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Fo(f)198 3414 y Fw(0)221 3447 y Fl(j)25 b(\025)g Fo(\025)h(>)f Fs(1)30 b(on)h(the)f(circle)g(\(see)h(App)s(endix)d(A)i(in)f([23)r (]\).)118 3690 y Fp(2.2)112 b(Con\014guration)38 b(space)118 3862 y Fs(W)-8 b(e)29 b(tak)m(e)f(\012)f(an)g(index)f(set)i(and)e (de\014ne)h(the)g(con\014guration)g(space)h(of)f(our)g(dynamical)e (system)j(as)f(the)118 3975 y(pro)s(duct)i(of)i(circles)e(:)1302 4096 y Fo(S)1358 4110 y Fn(\012)1438 4096 y Fs(=)1542 4010 y Fh(Y)1534 4207 y Fg(p)p Fw(2)p Fn(\012)1682 4096 y Fo(S)1743 4059 y Fn(1)1823 4096 y Fl(\032)c Fo(A)1987 4110 y Fn(\012)2068 4096 y Fs(=)2172 4010 y Fh(Y)2164 4207 y Fg(p)p Fw(2)p Fn(\012)2312 4096 y Fo(A)p Fs([)p Fo(\032)p Fs(])118 4354 y(\012)39 b(can)h(b)s(e)f(quite)g(general)g (and)g(could)f(ev)m(en)j(b)s(e)d(uncoun)m(table.)68 b(But)39 b(our)g(main)g(in)m(terest)g(will)e(b)s(e)118 4467 y(\012)i(=)g Fm(Z)398 4434 y Fg(d)434 4467 y Fs(.)66 b(F)-8 b(or)40 b(this)d(case,)43 b(some)c(spatial)f(b)s(eha)m(viour)f(can)i(b)s(e)g (studied)e(\(see)i([23)r(])g(or)f([3)q(])h(for)g(suc)m(h)118 4580 y(applications\).)118 4823 y Fp(2.3)112 b(Spaces)39 b(of)f(coupling)e(and)i(observ)-6 b(ables)118 4995 y Fs(Let)27 b Fl(F)36 b Fs(b)s(e)26 b(the)h(set)g(of)f(\014nite)g (subsets)g(of)g(\012,)h(con)m(taining)f(the)h(empt)m(y)g(set.)40 b(F)-8 b(or)27 b(all)f(\003)f Fl(2)g(F)9 b Fs(,)28 b(w)m(e)f(denote)118 5108 y Fo(S)174 5122 y Fn(\003)254 5108 y Fs(=)352 5040 y Fh(Q)438 5135 y Fg(p)p Fw(2)p Fn(\003)589 5108 y Fo(S)650 5075 y Fn(1)731 5108 y Fl(\032)g Fo(A)897 5122 y Fn(\003)978 5108 y Fs(=)1075 5040 y Fh(Q)1161 5135 y Fg(p)p Fw(2)p Fn(\003)1312 5108 y Fo(A)p Fs([)p Fo(\032)p Fs(].)45 b(W)-8 b(e)33 b(call)d Fo(E)1939 5122 y Fn(\003)2024 5108 y Fs(the)i(set)g(of)f(functions)g(whic)m(h)f(are)i(con)m(tin)m (uous)118 5221 y(in)d Fo(A)292 5235 y Fn(\003)376 5221 y Fs(and)h(holomorphic)e(in)h(its)h(in)m(terior.)1867 5517 y(2)p eop %%Page: 3 3 3 2 bop 118 162 a Fs(F)-8 b(or)36 b Fo(K)j Fl(\032)32 b Fs(\003,)k(w)m(e)g(denote)f Fo(j)1102 176 y Fn(\003)p Fg(;K)1287 162 y Fs(:)48 b Fo(E)1427 176 y Fg(K)1528 162 y Fl(!)33 b Fo(E)1719 176 y Fn(\003)1807 162 y Fs(and)h Fo(j)2025 176 y Fn(\003)2126 162 y Fs(:)48 b Fo(E)2266 176 y Fn(\003)2352 162 y Fl(!)32 b Fo(C)7 b Fs(\()p Fo(A)2650 176 y Fn(\012)2706 162 y Fs(\))35 b(the)g(natural)e(injections,)118 275 y(then)i(de\014ne)g Fo(E)5 b Fs(\()p Fo(A)773 289 y Fn(\012)829 275 y Fs(\))36 b(as)g(the)g(closure)f(of)h Fl([)1658 289 y Fn(\003)p Fw(2F)1815 275 y Fo(j)1852 289 y Fn(\003)1905 275 y Fs(\()p Fo(E)2007 289 y Fn(\003)2061 275 y Fs(\).)57 b Fo(E)5 b Fs(\()p Fo(A)2353 289 y Fn(\012)2409 275 y Fs(\))36 b(is)e(in)h(fact)h(the)g(space)g(of)g(w)m(eakly)118 388 y(holomorphic)29 b(con)m(tin)m(uous)h(functions)f(on)h Fo(A)1680 402 y Fn(\012)1765 388 y Fs(\(see)i(App)s(endix)27 b(B)k(of)g([23)q(]\).)259 551 y(W)-8 b(e)35 b(w)m(an)m(t)f(to)g(con)m (trol)g(the)g(spatial)f(expansion)f(of)i(the)f(functions)f(whic)m(h)g (will)f(pla)m(y)i(the)h(role)f(of)118 664 y(coupling)c(and)h(observ)-5 b(ables.)39 b(F)-8 b(or)32 b(this,)d(w)m(e)i(c)m(ho)s(ose)g(a)g (parameter)g(0)26 b Fo(<)f(\022)i Fl(\024)e Fs(1)31 b(and)e(de\014ne:) 395 940 y Fo(H)471 955 y Fg(\022)535 940 y Fs(=)631 785 y Fh(\()704 940 y Fo(\036)c Fl(2)g Fo(E)5 b Fs(\()p Fo(A)1044 954 y Fn(\012)1100 940 y Fs(\))41 b(:)g Fo(\036)25 b Fs(=)1428 854 y Fh(X)1417 1051 y Fn(\003)p Fw(2F)1585 940 y Fo(j)1622 954 y Fn(\003)1676 940 y Fo(\036)1730 954 y Fn(\003)1813 940 y Fs(with)k Fo(\036)2074 954 y Fn(\003)2153 940 y Fl(2)c Fo(E)2306 954 y Fn(\003)2389 940 y Fs(and)2592 854 y Fh(X)2581 1051 y Fn(\003)p Fw(2F)2749 940 y Fo(\022)2795 903 y Fw(\000j)p Fn(\003)p Fw(j)2942 940 y Fl(j)p Fo(\036)3021 954 y Fn(\003)3074 940 y Fl(j)h Fo(<)f Fl(1)3312 785 y Fh(\))118 1227 y Fs(with,)k(for)i Fo(\036)25 b Fl(2)g Fo(H)731 1242 y Fg(\022)800 1227 y Fs(:)382 1503 y Fl(j)p Fo(\036)p Fl(j)486 1530 y Fg(\022)551 1503 y Fs(=)g(inf)772 1348 y Fh(\()857 1417 y(X)846 1614 y Fn(\003)p Fw(2F)1014 1503 y Fo(\022)1060 1466 y Fw(\000j)p Fn(\003)p Fw(j)1206 1503 y Fl(j)p Fo(\036)1285 1517 y Fn(\003)1339 1503 y Fl(j)40 b Fs(:)h(\()p Fo(\036)1559 1517 y Fn(\003)1613 1503 y Fs(\))1648 1530 y Fn(\003)p Fw(2F)1836 1503 y Fs(suc)m(h)30 b(that)h Fo(\036)2292 1517 y Fn(\003)2370 1503 y Fl(2)25 b Fo(E)2523 1517 y Fn(\003)2606 1503 y Fs(and)30 b Fo(\036)c Fs(=)2969 1417 y Fh(X)2959 1614 y Fn(\003)p Fw(2F)3127 1503 y Fo(j)3164 1517 y Fn(\003)3217 1503 y Fo(\036)3271 1517 y Fn(\003)3324 1348 y Fh(\))118 1795 y Fs(Then)32 b(\()p Fo(H)469 1810 y Fg(\022)508 1795 y Fo(;)15 b Fl(j)21 b(\001)f(j)664 1822 y Fg(\022)703 1795 y Fs(\))34 b(is)d(a)j Fo(\022)s Fs(-p)s(enalized)c(inductiv)m(e)h(limit)g(of)i(the)g(spaces)g Fo(E)2652 1809 y Fn(\003)2705 1795 y Fs(.)48 b(This)31 b(de\014nes)h(a)h(Banac)m(h)118 1908 y(algebra.)49 b(If)33 b Fo(\022)f(<)e Fs(1,)k(functions)e(of)h Fo(H)1431 1923 y Fg(\022)1503 1908 y Fs(dep)s(end)f(w)m(eakly)h(of)g(big)g(sets)g (\003.)49 b(F)-8 b(or)34 b Fo(\022)e Fs(=)e(1,)k Fo(H)3225 1922 y Fn(1)3294 1908 y Fs(=)c Fo(E)5 b Fs(\()p Fo(A)3570 1922 y Fn(\012)3626 1908 y Fs(\))118 2021 y(and)30 b Fl(j)20 b(\001)h(j)411 2047 y Fg(\022)475 2021 y Fs(=)k Fl(j)c(\001)f(j)687 2047 y Fw(1)762 2021 y Fs(.)41 b(W)-8 b(e)31 b(denote)g Fo(H)1360 1988 y Fg(r)1353 2048 y(\022)1429 2021 y Fs(the)f(set)h(of)g(real-analytic)f(maps)f(of)i Fo(H)2771 2036 y Fg(\022)2810 2021 y Fs(.)118 2264 y Fp(2.4)112 b(Coupled)38 b(maps)118 2436 y Fs(W)-8 b(e)32 b(can)e(no)m(w)h(de\014ne)e(the)i(class)f(of)h(dynamical)d(systems)j(w) m(e)g(w)m(an)m(t)g(to)g(study:)118 2623 y Fk(De\014nition)k(2.2.)43 b Fj(F)-7 b(or)34 b Fo(\032)25 b(>)g Fs(0)p Fj(,)33 b Fo(\025)26 b(>)f Fs(1)p Fj(,)33 b Fs(0)26 b Fo(<)f(\022)j Fl(\024)d Fs(1)33 b Fj(and)h Fs(0)26 b Fl(\024)f Fo(\024)h(<)f Fl(1)p Fj(,)33 b(we)g(take)g Fs(\()p Fo(f)3026 2637 y Fg(p)3065 2623 y Fs(\))3100 2654 y Fg(p)p Fw(2)p Fn(\012)3271 2623 y Fj(exp)-5 b(anding)118 2736 y(maps)34 b(fr)-5 b(om)34 b Fl(E)8 b Fs(\()p Fo(\032;)15 b(\025)p Fs(\))p Fj(,)34 b(and)f Fs(\()p Fo(g)1152 2750 y Fg(p)1193 2736 y Fs(\))1228 2767 y Fg(p)p Fw(2)p Fn(\012)1398 2736 y Fj(c)-5 b(oupling)34 b(maps)g(fr)-5 b(om)33 b Fo(H)2288 2703 y Fg(r)2281 2764 y(\022)2359 2736 y Fj(such)g(that)h Fl(j)p Fo(g)2820 2750 y Fg(p)2860 2736 y Fl(j)2885 2767 y Fg(\022)2949 2736 y Fo(<)25 b(\024)p Fj(.)118 2866 y(We)32 b(de\014ne)h(the)g(asso)-5 b(ciate)g(d)35 b Fi(coupled)c (analytic)f(map)i Fj(as)h Fo(F)2136 2880 y Fn(\012)2217 2866 y Fs(=)25 b(\()p Fo(F)2406 2880 y Fg(p)2447 2866 y Fs(\))2482 2897 y Fg(p)p Fw(2)p Fn(\012)2660 2866 y Fs(:)41 b Fo(A)2794 2880 y Fn(\012)2874 2866 y Fl(!)25 b(C)3043 2833 y Fn(\012)3099 2866 y Fj(,)32 b(wher)-5 b(e)34 b(:)1265 3071 y Fo(F)1323 3085 y Fg(p)1363 3071 y Fs(\()p Fo(z)t Fs(\))26 b(=)f Fo(f)1646 3085 y Fg(p)1685 3071 y Fs(\()p Fo(z)1762 3085 y Fg(p)1802 3071 y Fs(\))c(+)f Fo(g)1992 3085 y Fg(p)2032 3071 y Fs(\()p Fo(z)t Fs(\))94 b Fl(8)p Fo(p)24 b Fl(2)h Fs(\012)118 3275 y Fj(We)32 b(denote)i Fo(C)7 b(M)j Fs([)p Fo(\032;)15 b(\025;)g(\022)s(;)g(\024)p Fs(])33 b Fj(the)g(sp)-5 b(ac)g(e)34 b(of)f(al)5 b(l)33 b(such)g(c)-5 b(ouple)g(d)34 b(analytic)g(maps.)118 3561 y Ft(3)135 b(Results)118 3764 y Fs(F)-8 b(or)31 b(all)f(observ)-5 b(able)29 b Fo(b)c Fl(2)g Fo(C)7 b Fs(\()p Fo(S)1163 3778 y Fn(\012)1218 3764 y Fs(\))31 b(and)e(all)h Fo(T)38 b Fl(\025)25 b Fs(1,)31 b(w)m(e)f(write:)1563 4055 y Fo(S)1619 4069 y Fg(T)1674 4055 y Fo(b)25 b Fs(=)1834 3941 y Fg(T)10 b Fw(\000)p Fn(1)1839 3968 y Fh(X)1847 4163 y Fg(t)p Fn(=0)1991 4055 y Fo(b)20 b Fl(\016)g Fo(F)2186 4017 y Fg(t)118 4340 y Fs(In)37 b([23)q(],)j(an)e(ergo)s(dic)g(theorem) g(for)f(the)h(random)f(v)-5 b(ariables)37 b Fo(S)2356 4354 y Fg(T)2411 4340 y Fo(b)g Fs(under)g(Leb)s(esgue)g(measure)h(and) 118 4453 y(deca)m(y)44 b(of)f(correlations)g(for)f(the)i(limit)c (measure)j(are)g(pro)m(v)m(ed)h(under)d(the)i(assumption)f(that)h(the) 118 4565 y(coupling)29 b(is)g(w)m(eak)i(enough:)118 4753 y Fk(Theorem)g(3.1)i(\(Th.)45 b(2.1)32 b(of)g([23)q(]\).)39 b Fj(F)-7 b(or)32 b(al)5 b(l)30 b Fo(\032)c(>)f Fs(0)p Fj(,)31 b Fo(\025)25 b(>)g Fs(1)p Fj(,)31 b(ther)-5 b(e)31 b(exists)g Fo(\022)2884 4767 y Fn(0)2923 4753 y Fs(\()p Fo(\032;)15 b(\025)p Fs(\))26 b Fl(2)f Fs(\(0)p Fo(;)3376 4717 y Fn(1)p 3376 4732 36 4 v 3376 4784 a(3)3422 4753 y Fs(\))30 b Fj(such)118 4866 y(that)k(for)f Fo(\022)28 b(<)d(\022)659 4880 y Fn(0)730 4866 y Fj(ther)-5 b(e)34 b(is)e Fo(\024)26 b(>)f Fs(0)33 b Fj(for)g(which)h(the)f(fol)5 b(lowing)33 b(holds)i(for)e(every)f Fo(F)39 b Fl(2)24 b Fo(C)7 b(M)j Fs([)p Fo(\032;)15 b(\025;)g(\022)s(;)g(\024)p Fs(])p Fj(:)226 5054 y(1.)45 b(Ther)-5 b(e)34 b(exists)f(a)g(natur)-5 b(al)34 b(pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(e)34 b Fo(\027)k Fj(invariant)c(under)f Fo(F)13 b Fj(,)33 b(i.e.)41 b Fo(F)3143 5021 y Fw(\003)3182 5054 y Fo(\027)31 b Fs(=)25 b Fo(\027)6 b Fj(,)1867 5517 y Fs(3)p eop %%Page: 4 4 4 3 bop 226 162 a Fj(2.)45 b(F)-7 b(or)33 b(al)5 b(l)32 b Fo(b)26 b Fl(2)e Fo(C)7 b Fs(\()p Fo(S)956 176 y Fn(\012)1011 162 y Fs(\))p Fj(,)32 b Fo(m)1186 129 y Fn(\012)1241 162 y Fj(-almost)h(every)f Fo(x)f Fj(\(with)i Fo(m)e Fj(the)h(L)-5 b(eb)g(esgue)32 b(me)-5 b(asur)g(e)33 b(on)f(the)g(cir)-5 b(cle\),)1598 405 y Fs(lim)1565 466 y Fg(T)10 b Fw(!1)1793 344 y Fs(1)p 1782 384 66 4 v 1782 468 a Fo(T)1858 405 y(S)1914 419 y Fg(T)1969 405 y Fo(b)25 b Fs(=)2129 282 y Fh(Z)2180 488 y Fg(S)2223 499 y Ff(\012)2289 405 y Fo(b)15 b(d\027)1110 b Fs(\(1\))226 696 y Fj(3.)45 b(Ther)-5 b(e)34 b(exists)f Fo(\015)d(>)25 b Fs(1)33 b Fj(and)h Fo(\022)27 b(<)e(#)g(<)g Fs(1)33 b Fj(such)g(that)h(for)f(al)5 b(l)33 b Fo(b)25 b Fl(2)g Fo(H)2597 711 y Fg(\022)2636 696 y Fj(,)32 b Fo(a)25 b Fl(2)g Fo(H)2931 711 y Fg(#)3009 696 y Fj(and)34 b Fo(T)k Fl(\025)25 b Fs(1)p Fj(,)973 810 y Fh(\014)973 864 y(\014)973 919 y(\014)973 974 y(\014)1003 818 y(Z)1054 1024 y Fg(S)1097 1035 y Ff(\012)1164 942 y Fo(b)20 b Fl(\016)g Fo(F)1359 904 y Fg(T)1435 942 y Fl(\001)g Fo(a)15 b(d\027)27 b Fl(\000)1753 818 y Fh(Z)1803 1024 y Fg(S)1846 1035 y Ff(\012)1913 942 y Fo(b)15 b(d\027)2080 818 y Fh(Z)2131 1024 y Fg(S)2174 1035 y Ff(\012)2240 942 y Fo(a)g(d\027)2401 810 y Fh(\014)2401 864 y(\014)2401 919 y(\014)2401 974 y(\014)2457 942 y Fl(\024)25 b Fs(2)p Fl(j)p Fo(b)p Fl(j)2688 969 y Fg(\022)2727 942 y Fl(j)p Fo(a)p Fl(j)2826 969 y Fg(#)2872 942 y Fo(\015)2924 904 y Fw(\000)p Fg(T)3545 942 y Fs(\(2\))259 1232 y(These)33 b(prop)s(erties)d(are)j(consequences)h(of)e(a)h(more)g(tec)m(hnical)f (result,)h(the)f(fact)i(that)f(a)g(transfer)118 1345 y(op)s(erator)40 b(asso)s(ciated)g(to)g Fo(F)53 b Fs(exists)39 b(on)h(a)f(w)m(ell)g(c)m(hosen)h(Banac)m(h)h(space)f(and)f(has)g(a)h (sp)s(ectral)f(gap)118 1458 y(b)s(elo)m(w)30 b(1,)i(whic)m(h)e(is)g (the)i(simple)d(maximal)h(eigen)m(v)-5 b(alue.)42 b(They)31 b(are)g(really)f(an)h(in\014nite)e(dimensional)118 1571 y(v)m(ersion)h(of)g(classical)g(single)f(site)h(results.)259 1781 y(Our)38 b(metho)s(d)g(consists)g(in)f(generalizing)h(the)h (construction)f(of)h(this)e(op)s(erator)i(to)h(its)e(p)s(ertur-)118 1894 y(bations)h(b)m(y)h(p)s(oten)m(tials)f(and)g(then)g(extending)g (the)h(sp)s(ectral)e(gap)i(b)m(y)g(p)s(erturbation)e(theory)h(\(see)118 2007 y(Theorem)30 b(4.1)i(and)d(its)h(pro)s(of)g(Section)g(5)h(for)f (more)g(details\).)259 2120 y(W)-8 b(e)30 b(impro)m(v)m(e)e(the)h (result)e(of)i([23)q(])g(with)e(the)h(follo)m(wing)f(large)i (deviations)e(upp)s(er)g(b)s(ound,)g(and)g(an)118 2233 y(asso)s(ciated)k(partial)e(lo)m(w)m(er)i(b)s(ound)d(\(see)j(Thorem)f (4.2)i(for)e(a)g(more)h(precise)f(statemen)m(t\):)118 2418 y Fk(Theorem)36 b(3.2.)43 b Fj(Under)35 b(the)g(same)g(c)-5 b(onditions)36 b(on)f(the)g(p)-5 b(ar)g(ameters)37 b(as)e(in)f(The)-5 b(or)g(em)36 b(3.1,)g(for)f(al)5 b(l)118 2531 y Fo(u)25 b Fl(2)g Fo(H)364 2498 y Fg(r)357 2559 y(\022)402 2531 y Fj(,)33 b(ther)-5 b(e)33 b(exists)g(a)g(lower)h(semi-c)-5 b(ontinuous,)33 b(c)-5 b(onvex)33 b(and)h(non-ne)-5 b(gative)33 b(function)g Fo(I)3359 2545 y Fg(u)3429 2531 y Fs(:)26 b Fm(R)34 b Fl(!)118 2644 y Fm(R)29 b Fl([)20 b(f)p Fs(+)p 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3350 y Fj(2.)45 b(F)-7 b(or)34 b(al)5 b(l)33 b Fo(x)25 b Fl(2)g Fs(\()q Fo(a)892 3364 y Fg(u)937 3350 y Fo(;)15 b(b)1016 3364 y Fg(u)1061 3350 y Fs(\))33 b Fj(and)h Fo(\016)29 b(>)c Fs(0)p Fj(:)689 3601 y Fs(lim)15 b(inf)719 3662 y Fg(T)10 b Fw(!1)976 3540 y Fs(1)p 966 3580 66 4 v 966 3664 a Fo(T)1057 3601 y Fs(log)17 b Fo(m)1270 3564 y Fn(\012)1340 3473 y Fh(\022)1407 3601 y Fo(z)44 b Fs(:)1569 3540 y Fo(S)1625 3554 y Fg(T)1680 3540 y Fo(u)p Fs(\()p Fo(z)t Fs(\))p 1569 3580 280 4 v 1676 3664 a Fo(T)1884 3601 y Fl(2)25 b Fo(B)5 b Fs(\()p Fo(x;)15 b(\016)s Fs(\))2249 3473 y Fh(\023)2342 3601 y Fl(\025)25 b(\000)p Fo(I)2549 3615 y Fg(u)2594 3601 y Fs(\()p Fo(x)p Fs(\))344 b(\(Lo)m(w)m(er)32 b(Bound\))118 3889 y Fj(This)e(implies)h (in)e(p)-5 b(articular)32 b(that)f(the)g(c)-5 b(onver)g(genc)g(e)30 b(in)g(\(1\))g(is)g(exp)-5 b(onential,)32 b(which)f(me)-5 b(ans)31 b(that)g(for)118 4002 y(al)5 b(l)33 b Fo(A)26 b Fl(2)f Fm(R)41 b Fj(such)32 b(that)918 3929 y Fh(R)961 4034 y Fg(S)1004 4045 y Ff(\012)1071 4002 y Fo(u)15 b(d\027)31 b Fl(62)1371 3979 y Fs(\026)1347 4002 y Fo(A)p Fj(:)1084 4274 y Fs(lim)15 b(sup)1127 4352 y Fg(T)10 b Fw(!1)1398 4212 y Fs(1)p 1388 4253 66 4 v 1388 4336 a Fo(T)1478 4274 y Fs(log)17 b Fo(m)1691 4236 y Fn(\012)1761 4146 y Fh(\032)1829 4274 y Fo(z)45 b Fs(:)1992 4212 y Fo(S)2048 4226 y Fg(T)2102 4212 y Fo(u)p Fs(\()p Fo(z)t Fs(\))p 1992 4253 280 4 v 2099 4336 a Fo(T)2307 4274 y Fl(2)24 b Fo(A)2460 4146 y Fh(\033)2554 4274 y Fo(<)h Fs(0)850 b(\(3\))259 4521 y(Moreo)m(v)m(er,)48 b(w)m(e)43 b(obtain)f(new)g (probabilistic)d(results)j(for)g(the)h(random)e(v)-5 b(ariables)42 b Fo(S)3285 4535 y Fg(T)3339 4521 y Fo(u)h Fs(under)118 4634 y(Leb)s(esgue)30 b(measure,)h(namely)e(a)i(cen)m (tral)g(limit)d(Theorem)i(and)g(a)h(mo)s(derate)g(deviations)e (principle:)118 4819 y Fk(Theorem)38 b(3.3.)44 b Fj(Supp)-5 b(ose)37 b(the)f(hyp)-5 b(otheses)38 b(of)e(The)-5 b(or)g(em)38 b(3.1)f(ar)-5 b(e)36 b(satis\014e)-5 b(d.)53 b(F)-7 b(or)37 b(every)e Fo(u)c Fl(2)g Fo(H)3596 4786 y Fg(r)3589 4847 y(\022)3634 4819 y Fj(,)118 4932 y(we)i(write)g Fo(m)564 4946 y Fg(u)634 4932 y Fs(=)730 4859 y Fh(R)773 4964 y Fg(S)816 4975 y Ff(\012)883 4932 y Fo(u)15 b(d\027)6 b Fj(.)42 b(Then)33 b(the)g(limit)1343 5220 y Fs(lim)1311 5280 y Fg(T)10 b Fw(!1)1518 5096 y Fh(Z)1568 5302 y Fg(S)1611 5313 y Ff(\012)1678 5092 y Fh(\022)1755 5158 y Fo(S)1811 5172 y Fg(T)1866 5158 y Fo(u)20 b Fl(\000)g Fo(T)13 b(m)2175 5172 y Fg(u)p 1755 5199 465 4 v 1916 5217 a Fl(p)p 1992 5217 66 4 v 77 x Fo(T)2229 5092 y Fh(\023)2296 5114 y Fn(2)2351 5220 y Fo(d\027)q(;)1867 5517 y Fs(4)p eop %%Page: 5 5 5 4 bop 118 162 a Fj(exists)33 b(and)h(is)e(non)h(ne)-5 b(gative.)42 b(We)33 b(denote)g(it)g Fo(\033)1812 129 y Fn(2)1809 185 y Fg(u)1886 162 y Fj(and)h(have)f(the)g(fol)5 b(lowing)33 b(c)-5 b(ondition:)605 367 y Fo(\033)660 329 y Fn(2)657 389 y Fg(u)727 367 y Fs(=)25 b(0)187 b Fj(i\013)e Fl(9)15 b Fo(v)28 b Fl(2)d Fo(L)1610 329 y Fn(2)1649 367 y Fs(\()p Fo(\027)6 b Fs(\))59 b Fj(such)32 b(that)59 b Fo(u)26 b Fs(=)f Fo(v)e Fl(\000)d Fo(v)k Fl(\016)c Fo(F)71 b Fj(in)33 b Fo(L)3014 329 y Fn(2)3053 367 y Fs(\()p Fo(\027)6 b Fs(\))371 b(\(4\))118 621 y Fj(F)-7 b(or)34 b Fo(u)e Fj(such)h(that)h Fo(\033)820 588 y Fn(2)817 643 y Fg(u)887 621 y Fo(>)25 b Fs(0)p Fj(,)33 b(we)g(have:)1186 751 y Fh(\022)1263 818 y Fo(S)1319 832 y Fg(T)1374 818 y Fo(u)20 b Fl(\000)g Fo(T)13 b(m)1683 832 y Fg(u)p 1263 858 465 4 v 1376 876 a Fl(p)p 1452 876 66 4 v 77 x Fo(T)f(\033)1569 967 y Fg(u)1737 751 y Fh(\023)1804 774 y Fw(\003)1859 805 y Fh(\000)1901 879 y Fo(m)1981 842 y Fn(\012)2035 805 y Fh(\001)2128 828 y Fe(L)l(aw)2117 879 y Fl(\000)-15 b(!)40 b(N)13 b Fs(\(0)p Fo(;)i Fs(1\))818 b(\(CL)-8 b(T\))118 1145 y Fj(and)34 b(for)f(al)5 b(l)580 1109 y Fn(1)p 580 1124 36 4 v 580 1177 a(2)650 1145 y Fo(<)25 b(\013)h(<)f Fs(1)p Fj(,)33 b Fo(A)25 b Fl(\032)g Fm(R)42 b Fj(Bor)-5 b(el)33 b(set:)221 1438 y Fl(\000)i Fs(inf)312 1544 y Fg(x)p Fw(2)410 1486 y Fr(\016)399 1544 y Fg(A)502 1377 y Fo(x)554 1344 y Fn(2)p 477 1417 143 4 v 477 1501 a Fs(2)p Fo(\033)577 1474 y Fn(2)574 1523 y Fg(u)655 1438 y Fl(\024)25 b Fs(lim)15 b(inf)781 1499 y Fg(T)10 b Fw(!1)1125 1377 y Fs(1)p 1028 1417 241 4 v 1028 1501 a Fo(T)1094 1474 y Fn(2)p Fg(\013)p Fw(\000)p Fn(1)1294 1438 y Fs(log)16 b Fo(m)1506 1401 y Fn(\012)1576 1310 y Fh(\022)1643 1438 y Fo(z)45 b Fs(:)1805 1377 y Fo(S)1861 1391 y Fg(T)1916 1377 y Fo(u)p Fs(\()p Fo(z)t Fs(\))21 b Fl(\000)f Fo(T)13 b(m)2342 1391 y Fg(u)p 1805 1417 582 4 v 2039 1501 a Fo(T)2105 1474 y Fg(\013)2422 1438 y Fl(2)25 b Fo(A)2576 1310 y Fh(\023)754 1728 y Fl(\024)g Fs(lim)15 b(sup)893 1806 y Fg(T)10 b Fw(!1)1251 1667 y Fs(1)p 1154 1707 241 4 v 1154 1790 a Fo(T)1220 1764 y Fn(2)p Fg(\013)p Fw(\000)p Fn(1)1419 1728 y Fs(log)17 b Fo(m)1632 1691 y Fn(\012)1702 1600 y Fh(\022)1769 1728 y Fo(z)45 b Fs(:)1931 1667 y Fo(S)1987 1681 y Fg(T)2042 1667 y Fo(u)p Fs(\()p Fo(z)t Fs(\))21 b Fl(\000)f Fo(T)13 b(m)2468 1681 y Fg(u)p 1931 1707 582 4 v 2164 1790 a Fo(T)2230 1764 y Fg(\013)2548 1728 y Fl(2)25 b Fo(A)2702 1600 y Fh(\023)2794 1728 y Fl(\024)g(\000)k Fs(inf)2976 1798 y Fg(x)p Fw(2)3081 1781 y Fn(\026)3063 1798 y Fg(A)3166 1667 y Fo(x)3218 1634 y Fn(2)p 3141 1707 143 4 v 3141 1790 a Fs(2)p Fo(\033)3241 1764 y Fn(2)3238 1813 y Fg(u)3376 1728 y Fs(\(MDP\))118 1978 y Fj(R)-5 b(emark.)41 b Fs(All)27 b(results)g(ab)s(o)m(v)m(e)j(are)f(giv)m(en)f(with)f(Leb)s(esgue)h (measure)g(as)h(initial)d(probabilit)m(y)-8 b(.)38 b(In)27 b(fact,)118 2091 y(they)e(remain)f(true)g(taking)h(measures)g(in)e(the) i(Banac)m(h)h(space)g(on)e(whic)m(h)g(our)g(op)s(erators)h(act)h (\(exactly)118 2204 y(on)j(the)g(subset)f(of)h(this)f(Banac)m(h)i (space)g(whic)m(h)d(con)m(tains)i(probabilities,)d(denoted)j Fl(M)3127 2160 y Fg(p)3127 2233 y(#)3174 2204 y Fs(,)g(see)h(Section) 118 2317 y(4.1\).)45 b(W)-8 b(e)32 b(will)d(pro)m(v)m(e)j(our)e (results)g(in)g(this)g(more)h(general)h(con)m(text.)44 b(The)31 b(same)h(generalization)f(for)118 2430 y(the)g(ergo)s(dic)f (theorem)g(\(1\))i(is)d(v)-5 b(alid)29 b(and)h(the)g(pro)s(of)g(of)g ([23)r(])g(adapted)h(in)e(a)h(simple)f(w)m(a)m(y)-8 b(.)118 2716 y Ft(4)135 b(Use)45 b(of)h(the)f(sp)t(ectral)g(gap)118 2919 y Fs(In)26 b(this)g(section,)h(w)m(e)h(will)c(pro)m(v)m(e)j (Theorems)f(3.2)i(and)e(3.3)i(giv)m(en)f(an)f(in)m(termediate)h(result) e(\(Theorem)118 3032 y(4.1\))30 b(on)f(the)g(sp)s(ectral)f(gap)h(for)g (p)s(erturb)s(ed)d(op)s(erators.)40 b(W)-8 b(e)30 b(use)f(in)e(these)i (pro)s(ofs)f(the)h(same)g(t)m(yp)s(e)g(of)118 3145 y(metho)s(ds)h(as)g (in)f(the)i(pap)s(ers)e(of)i(J.)f(Rousseau-Egele)h([21)q(])g(or)f(A.)h (Broise)f([7)q(].)118 3388 y Fp(4.1)112 b(Space)39 b(of)e(densities)118 3560 y Fs(F)-8 b(or)31 b Fo(K)h Fl(\032)25 b Fs(\003,)31 b(let)f Fo(\031)788 3574 y Fg(K)q(;)p Fn(\003)961 3560 y Fs(:)41 b Fo(E)1094 3574 y Fn(\003)1172 3560 y Fl(!)26 b Fo(E)1356 3574 y Fg(K)1454 3560 y Fs(b)s(e)k(the)h(pro)5 b(jection)30 b(de\014ned)f(b)m(y:)1053 3810 y Fo(\031)1105 3824 y Fg(K)q(;)p Fn(\003)1238 3810 y Fo(\036)1292 3824 y Fn(\003)1345 3810 y Fs(\()p Fo(z)1422 3824 y Fg(K)1491 3810 y Fs(\))d(=)1648 3687 y Fh(Z)1698 3893 y Fg(S)1741 3907 y Ff(\003)p Fr(n)p Fd(K)1893 3810 y Fo(\036)1947 3824 y Fn(\003)2000 3810 y Fs(\()p Fo(z)2077 3824 y Fn(\003)2130 3810 y Fs(\))15 b Fo(m)2260 3773 y Fn(\003)p Fw(n)p Fg(K)2413 3810 y Fs(\()p Fo(dz)2537 3829 y Fn(\003)p Fw(n)p Fg(K)2691 3810 y Fs(\))118 4082 y(If)30 b(\003)25 b(=)g(\012,)31 b(w)m(e)f(will)e(note)j Fo(\031)1074 4096 y Fg(K)1168 4082 y Fs(=)25 b Fo(\031)1316 4096 y Fg(K)q(;)p Fn(\012)1450 4082 y Fs(.)259 4245 y(F)-8 b(ollo)m(wing)40 b([23)q(],)k(w)m(e)c (de\014ne)g(no)m(w)g(the)h(Banac)m(h)h(space)e(on)h(whic)m(h)e(our)h (op)s(erators)g(w)m(ork.)71 b(W)-8 b(e)118 4358 y(need)29 b(to)i(tak)m(e)g(it)e(su\016cien)m(tly)f(large,)i(and)f(sp)s (eci\014cally)e(not)j(included)c(in)i Fo(L)2752 4325 y Fn(1)2792 4358 y Fs(\()p Fo(dm)2954 4325 y Fn(\012)3009 4358 y Fs(\).)41 b(Indeed,)29 b(in)f(the)118 4471 y(uncoupled)j(case)i (\(when)f(the)h(couplings)e Fo(g)1628 4485 y Fg(p)1700 4471 y Fs(are)i(zero\),)h(w)m(e)f(kno)m(w)g(that)g(the)g(natural)f (measure)g(will)118 4584 y(b)s(e)f(the)i(in\014nite)c(pro)s(duct)i(of)h (the)g(SRB)g(measures)g Fo(h)1975 4598 y Fg(p)2015 4584 y Fo(dm)g Fs(for)g(the)g(single)f(site)g(functions)g Fo(f)3335 4598 y Fg(p)3374 4584 y Fs(,)i(whic)m(h)118 4697 y(will)f(generally)j(not)g(b)s(e)f(absolutely)g(con)m(tin)m(uous)h (with)e(resp)s(ect)i(to)h(Leb)s(esgue)f(measure.)54 b(T)-8 b(o)36 b(get)g(a)118 4810 y(large)31 b(enough)f(space,)h(w)m(e)g(c)m (ho)s(ose)g(a)g(parameter)g(0)25 b Fo(<)g(\022)j Fl(\024)d Fs(1)61 b(and)30 b(de\014ne:)334 5059 y Fl(M)443 5074 y Fg(\022)508 5059 y Fs(=)604 4931 y Fh(\032)672 5059 y Fo(\036)25 b Fs(=)g(\()p Fo(\036)936 5073 y Fn(\003)990 5059 y Fs(\))1025 5073 y Fn(\003)p Fw(2F)1223 5059 y Fs(:)41 b Fo(\031)1341 5074 y Fn(\003)p Fg(;)p Fn(\003)1459 5055 y Fr(0)1484 5059 y Fo(\036)1538 5074 y Fn(\003)1587 5055 y Fr(0)1639 5059 y Fs(=)25 b Fo(\036)1789 5073 y Fn(\003)1868 5059 y Fl(8)p Fs(\003)g Fl(\032)g Fs(\003)2166 5021 y Fw(0)2219 5059 y Fs(and)30 b Fl(k)p Fo(\036)p Fl(k)2541 5086 y Fg(\022)2605 5059 y Fs(=)j(sup)2701 5137 y Fn(\003)p Fw(2F)2870 5059 y Fo(\022)2916 5021 y Fw(j)p Fn(\003)p Fw(j)3007 5059 y Fl(j)p Fo(\036)3086 5073 y Fn(\003)3140 5059 y Fl(j)25 b Fo(<)g Fl(1)3377 4931 y Fh(\033)1867 5517 y Fs(5)p eop %%Page: 6 6 6 5 bop 118 162 a Fs(\()p Fl(M)262 177 y Fg(\022)302 162 y Fo(;)15 b Fl(k)21 b(\001)f(k)499 189 y Fg(\022)538 162 y Fs(\))36 b(is)f(a)h(Banac)m(h)h(space)f(and)f(a)i Fo(H)1705 177 y Fg(\022)1743 162 y Fs(-mo)s(dule:)51 b Fo(g)38 b Fs(=)2327 94 y Fh(P)2423 189 y Fn(\003)2472 170 y Fr(0)2494 189 y Fw(2F)2617 162 y Fo(g)2660 178 y Fn(\003)2709 159 y Fr(0)2772 162 y Fs(elemen)m(t)e(of)g Fo(H)3298 177 y Fg(\022)3373 162 y Fs(acts)g(on)118 275 y Fo(\036)26 b Fs(=)e(\()p Fo(\036)382 289 y Fn(\003)436 275 y Fs(\))471 289 y Fn(\003)p Fw(2F)659 275 y Fs(to)31 b(get)h Fo(g)23 b Fl(\003)e Fo(\036)k Fl(2)g(M)1328 290 y Fg(#)1405 275 y Fs(de\014ned)k(b)m(y:)993 491 y(\()q Fo(g)23 b Fl(\003)e Fo(\036)p Fs(\))1251 518 y Fn(\003)1329 491 y Fs(=)1447 404 y Fh(X)1425 603 y Fn(\003)1474 584 y Fr(0)1496 603 y Fw(2F)1615 491 y Fo(\031)1667 506 y Fn(\003)p Fg(;)p Fn(\003)p Fw([)p Fn(\003)1881 487 y Fr(0)1922 417 y Fh(\000)1964 491 y Fo(j)2001 506 y Fn(\003)p Fw([)p Fn(\003)2146 487 y Fr(0)2169 506 y Fg(;)p Fn(\003)2238 487 y Fr(0)2264 491 y Fs(\()p Fo(g)2342 506 y Fn(\003)2391 487 y Fr(0)2418 491 y Fs(\))g Fl(\001)f Fo(\036)2573 506 y Fn(\003)p Fw([)p Fn(\003)2718 487 y Fr(0)2745 417 y Fh(\001)118 782 y Fs(and)30 b(the)g(follo)m(wing)f(b)s(ound)g(holds:) 39 b Fl(k)p Fo(g)24 b Fl(\003)d Fo(\036)p Fl(k)1666 808 y Fg(\022)1731 782 y Fl(\024)k(j)p Fo(g)s Fl(j)1924 808 y Fg(\022)1963 782 y Fl(k)p Fo(\036)p Fl(k)2108 808 y Fg(\022)2147 782 y Fs(.)118 894 y Fl(M)227 909 y Fg(\022)299 894 y Fs(con)m(tains)34 b(the)f(uncoupled)e(natural)i(measure)g Fl(\012)2005 908 y Fg(p)p Fw(2)p Fn(\012)2142 894 y Fs(\()p Fo(h)2229 908 y Fg(p)2269 894 y Fo(dm)p Fs(\),)i(for)e Fo(\022)2679 861 y Fw(\000)p Fn(1)2802 894 y Fo(>)d Fs(sup)3040 916 y Fg(p)p Fw(2)p Fn(\012)3193 894 y Fl(j)p Fo(h)3270 908 y Fg(p)3310 894 y Fl(j)3335 925 y Fw(1)3410 894 y Fs(.)49 b(This)118 1007 y(measure)33 b(is)f(represen)m(ted)h(b)m(y)g Fo(\036)d Fs(=)g(\()p Fo(\036)1458 1021 y Fn(\003)1537 1007 y Fs(=)1633 939 y Fh(Q)1719 1034 y Fg(p)p Fw(2)p Fn(\003)1869 1007 y Fo(h)p Fs(\()p Fo(z)1998 1021 y Fg(p)2039 1007 y Fs(\)\))2110 1051 y Fn(\003)p Fw(2F)2267 1007 y Fs(,)k(although)e(it)h(is)f(not)i(absolutely)e(con-)118 1120 y(tin)m(uous)k(with)f(resp)s(ect)h(to)i(Leb)s(esgue)e(measure.)60 b(More)37 b(generally)-8 b(,)38 b(if)d(w)m(e)j(consider)d(the)i(follo)m (wing)118 1233 y(subset)30 b(of)g Fl(M)609 1248 y Fg(\022)649 1233 y Fs(:)953 1388 y Fl(M)1062 1351 y Fg(m)1062 1411 y(\022)1154 1388 y Fs(=)1250 1260 y Fh(\032)1318 1388 y Fo(\036)c Fl(2)f(M)1593 1403 y Fg(\022)1672 1388 y Fs(:)49 b(sup)1738 1466 y Fn(\003)p Fw(2F)1906 1264 y Fh(Z)1957 1471 y Fg(S)2000 1482 y Ff(\003)2064 1388 y Fl(j)p Fo(\036)2143 1402 y Fn(\003)2197 1388 y Fs(\()p Fo(z)2274 1402 y Fn(\003)2328 1388 y Fs(\))p Fl(j)15 b Fo(dz)2492 1402 y Fn(\003)2571 1388 y Fo(<)25 b Fl(1)2758 1260 y Fh(\033)118 1606 y Fs(then)30 b(ev)m(ery)h Fo(\036)26 b Fl(2)f(M)840 1573 y Fg(m)840 1634 y(\022)937 1606 y Fs(can)30 b(b)s(e)g(seen)h(as)f(a)h(measure)f(on)g Fo(S)2148 1620 y Fn(\012)2234 1606 y Fs(de\014ned)f(b)m(y)190 1731 y Fh(Z)240 1937 y Fg(S)283 1948 y Ff(\012)350 1855 y Fo(g)19 b(d\036)26 b Fs(=)f Fo(\036)p Fs(\()p Fo(g)s Fs(\))h(=)47 b(lim)927 1915 y Fn(\003)p Fw(!)p Fn(\012)1113 1731 y Fh(Z)1163 1937 y Fg(S)1206 1948 y Ff(\003)1271 1855 y Fo(g)1314 1869 y Fn(\003)1368 1855 y Fo(\036)1422 1869 y Fn(\003)1475 1855 y Fo(dm)1602 1817 y Fn(\003)1746 1855 y Fl(8)p Fo(g)28 b Fl(2)d Fo(C)7 b Fs(\()p Fo(S)2117 1869 y Fn(\012)2172 1855 y Fs(\))31 b(and)e Fo(g)2457 1869 y Fn(\003)2536 1855 y Fl(2)c Fo(C)7 b Fs(\()p Fo(S)2785 1869 y Fn(\003)2838 1855 y Fs(\))30 b(suc)m(h)g(that)h Fo(g)3348 1869 y Fn(\003)3427 1855 y Fl(!)25 b Fo(g)118 2109 y Fs(All)44 b(these)h(measures)f(ha)m(v)m(e)i(\014nite)e (marginals)f(on)i Fo(S)2030 2123 y Fn(\003)2128 2109 y Fs(whic)m(h)e(are)i(absolutely)f(con)m(tin)m(uous)h(with)118 2222 y(resp)s(ect)e(to)g Fo(m)645 2189 y Fn(\003)698 2222 y Fs(,)j(with)41 b(densit)m(y)h Fo(\036)1368 2236 y Fn(\003)1467 2222 y Fl(2)j Fo(E)5 b Fs(\()p Fo(A)1748 2236 y Fn(\003)1802 2222 y Fs(\).)77 b(W)-8 b(e)44 b(will)c(denote)j Fl(M)2706 2178 y Fg(p)2706 2251 y(\022)2788 2222 y Fs(the)g(set)g(of)g (probabilit)m(y)118 2335 y(measures)30 b(in)f Fl(M)723 2302 y Fg(m)723 2363 y(\022)790 2335 y Fs(.)118 2578 y Fp(4.2)112 b(Sp)s(ectral)37 b(gap)h(for)g(p)s(erturb)s(ed)g(transfer) g(op)s(erators)118 2750 y Fs(W)-8 b(e)31 b(state)h(no)m(w)e(the)g (existence)g(and)f(the)i(prop)s(ert)m(y)e(of)h(sp)s(ectral)f(gap)i(for) e(p)s(erturb)s(ed)f(tranfer)h(op)s(era-)118 2863 y(tors:)118 3048 y Fk(Theorem)37 b(4.1.)44 b Fj(F)-7 b(or)37 b Fo(F)43 b Fl(2)30 b Fo(C)7 b(M)j Fs([)p Fo(\032;)15 b Fs(\003)p Fo(;)g(\022)s(;)g(\024)p Fs(])p Fj(,)37 b(whose)f(p)-5 b(ar)g(ameters)38 b(satisfy)f(c)-5 b(onditions)37 b(of)e(The)-5 b(or)g(em)118 3161 y(3.1)32 b(and)f(with)h Fo(#)p Fj(,)f Fo(\015)k Fj(and)d Fo(\027)37 b Fj(as)31 b(in)g(this)g(r)-5 b(esult,)32 b(ther)-5 b(e)31 b(exists)h(for)f(al)5 b(l)31 b Fo(T)38 b Fl(\025)25 b Fs(1)31 b Fj(an)h(analytic)g(functional:)1339 3363 y Fo(M)1437 3326 y Fn(\()p Fg(T)10 b Fn(\))1587 3363 y Fs(:)41 b Fo(H)1729 3378 y Fg(\022)1793 3363 y Fl(\000)-16 b(!)25 b Fo(L)p Fs(\()p Fl(M)2170 3378 y Fg(#)2217 3363 y Fo(;)15 b Fl(M)2366 3378 y Fg(\022)2405 3363 y Fs(\))1105 b(\(5\))1715 3518 y Fo(u)26 b Fl(7\000)-16 b(!)25 b Fo(M)2062 3480 y Fn(\()q Fg(T)10 b Fn(\))2052 3540 y Fg(u)118 3720 y Fj(satisfying:)456 3922 y Fl(\017)26 b Fj(Ther)-5 b(e)33 b(exists)g Fo(T)1090 3936 y Fn(0)1155 3922 y Fl(\025)25 b Fs(1)33 b Fj(such)g(that)h Fo(M)1820 3885 y Fn(\()p Fg(T)10 b Fn(\))1810 3945 y Fg(u)1955 3922 y Fl(2)25 b Fo(L)p Fs(\()p Fl(M)2247 3937 y Fg(#)2293 3922 y Fs(\))126 b Fj(if)32 b Fo(T)38 b Fl(\025)25 b Fo(T)2782 3936 y Fn(0)456 4087 y Fl(\017)h(k)p Fo(M)670 4049 y Fn(\()q Fg(T)10 b Fn(\))660 4109 y Fg(u)781 4087 y Fl(k)25 b(\024)g Fo(e)989 4049 y Fg(T)10 b Fw(j)p Fg(u)p Fw(j)1120 4070 y Fd(\022)1552 4087 y Fl(\017)46 b(k)p Fo(M)1786 4049 y Fn(\()q Fg(T)10 b Fn(\))1776 4109 y Fg(u)1916 4087 y Fl(\000)20 b Fo(M)2105 4039 y Fn(\()q Fg(T)10 b Fn(\))2095 4113 y(0)2215 4087 y Fl(k)26 b(\024)f Fo(e)2424 4049 y Fg(T)10 b Fw(j)p Fg(u)p Fw(j)2555 4070 y Fd(\022)2615 4087 y Fl(\000)20 b Fs(1)794 b(\(6\))456 4241 y Fl(\017)26 b Fo(M)625 4204 y Fn(\()q Fg(t)p Fn(\))615 4264 y Fg(u)730 4241 y Fl(\016)21 b Fo(M)894 4204 y Fn(\()p Fg(T)10 b Fn(\))884 4264 y Fg(u)1029 4241 y Fs(=)25 b Fo(M)1223 4204 y Fn(\()p Fg(t)p Fn(+)p Fg(T)10 b Fn(\))1213 4264 y Fg(u)1539 4241 y Fj(for)33 b Fo(t)25 b Fl(\025)g Fs(1)p Fo(;)15 b(T)39 b Fl(\025)25 b Fo(T)2164 4255 y Fn(0)3545 4241 y Fs(\(7\))456 4396 y Fl(\017)h Fo(M)625 4358 y Fn(\()q Fg(T)10 b Fn(\))615 4418 y Fg(u)750 4396 y Fs(\()q Fl(M)895 4358 y Fg(m)895 4419 y(#)961 4396 y Fs(\))26 b Fl(\032)f(M)1227 4358 y Fg(m)1227 4419 y(\022)3545 4396 y Fs(\(8\))456 4587 y Fl(\017)542 4463 y Fh(Z)593 4669 y Fg(S)636 4680 y Ff(\012)702 4587 y Fo(b)c Fl(\016)f Fo(F)898 4549 y Fg(T)969 4587 y Fs(exp)15 b(\()p Fo(S)1214 4601 y Fg(T)1269 4587 y Fo(u)p Fs(\))30 b Fo(d\036)c Fs(=)1609 4463 y Fh(Z)1660 4669 y Fg(S)1703 4680 y Ff(\012)1769 4587 y Fo(b)15 b(d)1885 4486 y Fh(\020)1940 4587 y Fo(M)2038 4549 y Fn(\()q Fg(T)10 b Fn(\))2028 4609 y Fg(u)2148 4587 y Fo(\036)2202 4486 y Fh(\021)2458 4587 y Fl(8)15 b Fo(b)25 b Fl(2)g Fo(C)7 b Fs(\()p Fo(S)2837 4601 y Fn(\012)2891 4587 y Fs(\))p Fo(;)31 b(\036)26 b Fl(2)e(M)3256 4549 y Fg(m)3256 4610 y(#)3545 4587 y Fs(\(9\))118 4876 y Fj(Mor)-5 b(e)g(over,)40 b(for)e(al)5 b(l)38 b Fo(\016)f(<)1029 4835 y Fn(1)p Fw(\000)p Fg(\015)1159 4812 y Fr(\000)p Fd(T)1243 4827 y Ff(0)p 1029 4855 254 4 v 1138 4908 a Fn(3)1293 4876 y Fj(,)h(ther)-5 b(e)38 b(exists)g Fo(\032)c(>)g Fs(0)j Fj(such)h(that)h(if)e Fl(j)p Fo(u)p Fl(j)2710 4903 y Fg(\022)2783 4876 y Fo(<)d(\032)p Fj(,)k(we)g(c)-5 b(an)38 b(write)g(for)118 4989 y Fo(k)29 b Fl(\025)c Fs(1)p Fj(:)1360 5102 y Fo(M)1458 5064 y Fn(\()q Fg(k)r(T)1566 5073 y Ff(0)1600 5064 y Fn(\))1448 5124 y Fg(u)1657 5102 y Fs(=)g Fo(\025)1806 5064 y Fg(k)r(T)1886 5073 y Ff(0)1939 5102 y Fs(\()q Fo(u)p Fs(\))15 b Fo(Q)2149 5116 y Fg(u)2214 5102 y Fs(+)20 b Fo(R)2375 5064 y Fg(k)2374 5124 y(u)3500 5102 y Fs(\(10\))118 5268 y Fj(with,)33 b(for)h Fo(D)564 5283 y Fg(\022)603 5268 y Fs(\(0)p Fo(;)15 b(\032)p Fs(\))34 b Fj(the)f(b)-5 b(al)5 b(l)33 b(of)g(r)-5 b(adius)34 b Fo(\032)f Fj(ar)-5 b(ound)34 b Fs(0)f Fj(in)g Fo(H)2188 5283 y Fg(\022)2226 5268 y Fj(:)1867 5517 y Fs(6)p eop %%Page: 7 7 7 6 bop 254 162 a Fl(\017)46 b Fo(\025)63 b Fs(:)g Fo(u)48 b Fl(2)f Fo(D)832 177 y Fg(\022)871 162 y Fs(\(0)p Fo(;)15 b(\032)p Fs(\))50 b Fl(7\000)-16 b(!)48 b Fo(\025)p Fs(\()p Fo(u)p Fs(\))g Fl(2)f Fm(C)69 b Fj(is)45 b(analytic)h(and)g (satis\014es)g Fo(\025)2821 129 y Fg(T)2862 138 y Ff(0)2916 162 y Fs(\()p Fo(u)p Fs(\))i Fl(2)f Fo(D)s Fs(\(1)p Fo(;)15 b(\016)s Fs(\))47 b Fj(and)345 275 y Fo(\025)p Fs(\(0\))27 b(=)e(1)p Fj(,)254 458 y Fl(\017)46 b Fo(Q)41 b Fs(:)f Fo(u)26 b Fl(2)e Fo(D)761 473 y Fg(\022)801 458 y Fs(\(0)p Fo(;)15 b(\032)p Fs(\))27 b Fl(7\000)-16 b(!)25 b Fo(Q)1273 472 y Fg(u)1343 458 y Fl(2)g Fo(L)p Fs(\()p Fl(M)1635 473 y Fg(#)1682 458 y Fs(\))31 b Fj(is)f(analytic)i(and)f(satis\014es)h Fo(Q)2771 425 y Fn(2)2771 480 y Fg(u)2841 458 y Fs(=)25 b Fo(Q)3009 472 y Fg(u)3054 458 y Fj(,)30 b Fo(Q)3184 472 y Fn(0)3249 458 y Fs(=)25 b Fo(\027)6 b(\031)3448 477 y Fw(;)3517 458 y Fj(and)345 571 y Fl(k)p Fo(Q)462 585 y Fg(u)528 571 y Fl(\000)20 b Fo(\027)6 b(\031)722 589 y Fw(;)761 571 y Fl(k)25 b(\024)g Fo(\016)970 538 y Fn(2)1011 571 y Fj(,)254 754 y Fl(\017)46 b Fo(R)c Fs(:)e Fo(u)25 b Fl(2)g Fo(D)759 769 y Fg(\022)799 754 y Fs(\(0)p Fo(;)15 b(\032)p Fs(\))26 b Fl(7\000)-15 b(!)25 b Fo(R)1268 776 y Fg(u)1338 754 y Fl(2)g Fo(L)p Fs(\()p Fl(M)1630 769 y Fg(#)1677 754 y Fs(\))30 b Fj(is)g(analytic)i(and)f (satis\014es)g(Sp)p Fs(\()p Fo(R)2893 776 y Fg(u)2939 754 y Fs(\))26 b Fl(\032)f Fo(D)s Fs(\(0)p Fo(;)15 b(\015)3346 721 y Fw(\000)p Fg(T)3442 730 y Ff(0)3497 754 y Fs(+)g Fo(\016)s Fs(\))345 867 y Fj(and)34 b Fl(k)p Fo(R)637 834 y Fg(k)636 889 y(u)681 867 y Fl(k)26 b(\024)f Fs(\()p Fo(\015)935 834 y Fw(\000)p Fg(T)1031 843 y Ff(0)1090 867 y Fs(+)20 b(2)p Fo(\016)s Fs(\))1304 834 y Fg(k)1349 867 y Fj(.)118 1042 y(R)-5 b(emark.)42 b Fs(The)30 b(imp)s(ortan)m(t)f (fact)j(in)d(these)h(estimates)h(is)f(that)h(they)f(imply)e(for)j(suc)m (h)f Fo(u)p Fs(:)1538 1286 y(lim)1512 1347 y Fg(k)r Fw(!1)1801 1224 y Fl(k)p Fo(R)1916 1191 y Fg(k)1915 1247 y(u)1961 1224 y Fl(k)p 1716 1265 375 4 v 1716 1348 a(j)p Fo(\025)1794 1322 y Fg(k)r(T)1874 1331 y Ff(0)1928 1348 y Fs(\()p Fo(u)p Fs(\))16 b Fl(j)2127 1286 y Fs(=)24 b(0)118 1524 y(so)31 b(that)g Fo(\025)p 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a Fo(n)p Fs(!)1981 2504 y Fo(@)2034 2466 y Fg(n)2082 2504 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))118 2801 y(where)30 b(in)f(fact)i Fo(@)719 2768 y Fn(0)759 2801 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))27 b(=)e Fo(\025)p Fs(\(0\))h(=)f(1)31 b(and)f Fo(@)1738 2768 y Fg(n)1785 2801 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))27 b(=)2206 2765 y Fg(@)2247 2742 y Fd(n)p 2188 2780 120 4 v 2188 2832 a Fg(@)t(z)2265 2813 y Fd(n)2317 2724 y Fh(\014)2317 2778 y(\014)2347 2837 y Fg(z)s Fn(=0)2493 2801 y Fo(\025)p Fs(\()p Fo(z)t(u)p Fs(\).)259 2959 y(The)i(k)m(ey)i (of)f(our)f(probabilistic)d(study)j(is)g(the)h(iden)m(ti\014cation)e (of)i(the)g(\014rst)e(t)m(w)m(o)j(deriv)-5 b(ativ)m(es)30 b(of)f Fo(\025)118 3072 y Fs(in)g(real-analytic)h(directions)f(with)g (statistical)h(estimates)h(of)g(the)f(system.)118 3247 y Fk(Prop)s(osition)36 b(4.1.)42 b Fj(F)-7 b(or)34 b(every)f Fo(u)25 b Fl(2)g Fo(H)1544 3214 y Fg(r)1537 3275 y(\022)1582 3247 y Fj(,)32 b(we)h(have)g(the)g(two)h(fol)5 b(lowing)33 b(identities:)193 3508 y Fo(@)246 3471 y Fn(1)286 3508 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))27 b(=)669 3385 y Fh(Z)719 3591 y Fg(S)762 3602 y Ff(\012)829 3508 y Fo(u)15 b(d\027)31 b Fs(=)25 b Fo(m)1195 3522 y Fg(u)1426 3508 y Fo(@)1479 3471 y Fn(2)1519 3508 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))27 b(=)58 b(lim)1902 3569 y Fg(T)10 b Fw(!1)2109 3385 y Fh(Z)2159 3591 y Fg(S)2202 3602 y Ff(\012)2269 3380 y Fh(\022)2346 3447 y Fo(S)2402 3461 y Fg(T)2457 3447 y Fo(u)20 b Fl(\000)g Fo(T)13 b(m)2766 3461 y Fg(u)p 2346 3488 465 4 v 2507 3506 a Fl(p)p 2583 3506 66 4 v 77 x Fo(T)2820 3380 y Fh(\023)2887 3403 y Fn(2)2942 3508 y Fo(d\027)31 b Fs(=)25 b Fo(\033)3216 3471 y Fn(2)3213 3531 y Fg(u)3283 3508 y Fl(\025)g Fs(0)76 b(\(11\))118 3763 y Fj(R)-5 b(emark:)78 b Fs(The)24 b(iden)m(ti\014cations)f(of)i Fo(@)1437 3730 y Fn(1)1477 3763 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))26 b(and)e Fo(@)1987 3730 y Fn(2)2027 3763 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))26 b(with)e(the)g(mean)h (and)f(the)h(asymptotic)118 3876 y(v)-5 b(ariance)30 b(of)g Fo(u)g Fs(under)f(the)h(equilibrium)25 b(state)32 b Fo(\027)j Fs(are)c(natural)e(result)g(in)f(view)i(of)g(classical)f (thermo-)118 3989 y(dynamic)f(formalism)g(results)g(\(see)i([22)q(]\):) 41 b Fo(\025)p Fs(\()p Fo(u)p Fs(\),)31 b(in)d(the)h(domain)g(where)g (it)g(is)f(de\014ned,)h(really)f(pla)m(ys)118 4102 y(the)j(role)f(of)g (a)h(top)s(ological)f(pressure.)118 4287 y Fj(Pr)-5 b(o)g(of.)295 4400 y(-)37 b(Identi\014c)-5 b(ation)39 b(of)f Fo(@)1090 4367 y Fn(1)1130 4400 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))p Fj(.)58 b Fs(W)-8 b(e)37 b(will)c(decomp)s(ose)j(eac)m(h)h Fo(T)47 b Fl(\025)34 b Fs(1)i(as)g Fo(T)47 b Fs(=)34 b Fo(k)s(T)3199 4414 y Fn(0)3262 4400 y Fs(+)3374 4377 y(~)3357 4400 y Fo(T)12 b Fs(,)38 b(with)118 4513 y(0)26 b Fl(\024)303 4490 y Fs(~)285 4513 y Fo(T)38 b(<)25 b(T)525 4527 y Fn(0)564 4513 y Fs(,)31 b(and)f(write:)557 4624 y Fh(Z)608 4830 y Fg(S)651 4841 y Ff(\012)718 4748 y Fs(exp)872 4619 y Fh(\022)959 4686 y Fs(1)p 949 4727 V 949 4810 a Fo(T)1024 4748 y(S)1080 4762 y Fg(T)1135 4748 y Fo(u)1187 4619 y Fh(\023)1284 4748 y Fo(d\027)i Fs(=)1504 4624 y Fh(Z)1554 4830 y Fg(S)1597 4841 y Ff(\012)1664 4748 y Fs(exp)1818 4619 y Fh(\022)1885 4748 y Fo(S)1941 4763 y Fg(k)r(T)2021 4772 y Ff(0)2074 4647 y Fh(\020)2145 4686 y Fo(u)p 2138 4727 V 2138 4810 a(T)2214 4647 y Fh(\021)2288 4748 y Fs(+)2399 4686 y(1)p 2389 4727 V 2389 4810 a Fo(T)2465 4748 y(S)2535 4761 y Fn(~)2521 4778 y Fg(T)2591 4647 y Fh(\020)2645 4748 y Fo(u)21 b Fl(\016)f Fo(F)2854 4710 y Fg(k)r(T)2934 4719 y Ff(0)2973 4647 y Fh(\021)3027 4619 y(\023)3124 4748 y Fo(d\027)118 5010 y Fs(W)-8 b(e)32 b(ha)m(v)m(e)f(then)f(a)h(uniform)d(estimate)j(for)g(the)f(term)h(with) 2210 4987 y(~)2193 5010 y Fo(T)12 b Fs(:)688 5227 y(exp)842 5099 y Fh(\022)909 5227 y Fl(\000)990 5166 y Fo(T)1043 5180 y Fn(0)p 990 5206 93 4 v 1003 5289 a Fo(T)1092 5227 y Fl(j)p Fo(u)p Fl(j)1194 5241 y Fw(1)1269 5099 y Fh(\023)1361 5227 y Fl(\024)25 b Fs(exp)1611 5099 y Fh(\022)1698 5166 y Fs(1)p 1688 5206 66 4 v 1688 5289 a Fo(T)1764 5227 y(S)1834 5240 y Fn(~)1820 5257 y Fg(T)1890 5126 y Fh(\020)1944 5227 y Fo(u)c Fl(\016)f Fo(F)2153 5190 y Fg(k)r(T)2233 5199 y Ff(0)2272 5126 y Fh(\021)2326 5099 y(\023)2418 5227 y Fl(\024)25 b Fs(exp)2668 5099 y Fh(\022)2745 5166 y Fo(T)2798 5180 y Fn(0)p 2745 5206 93 4 v 2758 5289 a Fo(T)2847 5227 y Fl(j)p Fo(u)p Fl(j)2949 5241 y Fw(1)3025 5099 y Fh(\023)3500 5227 y Fs(\(12\))1867 5517 y(7)p eop %%Page: 8 8 8 7 bop 118 170 a Fs(F)-8 b(or)32 b(the)g(remaining)d(term,)j(if)e Fo(T)40 b(>)1393 121 y Fw(j)p Fg(u)p Fw(j)1473 142 y Fd(\022)p 1393 149 116 4 v 1433 201 a Fg(\032)1518 170 y Fs(,)32 b(w)m(e)g(apply)e(the)h(iden)m(tit)m(y)g(\(9\))h(and)f(the)h (sp)s(ectral)e(decomp)s(o-)118 320 y(sition)f(\(10\))j(to)f Fo(M)767 272 y Fn(\()q Fg(k)r(T)875 281 y Ff(0)909 272 y Fn(\))771 325 y Fd(u)p 767 337 45 3 v 767 378 a(T)971 320 y Fs(to)g(get:)656 479 y Fh(Z)707 685 y Fg(S)750 696 y Ff(\012)816 603 y Fs(exp)970 474 y Fh(\022)1057 541 y Fs(1)p 1047 582 66 4 v 1047 665 a Fo(T)1123 603 y(S)1179 618 y Fg(k)r(T)1259 627 y Ff(0)1297 603 y Fo(u)1349 474 y Fh(\023)1446 603 y Fo(d\027)h Fs(=)25 b Fo(\031)1718 621 y Fw(;)1772 502 y Fh(\020)1826 603 y Fo(M)1924 555 y Fn(\()q Fg(k)r(T)2032 564 y Ff(0)2066 555 y Fn(\))1928 607 y Fd(u)p 1924 619 45 3 v 1924 660 a(T)2098 603 y Fs(\()p Fo(\027)6 b Fs(\))2219 502 y Fh(\021)1570 834 y Fs(=)25 b Fo(\025)1719 796 y Fg(k)r(T)1799 805 y Ff(0)1852 733 y Fh(\020)1923 772 y Fo(u)p 1916 813 66 4 v 1916 896 a(T)1992 733 y Fh(\021)2061 834 y Fo(\031)2113 852 y Fw(;)2168 733 y Fh(\020)2222 834 y Fo(Q)2308 824 y Fd(u)p 2304 836 45 3 v 2304 877 a(T)2363 834 y Fs(\()p Fo(\027)6 b Fs(\))2484 733 y Fh(\021)2559 834 y Fs(+)19 b Fo(\031)2701 852 y Fw(;)2756 733 y Fh(\020)2810 834 y Fo(R)2880 796 y Fg(k)2893 832 y Fd(u)p 2889 844 V 2889 886 a(T)2948 834 y Fs(\()p Fo(\027)6 b Fs(\))3069 733 y Fh(\021)3500 834 y Fs(\(13\))118 1052 y(W)-8 b(e)32 b(can)e(no)m(w)h(ev)-5 b(aluate)31 b(the)f(limit)f(as)h Fo(T)43 b Fs(tends)30 b(to)h(in\014nit)m(y)d(of)j(eac)m(h)g(term)g(in)e (this)g(expression:)625 1325 y Fo(\025)678 1288 y Fg(k)r(T)758 1297 y Ff(0)812 1224 y Fh(\020)883 1264 y Fo(u)p 876 1304 66 4 v 876 1387 a(T)952 1224 y Fh(\021)1031 1325 y Fs(=)1127 1197 y Fh(\022)1194 1325 y Fs(1)21 b(+)1371 1264 y(1)p 1361 1304 V 1361 1387 a Fo(T)1436 1325 y(@)1489 1288 y Fn(1)1529 1325 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))22 b(+)e Fo(o)1961 1197 y Fh(\022)2048 1264 y Fs(1)p 2038 1304 V 2038 1387 a Fo(T)2114 1197 y Fh(\023)o(\023)2247 1220 y Fg(k)r(T)2327 1229 y Ff(0)2391 1325 y Fl(\000)-15 b(!)25 b Fs(exp)2717 1251 y Fh(\000)2758 1325 y Fo(@)2811 1288 y Fn(1)2851 1325 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))3111 1251 y Fh(\001)118 1554 y Fs(b)s(ecause)36 b(the)h(deriv)-5 b(ativ)m(es)35 b Fo(@)1134 1521 y Fg(n)1182 1554 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))38 b(are)e Fo(n)p Fs(-homogeneous)h(and)2477 1516 y Fg(k)r(T)2557 1525 y Ff(0)p 2477 1533 115 4 v 2508 1585 a Fg(T)2636 1554 y Fl(!)e Fs(1.)58 b(It)37 b(will)c(b)s(e)j(the)g(main)118 1667 y(term)31 b(in)e(our)g(estimation.)118 1827 y(W)-8 b(e)32 b(con)m(trol)e(the)h(t)m(w)m(o)h(others:)1028 1920 y Fh(\014)1028 1975 y(\014)1028 2029 y(\014)1058 2025 y Fo(\031)1110 2043 y Fw(;)1164 1924 y Fh(\020)1219 2025 y Fo(Q)1305 2015 y Fd(u)p 1300 2027 45 3 v 1300 2068 a(T)1359 2025 y Fs(\()p Fo(\027)6 b Fs(\))1480 1924 y Fh(\021)1555 2025 y Fl(\000)20 b Fs(1)1691 1920 y Fh(\014)1691 1975 y(\014)1691 2029 y(\014)1747 2025 y Fl(\024)1843 1920 y Fh(\015)1843 1975 y(\015)1843 2029 y(\015)1893 2025 y Fo(Q)1979 2015 y Fd(u)p 1975 2027 V 1975 2068 a(T)2054 2025 y Fl(\000)g Fo(Q)2217 2039 y Fn(0)2256 1920 y Fh(\015)2256 1975 y(\015)2256 2029 y(\015)2322 2025 y Fl(k)p Fo(\027)6 b Fl(k)2463 2051 y Fg(#)2535 2025 y Fl(\000)-16 b(!)26 b Fs(0)118 2240 y(b)m(y)k(con)m(tin)m(uit)m (y)h(of)f Fo(Q)849 2254 y Fg(u)894 2240 y Fs(,)h(and:)795 2370 y Fh(\014)795 2425 y(\014)795 2479 y(\014)825 2475 y Fo(\031)877 2493 y Fw(;)932 2374 y Fh(\020)986 2475 y Fo(R)1056 2437 y Fg(k)1069 2473 y Fd(u)p 1065 2485 V 1065 2526 a(T)1124 2475 y Fs(\()p Fo(\027)6 b Fs(\))1245 2374 y Fh(\021)1299 2370 y(\014)1299 2425 y(\014)1299 2479 y(\014)1355 2475 y Fl(\024)1451 2370 y Fh(\015)1451 2425 y(\015)1451 2479 y(\015)1501 2475 y Fo(R)1584 2473 y Fd(u)p 1580 2485 V 1580 2526 a(T)1639 2370 y Fh(\015)1639 2425 y(\015)1639 2479 y(\015)1689 2396 y Fg(k)1732 2475 y Fl(k)p Fo(\027)g Fl(k)1874 2501 y Fg(#)1945 2475 y Fl(\024)2041 2401 y Fh(\000)2083 2475 y Fo(\015)2135 2437 y Fw(\000)p Fg(T)2231 2446 y Ff(0)2290 2475 y Fs(+)20 b(2)p Fo(\016)2469 2401 y Fh(\001)2512 2418 y Fg(k)2554 2475 y Fl(k)p Fo(\027)6 b Fl(k)2696 2501 y Fg(#)2767 2475 y Fl(\000)-15 b(!)25 b Fs(0)118 2695 y(W)-8 b(e)32 b(get,)f(using)e(estimate)i(\(12\))h(and)e(inserting)f(previous)g (limits)f(in)h(\(13\):)582 2942 y(lim)549 3003 y Fg(T)10 b Fw(!1)756 2818 y Fh(Z)807 3024 y Fg(S)850 3035 y Ff(\012)917 2942 y Fs(exp)1071 2814 y Fh(\022)1158 2880 y Fs(1)p 1147 2921 66 4 v 1147 3004 a Fo(T)1223 2942 y(S)1279 2956 y Fg(T)1334 2942 y Fo(u)1386 2814 y Fh(\023)1483 2942 y Fo(d\027)32 b Fs(=)57 b(lim)1702 3003 y Fg(T)10 b Fw(!1)1910 2942 y Fo(\031)1962 2960 y Fw(;)2016 2841 y Fh(\020)2070 2942 y Fo(M)2168 2894 y Fn(\()q Fg(k)r(T)2276 2903 y Ff(0)2310 2894 y Fn(\))2172 2947 y Fd(u)p 2169 2959 45 3 v 2169 3000 a(T)2342 2942 y Fs(\()p Fo(\027)c Fs(\))2463 2841 y Fh(\021)2543 2942 y Fs(=)25 b(exp)2793 2868 y Fh(\000)2834 2942 y Fo(@)2887 2904 y Fn(1)2927 2942 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))3187 2868 y Fh(\001)3500 2942 y Fs(\(14\))259 3244 y(On)30 b(the)h(other)h(hand,) e(\(2\))i(implies)c(that)j Fo(\027)37 b Fs(is)30 b(mixing,)f(hence)i (ergo)s(dic,)g(whic)m(h)f(giv)m(es)h(the)g(limit,)118 3357 y(b)s(ecause)f Fo(u)h Fs(is)e(b)s(ounded:)1039 3582 y(lim)1006 3642 y Fg(T)10 b Fw(!1)1214 3458 y Fh(Z)1264 3664 y Fg(S)1307 3675 y Ff(\012)1374 3582 y Fs(exp)1528 3453 y Fh(\022)1615 3520 y Fs(1)p 1605 3561 66 4 v 1605 3644 a Fo(T)1680 3582 y(S)1736 3596 y Fg(T)1791 3582 y Fo(u)1843 3453 y Fh(\023)1941 3582 y Fo(d\027)31 b Fs(=)25 b(exp)2314 3453 y Fh(\022)2381 3458 y(Z)2431 3664 y Fg(S)2474 3675 y Ff(\012)2541 3582 y Fo(u)15 b(d\027)2706 3453 y Fh(\023)3500 3582 y Fs(\(15\))118 3836 y(And)30 b(w)m(e)g(can)h(iden)m(tify)e(b)s(oth)h(RHS)f(in)h(\(14\))h(and)f (\(15\))i(to)f(get:)1489 4076 y Fo(@)1542 4038 y Fn(1)1582 4076 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))27 b(=)1965 3952 y Fh(Z)2015 4158 y Fg(S)2058 4169 y Ff(\012)2125 4076 y Fo(u)15 b(d\027)295 4433 y Fj(-)23 b(Identi\014c)-5 b(ation)26 b(of)e Fo(@)1049 4400 y Fn(2)1088 4433 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))p Fj(.)39 b Fs(It)21 b(is)f(enough)g(to)i (sho)m(w)f(that)g(for)f Fo(u)26 b Fl(2)f Fo(H)2770 4400 y Fg(r)2763 4461 y(\022)2828 4433 y Fs(suc)m(h)c(that)g Fo(@)3264 4400 y Fn(1)3304 4433 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))27 b(=)118 4473 y Fh(R)161 4578 y Fg(S)204 4589 y Ff(\012)271 4546 y Fo(u)15 b(d\027)31 b Fs(=)25 b(0,)31 b(w)m(e)g(ha)m(v)m(e:)1267 4745 y(lim)1234 4806 y Fg(T)10 b Fw(!1)1441 4622 y Fh(Z)1491 4828 y Fg(S)1534 4839 y Ff(\012)1601 4617 y Fh(\022)1678 4684 y Fo(S)1734 4698 y Fg(T)1789 4684 y Fo(u)p 1678 4725 163 4 v 1689 4743 a Fl(p)p 1764 4743 66 4 v 1764 4820 a Fo(T)1851 4617 y Fh(\023)1918 4640 y Fn(2)1972 4745 y Fo(d\027)32 b Fs(=)24 b Fo(@)2244 4708 y Fn(2)2284 4745 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))118 4961 y(And)30 b(for)g(this,)f Fo(u)i Fs(b)s(eing)e(b)s(ounded,)f(w)m(e)j(kno)m(w)f(that)h(w)m(e)g (can)g(write:)317 5096 y Fh(Z)368 5302 y Fg(S)411 5313 y Ff(\012)477 5092 y Fh(\022)554 5158 y Fo(S)610 5172 y Fg(T)665 5158 y Fo(u)p 554 5199 163 4 v 565 5217 a Fl(p)p 641 5217 66 4 v 77 x Fo(T)727 5092 y Fh(\023)794 5114 y Fn(2)849 5220 y Fo(d\027)g Fs(=)1104 5158 y Fo(@)1157 5125 y Fn(2)p 1088 5199 126 4 v 1088 5282 a Fo(@)5 b(t)1174 5256 y Fn(2)1223 5088 y Fh(\014)1223 5143 y(\014)1223 5197 y(\014)1223 5252 y(\014)1254 5310 y Fg(t)p Fn(=0)1388 5096 y Fh(Z)1439 5302 y Fg(S)1482 5313 y Ff(\012)1549 5220 y Fs(exp)1703 5092 y Fh(\022)1834 5158 y Fo(t)p 1780 5199 142 4 v 1780 5217 a Fl(p)p 1855 5217 66 4 v 1855 5294 a Fo(T)1931 5220 y(S)1987 5234 y Fg(T)2042 5220 y Fo(u)2094 5092 y Fh(\023)2191 5220 y Fo(d\027)31 b Fs(=)2447 5158 y Fo(@)2500 5125 y Fn(2)p 2430 5199 126 4 v 2430 5282 a Fo(@)5 b(t)2516 5256 y Fn(2)2566 5088 y Fh(\014)2566 5143 y(\014)2566 5197 y(\014)2566 5252 y(\014)2596 5310 y Fg(t)p Fn(=0)2731 5220 y Fo(\031)2783 5238 y Fw(;)2838 5092 y Fh(\022)2904 5220 y Fo(M)3002 5172 y Fn(\()q Fg(T)10 b Fn(\))3020 5234 y Fd(tu)p 3003 5246 95 3 v 3003 5254 a Fr(p)p 3052 5254 45 3 v 3052 5296 a Fd(T)3112 5220 y Fs(\()p Fo(\027)c Fs(\))3233 5092 y Fh(\023)3500 5220 y Fs(\(16\))1867 5517 y(8)p eop %%Page: 9 9 9 8 bop 259 192 a Fs(F)-8 b(or)46 b Fo(T)62 b(>)672 91 y Fh(\020)736 143 y Fw(j)p Fg(t)p Fw(jj)p Fg(u)p Fw(j)881 164 y Fd(\022)p 736 171 180 4 v 808 223 a Fg(\032)926 91 y Fh(\021)980 114 y Fn(2)1020 192 y Fs(,)48 b(w)m(e)e(write)e(again) g Fo(T)62 b Fs(=)49 b Fo(k)s(T)2083 206 y Fn(0)2153 192 y Fs(+)2271 169 y(~)2253 192 y Fo(T)58 b Fs(with)43 b(0)50 b Fl(\024)2818 169 y Fs(~)2800 192 y Fo(T)62 b(<)49 b(T)3088 206 y Fn(0)3172 192 y Fs(and)c(use)f(the)118 328 y(comp)s(osition)29 b(rule)g(\(7\))j(and)d(the)i(sp)s(ectral)f(decomp)s(osition)f(\(10\))j (to)f(get:)380 578 y Fo(\031)432 597 y Fw(;)487 450 y Fh(\022)553 578 y Fo(M)651 531 y Fn(\()q Fg(T)10 b Fn(\))669 592 y Fd(tu)p 652 604 95 3 v 652 613 a Fr(p)p 701 613 45 3 v 701 655 a Fd(T)761 578 y Fs(\()p Fo(\027)c Fs(\))882 450 y Fh(\023)975 578 y Fs(=)25 b Fo(\025)1124 541 y Fg(k)r(T)1204 550 y Ff(0)1258 450 y Fh(\022)1363 517 y Fo(tu)p 1334 558 142 4 v 1334 576 a Fl(p)p 1410 576 66 4 v 77 x Fo(T)1486 450 y Fh(\023)1568 578 y Fo(\031)1620 597 y Fw(;)1674 450 y Fh(\022)1741 578 y Fo(M)1839 524 y Fs(\()1888 502 y Fn(~)1875 518 y Fg(T)1926 524 y Fs(\))1856 592 y Fd(tu)p 1839 604 95 3 v 1839 613 a Fr(p)p 1889 613 45 3 v 42 x Fd(T)1985 578 y Fl(\016)c Fo(Q)2150 577 y Fd(tu)p 2133 589 95 3 v 2133 598 a Fr(p)p 2183 598 45 3 v 42 x Fd(T)2241 578 y Fs(\()p Fo(\027)6 b Fs(\))2362 450 y Fh(\023)2450 578 y Fs(+)20 b Fo(\031)2593 597 y Fw(;)2647 450 y Fh(\022)2714 578 y Fo(M)2812 524 y Fs(\()2861 502 y Fn(~)2847 518 y Fg(T)2898 524 y Fs(\))2829 592 y Fd(tu)p 2812 604 95 3 v 2812 613 a Fr(p)p 2862 613 45 3 v 42 x Fd(T)2958 578 y Fl(\016)h Fo(R)3094 541 y Fg(k)3120 586 y Fd(tu)p 3103 598 95 3 v 3103 606 a Fr(p)p 3152 606 45 3 v 3152 648 a Fd(T)3211 578 y Fs(\()p Fo(\027)6 b Fs(\))3332 450 y Fh(\023)118 824 y Fs(W)-8 b(e)32 b(w)m(an)m(t)f(to)g (deriv)m(e)f(t)m(wice)h(this)e(last)h(expression:)239 1014 y Fo(@)292 981 y Fn(2)p 222 1055 126 4 v 222 1138 a Fo(@)5 b(t)308 1112 y Fn(2)358 944 y Fh(\014)358 998 y(\014)358 1053 y(\014)358 1107 y(\014)388 1166 y Fg(t)p Fn(=0)523 947 y Fh(\022)590 1076 y Fo(\025)643 1038 y Fg(k)r(T)723 1047 y Ff(0)776 947 y Fh(\022)882 1014 y Fo(tu)p 853 1055 142 4 v 853 1073 a Fl(p)p 929 1073 66 4 v 77 x Fo(T)1005 947 y Fh(\023)1087 1076 y Fo(\031)1139 1094 y Fw(;)1193 947 y Fh(\022)1260 1076 y Fo(M)1358 1021 y Fs(\()1407 999 y Fn(~)1394 1015 y Fg(T)1445 1021 y Fs(\))1375 1089 y Fd(tu)p 1358 1101 95 3 v 1358 1110 a Fr(p)p 1408 1110 45 3 v 42 x Fd(T)1504 1076 y Fl(\016)21 b Fo(Q)1669 1075 y Fd(tu)p 1652 1087 95 3 v 1652 1095 a Fr(p)p 1702 1095 45 3 v 42 x Fd(T)1760 1076 y Fs(\()p Fo(\027)6 b Fs(\))1881 947 y Fh(\023\023)2041 1076 y Fs(=)25 b(\()p Fo(v)s(w)r Fs(\))2321 1038 y Fw(00)2365 1076 y Fs(\(0\))2041 1278 y(=)g Fo(v)2184 1241 y Fw(00)2226 1278 y Fs(\(0\))p Fo(w)r Fs(\(0\))e(+)d(2)p Fo(v)2729 1241 y Fw(0)2753 1278 y Fs(\(0\))p Fo(w)2935 1241 y Fw(0)2960 1278 y Fs(\(0\))i(+)e Fo(v)s Fs(\(0\))p Fo(w)3417 1241 y Fw(00)3461 1278 y Fs(\(0\))118 1513 y(with)39 b Fo(v)s Fs(\()p Fo(t)p Fs(\))44 b(=)e Fo(\025)695 1480 y Fg(k)r(T)775 1489 y Ff(0)828 1412 y Fh(\020)914 1477 y Fg(tu)p 893 1492 110 4 v 893 1502 a Fw(p)p 952 1502 51 3 v 56 x Fg(T)1012 1412 y Fh(\021)1067 1513 y Fs(,)99 b(so)41 b(that)g Fo(v)s Fs(\(0\))j(=)e(1,)i Fo(v)2000 1480 y Fw(0)2024 1513 y Fs(\(0\))f(=)f(0,)i(and)c Fo(v)2643 1480 y Fw(00)2686 1513 y Fs(\(0\))k(=)2968 1476 y Fg(k)r(T)3048 1485 y Ff(0)p 2968 1492 115 4 v 3000 1544 a Fg(T)3092 1513 y Fo(@)3145 1480 y Fn(2)3185 1513 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\),)45 b(and)118 1712 y Fo(w)r Fs(\()p Fo(t)p Fs(\))26 b(=)f Fo(\031)462 1731 y Fw(;)517 1584 y Fh(\022)584 1712 y Fo(M)682 1658 y Fs(\()731 1635 y Fn(~)717 1652 y Fg(T)768 1658 y Fs(\))699 1726 y Fd(tu)p 682 1738 95 3 v 682 1746 a Fr(p)p 732 1746 45 3 v 43 x Fd(T)828 1712 y Fl(\016)20 b Fo(Q)992 1711 y Fd(tu)p 975 1723 95 3 v 975 1732 a Fr(p)p 1025 1732 45 3 v 42 x Fd(T)1084 1712 y Fs(\()p Fo(\027)6 b Fs(\))1205 1584 y Fh(\023)1272 1712 y Fs(,)76 b(so)31 b(that)g Fo(w)r Fs(\(0\))c(=)d(1,)32 b(and)172 2027 y Fo(w)239 1989 y Fw(00)282 2027 y Fs(\(0\))26 b(=)539 1965 y(1)p 529 2006 66 4 v 529 2089 a Fo(T)605 2027 y(\031)657 2045 y Fw(;)711 1899 y Fh(\022)778 2027 y Fo(M)876 1973 y Fs(\()926 1950 y Fn(~)912 1967 y Fg(T)963 1973 y Fs(\))866 2053 y Fn(0)1022 2027 y Fl(\016)21 b Fo(@)1141 1989 y Fn(2)1181 2027 y Fo(Q)p Fs(\(0;)15 b Fo(u)p Fs(\))22 b(+)d(2)p Fo(@)1670 1989 y Fn(1)1711 2027 y Fo(M)1809 1989 y Fn(\()1850 1972 y(~)1836 1989 y Fg(T)10 b Fn(\))1919 2027 y Fs(\(0;)15 b Fo(u)p Fs(\))21 b Fl(\016)g Fo(@)2266 1989 y Fn(1)2306 2027 y Fo(Q)p Fs(\(0;)15 b Fo(u)p Fs(\))22 b(+)d Fo(@)2750 1989 y Fn(2)2790 2027 y Fo(M)2888 1989 y Fn(\()2929 1972 y(~)2915 1989 y Fg(T)11 b Fn(\))2998 2027 y Fs(\(0;)k Fo(u)p Fs(\))22 b Fl(\016)f Fo(Q)3365 2041 y Fn(0)3404 1899 y Fh(\023)3486 2027 y Fs(\()p Fo(\027)6 b Fs(\))118 2273 y(whic)m(h)29 b(go)s(es)i(to)g(zero)g(when)f Fo(T)43 b Fs(go)s(es)31 b(to)g(in\014nit)m(y)-8 b(.)118 2386 y(In)30 b(the)g(same)h(w)m(a)m(y) 169 2575 y Fo(@)222 2542 y Fn(2)p 153 2616 126 4 v 153 2699 a Fo(@)5 b(t)239 2673 y Fn(2)288 2505 y Fh(\014)288 2560 y(\014)288 2614 y(\014)288 2669 y(\014)319 2727 y Fg(t)p Fn(=0)454 2509 y Fh(\022)521 2637 y Fo(\031)573 2655 y Fw(;)627 2509 y Fh(\022)694 2637 y Fo(M)792 2583 y Fs(\()841 2560 y Fn(~)827 2577 y Fg(T)878 2583 y Fs(\))809 2651 y Fd(tu)p 792 2663 95 3 v 792 2671 a Fr(p)p 842 2671 45 3 v 42 x Fd(T)938 2637 y Fl(\016)21 b Fo(R)1074 2599 y Fg(k)1100 2644 y Fd(tu)p 1083 2656 95 3 v 1083 2665 a Fr(p)p 1132 2665 45 3 v 1132 2707 a Fd(T)1191 2637 y Fs(\()p Fo(\027)6 b Fs(\))1312 2509 y Fh(\023\023)1472 2637 y Fs(=)1588 2575 y(1)p 1577 2616 66 4 v 1577 2699 a Fo(T)1653 2637 y(\031)1705 2655 y Fw(;)1760 2509 y Fh(\022)1827 2637 y Fo(M)1925 2583 y Fs(\()1974 2560 y Fn(~)1960 2577 y Fg(T)2011 2583 y Fs(\))1915 2663 y Fn(0)2071 2637 y Fl(\016)20 b Fo(@)2189 2599 y Fn(2)2229 2637 y Fo(R)2299 2599 y Fg(k)2341 2637 y Fs(\(0;)15 b Fo(u)p Fs(\))1497 2865 y(+)g(2)p Fo(@)1681 2828 y Fn(1)1721 2865 y Fo(M)1819 2828 y Fn(\()1860 2811 y(~)1846 2828 y Fg(T)10 b Fn(\))1929 2865 y Fs(\(0;)15 b Fo(u)p Fs(\))22 b Fl(\016)e Fo(@)2276 2828 y Fn(1)2316 2865 y Fo(R)2386 2828 y Fg(k)2428 2865 y Fs(\(0;)15 b Fo(u)p Fs(\))22 b(+)e Fo(@)2801 2828 y Fn(2)2841 2865 y Fo(M)2939 2828 y Fn(\()2980 2811 y(~)2966 2828 y Fg(T)10 b Fn(\))3049 2865 y Fs(\(0;)15 b Fo(u)p Fs(\))22 b Fl(\016)e Fo(R)3413 2828 y Fg(k)3412 2888 y Fn(0)3456 2764 y Fh(\021)3525 2865 y Fs(\()p Fo(\027)6 b Fs(\))118 3100 y(whic)m(h)29 b(go)s(es)i(to)g(zero)g(when)f Fo(T)43 b Fs(go)s(es)31 b(to)g(in\014nit)m(y)d(since)i(lim)2188 3115 y Fg(k)r Fw(!1)2387 3100 y Fo(R)2457 3067 y Fg(k)2525 3100 y Fs(=)25 b(0.)259 3261 y(Com)m(bining)j(all)i(these)g(results,)g(w)m(e)h(get)713 3513 y(lim)680 3573 y Fg(T)10 b Fw(!1)924 3451 y Fo(@)977 3418 y Fn(2)p 908 3492 126 4 v 908 3575 a Fo(@)5 b(t)994 3549 y Fn(2)1043 3381 y Fh(\014)1043 3435 y(\014)1043 3490 y(\014)1043 3544 y(\014)1073 3603 y Fg(t)p Fn(=0)1208 3513 y Fo(\031)1260 3531 y Fw(;)1315 3384 y Fh(\022)1382 3513 y Fo(M)1480 3465 y Fn(\()p Fg(T)10 b Fn(\))1497 3526 y Fd(tu)p 1480 3538 95 3 v 1480 3547 a Fr(p)p 1530 3547 45 3 v 42 x Fd(T)1590 3513 y Fs(\()p Fo(\027)c Fs(\))1711 3384 y Fh(\023)1803 3513 y Fs(=)58 b(lim)1899 3573 y Fg(T)10 b Fw(!1)2116 3451 y Fo(k)s(T)2219 3465 y Fn(0)p 2116 3492 143 4 v 2155 3575 a Fo(T)2269 3513 y(@)2322 3475 y Fn(2)2362 3513 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))27 b(=)e Fo(@)2798 3475 y Fn(2)2837 3513 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))118 3768 y(This,)29 b(together)j(with)d (equation)h(\(16\))i(implies)c(the)i(desired)f(equalit)m(y:)1041 4035 y Fo(@)1094 3997 y Fn(2)1134 4035 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))27 b(=)58 b(lim)1517 4095 y Fg(T)10 b Fw(!1)1724 3911 y Fh(Z)1775 4117 y Fg(S)1818 4128 y Ff(\012)1884 3907 y Fh(\022)1961 3973 y Fo(S)2017 3987 y Fg(T)2072 3973 y Fo(u)p 1961 4014 163 4 v 1972 4032 a Fl(p)p 2048 4032 66 4 v 77 x Fo(T)2134 3907 y Fh(\023)2201 3929 y Fn(2)2255 4035 y Fo(d\027)32 b Fs(=)25 b Fo(\033)2530 3997 y Fn(2)2527 4057 y Fg(u)2597 4035 y Fl(\025)g Fs(0)118 4288 y(and)30 b(giv)m(es)h(also)f(the)g(existence)h(of)g(the)f(limit.)p 3595 4288 4 62 v 3599 4230 55 4 v 3599 4288 V 3653 4288 4 62 v 118 4531 a Fp(4.4)112 b(Condition)36 b(for)i(p)s(ositivit)m(y)d (of)i Fc(\033)1755 4495 y Fn(2)1751 4556 y Fg(u)118 4703 y Fs(It)23 b(is)g(straigh)m(tforw)m(ard)g(that)h Fo(u)h Fs(=)g Fo(v)9 b Fl(\000)d Fo(v)j Fl(\016)d Fo(F)37 b Fs(implies)21 b Fo(\033)1976 4670 y Fn(2)1973 4725 y Fg(u)2043 4703 y Fs(=)k(0)e(b)s(ecause)h(in)e(this)g(case)i Fo(S)3047 4717 y Fg(T)3102 4703 y Fo(u)h Fs(=)g Fo(v)9 b Fl(\000)d Fo(v)j Fl(\016)d Fo(F)3580 4670 y Fg(T)3636 4703 y Fs(.)118 4864 y(F)-8 b(or)28 b(the)g(necessary)g(condition)e(in) g(\(4\),)j(w)m(e)f(ha)m(v)m(e)h(to)f(in)m(tro)s(duce)e(the)i(adjoin)m (t)f(of)h(the)f(comp)s(osition)f(b)m(y)118 4977 y Fo(F)13 b Fs(,)31 b Fo(P)38 b Fs(:)25 b Fo(L)453 4944 y Fn(2)493 4977 y Fs(\()p Fo(\027)6 b Fs(\))25 b Fl(!)h Fo(L)818 4944 y Fn(2)857 4977 y Fs(\()p Fo(\027)6 b Fs(\))31 b(de\014ned)e(b)m (y)855 5096 y Fh(Z)905 5302 y Fg(S)948 5313 y Ff(\012)1015 5220 y Fo(')20 b Fl(\016)h Fo(F)33 b Fl(\001)21 b Fo( )e(d\027)31 b Fs(=)1594 5096 y Fh(Z)1644 5302 y Fg(S)1687 5313 y Ff(\012)1754 5220 y Fo(')21 b Fl(\001)f Fs(\()p Fo(P)13 b( )s Fs(\))i Fo(d\027)189 b Fl(8)p Fo(';)15 b( )29 b Fl(2)c Fo(L)2764 5182 y Fn(2)2804 5220 y Fs(\()p Fo(\027)6 b Fs(\))1867 5517 y(9)p eop %%Page: 10 10 10 9 bop 118 162 a Fs(and)30 b(w)m(e)h(note)g(that)g(if)e Fo(u)c Fl(2)g Fo(C)7 b Fs(\()p Fo(S)1237 176 y Fn(\012)1292 162 y Fs(\))31 b(and)e Fo(g)g Fl(2)c Fo(H)1768 177 y Fg(\022)1807 162 y Fs(,)30 b(then)136 290 y Fh(Z)186 496 y Fg(S)229 507 y Ff(\012)296 413 y Fo(u)20 b Fl(\001)h Fo(P)485 376 y Fg(T)540 413 y Fo(g)d(d\027)32 b Fs(=)821 290 y Fh(Z)871 496 y Fg(S)914 507 y Ff(\012)981 413 y Fo(u)15 b(d)1110 313 y Fh(\020)1165 413 y Fo(M)1263 366 y Fn(\()p Fg(T)10 b Fn(\))1253 439 y(0)1373 413 y Fs(\()p Fo(g)24 b(?)d(\027)6 b Fs(\))1627 313 y Fh(\021)1706 413 y Fs(=)1802 285 y Fh(\022)1869 290 y(Z)1920 496 y Fg(S)1963 507 y Ff(\012)2029 413 y Fo(u)15 b(d\027)2194 285 y Fh(\023)2282 413 y Fl(\001)2327 285 y Fh(\022)2394 290 y(Z)2445 496 y Fg(S)2488 507 y Ff(\012)2554 413 y Fo(g)k(d\027)2714 285 y Fh(\023)2801 413 y Fs(+)2892 290 y Fh(Z)2943 496 y Fg(S)2986 507 y Ff(\012)3052 413 y Fo(u)c(d)3181 340 y Fh(\000)3224 413 y Fo(R)3294 376 y Fg(T)3293 436 y Fn(0)3348 413 y Fs(\()p Fo(g)25 b(?)20 b(\027)6 b Fs(\))3602 340 y Fh(\001)118 672 y Fs(W)-8 b(e)31 b(can)f(then)f(use)g(the)h(sp)s(ectral)f(gap)h(prop)s(ert)m(y)f (of)h Fo(M)2032 686 y Fn(0)2101 672 y Fs(\(see)g(Theorem)g(5.3\))h(to)f (get,)h(when)e Fo(m)3450 686 y Fg(u)3520 672 y Fs(=)c(0)118 785 y(and)30 b Fo(T)38 b Fl(\025)25 b Fo(T)535 799 y Fn(0)574 785 y Fs(:)1203 807 y Fh(\014)1203 861 y(\014)1203 916 y(\014)1203 970 y(\014)1234 815 y(Z)1284 1021 y Fg(S)1327 1032 y Ff(\012)1394 938 y Fo(u)20 b Fl(\001)g Fo(P)1582 901 y Fg(T)1637 938 y Fo(g)f(d\027)1797 807 y Fh(\014)1797 861 y(\014)1797 916 y(\014)1797 970 y(\014)1853 938 y Fl(\024)25 b(j)p Fo(u)p Fl(j)2051 965 y Fg(\022)2090 938 y Fo(\015)2142 901 y Fw(\000)p Fg(T)2252 938 y Fl(j)p Fo(g)s Fl(j)2350 965 y Fg(\022)2389 938 y Fl(k)p Fo(\027)6 b Fl(k)2530 965 y Fg(#)3500 938 y Fs(\(17\))118 1169 y(This)29 b(estimate)i(allo)m(ws)f(to)h(giv)m(e)g(another)f(expression) f(for)h Fo(\033)2224 1136 y Fn(2)2221 1191 y Fg(u)2266 1169 y Fs(.)41 b(W)-8 b(e)32 b(write:)662 1394 y(1)p 652 1435 66 4 v 652 1518 a Fo(T)743 1332 y Fh(Z)794 1538 y Fg(S)837 1549 y Ff(\012)903 1456 y Fs(\()q Fo(S)995 1470 y Fg(T)1049 1456 y Fo(u)p Fs(\))1137 1414 y Fn(2)1191 1456 y Fo(d\027)g Fs(=)1411 1332 y Fh(Z)1461 1538 y Fg(S)1504 1549 y Ff(\012)1571 1456 y Fo(u)1623 1418 y Fn(2)1677 1456 y Fo(d\027)27 b Fs(+)20 b(2)1947 1342 y Fg(T)10 b Fw(\000)p Fn(1)1952 1369 y Fh(X)1953 1567 y Fg(k)r Fn(=1)2103 1328 y Fh(\022)2170 1456 y Fs(1)21 b Fl(\000)2345 1394 y Fo(k)p 2337 1435 V 2337 1518 a(T)2413 1328 y Fh(\023)2495 1332 y(Z)2545 1538 y Fg(S)2588 1549 y Ff(\012)2655 1456 y Fo(u)f Fl(\001)h Fo(P)2844 1418 y Fg(k)2886 1456 y Fo(u)15 b(d\027)1315 1778 y Fs(=)25 b Fl(\000)1497 1654 y Fh(Z)1547 1860 y Fg(S)1590 1871 y Ff(\012)1657 1778 y Fo(u)1709 1740 y Fn(2)1763 1778 y Fo(d\027)i Fs(+)19 b(2)2032 1664 y Fg(T)10 b Fw(\000)p Fn(1)2038 1691 y Fh(X)2039 1889 y Fg(k)r Fn(=0)2189 1649 y Fh(\022)2256 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3875 y Fh(\023)2608 4003 y Fo(d\036)118 4249 y Fk(Prop)s(osition)45 b(4.3.)h Fj(F)-7 b(or)41 b(al)5 b(l)40 b(\014xe)-5 b(d)1440 4213 y Fn(1)p 1440 4228 36 4 v 1440 4280 a(2)1523 4249 y Fo(<)38 b(\013)g(<)g Fs(1)p Fj(,)j(for)f(al)5 b(l)40 b Fo(\036)e Fl(2)g(M)2540 4205 y Fg(p)2540 4278 y(#)2625 4249 y Fj(and)j(al)5 b(l)40 b Fo(u)e Fl(2)f Fo(H)3217 4216 y Fg(r)3210 4277 y(\022)3294 4249 y Fj(such)j(that)118 4362 y Fo(m)198 4376 y Fg(u)268 4362 y Fs(=)25 b(0)p Fj(,)33 b(we)g(have:)1603 4525 y Fs(\003)1666 4539 y Fg(\013)1716 4525 y Fs(\()p Fo(\014)5 b Fs(\))26 b(=)1974 4464 y Fo(\014)2030 4431 y Fn(2)p 1974 4504 96 4 v 1999 4588 a Fs(2)2079 4525 y Fo(\033)2134 4488 y Fn(2)2131 4548 y Fg(u)3500 4525 y Fs(\(19\))259 4758 y(The)35 b(analyticit)m(y)f (of)h(\003)1078 4772 y Fg(\013)1128 4758 y Fs(\()p Fo(\014)5 b Fs(\))36 b(allo)m(ws)e(to)h(apply)f(G\177)-45 b(artner-Ellis)33 b(Theorem)h(\(see)i(Theorem)f(I)s(I.6.1)118 4871 y(in)g([11)q(]\).)57 b(The)35 b(latter)h(sa)m(ys)h(that)1344 4832 y 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Fo(>)25 b Fs(0.)38 b(This)20 b(result)h(is)g (exactly)h(the)g(prop)s(ert)m(y)f(\(MDP\))j(of)e(Theorem)118 275 y(3.3.)118 388 y(If)34 b Fo(\033)268 355 y Fn(2)265 411 y Fg(u)341 388 y Fs(=)d(0,)k(then)f Fo(I)799 402 y Fg(\013)848 388 y Fs(\(0\))f(=)e(0)j(and)f Fo(I)1397 402 y Fg(\013)1447 388 y Fs(\()p Fo(x)p Fs(\))e(=)g(+)p Fl(1)j Fs(for)g(all)e Fo(x)g Fl(6)p Fs(=)f(0,)k(whic)m(h)e(corresp)s (onds)f(to)j(a)f(trivial)118 501 y(case.)118 738 y Fj(Pr)-5 b(o)g(of.)47 b Fs(W)-8 b(e)31 b(can)f(pro)s(ceed)g(as)g(for)g(the)g (cen)m(tral)g(limit)e(theorem)i(b)s(ecause)g Fo(T)2739 705 y Fn(1)p Fw(\000)p Fg(\013)2904 738 y Fl(!)25 b(1)30 b Fs(as)g Fo(T)38 b Fl(!)25 b(1)30 b Fs(so)118 851 y(that,)i(for)e Fo(T)43 b Fs(great)32 b(enough,)f(w)m(e)g(are)g(again)g(in)e(the)i (domain)e(where)h(Theorem)h(4.1)g(can)g(b)s(e)f(applied.)118 982 y(The)40 b(main)f(di\013erence)h(is)g(that)1278 909 y Fh(R)1320 1014 y Fg(S)1363 1025 y Ff(\012)1430 982 y Fs(exp)1584 881 y Fh(\020)1639 982 y Fo(\014)1725 943 y Fg(S)1768 954 y Fd(T)1817 943 y Fg(u)p 1705 961 174 4 v 1705 1016 a(T)1756 997 y Ff(1)p Fr(\000)p Fd(\013)1888 881 y Fh(\021)1973 982 y Fo(d\036)h Fs(div)m(erges)f(exp)s(onen)m (tially)e(fast)j(so)g(that)g(w)m(e)118 1128 y(need)30 b(the)h(factor)841 1092 y Fn(1)p 757 1107 204 4 v 757 1162 a Fg(T)808 1143 y Ff(2)p Fd(\013)p Fr(\000)p Ff(1)1001 1128 y Fs(to)g(rescale)g(it.)118 1307 y(F)-8 b(or)31 b Fo(T)38 b Fs(=)25 b Fo(k)s(T)571 1321 y Fn(0)631 1307 y Fs(+)740 1284 y(~)722 1307 y Fo(T)43 b Fs(with)29 b Fo(T)38 b(>)1212 1207 y Fh(\020)1276 1258 y Fw(j)p Fg(\014)s Fw(jj)p Fg(u)p Fw(j)1439 1279 y Fd(\022)p 1276 1287 198 4 v 1357 1339 a Fg(\032)1484 1207 y Fh(\021)1592 1202 y Ff(1)p 1548 1214 119 3 v 1548 1255 a(1)p Fr(\000)p Fd(\013)1681 1307 y Fs(,)30 b(w)m(e)h(denote)g Fo(u)2215 1321 y Fg(T)2295 1307 y Fs(=)2446 1266 y Fg(\014)s(u)p 2401 1287 174 4 v 2401 1341 a(T)2452 1322 y Ff(1)p Fr(\000)p Fd(\013)2585 1307 y Fs(.)40 b(Then:)637 1625 y(exp)791 1497 y Fh(\022)858 1625 y Fl(\000)967 1564 y Fo(\014)5 b(T)1076 1578 y Fn(0)p 939 1604 206 4 v 939 1688 a Fo(T)1005 1661 y Fn(1)p Fw(\000)p Fg(\013)1154 1625 y Fl(j)p Fo(u)p Fl(j)1256 1639 y Fw(1)1331 1497 y Fh(\023)1423 1625 y Fl(\024)25 b Fs(exp)1673 1470 y Fh( )1745 1625 y Fo(\014)1811 1560 y(S)1881 1573 y Fn(~)1867 1590 y Fg(T)1922 1560 y Fo(u)20 b Fl(\016)h Fo(F)2131 1527 y Fg(k)r(T)2211 1536 y Ff(0)p 1811 1604 439 4 v 1928 1688 a Fo(T)1994 1661 y Fn(1)p Fw(\000)p Fg(\013)2259 1470 y Fh(!)2356 1625 y Fl(\024)k Fs(exp)2606 1497 y Fh(\022)2712 1564 y Fo(\014)5 b(T)2821 1578 y Fn(0)p 2683 1604 206 4 v 2683 1688 a Fo(T)2749 1661 y Fn(1)p Fw(\000)p Fg(\013)2899 1625 y Fl(j)p Fo(u)p Fl(j)3001 1639 y Fw(1)3076 1497 y Fh(\023)118 1902 y Fs(and:)398 2009 y Fh(Z)448 2216 y Fg(S)491 2227 y Ff(\012)558 2133 y Fs(exp)712 2005 y Fh(\022)779 2133 y Fo(\014)845 2072 y(S)901 2087 y Fg(k)r(T)981 2096 y Ff(0)1019 2072 y Fo(u)p 845 2112 227 4 v 855 2195 a(T)921 2169 y Fn(1)p Fw(\000)p Fg(\013)1081 2005 y Fh(\023)1178 2133 y Fo(d\036)26 b Fs(=)f Fo(\031)1453 2152 y Fw(;)1507 2032 y Fh(\020)1562 2133 y Fo(M)1660 2096 y Fn(\()p Fg(k)r(T)1767 2105 y Ff(0)1801 2096 y Fn(\))1650 2156 y Fg(u)1691 2167 y Fd(T)1833 2133 y Fo(\036)1887 2032 y Fh(\021)1967 2133 y Fs(=)g Fo(\025)2116 2096 y Fg(k)r(T)2196 2105 y Ff(0)2249 2133 y Fs(\()q Fo(u)2337 2147 y Fg(T)2392 2133 y Fs(\))15 b Fo(\031)2494 2152 y Fw(;)2549 2133 y Fs(\()p Fo(Q)2656 2147 y Fg(u)2697 2158 y Fd(T)2749 2133 y Fo(\036)p Fs(\))21 b(+)f Fo(\031)3002 2152 y Fw(;)3056 2032 y Fh(\020)3111 2133 y Fo(R)3181 2096 y Fg(k)3180 2156 y(u)3221 2167 y Fd(T)3273 2133 y Fo(\036)3327 2032 y Fh(\021)118 2389 y Fs(with:)463 2565 y(1)p 365 2605 241 4 v 365 2689 a Fo(T)431 2662 y Fn(2)p Fg(\013)p Fw(\000)p Fn(1)631 2626 y Fs(log)764 2525 y Fh(\020)818 2626 y Fo(\025)871 2589 y Fg(k)r(T)951 2598 y Ff(0)1004 2626 y Fs(\()q Fo(u)1092 2640 y Fg(T)1147 2626 y Fs(\))1182 2525 y Fh(\021)1262 2626 y Fs(=)1368 2565 y Fo(k)s(T)1471 2579 y Fn(0)p 1368 2605 143 4 v 1406 2689 a Fo(T)1520 2626 y(T)1586 2589 y Fn(2)p Fw(\000)p Fn(2)p Fg(\013)1776 2626 y Fs(log)1909 2498 y Fh(\022)1976 2626 y Fs(1)g(+)2237 2565 y Fo(\014)2293 2532 y Fn(2)p 2142 2605 287 4 v 2142 2689 a Fs(2)p Fo(T)2253 2662 y Fn(2)p Fw(\000)p Fn(2)p Fg(\013)2438 2626 y Fo(@)2491 2589 y Fn(2)2531 2626 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))22 b(+)e Fo(o)2963 2498 y Fh(\022)3137 2565 y Fs(1)p 3040 2605 241 4 v 3040 2689 a Fo(T)3106 2662 y Fn(2)p Fw(\000)p Fn(2)p Fg(\013)3291 2498 y Fh(\023)o(\023)1443 2883 y Fl(\000)-15 b(!)1625 2822 y Fo(\014)1681 2789 y Fn(2)p 1625 2862 96 4 v 1650 2946 a Fs(2)1731 2883 y Fo(@)1784 2846 y Fn(2)1824 2883 y Fo(\025)p Fs(\(0;)15 b Fo(u)p Fs(\))26 b(=)2216 2822 y Fo(\014)2272 2789 y Fn(2)p 2216 2862 V 2241 2946 a Fs(2)2322 2883 y Fo(\033)2377 2846 y Fn(2)2374 2906 y Fg(u)2540 2883 y Fs(as)31 b Fo(T)38 b Fl(!)25 b(1)118 3108 y Fs(And)30 b(w)m(e)g(ha)m(v)m(e)i(for)e(the)h (remaining)d(term:)1254 3323 y(1)p 1156 3364 241 4 v 1156 3447 a Fo(T)1222 3421 y Fn(2)p Fg(\013)p Fw(\000)p 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Fh(\001)p 1651 3917 V 1673 4001 a Fo(\025)1726 3974 y Fg(k)r(T)1806 3983 y Ff(0)1859 4001 y Fs(\()p Fo(u)1946 4015 y Fg(T)2002 4001 y Fs(\))2068 3779 y Fh(\014)2068 3834 y(\014)2068 3888 y(\014)2068 3943 y(\014)2068 3997 y(\014)2124 3938 y Fl(\024)2220 3810 y Fh(\022)2296 3877 y Fo(\015)2348 3844 y Fw(\000)p Fg(T)2444 3853 y Ff(0)2503 3877 y Fs(+)20 b(2)p Fo(\016)p 2296 3917 388 4 v 2390 4001 a Fs(1)g Fl(\000)g Fo(\016)2694 3810 y Fh(\023)2761 3825 y Fg(k)2823 3938 y Fl(\001)h(k)p Fo(\036)p Fl(k)3014 3965 y Fg(#)3085 3938 y Fl(\000)-15 b(!)25 b Fs(0)259 4264 y(W)-8 b(e)32 b(get)f(in)e(conclusion)g(that)751 4519 y(\003)814 4533 y Fg(\013)863 4519 y Fs(\()p Fo(\014)5 b Fs(\))27 b(=)57 b(lim)1111 4579 y Fg(T)10 b Fw(!1)1426 4457 y Fs(1)p 1329 4498 241 4 v 1329 4581 a Fo(T)1395 4555 y Fn(2)p Fg(\013)p Fw(\000)p Fn(1)1594 4519 y Fs(log)1727 4395 y Fh(Z)1778 4601 y Fg(S)1821 4612 y Ff(\012)1887 4519 y Fs(exp)2041 4391 y Fh(\022)2274 4457 y Fo(\014)p 2118 4498 369 4 v 2118 4581 a(T)2184 4555 y Fn(1)p Fw(\000)p Fg(\013)2324 4581 y Fo(S)2380 4595 y Fg(T)2434 4581 y Fo(u)2496 4391 y Fh(\023)2594 4519 y Fo(d\036)25 b Fs(=)2826 4457 y Fo(\014)2882 4424 y Fn(2)p 2826 4498 96 4 v 2851 4581 a Fs(2)2932 4519 y Fo(\033)2987 4481 y Fn(2)2984 4541 y Fg(u)p 3595 4774 4 62 v 3599 4715 55 4 v 3599 4774 V 3653 4774 4 62 v 1844 5517 a Fs(12)p eop %%Page: 13 13 13 12 bop 118 162 a Fp(4.7)112 b(Pro)s(of)38 b(of)f(the)h(Large)g (deviations)e(result)118 334 y Fs(W)-8 b(e)24 b(cannot)g(pro)m(v)m(e)g (a)f(complete)g(large)h(deviations)e(principle)d(b)s(ecause)k(the)g (existence)h(of)f(the)g(sp)s(ectral)118 458 y(gap)33 b(for)f Fo(M)531 410 y Fn(\()q Fg(T)600 419 y Ff(0)634 410 y Fn(\))521 469 y Fg(u)698 458 y Fs(in)f(Theorem)h(4.1)i(is)d(v)-5 b(alid)31 b(only)g(for)h(small)f Fo(u)i Fs(and)e(the)i(scaling)e(tak)m (en)j(to)f(compute)118 571 y(the)i(log-Laplace)h(transform)e(is)g(not)h (the)h(same)f(as)g(for)g(mo)s(derate)g(deviations)f(\(it)h(corresp)s (onds)f(to)118 683 y(the)f(case)h Fo(\013)c 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4106 y Fg(\014)s(u)1055 4091 y Fo(\036)p Fs(\))20 b Fl(\000)g Fs(1)p Fl(j)26 b(\024)f Fo(\016)1490 4053 y Fn(2)1531 4091 y Fl(k)p Fo(\036)p Fl(k)1676 4118 y Fg(#)1934 4091 y Fs(and)2217 3905 y Fh(\014)2217 3959 y(\014)2217 4014 y(\014)2217 4068 y(\014)2217 4123 y(\014)2217 4177 y(\014)2257 3996 y Fo(\031)2309 4015 y Fw(;)2363 3896 y Fh(\020)2418 3996 y Fo(R)2488 3963 y Fg(k)2487 4024 y(\014)s(u)2575 3996 y Fs(\()p Fo(\036)p Fs(\))2699 3896 y Fh(\021)p 2257 4070 V 2323 4153 a Fo(\025)2376 4127 y Fg(k)r(T)2456 4136 y Ff(0)2509 4153 y Fs(\()q Fo(\014)5 b(u)p Fs(\))2764 3905 y Fh(\014)2764 3959 y(\014)2764 4014 y(\014)2764 4068 y(\014)2764 4123 y(\014)2764 4177 y(\014)2819 4091 y Fl(\000)-15 b(!)25 b Fs(0)p 3595 4389 4 62 v 3599 4331 55 4 v 3599 4389 V 3653 4389 4 62 v 259 4624 a(This)k(lo)s(cal)g(di\013eren)m(tiabilit)m (y)f(implies)g(the)i(follo)m(wing)f(partial)g(large)i(deviations)e (result:)118 4808 y Fk(Theorem)34 b(4.2.)42 b Fj(F)-7 b(or)34 b(al)5 b(l)33 b Fo(u)25 b Fl(2)g Fo(H)1312 4775 y Fg(r)1305 4836 y(\022)1383 4808 y Fj(and)33 b Fo(\036)26 b Fl(2)e(M)1833 4764 y Fg(p)1833 4838 y(#)1912 4808 y Fj(such)33 b(that)h Fl(k)p Fo(\036)p Fl(k)2450 4835 y Fg(#)2521 4808 y Fl(\024)25 b Fo(\016)2660 4775 y Fw(\000)p Fn(2)2755 4808 y Fj(,)33 b(we)g(de\014ne:)1386 5008 y Fo(I)1426 5022 y Fg(u)1471 5008 y Fs(\()p Fo(x)p Fs(\))26 b(=)g(sup)1715 5087 y Fg(\014)s Fw(2)p Fb(R)1868 5008 y Fs(\()q Fo(\014)5 b(x)20 b Fl(\000)g Fs(\003)2186 5022 y Fg(u)2231 5008 y Fs(\()p Fo(\014)5 b Fs(\)\))118 5268 y Fj(Then:)1844 5517 y Fs(13)p eop %%Page: 14 14 14 13 bop 226 162 a Fj(1.)45 b Fo(I)385 176 y Fg(u)463 162 y Fj(is)32 b(c)-5 b(onvex)34 b(and)f(lower)h(semi-c)-5 b(ontinuous,)33 b Fo(I)2012 176 y Fg(u)2057 162 y Fs(\()p Fo(x)p Fs(\))26 b(=)f(+)p Fl(1)32 b Fj(if)g Fl(j)p Fo(x)p Fl(j)26 b Fo(>)f Fl(j)p Fo(u)p Fl(j)2910 189 y Fw(1)2984 162 y Fj(,)33 b Fo(I)3085 176 y Fg(u)3130 162 y Fs(\()p Fo(x)p Fs(\))26 b Fl(\025)f Fs(0)33 b Fj(and:)1329 347 y Fo(I)1369 361 y Fg(u)1414 347 y Fs(\()p 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1118 a Fw(j)p Fg(u)p Fw(j)1322 1139 y Fd(\022)1367 985 y Fh(\021)1454 1085 y Fj(and)34 b Fo(\016)29 b(>)c Fs(0)p Fj(:)729 1356 y Fs(lim)15 b(inf)759 1417 y Fg(T)10 b Fw(!1)1017 1295 y Fs(1)p 1006 1335 66 4 v 1006 1418 a Fo(T)1097 1356 y Fs(log)17 b Fo(\036)1299 1228 y Fh(\022)1366 1356 y Fo(z)45 b Fs(:)1528 1295 y Fo(S)1584 1309 y Fg(T)1639 1295 y Fo(u)p Fs(\()p Fo(z)t Fs(\))p 1528 1335 280 4 v 1635 1418 a Fo(T)1843 1356 y Fl(2)25 b Fo(B)5 b Fs(\()p Fo(x;)15 b(\016)s Fs(\))2208 1228 y Fh(\023)2302 1356 y Fl(\025)25 b(\000)p Fo(I)2509 1370 y Fg(u)2554 1356 y Fs(\()p Fo(x)p Fs(\))384 b(\(Lo)m(w)m(er)32 b(Bound\))118 1624 y Fj(Pr)-5 b(o)g(of.)47 b Fs(1\))42 b(is)f(a)h(classical)e(result)h(of)g(large)h(deviations)e(theory)i (\(see)g(for)g(example)f(Lemma)h(2.2.5)118 1737 y(in)e([9)q(]\).)74 b(Equiv)-5 b(alence)40 b(\(21\))i(is)f(a)g(direct)g(consequence)h(of)f (Theorem)g(I)s(I.6.3)h(in)e([11)q(]:)63 b Fo(m)3347 1751 y Fg(u)3433 1737 y Fs(is)40 b(the)118 1850 y(di\013eren)m(tial)29 b(of)h(\003)739 1864 y Fg(u)815 1850 y Fs(at)h(p)s(oin)m(t)e(0,)i (hence)g(the)f(unique)f(minimizer)e(of)k Fo(I)2537 1864 y Fg(u)2582 1850 y Fs(.)118 1963 y(Both)36 b(b)s(ounds)d(of)i(large)g (deviations)g(are)g(obtained)g(b)m(y)g(applying)e(the)i(Theorem)g(of)g (G\177)-45 b(artner-Ellis)118 2076 y(\(see)31 b(Theorem)g(2.3.6)h(and)d (Lemma)i(2.3.9)h(in)d([9)q(]\).)p 3595 2076 4 62 v 3599 2018 55 4 v 3599 2076 V 3653 2076 4 62 v 118 2359 a Ft(5)135 b(P)l(erturb)t(ed)45 b(transfer)g(op)t(erators)h(and)f(sp)t(ectral)g (gap)118 2562 y Fs(In)d(this)g(section,)47 b(w)m(e)d(explain)d(ho)m(w)i (to)h(mo)s(dify)d([23)q(])j(to)g(obtain)e(p)s(erturb)s(ed)f(transfer)h (op)s(erators)118 2675 y(and)31 b(preserv)m(e)h(the)g(sp)s(ectral)f (gap)g(prop)s(ert)m(y)-8 b(.)45 b(This)29 b(analysis)i(giv)m(es)h(the)f (pro)s(of)g(of)h(the)g(in)m(termediate)118 2788 y(Theorem)e(4.1.)42 b(A)31 b(more)f(precise)g(construction)g(of)g(the)h(op)s(erators)f(is)g (giv)m(en)g(in)f(App)s(endix)f(A.)118 3028 y Fp(5.1)112 b(Finite)36 b(b)s(o)m(x)i(op)s(erators)118 3200 y Fs(The)29 b(construction)g(of)g(the)g(transfer)g(op)s(erators)g(is)g(w)m(ell)f (understo)s(o)s(d)f(b)m(y)i(lo)s(oking)f(at)i(restrictions)e(of)118 3313 y(the)35 b(coupled)f(map)g(to)h(\014nite)f(b)s(o)m(xes:)49 b(w)m(e)36 b(\014x)e(some)h(b)s(oundary)d(condition)i Fo(\030)i Fl(2)c Fo(S)3015 3327 y Fn(\012)3105 3313 y Fs(and)i(de\014ne)g(for)118 3426 y(all)29 b(\003)d Fl(2)f(F)9 b Fs(:)1395 3610 y Fo(F)1453 3624 y Fn(\003)1547 3610 y Fs(:)15 b Fo(A)1655 3624 y Fn(\003)1734 3610 y Fl(!)25 b(C)1903 3572 y Fn(\003)1587 3748 y Fo(z)1629 3762 y Fn(\003)1708 3748 y Fl(7!)g Fo(F)13 b Fs(\()p Fo(z)1972 3762 y Fn(\003)2046 3748 y Fl(_)20 b Fo(\030)2167 3769 y Fn(\003)2216 3751 y Fd(C)2271 3748 y Fs(\))p Fl(j)2331 3762 y Fn(\003)118 3932 y Fs(where)32 b Fo(z)425 3946 y Fn(\003)499 3932 y Fl(_)21 b Fo(\030)621 3954 y Fn(\003)670 3935 y Fd(C)758 3932 y Fs(denotes)32 b(the)h(p)s(oin)m(t)e Fo(w)g Fl(2)c Fo(S)1726 3946 y Fn(\012)1813 3932 y Fs(suc)m(h)32 b(that)h Fo(w)2284 3946 y Fg(i)2340 3932 y Fs(=)28 b Fo(z)2481 3946 y Fg(i)2542 3932 y Fs(for)j(all)g Fo(i)e Fl(2)f Fs(\003)k(and)f Fo(w)3297 3946 y Fg(i)3354 3932 y Fs(=)d Fo(\030)3493 3946 y Fg(i)3553 3932 y Fs(for)118 4045 y(all)h Fo(i)d Fl(2)f Fs(\003)450 4012 y Fg(C)509 4045 y Fs(.)118 4158 y(This)31 b(function)h Fo(F)745 4172 y Fn(\003)831 4158 y Fs(is)g(expanding)f(as)i(so)s(on)g(as)g Fo(\024)d(<)f Fs(\()p Fo(\025)22 b Fl(\000)g Fs(1\))p Fo(\032)33 b Fs(and)f(w)m(e)i(can)f(de\014ne)f(the)h(asso)s(ciated)118 4271 y(transfer)d(op)s(erator)h Fo(L)884 4285 y Fn(\003)977 4271 y Fs(:)41 b Fo(E)1110 4285 y Fn(\003)1188 4271 y Fl(!)25 b Fo(E)1371 4285 y Fn(\003)1455 4271 y Fs(as)30 b(follo)m(ws:)829 4392 y Fh(Z)879 4598 y Fg(S)922 4609 y Ff(\003)987 4516 y Fo(')21 b Fl(\016)f Fo(F)1190 4530 y Fn(\003)1264 4516 y Fl(\001)h Fo( )d(dm)1514 4478 y Fn(\003)1593 4516 y Fs(=)1689 4392 y Fh(Z)1739 4598 y Fg(S)1782 4609 y Ff(\003)1847 4516 y Fo(')j Fl(\001)f Fo(L)2034 4530 y Fn(\003)2087 4516 y Fs(\()p Fo( )s Fs(\))15 b Fo(dm)2361 4478 y Fn(\003)2507 4516 y Fl(8)p Fo(';)g( )29 b Fl(2)c Fo(E)2898 4530 y Fn(\003)3500 4516 y Fs(\(22\))118 4730 y(This)30 b(is)g(a)i(classical)f(to)s(ol)h(to)g(study)f (asymptotic)g(prop)s(erties)f(of)i(suc)m(h)f(dynamical)f(systems)i (\(see)g([1)q(])118 4843 y(for)e(an)h(extended)f(study)f(of)i(this)e (domain\).)118 4999 y(In)h(the)g(same)h(w)m(a)m(y)-8 b(,)32 b(for)e Fo(u)c Fl(2)e Fo(E)1187 5013 y Fn(\003)1241 4999 y Fs(,)30 b(w)m(e)h(can)g(de\014ne)e(a)i(p)s(erturb)s(ed)d(op)s (erator)i(b)m(y:)709 5096 y Fh(Z)759 5302 y Fg(S)802 5313 y Ff(\003)867 5220 y Fo(')21 b Fl(\016)g Fo(F)1071 5234 y Fn(\003)1144 5220 y Fl(\001)g Fo(e)1232 5182 y Fg(u)1297 5220 y Fl(\001)g Fo( )d(dm)1547 5182 y Fn(\003)1626 5220 y Fs(=)1722 5096 y Fh(Z)1772 5302 y Fg(S)1815 5313 y Ff(\003)1880 5220 y Fo(')j Fl(\001)f Fo(M)2093 5234 y Fn(\003)p Fg(;u)2207 5220 y Fs(\()p Fo( )s Fs(\))15 b Fo(dm)2481 5182 y Fn(\003)2626 5220 y Fl(8)p Fo(';)g( )30 b Fl(2)24 b Fo(E)3017 5234 y Fn(\003)1844 5517 y Fs(14)p eop %%Page: 15 15 15 14 bop 118 162 a Fs(Or,)30 b(equiv)-5 b(alen)m(tly:)1454 275 y Fo(M)1542 289 y Fn(\003)p Fg(;u)1655 275 y Fs(\()p Fo( )s Fs(\))27 b(=)e Fo(L)1972 289 y Fn(\003)2040 275 y Fs(\()p Fo(e)2117 238 y Fg(u)2183 275 y Fl(\001)20 b Fo( )s Fs(\))1175 b(\(23\))259 437 y(The)30 b(in)m(terest)h(of)f Fo(M)964 451 y Fn(\003)p Fg(;u)1108 437 y Fs(comes)h(from)f(the)h(form) m(ula:)892 598 y Fh(Z)943 805 y Fg(S)986 816 y Ff(\003)1050 722 y Fs(exp)1204 567 y Fh( )1276 609 y Fg(T)10 b Fw(\000)p Fn(1)1281 636 y Fh(X)1289 830 y Fg(t)p Fn(=0)1433 722 y Fo(u)20 b Fl(\016)h Fo(F)1642 685 y Fg(t)1629 745 y Fn(\003)1682 567 y Fh(!)1774 722 y Fl(\001)g Fo( )d(dm)2024 685 y Fn(\003)2103 722 y Fs(=)2199 598 y Fh(Z)2249 805 y Fg(S)2292 816 y Ff(\003)2357 722 y Fo(M)2455 685 y Fg(T)2445 745 y Fn(\003)p Fg(;u)2559 722 y Fs(\()p Fo( )s Fs(\))d Fo(dm)2833 685 y Fn(\003)118 1003 y Fs(whic)m(h)36 b(iden)m(ti\014es)h(the)g(Laplace)h(transform)f(of)1816 935 y Fh(P)1927 1003 y Fo(u)25 b Fl(\016)h Fo(F)2146 970 y Fg(t)2133 1030 y Fn(\003)2224 1003 y Fs(with)36 b(some)i(sp)s(ectral)f(c)m(haracteristic)h(of)118 1116 y Fo(M)206 1130 y Fn(\003)p Fg(;u)320 1116 y Fs(.)i(W)-8 b(e)31 b(th)m(us)e(need)g(an)g(in\014nite)e(dimensional)f(equiv)-5 b(alen)m(t)29 b(of)g(these)h(op)s(erators,)g(as)f(describ)s(ed)f(in)118 1229 y(Theorem)i(4.1.)259 1435 y(The)j(metho)s(d)g(to)h(construct)g (them)g(is)e(based)h(on)h(the)g(follo)m(wing)d(p)s(erturbativ)m(e)i (expansion,)g(de-)118 1548 y(riv)m(ed)d(from)g(Theorem)g(3.2)h(and)f (Lemma)g(3.4)i(of)e([23)r(]:)118 1727 y Fk(Theorem)k(5.1.)42 b Fj(If)33 b Fo(\024)25 b(<)g Fs(\()p Fo(\025)c Fl(\000)f Fs(1\))p Fo(\032)p Fj(,)33 b Fo(L)1485 1741 y Fn(\003)1570 1727 y Fj(has)h(the)f(inte)-5 b(gr)g(al)34 b(r)-5 b(epr)g(esentation:) 204 1999 y Fo(L)266 2013 y Fn(\003)319 1999 y Fs(\()p Fo( )s Fs(\)\()p Fo(!)543 2013 y Fn(\003)598 1999 y Fs(\))25 b(=)g Fl(\006)840 1875 y Fh(Z)891 2081 y Fn(\000)935 2092 y Ff(\003)1007 1913 y Fh(Y)999 2110 y Fg(p)p Fw(2)p Fn(\003)1146 1844 y Fh( )1218 1999 y Fo(k)s Fs(\()p Fo(!)1360 2013 y Fg(p)1400 1999 y Fo(;)15 b(f)1485 2013 y Fg(p)1525 1999 y Fs(\()p Fo(z)1602 2013 y Fg(p)1642 1999 y Fs(\)\))21 b(+)1838 1913 y Fh(X)1824 2110 y Fg(V)15 b Fw(2F)2000 1999 y Fo(\014)2051 2013 y Fg(p;V)2167 1999 y Fs(\()p Fo(!)2259 2013 y Fg(p)2299 1999 y Fo(;)g(z)2381 2013 y Fg(V)h Fw(\\)p Fn(\003)2559 1999 y Fl(_)j Fo(\030)2679 2021 y Fg(V)d Fw(\\)p Fn(\003)2832 2002 y Fd(C)2888 1999 y Fs(\))2923 1844 y Fh(!)3010 1999 y Fo( )s Fs(\()p Fo(z)3149 2013 y Fn(\003)3203 1999 y Fs(\))f Fo(\026)3308 1961 y Fn(\003)3362 1999 y Fs(\()p Fo(dz)3486 2013 y Fn(\003)3540 1999 y Fs(\))3500 2208 y(\(24\))118 2321 y Fj(wher)-5 b(e:)254 2500 y Fl(\017)46 b Fs(\000)402 2514 y Fn(\003)497 2500 y Fs(=)609 2432 y Fh(Q)695 2527 y Fg(p)p Fw(2)p Fn(\003)846 2500 y Fo(@)5 b(A)p Fs([)p Fo(\032)p Fs(])43 b Fj(and)f Fo(\026)1347 2467 y Fn(\003)1442 2500 y Fj(is)f(the)h (unique)f(holomorphic)k(di\013er)-5 b(ential)43 b(form)f(on)3372 2432 y Fh(Q)3457 2527 y Fg(p)p Fw(2)p Fn(\003)3608 2500 y Fl(C)345 2613 y Fj(which)34 b(extends)f Fo(m)1004 2627 y Fn(\003)1057 2613 y Fj(.)254 2798 y Fl(\017)46 b Fo(k)36 b Fj(is)d(the)g(p)-5 b(erio)g(dic)34 b(Cauchy)f(kernel:)1128 3025 y Fo(k)s Fs(\()p Fo(!)s(;)15 b(z)t Fs(\))27 b(=)1543 2964 y(1)p 1527 3005 77 4 v 1527 3088 a(2)p Fo(i)1629 3025 y Fs(cot)q(\()p Fo(z)e Fl(\000)20 b Fo(!)s Fs(\))25 b(=)2212 2964 y(1)p 2169 3005 132 4 v 2169 3088 a(2)p Fo(\031)s(i)2328 2939 y Fh(X)2326 3136 y Fg(n)p Fw(2)p Fb(Z)2654 2964 y Fs(1)p 2485 3005 383 4 v 2485 3088 a Fo(z)g Fl(\000)20 b Fo(!)j Fs(+)d Fo(n)254 3343 y Fl(\017)46 b Fo(\014)5 b Fs(\()p Fo(p;)15 b(V)21 b Fs(\))26 b Fj(ar)-5 b(e)26 b(we)-5 b(akly)26 b(holomorphic)j(functions)c(on)h Fo(D)2177 3357 y Fg(p;V)2319 3343 y Fs(=)e Fo(A)2482 3357 y Fg(p)2526 3343 y Fl(\002)t Fs(\000)2658 3357 y Fg(p)2702 3343 y Fl(\002)2777 3275 y Fh(Q)2862 3370 y Fg(q)r Fw(2)p Fg(V)16 b Fw(nf)p Fg(p)p Fw(g)3161 3343 y Fo(A)3229 3357 y Fg(q)3292 3343 y Fj(\(i.e.)39 b(c)-5 b(on-)345 3456 y(tinuous)34 b(in)f(al)5 b(l)33 b(variables)h(and)g (holomorphic)j(in)c Fo(w)2161 3470 y Fg(p)2227 3456 y Fl(2)26 b Fj(Int)p Fs(\()p Fo(A)2533 3470 y Fg(p)2573 3456 y Fs(\))33 b Fj(and)i Fo(z)2861 3475 y Fg(V)16 b Fw(nf)p Fg(p)p Fw(g)3089 3456 y Fl(2)26 b Fj(Int)p Fs(\()p Fo(A)3395 3475 y Fg(V)17 b Fw(nf)p Fg(p)p Fw(g)3598 3456 y Fs(\))p Fj(,)345 3569 y(se)-5 b(e)33 b(app)-5 b(endix)35 b(B)d(of)h([23)q(]\))g(such)f(that:)905 3733 y Fh(X)890 3930 y Fg(V)16 b Fw(2F)1066 3819 y Fo(\022)1112 3782 y Fw(\000j)p Fg(V)e Fw(j)1266 3819 y Fl(j)p Fo(\014)1342 3833 y Fg(p;V)1459 3819 y Fl(j)25 b(\024)g Fo(C)1670 3834 y Fg(\014)1743 3819 y Fs(=)2078 3758 y Fo(e)2120 3725 y Fn(2)p Fg(\031)r(\024)p 1849 3798 624 4 v 1849 3885 a Fo(e)1891 3859 y Fn(2)p Fg(\031)r Fn(\()p Fg(\025)p Fw(\000)p Fn(1\))p Fg(\032)2216 3885 y Fl(\000)20 b Fo(e)2349 3859 y Fn(2)p Fg(\031)r(\024)2502 3819 y Fl(\000)2832 3758 y Fs(1)p 2603 3798 504 4 v 2603 3885 a Fo(e)2645 3859 y Fn(2)p Fg(\031)r Fn(\()p Fg(\025)p Fw(\000)p Fn(1\))p Fg(\032)2970 3885 y Fl(\000)g Fs(1)345 4096 y Fj(and)901 4104 y Fh(Z)951 4310 y Fg(S)994 4318 y Fd(p)1050 4228 y Fo(\014)1101 4242 y Fg(p;V)1217 4228 y Fs(\()p Fo(!)1309 4242 y Fg(p)1349 4228 y Fo(;)15 b(z)1431 4247 y Fg(V)h Fw([f)p Fg(p)p Fw(g)1646 4228 y Fs(\))f Fo(d!)1800 4242 y Fg(p)1865 4228 y Fs(=)25 b(0)93 b Fl(8)p Fo(z)2192 4242 y Fg(p)2257 4228 y Fl(2)25 b Fs(\000)2400 4242 y Fg(p)2439 4228 y Fo(;)15 b(z)2521 4247 y Fg(V)i Fw(nf)p Fg(p)p Fw(g)2750 4228 y Fl(2)24 b Fo(A)2903 4247 y Fg(V)16 b Fw(nf)p Fg(p)p Fw(g)118 4532 y Fp(5.2)112 b(Existence)37 b(of)g(the)h(op)s(erators)118 4703 y Fs(W)-8 b(e)41 b(can)f(write)f(a)i (similar)c(in)m(tegral)j(represen)m(tation)g(for)f Fo(L)2251 4670 y Fg(T)2251 4730 y Fn(\003)2346 4703 y Fs(and)g(expand)g(it)h(b)m (y)g(in)m(terc)m(hange)g(of)118 4816 y(pro)s(ducts)c(and)h(sums.)61 b(This)36 b(giv)m(es)i(naturally)d(the)j(kind)e(of)h(con\014gurational) g(op)s(erators)h(w)m(e)g(need)118 4929 y(to)d(in)m(tro)s(duce)e(and)h (con)m(trol)g(to)h(de\014ne)e(an)i(in\014nite-dimensional)29 b(transfer)34 b(op)s(erator.)52 b(That)34 b(is)f(the)118 5042 y(metho)s(d)d(implemen)m(ted)f(b)m(y)h(H.H.)h(Rugh)f(in)f(Section) h(4)h(of)f([23)r(].)259 5155 y(T)-8 b(o)25 b(generalize)e(his)g (construction)g(to)i(p)s(erturb)s(ed)c(op)s(erators,)k(w)m(e)g(cannot,) h(for)d(tec)m(hnical)h(reasons,)118 5268 y(pro)s(ceed)34 b(as)g(in)e(\(23\))j(for)f(the)g(\014nite)f(dimensional)e(setting.)51 b(W)-8 b(e)35 b(ha)m(v)m(e)g(to)f(adapt)g(the)g(pro)s(of)f(of)h([23)q (])1844 5517 y(15)p eop %%Page: 16 16 16 15 bop 118 162 a Fs(with)29 b(some)j(additional)c(terms)j(corresp)s (onding)d(to)k(the)f(expansion)e(of)i(p)s(erturbations)e Fo(U)35 b Fl(2)26 b Fo(H)3446 177 y Fg(\022)3515 162 y Fs(and)118 275 y(to)32 b(con)m(trol)f(the)g(new)g(estimates.)43 b(This)29 b(allo)m(ws)h(to)i(construct)f(the)g(general)g(op)s(erators)g (describ)s(ed)e(in)118 388 y(the)i(follo)m(wing)e(result:)118 576 y Fk(Theorem)c(5.2.)35 b Fj(Supp)-5 b(ose)26 b(the)g(hyp)-5 b(otheses)27 b(of)f(The)-5 b(or)g(em)27 b(3.1)e(ar)-5 b(e)26 b(veri\014e)-5 b(d,)27 b(then)f(ther)-5 b(e)26 b(is)f Fo(\022)i(<)e(#)g(<)g Fs(1)118 689 y Fj(such)33 b(that)h(for)f(al)5 b(l)33 b Fo(T)38 b Fl(\025)25 b Fs(1)33 b Fj(we)g(have)g(a)g(multiline)-5 b(ar)34 b(functional:)1120 893 y Fl(L)1183 855 y Fn(\()p Fg(T)10 b Fn(\))1333 893 y Fs(:)490 b Fo(H)1931 855 y Fg(T)1924 916 y(\022)2011 893 y Fl(\000)-15 b(!)25 b Fo(L)p Fs(\()p Fl(M)2389 908 y Fg(#)2436 893 y Fo(;)15 b Fl(M)2585 908 y Fg(\022)2624 893 y Fs(\))1405 1058 y(\()p Fo(U)1502 1072 y Fn(0)1541 1058 y Fo(;)g(:)g(:)g(:)i(;)e(U)1805 1072 y Fg(T)10 b Fw(\000)p Fn(1)1951 1058 y Fs(\))25 b Fl(7\000)-15 b(!)25 b Fo(L)2245 1010 y Fn(\()p Fg(T)10 b Fn(\))2245 1091 y([)p Fg(U)2313 1100 y Ff(0)2347 1091 y Fg(;:::)n(;U)2493 1102 y Fd(T)e Fr(\000)p Ff(1)2620 1091 y Fn(])118 1272 y Fj(with)34 b(the)f(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties:)208 1487 y Fl(\017)26 b Fj(Ther)-5 b(e)33 b(exists)g Fo(T)842 1501 y Fn(0)907 1487 y Fl(\025)25 b Fs(1)33 b Fj(such)g(that)h Fo(L)1536 1439 y Fn(\()p Fg(T)10 b Fn(\))1536 1520 y([)p Fg(U)1604 1529 y Ff(0)1637 1520 y Fg(;:::)o(;U)1784 1531 y Fd(T)e Fr(\000)p Ff(1)1910 1520 y Fn(])1959 1487 y Fl(2)25 b Fo(L)p Fs(\()p Fl(M)2251 1502 y Fg(#)2298 1487 y Fs(\))93 b Fl(8)p Fo(T)37 b Fl(\025)25 b Fo(T)2716 1501 y Fn(0)208 1739 y Fl(\017)294 1635 y Fh(\015)294 1689 y(\015)294 1744 y(\015)345 1739 y Fo(L)407 1691 y Fn(\()p Fg(T)10 b Fn(\))407 1772 y([)p Fg(U)475 1781 y Ff(0)508 1772 y Fg(;:::)o(;U)655 1783 y Fd(T)e Fr(\000)p Ff(1)781 1772 y Fn(])805 1635 y Fh(\015)805 1689 y(\015)805 1744 y(\015)881 1739 y Fl(\024)977 1625 y Fg(T)i Fw(\000)p Fn(1)989 1653 y Fh(Y)989 1847 y Fg(t)p Fn(=0)1133 1739 y Fl(j)p Fo(U)1220 1753 y Fg(t)1250 1739 y Fl(j)1275 1766 y Fg(\022)3500 1739 y Fs(\(25\))208 1987 y Fl(\017)26 b Fo(L)341 1939 y Fn(\()p Fg(t)p Fn(+)p Fg(T)10 b Fn(\))341 2020 y([)p Fg(V)402 2029 y Ff(0)436 2020 y Fg(;:::)n(;V)575 2029 y Fd(t)p Fr(\000)p Ff(1)682 2020 y Fg(;U)750 2029 y Ff(0)783 2020 y Fg(;:::)o(;U)930 2031 y Fd(T)e Fr(\000)p Ff(1)1056 2020 y Fn(])1105 1987 y Fs(=)25 b Fo(L)1263 1939 y Fn(\()p Fg(t)p Fn(\))1263 2020 y([)p Fg(V)1324 2029 y Ff(0)1358 2020 y Fg(;:::)o(;V)1498 2029 y Fd(t)p Fr(\000)p Ff(1)1604 2020 y Fn(])1648 1987 y Fl(\016)20 b Fo(L)1775 1939 y Fn(\()p Fg(T)10 b Fn(\))1775 2020 y([)p Fg(U)1843 2029 y Ff(0)1877 2020 y Fg(;:::)o(;U)2024 2031 y Fd(T)e Fr(\000)p Ff(1)2150 2020 y Fn(])2392 1987 y Fj(if)32 b Fo(T)38 b Fl(\025)25 b Fo(T)2720 2001 y Fn(0)3500 1987 y Fs(\(26\))208 2159 y Fl(\017)h Fo(L)341 2121 y Fn(\()p Fg(T)10 b Fn(\))451 2159 y Fs(\()p Fl(M)595 2121 y Fg(m)595 2182 y(#)662 2159 y Fs(\))25 b Fl(\032)g(M)927 2121 y Fg(m)927 2182 y(\022)3500 2159 y Fs(\(27\))208 2394 y Fl(\017)294 2270 y Fh(Z)345 2476 y Fg(S)388 2487 y Ff(\012)454 2394 y Fo(b)c Fl(\016)f Fo(F)650 2356 y Fg(T)726 2394 y Fl(\001)771 2280 y Fg(T)10 b Fw(\000)p Fn(1)783 2308 y Fh(Y)784 2502 y Fg(t)p Fn(=0)927 2394 y Fo(U)989 2408 y Fg(t)1039 2394 y Fl(\016)21 b Fo(F)1176 2356 y Fg(t)1221 2394 y Fo(d\036)k Fs(=)1443 2270 y Fh(Z)1494 2476 y Fg(S)1537 2487 y Ff(\012)1604 2394 y Fo(b)15 b(d)1720 2293 y Fh(\020)1774 2394 y Fo(L)1836 2346 y Fn(\()p Fg(T)10 b Fn(\))1836 2427 y([)p Fg(U)1904 2436 y Ff(0)1938 2427 y Fg(;:::)o(;U)2085 2438 y Fd(T)e Fr(\000)p Ff(1)2211 2427 y Fn(])2235 2394 y Fo(\036)2289 2293 y Fh(\021)2544 2394 y Fl(8)15 b Fo(b)25 b Fl(2)g Fo(C)7 b Fs(\()p Fo(S)2923 2408 y Fn(\012)2978 2394 y Fs(\))p Fo(;)31 b(\036)25 b Fl(2)g(M)3343 2356 y Fg(m)3343 2417 y(#)3500 2394 y Fs(\(28\))118 2674 y Fj(Pr)-5 b(o)g(of.)47 b Fs(As)30 b(already)g(men)m(tioned,)f(this)g(pro)s(of)h(essen)m(tially)f(follo)m (ws)g(the)h(construction)f(of)i(P)m(art)f(IV)g(of)118 2787 y([23)q(].)41 b(W)-8 b(e)31 b(only)d(ha)m(v)m(e)j(to)f(add)f(in)f (con\014gurations)h(the)h(terms)f(corresp)s(onding)f(to)i(the)g(p)s (erturbations)118 2900 y Fo(U)180 2914 y Fg(i)208 2900 y Fs(.)48 b(W)-8 b(e)34 b(can)e(do)h(it)f(and)g(k)m(eep)h(go)s(o)s(d)g (estimates)g(uniformly)d(in)h Fo(U)39 b Fl(2)29 b Fo(H)2637 2915 y Fg(\022)2675 2900 y Fs(.)48 b(F)-8 b(or)33 b(comprehension,)f(w) m(e)118 3012 y(sk)m(etc)m(h)g(the)e(main)g(steps)g(and)g(mo)s (di\014cations)e(of)j(this)e(in)g(App)s(endix)f(A.)p 3595 3012 4 62 v 3599 2954 55 4 v 3599 3012 V 3653 3012 4 62 v 118 3211 a Fj(R)-5 b(emark:)42 b Fs(The)30 b(op)s(erator)g (constructed)h(in)e([23)q(])i(corresp)s(onds)e(to)i Fo(L)2485 3163 y Fn(\()p Fg(T)10 b Fn(\))2485 3244 y([1)p Fg(;:::)n(;)p Fn(1])2697 3211 y Fs(.)259 3374 y(Op)s(erators)27 b(of)g(Theorem)g(4.1) h(are)g(a)f(particular)f(case)i(of)f(the)h(general)f(op)s(erators)g (constructed)h(in)118 3487 y(Theorem)i(5.2.)42 b(F)-8 b(or)31 b Fo(u)25 b Fl(2)g Fo(H)1086 3502 y Fg(\022)1125 3487 y Fs(,)30 b(w)m(e)h(tak)m(e)1550 3702 y Fo(M)1648 3664 y Fn(\()p Fg(T)10 b Fn(\))1638 3724 y Fg(u)1783 3702 y Fs(=)25 b Fo(L)1941 3654 y Fn(\()p Fg(T)10 b Fn(\))1941 3735 y([)p Fg(e)1994 3716 y Fd(u)2034 3735 y Fg(;:::)o(;e)2166 3716 y Fd(u)2206 3735 y Fn(])118 3906 y Fs(W)-8 b(e)32 b(can)e(already)g(obtain)g(some)h(prop)s(erties)e(of)h(these)h(op)s (erators:)118 4093 y Fj(The)-5 b(or)g(em)35 b(4.1)e(-)f(First)h(p)-5 b(art.)48 b Fs(Since)28 b(it)g(is)h(the)g(comp)s(osition)f(of)h(the)h (analytic)e(function)g Fo(u)d Fl(7!)h Fo(e)3441 4060 y Fg(u)3515 4093 y Fs(and)118 4217 y(of)31 b(the)f(m)m(ultilinear)e (map)i Fl(L)1106 4184 y Fn(\()p Fg(T)10 b Fn(\))1215 4217 y Fs(,)31 b Fo(M)1369 4169 y Fn(\()p Fg(T)10 b Fn(\))1359 4229 y Fg(u)1509 4217 y Fs(is)30 b(analytic.)259 4341 y(W)-8 b(e)32 b(can)e(write)g(explicitly)e(the)j(series)e(expansion)g (of)i Fo(M)2233 4293 y Fn(\()q Fg(T)10 b Fn(\))2223 4353 y Fg(u)2373 4341 y Fs(around)30 b(a)g(p)s(oin)m(t)g Fo(u)p Fs(:)1316 4578 y Fo(M)1414 4530 y Fn(\()q Fg(T)10 b Fn(\))1404 4608 y Fg(u)p Fn(+)p Fg(h)1570 4578 y Fs(=)1667 4492 y Fh(X)1666 4688 y Fg(n)p Fw(\025)p Fn(0)1842 4517 y Fs(1)p 1824 4557 80 4 v 1824 4641 a Fo(n)p Fs(!)1914 4578 y Fo(@)1967 4541 y Fg(n)2014 4578 y Fo(M)2112 4541 y Fn(\()q Fg(T)g Fn(\))2222 4578 y Fs(\()p Fo(u)p Fs(;)15 b Fo(h)p Fs(\))p Fo(;)118 4871 y Fs(where)740 5004 y Fo(@)793 4966 y Fg(n)840 5004 y Fo(M)938 4966 y Fn(\()q Fg(T)10 b Fn(\))1048 5004 y Fs(\()p Fo(u)p Fs(;)15 b Fo(h)p Fs(\))27 b(=)1537 4917 y Fh(X)1385 5113 y Fg(n)1428 5122 y Ff(0)1462 5113 y Fg(;:::)o(;n)1604 5124 y Fd(T)8 b Fr(\000)p Ff(1)1730 5113 y Fw(\025)p Fn(0)1412 5154 y Fa(P)1486 5174 y Fd(T)g Fr(\000)p Ff(1)1486 5225 y Fd(t)p Ff(=0)1625 5204 y Fg(n)1668 5212 y Fd(t)1696 5204 y Fn(=)p Fg(n)2046 4942 y Fo(n)p Fs(!)p 1845 4983 481 4 v 1845 5066 a Fo(n)1900 5080 y Fn(0)1939 5066 y Fs(!)15 b Fl(\001)g(\001)g(\001)i Fo(n)2156 5080 y Fg(T)10 b Fw(\000)p Fn(1)2301 5066 y Fs(!)2336 5004 y Fo(L)2398 4966 y Fg(T)2398 5033 y Fn([)p Fg(h)2459 5011 y Fd(n)2497 5026 y Ff(0)2535 5033 y Fg(e)2568 5014 y Fd(u)2609 5033 y Fg(;:::)n(;h)2748 5002 y Fd(n)2786 5019 y(T)e Fr(\000)p Ff(1)2917 5033 y Fg(e)2950 5014 y Fd(u)2991 5033 y Fn(])3014 5004 y Fo(;)1844 5517 y Fs(16)p eop %%Page: 17 17 17 16 bop 118 162 a Fs(whic)m(h)36 b(is)f(an)i(elemen)m(t)g(of)g Fo(L)p Fs(\()p Fl(M)1273 177 y Fg(#)1320 162 y Fo(;)15 b Fl(M)1469 177 y Fg(\022)1508 162 y Fs(\))37 b(\(or)g Fo(L)p Fs(\()p Fl(M)1939 177 y Fg(#)1986 162 y Fs(\))g(if)f Fo(T)48 b Fl(\025)36 b Fo(T)2409 176 y Fn(0)2448 162 y Fs(\),)j(is)d(homogeneous)h(of)g(degree)h Fo(n)118 275 y Fs(and)30 b(satis\014es)g(the)g(b)s(ound)1248 284 y Fh(\015)1248 338 y(\015)1248 393 y(\015)1298 388 y Fo(@)1351 351 y Fg(n)1399 388 y Fo(M)1497 351 y Fn(\()p Fg(T)10 b Fn(\))1607 388 y Fs(\()p Fo(u)p Fs(;)15 b Fo(h)p Fs(\))1821 284 y Fh(\015)1821 338 y(\015)1821 393 y(\015)1898 388 y Fl(\024)25 b Fs(\()p Fo(T)13 b Fl(j)p Fo(h)p Fl(j)2198 415 y Fg(\022)2237 388 y Fs(\))2272 346 y Fg(n)2319 388 y Fo(e)2361 351 y Fg(T)d Fw(j)p Fg(u)p Fw(j)2492 371 y Fd(\022)118 563 y Fs(hence)31 b(w)m(e)f(con)m(trol)h(the)g (di\013erence)f(b)s(et)m(w)m(een)h(t)m(w)m(o)g(op)s(erators)g(b)m(y:) 1152 661 y Fh(\015)1152 715 y(\015)1152 770 y(\015)1203 765 y Fo(M)1301 717 y Fn(\()p Fg(T)10 b Fn(\))1291 795 y Fg(u)p Fn(+)p Fg(h)1452 765 y Fl(\000)19 b Fo(M)1640 728 y Fn(\()q Fg(T)10 b Fn(\))1630 788 y Fg(u)1750 661 y Fh(\015)1750 715 y(\015)1750 770 y(\015)1826 765 y Fl(\024)1922 664 y Fh(\020)1976 765 y Fo(e)2018 728 y Fg(T)g Fw(j)p Fg(h)p Fw(j)2150 748 y Fd(\022)2209 765 y Fl(\000)20 b Fs(1)2345 664 y Fh(\021)2415 765 y Fo(e)2457 728 y Fg(T)10 b Fw(j)p Fg(u)p Fw(j)2588 748 y Fd(\022)118 973 y Fs(Estimate)31 b(\(6\))g(is)e(the)i(particular)d(case)j(of)g (this)e(inequalit)m(y)f(around)h Fo(u)d Fs(=)f(0.)41 b(F)-8 b(orm)m(ulas)30 b(\(7\),)i(\(8\))f(and)118 1086 y(\(9\))h(are)e(easily)g(deduced)f(from)h(\(26\),)i(\(27\))g(and)e (\(28\).)p 3595 1086 4 62 v 3599 1028 55 4 v 3599 1086 V 3653 1086 4 62 v 118 1326 a Fp(5.3)112 b(Pro)s(of)38 b(of)f(the)h(Sp)s(ectral)e(gap)j(prop)s(ert)m(y)118 1510 y 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4979 y Fs(and)30 b(if)f(w)m(e)i(denote)643 5214 y Fo(Q)715 5228 y Fg(u)785 5214 y Fs(=)25 b Fl(\000)1005 5153 y Fs(1)p 962 5194 132 4 v 962 5277 a(2)p Fo(\031)s(i)1119 5091 y Fh(Z)1170 5297 y Fw(j)p Fg(\025)p Fw(\000)p Fn(1)p Fw(j)p Fn(=)p Fg(\016)1448 5214 y Fo(R)q Fs(\()p Fo(\025;)15 b(M)1734 5228 y Fg(u)1780 5214 y Fs(\))g Fo(d\025)26 b Fs(=)f Fl(\000)2176 5153 y Fs(1)p 2133 5194 V 2133 5277 a(2)p Fo(\031)s(i)2289 5091 y Fh(Z)2340 5297 y Fw(j)p Fg(\025)p Fw(\000)p Fn(1)p Fw(j)p Fn(=2)p Fg(\016)2654 5214 y Fo(R)q Fs(\()p Fo(\025;)15 b(M)2940 5228 y Fg(u)2985 5214 y Fs(\))g Fo(d\025)1844 5517 y Fs(17)p eop %%Page: 18 18 18 17 bop 118 162 a Fs(the)29 b(pro)5 b(jection)28 b(asso)s(ciated)h (to)g(the)g(sp)s(ectrum)e(of)i Fo(M)1983 176 y Fg(u)2056 162 y Fs(included)d(in)h Fo(D)s Fs(\(1)p Fo(;)15 b(\016)s Fs(\),)31 b(w)m(e)e(get)h(that)f Fo(Q)3403 176 y Fg(u)3476 162 y Fs(is)e(an)118 275 y(analytic)j(function)f(of)i Fo(u)f Fs(and)449 533 y Fl(k)p Fo(Q)566 547 y Fg(u)632 533 y Fl(\000)20 b 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y(op)s(erators.)118 4533 y Fp(A.1)112 b(Single)36 b(site)h(op)s(erators)118 4705 y Fs(F)-8 b(or)24 b Fo(f)319 4719 y Fg(p)383 4705 y Fl(2)h(E)8 b Fs(\()p Fo(\032;)15 b(\025)p Fs(\))24 b(an)e(expanding)g(map)g(on)h Fo(A)1688 4719 y Fg(p)1727 4705 y Fs(,)i(the)e(asso)s(ciated)g(transfer)f(op)s(erator)h Fo(L)3101 4720 y Fg(f)3135 4728 y Fd(p)3200 4705 y Fs(:)j Fo(E)3318 4719 y Fg(p)3383 4705 y Fl(\000)-16 b(!)26 b Fo(E)3622 4719 y Fg(p)118 4818 y Fs(can)31 b(b)s(e)f(written)f (\(this)h(is)f(a)i(particular)d(case)k(of)e(iden)m(tit)m(y)g(\(24\)\):) 1070 5064 y Fo(L)1132 5079 y Fg(f)1166 5087 y Fd(p)1206 5064 y Fo(\036)p Fs(\()p Fo(!)1352 5078 y Fg(p)1392 5064 y Fs(\))c(=)1549 4940 y Fh(Z)1599 5146 y Fn(\000)1643 5154 y Fd(p)1699 5064 y Fo(k)s Fs(\()p Fo(!)1841 5078 y Fg(p)1880 5064 y Fo(;)15 b(f)1965 5078 y Fg(p)2005 5064 y Fs(\()p Fo(z)2082 5078 y Fg(p)2122 5064 y Fs(\)\))p Fo(\036)p Fs(\()p Fo(z)2323 5078 y Fg(p)2364 5064 y Fs(\))g Fo(\026)2469 5026 y Fg(p)2509 5064 y Fs(\()p Fo(dz)2633 5078 y Fg(p)2674 5064 y Fs(\))1844 5517 y(18)p eop %%Page: 19 19 19 18 bop 118 162 a Fs(It)29 b(satis\014es)e Fo(l)574 176 y Fg(p)630 162 y Fl(\016)17 b Fo(L)754 177 y Fg(f)788 185 y Fd(p)853 162 y Fs(=)25 b Fo(l)976 176 y Fg(p)1044 162 y Fs(with)j Fo(l)1277 176 y Fg(p)1316 162 y Fs(\()p Fo(\036)p Fs(\))e(=)1562 89 y Fh(R)1605 194 y Fg(S)1648 202 y Fd(p)1704 162 y Fo(\036)p Fs(\()p Fo(z)1835 176 y Fg(p)1875 162 y Fs(\))15 b Fo(dz)2014 176 y Fg(p)2083 162 y Fs(and)28 b(enjo)m(ys)h(a)g(sp)s(ectral)e(gap)i(prop)s(ert)m(y)f (with)118 289 y(the)j(follo)m(wing)e(estimates,)i(uniformly)c(in)i Fo(f)1649 303 y Fg(p)1714 289 y Fl(2)c(E)8 b Fs(\()p Fo(\032;)15 b(\025)p Fs(\):)1032 427 y Fh(\015)1032 482 y(\015)1032 536 y(\015)1083 532 y Fo(L)1145 494 y Fg(T)1145 555 y(f)1179 563 y Fd(p)1219 427 y Fh(\015)1219 482 y(\015)1219 536 y(\015)1294 532 y Fl(\024)25 b Fo(c)1429 547 y Fg(h)1596 532 y Fs(and)1970 400 y Fh(\015)1970 455 y(\015)1970 509 y(\015)1970 564 y(\015)2030 532 y Fo(L)2092 494 y Fg(T)2092 555 y(f)2126 563 y Fd(p)2166 427 y Fh(\014)2166 482 y(\014)2166 536 y(\014)2196 595 y Fn(Ker)f Fg(l)2355 603 y Fd(p)2395 400 y Fh(\015)2395 455 y(\015)2395 509 y(\015)2395 564 y(\015)2471 532 y Fl(\024)h Fo(c)2606 546 y Fg(r)2644 532 y Fo(\021)2692 494 y Fg(T)3500 532 y Fs(\(30\))118 766 y(where)31 b Fo(c)421 781 y Fg(h)494 766 y Fl(\025)d Fs(1,)k Fo(c)734 780 y Fg(r)800 766 y Fo(>)c Fs(0)k(and)f Fo(\021)g(<)d Fs(1.)45 b(A)32 b(pro)s(of)f(of)h (these)g(results)f(can)h(b)s(e)f(found)g(in)f(App)s(endix)f(A)j(of)118 879 y([23)q(].)118 1120 y Fp(A.2)112 b(Con\014gurations)118 1292 y Fs(W)-8 b(e)32 b(de\014ne)d(what)i(a)f(branc)m(hing,)g(the)g (main)f(elemen)m(t)i(to)g(de\014ne)f(the)h(con\014gurations,)e(is:)118 1467 y Fk(De\014nition)39 b(A.1.)44 b Fj(A)35 b Fi(b)m(ranching)f(pair) h Fs(\()p Fo(S;)15 b(V)21 b Fs(\))36 b Fj(is)g(c)-5 b(omp)g(ose)g(d)38 b(by)e(a)g(subset)f Fo(S)h Fl(2)31 b(F)45 b Fj(and)36 b(a)g(function)118 1580 y Fo(V)61 b Fs(:)40 b Fo(S)31 b Fl(!)25 b(F)9 b Fj(.)42 b(We)32 b(denote)h Fo(V)21 b Fs([)p Fo(S)5 b Fs(])25 b(=)g Fo(S)h Fl([)19 b Fs(\()q Fl([)1664 1594 y Fg(p)p Fw(2)p Fg(S)1797 1580 y Fo(V)h Fs(\()p Fo(p)p Fs(\)\))q Fj(.)259 1755 y Fs(Giv)m(en)38 b Fo(K)44 b Fl(2)36 b(F)9 b Fs(,)40 b(\()p Fo(S;)15 b(V)21 b Fs(\))38 b(a)g(branc)m(hing)e(pair)h(suc)m(h)g(that)h Fo(S)k Fl(\032)37 b Fo(K)7 b Fs(,)39 b Fo(U)48 b Fl(2)36 b Fo(H)2925 1770 y Fg(\022)3001 1755 y Fs(and)h Fo(W)50 b Fl(2)37 b(F)9 b Fs(,)40 b(w)m(e)118 1868 y(de\014ne)30 b Fo(H)i Fs(=)25 b Fo(K)i Fl([)20 b Fo(V)g Fs([)p Fo(S)5 b Fs(])21 b Fl([)e Fo(W)43 b Fs(and)30 b(the)g(op)s(erator)h Fo(L)1946 1887 y Fg(K)q(;)p Fn(\()p Fg(S;V)14 b Fn(\))p Fg(;)p Fn(\()p Fg(U)o(;W)c Fn(\))2462 1868 y Fs(:)41 b Fo(E)2595 1882 y Fg(H)2687 1868 y Fl(!)25 b Fo(E)2870 1882 y Fg(K)2969 1868 y Fs(b)m(y:)201 2137 y Fo(L)263 2155 y Fg(K)q(;)p Fn(\()p Fg(S;V)14 b Fn(\))p Fg(;)p Fn(\()p Fg(U)o(;W)c Fn(\))738 2137 y Fs(\()p Fo(')832 2151 y Fg(H)901 2137 y Fs(\)\()p Fo(!)1028 2151 y Fg(K)1096 2137 y Fs(\))26 b(=)f Fl(\006)1339 2013 y Fh(Z)1389 2219 y Fg(S)1432 2233 y Fd(H)t Fr(n)p Fd(K)1595 2137 y Fo(m)1675 2099 y Fg(H)5 b Fw(n)p Fg(K)1842 2137 y Fs(\()p Fo(dz)1966 2155 y Fg(H)g Fw(n)p Fg(K)2134 2137 y Fs(\))2184 2013 y Fh(Z)2235 2219 y Fn(\000)2279 2230 y Fd(K)2363 2050 y Fh(Y)2357 2247 y Fg(p)p Fw(2)p Fg(S)2501 2137 y Fo(\014)2552 2155 y Fg(p;V)15 b Fn(\()p Fg(p)p Fn(\))2759 2137 y Fs(\()p Fo(!)2851 2151 y Fg(p)2891 2137 y Fo(;)g(z)2973 2155 y Fg(V)h Fn(\()p Fg(p)p Fn(\))p Fw([)p Fg(p)3207 2137 y Fs(\))1739 2398 y Fl(\002)1886 2311 y Fh(Y)1830 2513 y Fg(p)p Fw(2)p Fg(K)5 b Fw(n)p Fg(S)2074 2398 y Fo(k)s Fs(\()p Fo(!)2216 2412 y Fg(p)2256 2398 y Fo(;)15 b(f)2341 2412 y Fg(p)2380 2398 y Fs(\()p Fo(z)2457 2412 y Fg(p)2498 2398 y Fs(\)\))p Fo(U)2630 2412 y Fg(W)2711 2398 y Fs(\()p Fo(z)2788 2412 y Fg(W)2869 2398 y Fs(\))p Fo(')2963 2412 y Fg(H)3031 2398 y Fs(\()p Fo(z)3108 2412 y Fg(H)3176 2398 y Fs(\))g Fo(\026)3281 2360 y Fg(K)3350 2398 y Fs(\()p Fo(dz)3474 2412 y Fg(K)3543 2398 y Fs(\))118 2686 y(W)-8 b(e)32 b(then)e(ha)m(v)m(e)h(some)g(compatibilit)m(y)e(prop)s(erties)f (for)j(these)f(op)s(erators:)118 2862 y Fk(Lemma)j(A.1.)42 b Fj(We)32 b(have:)225 3172 y Fo(\031)277 3191 y Fg(K)5 b Fw(nf)p Fg(p)p Fw(g)p Fg(;K)570 3172 y Fo(L)632 3191 y Fg(K)q(;)p Fn(\()p Fg(S;V)14 b Fn(\))p Fg(;)p Fn(\()p Fg(U)o(;W)c Fn(\))1132 3172 y Fs(=)1228 2959 y Fh(8)1228 3040 y(>)1228 3068 y(<)1228 3231 y(>)1228 3259 y(:)1309 3041 y Fs(0)1160 b Fj(if)32 b Fo(p)25 b Fl(2)g Fo(S)1309 3177 y(L)1371 3195 y Fg(K)5 b Fw(nf)p Fg(p)p Fw(g)p Fg(;)p Fn(\()p Fg(S;V)15 b Fn(\))p Fg(;)p Fn(\()p Fg(U)o(;W)10 b Fn(\))2514 3177 y Fj(if)32 b Fo(p)25 b Fl(2)g Fs(\()p Fo(V)c Fs([)p Fo(S)5 b Fs(])20 b Fl([)g Fo(W)13 b Fs(\))20 b Fl(n)h Fo(S)1309 3312 y(L)1371 3331 y Fg(K)5 b Fw(nf)p Fg(p)p Fw(g)p Fg(;)p Fn(\()p Fg(S;V)15 b Fn(\))p Fg(;)p Fn(\()p Fg(U)o(;W)10 b Fn(\))2012 3312 y Fl(\016)21 b Fo(\031)2130 3331 y Fg(H)5 b Fw(nf)p Fg(p)p Fw(g)p Fg(;H)2514 3312 y Fj(if)32 b Fo(p)25 b Fl(2)g Fo(K)i Fl(n)20 b Fs(\()p Fo(V)h Fs([)p Fo(S)5 b Fs(])21 b Fl([)f Fo(W)13 b Fs(\))3500 3172 y(\(31\))118 3501 y Fj(and)34 b(the)f(sum)741 3604 y Fh(X)716 3801 y Fg(W)10 b Fw(2F)912 3690 y Fo(L)974 3709 y Fg(K)q(;)p Fn(\()p Fg(S;V)k Fn(\))p Fg(;)p Fn(\()p Fg(U)o(;W)c Fn(\))1469 3690 y Fl(\016)21 b Fo(\031)1587 3704 y Fg(H)1654 3690 y Fs(\()p Fo(\036)p Fs(\))26 b(=)f Fo(L)1962 3709 y Fg(K)q(;)p Fn(\()p Fg(S;V)15 b Fn(\))p Fg(;)p Fn(\(1)p Fg(;)p Fw(;)p Fn(\))2405 3690 y Fl(\016)21 b Fo(\031)2523 3709 y Fg(K)5 b Fw([)p Fg(V)15 b Fn([)p Fg(S)t Fn(])2780 3690 y Fs(\()p Fo(U)31 b(?)21 b(\036)p Fs(\))118 3967 y Fj(is)33 b(indep)-5 b(endent)34 b(of)f(the)g(de)-5 b(c)g(omp)g(osition)36 b Fo(U)f Fs(=)1754 3899 y Fh(P)1850 3994 y Fg(W)10 b Fw(2F)2050 3967 y Fo(U)2112 3981 y Fg(W)2225 3967 y Fj(of)33 b Fo(U)i Fl(2)25 b Fo(H)2591 3982 y Fg(\022)2629 3967 y Fj(.)118 4142 y(Pr)-5 b(o)g(of.)47 b Fs(F)-8 b(or)36 b(the)f(\014rst)f(part,)i(w)m(e)g(pro)s(ceed)e(as)i(for)e(Lemma)h(4.2)h (in)e([23)q(].)55 b(F)-8 b(or)36 b(the)f(second)g(part,)h(w)m(e)118 4255 y(comm)m(ute)c(sum)d(and)h(in)m(tegral,)g(obtaining)347 4430 y Fh(X)322 4627 y Fg(W)10 b Fw(2F)518 4517 y Fo(L)580 4535 y Fg(K)q(;)p Fn(\()p Fg(S;V)k Fn(\))p Fg(;)p Fn(\()p Fg(U)o(;W)c Fn(\))1075 4517 y Fl(\016)21 b Fo(\031)1193 4531 y Fg(H)1260 4517 y Fs(\()p Fo(\036)p Fs(\))26 b(=)f Fo(L)1568 4535 y Fg(K)q(;)p Fn(\()p Fg(S;V)14 b Fn(\))p Fg(;)p Fn(\(1)p Fg(;)p Fw(;)p Fn(\))2011 4517 y Fl(\016)2076 4361 y Fh( )2173 4430 y(X)2148 4627 y Fg(W)c Fw(2F)2344 4517 y Fo(\031)2396 4535 y Fg(K)5 b Fw([)p Fg(V)15 b Fn([)p Fg(S)t Fn(])p Fg(;K)5 b Fw([)p Fg(V)13 b Fn([)p Fg(S)t Fn(])p Fw([)p Fg(W)3051 4517 y Fs(\()p Fo(U)3148 4531 y Fg(W)3229 4517 y Fo(\036)3283 4531 y Fg(H)3350 4517 y Fs(\))3385 4361 y Fh(!)118 4788 y Fs(and)30 b(use)g(then)g(the)h (pro)5 b(jectivit)m(y)30 b(of)g Fo(\036)h Fs(and)f(the)g(de\014nition)e (of)j(the)f(mo)s(dule)f(pro)s(duct)g(on)h Fl(M)3388 4803 y Fg(\022)3428 4788 y Fs(.)p 3595 4788 4 62 v 3599 4730 55 4 v 3599 4788 V 3653 4788 4 62 v 259 5018 a(Giv)m(en)36 b Fo(T)47 b Fl(\025)34 b Fs(1,)k Fo(U)903 5032 y Fn(0)943 5018 y Fo(;)15 b(:)g(:)g(:)h(;)f(U)1206 5032 y Fg(T)10 b Fw(\000)p Fn(1)1387 5018 y Fl(2)34 b Fo(H)1558 5033 y Fg(\022)1596 5018 y Fs(,)k(w)m(e)e(w)m(an)m(t)h(to)g(construct)f(for) g(an)m(y)g Fo(K)41 b Fl(2)34 b(F)45 b Fs(an)36 b(op)s(erator)118 5145 y Fo(L)180 5097 y Fn(\()p Fg(T)10 b Fn(\))180 5178 y Fg(K)q(;)p Fn([)p Fg(U)329 5187 y Ff(0)362 5178 y Fg(;:::)n(;U)508 5189 y Fd(T)e Fr(\000)p Ff(1)635 5178 y Fn(])699 5145 y Fs(:)40 b Fl(M)873 5160 y Fg(#)945 5145 y Fl(!)25 b Fo(E)1128 5159 y Fg(K)1227 5145 y Fs(and)30 b(con)m(trol)g(its)g(norm.) 118 5268 y(W)-8 b(e)32 b(in)m(tro)s(duce)d(con\014gurations)h(and)f (asso)s(ciated)i(con\014gurational)f(op)s(erators)g(:)1844 5517 y(19)p eop %%Page: 20 20 20 19 bop 118 162 a Fk(De\014nition)35 b(A.2.)42 b Fj(A)32 b Fi(con\014guration)i Fj(on)f Fo(K)f Fl(2)24 b(F)42 b Fj(at)34 b(time)e Fo(T)38 b Fl(\025)25 b Fs(1)33 b Fj(is)g(the)g(choic)-5 b(e)33 b(of)g(:)254 349 y Fl(\017)46 b Fo(W)431 363 y Fg(T)10 b Fw(\000)p Fn(1)576 349 y Fo(;)15 b(:)g(:)g(:)i(W)824 363 y Fn(0)889 349 y Fl(2)24 b(F)9 b Fj(,)33 b(for)g(the)g(exp)-5 b(ansion)35 b(of)e(the)g(p)-5 b(erturb)g(ative)34 b(terms)f Fo(U)10 b Fj(,)254 536 y Fl(\017)46 b Fs(\()p Fo(S)436 550 y Fg(T)10 b Fw(\000)p Fn(1)582 536 y Fo(;)15 b(V)675 550 y Fg(T)10 b Fw(\000)p Fn(1)820 536 y Fs(\))p Fo(;)15 b(:)g(:)g(:)i(;)e Fs(\()p Fo(S)1148 550 y Fn(0)1188 536 y Fo(;)g(V)1281 550 y Fn(0)1321 536 y Fs(\))p Fj(,)33 b(br)-5 b(anching)34 b(p)-5 b(airs)34 b(for)f(the)g(exp)-5 b(ansion)34 b(of)f(the)g Fo(\014)3079 550 y Fg(p;V)3196 536 y Fj(,)254 723 y Fl(\017)46 b Fo(I)33 b Fl(2)25 b(F)41 b Fj(an)34 b Fi(initial)28 b(state)p Fj(,)118 910 y(such)36 b(that)h(if)e Fo(K)43 b Fj(is)36 b(exp)-5 b(ande)g(d)38 b(by)d Fo(K)1416 924 y Fg(T)1503 910 y Fs(=)30 b Fo(K)43 b Fj(and)36 b Fo(K)1980 924 y Fg(t)2041 910 y Fs(=)31 b Fo(K)2220 924 y Fg(t)p Fn(+1)2362 910 y Fl([)23 b Fo(V)2499 924 y Fg(t)2528 910 y Fs([)p Fo(S)2609 924 y Fg(t)2639 910 y Fs(])g Fl([)f Fo(W)2856 924 y Fg(t)2921 910 y Fj(for)36 b Fs(0)c Fl(\024)e Fo(t)h(<)g(T)13 b Fj(,)36 b(the)118 1022 y(fol)5 b(lowing)34 b(c)-5 b(onditions)34 b(ar)-5 b(e)34 b(satis\014e)-5 b(d:)1007 1225 y Fo(S)1063 1239 y Fg(t)1117 1225 y Fl(\032)25 b Fo(K)1290 1239 y Fg(t)p Fn(+1)1468 1225 y Fj(for)33 b Fs(0)26 b Fl(\024)f Fo(t)g(<)g(T)231 b Fj(and)127 b Fo(I)32 b Fl(\032)25 b Fo(K)2733 1239 y Fn(0)118 1428 y Fj(We)32 b(denote)i Fl(C)5 b Fs([)p Fo(K)q(;)15 b(T)e Fs(])34 b Fj(the)f(set)f(of)h(al)5 b(l)33 b(these)g(c)-5 b(on\014gur)g(ations.)259 1664 y Fs(T)d(o)31 b(eac)m(h)h(con\014guration)d Fo(C)j Fl(2)25 b(C)5 b Fs([)p Fo(K)q(;)15 b(T)e Fs(],)32 b(w)m(e)f(asso)s(ciate)g(a)g Fi(con\014gurational)f(op)s(erato)m(r)q Fs(:)635 1867 y Fo(L)697 1885 y Fg(K)q(;)p Fn([)p Fg(U)846 1894 y Ff(0)879 1885 y Fg(;:::)n(;U)1025 1896 y Fd(T)8 b Fr(\000)p Ff(1)1151 1885 y Fn(])1175 1867 y Fs([)p Fo(C)f Fs(])25 b(=)g Fo(L)1480 1829 y Fn(\()p Fg(T)10 b Fw(\000)p 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45 45 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 387.00 93.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 450 900 90 90 0 360 DrawEllipse gs col0 s gr % Polyline n 450 900 m 450 2205 l gs col0 s gr % Polyline n 900 900 m 900 2205 l gs col0 s gr % Polyline n 1350 900 m 1350 2205 l gs col0 s gr % Polyline n 1800 900 m 1800 2205 l gs col0 s gr % Polyline n 2250 900 m 2250 1800 l gs col0 s gr % Polyline n 2700 900 m 2700 1350 l gs col0 s gr % Polyline n 1350 900 m 2250 1350 l gs col0 s gr % Polyline n 1800 900 m 2250 1350 l gs col0 s gr % Polyline n 1350 1350 m 1800 1800 l gs col0 s gr % Polyline n 2700 1350 m 2250 1800 l gs col0 s gr % Polyline n 1800 1800 m 1350 2205 l gs col0 s gr % Polyline n 2250 1800 m 1350 2205 l gs col0 s gr % Polyline n 900 1800 m 1350 2205 l gs col0 s gr % Polyline n 5400 1440 m 4500 1035 l gs col0 s gr % Polyline n 5400 1440 m 5400 1035 l gs col0 s gr % Polyline n 4950 1710 m 4950 1485 l gs col0 s gr % 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3150 225 l gs col0 s gr % Polyline n 3600 585 m 3600 810 l 4005 585 l gs col0 s gr % Polyline n 4500 585 m 4500 810 l gs col0 s gr % Polyline n 5445 585 m 5850 990 l 5850 585 l gs col0 s gr % Polyline n 4950 585 m 4950 990 l 5400 1440 l gs col0 s gr % Polyline n 5400 1890 m 5850 1440 l 5850 990 l gs col0 s gr % Polyline n 5715 315 m 5715 450 l 5985 450 l 5985 315 l gs col0 s gr % Polyline n 3915 1215 m 3915 1350 l 4635 1350 l 4635 1215 l gs col0 s gr % Polyline n 4815 1665 m 4815 1800 l 5085 1800 l 5085 1665 l gs col0 s gr % Polyline n 3465 2205 m 3465 2340 l 5085 2340 l 5085 2205 l gs col0 s gr % Polyline gs clippath 1815 2370 m 1815 2310 l 1663 2310 l 1783 2340 l 1663 2370 l cp 435 2310 m 435 2370 l 587 2370 l 467 2340 l 587 2310 l cp eoclip n 450 2340 m 1800 2340 l gs col0 s gr gr % arrowhead n 587 2310 m 467 2340 l 587 2370 l col0 s % arrowhead n 1663 2370 m 1783 2340 l 1663 2310 l col0 s % Polyline gs clippath 2265 345 m 2265 285 l 2113 285 l 2233 315 l 2113 345 l cp 1335 285 m 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180.00 scf sf 4950 1935 m gs 1 -1 sc (e) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 180.00 scf sf 3600 2295 m gs 1 -1 sc (e) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 180.00 scf sf 4050 2295 m gs 1 -1 sc (e) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 180.00 scf sf 4950 2295 m gs 1 -1 sc (e) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 180.00 scf sf 4050 1305 m gs 1 -1 sc (e) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 180.00 scf sf 4050 585 m gs 1 -1 sc (e) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 180.00 scf sf 5850 405 m gs 1 -1 sc (e) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 180.00 scf sf 5850 585 m gs 1 -1 sc (e) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 180.00 scf sf 3600 585 m gs 1 -1 sc (e) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 150.00 scf sf 4500 585 m gs 1 -1 sc (i) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 150.00 scf sf 4950 585 m gs 1 -1 sc (i) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 150.00 scf sf 5400 585 m gs 1 -1 sc (i) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 180.00 scf sf 5400 1035 m gs 1 -1 sc (e) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 180.00 scf sf 0 540 m gs 1 -1 sc (0) col0 sh gr /Times-Roman ff 180.00 scf sf 0 990 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 180.00 scf sf 0 1440 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 180.00 scf sf 0 1890 m gs 1 -1 sc (3) col0 sh gr /Times-Roman ff 180.00 scf sf 0 2295 m gs 1 -1 sc (4) col0 sh gr /Times-Roman ff 180.00 scf sf 1800 270 m gs 1 -1 sc (I) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 180.00 scf sf 1080 2520 m gs 1 -1 sc (K) col0 sh gr $F2psEnd rs end %%EndDocument @endspecial 118 1588 a Fs(Figure)30 b(1:)41 b(An)30 b(example)g(of)h(con\014guration)e(\(left\))i(with)e(its)h(asso)s (ciated)h(tree)g(structure)f(\(righ)m(t\).)118 1701 y(Eac)m(h)h(circle) f(represen)m(ts)g(the)g(basis)f(of)i(a)f(branc)m(hing)f(pair,)h(eac)m (h)h(rectangle)g(represen)m(ts)f(the)h(p)s(ertur-)118 1814 y(bativ)m(e)38 b(term)g Fo(U)685 1828 y Fg(t)714 1814 y Fs(.)63 b(T)-8 b(o)38 b(eac)m(h)g(of)g(these)g(inner)e(p)s(oin)m (ts)g(and)h(to)i(the)f(b)s(ottom)f(p)s(oin)m(ts)g(are)h(asso)s(ciated) 118 1927 y(indep)s(enden)m(t)28 b(trees.)259 2276 y(Our)c(trees)h(are)g (exactly)g(the)g(same)g(as)g(in)e([23)q(].)39 b(W)-8 b(e)26 b(can)f(then)f(use)h(its)f(b)s(ounds)e(for)i(the)h(w)m(eigh)m (ts)g(of)118 2389 y(trees)j(under)e(the)i(condition)e(\(TR\))h(of)h ([23)q(].)40 b(W)-8 b(e)29 b(don't)e(write)g(this)f(condition)g(here,)i (but)f(just)g(notice)118 2502 y(that)k(for)f Fo(\032)g Fs(and)g Fo(\025)g Fs(giv)m(en,)g(there)h(exists)f Fo(\022)1580 2516 y Fn(0)1619 2502 y Fs(\()p Fo(\032;)15 b(\025)p Fs(\))26 b Fl(2)f Fs(\(0)p Fo(;)15 b Fs(1)p Fo(=)p Fs(3\))33 b(suc)m(h)d(that)h(for)f(all)f Fo(\022)e(<)e(\022)3140 2516 y Fn(0)3210 2502 y Fs(w)m(e)30 b(can)h(\014nd)118 2615 y Fo(\015)f(<)25 b(\021)339 2582 y Fw(\000)p Fn(1)464 2615 y Fs(\()p Fo(\021)34 b Fs(is)29 b(the)i(gap)f(of)g(the)h(simple)d (site)i(op)s(erator\))g(and)g Fo(\024)g Fs(suc)m(h)g(that)h(an)m(y)f Fo(F)39 b Fl(2)25 b Fo(C)7 b(M)j Fs([)p Fo(\032;)15 b(\025;)g(\022)s(;) g(\024)p Fs(])118 2728 y(satis\014es)35 b(\(TR\))g(with)f(this)g Fo(\015)5 b Fs(.)55 b(W)-8 b(e)36 b(write)f(b)s(elo)m(w)f(the)h (results)f(of)h(Lemmas)h(4.20)g(and)f(4.21)h(of)g([23)q(],)118 2841 y(whic)m(h)29 b(giv)m(e)i(these)g(b)s(ounds:)118 2997 y Fk(Prop)s(osition)40 b(A.3.)k Fj(The)36 b(size)g(of)g(a)g(tr)-5 b(e)g(e)36 b(is)g(the)g(sum)g(of)g(the)g(length)g(of)g(its)g(chains)h (adde)-5 b(d)38 b(to)e(the)118 3110 y(numb)-5 b(er)33 b(of)g(its)g(br)-5 b(anchings.)43 b(De\014ne:)1362 3286 y Fo(u)1414 3248 y Fg(T)1414 3308 y(p)1469 3286 y Fs(\()p Fo(s)p Fs(\))26 b(=)1779 3199 y Fh(X)1704 3401 y Fg(y)r Fw(2Y)6 b Fn([)p Fg(p;T)k Fn(])2001 3286 y Fl(k)p Fo(y)s Fl(k)p Fo(s)2182 3248 y Fe(size)26 b Fn(\()p Fg(y)r Fn(\))118 3564 y Fj(If)38 b(c)-5 b(ondition)41 b(\(TR\))e(is)f(satis\014e)-5 b(d)40 b(with)f Fo(\015)i Fl(2)36 b Fs(\(1)p Fo(;)15 b(\021)1878 3531 y Fw(\000)p Fn(1)1974 3564 y Fs(\))p Fj(,)40 b(then)f(ther)-5 b(e)39 b(exists)g Fo(#)d Fl(2)f Fs(\()p Fo(\022)s(;)15 b Fs(1\))39 b Fj(and)h Fo(T)3434 3578 y Fn(0)3509 3564 y Fl(\025)c Fs(1)118 3677 y Fj(such)d(that:)1394 3844 y Fo(u)1446 3806 y Fg(T)1446 3866 y(p)1501 3844 y Fs(\()p Fo(\015)5 b Fs(\))26 b Fl(\024)f Fo(\022)1791 3806 y Fw(\000)p Fn(1)3500 3844 y Fs(\(37\))1394 3995 y Fo(u)1446 3958 y Fg(T)1446 4018 y(p)1501 3995 y Fs(\()p Fo(\015)5 b Fs(\))26 b Fl(\024)f Fo(#)1799 3958 y Fw(\000)p Fn(1)2018 3995 y Fj(if)32 b Fo(T)38 b Fl(\025)25 b Fo(T)2346 4009 y Fn(0)118 4233 y Fp(A.4)112 b(Global)37 b(estimates)118 4404 y Fs(W)-8 b(e)32 b(deduce)e(from)g(Prop)s(ositions)e(A.2)j(and)f (A.3)h(ab)s(o)m(v)m(e)h(that:)118 4561 y Fk(Prop)s(osition)k(A.4.)688 4722 y Fh(X)597 4924 y Fg(C)5 b Fw(2C)t Fn([)p Fg(K)q(;T)10 b Fn(])925 4732 y Fh(\015)925 4786 y(\015)976 4809 y Fo(L)1038 4827 y Fg(K)q(;)p Fn([)p Fg(U)1187 4836 y Ff(0)1219 4827 y Fg(;:::)o(;U)1366 4838 y Fd(T)e Fr(\000)p Ff(1)1492 4827 y Fn(])1516 4809 y Fs([)p Fo(C)f Fs(])1638 4732 y Fh(\015)1638 4786 y(\015)1688 4850 y Fw(M)1772 4862 y Fd(#)1814 4850 y Fw(!)p Fg(E)1937 4861 y Fd(K)2024 4809 y Fl(\024)25 b Fo(\022)2166 4771 y Fw(\000j)p Fg(K)5 b Fw(j)2343 4695 y Fg(T)10 b Fw(\000)p Fn(1)2356 4722 y Fh(Y)2356 4917 y Fg(t)p Fn(=0)2499 4809 y Fl(j)p Fo(U)2586 4823 y Fg(t)2616 4809 y Fl(j)2641 4824 y Fg(\022)3500 4809 y Fs(\(38\))2024 5151 y Fl(\024)25 b Fo(#)2174 5113 y Fw(\000j)p Fg(K)5 b Fw(j)2352 5037 y Fg(T)10 b Fw(\000)p Fn(1)2364 5064 y Fh(Y)2364 5259 y Fg(t)p Fn(=0)2508 5151 y Fl(j)p Fo(U)2595 5165 y Fg(t)2625 5151 y Fl(j)2650 5166 y Fg(\022)2815 5151 y Fj(if)32 b Fo(T)38 b Fl(\025)25 b Fo(T)3143 5165 y Fn(0)1844 5517 y Fs(23)p eop %%Page: 24 24 24 23 bop 118 162 a Fj(Pr)-5 b(o)g(of.)47 b Fs(Because)32 b(of)e(the)h(injectivit)m(y)e(of)i(the)f(description)f(b)m(y)h(trees,)h (w)m(e)g(ha)m(v)m(e:)796 250 y Fh(X)705 452 y Fg(C)5 b Fw(2C)t Fn([)p Fg(K)q(;T)10 b Fn(])1033 259 y Fh(\015)1033 314 y(\015)1083 337 y Fo(L)1145 355 y Fg(K)q(;)p Fn([)p Fg(U)1294 364 y Ff(0)1327 355 y Fg(;:::)n(;U)1473 366 y Fd(T)e Fr(\000)p Ff(1)1600 355 y Fn(])1624 337 y Fs([)p Fo(C)f Fs(])1746 259 y Fh(\015)1746 314 y(\015)1796 377 y Fw(M)1880 389 y Fd(#)1922 377 y Fw(!)p Fg(E)2045 388 y Fd(K)1043 697 y Fl(\024)1230 611 y Fh(X)1139 812 y Fg(C)e Fw(2C)t Fn([)p Fg(K)q(;T)10 b Fn(])1482 611 y Fh(Y)1467 808 y Fg(p)p Fw(2)p Fg(K)1629 697 y Fl(k)p Fo(y)s Fs(\()p Fo(p;)15 b(T)e Fs(\))p Fl(k)2004 583 y Fg(T)d Fw(\000)p Fn(1)2018 611 y Fh(Y)2018 805 y Fg(t)p Fn(=0)2162 514 y Fh(2)2162 678 y(4)2222 697 y Fl(j)p Fo(U)2309 711 y Fg(t;W)2421 719 y Fd(t)2453 697 y Fl(j)2524 611 y Fh(Y)2493 808 y Fg(p)p Fw(2)p Fg(W)2643 816 y Fd(t)2686 697 y Fl(k)p Fo(y)s Fs(\()p Fo(p;)15 b(t)p Fs(\))p Fl(k)3013 514 y Fh(3)3013 678 y(5)1043 1039 y Fl(\024)1270 953 y Fh(X)1139 1150 y Fg(W)1206 1159 y Ff(0)1240 1150 y Fg(;:::)n(;W)1405 1161 y Fd(T)8 b Fr(\000)p Ff(1)1562 953 y Fh(Y)1547 1150 y Fg(p)p Fw(2)p Fg(K)1709 1039 y Fo(u)1761 1002 y Fg(T)1761 1062 y(p)1816 1039 y Fs(\(1\))1946 926 y Fg(T)i Fw(\000)p Fn(1)1960 953 y Fh(Y)1960 1147 y Fg(t)p Fn(=0)2103 1039 y Fl(j)p Fo(U)2190 1053 y Fg(t;W)2302 1061 y Fd(t)2334 1039 y Fl(j)2405 953 y Fh(Y)2374 1150 y Fg(p)p Fw(2)p Fg(W)2524 1158 y Fd(t)2567 1039 y Fo(u)2619 1002 y Fg(t)2619 1062 y(p)2659 1039 y Fs(\(1\))1043 1396 y Fl(\024)1154 1310 y Fh(Y)1139 1507 y Fg(p)p Fw(2)p Fg(K)1301 1396 y Fo(u)1353 1359 y Fg(T)1353 1419 y(p)1408 1396 y Fs(\(1\))1538 1283 y Fg(T)g Fw(\000)p Fn(1)1552 1310 y Fh(Y)1552 1504 y Fg(t)p Fn(=0)1695 1214 y Fh(0)1695 1377 y(@)1799 1310 y(X)1775 1507 y Fg(W)g Fw(2F)1971 1396 y Fl(j)p Fo(U)2058 1410 y Fg(t;W)2183 1396 y Fl(j)2245 1310 y Fh(Y)2223 1507 y Fg(p)p Fw(2)p Fg(W)2398 1396 y Fo(u)2450 1359 y Fg(t)2450 1419 y(p)2490 1396 y Fs(\(1\))2605 1214 y Fh(1)2605 1377 y(A)118 1667 y Fs(and)30 b(w)m(e)h(can)f (conclude)g(with)f(estimates)i(\(37\).)p 3595 1667 4 62 v 3599 1609 55 4 v 3599 1667 V 3653 1667 4 62 v 259 1885 a(These)f(b)s(ounds,)f(together)j(with)d(the)h(\014rst)g(part)g (of)g(Prop)s(osition)f(A.1)i(\(whic)m(h)e(assures)h(compat-)118 1998 y(ibilit)m(y)e(of)i(the)h(op)s(erators)f(constructed)h(for)f (di\013eren)m(t)g(subsets)g Fo(K)7 b Fs(\))30 b(mak)m(e)h(it)f(p)s (ossible)e(to)j(de\014ne:)938 2269 y Fo(L)1000 2221 y Fn(\()p Fg(T)10 b Fn(\))1000 2302 y([)p Fg(U)1068 2311 y Ff(0)1102 2302 y Fg(;:::)n(;U)1248 2313 y Fd(T)e Fr(\000)p Ff(1)1375 2302 y Fn(])1424 2269 y Fs(=)1520 2086 y Fh(0)1520 2249 y(@)1690 2182 y(X)1599 2383 y Fg(C)d Fw(2C)t Fn([)p Fg(K)q(;T)10 b Fn(])1927 2269 y Fo(L)1989 2287 y Fg(K)q(;)p Fn([)p Fg(U)2138 2296 y Ff(0)2171 2287 y Fg(;:::)n(;U)2317 2298 y Fd(T)e Fr(\000)p Ff(1)2444 2287 y Fn(])2468 2269 y Fs([)p Fo(C)f Fs(])2590 2086 y Fh(1)2590 2249 y(A)2669 2414 y Fg(K)e Fw(2F)118 2542 y Fs(as)29 b(an)g(op)s(erator)g(from)f Fl(M)1040 2557 y Fg(#)1115 2542 y Fs(to)i Fl(M)1334 2557 y Fg(\022)1373 2542 y Fs(,)f(or)g(to)g Fl(M)1755 2557 y Fg(#)1830 2542 y Fs(when)f Fo(T)38 b Fl(\025)25 b Fo(T)2306 2556 y Fn(0)2346 2542 y Fs(,)k(satisfying)e(the)i(announced)f(b)s(ound) 118 2655 y(\(25\).)118 2768 y(W)-8 b(e)40 b(see)f(also)g(that)g(this)f (op)s(erator)h(is)e(indep)s(enden)m(t)g(of)i(the)f(decomp)s(ositions)f Fo(U)2999 2782 y Fg(t)3068 2768 y Fs(=)3178 2700 y Fh(P)3274 2795 y Fg(W)10 b Fw(2F)3474 2768 y Fo(U)3536 2782 y Fg(t;W)118 2894 y Fs(writing,)29 b(with)675 2871 y(^)664 2894 y Fo(L)726 2861 y Fn(\()p Fg(t)p Fn(\))835 2894 y Fs(=)c Fo(L)993 2912 y Fg(K)1053 2923 y Fd(T)8 b Ff(+1)1178 2912 y Fg(;)p Fn(\()p Fg(S)1268 2920 y Fd(t)1296 2912 y Fg(;V)1357 2920 y Fd(t)1385 2912 y Fn(\))p Fg(;)p Fn(\(1)p Fg(;)p Fw(;)p Fn(\))1612 2894 y Fs(and)29 b Fo(K)1865 2908 y Fg(t)1925 2894 y Fs(the)i(corresp)s(onding)d(expansion:)216 3020 y Fh(0)216 3184 y(@)387 3116 y(X)296 3318 y Fg(C)5 b Fw(2C)t Fn([)p Fg(K)q(;T)10 b Fn(])624 3203 y Fo(L)686 3221 y Fg(K)q(;)p Fn([)p Fg(U)835 3230 y Ff(0)868 3221 y Fg(;:::)n(;U)1014 3232 y Fd(T)e Fr(\000)p Ff(1)1140 3221 y Fn(])1164 3203 y Fs([)p Fo(C)f Fs(])1286 3020 y Fh(1)1286 3184 y(A)1381 3203 y Fs(\()p Fo(\036)p Fs(\))351 3590 y(=)561 3504 y Fh(X)447 3705 y Fn(\()p Fg(S)517 3716 y Fd(T)h Fr(\000)p Ff(1)644 3705 y Fg(;V)705 3716 y Fd(T)741 3731 y Ff(1)780 3705 y Fn(\))833 3567 y Fs(^)822 3590 y Fo(L)884 3553 y Fn(\()p Fg(T)i Fw(\000)p Fn(1\))1100 3380 y Fh(2)1100 3540 y(6)1100 3599 y(4)1160 3408 y(0)1160 3571 y(@)1240 3590 y Fo(U)1302 3604 y Fg(T)g Fw(\000)p Fn(1)1467 3590 y Fo(?)1533 3408 y Fh(2)1533 3571 y(4)1708 3504 y(X)1594 3705 y Fn(\()p Fg(S)1664 3716 y Fd(T)e Fr(\000)p Ff(2)1791 3705 y Fg(;V)1852 3716 y Fd(T)1888 3731 y Ff(2)1927 3705 y Fn(\))1980 3567 y Fs(^)1969 3590 y Fo(L)2031 3553 y Fn(\()p Fg(T)i Fw(\000)p Fn(2\))2247 3590 y Fl(\001)15 b(\001)g(\001)2378 3567 y Fs(^)2368 3590 y Fo(L)2430 3553 y Fn(\(0\))2539 3517 y Fh(\002)2577 3590 y Fs(\()p Fo(U)2674 3604 y Fn(0)2734 3590 y Fo(?)21 b(\036)p Fs(\))2889 3617 y Fg(K)2949 3626 y 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b([23)q(],)g(b)s(ecause)f(in)f(this)g (case,)j(all)d(con\014gurational)h(op)s(erators)g(with)f(a)i Fo(W)2775 4417 y Fg(t)2830 4403 y Fl(6)p Fs(=)25 b Fl(;)31 b Fs(are)f(n)m(ull.)118 4516 y(All)21 b(other)h(prop)s(erties)f(of)h (these)g(op)s(erators)g(are)h(straigh)m(tforw)m(ard)f(adaptations)g(of) g(equiv)-5 b(alen)m(t)22 b(results)118 4629 y(in)29 b([23)q(].)118 4816 y Fj(A)-5 b(cknow)5 b(le)-5 b(dgements.)60 b Fs(The)36 b(author)g(thanks)g(G)m(\023)-43 b(erard)37 b(Ben)g(Arous,)h(his)d (Ph-D)h(advisor,)h(who)f(ha)m(v)m(e)118 4929 y(in)m(tro)s(duced)27 b(him)h(to)h(this)f(sub)5 b(ject)29 b(and)f(pro)m(vided)g(him)f (excellen)m(t)i(scien)m(ti\014c)f(supp)s(ort)g(and)g(material)118 5042 y(conditions)20 b(at)i(Lausanne.)38 b(He)22 b(thanks)f(also)h (Viviane)e(Baladi)h(for)g(fruitful)e(con)m(v)m(ersations)j(on)g(ergo)s (dic)118 5155 y(theory)-8 b(,)27 b(and)e(expresses)h(particular)d(ac)m (kno)m(wledgemen)m(ts)28 b(to)e(Hans-Henrik)f(Rugh)f(for)i(man)m(y)f (helpful)118 5268 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