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4082 y Fx(.)316 4298 y Fu(This)33 b(result)f(is)h(in)g(line)g(with)g(our)g(earlier)h(bound)f ([CE2])h(where)g(we)f(sho)n(wed)g(that,)i(e)o(xpressed)e(in)94 4417 y(the)26 b(units)e(of)i(the)f(present)h(paper)l(,)g(the)f(number)g(of)h (balls)f(of)g(radius)g Ft(")i Fu(in)e(L)2713 4374 y Fo(1)2823 4417 y Fu(needed)g(to)h(co)o(v)o(er)f(the)g(global)94 4537 y(attractor)g(\(when)g(restricted)g(to)f Fn(j)p Ft(x)p Fn(j)j(\024)h Ft(L)p Fu(\))d(has)g(a)g(bound)f(of)h(the)g(order)g(of)842 4722 y(e)o(xp)985 4641 y Fq(\000)1030 4722 y Ft(C)7 b Fr(\()p Ft(L=\016)1309 4749 y Fo(\003)1355 4722 y Fr(\))1394 4679 y Fz(d)p Fo(\000)p FA(1)1554 4722 y Fu(log)o Fr(\()p Fu(1)p Ft(=")p Fr(\))1905 4679 y FA(2)1945 4641 y Fq(\001)2013 4722 y Fn(\001)21 b Fu(e)o(xp)2205 4641 y Fq(\000)2250 4722 y Ft(C)7 b Fr(\()p Ft(L=\016)2529 4749 y Fo(\003)2575 4722 y Fr(\))2614 4679 y Fz(d)2677 4722 y Fu(log)o Fr(\()p Fu(1)p Ft(=")p Fr(\))3028 4641 y Fq(\001)3098 4722 y Ft(:)316 4907 y Fu(One)18 b(of)h(the)f(main)f (ingredients)g(of)i(the)f(proof)g(of)g(Eq.\(1.11\))g(is)g(the)g (\252dissipati)n(v)o(e)d(bound\272)j(of)g(Theorem)94 5027 y(5.1)25 b(belo)n(w)-6 b(.)34 b(In)25 b([CE2],)h(we)f(sho)n(wed)e(the)i(inequality)e ([CE2,)j(Eq.\(6.6\)]:)1563 5212 y(sup)1439 5304 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)p Fo(\000)p Fz(")1741 5284 y Ff(\000)p Fe(1)1842 5212 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b(t)p Fr(\))p Fn(j)42 b(\024)j Ft(C)7 b(")25 b(;)1090 b Fr(\()p Fu(1)p Ft(:)p Fu(12)p Fr(\))p eop %%Page: 5 5 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1722 b Fl(5)94 194 y Fu(when)26 b(0)i Fn(\024)h Ft(t)f Fn(\024)h Ft(\034)734 221 y Fo(\003)779 194 y Fu(,)c(and)h(sup)1137 221 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1361 194 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b Fu(0)p Fr(\))p Fn(j)26 b(\024)i Ft(")p Fu(.)38 b(W)-8 b(e)26 b(can)f(impro)o(v)o(e)f(this)g(bound)h(\(slightly\))e(as)j (follo)n(ws)94 314 y(and)f(write)g(it)f(in)h(natural)f(units)g(\(see)h (Eq.\(5.8\))g(for)g(a)g(precise)g(formulation\):)548 610 y(sup)469 701 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)p Fo(\000)p Fz(`)783 610 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b(\034)1070 637 y Fo(\003)1114 610 y Fr(\))p Fn(j)43 b(\024)i Ft(C)1424 529 y Fq(\000)1502 610 y Fu(sup)1470 701 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1689 610 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b Fu(0)p Fr(\))p Fn(j)j Fr(+)i(\()p Ft(\016)2251 637 y Fo(\003)2297 610 y Ft(=`)p Fr(\))p Fn(k)p Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b Fu(0)p Fr(\))p Fn(k)2803 637 y Fo(1)2885 529 y Fq(\001)o(\000)2988 543 y Ft(D)3070 570 y Fo(\003)p 2988 587 129 4 v 3005 679 a Ft(\027)3054 705 y Fo(\003)3128 610 y Fr(\))3167 567 y Fp(\()p FA(1)p Fp(+)p Fz(d)p Fp(\))p Fz(=)p FA(2)3472 610 y Ft(:)94 936 y Fu(Our)e(impro)o(v)o(ed)g(bound) g(in)g(this)g(paper)g(e)o(xploits)g(th)o(e)g(dissi)o(pation)f(as)h(follo)m (ws:)24 b(Let)15 b Ft(P)2987 963 y Fz(>)3069 936 y Fu(denote)g(a)g (localization)94 1055 y(of)34 b(the)f(F)o(ourier)h(transform)f(of)h Ft(w)i Fu(to)d(w)o(a)n(v)o(e)h(numbers)e Ft(k)37 b Fu(satisfying)32 b Fn(j)p Ft(k)s Fn(j)39 b Ft(>)i(k)2894 1082 y Fo(\003)2973 1055 y Fu(\(see)34 b(belo)n(w)f(for)g(a)h(more)94 1175 y(precise)26 b(de\256nition\).)35 b(Then,)209 1471 y(sup)129 1562 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)p Fo(\000)p Fz(`)443 1471 y Fn(j)p Ft(P)535 1498 y Fz(>)602 1471 y Ft(w)s Fr(\()p Ft(x;)17 b(\034)861 1498 y Fo(\003)906 1471 y Fr(\))p Fn(j)43 b(\024)i Ft(C)1216 1391 y Fq(\000)1299 1404 y Ft(M)1396 1431 y Fo(\003)p 1274 1448 193 4 v 1274 1540 a Ft(\027)1323 1566 y Fo(\003)1369 1540 y Ft(k)1424 1511 y FA(2)1421 1566 y Fo(\003)1479 1471 y Fr(\))1518 1428 y FA(1)p Fo(\000)p Fz(d=)p FA(4)1737 1391 y Fq(\000)1828 1471 y Fu(sup)1782 1562 y Fo(j)p Fz(x)p Fo(j)13 b(\024)h Fz(\025)2028 1471 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)j Fu(0)p Fr(\))p Fn(j)j Fr(+)j(\()p Ft(\016)2591 1498 y Fo(\003)2637 1471 y Ft(=`)p Fr(\))p Fn(k)p Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b Fu(0)p Fr(\))p Fn(k)3143 1498 y Fo(1)3224 1391 y Fq(\001\000)3327 1404 y Ft(D)3409 1431 y Fo(\003)p 3327 1448 129 4 v 3344 1540 a Ft(\027)3393 1566 y Fo(\003)3468 1471 y Fr(\))3507 1428 y Fp(\()p FA(1)p Fp(+)p Fz(d)p Fp(\))p Fz(=)p FA(2)3812 1471 y Ft(:)94 1798 y Fu(Thus,)25 b(if)f Ft(k)481 1825 y Fo(\003)552 1798 y Fu(is)g(suf)n (\256ciently)g(lar)n(ge,)h(the)g(high)f(frequenc)o(y)h(part)f(of)h Ft(w)j Fu(decays.)316 1917 y(Our)35 b(paper)h(is)f(or)n(ganized)g(as)h(follo) n(ws:)55 b(W)-8 b(e)36 b(\256rst)g(sho)n(w)e(that)h(the)g(topological)f (entrop)o(y)h(per)h(unit)94 2037 y(v)n(olume)29 b(is)h(de\256ned)g(and)f(is)h (bounded)f(by)g(the)h(e)o(xpansion)e(rate.)51 b(W)-8 b(e)30 b(then)g(sho)n(w)e(the)i(dissipati)n(v)o(e)d(bound)94 2156 y(mentioned)d(abo)o(v)o(e,)f(and)i(\256nally)f(sho)n(w)g(ho)n(w)f(it)i(can)g (be)f(used)h(to)f(pro)o(v)o(e)f(the)i(sampling)e(bound.)35 b(From)24 b(this,)94 2276 y(an)h(estimate)f(of)h(the)g(topological)e(entrop)o (y)h(per)i(unit)d(v)n(olume)h(can)h(be)g(deri)n(v)o(ed.)94 2462 y Fs(Ac)m(kno)m(wledgmen)m(ts.)40 b Fu(This)28 b(w)o(ork)g(w)o(as)h (partially)f(supported)f(by)i(the)f(F)o(onds)g(National)g(Suisse.)47 b(Our)94 2582 y(collaboration)26 b(w)o(as)h(also)f(made)g(possible)g(through) f(the)i(pleasant)f(atmosphere)g(at)h(the)f(IHES,)h(Bures-sur)n(-)94 2701 y(Yv)o(ette.)94 3060 y Fv(2.)46 b(Existence)g(of)f(the)g(T)-11 b(op)t(ological)46 b(En)l(trop)l(y)g(p)t(er)e(Unit)i(V)-11 b(olume)94 3259 y Fu(W)j(e)34 b(start)f(by)g(pro)o(ving)f(e)o(xistence)g(of)i (the)f(topological)e(entrop)o(y)i(per)h(unit)e(v)n(olume.)60 b(This)33 b(is)g(some)n(what)94 3379 y(similar)e(to)h(the)g(standard)g(proof) g(of)g(e)o(xistence)f(of)h(topological)f(entrop)o(y)g(\(see)i([AKM]\).)f(W)-8 b(e)33 b(de\256ne)f(the)94 3498 y(global)24 b(attractor)h Fk(A)g Fu(by)1462 3644 y Fk(A)p Fr(\()p Ft(B)5 b(;)17 b(\034)11 b Fr(\))42 b(=)1980 3550 y Fq(\\)1962 3764 y Fz(n)p Fo(\025)p FA(0)2125 3644 y Fr(\010)2197 3671 y Fz(n\034)2296 3644 y Fr(\()p Ft(B)5 b Fr(\))24 b Ft(:)1162 b Fr(\()p Fu(2)p Ft(:)p Fu(1)p Fr(\))94 3949 y Fu(Here,)35 b Ft(B)h Fu(is)31 b(a)h(ball)g(in)f(L)985 3906 y Fo(1)1101 3949 y Fu(and)h Fr(\010)1349 3976 y Fz(t)1415 3949 y Fu(is)g(the)f(semi-\257o)n(w)g(de\256ned)h(by)g(the)f(e)n(v)n(olution) f(Eq.\(1.1\).)56 b(It)32 b(can)g(be)94 4069 y(sho)n(wn)22 b(\(see)i([MS]\))g (that)e Fk(A)p Fr(\()p Ft(B)5 b(;)17 b(\034)11 b Fr(\))21 b Fu(is)i(in)l(v)n(ariant)f(and)h(that)g(it)f(does)h(not)g(depend)f(on)h(the)g (initial)f(ball)h Ft(B)k Fu(\(if)d(it)94 4188 y(is)e(lar)n(ge)g(enough\))g (nor)g(on)f(the)h(\(lar)n(ge)h(enough\))e(time)g Ft(\034)39 b Fn(\025)28 b Ft(\034)2203 4215 y FA(0)2242 4188 y Fr(\()p Ft(B)5 b Fr(\))p Fu(.)34 b(Thus,)22 b(we)g FB(de\256ne)g Fk(A)28 b Fr(=)g Fk(A)3392 4108 y Fq(\000)3437 4188 y Ft(B)5 b(;)17 b(\034)3607 4215 y FA(0)3645 4188 y Fr(\()p Ft(B)5 b Fr(\))3804 4108 y Fq(\001)3849 4188 y Fu(.)316 4324 y(F)o(or)20 b(an)o(y)g Ft(")28 b(>)g Fu(0)20 b(and)h(an)o(y)e(bounded)h(set)g Ft(Q)g Fu(in)g Fs(R)1990 4281 y Fz(d)2036 4324 y Fu(,)i(whose)d(boundary)h(has)g (Lebesgue)h(measure)f(0,)h(we)94 4444 y(de\256ne)27 b Fn(W)470 4401 y Fz(")462 4471 y(Q)557 4444 y Fu(as)g(the)f(set)g(of)g(all)h(\256nite)f (co)o(v)o(erings)f(of)h Fn(A)g Fu(by)g(open)g(sets)g(in)g(L)2732 4401 y Fo(1)2817 4444 y Fr(\()p Ft(Q)p Fr(\))f Fu(of)i(diameter)f(at)g(most)g Ft(")p Fu(.)94 4563 y(Note)i(that)f(by)g(the)g(compactness)g(of)g Fk(A)p Fn(j)1521 4590 y Fz(Q)1589 4563 y Fu(,)h(which)f(follo)n(ws)f(from)h (the)g(uniform)g(bounds)f(on)h(the)h(gradient)94 4683 y(by)d(Ascoli)f([MS],)h (such)g(\256nite)f(co)o(v)o(erings)g(e)o(xist.)316 4803 y(Let)f Ft(\034)38 b(>)28 b Fu(0)23 b(be)g(a)g(\256x)o(ed)g(time)f(step,)h(and)g(let) f Ft(T)42 b Fr(=)28 b Ft(n\034)34 b Fu(with)22 b Ft(n)28 b Fn(2)g Fs(Z)p Fu(.)35 b(F)o(or)23 b Fn(U)38 b(2)28 b(W)3150 4760 y Fz(")3142 4829 y(Q)3210 4803 y Fu(,)23 b(we)g(say)g(that)g(tw)o(o)94 4922 y(trajectories)28 b Ft(A)640 4949 y FA(1)707 4922 y Fu(and)g Ft(A)954 4949 y FA(2)1021 4922 y Fu(in)g Fn(A)f Fu(are)h Fn(U)10 b Fu(-dif)n(ferent)28 b(before)g(time)f Ft(T)42 b Fu(if)28 b(there)g(is)f(at)h(least)f(one)h Ft(j)6 b Fu(,)28 b(0)j Fn(\024)i Ft(j)k Fn(\024)32 b Ft(n)94 5042 y Fu(for)26 b(which)e(the)h(points)e Fr(\010)991 5069 y Fz(j)t(\034)1078 5042 y Fr(\()p Ft(A)1192 5069 y FA(1)1231 5042 y Fr(\))i Fu(and)g Fr(\010)1536 5069 y Fz(j)t(\034)1622 5042 y Fr(\()p Ft(A)1736 5069 y FA(2)1776 5042 y Fr(\))f Fu(do)h(not)f(belong)g(to)h(the)f(same)h(element)f(of)h Fn(U)10 b Fu(.)36 b(Let)1806 5291 y Ft(N)1886 5318 y Fz(T)6 b(;\034)2013 5291 y Fr(\()p Fn(U)k Fr(\))p eop %%Page: 6 6 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1722 b Fl(6)94 194 y Fu(be)34 b(the)f(lar)n(gest)f(number)h(of)g(trajectories)g (which)f(are)i(pairwise)f Fn(U)10 b Fu(-dif)n(ferent)33 b(before)g(time)g Ft(T)47 b Fu(\(and)33 b(con-)94 314 y(sidered)39 b(with)f(time-step)g Ft(\034)11 b Fu(.\))78 b(Note)38 b(that)h(this)e(number)i(is)f(\256nite)h (since)f(it)h(is)f(at)h(most)f Fr(\()p Fu(Card)h Fn(U)10 b Fr(\))3703 271 y Fz(T)f(=\034)3849 314 y Fu(.)94 433 y(Let)1405 553 y Ft(N)1485 580 y Fz(Q;T)d(;\034)t(;")1783 553 y Fr(=)95 b Fu(inf)1889 617 y Fo(U)7 b(2W)2085 593 y Fd(")2079 643 y(Q)2154 553 y Ft(N)2234 580 y Fz(T)f(;\034)2361 553 y Fr(\()p Fn(U)k Fr(\))25 b Ft(:)94 915 y Fs(Lemma)53 b(2.1.)80 b Fc(Let)39 b Ft(Q)1024 942 y FA(1)1103 915 y Fc(and)h Ft(Q)1366 942 y FA(2)1445 915 y Fc(be)f(tw)o(o)g(bounded)g(domains)f(with)h(boundary)g(of)g (zero)h(Lebesgue)94 1035 y(measure)26 b(and)f Ft(Q)700 1062 y FA(1)763 1035 y Fn(\\)d Ft(Q)930 1062 y FA(2)996 1035 y Fc(of)j(zero)h (Lebesgue)f(measure.)38 b(The)26 b(functions)e Ft(N)2760 1062 y Fz(Q;T)6 b(;\034)t(;")3056 1035 y Fc(satisfy)25 b(the)g(follo)n(wing)94 1154 y(bounds:)150 1274 y(i\))50 b Ft(N)341 1301 y Fz(Q;T)6 b(;\034)t(;")636 1274 y Fc(is)24 b(non-increasing)g(in)h Ft(")p Fc(.)122 1394 y(ii\))50 b Ft(N)341 1420 y Fz(Q;T)475 1438 y Fe(1)504 1420 y Fp(+)p Fz(T)612 1438 y Fe(2)642 1420 y Fz(;\034)t(;")800 1394 y Fn(\024)28 b Ft(N)985 1420 y Fz(Q;T)1119 1438 y Fe(1)1149 1420 y Fz(;\034)t(;")1304 1394 y Ft(N)1384 1420 y Fz(Q;T)1518 1438 y Fe(2)1548 1420 y Fz(;\034)t(;")1678 1394 y Fc(.)94 1513 y(iii\))50 b Ft(N)341 1540 y Fz(Q)404 1558 y Fe(1)433 1540 y Fo([)p Fz(Q)550 1558 y Fe(2)580 1540 y Fz(;T)6 b(;\034)t(;")815 1513 y Fn(\024)28 b Ft(N)1000 1540 y Fz(Q)1063 1558 y Fe(1)1093 1540 y Fz(;T)6 b(;\034)t(;")1325 1513 y Ft(N)1405 1540 y Fz(Q)1468 1558 y Fe(2)1498 1540 y Fz(;T)g(;\034)t(;")1705 1513 y Fc(.)94 1734 y Fs(Pro)s(of.)38 b Fu(i\))24 b(follo)n(ws)e(since)i Fn(W)1181 1691 y Fz(")1173 1761 y(Q)1265 1734 y Fu(is)f(non-decreasing)h(in)f Ft(")p Fu(.)36 b(F)o(or)24 b(a)g(gi)n(v)o(en)e Fn(U)38 b(2)28 b(W)2966 1691 y Fz(")2958 1761 y(Q)3027 1734 y Fu(,)c(we)g(ha)n(v)o(e)f (easily)g(from)94 1853 y(the)i(de\256nition)1223 1973 y Ft(N)1303 2000 y Fz(T)1350 2018 y Fe(1)1380 2000 y Fp(+)p Fz(T)1488 2018 y Fe(2)1518 2000 y Fz(;\034)1592 1973 y Fr(\()p Fn(U)10 b Fr(\))44 b Fn(\024)g Ft(N)1987 2000 y Fz(T)2034 2018 y Fe(1)2064 2000 y Fz(;\034)2138 1973 y Fr(\()p Fn(U)10 b Fr(\))25 b Ft(N)2393 2000 y Fz(T)2440 2018 y Fe(2)2470 2000 y Fz(;\034)2543 1973 y Fr(\()p Fn(U)10 b Fr(\))25 b Ft(:)94 2144 y Fu(Indeed,)k(if)f(we)g (consider)g(a)g(maximal)f(collection)g(of)h(trajectories)g Ft(A)2545 2171 y FA(1)2601 2144 y Ft(;)17 b(A)2721 2171 y FA(2)2760 2144 y Ft(;)33 b Fn(\001)17 b(\001)g(\001)d Ft(;)33 b(A)3089 2171 y Fz(N)3152 2189 y Fd(T)3193 2205 y Fe(1)3223 2189 y Fb(+)p Fd(T)3315 2205 y Fe(2)3346 2189 y Fd(;\034)3413 2171 y Fp(\()p Fo(U)7 b Fp(\))3538 2144 y Fu(,)29 b(we)f(can)94 2283 y(collect)d(with)g Ft(A)663 2310 y FA(1)727 2283 y Fu(all)g(the)g(remaining)f Ft(A)1503 2310 y Fz(j)1570 2283 y Fu(\()p Ft(j)34 b(>)29 b Fu(1\))c(whose)g(step)f Ft(\034)36 b Fu(orbit)25 b Fn(U)35 b Fu(coincides)25 b(with)f(that)g(of)i Ft(A)3710 2310 y FA(1)3774 2283 y Fu(up)94 2403 y(to)f Ft(T)255 2430 y FA(1)295 2403 y Fu(.)38 b(W)-8 b(e)25 b(kno)n(w)g(that)f(since)h(these)h(orbits)e(should)g (dif)n(fer)h(between)g Ft(T)2595 2430 y FA(1)2661 2403 y Fu(and)g Ft(T)2888 2430 y FA(2)2953 2403 y Fu(their)g(number)g(is)g(at)g(most)94 2522 y Ft(N)174 2549 y Fz(T)221 2567 y Fe(2)251 2549 y Fz(;\034)325 2522 y Fr(\()p Fn(U)10 b Fr(\))p Fu(.)35 b(W)-8 b(e)23 b(continue)f(with)h (the)f(remaining)h(orbits)f(and)h(this)f(leads)g(to)h(at)g(most)f Ft(N)3131 2549 y Fz(T)3178 2567 y Fe(1)3208 2549 y Fz(;\034)3281 2522 y Fr(\()p Fn(U)10 b Fr(\))23 b Fu(groups.)34 b(ii\))94 2642 y(no)n(w)24 b(follo)n(ws)g(immediately)-6 b(.)316 2762 y(In)26 b(order)h(to)f(pro)o(v)o(e)f(iii\),)h(we)g(consider)g Fn(U)1752 2788 y FA(1)1822 2762 y Fn(2)k(W)2024 2719 y Fz(")2016 2788 y(Q)2079 2806 y Fe(1)2141 2762 y Fu(and)c Fn(U)2373 2788 y FA(2)2443 2762 y Fn(2)k(W)2645 2719 y Fz(")2637 2788 y(Q)2700 2806 y Fe(2)2735 2762 y Fu(.)40 b(Since)27 b(we)f(are)h(using)f(the)g(L)3790 2719 y Fo(1)94 2881 y Fu(norm,)f(an)g(ar)n(gument)f(similar)g(to)g(the)h(one) f(abo)o(v)o(e)g(leads)h(to)1171 3079 y Ft(N)1251 3106 y Fz(T)6 b(;\034)1377 3079 y Fr(\()p Fn(U)1478 3106 y FA(1)1541 3079 y Fn(\002)22 b(U)1702 3106 y FA(2)1742 3079 y Fr(\))44 b Fn(\024)h Ft(N)2027 3106 y Fz(T)6 b(;\034)2154 3079 y Fr(\()p Fn(U)2255 3106 y FA(1)2295 3079 y Fr(\))25 b Ft(N)2439 3106 y Fz(T)6 b(;\034)2566 3079 y Fr(\()p Fn(U)2667 3106 y FA(2)2707 3079 y Fr(\))24 b Ft(:)94 3277 y Fu(W)-8 b(e)26 b(also)e(ha)n(v)o(e)h(easily)f (\(again)g(because)i(we)f(are)g(using)f(the)h(L)2252 3234 y Fo(1)2361 3277 y Fu(norm\))1463 3475 y Fn(W)1569 3432 y Fz(")1561 3502 y(Q)1624 3520 y Fe(1)1682 3475 y Fn(\002)d(W)1887 3432 y Fz(")1879 3502 y(Q)1942 3520 y Fe(2)2005 3475 y Fn(\032)28 b(W)2216 3432 y Fz(")2208 3502 y(Q)2271 3520 y Fe(1)2301 3502 y Fo([)p Fz(Q)2418 3520 y Fe(2)2477 3475 y Ft(:)94 3673 y Fu(The)d(claim)g (iii\))f(no)n(w)g(follo)n(ws)f(easily)-6 b(.)94 3843 y Fs(De\014nition.)35 b Fu(Henceforth,)25 b(we)f(shall)g(w)o(ork)g(with)f(domains)g(which)g(are)i (cubes,)f(and)g(we)h(denote)f Ft(Q)3669 3870 y Fz(L)3752 3843 y Fu(the)94 3963 y(cube)h(of)g(side)g Ft(L)f Fu(centered)i(at)f(the)f (origin.)94 4154 y Fs(Theorem)38 b(2.2.)e Fc(The)25 b(follo)n(wing)e(limit)g (e)o(xists)1123 4399 y Ft(h)45 b Fr(=)54 b Fu(lim)1347 4462 y Fz(")p Fo(!)p FA(0)1555 4399 y Fu(lim)1515 4463 y Fz(L)p Fo(!1)1789 4332 y Fu(1)p 1757 4376 115 4 v 1757 4467 a Ft(L)1825 4438 y Fz(d)1942 4399 y Fu(lim)1900 4463 y Fz(T)11 b Fo(!1)2156 4332 y Fu(1)p 2145 4376 73 4 v 2145 4467 a Ft(T)2246 4399 y Fu(log)16 b Ft(N)2470 4426 y Fz(Q)2533 4444 y Fd(L)2585 4426 y Fz(;T)6 b(;\034)t(;")2818 4399 y Ft(:)823 b Fr(\()p Fu(2)p Ft(:)p Fu(2)p Fr(\))94 4631 y Fc(Moreo)o(v)o(er)l(,)29 b Ft(h)h Fc(does)e(not)g(depend)h(on)f Ft(\034)11 b Fc(.)48 b(It)29 b(is)f(called)h(the)g(topological)e(entrop)o(y)h(per)i(unit)e(v)n(olume)f(of) i(the)94 4750 y(system.)94 4971 y Fs(Pro)s(of.)38 b Fu(From)23 b(ii\))g(of)h(the)f(Lemma)g(2.1,)g(and)h(the)f(usual)g(subadditi)n(vity)d(ar) n(gument,)j(we)h(conclude)f(that)g(the)94 5091 y(follo)n(wing)g(limit)h(e)o (xists)1330 5257 y Ft(h)1387 5205 y Fp(\()p FA(1)p Fp(\))1387 5286 y Fz(Q;\034)t(;")1624 5257 y Fr(=)87 b Fu(lim)1746 5321 y Fz(T)11 b Fo(!1)2003 5190 y Fu(1)p 1991 5234 V 1991 5325 a Ft(T)2092 5257 y Fu(log)16 b Ft(N)2316 5284 y Fz(Q;T)6 b(;\034)t(;")2611 5257 y Ft(:)p eop %%Page: 7 7 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1722 b Fl(7)94 194 y Fu(Moreo)o(v)o(er)l(,)24 b(from)h(i\))g(it)f(is)g (non-increasing)g(in)h Ft(")g Fu(and)g(from)f(iii\))g(it)h(satis\256es)1320 386 y Ft(h)1377 334 y Fp(\()p FA(1)p Fp(\))1377 415 y Fz(Q)1440 433 y Fe(1)1470 415 y Fo([)p Fz(Q)1587 433 y Fe(2)1617 415 y Fz(;\034)t(;")1791 386 y Fn(\024)45 b Ft(h)1970 334 y Fp(\()p FA(1)p Fp(\))1970 415 y Fz(Q)2033 433 y Fe(1)2063 415 y Fz(;\034)t(;")2216 386 y Fr(+)22 b Ft(h)2372 334 y Fp(\()p FA(1)p Fp(\))2372 415 y Fz(Q)2435 433 y Fe(2)2465 415 y Fz(;\034)t(;")2621 386 y Ft(:)94 562 y Fu(Therefore)k(the)f(follo)n(wing)e(limit)g(e)o(xists)1474 784 y Ft(h)1531 741 y Fp(\()p FA(2)p Fp(\))1531 811 y Fz(\034)t(;")1666 784 y Fr(=)68 b Fu(lim)1771 848 y Fz(L)p Fo(!1)2045 717 y Fu(1)p 2013 761 115 4 v 2013 852 a Ft(L)2081 823 y Fz(d)2139 784 y Ft(h)2196 732 y Fp(\()p FA(1)p Fp(\))2196 813 y Fz(Q)2259 831 y Fd(L)2312 813 y Fz(;\034)t(;")2467 784 y Ft(:)94 1019 y Fu(Moreo)o(v)o(er)l (,)24 b Ft(h)587 977 y Fp(\()p FA(2)p Fp(\))587 1046 y Fz(\034)t(;")719 1019 y Fu(is)h(non-increasing)f(in)g Ft(")p Fu(.)36 b(This)24 b(pro)o(v)o(es)g(that)g(the)h(limit)e(Eq.\(2.2\))h(e)o(xists.)316 1139 y(W)-8 b(e)38 b(no)n(w)f(sho)n(w)g(that)h(it)f(is)h(independent)f(of)h Ft(\034)11 b Fu(.)75 b(As)38 b(in)g(the)g(treatment)f(of)h(topological)f (entrop)o(y)94 1258 y(for)e(\256nite)g(dimensional)e(systems,)j(we)e(start)h (by)f(gi)n(ving)f(an)i(equi)n(v)n(alent)e(de\256nition.)64 b(Gi)n(v)o(en)33 b(a)i(positi)n(v)o(e)94 1378 y(number)22 b Ft(\020)7 b Fu(,)23 b(we)g(say)f(that)g(tw)o(o)g(trajectories)g Ft(A)1697 1405 y FA(1)1759 1378 y Fu(and)g Ft(A)2000 1405 y FA(2)2062 1378 y Fu(in)g Fn(A)g Fu(are)i Ft(\020)7 b Fu(-separated)22 b(in)g Ft(Q)h Fu(before)g(time)e Ft(T)37 b Fu(\(with)94 1498 y(time-steps)24 b Ft(\034)11 b Fu(\))25 b(if)g(there)g(e)o(xists)e(an)i(inte) o(ger)f Ft(j)33 b Fn(2)c Fu([0)p Ft(;)17 b(T)11 b(=\034)g Fu(])24 b(for)h(which)1265 1673 y Fn(k)p Fr(\010)1387 1700 y Fz(j)t(\034)1474 1673 y Fr(\()p Ft(A)1588 1700 y FA(1)1627 1673 y Fr(\))d Fn(\000)h Fr(\010)1860 1700 y Fz(j)t(\034)1946 1673 y Fr(\()p Ft(A)2060 1700 y FA(2)2100 1673 y Fr(\))p Fn(k)2189 1700 y FA(L)2232 1680 y Ff(1)2304 1700 y Fp(\()p Fz(Q)p Fp(\))2478 1673 y Fn(\025)45 b Ft(\020)32 b(:)94 1849 y Fu(W)-8 b(e)28 b(denote)e(by)h Ft(R)747 1876 y Fz(Q;T)6 b(;\034)960 1849 y Fr(\()p Ft(\020)h Fr(\))26 b Fu(the)h(maximum)e(of)i(the)g(cardinalities)e(of)i(the)g(sets)f(of)h (trajectories)g(which)f(are)94 1996 y(pairwise)g Ft(\020)7 b Fu(-dif)n(ferent)25 b(before)h(time)f Ft(T)14 b Fu(.)38 b(Since)26 b(the)g(co)o(v)o(erings)e(in)h Fn(W)2540 1948 y Fz(\020)2532 2025 y(Q)2626 1996 y Fu(are)h(of)g(diameter)f(less)g(than)g Ft(\020)7 b Fu(,)26 b(one)94 2115 y(has)1452 2235 y Ft(R)1528 2262 y Fz(Q;T)6 b(;\034)1741 2235 y Fr(\()p Ft(\020)h Fr(\))43 b Fn(\024)i Ft(N)2115 2262 y Fz(Q;T)6 b(;\034)t(;\020)f(=)p FA(2)2489 2235 y Ft(:)94 2396 y Fu(Let)31 b(no)n(w)e Fn(U)41 b Fu(be)31 b(a)f(\256nite)h(co)o(v)o(ering)e(of)h Fn(A)g Fu(by)h(balls)e(of)i (radius)f Ft(\020)37 b Fu(in)30 b(L)2581 2353 y Fo(1)2666 2396 y Fr(\()p Ft(Q)p Fr(\))p Fu(.)51 b(Then,)32 b(if)e(tw)o(o)g(trajectories)94 2515 y(dif)n(fer)25 b(on)g Fn(U)35 b Fu(at)25 b(some)f(time,)g(their)g (distance)h(is)f(at)h(least)f Ft(\020)7 b Fu(.)36 b(Therefore)1472 2691 y Ft(N)1552 2718 y Fz(Q;T)6 b(;\034)t(;)p FA(2)p Fz(\020)1905 2691 y Fn(\024)44 b Ft(R)2102 2718 y Fz(Q;T)6 b(;\034)2315 2691 y Fr(\()p Ft(\020)h Fr(\))25 b Ft(:)94 2867 y Fu(These)g(tw)o(o)g (estimates)e(imply)h(immediately)f(that)1025 3113 y(lim)1015 3177 y Fz(")p Fo(!)p FA(0)1184 3113 y Ft(h)1241 3070 y Fp(\()p FA(2)p Fp(\))1241 3140 y Fz(\034)t(;")1376 3113 y Fr(=)37 b Fu(lim)1481 3177 y Fz(")p Fo(!)p FA(0)1690 3113 y Fu(lim)1649 3177 y Fz(L)p Fo(!1)1924 3046 y Fu(1)p 1891 3090 V 1891 3181 a Ft(L)1959 3153 y Fz(d)2076 3113 y Fu(lim)2034 3177 y Fz(T)11 b Fo(!1)2279 3029 y Fu(log)16 b Ft(R)2499 3056 y Fz(Q)2562 3074 y Fd(L)2614 3056 y Fz(;T)6 b(;\034)2765 3029 y Fr(\()p Ft(")p Fr(\))p 2279 3090 610 4 v 2548 3181 a Ft(T)2926 3113 y(:)715 b Fr(\()p Fu(2)p Ft(:)p Fu(3)p Fr(\))94 3480 y Fs(Lemma)50 b(2.3.)72 b Fc(There)37 b(are)h(numbers)e Ft(\015)49 b(>)c Fu(0)p Fc(,)39 b Fr(\000)45 b Ft(>)g Fu(1)36 b Fc(and)h Ft(C)52 b(<)44 b Fn(1)37 b Fc(such)f(that)h(for)g(all)f Ft(L)p Fc(,)k Ft(")k(>)h Fu(0)94 3600 y Fc(satisfying)22 b Ft(L)28 b(>)g(C)7 b(")830 3557 y Fo(\000)p FA(1)956 3600 y Fc(and)24 b(for)f(all)g Ft(A)1461 3627 y FA(1)1524 3600 y Fc(and)h Ft(A)1767 3627 y FA(2)1830 3600 y Fc(in)f Fn(A)g Fc(for)g(which)g Fn(k)p Ft(A)2565 3627 y FA(1)2624 3600 y Fn(\000)c Ft(A)2795 3627 y FA(2)2834 3600 y Fn(k)2884 3627 y FA(L)2927 3607 y Ff(1)3000 3627 y Fp(\()p Fz(Q)3094 3645 y Fd(L)3146 3627 y Fp(\))3210 3600 y Fn(\024)28 b Ft(")p Fc(,)c(we)g(ha)n(v)o(e)f(for)94 3744 y(an)o(y)i Fu(0)i Fn(\024)h Ft(t)g Fn(\024)g Ft(C)691 3702 y Fo(\000)p FA(1)794 3744 y Ft("L)22 b Fn(\000)h Fu(1)p Fc(,)1007 3920 y Fn(k)p Fr(\010)1129 3947 y Fz(t)1164 3920 y Fr(\()p Ft(A)1278 3947 y FA(1)1317 3920 y Fr(\))f Fn(\000)g Fr(\010)1549 3947 y Fz(t)1584 3920 y Fr(\()p Ft(A)1698 3947 y FA(2)1738 3920 y Fr(\))p Fn(k)1827 3947 y FA(L)1870 3927 y Ff(1)1942 3947 y Fp(\()p Fz(Q)2036 3971 y Fd(L)p Ff(\000)p Fd(C)s(")2225 3956 y Ff(\000)p Fe(1)2308 3971 y Fb(\()p Fd(t)p Fb(+)p Fe(1)p Fb(\))2471 3947 y Fp(\))2551 3920 y Fn(\024)45 b Fr(\000)p Ft(e)2781 3877 y Fz(\015)t(t)2863 3920 y Ft(")25 b(;)94 4118 y Fc(with)g Fr(\000)i Ft(>)i Fu(1)p Fc(.)94 4329 y Fs(Pro)s(of.)40 b Fu(See)26 b(Theorem)e(5.1,)g(Eq.\(5.8\))h (belo)n(w)-6 b(.)34 b(In)25 b(f)o(act,)g(the)g(statement)f(w)o(as)h(already)g (sho)n(wn)e(in)i([CE2].)316 4448 y(Using)f(the)g(abo)o(v)o(e)g(estimate)g (one)h(has)g(easily)f(for)h Ft(\034)2122 4405 y Fo(0)2178 4448 y Ft(<)j(\034)11 b Fu(,)1016 4624 y Ft(R)1092 4651 y Fz(Q)1155 4669 y Fd(L)1207 4651 y Fz(;T)6 b(;\034)1353 4631 y Ff(0)1384 4624 y Fr(\()p Ft(")p Fr(\))44 b Fn(\024)h Ft(R)1750 4651 y Fz(Q)1813 4675 y Fd(L)p Fb(+)p Fd(C)s(")1999 4660 y Ff(\000)p Fe(1)2082 4675 y Fb(\()p Fd(\034)6 b Fb(+)p Fe(1)p Fb(\))2257 4651 y Fz(;T)g(;\034)2407 4624 y Fr(\()p Ft(")p Fr(\000)2554 4581 y Fo(\000)p FA(1)2657 4624 y Ft(e)2703 4581 y Fo(\000)p Fz(\015)t(\034)2861 4624 y Fr(\))25 b Ft(;)94 4821 y Fu(and)1016 4940 y Ft(R)1092 4967 y Fz(Q)1155 4985 y Fd(L)1207 4967 y Fz(;T)6 b(;\034)1357 4940 y Fr(\()p Ft(")p Fr(\))45 b Fn(\024)f Ft(R)1723 4967 y Fz(Q)1786 4991 y Fd(L)p Fb(+)p Fd(C)s(")1972 4976 y Ff(\000)p Fe(1)2055 4991 y Fb(\()p Fd(\034)6 b Fb(+)p Fe(1)p Fb(\))2230 4967 y Fz(;T)g(;\034)2376 4947 y Ff(0)2407 4940 y Fr(\()p Ft(")p Fr(\000)2554 4897 y Fo(\000)p FA(1)2657 4940 y Ft(e)2703 4897 y Fo(\000)p Fz(\015)t(\034)2861 4940 y Fr(\))25 b Ft(:)94 5150 y Fu(It)18 b(follo)n(ws)d(no)n(w)i(immediately)e(from)i (Eq.\(2.3\))g(that)f(lim)2042 5177 y Fz(")p Fo(!)p FA(0)2216 5150 y Ft(h)2273 5107 y Fp(\()p FA(2)p Fp(\))2273 5177 y Fz(";\034)2402 5150 y Fu(does)h(not)f(depend)h(on)g Ft(\034)11 b Fu(.)33 b(This)17 b(concludes)94 5270 y(the)25 b(proof)g(of)g(Theorem)f(2.2.)p eop %%Page: 8 8 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1722 b Fl(8)94 194 y Fv(3.)46 b(Upp)t(er)f(Bound)e(on)i(the)g(En)l(trop)l(y)h(p)t (er)f(Unit)g(V)-11 b(olume)94 383 y Fu(It)38 b(does)f(not)f(follo)n(w)g(from) h(what)g(w)o(as)h(said)e(in)h(the)g(pre)n(vious)f(section)h(that)g(we)g(ha)n (v)o(e)g(de\256ned)h(a)f(\256nite)94 503 y(number)-5 b(.)42 b(W)-8 b(e)27 b(no)n(w)f(gi)n(v)o(e)g(an)h(upper)g(bound.)41 b(W)-8 b(e)28 b(\256rst)f(observ)o(e)f(that)h(from)g(the)f(sub-additi)n(vity) f(ar)n(gument)94 623 y(we)h(ha)n(v)o(e)1043 768 y(lim)1001 832 y Fz(T)11 b Fo(!1)1258 701 y Fu(1)p 1247 745 73 4 v 1247 836 a Ft(T)1347 768 y Fu(log)16 b Ft(R)1567 795 y Fz(Q)1630 813 y Fd(L)1682 795 y Fz(;T)6 b(;\034)1833 768 y Fr(\()p Ft(")p Fr(\))44 b(=)h Fu(inf)2152 832 y Fz(T)2278 701 y Fu(1)p 2267 745 V 2267 836 a Ft(T)2368 768 y Fu(log)15 b Ft(N)2591 795 y Fz(Q)2654 813 y Fd(L)2707 795 y Fz(;T)6 b(;\034)t(;")2939 768 y Ft(:)94 986 y Fu(Therefore,)26 b(in)e(order)i(to)e(obtain)g(an)h(upper) g(bound,)f(we)h(can)g(\256x)g(a)g Ft(T)39 b Fu(and)25 b(v)n(ary)f Ft(L)p Fu(.)316 1105 y(Let)37 b Ft(H)569 1132 y Fz(")648 1105 y Fu(denote)f(the)h Ft(")p Fu(-entrop)o(y)g(per)g(unit)f(v)n(olume)g (de\256ned)h(in)f([KT],)h(\(see)g(also)g([CE2])g(for)g(the)94 1225 y(application)22 b(to)g(the)h(case)g(at)g(hand\).)35 b(This)22 b(means)h(that)f(we)h(\256rst)g(de\256ne)g Ft(N)2694 1252 y Fz(Q)2762 1225 y Fr(\()p Ft(")p Fr(\))g Fu(as)g(the)f(minimum)f(number)94 1344 y(of)29 b(balls)e(of)h(radius)g Ft(")h Fu(in)e(L)1046 1302 y Fo(1)1131 1344 y Fr(\()p Ft(Q)p Fr(\))g Fu(needed)i(to)e(co)o(v)o(er)h Fk(A)p Fn(j)2083 1371 y Fz(Q)2178 1344 y Fu(\(the)g(functions)f(on)h(the)g (attractor)g Fk(A)g Fu(restricted)94 1464 y(to)d Ft(Q)p Fu(\),)g(and)f(then) 1451 1652 y Ft(H)1534 1679 y Fz(")1621 1652 y Fr(=)85 b Fu(lim)1743 1716 y Fz(L)p Fo(!1)1985 1568 y Fu(log)16 b Ft(N)2209 1595 y Fz(Q)2272 1613 y Fd(L)2329 1568 y Fr(\()p Ft(")p Fr(\))p 1985 1629 469 4 v 2162 1720 a Ft(L)2230 1691 y Fz(d)2490 1652 y Ft(:)1151 b Fr(\()p Fu(3)p Ft(:)p Fu(1)p Fr(\))94 1870 y Fu(W)-8 b(e)26 b(de\256ne)f(the)g(upper)g(dimension)e(per)i(unit)f(v)n(olume) f Ft(d)2062 1897 y FA(up)2162 1870 y Fu(of)i Fn(A)f Fu(by)1472 2167 y Ft(d)1524 2194 y FA(up)1643 2167 y Fr(=)45 b Fu(lim)17 b(sup)1833 2252 y Fz(")p Fo(!)p FA(0)2194 2100 y Ft(H)2277 2127 y Fz(")p 2081 2144 351 4 v 2081 2236 a Fu(log)o Fr(\()p Fu(1)p Ft(=")p Fr(\))2469 2167 y Ft(:)94 2452 y Fu(Note)29 b(that)f(it)g(follo)n(ws)f(from)h([CE2])h(that)f Ft(d)1628 2479 y FA(up)1731 2452 y Fu(is)g(\256nite.)47 b(It)29 b(is)f(an)g(open)h (question)e(to)h(pro)o(v)o(e)f(that)h(the)h(limit)94 2571 y(e)o(xists.)35 b(Ho)n(we)n(v)o(er)l(,)23 b(we)i(ha)n(v)o(e)g(a)g(bound:)94 2763 y Fs(Theorem)38 b(3.1.)e Fc(The)25 b(topological)e(entrop)o(y)h(per)i (unit)e(v)n(olume)f(is)i(bounded)f(by)1729 2992 y Ft(h)62 b Fn(\024)f Ft(d)2038 3019 y FA(up)2129 2992 y Ft(\015)30 b(;)94 3221 y Fc(where)c Ft(\015)k Fc(is)24 b(the)h(constant)f(appearing)h(in)f (Lemma)g(2.3.)94 3456 y Fs(Remark.)39 b Fu(In)25 b(terms)f(of)h(the)f(v)n (ariables)h(which)f(scale)h(lik)o(e)f(the)h(system,)f(we)h(ha)n(v)o(e)f Ft(\015)33 b Fr(=)28 b Ft(M)3329 3483 y Fo(\003)3375 3456 y Fu(.)94 3633 y Fs(Pro)s(of.)41 b Fu(Let)26 b Ft(\021)34 b(>)c Fu(0)d(be)f(a)h(\256x)o(ed)f(number)-5 b(.)39 b(Let)26 b Ft(")1854 3660 y FA(0)1924 3633 y Ft(>)31 b Fu(0)26 b(be)g(small)g(enough)f(such)h (that)g(for)h(all)f Ft(")k Fn(2)g Fr(\()p Fu(0)p Ft(;)17 b(")3796 3660 y FA(0)3835 3633 y Fr(\))94 3752 y Fu(we)26 b(ha)n(v)o(e)1662 3832 y Ft(H)1745 3859 y Fz(")p 1549 3876 V 1549 3967 a Fu(log)o Fr(\()p Fu(1)p Ft(=")p Fr(\))1956 3899 y Fn(\024)45 b Ft(d)2130 3926 y FA(up)2226 3899 y Fr(+)23 b Ft(\021)29 b(:)94 4142 y Fu(F)o(or)c(a)g(\256x)o(ed)g Ft(")j Fn(2)g Fr(\()p Fu(0)p Ft(;)17 b(")896 4169 y FA(0)935 4142 y Fr(\))25 b Fu(let)g Ft(L)1192 4169 y Fz(\021)s(;")1329 4142 y Ft(>)j Fu(0)d(be)g(such)f(that)h(for)g(an)o (y)f Ft(L)j(>)i(L)2587 4169 y Fz(\021)s(;")2721 4142 y Fu(we)c(ha)n(v)o(e,) 1446 4302 y Fq(\014)1446 4362 y(\014)1446 4422 y(\014)1446 4482 y(\014)1446 4541 y(\014)1491 4393 y Fu(log)16 b Ft(N)1715 4420 y Fz(Q)1778 4438 y Fd(L)1835 4393 y Fr(\()p Ft(")p Fr(\))p 1491 4454 469 4 v 1668 4545 a Ft(L)1736 4516 y Fz(d)1993 4477 y Fn(\000)23 b Ft(H)2176 4503 y Fz(")2218 4302 y Fq(\014)2218 4362 y(\014)2218 4422 y(\014)2218 4482 y(\014)2218 4541 y(\014)2296 4477 y Fn(\024)44 b Ft(\021)29 b(:)316 4795 y Fu(W)-8 b(e)20 b(no)n(w)f(gi)n(v)o(e)f(an)i(upper)g(bound)e(on)i Ft(R)1667 4822 y Fz(Q)1730 4840 y Fd(L)1782 4822 y Fz(;T)6 b(;\034)1932 4795 y Fr(\()p Ft(\020)h Fr(\))p Fu(.)34 b(F)o(or)20 b Ft(")28 b Fn(2)g Fr(\()p Fu(0)p Ft(;)17 b(")2625 4822 y FA(0)2664 4795 y Fr(\))p Fu(,)k(choose)e(a)h(\256nite)g(co)o(v)o(ering)e Fn(U)30 b Fu(of)94 4941 y Fn(A)21 b Fu(in)g(L)355 4898 y Fo(1)439 4941 y Fr(\()p Ft(Q)557 4967 y Fz(L)p Fp(+\()p Fz(T)11 b Fp(+)p FA(1)p Fp(\))p Fz(")927 4948 y Ff(\000)p Fe(1)1015 4941 y Fr(\))21 b Fu(by)g(balls)f(of)i(radius)e Ft(")p Fr(\000)1882 4898 y Fo(\000)p FA(1)1985 4941 y Ft(e)2031 4898 y Fo(\000)p Fz(\015)t(T)2203 4941 y Ft(=)p Fu(2.)34 b(By)21 b([CE2])h(and)f(the)g(pre)n(vious)f (discussion,)94 5060 y(we)26 b(kno)n(w)d(that)i(we)g(can)g(\256nd)g(such)f(a) i(co)o(v)o(ering)d Fn(U)35 b Fu(of)25 b(cardinality)f(at)h(most)662 5289 y(e)o(xp)804 5209 y Fq(\000)850 5289 y Fr(\()p Ft(L)d Fr(+)g Ft(C)7 b(")1202 5246 y Fo(\000)p FA(1)1305 5289 y Fr(\()p Ft(T)36 b Fr(+)23 b Fu(1)p Fr(\)\))1666 5246 y Fz(d)1734 5289 y Fn(\001)e Fr(\()p Ft(\021)26 b Fr(+)d(\()p Ft(d)2088 5316 y FA(up)2184 5289 y Fr(+)g Ft(\021)t Fr(\)\()p Ft(\015)6 b(T)35 b Fr(+)22 b Fu(log)p Fr(\()p Fu(2)p Fr(\000)p Ft(")2990 5246 y Fo(\000)p FA(1)3092 5289 y Fr(\)\)\))3209 5209 y Fq(\001)3279 5289 y Ft(:)p eop %%Page: 9 9 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1722 b Fl(9)94 194 y Fu(Moreo)o(v)o(er)l(,)29 b(from)f(the)h(de\256nition)f(of)h Ft(\015)6 b Fu(,)28 b(if)h Ft(A)1707 221 y FA(1)1775 194 y Fu(and)g Ft(A)2023 221 y FA(2)2091 194 y Fu(belong)f(to)h(the)f(same)h (element)f(of)h Fn(U)39 b Fu(,)29 b(it)g(follo)n(ws)94 314 y(from)c(Lemma)f(2.3)h(that)1215 437 y(sup)1161 523 y FA(0)p Fo(\024)p Fz(t)p Fo(\024)p Fz(T)1425 437 y Fn(k)p Fr(\010)1547 464 y Fz(t)1582 437 y Fr(\()p Ft(A)1696 464 y FA(1)1735 437 y Fr(\))d Fn(\000)h Fr(\010)1968 464 y Fz(t)2003 437 y Fr(\()p Ft(A)2117 464 y FA(2)2156 437 y Fr(\))p Fn(k)2245 464 y FA(L)2288 444 y Ff(1)2360 464 y Fp(\()p Fz(Q)2454 482 y Fd(L)2506 464 y Fp(\))2587 437 y Fn(\024)44 b Ft(")26 b(:)94 697 y Fu(Therefore,)372 940 y Ft(R)448 967 y Fz(Q;T)6 b(;\034)661 940 y Fr(\()p Ft(")p Fr(\))45 b Fn(\024)f Fu(e)o(xp)1094 860 y Fq(\000)1139 940 y Fr(\()p Ft(L)22 b Fr(+)h Ft(C)7 b(")1492 898 y Fo(\000)p FA(1)1595 940 y Fr(\()p Ft(T)36 b Fr(+)22 b Fu(1)p Fr(\)\))1955 898 y Fz(d)2023 940 y Fn(\001)g Fr(\()p Ft(\021)k Fr(+)c(\()p Ft(d)2377 967 y FA(up)2474 940 y Fr(+)g Ft(\021)t Fr(\)\()p Ft(\015)6 b(T)35 b Fr(+)23 b Fu(log)o Fr(\()p Fu(2)p Fr(\000)p Ft(")3279 898 y Fo(\000)p FA(1)3382 940 y Fr(\)\)\))3499 860 y Fq(\001)3568 940 y Ft(:)94 1184 y Fu(It)i(follo)n(ws)f(that)777 1453 y(lim)737 1517 y Fz(L)p Fo(!1)1011 1386 y Fu(1)p 979 1430 115 4 v 979 1521 a Ft(L)1047 1493 y Fz(d)1121 1453 y Fu(inf)1150 1517 y Fz(T)1277 1386 y Fu(1)p 1266 1430 73 4 v 1266 1521 a Ft(T)1366 1453 y Fu(log)16 b Ft(R)1586 1480 y Fz(Q)1649 1498 y Fd(L)1701 1480 y Fz(;T)6 b(;\034)1852 1453 y Fr(\()p Ft(")p Fr(\))44 b Fn(\024)85 b Fu(lim)2142 1517 y Fz(L)p Fo(!1)2416 1386 y Fu(1)p 2384 1430 115 4 v 2384 1521 a Ft(L)2452 1493 y Fz(d)2533 1386 y Fu(1)p 2522 1430 73 4 v 2522 1521 a Ft(T)2623 1453 y Fu(log)15 b Ft(R)2842 1480 y Fz(Q)2905 1498 y Fd(L)2957 1480 y Fz(;T)6 b(;\034)3108 1453 y Fr(\()p Ft(")p Fr(\))1172 1657 y Fn(\024)45 b Ft(\021)26 b Fr(+)d(\()p Ft(d)1560 1684 y FA(up)1656 1657 y Fr(+)g Ft(\021)t Fr(\)\()p Ft(\015)k Fr(+)22 b Ft(T)2137 1614 y Fo(\000)p FA(1)2256 1657 y Fu(log)o Fr(\()p Fu(2)p Fr(\000)p Ft(")2580 1614 y Fo(\000)p FA(1)2683 1657 y Fr(\)\))i Ft(;)94 1900 y Fu(and)h(the)g(result)f(follo)n(ws)f(by)i(letting) f Ft(T)41 b Fn(!)28 b(1)c Fu(and)h(then)f Ft(\021)32 b Fn(!)27 b Fu(0.)94 2259 y Fv(4.)59 b(Lo)t(calization)h(in)e(Momen)l(tum)h(Space)f (and)g(Bounds)g(on)g(the)h(Semi-)94 2378 y(Group)94 2572 y Fu(In)34 b(this)f(section,)i(we)f(deal)g(with)e(some)h(simple)g(bounds)f(on)i (the)f(semi-group)g(generated)h(by)f Ft(D)s Fr(\001)p Fu(.)63 b(W)-8 b(e)94 2691 y(be)o(gin)31 b(by)h(constructing)f(the)g(localization)g (in)h(momentum)e(space.)57 b(Let)32 b Ft(\037)38 b Fn(\025)g Fu(0)32 b(be)g(a)h(smooth)d(function)94 2811 y(with)d(support)g(in)g Fn(j)p Ft(k)s Fn(j)k(\024)h Fu(2)c(and)g(which)f(is)g(equal)h(to)f(1)h(for)g Fn(j)p Ft(k)s Fn(j)j(\024)h Fu(1.)44 b(W)-8 b(e)28 b(also)f(assume)g Ft(\037)32 b Fn(\024)g Fu(1.)45 b(W)-8 b(e)28 b(shall)94 2930 y(denote)d Ft(\037)447 2957 y Fz(k)r(k)1522 2975 y Ff(\003)1597 2930 y Fr(=)k Fu(1)22 b Fn(\000)g Ft(\037)p Fr(\()p Ft(k)s(=k)2132 2957 y Fo(\003)2178 2930 y Fr(\))p Fu(.)36 b(W)-8 b(e)25 b(de\256ne)h(the)e(con)l(v)n(olution)f(operators)i Fk(G)3799 2957 y Fz(\034)3849 2930 y Fu(,)94 3050 y Fk(G)154 3077 y Fz(\034)t(;>)311 3050 y Fu(and)g Fk(G)540 3077 y Fz(\034)t(;<)696 3050 y Fu(by)883 3360 y Fk(G)943 3387 y Fz(\034)993 3360 y Fr(\()p Ft(x)p Fr(\))43 b(=)1293 3224 y Fq(Z)1409 3360 y Fu(d)p Ft(k)20 b(e)1577 3317 y Fz(ik)r(x)p Fo(\000)p Fz(D)r(k)1867 3274 y Fe(2)1898 3317 y Fz(\034)1973 3360 y Ft(;)801 3621 y Fk(G)861 3648 y Fz(\034)t(;>)993 3621 y Fr(\()p Ft(x)p Fr(\))43 b(=)1293 3485 y Fq(Z)1409 3621 y Fu(d)p Ft(k)20 b(e)1577 3578 y Fz(ik)r(x)1699 3621 y Ft(e)1745 3578 y Fo(\000)p Fz(D)r(k)1918 3535 y Fe(2)1949 3578 y Fz(\034)1999 3621 y Ft(\037)2061 3648 y Fz(k)r(>k)2209 3666 y Ff(\003)2282 3621 y Ft(;)801 3882 y Fk(G)861 3909 y Fz(\034)t(;<)993 3882 y Fr(\()p Ft(x)p Fr(\))43 b(=)1293 3747 y Fq(Z)1409 3882 y Fu(d)p Ft(k)20 b(e)1577 3839 y Fz(ik)r(x)1699 3882 y Ft(e)1745 3839 y Fo(\000)p Fz(D)r(k)1918 3796 y Fe(2)1949 3839 y Fz(\034)1999 3882 y Ft(\037)2061 3909 y Fz(k)r()2981 3882 y Fr(\()p Ft(x)p Fr(\))h Ft(:)94 4239 y Fs(Notation.)35 b Fu(The)23 b(constants)e Ft(B)1259 4197 y Fo(\003)1254 4266 y FA(0)1305 4239 y Fu(,)i Ft(B)1434 4197 y Fo(\003)1429 4266 y FA(1)1480 4239 y Ft(;)17 b(:)g(:)g(:)d(;)22 b Fu(do)h(only)f(depend)h(on)g(the)f(quotient)g Ft(D)3040 4266 y Fo(\003)3086 4239 y Ft(=\027)3185 4266 y Fo(\003)3254 4239 y Fu(\(something)g(lik)o(e)94 4359 y(the)k(condition)d(number)i(of)h(the)f (matrix)f Ft(D)s Fu(\),)i FB(b)n(ut)f(not)g(on)g(any)g(other)g(par)o(ameter)o (s)e(of)i(the)h(pr)l(oblem,)e(e)n(xcept)94 4479 y Ft(d)p Fu(.)42 b(W)-8 b(e)27 b(also)f(recall)h(that)f(the)h(constants)e Ft(C)1598 4505 y FA(0)1638 4479 y Ft(;)17 b(:)g(:)g(:)d(;)27 b Fu(are)g(numerical)f(f)o (actors)h(which)f(do)h(not)f(depend)g(on)h(an)o(y)94 4598 y(parameters)e(of)g (the)g(problem)f(e)o(xcept)g Ft(d)p Fu(.)316 4718 y(W)-8 b(e)25 b(no)n(w)f(state)h(and)f(pro)o(v)o(e)g(v)n(arious)g(estimates)g(on)g(these)h (k)o(ernels.)94 4909 y Fs(Lemma)43 b(4.1.)51 b Fc(F)o(or)30 b(e)n(v)o(ery)f Ft(p)34 b Fn(\025)h Fu(0)30 b Fc(there)g(is)f(a)h(constant)f Ft(B)2266 4866 y Fo(\003)2261 4936 y FA(0)2311 4909 y Fr(\()p Ft(p)p Fr(\))g Fc(such)h(that)f(for)h(all)f Ft(\034)46 b(>)35 b Fu(0)29 b Fc(one)h(has)f(the)94 5029 y(bound)873 5044 y Fq(\015)873 5103 y(\015)873 5163 y(\015)873 5223 y(\015)928 5052 y(Z)1044 5188 y Fu(d)p Ft(k)20 b(e)1212 5145 y Fz(ik)r(x)1335 5188 y Ft(e)1381 5145 y Fo(\000)p Fz(D)r(k)1554 5102 y Fe(2)1584 5145 y Fz(\034)1634 5044 y Fq(\015)1634 5103 y(\015)1634 5163 y(\015)1634 5223 y(\015)1734 5188 y Fn(\024)45 b Ft(B)1937 5145 y Fo(\003)1932 5215 y FA(0)1982 5188 y Fr(\()p Ft(p)p Fr(\))2551 5121 y Fu(1)p 2122 5165 910 4 v 2122 5279 a Fr(\()p Ft(\027)2210 5306 y Fo(\003)2255 5279 y Ft(\034)11 b Fr(\))2349 5250 y Fz(d=)p FA(2)2471 5198 y Fq(\000)2517 5279 y Fu(1)22 b Fr(+)2715 5231 y Fo(j)p Fz(x)p Fo(j)2808 5211 y Fe(2)p 2700 5256 152 4 v 2700 5313 a Fz(D)2765 5331 y Ff(\003)2807 5313 y Fz(\034)2864 5198 y Fq(\001)2909 5237 y Fz(p=)p FA(2)3068 5188 y Ft(:)573 b Fr(\()p Fu(4)p Ft(:)p Fu(1)p Fr(\))p eop %%Page: 10 10 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(10)94 194 y Fc(Furthermore,)21 b(con)l(v)n(olution)d(with)h Fk(G)1385 221 y Fz(\034)1455 194 y Fc(is)g(a)h(well-de\256ned)g(operator)g (on)f(bounded)g(functions)f(and)i(has)g(norm)94 314 y(\(as)26 b(a)f(map)f(from)h Fu(L)781 271 y Fo(1)890 314 y Fc(to)f(itself\))h(bounded)f (by)1665 559 y Fn(k)p Fk(G)1775 586 y Fz(\034)1825 559 y Fn(k)1875 586 y Fo(1)2003 559 y Fn(\024)45 b Ft(B)2206 516 y Fo(\003)2201 586 y FA(1)2276 559 y Ft(;)1365 b Fr(\()p Fu(4)p Ft(:)p Fu(2)p Fr(\))94 804 y Fc(for)26 b(some)e Ft(B)552 762 y Fo(\003)547 831 y FA(1)597 804 y Fc(.)94 1052 y Fs(Pro)s(of.)44 b Fu(F)o(or)30 b(simplicity)-6 b(,)28 b(we)i(write)f(the)g(proof)h(for)g(the)f(case)h(of)g Ft(d)k Fr(=)h Fu(1,)30 b(b)n(ut)f(with)g(a)h(distinction)e(of)h(the)94 1171 y(upper)c(and)g(lo)n(wer)f(bounds)g(\()p Ft(D)1192 1198 y Fo(\003)1238 1171 y Fu(,)h(resp.)36 b Ft(\027)1564 1198 y Fo(\003)1610 1171 y Fu(\))25 b(on)g(the)f(matrix)g Ft(D)s Fu(.)37 b(F)o(or)24 b(e)o(xample)945 1470 y Ft(x)1019 1334 y Fq(Z)1135 1470 y Fu(d)p Ft(k)c(e)1303 1427 y Fz(ik)r(x)1425 1470 y Ft(e)1471 1427 y Fo(\000)p Fz(D)r(k)1644 1384 y Fe(2)1675 1427 y Fz(\034)1769 1470 y Fr(=)1891 1334 y Fq(Z)2007 1470 y Fu(d)p Ft(k)2129 1389 y Fq(\000)2174 1470 y Fn(\000)p Ft(i@)2338 1497 y Fz(k)2388 1470 y Ft(e)2434 1427 y Fz(ik)r(x)2557 1389 y Fq(\001)2603 1470 y Ft(e)2649 1427 y Fo(\000)p Fz(D)r(k)2822 1384 y Fe(2)2852 1427 y Fz(\034)1769 1731 y Fr(=)45 b Fn(\000)p Ft(i)2019 1595 y Fq(Z)2136 1731 y Fu(d)p Ft(k)19 b(e)2303 1688 y Fz(ik)r(x)2426 1731 y Ft(e)2472 1688 y Fo(\000)p Fz(D)r(k)2645 1645 y Fe(2)2676 1688 y Fz(\034)2726 1731 y Fu(2)p Ft(D)s(k)s(\034)35 b(:)94 2020 y Fu(Thus,)25 b(in)f(this)g(case,)863 2169 y Fq(\014)863 2229 y(\014)863 2289 y(\014)863 2349 y(\014)897 2178 y(Z)1013 2314 y Fu(d)p Ft(k)19 b(e)1180 2271 y Fz(ik)r(x)1303 2314 y Ft(e)1349 2271 y Fo(\000)p Fz(D)r(k)1522 2228 y Fe(2)1553 2271 y Fz(\034)1603 2169 y Fq(\014)1603 2229 y(\014)1603 2289 y(\014)1603 2349 y(\014)1680 2314 y Fn(\024)45 b(j)p Ft(x)p Fn(j)1915 2271 y Fo(\000)p FA(1)2033 2178 y Fq(Z)2149 2314 y Fu(d)p Ft(k)19 b(e)2316 2271 y Fo(\000)p Fz(D)r(k)2489 2228 y Fe(2)2520 2271 y Fz(\034)2570 2314 y Fu(2)p Ft(D)s Fn(j)p Ft(k)s Fn(j)p Ft(\034)1680 2584 y Fn(\024)45 b Ft(C)1873 2611 y FA(0)1913 2503 y Fq(\000)1971 2516 y Ft(D)2053 2543 y Fo(\003)2099 2516 y Ft(\034)p 1971 2561 184 4 v 1986 2652 a Fn(j)p Ft(x)p Fn(j)2099 2623 y FA(2)2166 2503 y Fq(\001)2211 2541 y FA(1)p Fz(=)p FA(2)2485 2516 y Fu(1)p 2339 2561 343 4 v 2339 2654 a Fr(\()p Ft(\027)2427 2681 y Fo(\003)2472 2654 y Ft(\034)11 b Fr(\))2566 2625 y FA(1)p Fz(=)p FA(2)2694 2503 y Fq(\000)2751 2516 y Ft(D)2833 2543 y Fo(\003)p 2751 2561 129 4 v 2768 2652 a Ft(\027)2817 2679 y Fo(\003)2891 2503 y Fq(\001)2937 2541 y FA(1)p Fz(=)p FA(2)3078 2584 y Ft(:)3669 2450 y Fr(\()p Fu(4)p Ft(:)p Fu(3)p Fr(\))94 2891 y Fu(Using)27 b Ft(D)442 2918 y Fo(\003)489 2891 y Ft(=\027)588 2918 y Fo(\003)665 2891 y Fn(\025)32 b Fu(1,)d(the)e(generalization)h(to)f(arbitrary)h Ft(d)g Fu(and)f Ft(p)h Fu(is)g(easy)f(and)h(is)f(left)h(to)f(the)h(reader)-5 b(.)45 b(The)94 3011 y(second)29 b(assertion)g(follo)n(ws)e(by)i(taking)f Ft(p)33 b Fr(=)h Ft(d)25 b Fr(+)g Fu(2)k(and)g(inte)o(grating.)46 b(The)29 b(reader)h(can)g(also)e(check)i(that)94 3130 y Ft(B)175 3087 y Fo(\003)170 3157 y FA(1)249 3130 y Fr(=)e Ft(C)425 3157 y FA(1)465 3130 y Fr(\()p Ft(D)586 3157 y Fo(\003)632 3130 y Ft(=\027)731 3157 y Fo(\003)777 3130 y Fr(\))816 3087 y Fz(d)p Fp(+)p FA(2)958 3130 y Fu(,)d(for)g(some)f Ft(C)1455 3157 y FA(1)1495 3130 y Fu(.)94 3322 y Fs(Lemma)39 b(4.2.)g Fc(Let)25 b Ft(\034)40 b Fn(\025)29 b Fu(0)p Fc(.)38 b(F)o(or)25 b(all)g Ft(p)k Fn(\025)g Fu(0)d Fc(there)f(is)g(a)h(constant)f Ft(B)2552 3279 y Fo(\003)2547 3349 y FA(2)2597 3322 y Fr(\()p Ft(p)p Fr(\))h Fc(such)f(that)g(for)h(all)f Ft(z)33 b Fn(2)c Fs(C)3658 3279 y Fz(d)3730 3322 y Fc(one)94 3441 y(has)c(a)g(bound)398 3596 y Fq(\015)398 3656 y(\015)398 3716 y(\015)398 3775 y(\015)453 3605 y(Z)569 3740 y Fu(d)p Ft(k)20 b(e)737 3698 y Fz(ik)r(z)855 3740 y Ft(e)901 3698 y Fo(\000)p Fz(D)r(k)1074 3655 y Fe(2)1105 3698 y Fz(\034)1155 3740 y Ft(\037)1217 3767 y Fz(k)r()28 b Fu(0)23 b Fc(and)g(all)g Ft(\034)38 b(>)28 b Fu(0)p Fc(,)94 3183 y(one)d(has)g(the)g(follo)n(wing)d(bounds:)150 3302 y(i\))50 b(When)24 b Ft(\027)572 3329 y Fo(\003)618 3302 y Ft(\034)39 b(>)28 b Fu(1)p Ft(=k)961 3260 y FA(2)958 3329 y Fo(\003)1028 3302 y Fc(one)d(has)827 3508 y Fq(\015)827 3568 y(\015)827 3628 y(\015)827 3687 y(\015)882 3517 y(Z)999 3652 y Fu(d)p Ft(k)19 b(e)1166 3610 y Fz(ik)r(x)1289 3652 y Ft(e)1335 3610 y Fo(\000)p Fz(D)r(k)1508 3567 y Fe(2)1539 3610 y Fz(\034)1588 3652 y Ft(\037)1650 3679 y Fz(k)r(>k)1798 3697 y Ff(\003)1846 3508 y Fq(\015)1846 3568 y(\015)1846 3628 y(\015)1846 3687 y(\015)1946 3652 y Fn(\024)44 b Ft(B)2148 3610 y Fo(\003)2143 3679 y FA(3)2194 3652 y Fr(\()p Ft(p)p Fr(\))2588 3585 y Ft(e)2634 3542 y Fo(\000)p Fz(\027)2736 3560 y Ff(\003)2779 3542 y Fz(k)2823 3499 y Fe(2)2821 3569 y Ff(\003)2863 3542 y Fz(\034)8 b(=)p FA(2)p 2333 3630 910 4 v 2333 3743 a Fr(\()p Ft(\027)2421 3770 y Fo(\003)2467 3743 y Ft(\034)j Fr(\))2561 3715 y Fz(d=)p FA(2)2683 3663 y Fq(\000)2729 3743 y Fu(1)21 b Fr(+)2927 3695 y Fo(j)p Fz(x)p Fo(j)3020 3675 y Fe(2)p 2912 3720 152 4 v 2912 3778 a Fz(D)2977 3796 y Ff(\003)3019 3778 y Fz(\034)3076 3663 y Fq(\001)3121 3701 y Fz(p=)p FA(2)3280 3652 y Ft(:)361 b Fr(\()p Fu(4)p Ft(:)p Fu(8)p Fr(\))122 4022 y Fc(ii\))50 b(When)24 b Ft(\027)572 4049 y Fo(\003)618 4022 y Ft(\034)39 b Fn(\024)28 b Fu(1)p Ft(=k)961 3979 y FA(2)958 4049 y Fo(\003)1028 4022 y Fc(one)d(has)340 4186 y Fq(\015)340 4245 y(\015)340 4305 y(\015)340 4365 y(\015)395 4194 y(Z)511 4330 y Fu(d)p Ft(k)20 b(e)679 4287 y Fz(ik)r(x)801 4330 y Ft(e)847 4287 y Fo(\000)p Fz(D)r(k)1020 4244 y Fe(2)1051 4287 y Fz(\034)1101 4330 y Ft(\037)1163 4357 y Fz(k)r(>k)1311 4375 y Ff(\003)1359 4186 y Fq(\015)1359 4245 y(\015)1359 4305 y(\015)1359 4365 y(\015)1458 4330 y Fn(\024)45 b Ft(B)1661 4287 y Fo(\003)1656 4357 y FA(3)1706 4330 y Fr(\()p Ft(p)p Fr(\))p Ft(e)1880 4287 y Fo(\000)p Fz(\027)1982 4305 y Ff(\003)2024 4287 y Fz(k)2068 4244 y Fe(2)2066 4314 y Ff(\003)2108 4287 y Fz(\034)8 b(=)p FA(2)1453 4639 y Fn(\001)1503 4468 y Fq( )2023 4571 y Fu(1)p 1593 4616 910 4 v 1593 4730 a Fr(\()p Ft(\027)1681 4757 y Fo(\003)1727 4730 y Ft(\034)j Fr(\))1821 4701 y Fz(d=)p FA(2)1943 4649 y Fq(\000)1988 4730 y Fu(1)22 b Fr(+)2187 4681 y Fo(j)p Fz(x)p Fo(j)2280 4661 y Fe(2)p 2172 4707 152 4 v 2172 4764 a Fz(D)2237 4782 y Ff(\003)2279 4764 y Fz(\034)2336 4649 y Fq(\001)2381 4688 y Fz(p=)p FA(2)2537 4639 y Fr(+)3023 4571 y Fu(1)p 2649 4616 799 4 v 2649 4722 a Ft(k)2704 4679 y Fo(\000)p Fz(d)2701 4736 y Fo(\003)2812 4641 y Fq(\000)2858 4722 y Fu(1)g Fr(+)h Ft(k)3085 4693 y FA(2)3082 4749 y Fo(\003)3127 4722 y Fn(j)p Ft(x)p Fn(j)3240 4693 y FA(2)3279 4641 y Fq(\001)3325 4680 y Fz(p=)p FA(2)3459 4468 y Fq(!)3563 4639 y Ft(:)3669 4509 y Fr(\()p Fu(4)p Ft(:)p Fu(9)p Fr(\))94 5167 y Fs(Pro)s(of.)40 b Fu(Assume)24 b(\256rst)h(that)g Ft(\027)1193 5194 y Fo(\003)1239 5167 y Ft(\034)39 b Fn(\024)28 b Fu(1)p Ft(=k)1582 5124 y FA(2)1579 5194 y Fo(\003)1625 5167 y Fu(.)36 b(In)25 b(that)g(case,)h Ft(e)2237 5124 y Fz(\027)2277 5142 y Ff(\003)2319 5124 y Fz(k)2363 5081 y Fe(2)2361 5151 y Ff(\003)2403 5124 y Fz(\034)2481 5167 y Fn(\024)i Ft(e)p Fu(,)e(and)f(thus)f(it)g(suf)n (\256ces)h(to)g(produce)94 5286 y(a)35 b(bound)f(without)f(e)o(xponential)g (f)o(actor)-5 b(.)64 b(W)-8 b(e)35 b(can)g(write)f Ft(\037)2248 5313 y Fz(k)r(>k)2396 5331 y Ff(\003)2485 5286 y Fr(=)41 b Fu(1)27 b Fn(\000)h Ft(\037)2847 5313 y Fz(k)r(k)1209 511 y Ff(\003)1257 322 y Fq(\015)1257 382 y(\015)1257 442 y(\015)1257 501 y(\015)1356 467 y Fn(\024)45 b Ft(eB)1605 424 y Fo(\003)1600 493 y FA(0)1651 467 y Fr(\()p Ft(p)p Fr(\))2220 399 y Fu(1)p 1791 444 911 4 v 1791 557 a Fr(\()p Ft(\027)1879 584 y Fo(\003)1924 557 y Ft(\034)11 b Fr(\))2018 529 y Fz(d=)p FA(2)2140 477 y Fq(\000)2185 557 y Fu(1)22 b Fr(+)2384 509 y Fo(j)p Fz(x)p Fo(j)2477 489 y Fe(2)p 2369 534 152 4 v 2369 592 a Fz(D)2434 610 y Ff(\003)2476 592 y Fz(\034)2532 477 y Fq(\001)2578 516 y Fz(d=)p FA(2)2712 386 y Fq(\000)2770 399 y Ft(D)2852 426 y Fo(\003)p 2770 444 129 4 v 2787 535 a Ft(\027)2836 562 y Fo(\003)2910 386 y Fq(\001)2956 424 y Fz(p=)p FA(2)1351 785 y Fr(+)g Ft(eB)1577 742 y Fo(\003)1572 812 y FA(2)1623 785 y Fr(\()p Ft(p)p Fr(\))2052 718 y Fu(1)p 1763 762 629 4 v 1763 864 a Fr(\()p Ft(k)1857 821 y Fo(\000)p FA(2)1854 878 y Fo(\003)1981 864 y Fr(+)g Ft(\027)2129 890 y Fo(\003)2175 864 y Ft(\034)11 b Fr(\))2269 835 y Fz(d=)p FA(2)2765 718 y Fu(1)p 2415 762 750 4 v 2415 795 a Fq(\000)2461 876 y Fu(1)21 b Fr(+)2753 827 y Fo(j)p Fz(x)p Fo(j)2846 808 y Fe(2)p 2644 853 341 4 v 2644 923 a Fz(k)2688 890 y Ff(\000)p Fe(2)2686 938 y Ff(\003)2772 923 y Fp(+)p Fz(D)2898 941 y Ff(\003)2940 923 y Fz(\034)2997 795 y Fq(\001)3042 834 y Fz(p=)p FA(2)3176 704 y Fq(\000)3234 718 y Ft(D)3316 745 y Fo(\003)p 3234 762 129 4 v 3250 853 a Ft(\027)3299 880 y Fo(\003)3374 704 y Fq(\001)3444 785 y Ft(;)3619 656 y Fr(\()p Fu(4)p Ft(:)p Fu(10)p Fr(\))94 1137 y Fu(from)26 b(which)e(the)h(\256rst)h(assertion)e(follo)n(ws.)36 b(In)25 b(the)g(case)h Ft(\027)2149 1164 y Fo(\003)2195 1137 y Ft(\034)39 b Fn(\025)29 b Fu(1)p Ft(=k)2539 1094 y FA(2)2536 1164 y Fo(\003)2581 1137 y Fu(,)d(we)f(inte)o(grate)g(again)f(by)h(parts)g (and)94 1256 y(get)g(to)g(bound)f(an)g(e)o(xpression)g(of)h(the)g(form)f (\(we)h(w)o(ork)g(again)f(in)h(the)f(case)i Ft(d)h Fr(=)h Fu(1)d(only\):)255 1520 y Ft(x)329 1384 y Fq(Z)445 1520 y Fu(d)p Ft(k)19 b(e)612 1477 y Fz(ik)r(x)735 1520 y Ft(e)781 1477 y Fo(\000)p Fz(D)r(k)954 1434 y Fe(2)985 1477 y Fz(\034)1035 1520 y Ft(\037)1097 1547 y Fz(k)r(>k)1245 1565 y Ff(\003)763 1781 y Fr(=)45 b Fn(\000)p Ft(i)1013 1646 y Fq(Z)1096 1781 y Fu(d)p Ft(k)19 b(e)1263 1738 y Fz(ik)r(x)1386 1781 y Ft(e)1432 1738 y Fo(\000)p Fz(D)r(k)1605 1695 y Fe(2)1636 1738 y Fz(\034)1702 1700 y Fq(\000)1748 1781 y Fu(2)p Ft(D)s(k)s Fr(\()p Fu(1)j Fn(\000)g Ft(\037)p Fr(\()p Ft(k)s(=k)2406 1808 y Fo(\003)2452 1781 y Fr(\)\))g Fn(\000)g Ft(k)2706 1738 y Fo(\000)p FA(1)2703 1808 y Fo(\003)2808 1700 y Fq(\000)2854 1781 y Ft(@)2907 1808 y Fz(k)2956 1781 y Ft(\037)3018 1700 y Fq(\001)3064 1781 y Fr(\()p Ft(k)s(=k)3260 1808 y Fo(\003)3305 1781 y Fr(\))3344 1700 y Fq(\001)3431 1781 y Ft(:)3619 1651 y Fr(\()p Fu(4)p Ft(:)p Fu(11)p Fr(\))94 2040 y Fu(Therefore,)k(we)f(get,)g (as)g(in)f(\(4.3\),)371 2168 y Fq(\014)371 2228 y(\014)371 2288 y(\014)371 2348 y(\014)404 2177 y(Z)520 2313 y Fu(d)p Ft(k)c(e)688 2270 y Fz(ik)r(x)810 2313 y Ft(e)856 2270 y Fo(\000)p Fz(D)r(k)1029 2227 y Fe(2)1060 2270 y Fz(\034)1110 2313 y Ft(\037)1172 2340 y Fz(k)r(>k)1320 2358 y Ff(\003)1368 2168 y Fq(\014)1368 2228 y(\014)1368 2288 y(\014)1368 2348 y(\014)1445 2313 y Fn(\024)45 b(j)p Ft(x)p Fn(j)1680 2270 y Fo(\000)p FA(1)1781 2313 y Fk(O)p Fr(\()p Fu(1)p Fr(\))p Ft(e)2025 2270 y Fo(\000)p Fz(\027)2127 2288 y Ff(\003)2169 2270 y Fz(k)2213 2227 y Fe(2)2243 2270 y Fz(\034)8 b(=)p FA(2)2385 2177 y Fq(Z)2502 2313 y Fu(d)p Ft(k)19 b(e)2669 2270 y Fo(\000)p Fz(D)r(k)2842 2227 y Fe(2)2873 2270 y Fz(\034)8 b(=)p FA(2)3015 2313 y Fn(j)p Fu(2)p Ft(D)s(k)s(\034)j Fn(j)1440 2583 y Fr(+)39 b Fn(j)p Ft(x)p Fn(j)1669 2540 y Fo(\000)p FA(1)1770 2583 y Fk(O)p Fr(\()p Fu(1)p Fr(\))p Ft(e)2014 2540 y Fo(\000)p Fz(\027)2116 2558 y Ff(\003)2158 2540 y Fz(k)2202 2497 y Fe(2)2232 2540 y Fz(\034)8 b(=)p FA(2)2374 2447 y Fq(Z)2491 2583 y Fu(d)p Ft(k)19 b(e)2658 2540 y Fo(\000)p Fz(D)r(k)2831 2497 y Fe(2)2862 2540 y Fz(\034)8 b(=)p FA(2)3004 2498 y Fq(\014)3004 2558 y(\014)3037 2583 y Ft(k)3092 2540 y Fo(\000)p FA(1)3089 2610 y Fo(\003)3194 2583 y Ft(\037)3256 2540 y Fo(0)3285 2583 y Fr(\()p Ft(k)s(=k)3481 2610 y Fo(\003)3526 2583 y Fr(\))3565 2498 y Fq(\014)3565 2558 y(\014)1445 2844 y Fn(\024)45 b Ft(C)7 b(e)1691 2801 y Fo(\000)p Fz(\027)1793 2819 y Ff(\003)1836 2801 y Fz(k)1880 2758 y Fe(2)1909 2801 y Fz(\034)h(=)p FA(2)2035 2763 y Fq(\000)2093 2777 y Ft(D)2175 2803 y Fo(\003)2221 2777 y Ft(\034)p 2093 2821 184 4 v 2108 2912 a Fn(j)p Ft(x)p Fn(j)2221 2883 y FA(2)2288 2763 y Fq(\001)2333 2801 y FA(1)p Fz(=)p FA(2)2607 2777 y Fu(1)p 2461 2821 343 4 v 2461 2914 a Fr(\()p Ft(\027)2549 2941 y Fo(\003)2595 2914 y Ft(\034)j Fr(\))2689 2885 y FA(1)p Fz(=)p FA(2)2816 2763 y Fq(\000)2873 2777 y Ft(D)2955 2803 y Fo(\003)p 2873 2821 129 4 v 2890 2912 a Ft(\027)2939 2939 y Fo(\003)3013 2763 y Fq(\001)3059 2801 y FA(1)p Fz(=)p FA(2)1440 3138 y Fr(+)39 b Ft(C)7 b(e)1680 3095 y Fo(\000)p Fz(\027)1782 3113 y Ff(\003)1824 3095 y Fz(k)1868 3052 y Fe(2)1898 3095 y Fz(\034)h(=)p FA(2)2182 3071 y Fu(1)p 2036 3115 343 4 v 2036 3208 a Fr(\()p Ft(\027)2124 3235 y Fo(\003)2170 3208 y Ft(\034)j Fr(\))2264 3179 y FA(1)p Fz(=)p FA(2)2403 3071 y Ft(k)2458 3028 y Fo(\000)p FA(1)2455 3097 y Fo(\003)p 2403 3115 158 4 v 2425 3206 a Fn(j)p Ft(x)p Fn(j)1445 3410 y(\024)45 b Ft(C)7 b(e)1691 3367 y Fo(\000)p Fz(\027)1793 3385 y Ff(\003)1836 3367 y Fz(k)1880 3325 y Fe(2)1909 3367 y Fz(\034)h(=)p FA(2)2035 3330 y Fq(\000)2093 3343 y Ft(D)2175 3370 y Fo(\003)2221 3343 y Ft(\034)p 2093 3387 184 4 v 2108 3479 a Fn(j)p Ft(x)p Fn(j)2221 3450 y FA(2)2288 3330 y Fq(\001)2333 3367 y FA(1)p Fz(=)p FA(2)2607 3343 y Fu(1)p 2461 3387 343 4 v 2461 3480 a Fr(\()p Ft(\027)2549 3507 y Fo(\003)2595 3480 y Ft(\034)j Fr(\))2689 3452 y FA(1)p Fz(=)p FA(2)2816 3330 y Fq(\000)2873 3343 y Ft(D)2955 3370 y Fo(\003)p 2873 3387 129 4 v 2890 3479 a Ft(\027)2939 3506 y Fo(\003)3013 3330 y Fq(\001)3059 3367 y FA(1)p Fz(=)p FA(2)3200 3410 y Ft(:)94 3683 y Fu(The)25 b(generalization)g(to)f(arbitrary)h Ft(p)g Fu(and)g Ft(d)f Fu(is)h(easy)g(and)g(is)f(left)h(to)f(the)h(reader)-5 b(.)94 4026 y Fv(5.)46 b(The)e(Dissipativ)l(e)k(Bound)94 4215 y Fu(In)25 b(this)f(section,)g(we)h(consider)g(in)f(detail)h(the)f(equation) 2046 4172 y FA(1)1167 4431 y Fr(_)-59 b Ft(w)s Fr(\()p Ft(x;)17 b(t)p Fr(\))43 b(=)i Ft(D)s Fr(\001)p Ft(w)s Fr(\()p Ft(x;)17 b(t)p Fr(\))k(+)h Ft(M)11 b Fr(\()p Ft(x;)17 b(t)p Fr(\))p Ft(w)s Fr(\()p Ft(x;)g(t)p Fr(\))22 b Ft(;)836 b Fr(\()p Fu(5)p Ft(:)p Fu(1)p Fr(\))94 4647 y Fu(where)32 b Ft(w)i Fu(tak)o(es)d(v)n(alues)f (in)g Fs(R)1186 4604 y Fz(N)1262 4647 y Fu(.)54 b(Our)31 b(bounds)f(will)g(w) o(ork)h(in)f(dimensions)f Ft(d)36 b Fn(\024)h Fu(3.)54 b(W)-8 b(e)32 b(\256rst)f(state)f(the)94 4767 y(assumptions)23 b(of)i(the)g (Introduction)e(in)h(a)i(more)e(precise)h(form.)36 b(W)-8 b(e)25 b(\256rst)g(assume)f(that)1593 4982 y Fn(k)p Ft(M)11 b Fr(\()p Ft(x;)17 b(t)p Fr(\))p Fn(k)42 b(\024)j Ft(M)2278 5009 y Fo(\003)2348 4982 y Ft(;)1293 b Fr(\()p Fu(5)p Ft(:)p Fu(2)p Fr(\))p 94 5093 1200 4 v 241 5159 a Fy(1)316 5202 y Fx(F)o(or)20 b(simplicity)i(we)e (assume)i(isotrop)o(y)e(of)g(the)h(dif)n(fusion)f(in)g(the)h Fi(d)g Fx(components)g(of)f(the)h(coordinates,)g(b)n(ut)g(this)g(requirement) 94 5291 y(could)g(be)f(dropped)f(if)g(desired.)p eop %%Page: 13 13 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(13)94 194 y Fu(for)26 b(all)e Ft(x)k Fn(2)g Fs(R)625 151 y Fz(d)671 194 y Fu(,)d Ft(t)j Fn(2)g Fs(R)965 221 y Fp(+)1031 194 y Fu(.)36 b(W)-8 b(e)25 b(assume)f(further)h(that)1641 434 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b(t)p Fr(\))p Fn(j)42 b(\024)j Ft(Q)2230 461 y Fo(\003)2300 434 y Ft(;)1341 b Fr(\()p Fu(5)p Ft(:)p Fu(3)p Fr(\))94 675 y Fu(for)33 b(all)f Ft(x)p Fu(,)i Ft(t)p Fu(.)58 b(\(This)31 b(is)h(the)g(reason)g(for)h(the)f(choice)g (of)g Ft(Q)2186 702 y Fo(\003)2232 675 y Ft(=)p Fu(2)g(in)f(the)h(bound)g(on) g Ft(u)p Fu(.\))58 b(Here,)34 b(and)e(in)g(the)94 794 y(remainder)25 b(of)g(the)g(paper)l(,)g Fn(j)d(\001)g(j)i Fu(is)g(the)h Ft(l)1467 751 y FA(2)1507 794 y Fu(-norm)f(of)h(a)g(v)o(ector)f(in)h Fs(R)2413 751 y Fz(N)2488 794 y Fu(.)316 914 y(W)-8 b(e)38 b(will)e(\256x)i(the)f(constant)g Ft(k)1400 941 y Fo(\003)1483 914 y Fu(only)f(in)h(the)h(ne)o(xt)e(section.)73 b(But)37 b(we)h(will)f(w)o (ork)g(here)h(with)e(the)94 1033 y(follo)n(wing)23 b(\252comparisons)h(of)h (scales\272)g(which)f(will)g(be)h(essential)f(in)h(the)f(bounds:)1654 1274 y Ft(\034)1698 1300 y Fo(\003)1787 1274 y Fr(=)45 b Fu(1)p Ft(=)-6 b(M)2100 1300 y Fo(\003)2171 1274 y Ft(;)1470 b Fr(\()p Fu(5)p Ft(:)p Fu(4)p Fr(\))1653 1423 y Ft(\016)1701 1380 y FA(2)1697 1450 y Fo(\003)1787 1423 y Fr(=)45 b Ft(D)1991 1450 y Fo(\003)2037 1423 y Ft(\034)2081 1450 y Fo(\003)2152 1423 y Ft(;)1489 b Fr(\()p Fu(5)p Ft(:)p Fu(5)p Fr(\))1645 1572 y Ft(k)1700 1530 y FA(2)1697 1599 y Fo(\003)1787 1572 y Fn(\025)45 b Fu(1)p Ft(=)p Fr(\()p Ft(\027)2097 1599 y Fo(\003)2142 1572 y Ft(\034)2186 1599 y Fo(\003)2232 1572 y Fr(\))24 b Ft(:)1346 b Fr(\()p Fu(5)p Ft(:)p Fu(6)p Fr(\))316 1813 y Fu(W)-8 b(e)25 b(consider)g(ne)o(xt)f(the)g(inte)o(gral)g(representation)g(of)h Ft(w)2271 1840 y FA(1)2311 1813 y Fr(\()p Fn(\001)p Fr(\))i Fn(\021)h Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b(t)27 b Fr(=)h Ft(\034)2947 1840 y Fo(\003)2992 1813 y Fr(\))p Fu(:)181 2106 y Ft(w)s Fr(\()p Ft(x;)17 b(\034)440 2133 y Fo(\003)485 2106 y Fr(\))44 b(=)689 2025 y Fq(\000)735 2106 y Fk(G)795 2133 y Fz(\034)832 2151 y Ff(\003)901 2106 y Ft(?)22 b(w)1047 2025 y Fq(\001)1092 2106 y Fr(\()p Ft(x)p Fr(\))g(+)1349 1970 y Fq(Z)1448 2013 y Fz(\034)1485 2031 y Ff(\003)1404 2197 y FA(0)1548 2106 y Fu(d)p Ft(s)1662 2025 y Fq(\000)1707 2106 y Fk(G)1767 2133 y Fz(\034)1804 2151 y Ff(\003)1846 2133 y Fo(\000)p Fz(s)1973 2106 y Ft(?)2045 2025 y Fq(\000)2090 2106 y Ft(M)11 b Fr(\()p Fn(\001)p Ft(;)17 b(s)p Fr(\))p Ft(w)s Fr(\()p Fn(\001)p Ft(;)g(s)p Fr(\))2668 2025 y Fq(\001)l(\001)2755 2106 y Fr(\()p Ft(x)p Fr(\))44 b Fn(\021)h Ft(w)3130 2054 y Fp(\()p FA(1)p Fp(\))3127 2134 y FA(1)3254 2106 y Fr(+)23 b Ft(w)3428 2054 y Fp(\()p FA(2)p Fp(\))3425 2134 y FA(1)3555 2106 y Ft(:)86 b Fr(\()p Fu(5)p Ft(:)p Fu(7)p Fr(\))94 2432 y Fu(Using)21 b(the)f(decomposition)f Fk(G)1164 2459 y Fz(\034)1242 2432 y Fr(=)28 b Fk(G)1407 2459 y Fz(\034)t(;<)1552 2432 y Fr(+)13 b Fk(G)1702 2459 y Fz(\034)t(;>)1835 2432 y Fu(,)21 b(we)g(split)f Ft(w)2285 2380 y Fp(\()p FA(1)p Fp(\))2282 2461 y FA(1)2408 2432 y Fu(and)h Ft(w)2647 2380 y Fp(\()p FA(2)p Fp(\))2644 2461 y FA(1)2770 2432 y Fu(into)f(high)g(and)h (lo)n(w)f(frequenc)o(y)94 2552 y(parts:)945 2688 y Ft(w)1016 2715 y FA(1)p Fz(;<)1186 2688 y Fr(=)45 b Ft(w)1382 2637 y Fp(\()p FA(1)p Fp(\))1379 2717 y FA(1)p Fz(;<)1527 2688 y Fr(+)23 b Ft(w)1701 2637 y Fp(\()p FA(2)p Fp(\))1698 2717 y FA(1)p Fz(;<)1849 2688 y Ft(;)215 b(w)2163 2715 y FA(1)p Fz(;>)2334 2688 y Fr(=)44 b Ft(w)2529 2637 y Fp(\()p FA(1)p Fp(\))2526 2717 y FA(1)p Fz(;>)2675 2688 y Fr(+)22 b Ft(w)2848 2637 y Fp(\()p FA(2)p Fp(\))2845 2717 y FA(1)p Fz(;>)2996 2688 y Ft(:)94 2879 y Fu(Note)j(that)f Ft(w)558 2906 y Fz(<)651 2879 y Fu(has)h(F)o(ourier)f (components)g(in)g Fn(fj)p Ft(k)s Fn(j)j(\024)h Fu(2)p Ft(k)2129 2906 y Fo(\003)2175 2879 y Fn(g)c Fu(and)h Ft(w)2489 2906 y Fz(>)2582 2879 y Fu(has)f(components)g(in)g Fn(fj)p Ft(k)s Fn(j)j(\025)h Ft(k)3693 2906 y Fo(\003)3739 2879 y Fn(g)p Fu(.)94 3070 y Fs(Theorem)44 b(5.1.)54 b Fu(\(Dissipati)n(v)o(e)28 b(bound\))i Fc(In)h(dimension)e Ft(d)36 b Fn(\024)g Fu(3)p Fc(,)c(we)f(ha)n(v)o(e)g(the)f(follo)n(wing)f(bounds)h(for)94 3190 y(some)25 b(constants)e Ft(B)806 3147 y Fo(\003)801 3217 y FA(4)877 3190 y Fc(and)h Ft(B)1126 3147 y Fo(\003)1121 3217 y FA(5)1172 3190 y Fc(:)35 b(Upon)25 b(localizing)f(in)g(space,)h(we)g(ha)n (v)o(e)g(for)g Fu(0)i Fn(\024)h Ft(\034)39 b Fn(\024)28 b Ft(\034)3179 3217 y Fo(\003)3224 3190 y Fc(,)800 3433 y Fu(sup)720 3523 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)p Fo(\000)p Fz(`)1035 3433 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b(\034)11 b Fr(\))p Fn(j)42 b(\024)i Ft(B)1644 3390 y Fo(\003)1639 3460 y FA(4)1690 3352 y Fq(\000)1767 3433 y Fu(sup)1736 3523 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1954 3433 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b Fu(0)p Fr(\))p Fn(j)j Fr(+)j(\()p Ft(\016)2517 3460 y Fo(\003)2563 3433 y Ft(=`)p Fr(\))p Fn(k)p Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b Fu(0)p Fr(\))p Fn(k)3069 3460 y Fo(1)3150 3352 y Fq(\001)3221 3433 y Ft(:)3669 3475 y Fr(\()p Fu(5)p Ft(:)p Fu(8)p Fr(\))94 3749 y Fc(Localizing)24 b(in)h(position)e(space)i(and)g(momentum)e(space,)i (we)g(ha)n(v)o(e)1078 3986 y Fu(sup)998 4076 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)p Fo(\000)p Fz(`)1296 3986 y Fn(j)p Ft(w)1395 4013 y Fz(>)1462 3986 y Fr(\()p Ft(x;)17 b(\034)1647 4013 y Fo(\003)1692 3986 y Fr(\))p Fn(j)1048 4288 y(\024)45 b Ft(B)1251 4245 y Fo(\003)1246 4315 y FA(5)1313 4147 y Fq(\022)1386 4288 y Ft(e)1432 4245 y Fo(\000)p Fp(\()p Fz(\027)1565 4263 y Ff(\003)1607 4245 y Fz(k)1651 4202 y Fe(2)1649 4272 y Ff(\003)1691 4245 y Fz(=)l(M)1804 4263 y Ff(\003)1845 4245 y Fp(\))p Fz(=)p FA(2)1979 4288 y Fr(+)2079 4207 y Fq(\000)2162 4220 y Ft(M)2259 4247 y Fo(\003)p 2137 4265 193 4 v 2137 4356 a Ft(\027)2186 4383 y Fo(\003)2232 4356 y Ft(k)2287 4327 y FA(2)2284 4383 y Fo(\003)2341 4207 y Fq(\001)2387 4245 y FA(1)p Fo(\000)p Fz(d=)p FA(4)2606 4147 y Fq(\023)1318 4508 y Fn(\001)1385 4427 y Fq(\000)1476 4508 y Fu(sup)1430 4598 y Fo(j)p Fz(x)p Fo(j)13 b(\024)h Fz(\025)1676 4508 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)j Fu(0)p Fr(\))p Fn(j)j Fr(+)j(\()p Ft(\016)2239 4534 y Fo(\003)2285 4508 y Ft(=`)p Fr(\))p Fn(k)p Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b Fu(0)p Fr(\))p Fn(k)2791 4534 y Fo(1)2872 4427 y Fq(\001)2942 4508 y Ft(:)3669 4288 y Fr(\()p Fu(5)p Ft(:)p Fu(9)p Fr(\))94 4946 y Fs(Pro)s(of.)40 b Fu(W)-8 b(e)25 b(de\256ne,)g(as)g(in)g([CE1],)g(a)g(f)o(amily)f(of)h(space) g(cutof)n(f)f(functions:)35 b(Let)990 5234 y Ft( )1055 5261 y Fz(a)1103 5234 y Fr(\()p Ft(x)p Fr(\))43 b(=)1855 5167 y Ft(Z)p 1415 5211 956 4 v 1415 5304 a Fu(1)22 b Fr(+)h(\(\()p Ft(x)e Fn(\000)i Ft(a)p Fr(\))1935 5276 y FA(2)1974 5304 y Ft(=\016)2072 5276 y FA(2)2068 5331 y Fo(\003)2114 5304 y Fr(\))2153 5276 y FA(1)p Fp(+)p Fz(d=)p FA(2)2427 5234 y Fr(=)45 b Ft( )t Fr(\()p Ft(x)21 b Fn(\000)i Ft(a)p Fr(\))h Ft(;)p eop %%Page: 14 14 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(14)94 194 y Fu(where)21 b Ft(Z)35 b Fr(=)28 b Ft(\016)614 151 y Fo(\000)p Fz(d)610 221 y Fo(\003)723 113 y Fq(\016)797 114 y(R)880 194 y Fu(d)p Ft(x)p Fr(\()p Fu(1)12 b Fr(+)g Ft(x)1234 151 y FA(2)1273 194 y Fr(\))1312 151 y Fo(\000)p FA(1)p Fo(\000)p Fz(d=)p FA(2)1614 194 y Fu(is)19 b(chosen)h(such)g(that)2370 114 y Fq(R)2453 194 y Fu(d)p Ft(x)d( )t Fr(\()p Ft(x)p Fr(\))26 b(=)i Fu(1.)34 b(W)-8 b(e)20 b(start)g(with)f(a)i(bound)94 329 y(in)k(L)258 286 y FA(2)298 329 y Fu(:)94 520 y Fs(Lemma)39 b(5.2.)d Fc(There)25 b(is)g(a)g(constant)f Ft(B)1571 477 y Fo(\003)1566 547 y FA(6)1641 520 y Fc(such)g(that)h(the)f(solution)f(of)i (Eq.\(5.1\))g(satis\256es:)223 767 y Fu(sup)158 853 y FA(0)p Fo(\024)p Fz(t)p Fo(\024)p Fz(\034)384 871 y Ff(\003)521 767 y Fu(sup)442 858 y Fo(j)p Fz(a)p Fo(j\024)p Fz(\025)p Fo(\000)p Fz(`)755 631 y Fq(Z)871 767 y Fu(d)p Ft(x)17 b( )1060 794 y Fz(a)1107 767 y Fr(\()p Ft(x)p Fr(\))p Fn(j)p Ft(w)s Fr(\()p Ft(x;)g(t)p Fr(\))p Fn(j)1588 724 y FA(2)1670 767 y Fn(\024)45 b Ft(B)1873 724 y Fo(\003)1868 794 y FA(6)1918 686 y Fq(\000)1995 767 y Fu(sup)1964 858 y Fo(j)p Fz(x)p Fo(j)p Fz(<\025)2182 767 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b Fu(0)p Fr(\))p Fn(j)2542 724 y FA(2)2602 767 y Fr(+)23 b Fn(k)p Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b Fu(0)p Fr(\))p Fn(k)3077 724 y FA(2)3077 794 y Fo(1)3159 767 y Fr(\()p Ft(\016)3242 794 y Fo(\003)3288 767 y Ft(=`)p Fr(\))3419 724 y FA(2)3458 686 y Fq(\001)3528 767 y Ft(:)63 b Fr(\()p Fu(5)p Ft(:)p Fu(10)p Fr(\))94 1137 y Fs(Pro)s(of.)40 b Fu(Let)1381 1283 y Ft(X)1464 1310 y Fz(t)1543 1283 y Fr(=)1664 1147 y Fq(Z)1781 1283 y Fu(d)p Ft(x)17 b( )1970 1310 y Fz(a)2017 1283 y Fr(\()p Ft(x)p Fr(\))p Fn(j)p Ft(w)s Fr(\()p Ft(x;)g(t)p Fr(\))p Fn(j)2498 1240 y FA(2)2560 1283 y Ft(:)94 1495 y Fu(Then)25 b(we)g(ha)n(v)o(e,)g(from)f(the)h(equations)f(of)h(motion,)215 1736 y Ft(@)268 1762 y Fz(t)303 1736 y Ft(X)386 1762 y Fz(t)465 1736 y Fr(=)586 1600 y Fq(Z)703 1736 y Fu(d)p Ft(x)17 b( )892 1762 y Fz(a)939 1655 y Fq(\000)985 1736 y Ft(w)25 b Fn(\001)d Fr(\()p Ft(D)s Fr(\001)p Ft(w)j Fr(+)d Ft(M)11 b(w)s Fr(\))22 b(+)g(\()p Ft(D)s Fr(\001)p Ft(w)k Fr(+)c Ft(M)11 b(w)s Fr(\))22 b Fn(\001)f Ft(w)2644 1655 y Fq(\001)465 1997 y Fr(=)44 b Fn(\000)p Fu(2)730 1861 y Fq(Z)846 1997 y Fu(d)p Ft(x)17 b( )1035 2024 y Fz(a)1083 1997 y Fr(\()p Fn(r)p Ft(w)s Fr(\))22 b Fn(\001)f Ft(D)1471 2024 y FA(sym)1593 1997 y Fn(r)p Ft(w)k Fr(+)e Fu(2)1939 1861 y Fq(Z)2055 1997 y Fu(d)p Ft(x)17 b( )2244 2024 y Fz(a)2291 1997 y Ft(M)2388 2024 y FA(sym)2509 1997 y Fn(j)p Ft(w)s Fn(j)2639 1954 y FA(2)2700 1997 y Fn(\000)2800 1861 y Fq(Z)2916 1997 y Fu(d)p Ft(x)g Fr(\()p Fn(r)p Ft( )3227 2024 y Fz(a)3274 1997 y Fr(\))p Fn(r)p Ft(w)25 b Fn(\001)d Ft(D)s(w)28 b(:)94 2256 y Fu(Here,)f Ft(M)436 2283 y FA(sym)586 2256 y Fr(=)j(\()p Ft(M)i Fr(+)24 b Ft(M)1070 2213 y FA(T)1117 2256 y Fr(\))p Ft(=)p Fu(2)h(and)h Fn(k)p Ft(M)1598 2283 y FA(sym)1718 2256 y Fn(k)j(\024)h(k)p Ft(M)11 b Fn(k)28 b(\024)i Ft(M)2344 2283 y Fo(\003)2389 2256 y Fu(.)39 b(Observ)o(e)25 b(that)g(by)h(our)g(choice)g (of)f Ft( )30 b Fu(we)94 2375 y(ha)n(v)o(e)23 b Fn(jr)p Ft( )t Fn(j)j(\024)j Ft(C)713 2402 y FA(2)753 2375 y Ft( )t(=\016)916 2402 y Fo(\003)984 2375 y Fu(for)23 b(some)f(constant)g Ft(C)1782 2402 y FA(2)1845 2375 y Fu(independent)f(of)i Ft(\016)2504 2402 y Fo(\003)2550 2375 y Fu(.)36 b(Using)21 b(the)i(de\256nitions)e (\(1.3\))i(of)g Ft(\027)3828 2402 y Fo(\003)94 2495 y Fu(and)i Ft(D)345 2522 y Fo(\003)392 2495 y Fu(,)f(we)h(\256nd)218 2741 y Fn(j)p Ft(@)299 2768 y Fz(t)334 2741 y Ft(X)417 2768 y Fz(t)452 2741 y Fn(j)44 b(\024)g(\000)p Fu(2)p Ft(\027)821 2768 y Fo(\003)884 2605 y Fq(Z)1000 2741 y Fu(d)p Ft(x)17 b( )1189 2768 y Fz(a)1237 2741 y Fn(jr)p Ft(w)s Fn(j)1450 2698 y FA(2)1511 2741 y Fr(+)23 b Fu(2)1678 2605 y Fq(Z)1794 2741 y Fu(d)p Ft(x)17 b( )1983 2768 y Fz(a)2030 2741 y Ft(M)2127 2768 y Fo(\003)2172 2741 y Fn(j)p Ft(w)s Fn(j)2302 2698 y FA(2)2364 2741 y Fr(+)22 b Fu(2)p Ft(C)2584 2768 y FA(2)2641 2605 y Fq(Z)2757 2741 y Fu(d)p Ft(x)17 b Fr(\()p Ft(D)3002 2768 y Fo(\003)3048 2741 y Ft(=\016)3142 2768 y Fo(\003)3187 2741 y Fr(\))p Ft( )3291 2768 y Fz(a)3339 2741 y Fn(j)p Ft(w)25 b Fn(\001)d(r)p Ft(w)s Fn(j)i Ft(:)94 3000 y Fu(W)-8 b(e)24 b(polarize)g(the)f(term)g(containing)f Ft(w)g Fn(\001)d(r)p Ft(w)27 b Fu(and)c(use)h(the)f(identity)f Fr(\()p Ft(D)2627 3027 y Fo(\003)2673 3000 y Ft(=\016)2767 3027 y Fo(\003)2813 3000 y Fr(\))2852 2957 y FA(2)2891 3000 y Ft(=\027)2990 3027 y Fo(\003)3064 3000 y Fr(=)28 b Ft(M)3266 3027 y Fo(\003)3311 3000 y Fu(.)36 b(see)24 b(Eq.\(5.5\),)94 3119 y(Then)h(we)g(see)g(that)g(we)g(can)g(\256nd)g(a)g(constant)f Ft(C)1773 3146 y FA(3)1838 3119 y Fu(such)g(that)1252 3312 y Fn(j)p Ft(@)1333 3339 y Fz(t)1368 3312 y Ft(X)1451 3339 y Fz(t)1485 3312 y Fn(j)44 b(\024)h Ft(C)1750 3339 y FA(3)1790 3312 y Ft(M)1887 3339 y Fo(\003)1933 3232 y Fq(\000)1978 3312 y Fu(1)22 b Fr(+)h(\()p Ft(D)2271 3339 y Fo(\003)2317 3312 y Ft(=\027)2416 3339 y Fo(\003)2462 3312 y Fr(\))2501 3232 y Fq(\001)2546 3312 y Ft(X)2629 3339 y Fz(t)2689 3312 y Ft(:)902 b Fr(\()p Fu(5)p Ft(:)p Fu(11)p Fr(\))94 3505 y Fu(Coming)24 b(back)h(to)g(the)g(assumptions)d(of)j(Lemma)f(5.2,)h(we)g(see)g(that)f(when) h Fn(j)p Ft(a)p Fn(j)i(\024)h Ft(\025)22 b Fn(\000)h Ft(`)h Fu(we)h(ha)n(v)o(e)695 3749 y Ft(X)778 3776 y FA(0)861 3749 y Fr(=)983 3614 y Fq(Z)1099 3749 y Fu(d)p Ft(x)17 b( )1288 3776 y Fz(a)1336 3749 y Fn(j)p Ft(w)1435 3776 y FA(0)1475 3749 y Fn(j)1503 3707 y FA(2)1587 3749 y Fr(=)1708 3614 y Fq(Z)1764 3840 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1987 3749 y Fu(d)p Ft(x)g( )2176 3776 y Fz(a)2224 3749 y Fn(j)p Ft(w)2323 3776 y FA(0)2363 3749 y Fn(j)2391 3707 y FA(2)2452 3749 y Fr(+)2552 3614 y Fq(Z)2607 3840 y Fo(j)p Fz(x)p Fo(j\025)p Fz(\025)2831 3749 y Fu(d)p Ft(x)g( )3020 3776 y Fz(a)3067 3749 y Fn(j)p Ft(w)3166 3776 y FA(0)3206 3749 y Fn(j)3234 3707 y FA(2)1587 4034 y Fn(\024)76 b Fu(sup)1708 4124 y Fo(j)p Fz(x)p Fo(j)p Fz(<\025)1927 4034 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b Fu(0)p Fr(\))p Fn(j)2287 3991 y FA(2)2347 4034 y Fr(+)22 b Ft(C)2517 4061 y FA(4)2558 4034 y Fn(k)p Ft(w)2679 4061 y FA(0)2719 4034 y Fn(k)2769 3991 y FA(2)2769 4061 y Fo(1)2853 3953 y Fq(\000)2910 3967 y Ft(\016)2954 3994 y Fo(\003)p 2910 4011 91 4 v 2935 4102 a Ft(`)3012 3953 y Fq(\001)3058 3991 y FA(2)3123 4034 y Ft(:)94 4305 y Fu(Using)27 b(this)f(bound)g(on)h(the)g(initial)f (condition,)g(the)h(dif)n(ferential)f(inequality)g(\(5.11\),)i(and)f Ft(\034)3323 4332 y Fo(\003)3368 4305 y Ft(M)3465 4332 y Fo(\003)3541 4305 y Fr(=)32 b Fu(1,)27 b(the)94 4425 y(assertion)e(of)f(Lemma)h(5.2)f (follo)n(ws)f(with)i Ft(B)1665 4382 y Fo(\003)1660 4451 y FA(6)1738 4425 y Fr(=)j Fu(e)o(xp)1985 4344 y Fq(\000)2031 4425 y Ft(C)2102 4451 y FA(3)2142 4425 y Fr(\()p Fu(1)21 b Fr(+)i(\()p Ft(D)2473 4451 y Fo(\003)2519 4425 y Ft(=\027)2618 4451 y Fo(\003)2664 4425 y Fr(\)\))2742 4344 y Fq(\001)2787 4425 y Fu(.)316 4573 y(W)-8 b(e)26 b(be)o(gin)f(the)h(proof)g(of)g(\(5.9\).)39 b(W)-8 b(e)27 b(deal)f(\256rst)g(with)f(the)h(bound)f(on)g Ft(w)2827 4522 y Fp(\()p FA(2)p Fp(\))2824 4602 y FA(1)p Fz(;>)2951 4573 y Fr(\()p Ft(x)p Fr(\))p Fu(,)g(when)h Fn(j)p Ft(x)p Fn(j)j(\024)g Ft(\025)23 b Fn(\000)h Ft(`)p Fu(.)94 4693 y(Consider)l(,)h(for)g Ft(\034)685 4720 y Fo(\003)758 4693 y Fn(\025)j Ft(t)g Fn(\025)g Ft(s)g Fn(\025)g Fu(0,)c(the)h(quantity)824 4939 y Ft(Y)882 4966 y Fz(t;s)1022 4939 y Fr(=)1144 4803 y Fq(Z)1260 4939 y Fu(d)p Ft(y)20 b Fk(G)1439 4966 y Fz(t)p Fo(\000)p Fz(s;>)1660 4939 y Fr(\()p Ft(x)i Fn(\000)g Ft(y)t Fr(\))p Ft(M)11 b Fr(\()p Ft(y)t(;)17 b(s)p Fr(\))p Ft(w)s Fr(\()p Ft(y)t(;)g(s)p Fr(\))1022 5219 y(=)1144 5083 y Fq(Z)1260 5219 y Fu(d)p Ft(y)1391 5139 y Fk(G)1451 5166 y Fz(t)p Fo(\000)p Fz(s;>)1672 5139 y Fr(\()p Ft(x)22 b Fn(\000)g Ft(y)t Fr(\))p 1391 5196 590 4 v 1447 5216 a Fq(p)p 1547 5216 378 4 v 85 x Ft( )t Fr(\()p Ft(x)f Fn(\000)i Ft(y)t Fr(\))1992 5129 y Fq(p)p 2092 5129 V 90 x Ft( )t Fr(\()p Ft(x)e Fn(\000)i Ft(y)t Fr(\))o Ft(M)11 b Fr(\()p Ft(y)t(;)17 b(s)p Fr(\))p Ft(w)s Fr(\()p Ft(y)t(;)g(s)p Fr(\))j Ft(:)p eop %%Page: 15 15 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(15)94 194 y Fu(W)-8 b(e)26 b(consider)e(\256rst)h(the)g(case)g Ft(\027)1184 221 y Fo(\003)1230 194 y Ft(\034)39 b(>)28 b(k)1473 151 y Fo(\000)p FA(2)1470 221 y Fo(\003)1575 194 y Fu(:)36 b(Then,)24 b(by)h(the)f(Schw)o(arz)i(inequality)e(and)g(Eq.\(4.8\),)h(we)g (ha)n(v)o(e)650 475 y Ft(Y)730 433 y FA(2)708 502 y Fz(t;s)848 475 y Fn(\024)970 340 y Fq(Z)1086 475 y Fu(d)p Ft(y)1217 396 y Fn(j)p Fk(G)1305 423 y Fz(t)p Fo(\000)p Fz(s;>)1526 396 y Fr(\()p Ft(x)c Fn(\000)i Ft(y)t Fr(\))p Fn(j)1863 353 y FA(2)p 1217 453 685 4 v 1371 544 a Ft( )t Fr(\()p Ft(x)e Fn(\000)i Ft(y)t Fr(\))1914 475 y Ft(M)2022 433 y FA(2)2011 502 y Fo(\003)2078 340 y Fq(Z)2194 475 y Fu(d)p Ft(z)e( )2376 502 y Fz(x)2426 475 y Fr(\()p Ft(z)t Fr(\))p Fn(j)p Ft(w)s Fr(\()p Ft(z)t(;)c(s)p Fr(\))p Fn(j)2904 433 y FA(2)848 819 y Fn(\024)970 683 y Fq(Z)1086 819 y Fu(d)p Ft(y)1249 728 y(B)1330 685 y Fo(\003)1325 755 y FA(3)1375 728 y Fr(\()p Ft(p)p Fr(\))1503 685 y FA(2)1543 728 y Ft(e)1589 685 y Fo(\000)p Fz(\027)1691 703 y Ff(\003)1733 685 y Fz(k)1777 642 y Fe(2)1775 712 y Ff(\003)1817 685 y Fp(\()p Fz(t)p Fo(\000)p Fz(s)p Fp(\))2014 647 y Fq(\000)2072 679 y Fz(D)2137 697 y Ff(\003)p 2072 705 107 4 v 2085 762 a Fz(\027)2125 780 y Ff(\003)2191 647 y Fq(\001)2236 685 y FA(2)p Fz(p)p 1217 796 1133 4 v 1217 910 a Fr(\()p Ft(\027)1305 937 y Fo(\003)1351 910 y Fr(\()p Ft(t)k Fn(\000)i Ft(s)p Fr(\)\))1672 881 y Fz(d)1718 829 y Fq(\000)1763 910 y Fu(1)f Fr(+)1983 861 y Fo(j)p Fz(x)p Fo(\000)p Fz(y)s Fo(j)2181 841 y Fe(2)p 1947 887 300 4 v 1947 944 a Fz(D)2012 962 y Ff(\003)2054 944 y Fp(\()p Fz(t)p Fo(\000)p Fz(s)p Fp(\))2258 829 y Fq(\001)2304 868 y Fz(p)2374 751 y Fu(1)f Fr(+)i(\()p Fn(j)p Ft(x)f Fn(\000)g Ft(y)t Fn(j)2871 709 y FA(2)2910 751 y Ft(=\016)3008 709 y FA(2)3004 778 y Fo(\003)3050 751 y Fr(\))3089 709 y FA(1)p Fp(+)p Fz(d=)p FA(2)p 2374 796 934 4 v 2803 887 a Ft(Z)843 1137 y Fn(\001)38 b Ft(M)1017 1095 y FA(2)1006 1164 y Fo(\003)1073 1002 y Fq(Z)1189 1137 y Fu(d)p Ft(z)21 b( )1371 1164 y Fz(x)1421 1137 y Fr(\()p Ft(z)t Fr(\))p Fn(j)p Ft(w)s Fr(\()p Ft(z)t(;)c(s)p Fr(\))p Fn(j)1899 1095 y FA(2)1963 1137 y Ft(:)94 1383 y Fu(Since)26 b(we)f(deal)g(with)f Fn(j)p Ft(x)p Fn(j)j(\024)h Ft(\025)22 b Fn(\000)h Ft(`)p Fu(,)h(we)h(get)g (from)f(\(5.10\),)235 1680 y Ft(Y)315 1637 y FA(2)293 1707 y Fz(t;s)433 1680 y Fn(\024)45 b Ft(B)636 1637 y Fo(\003)631 1707 y FA(3)681 1680 y Fr(\()p Ft(p)p Fr(\))809 1637 y FA(2)849 1599 y Fq(\000)906 1613 y Ft(D)988 1640 y Fo(\003)p 906 1657 129 4 v 923 1748 a Ft(\027)972 1775 y Fo(\003)1046 1599 y Fq(\001)1092 1637 y FA(2)p Fz(p)1173 1680 y Ft(Z)1248 1637 y Fo(\000)p FA(1)1367 1545 y Fq(Z)1483 1680 y Fu(d)p Ft(y)1614 1613 y(e)1660 1570 y Fo(\000)p Fz(\027)1762 1588 y Ff(\003)1804 1570 y Fz(k)1848 1527 y Fe(2)1846 1597 y Ff(\003)1888 1570 y Fp(\()p Fz(t)p Fo(\000)p Fz(s)p Fp(\))p 1614 1657 472 4 v 1634 1748 a Ft(\027)1689 1720 y Fz(d)1683 1775 y Fo(\003)1736 1748 y Fr(\()p Ft(t)22 b Fn(\000)h Ft(s)p Fr(\))2019 1720 y Fz(d)2097 1599 y Fq(\000)2143 1680 y Fu(1)f Fr(+)2368 1613 y Fn(j)p Ft(x)g Fn(\000)h Ft(y)t Fn(j)2656 1570 y FA(2)p 2327 1657 411 4 v 2327 1748 a Ft(D)2409 1775 y Fo(\003)2455 1748 y Fr(\()p Ft(t)f Fn(\000)g Ft(s)p Fr(\))2749 1599 y Fq(\001)2794 1637 y Fo(\000)p Fz(p)2903 1599 y Fq(\000)2948 1680 y Fu(1)g Fr(+)3132 1613 y Fn(j)p Ft(x)g Fn(\000)g Ft(y)t Fn(j)3419 1570 y FA(2)p 3132 1657 327 4 v 3250 1748 a Ft(\016)3298 1720 y FA(2)3294 1775 y Fo(\003)3470 1599 y Fq(\001)3516 1637 y FA(1)p Fp(+)p Fz(d=)p FA(2)427 1952 y Fn(\001)39 b Ft(M)602 1909 y FA(2)591 1979 y Fo(\003)641 1952 y Ft(B)722 1909 y Fo(\003)717 1979 y FA(6)768 1871 y Fq(\000)813 1952 y Ft(K)905 1909 y FA(2)967 1952 y Fr(+)23 b Ft(Q)1146 1909 y FA(2)1146 1979 y Fo(\003)1191 1871 y Fq(\000)1249 1884 y Ft(\016)1293 1911 y Fo(\003)p 1249 1929 91 4 v 1273 2020 a Ft(`)1351 1871 y Fq(\001)1396 1909 y FA(2)1436 1871 y Fq(\001)1507 1952 y Ft(;)94 2180 y Fu(where)j Ft(K)34 b Fr(=)28 b Fu(sup)726 2207 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)949 2180 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b Fu(0)p Fr(\))p Fn(j)23 b Fu(and)i Ft(Q)1580 2207 y Fo(\003)1653 2180 y Fn(\025)j(k)p Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b Fu(0)p Fr(\))p Fn(k)2133 2207 y Fo(1)2216 2180 y Fu(.)35 b(Note)25 b(no)n(w)f(that)g(by)h(Eq.\(5.5\),)1433 2400 y Ft(\016)1481 2357 y FA(2)1477 2427 y Fo(\003)1567 2400 y Fr(=)45 b Ft(D)1771 2427 y Fo(\003)1817 2400 y Ft(\034)1861 2427 y Fo(\003)1951 2400 y Fn(\025)g Ft(D)2155 2427 y Fo(\003)2201 2400 y Fr(\()p Ft(t)22 b Fn(\000)h Ft(s)p Fr(\))h Ft(;)94 2595 y Fu(since)h Ft(\034)368 2621 y Fo(\003)441 2595 y Fn(\025)j Ft(t)g Fn(\025)g Ft(s)f Fn(\025)i Fu(0.)35 b(Therefore,)544 2893 y Ft(Y)624 2851 y FA(2)602 2920 y Fz(t;s)742 2893 y Fn(\024)45 b Ft(B)945 2851 y Fo(\003)940 2920 y FA(3)990 2893 y Fr(\()p Ft(p)p Fr(\))1118 2851 y FA(2)1158 2813 y Fq(\000)1215 2826 y Ft(D)1297 2853 y Fo(\003)p 1215 2870 129 4 v 1232 2962 a Ft(\027)1281 2989 y Fo(\003)1355 2813 y Fq(\001)1401 2851 y FA(2)p Fz(p)1482 2893 y Ft(Z)1557 2851 y Fo(\000)p FA(1)1676 2758 y Fq(Z)1792 2893 y Fu(d)p Ft(y)1923 2826 y(e)1969 2783 y Fo(\000)p Fz(\027)2071 2801 y Ff(\003)2113 2783 y Fz(k)2157 2740 y Fe(2)2155 2810 y Ff(\003)2197 2783 y Fp(\()p Fz(t)p Fo(\000)p Fz(s)p Fp(\))p 1923 2870 472 4 v 1943 2962 a Ft(\027)1998 2933 y Fz(d)1992 2989 y Fo(\003)2046 2962 y Fr(\()p Ft(t)21 b Fn(\000)i Ft(s)p Fr(\))2328 2933 y Fz(d)2406 2813 y Fq(\000)2452 2893 y Fu(1)f Fr(+)2678 2826 y Fn(j)p Ft(x)f Fn(\000)i Ft(y)t Fn(j)2965 2783 y FA(2)p 2636 2870 411 4 v 2636 2962 a Ft(D)2718 2989 y Fo(\003)2764 2962 y Fr(\()p Ft(t)f Fn(\000)g Ft(s)p Fr(\))3058 2813 y Fq(\001)3103 2851 y FA(1)p Fp(+)p Fz(d=)p FA(2)p Fo(\000)p Fz(p)737 3165 y Fn(\001)38 b Ft(M)911 3122 y FA(2)900 3192 y Fo(\003)950 3165 y Ft(B)1031 3122 y Fo(\003)1026 3192 y FA(6)1077 3084 y Fq(\000)1122 3165 y Ft(K)1214 3122 y FA(2)1276 3165 y Fr(+)23 b Ft(Q)1455 3122 y FA(2)1455 3192 y Fo(\003)1500 3084 y Fq(\000)1558 3098 y Ft(\016)1602 3125 y Fo(\003)p 1558 3142 91 4 v 1582 3233 a Ft(`)1660 3084 y Fq(\001)1706 3122 y FA(2)1745 3084 y Fq(\001)1816 3165 y Ft(:)94 3409 y Fu(T)-8 b(aking)25 b Ft(p)k Fr(=)f Ft(d)23 b Fr(+)f Fu(2,)k(inte)o(grating)d (o)o(v)o(er)i Ft(y)t Fu(,)f(and)i(using)e(again)h Ft(\016)2310 3366 y FA(2)2306 3435 y Fo(\003)2380 3409 y Fr(=)k Ft(D)2568 3435 y Fo(\003)2614 3409 y Ft(=)-6 b(M)2755 3435 y Fo(\003)2801 3409 y Fu(,)25 b(we)h(get,)f(for)g(some)g(constant)94 3528 y Ft(B)175 3485 y Fo(\003)170 3555 y FA(7)221 3528 y Fu(:)255 3827 y Ft(Y)335 3785 y FA(2)313 3854 y Fz(t;s)453 3827 y Fn(\024)45 b Ft(C)646 3854 y FA(5)686 3827 y Ft(B)767 3785 y Fo(\003)762 3854 y FA(3)812 3827 y Fr(\()p Ft(p)p Fr(\))940 3785 y FA(2)980 3747 y Fq(\000)1038 3760 y Ft(D)1120 3787 y Fo(\003)p 1038 3805 129 4 v 1054 3896 a Ft(\027)1103 3923 y Fo(\003)1178 3747 y Fq(\001)1223 3785 y FA(2)p Fz(d)p Fp(+)p FA(4)1401 3827 y Ft(\016)1449 3785 y Fz(d)1445 3854 y Fo(\003)1495 3827 y Ft(D)1580 3785 y Fz(d=)p FA(2)1577 3854 y Fo(\003)1703 3827 y Fr(\()p Ft(t)22 b Fn(\000)g Ft(s)p Fr(\))1985 3785 y Fz(d=)p FA(2)2119 3760 y Ft(e)2165 3717 y Fo(\000)p Fz(\027)2267 3735 y Ff(\003)2309 3717 y Fz(k)2353 3674 y Fe(2)2351 3744 y Ff(\003)2393 3717 y Fp(\()p Fz(t)p Fo(\000)p Fz(s)p Fp(\))p 2119 3805 472 4 v 2139 3896 a Ft(\027)2194 3867 y Fz(d)2188 3923 y Fo(\003)2241 3896 y Fr(\()p Ft(t)g Fn(\000)h Ft(s)p Fr(\))2524 3867 y Fz(d)2602 3827 y Ft(M)2710 3785 y FA(2)2699 3854 y Fo(\003)2749 3827 y Ft(B)2830 3785 y Fo(\003)2825 3854 y FA(6)2876 3747 y Fq(\000)2922 3827 y Ft(K)28 b Fr(+)23 b Ft(Q)3214 3854 y Fo(\003)3272 3760 y Ft(\016)3316 3787 y Fo(\003)p 3272 3805 91 4 v 3296 3896 a Ft(`)3374 3747 y Fq(\001)3419 3785 y FA(2)453 4149 y Fr(=)575 4068 y Fq(\000)621 4149 y Ft(B)702 4106 y Fo(\003)697 4176 y FA(7)747 4068 y Fq(\001)793 4106 y FA(2)833 4149 y Ft(M)941 4106 y FA(2)p Fo(\000)p Fz(d=)p FA(2)930 4176 y Fo(\003)1159 4068 y Fq(\000)1205 4149 y Ft(K)29 b Fr(+)23 b Ft(Q)1498 4176 y Fo(\003)1555 4082 y Ft(\016)1599 4109 y Fo(\003)p 1555 4126 V 1579 4218 a Ft(`)1657 4068 y Fq(\001)1703 4106 y FA(2)1843 4082 y Ft(e)1889 4039 y Fo(\000)p Fz(\027)1991 4057 y Ff(\003)2033 4039 y Fz(k)2077 3996 y Fe(2)2075 4066 y Ff(\003)2117 4039 y Fp(\()p Fz(t)p Fo(\000)p Fz(s)p Fp(\))p 1788 4126 582 4 v 1788 4242 a Ft(\027)1843 4191 y Fz(d=)p FA(2)1837 4256 y Fo(\003)1965 4242 y Fr(\()p Ft(t)f Fn(\000)h Ft(s)p Fr(\))2248 4213 y Fz(d=)p FA(2)2406 4149 y Ft(:)3619 3979 y Fr(\()p Fu(5)p Ft(:)p Fu(12)p Fr(\))94 4420 y Fu(T)-8 b(aking)26 b(the)h(square)f(root)g(of)h(this)e (bound,)h(inte)o(grating)f(o)o(v)o(er)g Ft(s)p Fu(,)i(\(and)f(using)g(at)g (this)g(point)f(the)h(hypothesis)94 4539 y Ft(d)i Fn(\024)g Fu(3\))d(we)g(get)g(for)g(all)f Ft(\034)39 b Fn(2)28 b Fu([0)p Ft(;)17 b(\034)1289 4566 y Fo(\003)1334 4539 y Fu(])25 b(and)f(all)h Fn(j)p Ft(x)p Fn(j)i(\024)h Ft(\025)22 b Fn(\000)h Ft(`)p Fu(,)253 4838 y Fn(j)p Ft(w)355 4786 y Fp(\()p FA(2)p Fp(\))352 4860 y Fz(>)457 4838 y Fr(\()p Ft(x;)17 b(\034)11 b Fr(\))p Fn(j)43 b(\024)884 4702 y Fq(Z)984 4745 y Fz(\034)940 4929 y FA(0)1050 4838 y Fu(d)p Ft(s)17 b Fn(j)p Ft(Y)1250 4865 y Fz(\034)t(;s)1356 4838 y Fn(j)44 b(\024)1550 4702 y Fq(Z)1649 4745 y Fo(1)1605 4929 y FA(0)1750 4838 y Fu(d)p Ft(s)17 b(B)1945 4795 y Fo(\003)1940 4865 y FA(7)1990 4838 y Ft(M)2098 4795 y FA(1)p Fo(\000)p Fz(d=)p FA(4)2087 4865 y Fo(\003)2316 4838 y Fr(\()p Ft(\027)2404 4865 y Fo(\003)2450 4838 y Ft(k)2505 4795 y FA(2)2502 4865 y Fo(\003)2548 4838 y Fr(\))2587 4795 y Fz(d=)p FA(4)2709 4757 y Fq(\000)2754 4838 y Ft(K)29 b Fr(+)23 b Ft(Q)3047 4865 y Fo(\003)3104 4771 y Ft(\016)3148 4797 y Fo(\003)p 3104 4815 91 4 v 3129 4906 a Ft(`)3207 4757 y Fq(\001)3287 4771 y Ft(e)3333 4728 y Fo(\000)p Fz(\027)3435 4746 y Ff(\003)3478 4728 y Fz(k)3522 4685 y Fe(2)3520 4755 y Ff(\003)3562 4728 y Fz(s=)p FA(2)p 3264 4815 440 4 v 3264 4908 a Fr(\()p Ft(\027)3352 4935 y Fo(\003)3398 4908 y Ft(k)3453 4879 y FA(2)3450 4935 y Fo(\003)3496 4908 y Ft(s)p Fr(\))3582 4879 y Fz(d=)p FA(4)763 5111 y Fn(\024)44 b Ft(B)965 5069 y Fo(\003)960 5138 y FA(8)1011 5031 y Fq(\000)1094 5044 y Ft(M)1191 5071 y Fo(\003)p 1068 5089 193 4 v 1068 5180 a Ft(\027)1117 5207 y Fo(\003)1163 5180 y Ft(k)1218 5151 y FA(2)1215 5207 y Fo(\003)1273 5031 y Fq(\001)1319 5069 y FA(1)p Fo(\000)p Fz(d=)p FA(4)1538 5031 y Fq(\000)1584 5111 y Ft(K)29 b Fr(+)22 b Ft(Q)1876 5138 y Fo(\003)1934 5044 y Ft(\016)1978 5071 y Fo(\003)p 1934 5089 91 4 v 1958 5180 a Ft(`)2036 5031 y Fq(\001)2106 5111 y Ft(:)3619 5291 y Fr(\()p Fu(5)p Ft(:)p Fu(13)p Fr(\))p eop %%Page: 16 16 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(16)94 199 y Fu(This)19 b(completes)f(the)h(study)f(of)h(the)g(contrib)n (ution)e(of)i Ft(w)2013 147 y Fp(\()p FA(2)p Fp(\))2010 227 y FA(1)p Fz(;>)2155 199 y Fu(to)g(the)f(bound)h(of)g(Eq.\(5.9\),)g(when)g Ft(\027)3418 225 y Fo(\003)3464 199 y Ft(\034)39 b(>)28 b Fu(1)p Ft(=k)3807 156 y FA(2)3804 225 y Fo(\003)3849 199 y Fu(.)316 340 y(W)-8 b(e)27 b(ne)o(xt)f(deal)h(with)f(the)g(case)i Ft(\027)1463 367 y Fo(\003)1509 340 y Ft(\034)41 b Fn(\024)31 b Fu(1)p Ft(=k)1857 298 y FA(2)1854 367 y Fo(\003)1899 340 y Fu(.)42 b(The)27 b(contrib)n(ution)e (corresponding)h(to)g(the)h(\256rst)f(term)94 460 y(of)f(Eq.\(4.9\))e(is)h (treated)g(as)h(before)f(noting)f(that)h(in)f(Eq.\(5.13\))h(we)g(actually)g (inte)o(grate)f(o)o(v)o(er)h(all)f Ft(\034)11 b Fu(,)24 b(and)h(that)94 579 y(the)30 b(inequality)e Ft(\027)723 606 y Fo(\003)769 579 y Ft(\034)45 b(>)35 b Fu(1)p Ft(=k)1125 537 y FA(2)1122 606 y Fo(\003)1196 579 y Fu(w)o(as)30 b(not)f(used)g(an)o(ywhere)g(e)o(xcept)g (for)h(being)f(able)h(to)f(use)g(\(4.8\))h(which)f(is)94 699 y(the)e(same)g(as)g(the)f(\256rst)h(term)f(in)h(\(4.9\).)41 b(Thus,)27 b(we)g(consider)f(here)i(only)e(the)g(second)h(term.)41 b(Although)25 b(the)94 819 y(bounds)j(are)h(quite)e(similar)g(to)h(the)g(pre) n(vious)f(case,)j(it)d(might)g(be)i(better)f(to)g(actually)f(spell)h(them)f (out.)46 b(By)94 938 y(the)25 b(Schw)o(arz)h(inequality)d(and)i(Eq.\(4.9\),)g (we)g(no)n(w)f(bound)664 1296 y Ft(W)772 1254 y FA(2)758 1323 y Fz(t;s)898 1296 y Fr(=)1020 1161 y Fq(Z)1136 1296 y Fu(d)p Ft(y)1267 1205 y(B)1348 1162 y Fo(\003)1343 1232 y FA(3)1394 1205 y Fr(\()p Ft(p)p Fr(\))1522 1162 y FA(2)1561 1205 y Ft(e)1607 1162 y Fo(\000)p Fz(\027)1709 1180 y Ff(\003)1751 1162 y Fz(k)1795 1120 y Fe(2)1793 1189 y Ff(\003)1835 1162 y Fp(\()p Fz(t)p Fo(\000)p Fz(s)p Fp(\))2033 1125 y Fq(\000)2090 1156 y Fz(D)2155 1174 y Ff(\003)p 2090 1182 107 4 v 2103 1240 a Fz(\027)2143 1257 y Ff(\003)2209 1125 y Fq(\001)2255 1162 y FA(2)p Fz(p)p 1267 1273 1069 4 v 1336 1376 a Ft(k)1391 1334 y Fo(\000)p FA(2)p Fz(d)1388 1391 y Fo(\003)1534 1296 y Fq(\000)1580 1376 y Fu(1)e Fr(+)g Ft(k)1806 1348 y FA(2)1803 1403 y Fo(\003)1849 1376 y Fn(j)p Ft(x)g Fn(\000)g Ft(y)t Fn(j)2136 1348 y FA(2)2175 1296 y Fq(\001)2221 1335 y Fz(p)2359 1229 y Fu(1)g Fr(+)h(\()p Fn(j)p Ft(x)e Fn(\000)i Ft(y)t Fn(j)2857 1186 y FA(2)2896 1229 y Ft(=\016)2994 1186 y FA(2)2990 1256 y Fo(\003)3036 1229 y Fr(\))3075 1186 y FA(1)p Fp(+)p Fz(d=)p FA(2)p 2359 1273 934 4 v 2789 1365 a Ft(Z)893 1587 y Fn(\001)38 b Ft(M)1067 1544 y FA(2)1056 1614 y Fo(\003)1123 1451 y Fq(Z)1240 1587 y Fu(d)p Ft(z)21 b( )1422 1614 y Fz(x)1472 1587 y Fr(\()p Ft(z)t Fr(\))p Fn(j)p Ft(w)s Fr(\()p Ft(z)t(;)c(s)p Fr(\))p Fn(j)1950 1544 y FA(2)2013 1587 y Ft(:)94 1873 y Fu(Since)26 b(we)f(still)e(deal)i(with)f Fn(j)p Ft(x)p Fn(j)j(\024)h Ft(\025)23 b Fn(\000)f Ft(`)p Fu(,)j(we)g(get)f (from)h(\(5.10\),)221 2211 y Ft(W)329 2168 y FA(2)315 2238 y Fz(t;s)456 2211 y Fn(\024)45 b Ft(B)659 2168 y Fo(\003)654 2238 y FA(3)704 2211 y Fr(\()p Ft(p)p Fr(\))832 2168 y FA(2)872 2130 y Fq(\000)929 2144 y Ft(D)1011 2171 y Fo(\003)p 929 2188 129 4 v 946 2279 a Ft(\027)995 2306 y Fo(\003)1070 2130 y Fq(\001)1115 2168 y FA(2)p Fz(p)1196 2211 y Ft(Z)1271 2168 y Fo(\000)p FA(1)1390 2075 y Fq(Z)1506 2211 y Fu(d)p Ft(y)1637 2144 y(e)1683 2101 y Fo(\000)p Fz(\027)1785 2119 y Ff(\003)1827 2101 y Fz(k)1871 2058 y Fe(2)1869 2128 y Ff(\003)1911 2101 y Fp(\()p Fz(t)p Fo(\000)p Fz(s)p Fp(\))p 1637 2188 472 4 v 1774 2291 a Ft(k)1829 2249 y Fo(\000)p FA(2)p Fz(d)1826 2305 y Fo(\003)2121 2130 y Fq(\000)2166 2211 y Fu(1)22 b Fr(+)h Ft(k)2393 2168 y FA(2)2390 2238 y Fo(\003)2435 2211 y Fn(j)p Ft(x)f Fn(\000)h Ft(y)t Fn(j)2723 2168 y FA(2)2762 2130 y Fq(\001)2807 2168 y Fo(\000)p Fz(p)2916 2130 y Fq(\000)2961 2211 y Fu(1)f Fr(+)3145 2144 y Fn(j)p Ft(x)g Fn(\000)g Ft(y)t Fn(j)3432 2101 y FA(2)p 3145 2188 327 4 v 3263 2279 a Ft(\016)3311 2251 y FA(2)3307 2306 y Fo(\003)3483 2130 y Fq(\001)3529 2168 y FA(1)p Fp(+)p Fz(d=)p FA(2)451 2483 y Fn(\001)38 b Ft(M)625 2440 y FA(2)614 2510 y Fo(\003)664 2483 y Ft(B)745 2440 y Fo(\003)740 2510 y FA(6)791 2402 y Fq(\000)836 2483 y Ft(K)928 2440 y FA(2)990 2483 y Fr(+)23 b Ft(Q)1169 2440 y FA(2)1169 2510 y Fo(\003)1214 2402 y Fq(\000)1272 2415 y Ft(\016)1316 2442 y Fo(\003)p 1272 2460 91 4 v 1296 2551 a Ft(`)1374 2402 y Fq(\001)1420 2440 y FA(2)1460 2402 y Fq(\001)1530 2483 y Ft(:)94 2747 y Fu(Note)i(no)n(w)f(that)g(by)h(Eq.\(5.5\),)1528 2866 y Ft(\016)1576 2824 y FA(2)1572 2893 y Fo(\003)1663 2866 y Fr(=)44 b Ft(D)1866 2893 y Fo(\003)1913 2866 y Ft(=)-6 b(M)55 b Fn(\025)45 b Ft(k)2286 2824 y Fo(\000)p FA(2)2283 2893 y Fo(\003)2413 2866 y Ft(:)94 3054 y Fu(This)25 b(leads)f(to)671 3317 y Ft(W)779 3274 y FA(2)765 3344 y Fz(t;s)906 3317 y Fn(\024)45 b Fr(\()p Ft(B)1148 3274 y Fo(\003)1143 3344 y FA(3)1193 3317 y Fr(\()p Ft(p)p Fr(\)\))1360 3274 y FA(2)1399 3236 y Fq(\000)1457 3249 y Ft(D)1539 3276 y Fo(\003)p 1457 3294 129 4 v 1473 3385 a Ft(\027)1522 3412 y Fo(\003)1597 3236 y Fq(\001)1643 3274 y FA(2)p Fz(p)1724 3317 y Ft(Z)1799 3274 y Fo(\000)p FA(1)1917 3181 y Fq(Z)2034 3317 y Fu(d)p Ft(y)19 b(k)2207 3274 y FA(2)p Fz(d)2204 3344 y Fo(\003)2289 3236 y Fq(\000)2334 3317 y Fu(1)j Fr(+)h Ft(k)2561 3274 y FA(2)2558 3344 y Fo(\003)2604 3317 y Fn(j)p Ft(x)e Fn(\000)i Ft(y)t Fn(j)2891 3274 y FA(2)2930 3236 y Fq(\001)2976 3274 y FA(1)p Fp(+)p Fz(d=)p FA(2)p Fo(\000)p Fz(p)900 3588 y Fn(\001)39 b Ft(M)1075 3546 y FA(2)1064 3615 y Fo(\003)1114 3588 y Ft(B)1195 3546 y Fo(\003)1190 3615 y FA(6)1241 3508 y Fq(\000)1286 3588 y Ft(K)1378 3546 y FA(2)1440 3588 y Fr(+)23 b Ft(Q)1619 3546 y FA(2)1619 3615 y Fo(\003)1664 3508 y Fq(\000)1722 3521 y Ft(\016)1766 3548 y Fo(\003)p 1722 3566 91 4 v 1746 3657 a Ft(`)1824 3508 y Fq(\001)1870 3546 y FA(2)1909 3508 y Fq(\001)1980 3588 y Ft(:)94 3858 y Fu(T)-8 b(aking)25 b Ft(p)i Fr(=)i Ft(d)22 b Fr(+)g Fu(2)j(and)g(inte)o(grating)e(o)o (v)o(er)h Ft(y)k Fu(we)d(get)g(this)f(time,)g(for)h(some)f Ft(B)2866 3815 y Fo(\003)2861 3885 y FA(9)2911 3858 y Fu(:)1115 4149 y Ft(W)1223 4106 y FA(2)1209 4176 y Fz(t;s)1349 4149 y Fn(\024)1471 4069 y Fq(\000)1517 4149 y Ft(B)1598 4106 y Fo(\003)1593 4176 y FA(9)1643 4069 y Fq(\001)1689 4106 y FA(2)1729 4149 y Ft(\016)1777 4106 y Fz(d)1773 4176 y Fo(\003)1823 4149 y Ft(k)1878 4106 y Fo(\000)p Fz(d)1875 4176 y Fo(\003)1987 4149 y Ft(k)2042 4106 y FA(2)p Fz(d)2039 4176 y Fo(\003)2123 4149 y Ft(M)2231 4106 y FA(2)2220 4176 y Fo(\003)2271 4069 y Fq(\000)2316 4149 y Ft(K)29 b Fr(+)23 b Ft(Q)2609 4176 y Fo(\003)2666 4082 y Ft(\016)2710 4109 y Fo(\003)p 2666 4126 V 2691 4218 a Ft(`)2768 4069 y Fq(\001)2814 4106 y FA(2)1349 4394 y Fr(=)1471 4313 y Fq(\000)1517 4394 y Ft(B)1598 4351 y Fo(\003)1593 4421 y FA(9)1643 4313 y Fq(\001)1689 4351 y FA(2)1729 4394 y Ft(M)1837 4351 y FA(2)1826 4421 y Fo(\003)1876 4394 y Fr(\()p Ft(\016)1959 4421 y Fo(\003)2005 4394 y Ft(k)2057 4421 y Fo(\003)2103 4394 y Fr(\))2142 4351 y Fz(d)2188 4313 y Fq(\000)2233 4394 y Ft(K)29 b Fr(+)23 b Ft(Q)2526 4421 y Fo(\003)2583 4327 y Ft(\016)2627 4354 y Fo(\003)p 2583 4371 V 2608 4462 a Ft(`)2685 4313 y Fq(\001)2731 4351 y FA(2)2796 4394 y Ft(:)3619 4262 y Fr(\()p Fu(5)p Ft(:)p Fu(14)p Fr(\))94 4678 y Fu(T)-8 b(aking)25 b(the)f(square)h(root)g(of)g(this) e(bound,)h(inte)o(grating)f(o)o(v)o(er)h Ft(s)p Fu(,)h(and)g(noting)e(that)i Ft(\027)3009 4705 y Fo(\003)3055 4678 y Ft(k)3110 4635 y FA(2)3107 4705 y Fo(\003)3152 4678 y Ft(\034)39 b Fn(\024)28 b Fu(1,)c(we)i(get)826 4832 y Fq(Z)926 4875 y Fz(\034)881 5059 y FA(0)992 4968 y Fu(d)p Ft(s)17 b Fn(j)p Ft(W)1228 4995 y Fz(\034)t(;s)1334 4968 y Fn(j)44 b(\024)1527 4832 y Fq(Z)1627 4875 y Fz(\034)1583 5059 y FA(0)1693 4968 y Fu(d)p Ft(s)17 b(B)1888 4925 y Fo(\003)1883 4995 y FA(9)1933 4968 y Ft(M)2030 4995 y Fo(\003)2075 4968 y Fr(\()p Ft(\016)2158 4995 y Fo(\003)2204 4968 y Ft(k)2256 4995 y Fo(\003)2302 4968 y Fr(\))2341 4925 y Fz(d=)p FA(2)2463 4887 y Fq(\000)2508 4968 y Ft(K)29 b Fr(+)23 b Ft(Q)2801 4995 y Fo(\003)2858 4900 y Ft(\016)2902 4927 y Fo(\003)p 2858 4945 V 2883 5036 a Ft(`)2960 4887 y Fq(\001)1406 5236 y Fr(=)44 b Ft(B)1608 5193 y Fo(\003)1603 5263 y FA(9)1654 5155 y Fq(\000)1711 5169 y Ft(D)1793 5195 y Fo(\003)p 1711 5213 129 4 v 1728 5304 a Ft(\027)1777 5331 y Fo(\003)1852 5155 y Fq(\001)1897 5193 y Fz(d=)p FA(4)2019 5155 y Fq(\000)2102 5169 y Ft(M)2199 5195 y Fo(\003)p 2077 5213 193 4 v 2077 5304 a Ft(\027)2126 5331 y Fo(\003)2172 5304 y Ft(k)2227 5276 y FA(2)2224 5331 y Fo(\003)2282 5155 y Fq(\001)2327 5193 y FA(1)p Fo(\000)p Fz(d=)p FA(4)2547 5155 y Fq(\000)2592 5236 y Ft(K)29 b Fr(+)23 b Ft(Q)2885 5263 y Fo(\003)2942 5169 y Ft(\016)2986 5195 y Fo(\003)p 2942 5213 91 4 v 2967 5304 a Ft(`)3044 5155 y Fq(\001)3115 5236 y Ft(:)3619 5106 y Fr(\()p Fu(5)p Ft(:)p Fu(15)p Fr(\))p eop %%Page: 17 17 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(17)94 194 y Fu(This)25 b(completes)e(the)i(bound)f(for)h Ft(\027)1336 221 y Fo(\003)1382 194 y Ft(k)1437 151 y FA(2)1434 221 y Fo(\003)1480 194 y Ft(\034)38 b Fn(\024)28 b Fu(1.)316 335 y(The)36 b(contrib)n(ution)e(from)i Ft(w)1332 283 y Fp(\()p FA(1)p Fp(\))1329 364 y FA(1)p Fz(;>)1491 335 y Fu(is)g(easier)g(to)g(bound.) 69 b(Using)35 b(the)h(de\256nition)f(of)h Ft(w)3363 283 y Fp(\()p FA(1)p Fp(\))3360 364 y FA(1)p Fz(;>)3486 335 y Fu(,)j(we)d(split)94 477 y(the)29 b(con)l(v)n(olution)f(inte)o(gral)g(into)g(the)h(re)o(gion)f Fn(j)p Ft(y)t Fn(j)33 b(\024)h Ft(\025)c Fu(and)f(its)f(complement.)48 b(Because)30 b Ft(\027)3308 504 y Fo(\003)3354 477 y Ft(\034)3398 504 y Fo(\003)3477 477 y Fn(\025)k Ft(k)3643 434 y Fo(\000)p FA(2)3640 504 y Fo(\003)3774 477 y Fu(by)94 596 y(Eq.\(5.6\))23 b(we)g(can)f(use)h(the)f(bound)g(of)h(Eq.\(4.8\),)f(and)h(we)g(get)f(for)h (the)g(\256rst)f(contrib)n(ution)f(when)i Ft(p)k(>)h(d)18 b Fr(+)f Fu(1,)159 947 y Ft(B)240 904 y Fo(\003)235 974 y FA(3)285 947 y Fr(\()p Ft(p)p Fr(\))430 743 y Fq(\014)430 803 y(\014)430 862 y(\014)430 922 y(\014)430 982 y(\014)430 1042 y(\014)462 811 y(Z)518 1038 y Fo(j)p Fz(y)s Fo(j\024)p Fz(\025)639 947 y Fu(d)p Ft(y)1097 880 y(e)1143 837 y Fo(\000)p Fz(\027)1245 855 y Ff(\003)1287 837 y Fz(k)1331 794 y Fe(2)1329 864 y Ff(\003)1371 837 y Fz(\034)1408 855 y Ff(\003)1450 837 y Fz(=)p FA(2)p 803 924 1021 4 v 803 1038 a Fr(\()p Ft(\027)891 1065 y Fo(\003)937 1038 y Ft(\034)981 1065 y Fo(\003)1027 1038 y Fr(\))1066 1009 y Fz(d=)p FA(2)1187 957 y Fq(\000)1233 1038 y Fu(1)22 b Fr(+)1417 990 y Fo(j)p Fz(x)p Fo(\000)p Fz(y)s Fo(j)1615 970 y Fe(2)p 1417 1015 228 4 v 1438 1072 a Fz(D)1503 1090 y Ff(\003)1545 1072 y Fz(\034)1582 1090 y Ff(\003)1656 957 y Fq(\001)1702 996 y Fz(p=)p FA(2)1836 947 y Ft(w)s Fr(\()p Ft(y)t(;)17 b Fu(0)p Fr(\))2136 743 y Fq(\014)2136 803 y(\014)2136 862 y(\014)2136 922 y(\014)2136 982 y(\014)2136 1042 y(\014)2211 947 y Fn(\024)45 b Ft(B)2414 904 y Fo(\003)2409 974 y FA(10)2483 947 y Ft(e)2529 904 y Fo(\000)p Fz(\027)2631 922 y Ff(\003)2673 904 y Fz(k)2717 861 y Fe(2)2715 931 y Ff(\003)2757 904 y Fz(\034)2794 922 y Ff(\003)2836 904 y Fz(=)p FA(2)2964 947 y Fu(sup)2933 1038 y Fo(j)p Fz(y)s Fo(j\024)p Fz(\025)3149 947 y Fn(j)p Ft(w)s Fr(\()p Ft(y)t(;)17 b Fu(0)p Fr(\))p Fn(j)23 b Ft(:)63 b Fr(\()p Fu(5)p Ft(:)p Fu(16)p Fr(\))94 1301 y Fu(F)o(or)17 b(the)g(second)f(term,)i (where)f Fn(j)p Ft(y)t Fn(j)27 b(\025)h Ft(\025)p Fu(,)19 b(the)d (restriction)g(of)h(the)f(bound)g(to)h Fn(j)p Ft(x)p Fn(j)26 b(\024)j Ft(\025)t Fn(\000)t Ft(`)17 b Fu(implies)e Fn(j)p Ft(x)t Fn(\000)t Ft(y)t Fn(j)27 b(\025)i Ft(`)p Fu(,)94 1421 y(and)c(then)g(we)g(get)f(using)g Ft(\034)1032 1447 y Fo(\003)1105 1421 y Fr(=)k Fu(1)p Ft(=)-6 b(M)1401 1447 y Fo(\003)1447 1421 y Fu(:)774 1759 y Ft(B)855 1716 y Fo(\003)850 1786 y FA(3)900 1759 y Fr(\()p Ft(p)p Fr(\))1028 1584 y Fq(\014)1028 1644 y(\014)1028 1704 y(\014)1028 1764 y(\014)1028 1823 y(\014)1078 1623 y(Z)1133 1849 y Fo(j)p Fz(x)p Fo(\000)p Fz(y)s Fo(j\025)p Fz(`)1348 1759 y Fu(d)p Ft(y)1822 1691 y(e)1868 1648 y Fo(\000)p Fz(\027)1970 1666 y Ff(\003)2012 1648 y Fz(k)2056 1606 y Fe(2)2054 1675 y Ff(\003)2096 1648 y Fz(\034)2133 1666 y Ff(\003)2175 1648 y Fz(=)p FA(2)p 1529 1736 1021 4 v 1529 1850 a Fr(\()p Ft(\027)1617 1876 y Fo(\003)1663 1850 y Ft(\034)1707 1876 y Fo(\003)1752 1850 y Fr(\))1791 1821 y Fz(d=)p FA(2)1913 1769 y Fq(\000)1958 1850 y Fu(1)22 b Fr(+)2142 1801 y Fo(j)p Fz(x)p Fo(\000)p Fz(y)s Fo(j)2340 1781 y Fe(2)p 2142 1827 228 4 v 2163 1884 a Fz(D)2228 1902 y Ff(\003)2270 1884 y Fz(\034)2307 1902 y Ff(\003)2381 1769 y Fq(\001)2427 1808 y Fz(p=)p FA(2)2561 1759 y Ft(w)s Fr(\()p Ft(y)t(;)17 b Fu(0)p Fr(\))2861 1584 y Fq(\014)2861 1644 y(\014)2861 1704 y(\014)2861 1764 y(\014)2861 1823 y(\014)1089 2128 y Fn(\024)44 b Ft(B)1291 2085 y Fo(\003)1286 2155 y FA(11)1373 2061 y Fr(\()p Ft(D)1494 2088 y Fo(\003)1540 2061 y Ft(\034)1584 2088 y Fo(\003)1629 2061 y Fr(\))1668 2018 y FA(1)p Fz(=)p FA(2)p 1373 2105 411 4 v 1557 2197 a Ft(`)1795 2128 y Fr(\()p Ft(D)1916 2155 y Fo(\003)1962 2128 y Ft(\034)2006 2155 y Fo(\003)2052 2128 y Fr(\))2091 2085 y Fz(d=)p FA(2)2224 2061 y Ft(e)2270 2018 y Fo(\000)p Fz(\027)2372 2036 y Ff(\003)2415 2018 y Fz(k)2459 1975 y Fe(2)2457 2045 y Ff(\003)2499 2018 y Fz(\034)2536 2036 y Ff(\003)2577 2018 y Fz(=)p FA(2)p 2224 2105 434 4 v 2249 2198 a Fr(\()p Ft(\027)2337 2225 y Fo(\003)2383 2198 y Ft(\034)2427 2225 y Fo(\003)2472 2198 y Fr(\))2511 2170 y Fz(d=)p FA(2)2686 2128 y Fu(sup)2734 2210 y Fz(y)2841 2128 y Fn(j)p Ft(w)s Fr(\()p Ft(y)t(;)17 b Fu(0)p Fr(\))p Fn(j)1089 2410 y Fr(=)44 b Ft(B)1291 2367 y Fo(\003)1286 2436 y FA(11)1373 2342 y Ft(\016)1417 2369 y Fo(\003)p 1373 2387 91 4 v 1397 2478 a Ft(`)1475 2329 y Fq(\000)1532 2342 y Ft(D)1614 2369 y Fo(\003)p 1532 2387 129 4 v 1549 2478 a Ft(\027)1598 2505 y Fo(\003)1673 2329 y Fq(\001)1718 2367 y Fz(d=)p FA(2)1840 2410 y Ft(e)1886 2367 y Fo(\000)p Fz(\027)1988 2385 y Ff(\003)2031 2367 y Fz(k)2075 2324 y Fe(2)2073 2394 y Ff(\003)2115 2367 y Fz(\034)2152 2385 y Ff(\003)2193 2367 y Fz(=)p FA(2)2274 2410 y Ft(Q)2353 2436 y Fo(\003)2423 2410 y Ft(:)3619 2063 y Fr(\()p Fu(5)p Ft(:)p Fu(17)p Fr(\))94 2699 y Fu(Combining)24 b(Eqs.\(5.13\)\261\(5.17\),)f(the)i(inequality)e(\(5.9\))i (follo)n(ws.)316 2819 y(The)h(proof)h(of)f(Eq.\(5.8\))g(is)g(v)o(ery)g (similar)f(to)h(the)h(one)f(gi)n(v)o(en)f(abo)o(v)o(e,)h(and)g(we)h(indicate) e(just)h(the)g(fe)n(w)94 2938 y(modi\256cations)19 b(needed.)34 b(Instead)20 b(of)f(the)h(k)o(ernel)g Fk(G)1878 2965 y Fz(\034)t(;>)2029 2938 y Fu(we)g(no)n(w)f(use)h(the)f(k)o(ernel)h Fk(G)2976 2965 y Fz(\034)3026 2938 y Fu(,)h(and)e(we)h(will)f(call)h Ft(Z)3778 2965 y Fz(t;s)94 3078 y Fu(the)31 b(quantity)f(corresponding)g(to)g Ft(Y)1372 3105 y Fz(t;s)1500 3078 y Fu(b)n(ut)g(with)g Fk(G)1924 3105 y Fz(\034)2005 3078 y Fu(replaced)i(by)e Fk(G)2564 3105 y Fz(\034)t(;>)2696 3078 y Fu(.)54 b(Consider)31 b(\256rst)g Ft(\027)3401 3105 y Fo(\003)3447 3078 y Ft(\034)47 b(>)37 b Fu(1)p Ft(=k)3807 3036 y FA(2)3804 3105 y Fo(\003)3849 3078 y Fu(.)94 3198 y(Since)27 b(the)e(bound)g(\(4.8\))h(is)f(of)h(the)g(same)f (type)h(as)g(the)f(bound)g(\(4.1\),)h(b)n(ut)f(without)g(he)h(e)o(xponential) e(f)o(actor)l(,)94 3318 y(all)i(bounds)f(go)h(through)f(as)h(before)h(up)f (to)f(the)h(inequality)f(\(5.12\),)h(which)g(is)f(replaced)i(by)f(a)g (similar)f(one,)94 3437 y(b)n(ut)i(without)f(the)h(e)o(xponential)e(f)o (actor)-5 b(.)43 b(T)-8 b(aking)26 b(again)h(the)g(square)g(root)g(and)g (inte)o(grating)e(o)o(v)o(er)h Ft(s)p Fu(,)i(we)f(get)94 3557 y(for)f Ft(d)h Fn(\024)h Fu(3,)836 3716 y Fn(j)p Ft(w)938 3673 y Fp(\()p FA(2)p Fp(\))1040 3716 y Fr(\()p Ft(x;)17 b(\034)11 b Fr(\))p Fn(j)43 b(\024)1468 3581 y Fq(Z)1567 3623 y Fz(\034)1523 3807 y FA(0)1634 3716 y Fu(d)p Ft(s)17 b Fn(j)p Ft(Z)1844 3743 y Fz(\034)t(;s)1949 3716 y Fn(j)44 b(\024)2143 3581 y Fq(Z)2242 3623 y Fz(\034)2279 3641 y Ff(\003)2198 3807 y FA(0)2342 3716 y Fu(d)p Ft(s)17 b Fr(\()p Ft(B)2576 3673 y Fo(\003)2571 3743 y FA(7)2621 3716 y Fr(\))2660 3673 y FA(2)1340 3984 y Fn(\001)39 b Ft(M)1515 3941 y FA(1)p Fo(\000)p Fz(d=)p FA(4)1504 4011 y Fo(\003)1733 3984 y Fr(\()p Ft(\027)1821 4011 y Fo(\003)1867 3984 y Ft(k)1922 3941 y FA(2)1919 4011 y Fo(\003)1965 3984 y Fr(\))2004 3941 y Fz(d=)p FA(4)2126 3904 y Fq(\000)2171 3984 y Ft(K)29 b Fr(+)23 b Ft(Q)2464 4011 y Fo(\003)2521 3917 y Ft(\016)2565 3944 y Fo(\003)p 2521 3961 91 4 v 2546 4053 a Ft(`)2623 3904 y Fq(\001)2876 3917 y Fu(1)p 2681 3961 440 4 v 2681 4054 a Fr(\()p Ft(\027)2769 4081 y Fo(\003)2815 4054 y Ft(k)2870 4026 y FA(2)2867 4081 y Fo(\003)2912 4054 y Ft(s)p Fr(\))2998 4026 y Fz(d=)p FA(4)1346 4258 y Fn(\024)45 b Ft(B)1549 4215 y Fo(\003)1544 4285 y FA(12)1618 4258 y Ft(M)1726 4215 y FA(1)p Fo(\000)p Fz(d=)p FA(4)1715 4285 y Fo(\003)1945 4177 y Fq(\000)1990 4258 y Ft(K)29 b Fr(+)23 b Ft(Q)2283 4285 y Fo(\003)2340 4190 y Ft(\016)2384 4217 y Fo(\003)p 2340 4235 91 4 v 2365 4326 a Ft(`)2442 4177 y Fq(\001)2488 4258 y Ft(\034)2543 4215 y FA(1)p Fo(\000)p Fz(d=)p FA(4)2532 4285 y Fo(\003)1346 4502 y Fr(=)45 b Ft(B)1549 4460 y Fo(\003)1544 4529 y FA(12)1618 4422 y Fq(\000)1663 4502 y Ft(K)29 b Fr(+)23 b Ft(Q)1956 4529 y Fo(\003)2013 4435 y Ft(\016)2057 4462 y Fo(\003)p 2013 4480 V 2038 4571 a Ft(`)2116 4422 y Fq(\001)2186 4502 y Ft(:)3619 4101 y Fr(\()p Fu(5)p Ft(:)p Fu(18)p Fr(\))94 4752 y Fu(This)29 b(bounds)f(the)h(contrib)n(ution)e(of)i Ft(w)1469 4701 y Fp(\()p FA(2)p Fp(\))1466 4781 y FA(1)1601 4752 y Fu(to)f(Eq.\(5.8\))h(where)h Ft(\027)2385 4779 y Fo(\003)2431 4752 y Ft(\034)44 b(>)34 b Fu(1)p Ft(=k)2785 4710 y FA(2)2782 4779 y Fo(\003)2827 4752 y Fu(.)49 b(In)29 b(the)g(opposite)f(case,)j(we)94 4872 y(ar)n(gue)i(e)o (xactly)e(as)h(in)f(the)h(proof)f(of)h(\(5.9\),)i(since)d(the)h(e)o (xponential)e(f)o(actor)i(in)g(\(4.9\))g(w)o(as)f(an)o(yw)o(ay)h(of)f(no)94 5008 y(use)e(before.)47 b(Finally)-6 b(,)28 b(the)h(contrib)n(ution)d(of)j Ft(w)1762 4956 y Fp(\()p FA(1)p Fp(\))1759 5036 y FA(1)1892 5008 y Fu(is)f(bounded)g(in)g(e)o(xactly)g(the)g(same)g(w)o(ay)h(as)f(the)h (one)f(of)94 5150 y Ft(w)168 5099 y Fp(\()p FA(1)p Fp(\))165 5179 y FA(1)p Fz(;>)292 5150 y Fu(,)e(e)o(xcept)g(for)h(the)f(e)o(xponential) f(f)o(actor)l(,)i(and)g(we)f(get)h(the)f(bound)g(\(5.8\).)40 b(The)27 b(proof)f(of)g(Theorem)g(5.1)94 5270 y(is)f(complete.)p eop %%Page: 18 18 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(18)94 194 y Fv(6.)46 b(The)e(Sampling)94 383 y Fu(W)-8 b(e)26 b(assume)e(that)g Ft(w)813 410 y FA(1)881 383 y Fr(=)k Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b(t)27 b Fr(=)h Ft(\034)1384 410 y Fo(\003)1429 383 y Fr(\))d Fu(satis\256es)f(the)h(\252sampling)e (bound\272)1692 602 y Fn(j)p Ft(w)1791 629 y FA(1)1830 602 y Fn(j)1858 629 y Fz(\016)1894 647 y Ff(\003)1936 629 y Fz(;\025)2056 602 y Fn(\024)45 b Ft(")25 b(;)1392 b Fr(\()p Fu(6)p Ft(:)p Fu(1)p Fr(\))94 820 y Fu(where)1244 940 y Fn(j)p Ft(f)11 b Fn(j)1360 967 y Fz(\016)1396 985 y Ff(\003)1437 967 y Fz(;\025)1557 940 y Fn(\021)293 b Fu(sup)1679 1033 y Fo(f)p Fz(n)p Fo(2)p Fa(Z)1878 1013 y Fd(d)1944 1033 y FA(:)25 b Fz(\016)2024 1051 y Ff(\003)2066 1033 y Fo(j)p Fz(n)p Fo(j\024)p Fz(\025)p Fo(g)2330 940 y Fn(j)p Ft(f)11 b Fr(\()p Ft(n\016)2561 967 y Fo(\003)2606 940 y Fr(\))p Fn(j)24 b Ft(;)944 b Fr(\()p Fu(6)p Ft(:)p Fu(2)p Fr(\))94 1199 y Fu(with)25 b Ft(")44 b Fn(\024)h Ft(Q)588 1226 y Fo(\003)634 1199 y Fu(.)35 b(Furthermore,)25 b(we)g(assume)1573 1418 y(sup)1541 1508 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1759 1418 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b Fu(0)p Fr(\))p Fn(j)43 b(\024)h Ft(K)32 b(:)1241 b Fr(\()p Fu(6)p Ft(:)p Fu(3)p Fr(\))94 1712 y Fu(Then)25 b(we)g(ha)n(v)o(e)g(the)94 1903 y Fs(Theorem)42 b(6.1.)47 b Fc(Consider)29 b(the)f(solutions)f Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b(t)p Fr(\))27 b Fc(of)h(Eq.\(5.1\),)i(and)e(assume)g(that) g(the)h(bounds)e(\(5.2\),)94 2023 y(\(5.3\))34 b(hold.)60 b(There)34 b(are)g(constants)e Ft(B)1488 1980 y Fo(\003)1483 2050 y FA(13)1557 2023 y Fc(,)k Ft(B)1699 1980 y Fo(\003)1694 2050 y FA(14)1768 2023 y Fc(,)f(and)f Ft(B)2087 1980 y Fo(\003)2082 2050 y FA(15)2189 2023 y Fc(independent)e(of)i Ft(")p Fc(,)i Ft(K)7 b Fc(,)35 b(and)e Ft(\025)g Fc(such)g(that)g(the)94 2142 y(follo)n(wing)f(holds:)51 b(If)33 b(the)g(initial)f(condition)f(satis\256es)i(\(6.3\))g(and)g Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b(\034)2718 2169 y Fo(\003)2762 2142 y Fr(\))33 b Fc(satis\256es)f(Eq.\(6.1\))h(with)f(the)94 2262 y(de\256nition)24 b(Eq.\(5.5\))h(of)g Ft(\016)1005 2289 y Fo(\003)1051 2262 y Fc(,)g(then)1539 2381 y Fu(sup)1494 2472 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1696 2452 y Ff(0)1739 2381 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b(\034)2026 2408 y Fo(\003)2070 2381 y Fr(\))p Fn(j)44 b(\024)h Ft(K)2395 2339 y Fo(0)2447 2381 y Ft(;)94 2637 y Fc(where)1538 2756 y Ft(K)1630 2713 y Fo(0)1702 2756 y Fr(=)g Ft(B)1905 2713 y Fo(\003)1900 2783 y FA(13)1974 2756 y Ft(")23 b Fr(+)f Fu(3)p Ft(K)r(=)p Fu(4)i Ft(;)1238 b Fr(\()p Fu(6)p Ft(:)p Fu(4)p Fr(\))94 2936 y Fc(and)25 b Ft(\025)321 2894 y Fo(0)377 2936 y Fr(=)j Ft(\025)22 b Fn(\000)h Ft(B)743 2894 y Fo(\003)738 2963 y FA(14)834 2936 y Fn(\000)g Ft(B)1015 2894 y Fo(\003)1010 2963 y FA(15)1084 2936 y Ft(=K)7 b Fc(.)94 3166 y Fs(Remark.)42 b Fu(Since)29 b(3)p Ft(=)p Fu(4)49 b Ft(<)h Fu(1,)29 b(Eq.\(6.4\))f(sho)n(ws)f(that)h Ft(K)2145 3124 y Fo(0)2202 3166 y Fu(is)g(smaller)f(than)i Ft(K)35 b Fu(as)29 b(long)e(as)i Ft(K)39 b(>)34 b Fu(4)p Ft(B)3734 3124 y Fo(\003)3729 3193 y FA(13)3803 3166 y Ft(")p Fu(.)94 3286 y(\(In)g(f)o(act,)j(by)c(choosing)g(dif)n(ferent)g(constants\320in)f (particular)h(a)i(lar)n(ge)f Ft(k)2714 3313 y Fo(\003)2759 3286 y Fu(\320we)g(can)g(achie)n(v)o(e)f(an)o(y)g(ratio)94 3406 y Ft(\032)28 b(>)g Fu(0)d(instead)f(of)h(the)f(3)p Ft(=)p Fu(4.\))316 3525 y(A)f(corollary)g(of)h(the)f(proof)g(is)g(the)h(follo)n (wing)d(result)i(which)g(can)h(be)f(vie)n(wed)g(as)g(a)h(generalization)f(of) 94 3645 y(the)i(dissipati)n(v)o(e)d(bounds)i(of)h(Theorem)f(5.1.)94 3836 y Fs(Corollary)39 b(6.2.)d Fc(In)25 b(dimension)e Ft(d)28 b Fn(\024)g Fu(3)d Fc(we)g(ha)n(v)o(e)f(the)h(follo)n(wing)e(bounds)895 4101 y Fu(sup)815 4192 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)p Fo(\000)p Fz(`)1129 4101 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b(\034)1416 4128 y Fo(\003)1460 4101 y Fr(\))p Fn(j)44 b(\024)h Ft(B)1774 4058 y Fo(\003)1769 4128 y FA(16)1843 4101 y Fn(j)p Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b(\034)2101 4128 y Fo(\003)2145 4101 y Fr(\))p Fn(j)2212 4128 y Fz(\016)2248 4146 y Ff(\003)2289 4128 y Fz(;\025)2387 4101 y Fr(+)2499 4034 y Fu(3)p 2499 4078 50 4 v 2499 4170 a(4)2609 4101 y(sup)2577 4192 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)2796 4101 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)g Fu(0)p Fr(\))p Fn(j)1582 4396 y Fr(+)23 b Ft(B)1763 4353 y Fo(\003)1758 4422 y FA(17)1844 4328 y Ft(\016)1888 4355 y Fo(\003)p 1844 4373 91 4 v 1868 4464 a Ft(`)1946 4396 y Fn(k)p Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b Fu(0)p Fr(\))p Fn(k)2321 4422 y Fo(1)2428 4396 y Ft(:)3669 4240 y Fr(\()p Fu(6)p Ft(:)p Fu(5)p Fr(\))94 4752 y Fs(Pro)s(of.)38 b Fu(Our)23 b(proof)h(is)e(based)i(on)f (sampling)f(theory)h(for)g(functions)f(in)h(the)h(Bernstein)f(classes,)g (\(see,)h FB(e)o(.g)o(.,)94 4872 y Fu([B]\).)94 5047 y Fs(De\014nition.)36 b Fu(W)-8 b(e)26 b(call)e Fk(B)1061 5074 y Fz(S;\033)1212 5047 y Fu(the)g(Bernstein)h(class)g(of)g(entire)f(analytic)h(functions)e Ft(f)36 b Fu(bounded)24 b(by)1580 5291 y Fn(j)p Ft(f)11 b Fr(\()p Ft(z)t Fr(\))p Fn(j)44 b(\024)g Ft(S)6 b(e)2102 5248 y Fz(\033)s Fo(j)p FA(Im)14 b Fz(z)s Fo(j)2361 5291 y Ft(:)p eop %%Page: 19 19 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(19)94 194 y Fu(Our)25 b(\256rst)g(observ)n(ation)f(is)g(that)g Fk(G)1267 221 y Fz(\034)t(;<)1424 194 y Fu(maps)g(into)g(a)h(Bernstein)g (class)f Fk(B)2609 221 y Fz(B)s(;)p FA(2)p Fz(k)2773 239 y Ff(\003)2820 194 y Fu(:)94 386 y Fs(Lemma)39 b(6.3.)d Fc(The)25 b(con)l(v)n(olution)e(operator)i(with)f(k)o(ernel)1086 667 y Fk(G)1146 694 y Fz(\034)t(;<)1277 667 y Fr(\()p Ft(x)e Fn(\000)h Ft(y)t Fr(\))43 b(=)1752 531 y Fq(Z)1868 667 y Fu(d)p Ft(k)19 b(e)2035 624 y Fz(ik)r Fp(\()p Fz(x)p Fo(\000)p Fz(y)s Fp(\))2326 667 y Ft(e)2372 624 y Fo(\000)p Fz(D)r(k)2545 581 y Fe(2)2576 624 y Fz(\034)2625 667 y Ft(\037)2687 694 y Fz(k)r()g(N)41 b Fu(is)28 b(bounded)h(by)g Fk(O)p Fr(\()p Ft(N)1273 629 y Fo(\000)p FA(1)1375 672 y Fr(\))p Fn(k)p Ft(f)11 b Fn(k)1574 699 y Fo(1)1658 672 y Fu(.)49 b(In)30 b(the)f(second)g(e)o(xpression,)g(we)h(assume)f(for)g(the)g(moment)94 792 y(that)c Ft(x)i Fn(2)i Fr(\()p Ft(x)544 819 y FA(0)583 792 y Ft(;)17 b(x)685 819 y FA(2)724 792 y Fr(\))p Fu(.)36 b(Then)24 b(the)h(sum)f(o)o(v)o(er)g Fn(j)p Ft(n)p Fn(j)j Ft(>)h(N)36 b Fu(is)24 b(bounded)g(again)g(by)h Fk(O)p Fr(\()p Ft(N)2978 749 y Fo(\000)p FA(1)3080 792 y Fr(\))p Fn(k)p Ft(f)11 b Fn(k)3279 819 y Fo(1)3362 792 y Fu(.)94 983 y Fs(Prop)s(osition)39 b(6.6.)d Fc(Assume)24 b(that)g Ft(f)39 b Fn(2)28 b Fk(B)1706 1010 y Fz(S;\033)1831 983 y Fc(.)36 b(Assume)24 b(furthermore)h(that)f Ft(f)35 b Fc(satis\256es)25 b(the)g(bounds)1423 1216 y Fn(j)p Ft(f)11 b Fr(\()p Ft(x)1607 1243 y Fz(j)1648 1216 y Fr(\))p Fn(j)27 b Ft(<)h(\013)75 b Fc(for)58 b Fn(j)p Ft(j)6 b Fn(j)43 b(\024)i Ft(J)35 b(;)94 1461 y Fc(where)26 b Ft(x)420 1487 y Fz(j)489 1461 y Fr(=)j Ft(\031)t(j)6 b(=)p Fr(\()p Fu(2)p Ft(\033)t Fr(\))m Fc(.)36 b(Assume)24 b(furthermore)h(that)f Fn(k)p Ft(f)11 b Fn(k)2181 1487 y Fo(1)2292 1461 y Fn(\024)28 b Ft(S)2464 1418 y Fo(0)2492 1461 y Fc(.)36 b(Then,)25 b(one)f(has)h(the)g (bound)761 1773 y Fn(j)p Ft(f)11 b Fr(\()p Ft(x)p Fr(\))p Fn(j)42 b(\024)j Ft(C)1247 1800 y FA(6)1287 1773 y Ft(\013)23 b Fr(+)1485 1705 y Ft(C)1556 1732 y FA(7)1596 1705 y Ft(S)1663 1663 y Fo(0)p 1485 1750 206 4 v 1543 1841 a Ft(N)1778 1773 y Fc(for)33 b(all)58 b Fn(j)p Ft(x)p Fn(j)27 b(\024)h Ft(\031)t Fr(\()p Ft(J)k Fn(\000)22 b Ft(N)33 b Fn(\000)23 b Fu(1)p Fr(\))p Ft(=)p Fr(\()p Fu(2)p Ft(\033)t Fr(\))g Ft(:)94 2032 y Fc(The)30 b(constants)e Ft(C)750 2059 y FA(6)820 2032 y Fc(and)h Ft(C)1064 2059 y FA(7)1134 2032 y Fc(are)h(independent)f(of)g(the)h(parameters)f(of)h(the)f(problem.)49 b(W)-8 b(e)30 b(shall)f(assume)94 2152 y Ft(C)165 2178 y FA(6)233 2152 y Ft(>)f Fu(1)p Fc(.)94 2388 y Fs(Pro)s(of.)40 b Fu(W)-8 b(e)25 b(\256rst)g(bound)f Ft(f)1112 2345 y Fo(0)1139 2388 y Fr(\()p Fu(0)p Fr(\))h Fu(using)e(Eq.\(6.8\),)i(and)f(this)g(leads)h(to)709 2670 y Ft(\033)770 2627 y Fo(\000)p FA(1)872 2670 y Fn(j)p Ft(f)960 2627 y Fo(0)987 2670 y Fr(\()p Fu(0)p Fr(\))p Fn(j)43 b(\024)1345 2602 y Fu(4)p 1320 2647 101 4 v 1320 2738 a Ft(\031)1381 2709 y FA(2)1432 2670 y Ft(\013)1555 2575 y Fq(X)1513 2793 y Fo(j)p Fz(n)p Fo(j)p Fz()3059 4335 y Fu(.)35 b(Setting)21 b Ft(\025)3481 4293 y Fo(00)3560 4335 y Fr(=)28 b Ft(\025)16 b Fn(\000)g Ft(`)94 4455 y Fu(and)25 b(using)f(the)h(bound)f(Eq.\(5.9\),)g(we)h(get)838 4744 y(sup)782 4834 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)984 4814 y Ff(00)1049 4744 y Fn(j)p Ft(w)1148 4771 y FA(1)p Fz(;>)1274 4744 y Fr(\()p Ft(x)p Fr(\))p Fn(j)118 b(\024)45 b Ft(B)1758 4701 y Fo(\003)1753 4771 y FA(5)1820 4603 y Fq(\022)1893 4744 y Ft(e)1939 4701 y Fo(\000)p Fp(\()p Fz(\027)2072 4719 y Ff(\003)2115 4701 y Fz(k)2159 4658 y Fe(2)2157 4728 y Ff(\003)2199 4701 y Fz(=)l(M)2312 4719 y Ff(\003)2353 4701 y Fp(\))p Fz(=)p FA(2)2487 4744 y Fr(+)2586 4663 y Fq(\000)2669 4677 y Ft(M)2766 4703 y Fo(\003)p 2644 4721 193 4 v 2644 4812 a Ft(\027)2693 4839 y Fo(\003)2739 4812 y Ft(k)2794 4784 y FA(2)2791 4839 y Fo(\003)2849 4663 y Fq(\001)2894 4701 y FA(1)p Fo(\000)p Fz(d=)p FA(4)3113 4603 y Fq(\023)1475 4986 y Fn(\001)1541 4906 y Fq(\000)1632 4986 y Fu(sup)1587 5077 y Fo(j)p Fz(x)p Fo(j)13 b(\024)h Fz(\025)1833 4986 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)j Fu(0)p Fr(\))p Fn(j)j Fr(+)i(\()p Ft(\016)2395 5013 y Fo(\003)2441 4986 y Ft(=`)p Fr(\))p Fn(k)p Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b Fu(0)p Fr(\))p Fn(k)2947 5013 y Fo(1)3029 4906 y Fq(\001)3099 4986 y Ft(:)3619 4877 y Fr(\()p Fu(6)p Ft(:)p Fu(10)p Fr(\))p 94 5195 1200 4 v 233 5248 a Fg(\003)316 5291 y Fx(In)i(this)h(section,)h Fi(N)29 b Fx(denotes)20 b(just)g(an)g(inte)o(ger)m(,)g(and)g(not)g(the)g(number)g(of) f(components)h(of)f Fi(u)p Fx(.)p eop %%Page: 21 21 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(21)316 194 y Fu(W)-8 b(e)28 b(no)n(w)f(be)o(gin)g(\256xing)h(the)g (constants:)41 b(Recall)28 b(that)g Ft(Q)2310 221 y Fo(\003)2355 194 y Fu(,)h Ft(M)2506 221 y Fo(\003)2580 194 y Fu(and)f Ft(\016)2796 221 y Fo(\003)2874 194 y Fr(=)33 b Ft(D)3066 221 y Fo(\003)3112 194 y Ft(\034)3156 221 y Fo(\003)3229 194 y Fu(are)c(gi)n(v)o(en)e(by)g(the) 94 314 y(parameters)e(of)g(the)g(problem.)35 b(W)-8 b(e)25 b(ha)n(v)o(e)g(also)f(assumed,)g(as)h(a)g(hypothesis)e(of)i(Theorem)f(6.1,)g (that)1573 537 y(sup)1541 628 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1759 537 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b Fu(0)p Fr(\))p Fn(j)43 b(\024)h Ft(K)32 b(:)94 844 y Fu(W)-8 b(e)25 b(choose)e Ft(`)28 b Fr(=)g Ft(\016)768 871 y Fo(\003)814 844 y Ft(Q)893 871 y Fo(\003)938 844 y Ft(=K)7 b Fu(,)24 b(so)f(that)h(the)g(last)f(f)o(actor)h (in)g(\(6.10\))f(is)h(bounded)f(by)g(2)p Ft(K)7 b Fu(.)35 b(W)-8 b(e)25 b(ne)o(xt)e(choose)g Ft(k)3828 871 y Fo(\003)94 964 y Fu(so)i(lar)n(ge)g(that)1028 1140 y Ft(B)1109 1097 y Fo(\003)1104 1167 y FA(5)1171 999 y Fq(\022)1244 1140 y Ft(e)1290 1097 y Fo(\000)p Fp(\()p Fz(\027)1423 1115 y Ff(\003)1465 1097 y Fz(k)1509 1054 y Fe(2)1507 1124 y Ff(\003)1549 1097 y Fz(=)l(M)1662 1115 y Ff(\003)1703 1097 y Fp(\))p Fz(=)p FA(2)1837 1140 y Fr(+)1937 1059 y Fq(\000)2020 1072 y Ft(M)2117 1099 y Fo(\003)p 1995 1117 193 4 v 1995 1208 a Ft(\027)2044 1235 y Fo(\003)2090 1208 y Ft(k)2145 1179 y FA(2)2142 1235 y Fo(\003)2199 1059 y Fq(\001)2245 1097 y FA(1)p Fo(\000)p Fz(d=)p FA(4)2464 999 y Fq(\023)2582 1140 y Fn(\024)2771 1072 y Fu(1)p 2716 1117 161 4 v 2716 1208 a(8)p Ft(C)2837 1235 y FA(6)2913 1140 y Ft(:)678 b Fr(\()p Fu(6)p Ft(:)p Fu(11)p Fr(\))94 1376 y Fu(More)25 b(precisely)-6 b(,)24 b(we)h(let)739 1599 y Ft(k)791 1626 y Fo(\003)881 1599 y Fr(=)45 b Ft(C)1074 1626 y FA(8)1114 1599 y Fu(16)p Ft(C)1285 1626 y FA(6)1325 1519 y Fq(\000)1370 1599 y Ft(B)1451 1557 y Fo(\003)1446 1626 y FA(5)1497 1519 y Fq(\001)1542 1557 y FA(1)p Fz(=)p Fp(\()p FA(2)p Fo(\000)p Fz(d=)p FA(2)p Fp(\))1900 1599 y Fr(\()p Ft(M)2036 1626 y Fo(\003)2081 1599 y Ft(=\027)2180 1626 y Fo(\003)2226 1599 y Fr(\))2265 1557 y FA(1)p Fz(=)p FA(2)2424 1599 y Fn(\021)g Ft(B)2627 1557 y Fo(\003)2622 1626 y FA(21)2696 1599 y Fr(\()p Ft(M)2832 1626 y Fo(\003)2878 1599 y Ft(=\027)2977 1626 y Fo(\003)3022 1599 y Fr(\))3061 1557 y FA(1)p Fz(=)p FA(2)3202 1599 y Ft(;)389 b Fr(\()p Fu(6)p Ft(:)p Fu(12)p Fr(\))94 1823 y Fu(and)34 b(this)e(is)h(our)g(\256nal)h (choice)f(for)h Ft(k)1421 1850 y Fo(\003)1467 1823 y Fu(.)61 b(Clearly)-6 b(,)36 b(if)d Ft(C)2066 1850 y FA(8)2139 1823 y Fu(is)g(suf)n(\256ciently)g(lar)n(ge,)i(both)e(terms)g(in)g(the)g(sum)94 1943 y(\(6.11\))25 b(will)f(contrib)n(ute)g(less)g(than)h(1)p Ft(=)p Fr(\()p Fu(16)p Ft(C)1648 1970 y FA(6)1687 1943 y Fr(\))p Fu(.)94 2119 y Fs(Remark.)41 b Fu(It)27 b(is)f(at)h(this)f(point)g(crucial)h (that)g(the)g(construction)e(of)i(the)g(quantities)e Ft(B)3144 2076 y Fo(\003)3139 2146 y FA(0)3190 2119 y Ft(;)17 b(B)3316 2076 y Fo(\003)3311 2146 y FA(1)3360 2119 y Ft(;)g(:)g(:)g(:)e(;)26 b Fu(did)h FB(not)94 2238 y Fu(depend)g(on)g Ft(k)588 2265 y Fo(\003)634 2238 y Fu(,)g(since)g(all)f(these)h(constants)f(depend\320as)g (we)i(ha)n(v)o(e)e(said)h(before\320only)f(on)h(the)f(quotient)94 2358 y Ft(D)176 2385 y Fo(\003)223 2358 y Ft(=\027)322 2385 y Fo(\003)368 2358 y Fu(.)38 b(In)25 b(particular)l(,)h(for)f(problems)g (where)h Ft(u)f Fu(had)h(only)f(one)g(component,)g(this)f(w)o(ould)h(mean)g (that)g(the)94 2477 y Ft(B)175 2434 y Fo(\003)170 2504 y Fz(j)246 2477 y Fu(are)h(just)f(pure)g(numerical)g(f)o(actors,)g(since)g(then)g Ft(D)2018 2504 y Fo(\003)2093 2477 y Fr(=)k Ft(\027)2248 2504 y Fo(\003)2294 2477 y Fu(.)37 b(It)25 b(is)g(also)g(important)f(to)g(note)h (that)g Ft(k)3659 2504 y Fo(\003)3730 2477 y Fu(and)94 2597 y Ft(\016)138 2624 y Fo(\003)209 2597 y Fu(do)g FB(not)f Fu(depend)h(on)f (the)h(quantities)e Ft(\025)i Fu(or)g Ft(K)32 b Fu(which)24 b(occur)i(in)e(Theorem)h(6.1.)316 2716 y(Thus,)f(so)g(f)o(ar)l(,)i(with)e (our)h(choices)f(we)h(conclude)g(from)f(Eq.\(6.10\))h(that)1466 2987 y(sup)1409 3077 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1611 3057 y Ff(00)1677 2987 y Fn(j)p Ft(w)1776 3013 y FA(1)p Fz(;>)1902 2987 y Fr(\()p Ft(x)p Fr(\))p Fn(j)43 b(\024)2297 2919 y Fu(1)p 2242 2964 V 2242 3055 a(4)p Ft(C)2363 3082 y FA(6)2415 2987 y Ft(K)31 b(:)1060 b Fr(\()p Fu(6)p Ft(:)p Fu(13)p Fr(\))94 3286 y Fu(W)-8 b(e)26 b(ne)o(xt)e(observ)o(e)g(that)g Ft(w)1023 3312 y FA(1)p Fz(;<)1177 3286 y Fr(=)k Ft(w)1353 3312 y FA(1)1416 3286 y Fn(\000)22 b Ft(w)1586 3312 y FA(1)p Fz(;>)1737 3286 y Fu(and)j(so)f(\(6.13\))h(leads)g(to)658 3575 y Fn(j)p Ft(w)757 3601 y FA(1)p Fz(;<)883 3575 y Fr(\()p Ft(n\016)1026 3601 y Fo(\003)1072 3575 y Fr(\))p Fn(j)43 b(\024)i(j)p Ft(w)1403 3601 y FA(1)1443 3575 y Fr(\()p Ft(n\016)1586 3601 y Fo(\003)1632 3575 y Fr(\))p Fn(j)21 b Fr(+)i Fn(j)p Ft(w)1919 3601 y FA(1)p Fz(;>)2044 3575 y Fr(\()p Ft(n\016)2187 3601 y Fo(\003)2233 3575 y Fr(\))p Fn(j)44 b(\024)h(j)p Ft(w)2565 3601 y FA(1)2604 3575 y Fr(\()p Ft(n\016)2747 3601 y Fo(\003)2793 3575 y Fr(\))p Fn(j)22 b Fr(+)3049 3507 y Fu(1)p 2993 3552 V 2993 3643 a(4)p Ft(C)3114 3670 y FA(6)3166 3575 y Ft(K)32 b(;)308 b Fr(\()p Fu(6)p Ft(:)p Fu(14)p Fr(\))94 3866 y Fu(pro)o(vided)24 b Ft(\016)516 3893 y Fo(\003)562 3866 y Fn(j)p Ft(n)p Fn(j)44 b(\024)g Ft(\025)901 3824 y Fo(00)952 3866 y Fu(.)316 3986 y(By)24 b(construction,)e Ft(w)1068 4013 y FA(1)p Fz(;<)1218 3986 y Fu(has)h(a)h(F)o(ourier)f (transform)g(with)f(support)h(in)g Fn(j)p Ft(k)s Fn(j)k(\024)h Fu(2)p Ft(k)3137 4013 y Fo(\003)3206 3986 y Fu(and)23 b(furthermore,)94 4105 y(by)i(Lemma)f(6.4,)h(it)f(is)g(in)h Fk(B)1073 4132 y Fz(S;\033)1223 4105 y Fu(with)f Ft(S)34 b Fr(=)28 b Ft(B)1706 4063 y Fo(\003)1701 4132 y FA(19)1800 4105 y Fu(and)d Ft(\033)31 b Fr(=)d Fu(2)p Ft(k)2264 4132 y Fo(\003)2310 4105 y Fu(.)316 4237 y(W)-8 b(e)41 b(can)h(no)n(w)e(apply)g(Proposition)g(6.6)g(to)h(the)f (function)h Ft(f)60 b Fr(=)51 b Ft(w)2768 4264 y FA(1)p Fz(;<)2894 4237 y Fu(,)45 b(with)40 b Ft(S)3249 4195 y Fo(0)3327 4237 y Fr(=)50 b Ft(S)56 b Fr(=)51 b Ft(B)3780 4195 y Fo(\003)3775 4264 y FA(19)3849 4237 y Fu(.)94 4369 y(Choosing)41 b Ft(J)61 b Fr(=)52 b Fu([2)p Ft(\033)t(\025)965 4327 y Fo(00)1015 4369 y Ft(=\031)t Fu(])41 b(\(here,)47 b([)p Fn(\001)p Fu(])42 b(is)f(the)g(inte)o (ger)g(part\))h(we)g(conclude)f(that,)46 b(for)41 b Fn(j)p Ft(x)p Fn(j)51 b(\024)g Ft(\025)3717 4327 y Fo(0)3797 4369 y Fn(\021)94 4489 y Ft(\025)152 4446 y Fo(00)226 4489 y Fn(\000)22 b Ft(\016)369 4516 y Fo(\003)438 4489 y Fn(\000)g Ft(\031)t Fr(\()p Ft(N)32 b Fr(+)23 b Fu(1)p Fr(\))p Ft(=)p Fr(\()p Fu(2)p Ft(k)1129 4516 y Fo(\003)1174 4489 y Fr(\))p Fu(,)h(one)h(has)g(a)g(bound) 1110 4711 y Fn(j)p Ft(w)1209 4737 y FA(1)p Fz(;<)1335 4711 y Fr(\()p Ft(x)p Fr(\))p Fn(j)43 b(\024)i Ft(C)1734 4737 y FA(6)1774 4711 y Fn(j)p Ft(w)1873 4737 y FA(1)p Fz(;<)1999 4711 y Fn(j)2027 4737 y Fz(\016)2063 4755 y Ff(\003)2105 4737 y Fz(;\025)2176 4717 y Ff(00)2252 4711 y Fr(+)22 b Ft(C)2422 4737 y FA(7)2462 4711 y Ft(N)2553 4667 y Fo(\000)p FA(1)2655 4711 y Ft(B)2736 4667 y Fo(\003)2731 4737 y FA(19)2830 4711 y Ft(:)3619 4713 y Fr(\()p Fu(6)p Ft(:)p Fu(15)p Fr(\))94 4936 y Fu(It)j(is)g(useful)f(to)g(introduce)h Ft(B)1125 4894 y Fo(\003)1120 4963 y FA(22)1222 4936 y Fr(=)j Ft(C)1398 4963 y FA(7)1438 4936 y Ft(B)1519 4894 y Fo(\003)1514 4963 y FA(19)1613 4936 y Fu(Using)c(Eq.\(6.14\),)g(this)g(leads)h(to)887 5213 y(sup)841 5304 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1043 5284 y Ff(0)1087 5213 y Fn(j)p Ft(w)1186 5240 y FA(1)p Fz(;<)1312 5213 y Fr(\()p Ft(x)p Fr(\))p Fn(j)43 b(\024)i Ft(C)1711 5240 y FA(6)1751 5133 y Fq(\000)1796 5213 y Fn(j)p Ft(w)1895 5240 y FA(1)1935 5213 y Fn(j)1963 5240 y Fz(\016)1999 5258 y Ff(\003)2041 5240 y Fz(;\025)2112 5220 y Ff(00)2188 5213 y Fr(+)2355 5146 y Fu(1)p 2299 5190 V 2299 5282 a(4)p Ft(C)2420 5309 y FA(6)2472 5213 y Ft(K)2564 5133 y Fq(\001)2632 5213 y Fr(+)22 b Ft(N)2822 5171 y Fo(\000)p FA(1)2924 5213 y Ft(B)3005 5171 y Fo(\003)3000 5240 y FA(22)3100 5213 y Ft(:)3619 5230 y Fr(\()p Fu(6)p Ft(:)p Fu(16)p Fr(\))p eop %%Page: 22 22 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(22)94 194 y Fu(W)-8 b(e)26 b(ne)o(xt)e(choose)1672 340 y Ft(N)55 b Fr(=)1961 272 y Fu(1)p 1941 317 92 4 v 1941 408 a Ft(K)2044 340 y Fu(4)p Ft(B)2175 297 y Fo(\003)2170 367 y FA(22)2269 340 y Ft(:)1322 b Fr(\()p Fu(6)p Ft(:)p Fu(17)p Fr(\))94 546 y Fu(Then)25 b(we)g(get)1210 666 y(sup)1165 756 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1367 736 y Ff(0)1410 666 y Fn(j)p Ft(w)1509 692 y FA(1)p Fz(;<)1635 666 y Fr(\()p Ft(x)p Fr(\))p Fn(j)44 b(\024)g Ft(C)2034 692 y FA(6)2074 666 y Fn(j)p Ft(w)2173 692 y FA(1)2213 666 y Fn(j)2241 692 y Fz(\016)2277 710 y Ff(\003)2319 692 y Fz(;\025)2390 673 y Ff(00)2466 666 y Fr(+)22 b Ft(K)r(=)p Fu(2)i Ft(:)815 b Fr(\()p Fu(6)p Ft(:)p Fu(18)p Fr(\))94 921 y Fu(Combining)24 b(Eqs.\(6.18\))g(and)h(\(6.13\),)f(we) h(get)359 1188 y(sup)313 1279 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)515 1259 y Ff(0)559 1188 y Fn(j)p Ft(w)658 1215 y FA(1)698 1188 y Fr(\()p Ft(x)p Fr(\))p Fn(j)43 b(\024)90 b Fu(sup)1026 1279 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1228 1259 y Ff(0)1271 1188 y Fn(j)p Ft(w)1370 1215 y FA(1)p Fz(;<)1496 1188 y Fr(\()p Ft(x)p Fr(\))p Fn(j)21 b Fr(+)79 b Fu(sup)1780 1279 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1982 1259 y Ff(00)2048 1188 y Fn(j)p Ft(w)2147 1215 y FA(1)p Fz(;>)2272 1188 y Fr(\()p Ft(x)p Fr(\))p Fn(j)44 b(\024)g Ft(C)2671 1215 y FA(6)2712 1188 y Fn(j)p Ft(w)2811 1215 y FA(1)2850 1188 y Fn(j)2878 1215 y Fz(\016)2914 1233 y Ff(\003)2956 1215 y Fz(;\025)3027 1195 y Ff(0)3081 1188 y Fr(+)3192 1121 y Ft(K)p 3192 1165 V 3213 1257 a Fu(2)3318 1188 y Fr(+)3464 1121 y Ft(K)p 3430 1165 161 4 v 3430 1257 a Fu(4)p Ft(C)3551 1283 y FA(6)3628 1188 y Ft(:)94 1484 y Fu(Since)26 b(we)f(ha)n(v)o(e)f(tak)o(en)h Ft(C)1002 1511 y FA(6)1070 1484 y Ft(>)j Fu(1,)d(we)g(see)g(that)f(we)h(get)g(\256nally)1239 1703 y(sup)1194 1794 y Fo(j)p Fz(x)p Fo(j\024)p Fz(\025)1396 1774 y Ff(0)1439 1703 y Fn(j)p Ft(w)1538 1730 y FA(1)1578 1703 y Fr(\()p Ft(x)p Fr(\))p Fn(j)44 b(\024)g Ft(C)1977 1730 y FA(6)2017 1703 y Fn(j)p Ft(w)2116 1730 y FA(1)2156 1703 y Fn(j)2184 1730 y Fz(\016)2220 1748 y Ff(\003)2262 1730 y Fz(;\025)2333 1710 y Ff(0)2387 1703 y Fr(+)22 b Fu(3)p Ft(K)r(=)p Fu(4)i Ft(;)94 1998 y Fu(and)1047 2093 y Ft(\025)1105 2050 y Fo(0)1177 2093 y Fr(=)45 b Ft(\025)1357 2050 y Fo(00)1430 2093 y Fn(\000)23 b Ft(\016)1574 2120 y Fo(\003)1642 2093 y Fn(\000)f Ft(\031)t Fr(\()p Ft(N)33 b Fr(+)22 b Fu(1)p Fr(\))p Ft(=)p Fr(\()p Fu(2)p Ft(k)2333 2120 y Fo(\003)2378 2093 y Fr(\))1177 2243 y(=)45 b Ft(\025)22 b Fn(\000)h Ft(\016)1523 2269 y Fo(\003)1569 2243 y Ft(B)1650 2200 y Fo(\003)1645 2269 y FA(22)1719 2243 y Ft(=K)29 b Fn(\000)22 b Ft(\016)2026 2269 y Fo(\003)2095 2243 y Fn(\000)g Ft(\031)t Fr(\()p Ft(N)32 b Fr(+)23 b Fu(1)p Fr(\))p Ft(=)p Fr(\()p Fu(2)p Ft(k)2786 2269 y Fo(\003)2831 2243 y Fr(\))h Ft(:)94 2424 y Fu(From)33 b(the)g(de\256nition)f(of)h Ft(N)50 b Fr(=)39 b Fk(O)p Fr(\()p Fu(1)p Ft(=K)7 b Fr(\))32 b Fu(it)h(follo)n(ws)e (at)i(once)g(that)f(there)h(are)h(constants)e Ft(B)3445 2381 y Fo(\003)3440 2450 y FA(14)3547 2424 y Fu(and)h Ft(B)3805 2381 y Fo(\003)3800 2450 y FA(15)94 2543 y Fu(such)25 b(that)1460 2663 y Ft(\025)1518 2620 y Fo(0)1590 2663 y Fn(\025)45 b Ft(\025)22 b Fn(\000)h Ft(B)1973 2620 y Fo(\003)1968 2690 y FA(14)2064 2663 y Fn(\000)g Ft(B)2245 2620 y Fo(\003)2240 2690 y FA(15)2314 2663 y Ft(=K)32 b(:)94 2843 y Fu(The)25 b(proof)g(of)g(Theorem)f(6.1)h(is)f (complete.)94 3035 y Fs(Theorem)39 b(6.7.)f Fu(\(Sampling)25 b(bound\))g Fc(Consider)g(the)h(solutions)d Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b(t)p Fr(\))24 b Fc(of)i(Eq.\(5.1\),)f(and)h(assume)f(that) 94 3154 y(the)g(bounds)f(\(5.2\),)h(\(5.3\))f(hold.)35 b(There)26 b(are)f(constants)f Ft(B)2129 3111 y Fo(\003)2124 3181 y FA(23)2198 3154 y Fc(,)h Ft(B)2329 3111 y Fo(\003)2324 3181 y FA(24)2423 3154 y Fc(and)g Ft(B)2673 3111 y Fo(\003)2668 3181 y FA(25)2767 3154 y Fc(such)g(that)f(if)1517 3374 y Fn(j)p Ft(w)s Fr(\()p Ft(i\016)1736 3400 y Fo(\003)1782 3374 y Ft(;)17 b(t)k Fn(\000)i Ft(j)6 b(\034)2075 3400 y Fo(\003)2120 3374 y Fr(\))p Fn(j)44 b(\024)g Ft(")26 b(;)1167 b Fr(\()p Fu(6)p Ft(:)p Fu(19)p Fr(\))94 3593 y Fc(for)26 b(all)e Fn(j)p Ft(i)p Fn(j)j(\024)i Fr(\()p Ft(L)21 b Fr(+)i Ft(B)892 3550 y Fo(\003)887 3620 y FA(23)961 3593 y Ft(=")p Fr(\))p Ft(=\016)1190 3620 y Fo(\003)1261 3593 y Fc(and)i(all)f Fn(j)p Ft(j)6 b Fn(j)27 b(\024)h Ft(B)1870 3550 y Fo(\003)1865 3620 y FA(24)1956 3593 y Fu(log)o Fr(\()p Fu(1)p Ft(=")p Fr(\))p Fc(,)d(then)1527 3812 y Fu(sup)1492 3902 y Fo(j)p Fz(x)p Fo(j\024)p Fz(L)1718 3812 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b(t)p Fr(\))p Fn(j)42 b(\024)j Ft(B)2309 3769 y Fo(\003)2304 3839 y FA(25)2378 3812 y Ft(")25 b(:)1142 b Fr(\()p Fu(6)p Ft(:)p Fu(20)p Fr(\))94 4219 y Fs(Remark.)50 b Fu(It)37 b(will)g(be)g(seen)g(from)g(the)f(proof)h(that)g(it)f(suf)n (\256ces)h(to)g(sample)f(on)h(a)g(some)n(what)f(smaller)l(,)94 4339 y(non-rectangular)25 b(domain)f(for)h(\(6.20\))g(to)f(hold.)35 b(See)26 b(also)e(Fig.)h(1.)94 4514 y Fs(Pro)s(of.)44 b Fu(W)-8 b(e)30 b(obtain)f(the)g(result)g(of)h(Theorem)f(6.7)g(by)g(iteration)g(of)g (Theorem)h(6.1.)49 b(W)l(ith)29 b Ft(")h Fu(gi)n(v)o(en)e(as)i(in)94 4634 y(the)25 b(statement)f(of)h(Theorem)f(6.7,)h(let)f Ft(m)i Fu(be)f(the)f(smallest)g(inte)o(ger)g(for)h(which)1638 4853 y Fr(\()p Fu(3)p Ft(=)p Fu(4)p Fr(\))1866 4810 y Fz(m)1941 4853 y Ft(Q)2020 4880 y Fo(\003)2110 4853 y Fn(\024)44 b Ft(")26 b(;)94 5072 y Fu(and)f(note)g(that)f(then)1000 5291 y Ft(m)45 b Fn(\024)f Fu(log)p Fr(\()p Ft(Q)1499 5318 y Fo(\003)1544 5291 y Ft(=")p Fr(\))p Ft(=)17 b Fu(log)o Fr(\()p Fu(4)p Ft(=)p Fu(3)p Fr(\))k(+)i Fu(1)44 b Fn(\024)g Ft(C)2508 5318 y FA(9)2565 5291 y Fu(log)o Fr(\()p Fu(1)p Ft(=")p Fr(\))25 b Ft(;)p eop %%Page: 23 23 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(23)94 194 y Fu(when)32 b Ft(")37 b(<)552 155 y FA(1)p 552 171 35 4 v 552 228 a(2)599 194 y Fu(.)56 b(The)32 b(number)f Ft(m)h Fu(will)e(be)i(the)g(number)f(of)g(time)g(steps)g(needed)h(to)f(achie) n(v)o(e)g(a)h(precision)94 314 y Ft(B)175 271 y Fo(\003)170 341 y FA(25)245 314 y Ft(")38 b Fu(in)f(Eq.\(6.20\),)j(where)e(we)g(de\256ne) g Ft(B)1683 271 y Fo(\003)1678 341 y FA(25)1798 314 y Fr(=)46 b Fu(2)p Ft(B)2052 271 y Fo(\003)2047 341 y FA(13)2121 314 y Fu(,)41 b(and)d Ft(B)2450 271 y Fo(\003)2445 341 y FA(24)2564 314 y Fr(=)46 b Ft(C)2758 341 y FA(9)2798 314 y Fu(.)75 b(Let)37 b(no)n(w)g Ft(K)3360 341 y FA(0)3445 314 y Fr(=)46 b Ft(Q)3647 341 y Fo(\003)3730 314 y Fu(and)94 433 y Ft(K)179 460 y Fz(j)t Fp(+)p FA(1)345 433 y Fr(=)28 b Ft(B)531 390 y Fo(\003)526 460 y FA(13)600 433 y Ft(")23 b Fr(+)f Fu(3)p Ft(K)903 460 y Fz(j)945 433 y Ft(=)p Fu(4,)i(and)h(let)g Ft(L)1456 460 y Fz(m)1559 433 y Fr(=)j Ft(L)p Fu(,)d(and)f Ft(L)2018 460 y Fz(j)t Fo(\000)p FA(1)2185 433 y Fr(=)k Ft(L)2358 460 y Fz(j)2422 433 y Fr(+)23 b Ft(\016)2566 460 y Fo(\003)2634 433 y Fr(+)f Ft(B)2814 390 y Fo(\003)2809 460 y FA(15)2884 433 y Ft(=K)3019 460 y Fz(j)t Fo(\000)p FA(1)3157 433 y Fu(.)36 b(Assume)24 b(no)n(w)1517 683 y Fn(j)p Ft(w)s Fr(\()p Ft(i\016)1736 710 y Fo(\003)1782 683 y Ft(;)17 b(t)k Fn(\000)i Ft(j)6 b(\034)2075 710 y Fo(\003)2120 683 y Fr(\))p Fn(j)44 b(\024)g Ft(")26 b(;)1167 b Fr(\()p Fu(6)p Ft(:)p Fu(21)p Fr(\))94 932 y Fu(for)24 b(all)e Fn(j)p Ft(i)p Fn(j)28 b(\024)g Ft(L)647 959 y Fz(m)p Fo(\000)p Fz(i)813 932 y Ft(=\016)907 959 y Fo(\003)976 932 y Fu(and)23 b(for)g(all)f Ft(j)34 b Fr(=)28 b Fu(0)p Ft(;)17 b(:)g(:)g(:)d(;)j(m)p Fu(.)35 b(The)23 b(reader)h(can)f(check)h(easily)e(that)g(our)h (de\256nitions)94 1052 y(are)j(made)f(such)f(that)h(Theorem)f(6.1)h(applies)f (at)h(each)g(time)f(step)g(considered.)36 b(Since)25 b(we)g(ha)n(v)o(e)1462 1301 y Fn(k)p Ft(w)s Fr(\()p Fn(\001)p Ft(;)17 b(t)k Fn(\000)h Ft(j)6 b(\034)1945 1328 y Fo(\003)1990 1301 y Fr(\))p Fn(k)2079 1328 y Fo(1)2208 1301 y Fn(\024)44 b Ft(Q)2408 1328 y Fo(\003)2479 1301 y Ft(;)94 1551 y Fu(for)26 b(all)e Ft(j)6 b Fu(,)25 b(we)g(can)g (inducti)n(v)o(ely)d(bound,)i(for)h Ft(j)33 b Fr(=)c Fu(1)p Ft(;)17 b(:)g(:)g(:)d(;)j(m)p Fu(,)833 1800 y(sup)778 1891 y Fo(j)p Fz(x)p Fo(j\024)p Fz(L)988 1909 y Fd(j)1042 1800 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)g(t)j Fn(\000)j Fr(\()p Ft(m)f Fn(\000)h Ft(j)6 b Fr(\))p Ft(\034)1819 1827 y Fo(\003)1864 1800 y Fr(\))p Fn(j)44 b(\024)g Ft(C)2167 1827 y FA(6)2207 1800 y Ft(")23 b Fr(+)g Fu(3)p Ft(K)2511 1827 y Fz(j)t Fo(\000)p FA(1)2649 1800 y Ft(=)p Fu(4)44 b Fr(=)h Ft(K)3000 1827 y Fz(j)t Fp(+)p FA(1)3162 1800 y Ft(:)94 2157 y Fu(Note)25 b(no)n(w)f(that)g Ft(K)766 2184 y Fz(j)836 2157 y Fr(=)k Ft(Q)1020 2184 y Fo(\003)1065 2157 y Ft(\032)1117 2114 y Fz(j)1181 2157 y Fr(+)23 b Ft(C)1352 2184 y FA(6)1392 2157 y Ft(")p Fr(\()p Fu(1)f Fn(\000)g Ft(\032)1700 2114 y Fz(j)1742 2157 y Fr(\))p Ft(=)p Fr(\()p Fu(1)f Fn(\000)i Ft(\032)p Fr(\))p Fu(,)h(where)h Ft(\032)i Fr(=)i Fu(3)p Ft(=)p Fu(4)24 b(and)h(thus)1240 2523 y Ft(L)1308 2550 y Fz(m)p Fo(\000)p Fz(j)1527 2523 y Fr(=)45 b Ft(L)22 b Fr(+)h Ft(j)6 b(\016)1930 2550 y Fo(\003)1998 2523 y Fr(+)22 b Ft(B)2178 2480 y Fo(\003)2173 2550 y FA(15)2318 2394 y Fz(j)2264 2428 y Fq(X)2274 2641 y Fz(i)p Fp(=)p FA(1)2425 2523 y Ft(K)2517 2480 y Fo(\000)p FA(1)2510 2551 y Fz(m)p Fo(\000)p Fz(i)2700 2523 y Ft(:)94 2880 y Fu(Clearly)-6 b(,)25 b(there)g(is)g(a)g Ft(C)887 2907 y FA(10)989 2880 y Ft(>)j Fu(0)d(such)g(that)f Ft(K)1636 2907 y Fz(j)1705 2880 y Fn(\025)k Ft(C)1881 2907 y FA(10)1956 2880 y Ft(\032)2008 2837 y Fz(j)2050 2880 y Fu(,)d(so)f(that)g(we)i(get)1229 3246 y Ft(L)1297 3273 y Fz(m)p Fo(\000)p Fz(j)1516 3246 y Fr(=)44 b Ft(L)22 b Fr(+)h Ft(j)6 b(\016)1918 3273 y Fo(\003)1986 3246 y Fr(+)2098 3179 y Ft(B)2179 3136 y Fo(\003)2174 3205 y FA(15)p 2098 3223 151 4 v 2100 3314 a Ft(C)2171 3341 y FA(10)2330 3117 y Fz(j)2276 3151 y Fq(X)2286 3364 y Fz(i)p Fp(=)p FA(1)2537 3179 y Fu(1)p 2449 3223 227 4 v 2449 3314 a Ft(\032)2501 3286 y Fz(m)p Fo(\000)p Fz(j)2712 3246 y Ft(:)94 3582 y Fu(In)25 b(particular)l(,)g(we)g(can)g(\256nd)g(some)f Ft(C)1420 3609 y FA(11)1495 3582 y Fu(,)h(so)g(that)671 3902 y Ft(L)739 3929 y FA(0)823 3902 y Fn(\024)45 b Ft(L)22 b Fr(+)g Ft(m\016)1265 3929 y Fo(\003)1334 3902 y Fr(+)1445 3835 y Ft(B)1526 3792 y Fo(\003)1521 3862 y FA(15)p 1445 3880 151 4 v 1447 3971 a Ft(C)1518 3998 y FA(10)1688 3835 y Ft(\032)1740 3792 y Fo(\000)p Fz(m)p 1620 3880 326 4 v 1620 3971 a Ft(\032)1672 3942 y Fo(\000)p FA(1)1795 3971 y Fn(\000)h Fu(1)2001 3902 y Fn(\024)45 b Ft(L)22 b Fr(+)g Ft(C)2383 3929 y FA(9)2423 3902 y Ft(\016)2467 3929 y Fo(\003)2530 3902 y Fu(log)o Fr(\()p Fu(1)p Ft(=")p Fr(\))g(+)h Ft(C)3074 3929 y FA(11)3149 3902 y Ft(=")i(:)94 4214 y Fu(Note)j(that)f Ft(L)561 4241 y FA(0)628 4214 y Fu(is)g(the)h(width)f(of)g(the)h (\252earliest\272)g(bound)e(in)i(\(6.19\),)g(and)f(we)h(choose)f Ft(B)3167 4171 y Fo(\003)3162 4241 y FA(23)3268 4214 y Fr(=)32 b Ft(C)3448 4241 y FA(9)3488 4214 y Ft(\016)3532 4241 y Fo(\003)3602 4214 y Fr(+)24 b Ft(C)3774 4241 y FA(11)3849 4214 y Fu(.)94 4333 y(Since)i Ft(K)426 4360 y Fz(m)529 4333 y Fn(\024)i Ft(B)715 4291 y Fo(\003)710 4360 y FA(13)784 4333 y Ft(")e Fu(we)f(ha)n(v)o(e)f(sho)n (wn)g(that)g(the)h(bounds)e(of)i(Eq.\(6.21\))f(are)i(suf)n(\256cient)e(to)h (ensure)1502 4583 y(sup)1467 4673 y Fo(j)p Fz(x)p Fo(j\024)p Fz(L)1693 4583 y Fn(j)p Ft(w)s Fr(\()p Ft(x;)17 b(t)p Fr(\))p Fn(j)42 b(\024)j Fu(2)p Ft(B)2334 4540 y Fo(\003)2329 4610 y FA(13)2403 4583 y Ft(")25 b(:)1117 b Fr(\()p Fu(6)p Ft(:)p Fu(22)p Fr(\))94 4908 y Fu(The)25 b(proof)g(of)g(Theorem)f(6.7)h(is)f (completed.)p eop %%Page: 24 24 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(24)94 194 y Fv(7.)46 b(Outlo)t(ok)94 383 y Fu(In)28 b(this)d(section,)i (we)g(wish)g(to)f(discuss)g(potential)g(e)o(xperimental)f(aspects)i(of)g(our) g(results.)41 b(These)27 b(aspects)94 503 y(must)c(necessarily)g(rely)g(on)g (a)h(number)f(of)h(conjectures)f(about)g(the)g(system)f(under)i (consideration,)e(and)h(are)94 623 y(similar)k(in)g(spirit)f(to)h(the)g (discussion)e(found)i(in)g([ER].)h(While)f(we)g(ha)n(v)o(e)g(de\256ned)h (topological)e(entrop)o(y)h(in)94 742 y(Section)j(2,)h(we)f(no)n(w)f(need)h (to)g(address)f(the)h(question)e(of)i(entrop)o(y)g(relati)n(v)o(e)e(to)i(an)g (in)l(v)n(ariant)e(measure)i Ft(\026)p Fu(.)94 862 y(W)-8 b(e)23 b(shall)f(call)h(it)f Ft(h)760 889 y Fz(\026)813 862 y Fu(.)35 b(More)23 b(precisely)-6 b(,)22 b(let)g Ft(f)33 b Fu(be)23 b(a)g(continuous)d(map)j(of)f(a)h(compact)f(metric)g(space)h(and)f(let)94 981 y Ft(\026)27 b Fu(be)g(an)f Ft(f)11 b Fu(-in)l(v)n(ariant)25 b(non-atomic)g(er)n(godic)i(measure.)40 b(The)27 b(entrop)o(y)e Ft(h)2658 1008 y Fz(\026)2712 981 y Fr(\()p Ft(f)11 b Fr(\))26 b Fu(is)g(determined)f(as)i(follo)n(ws.)94 1101 y(F)o(or)e Ft(")j(>)h Fu(0)24 b(and)h(an)g(inte)o(ger)f Ft(n)k(>)g Fu(0)c(let)886 1293 y Ft(V)e Fr(\()p Ft(x;)17 b(";)g(n)p Fr(\))42 b(=)1461 1213 y Fq(\010)1519 1293 y Ft(y)1599 1209 y Fq(\014)1599 1268 y(\014)1660 1293 y Ft(d)p Fr(\()p Ft(f)1811 1250 y Fz(i)1844 1293 y Fr(\()p Ft(y)t Fr(\))p Ft(;)17 b(f)2080 1250 y Fz(i)2110 1293 y Fr(\()p Ft(x)p Fr(\)\))27 b Ft(<)h(")h(;)44 b Fu(0)27 b Fn(\024)h Ft(i)g(<)g(n)2972 1213 y Fq(\011)3055 1293 y Ft(:)94 1486 y Fu(One)d(has)g(\(see)g([BK],)h([Y]\):)1083 1721 y Ft(h)1140 1748 y Fz(\026)1194 1721 y Fr(\()p Ft(f)11 b Fr(\))43 b(=)i Fn(\000)27 b Fu(lim)1591 1785 y Fz(")p Fo(!)p FA(0)1760 1721 y Fu(lim)17 b(inf)1788 1781 y Fz(n)p Fo(!1)2058 1654 y Fu(1)p 2053 1698 60 4 v 2053 1790 a Ft(n)2142 1721 y Fu(log)f Ft(\026)p Fr(\()p Ft(V)22 b Fr(\()p Ft(x;)17 b(";)g(n)p Fr(\)\))23 b Ft(:)94 1946 y Fu(This)34 b(is)g(not)g(a)h(v)o(ery)f(con)l(v)o(enient)f(e)o (xpression)h(for)g(e)o(xplicit)f(computations.)63 b(As)34 b(for)h(the)g(case) g(of)f(\256nite)94 2065 y(dimension)e(\(see)h([GP]\))h(one)f(can)h(try)f(to)f (determine)h(instead)f(a)i(correlation)f(entrop)o(y)-6 b(.)59 b(In)33 b(our)g(case)h(we)94 2185 y(w)o(ould)f(also)h(lik)o(e)f(to)g(include) g(the)h(dependence)g(on)f(the)h(size)f(of)h(the)g(windo)n(w)e(in)h(which)g (the)h(system)f(is)94 2304 y(observ)o(ed.)i(One)22 b(is)g(naturally)g(lead)h (to)f(the)g(follo)n(wing)e(de\256nition)i(of)h(correlation)f(entrop)o(y)g (per)g(unit)g(length)94 2424 y Ft(K)179 2451 y FA(2)219 2424 y Fr(\()p Ft(\026)p Fr(\))p Fu(\320and)27 b(unit)g(time\320\(see)g([GP)-11 b(,)28 b(ER]\))g(for)g(a)g(measure)g Ft(\026)g Fu(which)f(is)h(space)g(and)g (time)f(in)l(v)n(ariant)f(and)94 2543 y(er)n(godic.)36 b(The)25 b(de\256nition)f(of)h(this)f(quantity)f(is:)185 2783 y Ft(K)270 2809 y FA(2)309 2783 y Fr(\()p Ft(\026)p Fr(\))44 b(=)h Fn(\000)704 2715 y Fu(1)p 702 2760 55 4 v 702 2851 a Ft(\034)795 2783 y Fu(lim)785 2846 y Fz(")p Fo(!)p FA(0)994 2783 y Fu(lim)954 2847 y Fz(L)p Fo(!1)1228 2715 y Fu(1)p 1196 2760 115 4 v 1196 2851 a Ft(L)1264 2822 y Fz(d)1376 2783 y Fu(lim)1339 2842 y Fz(n)p Fo(!1)1580 2715 y Fu(1)p 1575 2760 60 4 v 1575 2851 a Ft(n)554 3101 y Fu(log)698 2901 y Fq(0)698 3080 y(@)834 3101 y Fu(lim)786 3165 y Fz(N)7 b Fo(!1)1084 3034 y Fu(1)p 1044 3078 131 4 v 1044 3169 a Ft(N)1135 3141 y FA(2)1217 2976 y Fz(N)g Fo(\000)p FA(1)1229 3006 y Fq(X)1203 3221 y Fz(j;k)r Fp(=)p FA(0)1416 2976 y Fz(n)p Fo(\000)p FA(1)1425 3006 y Fq(Y)1427 3219 y Fz(i)p Fp(=)p FA(0)1579 3101 y Fr(\002)1656 2961 y Fq(\022)1730 3101 y Ft(")22 b Fn(\000)h(k)p Fr(\010)2020 3128 y Fp(\()p Fz(i)p Fp(+)p Fz(j)t Fp(\))p Fz(\034)2258 3101 y Fr(\()p Ft(A)p Fr(\))e Fn(\000)i Fr(\010)2604 3128 y Fp(\()p Fz(i)p Fp(+)p Fz(k)r Fp(\))p Fz(\034)2849 3101 y Fr(\()p Ft(A)p Fr(\))p Fn(k)3052 3128 y FA(L)3095 3108 y Ff(1)3167 3128 y Fp(\()p Fz(Q)3261 3146 y Fd(L)3313 3128 y Fp(\))3349 2961 y Fq(\023)3423 2901 y(1)3423 3080 y(A)3551 3101 y Ft(;)3669 2977 y Fr(\()p Fu(7)p Ft(:)p Fu(1)p Fr(\))94 3406 y Fu(where)j Fr(\002)f Fu(is)g(the)f(Hea)n (viside)h(function,)e(and)i Fr(\010)1747 3433 y Fz(t)1807 3406 y Fu(is)f(the)h(\257o)n(w)g(of)g(the)f(dynamics.)316 3526 y(It)30 b(is)g(an)h(open)f(question)f(to)h(pro)o(v)o(e)f(that)h(the)h(limits)d(in)i (the)g(abo)o(v)o(e)g(e)o(xpression)f(e)o(xist)g(for)i Ft(\026)f Fu(almost)94 3646 y(e)n(v)o(ery)22 b Ft(A)g Fu(\(e)o(xcept)g(the)g(limit)e(o) o(v)o(er)i Ft(")g Fu(since)g(the)g(quantity)f(is)g(increasing\).)35 b(Furthermore,)23 b(the)f(result)f(should)94 3765 y(also)i(be)g(independent)f (of)h(the)g(time)f(step)g Ft(\034)11 b Fu(.)35 b(But)23 b(let)g(us)f(assume)g (that)h(these)g(limits)e(e)o(xist.)33 b(In)23 b(that)g(case)g(our)94 3885 y(sampling)g(bound)h(Theorem)h(6.7)f(gi)n(v)o(es)f(us)h(a)h(constructi)n (v)o(e)d(handle)j(on)f(computing)f(the)h(r)-5 b(.h.s)24 b(of)h(Eq.\(7.1\).)94 4004 y(W)-8 b(e)26 b(recall)f(the)g(de\256nition)1306 4124 y Fn(j)p Ft(f)11 b Fn(j)1422 4151 y Fz(\016)n(;\025)1576 4124 y Fn(\021)273 b Fu(sup)1697 4216 y Fo(f)p Fz(n)p Fo(2)p Fa(Z)1896 4197 y Fd(d)1963 4216 y FA(:)25 b Fz(\016)s Fo(j)p Fz(n)p Fo(j\024)p Fz(\025)p Fo(g)2310 4124 y Fn(j)p Ft(f)11 b Fr(\()p Ft(n\016)t Fr(\))p Fn(j)23 b Ft(:)1006 b Fr(\()p Fu(7)p Ft(:)p Fu(2)p Fr(\))94 4494 y Fs(Theorem)45 b(7.1.)57 b Fc(Assuming)30 b(the)i(limits)e(in) h(\(7.1\))h(e)o(xist)f(for)h Ft(\026)p Fc(-almost)e(e)n(v)o(ery)i Ft(A)p Fc(,)h(we)f(ha)n(v)o(e)g(for)g(e)n(v)o(ery)94 4614 y Ft(\016)g Fn(2)c Fr(\()p Fu(0)p Ft(;)17 b(\016)442 4641 y Fo(\003)487 4614 y Fu(])25 b Fc(and)g(e)n(v)o(ery)f Ft(\034)39 b Fn(2)28 b Fr(\()p Fu(0)p Ft(;)17 b(\034)1311 4641 y Fo(\003)1355 4614 y Fu(])p Fc(:)297 4858 y Ft(K)382 4885 y FA(2)421 4858 y Fr(\()p Ft(\026)p Fr(\))44 b(=)h Fn(\000)816 4791 y Fu(1)p 814 4835 55 4 v 814 4927 a Ft(\034)907 4858 y Fu(lim)898 4922 y Fz(")p Fo(!)p FA(0)1107 4858 y Fu(lim)1066 4923 y Fz(L)p Fo(!1)1340 4791 y Fu(1)p 1308 4835 115 4 v 1308 4927 a Ft(L)1376 4898 y Fz(d)1489 4858 y Fu(lim)1451 4918 y Fz(n)p Fo(!1)1693 4791 y Fu(1)p 1688 4835 60 4 v 1688 4927 a Ft(n)667 5177 y Fu(log)811 4977 y Fq(0)811 5156 y(@)946 5177 y Fu(lim)898 5241 y Fz(N)7 b Fo(!1)1196 5109 y Fu(1)p 1156 5154 131 4 v 1156 5245 a Ft(N)1247 5216 y FA(2)1329 5052 y Fz(N)g Fo(\000)p FA(1)1341 5082 y Fq(X)1315 5297 y Fz(j;k)r Fp(=)p FA(0)1528 5052 y Fz(n)p Fo(\000)p FA(1)1538 5082 y Fq(Y)1539 5295 y Fz(i)p Fp(=)p FA(0)1691 5177 y Fr(\002)1768 5036 y Fq(\022)1842 5177 y Ft(")23 b Fn(\000)f(j)p Fr(\010)2110 5204 y Fp(\()p Fz(i)p Fp(+)p Fz(j)t Fp(\))p Fz(\034)2348 5177 y Fr(\()p Ft(A)p Fr(\))g Fn(\000)g Fr(\010)2694 5204 y Fp(\()p Fz(i)p Fp(+)p Fz(k)r Fp(\))p Fz(\034)2940 5177 y Fr(\()p Ft(A)p Fr(\))p Fn(j)3121 5204 y Fz(\016)n(;L)3237 5036 y Fq(\023)3311 4977 y(1)3311 5156 y(A)3439 5177 y Ft(:)3669 5052 y Fr(\()p Fu(7)p Ft(:)p Fu(3)p Fr(\))p eop %%Page: 25 25 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(25)94 194 y Fs(Remark.)40 b Fu(Note)25 b(that)g(the)h(sum)1297 119 y Fq(P)1402 151 y Fz(N)7 b Fo(\000)p FA(1)1402 224 y Fz(j;k)r Fp(=)p FA(0)1620 119 y Fq(Q)1714 151 y Fz(n)p Fo(\000)p FA(1)1714 224 y Fz(i)p Fp(=)p FA(0)1882 194 y Fr(\002\()p Ft(:)17 b(:)g(:)n Fr(\))26 b Fu(counts)f(the)g(number)h(of)f(pairs)h(of)g(points)e(in)h(the)94 314 y(sample)k(which)h(ha)n(v)o(e)f(a)h(\252distance\272)f(of)h(less)f(than)g Ft(")h Fu(in)f(embedding)g(dimension)f Ft(n)p Fu(,)i FB(wher)l(e)h(the)e (distance)94 433 y(is)h(measur)l(ed)f(with)h(the)f(discr)l(ete)h(sampling)e (step)h Ft(\016)34 b FB(o)o(ver)c(a)f(r)l(e)l(gion)h Fn(j)p Ft(x)p Fn(j)j(\024)i Ft(L)p FB(.)51 b Fu(In)30 b(this)e(sense,)j(our)f (result)94 553 y(says)k(that)g(for)h(PDE')-5 b(s)34 b(on)g(the)g(in\256nite)g (line,)i(measuring)d(the)i Ft(K)2433 580 y FA(2)2506 553 y Fu(entrop)o(y)f(can)h(be)f(done)h(by)f(the)g(usual)94 672 y(Grassber)n(ger)n (-Procaccia)27 b(algorithm)d([GP)-11 b(,)25 b(ER].)94 836 y Fs(Remark.)47 b Fu(If)35 b(we)f(choose)g Ft(\016)44 b Fr(=)d Ft(\016)1382 863 y Fo(\003)1462 836 y Fu(and)34 b Ft(\034)51 b Fr(=)41 b Ft(\034)1897 863 y Fo(\003)1942 836 y Fu(,)36 b(then)e(the)g (\252sampling)e(error\272)j(is)f(controlled)f(by)g(the)94 955 y(constants)d(of)g(the)g(proof)g(of)g(Theorem)g(6.7.)52 b(But)30 b(if)g(we)h(choose,)g(for)f(e)o(xample)g Ft(\034)45 b Fr(=)36 b Ft(\034)3216 982 y Fo(\003)3261 955 y Ft(=)p Fu(2,)c(the)e(number)94 1075 y(of)c(time)f(steps\320)p FB(i.e)o(.,)e Fu(the)j(f)o(actor)g Ft(B)1361 1032 y Fo(\003)1356 1102 y FA(24)1430 1075 y Fu(\320needed)f(in)g (Theorem)h(6.7)f(will)f(be)i(the)f(same)g(as)h(if)f Ft(M)3445 1102 y Fo(\003)3516 1075 y Fu(had)h(been)94 1194 y(replaced)h(by)f(2)p Ft(M)731 1221 y Fo(\003)803 1194 y Fu(in)g(the)g(original)g(assumptions.)38 b(So)27 b(clearly)-6 b(,)26 b(the)g(total)g(time)f(during)h(which)g(one)g (must)94 1314 y(measure)37 b(to)g(achie)n(v)o(e)f(a)h(gi)n(v)o(en)f(sampling) f(precision)h(can)h(not)f(be)h(shortened)g(by)f(sampling)f(at)i(shorter)94 1434 y(interv)n(als.)e(Similar)24 b(precautions)g(are)i(necessary)f(for)g (sampling)e(in)i(space.)94 1597 y Fs(Pro)s(of.)45 b Fu(In)30 b(vie)n(w)f(of)h(the)g(remark,)h(we)g(gi)n(v)o(e)d(the)i(proof)g(only)f(for)h (the)g(case)h Ft(\034)45 b Fr(=)36 b Ft(\034)3080 1624 y Fo(\003)3125 1597 y Fu(,)31 b Ft(\016)39 b Fr(=)d Ft(\016)3421 1624 y Fo(\003)3467 1597 y Fu(,)31 b(in)f(which)94 1717 y(case)e(the)f(constants)g Ft(B)922 1674 y Fo(\003)917 1744 y FA(23)991 1717 y Ft(;)17 b(:)g(:)g(:)d(;)27 b Fu(retain)g(their)g(meaning)g(from)g(earlier)g(parts)g (of)h(the)f(paper)-5 b(.)43 b(W)-8 b(e)27 b(ob)o(viously)94 1836 y(ha)n(v)o(e)898 2010 y(1)p 858 2055 131 4 v 858 2146 a Ft(N)949 2117 y FA(2)1031 1953 y Fz(N)7 b Fo(\000)p FA(1)1043 1983 y Fq(X)1017 2198 y Fz(j;k)r Fp(=)p FA(0)1230 1953 y Fz(n)p Fo(\000)p FA(1)1239 1983 y Fq(Y)1241 2196 y Fz(i)p Fp(=)p FA(0)1393 2078 y Fr(\002)1470 1937 y Fq(\022)1544 2078 y Ft(")22 b Fn(\000)h(j)p Fr(\010)1812 2105 y Fp(\()p Fz(i)p Fp(+)p Fz(j)t Fp(\))p Fz(\034)2037 2123 y Ff(\003)2084 2078 y Fr(\()p Ft(A)p Fr(\))e Fn(\000)i Fr(\010)2430 2105 y Fp(\()p Fz(i)p Fp(+)p Fz(k)r Fp(\))p Fz(\034)2662 2123 y Ff(\003)2709 2078 y Fr(\()p Ft(A)p Fr(\))p Fn(j)2890 2105 y Fz(\016)2926 2123 y Ff(\003)2966 2105 y Fz(;L)3050 1937 y Fq(\023)676 2443 y Fn(\025)850 2376 y Fu(1)p 810 2420 V 810 2512 a Ft(N)901 2483 y FA(2)984 2319 y Fz(N)7 b Fo(\000)p FA(1)995 2349 y Fq(X)969 2563 y Fz(j;k)r Fp(=)p FA(0)1182 2319 y Fz(n)p Fo(\000)p FA(1)1192 2349 y Fq(Y)1193 2562 y Fz(i)p Fp(=)p FA(0)1345 2443 y Fr(\002)1422 2303 y Fq(\022)1496 2443 y Ft(")23 b Fn(\000)f(k)p Fr(\010)1786 2470 y Fp(\()p Fz(i)p Fp(+)p Fz(j)t Fp(\))p Fz(\034)2011 2488 y Ff(\003)2058 2443 y Fr(\()p Ft(A)p Fr(\))f Fn(\000)i Fr(\010)2404 2470 y Fp(\()p Fz(i)p Fp(+)p Fz(k)r Fp(\))p Fz(\034)2636 2488 y Ff(\003)2683 2443 y Fr(\()p Ft(A)p Fr(\))p Fn(k)2886 2470 y FA(L)2929 2450 y Ff(1)3000 2470 y Fp(\()p Fz(Q)3094 2488 y Fd(L)3147 2470 y Fp(\))3183 2303 y Fq(\023)3281 2443 y Ft(:)94 2720 y Fu(On)i(the)g(other)f(hand,)h(applying)f(Theorem)g(6.7,)g (we)h(ha)n(v)o(e)g(for)g Ft(n)j(>)g(B)2559 2677 y Fo(\003)2554 2747 y FA(24)2645 2720 y Fu(log)o Fr(\()p Fu(1)p Ft(=")p Fr(\))22 b(+)g Fu(1)j(and)g Ft(L)i(>)h(B)3642 2677 y Fo(\003)3637 2747 y FA(23)3712 2720 y Ft(=")p Fu(:)1227 2863 y Fz(n)p Fo(\000)p FA(1)1237 2893 y Fq(Y)1238 3106 y Fz(i)p Fp(=)p FA(0)1390 2988 y Fr(\002)1467 2847 y Fq(\022)1541 2988 y Ft(")23 b Fn(\000)f(j)p Fr(\010)1809 3015 y Fp(\()p Fz(i)p Fp(+)p Fz(j)t Fp(\))p Fz(\034)2034 3032 y Ff(\003)2081 2988 y Fr(\()p Ft(A)p Fr(\))f Fn(\000)i Fr(\010)2427 3015 y Fp(\()p Fz(i)p Fp(+)p Fz(k)r Fp(\))p Fz(\034)2659 3032 y Ff(\003)2706 2988 y Fr(\()p Ft(A)p Fr(\))p Fn(j)2887 3015 y Fz(\016)2923 3032 y Ff(\003)2964 3015 y Fz(;L)3047 2847 y Fq(\023)572 3338 y Fn(\024)886 3214 y Fz(n)p Fo(\000)p FA(1)895 3243 y Fq(Y)694 3462 y Fz(i)p Fp(=)p FA([)p Fz(B)869 3438 y Ff(\003)866 3487 y Fe(24)935 3462 y FA(log)q Fp(\()p FA(1)p Fz(=")p Fp(\))p FA(])1240 3338 y Fr(\002)1317 3198 y Fq(\022)1391 3338 y Ft(B)1472 3295 y Fo(\003)1467 3365 y FA(25)1541 3338 y Ft(")g Fn(\000)f(k)p Fr(\010)1831 3365 y Fp(\()p Fz(i)p Fp(+)p Fz(j)t Fp(\))p Fz(\034)2056 3383 y Ff(\003)2103 3338 y Fr(\()p Ft(A)p Fr(\))f Fn(\000)i Fr(\010)2449 3365 y Fp(\()p Fz(i)p Fp(+)p Fz(k)r Fp(\))p Fz(\034)2681 3383 y Ff(\003)2728 3338 y Fr(\()p Ft(A)p Fr(\))p Fn(k)2931 3365 y FA(L)2974 3345 y Ff(1)3045 3365 y Fp(\()p Fz(Q)3139 3383 y Fd(L)p Ff(\000)p Fd(B)3295 3366 y Ff(\003)3293 3415 y Fe(23)3347 3383 y Fd(=")3423 3365 y Fp(\))3459 3198 y Fq(\023)453 3756 y Fn(\024)575 3613 y Fz(n)p Fo(\000)p FA(1)p Fo(\000)p FA([)p Fz(B)869 3570 y Ff(\003)866 3640 y Fe(24)935 3613 y FA(log)q Fp(\()p FA(1)p Fz(=")p Fp(\))p FA(])835 3661 y Fq(Y)837 3874 y Fz(i)p Fp(=)p FA(0)1240 3756 y Fr(\002)1317 3615 y Fq(\022)1391 3756 y Ft(B)1472 3713 y Fo(\003)1467 3783 y FA(25)1541 3756 y Ft(")g Fn(\000)f(k)p Fr(\010)1831 3783 y Fp(\()p Fz(i)p Fp(+)p Fz(j)t Fp(+)p FA([)p Fz(B)2135 3759 y Ff(\003)2132 3808 y Fe(24)2201 3783 y FA(log)q Fp(\()p FA(1)p Fz(=")p Fp(\))p FA(])p Fp(\))p Fz(\034)2558 3800 y Ff(\003)2604 3756 y Fr(\()p Ft(A)p Fr(\))1613 4069 y Fn(\000)h Fr(\010)1785 4096 y Fp(\()p Fz(i)p Fp(+)p Fz(k)r Fp(+)p FA([)p Fz(B)2096 4072 y Ff(\003)2093 4121 y Fe(24)2162 4096 y FA(log)p Fp(\()p FA(1)p Fz(=")p Fp(\))p FA(])p Fp(\))p Fz(\034)2518 4114 y Ff(\003)2565 4069 y Fr(\()p Ft(A)p Fr(\))p Fn(k)2768 4096 y FA(L)2811 4076 y Ff(1)2882 4096 y Fp(\()p Fz(Q)2976 4114 y Fd(L)p Ff(\000)p Fd(B)3132 4097 y Ff(\003)3130 4146 y Fe(23)3184 4114 y Fd(=")3259 4096 y Fp(\))3295 3929 y Fq(\023)3393 4069 y Ft(:)94 4289 y Fu(Therefore,)214 4476 y(1)p 173 4521 V 173 4612 a Ft(N)264 4583 y FA(2)347 4419 y Fz(N)7 b Fo(\000)p FA(1)359 4449 y Fq(X)333 4663 y Fz(j;k)r Fp(=)p FA(0)546 4419 y Fz(n)p Fo(\000)p FA(1)555 4449 y Fq(Y)557 4662 y Fz(i)p Fp(=)p FA(0)709 4543 y Fr(\002)786 4403 y Fq(\022)860 4543 y Ft(")22 b Fn(\000)h(j)p Fr(\010)1128 4570 y Fp(\()p Fz(i)p Fp(+)p Fz(j)t Fp(\))p Fz(\034)1353 4588 y Ff(\003)1399 4543 y Fr(\()p Ft(A)p Fr(\))f Fn(\000)g Fr(\010)1745 4570 y Fp(\()p Fz(i)p Fp(+)p Fz(k)r Fp(\))p Fz(\034)1977 4588 y Ff(\003)2024 4543 y Fr(\()p Ft(A)p Fr(\))p Fn(j)2205 4570 y Fz(\016)2241 4588 y Ff(\003)2282 4570 y Fz(;L)2365 4403 y Fq(\023)344 4945 y Fn(\024)518 4878 y Fu(1)p 478 4922 V 478 5014 a Ft(N)569 4985 y FA(2)651 4821 y Fz(N)7 b Fo(\000)p FA(1)663 4851 y Fq(X)637 5065 y Fz(j;k)r Fp(=)p FA(0)850 4803 y Fz(n)p Fo(\000)p FA(1)p Fo(\000)p FA([)p Fz(B)1144 4760 y Ff(\003)1141 4830 y Fe(24)1210 4803 y FA(log)q Fp(\()p FA(1)p Fz(=")p Fp(\))p FA(])1110 4851 y Fq(Y)1112 5064 y Fz(i)p Fp(=)p FA(0)1515 4945 y Fr(\002)1592 4805 y Fq(\022)1666 4945 y Ft(B)1747 4902 y Fo(\003)1742 4972 y FA(25)1816 4945 y Ft(")23 b Fn(\000)f(k)p Fr(\010)2106 4972 y Fp(\()p Fz(i)p Fp(+)p Fz(j)t Fp(\))p Fz(\034)2331 4990 y Ff(\003)2378 4945 y Fr(\()p Ft(A)p Fr(\))f Fn(\000)i Fr(\010)2724 4972 y Fp(\()p Fz(i)p Fp(+)p Fz(k)r Fp(\))p Fz(\034)2956 4990 y Ff(\003)3003 4945 y Fr(\()p Ft(A)p Fr(\))p Fn(k)3206 4972 y FA(L)3249 4952 y Ff(1)3320 4972 y Fp(\()p Fz(Q)3414 4990 y Fd(L)p Ff(\000)p Fd(B)3570 4973 y Ff(\003)3568 5022 y Fe(23)3622 4990 y Fd(=")3698 4972 y Fp(\))3734 4805 y Fq(\023)388 5263 y Fr(+)520 5195 y Fu(1)p 500 5240 91 4 v 500 5331 a Ft(N)603 5263 y Fk(O)673 5182 y Fq(\000)719 5263 y Fu(1)f Fr(+)g Fu(log)p Fr(\()p Fu(1)p Ft(=")p Fr(\))1242 5182 y Fq(\001)1287 5220 y FA(2)1352 5263 y Ft(:)p eop %%Page: 26 26 bop 94 -45 a Fm(Topological)39 b(Entr)n(opy)f(per)e(Unit)i(V)m(olume)1672 b Fl(26)94 194 y Fu(The)32 b(result)g(no)n(w)f(follo)n(ws)f(by)i(taking)f (the)g(limits)g Ft(N)48 b Fn(!)37 b(1)p Fu(,)c Ft(n)38 b Fn(!)f(1)p Fu(,)c Ft(L)k Fn(!)g(1)32 b Fu(\(in)g(that)f(order\))h(for)h Ft(")94 314 y Fu(\256x)o(ed,)25 b(and)g(then)f(letting)g Ft(")h Fu(tend)g(to)f(zero.)94 672 y Fv(References)94 921 y Fx([AKM])91 b(Adler)m(,)24 b(R.,)f(K)m(onheim,)h(A.G.)e(and)h(McAndre)n(w)-5 b(,)23 b(M.H.:)34 b(T)-6 b(opological)23 b(Entrop)o(y.)38 b(T)m(rans.)f(Am.)h (Math.)h(Soc.)23 b Fw(114)p Fx(,)g(390)426 1011 y(\(1965\).)94 1101 y([B])225 b(Boas,)21 b(R.P)-9 b(.:)29 b Fh(Entir)m(e)20 b(Functions)p Fx(,)g(Ne)n(w)f(Y)-9 b(ork:)28 b(Academic)21 b(Press)g(\(1954\).)94 1190 y([BK])167 b(Brin,)21 b(M.)f(and)g(Katok,)g(A.:) 29 b(On)20 b(local)h(entrop)o(y.)30 b(In)20 b Fh(Geometric)g(Dynamics)h Fx(\(Rio)f(de)g(Janeiro,)h(1981\),)e(Lecture)i(Notes)f(in)426 1280 y(Mathematics)i Fw(1007)p Fx(.)28 b(30\26138)19 b(\(1983\).)94 1370 y([CJT])145 b(Cockb)n(urn,)16 b(B.,)h(Jones,)f(D.A.)e(and)i(T)m(iti,)g (E.:)27 b(Estimating)16 b(the)f(number)g(of)g(asymptotic)h(de)o(grees)g(of)f (freedom)g(for)f(nonlinear)426 1459 y(dissipati)n(v)o(e)21 b(systems.)29 b(Math.)20 b(Comput.)g Fw(66)p Fx(,)g(1073\2611087)e(\(1997\).) 94 1549 y([C])225 b(Collet,)21 b(P)-9 b(.:)29 b(Thermodynamic)20 b(limit)h(of)e(the)h(Ginzb)n(ur)o(g-Landau)f(equation.)29 b(Nonlinearity)20 b Fw(7)p Fx(,)g(1175\2611190)e(\(1994\).)94 1639 y([CE1])136 b(Collet,)30 b(P)-9 b(.)28 b(and)f(Eckmann,)i(J.-P)-9 b(.:)43 b(The)27 b(time-dependent)h(amplitude)g(equation)g(for)e(the)h (Swift-Hohenber)o(g)e(problem.)426 1728 y(Commun.)20 b(Math.)g(Phys.)g Fw(132)p Fx(,)f(139\261153)g(\(1990\).)94 1818 y([CE2])136 b(Collet,)31 b(P)-9 b(.)29 b(and)f(Eckmann,)i(J.-P)-9 b(.:)46 b(Extensi)n(v)o(e)28 b(properties)g(of)g(the)g(comple)o(x)h(Ginzb)n(ur)o (g-Landau)d(equation.)54 b(Preprint.)426 1908 y(\(1998\).)94 1997 y([ER])176 b(Eckmann,)19 b(J.-P)-9 b(.)17 b(and)h(Ruelle,)h(D.:)27 b(Er)o(godic)17 b(theory)g(of)g(chaos)h(and)g(strange)g(attractors.)28 b(Re)n(v)-5 b(.)18 b(Mod.)f(Phys.)g Fw(57)p Fx(,)h(617\261656)426 2087 y(\(1985\).)94 2177 y([ER2])136 b(Eckmann,)22 b(J.-P)-9 b(.)21 b(and)g(Ruelle,)i(D.:)30 b(Fundamental)23 b(limitations)f(for)e (estimating)i(dimensions)f(and)h(Liapuno)o(v)e(e)o(xponents)426 2266 y(in)g(dynamical)h(systems.)29 b(Physica)21 b Fw(D56)p Fx(,)e(185\261187)g(\(1992\).)94 2356 y([GP])176 b(Grassber)o(ger)m(,)16 b(P)-9 b(.)17 b(and)g(I.)f(Procaccia:)29 b(Estimating)17 b(the)g(K)m (olmogoro)o(v)e(entrop)o(y)i(from)e(a)i(chaotic)h(signal.)28 b(Phys.)16 b(Re)n(v)-5 b(.)17 b Fw(A28)p Fx(,)426 2446 y(2591)j(\(1983\).)94 2535 y([GV])162 b(Ginibre,)19 b(J.)g(and)h(G.)f(V)-9 b(elo:)28 b(The)20 b(Cauchy)f(problem)g(in)g(local)h(spaces)h(for)d(the)i(comple)o(x)f (Ginzb)n(ur)o(g-Landau)f(equation.)i(II:)426 2625 y(contraction)h(methods.)29 b(Commun.)19 b(Math.)h(Phys.)g Fw(187)p Fx(,)f(45\26179)g(\(1997\).)94 2735 y([KT])171 b(K)m(olmogoro)o(v)-5 b(,)17 b(A.N.)f(and)i(T)m(ikhomiro)o(v) -5 b(,)16 b(V)-10 b(.M.:)27 b Fi(")p Fx(-entrop)o(y)17 b(and)h Fi(")p Fx(-capacity)g(of)f(sets)h(in)g(functional)g(spaces)3441 2693 y Fy(1)3477 2735 y Fx(.)28 b(In)16 b Fh(Selected)426 2825 y(W)-7 b(orks)21 b(of)f(A.N)f(K)m(olmo)o(gor)l(o)o(v)-6 b(,)20 b(V)-9 b(ol)20 b(III)p Fx(,)f(\(Shirayaye)n(v)-5 b(,)19 b(A.N.,)g(ed.\).)29 b(Dordrecht,)19 b(Kluv)o(er)g(\(1993\).)94 2915 y([MS])163 b(Mielk)o(e,)26 b(A.)d(and)h(Schneider)m(,)i(G.:)36 b(Attractors)23 b(for)g(modulation)h(equations)g(on)g(unbounded)f(domains\320e)o(xistence)i (and)426 3004 y(comparison.)k(Nonlinearity)20 b Fw(8)p Fx(,)g(743\261768)e (\(1995\).)94 3094 y([Y])220 b(Y)-9 b(oung,)19 b(L.S.:)29 b(Dimension,)20 b(entrop)o(y)g(and)g(L)l(yapuno)o(v)e(e)o(xponents.)29 b(Er)o(g.)18 b(Th.)i(Dyn.)f(Sys.)h Fw(2)p Fx(,)f(109\261124)g(\(1982\).)p 94 5182 1200 4 v 241 5248 a Fy(1)316 5291 y Fx(The)h(v)o(ersion)f(in)h(this)g (collection)h(is)f(more)g(complete)h(than)f(the)h(original)e(paper)h(of)g (Uspekhi)f(Mat.)h(Nauk,)g Fw(14)p Fx(,)f(3\26186)h(\(1959\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF