the)i(time)f(in)n(terv)l(al)h([0)p Fr(;)2206 2152 y Fo(2)p Fk(\031)p 2178 2163 110 3 v 2178 2202 a Fi(j)p Fk(a)p Fi(\000)p Fk(b)p Fi(j)2294 2178 y Ft(].)308 2272 y(Then)24 b(an)g(in)n(tersection)h(of)f(the)h(\015o)n (w)e(line)j(with)d Fr(f)1587 2247 y Fi(\000)p Fo(1)1651 2272 y Ft(\([0)p Fr(;)11 b Fq(1)p Ft(\)\))25 b(o)r(ccurs)f(p)r(erio)r (di-)308 2351 y(cally)g(after)e(time)h(in)n(terv)l(als)g(of)g(length) 1351 2324 y Fo(2)p Fk(\031)p 1323 2335 V 1323 2374 a Fi(j)p Fk(a)p Fi(\000)p Fk(b)p Fi(j)1439 2351 y Ft(.)32 b(The)22 b(in)n(tersection)h(is)g(p)r(ositiv)n(e)g(if)308 2438 y Fr(a)c(>)g(b)j Ft(and)f(negativ)n(e)i(if)f Fr(a)c(<)h(b)p Ft(.)30 b(Th)n(us)21 b(w)n(e)i(ha)n(v)n(e)f(sho)n(wn)1017 2578 y(lk\(\()p Fr(z)s(;)11 b(w)r Ft(\))p Fr(;)g(N)c Ft(\))18 b(=)1470 2532 y Fr(a)d Fq(\000)g Fr(b)p 1470 2562 146 3 v 1506 2624 a Ft(2)p Fr(\031)308 2709 y Ft(if)23 b(\()p Fr(z)s(;)11 b(w)r Ft(\))18 b Fq(62)h Fr(N)7 b Ft(.)376 2788 y(F)-6 b(or)20 b(\015o)n(w)f(lines)i(with)e(starting)g(p) r(oin)n(t)g(in)h Fr(N)27 b Ft(the)19 b(linking)h(n)n(um)n(b)r(er)e (with)i Fr(N)26 b Ft(is)308 2867 y(not)c(w)n(ell-de\014ned.)376 2945 y(F)-6 b(or)22 b(the)g(asymptotic)e(linking)i(n)n(um)n(b)r(er)f (of)h Fr(aH)1571 2955 y Fo(1)1612 2945 y Ft(+)15 b Fr(bH)1761 2955 y Fo(2)1810 2945 y Ft(with)21 b Fr(N)29 b Ft(w)n(e)22 b(\014nd)590 3086 y(lk)q(\()p Fr(aH)760 3096 y Fo(1)800 3086 y Ft(+)16 b Fr(bH)950 3096 y Fo(2)976 3086 y Fr(;)11 b(N)c Ft(\))19 b(=)1187 3040 y Fr(a)14 b Fq(\000)i Fr(b)p 1187 3070 V 1223 3132 a Ft(2)p Fr(\031)1339 3086 y Ft(\(v)n(ol\()p Fr(S)1520 3058 y Fo(3)1545 3086 y Ft(\)\))1597 3058 y Fo(2)1642 3086 y Ft(=)j(2\()p Fr(a)c Fq(\000)g Fr(b)p Ft(\))p Fr(\031)1981 3058 y Fo(3)2030 3086 y Fr(:)343 3244 y Ft(4.)33 b Fs(Linking)27 b(numbers)f(between)f(vector)f(fields)h (and)i(measured)1189 3322 y(f)o(olia)l(tions)376 3440 y Ft(As)32 b(b)r(efore,)i(w)n(e)e(assume)e(that)h Fr(M)38 b Ft(is)32 b(a)g(closed)g(orien)n(ted)g(manifold)e(with)308 3519 y Fr(b)336 3529 y Fo(1)363 3519 y Ft(\()p Fr(M)7 b Ft(\))29 b(=)i(0,)g Fr(\026)d Ft(is)h(a)g(v)n(olume)f(form)g(on)g Fr(M)7 b Ft(,)31 b(and)d Fr(X)33 b Ft(a)c(v)n(ector)g(\014eld)g(that)e (is)p eop %%Page: 12 12 12 11 bop 308 159 a Fm(12)560 b(D.)18 b(K)o(OTSCHICK)e(AND)h(T.)g(V)o (OGEL)308 294 y Ft(div)n(ergence-free)27 b(with)d(resp)r(ect)h(to)g Fr(\026)p Ft(,)h(i.)g(e.)g(suc)n(h)f(that)f Fr(L)1789 304 y Fk(X)1834 294 y Fr(\026)g Ft(=)g(0.)40 b(Then)24 b(the)308 373 y(\()p Fr(n)9 b Fq(\000)g Ft(1\)-form)18 b Fr(\021)j Ft(=)e Fr(i)824 383 y Fk(X)869 373 y Fr(\026)f Ft(is)i(closed,)g(and)f(our)f(assumptions)f(imply)h(that)g Fr(\021)j Ft(m)n(ust)308 451 y(b)r(e)h(exact.)30 b(Let)22 b Fr(\013)h Ft(b)r(e)f(a)g(primitiv)n(e.)376 530 y(W)-6 b(e)15 b(generalise)h(the)e(discussion)h(in)g(section)g(3)f(b)n(y)h (replacing)g(the)f(submanifold)308 609 y Fr(N)27 b Ft(b)n(y)19 b(an)g(orien)n(ted)g(co)r(dimension)g(2)g(foliation)h Fq(F)27 b Ft(with)19 b(a)g(holonom)n(y-in)n(v)l(arian)n(t)308 687 y(transv)n(erse)j(measure)f Fr(\027)t Ft(.)29 b(This)22 b(de\014nes)g(a)g(curren)n(t)830 796 y Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\))c(:)23 b(\012)1151 769 y Fk(n)p Fi(\000)p Fo(2)1243 796 y Ft(\()p Fr(M)7 b Ft(\))18 b Fq(\000)-10 b(!)19 b Fp(R)1322 940 y Fr(!)i Fq(7\000)-10 b(!)1512 848 y Fj(Z)1549 1001 y Fk(M)1614 940 y Fr(!)17 b Fq(^)e Fr(\027)26 b(;)308 1084 y Ft(where)981 1092 y Fj(Z)1018 1245 y Fk(M)1083 1184 y Fr(!)18 b Fq(^)d Fr(\027)22 b Ft(=)1327 1092 y Fj(Z)1365 1245 y Fk(T)1402 1184 y Ft(\()1428 1092 y Fj(Z)1465 1245 y Fi(F)1518 1184 y Fr(!)r Ft(\))p Fr(d\027)308 1314 y Ft(is)i(de\014ned)e(b)n(y)g (decomp)r(osing)g Fr(!)k Ft(using)c(a)h(partition)f(of)h(unit)n(y)g (sub)r(ordinate)e(to)308 1392 y(a)i(\014nite)g(atlas)g(of)h(foliation)f (c)n(harts)g(for)g(\()p Fr(M)s(;)11 b Fq(F)c Ft(\),)24 b(in)n(tegrating)f(the)g(summands)308 1471 y(o)n(v)n(er)29 b(the)g(plaques)f(of)h Fq(F)36 b Ft(in)29 b(the)g(c)n(harts,)h(and)e (then)g(in)n(tegrating)h(the)f(result)308 1550 y(o)n(v)n(er)22 b(the)g(transv)n(ersals)f Fr(T)31 b Ft(using)21 b Fr(\027)t Ft(.)30 b(The)21 b(double)h(in)n(tegral)g(is)g(indep)r(enden)n(t)f(of) 308 1628 y(the)i(c)n(hoices)h(of)f(c)n(harts)f(and)h(partition)f(of)h (unit)n(y)f(b)r(ecause)h Fr(\027)k Ft(w)n(as)22 b(assumed)g(to)308 1707 y(b)r(e)g(holonom)n(y-in)n(v)l(arian)n(t.)28 b(See)23 b([13,)f(4])h(for)f(the)f(details)i(of)f(this)g(construction.)376 1786 y(The)c(curren)n(t)f Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\))18 b(is)h(closed,)h(and)d(is)i(called)g(the)f(Ruelle{Sulliv)l(an) h(cycle)308 1864 y(of)k(the)g(in)n(v)l(arian)n(t)g(measure)e Fr(\027)t Ft(.)32 b(It)23 b(will)g(pla)n(y)g(the)g(role)g(of)g(the)f (submanifold)g Fr(N)308 1943 y Ft(in)30 b(section)g(3.)52 b(The)30 b(assumption)d(that)i Fr(N)36 b Ft(b)r(e)30 b(n)n(ull-homologous)e(o)n(v)n(er)i Fp(R)h Ft(is)308 2022 y(then)19 b(replaced)h(b)n(y)f(the)h(assumption)d(that)i(the)g (Ruelle{Sulliv)l(an)i(cycle)g(is)f(n)n(ull-)308 2100 y(homologous:)29 b([)p Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\)])19 b(=)g(0)g Fq(2)g Fr(H)1215 2110 y Fk(n)p Fi(\000)p Fo(2)1307 2100 y Ft(\()p Fr(M)s(;)11 b Fp(R)p Ft(\).)376 2179 y(The)21 b(Ruelle{Sulliv)l(an)i(cycle)g Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\))22 b(is)g(con)n(tin)n(uous)f(with)h (resp)r(ect)f(to)h(the)308 2257 y Fr(C)360 2233 y Fo(0)387 2257 y Ft({top)r(ology)15 b(on)f(con)n(tin)n(uous)h(forms.)26 b(In)15 b(general)g(it)g(is)h(not)e(p)r(ossible)h(to)g(extend)308 2336 y(its)30 b(domain)f(of)h(de\014nition)f(to)h Fr(L)1169 2312 y Fo(1)1195 2336 y Ft({forms.)52 b(Nev)n(ertheless,)34 b(it)c(is)g(p)r(ossible)g(to)308 2415 y(de\014ne)e Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\))28 b(on)g(in)n(tegrable)g(double)f (forms)g Fr(F)9 b Ft(\()p Fr(x;)i(y)r Ft(\))28 b(with)g(the)f(prop)r (ert)n(y)308 2493 y(that)1042 2502 y Fj(Z)1079 2655 y Fk(y)r Fi(2)p Fk(M)1201 2594 y Fq(j)p Fr(F)9 b Ft(\()p Fr(x;)i(y)r Ft(\))p Fq(j)1444 2604 y Fk(y)1471 2594 y Fr(\026)p Ft(\()p Fr(y)r Ft(\))308 2732 y(is)30 b(a)f(con)n(tin)n(uous) f(form)g(in)h(the)g(v)l(ariable)h Fr(x)g Fq(2)h Fr(M)7 b Ft(.)50 b(Here)30 b(w)n(e)f(expand)g Fr(F)38 b Ft(as)308 2811 y(a)31 b(double)g(form)f(in)h(the)g(v)l(ariables)h Fr(x)f Ft(and)f Fr(y)r Ft(,)k(and)d(tak)n(e)g(the)g Fr(x)p Ft(-comp)r(onen)n(t)308 2890 y(m)n(ultiplied)25 b(b)n(y)g(the)g(norm)f (of)h(the)g Fr(y)r Ft(-comp)r(onen)n(t)g(with)f(resp)r(ect)i(to)f(our)f (\014xed)308 2968 y(Riemannian)j(metric.)46 b(Th)n(us)27 b Fq(j)p Fr(F)9 b Ft(\()p Fr(x;)i(y)r Ft(\))p Fq(j)1346 2978 y Fk(y)1401 2968 y Ft(is)28 b(a)g(di\013eren)n(tial)f(form)g(whic) n(h)h(is)g(a)308 3047 y(pro)r(duct)d(of)g Fr(dx)710 3057 y Fk(i)729 3047 y Ft(,)i(but)e(whose)g(co)r(e\016cien)n(t)i(function)e (also)h(dep)r(ends)f(on)h Fr(y)r Ft(.)41 b(As)308 3126 y(w)n(e)26 b(in)n(tegrate)e(along)i(the)e(lea)n(v)n(es)j(of)e Fq(F)7 b Ft(,)26 b(w)n(e)f(are)h(only)e(concerned)i(with)e(those)308 3204 y(summands)17 b(of)k Fr(F)9 b Ft(\()p Fr(x;)i(y)r Ft(\))20 b(whose)f(degree)i(in)f(the)f(v)l(ariable)i Fr(x)e Ft(equals)i(the)e(rank)h(of)308 3283 y(the)26 b(foliation,)h(and)f(w)n(e)g(require)f Fq(j)p Fr(F)9 b Ft(\()p Fr(x;)i(y)r Ft(\))p Fq(j)1409 3293 y Fk(y)1463 3283 y Ft(to)25 b(b)r(e)h(either)g(zero)g(or)f(to)h(induce)308 3362 y(the)f(giv)n(en)h(orien)n(tation)f(when)g(restricted)g(to)h(the)f (leaf)h(of)g Fq(F)33 b Ft(through)23 b Fr(x)p Ft(.)40 b(By)308 3440 y(Prop)r(osition)25 b(1)g(the)f(linking)i(form)d Fr(L)i Ft(satis\014es)h(this)e(in)n(tegrabilit)n(y)i(condition.)308 3519 y(The)d(form)f Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\)\()p Fr(F)e Ft(\()p Fr(x;)i(y)r Ft(\)\))22 b(is)h(obtained)g(b)n (y)f(p)r(erforming)g(the)g(in)n(tegrations)p eop %%Page: 13 13 13 12 bop 677 159 a Fm(LINKING)23 b(NUMBERS)f(OF)h(MEASURED)g(F)o(OLIA) l(TIONS)316 b(13)308 294 y Ft(in)31 b(the)g(Ruelle{Sulliv)l(an)h(cycle) h(with)d(resp)r(ect)h(to)g(the)g(\014rst)f(v)l(ariable.)58 b(The)308 373 y(result)22 b(is)g(an)g(in)n(tegrable)g(form)f(as)h(can)g (b)r(e)g(sho)n(wn)f(with)h(F)-6 b(ubini's)22 b(theorem)308 439 y Fj(Z)346 592 y Fk(y)r Fi(2)p Fk(M)467 474 y Fj(\014)467 514 y(\014)490 531 y Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(x)p Ft(\)\))805 477 y Fj(\000)835 531 y Fr(F)e Ft(\()p Fr(x;)i(y)r Ft(\))1040 477 y Fj(\001)1071 474 y(\014)1071 514 y(\014)1108 531 y Fr(\026)p Ft(\()p Fr(y)r Ft(\))18 b Fq(\024)h Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(x)p Ft(\)\))1649 436 y Fj(\022)1698 439 y(Z)1735 592 y Fk(y)r Fi(2)p Fk(M)1857 531 y Fq(j)p Fr(F)e Ft(\()p Fr(x;)i(y)r Ft(\))p Fq(j)2100 541 y Fk(y)2150 531 y Fr(\026)p Ft(\()p Fr(y)r Ft(\))2276 436 y Fj(\023)2351 531 y Fr(:)376 699 y Ft(W)-6 b(e)22 b(de\014ne)g(a)g(Hopf-t)n(yp)r(e)g(in)n(tegral)g (for)g Fr(X)27 b Ft(and)21 b(\()p Fq(F)7 b Fr(;)k(\026)p Ft(\))22 b(b)n(y)g(setting)764 855 y Fr(H)5 b Ft(\()p Fr(X)r(;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))18 b(=)1171 763 y Fj(Z)1209 916 y Fk(M)1273 855 y Fr(\013)e Fq(^)f Fr(\027)22 b Ft(=)d Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\)\()p Fr(\013)p Ft(\))22 b Fr(:)308 1010 y Ft(This)34 b(is)h(indep)r(enden)n(t)e(of)h(the)g(c)n(hoice)h(made)e(for)h Fr(\013)h Ft(b)r(ecause)f(the)g(Ruelle{)308 1088 y(Sulliv)l(an)22 b(cycle)i(is)e(assumed)f(to)h(b)r(e)f(n)n(ull-homologous.)376 1167 y(W)-6 b(e)17 b(w)n(an)n(t)e(to)h(in)n(terpret)g(this)g(in)n (tegral)g(as)g(an)g(a)n(v)n(erage)h(of)g(asymptotic)e(linking)308 1246 y(n)n(um)n(b)r(ers)20 b(of)h(\015o)n(w)g(lines)g(of)h Fr(X)k Ft(with)21 b Fq(F)7 b Ft(.)29 b(T)-6 b(o)21 b(do)g(so,)h(w)n(e)f (need)g(again)h(a)f(suitable)308 1324 y(system)27 b(of)h(short)f (paths.)46 b(W)-6 b(e)29 b(use)f(the)f(same)g(notation)g(for)h(the)f (paths)g(and)308 1403 y(closed-up)22 b(\015o)n(w)g(lines)h(as)f(b)r (efore.)308 1503 y Fw(De\014nition)30 b(11.)f Ft(A)c Fg(system)i(of)f(short)f(p)m(aths)f Ft(in)h Fr(M)32 b Ft(is)25 b(a)g(set)g(\006)f(of)h(piecewise)308 1581 y(di\013eren)n (tiable)d(paths)f(with)h(the)f(follo)n(wing)i(prop)r(erties:)409 1681 y(\(1\))k(F)-6 b(or)24 b(ev)n(ery)g(pair)f(of)h(p)r(oin)n(ts)f Fr(p;)11 b(q)25 b Fq(2)c Fr(M)31 b Ft(there)23 b(is)h(exactly)g(one)g (orien)n(ted)521 1760 y(path)d Fr(\033)r Ft(\()p Fr(p;)11 b(q)r Ft(\))20 b Fq(2)e Ft(\006)k(ha)n(ving)g(starting)f(p)r(oin)n(t)g Fr(p)h Ft(and)f(end)h(p)r(oin)n(t)g Fr(q)r Ft(.)409 1838 y(\(2\))27 b(The)17 b(paths)f(dep)r(end)g(con)n(tin)n(uously)h(on)g (their)g(starting)f(and)h(end)f(p)r(oin)n(ts)521 1917 y(almost)21 b(ev)n(erywhere.)409 1995 y(\(3\))27 b(The)22 b(limit)308 2151 y(\(9\))345 b(lim)728 2192 y Fk(t)p Fi(!1)858 2106 y Ft(1)p 858 2136 34 3 v 863 2198 a Fr(t)909 2060 y Fj(Z)947 2212 y Fk(y)r Fi(2)p Fk(\033)r Fo(\()p Fk(\036)1081 2218 y Fe(t)1099 2212 y Fo(\()p Fk(x)p Fo(\))p Fk(;x)p Fo(\))1235 2151 y Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\)\()p Fr(L)p Ft(\()p Fr(p;)g(y)r Ft(\)\))18 b(=)h(0)521 2324 y(exists)j(in)g(the)g Fr(L)939 2300 y Fo(1)966 2324 y Ft({sense.)376 2424 y(In)28 b(the)h(case)g(when) g(the)f(holonom)n(y-in)n(v)l(arian)n(t)g(measure)g Fr(\027)33 b Ft(is)c(giv)n(en)g(b)n(y)g(a)308 2503 y(smo)r(oth)21 b(di\013eren)n(tial)h(form)e Fr(\014)t Ft(,)j(the)e(Ruelle{Sulliv)l(an) i(cycle)g(is)g(giv)n(en)f(b)n(y)964 2655 y Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\)\()p Fr(!)r Ft(\))19 b(=)1373 2563 y Fj(Z)1410 2716 y Fk(M)1475 2655 y Fr(!)f Fq(^)c Fr(\014)26 b(:)308 2814 y Ft(The)f(pro)r(of)g(of)g(Theorem)f(4)h(in)g ([14)q(])g(generalises)i(v)n(erbatim)d(to)g(this)h(situation)308 2892 y(and)d(giv)n(es:)308 2992 y Fw(Theorem)j(12.)k Fg(If)c(the)f(tr)m(ansverse)g(me)m(asur)m(e)g(is)g(given)h(by)g(a)f (smo)m(oth)e(holono-)308 3071 y(my-invariant)j Ft(2)p Fg({form)e Fr(\014)t Fg(,)j(then)e(a)g(set)h(of)g(length-minimizing)g (ge)m(o)m(desics)g(is)g(a)308 3149 y(system)f(of)g(short)e(p)m(aths.) 376 3249 y Ft(More)g(generally)-6 b(,)24 b(w)n(e)e(ha)n(v)n(e:)308 3348 y Fw(Theorem)29 b(13.)h Fg(L)m(et)d Fq(F)34 b Fg(b)m(e)28 b(an)e(oriente)m(d)h(foliation)g(with)f(an)h(arbitr)m(ary)f(holo-)308 3427 y(nomy-invariant)i(tr)m(ansverse)h(me)m(asur)m(e)g Fr(\027)t Fg(.)45 b(Then)29 b(ther)m(e)g(exists)g(a)g(system)g(of)308 3506 y(short)23 b(p)m(aths.)p eop %%Page: 14 14 14 13 bop 308 159 a Fm(14)560 b(D.)18 b(K)o(OTSCHICK)e(AND)h(T.)g(V)o (OGEL)308 294 y Fg(Pr)m(o)m(of.)27 b Ft(W)-6 b(e)18 b(w)n(an)n(t)g(to)f (generalise)i(the)e(pro)r(of)g(of)h(Theorem)f(4.)28 b(W)-6 b(e)18 b(use)g(the)f(nota-)308 373 y(tion)e(in)n(tro)r(duced)e(there,)k (except)d(that)g(w)n(e)h(no)n(w)f(de\014ne)g Fr(C)5 b Ft(\()p Fr(y)r Ft(\))19 b(=)g Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\))2216 318 y Fj(\000)2247 373 y Fr(L)p Ft(\()p Fr(p;)g(y)r Ft(\))2440 318 y Fj(\001)2471 373 y Ft(.)308 451 y(If)27 b(w)n(e)f(can)g(satisfy)g(the)g(t)n(w)n(o)g (conditions)g(for)g(the)g(c)n(hoice)i(of)e(the)g(base)g(p)r(oin)n(ts) 308 530 y Fr(u)346 540 y Fk(i)385 530 y Fq(2)19 b Fr(U)494 540 y Fk(i)536 530 y Ft(\(with)j(the)g(generalised)h(de\014nition)f(of) h Fr(C)5 b Ft(\()p Fr(y)r Ft(\)\))22 b(in)h(the)f(pro)r(of)g(of)h (Theo-)308 609 y(rem)f(4,)h(then)f(w)n(e)h(can)g(construct)e(a)i (system)f(of)h(short)e(paths)h(\006)g(just)h(as)f(in)h(the)308 687 y(pro)r(of)f(of)g(Theorem)f(4.)376 766 y(Fix)i(arbitrary)f(p)r(oin) n(ts)k(~)-37 b Fr(u)1014 776 y Fk(i)1053 766 y Fq(2)21 b Fr(U)1164 776 y Fk(i)1206 766 y Ft(for)i(all)h(1)c Fq(\024)h Fr(i)f Fq(\024)h Fr(r)k Ft(and)e(paths)h(~)-35 b Fr(\015)2043 776 y Fk(ij)2107 766 y Ft(without)308 845 y(self)31 b(in)n(tersection)f(joining)j(~)-37 b Fr(u)1057 855 y Fk(i)1105 845 y Ft(and)33 b(~)-37 b Fr(u)1279 855 y Fk(j)1304 845 y Ft(.)52 b(F)-6 b(or)30 b(eac)n(h)g(pair)f Fr(i)i Fq(6)p Ft(=)h Fr(j)t Ft(,)g(extend)d(the)308 923 y(v)n(elo)r(cit)n(y)d(v)n(ector)f(\014eld)g(along)j(~)-35 b Fr(\015)1116 933 y Fk(ij)1182 923 y Ft(to)25 b(a)g(v)n(ector)g (\014eld)g(on)g Fr(M)32 b Ft(whose)24 b(time-one-)308 1002 y(\015o)n(w)f(transp)r(orts)f(a)h(small)g(ball)h(con)n(tained)g (in)f Fr(U)1551 1012 y Fk(i)1593 1002 y Ft(around)k(~)-37 b Fr(u)1857 1012 y Fk(i)1899 1002 y Ft(to)23 b(another)f(ball)308 1081 y(con)n(tained)h(in)f Fr(U)730 1091 y Fk(j)777 1081 y Ft(around)k(~)-37 b Fr(u)1040 1091 y Fk(j)1064 1081 y Ft(.)31 b(Because)23 b Fr(C)5 b Ft(\()p Fr(y)r Ft(\))23 b(is)g(an)f(in)n(tegrable)h(form)e(on)h Fr(M)7 b Ft(,)308 1159 y(for)19 b Fr(\026)p Ft({almost)g(ev)n(ery)g(starting)f(p)r(oin)n (t)h(in)g(a)h(ball)g(around)i(~)-37 b Fr(u)1786 1169 y Fk(i)1823 1159 y Ft(the)20 b(in)n(tegral)f(in)g(\(6\))308 1238 y(exists.)47 b(Hence)29 b(for)e(almost)g(ev)n(ery)h(c)n(hoice)g (of)g Fr(u)1560 1248 y Fk(i)1606 1238 y Ft(in)g(the)f(ball)h(around)i (~)-37 b Fr(u)2209 1248 y Fk(i)2256 1238 y Ft(w)n(e)308 1317 y(meet)22 b(the)f(\014rst)g(condition.)30 b(The)21 b Fr(\015)1204 1327 y Fk(ij)1268 1317 y Ft(are)h(the)f(time-one-\015o)n (wlines.)376 1395 y(Also)27 b(the)g(second)g(condition)f(for)h(the)f Fr(u)1422 1405 y Fk(i)1468 1395 y Ft(is)h(satis\014ed)f(outside)h(of)g (a)g(set)g(of)308 1474 y(measure)i(zero.)52 b(T)-6 b(o)29 b(see)h(this,)h(recall)g(that)d(the)h(Ruelle{Sulliv)l(an)h(cycle)h(can) 308 1553 y(b)r(e)22 b(represen)n(ted)f(as)g(a)h(\014nite)f(sum)f(suc)n (h)i(that)e(ev)n(ery)i(summand)c(is)k(an)g(in)n(tegral)308 1631 y(with)k(resp)r(ect)h(to)f(a)h(pro)r(duct)e(measure.)42 b(Apply)27 b(F)-6 b(ubini's)27 b(theorem)e(and)h(the)308 1710 y(triangle)c(inequalit)n(y)h(to)e(the)h(in)n(tegral)308 1910 y(\(10\))493 1819 y Fj(Z)531 1971 y Fk(u)p Fi(2)p Fk(U)623 1978 y Fe(i)654 1815 y Fj(\022)704 1819 y(Z)741 1971 y Fi(f)p Fk(x)p Fi(j)p Fk(n)p Fo(\()p Fk(x)p Fo(\)=)p Fk(i)p Fi(g)990 1819 y Fj(Z)1027 1971 y Fk(y)r Fi(2)p Fk(\033)r Fo(\()p Fk(x;u)p Fo(\))1232 1853 y Fj(\014)1232 1893 y(\014)1254 1910 y Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\))1564 1856 y Fj(\000)1595 1910 y Fr(L)p Ft(\()p Fr(p;)g(y)r Ft(\))1788 1856 y Fj(\001)1819 1853 y(\014)1819 1893 y(\014)1875 1910 y Fr(\026)p Ft(\()p Fr(x)p Ft(\))2005 1815 y Fj(\023)2064 1910 y Fr(\026)p Ft(\()p Fr(u)p Ft(\))327 2111 y Fq(\024)19 b Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\))719 2016 y Fj(\022)768 2019 y(Z)836 2037 y Fo(1)806 2171 y Fk(s)p Fo(=0)903 2019 y Fj(Z)941 2171 y Fk(u)p Fi(2)p Fk(U)1033 2178 y Fe(i)1064 2019 y Fj(Z)1102 2171 y Fi(f)p Fk(x)p Fi(j)p Fk(n)p Fo(\()p Fk(x)p Fo(\)=)p Fk(i)p Fi(g)1350 2033 y Fj(\014)1350 2074 y(\014)1350 2114 y(\014)1372 2111 y Fr(i)c Fe(@)r(\033)p 1402 2119 50 3 v 1405 2147 a(@)r(s)1461 2111 y Fr(L)1506 2056 y Fj(\000)1537 2111 y Fr(p;)k(\033)r Ft(\()p Fr(x;)g(u)p Ft(\)\()p Fr(s)p Ft(\))1879 2056 y Fj(\001)1909 2033 y(\014)1909 2074 y(\014)1909 2114 y(\014)1942 2111 y Fr(\026)p Ft(\()p Fr(x)p Ft(\))21 b Fr(\026)p Ft(\()p Fr(u)p Ft(\))p Fr(ds)2288 2016 y Fj(\023)2363 2111 y Fr(:)308 2299 y Ft(If)33 b(the)g(last)g (expression)g(is)g(w)n(ell-de\014ned,)k(then)32 b(b)n(y)h(F)-6 b(ubini's)33 b(theorem)f(the)308 2378 y(expression)27 b(obtained)g(b)n(y)g(dropping)f(the)h(in)n(tegration)g(with)f(resp)r (ect)i(to)f Fr(u)f Ft(is)308 2457 y(w)n(ell-de\014ned)k(for)g Fr(\026)p Ft({almost)f(ev)n(ery)h(c)n(hoice)h(of)f Fr(u)i Fq(2)g Fr(U)1741 2467 y Fk(i)1760 2457 y Ft(.)54 b(Th)n(us)29 b(w)n(e)h(ha)n(v)n(e)g(to)308 2535 y(sho)n(w)17 b(that)g(the)g(form)g (w)n(e)h(apply)f(the)g(Ruelle{Sulliv)l(an)i(cycle)g(to)e(is)h(con)n (tin)n(uous.)308 2614 y(Consider)308 2776 y(\(11\))653 2684 y Fj(Z)691 2837 y Fk(y)r Fi(2)p Fk(U)781 2844 y Fe(i)812 2684 y Fj(Z)850 2837 y Fi(f)p Fk(x)p Fi(j)p Fk(n)p Fo(\()p Fk(x)p Fo(\)=)p Fk(i)p Fi(g)1098 2698 y Fj(\014)1098 2739 y(\014)1098 2779 y(\014)1120 2776 y Fr(i)7 b Fe(@)r(\033)p 1150 2784 V 1153 2812 a(@)r(s)1209 2776 y Fr(L)1254 2721 y Fj(\000)1285 2776 y Fr(p;)k(\033)r Ft(\()p Fr(x;)g(y)r Ft(\)\()p Fr(s)p Ft(\))1623 2721 y Fj(\001)1654 2698 y(\014)1654 2739 y(\014)1654 2779 y(\014)1709 2776 y Fr(\026)p Ft(\()p Fr(x)p Ft(\))21 b Fr(\026)p Ft(\()p Fr(y)r Ft(\))308 2968 y(with)29 b Fr(p)j Fq(2)f Fr(M)7 b Ft(.)53 b(F)-6 b(or)29 b(\014xed)g Fr(s)j Fq(2)g Ft([0)p Fr(;)11 b Ft(1])30 b(the)g(in)n(tegrand)e(has)i (at)f(most)f(a)i(p)r(ole)308 3047 y(of)f(order)e Fr(n)20 b Fq(\000)g Ft(1)28 b(along)g(an)g Fr(n)p Ft(-dimensional)g (submanifold)f(\(the)h(solutions)g(of)308 3126 y Fr(\033)r Ft(\()p Fr(x;)11 b(y)r Ft(\)\()p Fr(s)p Ft(\))18 b(=)h Fr(p)p Ft(\))h(in)f(the)g(2)p Fr(n)p Ft(-dimensional)g(pro)r(duct)f (manifold)h Fr(U)1952 3136 y Fk(i)1980 3126 y Fq(\002)10 b(f)p Fr(x)p Fq(j)p Fr(n)p Ft(\()p Fr(x)p Ft(\))17 b(=)308 3204 y Fr(i)p Fq(g)p Ft(.)29 b(This)21 b(sho)n(ws)h(that)e(the)h(in)n (tegral)h(\(11\))f(is)h(w)n(ell-de\014ned.)30 b(F)-6 b(urthermore,)20 b(the)308 3283 y(in)n(tegral)25 b(dep)r(ends)f(con)n (tin)n(uously)g(on)h Fr(s)f Ft(and)g Fr(p)p Ft(.)38 b(In)24 b(particular,)h(it)g(do)r(es)f(exist)308 3362 y(for)i(the)f(b)r (oundary)f(v)l(alues)i Fr(s)f Ft(=)g(0)g(and)g Fr(s)g Ft(=)g(1.)40 b(This)26 b(sho)n(ws)f(that)f(w)n(e)i(apply)308 3440 y(the)18 b(Ruelle{Sulliv)l(an)h(cycle)g(to)e(a)h(con)n(tin)n(uous) g(form)f(and)g(thereb)n(y)g(justi\014es)h(the)308 3519 y(application)k(of)h(the)e(Ruelle{Sulliv)l(an)i(cycle)g(in)f(\(10\).)p eop %%Page: 15 15 15 14 bop 677 159 a Fm(LINKING)23 b(NUMBERS)f(OF)h(MEASURED)g(F)o(OLIA) l(TIONS)316 b(15)376 294 y Ft(So,)34 b(if)e(w)n(e)g(c)n(ho)r(ose)g(\()p Fr(u)958 304 y Fo(1)984 294 y Fr(;)11 b(:)g(:)g(:)i(;)e(u)1169 304 y Fk(r)1195 294 y Ft(\))35 b Fq(2)g Fr(U)1381 304 y Fo(1)1429 294 y Fq(\002)22 b Fr(:)11 b(:)g(:)23 b Fq(\002)f Fr(U)1721 304 y Fk(r)1778 294 y Ft(outside)31 b(of)h(a)g(set)g(of)308 373 y(measure)e(zero,)j(w)n(e)d(obtain)g(a)g(system)f(of)i(short)e (paths)g(as)i(in)f(the)g(pro)r(of)g(of)308 451 y(Theorem)21 b(4.)1640 b Ff(\003)308 589 y Fg(R)m(emark)39 b Ft(14)p Fg(.)33 b Ft(So)e(far)g(w)n(e)g(ha)n(v)n(e)h(only)e(considered)i (non{singular)e(foliations.)308 668 y(In)d(the)g(next)g(section,)j(w)n (e)d(will)h(also)g(w)n(an)n(t)f(to)g(use)g(singular)h(foliations.)46 b(W)-6 b(e)308 747 y(therefore)21 b(p)r(oin)n(t)f(out)g(that,)h(as)g (in)f([14)q(],)i(Theorem)d(12)i(applies)g(equally)g(w)n(ell)h(to)308 825 y(singular)f(foliations)g(with)f(a)g(holonom)n(y-in)n(v)l(arian)n (t)f(smo)r(oth)g(di\013eren)n(tial)i(form.)376 904 y(Theorem)29 b(13)h(can)g(sometimes)f(b)r(e)h(applied)g(to)g(singular)g(foliations.) 54 b(F)-6 b(or)308 983 y(example,)33 b(this)d(can)g(b)r(e)g(done)g(if)h (the)f(supp)r(ort)e(of)j(the)f(transv)n(erse)g(measure)308 1061 y(has)d(a)g(neigh)n(b)r(ourho)r(o)r(d)e(to)i(whic)n(h)g(the)f (foliation)i(extends)f(in)g(a)g(nonsingular)308 1140 y(w)n(a)n(y)-6 b(.)30 b(In)22 b(the)f(situation)h(of)g(the)g(previous)f (section,)i(a)f(single)h(n)n(ull-homologous)308 1219 y(submanifold)g Fr(N)28 b Fq(\032)23 b Fr(M)30 b Ft(can)24 b(alw)n(a)n(ys)g(b)r(e)g(extended)f(to)g(a)h(smo)r(oth)e(foliation)j (of)308 1297 y(a)i(whole)f(neigh)n(b)r(ourho)r(o)r(d)e(of)i Fr(N)7 b Ft(,)28 b(and)d(w)n(e)i(can)f(tak)n(e)g(the)g(measure)f(giv)n (en)i(b)n(y)308 1376 y Fr(N)7 b Ft(,)27 b(with)d(supp)r(ort)g Fr(N)7 b Ft(.)40 b(Then)24 b(the)h(ab)r(o)n(v)n(e)h(theorem)e(can)h(b)r (e)g(used)g(instead)h(of)308 1455 y(Theorem)21 b(4.)376 1545 y(No)n(w)28 b(the)h(linking)f(n)n(um)n(b)r(er)f(of)h Fr(\015)t Ft(\()p Fr(x;)11 b(t)p Ft(\))28 b(with)g Fq(F)35 b Ft(is)29 b(de\014ned)f(b)n(y)g(generalis-)308 1624 y(ing)23 b(\(2\))f(as)g(follo)n(ws:)308 1727 y Fw(De\014nition)i(15.)i Ft(The)20 b(linking)g(n)n(um)n(b)r(er)f(lk\()p Fr(\015)t(;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))20 b(of)g(a)g(\(n)n(ull-homologous\))308 1806 y(closed)g(lo)r(op)f Fr(\015)k Ft(in)d Fr(M)26 b Ft(with)18 b(the)h(measured)f(foliation)i(\()p Fq(F)7 b Fr(;)k(\027)t Ft(\))19 b(is)h(the)f(ev)l(aluation)308 1884 y(of)33 b(the)g(Ruelle{Sulliv)l(an)g(cycle)h Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\))33 b(on)f(a)h(\()p Fr(n)22 b Fq(\000)h Ft(2\)-form)31 b Fr(\013)i Ft(with)f(the)308 1963 y(prop)r(ert)n(y)21 b(that)g Fr(d\013)i Ft(is)f(P)n(oincar)n(\023) -31 b(e)22 b(dual)g(to)g Fr(\015)t Ft(.)376 2066 y(As)g(the)g (Ruelle{Sulliv)l(an)g(cycle)i(is)e(assumed)e(to)i(b)r(e)g(n)n (ull-homologous,)f(this)308 2144 y(ev)l(aluation)i(is)f(indep)r(enden)n (t)f(of)h(the)g(c)n(hoice)h(of)f Fr(\013)p Ft(.)376 2223 y(The)f(follo)n(wing)i(is)f(the)g(adaption)f(of)h(Prop)r(osition)g(6)g (to)f(this)h(situation:)308 2326 y Fw(Prop)r(osition)32 b(16.)e Fg(L)m(et)f Fr(\013)p Ft(\()p Fr(x;)11 b(t)p Ft(\))27 b Fg(b)m(e)i Ft(\()p Fr(n)18 b Fq(\000)h Ft(2\))p Fg(-forms)27 b(with)h(the)g(pr)m(op)m(erty)e(that)308 2405 y Fr(d\013)p Ft(\()p Fr(x;)11 b(t)p Ft(\))24 b Fg(ar)m(e)f(Poinc)m (ar)n(\023)-32 b(e)23 b(duals)g(for)h Fr(\015)t Ft(\()p Fr(x;)11 b(t)p Ft(\))p Fg(.)29 b(Then)24 b(the)g(limit)465 2564 y Ft(lk\()p Fr(x;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))19 b(=)30 b(lim)847 2605 y Fk(t)p Fi(!1)978 2519 y Ft(1)p 978 2549 34 3 v 983 2611 a Fr(t)1018 2564 y Ft(lk\()p Fr(\015)t Ft(\()p Fr(x;)11 b(t)p Ft(\))p Fr(;)g Fq(F)c Fr(;)k(\027)t Ft(\))18 b(=)30 b(lim)1542 2605 y Fk(t)p Fi(!1)1673 2519 y Ft(1)p 1673 2549 V 1678 2611 a Fr(t)1713 2564 y(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\)\()p Fr(\013)p Ft(\()p Fr(x;)g(t)p Ft(\)\))308 2719 y Fg(exists)29 b(in)g(the)f Fr(L)738 2695 y Fo(1)765 2719 y Fg(-sense.)45 b(It)29 b(is)g(an)f(inte)m(gr)m(able)g(function)i(on)e Fr(M)35 b Fg(which)28 b(do)m(es)308 2798 y(not)c(dep)m(end)f(on)g(the)h(chosen) f(system)h(of)g(short)e(p)m(aths.)308 2936 y(Pr)m(o)m(of.)27 b Ft(By)22 b(the)f(de\014nition)h(of)g(lk\()p Fr(x;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))22 b(and)f(Prop)r(osition)h(1)g(w)n(e)g (\014nd)445 3096 y(lk\()p Fr(x;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))19 b(=)j(lim)819 3136 y Fk(t)p Fi(!1)950 3050 y Ft(1)p 950 3080 V 955 3142 a Fr(t)990 3096 y Ft(lk\()p Fr(\015)t Ft(\()p Fr(x;)11 b(t)p Ft(\))p Fr(;)g Fq(F)c Fr(;)k(\027)t Ft(\))18 b(=)30 b(lim)1515 3136 y Fk(t)p Fi(!1)1645 3050 y Ft(1)p 1645 3080 V 1650 3142 a Fr(t)1685 3096 y(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\)\()p Fr(\013)p Ft(\()p Fr(x;)g(t)p Ft(\)\))757 3266 y(=)22 b(lim)819 3306 y Fk(t)p Fi(!1)950 3220 y Ft(1)p 950 3250 V 955 3312 a Fr(t)990 3266 y(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\))1311 3171 y Fj(\022)1360 3174 y(Z)1398 3327 y Fk(y)r Fi(2)p Fk(M)1519 3266 y Fr(L)p Ft(\()p Fr(p;)g(y)r Ft(\))16 b Fq(^)f Fr(d\013)p Ft(\()p Fr(x;)c(t)p Ft(\)\()p Fr(y)r Ft(\))2093 3171 y Fj(\023)2176 3266 y Fr(:)308 3440 y Ft(The)28 b(harmonic)f(and)g(exact)h(terms)f(in) h(\(4\))f(do)g(not)h(con)n(tribute)f(b)r(ecause)h(the)308 3519 y(Ruelle{Sulliv)l(an)c(cycle)g(is)g(assumed)d(to)i(b)r(e)g(n)n (ull-homologous.)31 b(If)23 b Fr(p)g Ft(do)r(es)g(not)p eop %%Page: 16 16 16 15 bop 308 159 a Fm(16)560 b(D.)18 b(K)o(OTSCHICK)e(AND)h(T.)g(V)o (OGEL)308 294 y Ft(lie)23 b(on)f Fr(\015)t Ft(\()p Fr(x;)11 b(t)p Ft(\),)21 b(and)h(hence)g(for)g(almost)f(ev)n(ery)h Fr(x)d Fq(2)f Fr(M)7 b Ft(,)22 b(w)n(e)h(ha)n(v)n(e)539 462 y(lk\()p Fr(x;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))19 b(=)j(lim)914 503 y Fk(t)p Fi(!1)1044 417 y Ft(1)p 1044 447 34 3 v 1049 509 a Fr(t)1084 462 y(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\))1405 367 y Fj(\022)1455 371 y(Z)1492 523 y Fk(y)r Fi(2)p Fk(\015)s Fo(\()p Fk(x;t)p Fo(\))1685 462 y Fr(L)p Ft(\()p Fr(p;)g(y)r Ft(\))1878 367 y Fj(\023)851 658 y Ft(=)22 b(lim)914 698 y Fk(t)p Fi(!1)1044 612 y Ft(1)p 1044 642 V 1049 704 a Fr(t)1084 658 y(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\))1405 562 y Fj(\022)1455 566 y(Z)1492 718 y Fk(y)r Fi(2)p Fk(\036)p Fo(\()p Fk(x;t)p Fo(\))1686 658 y Fr(L)p Ft(\()p Fr(p;)g(y)r Ft(\))1879 562 y Fj(\023)851 857 y Ft(=)22 b(lim)914 897 y Fk(t)p Fi(!1)1044 811 y Ft(1)p 1044 841 V 1049 903 a Fr(t)1084 857 y(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\))1405 761 y Fj(\022)1455 765 y(Z)1522 783 y Fk(t)1492 917 y Fo(0)1553 857 y Fr(i)1576 867 y Fk(X)1621 857 y Fr(L)p Ft(\()p Fr(p;)g(\036)1793 867 y Fk(s)1819 857 y Ft(\()p Fr(x)p Ft(\)\))p Fr(ds)2000 761 y Fj(\023)2081 857 y Fr(:)308 1021 y Ft(The)24 b(second)h(equalit)n (y)g(is)f(true)g(b)r(ecause)h(of)f(the)h(de\014nition)e(of)i(the)f (system)g(of)308 1100 y(short)h(paths.)39 b(The)25 b(\015o)n(w)g(of)g Fr(X)31 b Ft(preserv)n(es)25 b(the)g(v)n(olume)g(form)f Fr(\026)h Ft(and)g(w)n(e)h(can)308 1178 y(apply)f(the)g(mean)g(ergo)r (dic)h(theorem.)38 b(Hence,)28 b(the)d(limit)g(on)h(the)f(righ)n(t)g (hand)308 1257 y(side)e(of)584 1412 y(lk\()p Fr(x;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))19 b(=)j(lim)958 1453 y Fk(t)p Fi(!1)1089 1367 y Ft(1)p 1089 1397 V 1094 1458 a Fr(t)1140 1320 y Fj(Z)1207 1338 y Fk(t)1177 1473 y Fo(0)1239 1412 y Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\))1549 1358 y Fj(\000)1579 1412 y Fr(i)1602 1422 y Fk(X)1647 1412 y Fr(L)p Ft(\()p Fr(p;)g(\036)1819 1422 y Fk(s)1844 1412 y Ft(\()p Fr(x)p Ft(\)\))1960 1358 y Fj(\001)1990 1412 y Fr(ds)308 1580 y Ft(exists)23 b(in)f(the)f Fr(L)726 1556 y Fo(1)753 1580 y Fq(\000)p Ft(sense)i(and)e(represen)n (ts)h(an)g(in)n(tegrable)g(function)g(on)f Fr(M)7 b Ft(.)30 b(It)308 1659 y(do)r(es)22 b(not)g(dep)r(end)f(on)g(the)h(system)f(of)h (short)f(paths.)629 b Ff(\003)376 1796 y Ft(Using)18 b(this,)g(w)n(e)g(can)g(\014nally)f(de\014ne)h(the)f(a)n(v)n(erage)h (asymptotic)e(linking)i(n)n(um-)308 1875 y(b)r(er)k(of)g(the)f(v)n (ector)h(\014eld)g Fr(X)k Ft(with)c(the)f(measured)g(foliation)h(\()p Fq(F)7 b Fr(;)k(\027)t Ft(\))22 b(b)n(y)f(setting)881 2037 y(lk)q(\()p Fr(X)r(;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))18 b(=)1282 1945 y Fj(Z)1320 2098 y Fk(M)1384 2037 y Ft(lk)q(\()p Fr(x;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))p Fr(\026)21 b(:)308 2199 y Ft(Theorem)g(7)h(generalises)h(as)f(follo)n(ws:)308 2301 y Fw(Theorem)h(17.)j Fg(L)m(et)d Fr(M)28 b Fg(b)m(e)23 b(a)e(close)m(d)h(oriente)m(d)g Fr(n)p Fg(-manifold)f(with)h Fr(b)2040 2311 y Fo(1)2066 2301 y Ft(\()p Fr(M)7 b Ft(\))18 b(=)h(0)p Fg(.)308 2380 y(L)m(et)25 b Fr(X)k Fg(b)m(e)24 b(a)g(diver)m(genc)m(e-fr)m(e)m(e)h(ve)m(ctor)f(\014eld)g(on)g Fr(M)7 b Fg(,)24 b(and)g Fq(F)31 b Fg(an)24 b(oriente)m(d)g(c)m(o)m (di-)308 2458 y(mension)d Ft(2)g Fg(foliation)g(with)f(a)h(tr)m (ansverse)g(me)m(asur)m(e)f Fr(\027)25 b Fg(whose)c(R)n(uel)s(le{Sul)s (livan)308 2537 y(cycle)k(is)e(nul)s(l-homolo)m(gous.)29 b(Then)24 b(the)f(aver)m(age)h(asymptotic)f(linking)h(numb)m(er)308 2616 y(of)g(the)f(orbits)h(of)f Fr(X)29 b Fg(with)23 b Ft(\()p Fq(F)7 b Fr(;)k(\027)t Ft(\))24 b Fg(exists)g(and)f(e)m (quals)h(a)f(Hopf-typ)m(e)g(inte)m(gr)m(al:)939 2742 y Ft(lk\()p Fr(X)r(;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))19 b(=)g Fr(H)5 b Ft(\()p Fr(X)r(;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))23 b Fr(:)308 2880 y Fg(Pr)m(o)m(of.)k Ft(W)-6 b(e)24 b(use)g(the)f(calculations)i(in)e(the)h(pro)r(of)f(of)h(Prop)r (osition)e(16)i(and)f(the)308 2959 y(mean)g(ergo)r(dic)h(theorem.)34 b(Since)24 b Fr(b)1207 2969 y Fo(1)1233 2959 y Ft(\()p Fr(M)7 b Ft(\))21 b(=)h(0,)j Fr(b)1554 2969 y Fk(n)p Fi(\000)p Fo(1)1647 2959 y Ft(\()p Fr(M)7 b Ft(\))23 b(also)h(v)l(anishes.)35 b(The)308 3037 y(\()p Fr(n)6 b Fq(\000)g Ft(1\){form)16 b Fr(i)703 3047 y Fk(X)748 3037 y Fr(\026)i Ft(is)g(closed)g(and)f(hence)g(exact.)29 b(Cho)r(ose)17 b(an)g(\()p Fr(n)6 b Fq(\000)g Ft(2\){form)16 b Fr(\013)2285 3047 y Fk(X)308 3116 y Ft(suc)n(h)f(that)f Fr(d\013)662 3126 y Fk(X)727 3116 y Ft(=)19 b Fr(i)820 3126 y Fk(X)865 3116 y Fr(\026)p Ft(.)27 b(By)15 b(the)g(mean)f(ergo)r (dic)h(theorem)f(and)g(Prop)r(osition)h(16)645 3274 y(lk\()p Fr(X)r(;)c Fq(F)c Fr(;)k(\027)t Ft(\))19 b(=)1039 3183 y Fj(Z)1076 3335 y Fk(x)p Fi(2)p Fk(M)1199 3274 y Ft(lk)q(\()p Fr(x;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))p Fr(\026)p Ft(\()p Fr(x)p Ft(\))976 3455 y(=)1039 3363 y Fj(Z)1076 3516 y Fk(x)p Fi(2)p Fk(M)1199 3455 y Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\)\()p Fr(i)1558 3465 y Fk(X)1603 3455 y Fr(L)p Ft(\()p Fr(p;)g(x)p Ft(\)\))p Fr(\026)p Ft(\()p Fr(x)p Ft(\))19 b Fr(:)p eop %%Page: 17 17 17 16 bop 677 159 a Fm(LINKING)23 b(NUMBERS)f(OF)h(MEASURED)g(F)o(OLIA) l(TIONS)316 b(17)308 294 y Ft(Lo)r(cally)-6 b(,)21 b(the)d (Ruelle{Sulliv)l(an)i(cycle)f(is)g(giv)n(en)g(b)n(y)f(a)h(pro)r(duct)e (measure,)h(hence)308 373 y(w)n(e)k(can)g(apply)g(F)-6 b(ubini's)23 b(theorem.)28 b(W)-6 b(e)23 b(\014nd)604 543 y(lk\()p Fr(X)r(;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))19 b(=)p Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\))1307 448 y Fj(\022)1356 451 y(Z)1394 603 y Fk(x)p Fi(2)p Fk(M)1517 543 y Fr(i)1540 553 y Fk(X)1585 543 y Fr(L)p Ft(\()p Fr(p;)g(x)p Ft(\))k Fq(^)g Fr(\026)p Ft(\()p Fr(x)p Ft(\))1987 448 y Fj(\023)935 730 y Ft(=)p Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\))1307 635 y Fj(\022)1356 638 y(Z)1394 790 y Fk(x)p Fi(2)p Fk(M)1517 730 y Fr(L)p Ft(\()p Fr(p;)g(x)p Ft(\))k Fq(^)g Fr(i)1812 740 y Fk(X)1857 730 y Fr(\026)p Ft(\()p Fr(x)p Ft(\))1987 635 y Fj(\023)935 870 y Ft(=)p Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\027)t Ft(\()p Fr(p)p Ft(\)\)\()p Fr(\013)1364 880 y Fk(X)1409 870 y Ft(\))18 b(=)h Fr(H)5 b Ft(\()p Fr(X)r(;)11 b Fq(F)c Fr(;)k(\027)t Ft(\))22 b Fr(;)308 1002 y Ft(where)28 b(the)g(p)r(en)n(ultimate)f(equalit)n(y)h (is)h(due)f(to)g(Prop)r(osition)f(1)h(and)g(Stok)n(es's)308 1080 y(theorem.)1716 b Ff(\003)308 1224 y Fg(R)m(emark)29 b Ft(18)p Fg(.)d Ft(The)19 b(discussion)h(in)g(this)f(section)h (reduces)f(to)g(that)g(of)h(the)f(previ-)308 1303 y(ous)f(section)h(in) f(the)g(case)h(that)e(the)h(in)n(v)l(arian)n(t)g(measure)f Fr(\027)22 b Ft(is)c(giv)n(en)h(b)n(y)f(a)g(closed)308 1381 y(leaf)23 b Fr(N)7 b Ft(.)30 b(The)21 b(rest)h(of)g(the)g (foliation)h Fq(F)29 b Ft(then)21 b(pla)n(ys)h(no)g(role.)308 1473 y Fg(R)m(emark)39 b Ft(19)p Fg(.)32 b Ft(F)-6 b(or)30 b(a)h(smo)r(oth)d(foliation)j(with)e(a)i(holonom)n(y-in)n(v)l(arian)n (t)d(mea-)308 1552 y(sure)21 b(giv)n(en)h(b)n(y)e(a)i(smo)r(oth)d (exact)i(2-form,)g(one)g(can)g(pro)n(v)n(e)g(Theorem)f(17)h(using)308 1631 y(Arnold's)g(de\014nition)f(of)h(a)f(system)g(of)h(short)e(paths)h (and)g(the)g(Birkho\013)g(ergo)r(dic)308 1709 y(theorem)c(as)i(in)f ([1],)i(rather)d(than)h(the)g(mean)f(ergo)r(dic)i(theorem)e(as)h(ab)r (o)n(v)n(e.)28 b(This)308 1788 y(w)n(as)22 b(done)g(b)n(y)g(Khesin)g (in)g([8].)376 1880 y(Here)h(are)f(t)n(w)n(o)g(examples)f(for)h (linking)h(n)n(um)n(b)r(ers)d(b)r(et)n(w)n(een)i(measured)f(folia-)308 1959 y(tions)h(and)f(div)n(ergence-free)j(v)n(ector)e(\014elds.)308 2051 y Fg(Example)34 b Ft(20)p Fg(.)d Ft(Let)e Fq(F)35 b Ft(b)r(e)28 b(the)f(Reeb)h(foliation)h(on)e Fr(S)1705 2026 y Fo(3)1732 2051 y Ft(.)47 b(It)28 b(has)f(exactly)i(one)308 2129 y(closed)e(leaf,)i(whic)n(h)d(is)h(di\013eomorphic)e(to)h Fr(T)1468 2105 y Fo(2)1495 2129 y Ft(.)43 b(All)27 b(other)f(lea)n(v)n (es)h(are)g(di\013eo-)308 2208 y(morphic)c(to)i Fp(R)698 2183 y Fo(2)750 2208 y Ft(and)f(ha)n(v)n(e)h(linear)f(gro)n(wth.)36 b(By)24 b(a)h(construction)e(going)i(bac)n(k)308 2286 y(to)18 b(Plan)n(te,)i(cf.)f([4)q(,)g(13],)g(an)n(y)g(leaf)g Fq(L)g Ft(of)f(sub)r(exp)r(onen)n(tial)g(gro)n(wth)f(de\014nes)i(an)f (in-)308 2365 y(v)l(arian)n(t)i(measure)g Fr(\026)826 2375 y Fi(L)881 2365 y Ft(with)g(supp)r(ort)f(con)n(tained)h(on)g(a)g (union)g(of)h(minimal)e(sets)308 2444 y(in)p 383 2390 47 3 v 20 w Fq(L)q Ft(.)29 b(F)-6 b(or)20 b(the)g(Reeb)g(foliation)h (one)f(can)g(sho)n(w)g(easily)g(that,)g(up)g(to)f(a)i(factor)f(of)308 2522 y(2)p Fr(\031)r Ft(,)k(the)d(measure)h Fr(\027)826 2532 y Fi(L)883 2522 y Ft(de\014ned)f(b)n(y)h(an)g(op)r(en)f(leaf)i (equals)g(that)e(de\014ned)g(b)n(y)h(the)308 2601 y(unique)g(closed)h (leaf)f Fr(T)888 2577 y Fo(2)915 2601 y Ft(.)376 2680 y(Consider)i(no)n(w)g(the)h(pro)r(duct)e(of)i(t)n(w)n(o)g(Reeb)f (foliations)i(on)e Fr(S)1962 2655 y Fo(3)2005 2680 y Fq(\002)18 b Fr(S)2120 2655 y Fo(3)2146 2680 y Ft(.)38 b(This)308 2758 y(foliation)25 b Fq(F)f(\002)16 b(F)32 b Ft(has)24 b(co)r(dimension)f(t)n(w)n(o)h(and)f(con)n(tains)h(exactly) h(one)f(closed)308 2837 y(leaf)h(di\013eomorphic)d(to)i Fr(T)978 2813 y Fo(4)1004 2837 y Ft(.)35 b(If)24 b Fq(L)1171 2847 y Fo(1)1222 2837 y Ft(and)f Fq(L)1398 2847 y Fo(2)1448 2837 y Ft(are)h(op)r(en)f(lea)n(v)n(es)i(of)f Fq(F)7 b Ft(,)25 b(then)e(the)308 2916 y(leaf)30 b Fq(L)484 2926 y Fo(1)531 2916 y Fq(\002)20 b(L)649 2926 y Fo(2)706 2916 y Ft(in)29 b(the)f(pro)r(duct)g(foliation)i(de\014nes)e(the)h (holonom)n(y-in)n(v)l(arian)n(t)308 2994 y(transv)n(erse)24 b(measure)g Fr(\027)916 3004 y Fi(L)949 3010 y Fh(1)991 2994 y Fq(\002)18 b Fr(\027)1094 3004 y Fi(L)1127 3010 y Fh(2)1177 2994 y Ft(for)24 b Fq(F)g(\002)17 b(F)7 b Ft(.)38 b(By)24 b(the)g(discussion)h(ab)r(o)n(v)n(e)g(w)n(e)308 3073 y(ha)n(v)n(e)d(the)g(equalit)n(y)1019 3206 y Fr(\027)1052 3216 y Fi(L)1085 3222 y Fh(1)1126 3206 y Fq(\002)15 b Fr(\027)1226 3216 y Fi(L)1259 3222 y Fh(2)1303 3206 y Ft(=)1413 3161 y(1)p 1380 3191 100 3 v 1380 3253 a(4)p Fr(\031)1452 3233 y Fo(2)1486 3206 y Fr(\027)1519 3218 y Fk(T)1554 3205 y Fh(4)1602 3206 y Fr(:)308 3348 y Ft(Th)n(us)28 b(the)f(linking)h(n)n(um)n(b)r(er)f(of)h(a)g(div)n(ergence-free)h(v)n (ector)f(\014eld)g(with)f(\()p Fq(F)f(\002)308 3427 y(F)7 b Fr(;)k(\027)425 3437 y Fi(L)458 3443 y Fh(1)507 3427 y Fq(\002)23 b Fr(\027)615 3437 y Fi(L)648 3443 y Fh(2)673 3427 y Ft(\))33 b(is,)j(up)c(to)g(a)h(factor)g(of)g(4)p Fr(\031)1451 3403 y Fo(2)1478 3427 y Ft(,)j(the)c(same)g(as)h(that)f (with)g(the)308 3506 y(submanifold)21 b Fr(T)729 3481 y Fo(4)774 3506 y Fq(\032)e Fr(S)890 3481 y Fo(3)931 3506 y Fq(\002)d Fr(S)1044 3481 y Fo(3)1092 3506 y Ft(giv)n(en)22 b(b)n(y)g(the)g(closed)h(leaf.)p eop %%Page: 18 18 18 17 bop 308 159 a Fm(18)560 b(D.)18 b(K)o(OTSCHICK)e(AND)h(T.)g(V)o (OGEL)308 294 y Fg(Example)33 b Ft(21)p Fg(.)d Ft(Next)d(w)n(e)g (consider)g(a)g(singular)f(foliation)h(on)g Fp(C)12 b Fr(P)2001 270 y Fo(2)2032 294 y Ft(.)43 b(In)26 b(order)308 373 y(to)h(obtain)g(a)g(n)n(ull-homologous)f(Ruelle{Sulliv)l(an)i (cycle)h(w)n(e)e(will)h(use)g(signed)308 451 y(measures,)22 b(rather)f(than)g(measures.)29 b(Consider)21 b(the)h(a\016ne)g(em)n(b)r (edding)1121 593 y Fp(C)1166 565 y Fo(2)1215 593 y Fr(,)-11 b Fq(!)20 b Fp(C)12 b Fr(P)1405 565 y Fo(2)1001 695 y Ft(\()p Fr(z)1058 705 y Fo(0)1084 695 y Fr(;)f(z)1144 705 y Fo(1)1171 695 y Ft(\))18 b Fq(7!)i Ft([1)f(:)g Fr(z)1440 705 y Fo(0)1485 695 y Ft(:)g Fr(z)1553 705 y Fo(1)1579 695 y Ft(])k Fr(:)308 843 y Ft(W)-6 b(e)26 b(foliate)f Fp(C)672 819 y Fo(2)727 843 y Ft(b)n(y)f(the)g(pro)r(duct)g (foliation)h Fp(C)33 b Fq(\002)17 b Fp(C)42 b Ft(and)24 b(consider)g(the)g(lea)n(v)n(es)308 922 y Fq(L)354 932 y Fo(1)421 922 y Ft(=)39 b Fq(f)p Ft(1)p Fq(g)23 b(\002)g Fp(C)51 b Ft(and)33 b Fq(L)979 932 y Fi(\000)p Fo(1)1083 922 y Ft(=)39 b Fq(f\000)p Ft(1)p Fq(g)23 b(\002)h Fp(C)12 b Ft(.)70 b(Although)33 b(these)h(lea)n(v)n(es)i(are)308 1001 y(not)23 b(closed,)i(the)f(coun)n(ting)f(measure)g(is)h(a)f(w)n (ell-de\014ned)h(holonom)n(y-in)n(v)l(arian)n(t)308 1079 y(transv)n(erse)31 b(measure)g(since)h(the)f(closure)h(of)g Fq(L)1552 1089 y Fo(1)1579 1079 y Ft(,)i(resp)r(ectiv)n(ely)f Fq(L)2047 1089 y Fi(\000)p Fo(1)2111 1079 y Ft(,)h(is)e(the)308 1158 y(union)27 b(of)h(the)f(leaf)h(with)f Fq(f)p Ft([0)h(:)g(0)g(:)h (1])p Fq(g)p Ft(.)46 b(F)-6 b(rom)27 b(no)n(w)g(on)g(w)n(e)g(equip)h Fq(L)2148 1168 y Fo(1)2202 1158 y Ft(with)308 1237 y(the)i(coun)n(ting) g(measure)f(and)h Fq(L)1156 1247 y Fi(\000)p Fo(1)1250 1237 y Ft(with)g(the)g(negativ)n(e)h(coun)n(ting)e(measure,)308 1315 y(i.)23 b(e.)f(the)f(signed)h(measure)e(whic)n(h)i(coun)n(ts)f (eac)n(h)h(in)n(tersection)g(p)r(oin)n(t)f(with)g Fq(\000)p Ft(1.)308 1394 y(W)-6 b(e)23 b(denote)f(this)f(signed)h(measure)f(b)n (y)h Fr(\016)1343 1404 y Fo(1)1385 1394 y Fq(\000)15 b Fr(\016)1481 1404 y Fi(\000)p Fo(1)1546 1394 y Ft(.)376 1473 y(The)21 b(corresp)r(onding)g(Ruelle{Sulliv)l(an)i(cycle)g(is)682 1614 y Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\016)873 1624 y Fo(1)915 1614 y Fq(\000)k Fr(\016)1011 1624 y Fi(\000)p Fo(1)1075 1614 y Ft(\))k(:)g(\012)1205 1586 y Fo(2)1231 1614 y Ft(\()p Fr(M)7 b Ft(\))18 b Fq(\000)-11 b(!)20 b Fp(R)1310 1758 y Fr(!)h Fq(7\000)-11 b(!)1500 1666 y Fj(Z)1537 1818 y Fi(L)1570 1824 y Fh(1)1607 1758 y Fr(!)18 b Fq(\000)1733 1666 y Fj(Z)1771 1818 y Fi(L)1804 1824 y Fd(\000)p Fh(1)1873 1758 y Fr(!)25 b(:)308 1948 y Ft(Here)e Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\016)655 1958 y Fo(1)697 1948 y Fq(\000)16 b Fr(\016)794 1958 y Fi(\000)p Fo(1)858 1948 y Ft(\))22 b(is)g(n)n(ull-homologous)f(b)n(y)h (construction.)376 2027 y(Consider)g Fr(N)28 b Ft(=)20 b Fq(f)p Ft(\()p Fr(z)s(;)11 b(w)r Ft(\))18 b Fq(2)h Fp(C)1129 2003 y Fo(2)1178 2027 y Fq(\032)g Fp(C)13 b Fr(P)1346 2003 y Fo(2)1376 2027 y Fq(j)p Fr(z)22 b Fq(2)c Ft([)p Fq(\000)p Ft(1)p Fr(;)11 b Ft(1])21 b Fq(\032)e Fp(R)p Fq(g)q Ft(.)33 b(The)23 b(b)r(oundary)308 2106 y(of)31 b(this)f(submanifold)e(of)j Fp(C)1037 2081 y Fo(2)1098 2106 y Ft(is)g(exactly)f(the)g(union)g(of)g(the)g(t)n(w)n(o)g (lea)n(v)n(es)i Fq(L)2304 2116 y Fo(1)308 2184 y Ft(and)24 b Fq(L)485 2194 y Fi(\000)p Fo(1)550 2184 y Ft(,)h(the)g(latter)f(with) g(the)g(rev)n(ersed)h(orien)n(tation.)37 b(The)25 b(corresp)r(onding) 308 2263 y(curren)n(t)1051 2404 y(\012)1099 2377 y Fo(3)1126 2404 y Ft(\()p Fp(C)12 b Fr(P)1248 2377 y Fo(2)1278 2404 y Ft(\))19 b Fq(\000)-11 b(!)19 b Fp(R)1260 2548 y Fr(!)j Fq(7\000)-11 b(!)1450 2456 y Fj(Z)1488 2609 y Fk(N)1544 2548 y Fr(!)308 2728 y Ft(has)22 b(the)g(b)r(oundary)e Fr(C)5 b Ft(\()p Fq(F)i Fr(;)k(\016)1028 2738 y Fo(1)1070 2728 y Fq(\000)16 b Fr(\016)1167 2738 y Fi(\000)p Fo(1)1231 2728 y Ft(\).)376 2806 y(Let)31 b Fr(\026)f Ft(b)r(e)h(a)f(v)n(olume)g (form)g(on)g Fp(C)13 b Fr(P)1333 2782 y Fo(2)1394 2806 y Ft(and)30 b Fr(X)36 b Ft(a)31 b(div)n(ergence{free)h(v)n(ector)308 2885 y(\014eld.)e(Let)20 b Fr(\013)i Ft(b)r(e)e(a)h(primitiv)n(e)f(of)h Fr(i)1177 2895 y Fk(X)1222 2885 y Fr(\026)p Ft(.)29 b(Then)20 b(the)g(linking)h(n)n(um)n(b)r(er)e(of)i Fr(X)k Ft(with)308 2964 y(\()p Fq(F)7 b Fr(;)k(\016)447 2974 y Fo(1)490 2964 y Fq(\000)k Fr(\016)586 2974 y Fi(\000)p Fo(1)650 2964 y Ft(\))22 b(is)814 3141 y(lk\()p Fr(X)r(;)11 b Fq(F)c Fr(;)k(\016)1092 3151 y Fo(1)1135 3141 y Fq(\000)k Fr(\016)1231 3151 y Fi(\000)p Fo(1)1295 3141 y Ft(\))k(=)1410 3049 y Fj(Z)1447 3202 y Fi(L)1480 3208 y Fh(1)1517 3141 y Fr(\013)d Fq(\000)1642 3049 y Fj(Z)1680 3202 y Fi(L)1713 3208 y Fd(\000)p Fh(1)1782 3141 y Fr(\013)1340 3330 y Ft(=)1410 3238 y Fj(Z)1447 3390 y Fk(N)1504 3330 y Fr(i)1527 3340 y Fk(X)1572 3330 y Fr(\026)22 b(:)308 3506 y Ft(This)g(is)g (exactly)h(the)e(\015ux)h(of)g Fr(X)27 b Ft(through)20 b Fr(N)7 b Ft(.)p eop %%Page: 19 19 19 18 bop 677 159 a Fm(LINKING)23 b(NUMBERS)f(OF)h(MEASURED)g(F)o(OLIA) l(TIONS)316 b(19)494 294 y Ft(5.)33 b Fs(Godbillon-Vey)26 b(inv)-6 b(ariants)28 b(as)d(linking)i(numbers)376 412 y Ft(Recall)22 b(that)e(the)h(tangen)n(t)g(distribution)f(of)h(a)h(smo) r(oth)d(co)r(dimension)i(1)g(foli-)308 491 y(ation)h Fq(F)28 b Ft(with)21 b(trivial)h(normal)e(bundle)h(on)h(a)f(manifold)g Fr(M)29 b Ft(is)22 b(the)f(k)n(ernel)h(of)g(a)308 569 y(non-v)l(anishing)g(1-form)e Fr(\013)j Ft(whic)n(h,)f(b)n(y)g(the)f(F) -6 b(rob)r(enius)22 b(theorem,)f(satis\014es)308 684 y(\(12\))732 b Fr(d\013)19 b Ft(=)g Fr(\013)c Fq(^)g Fr(\014)308 799 y Ft(for)26 b(some)f(1-form)g Fr(\014)t Ft(.)42 b(The)25 b(3-form)g Fr(\014)d Fq(^)c Fr(d\014)30 b Ft(is)c(closed,)i(and)d(its)h(cohomology)308 877 y(class)20 b Fr(GV)15 b Ft(\()p Fq(F)7 b Ft(\))18 b Fq(2)h Fr(H)814 853 y Fo(3)841 877 y Ft(\()p Fr(M)s(;)11 b Fp(R)p Ft(\))20 b(is)g(indep)r(enden)n(t)e(of)h(the)f(c)n(hoices)i(made)e(for)h Fr(\013)g Ft(and)308 956 y Fr(\014)t Ft(.)30 b(This)22 b(is)g(the)g(Go)r(dbillon-V)-6 b(ey)23 b(in)n(v)l(arian)n(t)e([7)q(])h (of)g Fq(F)7 b Ft(.)376 1035 y(In)22 b(the)g(case)h(that)f Fr(M)29 b Ft(is)23 b(closed,)h(orien)n(ted)e(and)g(3-dimensional,)g Fr(GV)15 b Ft(\()p Fq(F)7 b Ft(\))22 b(is)308 1113 y(equiv)l(alen)n(t)h (to)f(the)f(Hopf)i(in)n(tegral)308 1260 y(\(13\))1153 1168 y Fj(Z)1190 1321 y Fk(M)1255 1260 y Fr(\014)c Fq(^)c Fr(d\014)26 b(:)308 1409 y Ft(If)k(w)n(e)g(c)n(ho)r(ose)f(an)h (arbitrary)e(v)n(olume)h(form)f Fr(\026)h Ft(on)h Fr(M)36 b Ft(and)29 b(de\014ne)g(a)h(v)n(ector)308 1488 y(\014eld)22 b Fr(X)k Ft(in)21 b Fr(M)28 b Ft(b)n(y)21 b(the)g(form)n(ula)g Fr(i)1169 1498 y Fk(X)1214 1488 y Fr(\026)d Ft(=)h Fr(d\014)t Ft(,)j(then)f Fr(X)26 b Ft(is)c(div)n(ergence-free)g(with)308 1566 y(resp)r(ect)k(to)f Fr(\026)p Ft(,)h(and)f(\(13\))g(is)h(just)f (the)g(Hopf)g(in)n(v)l(arian)n(t)h Fr(H)5 b Ft(\()p Fr(X)r(;)11 b(X)5 b Ft(\).)39 b(If)26 b Fr(M)32 b Ft(is)26 b(a)308 1645 y Fp(R)p Ft(-homology)f(sphere,)g(then)e(b)n(y)h(the)g(\\helicit)n (y)h(theorem")e(due)h(to)g(Arnold)g([1])308 1724 y(and)h(the)g(second)g (author)f([14],)j(this)e(Hopf)g(in)n(v)l(arian)n(t)g(can)g(b)r(e)g(in)n (terpreted)g(as)308 1802 y(the)d(a)n(v)n(erage)h(asymptotic)d (self-linking)j(n)n(um)n(b)r(er)d(of)j(the)e(orbits)h(of)g Fr(X)5 b Ft(.)376 1881 y(Assume)25 b(no)n(w)g(that)f(w)n(e)i(ha)n(v)n (e)f(a)g(smo)r(oth)f(1-parameter)g(family)h(of)g(smo)r(oth)308 1960 y(co)r(dimension)30 b(1)g(foliations)g Fq(F)1096 1970 y Fk(t)1147 1960 y Ft(with)f(trivial)h(normal)f(bundles.)53 b(Then)29 b(\(12\))308 2038 y(still)h(holds,)i(but)c(no)n(w)h Fr(\013)h Ft(and)f Fr(\014)k Ft(are)d(functions)f(of)g(the)g(parameter) f Fr(t)j Fq(2)h Fp(R)p Ft(.)308 2117 y(W)-6 b(e)29 b(denote)f(the)g (time)g(deriv)l(ativ)n(es)h(b)n(y)f(a)h(dot.)48 b(It)28 b(w)n(as)g(sho)n(wn)g(b)n(y)g(the)g(\014rst)308 2198 y(author)18 b([9)q(])h(that)f(for)i(ev)n(ery)f Fr(t)g Ft(the)g(4-form)f(\()1422 2180 y(_)1405 2198 y Fr(\014)13 b Fq(^)c Fr(\014)14 b Fq(^)9 b Fr(d\014)t Ft(\)\()p Fr(t)p Ft(\))19 b(is)g(closed,)i(and)e(that)308 2277 y(its)24 b(cohomology)f(class)h Fr(T)9 b(GV)15 b Ft(\()p Fq(F)1148 2287 y Fk(t)1168 2277 y Ft(\))21 b Fq(2)h Fr(H)1342 2252 y Fo(4)1368 2277 y Ft(\()p Fr(M)s(;)11 b Fp(R)p Ft(\))26 b(is)e(a)f(w)n(ell-de\014ned)h(in)n(v)l(arian)n(t)308 2355 y(of)e(the)g(family)g Fq(F)747 2365 y Fk(t)789 2355 y Ft(that)f(is)i(indep)r(enden)n(t)e(of)h(c)n(hoices.)376 2434 y(In)g(the)h(case)h(that)e Fr(M)30 b Ft(is)23 b(closed,)h(orien)n (ted)f(and)g(4-dimensional,)g Fr(T)9 b(GV)15 b Ft(\()p Fq(F)2285 2444 y Fk(t)2305 2434 y Ft(\))308 2513 y(is)23 b(equiv)l(alen)n(t)f(to)g(the)g(in)n(tegral)308 2659 y(\(14\))1057 2567 y Fj(Z)1095 2720 y Fk(M)1176 2641 y Ft(_)1159 2659 y Fr(\014)d Fq(^)c Fr(\014)k Fq(^)c Fr(d\014)t Ft(\()p Fr(t)p Ft(\))22 b Fr(:)308 2808 y Ft(W)-6 b(e)31 b(w)n(an)n(t)e(to)h(giv)n(e)h(an)f(in)n(terpretation)f (of)h(this)g(as)g(an)g(a)n(v)n(erage)g(asymptotic)308 2887 y(linking)24 b(n)n(um)n(b)r(er)e(of)h(a)h(suitable)g(v)n(ector)f (\014eld)h(and)f(a)g(measured)f(co)r(dimension)308 2966 y(2)g(foliation.)376 3044 y(Cho)r(ose)14 b(an)g(arbitrary)f(v)n(olume)i (form)e Fr(\026)h Ft(on)h Fr(M)21 b Ft(and)14 b(de\014ne)h(a)f (time-dep)r(enden)n(t)308 3126 y(v)n(ector)22 b(\014eld)g Fr(X)27 b Ft(b)n(y)22 b(the)g(form)n(ula)f Fr(i)1202 3136 y Fk(X)1247 3126 y Fr(\026)d Ft(=)h Fr(d)p Ft(\()1452 3108 y(_)1435 3126 y Fr(\014)g Fq(^)c Fr(\014)t Ft(\).)376 3204 y(Di\013eren)n(tiating)33 b(\(12\))o(,)j(w)n(e)c(see)h(that)f Fr(d\014)26 b Fq(^)c Fr(d\014)41 b Ft(=)36 b(0)d(b)r(ecause)f Fr(d\014)37 b Ft(is)c(de-)308 3283 y(comp)r(osable.)43 b(Th)n(us,)27 b(on)f(the)g(op)r(en)h(set)f(in)h Fr(M)33 b Ft(where)26 b Fr(d\014)31 b Ft(do)r(es)c(not)e(v)l(anish,)308 3362 y(its)j(k)n(ernel)g(is)h(a)f(2-dimensional)f(distribution)f(whic)n (h)i(is)g(in)n(tegrable)g(b)r(ecause)308 3440 y(the)i(de\014ning)g (form)f(is)h(closed.)56 b(W)-6 b(e)30 b(denote)g(b)n(y)g Fq(G)35 b Ft(the)30 b(singular)g(co)r(dimen-)308 3519 y(sion)f(2)g(foliation)g(de\014ned)f(b)n(y)g Fr(d\014)t Ft(.)50 b(Note)29 b(that)f(the)g(exact)h(form)e Fr(d\014)33 b Ft(de\014nes)p eop %%Page: 20 20 20 19 bop 308 159 a Fm(20)560 b(D.)18 b(K)o(OTSCHICK)e(AND)h(T.)g(V)o (OGEL)308 294 y Ft(an)25 b(in)n(v)l(arian)n(t)g(transv)n(ersal)g (measure)f(for)h Fq(G)k Ft(whose)c(Ruelle{Sulliv)l(an)h(cycle)g(is)308 373 y(n)n(ull-homologous.)32 b(No)n(w)23 b(w)n(e)h(recognise)f(\(14\))g (as)g(the)g(Hopf-t)n(yp)r(e)f(in)n(tegral)i(as-)308 451 y(so)r(ciated)j(to)e(the)h(v)n(ector)g(\014eld)g Fr(X)32 b Ft(\(at)25 b(time)h Fr(t)p Ft(\))f(and)h(the)f(measured)g(foliation) 308 530 y(\()p Fq(G)t Fr(;)11 b(d\014)t Ft(\))23 b(\(at)e(the)h(same)f (time)g Fr(t)p Ft(\).)376 609 y(T)-6 b(o)19 b(in)n(terpret)f(this)h(as) g(an)g(a)n(v)n(erage)g(asymptotic)f(linking)h(n)n(um)n(b)r(er)f (according)308 687 y(to)f(Theorem)f(17,)j(w)n(e)e(need)g(to)g(assume)f Fr(b)1329 697 y Fo(1)1356 687 y Ft(\()p Fr(M)7 b Ft(\))18 b(=)h(0.)28 b(As)18 b Fr(M)24 b Ft(is)18 b(closed,)h(orien)n(ted)308 766 y(and)f(4-dimensional,)h(it)g(follo)n(ws)g(that)f(the)g(Euler)g(c)n (haracteristic)i(of)e Fr(M)26 b Ft(is)19 b(p)r(osi-)308 845 y(tiv)n(e,)g(and)d(so)h(there)f(cannot)g(b)r(e)h(an)n(y)g (non-singular)e(co)r(dimension)h(1)h(foliation)h Fq(F)308 923 y Ft(on)i Fr(M)7 b Ft(.)29 b(Ho)n(w)n(ev)n(er,)21 b(as)f(w)n(e)g(ha)n(v)n(e)g(ended)g(up)f(with)g(only)h(a)g(singular)f (foliation)i(for)308 1002 y Fq(G)t Ft(,)g(there)e(is)h(no)g(harm)e(in)h (allo)n(wing)i Fq(F)26 b Ft(to)20 b(b)r(e)f(singular)h(as)f(w)n(ell.)30 b(So)20 b(w)n(e)f(just)h(as-)308 1081 y(sume)c(that)g Fq(F)658 1091 y Fk(t)696 1081 y Ft(is)h(the)g(k)n(ernel)g(of)g(a)g (time-dep)r(enden)n(t)f(1-form)g Fr(\013)h Ft(satisfying)h(\(12\))o(,) 308 1159 y(but)i(allo)n(w)h Fr(\013)g Ft(to)g(ha)n(v)n(e)g(zero)r(es.) 30 b(Under)20 b(certain)h(tec)n(hnical)h(assumptions)d(on)h Fr(\013)p Ft(,)308 1238 y(cf.)31 b(the)f(app)r(endix,)i(the)e (de\014nition)g(of)g Fr(T)9 b(GV)15 b Ft(\()p Fq(F)1577 1248 y Fk(t)1597 1238 y Ft(\))30 b(go)r(es)h(through)d(as)j(in)f(the) 308 1317 y(non-singular)18 b(case,)j(and)e(if)g Fr(b)1050 1327 y Fo(1)1077 1317 y Ft(\()p Fr(M)7 b Ft(\))18 b(=)h(0,)h(then)e (according)h(to)g(Theorem)f(17)h(the)308 1395 y(in)n(tegral)i(of)g Fr(T)9 b(GV)15 b Ft(\()p Fq(F)848 1405 y Fk(t)868 1395 y Ft(\))20 b(o)n(v)n(er)h Fr(M)27 b Ft(is)21 b(the)g(a)n(v)n(erage)g (asymptotic)e(linking)i(n)n(um)n(b)r(er)308 1474 y(lk\()p Fr(X)r(;)11 b Fq(G)t Fr(;)g(d\014)t Ft(\),)18 b(with)e Fr(X)21 b Ft(and)15 b Fq(G)20 b Ft(de\014ned)c(as)g(ab)r(o)n(v)n(e.)28 b(W)-6 b(e)16 b(can)g(apply)g(Theorem)f(17)308 1553 y(to)24 b(the)f(singular)h(foliation)g Fq(G)k Ft(b)r(ecause)c(the)g(holonom)n (y-in)n(v)l(arian)n(t)e(measure)h(is)308 1631 y(giv)n(en)g(b)n(y)e(a)h (smo)r(oth)f(form,)g(see)h(Remark)f(14.)376 1710 y(If)15 b Fq(F)483 1720 y Fk(t)520 1710 y Ft(is)h(a)g(1-parameter)e(family)h (of)h(co)r(dimension)f Fr(q)k Ft(foliations)d(on)g Fr(M)22 b Ft(de\014ned)308 1789 y(b)n(y)e(a)g(decomp)r(osable)f Fr(q)r Ft(-from)g Fr(\013)p Ft(,)i(then)e(setting)h Fr(d\013)f Ft(=)g Fr(\013)11 b Fq(^)f Fr(\014)24 b Ft(as)c(ab)r(o)n(v)n(e,)h(w)n (e)f(can)308 1870 y(consider)588 1852 y(_)571 1870 y Fr(\014)j Fq(^)c Fr(\014)k Fq(^)c Ft(\()p Fr(d\014)t Ft(\))946 1845 y Fk(q)972 1870 y Ft(.)47 b(It)27 b(w)n(as)h(pro)n(v)n (ed)g(in)f([9)q(])h(that)f(this)h(is)g(closed,)i(and)308 1948 y(that)e(its)g(cohomology)g(class)h Fr(T)9 b(GV)15 b Ft(\()p Fq(F)1312 1958 y Fk(t)1332 1948 y Ft(\))29 b Fq(2)h Fr(H)1522 1924 y Fo(2)p Fk(q)r Fo(+2)1632 1948 y Ft(\()p Fr(M)s(;)11 b Fp(R)p Ft(\))30 b(is)f(a)f(w)n(ell-de\014ned) 308 2027 y(in)n(v)l(arian)n(t)23 b(of)g(the)g(family)f Fq(F)1027 2037 y Fk(t)1048 2027 y Ft(.)32 b(One)23 b(migh)n(t)f(b)r(e)h (tempted)e(to)h(think)g(that)g(when)308 2106 y Fr(M)35 b Ft(is)28 b(closed)g(orien)n(ted)f(of)h(dimension)e(2)p Fr(q)c Ft(+)d(2,)29 b(then)e(this)g(in)n(v)l(arian)n(t)g(should)308 2184 y(alw)n(a)n(ys)21 b(b)r(e)g(an)g(a)n(v)n(erage)g(asymptotic)f (linking)h(n)n(um)n(b)r(er,)e(as)i(w)n(as)g(pro)n(v)n(ed)g(ab)r(o)n(v)n (e)308 2263 y(for)26 b(the)g(case)h Fr(q)h Ft(=)e(1.)42 b(Ho)n(w)n(ev)n(er,)29 b(this)d(do)r(es)g(not)f(seem)h(to)g(b)r(e)g (the)f(case.)43 b(W)-6 b(e)308 2342 y(alw)n(a)n(ys)21 b(ha)n(v)n(e)g(\()p Fr(d\014)t Ft(\))796 2317 y Fk(q)r Fo(+1)901 2342 y Ft(=)e(0,)j(sho)n(wing)e(that)g Fr(d\014)25 b Ft(has)20 b(at)g(least)i(a)e(2-dimensional)308 2420 y(k)n(ernel.)38 b(If)24 b Fr(T)9 b(GV)16 b Ft(\()p Fq(F)832 2430 y Fk(t)852 2420 y Ft(\))24 b(do)r(esn't)g(v)l(anish,)h(then)f(\()p Fr(d\014)t Ft(\))1639 2396 y Fk(q)1688 2420 y Fq(6)p Ft(=)f(0)i(on)f(an)g(op)r(en)g(set)h(in)308 2499 y Fr(M)7 b Ft(,)31 b(so)d(that)g(the)g(k)n(ernel)h(of)g Fr(d\014)k Ft(is)c(exactly)f(of)h(rank)f(2)g(\(on)g(this)h(op)r(en)f(set\).)308 2578 y(This)e(means)f(that)g(whenev)n(er)h Fr(q)i(>)e Ft(1,)h(the)f(co)r(dimension)f(of)h(the)g(cok)n(ernel)h(of)308 2656 y Fr(d\014)c Ft(is)c(strictly)f(larger)h(than)e(2,)j(and)e(so)g (there)g(can)h(b)r(e)f(no)g(linking)h(n)n(um)n(b)r(er)d(with)308 2735 y(1-dimensional)22 b(\015o)n(w)f(lines.)1152 2929 y Fs(Appendix:)320 3008 y(Singular)26 b(f)o(olia)l(tions)f(and)i(f)o (orms)d(with)j(the)e(division)i(pr)o(oper)l(ty)376 3126 y Ft(In)f(the)h(last)g(section)h(w)n(e)f(ha)n(v)n(e)g(had)g(to)f (consider)h(co)r(dimension)g(one)g(folia-)308 3204 y(tions)e(de\014ned) f(b)n(y)h(one-forms)f Fr(\013)h Ft(whic)n(h)g(ha)n(v)n(e)g(zero)r(es.) 39 b(In)25 b(this)g(app)r(endix)f(w)n(e)308 3283 y(explain)g(ho)n(w)e (the)h(de\014nitions)g(of)g(the)g(Go)r(dbillon-V)-6 b(ey)24 b(in)n(v)l(arian)n(t)f Fr(GV)38 b Ft(and)23 b(of)308 3362 y(the)g(in)n(v)l(arian)n(t)h Fr(T)9 b(GV)38 b Ft(of)24 b(families)g(extend)f(to)g(this)h(situation.)33 b(As)24 b(in)g(the)f(rest)308 3440 y(of)f(this)g(pap)r(er,)g(all)g(forms)f(are) h(smo)r(oth)f(of)h(class)g Fr(C)1614 3416 y Fi(1)1665 3440 y Ft(.)376 3519 y(F)-6 b(ollo)n(wing)23 b(Moussu)f([10],)h(w)n(e)f (consider)g(the)g(follo)n(wing:)p eop %%Page: 21 21 21 20 bop 677 159 a Fm(LINKING)23 b(NUMBERS)f(OF)h(MEASURED)g(F)o(OLIA) l(TIONS)316 b(21)308 294 y Fw(De\014nition)30 b(22.)f Ft(A)d(one-form)d Fr(\013)j Ft(has)e(the)h(division)g(prop)r(ert)n(y)f (if)h Fr(!)20 b Fq(^)c Fr(\013)25 b Ft(=)f(0)308 373 y(for)f(a)g(smo)r(oth)e Fr(!)i Fq(2)d Ft(\012)877 348 y Fk(k)905 373 y Ft(\()p Fr(M)7 b Ft(\))22 b(with)g(0)f Fr(<)f(k)j(<)d Ft(dim)o(\()p Fr(M)7 b Ft(\))22 b(implies)h Fr(!)g Ft(=)e Fr(\013)16 b Fq(^)f Fr(\014)27 b Ft(for)308 451 y(some)22 b(smo)r(oth)e Fr(\014)j Fq(2)18 b Ft(\012)878 427 y Fk(k)q Fi(\000)p Fo(1)968 451 y Ft(\()p Fr(M)7 b Ft(\).)376 552 y(Note)23 b(that)f(the)g(division)h(prop)r(ert)n(y)f (then)g(implies)h(that)f Fr(\014)k Ft(is)e(unique)e(up)g(to)308 631 y(the)g(addition)f(of)i(m)n(ultiples)e(of)h Fr(\013)p Ft(.)376 710 y(No)n(where)27 b(v)l(anishing)g(1-forms)f(ha)n(v)n(e)h (the)g(division)g(prop)r(ert)n(y)-6 b(,)28 b(as)f(do)f(some)308 788 y(classes)g(of)e(singular)h(1-forms.)35 b(F)-6 b(or)25 b(example,)g(it)f(is)h(easy)g(to)f(see)h(that)e(if)i(near)308 867 y(ev)n(ery)i(p)r(oin)n(t)f(where)g(it)g(v)l(anishes,)i Fr(\013)f Ft(is)f(lo)r(cally)i(the)e(di\013eren)n(tial)g(of)h(a)f (Morse)308 945 y(function,)18 b(then)f(it)g(has)g(the)f(division)i (prop)r(ert)n(y)-6 b(.)27 b(More)18 b(generally)-6 b(,)19 b(Moussu)e([10])308 1024 y(pro)n(v)n(ed)33 b(that)f(1-forms)f(with)i (only)f(algebraically)i(isolated)g(zeros)f(ha)n(v)n(e)g(the)308 1103 y(division)24 b(prop)r(ert)n(y)-6 b(.)33 b(Algebraic)25 b(isolation)f(of)g(the)f(zeros)h(means)f(that)f(in)i(lo)r(cal)308 1181 y(co)r(ordinates)e(at)g(a)g(zero)g(the)g(co)r(e\016cien)n(t)h (functions)f(of)g(the)g(one-form)f(span)g(an)308 1260 y(ideal)i(of)f(\014nite)g(co)r(dimension)f(in)h(the)g(algebra)g(of)g (germs)f(of)h(functions.)376 1339 y(The)27 b(calculation)h(sho)n(wing)f (the)g(w)n(ell-de\014nedness)g(of)h(the)f(Go)r(dbillon-V)-6 b(ey)308 1417 y(in)n(v)l(arian)n(t)17 b([7])g(of)f(a)h(co)r(dimension)e (one)i(foliation)g(do)r(es)f(not)g(require)g(the)g(de\014ning)308 1496 y(form)g Fr(\013)i Ft(to)e(b)r(e)h(non-zero,)h(but)e(rather)h (requires)f(only)h(that)f(it)i(ha)n(v)n(e)f(the)f(division)308 1575 y(prop)r(ert)n(y)-6 b(.)29 b(Th)n(us)21 b(w)n(e)h(ha)n(v)n(e:)308 1675 y Fw(Theorem)28 b(23.)h Fg(L)m(et)d Fr(\013)g Fg(b)m(e)g(a)g(smo)m (oth)e Ft(1)p Fg(-form)h(with)h Fr(\013)17 b Fq(^)g Fr(d\013)23 b Ft(=)h(0)p Fg(,)i(and)g Fq(F)33 b Fg(its)308 1754 y(kernel)27 b(foliation.)39 b(If)27 b Fr(\013)g Fg(has)f(the)g(division)h(pr)m(op)m (erty)e(then)h(ther)m(e)h(is)f(a)g Ft(1)p Fg(-form)308 1833 y Fr(\014)i Fg(with)308 1955 y Ft(\(15\))711 b Fr(d\013)19 b Ft(=)g Fr(\013)d Fq(^)e Fr(\014)28 b(:)308 2078 y Fg(The)17 b Ft(3)p Fg(-form)g Fr(\014)t Fq(^)p Fr(d\014)k Fg(is)c(close)m(d)f (and)g(its)h(c)m(ohomolo)m(gy)e(class)i Fr(GV)e Ft(\()p Fq(F)7 b Ft(\))19 b Fq(2)f Fr(H)2128 2054 y Fo(3)2155 2078 y Ft(\()p Fr(M)s(;)11 b Fp(R)p Ft(\))308 2157 y Fg(is)20 b(indep)m(endent)g(of)f(the)h(choic)m(es)g(made)f(for)g Fr(\013)h Fg(and)f Fr(\014)t Fg(,)i(as)e(long)h(as)f Fr(\013)h Fg(is)g(change)m(d)308 2236 y(only)k(by)g(multiplic)m(ation)f (with)g(a)g(nowher)m(e)g(vanishing)i(function.)376 2336 y Ft(This)c(generalises)j(to)d(families)i(in)f(the)f(follo)n(wing)i(w)n (a)n(y:)308 2437 y Fw(Theorem)28 b(24.)h Fg(L)m(et)d Fr(\013)h Fg(b)m(e)f(a)g(smo)m(oth)f Ft(1)p Fg(-form)g(on)h Fr(M)33 b Fg(with)26 b Fr(\013)17 b Fq(^)g Fr(d\013)24 b Ft(=)f(0)j Fg(and)308 2516 y(which)e(dep)m(ends)f(smo)m(othly)f(on)h (a)g(p)m(ar)m(ameter)f Fr(t)d Fq(2)g Fp(R)p Fg(.)32 b(L)m(et)24 b Fq(F)1842 2526 y Fk(t)1886 2516 y Fg(b)m(e)g(the)f(family)h(of)308 2594 y(kernel)k(foliations.)42 b(If)27 b(for)g(every)i Fr(t)e Fg(the)g(form)g Fr(\013)h Fg(has)e(the)h(division)h(pr)m(op)m (erty,)308 2676 y(then)19 b(the)g Ft(4)p Fg(-forms)f Ft(\()835 2658 y(_)819 2676 y Fr(\014)8 b Fq(^)t Fr(\014)g Fq(^)t Fr(d\014)t Ft(\)\()p Fr(t)p Ft(\))19 b Fg(ar)m(e)f(close)m(d,)h (and)g(their)g(c)m(ohomolo)m(gy)e(classes)308 2754 y Fr(T)9 b(GV)15 b Ft(\()p Fq(F)535 2764 y Fk(t)556 2754 y Ft(\))j Fq(2)h Fr(H)724 2730 y Fo(4)750 2754 y Ft(\()p Fr(M)s(;)11 b Fp(R)p Ft(\))19 b Fg(ar)m(e)d(indep)m(endent)h(of)g(the)g (choic)m(es)g(made)g(for)g Fr(\013)g Fg(and)f Fr(\014)t Fg(,)308 2833 y(as)k(long)f(as)h Fr(\013)g Fg(is)g(change)m(d)f(only)h (by)g(multiplic)m(ation)f(with)h(a)f(nowher)m(e)g(vanishing)308 2912 y(function.)308 3047 y(Pr)m(o)m(of.)27 b Ft(Once)j(w)n(e)f(ha)n(v) n(e)g(the)f(forms)g Fr(\014)33 b Ft(and)1480 3029 y(_)1463 3047 y Fr(\014)t Ft(,)e(c)n(hec)n(king)f(the)e(claims)h(in)g(the)308 3126 y(theorem)21 b(is)h(done)g(exactly)g(as)g(in)g(the)g(nonsingular)f (case,)i(compare)e([9)q(].)376 3204 y(In)i(the)g(case)h(when)f Fr(\013)g Ft(has)h(no)f(zeros,)h(it)g(is)f(easy)h(to)f(see)h(that)f(if) h Fr(\013)f Ft(dep)r(ends)308 3283 y(smo)r(othly)d(on)h Fr(t)p Ft(,)h(then)f(one)g(can)g(also)h(c)n(ho)r(ose)g Fr(\014)j Ft(in)d(\(15\))f(to)g(dep)r(end)f(smo)r(othly)308 3362 y(on)29 b Fr(t)p Ft(.)50 b(In)29 b(the)f(singular)h(case,)j(the)c (smo)r(oth)f(dep)r(endence)i(of)g Fr(\014)k Ft(on)c Fr(t)g Ft(is)g(less)308 3440 y(ob)n(vious.)46 b(Ho)n(w)n(ev)n(er,)31 b(if)d Fr(\013)p Ft(\(0\))f(has)g(only)h(algebraically)h(isolated)f (zeros,)h(then)308 3519 y(the)g(same)f(is)h(true)f(for)h Fr(\013)p Ft(\()p Fr(t)p Ft(\))g(for)f(all)i Fr(t)e Ft(su\016cien)n (tly)i(close)g(to)e(0,)j(and)d(in)h(this)p eop %%Page: 22 22 22 21 bop 308 159 a Fm(22)560 b(D.)18 b(K)o(OTSCHICK)e(AND)h(T.)g(V)o (OGEL)308 294 y Ft(case)26 b Fr(\014)j Ft(can)24 b(b)r(e)h(c)n(hosen)g (to)g(dep)r(end)f(smo)r(othly)f(on)h Fr(t)h Ft(b)n(y)g(adapting)e (Moussu's)308 373 y(argumen)n(t)d([10)q(].)376 451 y(One)j(can)f(also)h (giv)n(e)g(an)g(alternativ)n(e)f(treatmen)n(t)f(for)h(all)i(divisible)f (forms)f(as)308 530 y(follo)n(ws.)31 b(Recall)23 b(from)d([9)q(])i(the) g(iden)n(tit)n(y)308 643 y(\(16\))707 625 y(_)691 643 y Fr(\014)d Fq(^)c Fr(\014)k Fq(^)14 b Fr(d\014)23 b Ft(=)d Fr(d)p Ft(\()13 b(_)-31 b Fr(\013)15 b Fq(^)g Fr(\014)k Fq(^)c Fr(\015)t Ft(\))f Fq(\000)30 b Ft(_)-32 b Fr(\013)15 b Fq(^)g Fr(\014)k Fq(^)c Fr(\013)g Fq(^)g Fr(\016)25 b(;)308 749 y Ft(where)19 b Fr(\015)j Ft(and)c Fr(\016)k Ft(are)d(1-forms)e(constructed)h(b)n(y)g(successiv)n(e)i (exterior)f(di\013eren)n(ti-)308 827 y(ation)f(of)23 b(\(15\))18 b(and)f(the)h(division)h(prop)r(ert)n(y)e(of)h Fr(\013)p Ft(.)29 b(This)18 b(sho)n(ws)g(that)f Fr(T)9 b(GV)15 b Ft(\()p Fq(F)2285 837 y Fk(t)2305 827 y Ft(\))308 906 y(can)23 b(b)r(e)f(de\014ned)g(using)h(only)f(the)g(smo)r(oth)f (dep)r(endence)i(of)g Fr(\013)g Ft(on)f Fr(t)p Ft(,)h(and)f(w)n(ell-) 308 985 y(de\014nedness)g(follo)n(ws)g(as)h(in)f([9])g(using)g(the)g (division)g(prop)r(ert)n(y)-6 b(.)366 b Ff(\003)1123 1118 y Fs(References)336 1225 y Fv(1.)28 b(V.)55 b(I.)g(Arnold,)64 b Fn(The)53 b(asymptotic)g(Hopf)f(invariant)g(and)g(its)i(applic)m (ations)p Fv(,)408 1293 y(Sel.)19 b(Math.)g(So)n(v.)g Fc(5)g Fv(\(1986\),)f(327{345.)336 1360 y(2.)28 b(V.)33 b(I.)f(Arnold)i(and)f(B.)f(A.)h(Khesin,)j Fb(T)-5 b(op)r(ological)35 b(Metho)r(ds)e(in)h(Hydro)r(dynamics)p Fv(,)408 1427 y(Springer)20 b(V)-5 b(erlag)20 b(1998.)336 1495 y(3.)28 b(R.)f(Bott)g(and)h(L.)f(W.)g(T)-5 b(u,)29 b Fb(Di\013eren)n(tial)g(F) -5 b(orms)27 b(in)h(Algebraic)h(T)-5 b(op)r(ology)p Fv(,)29 b(Springer)408 1562 y(V)-5 b(erlag)20 b(1982.)336 1630 y(4.)28 b(A.)15 b(Candel)g(and)h(L.)e(Conlon,)i Fb(F)-5 b(oliations)18 b(I)p Fv(,)c(Amer.)h(Math.)g(So)r(c.)f(Pro)n(vidence,)j (R.I.)e(2000.)336 1697 y(5.)28 b(G.)d(Con)n(treras)g(and)h(R.)f (Iturriaga,)i Fn(A)o(ver)m(age)d(linking)i(numb)m(ers)p Fv(,)f(Ergo)r(d.)g(Th.)g(&)f(Dy-)408 1765 y(nam.)19 b(Sys.)f Fc(19)h Fv(\(1999\),)g(1425{1435.)336 1832 y(6.)28 b(N.)f(Dunford)g (and)g(J.)g(T.)e(Sc)n(h)n(w)n(artz,)30 b Fb(Linear)e(op)r(erators)e(I)p Fv(,)h(In)n(terscience)h(Publishers)408 1899 y(1958.)336 1967 y(7.)g(C.)i(Go)r(dbillon)j(and)f(J.)e(V)-5 b(ey)g(,)34 b Fn(Un)d(invariant)e(des)i(feuil)s(letages)g(de)g(c)m(o)m(dimension)f Fv(1,)408 2034 y(C.)18 b(R.)h(Acad.)f(Sci.)i(P)n(aris)g Fc(273)f Fv(\(1971\),)g(92{95.)336 2102 y(8.)28 b(B.)34 b(A.)h(Khesin,)k Fn(Er)m(go)m(dic)34 b(interpr)m(etation)e(of)i(inte)m (gr)m(al)g(hydr)m(o)m(dynamic)g(invariants)p Fv(,)408 2169 y(J.)18 b(Geom.)h(and)g(Ph)n(ysics)h Fc(9)f Fv(\(1992\),)f (101{110.)336 2236 y(9.)28 b(D.)k(Kotsc)n(hic)n(k,)37 b Fn(Go)m(dbil)s(lon-V)l(ey)32 b(invariants)f(for)h(families)i(of)e (foliations)p Fv(,)j(Preprin)n(t)408 2304 y(arXiv:math.GT/0111137.)308 2371 y(10.)28 b(R.)22 b(Moussu,)h Fn(Sur)g(l'existenc)m(e)f(d'int)o (\023)-27 b(egr)m(ales)21 b(pr)m(emi)o(\022)-27 b(er)m(es)21 b(p)m(our)h(un)g(germe)g(de)h(forme)f(de)408 2439 y(Pfa\013)p Fv(,)d(Ann.)g(Inst.)g(F)-5 b(ourier)19 b Fc(26)h Fv(\(1976\),)e (171{220.)308 2506 y(11.)28 b(G.)19 b(de)f(Rham,)h Fb(Di\013eren)n (tiable)i(manifolds)p Fv(,)f(Springer)h(V)-5 b(erlag)19 b(1984.)308 2574 y(12.)28 b(T.)42 b(Rivi)n(\022)-27 b(ere,)50 b Fn(High-dimensional)41 b(helicities)h(and)f(rigidity)h(of)g(linke)m (d)g(foliations)p Fv(,)408 2641 y(Preprin)n(t.)308 2708 y(13.)28 b(D.)21 b(Sulliv)m(an,)i Fn(Cycles)f(for)g(the)f(dynamic)m(al) h(study)g(of)g(foliate)m(d)f(manifolds)h(and)f(c)m(omplex)408 2776 y(manifolds)p Fv(,)d(In)n(v)n(en)n(t.)i(math.)e Fc(36)h Fv(\(1976\),)g(225{255.)308 2843 y(14.)28 b(T.)21 b(V)-5 b(ogel,)23 b Fn(On)f(the)g(asymptotic)g(linking)g(numb)m(er)p Fv(,)f(Pro)r(c.)g(Amer.)g(Math.)h(So)r(c.)f(\(to)h(ap-)408 2911 y(p)r(ear\).)376 3037 y Fu(Ma)l(thema)l(tisches)17 b(Institut,)f(Lud)o(wig-Maximilians-Universit)1978 3032 y(\177)1976 3037 y(at)h(M)2118 3032 y(\177)2116 3037 y(unchen,)308 3104 y(Theresienstr.)23 b(39,)e(80333)h(M)1060 3099 y(\177)1058 3104 y(unchen,)h(Germany)376 3172 y Fn(E-mail)d(addr)m(ess)5 b Fv(:)23 b Fa(dieter@me)q(mb)q(er)q(.a)q(ms)q (.o)q(rg)376 3286 y Fu(Ma)l(thema)l(tisches)17 b(Institut,)f(Lud)o (wig-Maximilians-Universit)1978 3281 y(\177)1976 3286 y(at)h(M)2118 3281 y(\177)2116 3286 y(unchen,)308 3354 y(Theresienstr.)23 b(39,)e(80333)h(M)1060 3349 y(\177)1058 3354 y(unchen,)h(Germany)376 3421 y Fn(E-mail)d(addr)m(ess)5 b Fv(:)23 b Fa(thomas.vo)q(ge)q(l@)q(ma)q(th)q(em)q(at)q(ik)q(.u)q(ni-) q(mu)q(en)q(ch)q(en)q(.d)q(e)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF