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Fn(0)2965 5052 y Fp(\()p Fq(B)5 b Fp(\))p FF(.)70 b(Thus,)39 b(we)e Ft(de\256ne)94 5172 y Fo(G)d Fp(=)28 b Fo(G)6 b Fp(\()p Fq(B)f(;)17 b(T)d Fp(\))p FF(.)35 b(It)24 b(is)h(made)f(up)h(of)f (functions)g(which)g(e)o(xtend)g(to)g(bounded)g(analytic)g(functions)g(in)g (a)h(strip.)94 5291 y(Its)h(width)e(and)i(the)f(bound)g(on)g(the)h(functions) e(only)h(depend)g(on)h(the)f(parameters)h(of)f(the)h(problem.)37 b(These)p eop %%Page: 3 3 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)857 b Fk(3)94 194 y FF(f)o(acts)35 b(can)f(be)h(found)e(scattered)h(in)g(the)g(literature,)i(b)n(ut)e(are)h (\252well-kno)n(wn,)-7 b(\272)35 b(see,)i Ft(e)o(.g)o(.)p FF(,)e([C].)g(The)f (set)g Fo(G)94 314 y FF(probably)e(lacks)h(properties)f(i\))h(and)f(ii\))h (abo)o(v)o(e)e(in)i(the)f(topology)f(of)i(uniformly)e(continuous)g (functions.)94 433 y(W)-8 b(e)33 b(will)f(ne)n(v)o(ertheless)f(call)h(it)g(a) 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b(CGL)g(satis\256es)f(a)h(bound)1438 4739 y Fq(H)1521 4766 y Fr(")1564 4739 y Fp(\()p Fo(G)6 b Fp(\))43 b Fo(\024)i FF(const)p Fq(:)16 b FF(log)o Fp(\()p FF(1)p Fq(=")p Fp(\))25 b Fq(;)1138 b Fp(\()p FF(4)p Fq(:)p FF(1)p Fp(\))94 4947 y Fh(where)26 b(the)e(constant)g(depends)h(only)f(on)g(the)h(parameters) g(of)g(the)g(equation.)94 5172 y Fs(Remark.)41 b FF(The)28 b(reader)g(should)f(note)g(that)g(this)f(result)h(contains)g(ne)n(w)g (information)f(about)h(the)g(set)g Fo(G)34 b FF(of)94 5291 y(limiting)20 b(states.)34 b(It)22 b(is)f(for)h(e)o(xample)e(well)i(kno)n(wn) e(that)h(the)h(solutions)d(of)j(CGL)g(are)g(analytic)f(and)h(bounded)p eop %%Page: 5 5 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)857 b Fk(5)94 194 y FF(in)30 b(a)g(strip,)h(that)e(is,)i(the)o(y)e(are)i(in)f(the)g(class)f Fj(S)1701 221 y Fr(h)1753 194 y Fp(\()p Fq(C)7 b Fp(\))30 b FF(for)g(some)f Fq(h)36 b(>)f FF(0)30 b(and)g(some)f Fq(C)43 b(<)35 b Fo(1)p 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Fq(S)p 2642 1084 75 4 v 6 w(w)2716 1058 y Fv(\001)2784 1139 y Fp(+)h Fq(cc)i(;)94 1416 y FF(where)h Fq(cc)f FF(denotes)f(the)h(comple)o(x)e(conjugate.)35 b(Inte)o(grating)24 b(by)g(parts)h(we)g(get)679 1698 y Fq(X)52 b Fp(=)45 b Fo(\000)p Fp(\()p FF(1)22 b Fp(+)h Fq(i\013)p Fp(\))1378 1562 y Fv(Z)1494 1698 y Fq(dx)17 b(')1685 1725 y Fr(a)1732 1698 y Fo(jr)p Fq(w)s Fo(j)1945 1655 y Fn(2)2007 1698 y Fo(\000)22 b Fp(\()p FF(1)g Fp(+)h Fq(i\013)p Fp(\))2471 1562 y Fv(Z)2587 1698 y Fq(dx)37 b Fp(\026)-70 b Fq(w)r Fp(\()p Fo(r)p Fq(')2973 1725 y Fr(a)3043 1698 y Fo(\001)22 b(r)p Fq(w)s Fp(\))809 1959 y(+)925 1824 y Fv(Z)1041 1959 y Fq(dx)17 b(')1232 1986 y Fr(a)1300 1959 y Fp(\026)-70 b Fq(w)1354 1879 y Fv(\000)1399 1959 y Fq(R)q(w)25 b Fp(+)e Fq(S)j Fp(\026)-70 b Fq(w)1812 1879 y Fv(\001)1880 1959 y Fp(+)23 b Fq(cc)i(:)3669 1829 y Fp(\()p FF(6)p Fq(:)p FF(8)p Fp(\))94 2244 y FF(By)g(the)f(choice)h(of)f Fq(')838 2271 y Fr(a)911 2244 y FF(we)h(ha)n(v)o(e)f Fo(jr)p Fq(')1437 2271 y Fr(a)1485 2244 y Fp(\()p Fq(x)p Fp(\))p Fo(j)j(\024)h FF(const)p Fq(:)o(')2083 2271 y Fr(a)2132 2244 y Fp(\()p Fq(x)p Fp(\))p FF(,)23 b(uniformly)g(in)h Fq(x)h FF(and)f Fq(a)p FF(.)36 b(Therefore)25 b Fq(X)32 b FF(can)94 2364 y(be)25 b(bounded)f(abo)o(v)o(e)g (by)378 2646 y Fq(X)52 b Fo(\024)45 b(\000)p FF(2)779 2510 y Fv(Z)895 2646 y Fq(dx)17 b(')1086 2673 y Fr(a)1134 2646 y Fo(jr)p Fq(w)s Fo(j)1347 2603 y Fn(2)1408 2646 y Fp(+)23 b Fj(O)1578 2673 y Fr(\013;\014)1707 2646 y Fp(\()p FF(1)p Fp(\))1852 2510 y Fv(Z)1967 2646 y Fq(dx)17 b(')2158 2673 y Fr(a)2206 2646 y Fo(j)p Fq(w)s Fo(jjr)p Fq(w)s Fo(j)k Fp(+)i Fj(O)2740 2673 y Fr(\013;\014)2869 2646 y Fp(\()p FF(1)p Fp(\))3014 2510 y Fv(Z)3130 2646 y Fq(dx)17 b(')3321 2673 y Fr(a)3368 2646 y Fo(j)p Fq(w)s Fo(j)3498 2603 y Fn(2)3562 2646 y Fq(:)94 2923 y FF(By)26 b(polarization,)d(and)i(using)f(that)g Fq(')1410 2950 y Fr(a)1486 2923 y Fq(>)k FF(0,)d(we)g(get)g(a)g(bound)715 3210 y Fq(@)768 3237 y Fr(t)820 3075 y Fv(Z)936 3210 y Fq(dx)17 b(')1127 3237 y Fr(a)1175 3210 y Fo(j)p Fq(w)s Fo(j)1305 3167 y Fn(2)1388 3210 y Fo(\024)45 b(\000)1604 3075 y Fv(Z)1720 3210 y Fq(dx)17 b(')1911 3237 y Fr(a)1959 3210 y Fo(jr)p Fq(w)s Fo(j)2172 3167 y Fn(2)2233 3210 y Fp(+)23 b Fj(O)2403 3237 y Fr(\013;\014)2533 3210 y Fp(\()p FF(1)p Fp(\))2678 3075 y Fv(Z)2793 3210 y Fq(dx)17 b(')2984 3237 y Fr(a)3032 3210 y Fo(j)p Fq(w)s Fo(j)3162 3167 y Fn(2)3226 3210 y Fq(:)415 b Fp(\()p FF(6)p Fq(:)p FF(9)p Fp(\))94 3490 y FF(Therefore)26 b(we)f(see)g(that)f(there)h(is)g(a)g(constant)f Fq(C)32 b FF(which)24 b(depends)h(only)f(on)g Fq(\013)h FF(and)g Fq(\014)5 b FF(,)25 b(for)g(which)f(we)h(ha)n(v)o(e)94 3609 y(the)g(dif)n(ferential)f(inequality) 894 3891 y Fq(@)947 3918 y Fr(t)998 3755 y Fv(Z)1114 3891 y Fq(dx)17 b(')1305 3918 y Fr(a)1353 3891 y Fp(\()p Fq(x)p Fp(\))p Fo(j)p Fq(w)s Fp(\()p Fq(t;)g(x)p Fp(\))p Fo(j)1834 3848 y Fn(2)1916 3891 y Fo(\024)44 b Fq(C)2132 3755 y Fv(Z)2249 3891 y Fq(dx)17 b(')2440 3918 y Fr(a)2487 3891 y Fp(\()p Fq(x)p Fp(\))g Fo(j)p Fq(w)s Fp(\()p Fq(t;)g(x)p Fp(\))p Fo(j)2985 3848 y Fn(2)3047 3891 y Fq(:)544 b Fp(\()p FF(6)p Fq(:)p FF(10)p Fp(\))94 4193 y FF(Since)26 b Fq(w)s Fp(\()p FF(0)p Fq(;)17 b(x)p Fp(\))23 b FF(is)i(bounded)f(on)g Fs(R)1338 4150 y Fr(d)1409 4193 y FF(and)h(small)f(on)g Fq(Q)2022 4219 y Fr(L)2082 4193 y FF(,)h(we)g(ha)n(v)o(e)f(for)h Fq(`)j(>)g FF(1)p Fq(=")p FF(,)1085 4480 y(sup)1006 4565 y Fr(a)p Fi(2)p Fr(Q)1166 4576 y Fz(L)p Fd(\000)p Fz(`)1319 4344 y Fv(Z)1435 4480 y Fq(dx)17 b(')1626 4506 y Fr(a)1674 4480 y Fp(\()p Fq(x)p Fp(\))p Fo(j)p Fq(w)s Fp(\()p FF(0)p Fq(;)g(x)p Fp(\))p Fo(j)2169 4437 y Fn(2)2250 4480 y Fo(\024)45 b Fj(O)p Fp(\()p FF(1)22 b Fp(+)h Fq(K)2745 4437 y Fn(2)2785 4480 y Fp(\))p Fq(")2870 4437 y Fn(2)2934 4480 y Fq(:)94 4810 y FF(T)-8 b(o)24 b(see)g(this,)f(split)g(the)h(inte)o (gration)e(re)o(gion)h(into)g Fq(Q)1896 4837 y Fr(L)1979 4810 y FF(and)h Fs(R)2233 4767 y Fr(d)2299 4810 y Fo(n)c Fq(Q)2448 4837 y Fr(L)2508 4810 y FF(.)35 b(In)24 b(the)g(\256rst)g(re)o(gion,)f Fq(w)k FF(is)c(small)g(and)94 4930 y(in)i(the)g(second)f(re)o(gion)g(the)h (inte)o(gral)f(of)g Fq(')1570 4956 y Fr(a)1644 4930 y FF(is)g(small)g(and)h Fo(j)p Fq(w)s Fo(j)i(\024)h Fq(K)7 b FF(.)35 b(Using)24 b(Eq.\(6.10\),)g(we)h (\256nd)643 5217 y(sup)600 5302 y Fn(0)p Fi(\024)p Fr(t)p Fi(\024)p Fn(1)920 5217 y FF(sup)841 5302 y Fr(a)p Fi(2)p Fr(Q)1001 5313 y Fz(L)p Fd(\000)p Fz(`)1154 5081 y Fv(Z)1270 5217 y Fq(dx)17 b(')1461 5243 y Fr(a)1509 5217 y Fp(\()p Fq(x)p Fp(\))g Fo(j)p Fq(w)s Fp(\()p Fq(t;)g(x)p Fp(\))p Fo(j)2007 5174 y Fn(2)2088 5217 y Fo(\024)45 b Fq(e)2256 5174 y Fr(C)2323 5217 y Fj(O)2393 5243 y Fr(\013;\014)2523 5217 y Fp(\()p FF(1)p Fp(\))p Fq(")2697 5174 y Fn(2)2781 5217 y Fp(=)f Fj(O)2972 5243 y Fr(\013;\014)3102 5217 y Fp(\()p FF(1)p Fp(\))p Fq(")3276 5174 y Fn(2)3341 5217 y Fq(:)p eop %%Page: 9 9 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)857 b Fk(9)94 194 y FF(Thus)25 b(we)g(ha)n(v)o(e)f(sho)n(wn)g(Eq.\(6.5\).)316 314 y(W)-8 b(e)19 b(ne)o(xt)g(bound)f(the)h(solutions)e(in)h(L)1600 271 y Fi(1)1685 314 y FF(.)34 b(Let)19 b Fq(G)1974 341 y Fr(t)2028 314 y FF(denote)g(the)g (con)l(v)n(olution)e(k)o(ernel)i(of)g(the)g(semigroup)94 433 y(generated)26 b(by)e(the)h(operator)g Fp(\()p FF(1)c Fp(+)i Fq(i\013)p Fp(\)\001)p FF(.)36 b(W)-8 b(e)25 b(ha)n(v)o(e)400 725 y Fq(w)s Fp(\()p Fq(t;)33 b Fo(\001)17 b Fp(\))43 b(=)i Fq(G)937 752 y Fr(t)994 725 y Fq(?)22 b(w)1137 752 y Fn(0)1199 725 y Fp(+)1299 590 y Fv(Z)1399 614 y Fr(t)1354 816 y Fn(0)1417 725 y Fq(ds)17 b(G)1611 752 y Fr(t)p Fi(\000)p Fr(s)1768 725 y Fq(?)1840 645 y Fv(\000)1885 725 y Fq(R)q Fp(\()p Fq(s;)32 b Fo(\001)17 b Fp(\))p Fq(w)s Fp(\()p Fq(s;)31 b Fo(\001)17 b Fp(\))22 b(+)g Fq(S)6 b Fp(\()p Fq(s;)32 b Fo(\001)17 b Fp(\))j(\026)-70 b Fq(w)r Fp(\()p Fq(s;)33 b Fo(\001)17 b Fp(\))3217 645 y Fv(\001)3286 725 y Fq(:)305 b Fp(\()p FF(6)p Fq(:)p FF(11)p Fp(\))94 997 y FF(W)-8 b(e)26 b(\256rst)f(bound)f(the)g(term)1294 1117 y Fq(Y)1352 1144 y Fr(t;s)1492 1117 y Fo(\021)45 b Fq(G)1692 1144 y Fr(t)p Fi(\000)p Fr(s)1850 1117 y Fq(?)1922 1036 y Fv(\000)1967 1117 y Fq(R)q Fp(\()p Fq(s;)32 b Fo(\001)17 b Fp(\))p Fq(w)s Fp(\()p Fq(s;)31 b Fo(\001)17 b Fp(\))2577 1036 y Fv(\001)2647 1117 y Fq(:)94 1299 y FF(W)-8 b(e)26 b(re)n(write)e(it)h(as)790 1562 y Fq(Y)848 1589 y Fr(t;s)944 1562 y Fp(\()p Fq(x)p Fp(\))44 b(=)1244 1427 y Fv(Z)1361 1562 y Fq(dy)1493 1488 y(G)1571 1515 y Fr(t)p Fi(\000)p Fr(s)1707 1488 y Fp(\()p Fq(x)22 b Fo(\000)g Fq(y)t Fp(\))p 1493 1539 522 4 v 1518 1559 a Fv(p)p 1617 1559 374 4 v 1617 1645 a Fq(')p Fp(\()p Fq(x)g Fo(\000)h Fq(y)t Fp(\))2044 1472 y Fv(p)p 2144 1472 V 90 x Fq(')p Fp(\()p Fq(x)f Fo(\000)g Fq(y)t Fp(\))16 b Fq(R)q Fp(\()p Fq(s;)h(y)t Fp(\))p Fq(w)s Fp(\()p Fq(s;)g(y)t Fp(\))j Fq(:)94 1855 y FF(By)26 b(the)e(Schw)o(arz)i(inequality)-6 b(,)23 b(we)i(get)g(a)g(bound)709 2154 y Fq(Y)789 2111 y Fn(2)767 2181 y Fr(t;s)908 2154 y Fo(\024)1029 2019 y Fv(Z)1146 2154 y Fq(dy)1278 2080 y Fo(j)p Fq(G)1384 2107 y Fr(t)p Fi(\000)p Fr(s)1519 2080 y Fo(j)1547 2037 y Fn(2)1587 2080 y Fp(\()p Fq(x)d Fo(\000)g Fq(y)t Fp(\))p 1278 2131 618 4 v 1400 2223 a Fq(')p Fp(\()p Fq(x)g Fo(\000)h Fq(y)t Fp(\))1946 2154 y Fo(\001)f Fj(O)2066 2181 y Fr(\013;\014)2196 2154 y Fp(\()p FF(1)p Fp(\))2341 2019 y Fv(Z)2456 2154 y Fq(dz)f(')2640 2181 y Fr(x)2690 2154 y Fp(\()p Fq(z)t Fp(\))p Fo(j)p Fq(w)s Fp(\()p Fq(s;)c(z)t Fp(\))p Fo(j)3168 2111 y Fn(2)3232 2154 y Fq(:)359 b Fp(\()p FF(6)p Fq(:)p FF(12)p Fp(\))94 2451 y FF(Using)30 b(Eq.\(6.5\),)i(the)e(second)h(f)o(actor)g(in)f(\(6.12\))g(is)h (bounded)e(by)i Fj(O)p Fp(\()p Fq(")2605 2408 y Fn(2)2645 2451 y Fp(\))p FF(.)53 b(The)31 b(comple)o(x)e(heat)i(k)o(ernel)g Fq(G)94 2570 y FF(can)26 b(be)f(bounded)f(as)h(follo)n(ws:)94 2762 y Fs(Lemma)39 b(6.4.)d Fh(F)o(or)25 b(e)n(v)o(ery)f Fq(n)j(>)i FF(0)24 b Fh(there)h(is)g(a)g(constant)f Fq(C)2199 2789 y Fr(n)2278 2762 y Fh(such)h(that)1338 3037 y Fo(j)p Fq(G)1444 3064 y Fr(t)1479 3037 y Fp(\()p Fq(z)t Fp(\))p Fo(j)44 b(\024)2027 2970 y Fq(C)2098 2997 y Fr(n)p 1812 3015 556 4 v 1812 3108 a Fp(\()p FF(1)22 b Fp(+)h Fq(z)2073 3079 y Fn(2)2113 3108 y Fq(=t)p Fp(\))2238 3079 y Fr(n=)p Fn(2)2462 2970 y FF(1)p 2408 3015 159 4 v 2408 3108 a Fq(t)2444 3079 y Fr(d=)p Fn(2)2603 3037 y Fq(;)988 b Fp(\()p FF(6)p Fq(:)p FF(13)p Fp(\))94 3313 y Fh(and)1277 3466 y Fo(jr)p Fq(G)1466 3493 y Fr(t)1501 3466 y Fp(\()p Fq(z)t Fp(\))p Fo(j)44 b(\024)1889 3398 y FF(1)p 1835 3443 V 1835 3536 a Fq(t)1871 3507 y Fr(d=)p Fn(2)2194 3398 y Fq(C)2265 3425 y Fr(n)p 2017 3443 480 4 v 2017 3534 a Fp(\()p FF(1)22 b Fp(+)g Fq(z)2277 3505 y Fn(2)2318 3534 y Fq(=t)p Fp(\))2443 3505 y Fr(n)2521 3398 y Fo(j)p Fq(z)t Fo(j)p 2521 3443 107 4 v 2556 3534 a Fq(t)2664 3466 y(:)927 b Fp(\()p FF(6)p Fq(:)p FF(14)p Fp(\))94 3757 y Fs(Pro)s(of.)40 b FF(Use)25 b(the)f(stationary)g (phase)h(method)f([H].)316 3876 y(Using)19 b(this)g(lemma,)i(the)f(\256rst)g (f)o(actor)h(in)f(\(6.12\))g(is)f(bounded)h(for)g Fq(t)12 b Fo(\000)g Fq(s)28 b(<)g FF(1)20 b(and)h(for)f Fq(n)g FF(lar)n(ge)h(enough,)94 3996 y(by)129 4159 y Fv(Z)246 4295 y Fq(dy)378 4221 y Fo(j)p Fq(G)484 4248 y Fr(t)p Fi(\000)p Fr(s)619 4221 y Fo(j)647 4178 y Fn(2)687 4221 y Fp(\()p Fq(x)h Fo(\000)g Fq(y)t Fp(\))p 378 4272 618 4 v 500 4363 a Fq(')p Fp(\()p Fq(x)g Fo(\000)h Fq(y)t Fp(\))1052 4295 y Fo(\024)1174 4159 y Fv(Z)1290 4295 y Fq(dy)1635 4227 y(C)1706 4254 y Fr(n)p 1406 4272 584 4 v 1406 4305 a Fv(\000)1452 4386 y FF(1)f Fp(+)1635 4337 y Ff(\()p Fr(x)p Fi(\000)p Fr(y)s Ff(\))1847 4317 y Fg(2)p 1635 4363 243 4 v 1692 4420 a Fr(t)p Fi(\000)p Fr(s)1890 4305 y Fv(\001)1935 4326 y Fr(n)2169 4227 y FF(1)p 2030 4272 329 4 v 2030 4363 a Fp(\()p Fq(t)g Fo(\000)h Fq(s)p Fp(\))2313 4334 y Fr(d)2370 4295 y Fp(\()p FF(1)f Fp(+)h Fo(j)p Fq(x)f Fo(\000)g Fq(y)t Fo(j)2868 4252 y Fn(4)2907 4295 y Fp(\))2946 4252 y Fr(d)3037 4295 y Fo(\024)44 b Fj(O)3228 4214 y Fv(\000)3274 4295 y Fp(\()p Fq(t)22 b Fo(\000)h Fq(s)p Fp(\))3557 4252 y Fi(\000)p Fr(d=)p Fn(2)3741 4214 y Fv(\001)3811 4295 y Fq(:)94 4609 y FF(Inserting)i(in)f(\(6.12\),)g(and)h(inte)o(grating)e (o)o(v)o(er)h Fq(s)p FF(,)h(we)g(get)g(the)f(bound)1584 4762 y Fv(Z)1684 4787 y Fr(t)1639 4988 y Fn(0)1702 4898 y Fq(ds)17 b(Y)1876 4925 y Fr(t;s)2016 4898 y Fo(\024)44 b Fj(O)p Fp(\()p Fq(")p Fp(\))26 b Fq(;)94 5172 y FF(pro)o(vided)k Fq(d)37 b(<)f FF(4.)55 b(The)31 b(term)g(in)l(v)n(olving)e Fq(S)37 b FF(is)30 b(bounded)h(in)f(the)h(same)g(manner)-5 b(.)54 b(The)31 b(inhomogeneous)94 5291 y(term)k(in)g(\(6.11\))g(is)g(bounded)f(by)h(splitting)e(the)i(con)l(v)n (olution)f(inte)o(gral)g(into)g(the)h(re)o(gions)f Fq(y)46 b Fo(2)c Fq(Q)3635 5318 y Fr(L)3730 5291 y FF(and)p eop %%Page: 10 10 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(10)94 194 y Fq(y)45 b Fo(2)c Fs(R)381 151 y Fr(d)454 194 y Fo(n)28 b Fq(Q)611 221 y Fr(L)670 194 y FF(.)64 b(The)34 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Fp(\()p Fq(s;)32 b Fo(\001)17 b Fp(\))k(+)i Fq(S)6 b Fp(\()p Fq(s;)32 b Fo(\001)17 b Fp(\))i(\026)-69 b Fq(w)r Fp(\()p Fq(s;)32 b Fo(\001)17 b Fp(\))3485 769 y Fv(\001)3555 849 y Fq(:)94 1129 y FF(W)-8 b(e)23 b(deal)e(\256rst)h(with)f(the)g(inhomogeneous)f(term.)34 b(Using)21 b(the)h(same)f(splitting)e(as)j(before,)h(and)e(Lemma)h(6.4,)94 1249 y(we)k(get)1272 1368 y(sup)1192 1454 y Fr(x)p Fi(2)p Fr(Q)1354 1465 y Fz(L)p Fd(\000)p Fz(`)1507 1368 y Fo(j)1535 1288 y Fv(\000)1580 1368 y Fp(\()p Fo(r)p Fq(G)1780 1395 y Fr(t)p Ff(=)p Fn(1)1911 1368 y Fp(\))c Fq(?)g(w)2115 1395 y Fn(0)2155 1288 y Fv(\001)2201 1368 y Fp(\()p Fq(x)p Fp(\))p Fo(j)43 b(\024)i Fj(O)p Fp(\()p Fq(")p Fp(\))26 b Fq(:)94 1627 y FF(The)f(homogeneous)f(term)g Fq(I)33 b FF(in)l(v)n(olving)23 b Fq(R)i FF(is:)1416 1924 y Fq(I)52 b Fp(=)1634 1788 y Fv(Z)1734 1813 y Fr(t)1689 2014 y Fn(0)1752 1924 y Fq(ds)17 b Fo(r)p Fq(G)2029 1950 y Fr(t)p Fi(\000)p Fr(s)2186 1924 y Fq(?)2258 1843 y Fv(\000)2303 1924 y Fq(w)s(R)2454 1843 y Fv(\001)2524 1924 y Fq(:)94 2204 y FF(W)-8 b(e)26 b(w)o(ant)e(to)h(bound)f Fq(I)32 b FF(for)25 b Fq(t)j Fp(=)g FF(1)d(and)g(re)n(write)f(it)g(as)94 2490 y Fq(I)52 b Fp(=)312 2354 y Fv(Z)412 2379 y Fn(1)p Fr(=)p Fn(2)367 2581 y(0)511 2490 y Fq(ds)17 b Fo(r)p Fq(G)788 2517 y Fn(1)p Fi(\000)p Fr(s)938 2490 y Fq(?)999 2409 y Fv(\000)1045 2490 y Fq(w)s(R)1196 2409 y Fv(\001)1252 2490 y Fp(+)1340 2354 y Fv(Z)1440 2379 y Fn(1)1396 2581 y(1)p Fr(=)p Fn(2)1495 2490 y Fq(ds)g(G)1689 2517 y Fn(1)p Fi(\000)p Fr(s)1839 2490 y Fq(?)1900 2409 y Fv(\000)1946 2490 y Fq(w)s Fo(r)p Fq(R)2180 2409 y Fv(\001)2236 2490 y Fp(+)2324 2354 y Fv(Z)2425 2379 y Fn(1)2380 2581 y(1)p Fr(=)p Fn(2)2479 2490 y Fq(ds)g(G)2673 2517 y Fn(1)p Fi(\000)p Fr(s)2824 2490 y Fq(?)2885 2409 y Fv(\000)2931 2490 y Fq(R)q Fo(r)p Fq(w)3165 2409 y Fv(\001)3254 2490 y Fp(=)45 b Fq(I)3420 2517 y Fn(1)3470 2490 y Fp(+)11 b Fq(I)3602 2517 y Fn(2)3654 2490 y Fp(+)g Fq(I)3786 2517 y Fn(3)3846 2490 y Fq(:)94 2788 y FF(The)22 b(term)f Fq(I)519 2814 y Fn(2)581 2788 y FF(is)g(bounded)f(in)i(the)f(same)g(w)o(ay)h(as)f(the) h(inte)o(gral)e(of)i Fq(Y)2424 2814 y Fr(t;s)2520 2788 y FF(.)35 b(T)-8 b(o)21 b(bound)g(the)g(term)h Fq(I)3367 2814 y Fn(1)3428 2788 y FF(we)g(observ)o(e)94 2924 y(that)j(there)g(is)f(no)h(singularity)e (in)h(the)h(k)o(ernel)g(\(6.14\),)f(since)h Fq(s)i(<)2393 2885 y Fn(1)p 2393 2901 35 4 v 2393 2958 a(2)2440 2924 y FF(,)e(and)g (furthermore,)1417 3155 y Fo(jr)p Fq(G)1606 3182 y Fn(1)p Fi(\000)p Fr(s)1746 3155 y Fp(\()p Fq(z)t Fp(\))p Fo(j)44 b(\024)h FF(const)p Fq(:)o(')p Fp(\()p Fq(z)t Fp(\))25 b Fq(:)94 3386 y FF(Then)g(the)g(Schw)o (arz)h(inequality)d(and)i(the)g(results)f(on)g Fq(w)k FF(yield)1736 3616 y Fq(I)1780 3643 y Fn(1)1864 3616 y Fo(\024)45 b Fj(O)p Fp(\()p Fq(")p Fp(\))25 b Fq(:)1386 b Fp(\()p FF(6)p Fq(:)p FF(15)p Fp(\))94 3847 y FF(Finally)-6 b(,)24 b(consider)h Fq(I)827 3874 y Fn(3)866 3847 y FF(.)36 b(Inte)o(grating)24 b(Eq.\(6.9\))g(o)o(v)o(er) g Fq(s)h FF(from)f(0)h(to)2417 3808 y Fn(1)p 2417 3824 V 2417 3881 a(2)2464 3847 y FF(,)g(we)g(ha)n(v)o(e)499 4030 y Fv(Z)599 4055 y Fn(1)p Fr(=)p Fn(2)555 4257 y(0)698 4166 y Fq(ds)830 4030 y Fv(Z)946 4166 y Fq(dx)17 b(')1137 4193 y Fr(a)1185 4166 y Fp(\()p Fq(x)p Fp(\))g Fo(jr)p Fq(w)s Fp(\()p Fq(s;)g(x)p Fp(\))p Fo(j)1777 4123 y Fn(2)824 4461 y Fo(\024)45 b Fj(O)p Fp(\()p FF(1)p Fp(\))1161 4325 y Fv(Z)1260 4350 y Fn(1)p Fr(=)p Fn(2)1216 4551 y(0)1359 4461 y Fq(ds)1491 4325 y Fv(Z)1607 4461 y Fq(dx)17 b(')1798 4487 y Fr(a)1846 4461 y Fp(\()p Fq(x)p Fp(\))g Fo(j)p Fq(w)s Fp(\()p Fq(s;)g(x)p Fp(\))p Fo(j)2355 4418 y Fn(2)2413 4461 y Fp(+)2513 4325 y Fv(Z)2629 4461 y Fq(dx)g(')2820 4487 y Fr(a)2868 4461 y Fp(\()p Fq(x)p Fp(\))g Fo(j)p Fq(w)s Fp(\()p FF(0)p Fq(;)g(x)p Fp(\))p Fo(j)3380 4418 y Fn(2)3442 4461 y Fq(:)94 4763 y FF(Our)34 b(pre)n(vious)d(bounds)h(sho)n(w)g(that)h (the)g(r)-5 b(.h.s.)32 b(is)h(bounded)f(by)h Fj(O)p Fp(\()p Fq(")2544 4720 y Fn(2)2585 4763 y Fp(\))p FF(.)60 b(Therefore)34 b(there)f(is)g(a)h(v)n(alue)e(of)94 4883 y Fq(s)141 4840 y Fi(\003)215 4883 y Fo(2)c Fp(\()p FF(0)p Fq(;)454 4843 y Fn(1)p 454 4860 V 454 4917 a(2)500 4883 y Fp(\))d FF(for)g(which)1243 4927 y Fv(Z)1359 5063 y Fq(dx)17 b(')1550 5089 y Fr(a)1598 5063 y Fp(\()p Fq(x)p Fp(\))g Fo(jr)p Fq(w)s Fp(\()p Fq(s)2021 5020 y Fi(\003)2065 5063 y Fq(;)g(x)p Fp(\))p Fo(j)2234 5020 y Fn(2)2317 5063 y Fo(\024)45 b Fj(O)p Fp(\()p Fq(")2594 5020 y Fn(2)2634 5063 y Fp(\))25 b Fq(:)893 b Fp(\()p FF(6)p Fq(:)p FF(16)p Fp(\))94 5291 y FF(Furthermore,)25 b(we)g(ha)n(v)o(e)p eop %%Page: 11 11 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(11)94 194 y Fs(Lemma)39 b(6.5.)d Fh(W)-8 b(e)25 b(ha)n(v)o(e)g(the)f(bounds)129 458 y Fq(@)182 485 y Fr(t)234 323 y Fv(Z)350 458 y Fq(dx)17 b(')541 485 y Fr(a)589 458 y Fp(\()p Fq(x)p Fp(\))g Fo(jr)p Fq(w)s Fp(\()p Fq(t;)g(x)p Fp(\))p Fo(j)1170 415 y Fn(2)1251 458 y Fo(\024)45 b Fj(O)p Fp(\()p FF(1)p Fp(\))1588 323 y Fv(Z)1704 458 y Fq(dx)17 b(')1895 485 y Fr(a)1942 458 y Fp(\()p Fq(x)p Fp(\))g Fo(jr)p Fq(w)s Fp(\()p Fq(t;)g(x)p Fp(\))p Fo(j)2523 415 y Fn(2)2583 458 y Fp(+)22 b Fj(O)p Fp(\()p FF(1)p Fp(\))2897 323 y Fv(Z)3013 458 y Fq(dx)17 b(')3204 485 y Fr(a)3252 458 y Fp(\()p Fq(x)p Fp(\))g Fo(j)p Fq(w)s Fp(\()p Fq(t;)g(x)p Fp(\))p Fo(j)3750 415 y Fn(2)3811 458 y Fq(:)3619 629 y Fp(\()p FF(6)p Fq(:)p FF(17)p Fp(\))94 867 y Fs(Pro)s(of.)40 b FF(W)-8 b(e)25 b(start)f(with)422 1131 y Fq(@)475 1158 y Fr(t)527 995 y Fv(Z)643 1131 y Fq(dx)17 b(')834 1158 y Fr(a)898 1131 y Fo(jr)p Fq(w)s Fo(j)1111 1088 y Fn(2)1195 1131 y Fp(=)1317 995 y Fv(Z)1433 1131 y Fq(dx)g(')1624 1158 y Fr(a)1688 1131 y Fo(r)k Fp(\026)-71 b Fq(w)25 b Fo(\001)d Fq(@)1970 1158 y Fr(t)2005 1131 y Fo(r)p Fq(w)j Fp(+)e Fq(cc)1195 1401 y Fp(=)1317 1266 y Fv(Z)1433 1401 y Fq(dx)17 b(')1624 1428 y Fr(a)1688 1401 y Fo(r)k Fp(\026)-71 b Fq(w)25 b Fo(\001)d(r)2000 1261 y Fv(\022)2074 1401 y Fp(\()p FF(1)f Fp(+)i Fq(i\013)p Fp(\)\001)p Fq(w)i Fp(+)e Fq(R)q(w)h Fp(+)f Fq(S)j Fp(\026)-70 b Fq(w)3113 1261 y Fv(\023)3208 1401 y Fp(+)23 b Fq(cc)1195 1680 y Fp(=)45 b Fo(\000)1411 1545 y Fv(Z)1527 1680 y Fq(dx)17 b(')1718 1707 y Fr(a)1782 1680 y Fp(\001)k(\026)-71 b Fq(w)1939 1540 y Fv(\022)2013 1680 y Fp(\()p FF(1)21 b Fp(+)i Fq(i\013)p Fp(\)\001)p Fq(w)i Fp(+)e Fq(R)q(w)h Fp(+)f Fq(S)j Fp(\026)-70 b Fq(w)3052 1540 y Fv(\023)1239 1959 y Fo(\000)1339 1823 y Fv(Z)1455 1959 y Fq(dx)17 b Fp(\()p Fo(r)p Fq(')1768 1986 y Fr(a)1854 1959 y Fo(\001)22 b(r)e Fp(\026)-70 b Fq(w)s Fp(\))2100 1819 y Fv(\022)2173 1959 y Fp(\()p FF(1)22 b Fp(+)g Fq(i\013)p Fp(\)\001)p Fq(w)k Fp(+)c Fq(R)q(w)j Fp(+)e Fq(S)j Fp(\026)-70 b Fq(w)3212 1819 y Fv(\023)3308 1959 y Fp(+)23 b Fq(cc)i(:)94 2248 y FF(Using)i(again)f(the)h(e)o(xplicit)f(form)h(of)g Fq(')1472 2275 y Fr(a)1520 2248 y FF(,)h(completing)d(the)i(square)h(and)f (polarization,)f(as)h(in)g(the)g(proof)g(of)94 2368 y(Eq.\(6.10\),)e(the)f (assertion)g(follo)n(ws.)316 2487 y(W)-8 b(e)15 b(continue)g(with)g(the)g (proof)g(of)g(Lemma)g(6.3.)29 b(Let)15 b Fq(s)27 b Fo(2)h Fp(\()2345 2448 y Fn(1)p 2345 2464 35 4 v 2345 2522 a(2)2392 2487 y Fq(;)17 b FF(1])e(and)g Fq(T)2752 2514 y Fr(s)2821 2487 y Fp(=)2926 2407 y Fv(R)3009 2487 y Fq(dx)i(')3200 2514 y Fr(a)3248 2487 y Fp(\()p Fq(x)p Fp(\))p Fo(jr)p Fq(w)s Fp(\()p Fq(s;)g(x)p Fp(\))p Fo(j)3823 2444 y Fn(2)3860 2487 y FF(.)94 2621 y(Then)29 b(we)g(inte)o(grate)f(the)g(dif)n(ferential)g(inequality)f(\(6.17\))i(which)f (reads)h Fq(@)2729 2648 y Fr(t)2764 2621 y Fq(T)2822 2648 y Fr(t)2890 2621 y Fo(\024)k Fj(O)p Fp(\()p FF(1)p Fp(\))p Fq(T)3256 2648 y Fr(t)3316 2621 y Fp(+)24 b Fj(O)p Fp(\()p Fq(")3572 2578 y Fn(2)3613 2621 y Fp(\))k FF(from)94 2741 y Fq(s)141 2698 y Fi(\003)212 2741 y FF(to)c Fq(s)p FF(.)36 b(This)24 b(yields,)g(using)g(\(6.16\),)213 2890 y Fv(Z)329 3025 y Fq(dx)17 b(')520 3052 y Fr(a)568 3025 y Fp(\()p Fq(x)p Fp(\))p Fo(jr)p Fq(w)s Fp(\()p Fq(s;)g(x)p Fp(\))p Fo(j)1143 2983 y Fn(2)1224 3025 y Fo(\024)45 b FF(e)o(xp)1488 2945 y Fv(\000)1534 3025 y Fj(O)p Fp(\()p FF(1)p Fp(\)\()p Fq(s)21 b Fo(\000)i Fq(s)1986 2983 y Fi(\003)2031 3025 y Fp(\))2070 2945 y Fv(\001)2132 2890 y(Z)2249 3025 y Fq(dx)17 b(')2440 3052 y Fr(a)2487 3025 y Fp(\()p Fq(x)p Fp(\))p Fo(jr)p Fq(w)s Fp(\()p Fq(s)2893 2983 y Fi(\003)2938 3025 y Fq(;)g(x)p Fp(\))p Fo(j)3107 2983 y Fn(2)3167 3025 y Fp(+)23 b Fj(O)p Fp(\()p Fq(")3422 2983 y Fn(2)3462 3025 y Fp(\))1224 3241 y Fo(\024)45 b Fj(O)p Fp(\()p Fq(")1501 3198 y Fn(2)1541 3241 y Fp(\))25 b Fq(:)3619 3108 y Fp(\()p FF(6)p Fq(:)p FF(18)p Fp(\))94 3470 y FF(Using)f(this)g(bound,)g(we)h(re)n(write)714 3777 y Fq(I)758 3804 y Fn(3)842 3777 y Fp(=)963 3641 y Fv(Z)1063 3666 y Fn(1)1019 3868 y(1)p Fr(=)p Fn(2)1118 3777 y Fq(ds)1249 3641 y Fv(Z)1366 3777 y Fq(dy)1498 3703 y(G)1576 3730 y Fn(1)p Fi(\000)p Fr(s)1716 3703 y Fp(\()p Fq(x)d Fo(\000)h Fq(y)t Fp(\))p 1498 3754 527 4 v 1525 3774 a Fv(p)p 1625 3774 374 4 v 85 x Fq(')p Fp(\()p Fq(x)f Fo(\000)g Fq(y)t Fp(\))2054 3687 y Fv(p)p 2153 3687 V 2153 3777 a Fq(')p Fp(\()p Fq(x)g Fo(\000)h Fq(y)t Fp(\))o Fq(R)q Fp(\()p Fq(s;)17 b(y)t Fp(\))p Fo(r)p Fq(w)s Fp(\()p Fq(s;)g(y)t Fp(\))j Fq(:)94 4084 y FF(Using)k(the)h (Schw)o(arz)h(inequality)d(as)i(in)g(Eq.\(6.12\),)f(we)h(get)g(a)g(bound)1736 4321 y Fq(I)1780 4348 y Fn(3)1864 4321 y Fo(\024)45 b Fj(O)p Fp(\()p Fq(")p Fp(\))25 b Fq(:)94 4558 y FF(Combining)i(the)h(bounds)f(on)h Fq(I)1209 4585 y Fn(1)1248 4558 y FF(,)h Fq(I)1346 4585 y Fn(2)1414 4558 y FF(and)f Fq(I)1630 4585 y Fn(3)1698 4558 y FF(completes)f(the)h(proof) f(of)i(Eq.\(6.7\).)45 b(The)28 b(proof)g(of)g(Lemma)94 4677 y(6.3)d(is)f(complete.)316 4797 y(Lemma)30 b(6.3)h(gi)n(v)o(es)f(us)h (control)f(o)o(v)o(er)g(the)h(e)n(v)n(olution)f(of)h(dif)n(ferences)g(in)g Fo(G)6 b FF(,)33 b(when)e(the)o(y)f(are)i(small)94 4916 y(in)e Fo(G)6 b(j)295 4943 y Fr(Q)358 4953 y Fz(L)415 4916 y FF(.)52 b(W)-8 b(e)30 b(shall)f(no)n(w)g(use)h(this)f(information)g(to)g(study)g(the) h(deformation)f(of)h(balls)f(co)o(v)o(ering)g Fo(G)6 b(j)3729 4943 y Fr(Q)3792 4953 y Fz(L)3849 4916 y FF(.)94 5036 y(T)-8 b(o)29 b(formulate)g(the)f(ne)o(xt)g(result)h(we)g(need)g(the)g(follo)n(wing) e(notation:)42 b(Consider)29 b(the)f(uni)n(v)o(ersal)g(attracting)94 5172 y(set)34 b Fo(G)6 b FF(.)62 b(The)34 b(quantity)e Fq(N)1031 5120 y Ff(\()p Fr(t)p Ff(\))1020 5201 y Fr(B)1128 5172 y Fp(\()p Fq(")p Fp(\))i FF(denotes)f(the)g(number)g(of)h(balls)e(of)i(radius)f Fq(")h FF(needed)g(to)f(co)o(v)o(er)g(the)g(set)94 5291 y Fp(\002)171 5248 y Fr(t)207 5291 y Fp(\()p Fo(G)6 b Fp(\))p Fo(j)378 5318 y Fr(B)446 5291 y FF(,)25 b(in)f(L)659 5248 y Fi(1)743 5291 y Fp(\()p Fq(B)5 b Fp(\))p FF(,)24 b(where)i Fp(\002)1297 5248 y Fr(t)1357 5291 y FF(is)e(the)h(semi-\257o)n(w)f(de\256ned)h(by)g(the)f(CGL) i(equation.)p eop %%Page: 12 12 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(12)94 194 y Fs(Prop)s(osition)42 b(6.6.)k Fh(There)29 b(are)g(constants)e Fq(c)32 b(<)h Fo(1)28 b Fh(and)g Fq(D)s Fh(,)h Fq(D)2457 221 y Fn(1)2530 194 y Fq(<)k Fo(1)27 b Fh(such)h(that)g(for)g(all)g(suf)n(\256ciently)94 314 y(small)c Fq(")k(>)h FF(0)24 b Fh(and)h(all)g Fq(L)i(>)h FF(3)p Fq(=")d Fh(one)g(has)g(the)f(bound)993 605 y Fq(N)1084 553 y Ff(\()p Fr(t)p Ff(+)p Fn(1)p Ff(\))1073 634 y Fr(Q)1136 644 y Fz(L)1277 605 y Fp(\()p Fq("=)p FF(2)p Fp(\))44 b Fo(\024)1667 494 y Fv(\020)1740 537 y Fq(c)p 1738 582 47 4 v 1738 673 a(")1797 494 y Fv(\021)1856 515 y Fr(D)1921 525 y Fg(1)1951 515 y Fr(L)2006 485 y Fz(d)p Fd(\000)p Fg(1)2127 515 y Fr(")2165 485 y Fd(\000)p Fc(\()p Fg(1)p Fc(+)p Fz(d)p Fc(\))2412 605 y Fq(D)2497 562 y Fr(L)2552 532 y Fz(d)2599 605 y Fq(N)2690 553 y Ff(\()p Fr(t)p Ff(\))2679 634 y Fr(Q)2742 644 y Fz(L)2799 605 y Fp(\()p Fq(")p Fp(\))25 b Fq(:)643 b Fp(\()p FF(6)p Fq(:)p FF(19)p Fp(\))316 849 y FF(Before)26 b(we)f(pro)o(v)o(e)f(this)g(proposition,)e(we)j(need)g(a)g (geometric)g(lemma:)94 1041 y Fs(Lemma)44 b(6.7.)51 b Fh(Let)30 b Fq(Q)g Fh(be)g(a)g(set)g(of)g(diameter)g Fq(r)i Fh(in)e Fs(R)2109 998 y Fr(d)2185 1041 y Fh(and)g(assume)g(that)f Fj(F)h Fh(is)g(a)g(f)o(amily) f(of)h(comple)o(x)94 1160 y(functions)24 b Fq(f)36 b Fh(on)24 b Fq(Q)h Fh(which)f(satisfy)g(the)h(bounds)1503 1379 y Fo(j)p Fq(f)11 b Fo(j)44 b(\024)g Fq(a)25 b(;)116 b Fo(jr)p Fq(f)11 b Fo(j)43 b(\024)i Fq(b)24 b(;)94 1597 y Fh(with)h Fq(br)30 b Fo(\024)e Fq(c=)p FF(2)p Fh(.)35 b(Then)25 b(one)g(can)g(co)o(v)o(er)f Fj(F)i Fh(with)e(not)g(more)g(than)1828 1815 y Fp(\()p FF(4)p Fq(a=c)p Fp(\))2102 1773 y Fn(2)94 2034 y Fh(balls)h(of)f(radius)h Fq(c)g Fh(in)f FF(L)915 1991 y Fi(1)1000 2034 y Fp(\()p Fq(Q)p Fp(\))p Fh(.)94 2264 y Fs(Pro)s(of.)38 b FF(On)23 b(a)g(disk)g(in)f Fs(R)1029 2221 y Fr(d)1099 2264 y FF(of)h(diameter)g Fq(r)s FF(,)g(the)g(function)g Fq(f)33 b FF(v)n(aries)23 b(no)g(more)g(than)g Fq(br)i FF(which)e(is)g(bounded)94 2383 y(by)29 b Fq(c=)p FF(2.)48 b(On)29 b(the)g(other)g(hand,)g(one)g(can)g(\256nd)h(a)f(set)g Fj(S)g FF(of)g Fp(\()p FF(4)p Fq(a=c)p Fp(\))2430 2341 y Fn(2)2497 2383 y FF(comple)o(x)f(numbers)g(of)h(modulus)f(less)94 2503 y(than)21 b Fq(a)g FF(such)g(that)g(e)n(v)o(ery)g(comple)o(x)e(number)i(of)g (modulus)f(less)h(than)f Fq(a)h FF(is)g(within)f Fq(c=)p FF(2)h(of)g Fj(S)p FF(.)35 b(Since)22 b Fq(f)32 b FF(v)n(aries)94 2622 y(less)25 b(than)f Fq(c=)p FF(2)h(one)g(can)g(\256nd)g(a)g(constant)f (function)g Fq(f)1988 2580 y Fi(\003)2058 2622 y FF(with)g(v)n(alue)g(in)h Fj(S)g FF(such)f(that)h(sup)3200 2649 y Fr(Q)3284 2622 y Fo(j)p Fq(f)33 b Fo(\000)22 b Fq(f)3553 2580 y Fi(\003)3598 2622 y Fo(j)28 b Fq(<)g(c)p FF(.)94 2832 y Fs(Pro)s(of)48 b(of)e(Prop)s(osition)h (6.6.)f FF(By)34 b(de\256nition)e(we)h(can)h(\256nd,)h(for)f(e)n(v)o(ery)e Fq(t)39 b Fo(\025)h FF(0,)35 b Fq(N)3303 2780 y Ff(\()p Fr(t)p Ff(\))3292 2861 y Fr(Q)3355 2871 y Fz(L)3412 2832 y Fp(\()p Fq(")p Fp(\))f FF(balls)e(of)94 2989 y(radius)c Fq(")h FF(in)e(L)607 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y(Eq.\(6.20\):)1399 1249 y Fq(D)48 b Fp(=)c Fj(O)1720 1169 y Fv(\000)1767 1249 y FF(64)p Fq(c)1910 1198 y Fn(1)p Fr(=)13 b Fn(min)p Ff(\()p Fn(1)p Fr(;)p Fn(1)p Fr(=)p Ff(\()p Fn(4)p Fr(c)2375 1208 y Fg(2)2404 1198 y Ff(\)\))1910 1278 y Fn(1)2471 1169 y Fv(\001)2542 1249 y Fq(;)94 1435 y FF(where)21 b Fq(c)401 1462 y Fn(1)461 1435 y FF(and)e Fq(c)667 1462 y Fn(2)727 1435 y FF(can)i(be)e(found)h(in)f(the)h(proof)g(of)g(Lemma)f(6.3.)34 b(Note)19 b(also)h(that)f(there)h(is)f(a)h(crosso)o(v)o(er)f(point)94 1555 y(\(for)29 b(our)f(bound\))g(between)g(the)g(beha)n(vior)g(described)g (in)f(Theorem)h(4.2,)h(and)f(the)g(di)n(v)o(er)n(gence)f(described)94 1674 y(in)e(\(3.2\),)g(at)f(about)h Fq(")j Fp(=)g Fq(L)1028 1632 y Fi(\000)p Fn(1)p Fr(=)p Ff(\()p Fn(1)p Ff(+)p Fr(d)p Ff(\))1371 1674 y FF(.)94 2028 y Fu(7.)46 b(Lo)l(w)l(er)g(Bound)e(on)g(the)i Fe(")p Fu(-en)l(trop)l(y)f(p)t(er)f(Unit)i(Length)94 2217 y FF(In)29 b(this)e(section,)h(we)g(construct)g(a)g(lo)n(wer)g(bound)f(on)h Fq(H)2065 2244 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4915 y Fj(E)1942 4942 y Fr(b)1982 4915 y Fp(\()p Fq(\021)t Fp(\))2113 4835 y Fv(\001)p 1704 4966 462 4 v 1716 5027 a Fn(2)p Fr(b)p 1716 5043 70 4 v 1726 5100 a(\031)1814 5066 y FF(log)o Fp(\()p FF(1)p Fq(=")p Fp(\))2221 4989 y(=)45 b FF(1)25 b Fq(;)1223 b Fp(\()p FF(7)p Fq(:)p FF(2)p Fp(\))94 5291 y FF(so)25 b(there)g(are)h (\252man)o(y\272)e(such)g(functions.)35 b(\(See)26 b([KT],)f(Theorem)f(XXII)i (and)e(be)o(ginning)f(of)i Fo(x)p FF(3\).)p eop %%Page: 15 15 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(15)316 194 y FF(W)-8 b(e)30 b(w)o(ant)f(to)h(use)f(the)h(set)f Fj(E)1321 221 y Fr(b)1361 194 y Fp(\()p Fq(\021)t Fp(\))g FF(as)g(the)h(starting)f(point)f(for)i(the)f (construction)g(of)g(a)h(set)g(in)f Fo(G)36 b FF(with)94 314 y(positi)n(v)o(e)16 b Fq(")p FF(-entrop)o(y)-6 b(.)33 b(Thus,)18 b(we)g(w)o(ant)f(to)g(e)n(v)n(olv)o(e)f Fj(E)1879 341 y Fr(b)1919 314 y Fp(\()p Fq(\021)t Fp(\))h Ft(forwar)l(d)g 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FF(are)h(separated)g(almost)e(as)94 3619 y(much.)58 b(Therefore,)34 b(kno)n(wing)d(that)h(the)g(set)g Fj(E)1769 3646 y Fr(b)1808 3619 y Fp(\()p Fq(\021)t Fp(\))g FF(of)g Fq(f)43 b FF(has)32 b(positi)n(v)o(e)e Fq(")p FF(-entrop)o(y)i(implies)f(that)g(the)h (set)94 3738 y Fq(S)6 b Fp(\()p Fj(E)261 3765 y Fr(b)301 3738 y Fp(\()p Fq(\021)t Fp(\)\))p FF(\320which)26 b(is)h(in)g(the)g(global)f (attracting)h(set\320also)f(has)i(positi)n(v)o(e)d Fq(")p FF(-entrop)o(y)-6 b(,)27 b(as)h(we)f(shall)g(sho)n(w)94 3858 y(later)-5 b(.)94 4210 y Fs(7.2.)39 b(The)e(regularized)f(linear)h(ev)m(olution)94 4399 y FF(In)21 b(this)f(subsection,)g(we)g(construct)g(a)h(some)n(what)e (more)h(re)o(gular)h(representation)f(of)g Fp(\002)3131 4356 y Fr(T)3131 4426 y Fn(0)3194 4399 y FF(,)i(which)e(is)g(needed)94 4519 y(because)26 b(we)f(consider)f(ne)o(gati)n(v)o(e)f Fq(T)14 b FF(.)316 4638 y(W)-8 b(e)22 b(consider)e(the)i(class)f Fj(E)1246 4665 y Fr(b)1285 4638 y Fp(\()p Fq(\021)t Fp(\))p FF(,)h(with)e Fq(b)28 b Fp(=)g FF(1)p Fq(=)p FF(3.)34 b(It)21 b(is)g(clear)h(from)f(the)g (P)o(ale)o(y-W)l(iener)n(-Schw)o(artz)h([S])94 4773 y(theorem)k(that)f(the)g (functions)f Fq(f)39 b Fo(2)29 b Fj(E)1409 4800 y Fr(b)1449 4773 y Fp(\()p Fq(\021)t Fp(\))c FF(ha)n(v)o(e)g(a)h(F)o(ourier)f(transform) 2634 4747 y Fv(b)2615 4773 y Fq(f)11 b Fp(\()p Fq(k)s Fp(\))28 b(=)2942 4693 y Fv(R)3025 4773 y Fq(dx)17 b(e)3197 4731 y Fr(ik)r(x)3319 4773 y Fq(f)11 b Fp(\()p Fq(x)p Fp(\))25 b FF(which)g(is)94 4912 y(a)30 b(distrib)n(ution)c(with)i(support)g(in)g([)p Fo(\000)p Fq(b;)17 b(b)p FF(].)47 b(If)1754 4886 y Fv(b)1735 4912 y Fq(f)40 b FF(were)29 b(a)g(function,)g(we)g(could)f(freely)h(go)g(back)g(and)f(forth) 94 5032 y(between)34 b Fq(k)s FF(-space)f(and)g Fq(x)p FF(-space.)62 b(T)-8 b(o)33 b(deal)g(with)f(this)g(problem,)j(we)e(use)g(a)h(re)o (gularizing)e(de)n(vice.)60 b(Let)94 5162 y Fq(c)39 b(>)g(b)33 b FF(and)f(let)694 5135 y Fv(b)677 5162 y Fq( )k FF(be)d(a)g(positi)n(v)o(e)d Fo(C)1384 5119 y Fi(1)1501 5162 y FF(function)i(with)g(support)g(in)g([)p Fo(\000)p Fq(c;)17 b(c)p FF(])33 b(and)g(equal)f(to)h(1)f(on)g([)p Fo(\000)p Fq(b;)17 b(b)p FF(].)94 5291 y(W)-8 b(e)29 b(shall)e(tak)o(e)g Fq(b)32 b Fp(=)g FF(1)p Fq(=)p FF(3,)c Fq(c)k Fp(=)g FF(1)p Fq(=)1332 5208 y Fo(p)p 1414 5208 50 4 v 1414 5291 a FF(3.)45 b(Clearly)1872 5265 y Fv(b)1855 5291 y Fq( )s Fp(\()p Fq(k)s Fp(\))2074 5265 y Fv(b)2056 5291 y Fq(f)10 b Fp(\()p Fq(k)s Fp(\))31 b(=)2407 5265 y Fv(b)2389 5291 y Fq(f)10 b Fp(\()p Fq(k)s Fp(\))27 b FF(\(as)h(a)g(distrib)n(ution\))e(and)h(therefore)p eop %%Page: 16 16 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(16)94 194 y Fq( )18 b(?)c(f)38 b Fp(=)28 b Fq(f)k FF(\(in)21 b Fq(x)p FF(-space\),)i(where)e Fq(?)g FF(denotes)g(the)g(con)l(v)n(olution)e(product.)34 b(W)-8 b(e)22 b(de\256ne)g(a)f(re)o(gularized)g(linear)94 314 y(e)n(v)n(olution)i(k) o(ernel)1157 460 y Fq(g)1205 487 y Fr(T)1268 460 y Fp(\()p Fq(x)p Fp(\))43 b(=)1568 325 y Fv(Z)1684 460 y Fq(dk)20 b(e)1854 418 y Fr(ik)r(x)1994 434 y Fv(b)1977 460 y Fq( )s Fp(\()p Fq(k)s Fp(\))p Fq(e)2224 418 y Fr(T)11 b Ff(\()p Fn(1)p Fi(\000)p Ff(\()p Fn(1)p Ff(+)p Fr(i\013)p Ff(\))p Fr(k)2692 388 y Fg(2)2722 418 y Ff(\))2783 460 y Fq(;)94 687 y FF(and)25 b(then)g(we)g(de\256ne)1389 726 y Fv(\000)1435 807 y Fp(\002)1512 764 y Fr(T)1512 834 y Fn(0)p Fr(; )1631 807 y Fq(f)1691 726 y Fv(\001)1736 807 y Fp(\()p Fq(x)p Fp(\))44 b Fo(\021)2037 726 y Fv(\000)2083 807 y Fq(g)2131 834 y Fr(T)2215 807 y Fq(?)22 b(f)2347 726 y Fv(\001)2392 807 y Fp(\()p Fq(x)p Fp(\))i Fq(:)94 990 y FF(This)17 b(is)f(our)h(re)o (gularized)f(representation)g(of)h(the)g(linear)g(e)n(v)n(olution.)31 b(By)17 b(construction,)g(it)f(has)h(the)g(property:)94 1110 y(If)26 b Fq(f)38 b Fo(2)28 b Fj(E)428 1137 y Fr(b)468 1110 y Fp(\()p Fq(\021)t Fp(\))p FF(,)c(then)1647 1229 y Fp(\002)1724 1186 y Fr(T)1724 1256 y Fn(0)p Fr(; )1843 1229 y Fq(f)55 b Fp(=)45 b(\002)2146 1186 y Fr(T)2146 1256 y Fn(0)2209 1229 y Fq(f)36 b(;)1347 b Fp(\()p FF(7)p Fq(:)p FF(3)p Fp(\))94 1413 y FF(as)23 b(a)g(distrib)n(ution.)32 b(But,)23 b(as)g(we)g(shall)e(see)i (belo)n(w)-6 b(,)22 b(the)g(l.h.s.)f(is)i(a)f(well)g(de\256ned)h(function)f (and)g(thus)g(we)h(can)94 1532 y(use)j(either)g(of)g(the)g(de\256nitions,)e (whiche)n(v)o(er)h(is)h(more)f(con)l(v)o(enient.)38 b(Henceforth,)26 b(we)g(use)g(the)g(notation)e Fq(f)3839 1559 y Fr(t)94 1652 y FF(for)i Fp(\002)313 1609 y Fr(t)313 1679 y Fn(0)353 1652 y Fq(f)38 b Fp(=)28 b(\002)622 1609 y Fr(T)622 1679 y Fn(0)p Fr(; )741 1652 y Fq(f)11 b FF(.)94 2002 y Fs(7.3.)39 b(Pro)s(of)g(of)g(the)e (\014rst)h(part)f(of)h(Theorem)g(7.2)94 2191 y FF(This)33 b(theorem)f(is)h (relati)n(v)o(ely)f(con)l(v)o(entional,)h(b)n(ut)f(tedious,)i(to)f(pro)o(v)o (e.)60 b(W)-8 b(e)33 b(will)f(therefore)i(only)e(sk)o(etch)94 2311 y(the)d(standard)f(estimates)f(and)h(describe)h(in)f(detail)g(only)f (the)h(general)h(sequence)g(of)f(estimates)f(which)h(are)94 2430 y(needed.)316 2550 y(W)-8 b(e)22 b(be)o(gin)g(the)g(proof)g(of)g(the)g (\256rst)g(part)g(of)g(Theorem)g(7.2)g(with)f(a)i(study)e(of)h Fp(\002)3016 2507 y Fr(t)3052 2550 y FF(.)35 b(First)22 b(we)g(w)o(ould)g (lik)o(e)94 2669 y(to)j(pro)o(v)o(e)f(that)g Fp(\002)697 2626 y Fr(t)733 2669 y Fp(\()p Fq(f)821 2696 y Fi(\000)p Fr(T)945 2669 y Fp(\))e Fo(\000)h Fq(f)1155 2696 y Fr(t)p Fi(\000)p Fr(T)1334 2669 y FF(remains)i(small)f(for)h(0)i Fo(\024)h Fq(t)g Fo(\024)g Fq(T)14 b FF(.)94 2861 y Fs(Lemma)38 b(7.3.)f Fh(F)o(or)24 b Fq(\021)29 b Fh(small)23 b(enough,)h(there)h(is)f(a)h Fq(\032)i(>)h FF(0)d Fh(such)f(that)g(for)h(an)o(y)f Fq(T)42 b(>)28 b FF(0)c Fh(and)h(an)o(y)f Fq(t)j Fo(2)i FF([0)p Fq(;)17 b(T)d FF(])94 2980 y Fh(we)26 b(ha)n(v)o(e)e(for)h(all)g Fq(f)38 b Fo(2)28 b Fj(E)953 3007 y Fr(b)993 2980 y Fp(\()p Fq(\021)t Fp(\))p Fh(,)c(the)h(bound)1208 3219 y Fo(k)p Fp(\002)1335 3176 y Fr(t)1370 3219 y Fp(\()p Fq(f)1458 3246 y Fi(\000)p Fr(T)1583 3219 y Fp(\))d Fo(\000)g Fq(f)1792 3246 y Fr(t)p Fi(\000)p Fr(T)1947 3219 y Fo(k)1997 3246 y Fi(1)2126 3219 y Fo(\024)45 b Fq(\021)2301 3176 y Fn(2)2341 3219 y Fq(e)2387 3176 y Fi(\000)p Fr(\032)p Ff(\()p Fr(T)11 b Fi(\000)p Fr(t)p Ff(\))2733 3219 y Fq(:)94 3502 y Fs(Pro)s(of.)40 b FF(First)24 b(observ)o(e)h(that)f(by)h(assumption)d Fo(k)p Fq(f)11 b Fo(k)1921 3529 y Fi(1)2033 3502 y Fo(\024)28 b Fq(\021)t FF(.)36 b(By)25 b(de\256nition,)f(we)h(ha)n(v)o(e)766 3700 y Fv(\000)812 3781 y Fp(\002)889 3738 y Fi(\000)p Fr(T)889 3809 y Fn(0)1014 3781 y Fq(f)1074 3700 y Fv(\001)1119 3781 y Fp(\()p Fq(x)p Fp(\))44 b(=)1420 3645 y Fv(Z)1536 3781 y Fq(dy)20 b(dk)f(e)1826 3738 y Fr(ik)r Ff(\()p Fr(x)p Fi(\000)p Fr(y)s Ff(\))2117 3781 y Fq(e)2163 3738 y Fi(\000)p Fr(T)11 b Ff(\()p Fn(1)p Fi(\000)p Fr(k)2455 3708 y Fg(2)2485 3738 y Ff(\()p Fn(1)p Ff(+)p Fr(i\013)p Ff(\)\))2777 3754 y Fv(b)2760 3781 y Fq( )s Fp(\()p Fq(k)s Fp(\))p Fq(f)g Fp(\()p Fq(y)t Fp(\))23 b Fq(:)94 4081 y FF(Since)358 4055 y Fv(b)341 4081 y Fq( )28 b FF(is)c(smooth)g(and)h(supported)e(in)i Fo(j)p Fq(k)s Fo(j)i(\024)h Fq(c)p FF(,)d(we)g(get)g(from)f(this)g(the)h(easy)g(b)n (ut)f(useful)g(bound)1397 4329 y Fo(k)p Fq(f)1496 4356 y Fi(\000)p Fr(T)1621 4329 y Fo(k)1671 4356 y Fi(1)1799 4329 y Fo(\024)45 b Fj(O)p Fp(\()p Fq(\021)t Fp(\))p Fq(e)2168 4286 y Fi(\000)p Ff(\()p Fn(1)p Fi(\000)p Fr(c)2394 4256 y Fg(2)2425 4286 y Ff(\))p Fr(T)2543 4329 y Fq(:)1098 b Fp(\()p FF(7)p Fq(:)p FF(4)p Fp(\))94 4566 y FF(Using)24 b(Eq.\(7.3\),)h(we)g(see)g(that)f Fp(\002)1278 4523 y Fr(t)1278 4593 y Fn(0)p Fr(; )1397 4566 y Fq(f)1446 4593 y Fi(\000)p Fr(T)1599 4566 y Fp(=)k Fq(f)1753 4593 y Fr(t)p Fi(\000)p Fr(T)1933 4566 y FF(satis\256es)1239 4813 y Fq(@)1292 4840 y Fr(t)1327 4813 y Fq(f)1376 4840 y Fr(t)p Fi(\000)p Fr(T)1575 4813 y Fp(=)45 b(\()p FF(1)22 b Fp(+)g Fq(i\013)p Fp(\))p Fq(@)2103 4770 y Fn(2)2097 4840 y Fr(x)2147 4813 y Fq(f)2196 4840 y Fr(t)p Fi(\000)p Fr(T)2373 4813 y Fp(+)h Fq(f)2522 4840 y Fr(t)p Fi(\000)p Fr(T)2702 4813 y Fq(:)94 5050 y FF(W)-8 b(e)26 b(let)e Fq(v)32 b Fp(=)c(\002)636 5007 y Fr(t)672 5050 y Fp(\()p Fq(f)760 5077 y Fi(\000)p Fr(T)884 5050 y Fp(\))22 b Fo(\000)h Fq(f)1094 5077 y Fr(t)p Fi(\000)p Fr(T)1248 5050 y FF(,)i(and)g(then)g(we)g(\256nd)731 5291 y Fq(@)784 5318 y Fr(t)819 5291 y Fq(v)48 b Fp(=)d(\()p FF(1)22 b Fp(+)g Fq(i\013)p Fp(\))p Fq(@)1443 5248 y Fn(2)1437 5318 y Fr(x)1487 5291 y Fq(v)k Fp(+)d Fq(v)j Fo(\000)d Fp(\()p FF(1)e Fp(+)i Fq(i\014)5 b Fp(\)\()p Fq(v)26 b Fp(+)d Fq(f)2441 5318 y Fr(t)p Fi(\000)p Fr(T)2596 5291 y Fp(\))p Fo(j)p Fp(\()p Fq(v)i Fp(+)e Fq(f)2924 5318 y Fr(t)p Fi(\000)p Fr(T)3079 5291 y Fp(\))p Fo(j)3146 5248 y Fn(2)3210 5291 y Fq(:)p eop %%Page: 17 17 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(17)94 194 y FF(W)-8 b(e)26 b(write)e(this)g(as)h(an)g(inte)o(gral)f(equation)g(using)g Fq(v)t Fp(\()p FF(0)p Fq(;)17 b(x)p Fp(\))26 b(=)i FF(0.)36 b(W)-8 b(e)25 b(get)565 496 y Fq(v)t Fp(\()p 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Fq(t)p FF(,)h Fq(T)14 b FF(\))31 b(in)f(this)f(space)i(and)g(therefore)g(a)94 3427 y(unique)f(solution)f Fq(v)34 b FF(for)d(the)f(Eq.\(7.5\).)52 b(Furthermore,)32 b(the)f(asserted)f(bounds)f(of)i(Lemma)f(7.3)g(follo)n(w)f (at)94 3547 y(once.)36 b(W)-8 b(e)26 b(lea)n(v)o(e)e(the)h(\(tri)n(vial\))f (details)g(to)g(the)h(reader)-5 b(.)36 b(The)25 b(proof)g(of)g(Lemma)f(7.3)g (is)h(complete.)316 3667 y(W)-8 b(e)24 b(no)n(w)f(come)h(to)f(the)h(proof)g (of)g(con)l(v)o(er)n(gence)g(of)g Fp(\002)2201 3624 y Fr(T)2264 3667 y Fq(f)2313 3693 y Fi(\000)p Fr(T)2438 3667 y FF(,)g(as)g Fq(T)41 b Fo(!)28 b(1)p FF(.)35 b(W)-8 b(e)24 b(shall)f(sho)n(w)g(that)g(the) 94 3786 y(deri)n(v)n(ati)n(v)o(e)d(of)h(this)g(quantity)f(is)h(inte)o(grable) g(in)g Fq(T)14 b FF(.)35 b(W)-8 b(e)22 b(recall)g(that)f(if)h(we)f(ha)n(v)o (e)h(a)g(v)o(ector)f(\256eld)h Fq(X)29 b FF(with)20 b(\257o)n(w)94 3906 y Fq(')159 3933 y Fr(t)220 3906 y FF(then)1472 4012 y Fq(d)p 1454 4056 88 4 v 1454 4147 a(dt)1553 4079 y(')1618 4106 y Fr(t)1654 4079 y Fp(\()p Fq(x)p Fp(\))43 b(=)i FF(D)p Fq(')2091 4106 y Fr(t)2126 4079 y FF([)p Fq(x)p FF(])p Fq(X)8 b Fp(\()p Fq(x)p Fp(\))24 b Fq(:)94 4293 y FF(W)-8 b(e)27 b(use)f(throughout)e(the)i (notation)f(D)p Fq(F)14 b FF([)p Fq(x)p FF(])26 b(for)g(the)g(deri)n(v)n(ati) n(v)o(e)d(of)j Fq(F)40 b FF(e)n(v)n(aluated)25 b(at)h Fq(x)p FF(;)h(this)e(is)g(usually)g(an)94 4412 y(operator)-5 b(.)36 b(In)25 b(our)f(case,)i(we)f(get)209 4626 y Fq(d)p 173 4671 124 4 v 173 4762 a(dT)309 4694 y Fp(\002)386 4651 y Fr(T)449 4694 y Fp(\()p Fq(f)537 4721 y Fi(\000)p Fr(T)662 4694 y Fp(\))44 b(=)h FF(D)p Fp(\002)1016 4651 y Fr(T)1079 4694 y FF([)p Fq(f)1161 4721 y Fi(\000)p Fr(T)1286 4694 y FF(])739 4946 y Fo(\001)789 4806 y Fv(\022)863 4946 y Fp(\()p FF(1)21 b Fp(+)i Fq(i\013)p Fp(\))p Fq(@)1269 4904 y Fn(2)1263 4973 y Fr(x)1313 4946 y Fq(f)1362 4973 y Fi(\000)p Fr(T)1509 4946 y Fp(+)g Fq(f)1658 4973 y Fi(\000)p Fr(T)1805 4946 y Fo(\000)f Fp(\()p FF(1)g Fp(+)h Fq(i\014)5 b Fp(\))p Fq(f)2298 4973 y Fi(\000)p Fr(T)2423 4946 y Fo(j)p Fq(f)2500 4973 y Fi(\000)p Fr(T)2625 4946 y Fo(j)2653 4904 y Fn(2)2714 4946 y Fo(\000)23 b Fp(\()p FF(1)f Fp(+)g Fq(i\013)p Fp(\))p Fq(@)3220 4904 y Fn(2)3214 4973 y Fr(x)3264 4946 y Fq(f)3313 4973 y Fi(\000)p Fr(T)3460 4946 y Fo(\000)h Fq(f)3609 4973 y Fi(\000)p Fr(T)3734 4806 y Fv(\023)745 5171 y Fp(=)45 b Fo(\000)p Fp(\()p FF(1)22 b Fp(+)h Fq(i\014)5 b Fp(\))p FF(D)p Fp(\002)1438 5129 y Fr(T)1501 5171 y FF([)p Fq(f)1583 5198 y Fi(\000)p Fr(T)1708 5171 y FF(])1741 5091 y Fv(\000)1787 5171 y Fq(f)1836 5198 y Fi(\000)p Fr(T)1961 5171 y Fo(j)p Fq(f)2038 5198 y Fi(\000)p Fr(T)2163 5171 y Fo(j)2191 5129 y Fn(2)2230 5091 y Fv(\001)2301 5171 y Fq(:)3669 5291 y Fp(\()p FF(7)p Fq(:)p FF(8)p Fp(\))p eop %%Page: 18 18 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(18)94 194 y FF(W)-8 b(e)24 b(w)o(ant)e(to)g(pro)o(v)o(e)g(that)g(this)g(quantity)f(is)h(inte)o (grable)g(o)o(v)o(er)g Fq(T)14 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1091 1406 y Fq(@)1144 1432 y Fr(t)1179 1406 y Fq(w)47 b Fp(=)e(\()p FF(1)22 b Fp(+)g Fq(i\013)p Fp(\))p Fq(@)1825 1363 y Fn(2)1819 1432 y Fr(x)1870 1406 y Fq(w)j Fp(+)d Fq(w)j Fp(+)e Fq(R)2337 1432 y Fr(\014)2415 1406 y Fq(w)j Fp(+)c Fq(S)2672 1432 y Fr(\014)p 2751 1351 75 4 v 2751 1406 a Fq(w)28 b(;)741 b Fp(\()p FF(7)p Fq(:)p FF(10)p Fp(\))94 1624 y FF(with)25 b(initial)e(condition)g Fq(w)s Fp(\()p Fq(t)28 b Fp(=)g FF(0)p Fq(;)17 b Fo(\001)p Fp(\))26 b(=)i Fq(w)1601 1651 y Fn(0)1641 1624 y Fp(\()p Fo(\001)p Fp(\))p FF(.)35 b(The)25 b(coef)n(\256cients)g Fq(R)2545 1651 y Fr(\014)2623 1624 y FF(and)g Fq(S)2853 1651 y Fr(\014)2932 1624 y FF(are)g(gi)n(v)o(en)f(by)1170 1913 y Fq(R)1246 1940 y Fr(\014)1299 1913 y Fp(\()p Fq(t;)17 b(x)p Fp(\))43 b(=)i Fo(\000)p FF(2)p Fp(\()p FF(1)22 b Fp(+)h Fq(i\014)5 b Fp(\))p Fo(j)p Fp(\002)2257 1870 y Fr(t)2293 1913 y Fp(\()p Fq(f)2381 1940 y Fi(\000)p Fr(T)2505 1913 y Fp(\)\()p Fq(x)p Fp(\))p Fo(j)2707 1870 y Fn(2)2771 1913 y Fq(;)94 2181 y FF(and)1168 2330 y Fq(S)1229 2357 y Fr(\014)1282 2330 y Fp(\()p Fq(t;)17 b(x)p Fp(\))43 b(=)i Fo(\000)p Fp(\()p FF(1)22 b Fp(+)h Fq(i\014)5 b Fp(\))2102 2249 y Fv(\000)2148 2330 y Fp(\002)2225 2287 y Fr(t)2260 2330 y Fp(\()p Fq(f)2348 2357 y Fi(\000)p Fr(T)2473 2330 y Fp(\)\()p Fq(x)p Fp(\))2647 2249 y Fv(\001)2692 2269 y Fn(2)2773 2330 y Fq(:)94 2549 y FF(The)39 b(assertion)f(of)g(Lemma)g(7.4)g (follo)n(ws)f(no)n(w)-6 b(,)41 b(using)c(a)i(contraction)f(ar)n(gument,)k(as) c(in)g(the)h(study)e(of)94 2668 y(Eq.\(7.5\),)25 b(from)f(Lemma)h(7.3)f(and)h (the)f(pre)n(vious)g(formula.)35 b(The)25 b(details)f(are)i(again)e(left)h (to)f(the)h(reader)-5 b(.)316 2788 y(As)42 b(a)h(consequence,)k(combining)41 b(the)i(inequalities)e(\(7.4\))h(and)h(\(7.9\),)k(the)42 b(right)g(hand)h (side)f(of)94 2907 y(Eq.\(7.8\))28 b(is)g(e)o(xponentially)e(small)h(in)h Fq(T)42 b FF(and)28 b(therefore)h(inte)o(grable)e(and)h(we)g(ha)n(v)o(e)g(a)g (limit.)44 b(So)28 b(our)g(map)94 3027 y Fq(S)j FF(is)25 b(well-de\256ned)g (by)1181 3176 y Fq(S)6 b Fp(\()p Fq(f)11 b Fp(\))43 b(=)87 b FF(lim)1551 3240 y Fr(T)11 b Fi(!1)1784 3176 y Fp(\002)1861 3133 y Fr(T)1924 3176 y Fp(\()p Fq(f)2012 3203 y Fi(\000)p Fr(T)2137 3176 y Fp(\))44 b(=)h Fq(f)32 b Fp(+)23 b Fq(Z)7 b Fp(\()p Fq(f)k Fp(\))24 b Fq(;)94 3426 y FF(where)1303 3625 y Fq(Z)7 b Fp(\()p Fq(f)k Fp(\))44 b(=)86 b FF(lim)1681 3689 y Fr(T)11 b Fi(!1)1915 3489 y Fv(Z)2014 3514 y Fr(T)1970 3716 y Fn(0)2060 3625 y Fq(dt)2195 3557 y(d)p 2177 3602 88 4 v 2177 3693 a(dt)2277 3625 y Fp(\002)2354 3582 y Fr(t)2389 3625 y Fp(\()p Fq(f)2477 3652 y Fi(\000)p Fr(t)2574 3625 y Fp(\))25 b Fq(;)94 3915 y FF(and)j(in)g(f)o(act)g(we)g(ha)n(v)o(e)g(pro)o(v)o(en)f (that)g(this)g(last)g(term)h(is)g(of)g(order)g Fq(\021)2430 3872 y Fn(2)2498 3915 y FF(\(in)f(reality)h Fq(\021)2972 3872 y Fn(3)3012 3915 y FF(\).)45 b(This)27 b(completes)g(the)94 4035 y(proof)34 b(of)f(the)g(\256rst)g(part)h(of)f(Theorem)g(7.2.)61 b(It)33 b(remains)g(to)f(pro)o(v)o(e)h(that)f(it)h(is)g(Lipshitz)f(and)h(to)g (estimate)94 4154 y(its)27 b(Lipshitz)e(constant)h(in)g(L)1093 4111 y Fi(1)1178 4154 y FF(.)42 b(This)26 b(will)g(be)h(done)f(in)h(the)f(ne) o(xt)g(subsection,)g(together)h(with)f(some)g(e)n(v)o(en)94 4274 y(more)f(detailed)f(information)g(on)g Fq(Z)32 b FF(which)25 b(we)g(need)g(later)-5 b(.)p eop %%Page: 19 19 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(19)94 194 y Fs(7.4.)39 b(Pro)s(of)g(of)g(the)e(second)g(part)g(of)i(Theorem)e(7.2)94 383 y FF(In)29 b(this)f(subsection,)g(we)h(pro)o(v)o(e)e(the)i(second)f(part) h(of)g(Theorem)f(7.2,)h(in)f(f)o(act)h(e)n(v)o(en)f(more.)47 b(W)-8 b(e)29 b(\256rst)g(need)94 503 y(some)c(notation:)94 676 y Fs(Remark.)31 b FF(It)18 b(will)g(be)g(more)h(con)l(v)o(enient)e(to)h (w)o(ork)g(with)g(the)g(interv)n(als)f([)p Fo(\000)p Fq(L;)g(L)p FF(])i(instead)e(of)i([)p Fo(\000)p Fq(L=)p FF(2)p Fq(;)e(L=)p FF(2])94 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Fp(\002)2163 5197 y Fr(t)2198 5240 y FF([)p Fq(f)2280 5266 y Fi(\000)p Fr(t)2377 5240 y FF(])2427 5159 y Fv(\000)2472 5240 y Fq(f)2532 5197 y Fn(3)2521 5266 y Fi(\000)p Fr(t)2619 5159 y Fv(\001)2706 5240 y Fq(:)p eop %%Page: 20 20 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(20)94 194 y FF(From)25 b(this)f(formula)g(we)i(ha)n(v)o(e)744 459 y(D)p Fq(Z)884 486 y Fr(T)947 459 y FF([)p Fq(f)11 b FF(])p Fq(w)47 b Fp(=)e Fo(\000)1407 324 y Fv(Z)1507 348 y Fr(T)1462 550 y Fn(0)1553 459 y Fq(dt)17 b FF(D)1730 416 y Fn(2)1769 459 y Fp(\002)1846 416 y Fr(t)1882 459 y FF([)p Fq(f)1964 486 y Fi(\000)p Fr(t)2061 459 y FF(])p Fp(\()p Fq(f)2193 416 y Fn(3)2182 486 y Fi(\000)p Fr(t)2279 459 y Fq(;)g(w)s Fp(\))1186 749 y Fo(\000)39 b FF(3)1369 614 y Fv(Z)1468 638 y Fr(T)1424 840 y Fn(0)1514 749 y Fq(dt)17 b FF(D)p Fp(\002)1768 706 y Fr(t)1803 749 y FF([)p Fq(f)1885 776 y Fi(\000)p Fr(t)1982 749 y FF(])2032 669 y Fv(\000)2078 749 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1937 y Fd(\003)1056 1957 y Fn(])1096 1872 y Fo(k)p Fq(y)1195 1899 y Fr(t)1230 1872 y Fo(k)1280 1899 y Fn(L)1323 1879 y Fd(1)1395 1899 y Ff(\()p Fa(S)1476 1879 y Fd(0)1503 1899 y Ff(\))1567 1872 y Fo(\024)j Fj(O)p Fp(\()p FF(1)p Fp(\))p Fo(k)p Fq(v)1968 1899 y Fn(0)2008 1872 y Fo(k)2058 1899 y Fn(L)2101 1879 y Fd(1)2173 1899 y Ff(\()p Fa(B)p Ff(\))2326 1872 y Fp(+)23 b Fj(O)p Fp(\()p Fq(")2581 1829 y Fn(2)2621 1872 y Fp(\))p Fo(k)p Fq(v)2758 1899 y Fn(0)2798 1872 y Fo(k)2848 1899 y Fn(L)2891 1879 y Fd(1)2963 1899 y Ff(\()p Fa(R)p Ff(\))3122 1872 y Fq(:)94 2102 y FF(W)-8 b(e)23 b(can)f(no)n(w)f(undo) h(the)g(ef)n(fect)g(of)g(the)g(e)o(xponential)f(of)h(Eq.\(7.16\).)34 b(If)22 b(we)h(furthermore)f(replace)g Fq(v)3568 2129 y Fn(0)3631 2102 y FF(by)f(the)94 2221 y(initial)i(data)h Fq(f)601 2178 y Fn(2)590 2248 y Fi(\000)p Fr(t)687 2221 y Fq(w)758 2248 y Fi(\000)p Fr(t)856 2221 y FF(,)g(and)g(use)g(the)f(information)g(we)h(ha)n(v) o(e)g(on)f Fq(f)2402 2178 y Fn(2)2391 2248 y Fi(\000)p 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2916 y(D)1683 2873 y Fn(2)1723 2916 y Fp(\002)1800 2873 y Fr(t)1835 2916 y FF([)p Fq(f)11 b FF(])p Fp(\()p Fq(w)2071 2943 y Fn(1)2111 2916 y Fq(;)17 b(w)2227 2943 y Fn(2)2266 2916 y Fp(\))25 b Fq(:)94 3080 y FF(Again,)f(this)g(is)h(a)g (function)f Fq(z)29 b FF(which)c(is)f(a)h(solution)e(of)731 3264 y Fq(@)784 3290 y Fr(t)819 3264 y Fq(z)32 b Fp(=)d Fq(@)1062 3221 y Fn(2)1056 3290 y Fr(x)1106 3264 y Fq(z)d Fp(+)d Fq(z)k Fo(\000)c FF(3)1501 3183 y Fv(\000)1546 3264 y Fp(\002)1623 3221 y Fr(t)1659 3264 y Fq(f)1719 3183 y Fv(\001)1764 3203 y Fn(2)1804 3264 y Fq(z)j Fo(\000)d FF(6)p Fp(\002)2103 3221 y Fr(t)2138 3264 y Fq(f)33 b Fo(\001)22 b FF(D)p Fp(\002)2419 3221 y Fr(t)2454 3264 y FF([)p Fq(f)11 b FF(])p Fq(w)2651 3290 y Fn(1)2714 3264 y Fo(\001)21 b FF(D)p Fp(\002)2912 3221 y Fr(t)2948 3264 y FF([)p Fq(f)11 b FF(])p Fq(w)3145 3290 y Fn(2)3210 3264 y Fq(;)381 b Fp(\()p FF(7)p Fq(:)p FF(19)p Fp(\))94 3428 y FF(with)39 b(initial)g(data)h Fq(z)53 b Fp(=)c FF(0,)43 b(which)c(is)g(the) 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y FF(which)i(are)g(global\),)f(we)i(ha)n(v)o(e)e(for)h(an)o(y)f(tw)o(o)h (functions)e Fq(f)11 b FF(,)25 b Fq(f)2239 4265 y Fi(0)2291 4308 y FF(in)g Fj(E)2455 4334 y Fr(b)2494 4308 y Fp(\()p Fq(\021)t Fp(\))g FF(the)f(inequality)1169 4472 y Fo(k)p Fq(Z)7 b Fp(\()p Fq(f)k Fp(\))21 b Fo(\000)i Fq(Z)7 b Fp(\()p Fq(f)1727 4429 y Fi(0)1754 4472 y Fp(\))p Fo(k)1843 4499 y Fi(1)1971 4472 y Fo(\024)45 b Fj(O)p Fp(\()p Fq(\021)t Fp(\))p Fo(k)p Fq(f)33 b Fo(\000)22 b Fq(f)2585 4429 y Fi(0)2613 4472 y Fo(k)2663 4499 y Fi(1)2772 4472 y Fq(:)94 4636 y FF(Therefore,)1053 4755 y Fo(k)p Fq(S)6 b Fp(\()p Fq(f)11 b Fp(\))21 b Fo(\000)h Fq(S)6 b Fp(\()p Fq(f)1594 4712 y Fi(0)1621 4755 y Fp(\))p Fo(k)1710 4782 y Fi(1)1838 4755 y Fo(\025)45 b Fp(\()p FF(1)22 b Fo(\000)h Fj(O)p Fp(\()p Fq(\021)t Fp(\)\))p Fo(k)p Fq(f)32 b Fo(\000)22 b Fq(f)2701 4712 y Fi(0)2729 4755 y Fo(k)2779 4782 y Fi(1)2888 4755 y Fq(:)94 4911 y FF(Basically)-6 b(,)23 b(we)g(w)o(ant)f(to)h(use)f (Eq.\(7.12\))g(to)h(sho)n(w)e(that)i(if)f Fq(f)34 b FF(and)22 b Fq(f)2412 4868 y Fi(0)2463 4911 y FF(dif)n(fer)g(by)h(at)f(least)h Fq(")g FF(some)n(where)f(on)h Fs(B)94 5031 y FF(this)k(implies)g(that)g Fq(S)6 b Fp(\()p Fq(f)11 b Fp(\))27 b FF(and)g Fq(S)6 b Fp(\()p Fq(f)1334 4988 y Fi(0)1361 5031 y Fp(\))28 b FF(dif)n(fer)g(by)f(at)h(least)f Fq("=)p FF(2)h(some)n(where)f(on)h Fs(S)p FF(.)44 b(While)28 b(this)f(is)g(not)g(true)94 5150 y(in)d(general,)g(we)f(will)g(see)h(in)f (the)g(ne)o(xt)g(section)g(that)g(it)f(must)h(be)g(true)h(for)g(enough)f (functions)f(among)h(those)94 5270 y(which)f(form)f(the)h(centers)g(of)f(the) h(balls)f(which)g(co)o(v)o(er)g Fo(G)6 b FF(.)35 b(This)21 b(will)g(be)h(e)o(xploited)e(in)h(the)h(ne)o(xt)e(subsection.)p eop %%Page: 23 23 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(23)94 194 y Fs(7.5.)39 b(Pro)s(of)g(of)g(Theorem)e(7.1)94 383 y FF(The)32 b(idea)f(of)g(the)g(proof) f(is)h(to)g(sho)n(w)f(that)g(because)i Fq(S)6 b Fp(\()p Fq(f)11 b Fp(\))35 b Fo(\031)h Fq(f)42 b FF(and)31 b(because)g(the)g Fq(")p FF(-entrop)o(y)g(of)g(the)g(set)94 503 y Fj(E)155 530 y Fr(b)195 503 y Fp(\()p Fq(\021)283 530 y Fi(\003)329 503 y Fp(\))c FF(of)g Fq(f)38 b FF(is)27 b Fj(O)756 422 y Fv(\000)802 503 y FF(log)o Fp(\()p FF(1)p Fq(=")p Fp(\))1153 422 y Fv(\001)1198 503 y FF(,)h(the)f(same)g(will)f(hold)h(for)g(the)g(set)g Fq(S)2516 422 y Fv(\000)2561 503 y Fj(E)2622 530 y Fr(b)2662 503 y Fp(\()p Fq(\021)2750 530 y Fi(\003)2796 503 y Fp(\))2835 422 y Fv(\001)2881 503 y FF(.)42 b(Here,)29 b(and)e(in)f(the)h(sequel)94 623 y(we)e(\256x)g Fq(\021)414 649 y Fi(\003)485 623 y FF(to)f(the)h(v)n(alue)f(found)g(as)h(a)g (bound)e(in)i(Theorem)f(7.2.)35 b(Basically)-6 b(,)24 b(we)h(are)h(going)d (to)h(sho)n(w)g(that)g(if)94 742 y Fo(k)p Fq(f)33 b Fo(\000)23 b Fq(f)386 699 y Fi(0)413 742 y Fo(k)463 769 y Fi(1)575 742 y Fo(\025)28 b Fq(")p FF(,)d(then)g(not)f(only)1431 946 y Fo(k)p Fq(S)6 b Fp(\()p Fq(f)11 b Fp(\))21 b Fo(\000)i Fq(S)6 b Fp(\()p Fq(f)1973 904 y Fi(0)2000 946 y Fp(\))p Fo(k)2089 973 y Fi(1)2217 946 y Fo(\025)45 b Fq("=)p FF(2)25 b Fq(;)1081 b Fp(\()p FF(7)p Fq(:)p FF(20)p Fp(\))94 1151 y FF(b)n(ut)25 b(also)f(that)g(we)i(can)f (\256nd)g(enough)f(functions)g(for)h(which)f(sup)2350 1178 y Fr(x)p Fi(2)p Fa(B)2535 1151 y Fo(j)p Fq(f)11 b Fp(\()p Fq(x)p Fp(\))20 b Fo(\000)j Fq(f)2938 1108 y Fi(0)2965 1151 y Fp(\()p Fq(x)p Fp(\))p Fo(j)k Fq(>)h(")e FF(and)1284 1355 y(sup)1278 1441 y Fr(x)p Fi(2)p Fa(S)1444 1355 y Fo(j)p Fq(S)6 b Fp(\()p Fq(f)11 b Fp(\)\()p Fq(x)p Fp(\))20 b Fo(\000)j Fq(S)6 b Fp(\()p Fq(f)2098 1312 y Fi(0)2125 1355 y Fp(\)\()p Fq(x)p Fp(\))p Fo(j)43 b(\025)h Fq("=)p FF(4)25 b Fq(:)929 b Fp(\()p FF(7)p Fq(:)p FF(21)p Fp(\))94 1626 y FF(Here,)26 b(we)f(shall)f(choose)h Fq(`)i Fo(\025)h FF(1)p Fq(=")p FF(.)316 1745 y(Note)k(that)f(we)h(cannot)g (pro)o(v)o(e)f(\(7.21\))h(for)g Ft(individual)e FF(pairs)i(of)g(functions,)g (b)n(ut)g(only)f(for)h(a)g(\(lar)n(ge)94 1865 y(enough\))j(subset)g(of)g (them.)66 b(The)35 b(mechanism)f(responsible)g(for)i(that)e(is)h(a)h(\252cro) n(wding)e(lemma\272)h(in)f(the)94 1984 y(follo)n(wing)23 b(setting:)34 b(Let)24 b Fj(S)h FF(be)f(a)h(set)f(of)h Fq(N)38 b Fo(\035)28 b FF(1)c(functions)f(which)h(are)h(pairwise)f(at)h(a)f(distance)g(at)h(least) f Fq(\013)94 2104 y FF(from)i(each)g(other)l(,)g(when)f(considered)h(on)f(a)h (set)g Fq(I)1849 2131 y Fn(big)1968 2104 y FF(which)f(is)h(a)g(\256nite)f (union)g(of)h(interv)n(als.)37 b(Let)25 b Fq(I)3598 2131 y Fn(small)3780 2104 y FF(be)94 2223 y(another)g(\256nite)g(union)f(of)h (interv)n(als)e(contained)h(in)h Fq(I)1932 2250 y Fn(big)2026 2223 y FF(.)94 2415 y Fs(Lemma)39 b(7.6.)d Fh(Under)25 b(the)f(abo)o(v)o(e)g (assumptions)f(at)i(least)f(one)h(of)g(the)g(follo)n(wing)d(alternati)n(v)o (es)i(holds:)161 2534 y(\261)50 b(At)24 b(least)h Fq(N)684 2492 y Fn(1)p Fr(=)p Fn(2)799 2534 y Fq(=)p FF(2)g Fh(functions)e(in)i Fj(S)g Fh(dif)n(fer)f(pairwise)h(by)f Fq(\013)h Fh(on)g Fq(I)2489 2561 y Fn(big)2605 2534 y Fo(n)d Fq(I)2721 2561 y Fn(small)2877 2534 y Fh(.)161 2682 y(\261)50 b(At)24 b(least)h Fq(N)684 2639 y Fn(1)p Fr(=)p Fn(2)824 2682 y Fh(functions)f(in)g Fj(S)h Fh(dif)n(fer)g(pairwise)f(by)h Fq(\013=)p FF(3)f Fh(on)h Fq(I)2489 2709 y Fn(small)2645 2682 y Fh(.)94 2905 y Fs(Remark.)37 b FF(W)-8 b(e)24 b(can)g(symmetrize)f(the)g(statement.)35 b(W)-8 b(e)24 b(formulate)f(this)f(as)i(a)g(corollary)f(for)h(further)g(use:)94 3097 y Fs(Corollary)39 b(7.7.)d Fh(Under)24 b(the)h(abo)o(v)o(e)e (assumptions)g(at)h(least)g(one)h(of)g(the)f(follo)n(wing)f(alternati)n(v)o (es)f(holds:)161 3216 y(\261)50 b(At)24 b(least)h Fq(N)684 3174 y Fn(1)p Fr(=)p Fn(2)799 3216 y Fq(=)p FF(2)g Fh(functions)e(in)i Fj(S)g Fh(dif)n(fer)f(pairwise)h(by)f Fq(\013=)p FF(3)h Fh(on)f Fq(I)2588 3243 y Fn(big)2705 3216 y Fo(n)d Fq(I)2820 3243 y Fn(small)2976 3216 y Fh(.)161 3364 y(\261)50 b(At)24 b(least)h Fq(N)684 3321 y Fn(1)p Fr(=)p Fn(2)799 3364 y Fq(=)p FF(2)g Fh(functions)e(in)i Fj(S)g Fh(dif)n(fer)f(pairwise)h(by)f Fq(\013=)p FF(3)h Fh(on)f Fq(I)2588 3391 y Fn(small)2744 3364 y Fh(.)94 3587 y Fs(Pro)s(of.)40 b FF(W)-8 b(e)25 b(\256rst)g(need)g(the)g(follo)n (wing)d(auxiliary)94 3779 y Fs(Lemma)46 b(7.8.)58 b Fh(Let)32 b Fj(E)f Fh(be)i(a)f(set)g(of)g Fq(M)1570 3736 y Fn(2)1647 3779 y Fq(>)39 b FF(4)32 b Fh(points)e(in)i(a)g(metric)g(space.)58 b(Assume)31 b(that)h(for)g(a)h(gi)n(v)o(en)94 3898 y Fq(\032)28 b(>)g FF(0)c Fh(we)g(can)g(\256nd)g(in)f Fj(E)h Fh(no)f(more)h(than)f Fq(M)35 b Fh(points)22 b(which)i(are)g(pairwise)g(at)f(a)i(distance)e(at)h (least)f Fq(\032)p Fh(.)35 b(Then)94 4018 y(there)26 b(is)e(a)h(point)f Fq(x)766 4045 y Fi(\003)836 4018 y Fh(in)h Fj(E)f Fh(such)h(that)f(at)h (least)g Fq(M)5 b(=)p FF(2)24 b Fh(points)g(of)g Fj(E)h Fh(are)h(within)d(a)i (distance)g Fq(\032)f Fh(of)h Fq(x)3487 4045 y Fi(\003)3533 4018 y Fh(.)94 4242 y Fs(Pro)s(of.)35 b FF(Let)21 b Fj(E)653 4268 y Fn(0)714 4242 y FF(be)g(a)g(maximal)f(set)h(of)g(points)f(in)g Fj(E)h FF(with)f(pairwise)h(distance)g(at)g(least)g Fq(\032)p FF(.)34 b(By)21 b(assumption,)94 4361 y(the)31 b(cardinality)g(of)g Fj(E)880 4388 y Fn(0)951 4361 y FF(satis\256es)f Fo(j)p Fj(E)1386 4388 y Fn(0)1425 4361 y Fo(j)37 b(\024)f Fq(M)11 b FF(.)55 b(Adding)30 b(an)o(y)g(point)g Fq(x)2587 4388 y Fn(0)2664 4361 y Fo(2)36 b Fj(E)26 b Fo(n)f Fj(E)2989 4388 y Fn(0)3060 4361 y FF(to)31 b Fj(E)3230 4388 y Fn(0)3269 4361 y FF(,)i(we)e(can)h(\256nd)f(a) 94 4481 y(point)h Fq(x)389 4438 y Fi(0)389 4508 y Fn(0)467 4481 y Fo(2)38 b Fj(E)632 4508 y Fn(0)704 4481 y FF(such)32 b(that)g Fq(d)p Fp(\()p Fq(x)1249 4508 y Fn(0)1289 4481 y Fq(;)17 b(x)1391 4438 y Fi(0)1391 4508 y Fn(0)1430 4481 y Fp(\))38 b Fq(<)h(\032)p FF(,)33 b(where)g Fq(d)g FF(is)f(the)g(distance.)58 b(W)-8 b(e)33 b(continue)e(in)h(this)g(f)o(ashion)94 4600 y(with)f(e)n(v)o (ery)g(point)f Fq(x)845 4627 y Fr(j)919 4600 y FF(of)h Fj(E)25 b Fo(n)h Fj(E)1256 4627 y Fn(0)1295 4600 y FF(,)33 b(\256nding)e(a)h(partner) g Fq(x)2114 4557 y Fi(0)2114 4627 y Fr(j)2187 4600 y FF(in)f Fj(E)2357 4627 y Fn(0)2428 4600 y FF(with)f Fq(d)p Fp(\()p Fq(x)2784 4627 y Fr(j)2826 4600 y Fq(;)17 b(x)2928 4557 y Fi(0)2928 4627 y Fr(j)2969 4600 y Fp(\))37 b Fq(<)g(\032)p FF(.)55 b(There)32 b(are)g(thus)94 4740 y Fo(j)p Fj(E)24 b Fo(n)f Fj(E)341 4767 y Fn(0)381 4740 y Fo(j)31 b Fp(=)h Fq(M)657 4698 y Fn(2)720 4740 y Fo(\000)25 b Fq(M)38 b FF(choices)28 b(of)f Fq(x)1451 4698 y Fi(0)1451 4767 y Fr(j)1493 4740 y FF(.)45 b(But)28 b(since)f(there)h (are)h(at)f(most)e Fq(M)39 b FF(points)26 b(in)h Fj(E)3238 4767 y Fn(0)3278 4740 y FF(,)h(there)g(must)f(be)94 4881 y(at)k(least)g(one)f (point)g(in)g Fj(E)990 4908 y Fn(0)1060 4881 y FF(which)g(has)h(at)g(least)f Fp(\()p Fq(M)1961 4838 y Fn(2)2026 4881 y Fo(\000)c Fq(M)11 b Fp(\))p Fq(=)-6 b(M)40 b FF(partners.)54 b(Clearly)-6 b(,)32 b(this)d(point)h(can)h(be)94 5000 y(chosen)25 b(as)g Fq(x)561 5027 y Fi(\003)607 5000 y FF(.)36 b(Since)25 b Fp(\()p Fq(M)1061 4957 y Fn(2)1122 5000 y Fo(\000)e Fq(M)11 b Fp(\))p Fq(=)-6 b(M)37 b(>)29 b(M)5 b(=)p FF(2,)24 b(the)h(proof)f(of)h(Lemma)g(7.8)f(is)g (complete.)94 5172 y Fs(Pro)s(of)44 b(of)e(Lemma)g(7.6.)h FF(W)-8 b(e)29 b(assume)f(that)g(the)h(second)g(alternati)n(v)o(e)e(does)i(not)f (hold)g(and)h(sho)n(w)f(that)94 5291 y(then)f(the)f(\256rst)g(must)g(hold.)40 b(If)27 b(the)f(second)g(alternati)n(v)o(e)f(does)h(not)g(hold,)g(then)g(we)h (can)g(apply)f(Lemma)f(7.8)p eop %%Page: 24 24 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(24)94 194 y FF(with)26 b Fp(\()p Fq(M)33 b Fp(+)23 b FF(2)p Fp(\))656 151 y Fn(2)724 194 y Fq(>)30 b(N)40 b Fo(\025)29 b Fp(\()p Fq(M)k Fp(+)24 b FF(1)p Fp(\))1416 151 y Fn(2)1481 194 y FF(and)i Fq(\032)i Fp(=)i Fq(\013=)p FF(3)c(on)f(the)h(set)f Fj(S)h FF(of)g(functions)f(with)g (the)h(sup)f(norm)g(on)94 314 y Fq(I)138 341 y Fn(small)316 314 y FF(and)c(conclude)g(that)g(there)h(is)f(a)h(function,)f Fq(f)1847 271 y Fi(\003)1892 314 y 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FF(:)34 b(Let)21 b Fq(N)2428 1619 y Fn([)p Fi(\000)p Fr(L;L)p Fn(])2675 1592 y Fp(\()p Fq(")p Fp(\))g FF(denote)g(again)f(the)h(minimum)94 1712 y(number)35 b(of)g(balls)g(of)g(radius)f Fq(")i FF(in)f(L)1428 1669 y Fi(1)1512 1712 y Fp(\()p FF([)p Fo(\000)p Fq(L;)17 b(L)p FF(])p Fp(\))34 b FF(needed)i(to)e(co)o(v)o(er)h Fj(E)2693 1738 y Fr(b)2733 1712 y Fp(\()p Fq(\021)2821 1738 y Fi(\003)2867 1712 y Fp(\))f FF(\(restricted)i(to)e([)p Fo(\000)p Fq(L;)17 b(L)p FF(]\).)94 1831 y(Then)25 b(we)g(kno)n(w)f(that)1122 1999 y(lim)1112 2062 y Fr(")p Fi(!)p Fn(0)1443 1931 y FF(1)p 1293 1976 351 4 v 1293 2067 a(log)o Fp(\()p FF(1)p Fq(=")p Fp(\))1713 1999 y FF(lim)1672 2063 y Fr(L)p Fi(!1)1914 1919 y FF(log)16 b Fq(N)2138 1946 y Fn([)p Fi(\000)p Fr(L;L)p Fn(])2385 1919 y Fp(\()p Fq(")p Fp(\))p 1914 1976 595 4 v 2153 2067 a FF(2)p Fq(L)2565 1999 y Fp(=)2699 1931 y FF(2)p Fq(b)p 2699 1976 93 4 v 2715 2067 a(\031)2828 1999 y(:)94 2249 y FF(This)i(leads)g(to)g(upper)g(and)h(lo)n(wer) 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Fo(\035)g Fq(`)29 b Fh(on)g(can)h(\256nd)f(in)h Fj(S)2510 3818 y Fn(1)2579 3792 y Fh(a)g(set)f Fj(S)2847 3818 y Fn(2)2917 3792 y Fh(of)g(at)h(least)f Fq(N)3423 3818 y Fn(2)3497 3792 y Fp(=)3621 3752 y Fn(1)p 3621 3769 35 4 v 3621 3826 a(2)3668 3792 y Fq(N)3759 3740 y Fn(1)p Fr(=)p Fn(2)3748 3820 y(1)94 3911 y Fh(functions)24 b(which)h(dif)n(fer)f(pairwise)h(by)f Fq("=)p FF(3)h Fh(on)g FF(L)1846 3868 y Fi(1)1930 3911 y Fp(\()p Fs(S)p Fp(\))p Fh(.)94 4153 y Fs(Pro)s(of.)59 b FF(W)-8 b(e)45 b(apply)f(Corollary)g(7.7)h(with)f Fq(I)1765 4180 y Fn(big)1914 4153 y Fp(=)55 b FF([)p Fo(\000)p Fq(L;)17 b(L)p FF(])45 b(and)f Fq(I)2647 4180 y Fn(small)2858 4153 y Fp(=)56 b FF([)p Fo(\000)p Fq(L)34 b Fp(+)f Fq(`;)17 b(L)32 b Fo(\000)i Fq(`)p FF(])44 b(and)94 4301 y(with)c Fq(N)60 b Fp(=)50 b FF([)p Fp(\()p FF(1)p Fq(=")p Fp(\))836 4258 y Fr(\033)882 4268 y Fd(\003)923 4258 y Fr(L)p Ff(\()p Fn(1)p Fi(\000)p Fr(\016)s Ff(\))p Fr(=)p Fn(2)1257 4301 y FF(])40 b(and)g Fq(\013)49 b Fp(=)h Fq(")p FF(.)83 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Fq(\016)t Fp(\))g(+)g FF(1)94 5291 y(the)25 b(inequality)f Fq(N)745 5318 y Fn(3)812 5291 y Fq(>)k(N)997 5318 y Fn(4)1037 5291 y FF(.)36 b(This)24 b(is)g(a)i(contradiction)d(and)i(the)g(proof)f(of)h(Lemma)g(7.9)f(is)g (complete.)p eop %%Page: 25 25 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(25)316 194 y FF(Continuing)22 b(the)h(proof)g(of)g(Theorem)g(7.1,)g(we)h(tak)o(e)f(the)g(set)g Fj(S)2495 221 y Fn(2)2558 194 y FF(of)h Fq(N)2745 221 y Fn(2)2808 194 y FF(functions)e(among)g(the)h(initial)94 314 y(ones)36 b(which)g(dif)n(fer)f(pairwise)h(at)g(least)f(by)h Fq("=)p FF(3)g(on)f Fs(S)p FF(.)69 b(Note)36 b(that)f(this)g(is)h(dif)n(ferent)f (from)h(looking)e(at)94 433 y(functions)f(which)g(dif)n(fer)g(by)h Fq("=)p FF(3)f(only)g(on)g(that)h(interv)n(al)e(because)i(in)f Fj(S)2718 460 y Fn(2)2792 433 y FF(we)g(ha)n(v)o(e)h(some)f(information)94 553 y(outside,)28 b(namely)g(that)f(the)h(functions)f(dif)n(fer)h(by)g(at)g 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Fq(N)174 2118 y Fn(5)251 2091 y FF(of)36 b(the)g Fq(S)6 b Fp(\()p Fq(f)683 2118 y Fr(i)716 2091 y Fp(\))36 b FF(which)g(dif)n(fer)g(pairwise)g(by)g(at)g (least)g Fq("=)p FF(12)h(on)f Fs(S)p FF(,)j(there)d(is)g(an)g Fq(f)3188 2049 y Fi(\003\003)3311 2091 y FF(such)g(that)g(in)g(a)94 2211 y(neighborhood)31 b(of)h(radius)f Fq("=)p FF(36)h(around)g Fq(S)6 b Fp(\()p Fq(f)1773 2168 y Fi(\003\003)1859 2211 y Fp(\))31 b FF(we)i(can)f(\256nd)g(at)g(least)f Fq(N)2833 2238 y Fn(5)2905 2211 y FF(of)h(the)g(other)g Fq(S)6 b Fp(\()p Fq(f)11 b Fp(\))p FF(.)56 b(This)94 2330 y(implies)24 b(that)g(we)h(ha)n(v)o(e)g(a)g (sub-collection)e Fo(f)p Fq(f)1695 2357 y Fr(i)1728 2330 y Fo(g)i FF(of)g Fq(N)1991 2357 y Fn(5)2055 2330 y FF(functions)f(for)h(which) 1245 2580 y(sup)1239 2666 y Fr(x)p Fi(2)p Fa(S)1405 2580 y Fo(j)p Fq(S)6 b Fp(\()p Fq(f)1588 2607 y Fr(j)1629 2580 y Fp(\)\()p Fq(x)p Fp(\))21 b Fo(\000)i Fq(S)6 b Fp(\()p Fq(f)2079 2607 y Fr(j)2116 2587 y Fd(0)2147 2580 y Fp(\)\()p Fq(x)p Fp(\))p 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Fd(0)1250 3710 y Fp(\()p Fq(x)p Fp(\))p Fo(j)27 b Fq(<)h("=)p FF(36)22 b Fp(+)h Fo(O)s Fp(\()p Fq(")2030 3667 y Fn(2)2070 3710 y Fp(\))f(+)g Fj(O)p Fp(\()p Fq(\021)2388 3737 y Fi(\003)2435 3710 y Fp(\))28 b FF(sup)2491 3796 y Fr(x)p Fi(2)p Fa(B)2669 3710 y Fo(j)p Fq(f)2746 3737 y Fr(j)2788 3710 y Fp(\()p Fq(x)p Fp(\))21 b Fo(\000)i Fq(f)3093 3737 y Fr(j)3130 3717 y Fd(0)3162 3710 y Fp(\()p Fq(x)p Fp(\))p Fo(j)h Fq(:)94 4018 y FF(No)n(w)h(if)1060 4148 y(sup)1003 4238 y Fr(x)p Fi(2)p Fa(B)p Fi(n)p Fa(S)1273 4148 y Fo(j)p Fq(f)1350 4175 y Fr(j)1391 4148 y Fp(\()p Fq(x)p Fp(\))d Fo(\000)g Fq(f)1696 4175 y Fr(j)1733 4155 y Fd(0)1765 4148 y Fp(\()p Fq(x)p Fp(\))p Fo(j)43 b(\024)51 b FF(sup)2093 4234 y Fr(x)p Fi(2)p Fa(S)2259 4148 y Fo(j)p Fq(f)2336 4175 y Fr(j)2377 4148 y Fp(\()p Fq(x)p Fp(\))22 b Fo(\000)g Fq(f)2682 4175 y Fr(j)2719 4155 y Fd(0)2751 4148 y Fp(\()p Fq(x)p Fp(\))p Fo(j)i Fq(;)94 4423 y FF(the)h(pre)n(vious)f(inequality)f(implies)883 4672 y(sup)878 4758 y Fr(x)p Fi(2)p Fa(S)1043 4672 y Fo(j)p Fq(f)1120 4699 y Fr(j)1162 4672 y Fp(\()p Fq(x)p Fp(\))e Fo(\000)i Fq(f)1467 4699 y Fr(j)1504 4679 y Fd(0)1536 4672 y Fp(\()p Fq(x)p Fp(\))p Fo(j)43 b Fq(<)i Fp(\()p FF(1)22 b Fp(+)g Fj(O)p Fp(\()p Fq(\021)2232 4699 y Fi(\003)2279 4672 y Fp(\)\))2357 4629 y Fi(\000)p Fn(1)2458 4672 y Fp(\()p Fq("=)p FF(6)g Fp(+)h Fj(O)p Fp(\()p Fq(")2920 4629 y Fn(2)2961 4672 y Fp(\)\))h Fq(:)94 4980 y FF(Combining)g(the)g(last)h(tw)o(o)f(inequalities)f(we)i(ha)n (v)o(e)858 5230 y(sup)846 5316 y Fr(x)p Fi(2)p Fa(B)1025 5230 y Fo(j)p Fq(f)1102 5257 y Fr(j)1144 5230 y Fp(\()p Fq(x)p Fp(\))c Fo(\000)i Fq(f)1449 5257 y Fr(j)1486 5237 y Fd(0)1518 5230 y Fp(\()p Fq(x)p Fp(\))p Fo(j)43 b Fq(<)i Fp(\()p FF(1)22 b Fp(+)g Fj(O)p Fp(\()p Fq(\021)2214 5257 y Fi(\003)2261 5230 y Fp(\)\))2339 5187 y Fi(\000)p Fn(1)2440 5230 y Fp(\()p Fq("=)p FF(36)g Fp(+)h Fj(O)p Fp(\()p Fq(")2952 5187 y Fn(2)2992 5230 y Fp(\)\))i Fq(;)p eop %%Page: 26 26 bop 94 -45 a Fl(Extensive)37 b(Pr)n(oper)-7 b(ties)36 b(of)j(the)e(Ginzbur)n (g-Land)n(a)n(u)g(Equa)-7 b(tion)807 b Fk(26)94 194 y FF(and)37 b(we)g(ha)n(v)o(e)f(a)h(contradiction)f(since)g(the)h(distance)f(should)f(be) i(at)g(least)f Fq(")p FF(.)72 b(\(It)37 b(is)f(here)h(that)f(we)h(use)94 314 y(the)31 b(additional)f(information)f(we)i(ha)n(v)o(e)g(on)f(the)h(set)g Fj(S)2020 341 y Fn(2)2091 314 y FF(of)g Fq(N)2285 341 y Fn(2)2356 314 y FF(functions)e(constructed)i(in)f(Lemma)g(7.9.\))94 433 y(Therefore)c(we)f(conclude)g(that)1060 672 y(sup)1003 763 y Fr(x)p Fi(2)p Fa(B)p Fi(n)p Fa(S)1273 672 y Fo(j)p Fq(f)1350 699 y Fr(j)1391 672 y Fp(\()p Fq(x)p Fp(\))d Fo(\000)g Fq(f)1696 699 y Fr(j)1733 679 y Fd(0)1765 672 y Fp(\()p Fq(x)p Fp(\))p Fo(j)43 b Fq(>)51 b FF(sup)2093 758 y Fr(x)p Fi(2)p Fa(S)2259 672 y Fo(j)p Fq(f)2336 699 y Fr(j)2377 672 y Fp(\()p Fq(x)p Fp(\))22 b Fo(\000)g Fq(f)2682 699 y Fr(j)2719 679 y Fd(0)2751 672 y Fp(\()p Fq(x)p Fp(\))p Fo(j)i Fq(;)94 987 y FF(b)n(ut)29 b(since)f(the)h(sup)f(o)o(v)o(er)g(the)h(whole)f(interv)n(al)g(must)g(be)h Fq(")g FF(we)g(conclude)f(that)h(the)f(sup)h(on)f(the)h(l.h.s.)f(is)g(at)94 1122 y(least)22 b Fq(")p FF(.)36 b(Applying)21 b(again)g(Corollary)i(7.7)e (we)i(can)g(\256nd)f(among)f(the)i Fo(f)p Fq(f)2607 1149 y Fr(i)2639 1122 y Fo(g)f FF(at)h(least)3023 1083 y Fn(1)p 3023 1099 35 4 v 3023 1157 a(2)3069 1122 y Fq(N)3160 1070 y Fn(1)p Fr(=)p Fn(2)3149 1153 y(5)3298 1122 y FF(functions)e(such)94 1242 y(that)k(on)f(one)g(of)h(the)f(interv)n(als)f([)p Fo(\000)p Fq(L;)17 b Fo(\000)p Fq(L)22 b Fp(+)g Fq(`)p FF(])i(or)h([)p Fq(L)c Fo(\000)h Fq(`;)17 b(L)p FF(])24 b(of)g Fs(B)e Fo(n)g Fs(S)i FF(the)o(y)g(are)h(pairwise)f(at)h(a)g(distance)94 1361 y(at)i(least)g Fq("=)p FF(36.)42 b(As)26 b(before)h(this)f(leads)h(to)f(a)i (contradiction)d(if)i Fq(L)j Fo(\035)g Fq(`)d FF(because)g(there)g(should)e (be)i(at)g(most)94 1491 y Fq(")140 1449 y Fi(\000)p Fr(`\033)281 1459 y Fd(\003)324 1449 y Ff(\()p Fn(1)p Ff(+)p Fr(\016)s 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(strange)g(attractors.)45 b(Re)n(v)-5 b(.)26 b(Mod.)f(Phys.)g FE(57)p FD(,)h(617\261656)307 2727 y(\(1985\).)94 2816 y([G])101 b(Gallay)-5 b(,)31 b(Th.:)45 b(A)28 b(center)n(-stable)h(manifold)g(theorem)f (for)f(dif)n(ferential)i(equations)f(in)h(Banach)g(spaces.)55 b(Commun.)28 b(Math.)307 2906 y(Phys.)19 b FE(152)p FD(,)h(249\261268)f (\(1993\).)94 2996 y([GH])43 b(Ghidaglia,)19 b(J.M.)f(and)g(B.)g(H)1106 2995 y(\302)1102 2996 y(eron:)28 b(Dimension)18 b(of)f(the)i(attractors)g (associated)g(to)g(the)f(Ginzb)n(ur)o(g-Landau)f(partial)i(dif)n(ferential) 307 3085 y(equation.)29 b(Physica)21 b FE(28D)p FD(,)e(282\261304)g (\(1987\).)94 3175 y([HS])57 b(Hocking,)24 b(L.M.)f(and)g(K.)g(Ste)n(w)o (artson:)37 b(On)22 b(the)i(nonlinear)g(response)f(of)g(a)h(mar)o(ginally)f (unstable)h(plane)g(parallel)h(\257o)n(w)e(to)h(a)307 3265 y(tw)o(o-dimensional)c(disturbance.)29 b(Proc.)20 b(R.)g(Soc.)g(Lond.)f FE(A.)33 b(326)p FD(,)19 b(289\261313)g(\(1972\).)94 3354 y([H])101 b(H)372 3353 y(\310)365 3354 y(ormander)m(,)22 b(L.:)35 b FA(The)23 b(Analysis)g(of)g(Linear)g(P)-6 b(artial)23 b(Dif)o(fer)m(ential)g(Equations) p FD(,)g(Berlin,)h(Heidelber)o(g,)f(Ne)n(w)f(Y)-9 b(ork,)23 b(Springer)307 3444 y(\(1983\2611985\).)94 3551 y([KT])52 b(K)m(olmogoro)o(v) -5 b(,)26 b(A.N.)f(and)h(V)-10 b(.M.)25 b(T)m(ikhomiro)o(v:)39 b FB(")p FD(-entrop)o(y)26 b(and)g FB(")p FD(-capacity)h(of)f(sets)g(in)g (functional)g(spaces)3413 3508 y Fm(1)3449 3551 y FD(.)47 b(In)25 b FA(Selected)307 3641 y(W)-7 b(orks)20 b(of)g(A.N)g(K)m(olmo)o(gor)l(o)o(v) -6 b(,)20 b(V)-9 b(ol)20 b(III)p FD(,)e(\(Shirayaye)n(v)-5 b(,)20 b(A.N.,)f(ed.\).)28 b(Dordrecht,)19 b(Kluv)o(er)g(\(1993\).)94 3731 y([M])88 b(Mattila,)17 b(P)-9 b(.:)27 b FA(Geometry)14 b(of)h(Sets)g(and)f(Measur)m(es)i(in)e(Euclidean)h(Spaces)p FD(,)h(Cambridge,)g(Cambridge)f(Uni)n(v)o(ersity)f(Press)h(\(1995\).)94 3820 y([MS])44 b(Mielk)o(e,)39 b(A.)33 b(and)i(Schneider)f(G.:)57 b(Attractors)34 b(for)f(modulation)i(equations)f(on)g(unbounded)g (domains\320e)o(xistence)h(and)307 3910 y(comparison.)29 b(Nonlinearity)19 b FE(8)p FD(,)h(743\261768)f(\(1995\).)94 4000 y([R])106 b(Ruelle,)21 b(D.:)28 b FA(Statistical)21 b(Mec)o(hanics)p FD(,)g(Ne)n(w)e(Y)-9 b(ork,)19 b(Benjamin)i(\(1969\).)94 4089 y([S])115 b(Schw)o(artz,)21 b(L.:)28 b FA(Th)851 4088 y(\302)846 4089 y(eorie)21 b(des)f(Distrib)n (utions)p FD(,)g(P)o(aris,)g(Hermann)g(\(1950\).)94 4179 y([T])110 b(T)-6 b(emam,)38 b(R.:)56 b FA(In\256nite-dimensional)32 b(dynamical)i (systems)g(in)f(mec)o(hanics)h(and)f(physics)p FD(,)38 b(Ne)n(w-Y)-9 b(ork,)35 b(Springer)n(-V)-9 b(erlag)307 4269 y(\(1988\).)p 94 5182 1200 4 v 241 5248 a Fm(1)316 5291 y FD(The)20 b(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 FE(14)p FD(,)f(3\26186)h(\(1959\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF