Content-Type: multipart/mixed; boundary="-------------9905091039821" This is a multi-part message in MIME format. ---------------9905091039821 Content-Type: text/plain; name="99-156.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-156.keywords" decay of correlations, tower, transfer operator, projective metrics ---------------9905091039821 Content-Type: application/postscript; name="new2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="new2.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.55a Copyright 1986, 1994 Radical Eye Software %%Title: new2.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips new2 -o %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.05.09:1730 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N 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y(b)q(ounded)17 b(distortion)g(prop)q(ert)o(y:)k Fn(9)p Fo(C)c(>)d Fq(0)i(suc)o(h)g (that)h Fn(8)p Fo(x;)8 b(y)16 b Fq(in)g(the)g(same)f(\003)1444 1352 y Fv(`)1460 1345 y Fo(;)444 1468 y Fn(8)p Fo(k)f Fn(2)g Fm(N)p Fo(;)633 1396 y Fg(\014)633 1426 y(\014)633 1456 y(\014)633 1486 y(\014)655 1434 y Fn(L)689 1416 y Fv(k)689 1447 y Ft(\010)717 1434 y Fc(1)p Fq(\()p Fo(x)p Fq(\))p 655 1457 157 2 v 656 1503 a Fn(L)690 1486 y Fv(k)690 1516 y Ft(\010)718 1503 y Fc(1)p Fq(\()p Fo(y)r Fq(\))828 1468 y Fn(\000)c Fq(1)901 1396 y Fg(\014)901 1426 y(\014)901 1456 y(\014)901 1486 y(\014)932 1468 y Fn(\024)k Fo(C)t(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))p Fo(:)0 1590 y Fq(Let)18 b(\010)g(satisfy)f(the)g(\\w)o(eak)h(con)o(tribution)f(of)h(in\014nit)o (y)e(condition".)26 b(F)l(or)18 b(an)o(y)f Fo(k)h Fn(2)e Fm(N)p Fq(,)0 1650 y Fo(n)e Fn(2)g Fm(N)j Fq(and)g Fo(x)d Fn(2)g Fq(\003)360 1657 y Fv(n)383 1650 y Fq(,)i(the)g(b)q(ounded)h (distortion)f(prop)q(ert)o(y)h(giv)o(es:)156 1767 y Fn(L)190 1746 y Fv(k)190 1779 y Ft(\010)218 1767 y Fc(1)p Fq(\()p Fo(x)p Fq(\))42 b Fn(\024)f Fo(C)541 1733 y Fq(1)p 478 1755 150 2 v 478 1801 a Fo(m)521 1787 y Fk(0)533 1801 y Fq(\(\003)586 1808 y Fv(n)609 1801 y Fq(\))641 1699 y Fg(Z)632 1841 y Ft(\003)656 1845 y Fh(n)699 1767 y Fn(L)733 1746 y Fv(k)733 1779 y Ft(\010)761 1767 y Fc(1)p Fo(dm)858 1746 y Fk(0)355 1934 y Fn(\024)g Fo(C)541 1901 y Fq(1)p 478 1923 V 478 1968 a Fo(m)521 1954 y Fk(0)533 1968 y Fq(\(\003)586 1975 y Fv(n)609 1968 y Fq(\))641 1866 y Fg(Z)640 2008 y Ft(\001)699 1934 y Fn(L)733 1914 y Fv(k)733 1947 y Ft(\010)761 1934 y Fc(1)p Fo(dm)858 1914 y Fk(0)355 2096 y Fq(=)h Fo(C)541 2062 y Fq(1)p 478 2084 V 478 2130 a Fo(m)521 2115 y Fk(0)533 2130 y Fq(\(\003)586 2137 y Fv(n)609 2130 y Fq(\))649 2096 y(b)q(ecause)17 b Fo(m)873 2075 y Fk(0)900 2096 y Fq(is)f(a)h(conformal)e(probabilit)o (y)o Fo(:)0 2222 y Fq(So,)h(it)e(exists)h Fo(M)314 2204 y Fk(0)340 2222 y Fo(>)f Fq(0)h(suc)o(h)g(that)h(sup)719 2234 y Fv(x)p Fk(2)p Ft(\003)787 2238 y Fh(n)818 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350 y Ff(Sketch)20 b(of)e(pr)n(o)n(of)f(of)h(the)n(or)n(em)f (1.2)h Fq(:)47 b(F)l(rom)16 b(lemm)o(a)f(1.3,)i(there)f(exists)h Fo(M)k(>)15 b Fq(0)i(suc)o(h)47 410 y(that)i Fn(kL)214 392 y Fv(k)214 422 y Ft(\010)241 410 y Fc(1)p Fn(k)295 417 y Fk(1)350 410 y Fn(\024)d Fo(M)23 b Fq(for)c(all)e Fo(k)i Fn(2)e Fm(N)p Fq(.)27 b(It)18 b(is)g(then)f(easy)i(to)f(see)f (that)i Fn(L)1402 417 y Ft(\010)1448 410 y Fq(acts)f(on)g Fo(L)p Fq(.)47 470 y(Indeed,)d(let)h Fo(f)21 b Fq(b)q(elong)c(to)g Fo(L)f Fq(w)o(e)g(ha)o(v)o(e)f(the)h(follo)o(wing)g(inequalit)o(y:)k Fn(8)p Fo(k)14 b Fn(2)g Fm(N)96 589 y Fn(jL)144 568 y Fv(k)144 601 y Ft(\010)172 589 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g(L)362 568 y Fv(k)362 601 y Ft(\010)390 589 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))p Fn(j)14 b(\024)f Fo(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))16 b(Ct)g([)p Fn(k)p Fo(f)5 b Fn(k)881 596 y Fk(1)929 589 y Fq(+)11 b Fo(L)p Fq(\()p Fo(f)5 b Fq(\)])17 b Fn(8)p Fo(x;)22 b(y)c Fq(in)e(the)g(same)f (\015o)q(or)q Fo(:)47 708 y Fq(Ascoli)21 b(theorem)f(on)i(separable)f (spaces)h(imply)d(that)j(there)f(exists)g(a)g(sequence)g Fo(n)1645 715 y Fv(k)47 768 y Fq(suc)o(h)d(that)g Fo(Q)305 775 y Fv(n)326 781 y Fh(k)363 768 y Fq(=)434 748 y Ft(1)p 423 756 41 2 v 423 785 a Fv(n)444 791 y Fh(k)476 730 y Fg(P)529 743 y Fv(n)550 749 y Fh(k)569 743 y Fk(\000)p Ft(1)529 782 y Fv(i)p Ft(=0)625 768 y Fn(L)659 750 y Fv(i)659 780 y Ft(\010)687 768 y Fc(1)f Fq(con)o(v)o(erges,)g (uniformly)f(on)i(compact)f(subsets)h(of)47 834 y(\001)d(and)h(in)f Fo(L)286 816 y Ft(1)306 834 y Fq(\()p Fo(m)368 816 y Fk(0)379 834 y Fq(\),)g(to)g(some)f(limit)f Fo(h)750 841 y Ft(\010)791 834 y Fn(2)h Fo(L)p Fq(.)21 b(Using)15 b(the)g(fact)h(that)f Fo(m)1368 816 y Fk(0)1395 834 y Fq(is)g(conformal,)47 895 y(w)o(e)g(get)h(that)f(this)h(limit)c(is)j (non)h(zero)f(\(indeed,)g Fo(m)987 876 y Fk(0)998 895 y Fq(\()p Fo(h)1045 902 y Ft(\010)1073 895 y Fq(\))e(=)h(1)i(b)o(y)f (Leb)q(esgue)h(theorem\))47 955 y(and)h(satis\014es)g Fn(L)356 962 y Ft(\010)384 955 y Fo(h)412 962 y Ft(\010)453 955 y Fq(=)d Fo(h)533 962 y Ft(\010)561 955 y Fq(.)21 b(Let)16 b Fo(\026)e Fq(=)g Fo(h)806 962 y Ft(\010)834 955 y Fo(m)877 937 y Fk(0)888 955 y Fq(,)i(this)g(is)g(a)h Fo(F)7 b Fq(-in)o(v)m(arian)o(t)15 b(probabilit)o(y)l(.)47 1015 y(If)k(\(\001)p Fo(;)8 b(F)f Fq(\))18 b(is)h(irreducible)e(then)i ([A,D,U)n(])f(\(theorem)g(2.5\))h(implies)d(that)k Fo(\026)f Fq(=)g Fo(hm)f Fq(is)47 1075 y(ergo)q(dic)f(and)f Fo(h)e(>)g Fq(0.)47 1135 y(Using)19 b([A,D,U)n(])g(theorem)e(3.2)j(\(see)f(also)g (Y)l(oung's)h(argumen)o(ts)e([Y2)o(]\),)h(ap)q(erio)q(dicit)o(y)47 1196 y(of)g(the)f(system)g(implies)e(that)j(\()p Fo(F)q(;)8 b(\026)p Fq(\))18 b(is)h(exact.)27 b(So)19 b Fo(\026)g Fq(is)g(mixing)d(this)j(implies)d(that)47 1256 y Fn(L)81 1238 y Fv(k)81 1268 y Ft(\010)109 1256 y Fo(f)30 b Fq(con)o(v)o(erges) 23 b(to)i Fo(h)485 1263 y Ft(\010)512 1256 y Fo(m)555 1238 y Fk(0)567 1256 y Fq(\()p Fo(f)5 b Fq(\))24 b(uniformly)f(on)h (eac)o(h)g(compact)f(subset)i(of)f(\001)g(\(an)o(y)47 1316 y(accum)o(ulation)15 b(p)q(oin)o(t)h(of)h Fn(L)564 1298 y Fv(k)564 1328 y Ft(\010)591 1316 y Fo(f)22 b Fq(m)o(ust)15 b(b)q(e)h Fo(h)852 1323 y Ft(\010)880 1316 y Fo(m)923 1298 y Fk(0)934 1316 y Fq(\()p Fo(f)5 b Fq(\))17 b(b)q(ecause)f(of)h (mixing\).)193 b Fb(\004)47 1439 y Fq(Recall)15 b(that)i Fn(L)334 1446 y Ft(0)370 1439 y Fq(is)f(the)g(transfer)h(op)q(erator)g (asso)q(ciated)g(to)g(the)f(p)q(oten)o(tial)g Fn(\000)8 b Fq(log)h Fo(J)c(F)i Fq(.)47 1567 y Fl(THEOREM)18 b(1.4)24 b Fp(The)d(op)q(erator)g Fn(L)794 1574 y Ft(0)834 1567 y Fp(admits)e(a)i(non)g(zero)f(p)q(ositiv)o(e)f(\014xed)h(p)q(oin)o(t) 47 1627 y Fo(h)75 1634 y Ft(0)109 1627 y Fn(2)14 b Fo(L)p Fp(.)47 1687 y(If)h Fq(\(\001)p Fo(;)8 b(F)f Fq(\))14 b Fp(is)h(irreducible)f(then)h Fo(h)676 1694 y Ft(0)709 1687 y Fo(>)f Fq(0)p Fp(,)i(it)f(is)g(unique)f(up)i(to)g(a)f(m)o (ultiplicati)o(v)o(e)d(factor)47 1747 y(and)17 b(the)f(in)o(v)m(arian)o (t)g(measure)f Fo(\026)f Fq(=)g Fo(h)744 1754 y Ft(0)763 1747 y Fo(m)806 1754 y Ft(0)842 1747 y Fp(is)i(ergo)q(dic.)47 1808 y(If)g Fq(\(\001)p Fo(;)8 b(F)f Fq(\))14 b Fp(is)i(ap)q(erio)q (dic)g(then)g Fo(\026)g Fp(is)g(mixing.)j(Moreo)o(v)o(er)14 b Fn(L)1142 1789 y Fv(n)1142 1820 y Ft(0)1166 1808 y Fo(f)21 b Fp(con)o(v)o(erges)15 b(to)i Fo(h)1517 1815 y Ft(0)1536 1808 y Fo(m)1579 1815 y Ft(0)1599 1808 y Fq(\()p Fo(f)5 b Fq(\))47 1868 y Fp(uniformly)14 b(on)j(eac)o(h)f (compact)f(subset)i(of)f Fq(\001)g Fp(and)h(in)f Fo(L)1089 1850 y Ft(1)1109 1868 y Fq(\()p Fo(m)1171 1875 y Ft(0)1190 1868 y Fq(\))g Fp(for)h(an)o(y)f Fo(f)j Fn(2)14 b Fo(C)1517 1875 y Fv(u)1540 1868 y Fq(\(\001\))p Fp(.)47 1928 y(F)l(or)h(an)o(y)f (increasing)g(sequence)f Fq(\()p Fo(v)692 1935 y Fv(n)715 1928 y Fq(\))734 1935 y Fv(n)p Fk(2)p Fj(N)819 1928 y Fp(suc)o(h)h(that)1031 1891 y Fg(P)1083 1942 y Fv(`)p Fk(2)p Fj(N)1156 1928 y Fo(m)1199 1935 y Ft(0)1218 1928 y Fq(\(\003)1271 1935 y Fv(`)1287 1928 y Fq(\))7 b Fn(\001)g Fo(v)1358 1935 y Fv(`)1388 1928 y Fo(<)14 b Fn(1)p Fp(,)g(for)g(an)o(y) 47 1988 y Fo(f)25 b Fn(2)20 b Fo(L)g Fp(and)g Fo(g)i Fp(suc)o(h)d(that)h Fo(g)r(v)h Fn(2)f Fo(L)723 1970 y Ft(1)743 1988 y Fq(\()p Fo(m)805 1995 y Ft(0)824 1988 y Fq(\))g Fp(\()p Fo(g)r(v)h Fp(is)e(de\014ned)h(b)o(y)f Fo(g)r(v)r Fq(\()p Fo(x)p Fq(\))f(=)i Fo(g)r Fq(\()p Fo(x)p Fq(\))p Fo(v)1555 1995 y Fv(`)1590 1988 y Fp(if)f Fo(x)47 2048 y Fp(b)q(elong)e(to)g Fq(\003)297 2055 y Fv(`)313 2048 y Fp(\),)f(there)f(exists)h Fq(0)e Fo(<)g(\015)j(<)d Fq(1)i Fp(suc)o(h)g(that)422 2167 y Fo(c)443 2174 y Fv(n)466 2167 y Fq(\()p Fo(f)s(;)8 b(g)r Fq(\))14 b Fn(\024)f Fp(Ct)25 b Fq(max)o([\()p Fo(v)871 2174 y Fv(n)893 2167 y Fq(\))912 2147 y Fk(\000)p Ft(1)959 2167 y Fo(;)g(\015)1026 2147 y Fv(n)1049 2167 y Fq(])p Fn(k)p Fo(f)5 b Fn(k)16 b(k)p Fo(g)r(v)r Fn(k)1259 2174 y Ft(1)1278 2167 y Fo(:)47 2286 y Fp(W)l(e)f(also)g(estimate)d(the)j(sp)q(eed)f(of)h(con)o(v)o (ergence)e(of)i Fn(L)1044 2268 y Fv(n)1044 2298 y Ft(0)1068 2286 y Fo(f)20 b Fp(to)15 b Fo(h)1198 2293 y Ft(0)1218 2286 y Fo(m)1261 2293 y Ft(0)1280 2286 y Fq(\()p Fo(f)5 b Fq(\))15 b Fp(in)f(the)h(uniform)47 2346 y(norm)h(on)g(eac)o(h)g (\015o)q(or.)47 2406 y(In)g(particular,)47 2466 y Fn(\017)21 b Fp(if)g(there)f(exist)h(constan)o(ts)g Fq(0)i Fo(<)f(e)743 2473 y Ft(1)783 2466 y Fp(and)g Fq(0)h Fo(<)f(\022)h(<)f Fq(1)f Fp(suc)o(h)g(that)h(for)f(all)g Fo(`)h Fn(2)h Fm(N)p Fp(,)47 2527 y Fo(m)p Fq(\(\003)143 2534 y Fv(`)159 2527 y Fq(\))14 b Fn(\024)g Fo(e)268 2534 y Ft(1)287 2527 y Fo(\022)311 2509 y Fv(`)342 2527 y Fp(then)h(there)f(exists)g Fq(0)g Fo(<)g(\015)i(<)e Fq(1)h Fp(suc)o(h)g(that)g(for)f(an)o(y)h Fo(\022)1330 2509 y Fk(0)1355 2527 y Fo(<)f(\022)q Fp(,)h(for)g Fo(g)h Fp(suc)o(h)47 2592 y(that)158 2570 y Fv(g)p 158 2580 18 2 v 158 2609 a(\022)195 2592 y Fn(2)e Fo(L)275 2574 y Ft(1)295 2592 y Fq(\()p Fo(m)p Fq(\))i Fp(\(where)562 2570 y Fv(g)p 556 2580 29 2 v 556 2609 a(\022)573 2600 y Fd(0)590 2592 y Fq(\()p Fo(x)p Fq(\))d(=)726 2568 y Fv(g)q Ft(\()p Fv(x)p Ft(\))p 726 2580 66 2 v 738 2609 a Fv(\022)755 2600 y Fd(0)p Fh(`)813 2592 y Fp(for)j Fo(x)e Fn(2)g Fq(\003)1010 2599 y Fv(`)1026 2592 y Fp(\),)500 2729 y Fo(c)521 2736 y Fv(n)545 2729 y Fq(\()p Fo(f)s(;)8 b(g)r Fq(\))13 b Fn(\024)30 b Fp(Ct)16 b Fq(\()p Fo(\022)852 2709 y Fk(0)864 2729 y Fq(\))g Fo(\015)927 2709 y Fv(n)967 2729 y Fn(k)p Fo(f)5 b Fn(k)1071 2672 y Fg(\015)1071 2702 y(\015)1071 2732 y(\015)1109 2696 y Fo(g)p 1103 2718 36 2 v 1103 2763 a(\022)1127 2749 y Fk(0)1144 2672 y Fg(\015)1144 2702 y(\015)1144 2732 y(\015)1172 2765 y Ft(1)1200 2729 y Fo(;)p eop %%Page: 8 8 8 7 bop 0 133 a Fq(8)487 b Fp(V)o(\023)-23 b(eronique)16 b(Maume-Desc)o(hamps)0 286 y Fn(\017)h Fp(if)g(there)g(exist)f(constan) o(ts)i Fq(0)e Fo(<)g(e)664 293 y Ft(1)683 286 y Fp(,)h Fq(0)f Fo(<)g(\022)h(<)e Fq(1)j Fp(and)g Fq(0)e Fo(<)g(\014)i(<)d Fq(1)j Fp(suc)o(h)f(that)h(for)f(all)0 354 y Fo(`)d Fn(2)g Fm(N)p Fp(,)j Fo(m)p Fq(\(\003)243 361 y Fv(`)259 354 y Fq(\))d Fn(\024)f Fo(e)367 361 y Ft(1)387 354 y Fo(\022)411 336 y Fv(`)425 325 y Fh(\014)464 354 y Fp(then)j(for)h(an)o(y)f Fo(\014)773 336 y Fk(0)798 354 y Fo(<)e(\014)s Fp(,)h(let)g Fo(v)1004 361 y Fv(`)1034 354 y Fq(=)f Fo(\022)1110 336 y Fk(\000)p Fv(`)1151 325 y Fh(\014)1169 314 y Fd(0)1186 354 y Fp(,)461 477 y Fo(c)482 484 y Fv(n)505 477 y Fq(\()p Fo(f)s(;)8 b(g)r Fq(\))14 b Fn(\024)f Fp(Ct)k Fq(\()p Fo(\014)804 456 y Fk(0)815 477 y Fq(\))p Fo(\022)858 456 y Fv(n)879 444 y Fh(\014)897 434 y Fd(0)929 477 y Fn(k)p Fo(f)5 b Fn(k)17 b(k)p Fo(g)r(v)r Fn(k)1126 484 y Ft(1)1145 477 y Fo(;)0 646 y Fn(\017)12 b Fp(if)f(there)g(exist)g (constan)o(ts)i Fq(0)h Fo(<)g(e)633 653 y Ft(1)652 646 y Fp(,)e Fo(\014)k(>)e Fq(1)e Fp(suc)o(h)g(that)g(for)g(all)g Fo(`)i Fn(2)g Fm(N)p Fp(,)f Fo(m)p Fq(\(\003)1390 653 y Fv(`)1406 646 y Fq(\))h Fn(\024)g Fo(e)1515 653 y Ft(1)1534 646 y Fo(`)1554 628 y Fk(\000)p Fv(\014)1605 646 y Fp(,)0 728 y(then)i(for)h(an)o(y)f Fo(\015)h(>)c Fq(1)p Fp(,)k(let)e Fo(v)520 735 y Fv(`)550 728 y Fq(=)602 673 y 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y(that)f(the)f(deca)o(y)f(of)h(correlations)g(is)g(of)h(the)f (form)f Fo(c)986 1194 y Fv(n)1009 1187 y Fq(\()p Fo(f)s(;)8 b(g)r Fq(\))17 b Fn(\024)g Fq(\()p Fo(v)1237 1194 y Fv(n)1260 1187 y Fq(\))1279 1169 y Fk(\000)p Ft(1)1326 1187 y Fo(C)t Fq(\()p Fo(f)s(;)8 b(g)r Fq(\))17 b Fp(where)0 1247 y(the)h(dep)q (endence)g(of)g Fo(C)t Fq(\()p Fo(f)s(;)8 b(g)r Fq(\))18 b Fp(on)h Fo(g)h Fp(is)f(giv)o(en)e(b)o(y)h Fn(k)p Fo(g)r(v)r Fn(k)1041 1254 y Ft(1)1060 1247 y Fp(.)28 b(This)18 b(kind)g(of)h(prop) q(ert)o(y)f(is)0 1307 y(v)o(ery)c(useful)i(to)g(study)f(asymptotic)g (la)o(ws)g(of)h(successiv)o(e)e(return)i(times)d(\(see)j([C,G,S)o(],)0 1368 y([G,S)o(],)g([Sau],)f([P]\))h(and)h(w)o(as)f(not)h(obtained)g (using)f(coupling)g(metho)q(ds)g(as)h(in)f([Y2)o(].)0 1461 y Fq(Existence)h(and)i(mixing)d(prop)q(erties)i(of)g(an)h(in)o(v)m (arian)o(t)f(measure)e(absolutely)i(con)o(tin-)0 1521 y(uous)i(with)f(resp)q(ect)g(to)h Fo(m)507 1528 y Ft(0)545 1521 y Fq(follo)o(ws)f(form)g(theorem)e(1.2.)31 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y(it)i(to)h(estimate)e(the)h(correlation)g(deca)o (y)g(for)h(random)f(dynamical)e(systems.)30 b(Then,)0 2379 y(this)19 b(strategy)g(had)h(b)q(een)f(used)h(b)o(y)e(man)o(y)g (authors.)31 b(Let)19 b(us)h(men)o(tion)d(C.)i(Liv)o(erani)0 2440 y([Li1])i(and)i(M.)e(Viana)h([V)o(])f(for)h(Anoso)o(v)g(and)h (Axiom)d(A)h(di\013eomorphisms.)36 b(They)0 2500 y(used)18 b(Birkho\013)f(cones)h(to)g(obtain)g(exp)q(onen)o(tial)f(deca)o(y)g(of) h(correlations.)26 b(In)17 b([K,M,S)n(])0 2560 y(the)k(Birkho\013)h (cones)f(tec)o(hniques)f(w)o(ere)h(used)g(in)h(a)g(di\013eren)o(t)e(w)o (a)o(y)h(to)h(obtain)g(sub-)0 2620 y(exp)q(onen)o(tial)16 b(deca)o(y)g(of)h(correlations.)22 b(The)16 b(w)o(a)o(y)h(w)o(e)f(use)g (cone)h(tec)o(hniques)e(to)i(pro)o(v)o(e)0 2680 y(theorem)j(1.4)i (follo)o(ws)f(some)g(ideas)h(of)f(P)l(.)h(F)l(errero)f(and)h(B.)f(Sc)o (hmitt)e(\([F,S2)o(]\))i(and)0 2741 y([Ma1,)16 b(Ma2].)p eop %%Page: 9 9 9 8 bop 558 133 a Fp(Mixing)15 b(prop)q(erties)h(on)h(to)o(w)o(ers)510 b Fq(9)47 347 y(Let)18 b(us)h(recall)e(de\014nitions)g(and)i(prop)q (erties)f(of)g(cones)g(and)h(pro)s(jectiv)o(e)d(metrics)g(\(see)47 407 y([Li1])g(for)h(a)f(more)f(complete)f(presen)o(tation\).)47 527 y(Let)j Fo(B)h Fq(b)q(e)f(a)f(v)o(ector)g(space)g(and)h Fo(C)g Fn(\032)d Fo(B)19 b Fq(a)d(cone)g(with)h(the)f(follo)o(wing)f (prop)q(erties.)120 587 y Fn(\017)24 b Fo(C)c Fq(is)c(con)o(v)o(ex,)120 683 y Fn(\017)24 b Fo(C)15 b Fn(\\)c(\000)p Fo(C)17 b Fq(=)d Fn(f)p Fq(0)p Fn(g)p Fq(,)120 779 y Fn(\017)24 b Fq(if)11 b Fo(\013)240 786 y Fv(n)274 779 y Fq(is)g(a)h(sequence)e (of)h(real)g(n)o(um)o(b)q(ers)e(suc)o(h)i(that)g Fo(\013)1118 786 y Fv(n)1156 779 y Fn(!)i Fo(\013)f Fq(and)f Fo(x)p Fn(\000)p Fo(\013)1449 786 y Fv(n)1473 779 y Fo(y)16 b Fn(2)e Fo(C)g Fn(8)p Fo(n)169 840 y Fq(then)i Fo(x)11 b Fn(\000)g Fo(\013y)16 b Fn(2)e Fo(C)t Fq(.)20 b(This)d(prop)q(ert)o (y)f(is)g(called)f(\\in)o(tegral)h(closure".)47 924 y(F)l(or)h(suc)o(h) g(a)h(cone,)e(the)h(pseudo-metric)e Fo(\022)836 931 y Fv(C)882 924 y Fq(on)j Fo(C)i Fq(is)d(de\014ned)g(in)g(the)f(follo)o (wing)h(w)o(a)o(y)l(.)47 984 y(Let)g Fo(x;)8 b(y)15 b Fn(2)f Fo(C)t Fq(,)393 1074 y Fo(\026)p Fq(\()p Fo(x;)8 b(y)r Fq(\))14 b(=)f(inf)t Fn(f)p Fo(\014)j(>)d Fq(0)k(suc)o(h)f(that)h Fo(\014)s(x)10 b Fn(\000)h Fo(y)k Fn(2)f Fo(C)t Fn(g)p Fo(;)386 1164 y(\025)p Fq(\()p Fo(x;)8 b(y)r Fq(\))13 b(=)h(sup)p Fn(f)p Fo(\013)g(>)g Fq(0)j(suc)o(h)f(that)g Fo(y)d Fn(\000)e Fo(\013x)j Fn(2)g Fo(C)t Fn(g)p Fo(;)47 1243 y Fq(with)k(the)f(con)o(v)o(en)o(tion:)23 b Fo(\026)p Fq(\()p Fo(x;)8 b(y)r Fq(\))15 b(=)h Fn(1)i Fq(and)g Fo(\025)p Fq(\()p Fo(x;)8 b(y)r Fq(\))16 b(=)g(0)i(if)f(the)g(corresp)q (onding)i(sets)47 1303 y(are)d(empt)o(y)l(.)j(Let)d Fo(\022)402 1310 y Fv(C)432 1303 y Fq(\()p Fo(x;)8 b(y)r Fq(\))13 b(=)g(log)687 1281 y Fv(\026)p 687 1292 22 2 v 687 1320 a(\025)713 1303 y Fq(.)21 b Fo(\022)771 1310 y Fv(C)817 1303 y Fq(is)15 b(called)g(pseudo-metric)g(b)q(ecause)h(it)g(is)g(not) 47 1363 y(necessarily)f(\014nite.)20 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Fq(\))p Fo(:)47 1878 y Fl(THEOREM)18 b(2.1)24 b Fq([Bi1)o(])16 b Fp(F)l(or)h(an)o(y)f Fo(f)s(;)8 b(g)18 b Fp(in)d Fo(C)t Fp(,)h(w)o(e)g(ha)o(v)o(e:)490 1997 y Fo(\022)513 2004 y Fv(C)540 1995 y Fd(0)553 1997 y Fq(\()p Fo(P)7 b(f)s(;)h(P)f(g)r Fq(\))14 b Fn(\024)g Fq(tanh)914 1926 y Fg(\022)955 1963 y Fq(\001)p 955 1985 41 2 v 963 2031 a(4)1001 1926 y Fg(\023)1046 1997 y Fo(\022)1069 2004 y Fv(C)1098 1997 y Fq(\()p Fo(f)s(;)8 b(g)r Fq(\))p Fo(:)47 2115 y Fq(This)13 b(theorem)f(implies)e(that)k Fo(P)21 b Fq(:)13 b Fo(C)18 b Fn(!)13 b Fo(C)841 2097 y Fk(0)866 2115 y Fq(is)g(alw)o(a)o(ys)f(a)i(con)o(traction)f(\(in)g (wide)f(sense\))47 2176 y(for)17 b(the)f(pro)s(jectiv)o(e)e(metrics.)19 b(If)d(\001)d Fo(<)h Fn(1)i Fq(then)g(it)g(is)g(a)h(strict)e(con)o (traction.)47 2236 y(The)h(follo)o(wing)f(result)g(relies)f(the)i (metric)d Fo(\022)869 2243 y Fv(C)914 2236 y Fq(to)j(certain)e(norms)h (on)h Fo(B)s Fq(.)21 b(A)15 b(norm)g Fn(k)g(k)47 2296 y Fq(on)g Fo(B)j Fq(is)c(a)h 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Fp(.)j(F)l(or)16 b(an)o(y)g Fo(f)22 b Fp(and)17 b Fo(g)h Fp(in)e Fo(C)k Fp(suc)o(h)c(that)h Fo(\032)p Fq(\()p Fo(f)5 b Fq(\))14 b(=)g Fo(\032)p Fq(\()p Fo(g)r Fq(\))g Fn(6)p Fq(=)g(0)i Fp(w)o(e)g(ha)o(v)o(e:)463 2728 y Fn(k)p Fo(f)g Fn(\000)11 b Fo(g)r Fn(k)j(\024)f Fq(\()p Fo(e)736 2708 y Fv(\022)q Ft(\()p Fv(f)r(;g)q Ft(\))840 2728 y Fn(\000)e Fq(1\))d(min)o(\()p Fn(k)p Fo(f)d Fn(k)p Fo(;)j Fn(k)p Fo(g)r Fn(k)p Fq(\))p Fo(:)p eop %%Page: 10 10 10 9 bop 0 133 a Fq(10)488 b Fp(V)o(\023)-23 b(eronique)15 b(Maume-Desc)o(hamps)381 286 y Fr(3)82 b(Pro)r(of)27 b(of)g(theorem)g(1.4)0 440 y Fq(W)l(e)16 b(will)f(no)o(w)i(pro)o(v)o(e) e(theorem)g(1.4.)21 b(Let)c(us)f(b)q(egin)h(with)f(the)g(follo)o(wing)g (remark.)0 560 y Fl(Remark)g(3.1)24 b Fp(Let)16 b Fo(v)f Fq(=)f(\()p Fo(v)529 567 y Fv(`)545 560 y Fq(\))564 567 y Fv(`)p Fk(2)p Fj(N)643 560 y Fp(b)q(e)i(an)g(increasing)f(sequence)f (of)i(p)q(ositiv)o(e)e(n)o(um)o(b)q(ers)0 620 y(suc)o(h)i(that:)591 633 y Fg(X)597 739 y Fv(`)p Fk(2)p Fj(N)672 680 y Fo(m)715 687 y Ft(0)734 680 y Fq(\(\003)787 687 y Fv(`)803 680 y Fq(\))11 b Fn(\001)g Fo(v)882 687 y Fv(`)912 680 y Fo(<)j Fn(1)p Fo(:)0 811 y Fp(W)l(e)21 b(will)g(consider)g(suc)o(h)h(a) g Fo(v)h Fp(as)f(a)g(function)g(on)g Fq(\001)p Fp(:)32 b Fo(v)r Fq(\()p Fo(x)p Fq(\))22 b(=)h Fo(v)1258 818 y Fv(`)1296 811 y Fp(if)e Fo(x)g Fp(b)q(elongs)i(to)0 871 y Fq(\003)34 878 y Fv(`)50 871 y Fp(.)28 b(Let)18 b Fn(L)h Fp(b)q(e)f(the)g(transfer)h(op)q(erator)g(asso)q(ciated)g(to)g (the)f(c)o(hange)g(of)h(p)q(oten)o(tial)f Fo(v)r Fp(:)0 931 y Fn(L)p Fo(f)24 b Fq(=)17 b Fo(v)163 913 y Fk(\000)p Ft(1)210 931 y Fn(L)244 938 y Ft(0)264 931 y Fq(\()p Fo(f)5 b(v)r Fq(\))p Fp(.)28 b(The)19 b(measure)e Fo(m)h Fq(=)f Fo(v)r(m)879 938 y Ft(0)917 931 y Fp(is)h(conformal)g(for)h Fn(L)g Fp(and)g(it)f(follo)o(ws)0 991 y(easily)12 b(from)g(\(1.1\))i (and)f(lemma)d(1.3)k(that)f Fq(sup)854 1003 y Fv(k)q Fk(2)p Fj(N)931 991 y Fn(kL)990 973 y Fv(k)1012 991 y Fc(1)p Fn(k)1066 998 y Fk(1)1117 991 y Fo(<)h Fn(1)p Fp(.)20 b(Moreo)o(v)o(er,)12 b Fo(h)i Fq(=)1576 971 y Fv(h)1596 976 y Fi(0)p 1576 980 38 2 v 1586 1009 a Fv(v)0 1052 y Fp(is)i(an)h(eigen)o(v)o(ector)d(for)j Fn(L)p Fp(.)k(Theorem)15 b(1.2)i(implies)c(that)k(for)g(an)o(y)f Fo(f)j Fn(2)14 b Fo(C)1362 1059 y Fv(u)1384 1052 y Fq(\(\001\))p Fp(,)302 1153 y Fn(L)336 1132 y Fv(k)357 1153 y Fo(f)20 b Fn(\000)-9 b(!)14 b Fo(hm)p Fq(\()p Fo(f)5 b Fq(\))16 b Fp(uniformly)e(on)j(eac)o(h)f(compact)f(set)p Fo(:)0 1254 y Fq(Throughout)23 b(this)e(section,)h(w)o(e)f(\014x)g(a)h (strictly)e(increasing)h(sequence)f Fo(v)k Fq(=)f(\()p Fo(v)1520 1261 y Fv(`)1536 1254 y Fq(\))1555 1261 y Fv(`)p Fk(2)p Fj(N)0 1314 y Fq(suc)o(h)17 b(that)h Fo(m)p Fq(\(\001\))d Fo(<)g Fn(1)i Fq(and)h(the)f(sequence)f(\()903 1292 y Fv(v)920 1298 y Fh(`)p 883 1302 71 2 v 883 1331 a Fv(v)900 1337 y Fh(`)p Fi(+1)959 1314 y Fq(\))978 1321 y Fv(`)p Fk(2)p Fj(N)1059 1314 y Fq(is)h(also)h(increasing.)24 b(W)l(e)17 b(will)0 1382 y(assume)i(that)g Fo(m)h Fq(and)g Fo(h)f Fq(are)g(normalized)f(\()p Fo(m)p Fq(\(\001\))g(=)h(1)h(and)g Fo(m)p Fq(\()p Fo(h)p Fq(\))f(=)g(1\).)31 b(W)l(e)19 b(will)0 1442 y(w)o(ork)d(with)g(the)g(transfer)h(op)q(erator)g Fn(L)p Fq(.)0 1502 y(As)g(usual)h(in)f(these)g(topics,)g(the)g(main)f (to)q(ol)i(is)g(a)f(\\Lasota-Y)l(ork)o(e")j(t)o(yp)q(e)c(inequalit)o(y) l(.)0 1562 y(Before)f(pro)o(ving)i(it,)e(let)g(us)i(study)f(a)h(bit)f (the)g(distance)g(on)h(\001.)184 1705 y Fe(3.1)66 b(Some)22 b(prop)r(erties)i(of)d(the)i(distance)g(on)f Fz(\001)0 1799 y Fq(Let)15 b(us)g(consider)f(the)h(follo)o(wing)f(function)h(on)g (\001:)20 b Fo(\032)p Fq(\()p Fo(x)p Fq(\))14 b(=)g Fo(\014)j Fq(if)d Fo(x)g Fq(b)q(elongs)i(to)f(\003)1506 1806 y Ft(0)1540 1799 y Fq(and)0 1906 y Fo(\032)p Fq(\()p Fo(x)p Fq(\))f(=)f(1)k(otherwise.)k(F)l(or)16 b Fo(k)33 b Fn(2)14 b Fm(N)p Fq(,)i(w)o(e)g(will)f(denote)i(b)o(y)e Fo(\032)1104 1888 y Ft(\()p Fv(k)q Ft(\))1153 1906 y Fq(\()p Fo(x)p Fq(\))f(=)1307 1844 y Fv(k)1284 1859 y Fg(Y)1288 1964 y Fv(i)p Ft(=0)1356 1906 y Fo(\032)p Fq(\()p Fo(F)1439 1886 y Fv(i)1453 1906 y Fo(x)p Fq(\).)21 b(The)0 2016 y(relation)16 b(b)q(et)o(w)o(een)f Fo(\032)392 1998 y Ft(\()p Fv(k)q Ft(\))457 2016 y Fq(and)i(the)f(distance)g(on)h(\001)f (is)g(the)g(follo)o(wing:)0 2076 y(if)f Fo(x)h Fq(and)g Fo(y)i Fq(are)d(in)h(\003)395 2083 y Ft(0)415 2076 y Fq(,)f(their)g(preimages)g(b)o(y)g Fo(F)894 2058 y Fv(k)930 2076 y Fq(are)h(paired.)21 b(If)15 b Fo(x)1255 2058 y Fk(0)1283 2076 y Fq(is)g(a)h(preimage)f(of)0 2136 y Fo(x)p Fq(,)g(w)o(e)g(will)g(denote)h(b)o(y)f Fo(y)470 2118 y Fk(0)497 2136 y Fq(the)g(preimage)f(of)i Fo(y)i Fq(whic)o(h)d(b)q (elongs)h(to)g Fo(C)1292 2143 y Fv(k)1313 2136 y Fq(\()p Fo(x)1360 2118 y Fk(0)1372 2136 y Fq(\).)21 b(W)l(e)15 b(ha)o(v)o(e:)526 2237 y Fo(d)p Fq(\()p Fo(x)598 2217 y Fk(0)610 2237 y Fo(;)8 b(y)658 2217 y Fk(0)669 2237 y Fq(\))14 b(=)g Fo(\032)779 2217 y Ft(\()p Fv(k)q Ft(\))827 2237 y Fq(\()p Fo(x)874 2217 y Fk(0)886 2237 y Fq(\))d Fn(\001)g Fo(d)p Fq(\()p Fo(x;)d(y)r Fq(\))p 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y(ro\013)s('s)g(theorem)e(applied)h(to)g(log)10 b Fo(\032)p Fq(.)925 b Fb(\004)p eop %%Page: 11 11 11 10 bop 558 133 a Fp(Mixing)15 b(prop)q(erties)h(on)h(to)o(w)o(ers) 510 b Fq(11)47 286 y Fl(Remark)17 b(3.3)24 b Fp(A)19 b(\014ner)g(analysis)h(on)g Fo(\032)809 268 y Ft(\()p Fv(k)q Ft(\))877 286 y Fp(should)g(allo)o(w)g(one)f(to)h(a)o(v)o(oid)f (the)g(use)h(of)47 347 y(Birkho\013)s('s)c(and)h(Egoro\013)s('s)i (theorems)c(and)h(so)h(giv)o(e)f(a)g(constructiv)o(e)f(b)q(ound)j(for)e Fo(k)1602 354 y Ft(0)1622 347 y Fp(.)319 487 y Fe(3.2)66 b(A)22 b(Lasota-Y)-6 b(ork)n(e)23 b(t)n(yp)r(e)g(inequalit)n(y)47 570 y Fq(W)l(e)16 b(are)h(no)o(w)f(in)g(p)q(osition)h(to)g(pro)o(v)o(e) e(a)i(Lasota-Y)l(ork)o(e)g(t)o(yp)q(e)f(inequalit)o(y)l(.)47 691 y Fl(LEMMA)j(3.4)24 b Fp(F)l(or)18 b(an)o(y)g Fo(")f(>)g Fq(0)p Fp(,)h(there)f(exists)h Fo(C)i(>)d Fq(0)p Fp(,)h Fo(N)23 b Fn(2)17 b Fm(N)i Fp(and)f Fo(k)1431 698 y Ft(0)1468 691 y Fn(2)f Fm(N)i Fp(suc)o(h)47 751 y(that)e(for)f(all)g Fo(k)g(>)e(k)413 758 y Ft(0)433 751 y Fp(,)i(for)g(all)g Fo(f)j Fn(2)14 b Fo(L)p Fp(,)i(w)o(e)g(ha)o(v)o(e:)107 830 y(1.)24 b(for)17 b Fo(`)d Fn(\025)g Fo(k)r Fp(,)h Fo(x)h Fp(and)h Fo(y)h Fp(in)e Fq(\003)659 837 y Fv(`)513 879 y Fg(\014)513 909 y(\014)530 922 y Fn(L)564 901 y Fv(k)585 922 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))12 b Fn(\000)e(L)775 901 y Fv(k)797 922 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))890 879 y Fg(\014)890 909 y(\014)920 922 y Fn(\024)978 888 y Fo(v)1002 895 y Fv(`)p Fk(\000)p Fv(k)p 978 910 87 2 v 1001 956 a Fo(v)1025 963 y Fv(`)1070 922 y Fo(d)p Fq(\()p Fo(x;)j(y)r Fq(\))p Fo(L)p Fq(\()p Fo(f)d Fq(\))p Fo(;)107 1042 y Fp(2.)24 b(for)17 b Fo(N)i Fn(\024)13 b Fo(`)i(<)e(k)r Fp(,)j Fo(x)g Fp(and)h Fo(y)h Fp(in)e Fq(\003)769 1049 y Fv(`)421 1083 y Fg(\014)421 1113 y(\014)437 1125 y Fn(L)471 1104 y Fv(k)493 1125 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g(L)683 1104 y Fv(k)705 1125 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))798 1083 y Fg(\014)798 1113 y(\014)828 1125 y Fn(\024)14 b Fo(")i(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))g([)o Fo(L)p Fq(\()p Fo(f)d Fq(\))11 b(+)g(sup)e Fn(j)p Fo(f)c Fn(j)p Fq(])j Fo(;)107 1225 y Fp(3.)24 b(for)17 b Fo(`)d(<)g(N)5 b Fp(,)16 b Fo(x)g Fp(and)h Fo(y)g Fp(in)f Fq(\003)675 1232 y Fv(`)405 1266 y Fg(\014)405 1296 y(\014)422 1308 y Fn(L)456 1288 y Fv(k)478 1308 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g(L)668 1288 y Fv(k)689 1308 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))782 1266 y Fg(\014)782 1296 y(\014)813 1308 y Fn(\024)13 b Fo(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))g([)p Fo("L)p Fq(\()p Fo(f)d Fq(\))11 b(+)g Fo(C)h Fq(sup)c Fn(j)p Fo(f)d Fn(j)p Fq(])j Fo(:)47 1408 y Ff(Pr)n(o)n(of)h Fq(:)45 b(Let)17 b Fo(f)22 b Fq(b)q(elong)17 b(to)g Fo(L)p Fq(.)23 b(Item)14 b(1)p Fo(:)j Fq(directly)e(follo)o(ws)i(from)e(the)i(de\014nition)f(of) h Fn(L)47 1468 y Fq(and)g Fo(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\).)47 1529 y(Let)15 b Fo(x)f Fq(and)g Fo(y)i Fq(b)q(e)f(in)f(\003) 461 1536 y Fv(`)477 1529 y Fq(,)g Fo(`)h(<)e(k)r Fq(,)i(let)e(us)i (denote)f(b)o(y)g Fo(x)1025 1510 y Ft(0)1058 1529 y Fn(2)g Fq(\003)1139 1536 y Ft(0)1173 1529 y Fq(\(resp.)20 b Fo(y)1339 1510 y Ft(0)1359 1529 y Fq(\))14 b(the)g(preimage)47 1589 y(of)j Fo(x)f Fq(\(resp.)21 b Fo(y)r Fq(\))16 b(b)o(y)f Fo(F)455 1571 y Fv(k)q Fk(\000)p Fv(`)518 1589 y Fq(.)21 b(W)l(e)16 b(ha)o(v)o(e:)327 1641 y Fg(\014)327 1671 y(\014)344 1683 y Fn(L)378 1663 y Fv(k)400 1683 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)f(L)589 1663 y Fv(k)611 1683 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))704 1641 y Fg(\014)704 1671 y(\014)734 1683 y Fq(=)791 1650 y Fo(v)815 1657 y Ft(0)p 791 1672 44 2 v 793 1718 a Fo(v)817 1725 y Fv(`)848 1641 y Fg(\014)848 1671 y(\014)864 1683 y Fn(L)898 1663 y Fv(k)q Fk(\000)p Fv(`)962 1683 y Fo(f)g Fq(\()p Fo(x)1038 1663 y Ft(0)1058 1683 y Fq(\))11 b Fn(\000)g(L)1172 1663 y Fv(k)q Fk(\000)p Fv(`)1235 1683 y Fo(f)5 b Fq(\()p Fo(y)1309 1663 y Ft(0)1329 1683 y Fq(\))1348 1641 y Fg(\014)1348 1671 y(\014)1373 1683 y Fo(:)179 b Fq(\(3.4\))47 1789 y(Let)18 b(us)f(consider)g Fo(x)g Fq(and)h Fo(y)h Fq(in)d(\003)665 1796 y Ft(0)685 1789 y Fq(,)h(their)g(paired)g(preimages)e(will)i(b)q (e)g(denoted)g(b)o(y)g Fo(x)1655 1771 y Fk(0)47 1849 y Fq(and)g Fo(y)168 1831 y Fk(0)179 1849 y Fq(.)367 1957 y Fn(jL)415 1936 y Fv(k)436 1957 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g(L)626 1936 y Fv(k)648 1957 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))p Fn(j)13 b Fq(=)835 1923 y(1)p 825 1945 V 825 1991 a Fo(v)849 1998 y Ft(0)873 1957 y Fn(j)922 1909 y Fg(X)895 2019 y Fv(F)922 2010 y Fh(k)942 2019 y Fv(x)962 2010 y Fd(0)973 2019 y Ft(=)p Fv(x)1066 1923 y Fo(v)r Fq(\()p Fo(x)1139 1905 y Fk(0)1150 1923 y Fq(\))p 1034 1945 169 2 v 1034 1991 a Fo(J)5 b(F)1105 1976 y Fv(k)1125 1991 y Fq(\()p Fo(x)1172 1976 y Fk(0)1183 1991 y Fq(\))1207 1957 y Fo(f)g Fq(\()p Fo(x)1283 1936 y Fk(0)1295 1957 y Fq(\))11 b Fn(\000)1413 1923 y Fo(v)r Fq(\()p Fo(y)1484 1905 y Fk(0)1494 1923 y Fq(\))p 1380 1945 167 2 v 1380 1991 a Fo(J)5 b(F)1451 1976 y Fv(k)1471 1991 y Fq(\()p Fo(y)1516 1976 y Fk(0)1527 1991 y Fq(\))1551 1957 y Fo(f)g Fq(\()p Fo(y)1625 1936 y Fk(0)1637 1957 y Fq(\))p Fn(j)408 2112 y(\024)503 2078 y Fq(1)p 494 2100 44 2 v 494 2146 a Fo(v)518 2153 y Ft(0)576 2064 y Fg(X)550 2174 y Fv(F)577 2165 y Fh(k)596 2174 y Fv(x)616 2165 y Fd(0)627 2174 y Ft(=)p Fv(x)721 2078 y Fo(v)r Fq(\()p Fo(x)794 2060 y Fk(0)805 2078 y Fq(\))p 688 2100 169 2 v 688 2146 a Fo(J)g(F)759 2131 y Fv(k)779 2146 y Fq(\()p Fo(x)826 2131 y Fk(0)838 2146 y Fq(\))862 2112 y Fn(j)p Fo(f)g Fq(\()p Fo(x)952 2091 y Fk(0)963 2112 y Fq(\))11 b Fn(\000)g Fo(f)5 b Fq(\()p Fo(y)1117 2091 y Fk(0)1129 2112 y Fq(\))p Fn(j)409 2271 y Fq(+)519 2237 y(1)p 510 2259 44 2 v 510 2305 a Fo(v)534 2312 y Ft(0)593 2223 y Fg(X)566 2333 y Fv(F)593 2324 y Fh(k)613 2333 y Fv(x)633 2324 y Fd(0)644 2333 y Ft(=)p Fv(x)737 2237 y Fo(v)r Fq(\()p Fo(x)810 2219 y Fk(0)821 2237 y Fq(\))p 704 2259 169 2 v 704 2305 a Fo(J)g(F)775 2290 y Fv(k)796 2305 y Fq(\()p Fo(x)843 2290 y Fk(0)854 2305 y Fq(\))878 2271 y Fn(j)p Fo(f)g Fq(\()p Fo(y)966 2250 y Fk(0)977 2271 y Fq(\))p Fn(j)1035 2198 y Fg(\014)1035 2228 y(\014)1035 2258 y(\014)1035 2288 y(\014)1056 2237 y Fo(J)g(F)1127 2219 y Fv(k)1147 2237 y Fq(\()p Fo(x)1194 2219 y Fk(0)1206 2237 y Fq(\))p 1056 2259 V 1057 2305 a Fo(J)g(F)1128 2290 y Fv(k)1148 2305 y Fq(\()p Fo(y)1193 2290 y Fk(0)1205 2305 y Fq(\))1241 2271 y Fn(\000)11 b Fq(1)1315 2198 y Fg(\014)1315 2228 y(\014)1315 2258 y(\014)1315 2288 y(\014)408 2407 y Fn(\024)42 b Fo(L)p Fq(\()p Fo(f)5 b Fq(\))p Fo(d)p Fq(\()p Fo(x;)j(y)r Fq(\))762 2360 y Fg(X)736 2469 y Fv(F)763 2460 y Fh(k)782 2469 y Fv(x)802 2460 y Fd(0)813 2469 y Ft(=)p Fv(x)869 2407 y Fo(v)r Fq(\()p Fo(x)942 2386 y Fk(0)952 2407 y Fq(\))p Fo(J)d(F)1042 2386 y Fv(k)1063 2407 y Fq(\()p Fo(x)1110 2386 y Fk(0)1121 2407 y Fq(\))1140 2386 y Fk(\000)p Ft(1)1187 2407 y Fo(\032)1212 2386 y Ft(\()p Fv(k)q Ft(\))1261 2407 y Fq(\()p Fo(x)1308 2386 y Fk(0)1319 2407 y Fq(\))736 2493 y Fg(|)p 758 2493 257 6 v 257 w({z)p 1059 2493 V 257 w(})1014 2538 y Ft(\(1\))409 2621 y Fq(+)42 b Fo(C)19 b(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))g(sup)h Fn(j)p Fo(f)c Fn(j)863 2573 y Fg(X)837 2683 y Fv(F)864 2674 y Fh(k)883 2683 y Fv(x)903 2674 y Fd(0)914 2683 y Ft(=)p Fv(x)970 2621 y Fo(v)r Fq(\()p Fo(x)1043 2600 y Fk(0)1054 2621 y Fq(\))p Fo(J)g(F)1144 2600 y Fv(k)1164 2621 y Fq(\()p Fo(x)1211 2600 y Fk(0)1222 2621 y Fq(\))1241 2600 y Fk(\000)p Ft(1)837 2707 y Fg(|)p 859 2707 181 6 v 181 w({z)p 1084 2707 V 181 w(})1040 2752 y Ft(\(2\))1297 2621 y Fo(:)p eop %%Page: 12 12 12 11 bop 0 133 a Fq(12)488 b Fp(V)o(\023)-23 b(eronique)15 b(Maume-Desc)o(hamps)0 286 y Fq(The)h(term)f(\(2\))h(is)g(b)q(ounded)i (ab)q(o)o(v)o(e)e(b)o(y)g Fn(kL)809 268 y Fv(k)830 286 y Fc(1)p Fn(k)884 293 y Fk(1)935 286 y Fn(\024)e Fo(M)5 b Fq(.)0 347 y(Let)20 b(us)g(turn)g(our)g(atten)o(tion)g(to)g(term)e (\(1\).)32 b(Let)20 b Fo(\021)i(>)d Fq(0)i(to)f(b)q(e)g(\014xed)f (later,)h(let)f(\012)1598 354 y Fv(\021)0 407 y Fq(b)q(e)f(giv)o(en)e (b)o(y)h(lemma)d(3.2)k(and)g(assume)f Fo(k)h Fn(\025)e Fo(k)892 414 y Ft(0)912 407 y Fq(\()p Fo(\021)r Fq(\).)24 b(Recall)16 b(\(1.2\):)24 b(for)18 b Fo(z)h Fq(suc)o(h)f(that)0 467 y Fo(F)39 449 y Fv(k)60 467 y Fq(\()p Fo(z)r Fq(\))i Fn(2)g Fq(\003)230 474 y Ft(0)250 467 y Fq(,)g Fo(J)5 b(F)355 449 y Fv(k)375 467 y Fq(\()p Fo(z)r Fq(\))438 449 y Fk(\000)p Ft(1)505 467 y Fn(\024)20 b Fo(C)t(m)646 474 y Ft(0)665 467 y Fq(\()p Fo(C)719 474 y Fv(k)740 467 y Fq(\()p Fo(z)r Fq(\)\))g(and)h(remark)d(that)i Fo(\032)1245 449 y Ft(\()p Fv(k)q Ft(\))1314 467 y Fq(is)g(constan)o(t) g(on)0 527 y(eac)o(h)c Fo(k)r Fq(-cylinder.)k(Using)c(lemm)o(a)e(3.2)i (w)o(e)g(get:)48 628 y(\(1\))42 b Fn(\024)288 581 y Fg(X)268 688 y Fh(F)291 681 y(k)310 688 y(x)327 681 y Fd(0)339 688 y Fi(=)p Fh(x)237 720 y(\032)253 711 y Fi(\()p Fh(k)q Fi(\))296 720 y(\()p Fh(x)325 713 y Fd(0)337 720 y Fi(\))p Fd(\024)p Fh(\015)391 713 y(k)423 628 y Fo(v)r Fq(\()p Fo(x)496 607 y Fk(0)507 628 y Fq(\))p Fo(J)5 b(F)597 607 y Fv(k)617 628 y Fq(\()p Fo(x)664 607 y Fk(0)675 628 y Fq(\))694 607 y Fk(\000)p Ft(1)741 628 y Fo(\032)766 607 y Ft(\()p Fv(k)q Ft(\))815 628 y Fq(\()p Fo(x)862 607 y Fk(0)874 628 y Fq(\))11 b(+)987 581 y Fg(X)967 688 y Fh(F)990 681 y(k)1009 688 y(x)1026 681 y Fd(0)1038 688 y Fi(=)p Fh(x)958 716 y(x)975 709 y Fd(0)986 716 y(2)p Fi(\(\012)1041 720 y Fh(\021)1060 716 y Fi(\))1072 709 y Fh(c)1101 628 y Fo(v)r Fq(\()p Fo(x)1174 607 y Fk(0)1185 628 y Fq(\))p Fo(J)5 b(F)1275 607 y Fv(k)1295 628 y Fq(\()p Fo(x)1342 607 y Fk(0)1353 628 y Fq(\))1372 607 y Fk(\000)p Ft(1)1419 628 y Fo(\032)1444 607 y Ft(\()p Fv(k)q Ft(\))1493 628 y Fq(\()p Fo(x)1540 607 y Fk(0)1552 628 y Fq(\))152 792 y Fn(\024)41 b Fo(\015)260 771 y Fv(k)282 792 y Fo(M)16 b Fq(+)11 b Fo(C)t(m)p Fq(\(\012)530 771 y Fv(c)530 804 y(\021)551 792 y Fq(\))i Fn(\024)h Fo(C)t Fq(\()p Fo(\015)722 771 y Fv(k)754 792 y Fq(+)d Fo(\021)r Fq(\))p Fo(:)0 886 y Fq(W)l(e)19 b(ha)o(v)o(e)f(pro)o(v)o(ed) g(that)i(for)f Fo(x)g Fq(and)g Fo(y)i Fq(in)d(\003)831 893 y Ft(0)851 886 y Fq(,)h(for)h(an)o(y)e Fo(\021)j(>)d Fq(0,)i(pro)o(vided)e Fo(k)j Fq(is)e(large)0 947 y(enough,)222 1041 y Fn(jL)270 1021 y Fv(k)291 1041 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g(L)481 1021 y Fv(k)503 1041 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))p Fn(j)13 b(\024)h Fo(C)t(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))862 1001 y Fg(\002)882 1041 y Fq(\()p Fo(\015)929 1021 y Fv(k)961 1041 y Fq(+)j Fo(\021)r Fq(\))p Fo(L)p Fq(\()p Fo(f)5 b Fq(\))12 b(+)f(sup)d Fn(j)p Fo(f)d Fn(j)1355 1001 y Fg(\003)1384 1041 y Fo(:)0 1136 y Fq(Remark)15 b(that)h(w)o(e)g(ha)o(v)o(e)g(the)g(follo)o(wing)f (w)o(eak)h(inequalit)o(y)f(for)h(all)g Fo(k)r Fq(:)327 1231 y Fn(jL)375 1210 y Fv(k)396 1231 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g(L)586 1210 y Fv(k)608 1231 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))p Fn(j)13 b(\024)h Fo(C)t(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))g([)o Fo(L)p Fq(\()p Fo(f)d Fq(\))11 b(+)g(sup)e Fn(j)p Fo(f)c Fn(j)p Fq(])0 1326 y(Let)16 b Fo(")g Fq(b)q(e)h(\014xed,)e(c)o(ho)q(ose)i(:)73 1414 y Fn(\017)24 b Fo(\021)18 b Fq(suc)o(h)e(that)h Fo(C)t(\021)e(<)f("=)p Fq(2,)i(let)g Fo(k)707 1421 y Ft(1)740 1414 y Fq(=)e Fo(k)817 1421 y Ft(0)837 1414 y Fq(\()p Fo(\021)r Fq(\))i(b)q(e)h(giv)o(en)e(b)o(y)h(lemm)o(a)e(3.2,) 73 1512 y Fn(\017)24 b Fo(N)19 b(>)14 b Fq(0)i(suc)o(h)g(that)h Fo(C)537 1492 y Fv(v)554 1497 y Fi(0)p 532 1500 47 2 v 532 1529 a Fv(v)549 1535 y Fh(N)596 1512 y Fo(<)d(")p Fq(,)73 1617 y Fn(\017)24 b Fo(k)147 1624 y Ft(0)181 1617 y Fo(>)14 b(k)258 1624 y Ft(1)289 1617 y Fq(+)d Fo(N)21 b Fq(and)c(for)g Fo(k)f(>)d(k)685 1624 y Ft(0)716 1617 y Fn(\000)e Fo(N)5 b Fq(,)16 b Fo(\015)868 1599 y Fv(k)904 1617 y Fo(<)d("=)p Fq(2.)0 1706 y(Let)j Fo(k)g(>)e(k)205 1713 y Ft(0)241 1706 y Fq(and)j Fo(N)i Fn(\024)14 b Fo(`)g(<)g(k)r Fq(,)i(the)g(c)o(hoices)f(ab)q(o)o(v)o(e)h(and)h(\(3.4\))g(giv)o(e)5 1770 y Fg(\014)5 1800 y(\014)22 1812 y Fn(L)56 1791 y Fv(k)77 1812 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g(L)267 1791 y Fv(k)289 1812 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))382 1770 y Fg(\014)382 1800 y(\014)412 1812 y Fn(\024)470 1778 y Fo(v)494 1785 y Ft(0)p 470 1801 44 2 v 472 1846 a Fo(v)496 1853 y Fv(`)518 1812 y Fo(C)t(d)p Fq(\()p Fo(x;)j(y)r Fq(\))g([)o Fo(L)p Fq(\()p Fo(f)d Fq(\))11 b(+)g(sup)e Fn(j)p Fo(f)c Fn(j)p Fq(])14 b Fn(\024)f Fo("d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))g([)o Fo(L)p Fq(\()p Fo(f)d Fq(\))12 b(+)f(sup)d Fn(j)p Fo(f)d Fn(j)q Fq(])i Fo(:)0 1927 y Fq(Let)16 b Fo(`)f Fn(\024)e Fo(N)5 b Fq(,)16 b(w)o(e)g(ha)o(v)o(e:)151 1988 y Fg(\014)151 2018 y(\014)167 2031 y Fn(L)201 2010 y Fv(k)223 2031 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g(L)413 2010 y Fv(k)434 2031 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))527 1988 y Fg(\014)527 2018 y(\014)585 2031 y Fn(\024)671 1997 y Fo(v)695 2004 y Ft(0)p 671 2019 V 673 2065 a Fo(v)697 2072 y Fv(`)719 2031 y Fo(C)t(d)p Fq(\()p Fo(x;)j(y)r Fq(\))905 1990 y Fg(\002)925 2031 y Fq(\()p Fo(\015)972 2010 y Fv(k)q Fk(\000)p Fv(`)1046 2031 y Fq(+)j Fo(\021)r Fq(\))p Fo(L)p Fq(\()p Fo(f)5 b Fq(\))12 b(+)f(sup)d Fn(j)p Fo(f)d Fn(j)1439 1990 y Fg(\003)585 2126 y Fn(\024)42 b Fo(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\)[)p Fo("L)p Fq(\()p Fo(f)d Fq(\))10 b(+)h Fo(C)t Fn(k)p Fo(f)5 b Fn(k)1119 2133 y Fk(1)1156 2126 y Fq(])p Fo(:)1580 2221 y Fb(\004)413 2363 y Fe(3.3)66 b(Construction)23 b(of)f(cones)0 2452 y Fq(W)l(e)15 b(turn)g(out)g(to)g (the)g(hearth)g(of)g(the)g(pro)q(of)h(of)f(theorem)f(1.4.)21 b(Let)15 b(us)g(\014rst)g(in)o(tro)q(duce)0 2512 y(a)f(\014nite)g (partition)f(of)i(\001.)20 b(Let)14 b Fo(\021)i Fq(and)e Fo(t)g Fq(b)q(e)g(giv)o(en,)f(let)g Fo(s)h Fq(b)q(e)g(greater)g(than)g Fo(k)1410 2519 y Ft(0)1430 2512 y Fq(\()p Fo(\021)r Fq(\))g(giv)o(en)0 2572 y(b)o(y)i(lemm)o(a)e(3.2.)21 b(Let)c Fn(P)j Fq(b)q(e)c(the)g (\014nite)g(partition)g Fn(P)i Fq(=)13 b Fn(P)1082 2579 y Ft(1)1113 2572 y Fn([)e(P)1192 2579 y Ft(2)1228 2572 y Fq(de\014ned)16 b(b)o(y:)73 2688 y Fn(\017)24 b(P)157 2695 y Ft(2)190 2688 y Fq(=)242 2586 y Fg(8)242 2631 y(<)242 2721 y(:)314 2641 y([)286 2749 y Fv(R)p Ft(\()p Fv(j)r Ft(\))p Fk(\025)p Fv(t)408 2641 y Fg([)405 2747 y Fv(`)p Fk(2)p Fj(N)475 2688 y Fq(\003)509 2665 y Fv(j)509 2702 y(`)555 2688 y Fn([)j(f)p Fo(x)16 b(=)h(\032)750 2668 y Ft(\()p Fv(s)p Ft(\))796 2688 y Fq(\()p Fo(x)p Fq(\))c Fo(>)h(\015)955 2668 y Fv(s)973 2688 y Fn(g)998 2586 y Fg(9)998 2631 y(=)998 2721 y(;)1056 2688 y Fq(=:)g Fn(f)p Fo(P)1178 2695 y Ft(2)1198 2688 y Fn(g)p Fq(,)p eop %%Page: 13 13 13 12 bop 558 133 a Fp(Mixing)15 b(prop)q(erties)h(on)h(to)o(w)o(ers) 510 b Fq(13)120 286 y Fn(\017)24 b(P)204 293 y Ft(1)240 286 y Fq(is)16 b(the)g(partition)g(of)h Fo(P)670 268 y Fv(c)663 299 y Ft(2)703 286 y Fq(in)o(to)f Fo(s)p Fq(-cylinders.)47 391 y(Because)f(of)g(lemma)e(3.2,)i(the)g Fo(m)g Fq(measure)f(of)i Fo(P)951 398 y Ft(2)986 391 y Fq(is)f(as)h(small)e(as)i(w)o(e)f(w)o(an) o(t,)g(pro)o(vided)47 451 y Fo(t)h Fq(is)g(large)g(enough)h(and)g Fo(\021)h Fq(is)e(small)f(enough.)22 b(W)l(e)16 b(will)f(denote)h(b)o (y)g Fo(D)1359 458 y Ft(1)1395 451 y Fq(the)g(diameter)47 511 y(of)j Fn(P)140 518 y Ft(1)177 511 y Fq(\(whic)o(h)e(is)h(smaller)e (than)i Fo(\015)701 493 y Fv(s)738 511 y Fq(b)o(y)f(de\014nition\))h (and)g(b)o(y)g Fo(D)1251 518 y Ft(2)1289 511 y Fq(the)g Fo(m)f Fq(measure)g(of)47 572 y Fo(P)78 579 y Ft(2)98 572 y Fq(.)47 632 y(Theorem)i(1.2)i(implies)d(the)i(follo)o(wing)g (mixing)f(prop)q(ert)o(y:)29 b(for)21 b Fo(\013)g(<)g Fq(1)g Fo(<)g(\013)1525 614 y Fk(0)1558 632 y Fq(there)47 692 y(exists)16 b Fo(q)204 699 y Ft(0)240 692 y Fq(suc)o(h)g(that)g (for)h Fo(k)f(>)d(q)644 699 y Ft(0)664 692 y Fq(,)j(for)g Fo(P)24 b Fq(and)16 b Fo(P)955 674 y Fk(0)983 692 y Fq(in)g Fn(P)t Fq(,)587 832 y Fo(\013)e(<)689 798 y(m)p Fq(\()p Fo(P)j Fn(\\)12 b Fo(F)883 780 y Fk(\000)p Fv(k)931 798 y Fo(P)969 780 y Fk(0)981 798 y Fq(\))p 689 821 312 2 v 726 866 a Fo(m)p Fq(\()p Fo(P)7 b Fq(\))p Fo(\026)p Fq(\()p Fo(P)931 852 y Fk(0)943 866 y Fq(\))1019 832 y Fo(<)13 b(\013)1101 812 y Fk(0)1113 832 y Fo(:)439 b Fq(\(3.5\))47 969 y(Let)19 b Fo(a)p Fq(,)g Fo(b)p Fq(,)g Fo(c)g Fq(and)g Fo(d)g Fq(b)q(e)g(p)q(ositiv)o(e)f(n)o(um)o(b)q(ers,)g Fo(k)j Fq(and)e Fo(j)j Fq(in)o(tegers)c(to)h(b)q(e)g(c)o(hosen)g (later.)47 1029 y Fn(C)s Fq(\()p Fo(a;)8 b(b;)g(c;)g(d;)g(j)s Fq(\))16 b(is)g(the)g(cone)g(of)g(functions)h(in)f Fo(L)g Fq(suc)o(h)g(that:)107 1158 y(1.)24 b(0)14 b Fn(\024)306 1124 y Fq(1)p 265 1147 106 2 v 265 1192 a Fo(\026)p Fq(\()p Fo(P)7 b Fq(\))384 1090 y Fg(Z)384 1232 y Fv(P)442 1158 y Fo(f)e(dm)14 b Fn(\024)g Fo(a)640 1090 y Fg(Z)698 1158 y Fo(f)5 b(dm)16 b Fn(8)p Fo(P)k Fn(2)14 b(P)t Fq(,)107 1320 y(2.)24 b(for)17 b Fo(`)d Fn(\024)g Fo(j)s(k)r Fq(,)h(for)i Fo(x)f Fq(and)h Fo(y)h Fq(in)d(\003)756 1327 y Fv(`)773 1320 y Fq(,)578 1453 y Fn(j)p Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g Fo(f)5 b Fq(\()p Fo(y)r Fq(\))p Fn(j)13 b(\024)h Fo(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))p Fo(b)1089 1385 y Fg(Z)1146 1453 y Fo(f)d(dm;)107 1642 y Fq(3.)24 b(for)17 b Fo(`)d Fn(\024)g Fo(j)s(k)r Fq(,)h(for)i Fo(x)c Fn(2)i Fq(\003)608 1649 y Fv(`)635 1642 y Fn(\\)c Fo(P)710 1649 y Ft(2)747 1642 y Fn(j)p Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fn(j)13 b(\024)h Fo(c)965 1574 y Fg(Z)1023 1642 y Fo(f)5 b(dm)p Fq(,)107 1801 y(4.)24 b(for)18 b Fo(`)f(>)g(j)s(k)r Fq(,)g Fn(j)p Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fn(j)17 b(\024)618 1775 y Fv(v)635 1781 y Fh(k)q(j)p 618 1790 51 2 v 627 1818 a Fv(v)644 1824 y Fh(`)673 1801 y Fo(d)706 1733 y Fg(Z)765 1801 y Fo(f)5 b(dm)18 b Fq(and)g Fn(j)p Fo(f)5 b Fq(\()p Fo(x)p Fq(\))12 b Fn(\000)g Fo(f)5 b Fq(\()p Fo(y)r Fq(\))p Fn(j)16 b(\024)1332 1775 y Fv(v)1349 1781 y Fh(k)q(j)p 1332 1790 V 1341 1818 a Fv(v)1358 1824 y Fh(`)1387 1801 y Fo(d)1420 1733 y Fg(Z)1479 1801 y Fo(f)5 b(dm)p Fq(,)18 b(for)169 1883 y(all)e Fo(x;)24 b(y)15 b Fn(2)f Fq(\003)423 1890 y Fv(`)440 1883 y Fq(.)47 1987 y(Note)d(that)h(the)f(cone)g(also)h(dep)q(ends)g(on)g Fo(s)p Fq(,)g Fo(t)p Fq(,)f Fo(k)r Fq(.)20 b(W)l(e)11 b(will)f(c)o(ho)q(ose)i Fo(a)p Fq(,)g Fo(b)p Fq(,)f Fo(c)p Fq(,)h Fo(d)p Fq(,)g Fo(k)r Fq(,)g Fo(j)s Fq(,)g Fo(s)g Fq(and)47 2048 y Fo(t)i Fq(in)g(suc)o(h)g(a)h(w)o(a)o(y)e(that)i Fn(L)515 2029 y Fv(k)551 2048 y Fq(in)f(a)g(con)o(traction)g(from)g Fn(C)s Fq(\()p Fo(a;)8 b(b;)g(c;)g(j)s Fq(\))13 b(to)h Fn(C)s Fq(\()p Fo(a;)8 b(b;)g(c;)g(d;)g(j)h Fq(+)e(1\).)47 2108 y(Let)17 b(us)f(denote)g(b)o(y)g Fo(\016)444 2115 y Fv(j)476 2108 y Fq(=)564 2082 y Fv(v)581 2088 y Fh(k)q(j)p 533 2096 113 2 v 533 2125 a Fv(v)550 2132 y Fh(k)q Fi(\()p Fh(j)q Fi(+1\))664 2108 y Fo(<)e Fq(1.)47 2230 y Fl(Remark)j(3.5)24 b Fp(It)16 b(will)h(b)q(e)g(clear)f(from)g(the)h(pro)q(ofs)i(that)f(if) e(there)h(exists)f Fq(0)g Fo(<)g(\014)h(<)f Fq(1)47 2291 y Fp(suc)o(h)f(that)g Fq(0)g Fo(<)e(\016)372 2298 y Fv(j)404 2291 y Fo(<)h(\014)j Fp(then)e(condition)f(4)i(in)e(the)h(de\014nition) g(of)g(the)f(cone)h(is)g(useless.)47 2351 y(This)i(means)g(that)g(it)g (is)f(p)q(ossible)i(to)f(w)o(ork)g(with)g(the)g(cone)g Fn(C)s Fq(\()p Fo(a;)8 b(b;)g(c)p Fq(\))16 b Fp(of)h(functions)g(f)47 2411 y(suc)o(h)f(that:)107 2527 y(1.)24 b Fq(0)14 b Fn(\024)306 2494 y Fq(1)p 265 2516 106 2 v 265 2562 a Fo(\026)p Fq(\()p Fo(P)7 b Fq(\))384 2460 y Fg(Z)384 2601 y Fv(P)442 2527 y Fo(f)e(dm)14 b Fn(\024)g Fo(a)640 2460 y Fg(Z)698 2527 y Fo(f)5 b(dm)16 b Fn(8)p Fo(P)k Fn(2)14 b(P)t Fp(,)107 2723 y(2.)24 b Fo(L)p Fq(\()p Fo(f)5 b Fq(\))14 b Fn(\024)g Fo(b)365 2655 y Fg(Z)423 2723 y Fo(f)5 b(dm;)p eop %%Page: 14 14 14 13 bop 0 133 a Fq(14)488 b Fp(V)o(\023)-23 b(eronique)15 b(Maume-Desc)o(hamps)60 304 y(3.)24 b(for)16 b Fo(x)e Fn(2)g Fq(\003)319 311 y Fv(`)346 304 y Fn(\\)e Fo(P)422 311 y Ft(2)458 304 y Fn(j)p Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fn(j)14 b(\024)f Fo(c)676 237 y Fg(Z)735 304 y Fo(f)5 b(dm)p Fp(.)0 431 y Fl(PR)n(OPOSITION)19 b(3.6)24 b Fp(F)l(or)17 b(an)o(y)e Fq(0)g Fo(<)e(\033)j(<)e(\016)900 438 y Fv(j)917 431 y Fp(,)i(for)g(an)o(y)g(p)q(ositiv)o(e)f(n)o(um)o(b)q(ers)g Fo(a)1516 438 y Ft(0)1535 431 y Fp(,)h Fo(b)1586 438 y Ft(0)1605 431 y Fp(,)0 492 y Fo(c)21 499 y Ft(0)41 492 y Fp(,)k Fo(d)100 499 y Ft(0)120 492 y Fp(,)h(there)e(exist)h(an)g (in)o(teger)f Fo(k)664 499 y Ft(0)684 492 y Fp(,)i(p)q(ositiv)o(e)e(n)o (um)o(b)q(ers)g Fo(a)h Fn(\025)g Fo(a)1235 499 y Ft(0)1254 492 y Fp(,)h Fo(b)f Fn(\025)g Fo(b)1410 499 y Ft(0)1430 492 y Fp(,)g Fo(c)h Fn(\025)f Fo(c)1586 499 y Ft(0)1605 492 y Fp(,)0 552 y Fo(d)14 b Fn(\025)g Fo(d)117 559 y Ft(0)137 552 y Fp(,)i(in)o(tegers)f Fo(s)h Fp(and)h Fo(t)f Fp(suc)o(h)g(that)h(for)f(all)g Fo(k)g(>)e(k)991 559 y Ft(0)1011 552 y Fp(,)355 652 y Fn(L)389 631 y Fv(k)410 652 y Fn(C)s Fq(\()p Fo(a;)8 b(b;)g(c;)g(d;)g(j)s Fq(\))13 b Fn(\032)h(C)s Fq(\()p Fo(\033)r(a;)8 b(\016)895 659 y Fv(j)912 652 y Fo(b;)g(\016)977 659 y Fv(j)994 652 y Fo(c;)g(\016)1059 659 y Fv(j)1077 652 y Fo(d;)g(j)14 b Fq(+)d(1\))p Fo(:)0 751 y Fp(Moreo)o(v)o(er,)k(for)h Fo(f)j Fn(2)14 b(L)426 733 y Fv(k)448 751 y Fn(C)s Fq(\()p Fo(a;)8 b(b;)g(c;)g(d;)g(j)s Fq(\))p Fp(,)15 b(w)o(e)h(ha)o(v)o(e)509 841 y Fq(1)p 468 864 106 2 v 468 909 a Fo(\026)p Fq(\()p Fo(P)7 b Fq(\))587 807 y Fg(Z)587 949 y Fv(P)645 875 y Fo(f)e(dm)14 b Fn(\025)g Fo(A)i Fp(for)g(all)g Fo(P)21 b Fn(2)14 b(P)t Fo(;)0 1035 y Fp(where)i Fo(A)g Fp(can)g(b)q(e)h(c)o (hosen)f(arbitrarily)f(pro)o(vided)h(it)f(is)h(strictly)f(smaller)g (than)h Fq(1)p Fo(=)p Fq(2)p Fp(.)0 1141 y Fl(Remark)g(3.7)24 b Fp(In)19 b(fact,)f(it)g(is)h(clear)e(from)h(the)g(pro)q(of)i(that)f (mo)q(difying)e(sligh)o(tly)g(the)0 1201 y(argumen)o(t)d(the)h(constan) o(t)g Fo(A)g Fp(ab)q(o)o(v)o(e)g(ma)o(y)e(b)q(e)i(c)o(hosen)g(as)h (close)e(as)i(one)f(whic)o(h)f(from)g Fq(1)p Fp(.)0 1261 y(This)e(ma)o(y)f(b)q(e)h(imp)q(ortan)o(t)g(to)g(optimize)e(the)i(rate) h(of)f(con)o(v)o(ergence)f(in)h(the)g(exp)q(onen)o(tial)0 1321 y(case.)0 1412 y Fq(W)l(e)21 b(b)q(egin)h(with)g(the)f(follo)o (wing)h(lemm)o(a)d(whic)o(h)i(giv)o(es)g(a)h(con)o(trol)g(on)g(the)g (sup.)38 b(of)0 1472 y(functions)16 b(in)g Fn(C)s Fq(\()p Fo(a;)8 b(b;)g(c;)g(d;)g(j)s Fq(\).)20 b(When)c(there)f(is)h(no)h(am)o (biguit)o(y)l(,)c(w)o(e)i(will)g(forget)i(ab)q(out)0 1533 y(the)d(dep)q(endence)f(on)h Fo(a)p Fq(,)f Fo(b)p Fq(,)h Fo(c)g(d)g Fq(and)g Fo(j)j Fq(and)e(write)e Fn(C)k Fq(instead)d(of)g Fn(C)s Fq(\()p Fo(a;)8 b(b;)g(c;)g(d;)g(j)s Fq(\))13 b(or)h Fn(C)s Fq(\()p Fo(j)s Fq(\))0 1593 y(if)i(w)o(e)g(wish) g(to)h(emphasize)d(the)i(dep)q(endence)f(on)i Fo(j)s Fq(.)0 1698 y Fl(LEMMA)i(3.8)24 b Fp(F)l(or)c(an)o(y)f Fo(f)25 b Fn(2)20 b(C)s Fp(,)g(for)g Fo(`)g Fn(2)g Fm(N)h Fp(and)f Fo(P)27 b Fn(2)20 b(P)1156 1705 y Ft(1)1175 1698 y Fp(,)g(for)g Fo(x)g Fp(in)f Fq(\003)1429 1705 y Fv(`)1459 1698 y Fn(\\)13 b Fo(P)27 b Fp(w)o(e)0 1759 y(ha)o(v)o(e:)316 1870 y Fn(j)p Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fn(j)41 b(\024)612 1836 y Fq(1)p 565 1859 119 2 v 565 1904 a Fo(m)p Fq(\()p Fo(P)7 b Fq(\))697 1802 y Fg(Z)698 1944 y Fv(P)755 1870 y Fo(f)e(dm)12 b Fq(+)f(max)o([)p Fo(b;)d Fp(])1059 1893 y(.)1073 1870 y Fn(j)p Fo(P)f Fn(j)1147 1802 y Fg(Z)1206 1870 y Fo(f)e(dm)216 b Fq(\(3.6\))316 2033 y Fn(j)p Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fn(j)41 b(\024)g Fq([)p Fo(D)q(a)11 b Fq(+)g Fo(D)741 2040 y Ft(1)770 2033 y Fq(max)o([)p Fo(b;)23 b(d)p Fq(]])994 1966 y Fg(Z)1052 2033 y Fo(f)5 b(dm:)356 b Fq(\(3.7\))0 2169 y Fp(\(where)16 b Fo(D)g Fq(=)d(sup)341 2180 y Fv(P)5 b Fk(2P)439 2145 y Fv(\026)p Ft(\()p Fv(P)g Ft(\))p 434 2157 86 2 v 434 2186 a Fv(m)p Ft(\()p Fv(P)g Ft(\))525 2169 y Fp(\).)21 b(F)l(or)c(an)o(y)f Fo(f)j Fn(2)14 b(C)s Fp(,)i(for)h Fo(x)c Fn(2)h Fo(P)1102 2176 y Ft(2)1134 2169 y Fn(\\)d Fq(\003)1212 2176 y Fv(`)1228 2169 y Fp(,)16 b Fo(`)e Fn(2)g Fm(N)p Fp(,)318 2305 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))42 b Fn(\025)594 2272 y Fq(1)p 540 2294 132 2 v 540 2340 a Fo(m)p Fq(\()p Fo(P)633 2347 y Ft(2)653 2340 y Fq(\))685 2238 y Fg(Z)679 2379 y Fv(P)701 2384 y Fi(2)743 2305 y Fo(f)5 b(dm)12 b Fn(\000)f Fq(2)d(max)o([)p Fo(c;)g(d)p Fq(])1129 2238 y Fg(Z)1187 2305 y Fo(f)d(dm)318 2457 y(f)g Fq(\()p Fo(x)p Fq(\))42 b Fn(\024)58 b Fq(max)n([)p Fo(c;)8 b(d)p Fq(])746 2389 y Fg(Z)804 2457 y Fo(f)d(dm)1519 2381 y Fq(\(3.8\))0 2581 y Ff(Pr)n(o)n(of)k Fq(:)53 b(Let)19 b Fo(f)24 b Fq(b)q(elong)c(to)f Fn(C)s Fq(.)29 b(F)l(or)19 b Fo(x)f Fn(2)g Fo(P)25 b Fn(2)19 b(P)952 2588 y Ft(1)971 2581 y Fq(,)g(equations)g(\(3.6\))g(and)h(\(3.7\))f(for)0 2641 y Fo(`)14 b Fn(\024)g Fo(k)r(j)j Fq(directly)c(follo)o(w)h(from)f (de\014nitions.)21 b(Just)14 b(write)g(that)h(for)g Fo(x)f Fq(and)h Fo(y)h Fq(in)e Fo(P)21 b Fn(2)14 b(P)t Fq(,)303 2741 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))11 b Fn(\000)g Fo(L)p Fq(\()p Fo(f)5 b Fq(\))p Fo(d)p Fq(\()p Fo(x;)j(y)r Fq(\))14 b Fn(\024)f Fo(f)5 b Fq(\()p Fo(x)p Fq(\))14 b Fn(\024)g Fo(f)5 b Fq(\()p Fo(y)r Fq(\))11 b(+)g Fo(L)p Fq(\()p Fo(f)5 b Fq(\))p Fo(d)p Fq(\()p Fo(x;)j(y)r Fq(\))p eop %%Page: 15 15 15 14 bop 558 133 a Fp(Mixing)15 b(prop)q(erties)h(on)h(to)o(w)o(ers) 510 b Fq(15)47 286 y(and)17 b(in)o(tegrate)f(on)h Fo(P)24 b Fq(with)16 b(resp)q(ect)g(to)h Fo(m)f Fq(these)g(inequalities.)k(Let) d Fo(x;)24 b(y)16 b Fn(2)e Fo(P)1540 293 y Ft(2)1571 286 y Fn(\\)e Fq(\003)1650 293 y Fv(`)47 347 y Fq(with)k Fo(`)f Fn(\024)e Fo(k)r(j)s Fq(,)475 407 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))11 b Fn(\000)g Fq(2)23 b(sup)661 447 y Ft(\003)685 453 y Fh(`)701 447 y Fk(\\)p Fv(P)747 452 y Fi(2)772 407 y Fn(j)p Fo(f)5 b Fn(j)14 b(\024)g Fo(f)5 b Fq(\()p Fo(x)p Fq(\))13 b Fn(\024)29 b Fq(sup)1057 447 y Ft(\003)1081 453 y Fh(`)1097 447 y Fk(\\)p Fv(P)1143 452 y Fi(2)1168 407 y Fn(j)p Fo(f)5 b Fn(j)p Fo(;)47 525 y Fq(this)20 b(leads)f(to)h(\(3.8\))g(for)f Fo(`)h Fn(\024)f Fo(k)r(j)s Fq(.)31 b(Inequalities)18 b(for)h Fo(`)h(>)f(k)r(j)k Fq(follo)o(w)c(from)f(4)i(in)f(the)47 586 y(de\014nition)d(of)h(the)f (cone.)1114 b Fb(\004)47 706 y Fq(Let)17 b(us)f(b)q(egin)h(with)f(the)g (pro)q(of)h(of)g(prop)q(osition)g(3.6.)47 766 y Ff(Pr)n(o)n(of)g(of)h (pr)n(op)n(osition)e(3.6)p Fq(:)46 b(W)l(e)16 b(are)h(going)h(to)f(c)o (hec)o(k)e(ho)o(w)j(the)e(four)i(conditions)f(in)47 826 y(the)d(de\014nition)g(of)g Fn(C)k Fq(ev)o(olute)13 b(under)h(the)g (action)g(of)h Fn(L)1057 808 y Fv(k)1078 826 y Fq(.)21 b(First)14 b(of)g(all,)g(condition)g(4)p Fo(:)g Fq(of)47 887 y(the)d(cone)g(is)g(v)o(ery)f(easy)h(to)h(c)o(hec)o(k:)17 b(if)10 b Fo(x)k Fn(2)g Fq(\003)845 894 y Fv(`)861 887 y Fq(,)e Fo(`)i(>)g(k)r Fq(\()p Fo(j)t Fq(+)q(1\))d(then)g Fn(L)1276 868 y Fv(k)1298 887 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))14 b(=)1464 862 y Fv(v)1481 868 y Fh(`)p Fd(\000)p Fh(k)p 1464 875 74 2 v 1484 904 a Fv(v)1501 910 y Fh(`)1542 887 y Fo(f)5 b Fq(\()p Fo(x)1618 868 y Fk(\000)1647 887 y Fq(\))47 957 y(so)19 b Fn(jL)157 939 y Fv(k)178 957 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fn(j)18 b Fq(and)g Fn(jL)449 939 y Fv(k)471 957 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))12 b Fn(\000)g(L)663 939 y Fv(k)685 957 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))p Fn(j)17 b Fq(are)h(less)g(than)1104 928 y Fv(v)1121 935 y Fh(k)q Fi(\()p Fh(j)q Fi(+1\))p 1104 946 113 2 v 1144 974 a Fv(v)1161 980 y Fh(`)1222 957 y Fo(\016)1244 964 y Fv(j)1262 957 y Fo(d)1295 917 y Fg(R)1337 957 y Fo(f)5 b(dm)p Fq(,)18 b(for)g Fo(x)g Fq(and)47 1017 y Fo(y)h Fq(in)e(\003)182 1024 y Fv(`)198 1017 y Fq(,)g Fo(`)f(>)g(k)r(j)s Fq(.)24 b(It)17 b(just)g(remains)f(to) i(c)o(hec)o(k)d(the)i(\014rst)h(three)e(conditions.)25 b(Assume)47 1077 y(that)17 b Fo(d)g Fq(is)f(\014xed,)f Fo(b)h Fq(and)h Fo(c)f Fq(will)f(b)q(e)i(c)o(hosen)f(to)g(ha)o(v)o(e)g Fo(d)e Fn(\024)g Fq(min)n([)p Fo(b;)8 b(c)p Fq(].)47 1138 y(W)l(e)18 b(tak)o(e)f Fo(f)k Fn(2)16 b(C)s Fq(.)26 b(W)l(e)17 b(\014x)g(0)g Fo(<)f(\033)h(<)f(\016)774 1145 y Fv(j)810 1138 y Fq(and)i Fo(")e(<)g(\033)r Fq(,)h Fo(\013)f(<)g Fq(1)g Fo(<)g(\013)1286 1119 y Fk(0)1316 1138 y Fq(and)i Fo(q)1434 1145 y Ft(0)1470 1138 y Fn(2)e Fm(N)i Fq(suc)o(h)47 1198 y(that)f(\(3.5\))f(is)g(satis\014ed)h(for)g Fo(k)e(>)f(q)695 1205 y Ft(0)715 1198 y Fq(.)21 b(In)16 b(what)h(follo)o(ws,)e Fo(k)k Fq(is)d(greater)g(than)h Fo(q)1500 1205 y Ft(0)1519 1198 y Fq(.)47 1258 y(The)11 b(follo)o(wing)g(remark)f(will)g(b)q(e)h (used)g(man)o(y)f(times)f(in)i(the)g(pro)q(of:)20 b(if)10 b Fo(f)20 b Fn(2)14 b(C)g Fq(then)d(since)47 1318 y(w)o(e)h(c)o(ho)q (ose)h Fo(d)i Fn(\024)e Fq(max)o(\()p Fo(b;)20 b(c)p Fq(\))13 b(w)o(e)f(ha)o(v)o(e)f(that)i(sup)c Fn(j)p Fo(f)c Fn(j)14 b(\024)g Fo(c)1089 1278 y Fg(R)1130 1318 y Fo(f)5 b(dm)13 b Fq(and)g Fo(L)p Fq(\()p Fo(f)5 b Fq(\))15 b Fn(\024)e Fo(b)1527 1278 y Fg(R)1569 1318 y Fo(f)5 b(dm)47 1378 y Fq(\(use)16 b(also)h(that)g(if)f Fo(`)e(>)g(k)r(j)19 b Fq(then)667 1352 y Fv(v)684 1358 y Fh(k)q(j)p 667 1367 51 2 v 676 1395 a Fv(v)693 1401 y Fh(`)736 1378 y Fn(\024)14 b Fq(1\).)47 1438 y(Condition)j(1.)p 47 1446 266 2 v 47 1499 a(Let)g Fo(P)k Fn(2)14 b(P)t Fq(,)192 1593 y(1)p 151 1615 106 2 v 151 1661 a Fo(\026)p Fq(\()p Fo(P)7 b Fq(\))270 1559 y Fg(Z)270 1701 y Fv(P)328 1627 y Fn(L)362 1606 y Fv(k)384 1627 y Fo(f)e(dm)42 b Fq(=)648 1593 y(1)p 607 1615 V 607 1661 a Fo(\026)p Fq(\()p Fo(P)7 b Fq(\))750 1559 y Fg(Z)715 1705 y Fv(F)742 1695 y Fd(\000)p Fh(k)785 1705 y Fv(P)832 1627 y Fo(f)e(dm)14 b Fq(=)1003 1580 y Fg(X)995 1686 y Fv(P)1022 1676 y Fd(0)1033 1686 y Fk(2P)1138 1593 y Fq(1)p 1097 1615 V 1097 1661 a Fo(\026)p Fq(\()p Fo(P)7 b Fq(\))1271 1559 y Fg(Z)1205 1705 y Fv(P)1232 1695 y Fd(0)1244 1705 y Fk(\\)p Fv(F)1295 1695 y Fd(\000)p Fh(k)1337 1705 y Fv(P)1384 1627 y Fo(f)e(dm)523 1794 y Fq(=)618 1747 y Fg(X)602 1853 y Fv(P)629 1844 y Fd(0)641 1853 y Fk(2P)690 1858 y Fi(1)760 1761 y Fq(1)p 719 1783 V 719 1828 a Fo(\026)p Fq(\()p Fo(P)i Fq(\))893 1726 y Fg(Z)827 1872 y Fv(P)854 1863 y Fd(0)866 1872 y Fk(\\)p Fv(F)917 1863 y Fd(\000)p Fh(k)959 1872 y Fv(P)1006 1794 y Fo(f)e(dm)11 b Fq(+)1209 1761 y(1)p 1168 1783 V 1168 1828 a Fo(\026)p Fq(\()p Fo(P)c Fq(\))1343 1726 y Fg(Z)1276 1872 y Fv(P)1298 1877 y Fi(2)1316 1872 y Fk(\\)p Fv(F)1367 1863 y Fd(\000)p Fh(k)1410 1872 y Fv(P)1456 1794 y Fo(f)e(dm:)47 1969 y Fq(Using)16 b(lemma)e(3.8,)i(w)o(e)f(get)i(for)f Fo(P)699 1951 y Fk(0)725 1969 y Fn(2)e(P)807 1976 y Ft(1)826 1969 y Fq(:)131 2098 y Fo(m)p Fq(\()p Fo(F)232 2080 y Fk(\000)p Fv(k)279 2098 y Fo(P)19 b Fn(\\)11 b Fo(P)411 2080 y Fk(0)423 2098 y Fq(\))p 131 2120 312 2 v 168 2166 a Fo(\026)p Fq(\()p Fo(P)c Fq(\))p Fo(m)p Fq(\()p Fo(P)373 2151 y Fk(0)385 2166 y Fq(\))455 2031 y Fg(2)455 2121 y(4)488 2064 y(Z)483 2206 y Fv(P)510 2197 y Fd(0)546 2131 y Fo(f)e(d\026)13 b Fn(\000)d Fo(m)p Fq(\()p Fo(P)791 2111 y Fk(0)803 2131 y Fq(\))p Fo(D)862 2138 y Ft(1)882 2131 y Fo(b)911 2064 y Fg(Z)969 2131 y Fo(f)5 b(dm)1066 2031 y Fg(3)1066 2121 y(5)1130 2131 y Fn(\024)1244 2098 y Fq(1)p 1204 2120 106 2 v 1204 2166 a Fo(\026)p Fq(\()p Fo(P)i Fq(\))1377 2064 y Fg(Z)1311 2209 y Fv(P)1338 2200 y Fd(0)1350 2209 y Fk(\\)p Fv(F)1401 2200 y Fd(\000)p Fh(k)1444 2209 y Fv(P)1490 2131 y Fo(f)e(dm)161 2334 y Fq(1)p 120 2356 V 120 2401 a Fo(\026)p Fq(\()p Fo(P)i Fq(\))294 2299 y Fg(Z)228 2445 y Fv(P)255 2436 y Fd(0)266 2445 y Fk(\\)p Fv(F)317 2436 y Fd(\000)p Fh(k)360 2445 y Fv(P)407 2367 y Fo(f)e(dm)30 b Fn(\024)608 2334 y Fo(m)p Fq(\()p Fo(F)709 2315 y Fk(\000)p Fv(k)757 2334 y Fo(P)18 b Fn(\\)11 b Fo(P)888 2315 y Fk(0)900 2334 y Fq(\))p 608 2356 312 2 v 646 2401 a Fo(\026)p Fq(\()p Fo(P)c Fq(\))p Fo(m)p Fq(\()p Fo(P)851 2387 y Fk(0)863 2401 y Fq(\))932 2267 y Fg(2)932 2357 y(4)966 2299 y(Z)960 2442 y Fv(P)987 2432 y Fd(0)1024 2367 y Fo(f)e(d\026)12 b Fq(+)f Fo(m)p Fq(\()p Fo(P)1268 2347 y Fk(0)1279 2367 y Fq(\))p Fo(D)1338 2374 y Ft(1)1359 2367 y Fo(b)1388 2299 y Fg(Z)1446 2367 y Fo(f)5 b(dm)1543 2267 y Fg(3)1543 2357 y(5)1585 2367 y Fo(;)47 2517 y Fq(using)17 b(\(3.5\),)f(this)g (leads)g(to)60 2683 y Fo(\013)100 2582 y Fg(2)100 2672 y(4)133 2615 y(Z)128 2757 y Fv(P)155 2748 y Fd(0)191 2683 y Fo(f)5 b(dm)12 b Fn(\000)f Fo(\013)381 2662 y Fk(0)393 2683 y Fo(bD)454 2690 y Ft(1)474 2683 y Fo(m)p Fq(\()p Fo(P)574 2662 y Fk(0)585 2683 y Fq(\))612 2615 y Fg(Z)670 2683 y Fo(f)5 b(dm)767 2582 y Fg(3)767 2672 y(5)842 2683 y Fn(\024)968 2649 y Fq(1)p 928 2671 106 2 v 928 2717 a Fo(\026)p Fq(\()p Fo(P)i Fq(\))1101 2615 y Fg(Z)1035 2760 y Fv(P)1062 2751 y Fd(0)1074 2760 y Fk(\\)p Fv(F)1125 2751 y Fd(\000)p Fh(k)1168 2760 y Fv(P)1214 2683 y Fo(f)e(dm)p eop %%Page: 16 16 16 15 bop 0 133 a Fq(16)488 b Fp(V)o(\023)-23 b(eronique)15 b(Maume-Desc)o(hamps)410 305 y Fq(1)p 370 327 106 2 v 370 373 a Fo(\026)p Fq(\()p Fo(P)7 b Fq(\))543 271 y Fg(Z)477 417 y Fv(P)504 407 y Fd(0)516 417 y Fk(\\)p Fv(F)567 407 y Fd(\000)p Fh(k)610 417 y Fv(P)656 339 y Fo(f)e(dm)42 b Fn(\024)f Fo(\013)906 318 y Fk(0)927 239 y Fg(2)927 328 y(4)960 271 y(Z)955 413 y Fv(P)982 404 y Fd(0)1018 339 y Fo(f)5 b(dm)12 b Fq(+)f Fo(bD)1237 346 y Ft(1)1257 339 y Fo(m)p Fq(\()p Fo(P)1357 318 y Fk(0)1368 339 y Fq(\))1395 271 y Fg(Z)1453 339 y Fo(f)5 b(dm)1550 239 y Fg(3)1550 328 y(5)1592 339 y Fo(:)0 500 y Fq(F)l(or)16 b Fo(P)118 507 y Ft(2)139 500 y Fq(,)f(w)o(e)h(get:)113 621 y Fo(\013)153 554 y Fg(Z)147 695 y Fv(P)169 700 y Fi(2)211 621 y Fo(f)5 b(dm)11 b Fn(\000)g Fq(2)p Fo(cD)454 628 y Ft(2)475 621 y Fo(\013)506 601 y Fk(0)526 554 y Fg(Z)584 621 y Fo(f)5 b(dm)14 b Fn(\024)793 588 y Fq(1)p 753 610 V 753 655 a Fo(\026)p Fq(\()p Fo(P)7 b Fq(\))927 554 y Fg(Z)860 699 y Fv(P)882 704 y Fi(2)900 699 y Fk(\\)p Fv(F)951 690 y Fd(\000)p Fh(k)994 699 y Fv(P)1041 621 y Fo(f)e(dm)14 b Fn(\024)f Fo(\013)1235 601 y Fk(0)1247 621 y Fo(cD)1308 628 y Ft(2)1337 554 y Fg(Z)1395 621 y Fo(f)5 b(dm:)0 790 y Fq(Summing)18 b(these)i(inequalities)f(for)h (all)g Fo(P)29 b Fn(2)21 b Fo(P)906 797 y Ft(1)946 790 y Fq(and)g(for)g Fo(P)1155 797 y Ft(2)1175 790 y Fq(,)g(w)o(e)f(get)h (\(one)f(has)i(to)0 850 y(notice)16 b(that)g Fo(m)p Fq(\()p Fc(1)340 857 y Fv(P)362 862 y Fi(2)393 850 y Fn(\001)11 b Fo(f)5 b Fq(\))14 b Fn(\025)g Fq(0\):)162 975 y([)p Fo(\013)d Fn(\000)g Fo(\013)299 954 y Fk(0)311 975 y Fo(D)351 982 y Ft(1)371 975 y Fo(b)g Fn(\000)g Fq(2)p Fo(\013)508 954 y Fk(0)520 975 y Fo(cD)581 982 y Ft(2)601 975 y Fq(])623 907 y Fg(Z)681 975 y Fo(f)5 b(dm)42 b Fn(\024)946 941 y Fq(1)p 905 963 V 905 1009 a Fo(\026)p Fq(\()p Fo(P)7 b Fq(\))1024 907 y Fg(Z)1024 1049 y Fv(P)1082 975 y Fn(L)1116 954 y Fv(k)1138 975 y Fo(f)e(dm)557 1104 y Fq(1)p 516 1127 V 516 1172 a Fo(\026)p Fq(\()p Fo(P)i Fq(\))635 1070 y Fg(Z)635 1212 y Fv(P)693 1138 y Fn(L)727 1118 y Fv(k)749 1138 y Fo(f)47 b Fn(\024)41 b Fq([1)11 b(+)g Fo(D)1038 1145 y Ft(1)1058 1138 y Fo(b)g Fq(+)g Fo(cD)1200 1145 y Ft(2)1220 1138 y Fq(])p Fo(\013)1265 1118 y Fk(0)1302 1070 y Fg(Z)1360 1138 y Fo(f)5 b(dm)62 b Fq(\(3.9\))0 1296 y(Condition)17 b(2.)p 0 1304 266 2 v 0 1356 a(Let)11 b Fo(x)h Fq(and)g Fo(y)g Fq(b)q(elong)g(to)g(\003) 488 1363 y Fv(`)504 1356 y Fq(,)g Fo(`)j Fn(\024)e Fo(k)r Fq(\()p Fo(j)t Fq(+)q(1\))f(using)g(the)f(fact)g(that)h(if)f Fo(x)i Fn(2)h Fq(\003)1337 1363 y Fv(`)1354 1356 y Fq(,)e Fo(`)i Fn(\024)f Fo(k)r Fq(\()p Fo(j)t Fq(+)q(1\))0 1416 y(then)18 b Fo(F)152 1398 y Fv(k)173 1416 y Fo(x)201 1398 y Fk(0)230 1416 y Fq(=)g Fo(x)h Fq(with)f Fo(x)474 1398 y Fk(0)503 1416 y Fn(2)h Fq(\003)589 1423 y Fv(`)p Fk(\000)p Fv(k)670 1416 y Fq(and)g(lemma)d(3.4,)j(w)o(e)f(\014nd)h Fo(k)1221 1423 y Ft(0)1260 1416 y Fq(and)g Fo(N)k Fn(2)c Fm(N)g Fq(suc)o(h)0 1476 y(that)e(for)f Fo(k)g(>)e(k)298 1483 y Ft(0)318 1476 y Fq(,)126 1598 y Fn(jL)174 1577 y Fv(k)196 1598 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g(L)386 1577 y Fv(k)408 1598 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))p Fn(j)41 b(\024)641 1564 y Fo(v)665 1571 y Fv(`)p Fk(\000)p Fv(k)p 641 1586 87 2 v 664 1632 a Fo(v)688 1639 y Fv(`)733 1598 y Fo(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))p Fo(b)901 1530 y Fg(Z)958 1598 y Fo(f)d(dm)17 b Fq(if)f Fo(`)e Fn(\025)g Fo(k)556 1725 y Fn(\024)41 b Fo("d)p Fq(\()p Fo(x;)8 b(y)r Fq(\)[)p Fo(b)841 1657 y Fg(Z)898 1725 y Fo(f)d(dm)11 b Fq(+)g(sup)e Fn(j)p Fo(f)c Fn(j)p Fq(])16 b(if)g Fo(N)j Fn(\024)13 b Fo(`)i(<)e(k)556 1852 y Fn(\024)41 b Fo(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\)[)p Fo("b)841 1785 y Fg(Z)898 1852 y Fo(f)d(dm)11 b Fq(+)g Fo(C)h Fq(sup)d Fn(j)p Fo(f)c Fn(j)p Fq(])16 b(otherwise)p Fo(:)0 2007 y Fq(Lemma)e(3.8)j(giv)o(es)e(sup)9 b Fn(j)p Fo(f)c Fn(j)14 b(\024)f Fq(max)o([)p Fo(c;)24 b(bD)802 2014 y Ft(1)833 2007 y Fq(+)11 b Fo(aD)q Fq(])971 1939 y Fg(Z)1029 2007 y Fo(f)5 b(dm)17 b Fq(so)g(w)o(e)f(get:)192 2162 y Fn(jL)240 2141 y Fv(k)261 2162 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g(L)451 2141 y Fv(k)473 2162 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))p Fn(j)13 b(\024)651 2128 y Fo(v)675 2135 y Fv(`)p Fk(\000)p Fv(k)p 651 2150 V 674 2196 a Fo(v)698 2203 y Fv(`)743 2162 y Fo(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))p Fo(b)911 2094 y Fg(Z)968 2162 y Fo(f)d(dm)17 b Fq(if)e Fo(`)g Fn(\025)e Fo(k)256 b Fq(\(3.10\))233 2289 y Fn(\024)41 b Fo("d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))483 2221 y Fg(Z)541 2289 y Fo(f)d(dm)p Fq([)p Fo(b)11 b Fq(+)g(max)o(\()p Fo(c;)24 b(bD)963 2296 y Ft(1)994 2289 y Fq(+)11 b Fo(aD)q Fq(\)])16 b(if)g Fo(N)j Fn(\024)13 b Fo(`)i(<)e(k)69 b Fq(\(3.11\))233 2416 y Fn(\024)41 b Fo(d)p Fq(\()p Fo(x;)8 b(y)r Fq(\))460 2349 y Fg(Z)518 2416 y Fo(f)d(dm)p Fq([)p Fo("b)11 b Fq(+)g Fo(C)h Fq(max)n(\()p Fo(c;)24 b(bD)1009 2423 y Ft(1)1040 2416 y Fq(+)11 b Fo(aD)q Fq(\)])17 b(otherwise)o Fo(:)74 b Fq(\(3.12\))0 2538 y(Condition)17 b(3.)p 0 2546 266 2 v 0 2598 a(Let)f Fo(x)e Fn(2)g Fq(\003)210 2605 y Fv(`)237 2598 y Fn(\\)e Fo(P)313 2605 y Ft(2)349 2598 y Fq(with)k Fo(`)e Fn(\024)g Fo(k)r Fq(\()p Fo(j)g Fq(+)d(1\),)16 b(if)g Fo(`)e Fn(\025)g Fo(k)r Fq(,)i(w)o(e)g(ha)o(v)o (e)515 2723 y Fn(jL)563 2702 y Fv(k)585 2723 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fn(j)30 b(\024)797 2689 y Fo(v)821 2696 y Fv(`)p Fk(\000)p Fv(k)p 797 2711 87 2 v 821 2757 a Fo(v)845 2764 y Fv(`)889 2723 y Fo(c)935 2655 y Fg(Z)993 2723 y Fo(f)5 b(dm:)p eop %%Page: 17 17 17 16 bop 558 133 a Fp(Mixing)15 b(prop)q(erties)h(on)h(to)o(w)o(ers) 510 b Fq(17)47 286 y(Let)17 b Fo(`)d(>)g(k)r Fq(,)i(w)o(e)f(denote)i(b) o(y)e Fo(x)602 268 y Ft(0)638 286 y Fq(the)h(preimage)f(of)h Fo(x)g Fq(b)o(y)g Fo(F)1137 268 y Fv(`)1153 286 y Fq(,)g Fo(x)1211 268 y Ft(0)1244 286 y Fn(2)e Fq(\003)1325 293 y Ft(0)1344 286 y Fq(,)609 409 y Fn(L)643 389 y Fv(k)665 409 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))14 b(=)830 376 y Fo(v)854 383 y Ft(0)p 830 398 44 2 v 832 444 a Fo(v)856 451 y Fv(`)879 409 y Fn(L)913 389 y Fv(k)q Fk(\000)p Fv(`)976 409 y Fo(f)5 b Fq(\()p Fo(x)1052 389 y Ft(0)1072 409 y Fq(\))p Fo(:)437 b Fq(\(3.13\))47 544 y(F)l(or)17 b Fo(x)c Fn(2)h Fq(\003)257 551 y Ft(0)293 544 y Fq(and)j Fo(q)e Fn(2)f Fm(N)p Fq(,)47 683 y Fn(jL)95 662 y Fv(q)114 683 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fn(j)42 b(\024)369 635 y Fg(X)350 739 y Fh(F)373 732 y(q)391 739 y(x)408 732 y Fd(0)419 739 y Fi(=)p Fh(x)362 768 y(x)379 761 y Fd(0)390 768 y(2)p Fh(P)430 775 y Fi(2)510 649 y Fo(v)r Fq(\()p Fo(x)583 631 y Fk(0)594 649 y Fq(\))p 478 671 167 2 v 478 717 a Fo(J)5 b(F)549 703 y Fv(q)567 717 y Fq(\()p Fo(x)614 703 y Fk(0)625 717 y Fq(\))649 683 y Fo(f)g Fq(\()p Fo(x)725 662 y Fk(0)737 683 y Fq(\))11 b(+)840 635 y Fg(X)821 739 y Fh(F)844 732 y(q)862 739 y(x)879 732 y Fd(0)890 739 y Fi(=)p Fh(x)833 768 y(x)850 761 y Fd(0)865 768 y Fh(=)-20 b Fd(2)p Fh(P)901 775 y Fi(2)975 649 y Fo(v)r Fq(\()p Fo(x)1048 631 y Fk(0)1059 649 y Fq(\))p 949 671 155 2 v 949 717 a Fo(J)5 b(F)1020 703 y Fv(q)1038 717 y Fq(\()p Fo(x)p Fq(\))1109 683 y Fo(f)g Fq(\()p Fo(x)1185 662 y Fk(0)1196 683 y Fq(\))265 864 y Fn(\024)41 b Fo(c)374 797 y Fg(Z)432 864 y Fo(f)5 b(dm)17 b(C)617 817 y Fg(X)598 921 y Fh(F)621 914 y(q)638 921 y(x)655 914 y Fd(0)667 921 y Fi(=)p Fh(x)610 949 y(x)627 942 y Fd(0)638 949 y(2)p Fh(P)678 956 y Fi(2)721 864 y Fo(m)764 871 y Ft(0)783 864 y Fq(\()p Fo(C)837 871 y Fv(q)856 864 y Fq(\()p Fo(x)903 844 y Fk(0)915 864 y Fq(\)\))p Fo(v)r Fq(\()p Fo(x)1026 844 y Fk(0)1036 864 y Fq(\))11 b(+)1115 797 y Fg(Z)1173 864 y Fo(f)5 b(dm)p Fq([)p Fo(bD)1345 871 y Ft(1)1376 864 y Fq(+)11 b Fo(aD)q Fq(])17 b(using)f(\(3.7\))p Fo(:)47 1055 y Fq(So)h(w)o(e)f(ha)o(v)o(e)f(for)i Fo(x)c Fn(2)h Fq(\003)496 1062 y Fv(`)513 1055 y Fq(,)i Fo(`)e(<)g(k)r Fq(,)438 1190 y Fn(jL)486 1170 y Fv(k)508 1190 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fn(j)13 b(\024)h Fq([)p Fo(cC)t(D)797 1197 y Ft(2)827 1190 y Fq(+)d Fo(bD)937 1197 y Ft(1)968 1190 y Fq(+)g Fo(aD)q Fq(])1106 1122 y Fg(Z)1165 1190 y Fo(f)5 b(dm:)266 b Fq(\(3.14\))47 1326 y(W)l(e)16 b(are)h(no)o(w)f (in)g(p)q(osition)h(to)g(c)o(hose)f(the)g(parameters.)k(Chose:)120 1435 y Fn(\017)k Fo(d)14 b(>)g Fq(0)120 1560 y Fn(\017)24 b Fo(a)14 b(>)265 1526 y(\013)296 1508 y Fk(0)319 1526 y Fq(+)d(3)p Fo(=)p Fq(8)p Fo(\013)p 265 1548 208 2 v 354 1594 a(\033)478 1560 y Fq(,)120 1714 y Fn(\017)24 b Fo(c)14 b(>)261 1681 y Fq(1)d(+)g Fo(aD)p 261 1703 152 2 v 280 1748 a(\033)i Fn(\000)e Fo(")418 1714 y Fq(,)k Fo(c)f Fn(\025)g Fo(d)120 1873 y Fn(\017)24 b Fo(b)14 b(>)287 1839 y(C)t(c)p 261 1862 114 2 v 261 1907 a(\033)e Fn(\000)f Fo(")379 1873 y Fq(,)16 b Fo(b)d Fn(\025)h Fo(d)120 2001 y Fn(\017)24 b Fo(\021)18 b Fq(and)f Fo(t)f Fq(suc)o(h)g(that)h(2)p Fo(\013)611 1983 y Fk(0)623 2001 y Fo(cD)684 2008 y Ft(2)718 2001 y Fo(<)d(\013=)p Fq(4,)j(and)g Fo(C)t(D)1054 2008 y Ft(2)1087 2001 y Fo(<)d(")p Fq(,)120 2110 y Fn(\017)24 b Fo(s)g Fq(suc)o(h)f(that)g Fo(s)j(>)g(k)583 2117 y Ft(0)603 2110 y Fq(\()p Fo(\021)r Fq(\))d(\()p Fo(k)734 2117 y Ft(0)755 2110 y Fq(\()p Fo(\021)r Fq(\))g(is)g(giv)o 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Fq(1)p 519 2675 V 519 2721 a Fo(\026)p Fq(\()p Fo(P)7 b Fq(\))638 2619 y Fg(Z)638 2760 y Fv(P)696 2687 y Fn(L)730 2666 y Fv(k)752 2687 y Fo(f)e(dm)42 b Fn(\024)30 b Fo(\033)r(a)1024 2619 y Fg(Z)1081 2687 y Fo(f)5 b(dm;)p eop %%Page: 18 18 18 17 bop 0 133 a Fq(18)488 b Fp(V)o(\023)-23 b(eronique)15 b(Maume-Desc)o(hamps)0 286 y Fq(for)i Fo(x)e Fq(and)i Fo(y)h Fq(in)e(\003)346 263 y Fv(q)346 300 y(`)365 286 y Fq(,)g Fo(`)e Fn(\024)g Fo(k)r Fq(\()p Fo(j)g Fq(+)d(1\),)16 b(using)g(\(3.10,)h(3.11,)f(3.12\),)332 420 y Fn(jL)380 399 y Fv(k)402 420 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g(L)592 399 y Fv(k)613 420 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))p Fn(j)42 b(\024)f Fo(\016)864 427 y Fv(j)882 420 y Fo(b)911 352 y Fg(Z)969 420 y Fo(f)5 b(dm)17 b Fq(if)e Fo(`)g Fn(\025)e Fo(k)332 530 y Fn(jL)380 510 y Fv(k)402 530 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))11 b Fn(\000)g(L)592 510 y Fv(k)613 530 y Fo(f)5 b Fq(\()p Fo(y)r Fq(\))p Fn(j)42 b(\024)f Fo(\033)r(b)901 463 y Fg(Z)959 530 y Fo(f)5 b(dm)16 b Fq(otherwise)p Fo(:)0 662 y Fq(for)h(an)o(y)f Fo(x)d Fn(2)h Fo(P)286 669 y Ft(2)317 662 y Fn(\\)e Fq(\003)396 669 y Fv(`)412 662 y Fq(,)k Fo(`)e Fn(\024)g Fo(k)r Fq(\()p Fo(j)g Fq(+)d(1\),)16 b(using)h(\(3.13,)f(3.14\),)437 794 y Fn(jL)485 774 y Fv(k)506 794 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fn(j)42 b(\024)f Fo(\016)759 801 y Fv(j)777 794 y Fo(c)806 726 y Fg(Z)864 794 y Fo(f)5 b(dm)17 b Fq(if)f Fo(`)e Fn(\025)g Fo(k)437 905 y Fn(jL)485 884 y Fv(k)506 905 y Fo(f)5 b Fq(\()p Fo(x)p Fq(\))p Fn(j)42 b(\024)f Fo(\033)r(c)796 837 y Fg(Z)854 905 y Fo(f)5 b(dm)17 b Fq(otherwise)o Fo(:)1494 850 y Fq(\(3.15\))0 1034 y(This)f(concludes)g(the)g(pro)q(of)h(since)f Fo(\033)f(<)f(\016) 779 1041 y Fv(j)797 1034 y Fq(.)769 b Fb(\004)23 1094 y Fq(It)22 b(is)g(easily)g(seen)g(that)h(all)f(the)g(cones)h Fn(C)s Fq(\()p Fo(a;)8 b(b;)g(c;)g(d;)g(j)s Fq(\))21 b(satisfy)i(the)f(prop)q(erties)h(of)0 1154 y(section)g(2.)42 b(In)23 b(order)g(to)g(apply)g(results)g(from)f(section)h(2,)h(w)o(e)f 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Fp(b)q(elongs)f(to)h Fn(C)s Fq(\()p Fo(j)s Fq(\))0 2741 y Fp(for)e(all)e Fo(j)i Fn(2)d Fm(N)p Fp(.)22 b(Moreo)o(v)o(er,)15 b Fo(R)p Fq(\()p Fo(f)5 b Fq(\))15 b Fn(\024)e Fp(Ct)k Fn(k)p Fo(f)5 b Fn(k)p Fp(.)p eop %%Page: 19 19 19 18 bop 558 133 a Fp(Mixing)15 b(prop)q(erties)h(on)h(to)o(w)o(ers) 510 b Fq(19)47 286 y Ff(Pr)n(o)n(of)9 b Fq(:)42 b(Recall)15 b(that)h Fo(h)e Fq(=)577 266 y Fv(h)597 271 y Fi(0)p 577 275 38 2 v 587 304 a Fv(v)636 286 y Fq(where)h Fo(h)804 293 y Ft(0)837 286 y Fn(2)f Fo(L)i Fq(is)g(the)f(\014xed)g(p)q(oin)o(t) h(for)g Fn(L)1417 293 y Ft(0)1452 286 y Fq(\(see)f(pages)47 347 y(4,)22 b(7)g(and)f(10\).)37 b(Condition)21 b(4.)36 b(in)21 b(the)f(de\014nition)h(of)g(the)g(cone)g(is)g(satis\014ed)g (for)g Fo(h)47 407 y Fq(pro)o(vided)14 b Fo(d)g Fn(\025)g(k)p Fo(h)391 414 y Ft(0)410 407 y Fn(k)h Fq(\(recall)e Fo(m)p Fq(\()p Fo(h)p Fq(\))g(=)h(1\).)21 b(Condition)15 b(1.)21 b(is)14 b(satis\014ed)h(for)f Fo(h)h Fq(pro)o(vided)47 467 y Fo(a)k Fn(\025)g Fq(1.)31 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Fv(v)855 1698 y Fq(with)j Fo(f)5 b(h)1024 1705 y Ft(0)1057 1698 y Fn(2)15 b Fo(L)p Fq(.)428 b Fb(\004)15 1758 y Fq(W)l(e)14 b(conclude)g(this)h(section)f(with)h(the)f(estimations)g (for)h(functions)g(in)f Fo(L)h Fq(and)g(for)g(the)0 1819 y(initial)g(transfer)h(op)q(erator)i Fn(L)552 1826 y Ft(0)572 1819 y Fq(.)0 1923 y Fl(COR)n(OLLAR)-5 b(Y)20 b(3.14)k Fp(F)l(or)f(an)o(y)f Fo(f)31 b Fn(2)25 b Fo(L)e Fp(and)g Fo(g)i Fp(suc)o(h)e(that)g Fo(g)r(v)j Fn(2)f Fo(L)1402 1905 y Ft(1)1422 1923 y Fq(\()p Fo(m)1484 1930 y Ft(0)1504 1923 y Fq(\))p Fp(,)f(for)0 1983 y Fo(p)15 b Fn(2)f Fm(N)p Fp(,)k(and)f(an)o(y)f(increasing)g(sequence)g Fq(\()p Fo(v)814 1990 y Fv(`)830 1983 y Fq(\))849 1990 y Fv(`)p Fk(2)p Fj(N)930 1983 y Fp(suc)o(h)g(that)1146 1946 y Fg(P)1198 1998 y Fv(`)1223 1983 y Fo(v)1247 1990 y Fv(`)1263 1983 y Fo(m)1306 1990 y Ft(0)1326 1983 y Fq(\(\003)1379 1990 y Fv(`)1395 1983 y Fq(\))e Fo(<)h Fn(1)p Fp(,)h(for)0 2044 y(an)o(y)g Fo(`)e Fn(2)g Fm(N)348 2104 y Fq(sup)365 2144 y Ft(\003)389 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2731 y(for)17 b(the)f(sp)q(eed)g(of)g(con)o(v)o(ergence)f(in)h(the)g (uniform)f(norm.)p eop %%Page: 21 21 21 20 bop 558 133 a Fp(Mixing)15 b(prop)q(erties)h(on)h(to)o(w)o(ers) 510 b Fq(21)511 286 y Fe(4.1)66 b(Exp)r(onen)n(tial)24 b(deca)n(y)47 370 y Fq(If)f(there)f(exist)g(constan)o(ts)h(0)j Fo(<)f(e)714 377 y Ft(1)756 370 y Fq(and)e(0)j Fo(<)f(\022)h(<)f Fq(1)e(suc)o(h)f(that)i(for)f(all)f Fo(`)j Fn(2)h Fm(N)p Fq(,)47 430 y Fn(\024)14 b Fo(m)p Fq(\(\003)196 437 y Fv(`)212 430 y Fq(\))g Fn(\024)f Fo(e)320 437 y Ft(1)340 430 y Fo(\022)364 412 y Fv(`)396 430 y Fq(then)i(it)h(ma)o(y)e(b)q(e)h (tak)o(en)h Fo(v)882 437 y Fv(`)912 430 y Fq(=)d(\002)1001 412 y Fv(`)1033 430 y Fq(pro)o(vided)i(\002)1271 412 y Fk(\000)p Ft(1)1332 430 y Fo(<)f(\022)q Fq(.)21 b(In)16 b(this)f(sit-)47 490 y(uation,)e Fo(\016)231 497 y Fv(j)262 490 y Fn(\024)h Fq(\002)353 472 y Fk(\000)p Ft(1)414 490 y Fo(<)g Fq(1.)20 b(F)l(ollo)o(wing)11 b(remark)f(3.5,)i(it)f(is)g (su\016cien)o(t)f(to)i(w)o(ork)g(with)f(cones)47 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Fq(for)g(functions)h(in)e Fo(L)p Fq(.)522 1639 y Fe(4.3)66 b(P)n(olynomial)25 b(deca)n(y)47 1723 y Fq(If)15 b(there)f(exist)g(constan)o(ts)i(0)e Fo(<)g(e)660 1730 y Ft(1)679 1723 y Fq(,)h Fo(\014)h(>)e Fq(1)h(suc)o(h)g(that)g (for)g(all)f Fo(`)h Fn(2)f Fm(N)p Fq(,)h Fo(m)p Fq(\(\003)1437 1730 y Fv(`)1453 1723 y Fq(\))f Fn(\024)g Fo(e)1562 1730 y Ft(1)1581 1723 y Fo(`)1601 1705 y Fk(\000)p Fv(\014)1653 1723 y Fq(,)47 1788 y(it)k(ma)o(y)e(b)q(e)j(tak)o(e)e Fo(v)405 1795 y Fv(`)438 1788 y Fq(=)509 1768 y Fv(`)523 1757 y Fh(\014)q Fd(\000)p Fi(1)p 498 1777 98 2 v 498 1805 a Ft(\(ln)5 b Fv(`)p Ft(\))575 1796 y Fh(\015)618 1788 y Fq(pro)o(vided)17 b Fo(\015)j(>)d Fq(1.)27 b(W)l(e)18 b(obtain)h(deca)o(y)e(of)h(correlations)47 1866 y(in)113 1843 y Ft(\(ln)5 b Fv(n)p Ft(\))197 1831 y Fh(\015)p 113 1855 105 2 v 125 1884 a Fv(n)146 1874 y Fh(\014)q Fd(\000)p Fi(1)243 1866 y Fq(for)20 b(functions)h(in)f Fo(L)p Fq(.)33 b(L.-S.)21 b(Y)l(oung)f(obtained)h(deca)o(y)f(of)g (correlations)h(in)47 1927 y(1)p Fo(=n)124 1908 y Fv(\014)r Fk(\000)p Ft(1)194 1927 y Fq(.)575 2068 y Fe(4.4)66 b(Lo)n(w)n(er)22 b(b)r(ounds)47 2147 y Fq(W)l(e)c(use)h(a)g(mo)q(di\014cation)e(of)i (L.-S.)g(Y)l(oung's)f(argumen)o(t)g(\([Y2)o(]\).)27 b(Let)19 b Fo(f)k Fq(=)18 b Fo(h)12 b Fq(+)h Fo(c=v)1650 2154 y Fv(`)47 2209 y Fq(on)18 b(\003)150 2216 y Fv(`)184 2209 y Fq(with)g Fo(`)e Fn(\025)g Fq(1,)i Fo(c)f Fq(is)h(suc)o(h)f (that)h Fo(m)p Fq(\()p Fo(f)5 b Fq(\))16 b(=)g(1,)i(clearly)e Fo(f)21 b Fq(=)1248 2187 y Fv(f)1265 2192 y Fi(0)p 1248 2198 35 2 v 1256 2226 a Fv(v)1305 2209 y Fq(with)c Fo(f)1441 2216 y Ft(0)1477 2209 y Fn(2)f Fo(L)p Fq(.)25 b(W)l(e)47 2265 y(ha)o(v)o(e)16 b Fn(L)194 2247 y Fv(n)217 2265 y Fo(f)k Fq(=)13 b Fo(h)e Fq(+)g Fo(c=v)469 2272 y Fv(`)501 2265 y Fq(on)17 b(\003)603 2272 y Fv(`)635 2265 y Fq(for)g Fo(`)d Fn(\025)g Fo(n)p Fq(.)21 b(So,)16 b(for)g Fo(`)f Fn(\025)e Fo(n)p Fq(,)j Fn(L)1196 2247 y Fv(n)1220 2265 y Fo(f)g Fn(\000)10 b Fo(hm)p Fq(\()p Fo(f)5 b Fq(\))14 b(=)g Fo(c=v)1582 2272 y Fv(`)1615 2265 y Fq(on)47 2322 y(\003)81 2329 y Fv(`)98 2322 y Fq(.)27 b(This)19 b(implies)d(that)i (the)h(b)q(ound)g(giv)o(en)f(in)g(prop)q(osition)h(3.12)g(is)g(optimal) e(if)g(the)47 2378 y(ratio)g Fo(v)188 2385 y Fv(j)206 2378 y Fo(=v)254 2385 y Fv(j)r Ft(+1)333 2378 y Fq(go)q(es)g(to)g(1.)47 2526 y Fr(References)47 2628 y Fx([A,D,U])43 b(J.)22 b(AAR)o(ONSON,)g(M.)f(DENKER)h(&)g(M.)e(URBANSKI)k Fs(Er)n(go)n(dic)d (the)n(ory)i(for)245 2684 y(Markov)14 b(\014b)n(er)n(e)n(d)e(systems)g (and)i(p)n(ar)n(ab)n(olic)f(r)n(ational)g(maps.)f Fx(T)l(rans.)g(Amer.) f(Math.)245 2741 y(So)q(c.)k(\(1993\),)f Fy(337)h Fx(\(2\),)f(495-548.) p eop %%Page: 22 22 22 21 bop 0 133 a Fq(22)488 b Fp(V)o(\023)-23 b(eronique)15 b(Maume-Desc)o(hamps)0 286 y Fx([Ba,V])70 b(V.)18 b(BALADI,)g(M.)f (VIANA)i Fs(Str)n(ong)f(sto)n(chastic)g(stability)h(and)f(r)n(ate)h(of) g(mixing)198 343 y(for)e(unimo)n(dal)f(maps.)f Fx(Ann.)h(Sci.)800 331 y(\023)796 343 y(Ecole)g(Norm.)f(Sup.)g(\(1996\))f Fy(29)p Fx(,)h(4,)f(483-517.)0 433 y([Ba,Y])70 b(V.BALADI,)11 b(L.-S.)h(YOUNG)f Fs(On)h(the)h(sp)n(e)n(ctr)n(a)f(of)h(r)n(andomly)f (p)n(erturb)n(e)n(d)h(exp)n(and-)198 490 y(ing)j(maps.)f Fx(Comm.)f(Math.)g(Ph)o(ys.)h(\(1993\))e Fy(156)p Fx(,)i(2,)g(355-385.) 0 581 y([B,Y])93 b(M.)23 b(BENEDICKS)h(&)g(L.-S.)g(YOUNG)g Fs(De)n(c)n(ay)g(of)g(c)n(orr)n(elations)f(for)h(c)n(ertain)198 637 y(Henon)15 b(maps.)h Fx(\(1996\))d(preprin)o(t.)0 728 y([Bi1])105 b(G.)16 b(BIRKHOFF)i Fs(Extensions)e(of)i(Jentzch's)f (the)n(or)n(em.)g Fx(T.A.M.S.)e(\(1957\),)g Fy(85)p Fx(,)198 784 y(219-227.)0 875 y([Bi2])105 b(G.)14 b(BIRKHOFF)j Fs(L)n(attic)n(e)e(the)n(ory)i(\(3r)n(d)f(e)n(dition\).)e Fx(Amer.)h(Math.)f(So)q(c.)h(\(1967\).)0 965 y([Buz])95 b(J.)16 b(BUZZI)h Fs(Markov)h(extensions)d(for)j(multi-dimensional)f (dynamic)n(al)g(systems.)198 1022 y Fx(Israel)f(J.)f(Math.)f(\(to)g (app)q(ear\).)0 1112 y([Ca])116 b(A.A.)12 b(CASTR)o(O)h Fs(Backwar)n(d)i(inducing)e(and)h(exp)n(onential)f(de)n(c)n(ay)h(of)g (c)n(orr)n(elations)198 1169 y(for)19 b(p)n(artial)r(ly)f(hyp)n(erb)n (olic)g(attr)n(actors)g(with)h(mostly)f(c)n(ontr)n(acting)f(c)n(entr)n (al)g(dir)n(e)n(c-)198 1225 y(tion.)e Fx(PhD)g(Thesis)h(\(1998\).)0 1316 y([C])139 b(P)l(.)17 b(COLLET)h Fs(Statistics)f(of)i(closest)e(r)n (eturn)h(for)g(some)h(non)e(uniformly)h(hyp)n(er-)198 1372 y(b)n(olic)e(systems.)e Fx(Preprin)o(t)h(\(1999\))0 1463 y([C,G,S])52 b(P)l(.)16 b(COLLET,)g(A.)g(GAL)-5 b(VES,)16 b(B.)g(SCHMITT)g Fs(Unpr)n(e)n(dictability)g(of)h(the)h(o)n (c)n(cur-)198 1519 y(r)n(enc)n(e)13 b(time)h(of)g(a)h(long)e(laminar)h (p)n(erio)n(d)g(in)g(a)g(mo)n(del)g(of)h(temp)n(or)n(al)f (intermitency.)198 1576 y Fx(Ann.)h(Inst.)g(H.)g(P)o(oincar)o(\023)-21 b(e)15 b(Ph)o(ys.)f(Th)o(\023)-21 b(eor.)14 b(\(1992\),)f Fy(57)p Fx(,)i(3,)f(319-331.)0 1666 y([Ch])114 b(N.)15 b(CHERNO)o(V)h Fs(Statistic)n(al)f(pr)n(op)n(erties)i(of)f(pie)n(c)n (ewise)g(smo)n(oth)g(hyp)n(erb)n(olic)g(sys-)198 1723 y(tems)j(in)g(high)g(dimensions.)e Fx(Discrete)i(Con)o(tin.)f(Dynam.)g (Systems)g(\(1999\),)f Fy(5)p Fx(,)198 1779 y(2,)e(425-448.)0 1870 y([D])137 b(D.)15 b(DOLGOPY)l(A)l(T)i Fs(On)f(dynamics)g(of)h (mostly)g(c)n(ontr)n(acting)f(di\013e)n(omorphisms.)198 1926 y Fx(Preprin)o(t)f(\(1998\).)0 2017 y([F,S1])81 b(P)l(.)11 b(FERRER)o(O,)h(B.)f(SCHMITT)g Fs(R)o(uel)r(le)h(Perr)n(on)h (F)m(r)n(ob)n(enius)d(the)n(or)n(ems)j(and)f(pr)n(o-)198 2074 y(je)n(ctive)h(metrics.)f Fx(Collo)q(que)i(Math.)d(So)q(c.)i(J.)f (Boly)o(ai)h(Random)g(Fields.)g(Estergom)198 2130 y(\(Hungary\))h (\(1979\).)0 2221 y([F,S2])81 b(P)l(.)14 b(FERRER)o(O,)h(B.)g(SCHMITT)f Fs(On)h(the)h(r)n(ate)g(of)g(c)n(onver)n(genc)n(e)e(for)i(some)f(limit) 198 2277 y(r)n(atio)21 b(the)n(or)n(ems)f(r)n(elate)n(d)g(to)h (endomorphisms)g(with)g(a)f(non)g(r)n(e)n(gular)g(invariant)198 2334 y(density.)14 b Fx(Preprin)o(t)i(\(1994\).)0 2424 y([G,S])98 b(A.)22 b(GAL)-5 b(VES)23 b(&)g(B.)f(SCHMITT)g Fs(Ine)n(qualities)g(for)h(hitting)g(time)g(in)f(mixing)198 2481 y(dynamic)n(al)14 b(systems.)f Fx(Random)h(and)g(Computational)f (Dynamics)h(\(1997\),)e Fy(5)p Fx(,)i(4,)198 2537 y(337-347.)0 2628 y([H])138 b(F.)19 b(HOFBA)o(UER)h Fs(On)g(intrinsic)f(er)n(go)n (dicity)h(of)g(pie)n(c)n(ewise)f(monotonic)h(tr)n(ans-)198 2684 y(formations)c(with)g(p)n(ositive)f(entr)n(opy.)g Fx(Israel)g(J.)g(Math.)e(\(1979\),)g Fy(34)p Fx(,)i(1,)f(213-237;)198 2741 y(\(1981\),)f Fy(38)p Fx(,)i(11,)f(107-115.)p eop %%Page: 23 23 23 22 bop 558 133 a Fp(Mixing)15 b(prop)q(erties)h(on)h(to)o(w)o(ers) 510 b Fq(23)47 286 y Fx([K,N])90 b(KELLER,)19 b(T.)f(NO)o(WISKI)h Fs(Sp)n(e)n(ctr)n(al)f(the)n(ory,)i(zeta)f(functions)f(and)h(the)g (distri-)245 343 y(bution)c(of)f(p)n(erio)n(dic)h(p)n(oints)e(for)i (Col)r(let-Eckmann)e(maps.)h Fx(Comm.)e(Math.)g(Ph)o(ys.)245 399 y(\(1992\),)h Fy(149)p Fx(,)i(1,)g(31-69.)47 493 y([K,M,S])44 b(A.)13 b(K)o(OND)o(AH,)g(V.)g(MA)o(UME)f(&)h(B.)g (SCHMITT)g Fs(Vitesse)g(de)h(c)n(onver)n(genc)n(e)f(vers)245 550 y(l')o(\023)-22 b(etat)21 b(d')o(\023)-22 b(equlibr)n(e)21 b(p)n(our)h(des)f(dynamiques)g(markoviennes)f(non)g(h\177)-23 b(old)o(\023)h(eriennes.)245 606 y Fx(Ann.)16 b(Inst.)f(P)o(oincarr)o (\023)-21 b(e)14 b(Sec.)h(Prob.)g(Stat.)f(\(1997\))f Fy(33)j Fx(\(6\))e(675-695.)47 700 y([Li1])109 b(C.)20 b(LIVERANI)j Fs(De)n(c)n(ay)e(of)g(c)n(orr)n(elations.)e Fx(Ann.)i(of)f(Math.)g(\(1995\),)g Fy(142)g Fx(\(2\),)245 756 y(239-301)47 850 y([Li2])109 b(C.)12 b(LIVERANI)h Fs(Centr)n(al)f(limit)h(the)n(or)n(em)g(for)h(deterministic)f(systems.) d Fx(Pro)q(ceed-)245 907 y(ings)18 b(of)g(the)g(In)o(ternational)g (Congress)f(on)h(Dynamical)g(Systems,)g(Mon)o(tevideo)245 963 y(95,)c(Researc)o(h)i(Notes)f(in)h(Mathematics)f(series,)g (Pittman,)g(\(1997\).)47 1057 y([Ma1])84 b(V.)22 b(MA)o(UME-DESCHAMPS)g Fs(Pr)n(opri)o(\023)-22 b(et)o(\023)g(es)22 b(de)h(m)o(\023)-22 b(elange)22 b(p)n(our)i(des)e(syst)o(\022)-22 b(emes)245 1113 y(dynamiques)19 b(markoviens.)e Fx(PhD)h(Thesis,)h(Univ)o(ersit)o (\023)-21 b(e)18 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